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On fine differentiability properties of horizons and applications to Riemannian geometry

P. T. Chrusciel·J. H. G. Fu·G. J. Galloway·R. Howard
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Abstract

We study fine differentiability properties of horizons. We show that the set of end points of generators of a n-dimensional horizon H (which is included in a (n+1)-dimensional space-time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1\le k\le n+1 we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is ``almost a C^2 manifold of dimension n+1-k'': it can be covered, up to a set of vanishing (n+1-k)-dimensional Hausdorff measure, by a countable number of C^2 manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry.

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P. T. Chrusciel, J. H. G. Fu, G. J. Galloway, R. Howard. 2000-11-20. On fine differentiability properties of horizons and applications to Riemannian geometry. https://doi.org/10.1016/s0393-0440(01)00044-4

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