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Euclidean triangles have no hot spots
Annals of Mathematics ( IF 5.7 ) Pub Date : 2020-01-01 , DOI: 10.4007/annals.2020.191.1.3
Chris Judge 1 , Sugata Mondal 2
Affiliation  

We show that a second Neumann eigenfunction u of a Euclidean triangle has at most one (nonvertex) 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.

中文翻译:

欧几里得三角形没有热点

我们证明欧几里得三角形的第二个诺依曼特征函数 u 至多有一个(非顶点)临界点 p,如果 p 存在,那么它是莫尔斯指数 1 的非退化临界点。 使用这个我们推导出 (1) u 的极值仅在三角形的顶点处获得,并且 (2) 通用锐角三角形恰好有一个(非顶点)临界点,并且每个钝角三角形都没有(非顶点)临界点。这解决了平面中三角形的“热点”猜想。
更新日期:2020-01-01
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