Volume comparison with respect to scalar curvature

2016/09/28 by Yuan, Wei · 3 citations
#53C20 (Primary) #53C23 #53C24 (Secondary) #58J37 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.08849

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.

Cited by

• Critical metrics of the volume functional on complete manifolds
• Geometric inequalities for electrostatic systems with boundary
• The volume comparison of symmetric spaces of non-compact type of rank 1

Related

• Volume comparison theorem with respect to sigma-2 curvature
• Scalar Curvature Volume Comparison Theorems for Almost Rigid Sphere
• Volume comparison of conformally compact manifolds with scalar curvature R≥ -n(n-1)
• The volume comparison of symmetric spaces of non-compact type of rank 1
• Local volume comparison for Kahler manifolds