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Properly Immersed Surfaces in Hyperbolic 3-Manifolds by Meeks, William H.; Ramos, Álvaro K. is a scholarly article available to read on EtoBox.

What is Properly Immersed Surfaces in Hyperbolic 3-Manifolds about?

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 $N$.

Author

Meeks, William H.; Ramos, Álvaro K.

Published

2016

Language

EN