Optimal sweepouts of a Riemannian 2-sphere

TL;DR

This paper develops a method to refine sweepouts of Riemannian 2-spheres into simpler forms, leading to new insights into geodesic existence, isotopy conversion, and related geometric questions.

Contribution

It introduces a technique to produce sweepouts with simple or constant curves, impacting min-max geodesic theory and isotopy conversion methods.

Findings

Constructed sweepouts with simple or constant curves

Answered a question on min-max embedded geodesics

Extended homotopy-to-isotopy conversion results

Abstract

Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than L, we construct a second sweepout composed of curves of length less than L which are either constant curves or simple curves. This result, and the methods used to prove it, have several consequences; we answer a question of M. Freedman concerning the existence of min-max embedded geodesics, we partially answer a question due to N. Hingston and H.-B. Rademacher, and we also extend the results of [CL] concerning converting homotopies to isotopies in an effective way.