Ricci Flow of regions with curvature bounded below in dimension three

TL;DR

This paper establishes short-time smoothing estimates for three-dimensional Ricci flow solutions with initial curvature bounds, demonstrating how local geometric conditions influence the flow's regularization.

Contribution

It provides new curvature and volume estimates for Ricci flow in three dimensions with initial curvature bounds, extending understanding of local smoothing effects.

Findings

Short-time smoothing estimates depend on initial volume and distance to boundary.

Scaling arguments extend estimates to larger radii.

Results apply to complete solutions with bounded curvature.

Abstract

We consider smooth complete solutions to Ricci flow with bounded curvature on manifolds without boundary in dimension three. Assuming an open ball at time zero of radius one has curvature bounded from below by -1, then we prove estimates which show that compactly contained subregions of this ball will be smoothed out by the Ricci flow for a short but well defined time interval. The estimates we obtain depend only on the initial volume of the ball and the distance from the compact region to the boundary of the initial ball. Versions of these estimates for balls of radius r follow using scaling arguments.