{"id":"openai-math/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains","pdf":"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","url":"https://github.com/openai/math/tree/main/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026","tags":["spectral-geometry","neumann-eigenfunctions","hot-spots","pde"],"score":0,"title":"Strict hot spots and absence of interior critical points on smooth simply connected planar domains","domain":"Partial differential equations","family":"369","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AP","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.12622","date":"2026-10-08","kind":"web","site":"arXiv.org","text":"We give a reduction from a smooth-domain theorem excluding interior critical points of first positive Neumann eigenfunctions to a non-strict hot-spots conclusion on arbitrary bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue. The main…","added":"2026-10-24","title":"Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue Case","author":"Krzysztof Burdzy","record":"arxiv/2610.12622"}],"formalized":false,"family_title":"The hot spots conjecture for simply connected planar domains"}