Hot spots in convex hyperbolic planar domains with small eigenvalues

TL;DR

This paper proves a variant of Rauch's hot spots conjecture for hyperbolic planar domains, showing that on large convex domains, second Neumann eigenfunctions lack interior critical points.

Contribution

It establishes a new result for hyperbolic domains, extending hot spots conjecture to settings with small eigenvalues and convex geometry.

Findings

Second Neumann eigenfunctions have no interior critical points on large convex hyperbolic domains.

The paper confirms a variant of Rauch's hot spots conjecture in hyperbolic geometry.

Results apply to domains with sufficiently large area in the hyperbolic plane.

Abstract

We prove a variant of Rauch's hot spots conjecture for hyperbolic planar domains with small Neumann or mixed Dirichlet-Neumann eigenvalues. We conclude, for instance, that on bounded convex domains in the hyperbolic plane with sufficiently large area, second Neumann Laplace eigenfunctions have no interior critical points.