Long-Time Asymptotics of Homogeneous Ricci Flows

Definitions and mathematical statement

Let $M$ be a smooth manifold and let $g(t)$, $t\in[0,\infty)$, be a smooth family of Riemannian metrics satisfying the Ricci flow equation \[ \frac{\partial g(t)}{\partial t}=-2\operatorname{Ric}(g(t)). \] The flow is called immortal because it exists for every nonnegative time. Homogeneity means that, for each $t$, the isometry group of $(M,g(t))$ acts transitively on $M$.

Problem. For every such immortal homogeneous Ricci flow, determine the asymptotic behavior of $g(t)$ as $t\to\infty$.

Unresolved qualifications. This is an asymptotic classification question rather than a specified convergence conjecture. The supplied statement does not specify a normalization, a topology of convergence, whether to identify metrics by diffeomorphisms, or whether limits are to be pointed, subsequential, or taken along the entire flow. It also states no explicit dimension, compactness, connectedness, completeness, or curvature hypotheses. These omissions prevent formulation of a unique, more specific limit assertion from the supplied evidence alone.

The supplied dated assessment reports convergence of parabolic blow-down subsequences to expanding solitons and says that a unique full asymptotic description is not known in general. This assessment does not specify the hypotheses or convergence framework of the cited results; it is retained as supplied context and is not asserted here as an independently verified theorem.

Short English statement

Determine the long-time geometry of every homogeneous Ricci flow that exists for all $t\geq0$. The source leaves the normalization and meaning of asymptotic convergence unspecified. Its supplied status is partially solved, subject to the dated, unverified literature assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003151$, titled Long-Time Asymptotics of Homogeneous Ricci Flows; collection: Oberwolfach Reports Open Problems; exact source code: catalogue entry. The record gives proposed year 2016 and names no proposer.
• Supplied report citation: Geometrie (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/28, pp. 1577--1634. https://doi.org/10.4171/owr/2016/28. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Bohm--Lafuente. https://arxiv.org/abs/1701.00628.
• Supplied literature reference: Commentarii Mathematici Helvetici. https://doi.org/10.4171/CMH/364.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, attributes expanding-soliton limits of parabolic blow-down subsequences of immortal homogeneous Ricci flows to Bohm and Lafuente, and states that a unique full asymptotic description is not known in general. This is supplied dated curation derived from the OWR record, not independently verified; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified, with the qualification that only safe notation, typography, and TeX-delimiter cleanup was applied and that all hypotheses, quantifiers, and subquestions were preserved. This label does not establish mathematical correctness or independently verify the literature assessment.