Skinning measures in negative curvature and equidistribution of equidistant submanifolds

Abstract

AbstractLet$C$be a locally convex closed subset of a negatively curved Riemannian manifold$M$. We define the skinning measure${\sigma }_{C} $on the outer unit normal bundle to$C$in$M$by pulling back the Patterson–Sullivan measures at infinity, and give a finiteness result for${\sigma }_{C} $, generalizing the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to$C$equidistribute to the Bowen–Margulis measure${m}_{\mathrm{BM} } $on${T}^{1} M$, assuming only that${m}_{\mathrm{BM} } $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.