  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.
