Euclidean triangles have no hot spots

• 成果类型: Article
• 署名作者: Judge, Chris; Mondal, Sugata
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.191.1.3
• 发表日期: 2020
• 页码: 167-211
• 关键词: conjecture; eigenfunctions; domains

摘要

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.