Hot spots in domains of constant curvature

TL;DR

This paper extends hot spots conjecture results to constant curvature domains, proving it for certain triangles and analyzing eigenfunctions' critical points on polygons of constant curvature.

Contribution

It provides new proofs and results for hot spots conjecture analogues in constant curvature settings, including triangles and polygons, with eigenfunction critical point analysis.

Findings

Hot spots conjecture holds for non-acute geodesic triangles of negative curvature.

First mixed Dirichlet-Neumann eigenfunctions have no non-vertex critical points under certain conditions.

Second Neumann eigenfunctions have finitely many critical points in most constant curvature polygons.

Abstract

We prove constant-curvature analogues of several results regarding the hot spots conjecture in dimension two. Our main theorem shows that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. We also prove that, under certain circumstances, on constant (positive or negative) curvature triangles, first mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points. Moreover, we show that each of these eigenfunctions is monotonic with respect to some Killing field. Finally, we show that for general simply connected polygons of non-zero constant curvature--with exactly one family of exceptions--second Neumann eigenfunctions of the Laplacian have at most finitely many critical points.