Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the existence of minimal Heegaard surfaces

Daniel Ketover, Yevgeny Liokumovich, Antoine Song

TL;DR

The paper proves Pitts–Rubinstein’s conjecture by showing that a strongly irreducible Heegaard surface in a closed 3-manifold is isotopic to either an index ≤1 minimal surface or to the boundary of a tubular neighborhood of a non-orientable minimal surface with a vertical handle attached; the authors achieve this via a two-stage interpolation framework, a local min-max theory in cores with stable boundaries, and a neck-opening mechanism grounded in the Light Bulb Theorem. The construction yields precise topological control of the resulting minimal surfaces and provides a robust analytic alternative to combinatorial approaches, with several topological and geometric corollaries in hyperbolic, lens space, and positive scalar curvature settings. The methods combine refined area-control isotopies, stacking techniques near stable boundaries, and local min-max arguments to ensure interior minimality and index bounds, culminating in a broad applicability to the classification and finiteness of minimal Heegaard splittings. The results thus extend the landscape of minimal surface theory in 3-manifolds, offering both sharp existence statements and a toolkit for future analytic-topological investigations.

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.

On the existence of minimal Heegaard surfaces

TL;DR

The paper proves Pitts–Rubinstein’s conjecture by showing that a strongly irreducible Heegaard surface in a closed 3-manifold is isotopic to either an index ≤1 minimal surface or to the boundary of a tubular neighborhood of a non-orientable minimal surface with a vertical handle attached; the authors achieve this via a two-stage interpolation framework, a local min-max theory in cores with stable boundaries, and a neck-opening mechanism grounded in the Light Bulb Theorem. The construction yields precise topological control of the resulting minimal surfaces and provides a robust analytic alternative to combinatorial approaches, with several topological and geometric corollaries in hyperbolic, lens space, and positive scalar curvature settings. The methods combine refined area-control isotopies, stacking techniques near stable boundaries, and local min-max arguments to ensure interior minimality and index bounds, culminating in a broad applicability to the classification and finiteness of minimal Heegaard splittings. The results thus extend the landscape of minimal surface theory in 3-manifolds, offering both sharp existence statements and a toolkit for future analytic-topological investigations.

Abstract

Let be a strongly irreducible Heegaard surface in a closed oriented Riemannian -manifold. We prove that 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.

Paper Structure

This paper contains 26 sections, 43 theorems, 209 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Strongly irreducible Heegaard splittings
  3. Neck-pinch surgeries
  4. Tubular neighborhoods
  5. Surgeries on Heegaard splittings
  6. Interpolation: statement and outline of proof
  7. The case of connected stable minimal boundary.
  8. The case of multiple connected components.
  9. Interpolation I: deformation to a stacked surface
  10. Isotopy with surgeries.
  11. Removing the thin part of $\Gamma$
  12. Squeezing maps.
  13. Stacked surface.
  14. Choice of constants $\hat{r}>0$, $\hat{h}>0$, and open neighborhood $\Omega_{\hat{h}}$
  15. Choice of triangulation and constant $c$.
  16. ...and 11 more sections

Key Result

Theorem 1.1

If $H$ is a Heegaard surface of genus $g$ in an oriented Riemannian 3-manifold, then there exists a sequence of surfaces $\Sigma_i$ isotopic to $H$, and pairwise disjoint embedded minimal surfaces $\Gamma_1,...,\Gamma_k$ as well as positive integers $n_1,...,n_k$ so that Moreover, there holds where $\mathcal{O}$ denotes the subcollection of indices $j$ such that $\Gamma_j$ is orientable, $\mathc

Figures (8)

  • Figure 1: If $\Sigma^i_0 = \partial M$ and $\Sigma^i_{t_0- \varepsilon_i}$ is weakly close to $\partial M$ with multiplicity two, then $\{\Sigma^i_t\}_{t\in [0,t_0-\varepsilon_i]}$ sweeps out most of $M$.
  • Figure 2: The surface $\Gamma$ is within $\varepsilon$ (in varifold norm) from $\Sigma_1 +2 \Sigma_2$, where $\Sigma_1$ is a stable minimal surface of genus 2 and $\Sigma_2$ is a stable minimal sphere. We can isotope $\Gamma$ to $\Sigma_1$ while increasing its area by an arbitrarily small amount.
  • Figure 3: Two graphical sheets joined by a knotted neck. There is a homotopy, but no isotopy pushing the surface into the boundary of the cell, so $k_{ess} = 2$.
  • Figure 4: Deformation reducing the number of connected components of $\tilde{\Gamma} \cap C_{\hat{r}/2}(p_j^i)$ non-contractible in $C_{\hat{r}/2}(p_j^i)$.
  • Figure 5: Changing the over-under crossing in the proof of the Light Bulb Theorem.
  • ...and 3 more figures

Theorems & Definitions (85)

  • Theorem 1.1: Simon--Smith (1983)
  • Conjecture 1: J. Pitts and J.H. Rubinstein 1980s Rubinsteinnotes
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3: No Nesting - Lemma 2.2 in Scharlemann
  • Lemma 2.4
  • proof
  • ...and 75 more