Uniqueness of Conformal Ricci Flow and Backward Ricci Flow on Homogeneous 4-Manifolds
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In the first chapter we consider the question of uniqueness of conformal Ricci flow. We use an energy functional associated with this flow along closed manifolds with a metric of constant negative scalar curvature. Given initial conditions we use this functional to demonstrate the uniqueness of the solution to both the metric and the pressure function along conformal Ricci flow. In the next chapter we study backward Ricci flow of locally homogeneous geometries of 4-manifolds which admit compact quotients. We describe the longterm behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian geometry after suitable rescaling.