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

TL;DR

This paper explores the geometric structure of eigenfunction zero sets on Riemannian surfaces, modeling them as embedded graphs to estimate critical points and vanishing orders using graph theory and topology.

Contribution

It introduces a novel approach linking eigenfunction nodal sets to embedded graphs, providing bounds on critical points based on eigenvalue index and surface topology.

Findings

Nodal sets can be modeled as embedded metric graphs.

Bounds on critical points depend on eigenfunction index and Euler characteristic.

Method connects geometry, graph theory, and spectral analysis.

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-Poincar\'e characteristic of the surface.