Abstract

In this talk, we consider a sequence of compact n-dimensional Riemannian manifolds with marked point which converge to a smooth n-dimensional Riemannian manifold with marked point in the pointed Gromov-Hausdorff topology.  The goal here is to establish a volume convergence under some Ricci curvature bound.  In fact, we can remove the restriction on the smoothness and get a volume convergence of Hausdorff measure.

In the first talk, I will assume some estimate, and prove the volume convergence. In the second talk, I will show the detail estimates.

Ref: Xiaochun Rong, Degeneration of metrics under Ricci curvature bounded below.