Volume comparison with respect to scalar curvature - Applications to Bray and Schoen’s conjectures

[Math. Dept.]

January 11, 2019  15:00-16:00

W303  School of Mathematics

[seminar]20190111Wei Yuan-01.png

SPEAKER

袁伟(中山大学)

ABSTRACT

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 affirmative 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 affirmative answer to a conjecture proposed by Bray concerning the positive scalar curvature case. 

SUPPORTED BY

School of Mathematics, Sichuan University

LECTURE NOTES

TMCSC190111a_Wei Yuan_Volume comparison with respect to scalar curvature - Applications to Bray and Schoen’s conjectures