Volume comparison with respect to scalar curvature

TL;DR

This paper explores volume comparison theorems related to scalar curvature, establishing new results for metrics near special geometric structures and confirming parts of longstanding conjectures in differential geometry.

Contribution

It proves volume comparison for metrics near V-static and strictly stable Einstein metrics, providing partial answers to conjectures by Bray and Schoen.

Findings

Volume comparison holds for small geodesic balls near V-static metrics.

Volume comparison is valid near strictly stable Einstein metrics on closed manifolds.

A new proof of local rigidity near strictly stable Ricci-flat metrics is provided.

Abstract

In this article, we investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison holds for small geodesic balls of metrics near a V-static metric. For closed manifold, we prove the volume comparison for metrics near a strictly stable Einstein metric. As applications, we give a partial answer to a conjecture of Bray and recover a result of Besson, Courtois and Gallot, which partially confirms a conjecture of Schoen about closed hyperbolic manifold. Applying analogous techniques, we obtain a different proof of a local rigidity result due to Dai, Wang and Wei, which shows it admits no metric with positive scalar curvature near strictly stable Ricci-flat metrics.