Uniform Bounds across Strata of Flat Surfaces

Definitions and mathematical statement

The supplied record concerns strata of flat surfaces, but does not specify the type of flat surface, the definition of the strata, the quantity being bounded, or the formulas and hypotheses of the referenced bounds. Consequently, a complete mathematical statement cannot be recovered from this record alone.

Subject to recovering those definitions and bounds, the source asks for the following.

1. Extend the referenced lower bound to every hyperelliptic stratum, retaining its applicable hypotheses and determining its scope in each such stratum.
2. Establish corresponding upper and lower bounds for every stratum in the intended class of flat surfaces. The meaning of "corresponding," including the desired dependence on parameters, is not specified.
3. Determine whether the almost-everywhere qualification in the referenced lower bound can be removed while retaining the other hypotheses: does that lower bound hold for every surface in its intended domain? Alternatively, exhibit a surface satisfying the other hypotheses for which that lower bound fails.

Here "almost everywhere" requires a specified measure on a specified space; neither is supplied. The record also leaves unclear whether the third question concerns only the domain of the previously established lower bound or also the extensions requested above. No particular measure, normalization, genus range, or interpretation of hyperelliptic stratum is imposed in this draft.

Short English statement

The problem asks to extend an unspecified lower bound to all hyperelliptic strata, obtain corresponding bounds across all intended strata, and decide whether exceptional surfaces can violate the lower bound. The supplied excerpt omits the bounded quantity, the bounds themselves, and essential domain and measure conventions, so the formulation remains incomplete.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006028$, title "Uniform Bounds across Strata of Flat Surfaces," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2024.
• Supplied source citation: Riemann Surfaces: Random, Flat, and Hyperbolic Geometry (2025), Oberwolfach Reports, pp. 2369--2424, DOI 10.4171/owr/2024/41. The original excerpt identifies p. 2407. https://doi.org/10.4171/owr/2024/41.
• Supplied literature reference: Aulicino--Masur--Pan--Su, https://arxiv.org/abs/2601.15993. No title or specific theorem is supplied.
• Supplied literature reference: Aggarwal--Delecroix--Goujard--Zograf--Zorich, https://arxiv.org/abs/1810.05227. No title or specific theorem is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, reports upper bounds and lower bounds in genus two and recent restricted hyperelliptic results, and describes the all-hyperelliptic and all-strata questions as remaining open. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the editorial label corrected_verified and states that only clear formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. This supplied label does not establish mathematical correctness or completeness.