Which compact homogeneous spaces admit an invariant Einstein metric: completing the topological existence criteria of Wang–Ziller, Böhm, and Graev

Determining which compact homogeneous spaces $G/H$ admit a $G$-invariant Einstein metric is a major open problem of homogeneous Riemannian geometry. The only general sufficient conditions known are topological: disconnection of the Wang–Ziller graph, non-contractibility of Böhm's simplicial complex $\Delta_{G/H}$, and non-contractibility of Graev's nerve $X_{G/H}$, a compact semialgebraic set built from flags of non-toral intermediate subalgebras. This record concerns the open questions originating in Graev's paper 'The existence of invariant Einstein metrics on a compact homogeneous space' (Trans. Moscow Math. Soc. 73 (2012), 1–28). Two of the limitations recorded there — compactness of the defining family $\mathcal{K}$ and existence of a canonical triangulation of the polyhedron $|\mathcal{K}|$ — are discharged in all cases of interest and are not treated as open by the later literature; the surviving open core is: whether the simplicial-complex criterion is subsumed by the nerve criterion, whether the three topological criteria can be completed to a necessary-and-sufficient Lie-theoretic characterization of existence, and the existence question for classes of spaces on which all three criteria fail. This core is explicitly confirmed as open by Böhm–Kerr (2023) and Lauret–Will (2025), and the 2025–2026 continuation of the aligned-space program settles further subclasses without resolving it.

Background

A Riemannian metric $g$ is Einstein if its Ricci tensor satisfies $\mathrm{ric}(g)=\lambda g$ for a constant $\lambda$. On a compact manifold, Einstein metrics of unit volume are the critical points of the total scalar curvature (the Hilbert action), so existence questions become variational questions. For a compact homogeneous space $M=G/H$ — where $G$ is a compact Lie group acting transitively on $M$ and $H$ is a closed subgroup — the space $\mathcal{M}^G_1$ of unit-volume $G$-invariant metrics is a finite-dimensional manifold, and the Einstein equation becomes a system of algebraic equations on it; nevertheless, no general existence theory is known. By a theorem of Bochner, a $G$-invariant Einstein metric on a compact homogeneous space has $\lambda\ge 0$; $\lambda=0$ forces the metric to be flat, which happens exactly when $G/H$ is a torus, and $\lambda>0$ forces finite fundamental group (Myers). Wang and Ziller (1986) obtained the first non-existence results (e.g., compact homogeneous spaces other than tori with infinite fundamental group admit none) and the first global existence criterion; Böhm (2005) developed the main general non-existence technique via curvature assumptions on all invariant metrics.

All currently known general sufficient conditions are expressed through the intermediate subgroups (equivalently, intermediate Lie subalgebras $\mathfrak{k}$ with $\mathfrak{h}<\mathfrak{k}<\mathfrak{g}$). An intermediate subalgebra is called toral if it is an abelian extension of $\mathfrak{h}$; geometrically, if $\mathfrak{k}$ is the Lie algebra of a subgroup $K$, non-toral means the fibre $K/H$ of the fibration $K/H\to G/H\to G/K$ carries an invariant metric of positive scalar curvature, while toral fibres are flat tori. The three criteria are: (1) the Graph Theorem of Böhm–Wang–Ziller (2004), refining Wang–Ziller — if the graph of intermediate subgroups is disconnected, an invariant Einstein metric exists (originally under a semisimplicity assumption on $G$); (2) Böhm's theorem (2004) — if $G/H$ has finite fundamental group and the simplicial complex $\Delta_{G/H}$ is non-contractible, an invariant Einstein metric exists, where $\Delta_{G/H}$ is the flag complex (a simplicial complex whose simplices are the chains under inclusion) of connected intermediate subgroups containing $AH$, with $A$ a maximal torus of a complement of $H$ in the normalizer $N_G(H)$; (3) Graev's theorem (2012) — if the nerve $X_{G/H}$ is non-contractible, an invariant Einstein metric exists, where $X_{G/H}$ is the union, over all flags of non-toral intermediate subalgebras, of the simplices spanned by the corresponding orthogonal projections in $\mathrm{End}(\mathfrak{g})$; this nerve is always a compact semialgebraic set (a set defined by polynomial equalities and inequalities), and is non-contractible by convention when it is empty. Graev proved this by generalizing Böhm's deformation-retraction machinery — elementary contractible sets called 'butterflies' — from finite to compact semialgebraic families of subalgebras, and he extended the graph criterion to groups with non-trivial center via graphs $B_{WZ}$ and $B^{\min}_{WZ}$, whose vertices are the orbits, under the identity component of $N_G(H)$, of the non-toral 'almost semisimple' intermediate subalgebras $\mathfrak{l}=[\mathfrak{l},\mathfrak{l}]+\mathfrak{h}$ (for $B^{\min}_{WZ}$, the subalgebras spanned by finite sums of minimal such), two vertices being joined by an edge when the corresponding subalgebras are comparable; disconnection of either graph implies existence for arbitrary center $\mathfrak{z}(\mathfrak{g})$.

The state of the art: Böhm–Kerr (2006) classified low dimensions, showing every compact simply-connected homogeneous space of dimension at most 11 admits a homogeneous Einstein metric, a result optimal due to Wang–Ziller's examples. Böhm–Kerr (2023) gave the authoritative exposition of Graev's work (their paper is dedicated to Graev's memory) and posed the central open problems of the field. Lauret–Will (2025) settled the existence problem for the class of 'aligned' spaces — compact spaces $G/K$ whose third Betti number is maximal, $b_3(G/K)=s-1$ when $G$ has $s$ simple factors — in the two-factor case $G_1\times G_2/K$ with three pairwise inequivalent isotropy summands (the irreducible subspaces of the isotropy representation of $K$ on $\mathfrak{g}/\mathfrak{k}$), where the graph is always connected and both Böhm's complex and Graev's nerve are contractible, showing existence there is equivalent to a real root of an explicit quartic polynomial; this demonstrates concretely that the known criteria are strictly insufficient for any characterization of existence. Graev's own continuation, 'On the compactness of the set of invariant Einstein metrics' (2013), and recent counting results of Bettiol–Friedman (2025) address adjacent finiteness questions, not the characterization; Lauret–Will's continuation (2026) extends the Ricci-curvature computation to aligned spaces with many simple factors, enlarging the decided classes (including new non-existence families) without approaching a general criterion.

Problem Statement

Let $M=G/H$ be a compact homogeneous space, where $G$ is a compact connected Lie group and $H$ a connected closed subgroup. Recall the three attached objects: the graph of intermediate subgroups (whose disconnection implies existence of a $G$-invariant Einstein metric, by the Graph Theorem of Böhm–Wang–Ziller as extended by Graev to groups with non-trivial center); Böhm's simplicial complex $\Delta_{G/H}$, the flag complex of connected intermediate subgroups containing $AH$, where $A$ is a maximal torus of a complement of $H$ in $N_G(H)$; and Graev's nerve $X_{G/H}$, the compact semialgebraic union of the simplices spanned by the projections corresponding to flags of non-toral intermediate subalgebras, with the empty nerve non-contractible by convention. Non-contractibility of $X_{G/H}$ implies the existence of a $G$-invariant Einstein metric on $G/H$ (Graev's Theorem A); non-contractibility of $\Delta_{G/H}$ implies the same conclusion when $G/H$ has finite fundamental group (Böhm's Theorem B).

The open problem, with the single common objective of turning these sufficient conditions into a genuine characterization of existence, has three linked components:

(i) Prove or refute: for every compact homogeneous space $G/H$ as above, non-contractibility of $\Delta_{G/H}$ implies non-contractibility of $X_{G/H}$; that is, decide whether Böhm's Theorem B is subsumed by Graev's Theorem A. (Böhm–Kerr (2023) state that this 'is still open'; the reverse gap is known to occur, i.e., there are spaces with non-contractible nerve but contractible complex.)

(ii) Determine whether these criteria — the graph, the simplicial complex, and the nerve — can be completed to a necessary and sufficient Lie-theoretic condition for the existence of a $G$-invariant Einstein metric, given that there are compact homogeneous spaces with connected graph and contractible complex and nerve which nevertheless admit invariant Einstein metrics (for example, the aligned spaces classified by Lauret–Will).

(iii) For classes of compact homogeneous spaces on which all three known criteria fail — for instance, aligned spaces $G_1\times G_2/K$ beyond the subclasses settled by Lauret–Will (the three-summand spaces classified in 2025 and the further aligned families treated in 2026), or homogeneous spaces of compact non-simple Lie groups — decide existence or non-existence of invariant Einstein metrics.

The problem as a whole asks for a condition on the pair $(\mathfrak{g},\mathfrak{h})$, stated purely in Lie-theoretic and topological terms, that is necessary and sufficient for the existence of a $G$-invariant Einstein metric, and proved in both directions. The three components (i)–(iii) are independent sub-questions.

References

1. M. M. Graev, The existence of invariant Einstein metrics on a compact homogeneous space, Transactions of the Moscow Mathematical Society 73 (2012), 1–28. DOI: 10.1090/s0077-1554-2013-00202-2; arXiv:1110.3839. https://arxiv.org/abs/1110.3839
2. Christoph Böhm and Megan M. Kerr, Homogeneous Einstein metrics and butterflies, Annals of Global Analysis and Geometry 63 (2023), no. 4, article 29. DOI: 10.1007/s10455-023-09905-0; arXiv:2107.06609. https://arxiv.org/abs/2107.06609
3. Jorge Lauret and Cynthia Will, Einstein metrics on aligned homogeneous spaces with two factors, Journal of the London Mathematical Society 111 (2025), no. 3, article e70120. DOI: 10.1112/jlms.70120; arXiv:2408.00456. https://arxiv.org/abs/2408.00456
4. Christoph Böhm, Homogeneous Einstein metrics and simplicial complexes, Journal of Differential Geometry 67 (2004), 79–165. DOI: 10.4310/jdg/1099587730. https://doi.org/10.4310/jdg/1099587730
5. Christoph Böhm, McKenzie Wang and Wolfgang Ziller, A variational approach for compact homogeneous Einstein manifolds, Geometric and Functional Analysis 14 (2004), 681–733. DOI: 10.1007/s00039-004-0471-x. https://doi.org/10.1007/s00039-004-0471-x
6. McKenzie Y. Wang and Wolfgang Ziller, Existence and non-existence of homogeneous Einstein metrics, Inventiones Mathematicae 84 (1986), 177–194. DOI: 10.1007/BF01388738. https://doi.org/10.1007/BF01388738
7. Christoph Böhm, Non-existence of homogeneous Einstein metrics, Commentarii Mathematici Helvetici 80 (2005), 123–146. DOI: 10.4171/cmh/8. https://doi.org/10.4171/cmh/8
8. Christoph Böhm and Megan M. Kerr, Low-dimensional homogeneous Einstein manifolds, Transactions of the American Mathematical Society 358 (2006), 1455–1468. DOI: 10.1090/S0002-9947-05-04096-1. https://doi.org/10.1090/S0002-9947-05-04096-1
9. M. M. Graev, On the compactness of the set of invariant Einstein metrics, Annals of Global Analysis and Geometry 44 (2013), 471–500. DOI: 10.1007/s10455-013-9377-x. https://doi.org/10.1007/s10455-013-9377-x
10. Renato G. Bettiol and Hannah Friedman, Counting Homogeneous Einstein Metrics, preprint (2025), arXiv:2509.09830. https://arxiv.org/abs/2509.09830
11. Jorge Lauret and Cynthia Will, Einstein metrics on homogeneous spaces $H\times H/\Delta K$, Communications in Contemporary Mathematics 27 (2025), no. 9. DOI: 10.1142/s0219199725500105; arXiv:2402.13407. https://arxiv.org/abs/2402.13407
12. Jorge Lauret and Cynthia Will, Ricci curvature and Einstein metrics on aligned homogeneous spaces, The Journal of Geometric Analysis 36 (2026), no. 9, article 296. DOI: 10.1007/s12220-026-02544-4; arXiv:2410.13058. https://arxiv.org/abs/2410.13058