Monotone homotopies and contracting discs on Riemannian surfaces

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 $\gamma$ 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 $\le L$. If $ϵ>0$ and $q\in \gamma$, then there exists a homotopy that contracts $\gamma$ to $q$ over loops that are based at $q$ and have length bounded by $3L+2d+ϵ$, 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 $L+2d+ϵ$.