On the compactness of the set of invariant Einstein metrics

Abstract

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$. We will assume that the isotropy $H$-module $\mathfrak {g/h}$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $N=N(G,H)$, which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling). Using the moment map, we identify the space $\mathcal{M}_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$. We associate with a point ${x \in \partial N}$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $\bar{\mathcal{M}_1}= N$. As an application of the Aleksevsky--Kimel'fel'd theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $T\subset \partial N$ of solutions at the boundary together with its natural triangulation. Investigating the compactification $\bar{\mathcal{M}_1} $ of $\mathcal{M}_1$, we get an algebraic proof of the deep result by B\"ohm, Wang and Ziller about the compactness of the set $ \mathcal{E}_1 \subset \mathcal{M}_1$ of Einstein metrics. The original proof by B\"ohm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix we consider the non-symmetric K\"ahler homogeneous spaces $G/H$ with the second Betti number $b_2=1$. We write the normalized volumes $2,6,20,82,344$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.Comment: 25 pages, 4 figures. Some proofs, 3 references, and Appendix adde

Einstein equations for invariant metrics on flag spaces and their Newton polytopes

“…Under a restriction, we prove that Z d /L = (Z/2Z) n , where n = rank(Ω) (three proofs are given), and deduce the following result (mentioned without proof in [10]): We describe the vertices, the lattice L ⊂ Z d generated by the vertices, and the integer points p ∈ P ∩ Z d and p ∈ L ∩ P .…”

Cohomogeneity-one quasi-Einstein metrics

“…Firstly, µ = ∇R is a diffeomorphism from R −1 (1) to ∆, where ∆ ⊂ R n is a convex polytope depending on the homogeneous space G/H. This is Theorem 1 of [19], but note that the result is essentially a consequence of an un-numbered lemma in Section 4.2 of [15]. With this diffeomorphism, we can now treat r as a function from ∆ to R n , and it suffices to demonstrate that inf x∈∆ |r(x)| > 0.…”

“…This assumption is referred to as the monotypic condition, and it has appeared in, for example, [12], [20] and [19]. We search for solutions (g, u) of (1.1) such that u is a G-invariant function and g is a G-invariant Riemannian metric on G/H × (0, 1) having the form…”

“…This is because of the scaling properties of r and R. To establish (3.1), we make extensive use of results of [19]. Firstly, µ = ∇R is a diffeomorphism from R −1 (1) to ∆, where ∆ ⊂ R n is a convex polytope depending on the homogeneous space G/H.…”

“…With this diffeomorphism, we can now treat r as a function from ∆ to R n , and it suffices to demonstrate that inf x∈∆ |r(x)| > 0. Sections 4 and 5 of [19] explain how r can be extended continuously to Γ = ∂∆. This is done using contractions of Lie algebras.…”