Curvature Bounds for Surfaces in Hyperbolic 3-Manifolds

Abstract

AbstractA triangulation of a hyperbolic 3-manifold is 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 π1-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds.