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

Abstract

We show that on a compact Riemannian manifold with boundary there exists u ∈ C ∞ (M) such that, u |∂ M ≡ 0 and u solves the σ 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 σ k -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.

Conformally bending three-manifolds with boundary

“…Subsequently, the authors of this article proved the following, which can be viewed as a refinement of Lohkamp's result (see [5]):…”

“…-As we noted in [5], these results can also be viewed as scalar versions of the problem of constructing Poincaré-Einstein metrics with prescribed conformal infinity; see Section 6 of [5].…”

“…As in [5], we will consider a Dirichlet problem: let (M 3 , ∂M 3 , g) be a compact manifold with boundary; we want to solve…”

“…As in [5], we will use the continuity method to prove the existence of solutions to ( * ) . To this end, fix > 0 and for t ∈ [0, 1] define…”

“…In this Section we adapt the construction of [5] to give a subsolution of ( * ) . By the maximum principle of Proposition 3.3, this will give an a priori lower bound for solutions.…”