Asymptotically geodesic surfaces

TL;DR

This paper investigates the behavior of sequences of surfaces in hyperbolic 3-manifolds that become increasingly flat, revealing their density properties in finite volume cases and non-existence in infinite volume cases, with implications for the geometry of such manifolds.

Contribution

It establishes density results for asymptotically geodesic surfaces in finite volume hyperbolic 3-manifolds and shows their non-existence in infinite volume cases, extending understanding of surface limits in hyperbolic geometry.

Findings

Sequences are dense in the 2-plane Grassmann bundle for finite volume M.

Such sequences do not exist in infinite volume, geometrically finite M.

Examples show less constrained behavior in higher dimensions.

Abstract

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the 2-plane Grassmann bundle of M. When M has infinite volume and is geometrically finite, we show such sequences do not exist. As an application of the former, we obtain partial answers to the question of whether a negatively curved Riemannian 3-manifold that contains a sequence of asymptotically totally geodesic or totally umbilic surfaces must be hyperbolic. Finally, we give examples to show that if the dimension of M is greater than 3, the possible limiting behavior of asymptotically geodesic surfaces is less constrained than for totally geodesic surfaces.