Laplacian eigenvalue functionals and metric deformations on compact manifolds

By Ahmad El Soufi, S. Ilias. 2007

88 citations recorded by the source corpus.

Abstract In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on a compact manifold. We introduce a natural notion of the critical metric of such a functional and obtain necessary and sufficient conditions for a metric to be critical. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f9ed5c02cf48399be33d4cf64aa5b5e5950eaa90

Citations in the corpus

• An isoperimetric inequality for Laplace eigenvalues on the sphere (2017, unknown): “Theorem 3.3 (Nadirashvili [39], El Soufi, Ilias [19]) .”
• SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “This characterization is similar to that of extremizers of the eigenvalues of the Laplace operator on surfaces (see [85, 31, 32]).”