On the Topology of Black Hole Event Horizons in Higher Dimensions

In four dimensions the topology of the event horizon of an asymptotically flat stationary black hole is uniquely determined to be the two-sphere $S^2$. We consider the topology of event horizons in higher dimensions. First, we reconsider Hawking's theorem and show that the integrated Ricci scalar curvature with respect to the induced metric on the event horizon is positive also in higher dimensions. Using this and Thurston's geometric types classification of three-manifolds, we find that the only possible geometric types of event horizons in five dimensions are $S^3$ and $S^2 \times S^1$. In six dimensions we use the requirement that the horizon is cobordant to a four-sphere (topological censorship), Friedman's classification of topological four-manifolds and Donaldson's results on smooth four-manifolds, and show that simply connected event horizons are homeomorphic to $S^4$ or $S^2\times S^2$. We find allowed non-simply connected event horizons $S^3\times S^1$ and $S^2\times \Sigma_g$, and event horizons with finite non-abelian first homotopy group, whose universal cover is $S^4$. Finally, following Smale's results we discuss the classification in dimensions higher than six.Comment: 12 pages, minor edits 27/09/0

Universes inside a Λ\Lambda black hole

We address the question of universes inside a $\Lambda$ black hole which is described by a spherically symmetric globally regular solution to the Einstein equations with a variable cosmological term $\Lambda_{\mu\nu}$, asymptotically $\Lambda g_{\mu\nu}$ as $r\to 0$ with $\Lambda$ of the scale of symmetry restoration. Global structure of spacetime contains an infinite sequence of black and white holes, vacuum regular cores and asymptotically flat universes. Regular core of a $\Lambda$ white hole models the initial stages of the Universe evolution. In this model it starts from a nonsingular nonsimultaneous big bang, which is followed by a Kasner-type anisotropic expansion. Creation of a mass occurs mostly at the anisotropic stage of quick decay of the initial vacuum energy. We estimate also the probability of quantum birth of baby universes inside a $\Lambda$ black hole due to quantum instability of the de Sitter vacuum.Comment: REVTEX, 9 pages, 13 figures. To appear in Physics Letters

Energy dispersed large data wave maps in 2+1 dimensions

In this article we consider large data Wave-Maps from $\mathbb{R}^{2+1}$ into a compact Riemannian manifold $(\mathcal{M},g)$, and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. In a companion article we use these results in order to establish a full regularity theory for large data Wave-Maps.Comment: 89 page

Universes inside a Λ black hole

We address the question of universes inside a Λ black hole which is described by a spherically symmetric globally regular solution to the Einstein equations with a variable cosmological term Λμv, asymptotically Λgμv as r → 0 with Λ of the scale of symmetry restoration. Global structure of spacetime contains an infinite sequence of black and white holes, vacuum regular cores and asymptotically flat universes. Regular core of a Λ white hole models the initial stages of the Universe evolution. In this model it starts from a nonsingular nonsimultaneous big bang, which is followed by a Kasner-type anisotropic expansion. Creation of a mass occurs mostly at the anisotropic stage of quick decay of the initial vacuum energy. We estimate also the probability of quantum birth of baby universes inside a Λ black hole due to quantum instability of the de Sitter vacuum. © 2001 Elsevier Science B.V