Abstract: in an earlier work, we studied the mean curvature flow in Fuchsian manifold. The preservation of graphic property of the evolving surface plays a pivotal role for existence and then convergence. The more general case of almost Fuchsian manifold is the focus of the talk. The metric is still explicit in light of the classic work by Uhlenbeck, which looks slightly more complicated than the wrapped product metric for Fuchsian manifold. The calculation is more challenging. We have arrived at a similar conclusion for the mean curvature flow and are adapting it for the modified mean curvature flow. The program provides a way of constructing constant mean curvature surfaces and even a foliation as conjectured from topological point of view for 3-fold. This is based on joint works with Longzhi Lin and Zheng Huang.
