In Riemannian geometry, volume comparison theorem is one of the most fundamental results. The classic results concern volume comparison involving restrictions on Ricci curvature. In this talk, we will investigate the volume comparison with respect to scalar curvature. In particular, we show that one can only expect such results for small geodesic balls of metrics near V-static metrics. As for global results, we give volume comparison for metrics near Einstein metrics with certain restrictions. As applications, we give a partial answer to Schoen’s conjecture about hyperbolic manifolds, which recovers a result due to Besson-Courtois-Gallot with a different approach. We also provide a partial answer to a conjecture proposed by Bray concerning the positive scalar curvature case.

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