Liouville Quantum Gravity Above Central Charge One

Definitions and mathematical statement

Let LQG denote Liouville quantum gravity, let $\gamma$ denote its parameter in the source regime $0<\gamma\leq 2$, and let $c_M$ denote the matter central charge. The supplied record identifies this source regime with $c_M\leq 1$. The proposed target phase is \[ 1<c_M<25. \] The source regime includes $\gamma=2$, whereas both central-charge endpoints are excluded from the target phase. No correspondence between $\gamma$ and $c_M$ in the target phase is specified in the supplied record.

Problem as supplied. Given that a metric can already be constructed in the target phase, can the various other features of LQG established in the regime $0<\gamma\leq 2$, described in the record as $c_M\leq 1$, be extended to the phase $1<c_M<25$?

Unresolved formulation. The record does not identify the features to be extended, their hypotheses, the underlying LQG objects, or the meaning of an extension for each feature. Thus this is a research programme rather than a specified quantified conjecture. In particular, the supplied text does not determine whether a particular extension is requested for every $c_M\in(1,25)$, or whether different features have different parameter ranges. A precise theorem statement requires these choices to be supplied.

The supplied dated assessment reports constructions of weak and critical or supercritical LQG metrics and uniqueness across the relevant central-charge range, while describing the extension of every feature known in the subcritical regime to $1<c_M<25$ as remaining open. This assessment does not specify the hypotheses or range of each cited result and is not independently verified here.

Short English statement

The question asks whether additional properties of Liouville quantum gravity known for $0<\gamma\leq2$, associated in the record with $c_M\leq1$, extend to $1<c_M<25$, where the source says a metric is already available. The intended properties and their extension criteria are unspecified, so a precise individual conjecture cannot yet be extracted.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005139$, title: Liouville Quantum Gravity Above Central Charge One; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and supplies no proposer attribution.
• Supplied source citation: Universality: Random Matrices, Random Geometry and SPDEs (2023), Oberwolfach Reports, pp. 1501--1550, DOI 10.4171/owr/2022/27. https://doi.org/10.4171/owr/2022/27. The record separately gives proposed year 2022.
• Supplied literature reference: Ding--Gwynne, LQG metric uniqueness. https://doi.org/10.1112/plms.12492.
• Supplied literature reference: Pfeffer, weak LQG metrics. https://arxiv.org/abs/2104.04020.
• Supplied literature reference: Oberwolfach report. https://doi.org/10.4171/owr/2022/27.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially_solved and reports metric construction and uniqueness results while retaining a broader extension problem. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its verification note says that the official report was checked directly and that the cleaned statement preserves the mathematical target while removing extraction artifacts and source-dependent wording. This is a claim recorded in the supplied registry, not a source check performed for this draft.