2013/11/13 by Chambers, Gregory R., Rotman, Regina · 1 citation
#53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1311.2995
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.