View source for The hot spots conjecture for simply connected planar domains

'''The hot spots conjecture for simply connected planar domains''' is the subject of a manuscript dated 24 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). The hot spots conjecture, posed by J. Rauch in 1974, asserts that an eigenfunction for the first positive Neumann eigenvalue of a domain attains its maximum and minimum on the boundary; counterexamples are known for planar domains with holes (Burdzy and Werner, 1999), and Burdzy formulated a version for simply connected planar domains. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== The original result ==
* '''Claim.''' On every smooth bounded simply connected planar domain, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.
* '''Source.''' [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026 Manuscript folder] in the openai/math repository (commit <code>1581d4c</code>).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/369.md lean/docs/369.md] states that the formalization proves the interior statement on every nonempty smooth bounded simply connected planar domain: every nonzero eigenfunction in the first positive Neumann eigenspace has nonvanishing gradient throughout the interior, including when the eigenvalue is multiple.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-08 || Krzysztof Burdzy, [https://arxiv.org/abs/2610.12622 arXiv:2610.12622] || Extension || Removes boundary smoothness in the simple-eigenvalue case: a non-strict hot spots conclusion on arbitrary bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue, by exhaustion with conformal images of concentric disks. The smooth-domain theorem of the OpenAI manuscript is used as an external input. The abstract states that the paper was generated by ChatGPT with prompts from the author; the author's commentary in the paper describes the theorem as a corollary of the OpenAI results, and says that Rodrigo Bañuelos independently used ChatGPT to generate an almost identical theorem and proof, and that neither intends to submit the paper to a journal. || arXiv preprint, version 1.
|-
| 2026-10-10 || Jizhou Guo, [https://github.com/aster2024/openai-math-followups/tree/main/hot-spots-convex-planar aster2024/openai-math-followups], hot-spots-convex-planar || Use || For every bounded convex planar domain, with no smoothness assumption, every nonzero first Neumann eigenfunction attains its maximum and minimum on the boundary (the non-strict form), for simple or double eigenvalues; further statements on interior critical points and the nodal set. The README states that these are deduced from Theorem 1.1 of the OpenAI manuscript. || Paper and LaTeX source in a GitHub repository.
|-
| 2026-10-13 || Haiyun Deng, Xuyong Jiang and Xiaoping Yang, [https://arxiv.org/abs/2610.18546 arXiv:2610.18546] || Related || For every bounded convex planar domain, without boundary smoothness or eigenvalue simplicity, the maximum modulus of every nonzero second Neumann eigenfunction is attained only on the boundary, via a near-boundary rigidity theorem. The introduction cites the OpenAI manuscript's smooth simply connected theorem. || arXiv preprint, version 1.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]