Properly immersed surfaces in hyperbolic 3-manifolds

TL;DR

This paper investigates the properties of complete immersed surfaces with bounded mean curvature in hyperbolic 3-manifolds, establishing conditions for properness and describing the asymptotic behavior of their ends.

Contribution

It proves properness and total curvature results for immersed surfaces with bounded mean curvature in negatively curved 3-manifolds, and characterizes ends in finite volume hyperbolic 3-manifolds.

Findings

Surfaces with bounded mean curvature are proper in negatively curved 3-manifolds.

Total curvature of such surfaces equals 2π times their Euler characteristic.

Ends of surfaces in finite volume hyperbolic manifolds are asymptotic to totally umbilic annuli.

Abstract

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $\Sigma$ must be proper in $N$ and its total curvature must be equal to $2\pi \chi(\Sigma)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of $\Sigma$ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in $N$.