Volume comparison with respect to scalar curvature

Yuan, Wei

doi:10.2140/apde.2023.16.1

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Volume comparison with respect to scalar curvature

Wei Yuan

Vol. 16 (2023), No. 1, 1–34

DOI: 10.2140/apde.2023.16.1

Abstract

We investigate the volume comparison with respect to scalar curvature. In particular, we show the volume comparison holds for small geodesic balls of metrics near a $V$ -static metric. For closed manifolds, 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 manifolds. 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.

Dedicated to Nankai University on its 100th Anniversary (1919 - 2019)

Keywords

scalar curvature, volume comparison, $V$-static metric, stable Einstein metric, Bray's conjecture, Schoen's conjecture

Mathematical Subject Classification 2010

Primary: 53C20

Secondary: 53C23, 53C24, 58J37

Milestones

Received: 3 June 2017

Revised: 18 February 2021

Accepted: 4 May 2021

Published: 14 April 2023

Authors

Wei Yuan
Department of Mathematics Sun Yat-sen University Guangzhou China

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Vol. 16, No. 1, 2023