Informal Question about a Rescaled Einstein Limit

Definitions and mathematical statement

Let $\overline M$ be a smooth manifold with boundary and let $M=\operatorname{int}(\overline M)$. A boundary defining function is a smooth function $\rho:\overline M\to[0,\infty)$ such that \[ \rho^{-1}(0)=\partial\overline M,\qquad \rho>0\text{ on }M,\qquad \mathrm d\rho\ne0\text{ on }\partial\overline M. \] The supplied passage calls a metric $g$ Poincaré--Einstein when it is Einstein on the interior and $\rho^2g$ extends smoothly across the boundary. In Riemannian notation the Einstein condition is \[ \operatorname{Ric}(g)=\lambda g \] for a constant $\lambda$. The passage does not specify the dimension, a normalization or sign for $\lambda$, or completeness hypotheses.

To obtain the usual Riemannian conformal infinity, the extension $\overline g=\rho^2g$ must additionally be a nondegenerate Riemannian metric up to the boundary. This requirement is not explicit in the supplied wording. Under this requirement, write $\iota:\partial\overline M\hookrightarrow\overline M$ for the inclusion and set \[ h=\iota^*\overline g,\qquad [h]=\{e^{2f}h:f\in C^\infty(\partial\overline M,{\mathbb{R}})\}. \] If $\rho'=e^\omega\rho$ is another defining function near the boundary, then \[ \iota^*((\rho')^2g)=e^{2\omega|_{\partial\overline M}}h. \] Thus $[h]$ is independent of the defining function. It is called the conformal infinity, and $g$ is a filling of the boundary conformal manifold $(\partial\overline M,[h])$. The source instead says a filling of $(M,g)$; its intended boundary data require clarification.

Unresolved source question. The question is: "Is this metric new/interesting?" The record title and supplied curation describe the referent as a rescaled Einstein limit, but the supplied passage gives neither its formula nor the rescaling or limiting procedure. It also gives no mathematical criterion for "new" or "interesting." Consequently, no precise assertion about that metric can be recovered from this record.

Context retained from the passage. The source identifies the Poincaré model of the hyperbolic metric as Poincaré--Einstein. Its damaged typography appears to describe $g_m^+$ as a filling of $S^3/{\mathbb{Z}}_m$ and $g_m^-$ as its "cap." These symbols are retained only as a tentative transcription: the metrics, allowed values of $m$, quotient action, boundary conformal class, and meaning of "cap" are unspecified. The passage attributes fillings of this quotient to Page--Pope and asserts that their apparently different fillings must be isometric to the author's fillings. The hypotheses and identification needed to formulate that assertion are absent. It also mentions the role of such fillings in Witten's AdS/CFT correspondence and expresses hope that the described fillings could help determine renormalised volumes in the sense of reference [An]. These are supplied contextual claims and motivations, not additional problem statements or verified conclusions.

Short English statement

The record asks whether an unspecified rescaled Einstein limit is "new or interesting." It recalls conformal infinity and Poincaré--Einstein fillings, but does not define the metric or give a testable target. The supplied open label and unrecoverable statement status are registry classifications. This statement-only draft preserves the available context and requires human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30005873$, source code catalogue entry, title: Informal Question about a Rescaled Einstein Limit. The supplied record gives proposed year 2024 and no named proposer.
• Supplied report citation: Geometrie (2024), Oberwolfach Reports, pp. 1563--1614, DOI 10.4171/owr/2024/28. Source URL and literature-source label "original OWR report": https://doi.org/10.4171/owr/2024/28. The registry describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: "Oliveira--Sena-Dias Einstein metrics from Derdzinski duality": https://arxiv.org/abs/2306.17328.
• Supplied literature source: "Einstein metrics via Derdzinski duality": https://doi.org/10.1007/s12220-025-01975-9.
• The passage cites Definition 5 with reference [An2], Witten's AdS/CFT correspondence with [W], Page--Pope fillings with [PP], and renormalised volumes with [An]. The supplied record does not include bibliographic entries resolving these labels or identify the author of the passage.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the record open, describes the question as a novelty question rather than a precise mathematical assertion, and reports that related constructions are now published without establishing a canonical novelty classification. This is supplied dated curation, not an independently verified literature assessment. Its instruction to verify time-sensitive status and cited claims independently remains applicable; no source visit or verification is claimed here.