Finiteness of the moduli space of invariant Einstein metrics on a compact homogeneous space

For a compact homogeneous space $M=G/K$ with $M$ connected and finite fundamental group, the space of $G$ -invariant unit-volume Einstein metrics modulo the natural action of $\mathrm{Aut}(G/K)$ is known to be compact (Böhm–Wang–Ziller), and Böhm, Wang and Ziller conjectured it is finite, at least when the isotropy representation is multiplicity-free. Finiteness is equivalent to every $G$ -invariant Einstein metric being $G$ -rigid, i.e. isolated in the moduli space. Lauret's 2021 study of the stability of homogeneous Einstein manifolds re-flagged this as the main long-standing open question of the subject; it remains open in general, even for the full flag manifolds $\mathrm{SU}(n)/T$ with $n$ large, and no counterexample (a compact homogeneous space carrying an infinite family of pairwise inequivalent invariant Einstein metrics) is known. A 2025 preprint of Bettiol–Friedman confirms the conjecture on all generalized Wallach spaces via a combinatorial upper bound, underscoring that the general case is still unresolved.

Background

A Riemannian metric $g$ is Einstein when its Ricci tensor satisfies $\mathrm{Rc}(g)=\rho\,g$ for a constant $\rho$ . On a compact homogeneous space $M=G/K$ — a connected manifold on which a compact Lie group $G$ acts transitively, $K$ being the subgroup fixing a point — one studies the $G$ -invariant metrics, which form a finite-dimensional manifold $\mathcal{M}^G$ described entirely by linear algebra on the isotropy representation $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ , where $\mathfrak{g}$ is the Lie algebra of $G$ , $\mathfrak{k}$ that of $K$ , and $\mathfrak{p}$ is a complementary subspace on which $K$ acts. The subspace $\mathcal{M}^G_1$ consists of metrics of unit volume, and $\mathcal{E}^G_1\subset\mathcal{M}^G_1$ denotes the $G$ -invariant unit-volume Einstein metrics, a subset defined by an explicit finite system of algebraic equations in the metric parameters and the structure constants. The group $\mathrm{Aut}(G/K)$ of automorphisms of the homogeneous space acts on $\mathcal{M}^G$ , and the moduli space of invariant Einstein metrics is the quotient $\overline{\mathcal{E}}^G=\mathcal{E}^G_1/\mathrm{Aut}(G/K)$ ; a metric is called $G$ -rigid when its class is an isolated point of this quotient. Böhm, Wang and Ziller (2004) proved that, when $M$ has finite fundamental group, each connected component of $\mathcal{E}^G_1$ is compact and that only finitely many Einstein constants occur, so $\overline{\mathcal{E}}^G$ is compact; hence its finiteness is equivalent to the $G$ -rigidity of every $G$ -invariant Einstein metric on $M$ . They conjectured finiteness when the isotropy representation of $K$ on $\mathfrak{p}$ is multiplicity-free, i.e. decomposes into pairwise non-isomorphic irreducible summands, the case in which the infinitesimal deformation space of invariant metrics is as small as possible. Finiteness is known only in special classes: spaces whose isotropy representation has at most two summands (Wang–Ziller); the naturally reductive metrics of D'Atri–Ziller (metrics arising from $\mathrm{Ad}(K)$ -invariant inner products for which every geodesic is the orbit of a one-parameter subgroup of $G$ ); and generalized Wallach spaces (spaces whose isotropy complement splits into three $\mathrm{Ad}(K)$ -invariant summands whose pairwise brackets lie in the isotropy algebra). It is open already for generalized flag manifolds, notably the full flag manifold $\mathrm{SU}(n)/T$ ( $T$ a maximal torus, i.e. a maximal connected abelian subgroup of $\mathrm{SU}(n)$ ) for large $n$ . The question connects to the variational picture initiated by Hilbert: the invariant Einstein metrics are precisely the critical points of the total scalar curvature (the Hilbert action) restricted to $\mathcal{M}^G_1$ , so the finiteness question asks whether this finite-dimensional variational problem can have a continuum of geometrically distinct critical points, and it governs the long-time behavior of the homogeneous Ricci flow (the Ricci flow restricted to the finite-dimensional family of $G$ -invariant metrics), whose fixed points are exactly these metrics. Lauret (2021) computed the second variation of this functional via the Lichnerowicz Laplacian (the natural second-order elliptic operator on symmetric 2-tensors) acting on $G$ -invariant transverse-traceless tensors (trace-free, divergence-free symmetric 2-tensors), determining stability types in many classes, and recorded the finiteness of $\overline{\mathcal{E}}^G$ as the main long-standing open question of the subject; the same paper notes that on the compact simple groups $\mathrm{SU}(n)$ the Killing metric (the bi-invariant metric induced by the Killing form $\mathrm{tr}(\mathrm{ad}X\,\mathrm{ad}Y)$ ) is $G$ -rigid (Derdzinski–Gal) yet the general finiteness question is untouched. Most recently, Bettiol–Friedman (2025) gave an explicit combinatorial upper bound (a Bernstein–Kouchnirenko–Khovanskii polytope volume) on the number of isolated invariant Einstein metrics on multiplicity-free spaces and confirmed the conjecture on all generalized Wallach spaces, confirming that the general case remains open.

Problem Statement

For every compact homogeneous space $M=G/K$ (with $G$ a compact Lie group and $M$ connected with finite fundamental group), is the moduli space $\overline{\mathcal{E}}^G=\mathcal{E}^G_1/\mathrm{Aut}(G/K)$ of $G$ -invariant unit-volume Einstein metrics finite? By the compactness theorem of Böhm–Wang–Ziller this is equivalent to: is every $G$ -invariant Einstein metric on $M$ $G$ -rigid, i.e. an isolated point of the moduli space? Decide the question in full generality — either by a proof that $\overline{\mathcal{E}}^G$ is finite for every such $M$ (as conjectured by Böhm–Wang–Ziller at least in the multiplicity-free case), or by an explicit counterexample: a compact homogeneous space together with an infinite family of pairwise non-equivalent (modulo $\mathrm{Aut}(G/K)$ and homothety) $G$ -invariant unit-volume Einstein metrics.

References

Jorge Lauret, "On the stability of homogeneous Einstein manifolds", Asian Journal of Mathematics 26 (2022), no. 4, 555-584. DOI: 10.4310/ajm.2022.v26.n4.a3, https://doi.org/10.4310/ajm.2022.v26.n4.a3 (also arXiv:2105.06336, https://arxiv.org/abs/2105.06336)
Christoph Böhm, McKenzie Y. Wang, Wolfgang Ziller, "A variational approach for compact homogeneous Einstein manifolds", Geometric and Functional Analysis 14 (2004), no. 4. DOI: 10.1007/s00039-004-0471-x, https://doi.org/10.1007/s00039-004-0471-x
McKenzie Y. Wang, Wolfgang Ziller, "Existence and non-existence of homogeneous Einstein metrics", Inventiones Mathematicae 84 (1986), no. 1, 177-194. DOI: 10.1007/BF01388738, https://doi.org/10.1007/BF01388738
Christoph Böhm, "Homogeneous Einstein metrics and simplicial complexes", Journal of Differential Geometry 67 (2004), no. 1. DOI: 10.4310/jdg/1099587730, https://doi.org/10.4310/jdg/1099587730
J. E. D'Atri, W. Ziller, "Naturally reductive metrics and Einstein metrics on compact Lie groups", Memoirs of the American Mathematical Society 18 (1979), no. 215. DOI: 10.1090/memo/0215, https://doi.org/10.1090/memo/0215
Andrzej Derdzinski, Swiatosław R. Gal, "Indefinite Einstein metrics on simple Lie groups", Indiana University Mathematics Journal 63 (2014), no. 1, 165-212. DOI: 10.1512/iumj.2014.63.5191, https://doi.org/10.1512/iumj.2014.63.5191
Jorge Lauret, Cynthia E. Will, "On the stability of homogeneous Einstein manifolds II", Journal of the London Mathematical Society 106 (2022), no. 4, 3638-3669. DOI: 10.1112/jlms.12669, https://doi.org/10.1112/jlms.12669
Jorge Lauret, Cynthia E. Will, "Homogeneous Einstein metrics and local maxima of the Hilbert action", Journal of Geometry and Physics 178 (2022), 104544. DOI: 10.1016/j.geomphys.2022.104544, https://doi.org/10.1016/j.geomphys.2022.104544
Paul Schwahn, "The Lichnerowicz Laplacian on normal homogeneous spaces", Journal für die reine und angewandte Mathematik (Crelle's Journal) 814 (2024), 91-115. DOI: 10.1515/crelle-2024-0038, https://doi.org/10.1515/crelle-2024-0038
Mikhail R. Guzman, "Einstein metrics on the full flag $F(n)$ ", preprint (2024). arXiv:2411.09833, https://arxiv.org/abs/2411.09833
Anusha M. Krishnan, Vincent P. N. Wolff, Masoumeh Zarei, "Precompactness for homogeneous Einstein metrics on torus bundles", preprint (2026). arXiv:2606.27329, https://arxiv.org/abs/2606.27329
Renato G. Bettiol, Hannah Friedman, "Counting Homogeneous Einstein Metrics", preprint (2025). arXiv:2509.09830, https://arxiv.org/abs/2509.09830