On the existence of minimal Heegaard surfaces

Daniel Ketover; Yevgeny Liokumovich; Antoine Song

On the existence of minimal Heegaard surfaces

Differential Geometry 2025-12-02 v2 Geometric Topology

Authors: Daniel Ketover , Yevgeny Liokumovich , Antoine Song

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.

Keywords

heegaard surface minimal surface riemannian 3-manifold isotopy index

Cite

@article{arxiv.1911.07161,
  title  = {On the existence of minimal Heegaard surfaces},
  author = {Daniel Ketover and Yevgeny Liokumovich and Antoine Song},
  journal= {arXiv preprint arXiv:1911.07161},
  year   = {2025}
}

Comments

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

On the existence of minimal Heegaard surfaces

Abstract

Keywords

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