Gromov-Hausdorff limit of manifolds and some applications

Gromov-Hausdorff distance is a distance between two metric spaces, which was introduced by Gromov 1981. From Gromov's compactness theorem, we knew that any sequence of manifolds with uniform lower Ricci curvature bounds has a converging subsequence in Gromov-Hausdorff topology to a limit metric space. The limit metric space in general may not be a manifold. The structure of such limit metric space has been studied by Cheeger, Colding, Tian, Naber and many others since 1990. It turns out that such theory has powerful application in geometry. In fact, the resolution of Yau-Tian-Donaldson conjecture was largely relied on the development of the study of the limit metric space.
In the first part of the talk, we will discuss some recent progress of the Gromov-Hausdorff limit of a sequence of manifolds with Ricci curvature bounds. In the second part, we will discuss some applications based on the study of Gromov-Hausdorff limits.