Extremal metrics for the first eigenvalue of the Laplacian in a conformal class
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- by Ahmad El Soufi and Saïd Ilias PDF
- Proc. Amer. Math. Soc. 131 (2003), 1611-1618 Request permission
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.References
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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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1611-1618
- MSC (2000): Primary 58E11, 58J50, 35P15
- DOI: https://doi.org/10.1090/S0002-9939-02-06948-4
- MathSciNet review: 1950293