## Global Problems in Mathematical Relativity II

**Monday 9th October 2006 to Friday 20th October 2006**

11:00 to 12:00 |
The problem of stability for black hole spacetimes I review recent results on the behaviour of linear fields on black hole spacetime backgrounds with zero, positive, and negative cosmological constant and discuss the relation of this with the problem of stability for black hole spacetimes. |
INI 1 |

16:00 to 17:00 |
On the Dirichlet problem for the Einstein equations We show that the space of solutions to the (Riemannian)Einstein equations on a bounded domain is either empty or an infinite dimensional Banach manifold for which the map to the metric on the boundary is Fredholm, of index 0. The same result holds for metrics with compact "inner" boundary with (for instance) asymptotically flat ends. It also holds for the Einstein equations coupled to general matter fields, and in all dimensions. Applied to the static (or stationary) vacuum Einstein equations, the result is relevant to Bartnik's static extension conjecture, and generalizes results of P. Miao. |
INI 1 |

16:00 to 17:00 |
Uniqueness of solutions to wave equations with data on the horizon of a black hole It is well known that the problem of prescribing initial data on the boundary of a domain of dependence $\DD$ of solutions to a wave equation is not well posed in the complement of $\DD$. It is expected, however, that one still has uniqueness. In collaboration with Alexandru Ionescu we have been recently able to prove some uniqueness results both in the Minkowski space, as well as for the Schwarzschild and Kerr space-times in the domain of outer communication. |
INI 1 |

16:00 to 17:00 |
On stability of cosmological solutions to Einstein's equations coupled to a non-linear scalar field I'm going to discuss Einstein's equations coupled to a non-linear scalar field, the potential of which has a positive non-degenerate minimum, in the cosmological context. The question I will address is that of stability in the expanding direction. |
INI 1 |

16:00 to 17:00 | Formation of singularities for the wave map equation in 2+1 dimensions | INI 1 |

16:00 to 17:00 | Stability of solutions of the Einstein equations | INI 1 |

16:00 to 17:00 |
J Isenberg ([Oregon])Black hole rigidity in higher dimensions |
INI 1 |

16:15 to 17:15 |
GJ Galloway ([Miami])Rigidity of outermost MOTS and the topology of higher dimensional black holes In a talk given here last fall I presented joint work with Rick Schoen, in which we obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. We proved, for example, that, apart from certain exceptional circumstances, cross sections of the event horizon in stationary black hole spacetimes obeying a standard energy condition are of positive Yamabe type. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. In this talk I show how to rule out in this setting the possibility of any such exceptional circumstances (which might have permitted, e.g., toroidal cross sections). This follows from the main result to be discussed, which is a rigidity result for suitably outermost marginally outer trapped surfaces that are not of positive Yamabe type. |
INI 1 |