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Piotr T. Chrusciel - Research
Those are the research areas to which I have contributed:
1. Global structure of solutions of Einstein equations: global properties, cosmic censorship, asymptotics
2. Global Lorentzian geometry: the structure of horizons and the area theorem
3. The general relativistic constraint equations
4. The mass in general relativity
5. Classification of black hole space-times
6. Nonlinear partial differential equations
Here is a short description of some of my papers - it illustrates fairly well my research interests.
1. Global structure of solutions of Einstein equations: global structure of Robinson-Trautman space-times In the papers [14,15] I prove that all Robinson-Trautman space-times which describe asymptotically flat (in light-like directions) smooth metrics form black holes. This is the only existing result of this kind in the literature without any symmetry hypotheses (all the remaining results on black hole formation assume spherical symmetry). The result is obtained by proving global existence to the future of solutions of the Calabi-Robinson-Trautman equation [14], and by proving an asymptotic expansion for its solutions as time goes to infinity [15].
2. Global structure of solutions of Einstein equations: cosmic censorship In the paper [34] I prove, in collaboration with A. Rendall, that strong cosmic censorship holds in the class of locally spatially homogeneous metrics. This problem turns out to be rather subtle, because of the complicated, presumably chaotic, dynamical behavior of some metrics in this class ( e.g. the so called Bianchi IX metrics). In the paper [7], written in collaboration with B. Berger and V. Moncrief, I prove that strong cosmic censorship holds in the class of asymptotically flat space-times with cylindrical symmetry. This is the only existing result of this kind in which a non-linear problem is considered and in which no smallness hypotheses are made on the initial data. In [13] I prove that the existence of a curvature singularity is stable under non-linear perturbations of a class of Kasner metrics. Until [45] (where my results are generalised) this was the only existing result on non-linear stability of a curvature singularity. In [30] we give an exhaustive description of Cauchy horizons in Gowdy space-times; this result, together with [45], gives the first proof of SCC in this class of space-times.
3. Global structure of solutions of Einstein equations: asymptotics In the paper [12] I classify the asymptotic structures of Ashtekar-Hansen [4] type at spatial infinity under a geodesic regularity condition, and shown that the non-uniqueness of those structures does not lead to ambiguities in the definition of global quantities such as energy-momentum, etc. In the papers [3,2] (cf. also [1]), written in collaboration with L. Andersson and H. Friedrich, we establish existence of a large class of solutions of vacuum Einstein equations, the asymptotic properties of which correspond those conjectured by Bondi [8], Sachs [46] and Penrose [42] many years ago. Until the work in [3,2] the existence of such space-times was the main open problem in the understanding of the gravitational field in the radiating regime. A fundamental contribution to this problem has been made by Friedrich in [38], who shifted the problem to that of the construction of appropriate initial data of hyperboloidal type''. The construction of appropriate initial data has been performed in [3,2]. In the paper [32], written in collaboration with M. McCallum and D. Singleton, I show that the Bondi-Sachs analysis of the asymptotic behavior of the gravitational field in light-like directions can be generalized to include logarithmic expansions in , in a manner consistent with the Einstein vacuum equations. We find families of new constants of motion in the radiation regime.
4. Global Lorentzian geometry: the structure of horizons and the area theorem A classical theorem of Hawking and Ellis asserts that the area of sections of piecewise smooth horizons is non-increasing towards the future (under appropriate energy conditions and global restrictions on the space-times into consideration). Now in the paper [24], written in collaboration with G. Galloway, Cauchy and event horizons are constructed which are not differentiable on a dense set. Those examples make it clear that the hypothesis that the horizon is piecewise smooth made by Hawking and Ellis does not have any justification. In the paper [23], written in collaboration with E. Delay, G. Galloway, and R. Howard, we show that the area theorem holds without any supplementary hypotheses of differentiability of the event horizon. This requires overcoming various severe technical hurdles related to the low potential differentiability of the horizon under consideration. The above quoted papers also prove several new results concerning the structure of horizons.
5. The general relativistic constraint equations The initial data for Einstein equations are not aribrary, they have to satisfy a set of constraint equations. In a series of papers in collaboration with R. Beig, E. Delay, J. Isenberg, R. Mazzeo, D. Pollack and R. Schoen we show how to obtain new solutions of those equations by gluing old ones. This allows the construction of initial data sets with interesting properties: For example, those techniques can be used to prove existence of asymptotically simple vacuum space-times [22,21]; the existence of such space-times has been conjectured by Penrose in the 1960's, but was open until our work. We prove that one can glue together generic initial data sets [27,6], leading sometimes to black hole space-times with non-connected horizons [33]. The gluing can be used to construct spatially compact, vacuum, maximal globally hyperbolic space-times which do not contain any compact hypersurfaces with constant mean curvature [26]; this is very annoying, as CMC hypersurfaces provide a very convenient time function, whenever they exist.
6. The mass in general relativity In the paper [10] I prove that the ADM mass is a geometric invariant of an initial data set. While this result is often attributed to Bartnik [5] in the literature, the proof in [10] has been obtained independently, about at the same time, and is much simpler than Bartnik's proof. (The hypotheses of [10] are marginally stronger than those of [5].) In the paper [11] I show that the ADM mass is a geometric invariant of a boost-type domain in a space-time. This generalizes the results of [5,10] to a space-time setting. In the paper [28] I prove, in collaboration with J. Jezierski and Malcolm McCallum, that the Trautman-Bondi mass is (up to a multiplicative constant) the unique functional, in a large class of natural functionals, which is monotonous in time for all vacuum solutions of Einstein equations for which it is well defined. This is the first uniqueness result for the mass of the gravitational field in the radiating regime. In a series of classical papers [44,43], Penrose and collaborators have suggested proofs of the positive energy theorem using purely Lorentzian techniques. In [25] and in [20] we show that those methods can indeed be used to prove the result, assuming that the metrics are uniformly Schwarzschildian near infinity.
7. Classification of black hole space-times In the paper [18] I have finished the classification of static vacuum black holes which contain an asymptotically flat spacelike hypersurface with compact interior with a black hole inner boundary''. Recall that this problem has a long history, starting with the pioneering work of Israel [40]. The most complete result existing in the literature before my paper was that of Bunting and Masood-ul-Alam [9] who show, roughly speaking, that all appropriately regular such black holes which do not contain degenerate horizons belong to the Schwarzschild family. In the paper [18] I remove the condition of non-degeneracy of the event horizon, and show that the Schwarzschild black holes exhaust the family of all appropriately regular black hole space-times. (The paper [19] contains an improvement of similar previous results concerning the electro-vacuum black holes, under the restrictive condition that all degenerate components of the black hole carry charges of the same sign.) In the paper [17] I show, by constructing a counterexample, that one of the key theorems of the classification theory of black holes, due to Hawking (the so-called strong rigidity theorem''), was wrong. In [16] I give a corrected statement of the theorem, together with a proof. In the paper [36], written in collaboration with R. Wald, it is shown that the space-time exterior to a stationary black hole is simply connected, and that sections of stationary black holes have spherical topologies. This result fills a gap in a similar result by Hawking [39].
8. Nonlinear partial differential equations of hyperbolic type In the paper [35], written in collaboration with J. Shatah, global solvability of the Cauchy problem for the Yang-Mills equations on an arbitrary globally hyperbolic four dimensional Lorentzian manifold is proved. This was a classical open problem since the fundamental work of Eardley and Moncrief [37] (cf. also [41]), who have proved such a statement on four dimensional Minkowski space-time. In [31], written in collaboration with my PhD student O. Lengard, we prove propagation of polyhomogeneous singularities for a class of semi-linear wave equations.
9. Miscellaneous In the paper [29] an invariant definition of Yang Mills charges at spatial infinity is given.

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Piotr Chrusciel
2004-10-28