Flipping Heegaard splittings and minimal surfaces

TL;DR

This paper demonstrates that the number of genus g embedded minimal surfaces in the 3-sphere increases without bound as g grows, using min-max methods and Heegaard foliation techniques to produce minimal surfaces with controlled topology.

Contribution

It introduces a new approach to constructing infinitely many minimal surfaces of increasing genus in 3-manifolds via stabilization and flipping of Heegaard foliations.

Findings

Number of genus g minimal surfaces in S^3 tends to infinity as g increases.

Constructs minimal surfaces resembling doublings of Clifford torus with curvature blow-up.

Produces index at most 2 minimal surfaces with controlled topology in arbitrary 3-manifolds.

Abstract

We show that the number of genus $g$ embedded minimal surfaces in $\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as $g\rightarrow\infty$, and arise from a two-parameter min-max scheme in lens spaces. More generally, by stabilizing and flipping Heegaard foliations we produce index at most $2$ minimal surfaces with controlled topological type in arbitrary Riemannian three-manifolds.