Skinning measures in negative curvature and equidistribution of equidistant submanifolds

TL;DR

This paper introduces a new approach to skinning measures in negatively curved manifolds, proving their finiteness and equidistribution properties for equidistant hypersurfaces, under broad conditions.

Contribution

It generalizes previous work by establishing finiteness and equidistribution of skinning measures using novel methods, with results on the rate of convergence.

Findings

Finiteness of skinning measures under broad conditions

Equidistribution of skinning measures to Bowen-Margulis measure

Control on the rate of equidistribution with additional assumptions

Abstract

Let C be a locally convex subset of a negatively curved Riemannian manifold M. We define the skinning measure on the outer unit normal bundle to C in M by pulling back the Patterson-Sullivan measures at infinity, and give a finiteness result of skinning measures, generalising the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to C equidistribute to the Bowen-Margulis measure on the unit tangent bundle of M, assuming only that the Bowen-Margulis measure is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.