  This paper explores connections between Heegaard genus, minimal surfaces, and
pseudo-Anosov monodromies. Fixing a pseudo-Anosov map phi and an integer n, let
M_n be the 3-manifold fibered over S^1 with monodromy phi^n.
  JH Rubinstein showed that for a large enough n every minimal surface of genus
at most h in M_n is homotopic into a fiber; as a consequence Rubinstein
concludes that every Heegaard surface of genus at most h for M_n is standard,
that is, obtained by tubing together two fibers. We prove this result and also
discuss related results of Lackenby and Souto.
