Volume comparison with respect to scalar curvature

2016/09/28 by Wei Yuan, Yuan, Wei · 4 citations
Mathematics · #53C20 (Primary) #53C23 #53C24 (Secondary) #58J37 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C20 #msc:53C23 #msc:53C24 #msc:58J37

paper · pdf · doi:10.48550/arxiv.1609.08849

37 pages, new remarks 1.7, 1.10, 1.12, 1.14 added, Theorem C rewritten, new references added, some typos fixed

arxiv created 2021/02/19 · arxiv updated 2021/02/22

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.

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