A compactness theorem for rotationally symmetric Riemannian manifolds with positive scalar curvature

Abstract

Gromov and Sormani have conjectured the following compactness theorem on scalar curvature to hold. Given a sequence of compact Riemannian manifolds with nonnegative scalar curvature and bounded area of minimal surfaces, a subsequence is conjectured to converge in the intrinsic flat sense to a limit space, which has nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit space has an H-1 warping function which has nonnegative scalar curvature in a weak sense, and has Euclidean tangent cones almost everywhere.