Strict hot spots and absence of interior critical points on smooth simply connected planar domains

Follow-up works

• Krzysztof Burdzy, Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue Case (arXiv:2610.12622, 8 Oct 2026). Takes this paper's smooth-domain theorem as an unverified external input and, by exhaustion with conformal images of concentric disks, derives a non-strict hot-spots conclusion on bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue. [AI-generated per source]
• Haiyun Deng, Changfeng Gui, Xuyong Jiang, Xiaoping Yang, Ruofei Yao, Jun Zou, Symmetry and critical points of second Neumann eigenfunctions on isosceles trapezoids and kites (arXiv:2604.19003v4, 11 Oct 2026). Proves absence of interior critical points on isosceles trapezoids and kites; a "Declaration" added in v4 compares this paper's scope (smooth boundary, excluding polygons with corners) and its Green-kernel multiplier method with the authors' own.
• Jizhou Guo, First Neumann eigenfunctions of convex planar domains attain their extrema on the boundary (GitHub preprint, 2026-10-10). The author's repository describes a deduction from this paper's smooth-domain theorem of non-strict boundary extrema for every first-positive Neumann eigenfunction on bounded convex planar domains, allowing nonsmooth boundary and a double eigenvalue.
• Rami Atar and Krzysztof Burdzy, A starlike 3-dimensional counterexample to the hot spots conjecture (arXiv:2610.19908v1, 2026-10-14). Cites this planar theorem in its introduction and constructs a bounded starlike Lipschitz domain in three dimensions with a simple first positive Neumann eigenvalue and an eigenfunction whose unique global maximum is at the origin.

Comments and discussion

• Krzysztof Burdzy, remarks in Section 1 of arXiv:2610.12622 (arXiv, Oct 2026). Burdzy, who stated the simply connected conjecture in 2005, says he has not assessed this paper's proof and that "The OpenAI model that generated [3] missed the theorem presented in this note"; he also reports that Rodrigo Bañuelos used ChatGPT to generate an almost identical theorem and proof. [AI-generated per source]

Formalization

• mathcommunity/oai-math-revised, lean/docs/369.md (OpenAI math release Lean scope notes, 2026-10-06). States that the Lean formalization proves nonvanishing interior gradient for every nonzero first positive Neumann eigenfunction on nonempty smooth bounded simply connected planar domains, including the multiple-eigenvalue case.
• mathcommunity/oai-math-revised, lean/ComparatorChallenges/HotSpots.lean (release comparator statement). Gives the Lean 4/Mathlib statement OAI.StrictHotSpots.main_theorem: nonzero gradient, plus strict bounds by the boundary infimum and supremum, for nonzero u in the first Neumann eigenspace of an AdmissibleDomain.

Related prior literature mentioned by others

• Jaume de Dios Pont, Convex sets can have interior hot spots (arXiv:2412.06344, 2024-12-09). Cited by the follow-up works (Burdzy; Deng et al.; Atar–Burdzy) for its counterexamples on convex domains in sufficiently high dimensions, which is context for this paper's restriction to planar domains.

Possibly relevant literature (found by the agent, not checked)

• Jonathan Rohleder, Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains (arXiv:2306.12922, v4 Jun 2025). Introduces a variational principle on simply connected planar Lipschitz domains of the same kind as the tangent-vector-field principle in Section 3 of this paper, whose bibliography cites only Rohleder's arXiv:2106.05224 and the 2024 extended abstract.