arXiv subjects
Publications and source records attributed to William Breslin.
We show that a complete hyperbolic n-manifold has a geodesic triangulation such that the tetrahedra contained in the thick part are L-bilipschitz diffeomorphic to the standard Euclidean n-simplex, for some constant L depending only on the dimension and the constant used to define the thick-thin decomposition of M.
We show that if $\mathcal{L}$ is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of $\mathcal{L}$ are all in the interval $(-δ,δ)$ for a fixed $δ\in[0,1)$ and no complimentary region of $\mathcal{L}$ is an interval bundle over a surface, then each boundary leaf of $\mathcal{L}$ has a nontrivial fundamental group. We also prove existence of a fixed constant $δ_0 > 0$ such that if $\mathcal{L}$ is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of $\mathcal{L}$ are all in the interval $(-δ_0 ,δ_0)$ and no complimentary region of $\mathcal{L}$ is an interval bundle over a surface, then each boundary leaf of $\mathcal{L}$ has a noncyclic fundamental group.
For each $g \ge 2$, we prove existence of a computable constant $ε(g) > 0$ such that if $S$ is a strongly irreducible Heegaard surface of genus $g$ in a complete hyperbolic 3-manifold $M$ and $γ$ is a simple geodesic of length less than $ε(g)$ in $M$, then $γ$ is isotopic into $S$.
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.
We study principal curvatures of fibers and Heegaard surfaces smoothly embedded in hyperbolic 3-manifolds. It is well known that a fiber or a Heegaard surface in a hyperbolic 3-manifold cannot have principal curvatures everywhere less than one in absolute value. We show that given an upper bound on the genus of a minimally embedded fiber or Heegaard surface and a lower bound on the injectivity radius of the hyperbolic 3-manifold, there exists a $δ> 0$ such that the fiber or Heegaard surface must contain a point at which one of the principal curvatures is greater than $1 + δ$ in absolute value.