On the Laplace Operator Penalized by Mean Curvature

Let h = Σj=1d κj, where the κj are the principal curvatures of a d-dimensional hypersurface immersed in Rd+1, and let −Δ be the corresponding Laplace–Beltrami operator. We prove that the second eigenvalue of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ - \Delta - \frac{1}{d}{h^2}$\end{document} is strictly negative unless the surface is a sphere, in which case the second eigenvalue is zero. In particular this proves conjectures of Alikakos and Fusco.