Euclidean triangles have no hot spots

Abstract

We show that a second Neumann eigenfunction $u$ of a Euclidean triangle has at most one (non-vertex) critical point $p$, and if $p$ exists, then it is a non-degenerate critical point of Morse index 1. Using this we deduce that (1) the extremal values of $u$ are only achieved at a vertex of the triangle, and (2) a generic acute triangle has exactly one (non-vertex) critical point and that each obtuse triangle has no (non-vertex) critical points. This settles the ``hot spots" conjecture for triangles in the plane.