Following the program started in [24], we attempt to remove the analyticity assumption in the the well known Hawking-Carter-Robinson uniqueness result for regular stationary vacuum black holes. Unlike [24], which was based on a tensorial characterization of the Kerr solutions, due to Mars [29], we rely here on Hawking's original strategy, which is to reduce the case of general stationary space-times to that of stationary and axi-symmetric spacetimes for which the Carter-Robinson uniqueness result holds. In this reduction Hawking had to appeal to analyticity. Using a variant of the geometric Carleman estimates developed in [24], in this paper we show how to bypass analyticity in the case when the stationary vacuum space-time is a small perturbation of a given Kerr solution. Our perturbation assumption is expressed as a uniform smallness condition on the Mars-Simon tensor. The starting point of our proof is the new local rigidity theorem established in [2].

We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. The result extends a previous result of Friedrich, Rácz and Wald, see [FRW, Prop. B.1], which was limited to the domain of dependence of the bifurcate horizon. So far, the existence of a Killing vector-field in a full neighborhood has been proved only under the restrictive assumption of analyticity of the space-time. Using this result we provide the first unconditional proof that a stationary black-hole solution must possess an additional, rotational Killing field in an open neighborhood of the event horizon. This work is accompanied by a second paper, where we prove a uniqueness result for smooth stationary black-hole solutions which are close (in a very precise, geometric sense) to the Kerr family of solutions, for arbitrary 0 < a < m.

Abstract. We prove various uniqueness results from null infinity, for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. We find that the parts of infinity where we must impose a vanishing condition depend strongly on the background geometry. In particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than the ones in the Minkowski space-time. The results are nearly optimal in many respects. They can be considered analogues of uniqueness from infinity results for second order elliptic operators. This work is partly motivated by questions in general relativity.

Abstract. The main objective of this paper is to control the geometry of a future outgoing truncated null cone extending smoothly toward infinity in an Einstein-vacuum spacetime. In particular, we wish to do this under minimal regularity assumptions, namely, at the (weighted) L 2 -curvature level. We show that if the curvature flux and the data on an initial sphere of the cone are sufficiently close to the corresponding values in a standard Minkowski or Schwarzschild null cone, then we can obtain quantitative bounds on the geometry of the entire infinite cone. The same bounds also imply the existence of limits at infinity, along the null cone, of naturally scaled geometric quantities. In [1], we will apply these results in order to control various physical quantitiese.g., the Bondi energy and (linear and angular) momenta-associated with such infinite null cones in vacuum spacetimes.