Euclidean triangles have no hot spots

Judge, C. and Mondal, S. ORCID: https://orcid.org/0000-0002-2236-971X (2020) Euclidean triangles have no hot spots. Annals of Mathematics, 191 (1). pp. 167-211. ISSN 0003-486X doi: 10.4007/annals.2020.191.1.3

Abstract/Summary

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.

Additional Information	An Erratum for this article was published in January 2022 and can be found here: https://doi.org/10.4007/annals.2022.195.1.5
Item Type	Article
URI	https://centaur.reading.ac.uk/id/eprint/122774
Identification Number/DOI	10.4007/annals.2020.191.1.3
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Additional Information	An Erratum for this article was published in January 2022 and can be found here: https://doi.org/10.4007/annals.2022.195.1.5
Publisher	Priceton University
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