Extremal metrics for the eigenvalues of the Laplacian on manifolds with boundary

2023/05/10 by Eduardo Longa, Longa, Eduardo
Computer Science · Mathematics · #53C42 #58C40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2305.05897

openalex publication_date 2023/05/10 · openalex created_date 2023/05/12 · openalex updated_date 2026/07/28

Abstract

We show there are no extremal metrics for the eigenvalues of the Neumann Laplacian on any compact manifold. Nonetheless, we construct examples of conformally extremal metrics for the eigenvalues of this operator in any annulus and characterise these special metrics in the general case of a compact manifold of dimension n ≥ 2. As for the Dirichlet Laplacian, we prove non existence of extremal metrics on any compact surface.

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