Abstract

Given a closed subset Λ of the open unit ball B₁ ⊂ Rⁿ for n ≥ 3, we consider a complete Riemannian metric g on B ₁ ∖ Λ of constant scalar curvature equal to n(n− 1) and conformally related to the Euclidean metric. We prove that every closed Euclidean ball B ⊂ B₁ ∖ Λ is convex with respect to the metric g, assuming the mean curvature of the boundary ∂B₁ is nonnegative with respect to the inward normal.

Keywords

scalar curvature, locally conformally flat metric, convexity

Mathematical Subject Classification

Primary: 53A30, 53C21

Secondary: 52A20

Authors