通过Ricci流光滑化研究第一Betti数的刚性
Rigidity of the first Betti number via Ricci flow smoothing
AI总结：
本文证明了Colding-Gromov间隙定理的参数化版本：当Ricci曲率下有界的闭流形在Gromov-Hausdorff意义下接近低维有界几何流形且第一Betti数之差等于维数之差时，该流形微分同胚于环面丛，并提出了极小测地线扩散控制和Ricci流光滑化两个新技术工具。
摘要：
Colding-Gromov间隙定理断言，一个具有单位直径和最大第一Betti数的几乎非负Ricci曲率流形同胚于平坦环面。本文在Ricci曲率下有界的坍缩黎曼流形背景下证明了该定理的参数化版本：如果一个Ricci曲率一致下有界的闭流形在Gromov-Hausdorff意义下接近一个具有有界几何的（低维）流形，且它们的第一Betti数之差等于维数之差，则该闭流形微分同胚于具有有界几何流形上的环面丛。我们依赖两个新的技术工具：第一个是对初始数据沿短测地线段平行移动的极小测地线扩散的有效控制，第二个是针对某些Ricci曲率下有界的坍缩初始数据的Ricci流光滑化结果。
英文摘要：
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.
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