Local min-max surfaces and strongly irreducible minimal Heegaard splittings

Abstract

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 $R\ge 6$.

This preprint is superseded by arXiv:1911.07161 [math.DG]

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Cited by in corpus (4)

• Min-max theory for constant mean curvature hypersurfaces
• A proof of the multiplicity one conjecture for min-max minimal surfaces in arbitrary codimension
• Min-max minimal surfaces, horizons and electrostatic systems
• On the existence of unstable minimal Heegaard surfaces