Convexity in locally conformally flat manifolds with boundary

TL;DR

This paper proves that in certain conformally flat manifolds with boundary, Euclidean balls are convex under specific curvature conditions, extending understanding of geometric convexity in such manifolds.

Contribution

It establishes convexity of Euclidean balls in conformally flat manifolds with boundary and constant scalar curvature, under boundary mean curvature assumptions.

Findings

Euclidean balls are convex in the specified manifolds.

Convexity holds assuming nonnegative mean curvature of the boundary.

Results extend convexity properties to a class of conformally flat manifolds.

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.