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**Euclidean triangles have no hot spots.**
*(English)*
Zbl 1444.35119

Ann. Math. (2) 191, No. 1, 167-211 (2020); erratum ibid. 195, No. 1, 337-362 (2022).

The authors completely prove the hot spots conjecture for Euclidean triangles, that is, they show that any eigenfunction associated with the second (i.e., smallest nontrivial) eigenvalue of the Neumann Laplacian on a Euclidean triangle \(T\subset\mathbb{R}^2\) has no critical points in the interior of \(T\). The conjecture had been the subject of a polymath project and had only been proved for particular classes of triangles (see [B. Siudeja, Math. Z. 280, 783–806 (2015; Zbl 1335.35164)]).

The principal statement here (Theorem 1.1) is somewhat stronger: excluding the vertices, any second eigenfunction can have at most one critical point, which necessarily lies on a side of \(T\) and, in this case, the critical point is a nondegenerate critical point with Morse index one.

The proof is entirely analytic, no probabilistic arguments are used. The idea behind the proof is quite natural and elegant, although the details are relatively involved. The strategy consists of considering, for each given triangle \(T_0\), a family of triangles \(T_t\) joining \(T_0\) to a right isosceles triangle \(T_1\), where the second eigenfunction is known explicitly, and studying how critical points of the eigenfunction of \(T_t\) could disappear as \(t \to 1\).

The principal statement here (Theorem 1.1) is somewhat stronger: excluding the vertices, any second eigenfunction can have at most one critical point, which necessarily lies on a side of \(T\) and, in this case, the critical point is a nondegenerate critical point with Morse index one.

The proof is entirely analytic, no probabilistic arguments are used. The idea behind the proof is quite natural and elegant, although the details are relatively involved. The strategy consists of considering, for each given triangle \(T_0\), a family of triangles \(T_t\) joining \(T_0\) to a right isosceles triangle \(T_1\), where the second eigenfunction is known explicitly, and studying how critical points of the eigenfunction of \(T_t\) could disappear as \(t \to 1\).

Reviewer: James Bernard Kennedy (Lisboa)

### MSC:

35P05 | General topics in linear spectral theory for PDEs |

35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J25 | Boundary value problems for second-order elliptic equations |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |