Browse › Differential geometry › 350 Yau’s nodal bounds: surfaces and higher dimensions
Sharp nodal length on smooth surfaces
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Guiyu Xie, Cheng Zhang, Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces (arXiv:2610.28391, 2026-10-20). Proves a sharp linear upper bound for the interior nodal length of Steklov eigenfunctions on smooth compact surfaces with boundary; it lists this preprint as reference [26] among works on nodal sets of Laplace eigenfunctions on closed manifolds.
Formalization
- OpenAI math release, Lean scope note: Yau's nodal bounds: surfaces and higher dimensions (lean/docs/350.md, 6 October 2026). The release's scope note says the formalization covers the upper bound $\mathcal H^1(Z_u)\le C\sqrt\lambda$ for every nonzero real positive-eigenvalue eigenfunction on a fixed smooth closed connected surface, and that the known lower bound is not part of the formalized statement. [family-level]
- OpenAI math release, NodalLength.lean (comparator statement) (openai/math, lean/ComparatorChallenges, 2026). Comparator file with the Lean statement of the surface bound: there is $C\ge0$ such that every nonzero smooth eigenfunction with $\lambda>0$ has nodal length at most $C\sqrt\lambda$.
Possibly relevant literature (found by the agent, not checked)
- A. Logunov, E. Malinnikova, N. Nadirashvili, F. Nazarov, The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions (arXiv:2104.09012, 2021). Proves the sharp $C\sqrt\lambda$ nodal upper bound for Dirichlet eigenfunctions on bounded $C^1$ domains in $\mathbb R^n$, a comparable sharp upper bound in a different setting from the closed-surface Theorem 1.1; not in the paper's bibliography.
- A. Logunov, E. Malinnikova, Review of Yau's conjecture on zero sets of Laplace eigenfunctions (arXiv:1908.01639, 5 August 2019; CDM 2018 lectures). Survey of the smooth-case results, growth estimates and two-dimensional methods for zero sets, i.e. the state of the art that the introduction summarises before Theorem 1.1.
Digests and expositions
- heyprof, Yau’s nodal lines — Chladni on sphere and torus (3D) (VisuApp, undated). This interactive exposition illustrates nodal length on the round sphere and flat torus and places the family’s claimed surface bound alongside its higher-dimensional counterexamples. [family-level]
Elsewhere
- Emergent Mind: family 350
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Sharp-nodal-length-on-smooth-surfaces-September-23-2026,
author = {{OpenAI}},
title = {{Sharp nodal length on smooth surfaces}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf}{OAI:Sharp-nodal-length-on-smooth-surfaces-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/350.md. The Lean statements have not been checked by people against the paper's claims.
Family 350: Yau’s nodal bounds: surfaces and higher dimensions
Proves the sharp $C\sqrt\lambda$ upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S3 arbitrarily close to round. In dimension five, nodal measure can grow faster than $\lambda^{1/2+\varepsilon_0}$ for some fixed $\varepsilon_0\gt 0$, ruling out even arbitrarily small power losses.
Other papers in this family:
- Smooth counterexamples to Yau's nodal upper bound in dimensions three and four · September 23, 2026
- Power-law violations of Yau's nodal upper bound · September 23, 2026