Splitting of complete negative Einstein manifolds with cocompact connected Lie group symmetry: Conjecture 1.1 of Böhm–Lafuente without unimodularity

Böhm and Lafuente proved that a complete Einstein manifold with negative scalar curvature admitting a proper, isometric, cocompact action of a connected unimodular Lie group splits isometrically as a compact Einstein manifold times a symmetric space of non-compact type, and posed Conjecture 1.1: the same type of splitting (compact Einstein manifold times non-compact homogeneous Einstein manifold) should hold for actions of arbitrary connected Lie groups. Removing the unimodularity hypothesis is open: the weighted volume-density functional on the frame bundle, the cornerstone of the maximum-principle proof, is not currently definable for non-unimodular actions, and the orbit geometry in this regime (including whether non-product examples exist at all) is unknown. The problem is to prove or refute Conjecture 1.1 in the non-unimodular case.

Background

A Riemannian manifold $(M^n,g)$ is called Einstein if its Ricci tensor satisfies $\mathrm{Ric}_g=\lambda g$ for a constant $\lambda$. Outside the compact setting, no general structure theory of negative Einstein manifolds ($\lambda<0$) exists without symmetry assumptions, so a natural program is to understand those with 'cocompact symmetries': a connected Lie group $G$ acts properly (preimages of compact sets are compact — the analogue for continuous actions of a properly discontinuous group action), isometrically, and with compact orbit space $G\backslash M$. By a theorem of Bochner, the isometry group of a compact Riemannian manifold with negative Ricci curvature is finite, so a negative Einstein manifold admitting a nontrivial proper isometric action of a connected Lie group is necessarily non-compact. The orbit space is in general a non-smooth metric space, because $M$ contains singular and exceptional orbits; this non-smoothness is a central technical obstacle. A Lie group is unimodular if its left-invariant Haar measure is also right-invariant, equivalently $|\det(\mathrm{Ad}_g)|=1$ for all $g$; nilpotent, semisimple and reductive groups are unimodular, while many solvable groups (for instance the affine group of the line) are not. A symmetric space of non-compact type, written $L/K$ with $L$ semisimple and $K$ maximal compact, is a homogeneous space carrying canonical $L$-invariant metrics — the model examples of non-compact negative Einstein manifolds.

The transitive extreme of the program is the Alekseevskii conjecture (1975): every homogeneous Einstein manifold of negative scalar curvature is a simply connected solvmanifold, i.e. a solvable Lie group with a left-invariant metric, and hence diffeomorphic to $\mathbb{R}^n$; this was proved by Böhm and Lafuente. In arXiv:2307.13235 Böhm and Lafuente take the first step beyond transitivity: if a connected unimodular Lie group $L$ acts properly, isometrically and cocompactly on a complete negative Einstein manifold, then $L$ is semisimple and $M$ splits isometrically as $B\times L/K$, where $B$ is a compact Einstein manifold and $L/K$ is a symmetric space of non-compact type (Theorem A). In the Ricci-flat case they deduce from the Cheeger–Gromoll splitting theorem that the universal cover splits isometrically as $\bar{M}\times\mathbb{R}^k$ with $\bar{M}$ compact (Theorem B). The proof reduces the Einstein equation, via O'Neill's formulas for the Riemannian submersion (a submersion whose differential restricts to an isometry on the subspaces orthogonal to its fibers) onto the regular part of the orbit space, to a coupled elliptic system, and applies the maximum principle to functions associated with the orbits. Because the orbit space is not smooth, the argument is lifted to the orthonormal frame bundle (the bundle of orthonormal bases of the tangent spaces, on which the lifted group action is free and the quotient smooth); there, a key device is the $\beta^+$-weighted volume density, a weighted determinant of the orbit metrics — computed with respect to the pull-back of $g$, which is only positive semi-definite. At present this functional can be extended to such degenerate metrics only under unimodularity of the acting group, and this is precisely where the proof breaks down in general. The authors also use polar actions (actions whose horizontal distribution, orthogonal to the orbits, is integrable) and Lie-theoretic structure of semisimple groups (Iwasawa decompositions, minimal parabolic subgroups) to conclude that orbits are symmetric spaces. Note that compactness of the orbit space is genuinely necessary: the horospheres in real hyperbolic space $\mathbb{H}^n$ are the orbits of a proper isometric $\mathbb{R}^{n-1}$-action whose orbit space is $\mathbb{R}$, and there is no splitting.

Problem Statement

Decide Conjecture 1.1 of Böhm–Lafuente (arXiv:2307.13235) in the case where the acting group is not assumed to be unimodular.

Let $(M^n,g)$ be a complete Einstein manifold (that is, $\mathrm{Ric}_g=\lambda g$ for a constant $\lambda$) with negative scalar curvature ($\lambda<0$), which admits an action by a connected Lie group $G$ that is proper (preimages of compact sets under the map $G\times M\to M\times M$, $(g,x)\mapsto(x,g\cdot x)$, are compact), isometric, and cocompact (the orbit space $G\backslash M$ is compact). The group $G$ is arbitrary: it is not assumed to be unimodular (unimodular means the left-invariant Haar measure is also right-invariant, equivalently $|\det(\mathrm{Ad}_g)|=1$ for all $g\in G$), and it is not assumed to act transitively. Must $M$ split isometrically as a product

$$M \cong B \times H$$

where $B$ is a compact Einstein manifold and $H$ is a homogeneous Einstein manifold (a manifold on which some Lie group acts transitively by isometries, carrying an invariant Einstein metric) that is non-compact unless it reduces to a point; either factor is permitted to degenerate to a point?

An affirmative answer must establish the splitting for all such $(M,g,G)$. A negative answer must establish a counterexample: a complete negative Einstein manifold with a proper, isometric, cocompact action of a connected non-unimodular Lie group that admits no isometric splitting of the stated form. Note that compactness of the orbit space is part of the hypothesis and cannot be dropped (the horosphere action on real hyperbolic space shows this).

References

1. Christoph Böhm, Ramiro A. Lafuente, Non-compact Einstein manifolds with unimodular isometry group, arXiv:2307.13235 (2023); published in Journal of the European Mathematical Society (2025), DOI 10.4171/jems/1654. https://arxiv.org/abs/2307.13235
2. Christoph Böhm, Ramiro A. Lafuente, Non-compact Einstein manifolds with symmetry, Journal of the American Mathematical Society 36 (2023), no. 3, 591–651, DOI 10.1090/jams/1022, arXiv:2107.04210. https://arxiv.org/abs/2107.04210
3. Christoph Böhm, 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, arXiv:1811.12594. https://arxiv.org/abs/1811.12594
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