黎曼曲面上的单调同伦与收缩圆
Monotone homotopies and contracting discs on Riemannian surfaces
摘要：
我们证明了单调同伦的“胶合”定理，单调同伦是一个同伦通过简单的可收缩封闭曲线，本身是两两不相交。我们证明了两个具有适当重叠的单调同伦可以被一个单调同伦所取代。用于证明这一定理的思想在[CL 2]中被用来证明一个类似的结果，循环，这形成了一个关键的一步，他们的证明存在的极小曲面在完整的非紧流形的有限体积。我们还证明，如果单调同伦存在，则通过短曲线的不动点收缩存在。特别地，假设$γ$是黎曼曲面的简单闭曲线，并且存在覆盖$γ$所界定的由长度为$\leq L$的曲线组成的圆盘的单调收缩。如果$ε> 0$和$q \in γ$，则存在一个同伦，该同伦在以$q$为基础且长度以$3L + 2d + ε$为界的循环上将$γ$收缩为$q$，其中$d$是曲面的直径。如果曲面是一个圆盘，并且$γ$是该圆盘的边界，则该边界可以改进为$L + 2d + ε$。
英文摘要：
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 $\leq L$. If $ε> 0$ and $q \in γ$, 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 $L + 2d + ε$.
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