On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

TL;DR

This paper establishes a precise volume estimate for complete gradient Ricci solitons with a positive lower bound on scalar curvature, providing new insights into their geometric structure and scalar curvature behavior.

Contribution

It introduces a sharp volume estimate for such solitons and offers a direct elliptic proof of the scalar curvature estimate, extending understanding of Ricci solitons' geometry.

Findings

Sharp volume estimate for gradient Ricci solitons

Local lower bound for scalar curvature under Ricci flow

Manifold is Einstein if scalar curvature attains minimum

Abstract

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.