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. (Lean)

Manuscripts (3)

Sharp nodal length on smooth surfaces

September 23, 2026 · source · cite as \cite{OAI:Sharp-nodal-length-on-smooth-surfaces-September-23-2026}

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.

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

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026}

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.

Power-law violations of Yau's nodal upper bound

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026}

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.

Result 350 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.