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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class


Authors: Ahmad El Soufi and Saïd Ilias
Journal: Proc. Amer. Math. Soc. 131 (2003), 1611-1618
MSC (2000): Primary 58E11, 58J50, 35P15
Published electronically: December 6, 2002
MathSciNet review: 1950293
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Abstract: Let $M$ be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on $M$ to be extremal for $\lambda_1$ with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice $\Gamma$ of $\mathbb{R}^n$, the flat metric $g_{\Gamma}$ induced on $\mathbb{R}^n/\Gamma$ from the standard metric of $\mathbb{R}^n$ is extremal (in the previous sense). In the second part, we give, for any $\Gamma$, an upper bound of $\lambda_1$ on the conformal class of $g_{\Gamma}$ and exhibit a class of lattices $\Gamma$ for which the metric $g_{\Gamma}$ maximizes $\lambda_1$ on its conformal class.


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Additional Information

Ahmad El Soufi
Affiliation: Laboratoire de Mathematiques et Physique Theorique, Universite de Tours, Parc de Grandmont, 37200 Tours, France
Email: elsoufi@univ-tours.fr

Saïd Ilias
Affiliation: Laboratoire de Mathematiques et Physique Theorique, Universite de Tours, Parc de Grandmont, 37200 Tours, France
Email: ilias@univ-tours.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06948-4
PII: S 0002-9939(02)06948-4
Keywords: First eigenvalue of the Laplacian, extremal metrics, conformal classes, harmonic maps
Received by editor(s): January 5, 2000
Received by editor(s) in revised form: May 19, 2000
Published electronically: December 6, 2002
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2002 American Mathematical Society