Critical set of eigenfunctions of the Laplacian

2010/08/10 by Laurent Bakri, Bakri, Laurent
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #G.1.8 #Mathematical Dynamics and Fractals #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1008.1699

openalex publication_date 2010/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an upper bound for the (n-1)-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic Riemannian manifolds. This is the analog of H. Donnely and C. Fefferman result on nodal set of eigenfunctions.

Related

• Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure
• Eigenfunctions with prescribed nodal sets
• The Sharp Measure Upper Bound of the Nodal Sets of Neumann Laplace Eigenfunctions on C1,1 Domains
• The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions
• Distribution of the nodal sets of eigenfunctions on analytic manifolds