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

Abstract

In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial µ-entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case; espeically, the lower bound of the renormalized heat kernel, observed on a scale comparable to the initial diameter, matches with the lower bound of the renormalized volume ratio, giving the desired distance distortion estimate.

Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

“…Once the Ricci flow exists for a definite amount of time, for the purpose of smoothing, it is of key importance to compare the initial metric with the evolved metric. 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]:…”

Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

“…Once the Ricci flow exists for a definite amount of time, for the purpose of smoothing, it is of key importance to compare the initial metric with the evolved metric. 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 [45,23,81,24,1,2,25,52] for previous works on this topic in various settings. Very recently, based on the previous contributions, especially the local entropy theory developed in [83], 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]: Theorem 4.7.…”

“…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 [45,23,81,24,1,2,25,52] for previous works on this topic in various settings. Very recently, based on the previous contributions, especially the local entropy theory developed in [83], 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]: Theorem 4.7. Given a positive integer m, positive constants C0 , C R , T ≤ 1 and α ∈ (0, 1), there are constants C D ( C0 , C R , m) ≥ 1 and T D ( C0 , C R , m) ∈ (0, T ] such that for an m-dimensional complete Ricci flow (M, g(t)) defined for t ∈ [0, T ], if for some x 0 ∈ M and any t ∈ [0, T ] we have…”

Ricci flow smoothing for locally collapsing manifolds

“…In all the above mentioned results, the existence time lower bound of Ricci flows -which is critical for the smoothing purpose as we have discussed -depends on the uniform volume ratio lower bound of the initial metric: when the initial volume ratio at some point becomes smaller, the existence time of the Ricci flow becomes shorter. However, in many natural situations, especially for the purpose of smoothing a given initial metric by running the Ricci flow, there may be no uniform volume ratio lower bound to be assumed; and the purpose of the current paper, sequential to the previous work [30], is then to show that in certain cases when the initial data have a uniform Ricci curvature lower bound but without any uniform volume ratio lower bound, the Ricci flow could still be started locally for a definite period of time. Throughout the paper, for a compact subset K ⊂ M and R > 0, we will let B g (K, R) denote the geodesic R-neighborhood of K, i.e.…”