Open Problems and Questions about Geodesics — Question 47 (Guth)

Definitions and mathematical statement

The supplied record refers to two definitions, labelled (a) and (b), which are absent from the record. Write $I_{(b)}(M,g)$ provisionally for the invariant defined in the missing clause (b), and write $\mathcal S$ for the intended class of surfaces with metrics. Neither $I_{(b)}$ nor $\mathcal S$ can be defined precisely from the supplied material. The notation $\operatorname{area}(M,g)$ denotes the area of $M$ with respect to $g$.

The question attributed in context to Guth asks whether there exists a constant $C>0$, independent of the surface and its metric and, in particular, independent of its genus, such that \[ \forall (M,g)\in\mathcal S,\qquad I_{(b)}(M,g)\leq C\sqrt{\operatorname{area}(M,g)}. \] This is only a schematic statement of the intended question: the missing definition (b) and the missing hypotheses specifying $\mathcal S$ must be recovered before it constitutes a precise mathematical problem.

For context, the supplied source attributes to Balacheff and Sabourau, under the reference label [Bal-Sab2010], the bound \[ \operatorname{dias}(M,g)\leq C\sqrt{\operatorname{genus}(M)}\sqrt{\operatorname{area}(M,g)} \] for the diastole defined in the missing clause (a), and reports that its dependence on genus is optimal. This contextual bound concerns the invariant in (a); the question concerns the invariant in (b). The definition of that diastole, the hypotheses of the bound, and the intended genus range are not supplied. In particular, the displayed right-hand side vanishes at genus zero under the usual genus convention, so its applicability in that case requires clarification.

Short English statement

The question attributed in context to Guth asks whether the invariant from the missing definition (b) has an upper bound by a universal constant times the square root of area, independently of genus. Its definition and the allowed surfaces and metrics are absent, so the supplied record does not determine a complete precise statement.

Sources

• Local UnsolvedMath registry, permanent ID $800966$, source problem code catalogue entry; supplied title: Open Problems and Questions about Geodesics --- Question 47 (Guth). The registry labels the problem open; this is a supplied status, not an independently verified assessment.
• Burns and Matveev, Open Problems and Questions about Geodesics, source list dated 2019 in the record, Question 47: https://arxiv.org/abs/1308.5417. The registry lists Burns and Matveev as proposers and gives Guth as attribution/context. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these provenance claims have not been independently verified here.
• [Bal-Sab2010]: the supplied excerpt attributes the contextual diastolic inequality and optimal genus dependence to Balacheff and Sabourau. No full bibliographic entry or separate link for this reference is supplied.