The Böhm–Wang–Ziller Finiteness Conjecture: finiteness of invariant Einstein metrics on compact homogeneous spaces

On a compact homogeneous space $\mathsf{G}/\mathsf{H}$ whose isotropy representation splits into $\ell$ pairwise inequivalent irreducible summands, every $\mathsf{G}$-invariant metric is diagonal and the Einstein equation $\operatorname{Ric}=\lambda\,\mathrm{g}$ reduces to a system of Laurent polynomial equations. Böhm, Wang, and Ziller (2004) proved that the moduli space of unit-volume invariant Einstein metrics has finitely many compact components, and conjectured that in this pairwise-inequivalent ('simple spectrum') case the algebraic Einstein equations have only finitely many positive real solutions — equivalently, the space carries only finitely many $\mathsf{G}$-invariant Einstein metrics up to homothety and isometry. The conjecture is confirmed for $\ell=2$ summands, for all generalized Wallach spaces, and whenever the principal $A$-determinant of the scalar curvature does not vanish (which yields the explicit Bernstein–Khovanskii–Kushnirenko upper bound $D_{\ell-1}$ of Bettiol–Friedman, 2025), but it remains open in general — most notably for higher-rank full flag manifolds and all compact homogeneous spaces with $\ell\ge3$ pairwise inequivalent summands outside the known special classes.

Background

A Riemannian metric $\mathrm{g}$ on a manifold $M$ is Einstein if its Ricci tensor satisfies $\operatorname{Ric}_{\mathrm{g}}=\lambda\,\mathrm{g}$ for a constant $\lambda$ (the Einstein constant). On a compact homogeneous space $M=\mathsf{G}/\mathsf{H}$ — a compact manifold on which a compact Lie group $\mathsf{G}$ acts transitively with stabilizer $\mathsf{H}$ — the Einstein equation, a nonlinear PDE in general, becomes algebraic: a $\mathsf{G}$-invariant metric and its Ricci tensor are determined by their value at one point, and, fixing $\lambda=1$ (positive by compactness, and up to homothety, i.e., rescaling of the metric), the equation $\operatorname{Ric}=\mathrm{g}$ is equivalent to a system of Laurent polynomial equations, i.e., polynomial equations allowing negative powers of the variables. Not every compact homogeneous space admits a $\mathsf{G}$-invariant Einstein metric (Wang–Ziller, 1986); the question here concerns the cardinality of the set of such metrics on the spaces that do.

Fix a bi-invariant inner product $Q$ on the Lie algebra $\mathfrak g$ and decompose the $Q$-orthogonal complement $\mathfrak m$ of $\mathfrak h$ into irreducible real representations of $\mathsf H$ (the isotropy representation): $\mathfrak m=\mathfrak m_1\oplus\dots\oplus\mathfrak m_\ell$. When the summands are pairwise inequivalent — as happens, in particular, whenever $\operatorname{rank}\mathsf{G}=\operatorname{rank}\mathsf{H}$, e.g., on full flag manifolds $\mathsf{G}/\mathsf{T}$ with $\mathsf{T}$ a maximal torus — Schur-type arguments force every $\mathsf{G}$-invariant metric to be diagonal, $\mathrm{g}=x_1Q|_{\mathfrak m_1}+\dots+x_\ell Q|_{\mathfrak m_\ell}$ with $x_i>0$, and $\operatorname{Ric}_{\mathrm{g}}=\mathrm{g}$ is equivalent to the system

$$f_i(\mathbf x)=\frac{b_i}{2x_i}-\frac{1}{4d_i}\sum_{j,k=1}^{\ell}L_{ijk}\,\frac{2x_k^{2}-x_i^{2}}{x_ix_jx_k}-1=0,\qquad 1\le i\le\ell,$$

in the variables $\mathbf x=(x_i)\in(\mathbb R_+)^\ell$, where $d_i=\dim\mathfrak m_i$, the $b_i$ come from the Cartan–Killing form of $\mathfrak g$, and $L=(L_{ijk})$ is a symmetric tensor of structure constants of the decomposition. Positive solutions correspond bijectively to $\mathsf{G}$-invariant Einstein metrics with $\lambda=1$; the finite group $\mathsf{N}(\mathsf{H})/\mathsf{H}$ (the gauge group, the normalizer of $\mathsf H$ in $\mathsf G$ modulo $\mathsf H$) permutes equal-dimensional summands and identifies isometric solutions.

Two facts delimit the problem. First, finiteness can genuinely fail without the pairwise-inequivalence hypothesis: there are compact homogeneous spaces with equivalent isotropy summands carrying positive-dimensional families of invariant Einstein metrics (Böhm–Wang–Ziller, Example 6.10, as reported by Bettiol–Friedman), although in the known examples these families are orbits of a positive-dimensional gauge group. Second, Böhm–Wang–Ziller (2004), via the variational approach to the total scalar curvature functional on the finite-dimensional space of unit-volume invariant metrics, proved that the moduli space of invariant Einstein metrics has finitely many components, each compact (stated as Theorem 1.2 in Jablonski's survey); Graev (2013) later gave an algebraic proof of the compactness statement under the simple-spectrum hypothesis using Newton polytopes (the convex hulls of the exponent vectors of a polynomial's monomials). Compactness does not imply finiteness — a compact algebraic set may have positive-dimensional components — and the Finiteness Conjecture of Böhm–Wang–Ziller asserts that in the pairwise-inequivalent case the solution set is nevertheless finite.

The recent algebraic-geometric attack of Bettiol–Friedman (2025) recasts the system in Bernstein–Khovanskii–Kushnirenko (BKK) theory: the mixed volume of the equations' Newton polytopes equals the central Delannoy number $D_{\ell-1}=\sum_{k=0}^{\ell-1}2^{k}\binom{\ell-1}{k}^{2}$, which bounds the number of isolated complex solutions, and degeneracy of the system is controlled by the principal $A$-determinant $E_A(\operatorname{scal})$ of the scalar curvature (the product of the $A$-discriminants of all faces of the Newton polytope, in the sense of Gelfand–Kapranov–Zelevinsky). Nonvanishing of $E_A(\operatorname{scal})$ at a space's actual parameters forces the generic BKK count and hence finiteness; but many geometric systems, including those of full flag manifolds, lie on the BKK discriminant, so the bound is not achieved there and genericity arguments alone cannot settle the conjecture. Finiteness is currently confirmed for $\ell=2$, for all generalized Wallach spaces (spaces with $\ell=3$ summands satisfying $[\mathfrak m_i,\mathfrak m_i]\subset\mathfrak h$; at most $4$ invariant Einstein metrics each), and for every space with $E_A(\operatorname{scal})\ne0$; the general case is open.

Problem Statement

Prove or refute the Finiteness Conjecture of Böhm–Wang–Ziller (2004): for every $\ell\ge1$ and every compact homogeneous space $M=\mathsf{G}/\mathsf{H}$ whose isotropy representation decomposes into pairwise inequivalent irreducible real summands $\mathfrak m=\mathfrak m_1\oplus\dots\oplus\mathfrak m_\ell$ (as happens, in particular, when $\operatorname{rank}\mathsf{G}=\operatorname{rank}\mathsf{H}$), the associated homogeneous Einstein system

$$\frac{b_i}{2x_i}-\frac{1}{4d_i}\sum_{j,k=1}^{\ell}L_{ijk}\,\frac{2x_k^{2}-x_i^{2}}{x_ix_jx_k}-1=0,\qquad 1\le i\le\ell,$$

has only finitely many real solutions in the positive orthant $(\mathbb R_+)^{\ell}$; equivalently, $\mathsf{G}/\mathsf{H}$ carries only finitely many $\mathsf{G}$-invariant Einstein metrics up to homothety and isometry. The conjecture fails for some compact homogeneous space in this class exactly when its associated Einstein system has infinitely many positive real solutions in $(\mathbb R_+)^{\ell}$.

References

1. Renato G. Bettiol and Hannah Friedman, Counting Homogeneous Einstein Metrics, arXiv:2509.09830 (2025), DOI 10.48550/arXiv.2509.09830, https://arxiv.org/abs/2509.09830
2. Christoph Böhm, McKenzie Wang, and Wolfgang Ziller, A variational approach for compact homogeneous Einstein manifolds, Geometric and Functional Analysis 14 (2004), no. 4, 681–733, DOI 10.1007/s00039-004-0471-x, https://doi.org/10.1007/s00039-004-0471-x
3. Michael Jablonski, Survey: Homogeneous Einstein Manifolds, arXiv:2111.09782 (2021), https://arxiv.org/abs/2111.09782 (published as Homogeneous Einstein manifolds, Revista de la Unión Matemática Argentina 64 (2023), no. 2, 461–485, DOI 10.33044/revuma.3588, https://doi.org/10.33044/revuma.3588)
4. Michail 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 (preprint arXiv:1207.3034, https://arxiv.org/abs/1207.3034)
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6. Mikhail R. Guzman, Einstein metrics on the full flag F(n), arXiv:2411.09833 (2024), https://arxiv.org/abs/2411.09833
7. McKenzie Y. Wang and 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
8. Zhiqi Chen and Yuriĭ G. Nikonorov, Invariant Einstein metrics on generalized Wallach spaces, Science China Mathematics 62 (2019), no. 3, 569–584, DOI 10.1007/s11425-017-9150-3, https://doi.org/10.1007/s11425-017-9150-3
9. Marina Statha, Invariant metrics on homogeneous spaces with equivalent isotropy summands, Toyama Mathematical Journal 38 (2016), 35–60, arXiv:1603.06528, https://arxiv.org/abs/1603.06528