Hyperbolic Conformal Boundaries of Poincaré–Einstein Manifolds

Definitions and mathematical statement

A standard interpretation of the supplied question is as follows. Let $n\geq 2$ be a boundary dimension, and let $\overline M$ be a compact smooth $(n+1)$-dimensional manifold with nonempty boundary $N=\partial\overline M$ and interior $M$. A boundary defining function is a function $\rho\in C^\infty(\overline M)$ satisfying \[ \rho>0\text{ on }M,\qquad \rho=0\text{ on }N,\qquad {\,\mathrm{d}}\rho\neq0\text{ on }N. \] A Riemannian metric $g$ on $M$ is conformally compactifiable if, for such a defining function, $\overline g=\rho^2g$ extends to a nondegenerate Riemannian metric on $\overline M$. The source does not specify the required regularity of this extension. In the usual normalization, the Poincaré--Einstein condition is \[ \operatorname{Ric}_g=-n g. \] The conformal boundary, or boundary at infinity, is $(N,[\gamma])$, where $\gamma=\overline g|_{TN}$ and $[\gamma]$ is its conformal class. Thus the standard interpretation of a boundary of constant sectional curvature $-1$ is that this conformal class contains a metric $h$ satisfying \[ K_h(\sigma)=-1 \quad\text{for every }x\in N\text{ and every two-dimensional subspace }\sigma\subset T_xN. \] For each fixed $n\geq2$, the resulting existence assertion is \[ \begin{gathered} \exists\,\overline M,g,\rho,h:\quad (M,g)\text{ is complete},\qquad \operatorname{Ric}_g=-ng,\\ \overline g=\rho^2g\text{ extends as above},\qquad N=\partial\overline M\text{ is compact and connected},\\ h\in[\overline g|_{TN}],\qquad K_h\equiv-1. \end{gathered} \] The supplied problem asks whether such a manifold exists, or whether no such manifold is possible. Its dimension is unspecified: the record does not determine whether a particular $n$ is intended or whether existence in some dimension is sufficient. The restriction $n\geq2$ above makes the sectional-curvature condition nonvacuous; it is an explicit interpretive convention, not a supplied dimension hypothesis. The conformal-representative interpretation and the compactification regularity also require confirmation from the source. No existence or nonexistence conclusion is asserted here.

Short English statement

Does a complete conformally compactifiable Poincaré--Einstein manifold have a compact connected conformal boundary admitting a metric of constant sectional curvature $-1$? The supplied record leaves the dimension and compactification regularity unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local registry record, data/unsolvedmath/problems.json: permanent UnsolvedMath ID $30002692$, source problem code catalogue entry, title Hyperbolic Conformal Boundaries of Poincaré--Einstein Manifolds. No individual proposer is identified.
• Supplied report citation: Mini-Workshop: Einstein Metrics, Ricci Solitons and Ricci Flow under Symmetry Assumptions (2015), Oberwolfach Reports, pp. 2529--2568, DOI 10.4171/owr/2014/45. Supplied source URL: https://doi.org/10.4171/owr/2014/45. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The cited page range covers the report; no problem-specific page is supplied.
• Supplied literature source: Oberwolfach Report 45/2014, https://doi.org/10.4171/owr/2014/45. The exact supplied literature-check date is 2026-08-21. Its supplied status is open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. That assessment says the extracted statement lacks the mathematical question and therefore retains an open label under an uncertainty rule; this explanation conflicts with the explicit existence question present in the supplied record. Time-sensitive status and cited claims require independent verification.
• The record labels its statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied verification claims, not mathematical review performed for this draft. The proposed year and report identifier indicate 2014, while the supplied report citation gives 2015; the bibliographic relationship has not been independently checked.