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June 2019 Properly immersed surfaces in hyperbolic $3$-manifolds
William H. Meeks, Álvaro K. Ramos
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J. Differential Geom. 112(2): 233-261 (June 2019). DOI: 10.4310/jdg/1559786424


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 $N$.

Funding Statement

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


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.


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William H. Meeks. Álvaro K. Ramos. "Properly immersed surfaces in hyperbolic $3$-manifolds." J. Differential Geom. 112 (2) 233 - 261, June 2019.


Received: 9 September 2016; Published: June 2019
First available in Project Euclid: 6 June 2019

zbMATH: 07064404
MathSciNet: MR3960267
Digital Object Identifier: 10.4310/jdg/1559786424

Primary: 53A10
Secondary: 49Q05 , 53C42

Keywords: asymptotic injectivity radius , bounded mean curvature , Calabi–Yau problem , hyperbolic $3$-manifolds , Isoperimetric inequality

Rights: Copyright © 2019 Lehigh University

Vol.112 • No. 2 • June 2019
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