  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 $\leq L$. If $\epsilon > 0$ 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 $3L + 2d + \epsilon$, 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 $L + 2d + \epsilon$.
