Generic metrics, eigenfunctions and riemannian coverings of non compact manifolds

2010/01/14 by Samuel Tapie, Tapie, Samuel
Mathematics · #53C21 #58C40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1001.2506

openalex publication_date 2010/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M,g) be a non-compact riemannian n-manifold with bounded geometry at order k≥(n)/(2). We show that if the spectrum of the Laplacian starts with q+1 discrete eigenvalues isolated from the essential spectrum, and if the metric is generic for the \Cl C^k+2-strong topology, then the eigenvalues are distinct and their associated eigenfunctions are Morse. This generalizes to non-compact manifolds some arguments developped by K. Uhlenbeck. We deduce from this result that if Mⁿ has bounded geometry at order k≥(n)/(2) and has an isolated first eigenvalue for its Laplacian, then for any riemannian covering p : M'\ra M, we have λ₀(M) = sup_D λ₀(D), where D⊂ M' runs over all connected fundamental domains for p, and λ₀(D) is the bottom of the spectrum of D with Neumann boundary conditions.

Citations

• Differential equations on riemannian manifolds and their geometric applications

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