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

Abstract:

In this paper, we establish the non-positivity of the second eigenvalue of the Schrödinger 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 $\Sigma ^n$, which we will ask to be positive. If this eigenvalue is null, we will have a characterization of the sphere. This theorem generalizes the result of Harrell and Loss proved to the Laplace-Beltrame operator penalized by the square of the mean curvature.