Hot spots in domains of constant curvature

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.