Mass Curvature Relations in Asymptotically Hyperbolic Geometry

Definitions and mathematical statement

The supplied record concerns asymptotically hyperbolic manifolds and two objects called the conformal mass covariant and the Weyl mass covariant. It supplies neither definitions of these covariants nor precise hypotheses defining their existing domains or the intended broader setting.

Statement of the supplied research problem. Extend the conformal and Weyl mass covariants to broader classes of asymptotically hyperbolic manifolds, retaining their existing values wherever they are already defined, and establish a tight relation between a given mass covariant and an associated curvature operator. The edited statement describes the desired relation as sharp and asks for such a relation for each mass.

Unresolved mathematical specifications. A fully quantified formulation requires the dimension, geometric category, asymptotic model, decay and regularity assumptions, and boundary conditions; the definitions, normalizations, transformation laws, and original domains of the two mass covariants; and the associated curvature operators. None of these is specified in the supplied record. The meaning of a tight or sharp relation is also unspecified: no identity, inequality, equivalence, optimal constant, or equality condition is given. Consequently, this is a statement of the research target with unresolved qualifications, rather than a uniquely determined theorem or conjecture.

Short English statement

The problem asks for extensions of conformal and Weyl mass covariants in asymptotically hyperbolic geometry, together with precise relations to curvature operators. The supplied record leaves the covariants, admissible geometric hypotheses, operators, and meaning of sharpness undefined, so a complete mathematical formulation requires source clarification.

Sources

• Local UnsolvedMath registry: permanent ID $30003913$, title Mass Curvature Relations in Asymptotically Hyperbolic Geometry, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Mathematical General Relativity (2019), Oberwolfach Reports, pp. 2157--2251, DOI 10.4171/owr/2018/36. https://doi.org/10.4171/owr/2018/36. The supplied background attributes the problem to workshop open-problem discussions documented in this report. The citation gives 2019, while the DOI and proposed-year metadata contain 2018; this discrepancy is unresolved here.
• Supplied literature source, labeled Chrusciel--Herzlich 2003: https://arxiv.org/abs/math/0110035.
• Supplied literature source, labeled Oberwolfach Report 36: https://doi.org/10.4171/owr/2018/36.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that existing asymptotically hyperbolic mass constructions do not settle the requested tight link with a curvature operator. The background describes this assessment as literature triage derived from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This is a supplied editorial qualification, not an independent mathematical verification of this draft.