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Recent Progress on the Formation of Trapped Surfaces and Black Holes Junbin Li Department of Mathematics�� Sun Yat-sen University�� Guangzhou�� China Abstract In this paper we review the recent progress on the mathematical results on the formation of trapped surfaces and black holes in general relativity. 1 Preliminaries General relativity is a theory about gravity using the language of Riemannian geometry. A spacetime is a Lorentzian manifold satisfying the Einstein equations When the energy-momentum tensor T���� is set to be zero�� we call it the vacuum Einstein equations. In vacuum�� the Einstein equations read Ric���� = 0. The Minkowski space defined on R3+1 which is a flat and geodesically complete solution of the vacuum Einstein equations. One of the most important exact solutions of the vacuum Einstein equations is the family of Schwarzschild solutions (1.1) where the parameter M > 0 representing the mass. The Schwarzschild solutions are spherically symmetric and static�� and is the only family of vacuum solutions which are spherically symmetric by Birkhoff Theorem. This family of solutions describes the surrounding spacetime of a spherical star. It can be seen from the metric that r = 0 and r = 2M are both singularities of the Schwarzschild solutions. By direct computation�� we have �� so curvature blows up at r = 0 and hence the metric fails to be C2 when approaching r = 0. However�� the curvature remains bounded at r = 2M. In fact�� the Schwarzschild metric is smooth across r = 2M�� which is a null hypersurface. A surprising feature of Schwarzschild solutions is that there is a region�� denoted by B�� corresponding to r . 2M�� has the property that any future directed timelike or null curves with starting point in B cannot escape to the outside region r > 2M�� and in particular cannot escape to the future null infinity I+�� the future ideal boundary of the spacetime. Physically�� the future null infinity represents the location of faraway observers�� so the black hole region is a region invisible to faraway observers. The Schwarzschild solutions are the first and most important family of black hole solutions. It is a subfamily of a larger two-parameter family of black hole solutions �� where represents the mass and a represents the angular momentum per unit mass. When a = 0�� it reduces to the Schwarzschild solutions. Similar to the Schwarzschild solutions�� the region is the region outside the black hole. Kerr solutions represent stationary rotating black holes. For general asymptotically flat spacetime M�� Penrose first introduced the notion of the future null infinity I+ by conformal compacification. The future null infinity can be understood as the ideal future boundary of the spacetime�� where the spacetime becomes flat. The black hole region B�� can then be defined to be the region in M so that any timelike or null curves starting in B cannot end at I+. The (future) event horizon H+�� is defined to be the topological boundary of B in M. The family of Kerr black holes is stationary black hole�� which means that the black hole is already and always here. A central problem in general relativity is the problem of gravitational collapse: Can and how a black hole form in the evolution of the Einstein equations? 1.1 The initial value problem The evolution of the Einstein equations is formulated in terms of initial value problem. Let be a Cauchy initial data set in vacuum�� is a 3-dimensional Riemannian manifold�� is a two-tensor�� satisfying the constraint equations . (1.2) These equations are essentially the Gauss-Codazzi equations of �� embedding in M. Then the local well-posedness holds. Theorem 1.1(Choquet-Bruhat [9] and Geroch-Choquet-Bruhat [10]) Given a Cauchy initial data set of the vacuum Einstein equations�� there exists a unique maximal future development (M�� g)�� such that is a Cauchy hypersurface�� and is the past boundary of (M�� g)�� with �� being its first and second fundamental forms. Sometimes we consider characteristic problem instead of Cauchy problem with spacelike initial data�� that is�� (part of) the initial data is given on null hypersurface. This is because the constraint equations (1.2) are nonlinear system of elliptic equations and are difficult to solve. In the case of two null hypersurfaces intersecting transversally at a spacelike 2-surface�� we have Theorem 1.2 (Rendall [31]�� Luk [27]) Suppose that the characteristic initial data of the vacuum Einstein equations is prescribed on two null hypersurfaces C and C�� intersecting transversally at a spacelike 2-surface S. Then there exists a maximal future development (M�� g) such that (part of) C �� C is the pastboundary of (M�� g).1 One of the advantage of considering characteristic problem is the ��Gauss- Codazzi equations�� induced on C and C are simply ODEs along null generators of the null hypersurfaces. By simply integrating the ODEs�� the full initial