The Dirichlet Problem for the Prescribed Ricci Curvature Equation on Cohomogeneity One Manifolds

Let $M$ be a domain enclosed between two principal orbits on a cohomogeneity one manifold $M_1$. Suppose $T$ and $R$ are symmetric invariant (0,2)-tensor fields on $M$ and $\mathrm{\partial }M$, respectively. The paper studies the prescribed Ricci curvature equation $\mathrm{R}\mathrm{i}\mathrm{c}(G)=T$ for a Riemannian metric $G$ on $M$ subject to the boundary condition $G_{\mathrm{\partial }M}=R$ (the notation $G_{\partial M}$herestandsforthemetricinducedby$G$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.