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 $T$ 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.

16 pages

References in corpus (3)

• Gradient estimates for the heat equation under the Ricci flow
• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• Unique continuation results for Ricci curvature and applications