The Dirichlet Problem for Einstein Metrics on Cohomogeneity One Manifolds

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