Communications in Mathematical Physics

, Volume 195, Issue 3, pp 643–650

On the Laplace Operator Penalized by Mean Curvature

  • Evans M. Harrell II
  • Michael Loss
Article

DOI: 10.1007/s002200050406

Cite this article as:
Harrell II, E. & Loss, M. Commun. Math. Phys. (1998) 195: 643. doi:10.1007/s002200050406

Abstract:

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 \( - \Delta - \frac{1}{d}{h^2}\) 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Evans M. Harrell II
    • 1
  • Michael Loss
    • 1
  1. 1.School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA.¶E-mail: harrell@math.gatech.edu; loss@math.gatech.eduUSA