  We prove that for generic metrics on a 3-sphere, the minimal surface obtained
from the min-max procedure of Simon-Smith has index 1. We prove an analogous
result for minimal surfaces arising from strongly irreducible Heegaard
sweepouts in 3-manifolds. We also confirm a conjecture of Pitts-Rubinstein that
a strongly irreducible Heegaard splitting in a hyperbolic three-manifold can
either be isotoped to a minimal surface of index at most 1 or else after a
neck-pinch is isotopic to a one-sided minimal Heegaard surface.
