Monotone homotopies and contracting discs on Riemannian surfaces

A monotone homotopy is a homotopy composed of simple closed curves which are also pairwise disjoint. In this paper, we prove a "gluing" theorem for monotone homotopies; 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 [G. R. Chambers and Y. Liokumovich, Existence of minimal hypersurfaces in complete manifolds of finite volume, arXiv:1609.04058] 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 gamma is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which gamma bounds consisting of curves of length <= L. If epsilon > 0 and q is an element of gamma, then there exists a homotopy that contracts gamma to q over loops that are based at q and have length bounded by 3L + 2d + epsilon, where d is the diameter of the surface. If the surface is a disc, and if gamma is the boundary of this disc, then this bound can be improved to L + 2d + epsilon.