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Smooth counterexamples to Yau's nodal upper bound in dimensions three and four
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Abstract
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.
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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.
Follow-up works
- Guiyu Xie, Cheng Zhang, Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces (arXiv:2610.28391, 2026-10-20). Studies Steklov nodal sets on smooth surfaces; it cites this preprint as [27] and states that for $d\ge4$, taking products with an interval and flat tori, "the smooth Laplace counterexamples in [27] rule out both conjectured linear upper bounds" in the Steklov setting.
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 results give, in every smooth neighbourhood of the round metric on $S^3$, one fixed metric with eigenvalues tending to infinity and finite nodal measures of unbounded ratio to $\sqrt\lambda$, plus a second such fixed metric on $S^2\times T^2$. [family-level]
- OpenAI math release, SmoothYau.lean (comparator statement) (openai/math, lean/ComparatorChallenges, 2026). Comparator file stating the two results as Lean theorems
sphere_threeandsphere_two_torus_two, corresponding to Theorems 1.1 and 1.2.
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 a simple example in which the nodal Hausdorff measure of sub-Laplacian eigenfunctions is not bounded by $C\sqrt\lambda$, an earlier failure of the $\sqrt\lambda$ upper bound in a hypoelliptic, non-Riemannian setting; not cited in the paper.
- 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 contrasting analytic and smooth metrics and covering the smooth-case upper and lower bounds that form the "gap between polynomial growth and the conjectured square-root upper bound" discussed in the introduction.
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
- VibeMathed: Problem record
- Emergent Mind: family 350
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026,
author = {{OpenAI}},
title = {{Smooth counterexamples to Yau's nodal upper bound in dimensions three and four}},
howpublished = {OpenAI Math Release preprint
\href{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}{OAI:Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-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
- Power-law violations of Yau's nodal upper bound · September 23, 2026