The hot spots conjecture for smooth simply connected planar domains
Rauch's hot spots problem (1975) asks whether an eigenfunction for the first positive Neumann eigenvalue of a bounded domain attains its maximum and minimum only on the boundary. It fails in general: Burdzy and Werner (1999) gave a planar counterexample with two holes, and Burdzy (2005) one with a single hole. Burdzy conjectured that it holds for bounded simply connected planar domains; known cases include lip domains, triangles and symmetric convex domains. Does every first-positive Neumann eigenfunction on a bounded simply connected planar domain attain its extrema only on the boundary?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Spectral theory of the Neumann Laplacian
- Posed by
- Krzysztof Burdzy (simply connected planar form); the hot spots problem is due to Jeffrey Rauch
- Year posed
- 2005
- Years open
- 21y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 42 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: on every bounded simply connected planar domain with boundary, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so its global extrema lie only on the boundary; this holds even when the eigenvalue is multiple and with no convexity or symmetry assumption. It does not treat nonsmooth (for example Lipschitz or polygonal) simply connected domains, domains with holes, or higher dimensions.
What the AI did
The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 24, 2026. The paper says it builds on Rohleder's variational principle for tangent vector fields.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Burdzy's conjecture; it covers bounded simply connected planar domains with boundary only, which is why this is Partial. The challenge HotSpots is not in the formalization catalogue (lean/formalization.yaml); it is linked from lean/docs/369.md, its solution module exists at the pinned commit, and its statement was read here: for a nonempty open bounded simply connected planar domain with smooth boundary and any nonzero smooth up to the boundary in the first positive Neumann eigenspace (defined via the Rayleigh quotient), in and inside. This states the headline. Not rebuilt here.