Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfaces

TL;DR

This paper establishes sharp local smoothing estimates for the logarithmic fast diffusion equation and Ricci flow on surfaces, enabling better understanding of their behavior with rough initial data and geometric applications.

Contribution

It provides the first sharp local L^1-L^ smoothing estimate for these equations, improving previous L^p-L^ estimates and facilitating applications in geometric analysis.

Findings

Sharp local L^1-L^ smoothing estimate proved

Improves known L^p-L^ estimates for p > 1

Applications include handling rough initial data and asymptotics of cusp Ricci flow

Abstract

We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estimate for p larger than 1. It also has several applications in geometry, providing the missing step in order to pose the Ricci flow with rough initial data in the noncompact case, for example starting with a general noncompact Alexandrov surface, and giving the sharp asymptotics for the contracting cusp Ricci flow, as we show elsewhere.