This paper explores connections between Heegaard genus,
minimal surfaces, and pseudo-Anosov monodromies. Fixing
a pseudo-Anosov map φ and an integer n, let Mn
be the 3–manifold fibered over S1 with monodromy φn.
JH Rubinstein showed that for a large enough n every
minimal surface of genus at most h in Mn is homotopic
into a fiber; as a consequence Rubinstein concludes that
every Heegaard surface of genus at most h for Mn is
standard, that is, obtained by tubing together two fibers.
We prove this result and also discuss related results of
Lackenby and Souto.