Short geodesics in hyperbolic 3-manifolds

For each $g \ge 2$, we prove existence of a computable constant $\epsilon(g) > 0$ such that if $S$ is a strongly irreducible Heegaard surface of genus $g$ in a complete hyperbolic 3-manifold $M$ and $\gamma$ is a simple geodesic of length less than $\epsilon(g)$ in $M$, then $\gamma$ is isotopic into $S$.