The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds

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.