Metrics of Positive Ricci Curvature on Connected Sums: Projective Spaces, Products, and Plumbings
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Date
2019-09-18
Authors
Burdick, Bradley
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Publisher
University of Oregon
Abstract
The classification of simply connected manifolds admitting metrics of positive scalar curvature of initiated by Gromov-Lawson, at its core, relies on a careful geometric construction that preserves positive scalar curvature under surgery and, in particular, under connected sum. For simply connected manifolds admitting metrics of positive Ricci curvature, it is conjectured that a similar classification should be possible, and, in particular, there is no suspected obstruction to preserving positive Ricci curvature under connected sum. Yet there is no general construction known to take two Ricci-positive Riemannian manifolds and form a Ricci-positive metric on their connected sums. In this work, we utilize and extend Perelman’s construction of Ricci-positive metrics on connected sums of complex projective planes, to give an explicit construction of Ricci-positive metrics on connected sums given that the individual summands admit very specific Ricci- positive metrics, which we call core metrics. Working towards the new goal of constructing core metrics on manifolds known to support metrics of positive Ricci curvature: we show how to generalize Perelman’s construction to all projective spaces, we show that the existence of core metrics is preserved under iterated sphere bundles, and we construct core metrics on certain boundaries of plumbing disk bundles over spheres. These constructions come together to give many new examples of Ricci-positive connected sums, in particular on the connected sum of arbitrary products of spheres and on exotic projective spaces.
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Keywords
connected sums, manifolds with corners, positive Ricci curvature, Riemannian geometry, splines, surgery