Graph structure of the nodal set and bounds on the number of critical points of eigenfunctions on Riemannian manifolds

2024/09/18 by Hofmann, Matthias, Täufer, Matthias
#05C90 #58J50 #58K65 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.11800

Abstract

In this article, we illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory and the vanishing order of eigenfunctions. We identify the nodal set of an eigenfunction of the Laplacian (with smooth potential) on a compact, two-dmensional Riemannian manifolds, that is on Riemannian surfaces, as an embedded metric graph and then use tools from elementary graph theory in order to estimate the number of critical points in the nodal set of the k-th eigenfunction and the sum of vanishing orders at critical points in terms of k and the Euler-Poincaré characteristic of the surface.

Related

• Eigenfunctions with prescribed nodal sets
• Critical set of eigenfunctions of the Laplacian
• Neumann Domains on Graphs and Manifolds
• Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions
• The size of the nodal sets for the eigenfunctions of the smooth laplacian