Large-Genus Hyperbolic Surfaces Above the One-Quarter Spectral Threshold

Definitions and mathematical statement

A closed hyperbolic surface is a compact, connected, smooth surface without boundary, equipped with a Riemannian metric of constant Gaussian curvature $-1$. Under the usual interpretation of the source notation $\mathcal M_g$, restrict to orientable surfaces of genus $g\geq 2$, and let $\mathcal M_g$ denote their isometry classes. Orientability is implicit in this notation rather than explicit in the supplied corrected statement; this interpretation requires review against the report.

Use the nonnegative Laplace--Beltrami operator $\Delta_X=-\operatorname{div}_X\nabla_X$. Write its eigenvalues, counted with multiplicity, as \[ 0=\lambda_0(X)<\lambda_1(X)\leq\lambda_2(X)\leq\cdots. \] Thus $\lambda_1(X)$ is the first positive Laplace eigenvalue.

The supplied question is whether \[ \forall G\in{\mathbb{Z}},\ G\geq 2,\quad \exists g\in{\mathbb{Z}},\ g\geq G,\quad \exists X\in\mathcal M_g \quad\text{such that}\quad \lambda_1(X)>\frac14. \] The inequality is strict for each selected surface. No positive lower bound on $\lambda_1(X)-\tfrac14$ independent of genus is requested. The source calls the threshold asymptotically optimal, but supplies no precise formulation of that qualification; it is retained as source commentary, not an additional asserted theorem. The registry labels the problem open; that label is supplied curation, not an independently verified mathematical conclusion.

Short English statement

Can one find closed orientable hyperbolic surfaces with unbounded genus whose first positive Laplace eigenvalue exceeds $1/4$? The curvature normalization is $-1$, and the excess over $1/4$ may depend on the surface. Orientability follows the usual reading of the source notation $\mathcal M_g$ and remains subject to source review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006511$, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The project also identifies catalog/problems.json as a registry of additional existing records. This draft uses only the supplied complete record.
• Supplied report citation: Mini-Workshop: Hyperbolic meets Stochastic Geometry (2026), Oberwolfach Reports, pp. 3031--3062, DOI 10.4171/owr/2025/57. https://doi.org/10.4171/owr/2025/57. The original excerpt identifies p. 3051. No individual proposer is supplied. The record separately gives proposed year 2025.
• Mondal, On largeness and multiplicity of the first eigenvalue. https://arxiv.org/abs/1406.1080.
• Fortier Bourque--Petri, Linear programming bounds for hyperbolic surfaces. https://arxiv.org/abs/2302.02540.
• Wu--Xue, Random hyperbolic surfaces. https://arxiv.org/abs/2102.05581.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open and states that large-genus constructions approach the one-quarter threshold from below, while existence in arbitrarily large genus above that threshold remains unresolved. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry marks the statement corrected_verified and describes the repairs as changes to formula delimiters, notation, grammar, and typography preserving mathematical content and scope. This is a supplied verification claim, not independent source verification by this draft.