Lecture 3: Embedded mean-convex hypersurfaces in 3-manifolds

The lecture describes recent work with S. Brendle on embedded surfaces of positive mean curvature that move by mean curvature in a general Riemannian 3-manifold. We prove a general long-time existence and convergence result for a flow interrupted only by finitely many surgeries. As an application we construct canonical sweep-outs by 2-surfaces for asymptotically flat 3-manifolds arising as time-slices in Lorentzian manifolds that model isolated gravitating systems in General Relativity.