Heat kernel on Ricci shrinkers

TL;DR

This paper provides a detailed analysis of the heat kernel on Ricci shrinkers, offering sharper estimates and improved classical results, which are crucial for understanding short-time singularities in Ricci flows.

Contribution

It introduces new sharper estimates for the heat kernel on Ricci shrinkers and improves classical results like Sobolev inequalities and no-local-collapsing theorems for these flows.

Findings

Sharper heat kernel estimates for Ricci shrinkers

Improved Logarithmic Sobolev and Sobolev constant bounds

Enhanced no-local-collapsing and pseudo-locality theorems

Abstract

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estimate, the no-local-collapsing theorem, the pseudo-locality theorem and the strong maximum principle for curvature tensors, are essentially improved for Ricci flows induced by Ricci shrinkers. Our results provide many necessary tools to analyze short time singularities of the Ricci flows of general dimension.