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Volume comparison with respect to scalar curvature

Wei Yuan

Vol. 16 (2023), No. 1, 1–34

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)

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
Received: 3 June 2017
Revised: 18 February 2021
Accepted: 4 May 2021
Published: 14 April 2023
Wei Yuan
Department of Mathematics
Sun Yat-sen University

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