Existence of homogeneous Einstein metrics: a simplicial
complex via butterflies
Abstract: We consider the space of metrics of volume one on a fixed compact homogeneous space M = G/H with G and H compact. If the isotropy representation of H is irreducible, there is only one G-invariant metric, up to scaling, and it must be Einstein. If, more generally, H is maximal in G, Wang and Ziller proved there must be at least one G-invariant Einstein metric (1986). Indeed, the space of volume one G-invariant metrics can be described by a simplicial complex; its structure is governed by the set of all possible flags of subgroups of G containing the isotropy subgroup H. Böhm proved that when the associated simplicial complex is non-contractible, there must exist a G-invariant Einstein metric on M (2004). We will explore the relationship between the geometry of M=G/H and its algebraic structure. To describe the simplicial complex, we will introduce the notion of butterflies, due to Graev (2012). This project is joint with Christoph Böhm and Luigi Verdiani.