Homogeneous Einstein Metrics and Butterflies, part I

Abstract: Given a compact homogeneous space $G/H$, we can associate a "nerve" $X_{G/H}$, first introduced by M. M. Graev in 2012. The nerve $X_{G/H}$ is a purely Lie theoretical compact semi-algebraic set determined by the intermediate subgroups $K$ between $H$ and $G$.

Theorem [Gra] Let $G/H$ be a compact homogeneous space with $G,H$ connected. If the nerve $X_{G/H}$ is non-contractible, then $G/H$ admits a $G$-invariant Einstein metric.

The nerve $X_{G/H}$ of a compact homogeneous space $G/H$ can be described as follows: To each intermediate subalgebra $\mathfrak{k}$ with $\mathfrak{h}< \mathfrak{k} <\mathfrak{g}$ we associate a (self-adjoint) projection map $P$ on $\mathfrak{g}$ with $P^2=P$ and $\ker (P)=\mathfrak{k}$. Now for each flag $\varphi = (\mathfrak{k}_1<\cdots < \mathfrak{k}_r)$ of non-toral subalgebras (i.e., $\mathfrak{k}$ is not an abelian extension of $\mathfrak{h}$), we define the simplex $\Delta_\varphi^P$ as the convex hull of the corresponding projections on $\mathfrak{g}$. The nerve $X_{G/H}$ is the union of all such simplices. We will sketch the proof in the nerve $X_{G/H}$ framework.

This is joint work with Christoph Böhm.

differential geometry

Audience: researchers in the topic

Workshop on compact homogeneous Einstein manifolds

Series comments: The workshop aims at bringing together mathematicians that have contributed and are contributing to the study of Einstein metrics on the challenging class of all compact homogeneous spaces.