On the $L_r$-operators penalized by $(r+1)$-mean curvature

TL;DR

This paper proves the non-positivity of the second eigenvalue of a generalized Schrödinger operator on closed hypersurfaces, characterizing spheres when the eigenvalue is zero, extending previous results for the Laplace-Beltrami operator.

Contribution

It generalizes Evans and Loss's result to $L_r$-operators penalized by $(r+1)$-mean curvature, providing new spectral bounds and geometric characterizations.

Findings

Second eigenvalue of the operator is non-positive.

Sphere characterized by zero eigenvalue case.

Extension of previous Laplace-Beltrami results.

Abstract

In this paper, we establish the non-positivity of the second eigenvalue of the Schr\"odinger operator $-\textrm{div}\big( P_r \nabla\cdot\big) - W_r^2$ on a closed hypersurface $\Sigma^n$ of $\mathbb{R}^{n+1}$, where $W_r$ is a power of the $(r+1)$-th mean curvature of $\Sigma^n$. In the case that this eigenvalue is null we have a characterization of the sphere. This generalizes a result of Evans and Loss proved for the Laplace-Beltrame operator penalized by the square of the mean curvature.