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Black holes, marginally trapped surfaces and quasi-minimal surfaces. (English) Zbl 1194.53060
For physical reasons, a space-time is mathematically defined as a $4$-dimensional, smooth, connected, pseudo-Riemannian manifold with a smooth Lorentzian metric of signature $(-,+,+,+)$. The geometry of space-time in special relativity is described by the Minkowski spacetime $\Bbb E\sp{4}\sb 1$. Besides Minkowski space-time, there are two other space-times which are of constant curvature, the de Sitter space-time $\Bbb S\sp 4\sb 1$ and the anti-de Sitter space-times $\Bbb H\sp 4\sb 1$. De Sitter space-time can be defined as a hypersurface of Minkowski $5$-space. Einstein’s theory of general relativity predicts that when a large enough amount of mass is present in a sufficiently small region of space, all paths through space are warped inwards towards the center of the volume, preventing all matter and radiation within it from escaping. The theory has also important astrophysical applications. It points towards the existence of black holes. The black hole’s mass is proportional to the mass of the host galaxy. Black holes cannot be seen, because everything that falls into them, including light, is trapped. But the swift motions of gas and stars near an otherwise invisible object allows astronomers to calculate that it is a black hole and even to estimate its mass. In the theory of cosmic black holes, if there is a massive source inside the surface, then close enough to a massive enough source, the outgoing light rays may also be converging, a trapped surface. Everything inside is trapped. Nothing can escape, not even light. It is believed that there will be a marginally trapped surface, separating the trapped surfaces from the untrapped ones, where the outgoing light rays are instantaneously parallel. In [Phys. Rev. Lett. 14, 57--59 (1965; Zbl 0125.21206)] {\it R. Penrose} introduced the concept of trapped surfaces. In terms of the mean curvature vector field, a space-like surface in a $4$-dimensional space-time is marginally trapped if and only if its mean curvature vector field is light-like at each point on the surface. The surface of a black hole is the marginally trapped surface. Although many physicists are interested in marginally trapped surfaces, almost no classification results on marginally trapped surfaces are known from differential geometric point of view until the last few years. This article is an interesting survey on recent classification results on marginally trapped surfaces from differential geometric viewpoint, and on recent results on a closely related subject, like quasi-minimal surfaces in pseudo-Riemannian manifolds. The author presents various marginally trapped surfaces with positive relative nullity or parallel mean curvature vector in Minkowski space-time. A Lorentzian surface in a pseudo-Riemannian manifold is called quasi-minimal if its mean curvature vector is light-like at each point. Quasi-minimal surfaces with constant positive curvature, constant negative curvature, and with parallel mean curvature vectors are discussed.

53C50Lorentz manifolds, manifolds with indefinite metrics
83-02Research monographs (relativity)
53C40Global submanifolds (differential geometry)