Journal of Differential Geometry

Properly immersed surfaces in hyperbolic $3$-manifolds

William H. Meeks and Álvaro K. Ramos

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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


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