The Dirichlet Problem for Einstein Metrics on Cohomogeneity One Manifolds

TL;DR

This paper proves the existence of $G$-invariant Einstein metrics on cohomogeneity one manifolds with prescribed boundary metrics, under the assumption of inequivalent irreducible isotropy summands.

Contribution

It establishes a Dirichlet problem for Einstein metrics on cohomogeneity one manifolds with specific boundary conditions, assuming irreducible and inequivalent isotropy representations.

Findings

Existence of $G$-invariant Einstein metrics with prescribed boundary metrics.

Solution applies to manifolds with inequivalent irreducible isotropy summands.

Provides a method to solve the Einstein Dirichlet problem in this setting.

Abstract

Let $G/H$ be a compact homogeneous space, and let $\hat{g}_0$ and $\hat{g}_1$ be $G$-invariant Riemannian metrics on $G/H$. We consider the problem of finding a $G$-invariant Einstein metric $g$ on the manifold $G/H\times [0,1]$ subject to the constraint that $g$ restricted to $G/H\times \{0\}$ and $G/H\times \{1\}$ coincides with $\hat{g}_0$ and $\hat{g}_1$, respectively. By assuming that the isotropy representation of $G/H$ consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.