Linear growth of interior Steklov nodal volume
Note: The recorded paper treats Laplace eigenfunctions on closed manifolds, not Steklov eigenfunctions. The counterexample to the upper bound here follows from it through a product (cylinder) reduction made in the review of AIM — Open Applied Problems (15 October 2026), which has not been checked here.
Let $(M,g)$ be a smooth compact connected Riemannian manifold of dimension $d \ge 2$ with smooth nonempty boundary, and let $\nu$ be the outward unit normal. A Steklov eigenfunction is a real function $u \ne 0$ with $$\Delta_g u = 0 \text{ in } M, \qquad \partial_\nu u = \sigma u \text{ on } \partial M$$ for some $\sigma > 0$. Write $Z_u = \{x \in \operatorname{int} M : u(x) = 0\}$ for its nodal set in the interior and $\mathcal{H}^{d-1}_g$ for $(d-1)$-dimensional Hausdorff measure. The two-sided bound below is the Steklov analogue of Yau's conjecture for Laplace eigenfunctions. Polterovich, Sher and Toth proved sharp upper and lower bounds of this form for the nodal length on real-analytic Riemannian surfaces, and Zhu proved the upper bound on real-analytic manifolds of every dimension. On smooth manifolds, Sogge, Wang and Zhu proved $\mathcal{H}^{d-1}_g(Z_u) \ge c\,\sigma^{(2-d)/2}$.
Problem. Is it true that for every such manifold $(M,g)$ there are constants $0 < c_M \le C_M < \infty$, independent of $u$ and $\sigma$, such that $$c_M \sigma \le \mathcal{H}^{d-1}_g(Z_u) \le C_M \sigma$$ for every Steklov eigenfunction $u$ of $M$? Only the nodal set in the interior is counted; the zero set of the boundary trace has a different dimension.
Source: Girouard and Polterovich (2017), Open Problem 11, as the Steklov analogue of Yau's nodal conjecture; restated by Colbois, Girouard, Gordon and Sher (2024), Open Question 10.7. Stated by AIM — Open Applied Problems for the interior nodal set, (10.3) there, on smooth manifolds.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, DG
- Keywords
- AIM Open Applied Problems, nodal sets, Steklov eigenfunctions, Yau's conjecture
References
@article{girouard2017,
author = {Girouard, Alexandre and Polterovich, Iosif},
title = {Spectral geometry of the {S}teklov problem (survey article)},
journal = {J. Spectr. Theory},
volume = {7},
number = {2},
year = {2017},
pages = {321--359},
doi = {10.4171/JST/164},
eprint = {1411.6567},
archivePrefix = {arXiv},
}
@article{colbois2024,
author = {Colbois, Bruno and Girouard, Alexandre and Gordon, Carolyn and Sher, David},
title = {Some recent developments on the {S}teklov eigenvalue problem},
journal = {Rev. Mat. Complut.},
volume = {37},
number = {1},
year = {2024},
pages = {1--161},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
}
@article{polterovich2019,
author = {Polterovich, Iosif and Sher, David A. and Toth, John A.},
title = {Nodal length of {S}teklov eigenfunctions on real-analytic {R}iemannian surfaces},
journal = {J. Reine Angew. Math.},
volume = {754},
year = {2019},
pages = {17--47},
doi = {10.1515/crelle-2017-0018},
eprint = {1506.07600},
archivePrefix = {arXiv},
}
@article{zhu2020,
author = {Zhu, Jiuyi},
title = {Geometry and interior nodal sets of {S}teklov eigenfunctions},
journal = {Calc. Var. Partial Differential Equations},
volume = {59},
number = {5},
year = {2020},
pages = {150},
doi = {10.1007/s00526-020-01815-4},
}
@article{sogge2016,
author = {Sogge, Christopher D. and Wang, Xing and Zhu, Jiuyi},
title = {Lower bounds for interior nodal sets of {S}teklov eigenfunctions},
journal = {Proc. Amer. Math. Soc.},
volume = {144},
number = {11},
year = {2016},
pages = {4715--4722},
doi = {10.1090/proc/13067},
eprint = {1503.01091},
archivePrefix = {arXiv},
}
@misc{aim-steklov-bulk-nodal-volume,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Linear growth of interior Steklov nodal volume}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/110-steklov-bulk-nodal-volume.md},
note = {Entry 110 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove a sharp linear upper bound for the interior nodal length of Steklov eigenfunctions on smooth compact Riemannian surfaces with boundary, establishing the upper bound predicted by the Steklov analogue of Yau's conjecture in dimension two. On each fixed surface, we also show that the nodal length divided by the logarithm of the eigenvalue tends to infinity, uniformly over …
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 |
|---|---|---|---|
| 2026-10-26 | Open → Solution claimed | The registry's curators | Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces v1 recorded from arXiv:2610.28391v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00723, "Linear growth of interior Steklov nodal volume", clopen.solutions.