Speaker：Shicheng Xu（Capital Normal University）

Time：2022-03-30 14:00

Location：Tencent Meeting ID：631246238（No Password）

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) satisfiesλ_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. this is a joint work with yanyan niu.