On the existence of minimal Heegaard surfaces

2019/11/17 by Daniel Ketover, Ketover, Daniel, Yevgeny Liokumovich +3 · 11 citations
Mathematics · #Boundary (topology) #Combinatorics #Computer science #Conjecture #Curvature #Diffeomorphism #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Heegaard splitting #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Minimal surface #Pure mathematics #Scalar (mathematics) #Scalar curvature #Space (punctuation) #Surface (topology)

paper · open access · doi:10.48550/arxiv.1911.07161

published in CaltechAUTHORS (California Institute of Technology) (California Institute of Technology)

openalex publication_date 2019/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• A dichotomy for minimal hypersurfaces in manifolds thick at infinity

Cited by

• Mean curvature flow in homology and foliations of hyperbolic 3-manifolds
• Index one minimal surfaces in positively curved 3-manifolds
• Local min-max surfaces and strongly irreducible minimal Heegaard splittings
• On the existence of unstable minimal Heegaard surfaces

Related

• Local min-max surfaces and strongly irreducible minimal Heegaard splittings
• On the existence of unstable minimal Heegaard surfaces
• Almost normal Heegaard surfaces
• Critical Heegaard Surfaces
• Critical Heegaard surfaces obtained by amalgamation