Removing scalar curvature assumption for Ricci flow smoothing

TL;DR

This paper demonstrates that the scalar curvature bound is unnecessary for Ricci flow smoothing under certain conditions, broadening the applicability of short-time existence results and deriving related geometric and topological consequences.

Contribution

It shows the scalar curvature bound can be removed from previous assumptions, enabling new applications in Ricci flow smoothing and geometric analysis.

Findings

Scalar curvature bound is redundant for Ricci flow existence.

Established Ricci flow smoothing for measure space limits.

Proved Gromov-Hausdorff compactness and rigidity results.

Abstract

In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.