Large embedded balls and Heegaard genus in negative curvature

Abstract

We show if $M$ is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus $g$ then $g\ge \frac{1}{2}cosh\left(r\right)$ where $r$ denotes the radius of any isometrically embedded ball in $M$. Assuming an unpublished result of Pitts and Rubinstein improves this to $g\ge \frac{1}{2}cosh\left(r\right)+\frac{1}{2}.$ We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls.