Universal Steklov Isoperimetric Questions

Definitions and mathematical statement

The supplied excerpt asks about a fixed closed submanifold $\Sigma\subset{\mathbb{R}}^n$ of dimension $n-2$ and the quantity \[ \sup\{\sigma_1(\Omega):\Omega\subset{\mathbb{R}}^n\text{ is a submanifold with }\partial\Omega=\Sigma\}. \] It does not fully specify the admissible submanifolds or the convention for $\sigma_1$. The following is therefore a conditional reconstruction requiring confirmation against the report.

Fix an integer $n\geq2$ and a nonempty smooth compact embedded submanifold $\Sigma\subset{\mathbb{R}}^n$, without boundary, of dimension $n-2$. Under the interpretation that fillings are smooth compact connected embedded hypersurfaces, put \[ \mathcal A(\Sigma)=\{\Omega\subset{\mathbb{R}}^n:\Omega\text{ is a smooth compact connected embedded }(n-1)\text{-manifold},\ \partial\Omega=\Sigma\}. \] Equip each filling with its induced Euclidean metric $g$. For $\Omega\in\mathcal A(\Sigma)$, define $\sigma_1(\Omega)$ as the smallest strictly positive Steklov eigenvalue: the smallest $\sigma>0$ for which a nonzero smooth function $u$ satisfies \[ \Delta_g u=0\quad\text{in }\Omega\setminus\partial\Omega, \qquad \frac{\partial u}{\partial\nu}=\sigma u\quad\text{on }\partial\Omega, \] where $\nu$ is the outward unit conormal to the boundary within $\Omega$.

Assuming $\mathcal A(\Sigma)\ne\varnothing$, the finiteness question is whether, for every fixed such $\Sigma$, \[ \sup_{\Omega\in\mathcal A(\Sigma)}\sigma_1(\Omega)<\infty. \] Equivalently, does there exist a finite constant $C_\Sigma$, depending on the fixed embedded boundary but not on its filling, such that \[ \forall\Omega\in\mathcal A(\Sigma),\qquad \sigma_1(\Omega)\leq C_\Sigma? \] In particular, this bound would impose no upper bound on $\operatorname{Vol}_g(\Omega)$. The excerpt also asks whether a preceding inequality can omit interior volume, but that inequality is not supplied, so its precise additional content cannot be reconstructed.

The dimensional range, compactness, connectedness, regularity, induced metric, and spectral convention in this reconstruction require source confirmation. The supplied status information is contradictory: the open label and dated literature assessment describe an open finiteness question, whereas the supplied statement-verification field asserts that the report gives unbounded constructions answering both extracted questions negatively. No resolution of that conflict or mathematical conclusion is asserted here.

Short English statement

For a fixed embedded boundary $\Sigma\subset{\mathbb{R}}^n$ of dimension $n-2$, ask whether the first positive Steklov eigenvalue is bounded over all admissible hypersurfaces spanning $\Sigma$, independently of their interior volume. The admissibility assumptions require confirmation, the referenced preceding inequality is missing, and the supplied open and negatively answered descriptions conflict.

Sources

• Local UnsolvedMath registry, permanent ID $30005612$, title Universal Steklov Isoperimetric Questions, exact source problem code catalogue entry. The supplied record assigns proposed year 2023 and supplies no individual proposer.
• Supplied report citation: Geometric Spectral Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/36, pp. 2017--2106. https://doi.org/10.4171/owr/2023/36. This is also the supplied literature-source link labelled "original OWR report." The record attributes the material to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature-source attribution: Colbois--Girouard--Gittins, "prescribed-boundary Steklov spectrum." https://arxiv.org/abs/1711.06458.
• Literature checked at: 2026-08-22. This exact date records supplied dated curation, not independent verification. Its assessment states that prescribed-boundary Steklov bounds are proportional to interior volume and that the fixed-boundary finiteness question without volume remains explicitly open. The background expressly requires independent verification of time-sensitive status and cited claims.
• Conflicting supplied verification: the statement-verification field says that the official report answers both extracted questions negatively by unbounded constructions. The record nevertheless labels the problem "open" and its statement status "unrecoverable." These are supplied registry assertions, not independently verified mathematical facts.