The Dirichlet problem for the prescribed Ricci curvature equation on cohomogeneity one manifolds

Abstract

Let M be a domain enclosed between two principal orbits on a cohomogeneity one manifold $$M_1$$ . Suppose that T and R are symmetric invariant (0, 2)-tensor fields on M and $$\partial M$$ , respectively. The paper studies the prescribed Ricci curvature equation $${{\mathrm {Ric}}}(G)=T$$ for a Riemannian metric G on M subject to the boundary condition $$G_{\partial M}=R$$ (the notation $$G_{\partial M}$$ here stands for the metric induced by G on $$\partial M$$ ). Imposing a standard assumption on $$M_1$$ , we describe a set of requirements on T and R that guarantee global and local solvability.