The hot spots conjecture for smooth simply connected planar domains

Theorem 1.1: on every bounded simply connected planar domain with $C^\infty$ 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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The hot spots conjecture for smooth simply connected planar domains · Hot spots, smooth simply connected planar

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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VibeMathed record: The hot spots conjecture for smooth simply connected planar domains

specialist registry · passed

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 $C^\infty$ 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 $u$ smooth up to the boundary in the first positive Neumann eigenspace (defined via the Rayleigh quotient), $\nabla u\neq0$ in $\Omega$ and $\inf_{\partial\Omega}u<u<\sup_{\partial\Omega}u$ inside. This states the headline. Not rebuilt here.