Geodesics with bounded intersection number on surfaces are sparsely distributed
Geodesics with bounded intersection number on surfaces are sparsely distributed
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DOI：
10.1016/0040-9383(85)90056-4
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发表时间：
1985
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影响因子：
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通讯作者：
J. Birman;C. Series
中科院分区：
文献类型：
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作者：
J. Birman;C. Series
设M为一个具有负欧拉特征的曲面，可能有边界，该曲面要么紧致，要么由紧致曲面通过去除有限的点集而得到。设D为庞加莱圆盘。选取M的任意表示为U/I'，其中U _~ D为M和F c Isom (D)的泛覆盖空间。然后D上的庞加莱度规在M上推导出一个常负曲率的度规，U上的测地线投影到M上的测地线上，如果M上的测地线在两个方向上都是闭合且光滑的，或者是开放且无限长的，则称其为完备的。完整的测地线与不与0M相交的测地线重合。注意，如果M是通过去除有限个数的点形成顶点而从紧致曲面得到的，那么M上的完全开放测地线可能沿着顶点趋向无穷远。本文研究了最多有k个自交的完备测地线族，k i >0 0。我们的主要结果是：
LET M be a surface of negative Euler characteristic, possibly with boundary, which is either compact or obtained from a compact surface by removing a finite set of points. Let D be the Poincar~ disc. Choose any representation of M as U/I', where U _~ D is the universal covering space of M and F c Isom (D). Then the Poincar~ metric on D induces a metric of constant negative curvature on M and geodesics in U project to geodesics on M. A geodesic on M is said to be complete if it is either closed and smooth, or open and of infinite length in both directions. Complete geodesics coincide with those which never intersect 0M. Note that if M is obtained from a compact surface by removing a finite number of points to form cusps then a complete open geodesic on M might tend toward infinity along a cusp. In this paper we study the family G k of complete geodesics which have at most k transversal self-intersections, k i> 0. Our main results are: