Constructing monotone homotopies and sweepouts

2017/04/20 by Chambers, Erin Wolf, Chambers, Gregory R., de Mesmay, Arnaud +2 · 2 citations
#53C23 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.06175

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.

Cited by

• Sweeping x-monotone pseudolines
• Short Simple Orthogonal Geodesic Chords on a 2-Disk with Convex Boundary

Related

• Monotone homotopies and contracting discs on Riemannian surfaces
• Converting homotopies to isotopies and dividing homotopies in half in an effective way
• Optimal sweepouts of a Riemannian 2-sphere
• Contracting thin disks
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