# Constructing monotone homotopies and sweepouts

## TL;DR

This paper proves that homotopies of curves on Riemannian surfaces can be converted into monotone homotopies without increasing length, confirming a conjecture and enabling monotone sweepouts of spheres.

## Contribution

It establishes conditions under which homotopies can be made monotone without length increase, confirming a conjecture and extending to sweepouts of spheres.

## Key findings

- Homotopies can be converted to monotone homotopies without length increase.
- Any sweepout of a Riemannian 2-sphere can be replaced with a monotone sweepout.
- The results confirm a conjecture of Chambers and Rotman.

## Abstract

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.