Some Recent Progress in Classical General Relativity

One of the most difficult problems in classical relativity is to understand how and when singularities form. In the 1960’s, Hawking and Penrose proved that the existence of a closed trapped surface in an asymptotically flat spacelike hypersurface gives rise to a singularity in space-time. However, no proof based on pure partial differential equation arguments was found, and many questions remain unanswered. Given an initial data set (gij , pij) on a three-dimensional manifold so that gij is asymptotically Euclidean and pij (the induced second fundamental form in an embedding) falls off asymptotically, it is interesting to ask the following questions.