Some remarks on critical sets of Laplace eigenfunctions

Chris Judge, Sugata Mondal

Published 2022-04-25Version 1

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.