  Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that
there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is
either isotopic to a minimal surface of index at most one or isotopic to the
stable oriented double cover of a non-orientable minimal surface with a
vertical handle attached. In particular, this proves a result conjectured by
Rubinstein. Some consequences include the existence in any $\mathbb{R}P^3$ of
either a minimal torus or a minimal projective plane with stable universal
cover. In the case of positive scalar curvature, it is shown for spherical
space forms not diffeomorphic to $S^3$ or $\mathbb{R}P^3$ that any strongly
irreducible Heegaard splitting admits a minimal representative in its isotopy
class, and that there is a minimal Heegaard splitting of area less than $4\pi$
if $R\geq 6$.

Differential Geometry