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Let $(M,g)$ be a closed oriented Riemannian 3-manifold not diffeomorphic to the 3-sphere, and suppose that there is a strongly irreducible Heegaard splitting. H. Rubinstein conjectured that either there is a minimal surface of index at most one isotopic to $H$ or there is a non-orientable minimal surface such that the double cover with a vertical handle attached is isotopic to $H$. We will explain how to prove this, building on some ideas of Rubinstein. The key point is a version of min-max theory producing interior minimal surfaces when the ambient manifold has a non-empty stable minimal boundary. Some corollaries of the theorem include the existence in any $RP^3$ of either a minimal torus or a minimal projective plane with stable universal cover. Several consequences for metric with positive scalar curvature are also derived.

Joint work with Yevgeny Liokumovich and Dan Ketover.