2020/04/21 by Shaosai Huang, Bing Wang, Huang, Shaosai +1
Computer Science · Mathematics · #53C #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2004.09762
openalex publication_date 2020/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci curved manifold with unit diameter and maximal first Betti number is homeomorphic to the flat torus. In this paper, we prove a parametrized version of this theorem, in the context of collapsing Riemannian manifolds with Ricci curvature bounded below: if a closed manifold with Ricci curvature uniformly bounded below is Gromov-Hausdorff close to a (lower dimensional) manifold with bounded geometry, and has the difference of their first Betti numbers equal to the dimensional difference, then it is diffeomorphic to a torus bundle over the one with bounded geometry. We rely on two novel technical tools: the first is an effective control of the spreading of minimal geodesics with initial data parallel transported along a short geodesic segment, and the second is a Ricci flow smoothing result for certain collapsing initial data with Ricci curvature bounded below.