Laplacian eigenvalues functionals and metric deformations on compact manifolds

Abstract

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

References in corpus (2)

• Extremal eigenvalues of the Laplacian in a conformal class of metrics : the "conformal spectrum"
• Greatest least eigenvalue of the Laplacian on the Klein bottle

Cited by in corpus (17)

• Extremal spectral properties of Otsuki tori
• Generalized Lawson tori and Klein bottles
• Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian
• Non-maximality of known extremal metrics on torus and Klein bottle
• Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature
• From Steklov to Laplace: free boundary minimal surfaces with many boundary components
• Spectral properties of a family of minimal tori of revolution in five-dimensional sphere
• Extremal metrics for the Paneitz Operator on closed four-Manifolds
• Extremal spectral properties of Lawson tori
• Index of bipolar surfaces to Otsuki tori
• Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds
• Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces
• Extremal metrics for Laplace eigenvalues in perturbed conformal classes on products
• On the distribution of perturbations of propagated Schrödinger eigenfunctions
• Extremal Eigenvalues Of The Conformal Laplacian Under Sire-Xu Normalization
• Existence and classification of $S^1$-invariant free boundary annuli and Möbius bands in ${\mathbb{B}}^n$
• Extremal metrics for Laplace eigenvalues in perturbed conformal classes