Result 350, Differential geometry

Yau’s nodal bounds: surfaces and higher dimensions

Proves the sharp CλC\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 λ1/2+ε0\lambda^{1/2+\varepsilon_0} for some fixed ε0>0\varepsilon_0\gt 0, ruling out even arbitrarily small power losses.

Lean formalization Proof

The bigger picture

Why it matters

The places where a standing wave vanishes can trace complicated geometric patterns. These manuscripts claim that their size obeys a sharp rule on smooth surfaces, but can violate that rule in higher dimensions.

What changes?

The surface manuscript reports that every nonzero real Laplace eigenfunction, a standing-wave mode with positive eigenvalue lambda, on a fixed smooth closed (compact, boundaryless) connected surface has zero-set length at most C times the square root of lambda. C depends on the metric, not the mode. The known matching lower bound then completes Yau's conjecture here. In dimensions three and four, the manuscripts instead report sequences for fixed smooth metrics whose zero-set measure divided by the square root of lambda is unbounded.

What does that help mathematicians do?

In dimension five, the reported example uses one fixed smooth metric on the product of a four-sphere and a circle. A sequence of eigenfunctions has zero-set four-volume growing faster than lambda to the power one-half plus a fixed positive epsilon. This rules out rescuing the upper bound by allowing an arbitrarily small extra power. Researchers must therefore account for genuinely different growth rates, not merely adjust the constant multiplying the square root.

Are there practical applications?

The immediate value is foundational: it identifies limits on controlling wave zero sets from smooth geometry alone. The three-dimensional examples have metrics arbitrarily close to round on the three-sphere in the smooth topology; the four-dimensional example lies on a two-sphere times a two-dimensional torus. Even such proximity to round geometry therefore cannot guarantee the conjectured upper bound.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Sharp nodal length on smooth surfaces

September 23, 2026 26 pages

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 CλC\sqrt\lambda. Together with the known lower bound, this proves Yau's conjecture in this setting.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Smooth counterexamples to Yau's nodal upper bound in dimensions three and four

September 23, 2026 60 pages Main result formalized in Lean

We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on S2×T2S^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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Power-law violations of Yau's nodal upper bound

September 23, 2026 34 pages Main result formalized in Lean

We construct a smooth Riemannian metric on S4×S1S^4\times S^1 and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than λ1/2+ϵ0\lambda^{1/2+\epsilon_0} for one fixed ϵ0>0\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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/350.md.

Yau’s nodal bounds: surfaces and higher dimensions

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Yau's nodal-set conjecture predicts nodal size of order λ\sqrt\lambda for Laplace eigenfunctions of eigenvalue λ\lambda. The formalization proves the upper bound on every fixed smooth closed connected Riemannian surface: there is a surface-dependent constant CC such that the one-dimensional Hausdorff measure of the zero set of every nonzero real eigenfunction with λ>0\lambda>0 is at most CλC\sqrt\lambda. The known lower bound is not part of this selected statement.

The formalized results contradict Yau's proposed O(λ)O(\sqrt\lambda) upper bound for nodal measure in dimensions three and four. On S3S^3, every prescribed smooth neighborhood of the round metric contains one fixed smooth metric with an eigenfunction sequence of unbounded nodal-measure ratio. A second fixed metric on S2×T2S^2\times T^2 has the same property. In both cases the eigenvalues tend to infinity and the intrinsic nodal measures are finite.

Yau's nodal upper-bound conjecture predicts that the nodal measure of an eigenfunction is bounded by a constant times the square root of its eigenvalue. The formalized counterexample fixes one smooth metric on S4×S1S^4\times S^1 and a sequence of nonzero smooth eigenfunctions whose positive eigenvalues tend to infinity, while the intrinsic four-dimensional nodal measures divided by the square roots of the eigenvalues tend to infinity. The metric is fixed throughout the sequence.

Comparator links

Result Comparator statement
Sharp upper bound for nodal length on smooth surfaces NodalLength.lean
Nodal counterexamples in dimensions three and four SmoothYau.lean
Unbounded nodal ratio on S4×S1S^4\times S^1 YauCounterexample.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.