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

TL;DR

This paper investigates the boundary value problem for the prescribed Ricci curvature equation on cohomogeneity one manifolds, providing conditions for the existence of solutions under symmetry and boundary constraints.

Contribution

It establishes new criteria ensuring both local and global solvability of the prescribed Ricci curvature equation on cohomogeneity one manifolds with boundary.

Findings

Derived conditions for solvability of the Ricci curvature equation

Established criteria for boundary value problems on symmetric manifolds

Provided existence results under specific geometric assumptions

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.