Some remarks on critical sets of Laplace eigenfunctions

TL;DR

This paper investigates the structure of critical points of Laplace eigenfunctions, especially those with codimension 1, and establishes geometric conditions for eigenfunctions with infinitely many critical points.

Contribution

It provides new insights into the critical sets of Laplace eigenfunctions and characterizes polygons with infinitely many critical points of their eigenfunctions.

Findings

If a second Neumann eigenfunction of a simply connected polygon has infinitely many critical points, then the polygon must be a rectangle.

The study reveals the relationship between the critical set structure and the geometry of the domain.

It advances understanding of eigenfunction behavior in polygonal domains.

Abstract

We study the set of critical points of a solution to $\Delta u = \lambda \cdot u$ and in particular components of the critical set that have codimension 1. We show, for example, that if a second Neumann eigenfunction of a simply connected polygon $P$ has infinitely many critical points, then $P$ is a rectangle.