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# Curvature Bounds for Surfaces in Hyperbolic 3-Manifolds

Published:2010-07-06
Printed: Oct 2010
• William Breslin,
Department of Mathematics, University of Michigan, Ann Arbor, MI, U.S.A.
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## Abstract

A triangulation of a hyperbolic $3$-manifold is \emph{L-thick} if each tetrahedron having all vertices in the thick part of $M$ is $L$-bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant $L$ such that every complete hyperbolic $3$-manifold has an $L$-thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of $\pi_1$-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic $3$-manifolds.
 MSC Classifications: 57M50 - Geometric structures on low-dimensional manifolds