The dynamical Alekseevskii conjecture of Böhm and Lafuente asserts that if a homogeneous Riemannian manifold admits an immortal homogeneous Ricci flow (a flow existing for all positive times), then its universal cover is diffeomorphic to Rn\mathbb{R}^n. Together with Lafuente's theorem that spaces covered by Rn\mathbb{R}^n always have immortal flows, it yields an all-or-nothing law for homogeneous Ricci flow, and it implies the classical Alekseevskii conjecture on the topology of negative homogeneous Einstein manifolds. The conjecture is verified for symmetric spaces, compact homogeneous spaces, spaces whose isometry group has a compact normal semisimple factor, awesome metrics with semisimple transitive group, dimensions at most five (Billon 2025), and stable U⋉θVU\ltimes_\theta V-spaces with θ\theta-adapted standard metrics (Araujo 2026). This problem asks for a proof or refutation in the remaining cases — in particular in dimensions six and above — together with the directly linked question whether the space of invariant positive-scalar-curvature metrics on an arbitrary homogeneous space is always empty or contractible without assuming the conjecture.

Background

The Ricci flow is the geometric evolution equation dgdt=−2 Ric(g)\frac{dg}{dt}=-2\,\mathrm{Ric}(g) on a Riemannian metric g(t)g(t), where Ric\mathrm{Ric} is the Ricci curvature. A solution is called immortal if it exists for all times t≥0t\ge 0. On a homogeneous space — a manifold M=G/HM=G/H on which a Lie group GG acts transitively with compact stabilizer HH — every GG-invariant metric stays invariant under the flow, so the PDE reduces to an autonomous ODE on the finite-dimensional space of invariant inner products; an equivalent and often more tractable form is the bracket flow on the Lie algebra (Lauret 2013).

Lafuente (2015) proved that if the universal cover of a homogeneous space is diffeomorphic to Rn\mathbb{R}^n, then every homogeneous Ricci flow on it is immortal. Böhm and Lafuente (2018) conjectured the converse, now called the dynamical Alekseevskii conjecture: if a homogeneous space has an immortal homogeneous Ricci flow, then its universal cover is diffeomorphic to Rn\mathbb{R}^n. The conjecture gives an all-or-nothing law: either the universal cover is Rn\mathbb{R}^n and all homogeneous flows are immortal, or all homogeneous flows develop a singularity in finite time. Böhm and Lafuente (2018) also proved that any sequence of parabolic blow-downs of an immortal homogeneous Ricci flow subconverges to a homogeneous expanding Ricci soliton (a self-similar solution with Ric=λg+LXg\mathrm{Ric}=\lambda g+\mathcal{L}_X g, λ<0\lambda<0), which constrains but does not settle the topology.

The conjecture is the dynamical analogue of the classical Alekseevskii conjecture (1970s): every simply connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to Rn\mathbb{R}^n. Indeed, on a negative-Einstein metric the Ricci flow only rescales the metric, g(t)=(1−2λt)g0g(t)=(1-2\lambda t)g_0, and is therefore immortal, so the dynamical conjecture implies the classical one — a famous open problem (known in low dimensions by Arroyo–Lafuente 2017, and for Einstein metrics on Euclidean spaces by Böhm–Lafuente 2022, which shows they are solvmanifolds, i.e. simply connected solvable Lie groups with a left-invariant metric).

Verified cases of the dynamical conjecture: symmetric spaces; compact homogeneous spaces and coverings thereof, where Böhm (2015) also established Type-I singularity formation (a singularity whose curvature blows up at a uniformly bounded rate, in the sense ∣Rm∣≤C/(T−t)|\mathrm{Rm}|\le C/(T-t), as the extinction time TT is approached); spaces whose isometry group is, up to covering, a product with a compact semisimple factor — the same work of Araujo (2024) that gives a structure theorem for isometry groups of immortal homogeneous Ricci flows also settles all dimensions ≤4\le 4; awesome homogeneous metrics (those making the nilradical of a semisimple transitive group orthogonal to a complement) by Araujo (2024); dimension five by Billon (2025), via a systematic classification of five-dimensional homogeneous spaces and explicit ODE analysis of the two exceptional spaces SO(3)⋉R3/SO(2)SO(3)\ltimes\mathbb{R}^3/SO(2) and SL(2,C)/U(1)SL(2,\mathbb{C})/U(1); and, most recently, stable U⋉θVU\ltimes_\theta V-spaces (compact Lie algebra u\mathfrak{u} acting on an abelian ideal VV through a representation θ\theta with closed conjugation orbit) equipped with θ\theta-adapted standard metrics, by Araujo (2026), in arbitrary dimension.

A closely linked topological question concerns the space MScal>0G\mathcal{M}^{G}_{\mathrm{Scal}>0} of GG-invariant metrics of positive scalar curvature on M=G/HM=G/H. Bérard-Bergery (1978) showed this space is empty whenever the universal cover is diffeomorphic to Rn\mathbb{R}^n. Billon (2025, Proposition 5.1) proved that, assuming the dynamical Alekseevskii conjecture, MScal>0G\mathcal{M}^{G}_{\mathrm{Scal}>0} is empty or contractible for every homogeneous space, via a Ricci-flow deformation retraction onto the set of metrics with scalar curvature ≥1\ge 1 (using Lafuente's scalar-curvature growth for homogeneous flows); unconditionally this contractibility is known only for the settled classes, e.g. for Araujo's stable U⋉θVU\ltimes_\theta V-spaces (Theorem B, 2026).

Problem Statement

Prove or refute the dynamical Alekseevskii conjecture of Böhm–Lafuente in the cases that remain open: for every homogeneous space whose universal cover is not diffeomorphic to Rn\mathbb{R}^n and which is not covered by the settled classes (symmetric spaces; compact homogeneous spaces and their covers; isometry group a product with a compact semisimple factor up to covering; awesome metrics with semisimple transitive group; dimensions ≤5\le 5; stable U⋉θVU\ltimes_\theta V-spaces with θ\theta-adapted standard metrics) — in particular for simply connected homogeneous spaces of dimension n≥6n\ge 6 not covered by those criteria — determine whether every homogeneous Ricci flow develops a singularity in finite time, as the conjecture predicts. A single explicit counterexample — a homogeneous space with non-Euclidean universal cover, in a settled class or not, carrying an invariant metric whose homogeneous Ricci flow is immortal — refutes the conjecture. Second, establish, or disprove, unconditionally — i.e. without assuming the conjecture — that for every homogeneous space M=G/HM=G/H the space MScal>0G\mathcal{M}^{G}_{\mathrm{Scal}>0} of GG-invariant metrics of positive scalar curvature is empty or contractible.

References

  1. Maher Billon, "The dynamical Alekseevskii conjecture in dimension five" (2025), DOI 10.48550/arxiv.2507.17781, https://arxiv.org/abs/2507.17781
  2. Christoph Böhm and Ramiro A. Lafuente, "Immortal homogeneous Ricci flows", Inventiones mathematicae 212 (2018), 461–529, DOI 10.1007/s00222-017-0771-z, https://arxiv.org/abs/1701.00628
  3. Christoph Böhm, "On the long time behavior of homogeneous Ricci flows", Commentarii Mathematici Helvetici 90 (2015), 543–571, DOI 10.4171/cmh/364, https://doi.org/10.4171/cmh/364
  4. Ramiro A. Lafuente, "Scalar Curvature Behavior of Homogeneous Ricci Flows", The Journal of Geometric Analysis 25 (2015), 2313–2322, DOI 10.1007/s12220-014-9514-1, https://doi.org/10.1007/s12220-014-9514-1
  5. Roberto Araujo, "On the Isometry Group of Immortal Homogeneous Ricci Flows", The Journal of Geometric Analysis 34 (2024), article 173, DOI 10.1007/s12220-024-01609-6, https://arxiv.org/abs/2310.18182
  6. Roberto Araujo, "Long-time behavior of awesome homogeneous Ricci flows", Pacific Journal of Mathematics 331 (2024), 187–215, DOI 10.2140/pjm.2024.331.187, https://arxiv.org/abs/2312.16517
  7. Roberto Araujo, "Finite extinction time of a family of homogeneous Ricci flows" (arXiv:2507.05097, v2 dated 24 Feb 2026), DOI 10.48550/arxiv.2507.05097, https://arxiv.org/abs/2507.05097
  8. Romina M. Arroyo and Ramiro A. Lafuente, "The Alekseevskii conjecture in low dimensions", Mathematische Annalen 367 (2017), 283–309, DOI 10.1007/s00208-016-1386-1, https://doi.org/10.1007/s00208-016-1386-1
  9. Lionel Bérard-Bergery, "Sur la courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes", Annales scientifiques de l'École Normale Supérieure 11 (1978), 543–576, DOI 10.24033/asens.1356, https://doi.org/10.24033/asens.1356
  10. Jorge Lauret, "Ricci flow of homogeneous manifolds", Mathematische Zeitschrift 274 (2013), 373–403, DOI 10.1007/s00209-012-1075-z, https://doi.org/10.1007/s00209-012-1075-z
  11. Christoph Böhm and Ramiro A. Lafuente, "Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds", Geometry & Topology 26 (2022), 899–936, DOI 10.2140/gt.2022.26.899, https://doi.org/10.2140/gt.2022.26.899