Convexity in locally conformally flat manifolds with boundary

Marcos Petrúcio de A. Cavalcante

Vol. 234 (2008), No. 1, 23–31

DOI: 10.2140/pjm.2008.234.23

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 2000

Primary: 53A30, 53C21

Secondary: 52A20

Milestones

Received: 19 January 2007

Revised: 8 June 2007

Accepted: 5 September 2007

Published: 1 January 2008