Generalized Penrose inequalities

Research project P 14621 Generalized Penrose inequalities Helmuth URBANTKE 09.10.2000 The Penrose Inequality is a recent result in Mathematical Relativity. It holds for Riemannian 3-manifolds with non- negative Ricci scalar and an asymptotically flat region exterior to a compact minimal surface and reads: The total energy (the "ADM-mass") of every asymptotically flat region is bounded from below by a constant (or a function) times the square root of the area of the minimal surface (with different functions appearing in the literature). This result was conjectured by Penrose in about 1970 and proven shortly after by Geroch and Jang and Wald who imposed, however, the very stringent additional requirement of the existence of a smooth so-called "inverse mean curvature flow". General proofs were obtained recently via spinorial methods by Herzlich, and via other functional analytic means by Huisken and Ilmanen, as well as by Bray. The Riemannian 3-manifolds mentioned above may describe (maximal) 3-geometries in General Relativity which satisfy the energy condititon and which contain a "trapped surface" (a "black hole"). As Penrose pointed out, his inequality supports the so-called "cosmic censorship conjecture". The latter reads: Singularities, (which necessarily form in the future of asymptotically flat spacelike slices containing trapped surfaces), remain "behind" (as seen from infinity) the horizon during time evolution. Further, the Penrose Inequality supports the picture that spacetimes evolving from regular, "generic" vacuum data "settle down" to a member of the Kerr family by dispersing gravitational radiation. The proven versions of the Penrose Inequality are, however, not sturated for Kerr but precisely for the Schwarzschild metric. Accordingly the Penrose Inequality also finds an application as a mathematical tool in proving uniqueness of the Schwarzschild metric as static, asymptotically flat vacuum solution. In the present project we plan to find generalizations of the Penrose inequality which are unproven so far, or which are shown only under the unphysical assumption of the existence of a smooth inverse mean curvature flow. We plan to follow the above-metioned three approaches employed in the vacuum case, focussing on the spinorial approach. The projected generalizations should include in particular the electromagnetic field, a cosmological constant, and linear and angular momentum. This means that we expect to obtain better lower bounds on the total energy which will, in addition to the area of the trapped surface, also involve the electric charge, the cosmological constant, and the respective momenta. We also plan to strengthen a recently obtained "conformal version" of the positive mass theorem to get a bound which also involves the area of a trapped surface. We will first focus on obtaining these generalizations in the static and stationary cases. We expect that these cases yield useful insights to the general situation. Moreover, they should also have direct applications to uniqueness results for stationary black hole spacetimes with matter fields, a cosmological constant or angular momentum.