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 show english descriptions Geometric structures on low-dimensional manifolds 57M50 - Geometric structures on low-dimensional manifolds

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