Convexity in locally conformally flat manifolds with boundary

Abstract

of constant positive scalar curvature R(g) = n(n 1) and conformally related to the Euclidean metric . We will also assume that g has nonnegative boundary mean curvature. Here, and throughout this paper, second fundamental forms will be computed with respect to the inward unit normal vector. In this talk we prove Theorem 0.1. If B B1 is a standard Euclidean ball, then @B is convex with respect to the metric g. This theorem is motivated by an analogous one on the sphere due to R. Schoen [3]. He shows that if S n n 3, is closed and nonempty and g is a complete Riemannian metric on S n , conformal to the standard round metric g0 and with constant positive scalar curvature n(n 1), then every standard ball B S n is convex with respect to the metric g. Schoen used this geometrical result to prove the compactness of the set of solutions to the Yamabe problem in the locally conformally flat case. Later, D. Pollack also used Schoen’s theorem to prove a compactness result for the singular Yamabe problem on the sphere where the singular set is a finite collection of points