A central theme in Riemannian geometry is the interplay between Einstein metrics of negative scalar curvature and symmetry. Böhm and Lafuente's proof of the Alekseevskii conjecture (JAMS 2023) showed that every homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space, via a rigidity theorem (Theorem G) for non-unimodular groups acting properly, isometrically and cocompactly with a single orbit type: the nilradical acts polarly and the manifold is foliated by Einstein-solvmanifold leaves. Their follow-up (JEMS 2025) extends the theory to arbitrary, possibly singular compact orbit spaces for unimodular symmetry groups, forcing an isometric splitting into a compact Einstein factor and a symmetric space of non-compact type, and isolates the remaining general case as an explicit conjecture. This problem records that conjecture: must every complete Einstein manifold with negative scalar curvature admitting a proper, isometric, cocompact action of an arbitrary connected Lie group acting nontrivially split isometrically as the product of a compact Einstein manifold and a non-compact homogeneous Einstein space? The open content lies with non-unimodular symmetry groups, where for smooth orbit spaces only the foliation conclusion of Theorem G is known and for singular orbit spaces nothing is known; the authors identify the failure of the maximum principle on the non-smooth orbit space as the central unresolved obstacle.

Background

A Riemannian manifold (Mn,g)(M^n,g) is called Einstein if its Ricci tensor satisfies ric(g)=λg\mathrm{ric}(g)=\lambda g for some constant λ∈R\lambda\in\mathbb{R}; here we are concerned with the case λ<0\lambda<0 of negative scalar curvature. By a classical theorem of Bochner — on a compact manifold of negative Ricci curvature every Killing field vanishes — a manifold of negative Ricci curvature admitting a nontrivial connected group of isometries is necessarily non-compact. A connected Lie group GG acts properly, isometrically and cocompactly on (M,g)(M,g) if the action preserves the metric, has compact orbit space M/GM/G, and is proper (equivalently, for isometric actions on complete manifolds, the image of GG is a closed subgroup of the full isometry group). In general the orbit space is only a metric space, with strata coming from singular and exceptional orbits; it is a smooth manifold when all orbits are principal, i.e. when the action has a single orbit type.

The starting point of this problem is the resolution of the Alekseevskii conjecture by Böhm and Lafuente: every homogeneous Einstein manifold with negative scalar curvature (homogeneous meaning that the isometry group acts transitively) is diffeomorphic to a Euclidean space. Their proof deliberately ignores homogeneity: they study negative Einstein manifolds admitting a proper, isometric, cocompact action of a connected Lie group GG with a single orbit type. Two of their results matter here. Theorem F: a unimodular GG (one whose left Haar measure is also right-invariant; nilpotent, semisimple and reductive groups are unimodular) satisfying these hypotheses must be non-compact semisimple. Theorem G: if GG is non-unimodular, then the nilradical NN of GG (its maximal connected nilpotent normal subgroup) acts polarly — meaning there is an immersed submanifold meeting every NN-orbit orthogonally — and MM is foliated into pairwise locally isometric, minimal Einstein submanifolds Edim⁡N+1E^{\dim N+1}, which when MM is simply-connected are Einstein solvmanifolds: simply-connected solvable Lie groups carrying a left-invariant Einstein metric. In the simply-connected case one also has a diffeomorphism M≃Edim⁡N+1×P′M\simeq E^{\dim N+1}\times P' with M/N≃R×P′M/N\simeq\mathbb{R}\times P' — a topological, not isometric, product. Combined with the authors' earlier theorem that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, this reduces the classification of non-compact homogeneous Einstein spaces to the algebraic problem of classifying nilsolitons — left-invariant Ricci soliton metrics, i.e. metrics with ric(g)=λg+LXg\mathrm{ric}(g)=\lambda g+\mathcal{L}_X g for a vector field XX, on nilpotent Lie groups — a reduction rooted in Heber's structure theory of non-compact homogeneous Einstein spaces and in Lauret's theorem that Einstein solvmanifolds are standard.

The single-orbit-type hypothesis makes the orbit space smooth, which is what allows the Einstein equation to be treated as an elliptic system on it. In a follow-up work Böhm and Lafuente removed the smoothness assumption for unimodular symmetry groups: if a complete Einstein manifold with negative scalar curvature admits a proper, isometric, cocompact action of a connected unimodular Lie group LL — with the orbit space now allowed to be a genuinely singular compact metric space — then LL is semisimple and MM splits isometrically as B×L/KB\times L/K, where BB is a compact Einstein manifold and L/KL/K is a symmetric space of non-compact type (a quotient of a semisimple Lie group by a maximal compact subgroup, such as hyperbolic space). The authors stress that the failure of the maximum principle at the singular strata of the orbit space is not just a technicality, and that they overcame it only in the unimodular case, by passing to the frame bundle, on which the lifted action has a smooth quotient. They formulate the desired general statement, with no unimodularity and no smoothness assumption, as their Conjecture 1.1; settling it is the subject of this problem.

Problem Statement

Prove or disprove Conjecture 1.1 of Böhm and Lafuente ('Non-compact Einstein manifolds with unimodular isometry group', J. Eur. Math. Soc. 2025):

Let (Mn,g)(M^n,g) be a complete Einstein manifold with negative scalar curvature admitting a proper, isometric, cocompact action of a connected Lie group GG, with no assumption that all orbits are principal or that the orbit space G\MG\backslash M is smooth. Then MM splits isometrically as a product M=B×HM=B\times H, where BB is a compact Einstein manifold and HH is a non-compact homogeneous Einstein space (both factors then necessarily carrying the same negative Einstein constant as gg; in the homogeneous extreme BB is a point and the statement holds by construction).

The conjecture is read, as its authors read it, within their standing framework of complete, non-compact Einstein manifolds with cocompact symmetries: the acting group is assumed to act nontrivially, i.e. to have positive-dimensional image in the isometry group, so that by Bochner's theorem MM, and hence GG, is non-compact — a point the authors themselves stress. A degenerate literal reading under which the trivial action of a connected Lie group on an arbitrary compact negative Einstein manifold satisfies the hypotheses (and the conclusion then fails for any such manifold, since a product with a non-compact factor is non-compact) is not the conjecture's intended content and lies outside this problem.

The statement is known when GG is unimodular — in the stronger form that HH is a symmetric space of non-compact type (Böhm–Lafuente 2025) — and trivially when GG acts transitively. The open content is concentrated in non-unimodular GG: for smooth orbit spaces, the foliation and topological-product conclusion of Theorem G (Böhm–Lafuente 2023) falls short of the asserted isometric splitting with a compact factor, and for singular orbit spaces no structure result at all is currently available in the non-unimodular case. A resolution must settle the conjecture as stated, for arbitrary connected Lie groups acting nontrivially and arbitrary compact orbit spaces.

References

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  2. Christoph Böhm and Ramiro A. Lafuente, Non-compact Einstein manifolds with unimodular isometry group, Journal of the European Mathematical Society, published online 6 May 2025. DOI: 10.4171/jems/1654. arXiv:2307.13235. https://arxiv.org/abs/2307.13235
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