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Register a new userTotal squared mean curvature of immersed submanifolds in a negatively curved space
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Let $nge 2$ and $kge 1$ 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 $$lambda_1(M)le nleft(delta +frac{1}{operatorname{Vol} M}int_M |H|^2 operatorname{dvol}right).$$ 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.
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