  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
$$\lambda_1(M)\le n\left(\delta +\frac{1}{\operatorname{Vol} M}\int_M |H|^2
\operatorname{dvol}\right).$$ 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.
