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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds


Authors: Ahmad El Soufi, Evans M. Harrell II and Saïd Ilias
Journal: Trans. Amer. Math. Soc. 361 (2009), 2337-2350
MSC (2000): Primary 58J50, 58E11, 35P15
Published electronically: December 16, 2008
MathSciNet review: 2471921
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Abstract: We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group.

Among the consequences of this analysis are an extension of Reilly's inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.


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Additional Information

Ahmad El Soufi
Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France
Email: elsoufi@univ-tours.fr

Evans M. Harrell II
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: harrell@math.gatech.edu

Saïd Ilias
Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France
Email: ilias@univ-tours.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04780-6
PII: S 0002-9947(08)04780-6
Keywords: Spectrum, eigenvalue, Laplacian, Schr\"{o}dinger operator, Reilly inequality, Kohn Laplacian.
Received by editor(s): January 16, 2007
Published electronically: December 16, 2008
Article copyright: © Copyright 2008 by the authors