# A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvature

## TL;DR

This paper proves a conjecture about the convergence of sequences of rotationally symmetric Riemannian manifolds with positive scalar curvature, showing they converge to limit spaces with nonnegative scalar curvature and Euclidean tangent cones.

## Contribution

It establishes the conjecture for rotationally symmetric warped product manifolds, demonstrating the limit spaces have weakly nonnegative scalar curvature and Euclidean tangent cones.

## Key findings

- Limit spaces have $H^1$ warping functions with nonnegative scalar curvature.
- Limit spaces possess Euclidean tangent cones almost everywhere.
- Convergence occurs in the intrinsic flat sense for the specified class.

## Abstract

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have 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 spaces have $H^1$ warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost everywhere.