97-276 Harrell E.M. II, Loss M.

ON THE LAPLACE OPERATOR PENALIZED BY MEAN CURVATURE (25K, plain TeX) May 15, 97

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Abstract. Let $h=\sum_{j=1}^d \kappa_j$ where the $\kappa_j$ are the principal curvatures of a d-dimensional hypersurface immersed in $R^{d+1}$, and let $-\Delta$ be the corresponding Laplace--Beltrami operator. We prove that the second eigenvalue of $-\Delta - {1 \over d}h^2$ 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

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