Sharp upper diameter bounds for compact shrinking Ricci solitons

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.

13 pages, coefficient of Theorem 1.1 improved, accepted by AGAG