Prescribed Ricci Curvature on a Solid Torus

Details Prescribed Ricci Curvature on a Solid Torus Prescribed Ricci Curvature on a Solid Torus

2011-06-14

arxiv.org/abs/1106.2605v1

Mathematics

Differential Geometry

12 pages

Scientific paper

We investigate the prescribed Ricci curvature equation $\Ric(G)=T$ on a solid torus $\mathcal T$ 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 on $\mathcal T$. We assume $T$ is nondegenerate (in fact, even a lighter assumption would suffice). Our results then settle the questions of the existence and the uniqueness of solutions in the class of rotationally symmetric Riemannian metrics on a neighborhood of the boundary of $\mathcal T$. The paper concludes with a brief discussion of the Einstein equation on $\mathcal T$.

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