Extremal metrics for the first eigenvalue of the Laplacian in a conformal class期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Ahmad El Soufi Saï d Ilias

Institution:	Laboratoire de Mathematiques et Physique Theorique, Universite de Tours, Parc de Grandmont, 37200 Tours, France ; Laboratoire de Mathematiques et Physique Theorique, Universite de Tours, Parc de Grandmont, 37200 Tours, France

Abstract:	Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on 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.

Keywords:	First eigenvalue of the Laplacian extremal metrics conformal classes harmonic maps

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