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

Gursky, Matthew; Streets, Jeffrey; Warren, Micah
May 2011
Calculus of Variations & Partial Differential Equations;May2011, Vol. 41 Issue 1/2, p21
Academic Journal
We show that on a compact Riemannian manifold with boundary there exists $${u \in C^{\infty}(M)}$$ such that, u ≡ 0 and u solves the σ-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 σ-Ricci problem. By adopting results of (Mazzeo and Pacard, Pacific J. Math. 212(1), 169-185 (2003)), we show an interesting relationship between the complete metrics we construct and the existence of Poincaré-Einstein metrics. Finally we give a brief discussion of the corresponding questions in the case of positive curvature.


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