**Authors:**
Piotr T. Chrusciel,
Erwann Delay,
Gregory J. Galloway,
Ralph Howard

**Comments:** Here is the final version of our joint paper with E.Delay,
G.Galloway, and R.Howard on the area theorem, accepted for publication
in Annales Henri Poincare. The files 00* correspond to the original gr-qc/0001003
submission; the files area* are an update thereof, with several
corrections and extensions following comments by an anonymous
referee. In particular the original title "The Area Theorem" has
become "Regularity of Horizons and the Area Theorem"; further, it is
shown that Krolak's area theorems hold without differentiability
hypotheses on the horizon.

We prove that the area of sections of future event horizons in space-times satisfying the null energy condition is non-decreasing towards the future under any one of 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 ``H-regular'' Scri plus; 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 a theorem of Hawking, in which piecewise smoothness of the event horizon seems to have been assumed. No assumptions about the cosmological constant or its sign are made. 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.

**Listing of
Tue Dec 12 23:04:57 CET 2000: **
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Name | (Last modified) | Size |
---|---|---|

0001003.pdf | (May 27 2000) | 776438 |

0001003.ps | (May 27 2000) | 1335683 |

0001003v2.pdf | (May 27 2000) | 776597 |

0001003v2.ps | (May 27 2000) | 17699347 |

area12XII2000.dvi | (Dec 12 22:23) | 411020 |

area12XII2000.ps | (Dec 12 22:23) | 910385 |

area12XII2000.tex | (Dec 12 22:23) | 260185 |

- on http://xxx.lanl.gov/abs/gr-qc/9611032: our joint paper with Greg on "Nowhere differentiable horizons", which started all this buisiness;
- on http://xxx.lanl.gov/abs/gr-qc/9807059: my paper entitled "A remark on differentiability of Cauchy horizons", which gives a simpler proof of the Beem-Krolak theorem concerning the set of differentiable points of horizons; and
- on http://xxx.lanl.gov/abs/gr-qc/0011067: the paper in collaboration with with J.H.G. Fu, G.J. Galloway, and R. Howard, entitled On fine differentiability properties of horizons and applications to Riemannian geometry which, amongst others, settles the question of "how big is the set of end points of generators of horizons?" (the answer you might have guessed: rather small).

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