Geometrically and topologically random surfaces in a closed hyperbolic three manifold

TL;DR

This paper investigates the distribution and limiting behavior of random surfaces in closed hyperbolic 3-manifolds, revealing that topological limits tend to concentrate on geodesic loci, unlike geometric limits.

Contribution

It characterizes the invariant measures arising as limits of random minimal surfaces and distinguishes their topological versus geometric scarring properties.

Findings

Topological limits are totally scarred on geodesic loci if a geodesic subsurface exists.

Geometric limits are never totally scarred.

Invariant measures are described on the Grassmann bundle G(M).

Abstract

We study the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3-manifold M. In particular, we describe PSL(2,R) invariant measures on the Grassmann bundle G(M) which arise as limits of random minimal surfaces. It is showed that if M contains at least one totally geodesic subsurface then every topological limiting measure is totally scarring (i.e supported on the totally geodesic locus), while we prove that geometrical limiting measures are never totally scarring.