Some remarks on critical sets of Laplace eigenfunctions

Judge, C. and Mondal, S. ORCID: https://orcid.org/0000-0002-2236-971X (2025) Some remarks on critical sets of Laplace eigenfunctions. Annales mathématiques du Québec, 49 (1). pp. 155-163. ISSN 2195-4763 doi: 10.1007/s40316-024-00240-9

Abstract/Summary

We study the set of critical points of a solution to Delta u = lambda u, and in particular components of the critical set that have co-dimension 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.