{"id":"openai-math/Sharp-nodal-length-on-smooth-surfaces","pdf":"https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026","tags":["spectral-geometry","eigenfunctions","nodal-sets"],"score":0,"title":"Sharp nodal length on smooth surfaces","domain":"Differential geometry","family":"350","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most $`C\\sqrt\\lambda`$. Together with the known lower bound, this proves Yau's conjecture in this setting.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":false,"family_title":"Yau’s nodal bounds: surfaces and higher dimensions"}