## Journal of Differential Geometry

- J. Differential Geom.
- Volume 112, Number 2 (2019), 233-261.

### Properly immersed surfaces in hyperbolic $3$-manifolds

William H. Meeks and Álvaro K. Ramos

#### 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 integer multiplicity) to a totally umbilic annulus, properly embedded in $N$.

#### Note

This material is based upon work for the NSF under Award No. DMS-1309236. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF.

#### Note

Both authors were partially supported by CNPq-Brazil, grant no. 400966/2014-0.

#### Article information

**Source**

J. Differential Geom., Volume 112, Number 2 (2019), 233-261.

**Dates**

Received: 9 September 2016

First available in Project Euclid: 6 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1559786424

**Digital Object Identifier**

doi:10.4310/jdg/1559786424

**Mathematical Reviews number (MathSciNet)**

MR3960267

**Zentralblatt MATH identifier**

07064404

**Subjects**

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

**Keywords**

Calabi–Yau problem hyperbolic $3$-manifolds asymptotic injectivity radius bounded mean curvature isoperimetric inequality

#### Citation

Meeks, William H.; Ramos, Álvaro K. Properly immersed surfaces in hyperbolic $3$-manifolds. J. Differential Geom. 112 (2019), no. 2, 233--261. doi:10.4310/jdg/1559786424. https://projecteuclid.org/euclid.jdg/1559786424