The classification of singly periodic minimal surfaces with genus zero and scherk-type ends
The classification of singly periodic minimal surfaces with genus zero and scherk-type ends
复制标题
具有零亏格和舍克型末端的单周期极小曲面的分类
DOI：
10.1090/s0002-9947-06-04094-3
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发表时间：
2006
影响因子：
1.3
通讯作者：
M. Traizet
中科院分区：
文献类型：
--
作者：
Joaquín Pérez;M. Traizet
给定一个大于2的整数k，设S（k）是R3中的完备嵌入单周期极小曲面空间，其商亏格为零，有2k个Scherk型端点. S（k）中的曲面可以被证明是适当的，在这个条件下曲面的渐近几何是众所周知的。S（2）由单周期Scherk极小曲面的单参数族组成。本文证明了：对于任意k ≥ 3，通过映射给任意M ∈ S（k）分配一个边是其端点的通量向量的多边形（特殊多边形是一个两边长度为1，两边长度为k - 1的三角形），S（k）与凸酉非特殊多边形空间之间存在自然的一一对应.因此，S（k）简化为Karcher构造的鞍形塔。
Given an integer k > 2, let S(k) be the space of complete embedded singly periodic minimal surfaces in R 3 , which in the quotient have genus zero and 2k Scherk-type ends. Surfaces in S(k) can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is'well known. It is also known that S(2) consists of the 1-parameter family of singly periodic Scherk minimal surfaces. We prove that for each k ≥ 3, there exists a natural one-to-one correspondence between S(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M ∈ S(k) the polygon whose edges are the flux vectors at the ends of M (a special polygon is a parallelogram with two sides of length 1 and two sides of length k - 1). As consequence, S(k) reduces to the saddle towers constructed by Karcher.