 97276 Harrell E.M. II, Loss M.
 ON THE LAPLACE OPERATOR PENALIZED BY MEAN CURVATURE
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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 ddimensional hypersurface immersed in $R^{d+1}$,
and let $\Delta$ be the corresponding LaplaceBeltrami 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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