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

Leo Ivo S. Souza

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arXiv · 1611.02998

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

Leo Ivo S. Souza

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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 $Σ^n$ of $\mathbb{R}^{n+1}$, where $W_r$ is a power of the $(r+1)$-th mean curvature of $Σ^n$. 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.

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Leo Ivo S. Souza. 2016-11-09. On the $L_r$-operators penalized by $(r+1)$-mean curvature. https://arxiv.org/abs/1611.02998

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