Speaker: Shicheng XU (Capital Normal University)

Time： March 09, 10:00-11:00

Place： 腾讯会议 126-609-398 密码 202203

Title： Total squared mean curvature of immersed submanifolds in a negatively curved space

Abstract： Let n≥2 and k≥1 be two integers. Let M be an isometrically immersed closed submanifold of dimension n and co-dimension k, which is homotopic to a point, in a complete manifold N, where the sectional curvature of N is no more than δ<0. We prove that the total squared mean curvature of M in N and the first non-zero eigenvalue λ_1(M) satisfies λ_1(M)≤ n(δ +Vol^(-1)(M) ∫ |H|^2 dvol. The equality implies that M is minimally immersed in a geodesic sphere after lifted to the universal cover of N. This completely settles an open problem raised by E. Heintze in 1988.