THE GEOMETRY OF QUASI-SASAKI MANIFOLDS
by
ADAM P. WELLY
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2016
DISSERTATION APPROVAL PAGE
Student: Adam P. Welly
Title: The Geometry of quasi-Sasaki Manifolds
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Dr. Weiyong He Chair
Dr. Peter Gilkey Core Member
Dr. Peng Lu Core Member
Dr. Christopher Sinclair Core Member
Dr. Dietrich Belitz Institutional Representative
and
Scott Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2016
ii
c© 2016 Adam P. Welly
iii
DISSERTATION ABSTRACT
Adam P. Welly
Doctor of Philosophy
Department of Mathematics
June 2016
Title: The Geometry of quasi-Sasaki Manifolds
Let (M, g) be a quasi-Sasaki manifold with Reeb vector field ξ. Our goal is to
understand the structure of M when g is an Einstein metric. Assuming that the S1
action induced by ξ is locally free or assuming a certain non-negativity condition on
the transverse curvature, we prove some rigidity results on the structure of (M, g).
Naturally associated to a quasi-Sasaki metric g is a transverse Ka¨hler metric gT .
The transverse Ka¨hler-Ricci flow of gT is the normalized Ricci flow of the transverse
metric. Exploiting the transverse Ka¨hler geometry of (M, g), we can extend results
in Ka¨hler-Ricci flow to our transverse version. In particular, we show that a deep and
beautiful theorem due to Perleman has its counterpart in the quasi-Sasaki setting.
We also consider evolving a Sasaki metric g by Ricci flow. Unfortunately, if
g(0) is Sasaki then g(t) is not Sasaki for t > 0. However, in some instances g(t) is
quasi-Sasaki. We examine this and give some qualitative results and examples in the
special case that the initial metric is η-Einstein.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Adam P. Welly
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Western Washington University, Bellingham, WA
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2016, University of Oregon
Bachelor of Science, Mathematics, 2010, Western Washington University
AREAS OF SPECIAL INTEREST:
Differential Geometry
Geometric Analysis
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, Department of Mathematics, University of Oregon,
Eugene OR, 2010–2016
Peer Tutor, Department of Mathematics, Western Washington University,
Bellingham Wa, 2008–2010
v
ACKNOWLEDGEMENTS
I would like to thank my advisor Weiyong He for his encouragement and support,
as well as many pieces of valuable advice. I would also like to thank Joey Iverson for
providing the LATEX template for this document.
vi
Dedicated to my wife Allison and my son Owen.
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3. Transverse Ka¨hler Geometry . . . . . . . . . . . . . . . . . . . 12
1.4. Transverse Distance and Preferred Coordinates . . . . . . . . . 20
II. RIGIDITY OF QUASI-SASAKI-EINSTEIN METRICS . . . . . . . . . 25
2.1. The Ricci tensor and Einstein equation . . . . . . . . . . . . . 25
2.2. The Regular Case . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3. The quasi-Regular Case . . . . . . . . . . . . . . . . . . . . . . 39
2.4. Transverse Curvature Conditions . . . . . . . . . . . . . . . . . 42
2.5. Splitting the Contact Bundle . . . . . . . . . . . . . . . . . . . 45
2.6. The General Case with a Curvature Condition . . . . . . . . . 48
III. TRANSVERSE KA¨HLER-RICCI FLOW . . . . . . . . . . . . . . . . . 54
3.1. Transverse Ka¨hler-Ricci Flow . . . . . . . . . . . . . . . . . . 56
3.2. Perelman’s entropy functional on quasi-Sasaki manifolds . . . . 64
3.3. Bounds on Scalar Curvature and the Transverse Ricci Potential 74
viii
Chapter Page
3.4. Upper Bound on Diameter . . . . . . . . . . . . . . . . . . . . 85
3.5. Transverse Holomorphic Bisectional Curvature . . . . . . . . . 90
IV. THE RICCI FLOW OF A SASAKI METRIC . . . . . . . . . . . . . . 96
4.1. η-Einstein metrics . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2. Transverse Calabi-Yau Structure . . . . . . . . . . . . . . . . . 100
4.3. Transverse Fano Structure . . . . . . . . . . . . . . . . . . . . 102
4.4. Transverse Canonical Structure . . . . . . . . . . . . . . . . . 105
4.5. Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6. Questions and Directions for Future Research . . . . . . . . . . 108
V. APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
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CHAPTER I
INTRODUCTION
1.1. Overview
This dissertation is an exploration into the geometry of quasi-Sasaki manifolds.
As the name suggests, a quasi-Sasaki structure is similar to a Sasaki one, but is
more flexible than the latter. Indeed, Sasaki manifolds form a subset of the quasi-
Sasaki manifolds, but in general the two can have quite different features. For
example, cosymplectic manifolds are also quasi-Sasaki. A cosymplectic manifold has
an integrable contact bundle whereas the contact bundle of a Sasaki manifold is as far
from being integrable as possible. But like a Sasaki manifold, a quasi-Sasaki manifold
(M, g) comes equipped with a one-dimensional Riemannian foliation, Fξ, determined
by a nowhere vanishing vector field ξ. The space transverse to this foliation carries a
Ka¨hler structure. Dual to the vector field ξ (via the metric g) is a 1-form η. The kernel
of η is called the contact bundle. Perhaps the most significant difference between a
quasi-Sasaki and a true Sasaki structure has to do with the rank of the 2-form dη.
For a Sasaki structure, η ∧ (dη)n is a volume form and the transverse Ka¨hler metric
ω = dη. If a quasi-Sasaki manifold has (dη)n = 0, then it lacks a contact structure
and, even when (dη)n 6= 0, there may be no clear relationship between the transverse
Ka¨hler metric and the 2-form dη. In the next sections we will recall all of the necessary
definitions and elementary facts concerning quasi-Sasaki manifolds that will be used
throughout the remainder of this dissertation.
The pioneering work on quasi-Sasaki manifolds was iniated by Blair in [3] during
the late 1960’s. However, his work was not without errors. These errors were later
1
recognized by Tanno who then gave complete and correct revisions of Blair’s results in
[37]. The main results of these papers describe a splitting of quasi-Sasaki manifolds
by assuming the integrability of a certain almost product structure TM = P ⊕ Q
and positive definiteness of dη(·,Φ·) when restricted to P . These assumptions are
somewhat unnatural and unmotivated and we do not make them here. Rather, we
want to prove a splitting result like their’s, but under the assumption that the metric g
is Einstein. In fact, one of our motivating reasons for studying quasi-Sasaki manifolds
is the search for “new” Einstein metrics. A Riemannian metric g is called Einstein
if its Ricci tensor is proportional to g. That is, if there is a constant λ ∈ R such
that Ricg = λg. Differential geometers are interested in Einstein metrics for a wide
variety of reasons. Entire books have been written on the subject; the book by Besse
is a classic. Often times Einstein metrics are somehow the “best” or most desirable
metrics. They appear as critical points of certain geometrically defined functionals.
Physicists are interested in Einstein metrics for their role in the theory of general
relativity. Thus we are curious about the rigidity of quasi-Sasaki-Einstein metrics.
For example, we can create quasi-Sasaki-Einstein metrics by taking products of
Sasaki-Einstein and Ka¨hler-Einstein metrics, but we don’t consider these to be “new”
Einstein metrics. So the question is, are all quasi-Sasaki-Einstein metrics of this form
or are there more “interesting” examples possible? We investigate this in chapter II.
By assuming some regularity on the orbits of the Reeb vector field ξ, we can give a
precise description of the possible Einstein metrics. We define a transverse version
of non-negative quadratic orthogonal bisectional curvature (NQOBC) and show that
a quasi-Sasaki-Einstein metric satisfying this curvature conditions is rigid. We show
that M is locally the product of a Ka¨hler-Einstein manifold M0 and a full rank
quasi-Sasaki-Einstein manifold M1. Furthermore, the transverse Ka¨hler metric of M1
2
is a product of Ka¨hler-Einstein metrics. If it happens that the second basic Betti
number bB2 (M1) = 1, then the metric on M1 is simply a scaling of a Sasaki-Einstein
metric. Note that Sasaki metrics are not preserved by homotheties, but quasi-Sasaki
metrics are. Under the stronger assumption of positive or non-negative transverse
holomorphic bisectional curvature, we conjecture that something like the Frankel
and the generalized Frankel conjectures (which are actually no longer conjectures) in
Sasaki geometry ought to be true. See [20] and [21].
As we will see later in this chapter, a quasi-Sasaki manifold (M, g) has naturally
associated to it a transverse Ka¨hler metric gT . It is known by the work of Cao in [7]
that if the first Chern class of a Ka¨hler manifold is negative or null, then the Ka¨hler-
Ricci flow will converge to a Ka¨hler-Einstein metric. This gives us some motivation
to consider the transverse Ka¨hler-Ricci flow (TKRF). By exploiting the transverse
Ka¨hler geometry of (M, g), we can extend deep results in Ka¨hler-Ricci flow to our
transverse version. In particular, if the basic first Chern class is negative or null, we
can reproduce the results of Cao in [7] in our transverse setting. If M is compact and
the basic first Chern class is positive, then defining functionals which are monotonic
along the TKRF, we can mimic the work of Perleman in [34] and prove uniform
bounds on the diameter and scalar curvature along the flow. We also show that,
analogous to [2] and [29], positivity of transverse holomorphic bisectional curvature
is preserved along the TKRF. All of this and more is carried out in chapter III.
In chapter IV we start to explore Hamilton’s Ricci flow on a Sasaki manifold
(M, g). That is, we let the Sasaki metric g itself evolve by the Ricci flow:
∂g
∂t
= −2Ricg.
3
It is well-known that evolving a Ka¨hler metric by Ricci flow preserves the Ka¨hler
condition, but the Ricci flow does not preserve the Sasaki condition. However, we
will show that in the special case where the initial Sasaki metric is η-Einstein, that
there is a quasi-Sasaki structure associated to the evolving metric. The geometry of
an η-Einstein manifold is special enough that the Ricci flow equation can be reduced
to a system of two ordinary differential equations. The qualitative behavior of the
flow is discussed in each of the three cases determined by the transverse Ka¨hler-
Einstein structure. We then exhibit concrete examples of η-Einstein metrics where
the transverse metric gT satisfies RicTg = λg
T for λ = −1, 0, 4. We mention in passing
the classification theorem of three-dimensional Sasaki manifolds due to Geiges and
the uniformization theorem due to Belgun and how our examples tie in. We raise
some questions about the general case of evolving an arbitrary Sasaki metric by the
Ricci flow.
1.2. Preliminaries
In this section we will review the (Riemannian) basics of quasi-Sasaki geometry.
Throughout, M = M2n+1 will denote a smooth, oriented, connected, Riemannian
manifold of real dimension 2n + 1 for n ≥ 1. The metric will be denoted by g and
∇ will denote the Levi-Civita connection of g. We begin by defining a quasi-Sasaki
structure as a certain type of almost contact structure. This is the definition that
Blair gave in [3] and the one that we shall work with here.
Definition 1.2.1. An almost contact structure on M2n+1 is a triple (ξ, η,Φ) where ξ
is a nowhere vanishing vector field (called the Reeb vector field), η is a 1-form with
η(ξ) = 1 and Φ is a (1,1)-tensor satisfying Φ2 = −I + ξ ⊗ η.
4
We let Lξ ⊂ TM denote the line bundle generated by ξ. The contact bundle
D ⊂ TM is defined to be the kernel of η. We have the direct sum decomposition
TM = Lξ ⊕D.
The following theorem will be used frequently.
Theorem 1.2.2. Suppose M2n+1 has an almost contact structure (ξ, η,Φ). Then
Φξ = 0, the endomorphism Φ has rank 2n and η ◦ Φ = 0.
Proof. Since Φ2 = −I on D, we see that the rank of Φ is at least 2n. Now Φ2ξ = 0
implies that Φξ 6∈ D \ {0}. Thus Φξ = fξ for some f ∈ C∞(M). But we have that
0 = Φ2ξ = fΦξ = f 2ξ. Since ξ is nowhere vanishing, f = 0. Thus Φξ = 0 and this
implies that the rank of Φ is exactly 2n. For any vector field X ∈ TM , we now have
η(ΦX)ξ = Φ2ΦX + ΦX
= Φ(Φ2X +X)
= η(X)Φξ
= 0.
Therefore η ◦ Φ = 0.
Definition 1.2.3. An almost contact metric structure on M2n+1 is a quadruple
(ξ, η,Φ, g) where (ξ, η,Φ) is an almost contact structure and g is a Riemannian metric
on M that is compatible with Φ in the sense that Φ∗g = g − η ⊗ η.
If (ξ, η,Φ, g) is an almost contact metric structure, the metric compatibility with
Φ and Φξ = 0 implies that η = g(ξ, ·). Hence η and ξ are dual to each other via the
metric g. Given an almost contact structure, a compatible metric always exists.
5
Proposition 1.2.4. Given an almost contact structure (ξ, η,Φ) on M , there is a
Riemannian metric g such that (ξ, η,Φ, g) is an almost contact metric structure.
Proof. Let g˜ be any Riemannian metric on M . First define a metric g′ by
g′(X, Y ) := g˜
(
Φ2X,Φ2Y
)
+ η(X)η(Y ).
Then we have g′(X, ξ) = η(X) for all X ∈ TM . Now define a metric g by
g(X, Y ) :=
1
2
(g′(X, Y ) + g′(ΦX,ΦY ) + η(X)η(Y ))
and check that g(ΦX,ΦY ) = g(X, Y )− η(X)η(Y ) for all X, Y ∈ TM.
Let M2n+1 be equipped with an almost contact structure. Consider the manifold
M × R with the almost complex structure J : T (M × R)→ T (M × R), defined by
J
(
f1X + f2
d
dt
)
= f1ΦX − f2ξ + f1η(X) d
dt
. (1.1)
Here X ∈ TM , f1, f2 ∈ C∞(M × R) and d
dt
spans the tangent space to R. It is
straightforward to verify that J2 = −I.
Definition 1.2.5. An almost contact structure on M is called normal if the almost
complex structure J , defined by (1.1), is integrable.
Recall that an almost complex structure is called integrable if it is induced by
an actual complex structure. The celebrated theorem of Newlander and Nirenberg
asserts that an almost complex structure J is integrable if and only if the Nijenhuis
torsion tensor NJ ≡ 0. For a (1,1)-tensor T , the Nijenhuis torsion tensor NT is the
6
(1,2)-tensor defined by
NT (X, Y ) := T
2[X, Y ] + [TX, TY ]− T [TX, Y ]− T [X,TY ]. (1.2)
So if our almost product structure is normal, then NJ ≡ 0 and this implies that
for any vector fields X, Y ∈ TM ,
N (1)(X, Y ) := NΦ(X, Y ) + 2dη(X, Y )ξ = 0,
N (2)(X, Y ) := (LΦXη)(Y )− (LΦY η)(X) = 0,
N (3)(X) := (LξΦ)(X) = 0,
N (4)(X) := (Lξη)(X) = 0.
Conversely, the vanishing of these four tensor fields implies that NJ ≡ 0. In fact, the
vanishing of NJ is equivalent to the vanishing of N
(1). A proof of these facts and the
following proposition can be found in chapter 6 of [4].
Proposition 1.2.6. If N (1) vanishes then so do N (2), N (3) and N (4). Therefore, an
almost contact structure is normal if and only if N (1) ≡ 0.
An equation that we will use frequently, and one that holds for any 1-form η is
2dη(X, Y ) = Xη(Y )− Y η(X)− η([X, Y ]). (1.3)
The normality condition has some interesting consequences. We record a few of these
in the following lemma.
7
Lemma 1.2.7. Let (ξ, η,Φ) be a normal, almost contact structure on M . Then
dη(ξ, ·) = 0 and Φ∗dη = dη. Moreover, if g is a compatible metric then the integral
curves of ξ are geodesics.
Proof. Since N (4)(X) = (Lξη)(X) = 0 for any X ∈ TM , we have
2dη(ξ,X) = ξ(η(X))−X(η(ξ))− η([ξ,X])
= ξ(η(X))− η([ξ,X])
= (Lξη)(X) = 0.
Thus dη(ξ, ·) = 0. Cartan’s magic formula says that LΦXη = dιΦXη + ιΦXdη. Since
η ◦ Φ = 0, we get
(LΦXη)(Y ) = dη(ΦX, Y ).
From N (2)(X, Y ) = 0 and the above, we have dη(ΦX, Y ) = dη(ΦY,X). Now replacing
Y with ΦY and using dη(ξ, ·) = 0, we see that dη(ΦX,ΦY ) = −dη(Y,X) = dη(X, Y ).
For the last claim, notice that g(∇Xξ, ξ) = 12Xg(ξ, ξ) = 0. Again using N (4)(X) = 0,
we compute that for any X ∈ TM
g(X,∇ξξ) = ξg(X, ξ)− g(∇ξX, ξ) + g(∇Xξ, ξ)
= ξ(η(X))− η([ξ,X])
= (Lξη)(X) = 0.
Thus ∇ξξ = 0. Hence the integral curves of ξ are geodesics.
8
The fundamental 2-form associated to an almost contact metric structure
(ξ, η,Φ, g) is the 2-form ω defined for all X, Y ∈ TM by
ω(X, Y ) := g(X,ΦY ). (1.4)
From Φ∗g = g − η ⊗ η and η ◦ Φ = 0, it follows that Φ∗ω = ω. Note also that
ω(ξ, ·) = 0. Now we are ready to give the definition of a quasi-Sasaki structure.
Definition 1.2.8. A quasi-Sasaki structure is a normal, almost contact metric
structure whose fundamental 2-form is closed (i.e. dω = 0).
A smooth manifold equipped with a quasi-Sasaki structure is called a quasi-
Sasaki manifold. A fundamental difference between a Sasaki and a quasi-Sasaki
structure has to do with whether or not dη is non-degenerate on D. We can categorize
quasi-Sasaki structures according to their rank, which indicates how degenerate dη
is. The following definition was introduced in [3].
Definition 1.2.9. The rank of a quasi-Sasaki structure is 2p when (dη)p 6= 0 and
η ∧ (dη)p = 0. The rank is 2p+ 1 when η ∧ (dη)p 6= 0 and (dη)p+1 = 0.
A quasi-Sasaki structure of rank 1 is commonly called a cosymplectic structure.
In this case we have dη = 0. A quasi-Sasaki structure of rank 2n + 1 with ω = dη is
Sasaki. The next proposition follows easily from lemma 1.2.7.
Proposition 1.2.10. There are no quasi-Sasaki structures of even rank.
Proof. Since dη(ξ, ·) = 0, if (dη)p 6= 0 there are vector fieldsX1, . . . , X2p ∈ D such that
(dη)p(X1, . . . , X2p) 6= 0. Thus η∧(dη)p(ξ,X1, . . . , X2p) = (dη)p(X1, . . . , X2p) 6= 0.
9
Some facts about the geometry of a quasi-Sasaki manifold are contained in the
next lemma. Important is the fact that ξ is a Killing vector field (i.e. Lξg = 0). This
means that moving along the integral curves of ξ is an isometry of the metric.
Lemma 1.2.11. Let (ξ, η,Φ, g) be a quasi-Sasaki structure on M . Then Lξg = 0,
g(∇Xξ, Y ) = dη(X, Y ) and ∇ξΦ = 0.
Proof. Since N (3)(Y ) = (LξΦ)(Y ) = 0, we have Φ[ξ, Y ] = [ξ,ΦY ]. Thus we see that
(Lξω)(X, Y ) = (Lξg)(X,ΦY ).
The Lie derivative Lξω = dιξω + ιξdω = 0. Hence we have (Lξg)(X,ΦY ) = 0. Next,
using N (4)(X) = (Lξη)(X) = 0 and ∇ξξ = 0, one computes that
(Lξg)(X, η(Y )ξ) = 0.
The above shows that (Lξg)(X, (Φ + ξ ⊗ η)Y ) = 0. Since the map Φ + ξ ⊗ η is
non-singular, it follows that Lξg = 0. Therefore, ξ is a Klling vector field.
For a vector field Z, let θZ be the 1-form defined by θZ(Y ) := g(Y, Z). It is then
straightforward to verify that
(LZg)(X, Y ) + 2dθZ(X, Y ) = 2g(∇XZ, Y ).
Taking Z = ξ and using that ξ is a Killing vector field gives g(∇Xξ, Y ) = dη(X, Y ).
With this and the vanishing of tensors N (2) and N (3), a straightforward
computation, using that ∇ is Riemannian and torsion free, reveals that ∇ξΦ = 0.
Remark 1.2.12. The condition that ξ is a Killing vector field is equivalent to the
condition g(∇Xξ, Y ) + g(X,∇Y ξ) = 0 for all X, Y ∈ TM. Also, one can actually
10
show more than just ∇ξΦ = 0. In particular, lemma 6.1 of [4] shows that
g((∇XΦ)Y, Z) = dη(ΦY,X)η(Z)− dη(ΦZ,X)η(Y ). (1.5)
We end this section with some results which characterize cosymplectic manifolds
(those with dη = 0) among quasi-Sasak manifolds.
Lemma 1.2.13. A quasi-Sasaki manifold is cosymplectic if and only if ∇Φ = 0 if
and only if ∇ω = 0.
Proof. If ∇Φ = 0 then NΦ ≡ 0. By normality, 2dη(X, Y )ξ = −NΦ(X, Y ) = 0 for
all X, Y ∈ TM . Hence dη = 0. Conversely, if dη = 0 then from equation (1.5)
we conclude that ∇Φ = 0. The second if and only if statement follows easily from
ω = g(·,Φ·) and ∇g = 0.
Lemma 1.2.14. A quasi-Sasaki manifold is cosymplectic if and only if ∇η = 0.
Proof. Since ξ is a Killing vector field and η = g(ξ, ·), we compute that
(∇Xη)(Y ) = g(∇Xξ, Y ) = −g(∇Y ξ,X) = −(∇Y η)(X).
We can write equation (1.3) as
2dη(X, Y ) = (∇Xη)(Y )− (∇Y η)(X).
Combining these equations we have dη(X, Y ) = (∇Xη)(Y ) and the result follows.
Remark 1.2.15. The cosymplectic case is the only one where the contact subbundle
D is integrable. Indeed, it follows from equation (1.3) that a subbundle defined as
the kernel of a closed 1-form is involutive, hence integrable (see definition 5.0.2 and
11
theorem 5.0.3). Furthermore, from ∇η = 0 we see that D is a parallel distribution.
As TM = Lξ ⊕D and both Lξ and D are integrable in this case, it follows that M is
locally the product of a 1-dimensional manifold (line or circle) and a Ka¨hler manifold.
1.3. Transverse Ka¨hler Geometry
In this section we discuss the basics of Riemannian foliations and the associated
transverse geometry, as it applies to our study of quasi-Sasaki manifolds. For a more
thorough treatment of foliations, the interested reader should see [30] and/or [39].
Let (M, ξ, η,Φ, g) be a quasi-Sasaki manifold. Recall that Lξ is the line bundle
generated by ξ and D = ker(η). From Φ∗g = g − η ⊗ η and Φξ = 0 we see that the
decomposition TM = Lξ ⊕ D is orthogonal with respect to g. Hence D = L⊥ξ . The
integral submanifolds of Lξ, which are the integral curves of ξ, define the leaves of a
1-dimensional foliation of M , which we denote by Fξ. We let ν(Fξ) := TM/Lξ denote
the normal bundle of the foliation. Then we have the short exact sequence
0→ Lξ → TM → ν(Fξ)→ 0.
The metric provides a splitting σ : ν(Fξ)→ D of this short exact sequence. Explicitly,
for X¯ ∈ ν(Fξ) we have X¯ = X + Lξ for some X ∈ TM . Then σ(X¯) = X − g(X, ξ)ξ.
Clearly σ(X¯) ∈ D and it is straightforward to check that σ is a well-defined bijection.
Thus we may identify D and ν(Fξ). A transverse metric is a Riemannian metric on
the quotient bundle ν(Fξ). Since D ' ν(Fξ), we define the transverse metric gT by
gT := g|D.
12
The orthogonal projection piD : TM → D is given by piD(X) = X − η(X)ξ. Observe:
gT (X, Y ) = g(piD(X), piD(Y ))
= g(X, Y )− η(X)η(Y )
= Φ∗g(X, Y ).
Thus the transverse metric is related to the quasi-Sasaki metric via g = η⊗η+gT . We
want to know, under what conditions does the transverse metric reflect the geometry
of the leaf space M/Fξ? To answer this question, we need the next definition.
Definition 1.3.1. A vector field X ∈ TM is called foliate with respect to Fξ if
[X, ξ] ∈ Lξ. A Riemannian metric g is called bundle-like with respect to Fξ if for any
foliate vector fields X and Y which are orthogonal to Lξ, we have ξg(X, Y ) = 0.
Bundle-like metrics are important in the theory of foliations because they provide
globally defined transverse metrics which are locally the pull-back of a metric on the
local Riemannian quotient. The following is proposition 2.5.7 in [5].
Proposition 1.3.2. If g is bundle-like with respect to Fξ, then Fξ is a Riemannian
foliation. Conversely, if Fξ is a Riemannian foliation with transverse metric gT , then
there is a bundle-like metric g whose associated transverse metric is gT .
Thus we see that the transverse metric is locally the pull-back of a metric on
the local Riemannian quotient if and only if g is bundle-like. In this way, bundle-like
metrics reflect the geometry of the leaf space M/Fξ. The next lemma is immediate.
Lemma 1.3.3. Quasi-Sasaki metrics are bundle-like with respect to Fξ.
Proof. This follows easily from definition 1.3.1 and the fact that Lξg = 0.
13
The following definition is used to describe the behavior of the orbits of ξ.
Definition 1.3.4. The foliation Fξ is quasi-regular if each x ∈M has a neighborhood
Ux such that any leaf of Fξ passes through Ux at most mx ∈ Z+ times. If mx = 1 for
all x ∈ M , then Fξ is called regular. If for some point, no such m exists, then Fξ is
called irregular.
If all of the orbits of ξ are compact then Fξ is quasi-regular. In this case, ξ
generates an S1 action on M . If this action is free, then Fξ is regular. If the S1
action is only locally free, then Fξ is strictly quasi-regular. Assuming M is compact,
quasi-regularity implies that the orbits of ξ are all compact. Thus, if ξ has a non-
compact orbit, then Fξ is irregular. We will often say that M is regular (quasi-regular,
irregular) to mean that Fξ is regular (quasi-regular, irregular).
Example 1.3.5. A good example to keep in mind is the weighted Sasaki structure
on S3 = {(z0, z1) ∈ C2 : |z0|2 + |z1|2 = 1}. Pick real numbers a0, a1 > 0 and consider
the vector field
ξ =
√−1
∑
j
aj
(
zj
∂
∂zj
− z¯j ∂
∂z¯j
)
.
Restricting to S3, ξ is everywhere tangent to S3 and nowhere vanishing. The 1-form
η = −
√−1
2
∑
j
1
aj
(
z¯jdz
j − zjdz¯j
)
is a contact form with η(ξ) = 1. On D ⊂ TS3, we let Φ be the restriction of the
standard complex structure on C2 and extend it trivially in the ξ direction. Then
g = η ⊗ η + dη(Φ·, ·) defines a Sasaki-metric on S3. The integral curve of ξ through
the point (z0, z1) ∈ S3 is given by
(z0, z1) 7→ (e2piia0tz0, e2piia1tz1).
14
Choosing a0 = a1 = 1 gives the standard Sasaki structure on S
3. If a0 and a1
are chosen to be relatively prime integers, then Fξ is regular. If a0 and a1 are not
coprime or a0, a1 ∈ Q \ Z, then Fξ is quasi-regular. Suppose that a0a1 is irrational.
Orbits through points of the form (z0, 0) are circles as they return to (z0, 0) at time
t = a−10 . Similarly, orbits through (0, z1) are circles that return in time t = a
−1
1 .
However, the orbits through points (z0, z1) where z0z1 6= 0 are not compact. Since
a0
a1
is irrational, there is no time t such that both a0t and a1t are integers. Thus the
integral curve never returns to (z0, z1). The closure of these orbits is the torus T
2.
This example easily generalizes to higher odd dimensional spheres.
Adapting theorem 6.3.8 of [5] to the special case of a quasi-Sasaki manifold, we
get the following structure theorem. This important theorem will be used frequently
in the sequel. See chapter 4.3 of [5] for the definition of the orbifold cohomology. If the
orbifold is in fact a manifold, then the orbifold cohomology is the usual cohomology.
Theorem 1.3.6. Let (M, ξ, η,Φ, g) be a quasi-Sasaki manifold of rank 2p + 1 such
that the leaves of Fξ are all compact. Then
1. The leaf space M/Fξ is a Ka¨hler orbifold such that the canonical projection
pi : M → M/Fξ is an orbifold Riemannian submersion. The fibers are totally
geodesic submanifolds of M diffeomorphic to S1.
2. M is the total space of a principal S1-orbibundle over M/Fξ with connection
1-form η. The curvature form dη = pi∗Ω where Ω is a closed (1,1)-form of rank
p on M/Fξ such that 12pi [Ω] ∈ H2orb(M/Fξ,Z).
3. If Fξ is regular, then M is the total space of a principal S1-bundle over the
Ka¨hler manifold M/Fξ.
15
Now we will show that the transverse geometry of a quasi-Sasaki manifold has
a Ka¨hler structure. From Φ2 = −I + ξ ⊗ η, we have (Φ|D)2 = −I. Hence Φ|D is
an almost complex structure on D. We consider the complexified contact bundle
DC := D ⊗R C. Then Φ|D extends C-linearly to an almost complex structure on
DC, which we continue to call Φ. Let D1,0 and D0,1 denote the
√−1 and −√−1
eigenspaces of Φ. Then DC = D1,0 ⊕ D0,1. Complex conjugation is an R-linear
isomorphism between D1,0 and D0,1.
As the transverse metric is Φ-invariant (i.e. Φ∗gT = gT ), it defines a Hermitian
metric on D1,0. Explicitly, we C-linearly extend gT in both entries to a symmetric,
complex bilinear form on DC, which we continue to call gT . Then for any U, V ∈ D1,0,
we define the Hermitian metric h by
h(U, V ) := gT (U, V¯ ).
The map σ : D → D1,0 that sends X 7→ 1
2
(
X −√−1ΦX) is an isomorphism
of complex vector bundles (D,Φ) ' (D1,0,√−1). Under this isomorphism, a
straightforward computation reveals that
σ∗h =
1
2
(
gT +
√−1ω) .
It is well-known that the holomorphic tangent bundle of an almost complex
manifold is involutive if and only if the almost complex structure is integrable. In the
almost contact setting, we have a similar theorem.
Theorem 1.3.7. An almost contact structure is normal if and only if [D1,0, Lξ] ⊂ D1,0
and [D1,0,D1,0] ⊂ D1,0.
16
Proof. By proposition 1.2.6, normality is equivalent to NΦ + 2dη ⊗ ξ ≡ 0. Now let
X, Y ∈ D1,0. Using equation (1.3) and Φ2 = −I + ξ ⊗ η, we compute that
NΦ(X, Y ) + 2dη(X, Y )ξ = −2[X, Y ]− 2
√−1Φ[X, Y ],
NΦ(X, Y¯ ) + 2dη(X, Y¯ )ξ = 0,
NΦ(X, ξ) + 2dη(X, ξ)ξ = −[X, ξ]−
√−1Φ[X, ξ].
So NΦ+2dη⊗ξ ≡ 0 if and only if Φ[X, Y ] =
√−1[X, Y ] and Φ[X, ξ] = √−1[X, ξ].
Corollary 1.3.8. An almost contact structure is normal if and only if the Nijenhuis
torsion tensor NΦ vanishes on D1,0 ⊕ Lξ.
Proof. Since Φ∗dη = dη, we have dη(X, Y ) = 0 for X, Y ∈ D1,0. By lemma 1.2.7,
dη(X, ξ) = 0. The corollary now follows from the computation in theorem 1.3.7.
So for a quasi-Sasaki structure, the contact bundle D is endowed with a complex
structure Φ and a Hermitian metric whose Ka¨hler form is closed. Thus the data
(D,Φ, ω) gives M a transverse Ka¨hler structure. We define a connection ∇T on D by
∇TXY := piD (∇XY ) , X ∈ D,
∇Tξ Y := piD [ξ, Y ] .
One can check that ∇T is the unique, torsion-free connection on D with ∇TgT = 0.
For this reason we call ∇T the transverse Levi-Civita connection. We define the
transverse curvature operator for X, Y ∈ D by
RT (X, Y ) := ∇TX∇TY −∇TY∇TX −∇T[X,Y ].
17
With this we can then define the transverse Riemannian curvature tensor, the
transverse Ricci curvature and the transverse scalar curvature in the obvious way.
When computing transverse curvatures, we have all of the familiar formulas from
Ka¨hler geometry. Next we briefly mention the transverse de Rham cohomology and
its associated Hodge theory. For a detailed discussion, see [15] and [24].
Definition 1.3.9. A function f ∈ C∞(M) is called basic if df(ξ) = 0. A p-form
θ ∈ Ωp(M) is called basic if ιξθ = 0 and Lξθ = 0.
We let ΛpB(Fξ) be the sheaf of germs of basic p-forms and ΩpB(Fξ) the set of
global sections of ΛpB(Fξ). Observe that if θ is basic then so is dθ. This follows from
Lξθ = ιξdθ + dιξθ and Lξdθ = dLξθ. We set dB := d|ΩpB . Then dB : Ω
p
B → Ωp+1B is
well-defined, dB ◦ dB = 0 and we have a subcomplex of the de Rham complex:
0→ C∞B → Ω1B → · · · → Ω2nB → 0.
We denote the cohomology ring by H∗B(Fξ) and call it the basic cohomology ring.
There is a transverse Hodge theory for the basic cohomology ring and in many ways
it resembles the de Rham-Hodge theory of a Ka¨hler manifold. The transverse Hodge
star, ∗B, is defined in terms of the usual Hodge star by ∗Bα := ∗(η ∧ α). The adjoint
operator δB : Ω
p
B → Ωp−1B of dB is defined by
δB := − ∗B ◦dB ◦ ∗B.
We define the basic Hodge Laplacian (with respect to dB) by 4B := dBδB + δBdB.
We say that θ ∈ ΩpB is harmonic if 4Bθ = 0. The transverse Hodge theorem of [14]
states that every basic cohomology class has a unique harmonic representative. We
define the transverse tensor Laplacian by ∆T := ∇T∇T . For a basic function f , one
18
can check that
4Bf = −∆Tf = −gij¯∂i∂j¯f.
The transverse complex structure Φ induces a natural decomposition of the basic
p-forms into basic (i, j)-forms where i+ j = p:
ΩpB ⊗R C = ⊕i+j=pΩi,jB
and the exterior derivative splits as dB = ∂B + ∂¯B where
∂B : Ω
i,j
B → Ωi+1,jB and ∂¯B : Ωi,jB → Ωi,j+1B .
From d2B = 0 we get ∂
2
B = ∂¯
2
B = 0 and ∂B∂¯B + ∂¯B∂B = 0. As we did above, we can
define the basic Hodge Laplacians with respect to ∂B and ∂¯B. We can also define
a transverse Lefschetz operator L : θ 7→ θ ∧ ω. One can then recover the expected
transverse Ka¨hler identities. We set
dcB :=
√−1
2
(∂¯B − ∂B).
Then we have dBd
c
B =
√−1∂B∂¯B and dcB ◦ dcB = 0. The transverse Ricci form ρT is
the basic 2-form defined by
ρT := RicT (·,Φ·).
Just as in the Ka¨hler setting, one can show that
ρT = −√−1∂B∂¯B log(det(gT )).
19
Thus ρT is a dB closed (1,1)-form and so defines a basic cohomology class. The class
[ρT ] ∈ H2B(Fξ) is independent of the choice of transverse Ka¨hler metric and, as in the
Ka¨hler setting, 2pic1B = [ρ
T ], where c1B is the basic first Chern class.
1.4. Transverse Distance and Preferred Coordinates
First we introduce the transverse distance function that will be important for
us in chapter III. Let (M, g) be quasi-Sasaki and assume that M is compact and
quasi-regular. For a point x ∈M , let γx denote the orbit of ξ through x. Since γx is
compact for all x ∈M , we can make the following definition:
Definition 1.4.1. The transverse distance function dT is defined on M by
dT (x, y) := d(γx, γy)
where d is the geodesic distance function on M , determined by g. The transverse ball
of radius r > 0 about a point x ∈M is defined as
BT (x, r) := {y ∈M : dT (x, y) < r}.
The transverse diameter of M is the number ΘT defined as
ΘT := max
x,y∈M
dT (x, y).
The transverse distance function defines a metric on the leaf space M/Fξ. It
turns out to be the same as the geodesic distance function on M/Fξ, determined by
gT . In order to prove that dT is a metric, we need to show that it satisfies the triangle
inequality. To do so, we make use of the following proposition:
20
Proposition 1.4.2. For x, y ∈ M and p ∈ γx, there is q ∈ γy such that dT (x, y) =
d(p, q).
Proof. Since the curves γx and γy are compact, there are points x1 ∈ γx and y1 ∈ γy
such that dT (x, y) = d(x1, y1). Now let σ : [0, 1] → M be the length minimizing
geodesic with σ(0) = x1 and σ(1) = y1. Flowing along γx is an isometry of M . As γx
is compact, it is diffeomorphic to S1. Thus there is ζ ∈ S1 such that ζ(x1) = p. Let
q := ζ(y1) and σ˜ = ζ ◦ σ. Then σ˜ is a geodesic connecting p and q whose length is
the same as σ. Therefore, dT (x, y) = d(x1, y1) = d(p, q).
Corollary 1.4.3. The transverse distance function satisfies the triangle inequality.
For fixed x0 ∈M , r(x) := dT (x, x0) is a basic Lipschitz function.
Proof. Pick points x, y, z ∈M . By the previous proposition, there are points p ∈ γx,
q ∈ γy and s ∈ γz such that dT (x, y) = d(p, q) and dT (y, z) = d(q, s). Thus
dT (x, z) ≤ d(p, s) ≤ d(p, q) + d(q, s) = dT (x, y) + dT (y, z).
Let r be as above. By the triangle inequality, |r(x)−r(y)| ≤ dT (x, y) ≤ d(x, y). Thus
r is Lipschitz continuous with Lipschitz constant 1. Hence r is differentiable almost
everywhere and |∇r| ≤ 1 at any point of differentiability. Since r is constant along
the orbits of ξ, dr(ξ) = 0. Therefore, r is basic.
We conclude this chapter by introducing the local coordinates and local frame
that we will use in our computations in chapter II, III and IV. By normality, the
almost complex structure J , defined by (1.1), on M × R is integrable; hence it is
induced by a complex structure. This means that there are (real) local coordinates
(x0, y0, . . . , xn, yn) on M × R such that J∂xj = ∂yj and J∂yj = −∂xj .
21
We will identify M as a submanifold of M × R via M ' M × {0}. Let t be the
coordinate on R. Then the tangent bundle T (M×R) = TM⊕R[ d
dt
] = Lξ⊕D⊕R[ ddt ].
Observe that Jξ = d
dt
and J d
dt
= −ξ. Thus, Lξ ⊕ R[ ddt ] forms an involutive, complex
subbundle of T (M ×R). By the complex version of the Frobenius theorem, there are
local coordinates as above with ∂x0 = ξ and ∂y0 =
d
dt
. Thus y0 = t.
Now consider the complexification T (M × R) ⊗R C and C-linearly extend J .
For j = 0, . . . , n, put zj =
1
2
(xj −
√−1yj). Then (z0, . . . , zn, z¯0, . . . , z¯n) are local
coordinates on M × R with J∂zj =
√−1∂zj and J∂z¯j = −
√−1∂z¯j . We know that
∂z0 =
1
2
(ξ − √−1 d
dt
). For j ≥ 1, we can write ∂zj = Zj + fj ddt for some vector field
Zj ∈ TM ⊗R C and some complex-valued function fj. By the definition of J ,
J∂zj = ΦZj − fjξ + η(Zj)
d
dt
.
But from J∂zj =
√−1∂zj , we have
J∂zj =
√−1
(
Zj + fj
d
dt
)
.
Equating the two expression for J∂zj , we find that fj = −
√−1η(Zj) and
ΦZj =
√−1(Zj − η(Zj)ξ).
Setting t = 0, the local coordinates (z0, . . . , zn, z¯0, . . . , z¯n) on M ×R induce local
coordinates on M . Since t = y0, we get the local coordinates (x, z1, . . . , zn, z¯1, . . . , z¯n)
where x = x0. The coordinate vectors satisfy ∂x = ξ and Φ∂zj =
√−1(∂zj − η(∂zj)ξ).
We call these preferred local coordinates.
22
Since η is a real 1-form and dη is a basic (1,1)-form, in a small coordinate
neighborhood U we can write
η = dx−√−1(hjdzj − hj¯dz¯j)
where h : U → R is a (local) real basic function (i.e. ξh = ∂xh = 0). We use the
notation hj¯ = ∂z¯jh =
∂h
∂z¯j
= ∂h
∂zj
= ∂zjh = hj. Then, defining dηij¯ := hij¯, we have
dη =
√−1dηij¯dzi ∧ dz¯j.
Now, for j = 1, . . . , n, define vector fields
Xj := ∂zj +
√−1hjξ and Xj¯ := X¯j = ∂z¯j −
√−1hj¯ξ.
Notice that η(Xj) = η(X¯j) = 0, while ΦXj =
√−1Xj and ΦX¯j = −
√−1X¯j. Thus,
D1,0 = span{Xj} andD0,1 = span{X¯j}. We call the frame (ξ,X1, . . . , Xn, X¯1, . . . , X¯n)
a preferred local frame. Since dη(·, ξ) = 0, we see that
dη(Xi, X¯j) = dη(∂zi , ∂z¯j) =
√−1dηij¯.
Since the hj are basic functions, we compute that
[Xi, Xj] = [X¯i, X¯j] = [ξ,Xi] = [ξ, X¯j] = 0
and
[Xi, X¯j] = −2
√−1dηij¯ξ.
23
From the compatibility of Φ with g, in preferred local coordinates we can write
g = η ⊗ η + gTij¯dzi ⊗ dz¯j
where gTij¯ = g
T
ji¯
= gTj¯i. Since Lξg = Lξη = 0, the functions gTij¯ are basic. The transverse
metric and Ka¨hler form are, respectively,
gT = gTij¯dz
i ⊗ dz¯j and ω = √−1gTij¯dzi ∧ dz¯j.
Hence
gT (Xi, X¯j) = g
T (∂zi , ∂z¯j) = g
T
ij¯
and
ω(Xi, X¯j) = ω(∂zi , ∂z¯j) =
√−1gTij¯.
The transverse metric defines a pointwise norm on basic forms. If α = αidz
i and
β = βj¯dz¯
j are basic (1,0) and (0,1)-forms, respectively, then we define |α|2gT = gij¯αiαj
and |β|2gT = gij¯βj¯βi¯. We can then extend this to basic forms of type (p, q). For
example, the norm of a (1, 1)-form θ = θij¯dz
i ∧ dz¯j is given by |θ|2gT = gij¯gkl¯θil¯θjk¯.
At a point where gij¯ = δij, we see that |θ|2gT =
∑
i,k θik¯θik¯ =
∑
i,k |θik¯|2, as we would
expect. For any basic form θ, the norm with respect to the quasi-Sasaki metric g is
twice that of the norm with respect to the transverse metric gT ; |θ|2g = 2|θ|2gT .
24
CHAPTER II
RIGIDITY OF QUASI-SASAKI-EINSTEIN METRICS
2.1. The Ricci tensor and Einstein equation
A Sasaki metric g and dη are intimately related by g = η ⊗ η + dη(Φ·, ·). It is
not so clear, a priori, how a quasi-Sasaki metric and dη are related. In this section
we compute the Ricci tensor of a quasi-Sasaki metric. To better understand the
relationship between the metric and dη, we want to know how the Ricci tensor depends
on dη. Given that a quasi-Sasaki structure has a Riemannian foliation associated to
it, one could compute the Ricci tensor of g using curvature formulas involving the
O’Neill tensors A and T (see theorems 2.5.16 and 2.5.18 of [5]). However, these
curvature formulas will not give us what we want directly and to use them requires a
sufficiently large amount of computation anyway. Thus, we will present a barehanded
computation of the Ricci tensor and we will see the role that dη plays in the curvature.
We shall compute the Ricci tensor with respect to a preferred local frame {ξ,Xi, X¯i}.
Recall that {Xi} is a local frame for D1,0, gT (Xi, X¯j) = gTij¯ and [ξ,Xi] = [ξ, X¯i] = 0.
In lemma 1.2.7 we showed that ∇ξξ = 0. Now we observe that
∇Xiξ =
√−1gkj¯dηij¯Xk and ∇X¯iξ = ∇Xiξ = −
√−1gjk¯dηji¯X¯k. (2.1)
To see this, write∇Xiξ = αξ+βjXj+γkX¯k and use g(∇Xξ, Y ) = dη(X, Y ), keeping in
mind that dη is a basic (1,1)-form. Let piLξ : TM → Lξ be the orthogonal projection
onto Lξ. Proposition 2.5.14 in [5] asserts that for X, Y ∈ D,
piLξ (∇XY ) =
1
2
piLξ ([X, Y ]) . (2.2)
25
Writing [X, Y ] = αξ+βiXi +γ
iX¯i and applying η to both sides yields α = η([X, Y ]).
Hence piLξ([X, Y ]) = η([X, Y ])ξ. By equation (1.3), for X, Y ∈ D, we have
η([X, Y ]) = −2dη(X, Y ). Therefore, by (2.2) and the above,
piLξ (∇XY ) = −dη(X, Y )ξ. (2.3)
Now, as ∇XY = ∇TXY + piLξ (∇XY ) for X, Y ∈ D, equation (2.3) yields
∇XY = ∇TXY − dη(X, Y )ξ. (2.4)
Since the transverse metric is Ka¨hler and dη is a (1,1)-form, it follows from (2.4) that
∇XiXj = ∇TXiXj and ∇XiX¯j = −
√−1dηij¯ξ. (2.5)
Equations (2.1) and (2.5) along with ∇ξξ = 0 allow us to compute the curvature
tensor of a quasi-Sasaki structure with respect to our preferred frame. For notational
convenience, we will drop the superscript T from the transverse metric gT .
First we compute
R(Xi, ξ)ξ = ∇Xi∇ξξ −∇ξ∇Xiξ −∇[Xi,ξ]ξ
= 0−√−1gkj¯dηij¯∇ξXk − 0
= gkj¯gmp¯dηij¯dηkp¯Xm.
Hence we have g(R(Xi, ξ)ξ,Xl¯) = g
kj¯dηij¯dηkl¯ = g
kj¯dηij¯dηlk¯ and therefore,
Ric(ξ, ξ) = 2gil¯g(R(Xi, ξ)ξ,Xl¯) = 2g
il¯gkj¯dηij¯dηlk¯ = |dη|2g ≥ 0.
26
Next we will compute Ric(Xj, ξ). We start with
R(X¯i, ξ)Xj = ∇X¯i∇ξXj −∇ξ∇X¯iXj −∇[X¯i,ξ]Xj
=
√−1gkp¯∇X¯idηjp¯Xk −
√−1∇ξ(dηji¯ξ)− 0
=
√−1gkp¯ (X¯idηjp¯Xk + dηjp¯∇X¯iXk)− 0
=
√−1gkp¯ (X¯idηjp¯Xk +√−1dηjp¯dηki¯ξ) .
From equation (2.5) we have [Xi, Xj¯] = −2
√−1dηij¯ξ. Using this we compute that
R(X¯i, Xj)ξ = ∇X¯i∇Xjξ −∇Xj∇X¯iξ −∇[X¯i,Xj ]ξ
=
√−1gkp¯ (∇X¯idηjp¯Xk)+√−1gmq¯ (∇Xjdηmi¯Xq¯)− 2√−1dηji¯∇ξξ
=
√−1gkp¯ (X¯idηjp¯Xk + dηjp¯∇X¯iXk)+√−1gmq¯ (Xjdηmi¯Xq¯ + dηmi¯∇XjXq¯)− 0
=
√−1gkp¯ (X¯idηjp¯Xk +√−1dηjp¯dηki¯ξ)+√−1gmq¯ (Xjdηmi¯Xq¯ −√−1dηmi¯dηjq¯ξ) .
Now we can compute the Ricci curvature
Ric(Xj, ξ) = g
l¯i
(
g(R(Xl, Xj)ξ, X¯i) + g(R(X¯i, Xj)ξ,Xl)
)
= g l¯i
(
g(R(X¯i, ξ)Xj, Xl) + g(R(X¯i, Xj)ξ,Xl)
)
= 0 +
√−1g l¯i(∇Xjgmq¯dηmi¯)glq¯
=
√−1∇Xjgmi¯dηmi¯
= ∇XjtrgT (dη)
where trgT denotes the trace by the transverse metric g
T . For basic forms, trgT =
1
2
trg.
27
Lastly we want to compute Ric(X¯j, Xk). We start by computing R(Xi, X¯j)Xk.
Since we will eventually pair this with Xl¯, we will neglect any terms in the ξ direction.
R(Xi, X¯j)Xk = ∇Xi∇X¯jXk −∇X¯j∇XiXk −∇[Xi,X¯j ]Xk
= ∇Xi(
√−1dηkj¯ξ +∇TX¯jXk)−∇X¯j(∇TXiXk) + 2
√−1dηij¯∇ξXk
=
√−1dηkj¯∇Xiξ +∇Xi∇TX¯jXk −∇X¯j∇TXiXk − 2gmp¯dηij¯dηkp¯Xm
= −gmp¯dηkj¯dηip¯Xm +∇TXi∇TX¯jXk −∇TX¯j∇TXiXk − 2gmp¯dηij¯dηkp¯Xm.
As [ξ,Xk] = 0, ∇TξXk = piD([ξ,Xk]) = 0. Since [Xi, X¯j] = −2
√−1dηij¯ξ, it follows
that ∇T
[Xi,X¯j ]
Xk = 0. Thus the above can be written as
R(Xi, X¯j)Xk = R
T (Xi, X¯j)Xk − gmp¯
(
dηkj¯dηip¯ + 2dηij¯dηkp¯
)
Xm.
By a computation similar to the one above, modulo the terms in the ξ direction,
R(X¯l, X¯j)Xk = R
T (X¯l, X¯j)Xk + g
mp¯
(
dηkj¯dηml¯ − dηkl¯dηmj¯
)
X¯p.
Now we are ready to compute the Ricci curvature. Using the above we find
Ric(X¯j, Xk) = g
il¯
(
g(R(Xi, X¯j)Xk, Xl¯) + g(R(X¯l, X¯j)Xk, Xi)
)
+ g(R(ξ, X¯j)Xk, ξ)
= RicT (X¯j, Xk)− 2gil¯dηij¯dηkl¯.
The above computations prove our next proposition.
28
Proposition 2.1.1. With respect to a preferred frame, the Ricci tensor of a quasi-
Sasaki structure (ξ, η,Φ, g) satisfies
Ric(ξ, ξ) = 2gil¯gkj¯dηij¯dηkl¯ = |dη|2g, (2.6)
Ric(Xk, ξ) =
√−1∇Xkgij¯dηij¯ = ∇XktrgT (dη), (2.7)
Ric(Xi, X¯l) = Ric
T (Xi, X¯l)− 2gkj¯dηij¯dηkl¯. (2.8)
If our quasi-Sasaki structure is actually Sasaki, then the above equations should
yield the well-known formulae for the Ricci tensor of a Sasaki metric. In the Sasaki
setting, one has dηij¯ = gij¯. Substituting this into the above equations we get
Ric(ξ, ξ) = 2n,
Ric(Xk, ξ) = 0,
Ric(Xi, X¯l) = Ric
T (Xi, X¯l)− 2g(Xi, X¯l),
which is indeed correct.
Recall that an Einstein metric g is one where Ricg = λg for some constant
λ ∈ R. Proposition 2.1.1 gives us more information about the nature of a quasi-
Sasaki-Einstein metric. Since ξ has unit length, Lξ ⊥ D and g(Xi, X¯j) = gT (Xi, X¯j),
proposition 2.1.1 gives us our next result.
Proposition 2.1.2. A quasi-Sasaki metric is Einstein with Ricg = λg if and only if
|dη|2g = λ, (2.9)
∇XktrgT (dη) = 0, (2.10)
RicT (Xi, X¯l)− 2gkj¯dηij¯dηkl¯ = λgT (Xi, X¯l). (2.11)
29
Remark 2.1.3. The above proposition lends us some useful information. First we see
that λ ≥ 0 and that the norm of dη is constant over M . If λ > 0 (i.e. dη 6= 0),
then by Myers’s theorem M is compact. Since dη is basic, the second equation tells
us that the trace of dη is constant. Since dη is an exact (1,1)-form, this implies that
δ(dη) = 0 and therefore dη is harmonic. Finally, the trace of the third equation by
the transverse metric yields RT = λ(n+ 1) ≥ 0. Thus the transverse scalar curvature
is constant. This implies that the transverse Ricci form ρT is harmonic.
Proposition 2.1.4. If a Ka¨hler metric has constant scalar curvature then the Ricci
form is harmonic.
Proof. Let g = gαβ¯dz
α ⊗ dzβ¯ be a Ka¨hler metric and ρ = √−1Rαβ¯dzα ∧ dzβ¯ the
Ricci form. Recall that ρ is a closed (1,1)-form and Rαβ¯ = −∂α∂β¯ log det(g). Using
δ = −gklι∂k∇∂l we compute that
δρ =
√−1gλµ¯
(
Rαµ¯/λdz
α −Rλβ¯/µ¯dzβ¯
)
=
√−1gλµ¯
(
Rλµ¯/αdz
α −Rλµ¯/β¯dzβ¯
)
=
√−1
(
∂R
∂zα
dzα − ∂R
∂zβ¯
dzβ¯
)
.
So if the scalar curvature R is constant, then δρ = dρ = 0. Hence ρ is harmonic.
Remark 2.1.5. If the manifold is compact, then the converse to the above is true. That
is, a compact Ka¨hler metric with harmonic Ricci form has constant scalar curvature.
To see this, let Λ denote the dual Lefschetz operator and recall that Λρ = R and Λ
commutes with the Hodge Laplacian 4 = dδ + δd. So if ρ is harmonic then so is R.
Harmonic functions on compact manifolds are constant. Hence R is constant.
30
2.2. The Regular Case
Let (M, ξ, η,Φ, g) be a quasi-Sasaki manifold. In this section we assume that
the orbits of ξ are compact and the induced S1 action is free. Hence Fξ is regular.
By theorem 1.3.6, M is the total space of a principal circle bundle over the Ka¨hler
manifold B := M/Fξ. The transverse geometry of M is the Ka¨hler geometry of
(B, gT ). For a manifold B (not necessarily Ka¨hler), we let P(B, S1) denote the
collection of all principal circle bundles pi : P → B. We will quickly review some facts
about P(B, S1). For a reference to these facts, see chapter two of [4].
Theorem 2.2.1. There is a binary operation on P(B, S1) giving it the structure of
a group isomorphic to H2(B,Z).
For any integer m 6= 0, the cyclic group Zm := Z/mZ is a subgroup of S1. Thus,
for any P ∈ P(B, S1) there is a Zm action on P and P/Zm is a principal S1/Zm
bundle over B. Since S1/Zm ' S1, we can consider P/Zm ∈ P(B, S1).
Theorem 2.2.2. For P ∈ P(B, S1) and an integer m 6= 0, P/Zm ' mP. This implies
that if P is simply connected and |m| > 1, then P 6= mP ′ for any P ′ ∈ P(B, S1).
We let V := ker(pi∗) ⊂ TP and call it the vertical distribution. For θ ∈ S1,
we define θˆ : P → P by p 7→ θ · p. A connection on P ∈ P(B, S1) is a smooth
distribution H ⊂ TP such that TP = H ⊕ V and θˆ∗(H) = H for all θ ∈ S1. We call
H the horizontal distribution. A point p ∈ P induces a map pˆ : S1 → P by θ 7→ θ · p.
The image of pˆ is the orbit through p. It is diffeomorphic to pi−1(pi(p)) ' S1. At
1 ∈ S1, the differential of pˆ is a map pˆ∗ : R → TpP. The image of pˆ∗ is Vp. The
connection form of H is the S1 invariant 1-form η on P such that for X ∈ TpP ,
η(X) = t where pˆ∗(t) is the vertical part of X. Note that H = ker(η). For a principal
S1 bundle, the curvature form of the connection H is dη.
31
Lemma 2.2.3. Given a connection form η on P ∈ P(B, S1), there exists a unique,
closed 2-form Ω on B such that 1
2pi
[Ω] ∈ H2(B,Z) and dη = pi∗Ω. The cohomology
class of Ω is independent of the choice of connection form.
The cohomology class 1
2pi
[Ω] is called the characteristic class of the principal
circle bundle P . Theorem 2.2.1 shows that P(B, S1) is in bijection with H2(B,Z).
The above lemma and the following theorem due to Kobayashi in [28] show that
specifying a connection form η on P corresponds to choosing a representative of the
characteristic class. Principal circle bundles are classified by their characteristic class.
Theorem 2.2.4. Let Ω be a closed 2-form on B with 1
2pi
[Ω] ∈ H2(B,Z). Then there
is P ∈ P(B, S1) and a connection form η on P such that dη = pi∗Ω.
If g is Einstein with Ricg = λg, the first equation in proposition 2.1.2 shows that
λ ≥ 0. If λ = 0 then dη = 0 and by (2.11), RicgT = 0. Therefore, M is cosymplectic
and B is Calabi-Yau. We know already that M is locally the product of a circle and a
Ka¨hler manifold (see remark 1.2.15), but in this case we can say more. Since dη = 0,
the characteristic class 1
2pi
[Ω] = 0. This means that M is the trivial circle bundle over
B. Therefore, M = S1 ×B. This proves our next theorem.
Theorem 2.2.5. If (M, g) is a quasi-Sasaki manifold with Ricg = 0 and Fξ is regular
with compact leaves, then M = S1 ×B where B is a Calabi-Yau manifold and g is a
product metric.
Now we assume that g is Einstein with Ricg = λg for λ > 0. Then by Myers’s
theorem M is compact, in which case the regularity of Fξ is equivalent to the
assumption that the orbits of ξ are compact and the S1 action is free. The (1,1)-
tensor θ associated to dη is defined implicitly by
g(θ(X), Y ) = dη(X, Y ).
32
If we write θ(Xi) = θ
k
iXk, then θ
k
i =
√−1gkj¯dηij¯. With this and proposition 2.1.2,
the condition that g is Einstein with Ricg = λg is equivalent to
|dη|2g = λ,
4Bdη = 0,
RicgT (X, Y ) = λg
T (X, Y ) + 2gT (θ(X), θ(Y )).
From the third equation above we see that RicgT > 0. By theorem 1 of [27], B is
simply connected. Up to a factor of 2pi, the Ricci form of gT is a representative of
the first Chern class of B. Thus c1(B) > 0. Before we give the main theorem of
this section, we will prove the existence of quasi-Sasaki-Einstein metrics on certain
principal circle bundles. These examples illustrate the general case.
Let (Bnii , gi), i = 1, . . . , k, be Ka¨hler-Einstein manifolds with complex dimension
ni and c
1(Bi) > 0. We can write c
1(Bi) = qiαi where qi ∈ Z+ and αi ∈ H2(Bi,Z)
is an indivisible class. By scaling the metrics appropriately, we can assume that
Ricgi = qigi. Then we have 2pic
1(Bi) = [ρi] = qi[ωi] and, hence,
1
2pi
[ωi] = αi is an
indivisible, integral class. Let B = B1 × · · · × Bk and pii : B → Bi the projections.
Pick integers bi, not all zero, and set
Ω = b1pi
∗
1ω1 + · · ·+ bkpi∗kωk.
Then 0 6= 1
2pi
[Ω] ∈ H2(B,Z), so by theorem 2.2.4 there is a nontrivial principal circle
bundle pi : M → B and a connection form η on M such that dη = pi∗Ω.
Theorem 2.2.6. There exists a quasi-Sasaki-Einstein metric of positive scalar
curvature on the principal circle bundle described above.
33
The proof of the theorem is mostly the same as the proof of theorem 1.4 in [41].
We use the construction due to Wang and Ziller in [41] to produce the Einstein metrics
and then we show that they are in fact quasi-Sasaki. For the sake of completeness,
we include the construction here.
Proof. We consider a metric on M of the form g = a2η ⊗ η + pi∗h where
h = a1pi
∗
1g1 + · · ·+ akpi∗kgk
for constants a 6= 0 and ai > 0. The choice of a is inconsequential. We will show
that we can choose the ai so that g is Einstein, but first let us verify that this indeed
defines a quasi-Sasaki metric.
The 1-form ηa = aη determines the contact bundle D = ker(ηa) = ker(η). Since
it is the horizontal distribution of the connection, it is invariant under the S1 action
and TM = D ⊕ Lξ where Lξ = ker(pi∗) is the vertical distribution and ξ is a vector
field on M with η(ξ) = 1. Let ξa = a
−1ξ. Let J be the complex structure on B.
Define an endomorphism Φ : TM → TM by Φ := p˜i ◦ J ◦ pi∗, where p˜i denotes the
unique horizontal (with respect to η) lift of a vector field on B. Then Φξ = 0 and for
horizontal vector fields X ∈ D, we have Φ2X = −X. Therefore Φ2 = −I + ξa ⊗ ηa.
As h is invariant under J , it is easy to show that pi∗h(ΦX,ΦY ) = pi∗h(X, Y ). Hence
g(ΦX,ΦY ) = g(X, Y )− ηa(X)ηa(Y ).
Thus (ξa, ηa,Φ, g) is an almost contact metric structure on M . Since the base manifold
B is complex, a computation shows that the structure is normal. The fundamental
2-form ω := g(·,Φ·) = pi∗h(·,Φ·) = pi∗(a1pi∗1ω1 + · · ·+ akpi∗kωk), where ωi is the Ka¨hler
34
form of gi. Since each ωi is closed, it follows that dω = 0. Therefore, (ξa, ηa,Φ, g) is
quasi-Sasaki. Note that it is Sasaki precisely when ai = abi > 0 for all i = 1, . . . , k.
If the metric g is to satisfy Ricg = λg, then by proposition 2.1.2, we must have
λ = |dη|2g = 2a2
k∑
i=1
ni
(
bi
ai
)2
> 0. (2.12)
Since the Ka¨hler forms ωi are harmonic, we have that dη is harmonic. Thus equation
(2.10) is satisfied. The contact bundle has a natural splitting D = D1 ⊕ · · · ⊕ Dk
where pi∗Di = TBi. On Di we have dη = bipi∗i ω1. So to satisfy equation (2.11), for
i = 1, . . . , k, we need
qi
ai
= λ+ 2a2
(
bi
ai
)2
. (2.13)
Combining equations (2.12) and (2.13), for each j = 1, . . . , k, we have
(
k∑
i=1
ni
(
bi
ai
)2)(
qj
aj
− λ
)
= λ
(
bj
aj
)2
. (2.14)
Introducing the new variables sj :=
qj
λaj
, equation (2.14) becomes
(
k∑
i=1
ni
(
bi
qi
)2
s2i
)
(sj − 1) =
(
bj
qj
)2
s2j . (2.15)
Since not all bj = 0, this shows that sj ≥ 1 with equality if and only if bj = 0. We
can rearrange (2.15) to get
(
1 +
1
nj
− sj
) k∑
i=1
ni
(
bi
qi
)2
s2i =
1
nj
∑
i 6=j
ni
(
bi
qi
)2
s2i . (2.16)
35
This shows that sj ≤ 1 + 1nj with equality if and only if bi = 0 for all i 6= j.
Let us assume that all bj 6= 0; we will comment on the general case later. Then
1 < sj < 1 +
1
nj
≤ 2. If we multiply equation (2.15) by nj and then sum on j, we get
k∑
j=1
nj(sj − 1) = 1. (2.17)
Thus, solving the system (2.15) is equivalent to solving
(
bj
qj
)2 s2j
sj − 1 = c, (2.18)
for each j = 1, . . . , k and some constant c > 0, subject to the constraint (2.17).
On the interval (1, 2], the function f(s) = s
2
s−1 decreases monotonically with
f(s) → ∞ as s → 1+ and f(2) = 4. So we start by choosing c large enough that
c > 4
(
bj
qj
)2
for all j. Then there is a unique solution to (2.18) with 1 < sj < 2.
If c is extremely large, then all sj must be very close to 1. But then we will have∑k
j=1 nj(sj − 1) < 1. As c decreases, the sj increase monotonically. So we decrease
c until the largest sj = 2. Then we will have
∑k
j=1 nj(sj − 1) > 1 since nj ≥ 1.
Thus there is a unique value of c for which (2.18) has a solution with 1 < sj < 2 and∑k
j=1 nj(sj − 1) = 1. With the values of the sj in hand, we return to equation (2.12)
and solve for λ. Once we know sj and λ, we have aj and we are done.
Remark 2.2.7. If one of the bi, say b1 = 0, then on D1 we have dη = 0 and the bundle
over B1 is trivial so M splits as M
′×B1. Here pi′ : M ′ → B2× · · · ×Bk is a principal
S1 bundle with connection form η′ such that dη′ = pi′∗Ω. See section 2.5. We should
also note that if any of the bi = 0, then the quasi-Sasaki structure has rank < 2n+ 1.
36
Example 2.2.8. Suppose the base B = Bn1 consists of a single Ka¨hler-Einstein
manifold and consider the principal circle bundle pi : M → B with dη = bpi∗ω1, b 6= 0.
By theorem 2.2.6,
g = η ⊗ η + 2b
2(n+ 1)
q
pi∗g1
is quasi-Sasaki-Einstein with
Ricg =
nq2
2b2(n+ 1)2
g.
The quasi-Sasaki structure has full rank 2n + 1. If we set a = q
2b(n+1)
and ηa = aη,
then a2g = ηa ⊗ ηa + abpi∗1g. Since dηa = abpi∗ω1, we see that a2g is Sasaki-Einstein.
Example 2.2.9. Recall that CP n with the Fubini-Study metric ωFS is Ka¨hler-
Einstein with ρ = (n + 1)ωFS. It is well known that H
2(CP n,Z) ' Z and that
α = 1
2pi
[ωFS] is a generator. Thus c
1(CP n) = (n + 1)α and every principal circle
bundle pi : P → CP n has characteristic class 1
2pi
[Ω] = bα for some integer b. A classic
example is the Hopf fibration pi : S2n+1 → CP n. Since the total space S2n+1 is simply
connected, by theorem 2.2.2, b = ±1. We choose an orientation so that b = 1.
Let Bi = CP ni , i = 1, . . . , k, and B = B1 × · · · × Bk. Then H2(B,Z) ' ⊕ki=1Z
with generators αi ∈ H2(Bi,Z). Thus every principal circle bundle pi : P → B has
characteristic class 1
2pi
[Ω] =
∑k
i=1 biαi for some integers bi. The total space P is
simply connected if and only if the characteristic class is indivisible if and only if
gcd(b1, . . . , bk) = 1.
For a concrete example, take ni = 1 for i = 1, 2, 3, b1 = 1, b2 = −1 and b3 = 0.
Then qi = 2 and it is not too hard to solve the equations from theorem 2.2.6. Working
through them, we find that a1 = a2 = 3 and a3 =
9
2
. Thus the fundamental 2-form
37
ω = pi∗(3pi∗1ωFS + 3pi
∗
2ωFS +
9
2
pi∗3ωFS) and the metric g = η ⊗ η + ω(Φ·, ·) is quasi-
Sasaki-Einstein with Ricg =
4
9
g.
This example illustrates some interesting features of quasi-Sasaki-Einstein
metrics. First, the rank of this quasi-Sasaki structure is 5, whereas full rank is 7.
Thus this quasi-Sasaki structure is not a deformation of a Sasaki structure. However,
since b3 = 0, the total space P splits as P
′×B3 where pi′ : P ′ → B1×B2 is a principal
circle bundle with connection form η′ such that dη′ = pi′∗Ω. Thus g′ = η⊗η+ω′(Φ·, ·),
where ω′ = pi′∗(3pi∗1ωFS + 3pi
∗
2ωFS), is a quasi-Sasaki-Einstein metric on P
′ with
Ric′g =
4
9
g′. This quasi-Sasaki structure does indeed have full rank, but still it cannot
be a deformation of a Sasaki structure because dη′(Φ·, ·) is not positive definite; it is
negative definite over B2.
Now we will present the main theorem of this section. It is a consequence of
theorem 1 in [40].
Theorem 2.2.10. If (M, ξ, η,Φ, g) is a regular, quasi-Sasaki-Einstein manifold with
positive scalar curvature, then M is a principal circle bundle over a compact Ka¨hler
manifold B, where B is a product of Ka¨hler-Einstein manifolds and the metric g is
among those constructed in theorem 2.2.6.
Proof. Since g is Einstein with positive scalar curvature, M is compact. Then the
regularity of Fξ implies that the orbits of ξ are compact, so by theorem 1.3.6 M is a
principal circle bundle over a compact Ka¨hler manifold B with η as the connection
form. By lemma 1.2.7, dη(Φ·,Φ·) = dη. Thus the curvature form of the circle bundle
is of type (1,1). Now by theorem 1 in [40], B is a product of Ka¨hler-Einstein manifolds
and (M, g) is among those constructed in theorem 2.2.6.
38
Theorems 2.2.5 and 2.2.10 give us a clear picture of the possible quasi-Sasaki-
Einstein metrics in the regular case. As we have seen, they are very rigid and do not
offer us “new” Einstein metrics that we didn’t already know about.
2.3. The quasi-Regular Case
In this section we assume, more generally, that the orbits of ξ are compact and
the induced S1 action is locally free. Then Fξ is quasi-regular. By theorem 1.3.6, M is
the total space of a principal S1-orbibundle over the Ka¨hler orbifold B := M/Fξ with
connection form η and curvature form dη = pi∗Ω. Here Ω is a closed (1,1)-form with
1
2pi
[Ω] ∈ H2orb(B,Z). The treatment here closely follows that of the previous section.
We will not dive too deeply into the general theory of orbifolds and orbibundles, but
only reference pertinent information. For definitions and fundamental results, see [5]
and [26] and the references therein.
Many facts about principal circle bundles have an orbibundle counterpart. For
instance, theorem 4.3.15 of [5] says that the (isomorphism classes of) principal S1-
orbibundles over an orbifold B are in one-to-one correspondence with the elements of
H2orb(B,Z) and the bijection is given by the orbifold first Chern class. In particular,
the trivial orbibundle corresponds to the trivial orbifold cohomology class. Also, it is
not hard to produce orbibundle versions of lemma 2.2.3 and theorem 2.2.4. The same
argument as in the previous section gives us the orbifold version of theorem 2.2.5.
Theorem 2.3.1. If (M, g) is a quasi-Sasaki manifold with Ricg = 0 and Fξ has
compact leaves, then M = S1 × |B| where |B| is the underlying topological space of a
Calabi-Yau orbifold B.
Remark 2.3.2. Since M is a manifold, |B| must be a manifold too.
39
The orbifolds given by theorem 1.3.6 (i.e. those coming from the leaf space of a
quasi-regular quasi-Sasaki structure) are locally cyclic, meaning that all of the local
uniformizing groups are cyclic groups. Furthermore, given an orbifold B, theorem
4.3.15 of [5] also states that the total space of an S1-orbibundle over B is a smooth
manifold when all of the local uniformizing groups of B inject into S1. For these
reasons, we will only concern ourselves with locally cyclic orbifolds. As a sort of
converse to theorem 1.3.6, we have the following:
Proposition 2.3.3. Given a locally cyclic Ka¨hler orbifold B with 1
2pi
[Ω] ∈ H2orb(B,Z),
there is a quasi-Sasaki structure on the principal S1-orbibundle pi : M → B
corresponding to 1
2pi
[Ω] with connection form η such that dη = pi∗Ω.
Proof. Let ξ be the unit vector field along the fibers and consider the metric on M
given by g = η ⊗ η + pi∗gB where gB is a Ka¨hler metric on B. Define Φ = p˜i ◦ J ◦ pi∗
where J is the complex structure on B and p˜i is the horizontal lift with respect to η.
Then Φ2 = −I + ξ ⊗ η and Φ∗g = g − η ⊗ η. It follows that M is quasi-Sasaki.
We also have the orbi-analogue of theorem 2.2.6. Since the equations we must
solve are the same, the proof follows mutatis mutandis, and so we omit it.
Theorem 2.3.4. Let (Bnii , ωi), i = 1, . . . , k, be locally cyclic Ka¨hler-Einstein orbifolds
with c1orb(Bi) > 0. We can assume that the metrics have been normalized so that
1
2pi
[ωi] is an indivisible, integral class and c
1
orb(Bi) = qi
1
2pi
[ωi] for some qi ∈ Z+. Let
B = B1 × · · · × Bk and pii : B → Bi the projections. Pick integers bi, not all zero,
and set
Ω = b1pi
∗
1ω1 + · · ·+ bkpi∗kωk.
40
Then there is a quasi-Sasaki-Einstein metric with positive scalar curvature on the
principal S1-orbibundle pi : M → B corresponding to 1
2pi
[Ω] with connection form η
such that dη = pi∗Ω.
We conclude this section with the orbi-analogue of the rigidity theorem 2.2.10.
We will show that theorem 1 in [40] can be carried over to the orbifold setting and,
as in the previous section, it implies our theorem.
Theorem 2.3.5. If (M, g) is a quasi-regular, quasi-Sasaki-Einstein manifold with
positive scalar curvature, then M is a principal circle orbibundle over a compact
Ka¨hler orbifold B, where B is a product of Ka¨hler-Einstein orbifolds and the metric
g is among those constructed in theorem 2.3.4.
Proof. Since (M, g) is compact, quasi-regularity of Fξ implies that the orbits of ξ
are compact. By the structure theorem 1.3.6, M is a principal S1-orbibundle over
a compact Ka¨hler orbifold (B, gT ) with connection form η. By lemma 1.2.7, the
curvature form dη is of type (1,1). The conditions on dη and ρT imposed by the
Einstein equations are the same as the conditions on the curvature form and Ricci
form in [40]. Proposition 1 in [40] is proved by local computations, exploiting these
conditions. Thus we would have the same proposition in orbifold setting.
Proposition 2.3.6. Under the assumptions of the theorem, there is a global,
orthogonal decomposition TB = E1 ⊕ · · · ⊕ Er where the Ei are the eigenspaces of
the Ricci curvature. The eigenvalues are constant and any sum of the eigenspaces is
integrable and invariant under the complex structure.
The leaves of the foliation TB = E1⊕· · ·⊕Er have an induced Ka¨hler structure
and the Einstein equations for g show that they have positive Ricci curvature. Thus
the leaves are compact and simply connected by Myers’s theorem and Kobayashi’s
41
theorem in orbifold setting, respectively. By the same argument as in step 3 of
proposition 2 of [40] (using the local stability theorem of foliations with compact
leaves), we can find a local frame for E1 consisting of vector fields which are foliate
with respect to E⊥1 . Then we can perform the same local computation as in [40] to
show that both E1 and E
⊥
1 are totally geodesic. Then it follows that E1 and E
⊥
1
are parallel. Iterating this argument we find that each Ei is parallel. Then, since B
is compact (hence complete) and simply connected, by the de Rham decomposition
theorem for orbifolds (see [26]) B splits isometrically as a product of Ka¨hler orbifolds
B = B1 × · · · × Br according to the decomposition of TB. Hence B is a product of
Ka¨hler-Einstein orbifolds with c1orb(Bi) > 0. Then since the curvature form is of type
(1,1), it follows that the characteristic class of the S1-orbibundle must be of the form
in theorem 2.3.4.
2.4. Transverse Curvature Conditions
In the previous two sections we assumed that all of the orbits of ξ were compact.
We want to make no assumptions on the orbits of the Reeb field, but the irregular
case, where M is compact and ξ has noncompact orbits, is inherently more difficult
to deal with. The difficulty comes from the potential unruliness of the leaf space
M/Fξ. It may not have a tractable structure; for instance, it may not even be a
Hausdorff space! Since we are unable to work with the irregular case directly, to get
a handle on it we will assume that the transverse curvature satisfies a certain non-
negativity condition. In this section, we introduce the curvature conditions that we
will work with and mention a few of their topological implications. We first consider a
curvature condition known as non-negative quadratic orthogonal bisectional curvature
(NQOBC).
42
Definition 2.4.1. A Ka¨hler manifold of complex dimension n has NQOBC at a point
x ∈M if for any unitary frame {e1, . . . , en} of T 1,0x M and any real numbers α1, . . . , αn,
n∑
i,j=1
Ri¯ijj¯(αi − αj)2 ≥ 0.
We say that the manifold has NQOBC if it does so at every point x ∈M .
For more information about Ka¨hler manifolds with NQOBC, see the paper [8].
The primary reason why we are interested in this curvature condition is for the
following theorem:
Theorem 2.4.2. If a closed (compact with no boundary) Ka¨hler manifold M has
NQOBC, then every harmonic (1,1)-form is parallel.
Proof. The Bochner formula for (1,1)-forms on a Ka¨hler manifold is
4αij¯ = −∆αij¯ − 2Rij¯kl¯αlk¯ +Rik¯αkj¯ +Rkj¯αik¯.
Here 4 = dδ + δd is the Hodge laplacian and ∆ = tr(∇2) is the tensor laplacian. If
4α = 0, then the Bochner formula yields
αij¯∆αij¯ = −2Rij¯kl¯αlk¯αij¯ +Rik¯αkj¯αij¯ +Rkj¯αik¯αij¯. (2.19)
Since α is a (1,1)-form, A := α(J ·, ·) is a symmetric bilinear form that is invariant
under the complex structure J . Hence A defines a Hermitian form and thus there
is a local unitary frame {ei, ei} that diagonalizes A. So with respect to this frame,
43
αi¯i := α(ei, ei) are the only nonzero components of α and equation (2.19) becomes
αi¯i∆αi¯i = −2Ri¯ikk¯αkk¯αi¯i +Ri¯iα2i¯i +Rkk¯α2kk¯
= Ri¯ikk¯(αi¯i − αkk¯)2. (2.20)
Integrating equation (2.20) and using integration by parts, we get
∫
M
−|∇α|2 dV =
∫
M
Ri¯ikk¯(αi¯i − αkk¯)2 dV.
From this, the assumption of NQOBC implies that ∇α = 0.
If we assume that the transverse Ka¨hler metric of a quasi-Sasaki structure has
non-negative quadratic orthogonal transverse bisectional curvature, then the following
corollary is immediate.
Corollary 2.4.3. If the transverse Ka¨hler metric of a closed quasi-Sasaki manifold
has NQOBC, then every basic, harmonic (1,1)-form is transversely parallel. That is,
if α is a basic (1,1)-form and 4Bα = 0, then ∇Tα = 0.
The other curvature condition that we will consider is transverse holomorphic
bisectional curvature (THBC). This is just the holomorphic bisectional curvature of
the transverse Ka¨hler metric.
Definition 2.4.4. Let σ be a plane in Dp ⊂ TpM . We say that σ is Φ-invariant if
Φσ = σ. It follows that σ = span{Xp,ΦXp} for some 0 6= Xp ∈ Dp.
Definition 2.4.5. Let σ1 and σ2 be Φ-invariant planes in Dp ⊂ TpM . The transverse
holomorphic bisectional curvature (THBC) HT (σ1, σ2) is defined by
HT (σ1, σ2) := 〈RT (X,ΦX)ΦY, Y 〉,
44
where X ∈ σ1, Y ∈ σ2 and |X| = |Y | = 1.
It is a straightforward exercise to check that HT (σ1, σ2) is well-defined. That
is, HT (σ1, σ2) depends only on σ1 and σ2, not on the choice of X and Y . We say
that HT ≥ 0 at a point p ∈ M if HT (σ1, σ2) ≥ 0 for any two Φ-invariant planes
σ1, σ2 ⊂ Dp. We say that M has non-negative transverse holomorphic bisectional
curvature if HT ≥ 0 for all p ∈M .
Let σu = span{u,Φu} and σv = span{v,Φv} for some unit vectors u, v ∈ D.
Writing u = U+U¯ and v = V +V¯ where U = 1
2
(u−√−1Φu) and V = 1
2
(v−√−1Φv),
we have HT (σu, σv) = 4〈RT (U, U¯)V, V¯ 〉. So HT ≥ 0 if and only if 〈RT (U, U¯)V, V¯ 〉 ≥ 0
for all U, V ∈ D1,0.
In the Ka¨hler setting, positive and non-negative holomorphic bisectional
curvature has been studied extensively. For example, see any of [2], [18], [22], [29],
[35] or [42]. Positive and non-negative THBC has been studied in the Sasaki setting
in [20] and [21]. The following is a corollary of theorem 4 in [18]. Since the proof
would follow in exactly the same way, we shall omit it.
Corollary 2.4.6. If M is a compact, connected quasi-Sasaki manifold with positive
transverse bisectional curvature, then the second basic Betti number b2B = 1.
Note that NQOBC is a strictly weaker condition than non-negative holomorphic
bisectional curvature. Clearly NHBC implies NQOBC and indeed there are examples
of Ka¨hler manifolds with NQOBC that do not admit any Ka¨hler metrics with NHBC.
2.5. Splitting the Contact Bundle
In this section we show that the presence of transversely parallel (1,1)-forms
induces a splitting of the contact bundle of a quasi-Sasaki manifold. We begin with
a definition.
45
Definition 2.5.1. A subbundle D1 ⊂ D is called invariant or transversely parallel
if for all Y ∈ D1, ∇TXY ∈ D1 for any X ∈ TM . The contact bundle D is reducible
(with respect to gT ) if there are invariant subbundles D1 and D2 and an orthogonal
decomposition D = D1 ⊕ D2. We say that gT is (irreducible) reducible if D is (not)
reducible with respect to gT .
Let (M, g) be a quasi-Sasaki manifold. The fundamental 2-form ω (i.e. the
Ka¨hler form of the transverse metric gT ) is transversely parallel (i.e. ∇Tω = 0).
Suppose that α is another basic, transversely parallel (1,1)-form. Then for X, Y ∈ D,
define the (1,1)-tensor S by the equation
α(X, Y ) = ω(S(X), Y ). (2.21)
Notice that S is self-adjoint with respect to ω. Indeed,
ω(S(X), Y ) = α(X, Y ) = −α(Y,X) = −ω(S(Y ), X) = ω(X,S(Y )).
Hence S is a diagonalizable linear map on each contact space Dp ⊂ TpM . Since both
α and ω are transversely parallel, it follows from (2.21) that S is transversely parallel.
Therefore, the linear map Sp : Dp → Dp has the same set of eigenvalues for every
point p ∈ M . Let {c0, . . . , cr} be the distinct eigenvalues of S. As S is self-adjoint,
the eigenvalues ci ∈ R. If α is not a multiple of ω, then S is not a multiple of the
identity and thus r ≥ 1.
Let Di ⊂ D be the eigenbundle corresponding to the eigenvalue ci of S. Then
D = D0 ⊕ · · · ⊕ Dr (2.22)
46
and the distribution p 7→ Di(p) is transversely parallel. Indeed, let Y ∈ Di(p) and
X ∈ TpM . Since ∇TS = 0, we have S(∇TXY ) = ∇TXS(Y ) = ci∇TXY. Therefore
∇TXY ∈ Di(p) and Di is invariant in the sense of definition 2.5.1. The splitting (2.22)
is orthogonal with respect to ω. Indeed, if X ∈ Di and Y ∈ Dj and i 6= j, then
ciω(X, Y ) = ω(S(X), Y ) = ω(X,S(Y )) = cjω(X, Y ).
Since ci 6= cj, this shows that ω(X, Y ) = 0. Since ci is real, Di is closed under complex
conjugation (i.e. Di = Di). Since both α and ω are (1,1)-forms, they are Φ-invariant.
As S is self-adjoint, we get that
ω(X,S(ΦY )) = ω(S(X),ΦY ) = α(X,ΦY )
= −α(ΦX, Y )
= −ω(S(ΦX), Y ) = −ω(ΦX,S(Y )) = ω(X,ΦS(Y )).
Since ω is non-degenerate on D, this implies that S ◦Φ = Φ◦S. Hence, Di is invariant
under Φ (i.e. ΦDi = Di). With this we see that the splitting of D in (2.22) is
orthogonal with respect to gT and therefore, if α and ω are not proportional to each
other, D is reducible. Corresponding to this splitting, we write ω = ω0 ⊕ · · · ⊕ ωr,
α = α0 ⊕ · · · ⊕ αr and Φ = Φ0 ⊕ · · · ⊕ Φr, where
Φi(X) =

Φ(X) if X ∈ Di,
0 otherwise
.
On Di we have αi = ciωi. Since Di has a Ka¨her structure, Di has even real dimension.
Therefore, Di has a local frame {X1, . . . , Xpi , X¯1, . . . , X¯pi} where Xj ∈ D1,0.
47
Lemma 2.5.2. Let M be a compact quasi-Sasaki manifold with transverse NQOBC.
Then D is reducible if and only if b1,1B > 1.
Proof. If b1,1B > 1 then by the transverse Hodge theorem, we have at least two
distinct, basic, harmonic (1,1)-forms. With the assumption of transverse NQOBC,
these harmonic forms are transversely parallel (corollary 2.4.3) and as we just saw
above, this implies that D is reducible.
Conversely, suppose D = D1 ⊕ D2 and the Di are invariant with respect to gT .
Let ωi = ω|Di . Then ω = ω1 ⊕ ω2 and 0 6= [ωi]B ∈ H1,1B (M,Fξ). Furthermore, since
ωn = (ω1 ⊕ ω2)n 6= 0, [ω1]B and [ω2]B are independent. Thus b1,1B > 1.
2.6. The General Case with a Curvature Condition
In this section we assume that the quasi-Sasaki manifold (M, g) is Einstein with
Ricg = λg. Then dη is harmonic and, as we saw in remark 2.1.3, the transverse
scalar curvature RT = λ(n + 1) is constant. Proposition 2.1.4 then implies that the
transverse Ricci form ρT is harmonic. We also assume further that (M, g) satisfies a
transverse curvature condition at least as strong as NQOBC. Then by corollary 2.4.3,
dη and ρT are transversely parallel. Thus we can split the contact bundle as in the
previous section.
Let’s split the contact bundle with respect to dη. Let c0, . . . , cr, be the distinct
eigenvalues of S (the (1,1)-tensor form of dη). Then we have an orthogonal splitting
of D into transversely parallel distributions, D = D0⊕ · · · ⊕Dr. On each Di we have
dηi = ciωi. If we write ρ
T = ⊕ρTi according to this decomposition, then equation
(2.11) shows that ρTi = (λ + 2c
2
i )ωi ≥ 0. This implies that the splitting of D with
respect to ρT is the same as the splitting of D with respect to dη.
48
If our quasi-Sasaki structure has rank 2p + 1 for p < n, then ci = 0 for some
i. Reindexing if neccessary, we may assume that c0 = 0 and hence dη0 = 0. That
Di is transversely parallel implies that for any X, Y ∈ Di we have piD([X, Y ]) ∈ Di.
Since piLξ([X, Y ]) = −2dη(X, Y )ξ, we see that D0 and Lξ ⊕ Di are involutive, hence
integrable. Again using that the Di are transversely parallel, it is not hard to see that
Lξ ⊕ D1 ⊕ · · · ⊕ Dr is also integrable and thus M is locally a product manifold. In
fact, (M, g) is locally a Riemannian product, as we will see next.
Since gT and dη are both transversely parallel and we have the relations
cig
T (X,ΦiY ) = dηi(X, Y ), it follows that ∇TΦi = 0. We will show that Φ0, in
particular, is parallel with respect to the metric g. That is, ∇Φ0 = 0. Note that this
need not be the case for the other Φi. Remember that by lemma 1.2.13, ∇Φ = 0 if
and only if M is cosymplectic. First we prove the following useful proposition.
Proposition 2.6.1. If D = D0 ⊕ · · · ⊕ Dr admits a splitting such that dηi = ciωi,
then for Y ∈ D, ∇Y ξ = −
∑
i ciΦiY . In particular, if Y ∈ Di then ∇Y ξ = −ciΦY .
Proof. By lemma 1.2.11 we have dη(X, Y ) = −g(X,∇Y ξ). Since dη =
∑
i dηi and
dηi(X, Y ) = cig(X,ΦiY ), the above equation gives
∑
i
g(X, ciΦiY ) = −g(X,∇Y ξ).
Since X was arbitrary, the result follows.
Now we will show that ∇Φ0 = 0. Recall that the transverse Levi-Civita
connection ∇T is defined by
∇TXY := piD (∇XY ) , X ∈ D,
∇Tξ Y := piD [ξ, Y ] .
49
Lemma 2.6.2. The (1,1)-tensor Φ0 is parallel with respect to the metric g.
Proof. As Φ : TM → D and Φξ = 0, for X, Y ∈ D we have
0 = (∇TXΦ0)(Y ) = piD ((∇XΦ0)(Y )) = piD(∇XΦ0(Y )− Φ0(∇XY )).
This says that (∇XΦ0)(Y ) has no part in D. Does it have any part in Lξ? Well,
Φ0 : D0 → D0 and the term ∇XΦ0(Y ) ∈ D because the ξ part is dη(X,Φ0(Y ))ξ = 0
since dη = 0 on D0. Thus, for X, Y ∈ D, we have (∇XΦ0)(Y ) = 0. By proposition
2.6.1 and Φi ◦ Φj = 0 for i 6= j we compute that
(∇XΦ0)(ξ) = ∇XΦ0(ξ)− Φ0(∇Xξ)
=
∑
i
ciΦ0(ΦiX)
= 0.
Thus ∇XΦ0 = 0 for X ∈ D. By lemma 1.2.11, ∇ξΦ = 0 which implies that ∇ξΦ0 = 0
(we could also show this by direct computation). Therefore ∇Φ0 = 0.
Now let Φ˜ :=
∑r
i=1 Φi and D˜ :=
∑r
i=1Di. The map P := −Φ˜2 + ξ ⊗ η is
projection onto Lξ ⊕ D˜ and Q := −Φ20 is projection onto D0. Thus P 2 = P , Q2 = Q
and PQ = QP = 0. As in section ??, F := P − Q defines an almost product
structure. Since Φ = Φ0 + Φ˜ and Φ
2 = −I + ξ ⊗ η, we have F = I + 2Φ20. By
proposition 2.6.2, ∇Φ0 = 0 and this clearly implies that ∇Φ20 = 0. Hence ∇F = 0.
So by proposition 5.0.7, (M, g) is a locally decomposable Riemannian manifold. Thus,
locally, (M, g) is the Riemannian product of manifolds (M0, g0) and (M1, g1) whose
tangent spaces are (isomorphic to) D0 and Lξ ⊕ D˜ respectively. We will show that
(M0, g0) is Ka¨hler-Einstein and (M1, g1) is quasi-Sasaki-Einstein.
50
Pick a point x ∈ M and let M0 be a maximal integral submanifold of D0
containing the point x. Then dimR(M0) = 2(n − p). Define 2-forms ω0 and ω˜ by
ω0(X, Y ) = g(X,Φ0Y ) and ω˜(X, Y ) = g(X, Φ˜Y ). Then ω0 = ω− ω˜. Since ω is closed
and ω˜ =
∑r
i=1 c
−1
i dηi is proportional to dη, we have dω0 = 0. Because Φ0 has rank
2(n − p), ω0|D0 is non-degenerate. We have (Φ0|D0)2 = −I, ∇Φ0 = 0 and g|D0 is
compatible with Φ0. Therefore, (M0,Φ0|D0 , g|D0) is Ka¨hler. To see that the metric
on M0 is Einstein we refer back to equations (2.8) and (2.11). Since dη = 0 on D0,
these equations say precisely that g|D0 is an Einstein metric on M0
Let M1 be a maximal integral submanifold of Lξ ⊕ D˜ containing x. Then
dimR(M1) = 2p + 1. Since η ∧ (dη)p 6= 0 on Lξ ⊕ D˜, restricting η to Lξ ⊕ D˜ gives
M1 a contact structure. As Φ and Φ˜ agree on D˜ and both are zero on Lξ, it is easily
verified that the restriction of (ξ, η, Φ˜, g) to Lξ⊕D˜ gives M1 an almost contact metric
structure. Now we check the normality condition.
Recall that NΦ + ξ ⊗ 2dη = 0 and since ∇Φ0 = 0, we have NΦ0 = 0. Take
X, Y ∈ Lξ ⊕ D˜. Since Φ˜ = Φ− Φ0 and Φ0X = Φ0Y = 0, we compute that
NΦ˜(X, Y ) + 2dη(X, Y )ξ = Φ0[X,ΦY ] + Φ0[ΦX, Y ]−ΦΦ0[X, Y ]−Φ0Φ[X, Y ]. (2.23)
As the distribution Lξ ⊕ D˜ is integrable, it is involutive. Hence, each of the
bracket terms in (2.23) are in Lξ ⊕ D˜. Since Φ0 annihilates Lξ ⊕ D˜, it follows that
NΦ˜(X, Y ) + 2dη(X, Y )ξ = 0. Therefore, M1 has a normal almost contact metric
structure. Clearly, the fundamental 2-form ω˜ associated to this almost contact metric
structure is closed. Hence (ξ, η, Φ˜, g) restricted to Lξ ⊕ D˜ gives M1 a full rank quasi-
Sasaki structure. Moreover, the metric is Einstein. Summarizing, we have proved:
51
Theorem 2.6.3. Let M2n+1 be a quasi-Sasaki-Einstein manifold of rank 2p + 1 for
p < n with transverse NQOBC. Then (M, g) is locally the Riemannian product of
a Ka¨hler-Einstein manifold (M0, g0) with dimR(M0) = 2(n − p) and quasi-Sasaki-
Einstein manifold (M1, g1) of rank 2p+ 1. Moreover, the transverse Ka¨hler metric of
g1 is a product of transverse Ka¨hler-Einstein metrics.
Remark 2.6.4. That M is locally a product manifold does not require the transverse
curvature assumption. The Einstein equations alone are enough to get this and the
proof can go along the same lines as the proof of proposition 1 in [40]. However,
to get a local Riemannian product, the almost product structure must be parallel
(proposition 5.0.7). At this time, we don’t know if this can be proved solely on the
strength of the Einstein equations in the irregular case.
For the remainder of this section we may assume that our quasi-Sasaki structures
have full rank and if D is reducible, then none of the Di are integrable on their own.
If D is irreducible, then the transversely parallel forms ω, dη and ρT must all be
proportional to each other. That is, dη = cω and ρT = µω for some constants c and
µ. Since c 6= 0, our quasi-Sasaki structure is simply a scaling of a Sasaki-Einstein
structure. By replacing η with cη, replacing ξ with c−1ξ and scaling the transverse
metric by c2 we get a Sasaki-Einstein structure.
Combining corollaries 2.4.3, 2.4.6 and lemma 2.5.2 we get:
Theorem 2.6.5. A quasi-Sasaki-Einstein manifold with positive transverse holomorphic
bisectional curvature is, up to homothey, a Sasaki-Einstein manifold.
Now by [21] we get that
52
Corollary 2.6.6. A quasi-Sasaki-Einstein manifold with positive transverse
holomorphic bisectional curvature is diffeomorphic to S2n+1 and the metric can be
deformed to the round metric.
Lemma 2.6.7. A reducible quasi-Sasaki manifold with positive Ricci curvature is
quasi-regular.
Proof. Suppose that M is quasi-Sasaki with positive Ricci curvature and D = D1⊕D2
is reducible. Then the Di are transversely parallel and this implies that Lξ ⊕ Di is
an integrable subbundle of TM . Since the Di are Φ-invariant, using proposition 2.6.1
and equation (2.4), it is not hard to show that the leaves of Lξ ⊕ Di are totally
geodesic submanifolds of M . Hence the induced metric on them is complete. By
assumption, D1 and D2 have positive Ricci curvature. So by Myers’s theorem, the
leaves of Lξ ⊕ Di are compact. An integral curve of ξ is a transversal intersection
of a leaf from Lξ ⊕ D1 and a leaf from Lξ ⊕ D2. Thus the ξ orbits are compact and
therefore Fξ is quasi-regular.
From this lemma we see that an irregular quasi-Sasaki-Einstein manifold with
positive scalar curvature is irreducible. Hence we have the following corollary:
Corollary 2.6.8. An irregular quasi-Sasaki-Einstein manifold with positive scalar
curvature and transverse NQOBC is a scaling of a Sasaki-Einstein manifold.
53
CHAPTER III
TRANSVERSE KA¨HLER-RICCI FLOW
The aim of this chapter is to extend results in Ka¨hler-Ricci flow to quasi-Sasaki
manifolds. The transverse Ka¨hler-Ricci flow is the normalized Ka¨hler-Ricci flow of the
quasi-Sasaki manifold’s transverse Ka¨hler metric. We give a proof of its short-time
existence. The long-time existence follows from the work in [36]. In comparison with
[7], we can show that when the basic first Chern class c1B ≤ 0, the flow converges to a
transverse Ka¨hler-Einstein metric. This can be done just as in [36], but we choose not
to pursue it here. Rather, we are interested in the case that c1B > 0. In [31], Perelman
introduced hisW functional. TheW functional is monotone increasing along the Ricci
flow and this property has many important applications. In particular, for a Ka¨hler
manifold with positive first Chern class, the monotonicity of the W functional can
be used to prove uniform bounds on the scalar curvature and the diameter of the
manifold along the flow. This was first presented in [34].
We have two primary goals in this chapter. The first is to establish uniform
bounds on the transverse scalar curvature and the transverse diameter along the
transverse Ka¨hler-Ricci flow when c1B > 0. This will imply uniform bounds on the
scalar curvature and the diameter of the manifold along the flow. We first define
the WT functional, which is the analogue of the W functional for the quasi-Sasaki
setting. We show that it is monotonic along the flow and then we proceed in a way
similar to [34]. A subtle difference in our situation is that the WT functional is
only involved with basic functions. That is, functions which are constant in the ξ
direction. This poses some problems as the distance function on the manifold is not
basic. To deal with this, we utilize the transverse distance function from chapter I.
54
This approach was taken in [12] and [19], where these results were, independently,
established for Sasaki manifolds. The essential ingredient is really the transverse
Ka¨hler structure of the manifold. The additional contact structure afforded by Sasaki
manifolds is not required. Thus we can prove these results in the quasi-Sasaki setting
as well. On a compact Ka¨hler manifold, it is known that if the initial metric has
nonnegative bisectional curvature, then this property is preserved along the Ka¨hler-
Ricci flow. The second goal of this chapter is to show that nonnegativity of the
transverse (holomorphic) bisectional curvature, as defined in 2.4.5, is preserved along
the transverse Ka¨hler-Ricci flow.
This chapter is organized as follows: First we define the transverse Ka¨hler-
Ricci flow and prove its short-time existence. Next we give an evolving quasi-
Sasaki structure, compatible with the evolving transverse metric, and provide some
commentary on how the rank of this structure may change along the flow. Next
we define the WT and µT functionals and prove their monotonicity. After that we
show that the transverse scalar curvature and Ricci potential are bounded along the
flow by the diameter of the manifold. Then we prove a uniform upper bound on the
transverse diameter of the manifold along the flow. This implies a uniform bound on
the diameter of the manifold, and hence on the scalar curvature and Ricci potential.
The presentation in sections 3 and 4 is much like that in [34], but here we provide
more details for the arguments and computations. The last section is devoted to
studying the transverse Ka¨hler-Ricci flow when the initial metric has non-negative
(positive) transverse bisectional curvature. Some of the technical lemmas that get
used in this chapter are collected in the appendix.
55
3.1. Transverse Ka¨hler-Ricci Flow
Following the introduction of Hamilton’s Ricci flow, similar evolution equations
were considered on Ka¨hler manifolds. Since the 1980’s, Ka¨hler-Ricci flow (KRF) has
been a very fruitful area of mathematics. Many deep and beautiful results are known
and KRF is still an active area of research today. It is well-known in Ka¨hler geometry
via the Chern-Weil theory, that the first Chern class, c1, of a Ka¨hler manifold is
represented by a multiple of the Ricci form and that the de Rham cohomology class
of the Ricci form is independent of the Ka¨hler metric. Thus a necessary condition
for the existence of a Ka¨hler-Einstein metric ω is that c1 = λ[ω]. Hence c1 must be
positive definite, negative definite or null depending on whether λ is positive, negative
or zero respectively. The existence of Ka¨hler-Einstein metrics in the cases c1 < 0 and
c1 = 0 was proved independently by Aubin in [1] and Yau in [43]. In [7], Cao showed
that the normalized KRF converges to a Ka¨hler-Einstein metric when c1 ≤ 0. When
c1 > 0, things are more complicated and there are obstructions to the existence of
Ka¨hler-Einstein metrics, see [16]. However, in certain cases, it is known that the KRF
converges to a Ka¨hler-Ricci soliton when c1 > 0. See [33] and [38].
In [36], Smoczyk, Wang and Zhang defined the Sasaki-Ricci flow (SRF) as the
KRF of the underlying transverse Ka¨hler metric associated with a Sasaki structure.
They proved results analogous to those in [7]. The work in [36] on the SRF does
not rely heavily on the contact structure of a Sasaki manifold and many of their
results can be carried over to the quasi-Sasaki setting. Collins in [12] and He in [19],
independently, extended a remarkable result of Perelman in KRF with c1 > 0 (see
[34]) to the SRF with c1B > 0. In this chapter, we will do the same in the quasi-Sasaki
setting. In this section we define the flow and then prove its short time existence.
56
Definition 3.1.1. Let (ξ, η0,Φ0, g0) be a quasi-Sasaki structure with 2pic
1
B = κ[ω0]
for some constant κ. The transverse Ka¨hler-Ricci flow (TKRF) is given by
∂
∂t
gT = κgT − RicTg , gT (0) = gT0 . (3.1)
Let ρT := RicT (·,Φ·) be the transverse Ricci form. Recall that ρT is independent
of the transverse Ka¨hler metric and that the class [ρT ] = 2pic1B. As ω = g
T (·,Φ·),
equation (3.1) is equivalent to
∂
∂t
ω = κω − ρT , ω(0) = ω0. (3.2)
A transverse homothety, also called a D-homothety, is a deformation of a quasi-
Sasaki structure that scales the transverse metric. Given a quasi-Sasaki structure
(ξ, η,Φ, g) and λ > 0, we set
ξλ := λ
−1ξ, ηλ := λη and gλ := λ(λ− 1)η ⊗ η + λg.
Then (ξλ, ηλ,Φ, gλ) is a quasi-Sasaki structure and the transverse metric g
T
λ = λg
T .
So if 2pic1B = κ[ω], then by a D-homothety we may assume that κ = −1, 0 or 1,
depending on the sign of κ.
3.11. Short-Time Existence
To prove the short-time existence of a solution to equation (3.2), we reduce it
to a parabolic Monge-Ampe´re equation. Having done this, the standard theory of
parabolic equations guarantees the existence of a solution for at least a short time.
We will prove the case c1B > 0 since we shall assume this in the sequel. The cases
57
c1B = 0 and c
1
B < 0 can be proved similarly. We also note, although we don’t prove it
here, that equation (3.1) has a short-time solution even with no assumption on c1B.
We assume that [ω0] = 2pic
1
B and we know that 2pic
1
B = [ρ
T
0 ]. Thus, by the
transverse ∂∂¯-lemma of [15], there is a basic function f such that ω0 = ρ
T
0 +
√−1∂∂¯f .
Recall that ρT0 = −
√−1∂∂¯ log(det(ω0)) Thus
ω0 = −
√−1∂∂¯ log(det(ω0)) +
√−1∂∂¯f
= −√−1∂∂¯ log(e−f det(ω0))
= −√−1∂∂¯ log(det(e−f/nω0)).
We let Ω := det(e−f/nω0) so that we can write ω0 = −
√−1∂∂¯ log(Ω).
Next we consider the equation
∂φ
∂t
= log
det(ω0 +
√−1∂∂¯φ)
Ω
+ φ+ c(t), φ(0) ≡ 1, (3.3)
defined for all t ≥ 0 such that ω0 +
√−1∂∂¯φ > 0. Here c(t) is a time-dependent
constant. If φ solves (3.3), then it is easy to check that the metric ω := ω0 +
√−1∂∂¯φ
solves (3.2). Conversely, given a solution to (3.2), its basic cohomology class must
evolve as
∂
∂t
[ω] = [ω]− 2pic1B, [ω](0) = [ω0].
Solving this ODE, we find that [ω] = 2pic1B+e
t([ω0]−2pic1B). Since we assume that
[ω0] = 2pic
1
B, we have [ω(t)] = [ω0]. By the transverse ∂∂¯-lemma, there is φ ∈ C∞B (M)
such that
ω = ω0 +
√−1∂∂¯φ. (3.4)
58
Since ω(0) = ω0, we have that ∂∂¯φ(0) = 0. Differentiating the above, we get
ω − ρT = √−1∂∂¯ ∂φ
∂t
. (3.5)
Putting (3.4) into (3.5) and writing ρT = −√−1∂∂¯ log(det(ω)), we then have
√−1∂∂¯ ∂φ
∂t
=
√−1∂∂¯
(
log
det(ω)
Ω
+ φ
)
. (3.6)
Thus there is a time-dependent constant c(t) such that
∂φ
∂t
= log
det(ω)
Ω
+ φ+ c(t).
Hence φ solves (3.3). Therefore, equations (3.2) and (3.3) are equivalent.
Now consider the parabolic Monge-Ampe´re equation, given by
∂φ
∂t
= log
det(ω0ij¯ + φ,ij¯)
Ω
+ ξ2φ+ φ. (3.7)
Here φ is not assumed to be basic and the indices after the comma denote covariant
derivatives.
Lemma 3.1.2. Equation (3.7) preserves the property ξφ = 0.
Proof. The proof of this lemma is by a maximum principal argument and is the same
as the proof of Lemma 5.2 in [36]. See [36] for the details.
With this we can prove the short time existence of our flow.
Proposition 3.1.3. For some T > 0, equation (3.2) has a solution for t ∈ [0, T ).
59
Proof. Equation (3.7) is parabolic whenever we have
det(ω0ij¯ + φ,ij¯) > 0. (3.8)
By the standard parabolic theory, for any initial function φ(0) satisfying (3.8), there
is T > 0 and a solution, φ : M × [0, T )→ R, to (3.7) with det(ω0ij¯ + φ,ij¯) > 0 for all
t ∈ [0, T ).
Take φ(0) ≡ 1. Then (3.8) is satisfied and since ξφ(0) = 0, the solution φ is a
basic function by lemma 3.1.2. Recall that
φ,ij¯ := ∇2φ(Xi, Xj¯) = ∇dφ(Xi, Xj¯) = XiXj¯φ− dφ(∇XiXj¯).
By (2.5) we have ∇XiXj¯ = −dη(Xi, Xj¯)ξ. So when φ is basic we have φ,ij¯ = φij¯. Thus
our solution to (3.7) also solves (3.3) (with c(t) = 0). Hence, we have a solution to
(3.2) on [0, T ).
3.12. Long-Time Existence
The main theorem in [7] implies that for a compact Ka¨hler manifold (M, g0) with
c1 = κ[ω0], the Ka¨hler-Ricci flow
∂
∂t
g = κg − Ricg, g(0) = g0,
has a unique solution defined for all t ≥ 0 and when κ ≤ 0 the flow g(t) converges to
the unique Ka¨hler-Einstein metric in c1. Using this result, it is shown in [36] that for
60
a compact Sasaki manifold (M, η0, g0) with with c
1
B = κ[dη0], the Sasaki-Ricci flow
∂
∂t
gT = κgT − RicTg , gT (0) = gT0 ,
has a unique solution for t ∈ [0,∞) and when κ ≤ 0 the flow gT (t) converges to a
transverse Ka¨hler-Einstein metric.
In the Sasaki setting, dη is non-degenerate on D and it is the transverse Ka¨hler
form of the Sasaki metric. So the Sasaki-Ricci flow is really just our TKRF where
the initial structure is Sasaki (rank 2n + 1 quasi-Sasaki with ω = dη). To prove
the long-time existence of the flow, one needs uniform Ck-estimates for the metric
potential function φ. The estimates obtained in [36] do not rely on the fact that the
initial structure is Sasaki and thus can be carried over to the setting of our TKRF.
Therefore, the long-time existence of the TKRF follows from the work in [36], which,
in turn, follows from the work in [7]. We record this as a corollary.
Corollary 3.1.4. Let (M, ξ, η0,Φ, g0) be a compact quasi-Sasaki manifold with c
1
B =
κ[ω0]. Then the transverse Ka¨hler-Ricci flow
∂
∂t
gT = κgT − RicTg , gT (0) = gT0 ,
has a unique solution that exists for all t ≥ 0. When κ ≤ 0, the flow gT (t) converges
to a transverse Ka¨hler-Einstein metric.
3.13. The Evolving quasi-Sasaki Structure
Now that we have an evolving transverse Ka¨hler metric ωt = ω0 +
√−1∂∂¯φ(t)
that satisfies (3.2), we need an associated evolving quasi-Sasaki structure. We define
61
the evolving structure (ξ, ηt,Φt, gt) by
ηt := η0 + d
c
Bφ,
Φt := Φ0 − ξ ⊗ dcBφ ◦ Φ0,
gt := ηt ⊗ ηt + ωt(Φt·, ·).
As dcBφ is basic, it is clear that ηt(ξ) = η0(ξ) = 1. It is not hard to check that the
relations Φ2t = −I + ξ ⊗ ηt and gt(Φt·,Φt·) = gt − ηt ⊗ ηt hold for this evolving
structure. Hence, for each t ≥ 0, we have a quasi-Sasaki structure (ξ, ηt,Φt, gt) where
the transverse metric gTt evolves by TKRF.
It is natural to ask what happens to the rank of the quasi-Sasaki structure as
it evolves under the TKRF. If the initial structure is Sasaki (rank 2n + 1), then
dηt = ωt and thus the evolving structure remains Sasaki. However, the rank need
not be constant along the flow. If the initial structure is co-symplectic (rank 1),
then dη0 = 0. If the rank stays constant then dηt =
√−1∂∂¯φ = 0 for all t. This
implies that ω0 is a constant solution to (3.2). Hence ω0 is a transverse Ka¨hler-
Einstein metric. So if the initial structure is rank 1 and ω0 is not Einstein, then the
rank cannot remain constant. To construct such a manifold, simply take a compact
Ka¨hler manifold (M˜, J, g˜) with c1 > 0 which is not Einstein (these exist) and let
M = S1 × M˜ and g = dθ ⊗ dθ + g˜. Then {M,∂θ, dθ, J, g} is co-symplectic and ω is
not transverse Ka¨hler-Einstein.
If the initial structure has rank 2p + 1 with p < n, we would like to know what
happens to the rank of the evolving structure. If the initial structure is rank 1, then
the rank can only increase along the flow, but is this what happens in general?
62
We end this section with a few words on the volume form naturally associated to
a quasi-Sasaki manifold. Given a quasi-Sasaki structure on M , the top form η ∧ (ω)n
defines a volume form. If ωt = ω0 +
√−1∂∂¯φ evolves by transverse Ka¨hler-Ricci flow,
then as we noted above, ηt = η0 + d
c
Bφ is the evolving (almost) contact form. Since
dcBφ is basic,
dcBφ ∧ (ωt)n = 0,
because it is a basic form of degree 2n+ 1. In a preferred coordinate chart we have
gt = ηt ⊗ ηt + (gTij¯ + φij¯)dzi ⊗ dz¯j.
It follows that
(ωt)
n = c(n) det(gTij¯ + φij¯)dz
1 ∧ dz¯1 ∧ · · · ∧ dzn ∧ dz¯n,
where the constant
c(n) = n!
(
√−1)n
2n
.
Therefore, the evolving volume form
ηt ∧ (ωt)n = η0 ∧ (ωt)n
= c(n) det(gTij¯ + φij¯)dx ∧ dz1 ∧ dz¯1 ∧ · · · ∧ dzn ∧ dz¯n
= n!dVgt .
where dVgt is the Riemmanian volume form associated to gt.
63
3.2. Perelman’s entropy functional on quasi-Sasaki manifolds
In this section we develop the analog of Perelman’s W functional in the quasi-
Sasaki setting. Our transverse entropy functional WT can be viewed as the entropy
functionalW for the transverse Ka¨hler structure. We will show thatWT is monotonic
along the transverse Ka¨hler-Ricci flow. This will allow us, in the next section, to prove
a non-collapsing result which can then be used to prove bounds on the scalar curvature
and the diameter of the manifold along the TKRF, as in [34].
3.21. The transverse entropy functional
Let M be a quasi-Sasaki manifold. The WT functional is defined in terms of the
quasi-Sasaki metric g, but it really only depends on the induced transverse metric.
We make the following definition:
Definition 3.2.1. For a quasi-Sasaki manifold (M, ξ, η,Φ, g), a basic function f ∈
C∞B (M) and a real number τ > 0, the transverse entropy functionalWT is defined by
WT (g, f, τ) :=
∫
M
(τ(RT + |∇f |2) + f)e−fτ−n dV, (3.9)
where dV = η ∧ (ω)n.
Even though the WT functional is defined in terms of the metric g, it really
only depends on the induced transverse metric gT . For this reason we may write
WT (gT , ·, ·) to mean WT (g¯, ·, ·) where g¯ is any bundle-like metric inducing gT .
Since the function f in (3.9) is basic, the integrand of the WT functional only
involves the transverse Ka¨hler structure. Thus, when we compute the first variation
of WT we are essentially just computing as we would in the Ka¨hler setting, with all
of the familiar integration by parts formulas from Ka¨hler geometry at our disposal.
64
Proposition 5.0.10 and corollary 5.0.11 help make this precise. As noted in [17],
corollary 5.0.11 implies a general principle that will aid us greatly in our computations.
The principle is that if a result for a compact Ka¨hler manifold can be proved using
only Stokes theorem and integration by parts, then the same result holds for compact
quasi-Sasaki manifolds and the proof can go as in the Ka¨hler setting by inserting a
“∧η” in each line of the proof. See also proposition 4.6 in [19].
We consider a variation of the quasi-Sasaki structure that fixes ξ and the
transverse holomorphic structure. Let vij¯ = δg
T
ij¯, v = (g
T )ij¯vij¯, δf = h ∈ C∞B (M)
and δτ = σ. We assume that, at least locally, there is a basic function ψ such that
vij¯ = ∂i∂j¯ψ. Then we have vij¯,i = v,i.
Proposition 3.2.2. The first variation of WT on compact quasi-Sasaki manifolds is
given by
δWT (vij¯, h, σ) =
∫
M
(
σ(RT + ∆Tf)− τ〈vij¯, RTij¯ + fij¯〉+ h
)
τ−ne−f dV
+
∫
M
(v − h− nστ−1)(τ(2∆Tf − |∇Tf |2 +RT ) + f)τ−ne−f dV.
Before we prove the proposition, a few comments and observations are in order.
First, the Christoffel symbols of the transverse Levi-Civita connection in the ξ
direction, as well as those with mixed barred and unbarred indices, are all zero.
The others are given by the usual formula from Ka¨hler geometry,
Γkij = (g
T )kl¯
∂gT
il¯
∂zj
and Γk¯i¯j¯ = Γ
k
ij.
65
For a function f on M , we will use the notation fi :=
∂f
∂zi
= ∂if and fij¯ :=
∂2f
∂zi∂z¯j
=
∂i∂j¯f . Next, if f is a basic function, then
|∇f |2 = |∇Tf |2 = (gT )ij¯fifj¯
and
∆f = ∆Tf = (gT )ij¯∇i∇j¯f = (gT )ij¯fij¯.
Finally, for symmetric (0,2)-tensors Sij¯ and Tij¯, we have an inner product given by
〈Sij¯, Tij¯〉 := (gT )il¯(gT )kj¯Sij¯Tkl¯,
and we write |Sij¯|2 = 〈Sij¯, Sij¯〉. To ease notation, we will drop the superscript T
from the transverse connection, Laplacian and metric in what follows. As all of the
functions involved are basic, this should not cause any confusion. The presence of one
barred index and one unbarred index in the metric indicates that we are considering
the transverse Ka¨hler metric. Now for the proof:
Proof. From the relation gil¯gkl¯ = δ
k
i , we find that δg
ij¯ = −gil¯vkl¯gkj¯. Now the variation
of the transverse scalar curvature is
δRT = δ(gij¯RTij¯)
= −δ(gij¯∂i∂j¯ log(det(gTij¯)))
= −〈vij¯, RTij¯〉 −∆v.
66
Next we compute
δ|∇f |2 = δ(gij¯fifj¯)
= −〈vij¯, fifj¯〉+ gij¯(hifj¯ + fihj¯)
= −〈vij¯, fifj¯〉+ 2gT (∇f,∇h).
Recall that dV = η ∧ (ω)n = c(n) det(gTij¯)dx ∧ dz1 ∧ dz¯1 ∧ · · · ∧ dzn ∧ dz¯n. Hence
δdV = vdV and it follows that
δ(τ−ne−fdV ) = (v − h− nστ−1)τ−ne−fdV.
Putting these all together, we have
δWT (vij, h, σ) =
∫
M
(σ(RT + |∇f |2) + h)τ−ne−fdV
+
∫
M
τ(−〈vij¯, RTij¯ + fifj¯〉 −∆v + 2gT (∇f,∇h))τ−ne−fdV
+
∫
M
(τ(RT + |∇f |2) + f)(v − h− nστ−1)τ−ne−fdV.
Observe that ∆e−f = (|∇f |2−∆f)e−f . So, since M is closed, by Stokes’ theorem
we have ∫
M
(|∇f |2 −∆f)e−fdV = 0. (3.10)
67
Since f, h, v ∈ C∞B (M), we have the following integration by parts formulas:
∫
M
e−f∆v dV =
∫
M
v(|∇f |2 −∆f)e−f dV, (3.11)∫
M
〈vij¯, fifj¯〉e−fdV =
∫
M
(〈vij¯, fij¯〉 − v(∆f − |∇f |2))e−f dV, (3.12)∫
M
gT (∇f,∇h)e−fdV =
∫
M
h(|∇f |2 −∆f)e−f dV. (3.13)
Equations (3.11) and (3.13) are straightforward to verify. For a proof of (3.12),
see proposition 4.3 in [19]. Now, using equations (3.10)-(3.13) in our above expression
for δWT we get
δWT (vij¯, h, σ) =
∫
M
(
σ(RT + ∆f)− τ〈vij¯, RTij¯ + fij¯〉+ h
)
τ−ne−f dV
+
∫
M
(v − h− nστ−1)(τ(2∆f − |∇f |2 +RT ) + f)τ−ne−f dV.
If we choose h = v − nστ−1, then the second integral in the expression for
δWT vanishes. Now let vij¯ = gij¯ − RTij¯ − fij¯. Then v = n − RT − ∆f and h =
n− RT −∆f − nστ−1. Putting these choices of vij¯ and h into the above and noting
that 〈gij¯, RTij¯ + fij¯〉 = RT + ∆f , we find that
δWT (vij¯, h, σ) =
∫
M
(
τ |Rij¯ + fij¯|2 − (τ − σ + 1)(R + ∆f) + n− nστ−1
)
τ−ne−f dV.
68
Now setting σ = τ − 1, this becomes
δWT (vij¯, h, σ) =
∫
M
(
τ |Rij¯ + fij¯|2 − 2(R + ∆f) + nτ−1
)
τ−ne−f dV
=
∫
M
|Rij¯ + fij¯ − τ−1gij¯|2τ−n+1e−f dV ≥ 0. (3.14)
The next goal is to show that WT is monotonically increasing along the TKRF.
Let us outline one way to accomplish this.
(i) First show that for a diffeomorphism ϕ : M →M we have
WT (g, f, τ) =WT (ϕ∗g, ϕ∗f, τ). (3.15)
This is relatively straightforward.
(ii) Next find a smooth basic function f = f(t) satisfying
∂f
∂t
= nτ−1 −RT −∆f + |∇f |2.
This is a backward heat equation and for any time T > 0, if f(T ) ∈ C∞B (M),
then there is a unique basic solution on [0, T ]. To prove this, one can argue as
follows:
Proof (sketch). Pick T > 0 and consider the equation
∂w
∂s
= ∆w − (RT − nτ−1)w. (3.16)
This is a linear parabolic equation as long as gT remains positive definite. Since
gT is positive definite for all time along the TKRF, by the standard parabolic
69
theory, given an initial condition, a unique solution exists for s ∈ [0, T ]. If w(0)
is basic, then the solution remains basic. Indeed, as RT is basic, ξw satisfies
∂
∂s
(ξw) = ξ
∂w
∂s
= ξ∆w − ξ(RT + nτ−1)w = ∆(ξw)− (RT + nτ−1)(ξw).
So if ξw(0) = 0, then by the uniqueness of the solution, we must have ξw ≡ 0.
If w(0) ≥ 0, then w(s) ≥ 0. Furthermore, if w(0) is positive at some point
p ∈ M , then w(s) > 0. By the standard regularity theory for linear parabolic
equations, if we have w(0) ∈ W 1,2(M) then w(s) ∈ C∞(M). See proposition
4.9 in [19].
We can solve equation (3.16) with w(0) > 0 a basic function. Then we can
write w(s) = e−f(s) for some smooth function f . As w is basic for all s, so is f .
Let t = T − s. Then for t ∈ [0, T ], it’s not hard to check that f(t) solves the
backward heat equation
∂f
∂t
= nτ−1 −RT −∆Bf + |∇f |2.
(iii) Now consider the time-dependent diffeomorphisms ϕ(t, ·) : M →M having
∂
∂t
ϕ(t, p) = −∇f(t)|ϕ(t,p), ϕ(0, ·) = I.
That these diffeomorphisms exist is somewhat standard and a proof can be
found, for example, in lemma 3.15 of [10]. Next, direct computation shows that
if g evolves by TKRF, then g˜ := ϕ∗g induces a transverse metric which evolves
70
by
∂
∂t
g˜Tij = g˜ij¯ − R˜Tij¯ − f˜ij¯,
where R˜Tij¯ is the transverse Ricci tensor of g˜ and f˜ = ϕ
∗f is a basic function
satisfing
∂f˜
∂t
= nτ−1 − R˜T −∆f˜ .
(iv) Finally, letting τ evolve by
∂τ
∂t
= τ − 1,
the computations in (3.14) and (3.15) then imply that ∂W
T
∂t
≥ 0 along the
TKRF.
This outline is similar to the approach in [25] for the Ricci flow and the one taken
in [12] for the Sasaki-Ricci flow. We can also make this outline work for us, but there
are some technical difficulties involved. One difficulty encountered with this approach
is that the diffeomorphisms ϕ may not preserve the transverse holomorphic structure.
The pulled back metrics ϕ∗g may not be quasi-Sasaki as the induced transverse metric
may not even by Hermitian. This is not a terrible difficulty to overcome though. One
must simply define WT on a larger class of metrics. In particular, the class M of
Riemannian metrics on M for which ξ is a unit-length Killing vector field will suffice.
The metrics in M are bundle-like with respect to Fξ and thus induce transverse
metrics. But now one must check that WT has the same variational formula when
the metrics vary inM rather than as quasi-Sasaki metrics. This can be done and the
proof is the same as in the Riemannian setting. Then one must show that if g ∈ M
then ϕ∗g ∈ M, that ϕ∗ξ = ξ and that if f is basic then ϕ∗f is basic too. All of this
can be done but we will choose not to pursue it any further.
71
Another approach is to show by direct computation that ∂W
T
∂t
≥ 0 under the
coupled transverse Ka¨hler-Ricci flow

∂gT
∂t
= gT − RicTg
∂f
∂t
= nτ−1 −RT −∆f + |∇f |2
∂τ
∂t
= τ − 1
(3.17)
This approach takes advantage of the transverse Ka¨hler structure and we can remain
in the category of quasi-Sasaki metrics. We can prove the following proposition, which
is the same as proposition 4.4 in [19]:
Proposition 3.2.3. Under the coupled transverse Ka¨hler-Ricci flow (3.17), the WT
functional on quasi-Sasaki manifolds satisfies
∂WT
∂t
=
∫
M
|Rij¯ + fij¯ − τ−1gij¯|2τ−n+1e−f dV +
∫
M
|fij|2τ−n+1e−f dV ≥ 0. (3.18)
Moreover, WT is strictly increasing unless RTij¯ + fij¯ − τ−1gTij¯ = 0 and fij = 0. In this
case, ∇f is a real holomorphic vector field and gT is a transverse (gradient-shrinking)
Ka¨hler-Ricci soliton.
The proof involves a careful computation with several uses of integration by
parts. Some of the integration by parts formulas are not obvious. As the proof is the
same as in [19], we do not reproduce it here. Next we discuss the analog of Perelman’s
µ functional.
72
3.22. The transverse µ-functional
In the last subsection, we saw that if g evolves as ∂tgij¯ = gij¯ − RTij¯ − fij¯, f
evolves as ∂tf = nτ
−1 −RT −∆Bf and τ evolves as ∂tτ = τ − 1, then WT (g, f, τ) is
non-decreasing with t and gij¯∂tgij¯ − ∂tf − nτ ∂tτ = 0. Hence
∂
∂t
(τ−ne−fdV ) = 0.
This implies that
∫
M
τ−ne−fdV is constant and motivates our next definition.
Definition 3.2.4. The transverse µ-functional, µT , is defined by
µT (g, τ) := inf
f∈C∞B (M)
{WT (g, f, τ) :
∫
M
τ−ne−fdV = 1}.
As it is the case with the WT -functional, it is not surprising that the µT -
functional is also monotonically non-decreasing along the TKRF when ∂tτ = τ − 1.
We prove this next.
Proposition 3.2.5. If g(t) evolves by TKRF and τ satisfies ∂tτ = τ − 1, then
µT (g(t), τ(t)) is monotone non-decreasing in t. In particular, µT (g(t), 1) is non-
decreasing along the transverse Ka¨hler-Ricci flow.
Proof. Fix a time t1 > 0 and let ft1 ∈ C∞B (M) be normalized by
∫
M
τ−ne−ft1dV = 1.
Then solve the backward heat equation
∂f
∂t
= nτ−1 −RT −∆Bf + |∇f |2
73
for t ∈ [0, t1] with f(t1) = ft1 . We know that the solution f(t) remains basic. When
g evolves by TKRF and ∂tτ = τ − 1, we can compute that
∂
∂t
(τ−ne−fdV ) = (∆f − |∇f |2)τ−ne−fdV = −τ−n∆e−fdV.
Thus,
∂
∂t
∫
M
τ−ne−f dV = −
∫
M
τ−n∆e−f dV = 0.
This implies that
∫
M
τ−ne−f(t) dV = 1 for all t ∈ [0, t1]. Now chose 0 ≤ t0 < t1. By
the definition of µT and the monotonicity of WT , we get
µT (g(t0), τ(t0)) ≤ WT (g(t0), f(t0), τ(t0)) ≤ WT (g(t1), f(t1), τ(t1)).
Since ft1 = f(t1) was chosen arbitrarily, we conclude that
µT (g(t0), τ(t0)) ≤ inf
f(t1)
WT (g(t1), f(t1), τ(t1)) = µT (g(t1), τ(t1)).
3.3. Bounds on Scalar Curvature and the Transverse Ricci Potential
We assume there is a transverse Ka¨hler metric ω such that [ω] = 2pic1B. Then by
the transverse ∂∂¯-lemma, there is smooth function u(x, t) such that gTij¯−RTij¯ = ∂i∂j¯u.
We call u a transverse Ricci potential. We may assume that u is normalized so that∫
M
e−u dV = 1. If φ is a solution to (3.3), then
∂i∂j¯
∂φ
∂t
=
∂gTij¯
∂t
= gTij¯ −RTij¯ = ∂i∂j¯u.
74
So we can take u = ∂tφ. As R
T
ij¯ = −∂i∂j¯ log(det(gT )), we compute that
∂i∂j¯
∂u
∂t
=
∂
∂t
(gTij¯ −RTij¯)
= gTij¯ −RTij¯ + ∂i∂j¯gij¯
∂gTij¯
∂t
= ∂i∂j¯(u+ g
ij¯∂i∂j¯u)
= ∂i∂j¯(u+ ∆
Tu).
Thus, there is time-dependent constant A(t) such that
∂u
∂t
= u+ ∆Tu+ A(t). (3.19)
As ∂tdV = ∆
Tu dV , differentiating
∫
M
e−u dV = 1 with respect to t, we find that
A(t) = −
∫
M
ue−u dV.
We can use the monotonicity of the µT -functional to show that A is uniformly
bounded. Recall from proposition 3.2.5 that µT (g(t), 1) is non-decreasing in t.
Lemma 3.3.1. The function A(t) is uniformly bounded.
Proof. It is a routine calculus exercise to show that f(t) = te−t is bounded above by
e−1. From ∂tdV = ∆Tu dV , we see that the volume of the manifold is constant along
the TKRF. For simplicity, let us assume that Vol(M) =
∫
M
dV = 1. Then
A(t) = −
∫
M
ue−u dV ≥ −e−1
∫
M
dV = −e−1.
75
Tracing gTij¯ −RTij¯ = ∂i∂j¯u, we have ∆Tu = n−RT . Thus
WT (g(t), u(t), 1) =
∫
M
(n−∆Tu+ |∇u|2 + u)e−u dV
=
∫
M
∆T e−u dV +
∫
M
(n+ u)e−u dV = n− A(t).
Therefore,
A(t) = n−WT (g(t), u(t), 1) ≤ n− µT (g(t), 1) ≤ n− µT (g(0), 1).
Next we will see how the transverse scalar curvature evolves along the TKRF,
but first we recall a useful lemma. The proof is a standard calculus argument.
Lemma 3.3.2. If f = f(x, t) is a smooth function on M × [0, T ] for some T > 0
and f achieves its minimum (maximum) at (x0, t0), then either t0 = 0 or at the point
(x0, t0)
∂f
∂t
≤ 0 (≥ 0), ∇f = 0 and ∆f ≥ 0 (≤ 0).
Lemma 3.3.3. Along the TKRF, the transverse scalar curvature evolves by
∂RT
∂t
= −RT + |RicT |2 + ∆TRT
and is uniformly bounded from below.
Proof. Because of the transverse Ka¨hler structure, the Ricci tensor RTij¯ =
−∂i∂j¯ log(det(gij¯)). Thus we can write the scalar curvature as
RT = −gij¯∂i∂j¯ log(det(gij¯)).
76
From gil¯gkl¯ = δ
i
k, we compute ∂tg
ij¯ = −gil¯(∂tgkl¯)gkj¯. Therefore,
∂RT
∂t
= gil¯(∂tgkl¯)g
kj¯∂i∂j¯ log(det(gij¯))− gij¯∂i∂j¯gij¯∂tgij¯
= −gil¯(gkl¯ −RTkl¯)gkj¯RTij¯ − gij¯∂i∂j¯gij¯(gij¯ −RTij¯)
= −δikgkj¯RTij¯ + gil¯gkj¯RTkl¯RTij¯ − gij¯∂i∂j¯(n) + gij¯∂i∂j¯gij¯RTij¯
= −RT + |RicT |2 + ∆TRT .
Now fix a time T > 0 and suppose that RT (x, t) on M × [0, T ] takes its minimum at
(x0, t0). If t0 = 0, then we have
inf
M×[0,T ]
RT (x, t) ≥ inf
M
RT (x, 0).
Since RT is basic, ∆TRT = ∆RT . So if t0 6= 0, then by the above lemma, at (x0, t0),
0 ≥ ∂R
T
∂t
−∆TRT = −RT + |RicT |2 ≥ −RT .
So either way, as T > 0 was arbitrary, we have RT ≥ min{0, infM RT (x, 0)}.
Lemma 3.3.4. The function u(x, t) is uniformly bounded from below.
Proof. Fix T > 0. By lemmas 3.3.1 and 3.3.3, A(t)−RT is uniformly bounded above.
Set
C = max{0, sup
x∈M,t∈[0,T ]
n−RT + A(t)} <∞.
Suppose that u(x, t) is not uniformly bounded below. Then there is a point (x0, t0)
where u(x0, t0) +C < 0. Since u is smooth in x, there is a neighborhood U of x0 such
that
u(x, t0) + C < 0
77
for all x ∈ U . Recall that ∂tu = u+ ∆Tu+A(t) and ∆Tu = n−RT . Thus, for x ∈ U ,
∂u
∂t
|(x,t0) = u(x, t0) + n−RT (x, t0) + A(t0) < 0.
Since u is smooth in t, u(x, t) ≤ u(x, t0) for x ∈ U and t ≥ t0. From ∂tu < u+ C we
get
u(x, t) < (C + u(x, t0))e
t−t0 ≤ −C1et (3.20)
for x ∈ U , t ≥ t0 and some constant C1 > 0 that depends on t0.
Recall that ∂tφ = u. Integrating equation (3.20) from t0 to t gives
φ(x, t) ≤ φ(x, t0)− C1(et − et0) ≤ −C2et (3.21)
for x ∈ U , t ≥ t0 sufficiently large and C2 > 0. As u is normalized so that
∫
M
e−u dV =
1, −u cannot be very large. Thus there is a constant C3 > 0 such that
u(xt, t) := max
x∈M
u(x, t) ≥ −C3.
From ∂t(u− φ) = n−RT + A(t) ≤ C we get
u(xt, t)− φ(xt, t) ≤ max
x∈M
(u(x, 0)− φ(x, 0)) + Ct.
This implies there is a constant C4 so that
φ(x′t, t) := max
x∈M
φ(x, t) ≥ −C4 − Ct. (3.22)
78
Tracing gTij¯(t) = gij¯(0) + ∂i∂j¯φ, with the metric g(0) gives n + ∆g(0)φ =
gij¯(0)gTij¯(t) > 0. Recall that the volume of the manifold is constant along the TKRF
and we are assuming that vol(M) = 1. Let G0 be the Green’s function for ∆g(0). The
function φ−∫
M
φ dV0 integrates to zero with respect to dV0, so we may apply Green’s
formula to write
φ(x′t, t) =
∫
M
φ(y, t) dV0 −
∫
M
∆g(0)φ(y, t)G0(x
′
t, y) dV0. (3.23)
By (3.21), φ(x, t) ≤ −C2et on U × [t0,∞). Since −∆g(0)φ < n and G0(x, y) is
integrable, from (3.23) we get that for t ≥ t0
φ(x′t, t) ≤ vol(M \ U)φ(x′t, t)− vol(U)C2et + C˜.
Note that vol(M \ U) < 1. So for large enough t ≥ t0,
φ(x′t, t) ≤ −C5et + C6. (3.24)
Combining (3.22) and (3.24), we have
−C4 − Ct ≤ φ(x′t, t) ≤ −C5et + C6.
This is a contradiction for sufficiently large t as all of the constants involved are
independent of t > t0. Therefore, u(x, t) is uniformly bounded from below.
Lemma 3.3.5. There is a uniform constant C > 0 such that |∇u|2 ≤ C(u+ C).
Before we begin the proof, we need one new bit of notation. For a Ka¨hler
metric g, we define ∇f to be the vector field dual to ∂¯f . We define ∇¯f to be the
79
vector field dual to ∂f . In local coordinates, ∇f = gij¯fj¯dzi, ∇¯f = gij¯fidz¯j and
g(∇f, ∇¯f) = gij¯fifj¯ = |∇f |2. We make the analogous definition for a basic function
and the transverse Ka¨hler metric of a quasi-Sasaki structure.
Proof. Define the parabolic operator  := ∂t − ∆ and denote g(·, ·) = 〈·, ·〉. One
computes

(
f
h
)
=
f
h
− fh
h2
+
2<〈∇f, ∇¯h〉
h2
− 2f |∇h|
2
h3
,
where < denotes the real part of a complex number. From (3.19) we have
u = u+ A(t).
A local coordinate computation gives
∂t|∇u|2 = |∇u|2 + 〈Rij¯, uiuj¯〉+ 2<〈∇u, ∇¯∆u〉,
and the Bochner-Kodaira formula gives
∆|∇u|2 = |∇∇u|2 + |∇∇¯u|2 + 〈Rij¯, uiuj¯〉+ 2<〈∇u, ∇¯∆u〉.
Therefore
|∇u|2 = |∇u|2 − |∇∇u|2 − |∇∇¯u|2.
80
By lemma 3.3.4, there is a constant B > 0 such that u(x, t) + B > 0 for all
x ∈M and all t ∈ [0,∞). Now let H := |∇u|2/(u+ 2B). Then
H = |∇u|
2
u+ 2B
− |∇u|
2u
(u+ 2B)2
+
2<〈∇|∇u|2, ∇¯u〉
(u+ 2B)2
− 2|∇u|
2|∇u|2
(u+ 2B)3
=
|∇u|2 − |∇∇u|2 − |∇∇¯u|2
u+ 2B
− |∇u|
2(u+ A)
(u+ 2B)2
+
2<〈∇|∇u|2, ∇¯u〉
(u+ 2B)2
− 2|∇u|
4
(u+ 2B)3
=
−|∇∇u|2 − |∇∇¯u|2
u+ 2B
+
|∇u|2(2B − A)
(u+ 2B)2
+
2<〈∇|∇u|2, ∇¯u〉
(u+ 2B)2
− 2|∇u|
4
(u+ 2B)3
.
(3.25)
By the definition of H,
∇H = ∇|∇u|
2
u+ 2B
− |∇u|
2∇u
(u+ 2B)2
.
Hence
<〈∇H, ∇¯u〉
u+ 2B
=
<〈∇|∇u|2, ∇¯u〉
(u+ 2B)2
− |∇u|
4
(u+ 2B)3
. (3.26)
Next we observe that
|〈∇|∇u|2, ∇¯u〉|
(u+ 2B)2
≤ |∇u|
2(|∇∇u|+ |∇∇¯u|)
(u+ 2B)3/2(u+ 2B)1/2
≤ |∇u|
4
2(u+ 2B)3
+
|∇∇u|2 + |∇∇¯u|2
u+ 2B
. (3.27)
Choosing 0 < ε < 1/4 and combining (3.25), (3.26) and (3.27), yields
H ≤ |∇u|
2(2B − A)
(u+ 2B)2
+
(2− ε)<〈∇H, ∇¯u〉
u+ 2B
− ε|∇u|
4
2(u+ 2B)3
. (3.28)
81
If H is unbounded from above, then there is a time t0 such that
max
M×[0,t0]
H >
2B − A
ε/2
.
At the point where H achieves its maximum, ∇H = 0 and H ≥ 0. At this point,
(3.28) gives
0 ≤ H ≤ |∇u|
2
(u+ 2B)2
(
2B − A− ε
2
H
)
< 0.
This is a contradiction and therefore H is bounded from above. Thus there is a
constant C > 0 such that |∇u|2 ≤ C(u+ C).
Lemma 3.3.6. There is a uniform constant C > 0 such that RT ≤ C(u+ C).
Proof. From ∂i∂j¯u = g
T
ij¯ −RTij¯, ∆u = n−RT and ∂tRT = −RT + |RicT |2 + ∆TRT , we
compute
(∆u) = −RT
= RT − |RicT |2
= n−∆u− 〈gTij¯ − uij¯, gTij¯ − uij¯〉
= n−∆u− (n− 2∆u+ |uij¯|2)
= ∆u− |uij¯|2.
Let K := −∆u/(u+ 2B) where B > 0 is as in the previous lemma. We compute
K = −∆u
u+ 2B
+
∆uu
(u+ 2B)2
− 2<〈∇∆u, ∇¯u〉
(u+ 2B)2
+
2∆u|∇u|2
(u+ 2B)3
=
−∆u+ |uij¯|2
u+ 2B
+
∆u(u+ A)
(u+ 2B)2
− 2<〈∇∆u, ∇¯u〉
(u+ 2B)2
+
2∆u|∇u|2
(u+ 2B)3
=
|uij¯|2
u+ 2B
+
∆u(A− 2B)
(u+ 2B)2
− 2<〈∇∆u, ∇¯u〉
(u+ 2B)2
+
2∆u|∇u|2
(u+ 2B)3
. (3.29)
82
By the definition of K,
∇K = −∇∆u
u+ 2B
+
∆u∇u
(u+ 2B)2
.
Hence
<〈∇K, ∇¯u〉
u+ 2B
= −<〈∇∆u, ∇¯u〉
(u+ 2B)2
+
∆u|∇u|2
(u+ 2B)3
.
Combining this with (3.29) yields
K = |uij¯|
2
u+ 2B
+
∆u(A− 2B)
(u+ 2B)2
+
2<〈∇K, ∇¯u〉
u+ 2B
. (3.30)
Let H be as in the previous lemma. There we computed that
H = −|uij|
2 − |uij¯|2
u+ 2B
+
|∇u|2(2B − A)
(u+ 2B)2
+
2<〈∇H, ∇¯u〉
u+ 2B
.
Now let G := 2H +K. Then by (3.30) and the above, we have
G = −2|uij|
2 − |uij¯|2
u+ 2B
+
(2|∇u|2 −∆u)(2B − A)
(u+ 2B)2
+
2<〈∇G, ∇¯u〉
u+ 2B
. (3.31)
At the point where G achieves its maximum, ∇G = 0 and G ≥ 0. So at this
point, equation (3.31) yields
0 ≤ G ≤ −|uij¯|
2
u+ 2B
+
2|∇u|2(2B − A)
(u+ 2B)2
− ∆u(2B − A)
(u+ 2B)2
. (3.32)
In normal coordinates at the point where G is at a maximum, by Cauchy-Schwarz
(∆u)2 =
(∑
i
ui¯i
)2
≤ n
∑
i
u2i¯i = n|uij¯|2.
83
So by (3.32) we get
0 ≤ G ≤ 2|∇u|
2(2B − A)
(u+ 2B)2
− ∆u
u+ 2B
(
∆u
n
+
2B − A
u+ 2B
)
. (3.33)
Since inf u+ 2B > 0 and A(t) is uniformly bounded, (2B−A)/(u+ 2B) is uniformly
bounded from above. By the bound on |∇u|2 from the previous lemma, the first term
on the right hand side of (3.33) is uniformly bounded from above.
So if −∆u/(u+2B) becomes arbitrarily large, then ∆u −(u+2B) < 0 and the
right hand side of (3.33) goes negative. This is a contradiction. Thus −∆u/(u+ 2B)
is bounded from above. Hence there is a constant C > 0 such that −∆u ≤ C(u+C).
Since RT = n−∆u, the proof is complete.
Proposition 3.3.7. Let xt ∈M be such that u(xt, t) = miny∈M u(y, t). Then there is
a uniform constant C > 0 such that
u(y, t) ≤ C(d2t (xt, y) + 1),
|∇u| ≤ C(dt(xt, y) + 1),
RT (y, t) ≤ C(d2t (xt, y) + 1),
where dt is the distance function for the metric g(t).
We prove the bound on u. The bounds on |∇u| and RT then follow from lemmas
3.3.5 and 3.3.6 respectively. By lemma 3.3.4, u is uniformly bounded below. Thus
we may assume that u > 0. If not, then we just consider the function u˜ := u + C
where C > 0 is a constant. Then u˜ is a transverse Ricci potential and choosing C
large enough makes u˜ > 0.
84
Proof. The bound on |∇u|2 from lemma 3.3.5 implies that |∇√u| ≤ C0. So all of
the first derivatives of
√
u are uniformly bounded (recall that u is basic). Thus
√
u
is uniformly Lipschitz. Hence
|
√
u(y, t)−
√
u(z, t)| ≤ C0dt(y, z).
Now if u(xt, t) ≥ C(t) and C(t)→∞ as t→∞, then by the normalization of u
and the fact that the volume of the manifold is constant along the flow, we get
1 =
∫
M
e−u dV ≤ vol(M)e−C(t) → 0.
Contradiction. Thus u(xt, t) ≤ C1 for some constant C1 > 0. Therefore,
√
u(y, t) ≤ C0dt(y, xt) +
√
u(xt, t)
≤ C0dt(y, xt) +
√
C1.
This implies that u(y, t) ≤ Cd2t (y, xt) + C for some constant C > 0.
We see from this proposition that if we can bound the diameter of the manifold
along the TKRF, then we will have uniform bounds on the transverse Ricci potential
u and the transverse scalar curvature RT . We take this up in the next section.
3.4. Upper Bound on Diameter
In this section we will prove a uniform upper bound on the diameter of a quasi-
Sasaki manifold along the TKRF. This along with proposition 3.3.7 will provide
uniform bounds on u, ∇u and RT . We take an approach analogous to that of Sesum
and Tian in [34], but we must make some adjustments. The W T functional only
85
involves basic functions, but the distance function of a quasi-Sasaki manifold is not
basic. To deal with this, we work with the transverse distance function from definition
1.4.1.
We begin with a non-collapsing theorem. Since RT is bounded from below along
the TKRF (lemma 3.3.3), there is r0 > 0 small enough so that R
T (x, t) ≥ −r−20 for
all x and t.
Theorem 3.4.1. Let M be a quasi-Sasaki manifold whose ξ orbits are compact. Let
g(t) be a solution to the TKRF. Then there is a constant C > 0 such that for every
x ∈M , if RT ≤ Cr−2 on BTg(t)(x, r) for r ∈ (0, r0], then Vol(BTg(t)(x, r)) ≥ Cr2n.
Proof. We give a proof by contradiction. Suppose that the theorem is false. Then
there is a sequence of points and times (pk, tk) ∈M × [0,∞) with tk →∞ such that
RT ≤ Cr−2k on Bk := BTg(tk)(pk, rk) but r−2nk Vol(Bk)→ 0 as k →∞.
Let ψ : [0,∞)→ R be a plateau function such that ψ = 1 on [0, 1
2
], is decreasing
on [1
2
, 1] with bounded derivative and ψ = 0 on [1,∞). Let τk := r2k and define
wk(x) := e
Ckψ
(
r−1k d
T
g(tk)
(pk, x)
)
=: eCkψk(x),
where the constant Ck is chosen so that
∫
M
w2kτ
−n
k dVk = 1. Here dVk = ηtk ∧ (ωtk)n
is the volume form at time tk. Note that the support of wk is contained in Bk. Thus
1 =
∫
M
w2kτ
−n
k dVk = e
2Ckr−2nk
∫
M
ψ2
(
r−1k d
T
k (pk, x)
)
dVk ≤ e2Ckr−2nk vol(Bk).
Since 1 ≤ e2Ckr−2nk Vol(Bk) and r−2nk Vol(Bk)→ 0, it must be that Ck →∞ as k →∞.
Recall, WT (g, f, τ) = τ−n ∫
M
τ(RTw2 + 4|∇w|2) − w2 log(w2) dV, where w = e−f/2.
86
Now
WT (g(tk), wk, τk) = τ−nk
∫
M
r2k(R
Tw2k + 4|∇wk|2)− w2k log(w2k) dVk
≤ e2Ckτ−nk
∫
M
4|ψ′k|2 − ψ2k log(ψ2k) dVk + r2k max
Bk
RT − 2Ck,
Here we have used that dTg(tk) is Lipschitz continuous, hence differentiable almost
everywhere, with Lipschitz constant 1. Let Vk(r) := Vol(Bk(r)).
Notice that for t ∈ R, limt→0 t2 log(t2) = 0 and for t ∈ (0, 1], t2 log(t2) ≥ −1.
Since ψk takes its values in [0, 1] and ψ
′
k is bounded with support in [
1
2
, 1], there is
some constant C ′ > 0 such that
∫
M
4|ψ′k|2 − ψ2k log(ψ2k) dVk ≤ C ′(Vk(rk)− Vk(rk/2)).
By lemma 5.0.13, we may assume that Vk(rk) ≤ 5nVk(rk/2). Hence,
∫
M
4|ψ′k|2 − ψ2k log(ψ2k) dV gk ≤ C˜Vk(rk/2)
≤ C˜
∫
M
ψ2k dVk.
Combining this with the above, we find that
µT (g(tk), τk) ≤ WT (g(tk), wk, τk)
≤ C˜
∫
M
w2kτ
−n
k dVk + r
2
k max
Bk
RT − 2Ck
≤ C˜ + C − 2Ck → −∞.
By proposition 3.2.5, µ(g(t), 1 + (τ0 − 1)et) is non-decreasing in t. Now for each k,
set τ k0 := 1− (1− τk)e−tk . Then µ(g(0), τ k0 ) ≤ µ(g(tk), τk)→ −∞ as k → −∞. This
87
is a contradiction because µ(g(0), τ) is a continuous function of τ and τ k0 → 1 as
k →∞.
Let u be a transverse Ricci potential. Recall that u is a basic function. By
lemma 3.3.4, u is uniformly bounded from below. Let xt ∈M be such that u(xt, t) =
minx∈M u(x, t) and let dTt (y) = d
T
g(t)(xt, y). For k1 ≤ k2, we define the transverse
annulus
Bξ(k1, k2) := {y ∈M : 2k1 ≤ dTt (y) ≤ 2k2}.
Now we can prove the diameter bound.
Theorem 3.4.2. Let M be a quasi-Sasaki manifold whose ξ orbits are compact. Let
g(t) be a solution to the TKRF. Then there is a uniform constant C such that the
transverse diameter ΘTg(t) < C. This implies the diameter of M is uniformly bounded
along the flow.
Proof. Suppose that the transverse diameters are not uniformly bounded. Then there
is a sequence of times ti → ∞ such that ΘTg(ti) → ∞. Let εi > 0 be a sequence such
that εi → 0. By lemma 5.0.14, there are sequences ki1, ki2, ri1, ri2 and a uniform
constant C such that
1. Vi(k
i
1, k
i
2) := Volg(ti)(Bξ(k
i
1, k
i
2)) < εi
2. Vi(k
i
1, k
i
2) ≤ 210nVi(ki1 + 2, ki2 − 2)
3. ri1 ∈ [ki1, ki1 + 1], ri2 ∈ [ki2 − 1, ki2] and
∫
Bξ(r
i
1,r
i
2)
RT dVti ≤ CVi(ki1, ki2).
Now let ψi be a cut-off function such that ψi(t) = 1 for t ∈ [2ki1+2, 2ki2−2], ψi(t) = 0
for t ∈ (−∞, 2ri1 ] ∪ [2ri2 ,∞) and |ψ′(t)| < 2. Define functions wi(y) := eCiψi(dTti(y))
88
where the constant Ci is chosen so that
∫
M
w2i dVti = 1. Then we have
1 = e2Ci
∫
M
ψ2i dVti ≤ e2CiV1(ki1, ki2) < e2Ciεi.
Since εi → 0, it must be that Ci →∞.
Now we compute
WT (g(ti), wi, 1) =
∫
M
RTw2i + 4|∇wi|2 − w2i log(w2i ) dVti
= e2Ci
∫
M
RTψ2i + 4|ψ′i|2 − ψ2i log(ψi)2 − 2Ciψ2i dVti
= e2Ci
∫
M
RTψ2i + 4|ψ′i|2 − ψ2i log(ψ2i ) dVti − 2Ci. (3.34)
By lemma 5.0.14, we can estimate the first term in the integrand of (3.34) as
e2Ci
∫
M
RTψ2i ≤ e2Ci
∫
Bξ(r
i
1,r
i
2)
RT dVti
≤ e2CiCVi(ki1, ki2)
≤ e2CiC210nVi(ki1 + 2, ki2 − 2)
≤ C210n
∫
m
w2i dVti
= C210n
Since 0 ≤ ψi ≤ 1 and ψ′i is uniformly bounded, there is a uniform constant C such that
4|∇ψi|2 − ψ2i log(ψ2i ) ≤ C. Using lemma 5.0.14 again we can estimate the remaining
89
terms in (3.34). Note that Bξ(r
i
1, r
i
2) ⊆ Bξ(ki1, ki2).
e2Ci
∫
M
4|ψ′i|2 − ψ2i log(ψ2i ) dVg(ti) ≤ e2CiCVi(ki1, ki2)
≤ e2CiC210nVi(ki1 + 2, ki2 − 2)
≤ C210n
∫
M
w2i dVti
= C210n
Therefore
µT (g(0), 1) ≤ µT (g(ti), 1) ≤ WT (g(ti), wi, 1) ≤ C210n − 2Ci → −∞.
This contradicts lemma 5.0.12. Thus the transverse diameter is uniformly bounded
along the TKRF.
3.5. Transverse Holomorphic Bisectional Curvature
Recall the transverse holomorphic bisectional curvature defined in section 2.4.
In this section we will prove
Theorem 3.5.1. Let gT (t) be a solution to the TKRF. If the initial metric gT (0) has
nonnegative transverse holomorphic bisectional curvature then so does gT (t) for t ≥ 0.
If the initial metric has positive transverse holomorphic bisectional curvature at some
point x ∈M , then gT (t) has positive transverse holomorphic bisectional curvature for
t > 0.
To prove theorem 3.5.1, we need to compute the evolution of the transverse
holomorphic bisectional curvature along the TKRF. By standard computations in
90
Ka¨hler-Ricci flow,
∂
∂t
Ri¯ijj¯ = ∆Ri¯ijj¯ + F (Rm)i¯ijj¯ +Ri¯ijj¯,
where, for a tensor S with the same type as the curvature tensor,
F (S)i¯ijj¯ =
∑
k,l
(|Sik¯lj¯|2 − |Sik¯jl¯|2 + Si¯ilk¯Skl¯jj¯)−∑
k
< (Sik¯Ski¯jj¯ + Sjk¯Si¯ikj¯) .
See proposition 2.79 and corollary 2.82 in [9] for the computations in the Ka¨hler
setting. We can easily mimic these computations and derive a similar formula in the
quasi-Sasaki setting. We let HT = HTij := R
T
i¯ijj¯. Then we have
∂
∂t
HT = ∆THT + F (HT ) +HT .
Using Hamilton’s maximum principle for tensors, Bando (n ≤ 3) in [2] and Mok
(n ≥ 4) in [29] proved the analog of theorem 3.5.1 for the Ka¨hler-Ricci flow. To
apply this maximum principle, one must show that F has the null vector property.
That is, if there are nonzero X, Y ∈ TpM such that Sp(X, X¯, Y, Y¯ ) = 0, then
F (S)p(X, X¯, Y, Y¯ ) ≥ 0. We have the following:
Proposition 3.5.2. If S ≥ 0 and there exists X, Y ∈ D1,0, both non-zero, such that
Sp(X, X¯, Y, Y¯ ) = 0, then F (S)p(X, X¯, Y, Y¯ ) ≥ 0.
Proof. This is a local problem, so we choose a coordinate patch pi : U ⊂M → V ⊂ Cn
with p ∈ U . Then the transverse metric gT on U induces a Ka¨hler metric gV on V
which is evolving by Ka¨hler-Ricci flow. So F (RT ) restricted to V is the same as in the
Ka¨hler case. The result now follows from the analogous result in the Ka¨hler setting.
See [2], [29] or page 107 of [9].
Now we can prove theorem 3.5.1:
91
Proof. It suffices to prove the theorem for short time, say for t ∈ [0, t0] for some
t0 > 0. Then all metrics involved have bounded geometry. We define an auxiliary
tensor S in local coordinates by
Sij¯kl¯ :=
1
2
(
gTij¯g
T
kl¯ + g
T
il¯g
T
kj¯
)
.
Note that S has the same type as the curvature tensor, S > 0 at all points of M and
S is parallel, hence ∆S = 0. Then there is a constant C1 > 0 such that
−C1S ≤ ∂S
∂t
≤ C1S.
Since F is smooth and S > 0, there is a constant C2 > 0 such that
F (HT )− F (HT + fS) ≥ −C2|f |S
for f ∈ R with |f | ≤ 1. When f is a basic function, fS has the same type as the
transverse curvature tensor. Let f = f(t, x) be a basic function for all t and choose
ε > 0 small enough so that ε|f(t, x)| ≤ 1 for all (t, x). Then we compute
∂
∂t
(HT + εfS) = ∆HT + F (HT ) +HT + ε
∂f
∂t
S + εf
∂S
∂t
= F (HT )− F (HT + εfS) + F (HT + εfS) +HT + εfS
+ ∆(HT + εfS) + εS
(
∂f
∂t
−∆f
)
+ εf
(
∂S
∂t
− S
)
≥ F (HT + εfS) + ∆(HT + εfS) +HT + εfS
+ εS
(
∂f
∂t
−∆f − (C1 + 1 + C2)|f |
)
.
92
Now let f satisfy
∂f
∂t
−∆f − (C1 + 1 + C2)f = 1
with initial condition f(·, 0) ≡ 1. In local coordinates, ∆f = ξ2f + gij¯∂i∂j¯f . Since
ξgij¯ = 0, we have ξ(∆f) = ∆(ξf). Hence
∂(ξf)
∂t
= ∆(ξf) + (C1 + 1 + C2)ξf.
Since ξf(0) = 0, it follows from the uniqueness of the solution that ξf ≡ 0. Thus
f(·, t) is basic. By the maximum principle, f > 0 for t > 0. Therefore
∂
∂t
(HT + εfS) ≥ F (HT + εfS) + ∆(HT + εfS) +HT + εfS + εS. (3.35)
Since HT ≥ 0 at t = 0, we have HT + εfS > 0 at t = 0. In fact, HT + εfS > 0 for
all t. We give a proof by contradiction.
If not, then there is a first time t0 > 0, a point p ∈M and unit vectorsX, Y ∈ D1,0p
such that (HT + εfS)(X, X¯, Y, Y¯ ) = 0 at (t0, p). By proposition 3.5.2, at (t0, p),
F (HT + εfS)(X, X¯, Y, Y¯ ) ≥ 0.
Now choose a coordinate patch U about p ∈ M . Then pi : U → V ⊂ Cn is a
Riemannian submersion. Let XV = pi∗X, YV = pi∗Y . Extend XV and YV to a
normal neighborhood of (pi(p), t0) by parallel translation along radial geodesics such
that ∇XV = ∇YV = 0 at (pi(p), t0). Then extend XV and YV around t0 such that
93
∂tXV = ∂tYV = 0 at (pi(p), t0). Using (3.35) at (p, t0) we have
0 ≥ ∂
∂t
(
HT + εfS
)
(XV , X¯V , YV , Y¯V )
=
(
∂
∂t
(HT + εfS)
)
(XV , X¯V , YV , Y¯V )
≥ (∆(HT + εfS)) (XV , X¯V , YV , Y¯V ) + εS(XV , X¯V , YV , Y¯V )
= ∆
(
(HT + εfS)(XV , X¯V , YV , Y¯V )
)
+ εS(XV , X¯V , YV , Y¯V )
> 0.
Contradiction. Thus HT + εfS > 0 for all t. Letting ε → 0, we have HT ≥ 0
for all t. Now suppose that HT > 0 at some point in M when t = 0. Define
f0(p) := minH
T
p (X, X¯, Y, Y¯ ) for X, Y ∈ D1,0p with |X| = |Y | = 1. Then f0 ≥ 0
everywhere and f0 > 0 somewhere. As ξ acts on g isometrically, f0 is constant along
the orbits of ξ. Hence ξf0 = 0. As before, we compute that
∂
∂t
(HT − fS) ≥ F (HT − fS) + ∆(HT − fS) + S
(
−∂f
∂t
+ ∆f − (C1 + C2)|f |
)
.
Now let f satisfy
∂f
∂t
= ∆f − (C1 + C2)f
with initial condition f(·, 0) = f0. Again, since ξf0 = 0 we have ξf ≡ 0. If we let
f˜ = e(C1+C2)tf , then f˜ solves the heat equation
∂f˜
∂t
= ∆f˜ .
94
It is well known that a function f˜ satisfying the heat equation with nonnegative initial
condition is positive for t > 0, see [32]. Thus f > 0 for t > 0 and
∂
∂t
(HT − fS) ≥ F (HT − fS) + ∆(HT − fS).
By the definition of f0, we have H
T − fS ≥ 0 when t = 0. Arguing as above, we can
show that HT −fS ≥ 0 for t > 0. Since S > 0 and f > 0, it follows that HT > 0.
95
CHAPTER IV
THE RICCI FLOW OF A SASAKI METRIC
In this chapter we want to investigate the behavior of the Ricci flow when the
initial metric is Sasaki. We are not talking about the Sasaki-Ricci flow as in [12], [19]
or [36], which is just the transverse Ka¨hler-Ricci flow of a Sasaki manifold, but rather
the Ricci flow of the actual Sasaki metric. The Ricci flow of a Ka¨hler metric preserves
the Ka¨hler condition and this has been an extremely fruitful area of mathematical
research since the 1980s. Since Sasaki geometry is closely related to Ka¨hler geometry,
it seems reasonable to suspect that the Ricci flow of a Sasaki metric should be a
tractable subject, but one quickly runs into trouble; the Ricci flow need not preserve
the Sasaki condition. For example, consider a Sasaki-Einstein metric g0 with Einstein
constant λ. The Ricci flow
∂g
∂t
= −2Ricg with g(0) = g0 is given by g(t) = (1−2λt)g0.
Since Sasaki metrics are not preserved by homothety, g(t) is not Sasaki for t > 0.
However, a scaling of a Sasaki metric is a quasi-Sasaki metric. This gives us hope that
the Ricci flow does preserve some sort of structure related to the Sasaki geometry.
We begin with the following useful observation: Given a Sasaki structure
(ξ, η,Φ, g), we can create a two-parameter family of quasi-Sasaki structures. Given
A,B > 0, let ξB =
1√
B
ξ, ηB =
√
Bη and gA,B = ηB ⊗ ηB + AgT . Using the same
(1,1)-tensor Φ, it is easy to see that the (almost) contact structure is preserved.
A straighforward computation shows that the metric compatibility and normality
conditions are also preserved. However, gA,B(X,ΦY ) 6= dηB(X, Y ). Rather we have
gA,B(X,ΦY ) =
A√
B
dηB(X, Y ). Since Ka¨hler metrics are preserved by homothety, the
transverse metric gTA,B = Ag
T is Ka¨hler and thus (ξB, ηB,Φ, gA,B) defines a quasi-
Sasaki structure.
96
Definition 4.0.3. A quasi-Sasaki structure obtained via the above construction will
be called an (A,B)-deformation of a Sasaki structure.
By choosing A =
√
B, the deformed structure remains Sasaki. These type
of deformations are known in the literature as transverse or D-homotheties. Note
also that an (A,A)-deformation results in an honest scaling of the metric, but does
not preserve the Sasaki structure unless A = 1. From proposition 2.1.1 we get the
following corollary.
Corollary 4.0.4. If (ξ, η,Φ, g) is an (A,B)-deformation of a Sasaki structure, then
for X, Y ∈ D the Ricci tensor of g satisfies
Ric(ξ, ξ) = 2n
B
A2
Ric(X, ξ) = 0
Ric(X, Y ) = RicT (X, Y )− 2 B
A2
gT (X, Y ).
4.1. η-Einstein metrics
It is difficult (as far as the author can tell) to say much about the Ricci flow of
an arbitrary Sasaki metric. However, for a special class of Sasaki metrics known as
η-Einstein, we can actually say quite a lot. It is these metrics whose Ricci flow we
shall investigate.
Definition 4.1.1. A Sasaki metric g is η-Einstein if there are real numbers µ and ν
such that Ricci tensor of g satisfies
Ricg = µg + νη ⊗ η. (4.1)
97
Recall that for a Sasaki structure (ξ, η,Φ, g), η ∧ (dη)n is a volume form,
g(X,ΦY ) = dη(X, Y ) for all X, Y ∈ TM and for X, Y ∈ D the Ricci tensor satisfies
Ric(ξ, ξ) = 2n, (4.2)
Ric(X, ξ) = 0, (4.3)
Ric(X, Y ) = RicT (X, Y )− 2gT (X, Y ), (4.4)
From equations (4.1) and (4.2) we see that µ+ν = 2n. From equations (4.3) and
(4.4) we see that a Sasaki manifold is η-Einstein if and only if the transverse metric
is Ka¨hler-Einstein. Indeed, if RicT = λgT then Ric = (λ − 2)g + (2n + 2 − λ)η ⊗ η.
Conversely, if Ric = µg + νη ⊗ η then RicT = (µ + 2)gT . We should also note that
an η-Einstein manifold has constant scalar curvature. Tracing equation (4.1) we see
that the scalar curvature R = (2n + 1)µ + ν = 2n(µ + 1). An excellent reference for
the geometry of η-Einstein manifolds is the paper [6].
Let (ξ0, η0,Φ, g0) be η-Einstein. Then we can write g0 = η0 ⊗ η0 + gT0 where gT0
is transverse Ka¨hler-Einstein with RicT0 = λg
T
0 . Now consider an (A,B)-deformation
of this Sasaki structure. By corollary 4.0.4, for X, Y ∈ D, the Ricci tensor of the
deformed metric satisfies
Ric(ξ0, ξ0) = 2n
B2
A2
,
Ric(X, ξ0) = 0,
Ric(X, Y ) =
(
λ− 2B
A
)
gT0 (X, Y ).
98
Therefore, by the above equations, a metric of the form
g(t) := A(t)gT0 +B(t)η0 ⊗ η0 (4.5)
is the solution to the Ricci flow equation
∂g
∂t
= −2Ricg, g(0) = g0, (4.6)
with g0 an η-Einstein metric, if and only if A(t), B(t) > 0 satisfy
dA
dt
= 4
B
A
− 2λ, A(0) = 1, (4.7)
dB
dt
= −4nB
2
A2
, B(0) = 1. (4.8)
By standard existence and uniqueness theory for ODEs (e.g. the Picard-Lindelo¨f
theorem), the system (4.7), (4.8) has a unique solution that exists for at least a short
time. Note that if we try to impose the additional restriction that A =
√
B, then
the system has no solution. Hence the metric cannot evolve by Sasaki metrics in this
way. But, as we mentioned before, the metrics in (4.5) are quasi-Sasaki.
For our qualitative analysis of these differential equations, we will make use of
the conserved quantity
Λ :=
(
λ
2(n+ 1)
− B
A
)
B−
n+1
n . (4.9)
Indeed, a straightforward computation reveals that d
dt
Λ = 0. From the requirement
that A(0) = B(0) = 1, we find that Λ = λ
2(n+1)
− 1.
99
Since a Sasaki metric is η-Einstein if and only if the transverse metric is Ka¨hler-
Einstein, we naturally have three cases to consider. In the following sections we
investigate the Ricci flow equations (4.7) and (4.8) when the transverse Einstein
constant λ is zero, positive and negative, respectively. We want to understand the
behavior of the functions A and B in each case. Knowing their behavior, we can
determine how the curvature is changing along the Ricci flow. Computing as in
chapter II, the sectional curvature of a plane spanned by ξ and X ∈ D is
κ(X, ξ) =
B
A2
κ0(X, ξ), (4.10)
where κ0 denotes the sectional curvature with respect to the initial metric g0. The
sectional curvature of a plane spanned by X, Y ∈ D is
κ(X, Y ) =
1
A
κT0 (X, Y ) +O
(
B
A2
)
. (4.11)
4.2. Transverse Calabi-Yau Structure
When λ = 0 we have c1B = 0 and the transverse structure is Calabi-Yau. In this
case we can solve the Ricci flow equations explicitly. We compute
d
dt
(
B
A2
)
= −4(n+ 2)
(
B
A2
)2
. (4.12)
We solve this to get
B
A2
=
1
1 + 4(n+ 2)t
. (4.13)
Now we have
dB
dt
= −4n
(
B
A2
)
B =
−4n
1 + 4(n+ 2)t
B.
100
This yields
B(t) = (1 + 4(n+ 2)t)
−n
n+2 .
Then from equation (4.13) we get
A(t) = (1 + 4(n+ 2)t)
1
n+2 .
Therefore, the flow exists for all t ∈ [0,∞) and A(t)→∞ while B(t)→ 0 as t→∞.
In the limit, all sectional curvatures tend to zero and the metric degenerates to a flat
metric on the transverse space.
Example 4.2.1. For an example of an η-Einstein metric with transverse Calabi-Yau
structure, we consider the Heisenberg group H = H(3,R). It is a nilpotent Lie group
that can be identified algebraically as
H =


1 x z
0 1 y
0 0 1
 : x, y, z ∈ R
 .
We identify this Lie group topologically with R3 in the obvious way. We consider
the group action to be from the right. Hence, the action of right multiplication by
(a, b, c) is (x, y, z) 7→ (x + a, y + b, z + c + bx). The Lie algebra is generated by the
right invariant vector fields ξ = 2 ∂
∂z
, Y = 2 ∂
∂y
and X = 2( ∂
∂x
+ y ∂
∂z
) which satisfy the
bracket relations [ξ, Y ] = [ξ,X] = 0 and [Y,X] = 2ξ.
Define a 1-form η := 1
2
(dz − ydx). Note that η is invariant under the group
action. As η ∧ dη = 1
4
dx ∧ dy ∧ dz, we see that η is a contact form. We have
D := ker(η) = {X, Y }. The complex structure J : D → D defined by JX = Y and
101
JY = −X gives (D, dη, J) a Ka¨hler structure. This is just the standard, flat, Ka¨hler
structure on R2 = C.
Define a metric g0 := η ⊗ η + 14(dx⊗ dx + dy ⊗ dy). Then g0 is invariant under
the group action, {ξ, Y,X} is an orthonormal frame and (ξ, η,Φ, g0) gives H a Sasaki
structure. Here Φ is just the extension of J such that Φξ = 0. Explicitly we have
Φ = ∂
∂y
⊗ dx − ( ∂
∂x
+ y ∂
∂z
) ⊗ dy. The nonzero components of the Ricci tensor of g0
are Ric(ξ, ξ) = 2 and Ric(Y, Y ) = Ric(X,X) = −2. Therefore Ricg0 = −2g0 + 4η⊗ η.
Hence g0 is an η-Einstein metric. Under the Ricci flow, the curvature tends to zero
and the metric degenerates to a flat metric on R2.
Now define the compact quotient manifold N := H(3,R)/H(3,Z) where H(3,Z)
denotes matrices of the above form with x, y, z ∈ Z. Then N is a non-trivial circle
bundle over a complex torus and the Sasaki structure on H descends to a Sasaki
structure on N . The S1 fiber corresponds to the integral curves of the Reeb vector
field ξ = 2 ∂
∂z
. The Ricci flow on N behaves the same as the Ricci flow on H. Under
the flow, the S1 fiber shrinks to a point and we have convergence to a flat torus.
Remark 4.2.2. The behavior of the Ricci flow seen in this example is precisely the
behavior of the Ricci flow on any Nil 3-manifold with an invariant metric. See, for
example, chapter 1.6 of [11].
4.3. Transverse Fano Structure
When λ > 0 we have c1B > 0 and the transverse structure is Fano. In general,
the flow only exists for finite time t ∈ [0, T ), T < ∞, and both A(t), B(t) → 0 as
t→ T . We will consider three cases:
102
1. If λ > 2(n+ 1) then, from equation (4.9), Λ > 0. In this case we have
B
A
=
λ
2(n+ 1)
− ΛB n+1n < λ
2(n+ 1)
.
Then
dA
dt
= 4
B
A
− 2λ < 2λ
n+ 1
− 2λ < 0.
This shows that A(t)→ 0 in finite time. Since 0 < B < λ
2(n+1)
A, it follows that
B(t)→ 0 in finite time as well.
2. If λ = 2(n+ 1), then Λ = 0 and the solution is given by A(t) = B(t) = 1− 4nt,
as is easily checked. In this case the initial metric is Sasaki-Einstein. Notice
that the flow only exists for t ∈ [0, 1
4n
)
and both A(t), B(t)→ 0 as t→ 1
4n
.
3. If 0 < λ < 2(n+ 1), then Λ < 0. In this case we have
B
A
=
λ
2(n+ 1)
− ΛB n+1n > λ
2(n+ 1)
.
Then
dB
dt
= −4n
(
B
A
)2
<
−nλ2
(n+ 1)2
< 0.
This implies that B(t) → 0 in finite time. Since 0 < A < 2(n+1)
λ
B, it follows
that A(t)→ 0 in finite time as well.
In this transverse Fano case, we see that all the sectionals curvatures tend to infinity
as we approach the terminal time. This is to be expected (at least in one direction)
of a Ricci flow with a finite-time singularity. It would be interesting to determine the
terminal time explicitly in terms of λ and n in cases 1 and 3 above.
103
Example 4.3.1. The canonical example of an η-Einstein manifold with transverse
Fano structure is the classical Hopf fibration. This gives S3 the structure of a principle
circle bundle over CP 1 and the standard Sasaki structure on S3 (compare with
example 1.3.5). Writing S3 = {(w, z) ∈ C2 : |w|2 + |z|2 = 1}, the Hopf fibration
S1 ↪→ S3 → CP 1 is induced by the map pi : S3 → CP 1, where (w, z) 7→ [w : z].
Here [w : z] denotes the homogenous coordinates on CP 1. As [w : z] = [λw : λz] for
λ ∈ C∗, it is clear that the fibers of pi are circles S1 = {λ ∈ C : |λ| = 1}.
As we saw in chapter II, the collection of principal circle bundles over CP 1 has a
group structure isomorphic to H2(CP 1,Z) ' Z with generator 1
2pi
[ωFS]. The Fubini-
Study metric has constant holomorphic sectional curvature equal to 4. As in the
proof of theorem 2.2.6, there is a contact form η on S3 such that 1
2
dη = pi∗ωFS and
g := η ⊗ η + pi∗ωFS(J ·, ·)
defines a Sasaki metric. The metric g is in fact the round metric of constant sectional
curvature 1. So g is an Einstein (hence η-Einstein) metric with Ric = 2g. The Ricci
flow on S3 with initial metric g is g(t) := (1− 4t)g. The flow exists for t ∈ [0, 1
4
)
and
S3 shrinks to a point at the terminal time T = 1
4
. As t→ T the sectional curvatures
blows up.
Now fix ε ∈ (0, 1) and define a Sasaki metric gε via a D-homothetic deformation
of g. That is, gε := εg + (ε
2 − ε)η ⊗ η. Then g¯ := ε−1gε = εη ⊗ η + pi∗ωFS(J ·, ·)
and (S3, g¯) is the classical ε-collapsed Berger sphere. Let g¯(t) be the Ricci flow with
initial metric g¯. This flow is well-understood (see chapter 1.5 of [11]). The flow exists
on a maximal finite time interval [0, T ) and the metric becomes asymptotically round
as t → T . Thus, gε(ε−1t) = εg¯(ε−1t) is the Ricci flow on [0, εT ) with intitial metric
gε and it behaves the same as the flow g¯(t).
104
4.4. Transverse Canonical Structure
When λ < 0 we have c1B < 0 and we say that the transverse structure is canonical.
In this case the flow exists for all t ≥ 0, A(t) → ∞ as t → ∞ and B(t) tends to a
constant c(n, λ) ∈ (0, 1) as t→∞.
Since dB
dt
= −4nB2
A2
< 0 and B(0) = 1, we have 0 < B(t) ≤ 1. From (4.8) and
(4.9),
dB
dt
= −4n
(
λ
2(n+ 1)
− ΛB n+1n
)2
. (4.14)
Thus dB
dt
is uniformly bounded. Hence, the right hand side of (4.14) and its first
derivative are uniformly bounded. By a standard result in ODEs (see [23]), this
implies that (4.14) with the initial condition B(0) = 1 has a unique solution defined
for all t ≥ 0. Then it follows from (4.9) that equation (4.7) has a unique solution
defined for all t ≥ 0.
Since we must have A,B > 0, right away we see that dA
dt
= 4B
A
− 2λ ≥ −2λ > 0.
Thus
A(t) ≥ 1− 2λt.
Hence A(t)→∞ as t→∞. Now, from equation (4.9) we have
B
A
=
λ
2(n+ 1)
− ΛB n+1n ,
and it follows that
lim
t→∞
B(t) =
(
λ
2Λ(n+ 1)
) n
n+1
=
(
λ
λ− 2(n+ 1)
) n
n+1
.
As in the λ = 0 case, as t → ∞ all of the sectional curvatures tend to zero and the
metric degenerates to a flat metric on the transverse space.
105
Example 4.4.1. For an example of an η-Einstein metric with transvere canonical
structure, we consider M = B × R where B = {z ∈ C : |z| < 1}. Let ω be the
Bergman metric on B. We take
ω = 2i∂∂¯ log(1− |z|2) = 2i dz ∧ dz¯
(1− |z|2)2 .
Then (B,ω, J) is Ka¨hler with constant, negative, holomorphic sectional curvature.
Here J is the usual complex structure on C. Since ω is a real, closed 2-form on B,
there is some real 1-form α such that ω = dα. In particular,
α = i
zdz¯ − z¯dz
(1− |z|2) .
Let pi : M → B be the projection. Let t be the coordinate on R and define
1
2
η := pi∗α + dt. Then 1
2
dη = pi∗ω and we see that η is a contact form. The Reeb
vector field ξ = 1
2
∂
∂t
. We take Φ = p˜i ◦ J ◦ pi∗. Now define a metric
g0 := pi
∗ω(·, J ·) + η ⊗ η = 1
2
dη(·,Φ·) + η ⊗ η.
Then (ξ, η,Φ, g0) is a Sasaki structure on M .
In real coordinates (x, y), we have
α = 2
xdy − ydx
1− x2 − y2 and ω = 4
dx ∧ dy
(1− x2 − y2)2 .
Define vector fields
X :=
1− x2 − y2
2
(
∂
∂x
− η
(
∂
∂x
)
ξ
)
and Y :=
1− x2 − y2
2
(
∂
∂y
− η
(
∂
∂y
)
ξ
)
.
106
Then D = span{X, Y } and {ξ,X, Y } is an orthonormal frame for g. Working with
this frame, we compute the nonzero components of the Ricci tensor to be Ric(ξ, ξ) = 2
and Ric(X,X) = Ric(Y, Y ) = −3. Thus Ric = −3g + 5η ⊗ η. Hence, g is η-Einstein.
If we consider a metric of the form g = Api∗ω(·, J ·) + Bη ⊗ η, then we compute
that, with respect to the frame {ξ,X, Y }, the nonzero components of the Ricci tensor
of g are Ric(ξ, ξ) = 2B
2
A2
and Ric(X,X) = Ric(Y, Y ) = −1 − 2B
A
. Thus, we will have
the Ricci flow with inital metric g0 if we can solve
dA
dt
= 2 + 4
B
A
, A(0) = 1,
dB
dt
= −4B
2
A2
, B(0) = 1.
From the above we know that the solution exists for all t ≥ 0 and as t → ∞,
A(t)→∞, B(t)→
√
5
5
and the sectional curvatures all tend to zero.
Remark 4.4.2. There is a classification of 3-dimensional Sasaki manifolds due to
Geiges. The three examples we have presented in this chapter are the universal covers
of the geometries given in the classification. Our examples easily generalize to higher
dimensions, but in higher dimensions there is no such classification. In dimension 3,
the basic first Chern class is always signed or null. As one might expect, there is a
correspondence between the type of Sasaki structure (positive, negative or null c1B)
and the three geometries given in Geiges’ classification. A well-known theorem of
Belgun makes this precise. See [6] for more information about this.
107
4.5. Rigidity
To have a Ricci flow of the form (4.5), the initial Sasaki metric g0 must have
very special geometry. Indeed, we can prove the following:
Proposition 4.5.1. Let (ξ, η0,Φ, g0) be a Sasaki structure. Suppose that the Ricci
flow with initial metric g0 is given by g(t) = A(t)g
T
0 + B(t)η0 ⊗ η0. Then g0 is an
η-Einstein metric.
Proof. As we saw before, the metric g(t) provides a quasi-Sasaki structure and we
have Rict(ξ, ξ) = 2n
B2
A2
, Rict|D = RicTt − 2BAgT0 and Rict(X, ξ) = 0 for all X ∈ D.
Since gT (t) = A(t)gT0 is just a scaling of g
T
0 , we have Ric
T
t = Ric
T
0 . Hence, A satisfies
dA
dt
gT0 (X, Y ) = 4
B
A
gT0 (X, Y )− 2RicT0 (X, Y )
for all X, Y ∈ D. Rewrite the above as
2RicT0 (X, Y ) =
(
4
B
A
− dA
dt
)
gT0 (X, Y ).
Since the left-hand side is independent of time, 4B
A
− dA
dt
must be a constant, call
it 2δ. Then RicT0 = δg
T
0 . Thus, the initial transverse metric g
T
0 is Ka¨hler-Einstein.
Therefore, the initial Sasaki metric is η-Einstein.
4.6. Questions and Directions for Future Research
We have merely scratched the surface of the investigation into the Ricci flow of
a Sasaki metric and there are many questions still to be answered. We list just a few
of them below.
108
1. The Ricci flow with an initial Sasaki metric is not Sasaki for t > 0, but is it
quasi-Sasaki in general?
2. The author can show that, in general, the length of ξ with respect to the evolving
metric is decreasing. If the Ricci flow has a finite time singularity, does the
length of ξ going to zero characterize the singular time?
3. Is there a relationship between the Ricci flow of a Sasaki metric and the
transverse Ka¨hler-Ricci flow of a Sasaki metric?
109
CHAPTER V
APPENDIX
In chapter II we proved that under certain conditions a quasi-Sasaki-Einstein
manifold has a local Riemannian product structure. The proof of theorem 2.6.3 made
use of an integrable, in fact parallel, almost product structure. We decided to include
this short section here to review the necessary terminology and some criteria for
integrability.
Definition 5.0.1. An almost product structure on a smooth manifold M is a
(1,1)-tensor F (i.e. an endomorphism of the tangent bundle) such that F 2 = I.
If we let P = 1
2
(I + F ) and Q = 1
2
(I − F ), then
P 2 = P, Q2 = Q and PQ = QP = 0. (5.1)
Conversely, given (1,1)-tensors P and Q satisfying (5.1), setting F = P − Q
gives an almost product structure. The images of P and Q define complementary
distributions P and Q of TM . We will assume that P and Q have constant rank
p and q, respectively. That is, the dimension of the corresponding distribution is
constant for all x ∈ M . On the other hand, given two distributions P and Q such
that TM = P ⊕Q, setting P and Q to be their respective projections, we see that P
and Q satisfy (5.1).
Definition 5.0.2. Let D be a k-dimensional distribution on a smooth m-dimensional
manifold M . We say that D is involutive if for any X, Y ∈ D, the Lie bracket
[X, Y ] ∈ D.
110
We say that D is integrable if for each point x ∈ M there is a k-dimensional
submanifold N containing x such that TxN = Dx. We call N an integral submanifold
of D (through x).
We say that D is completely integrable if for each point x ∈ M , there is a
coordinate neighborhood (U, x1, . . . , xm) of x such that {∂x1 , . . . , ∂xk} is a local frame
for D. Setting xi = ci for i = k + 1, . . . ,m, and some constants ci gives an integral
submanifold of D.
The famous theorem of Frobenius says that these three notions are equivalent.
Theorem 5.0.3 (Frobenius). A distribution is involutive if and only if it is integrable
if and only if it is completely integrable.
Definition 5.0.4. An almost product structure F on M is integrable if the
distributions induced by P and Q are integrable in the sense of Frobenius. In this
case we say that M is a locally product manifold.
So if F = P − Q is an integrable almost product structure on M , then by
the Frobenius theorem, for each point x ∈ M , there is a coordinate neighborhood
(U, x1, . . . , xm) such that {∂x1 , . . . , ∂xp} is a local frame for P and {∂xp+1 , . . . , ∂xm} is
a local frame for Q. Setting xp+1, . . . , xm constant gives an integral submanifold of P
and setting x1, . . . , xp constant gives an integral submanifold of Q. Thus M is locally
the product of two manifolds whose tangent spaces are (isomorphic to) P and Q.
By proposition 3.1.4 in [13], the integrability of the almost product structure F is
equivalent to the vanishing of the Nijenhuis tensor NF defined by (1.2). Furthermore,
since NI = 0, one can check that NF = 2NP = −2NQ. Therefore:
Proposition 5.0.5. An almost product structure F = P −Q is integrable if and only
if NF = 0 if and only if NP = 0 if and only if NQ = 0.
111
Now let g be a Riemannian metric and ∇ the Levi-Civita connection of g. Recall
that ∇g = 0 and ∇XY −∇YX = [X, Y ]. If T is a parallel (1,1)-tensor (i.e. ∇T = 0),
then it is straightforward to check that the Nijenhuis tensor NT = 0. It is also easy
to check that the image of a parallel (1,1)-tensor is an involutive distribution. Since
∇F = 2∇P = −2∇Q, we see that if any of F , P or Q is parallel then the almost
product structure is integrable, but in this case even more is true.
Definition 5.0.6. Let (M, g) be a locally product manifold with TM = P ⊕ Q
an orthogonal decomposition. We say that (M, g) is a locally decomposable
Riemannian manifold if for any U, V ∈ P and X, Y ∈ Q we have Xg(U, V ) = 0
and Ug(X, Y ) = 0.
In this case the metric splits as an honest product metric corresponding to the
locally product structure. It is known by work of Yano that a necessary and sufficient
condition for a locally product manifold to be a locally decomposable Riemannian
manifold is ∇F = 0. Therefore we have the following:
Proposition 5.0.7. Let F = P −Q be an almost product structure on (M, g). Then
(M, g) is a locally decomposable Riemannian manifold if and only if ∇F = 0 if and
only if ∇P = 0 if and only if ∇Q = 0 if and only if P and Q are parallel distributions.
Remark 5.0.8. We remark that if F is an integrable almost product structure,
then there exists a torsion-free, affine connection (not necessarily the Levi-Civita
connection) with respect to which F is parallel. See the book [13] for reference.
In this section we have collected the technical lemmas and propositions used in
chapter III. We begin with
112
Lemma 5.0.9. Let x ∈M and V (r) := Vol(BT (x, r)). If Fξ is quasi-regular, then
lim
r→0
V (r)
V (r/2)
= 4n.
Proof. Recall that pi : M → Z is a principle S1-orbibundle over the orbifold Z =
M/Fξ. Pick a point x ∈ M and choose r > 0 small enough so that BT (x, r) is a
trivial S1 bundle over the geodesic ball pi(BT (x, r)) ⊂ Z. Since the set of orbifold
singularities in Z has measure zero, it does not contribute anything when we compute
volumes. Thus we may assume that pi(BT (x, r)) contains only smooth points. For
smooth points in Z, all S1 fibers have the same length, call it `. On BT (x, r) the
metric and volume form can be written as
g = η ⊗ η + pi∗h,
dV = η ∧ (pi∗ωh)n,
where h is a Ka¨hler metric on pi(BT (x, r)) and ωh is the Ka¨hler form. Hence
V (r) =
∫
BTg (x,r)
dV =
∫
S1×pi(BTg (x,r))
η ∧ (pi∗ωh)n = ` · Vol(pi(BTg (x, r)).
The result follows from this and the fact that for any geodesic ball B(r) ⊂ Z of radius
r,
lim
r→0
Vol(B(r))
Vol(B(r/2))
= 22n = 4n.
113
Proposition 5.0.10. Let M2n+1 be a compact quasi-Sasaki manifold without
boundary. Let α and β be basic forms with deg(α) + deg(β) = 2n− 1. Then
∫
M
dα ∧ β ∧ η + (−1)deg(α)
∫
M
α ∧ dβ ∧ η = 0.
Proof. We compute that
d(α ∧ β ∧ η) = dα ∧ β ∧ η + (−1)deg(α)α ∧ dβ ∧ η − α ∧ β ∧ dη.
The form α ∧ β ∧ dη is basic and of degree 2n + 1, thus α ∧ β ∧ dη = 0. The result
now follows from Stokes’ theorem.
Corollary 5.0.11. Let α be a basic (2n− 1)-form on a closed quasi-Sasaki manifold
M2n+1. Then ∫
M
dBα ∧ η = 0.
Proof. Take β = 1 in the above proposition.
Lemma 5.0.12. For any bundle-like metric g and τ > 0, we have µT (g, τ) > −∞.
Proof. Letting w := τ−n/2e−f/2, we can write the WT -functional as
WT (g, f, τ) =
∫
M
τ(RTw2 + 4|∇w|2)− 2(log(w) + n log(τ))w2 dV.
For a constant c > 0, it is not hard to see that WT (cgT , f, cτ) =WT (gT , f, τ). Hence
we may assume without loss of generality that τ = 1. Thus we need to show that
there is a constant C = C(g) such that
∫
M
RTw2 + 4|∇w|2 − 2w2 log(w) dV ≥ C.
114
Since infx∈M RT > −∞ and
∫
M
w2 dV = 1, we only have to show that
∫
M
4|∇w|2 − 2w2 log(w) dV ≥ C.
This follows from the log Sobolev inequality (see, for example, lemma 6.36 in [9]).
Lemma 5.0.13. Suppose there is a sequence rk > 0 and a sequence of points and
times (pk, tk) ∈M × [0,∞) with tk →∞ such that r−2nk Vk(rk)→ 0 as k →∞, where
Vk(rk) = Vol(B
T
g(tk)
(pk, rk)).
Then we may assume that Vk(rk) ≤ 5nVk(rk/2)
Proof. If it is not the case that Vk(rk) ≤ 5nVk(rk/2), then let rik = 2−irk for integers
i > 0. By lemma 5.0.9, we can choose i0 to be the least integer such that Vk(r
i0
k ) ≤
5nVk(r
i0
k /2). So for i < i0 we have Vk(r
i
k) > 5
nVk(r
i
k/2) = 5
nVk(r
i+1
k ). Iterating this
inequality, we get
Vk(r
i0
k ) < 5
−i0nVk(rk).
Therefore, as k →∞,
(ri0k )
−2nVk(r
i0
k ) <
(
4
5
)i0n
r−2nk Vk(rk)→ 0.
Thus we can replace rk with r
i0
k and we have Vk(r
i0
k ) ≤ 5nVk(ri0k /2).
Lemma 5.0.14. For any ε > 0 there are k1 < k2 such that if the transverse diameter
is sufficiently large then
1. Vol(Bξ(k1, k2)) < ε
2. Vol(Bξ(k1, k2)) ≤ 210nVol(Bξ(k1 + 2, k2 − 2))
115
3. There exists r1 ∈ [k1, k1 + 1], r2 ∈ [k2 − 1, k2] and a uniform constant C such
that ∫
Bξ(r1,r2)
RT dV ≤ CVol(Bξ(k1, k2)).
Proof. Let ε > 0 be given. As the manifold M is closed and ∂tdVt = ∆
Tu dVt =
∆u dVt, the volume of M is constant along the flow. So if the transverse diameter
becomes sufficiently large, then there is K big enough so that for all k2 > k1 ≥ K we
have Vol(Bξ(k1, k2)) < ε.
Take k2 > k1 ≥ K. If (2) holds then we are done. Suppose that (2) does not
hold. We then consider the annulus Bξ(k1 + 2, k2 − 2) and check to see if (2) holds
now. If it does then we are done, otherwise we repeat this step. Suppose that after
repeating this step m times, we are still unable to find k1 and k2 so that (1) and (2)
hold. Then we would have
Vol(Bξ(k1, k2)) > 2
10nmVol(Bξ(k1 + 2m, k2 − 2m)).
We can assume that k2−k1−4m is very close to 1. Then 2m is close to 12(k2−k1−1).
If we choose k  1 and set k1 = k/2 and k2 = 3k/2, then m is close to k/4, k1 + 2m
is close to k and k2 − 2m is close to k + 1.
The length of ξ is constant along the flow. Hence, the lengths of the integral
curves of ξ (which are closed geodesics) are constant along the flow. Thus they are
uniformly bounded above by some ` > 0. So for any p, q ∈ M we have dg(t)(p, q) ≤
dTg(t)(p, q) + `. This, along with proposition 3.3.7, gives R
T ≤ C22k on Bξ(k, k + 1)
for some uniform constant C. The annulus Bξ(k, k + 1) contains at least 2
2k disjoint
116
transverse balls of radius 2−k. Then by theorem 3.4.1 we have
Vol(Bξ(k, k + 1)) ≥
22k∑
i=1
Vol(BTg(t)(2
−k)) ≥ C22k2−2kn.
Combining this with the above we find that
ε > Vol(Bξ(k1, k2)) > 2
10nmVol(Bξ(k, k + 1)) ≥ C210nm22k2−2kn.
If k is large enough, then 10m > 2k and the above is a contradiction. Therefore (1)
and (2) hold.
To prove (3) we first define the transverse sphere STg (x, r) := {y ∈M : dTg (x, y) =
r}. Then d
dr
Vol(BTg (x, r)) = Vol(S
T
g (x, r)). Given k1  k2 such that (1) and (2) hold,
there is r1 ∈ [k1, k1 + 1] such that
Vol(STg(t)(x, 2
r1)) ≤ Vol(Bξ(k1, k2))
2k1−1
.
If not, then we would have
Vol(Bξ(k1, k1 + 1)) =
∫ 2k1+1
2k1
Vol(ST (x, r)) dr > 2Vol(Bξ(k1, k2)).
This is a contradiction since k1  k2. Similarly we can prove there is r2 ∈ [k2− 1, k2]
such that
Vol(STg(t)(x, 2
r2)) ≤ Vol(Bξ(k1, k2))
2k2−1
.
117
Using these volume estimates for the spheres and proposition 3.3.7, we have
∫
Bξ(r1,r2)
RT − n dV =
∫
Bξ(r1,r2)
−∆u dV
=
∫
ST (x,2r1 )
|∇u| dV +
∫
ST (x,2r2 )
|∇u| dV
≤ Vol(Bξ(k1, k2))
2k1−1
C2k1+1 +
Vol(Bξ(k1, k2))
2k2−1
C2k2
= CVol(Bξ(k1, k2)).
118
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