Yau’s nodal bounds: surfaces and higher dimensions

What OpenAI claims

Proves the sharp C√λ 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 λ^1/2+ε₀ for some fixed ε₀ > 0, ruling out even arbitrarily small power losses. (Lean)

Papers (3)

1. Sharp nodal length on smooth surfaces

   23 September 2026 · Preprint, not peer reviewed · PDF

   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√λ. Together with the known lower bound, this proves Yau's conjecture in this setting.

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

   23 September 2026 · Preprint, not peer reviewed · PDF

   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×𝕋² 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.

3. Power-law violations of Yau's nodal upper bound

   23 September 2026 · Preprint, not peer reviewed · PDF

   We construct a smooth Riemannian metric on S⁴× S¹ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than λ^1/2+ϵ₀ for one fixed ϵ₀ > 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.