Embeddedness of first Steklov maximisers with many boundary components
Let $M$ be a closed surface and $M_b$ the surface obtained from it by removing $b$ disjoint disks. Let $\lambda_1^*(M) = \sup_g \lambda_1(M, g) \operatorname{Area}(M, g)$ and $\sigma_1^*(M_b) = \sup_g \sigma_1(M_b, g)\, L(\partial M_b, g)$, the suprema running over all Riemannian metrics. Suppose that $\lambda_1^*(M)$ is realised by a metric, possibly with conical singularities, and that $\sigma_1^*(M_{b_j})$ is realised by a metric $g_j$ for an increasing sequence $(b_j)$; there are infinitely many such $M$. By Fraser and Schoen, there is a branched conformal free boundary minimal immersion $\tau_j \colon (M_{b_j}, g_j) \to \mathbb B^n$, isometric on the boundary, with $n$ depending only on $M$. Karpukhin and Stern proved that, after passing to a subsequence, these free boundary minimal surfaces converge in the varifold sense to a branched minimal surface $\tau \colon M \to \mathbb S^{n-1}$ whose induced metric realises $\lambda_1^*(M)$.
Problem. If the limiting minimal surface in $\mathbb S^{n-1}$ is embedded, are the free boundary minimal surfaces in $\mathbb B^n$ realising $\sigma_1^*(M_{b_j})$ embedded for all sufficiently large $j$?
Source: M. Karpukhin and D. Stern, From Steklov to Laplace: free boundary minimal surfaces with many boundary components, Duke Math. J. 173 (2024), Open Question 3. Listed as Open Question 5.41 in Colbois–Girouard–Gordon–Sher (2024).
The statement is adapted from doi.org/10.1007/s13163-023-00480-3, whose text is available under CC BY 4.0.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- SP
- Keywords
- Colbois–Girouard–Gordon–Sher Steklov problems, eigenvalue optimisation, free boundary minimal surfaces, Steklov eigenvalues
References
@article{karpukhin2024,
author = {Karpukhin, Mikhail and Stern, Daniel},
title = {From {S}teklov to {L}aplace: free boundary minimal surfaces with many boundary components},
journal = {Duke Mathematical Journal},
volume = {173},
number = {8},
pages = {1557--1629},
year = {2024},
doi = {10.1215/00127094-2023-0041},
eprint = {2109.11029},
archivePrefix = {arXiv}
}
@article{fraser2016,
author = {Fraser, Ailana and Schoen, Richard},
title = {Sharp eigenvalue bounds and minimal surfaces in the ball},
journal = {Inventiones Mathematicae},
volume = {203},
number = {3},
pages = {823--890},
year = {2016}
}
@misc{karpukhin2024b,
author = {Karpukhin, Mikhail and Kusner, Robert and McGrath, Peter and Stern, Daniel},
title = {Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization},
year = {2024},
eprint = {2402.13121},
archivePrefix = {arXiv},
note = {Maximisers on genus-one surfaces with any number of boundary components; embeddedness of those in $\mathbb{B}^6$ not addressed}
}
@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 = {Revista Matem{\'a}tica Complutense},
volume = {37},
number = {1},
pages = {1--161},
year = {2024},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
note = {Open Question 5.41}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01010, "Embeddedness of first Steklov maximisers with many boundary components", clopen.solutions.