Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature --- Part II

TL;DR

This paper investigates how the distance function behaves under Ricci flows with bounded scalar curvature, demonstrating its Hölder continuity and extendability up to singularities.

Contribution

It establishes the Hölder continuity of the distance function in Ricci flows with bounded scalar curvature and shows it can be extended continuously to singular times.

Findings

Distance function is 1/2-Hölder continuous on small time intervals

Distance function can be extended continuously up to singular time

Provides insights into the geometric behavior near singularities

Abstract

In this paper we analyze the behavior of the distance function under Ricci flows whose scalar curvature is uniformly bounded. We will show that on small time-intervals the distance function is $\frac12$-H\"older continuous in a uniform sense. This implies that the distance function can be extended continuously up to the singular time.