Existence of Complete conformal metrics of negative Ricci curvature on manifolds with boundary

Details Existence of Complete conformal metrics of negative Ricci curvature on manifolds with boundary Existence of Complete conformal metrics of negative Ricci curvature on manifolds with boundary

2009-07-27

arxiv.org/abs/0907.4641v1

Mathematics

Differential Geometry

25 pages, 2 figures

Scientific paper

We show that on a compact Riemannian manifold with boundary there exists $u \in C^{\infty}(M)$ such that, $u_{|\partial M} \equiv 0$ and $u$ solves the \sigma_k$-Ricci problem. In the case $k = n$ the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving the $\sigma_k$-Ricci problem. By adopting results of \cite{MP}, we show an interesting relationship between the complete metrics we construct and the existence of Poincar\'e-Einstein metrics. Finally we give a brief discussion of the corresponding questions in the case of positive curvature.

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