We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is non-decreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic space-time and there exists a conformal completion with a "Hregular" I + ; 3) the horizon is a black hole event horizon in a space-time which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends * Supported in part by KBN grant # 2 P03B 130 16. E-mail : Chrusciel@Univ-Tours.fr † Current adress: Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm. E-mail : Delay@gargan.math.Univ-Tours.fr ‡ Supported in part by NSF grant # DMS-9803566. E-mail : galloway@math.miami.edu § Supported in part by DoD Grant # N00014-97-1-0806 E-mail : howard@math.sc.edu 1 a theorem of Hawking, in which piecewise smoothness of the event horizon seems to have been assumed. We prove smoothness or analyticity of the relevant part of the event horizon when equality in the area inequality is attained -this has applications to the theory of stationary black holes, as well as to the structure of compact Cauchy horizons. In the course of the proof we establish several new results concerning the differentiability properties of horizons.
We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of Lükő on average chord lengths of closed curves.
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