Skinning measures in negative curvature and equidistribution of equidistant submanifolds

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.

32 pages. Revised version with important modifications to Section 4

References in corpus (3)

• Equidistribution and Counting for orbits of geometrically finite hyperbolic groups
• The Asymptotic distribution of circles in the orbits of Kleinian groups
• Counting, Mixing and Equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds

Cited by in corpus (2)

• Counting common perpendicular arcs in negative curvature
• Joint partial equidistribution of Farey rays in negatively curved manifolds and trees