Convexity in locally conformally flat manifolds with boundary

Abstract

Given a closed subset $\La$ of the open unit ball $B_1\subset \real^n$, $n \geq 3$, we will consider a complete Riemannian metric $g$ on $\bar{B_1} \setminus \La$ of constant scalar curvature equal to $n(n-1)$ and conformally related to the Euclidean metric. In this paper we prove that every closed Euclidean ball $\bar{B} \subset B_1\setminus \La$ is convex with respect to the metric $g$, assuming the mean curvature of the boundary $\partial B_1$ is nonnegative with respect to the inward normal.