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

Abstract

Let $n\ge 2$ and $k\ge 1$ 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 $N$ is no more than $\delta <0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue ${\lambda }_1(M)$ of $M$ satisfies $${\lambda }_1(M)\le n\left(\delta +\frac{1}{\mathrm{Vol}M}{\int }_M|H{|}^2\mathrm{dvol}\right).$$ 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.

13 pages