Sharp upper diameter bounds for compact shrinking Ricci solitons

TL;DR

This paper establishes a precise upper bound on the diameter of compact shrinking Ricci solitons using scalar curvature integrals and Perelman's entropy, with potential equality cases on round spheres.

Contribution

It introduces a sharp diameter bound for compact shrinking Ricci solitons based on scalar curvature and entropy, utilizing a new logarithmic Sobolev inequality and covering argument.

Findings

Sharp upper diameter bound in terms of scalar curvature integral and entropy

Equality cases potentially occur at round spheres

Method relies on a new logarithmic Sobolev inequality

Abstract

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.