Journal of Differential Geometry

Properly immersed surfaces in hyperbolic $3$-manifolds

William H. Meeks and Álvaro K. Ramos

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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 $N$.


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.


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

Article information

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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