Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	孙俊副教授，武汉大学
Inviter:
Title:	Volume Comparison Theorem With Respect to Scalar Curvature
Time & Venue:	2021.01.20 09:00-11:00 腾讯会议号：348 6898 7444
Abstract:	The classical volume comparison theorem says that if the Ricci curvature of a Riemannian manifold M is greater than or equal to (n-1), then its volume is less than or equal to that of the unit round sphere, with equality holds if and only if M is isometric to the round sphere. In this series of lectures, we will talk about volume comparison theorem with respect to scalar curvature. It consists of three parts: Part I: We will talk about Bray’s thesis (Stanford University, 1997), which conjectured that if the scalar curvature of M is greater than or equal to n(n-1), and the Ricci curvature of M is bounded from below by \epsilon, then the volume of M is less than or equal to that of the unit round sphere. Bray proved this conjecture when the dimension of M is 3. Part II: We will first recall Gursky-Vialovsky’s result, which provided an explicit lower bound 1/2 for \epsilon when the dimension of M is 3. Then we will talk about Brendle’s result, which shows the rigidity for \epsilon=1/2 when the dimension of M is 3. Part III: We will talk about some recent progress on the volume comparison theorem with respect to scalar curvature in high dimensions.