{"id":"openai-math/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four","pdf":"https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026","tags":["spectral-geometry","laplace-eigenfunctions","nodal-sets","riemannian-geometry"],"score":0,"title":"Smooth counterexamples to Yau's nodal upper bound in dimensions three and four","domain":"Differential geometry","family":"350","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on $`S^2\\times\\mathbb T^2`$ for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"family_title":"Yau’s nodal bounds: surfaces and higher dimensions"}