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.
released 2026-09-23 | 4 theorems · 11 lemmas · 18 proofs · 10,196 words |
PLAY LEVEL 1 »(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 $C\sqrt\lambda$. Together with the known lower bound, this proves Yau's conjecture in this setting.
released 2026-09-23 | 2 theorems · 25 lemmas · 37 proofs · 24,241 words |
PLAY LEVEL 2 »(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 $S^2\times\mathbb T^2$ 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.
released 2026-09-23 | 1 theorem · 10 lemmas · 15 proofs · 14,622 words |
PLAY LEVEL 3 »(pdf)
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.