For a bounded planar domain with insulated (Neumann) boundary, ask where the second Neumann eigenfunction of the Laplacian — the spatial profile of the slowest-decaying nonconstant mode of the heat equation — attains its hottest and coldest points. Rauch's 1974 hot spots conjecture, in its modern form (Conjecture 1.1 of Chen–Gui–Yao, arXiv:2311.12659), asserts that for every bounded simply connected planar domain, and in particular for every bounded convex planar domain, the global maximum and minimum of the second Neumann eigenfunction are attained only on the boundary. Despite a fifty-year effort and complete positive results for triangles, lip domains, and many symmetric and near-circular classes, the conjecture is open for general simply connected planar domains; it is known to fail for multiply connected planar domains and for convex sets in sufficiently high dimensions. The problem is to prove or refute the planar conjecture.

Background

Let Ω\Omega be a bounded connected domain in R2\mathbb{R}^2 and consider the Neumann eigenvalue problem −Δu=μu-\Delta u = \mu u in Ω\Omega with ∂u/∂ν=0\partial u/\partial\nu = 0 on ∂Ω\partial\Omega, where ν\nu is the outward unit normal; the Neumann (insulated) condition models a perfectly reflecting or thermally insulated boundary. The spectrum is a discrete sequence 0=μ1<μ2≤μ3≤⋯→∞0 = \mu_1 < \mu_2 \le \mu_3 \le \cdots \to \infty; the first eigenfunction is constant, and the second Neumann eigenfunction (an eigenfunction for μ2\mu_2, the first nonzero eigenvalue) dominates the long-time behavior of the heat equation on Ω\Omega after the constant (equilibrium) mode is subtracted. Rauch proposed in 1974 that the hottest and coldest points of an insulated body should, in this asymptotic sense, lie on the boundary: the maximum and minimum of the second Neumann eigenfunction should be attained only on ∂Ω\partial\Omega. Kawohl's 1985 monograph spread the problem, obtained the first positive results (cylinders, parallelepipeds, balls), and suggested the conjecture could fail in general. It does fail without topological restrictions: Burdzy and Werner (1999) constructed a planar annulus-type domain, and subsequent work of Bass–Burdzy and Burdzy (2005) produced further multiply connected counterexamples, so the conjecture is now posed for simply connected planar domains (a domain is simply connected if every closed curve in it can be contracted to a point, i.e. it has no holes), with convex planar domains as the flagship subcase. Positive results cover progressively wider but still special classes: domains with a line of symmetry (with technical assumptions) and obtuse triangles (Bañuelos–Burdzy, 1999); domains with two axes of symmetry (Jerison–Nadirashvili, 2000); lip domains, planar domains bounded by two Lipschitz graphs with Lipschitz constant 1 (Atar–Burdzy, 2004, by probabilistic coupling; re-proved variationally by Rohleder, arXiv:2106.05224); convex domains close to a ball (Miyamoto, 2009); acute triangles with smallest angle at most π/6\pi/6 (Siudeja, 2015); and finally all triangles (Judge–Mondal, Ann. of Math. 2020, with a 2022 erratum; sharpened by Chen–Gui–Yao, arXiv:2311.12659, who proved the extrema of the second Neumann eigenfunction of any triangle are attained exactly at the endpoints of the longest side, resolving the objective of Polymath 7, a 2012 large-scale online collaborative proof project centered on the hot spots conjecture for acute triangles). In higher dimensions the situation is now sharply delineated: Kennedy–Rohleder (arXiv:2410.00816) proved a strong form of the conjecture for higher-dimensional analogues of lip and doubly symmetric domains, while de Dios Pont (arXiv:2412.06344) disproved it for convex sets in all sufficiently large dimensions, and de Dios Pont–Hsu–Taylor (arXiv:2508.16321) quantified the optimal failure rate. The planar case — even for convex domains — remains the central open instance. A further caveat for general simply connected domains: μ2\mu_2 need not be simple, so the statement is required to hold for every eigenfunction belonging to μ2\mu_2.

Problem Statement

Prove or refute the planar hot spots conjecture in its modern form (Conjecture 1.1 of Chen–Gui–Yao, arXiv:2311.12659): for every bounded, simply connected planar domain Ω\Omega with Lipschitz boundary (in particular, for every bounded convex planar domain), every eigenfunction uu belonging to the second Neumann eigenvalue μ2\mu_2 (the first nonzero eigenvalue, counting multiplicity) attains its global maximum and its global minimum only on the boundary ∂Ω\partial\Omega; equivalently, there is no point pp in the interior of Ω\Omega with u(p)≥max⁡∂Ωuu(p) \ge \max_{\partial\Omega} u or u(p)≤min⁡∂Ωuu(p) \le \min_{\partial\Omega} u for any such uu. A single explicit simply connected (or convex) planar domain with a rigorous demonstration that one of its μ2\mu_2-eigenfunctions attains its global maximum or its global minimum at an interior point would refute the conjecture.

References

  1. Hongbin Chen, Changfeng Gui, Ruofei Yao. Uniqueness of critical points of the second Neumann eigenfunctions on triangles. arXiv:2311.12659 (2023; v2 December 2025). DOI: 10.48550/arXiv.2311.12659. https://arxiv.org/abs/2311.12659
  2. Krzysztof Burdzy, Wendelin Werner. A counterexample to the 'hot spots' conjecture. Annals of Mathematics (2) 149(1):309–317, 1999. DOI: 10.2307/121027. https://doi.org/10.2307/121027
  3. Rodrigo Bañuelos, Krzysztof Burdzy. On the 'hot spots' conjecture of J. Rauch. Journal of Functional Analysis 164(1):1–33, 1999. DOI: 10.1006/jfan.1999.3397. https://doi.org/10.1006/jfan.1999.3397
  4. David Jerison, Nikolai Nadirashvili. The 'hot spots' conjecture for domains with two axes of symmetry. Journal of the American Mathematical Society 13(4):741–772, 2000. DOI: 10.1090/S0894-0347-00-00346-5. https://doi.org/10.1090/S0894-0347-00-00346-5
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  6. Chris Judge, Sugata Mondal. Euclidean triangles have no hot spots. Annals of Mathematics (2) 191(1):167–211, 2020. DOI: 10.4007/annals.2020.191.1.3. https://doi.org/10.4007/annals.2020.191.1.3
  7. Chris Judge, Sugata Mondal. Erratum: Euclidean triangles have no hot spots. Annals of Mathematics (2) 195(1):337–362, 2022. DOI: 10.4007/annals.2022.195.1.5. https://doi.org/10.4007/annals.2022.195.1.5
  8. Bartłomiej Siudeja. Hot spots conjecture for a class of acute triangles. Mathematische Zeitschrift 280(3–4):783–806, 2015. DOI: 10.1007/s00209-015-1448-1. https://doi.org/10.1007/s00209-015-1448-1
  9. Yasuhito Miyamoto. The 'hot spots' conjecture for a certain class of planar convex domains. Journal of Mathematical Physics 50(10):103530, 2009. DOI: 10.1063/1.3251335. https://doi.org/10.1063/1.3251335
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  14. Lawford Hatcher. The hot spots conjecture for some non-convex polygons. arXiv:2405.19508 (2024). DOI: 10.48550/arXiv.2405.19508. https://arxiv.org/abs/2405.19508
  15. Lawford Hatcher. Hot spots in domains of constant curvature. arXiv:2508.13353 (2025). DOI: 10.48550/arXiv.2508.13353. https://arxiv.org/abs/2508.13353