Convexity in locally conformally flat manifolds with boundary

Abstract

Given a closed subset of the open unit ball B 1 ⊂ ‫ޒ‬ n for n ≥ 3, we consider a complete Riemannian metric g on B 1 \ of constant scalar curvature equal to n(n − 1) and conformally related to the Euclidean metric. We prove that every closed Euclidean ball B ⊂ B 1 \ is convex with respect to the metric g, assuming the mean curvature of the boundary ∂ B 1 is nonnegative with respect to the inward normal.