Genus-Free Steklov-Eigenvalue Bound

Definitions and mathematical statement

The supplied record concerns a compact, connected Riemannian manifold $(\Omega,g)$ of dimension $n\geq 2$ with boundary $\partial\Omega$. In the standard smooth setting with nonempty boundary, the Steklov eigenvalue problem is \[ \Delta_g u=0\quad\text{in }\Omega,\qquad \partial_{\nu_g}u=\sigma u\quad\text{on }\partial\Omega, \] where $u\not\equiv0$, $\Delta_g$ is the Laplace--Beltrami operator, and $\nu_g$ is the outward unit normal. Smoothness and nonemptiness of the boundary are standard conditions for this background formulation; the supplied excerpt does not explicitly specify them. The eigenvalues, counted with multiplicity, are written \[ 0=\sigma_0<\sigma_1\leq\sigma_2\leq\cdots\longrightarrow+\infty. \] Writing ${\,\mathrm{d}} V_g$ and ${\,\mathrm{d}} A_g$ for the Riemannian volume and boundary measures, respectively, the standard interpretation of the corrupted variational formula in the excerpt is \[ \sigma_1(\Omega,g)= \min_{\substack{f\in W^{1,2}(\Omega),\ \operatorname{Tr}f\not\equiv0\\ \int_{\partial\Omega}\operatorname{Tr}f\,{\,\mathrm{d}} A_g=0}} \frac{\displaystyle\int_\Omega |\nabla_g f|_g^2\,{\,\mathrm{d}} V_g} {\displaystyle\int_{\partial\Omega}|\operatorname{Tr}f|^2\,{\,\mathrm{d}} A_g}, \] where $W^{1,2}(\Omega)$ is the space of square-integrable functions with square-integrable weak first derivatives, and $\operatorname{Tr}f$ is the boundary trace. The nonzero-trace restriction ensures a positive denominator.

The extracted question specializes to a surface, hence to $n=2$, and refers to its genus $\gamma$. Its recoverable wording is:

Can the dependence on the genus $\gamma$ of the surface $\Omega$ be removed from inequality (3)?

Inequality (3) is absent from the supplied record. Consequently its normalization, constants, eigenvalue index, other parameters, and hypotheses cannot be specified, and a precise quantified bound cannot be recovered. The record also does not specify orientability or the convention for genus. The supplied background describes the target as a bound for normalized Steklov eigenvalues, while the excerpt discusses the spectral gap $\sigma_1$; this does not identify the missing inequality.

There is an unresolved conflict within the supplied evidence. The registry status and dated literature assessment label the question open, but the supplied statement-verification field says that the official report gives a counterexample and explicitly explains why the genus dependence cannot be removed, resolving the extracted question negatively. That field marks the statement as unrecoverable. Neither the missing bound nor the asserted counterexample is supplied, so this draft records the question and the conflicting qualifications without asserting an open conjecture or a negative theorem.

Short English statement

The extracted question asks whether genus can be omitted from a Steklov eigenvalue bound for surfaces. The referenced inequality (3) is missing, so the intended bound cannot be stated precisely. Supplied status information conflicts: dated curation calls the question open, while another supplied field reports a negative resolution in the source report. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID 30005611, title Genus-Free Steklov-Eigenvalue Bound, source problem code catalogue entry, in the supplied Oberwolfach Reports Open Problems collection. The supplied proposed year is 2023; no individual proposer is identified.
• Supplied primary-source citation: Geometric Spectral Theory (2024), Oberwolfach Reports, pp. 2017--2106, DOI 10.4171/owr/2023/36: https://doi.org/10.4171/owr/2023/36. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied literature-source label for this link is Oberwolfach Report 36.
• Supplied literature source, labelled Girouard and Polterovich primary bounds: https://arxiv.org/abs/1202.5108.
• Supplied literature source, labelled Current Steklov survey: https://dms.umontreal.ca/~iossif/steklov_surfaces.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that known upper bounds for normalized Steklov eigenvalues retain genus dependence and that removal of genus from the relevant bound was unresolved in the checked literature. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification. It conflicts with the supplied statement-verification field reporting a counterexample and negative resolution in the official report.
• The excerpt cites references [8] and [5] for background and open problems, but their bibliography entries are not supplied and cannot be identified from this record.