The sharp Li-Yau equality on Shrinking Ricci Solitons with applications

TL;DR

This paper proves a precise heat kernel equality on shrinking Ricci solitons, enabling classification of certain four-dimensional non-compact models arising in Ricci flow singularities.

Contribution

It establishes the sharp Li-Yau equality for conjugate heat kernels on shrinking Ricci solitons without curvature or volume restrictions, leading to new classification results.

Findings

Sharp Li-Yau equality holds on shrinking Ricci solitons

Classification of four-dimensional non-compact shrinking Ricci solitons

Applications to Type I singularity models in Ricci flow

Abstract

We prove that the sharp Li-Yau equality holds for the conjugate heat kernel on shrinking Ricci solitons without any curvature or volume assumptions. This quantity yields several estimates which allows us to classify four dimensional, non-compact shrinking Ricci solitons, which arise as Type I singularity models to the Ricci flow.