A Proof of the Collar Lemma
A Proof of the Collar Lemma
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领引理的证明
DOI：
10.1112/blms/13.2.141
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发表时间：
1981
影响因子：
0.9
通讯作者：
N. Halpern
中科院分区：
文献类型：
--
作者：
N. Halpern
设5是一个黎曼曲面，它不是在0，1或2点处穿孔的球面，或亏格为1的闭曲面。则S可以表示为上半平面模Fuchsian群。如果S没有边界曲线，则S上的内禀度量根据定义是庞加莱度量。若S有闭边界曲线，则内禀度量定义为Poincare度量在S的Schottky二重上对S的限制。这与关于SN的Poincare度量对S的限制相同，其中SN表示S的Nielsen扩张（参见[1]）。如果C是闭理想边界曲线或S的一个穿刺，则关于C的一个领被定义为S的一个双连通区域，该区域由C和一条Jordan曲线包围，若C是一条曲线，则该Jordan曲线正交于与C正交的测地弧束，若C是一个穿刺，则该Jordan曲线通过C。如果C是S的一个单闭测地线，则关于C的一个领是两个这样的双连通等面积区域的并。设S是S_N = U/G的曲面，其中U是上半平面，G是Fuchsian群. Mobius变换Zh-> elz将由Xi表示，并且变换Zh + 1由T表示。如果X {e，G和G不连续地作用在负真实的轴上，则线段{iy| 1^ y^ e1}是对应于Xt的S的边界曲线。围绕该边界曲线的面积为k的领是区域
Let 5 be a Riemann surface that is not the sphere punctured at 0, 1 or 2 points, or a closed surface of genus 1. Then S can be represented as the upper half plane modulo a Fuchsian group. If S has no boundary curves then the intrinsic metric on S is, by definition, the Poincare metric. If S has closed boundary curves then the intrinsic metric is defined as the restriction to S of the Poincare metric on the Schottky double of S. This is the same as the restriction to S of the Poincare metric on SN, where SN denotes the Nielsen extension of S (see [1]). All geometric terms will refer to the intrinsic metric.If C is a closed ideal boundary curve or a puncture ofS then a collar about C is defined to be a doubly connected region of S bounded by C and by a Jordan curve orthogonal to the pencil of geodesic arcs orthogonal to C if C is a curve and through C if C is a puncture. If C is a simple closed geodesic of S then a collar about C is the union of two such doubly connected regions of equal area. Let S be a surface with SN= U/G, where U is the upper half plane and G a Fuchsian group. The Mobius transformation z h-» elz will be denoted by X {and the transformation ZHZ+ 1 by T. If X {e G and G acts discontinuously on the negative real axis then the line segment {iy| 1^ y^ e1} is a boundary curve of S corresponding to Xt. A collar of area k around this boundary curve is the region