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.

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Authors

Chris Judge

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

Sugata Mondal

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