Euclidean triangles have no hot spots


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.


Chris Judge

Department of Mathematics, Indiana University, Bloomington, IN, USA

Sugata Mondal

School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India