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Power-law violations of Yau's nodal upper bound
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Abstract
We construct a smooth Riemannian metric on $S^4\times S^1$ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than $\lambda^{1/2+\epsilon_0}$ for one fixed $\epsilon_0\gt 0$. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.
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Notes
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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.
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 formalized counterexample fixes one smooth metric on $S^4\times S^1$ with eigenfunctions whose nodal measure divided by $\sqrt\lambda$ tends to infinity, a weaker statement than the manuscript's fixed positive power-law excess. [family-level]
- OpenAI math release, YauCounterexample.lean (comparator statement) (openai/math, lean/ComparatorChallenges, 2026). Comparator file stating unboundedness of $\mathcal H^4(Z_{u_k})/\sqrt{\lambda_k}$ on $S^4\times S^1$; it does not contain the fixed power-law excess $\lambda^{1/2+\epsilon_0}$ of Theorem 1.1.
Possibly relevant literature (found by the agent, not checked)
- S. Eswarathasan, C. Letrouit, Nodal sets of eigenfunctions of sub-Laplacians (arXiv:2301.11075, 2023). Gives an example in which sub-Laplacian nodal measure is not bounded by $C\sqrt\lambda$, an earlier failure of the $\sqrt\lambda$ bound outside the Laplace–Beltrami setting; not in the 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 Donnelly–Fefferman's analytic-case results and of Logunov's smooth-case polynomial bounds, the background against which the introduction places 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:Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026,
author = {{OpenAI}},
title = {{Power-law violations of Yau's nodal upper bound}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf}{OAI:Power-law-violations-of-Yaus-nodal-upper-bound-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:
- Sharp nodal length on smooth surfaces · September 23, 2026
- Smooth counterexamples to Yau's nodal upper bound in dimensions three and four · September 23, 2026