学术报告（胥世成 8.24）

摘  要： Let n≥2 and k≥1 be two integers. Let M be an isometrically immersed closed n-submanifold of co-dimension k that is homotopic to a point in a complete manifold N, where the sectional curvature of N ≤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/Vol M) ∫(|H|^2+δ) dv.The equality implies that M is minimally immersed in a metric 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 Y. Niu.