Metrics with Prescribed Ricci Curvature near the Boundary of a Manifold

TL;DR

This paper studies the existence and uniqueness of Riemannian metrics with prescribed Ricci curvature near the boundary of a manifold, with applications to symmetric metrics and Einstein equations on a solid torus.

Contribution

It provides new theorems on solving the prescribed Ricci curvature equation with boundary conditions, and explores applications to symmetric and Einstein metrics.

Findings

Established conditions for existence of solutions near the boundary.

Demonstrated uniqueness of solutions under certain boundary conditions.

Applied results to rotationally symmetric metrics and Einstein equations.

Abstract

Suppose $M$ is a manifold with boundary. Choose a point $o\in\partial M$. We investigate the prescribed Ricci curvature equation $\Ric(G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian metric. The letter $T$ in the right-hand side denotes a (0,2)-tensor. Our main theorems address the questions of the existence and the uniqueness of solutions. We explain, among other things, how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus $\mathcal T$. The paper concludes with a brief discussion of the Einstein equation on $\mathcal T$.