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/Htimes [0,1]) subject to the constraint that g restricted to (G{/}Htimes {0}) and (G/Htimes {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.