Monotone homotopies and contracting discs on Riemannian surfaces

Abstract

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [CL2] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume.We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that γ is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which γ bounds consisting of curves of length ≤ L. If ε > 0 and q ∈ γ, then there exists a homotopy that contracts γ to q over loops that are based at q and have length bounded by 3L + 2d + ε, where d is the diameter of the surface. If the surface is a disc, and if γ is the boundary of this disc, then this bound can be improved to L + 2d + ε.

On the complexity of optimal homotopies

“…Note that the ε-slack of Theorem 1 is also present here but is hidden in the open upper bound of L. In this theorem, as was observed by Chambers and Rotman [11], crediting Liokumovitch, the hypothesis that the manifold is entirely comprised between both curves is necessary: see [11, Figure 5] for a counter-example.…”

“…We begin by considering two curves forming the boundary of a discrete annulus, and study the homotopy between these boundaries of minimal height. Our article leverages on recent results in Riemannian geometry [10,11], and in particular on a companion article co-authored with Gregory Chambers and Regina Rotman [6] where we prove that in the Riemannian setting, such an optimal homotopy can be assumed to be very well behaved. Firstly, it can be assumed to be an isotopy, so that all the intermediate curves remain simple.…”

“…Probably the most extensively considered notion of optimality is the study of homotopies minimizing the area swept; see for example [25] for an overview of some variants of this problem, or [29] for a discussion of how minimum area homotopies and homologies are connected in higher dimensions. The notion of controlling the width of a homotopy has also been studied [5,23], and more recent work on minimal height homotopies [10,11] laid the foundation for our results in this work. On the computational side, the rise of Fréchet distance for measuring similarity between curves was a prime motivation for the notion of comparing two curves; see for example [1] for a survey.…”