Result 369, Partial differential equations

The hot spots conjecture for simply connected planar domains

Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.

Lean formalization Proof

The bigger picture

Why it matters

Where can the strongest and weakest parts of a basic diffusion pattern occur? The unreviewed manuscript reports that, for smooth, bounded planar regions without holes, these extremes must lie on the boundary.

What changes?

The claim covers every smooth, bounded, simply connected domain in two dimensions, meaning a planar region without holes. Neumann eigenfunctions are spatial patterns of the Laplace operator whose normal derivative vanishes at the boundary. For every nonzero eigenfunction associated with the first positive eigenvalue, the manuscript claims the gradient is nonzero everywhere inside. This includes cases where several independent eigenfunctions share that eigenvalue, not just cases with one independent pattern.

What does that help mathematicians do?

The absence of interior critical points is stronger than locating global maxima and minima. It also rules out interior saddle points, where the function rises in some directions and falls in others while its gradient vanishes. Consequently, researchers could deduce that every interior contour of constant value is locally a smooth curve, with no crossings or singular points. This constrains the geometry of these eigenfunctions.

Are there practical applications?

Its immediate value is foundational: it constrains the geometry of the slowest nonconstant spatial modes for diffusion with an insulating boundary. In the heat-equation interpretation, these modes describe the slowest possible decay toward equilibrium. The claim concerns those eigenfunctions, not arbitrary temperature profiles or every stage of heat flow.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

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

September 24, 2026 21 pages

We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, 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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026,
  author = {{OpenAI}},
  title = {{Strict hot spots and absence of interior critical points on smooth simply connected planar domains}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf}{OAI:Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/369.md.

The hot spots conjecture for simply connected planar domains

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The strict hot-spots conjecture asks whether extrema of a first nonconstant Neumann eigenfunction occur only on the boundary. The formalization proves the stronger 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. Consequently all global maxima and minima lie on the boundary. The conclusion applies to every eigenfunction even when the eigenvalue is multiple.

Comparator links

Result Comparator statement
Strict hot spots and absence of interior critical points HotSpots.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.