Monotone homotopies and contracting discs on Riemannian surfaces
Monotone homotopies and contracting discs on Riemannian surfaces
复制标题
黎曼曲面上的单调同伦和收缩圆
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
R. Rotman
中科院分区：
文献类型：
--
作者：
Gregory R. Chambers;R. Rotman
单调同伦是由两两不相交的简单闭曲线组成的同伦。在本文中，我们证明了单调同伦的一个“胶合”定理，我们表明，两个单调同伦，有适当的重叠可以替换为一个单一的单调同伦。用于证明这个定理的思想在[G. R. Chambers和Y. Liokumovich，Existence of minimal hypersurfaces in complete manifolds of finite volume，arXiv：1609.04058]证明了一个类似的结果循环，这形成了一个关键的一步，他们的证明存在极小曲面在完整的非紧流形的有限体积。我们还表明，如果单调同伦存在，那么通过短曲线的不动点收缩存在。特别地，假设[公式：见正文]是黎曼曲面的一条简单闭曲线，并且存在覆盖一个圆盘的单调收缩，该圆盘由长度为[公式：见正文]的曲线构成[公式：见正文]的边界。如果[公式：见正文]和[公式：见正文]，则存在一个同伦，它将[Formula：see text]收缩到[Formula：see text]上的循环，这些循环基于[Formula：see text]，长度由[Formula：see text]限定，其中[Formula：see text]是曲面的直径。如果曲面是一个圆盘，并且[Formula：see text]是这个圆盘的边界，那么这个边界可以改进为[Formula：see text]。
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 [Formula: see text] is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which [Formula: see text] bounds consisting of curves of length [Formula: see text]. If [Formula: see text] and [Formula: see text], then there exists a homotopy that contracts [Formula: see text] to [Formula: see text] over loops that are based at [Formula: see text] and have length bounded by [Formula: see text], where [Formula: see text] is the diameter of the surface. If the surface is a disc, and if [Formula: see text] is the boundary of this disc, then this bound can be improved to [Formula: see text].