We show if is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus then where denotes the radius of any isometrically embedded ball in . Assuming an unpublished result of Pitts and Rubinstein improves this to We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe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.
"Large embedded balls and Heegaard genus in negative curvature." Algebr. Geom. Topol. 4 (1) 31 - 47, 2004. https://doi.org/10.2140/agt.2004.4.31