Shrinking-Soliton Degeneration Report

Definitions and mathematical statement

A shrinking Ricci soliton, in the vector-field convention, is a Riemannian manifold $(M,g)$ equipped with a vector field $V$ and a constant $\lambda>0$ satisfying \[ \operatorname{Ric}_g+\tfrac12\mathcal L_Vg=\lambda g, \] where $\operatorname{Ric}_g$ is the Ricci tensor and $\mathcal L_Vg$ is the Lie derivative of the metric. In the gradient case, $V=\nabla_g f$ for a smooth function $f$, and the equation becomes $\operatorname{Ric}_g+\nabla_g^2f=\lambda g$. These definitions explain the terminology; the supplied extraction does not specify whether the intended solitons are gradient solitons.

The retained source question asks: given a sequence of shrinking Ricci solitons whose underlying metric spaces converge in the Gromov--Hausdorff sense to a metric space $(X,d)$, characterize the possible geometric and singularity structure of $X$.

Unresolved formulation. This is a broad characterization question, not a determinate mathematical assertion as supplied. The record does not specify dimension, compactness or completeness, normalization of the shrinking constants, quantitative geometric bounds, or whether convergence is ordinary Gromov--Hausdorff convergence of compact spaces or pointed convergence of noncompact spaces. It also gives no definition of the regular and singular loci or a precise structural conclusion to be established.

The supplied statement-verification note says that the official contribution uses this broad question as motivation and then states a compactness and singular-locus theorem under relevant volume and diameter bounds. Those hypotheses are omitted from the extraction, and their exact form and constants are not supplied. The record explicitly says that the extraction preserves no determinate residual open claim and promotes no separate clean statement. Consequently, neither that theorem nor a precise remaining conjecture can be reconstructed from the supplied evidence.

Short English statement

The retained question asks which geometric and singularity structures can occur in Gromov--Hausdorff limits of shrinking Ricci solitons. Essential hypotheses and the intended conclusion are missing. The supplied verification describes a related theorem with volume and diameter bounds, but identifies no precise remaining open problem.

Sources

• Supplied local UnsolvedMath registry record, data/unsolvedmath/problems.json, permanent ID $30002087$, title "Shrinking-Soliton Degeneration Report," exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2012.
• Supplied report citation: Geometrie (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/27, pp. 1639--1686. Supplied source URL: https://doi.org/10.4171/owr/2012/27. This is the report reference supplied by the registry; its contents have not been independently consulted for this draft.
• Supplied literature source, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/27.
• Literature assessment dated 2026-08-21: the supplied curation labels the record open because the extracted question does not define the relevant class of metric spaces sufficiently to support a determinate solved-status assessment. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the contribution uses the broad limit question as motivation and states a compactness and singular-locus theorem under relevant volume and diameter bounds. The extraction omits those hypotheses and retains no determinate residual open claim. The supplied statement status is unrecoverable; no clean statement is supplied, and the original record is retained for provenance.