Pure and Applied Mathematics Quarterly

Volume 14 (2018)

Number 3-4

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

Pages: 529 – 561

DOI: https://dx.doi.org/10.4310/PAMQ.2018.v14.n3.a5


Jiewon Park (Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Wenchuan Tian (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Changliang Wang (Max-Planck-Institut für Mathematik, Bonn, Germany)


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.


scalar curvature compactness, Sormani–Wenger intrinsic flat distance, rotationally symmetric manifolds

2010 Mathematics Subject Classification


Received 3 January 2019

Received revised 12 February 2019

Accepted 2 February 2019

Published 5 November 2019