On the existence of unstable minimal Heegaard surfaces

TL;DR

This paper demonstrates that generic metrics on a 3-sphere produce minimal surfaces with index 1 via min-max methods, and extends results to strongly irreducible Heegaard splittings, confirming a conjecture about their minimal surface representations.

Contribution

It establishes the index of minimal surfaces from min-max procedures for generic metrics and confirms a conjecture relating Heegaard splittings to minimal surfaces in hyperbolic 3-manifolds.

Findings

Minimal surfaces from min-max have index 1 for generic metrics on 3-spheres.

Strongly irreducible Heegaard splittings can be isotoped to minimal surfaces of index at most 1.

A conjecture relating Heegaard splittings and minimal surfaces in hyperbolic 3-manifolds is confirmed.

Abstract

We prove that for generic metrics on a 3-sphere, the minimal surface obtained from the min-max procedure of Simon-Smith has index 1. We prove an analogous result for minimal surfaces arising from strongly irreducible Heegaard sweepouts in 3-manifolds. We also confirm a conjecture of Pitts-Rubinstein that a strongly irreducible Heegaard splitting in a hyperbolic three-manifold can either be isotoped to a minimal surface of index at most 1 or else after a neck-pinch is isotopic to a one-sided minimal Heegaard surface.