Geometric Analysis and General Relativity

This article discusses methods of geometric analysis in general relativity, with special focus on the role of "critical surfaces" such as minimal surfaces, marginal surface, maximal surfaces and null surfaces.Comment: to appear in Elsevier Encyclopedia of Mathematical Physics, 200

Bel--Robinson energy and constant mean curvature foliations

An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an $L^{\infty}$ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.Comment: 9 page

The trapped region

I will discuss some recent results on marginally outer trapped surfaces, apparent horizons, and the trapped region. A couple of applications of the results developed for marginally outer trapped surfaces to coalescence of black holes and to the characterization of the trapped region are given.Comment: talk given at ERE08, Salamanca. To appear in proceedings of ERE08. v2: Minor typos fixe

Curvature estimates for stable marginally trapped surfaces

We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situations of physical insterest, such as dynamical horizons.Comment: 28 pages. This is a major rework of the previous version. It extends the curvature estimates to no longer require global area bounds. In addition some mistakes were correcte