Width Stability of Rotationally Symmetric Metrics.

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.