On the Structure of Nonsmooth Singularity Models of Ricci Flow

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The main goals of this work are to extend the structure theory of nonsmooth geometriclimits of certain classes of Ricci flow solutions, and to apply this theory to the study of finite-time singularities of Ricci flow. We will separately consider Ricci flows in the settings of Type-I Scalar curvature, volume-noncollapsed Ricci flow with a lower Ricci curvature bound, and general solutions of Kähler-Ricci flow. It is known that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci soliton with singularities of codimension 4. We prove that the entropy of a conjugate heat kernel based at the singular time converges to the soliton entropy of the singular soliton, and use this to characterize the singular set of the Ricci flow solution in terms of a heat kernel density function. This generalizes results previously only known with the stronger assumption of a Type-I curvature bound. We also show that in dimension 4, the singular Ricci soliton is smooth away from finitely many points, which are conical smooth orbifold singularities. Next, we show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a C0 orbifold at any finite-time singularity, so has an extension through the singularity via orbifold Ricci flow. Moreover, a Type-I blowup of the flow based at any orbifold point converges to a flat cone in the Gromov-Hausdorff sense, without passing to a subsequence. In addition, we prove Lp bounds for the curvature tensor on time-slices for any p < 2. In higher dimensions, we show that every singular point of the flow is a Type-II point, and that any tangent flow at a singular point is a static flow corresponding to a Ricci flat cone. Finally, in a joint work with Wangjian Jian, we improve the description of F-limits ofnoncollapsed Ricci flows in the Kähler setting. In particular, the singular strata S^k of such metric flows satisfy S^2j = S^2j+1. We also prove an analogous result for quantitative strata, and show that any tangent flow admits a nontrivial one-parameter action by isometries, which is locally free on the cone link in the static case. These results are established using parabolic regularizations of conjugate heat kernel potential functions based at almost-selfsimilar points, which may be of independent interest.

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232 pages


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Geometric Analysis; Ricci Flow


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Union Local


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Cao, Xiaodong

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Saloff-Coste, Laurent Pascal
Zhou, Xin

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Ph. D., Mathematics

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Doctor of Philosophy

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Government Document




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Attribution-NoDerivatives 4.0 International


dissertation or thesis

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