Local min-max surfaces and strongly irreducible minimal Heegaard splittings

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 $R\geq 6$.