The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds

Abstract

Let be a compact homogeneous space, and let and be G-invariant Riemannian metrics on . We consider the problem of finding a G-invariant Einstein metric g on the manifold subject to the constraint that g restricted to and coincides with and , respectively. By assuming that the isotropy representation of consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.