On fine differentiability properties of horizons and applications to Riemannian geometry

TL;DR

This paper investigates the detailed differentiability properties of horizons in Lorentzian geometry and applies these findings to enhance understanding of distance functions and cut loci in Riemannian geometry.

Contribution

It establishes measure-zero properties of horizon endpoints and characterizes the structure of convex hulls of generators using advanced geometric analysis.

Findings

Set of horizon endpoints has vanishing Hausdorff measure.

Convex hulls of generators are almost C^2 manifolds of specific dimensions.

Results provide new insights into the structure of distance functions and cut loci.

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.