On the existence of minimal Heegaard surfaces

Abstract

Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a long-standing conjecture of J. Pitts and J.H. Rubinstein.

v1: This preprint supersedes both arXiv:1706.01037 [math.DG] and arXiv:1709.09744 [math.DG]. v2: Main results unchanged, significant improvements in the exposition, several figures added