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

TL;DR

This paper establishes a sharp inequality relating the total squared mean curvature and the first non-zero eigenvalue for submanifolds in negatively curved spaces, resolving a long-standing open problem.

Contribution

It proves a precise inequality linking mean curvature and eigenvalues for immersed submanifolds in negatively curved manifolds, characterizing equality cases.

Findings

Derived an inequality connecting mean curvature and eigenvalues.

Characterized the equality case as minimal immersion in a sphere.

Resolved a problem posed by E. Heintze in 1988.

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