# Width Stability of Rotationally Symmetric Metrics

## TL;DR

This paper confirms a mathematical conjecture about geometric stability under rotational symmetry and extends related results in higher dimensions.

## Contribution

The paper proves a volume preserving intrinsic flat stability conjecture under rotational symmetry and extends rigidity theorems to higher dimensions.

## Key findings

- The stability conjecture of Marques and Neves is valid under rotational symmetry.
- A rigidity theorem is established for rotationally symmetric manifolds in dimensions at least three.
- Gromov–Hausdorff convergence is shown outside certain 'bad' sets and under non-negative Ricci curvature.

## Abstract

In 2018, Marques and Neves proposed a volume preserving intrinsic flat stability conjecture concerning their width rigidity theorem for the unit round 3-sphere. In this work, we establish the validity of this conjecture under the additional assumption of rotational symmetry. Furthermore, we obtain a rigidity theorem in dimensions at least three for rotationally symmetric manifolds, which is analogous to the width rigidity theorem of Marques and Neves. We also prove a volume preserving intrinsic flat stability result for this rigidity theorem. Lastly, we study variants of Marques and Neves’ stability conjecture. In the first, we show Gromov–Hausdorff convergence outside of certain “bad” sets. In the second, we assume non-negative Ricci curvature and show Gromov–Hausdorff stability.