Volume comparison with respect to scalar curvature

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.