Title: Compact homogeneous Einstein manifolds and butterflies
Abstract: (Joint work with Christoph Böhm.) When a
Riemannian manifold (M,g) has a constant Ricci curvature Ric(g) =
λ g we say g is an Einstein metric. How plentiful are these?
We know of no topological obstructions when dimension is 5 or
above. We have a lot of examples, but no broad existence theorems.
Suppose our space is homogeneous, M = G/H. In this setting,
G-invariant Einstein metrics are precisely the critical points of
scalar curvature restricted to the space of G-invariant metrics. In
2004 C. Böhm defined a powerful tool for detecting Einstein metrics:
a simplicial complex ΔG/H, is based purely on Lie theoretic
algebraic data for H < G. This work was extended by M. Graev in
2012, with a slightly different topological object, the nerve
XG/H.
Nerve Theorem (Graev): Let G/H be a compact homogeneous space
with G,H connected. If the nerve XG/H is non-contractible, then
G/H admits a G-invariant Einstein metric.
The nerve XG/H (or simplicial complex
ΔG/H) detects when G-invariant Einstein metrics
must exist for global reasons, identifying non-contractibility of the
subset of the domain of the scalar curvature where S(g) is
sufficiently large. We will describe the nerve for some examples
XG/H, some applications, and some open questions on existence and
non-existence in the homogeneous setting.