, Volume 195, Issue 3, pp 643-650

On the Laplace Operator Penalized by Mean Curvature

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Let , where the κ j are the principal curvatures of a d-dimensional hypersurface immersed in , and let −Δ be the corresponding Laplace–Beltrami operator. We prove that the second eigenvalue of 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.

Received: 11 December 1997 / Accepted: 22 December 1997