The local entropy along Ricci flow Part A: the no-local-collapsing theorems

Abstract

We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimate along the Ricci flow. It can be used to show the Gromov-Hausdorff convergence of the Kähler Ricci flow on each minimal projective manifold of general type.

Notes on Ricci flows with collapsinginitial data (I): Distance distortion

“…This result is parallel to Colding-Naber's Hölder continuity theorem for manifolds with a uniform Ricci curvature lower bound [13]. See the original work of Perelman [20] for a comparison geometry viewpoint of the Ricci flow, as well as the recent work of Bing Wang [26] for a nice explanation of the similarities between Ricci flows and manifolds with Ricci curvature lower bound.…”

“…We expect, in our future work, that locally collapsing initial data with Ricci curvature bounded below should also imply meaningful geometric structures on a positive-time slice. To indicate, we notice that the renormalized Sobolev inequality (1.2) we have used is only the global version of the locally valid estimate (2.1); on the other hand, the recent foundational work on local entropy by Bing Wang [26], provides the necessary technical tools that pass the initial local Sobolev constant estimate to positive-time slices. We would also like to mention the rencent work of Gang Tian and Zhenlei Zhang [24], and the book by Qi S. Zhang [32] for related efforts in this direction.…”

On a twisted conical Kähler–Ricci flow

“…Tmax ω 0 ), see Section 4. Therefore, by Wang's argument in [39,Theorem 8.2], we may obtain a uniform diameter upper bound along the twisted Kähler-Ricci flow (7.4) in this volume noncollapsing case (i.e. k = n).…”

Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

“…In general, the distance distortion estimate for Ricci flows is of central importance in the understanding of the geometry along the Ricci flows, and we refer the readers to [1,2,23,24,25,45,52,82] for previous works on this topic in various settings. Very recently, based on the previous contributions, especially the local entropy theory developed in [84], the distance distortion estimate for collapsing initial data [52], and the Hölder distance estimate for non-collapsing initial data in [49], we obtain the following Hölder distance estimate for collapsing initial data [54, Theorem A.1]:…”