Yau's sharp upper bound for nodal volume
Let $(M, g)$ be a closed connected $C^\infty$ Riemannian manifold of dimension $d \ge 2$. For a nonzero real eigenfunction $u$ with $-\Delta_g u = \lambda u$, $\lambda > 0$, let $Z_u = \{x \in M : u(x) = 0\}$ be its nodal set and $\mathcal{H}^{d-1}_g$ the $(d-1)$-dimensional Hausdorff measure for the metric $g$.
Yau conjectured that $c\sqrt{\lambda} \le \mathcal{H}^{d-1}_g(Z_u) \le C\sqrt{\lambda}$ with constants $c, C > 0$ depending only on $(M, g)$. Donnelly and Fefferman proved both bounds for real-analytic metrics. For smooth metrics, Logunov proved the lower bound, and in dimension $d \ge 3$ an upper bound $C\lambda^{\alpha}$ with an exponent $\alpha > 1/2$ depending only on $d$. Logunov, Malinnikova, Nadirashvili and Nazarov proved the bound $C\sqrt{\lambda}$ for Dirichlet eigenfunctions of the Laplacian on bounded domains in $\mathbb{R}^n$ with $C^1$ boundary.
Problem. Is it true that for every closed connected $C^\infty$ Riemannian manifold $(M, g)$ of dimension $d \ge 2$ there is a constant $C = C(M, g)$ such that every nonzero real eigenfunction $u$ with $-\Delta_g u = \lambda u$, $\lambda > 0$, satisfies $$\mathcal{H}^{d-1}_g(Z_u) \le C\sqrt{\lambda}?$$ The constant must be uniform over all eigenvalues and eigenfunctions, and no real-analyticity is assumed.
Source: Yau (1982), Problem section, as a two-sided bound c√λ ≤ H^{d-1}(Z_u) ≤ C√λ. Stated by AIM — Open Applied Problems for the upper bound on smooth closed manifolds.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, DG
- Keywords
- AIM Open Applied Problems, Laplace eigenfunctions, nodal sets, Yau's conjecture
References
@incollection{yau1982,
author = {Yau, Shing-Tung},
title = {Problem section},
booktitle = {Seminar on Differential Geometry},
series = {Ann. of Math. Stud.},
volume = {102},
publisher = {Princeton Univ. Press},
address = {Princeton, NJ},
year = {1982},
pages = {669--706},
doi = {10.1515/9781400881918-035},
}
@article{donnelly1988,
author = {Donnelly, Harold and Fefferman, Charles},
title = {Nodal sets of eigenfunctions on {R}iemannian manifolds},
journal = {Invent. Math.},
volume = {93},
number = {1},
year = {1988},
pages = {161--183},
doi = {10.1007/BF01393691},
}
@article{logunov2018upper,
author = {Logunov, Alexander},
title = {Nodal sets of {L}aplace eigenfunctions: polynomial upper estimates of the {H}ausdorff measure},
journal = {Ann. of Math. (2)},
volume = {187},
number = {1},
year = {2018},
pages = {221--239},
doi = {10.4007/annals.2018.187.1.4},
eprint = {1605.02587},
archivePrefix = {arXiv},
}
@article{logunov2018lower,
author = {Logunov, Alexander},
title = {Nodal sets of {L}aplace eigenfunctions: proof of {N}adirashvili's conjecture and of the lower bound in {Y}au's conjecture},
journal = {Ann. of Math. (2)},
volume = {187},
number = {1},
year = {2018},
pages = {241--262},
doi = {10.4007/annals.2018.187.1.5},
eprint = {1605.02589},
archivePrefix = {arXiv},
}
@article{logunov2021,
author = {Logunov, Alexander and Malinnikova, Eugenia and Nadirashvili, Nikolai and Nazarov, Fedor},
title = {The sharp upper bound for the area of the nodal sets of {D}irichlet {L}aplace eigenfunctions},
journal = {Geom. Funct. Anal.},
volume = {31},
number = {5},
year = {2021},
pages = {1219--1244},
doi = {10.1007/s00039-021-00581-5},
eprint = {2104.09012},
archivePrefix = {arXiv},
}
@misc{aim-yau-nodal-upper-bound,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Yau's sharp upper bound for nodal volume}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/011-yau-nodal-upper-bound.md},
note = {Entry 011 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on $S^2\times\T^2$ 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 …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00738, "Yau's sharp upper bound for nodal volume", clopen.solutions.