论文标题
第一个Betti数字通过RICCI流平滑的刚度
Rigidity of the first Betti number via Ricci flow smoothing
论文作者
论文摘要
冷 - 格罗莫夫差距定理断言，几乎非单独的弯曲歧管，单位直径和最大的第一贝蒂数字是平坦的圆环。在本文中，我们证明了该定理的参数化版本，在折叠的riemannian歧管折叠的背景下，下面有ricci曲率的曲率：如果下面有ricci弯曲的闭合歧管，下面均匀界限的是gromov-hausdorff，则是距离（较低的多维）差异，然后差异是偏差的差异，并与第一个差异相差，并且是第一个差异，并且是第一个差异，并且是差异的差异。与圆环捆绑包上的圆形束上有有界的几何形状。我们依靠两种新型的技术工具：第一个是对最小地球化学的传播的有效控制，其初始数据并行沿着短的地球段平行运输，第二个是RICCI流平滑结果，用于某些与RICCI曲线折叠的初始数据，以下是RICCI曲线。
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.
