Construction of higher genus minimal surfaces with one end and finite total curvature
Construction of higher genus minimal surfaces with one end and finite total curvature
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具有一端和有限总曲率的高属极小曲面的构造
DOI：
10.2748/tmj/1178225378
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发表时间：
1996
影响因子：
0.5
通讯作者：
Katsunori Sato
中科院分区：
文献类型：
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作者：
Katsunori Sato
我们证明了欧几里得3-空间中存在具有一个Enneper型末端和有限总曲率的完全最小曲面，其具有两个参数7、k并且属于jk，其中j和k是正整数。我们的主要问题是周期问题'，每个表面有j个周期需要被杀死。我们证明这些时期可以同时被消灭。介绍。最近，已经发现了许多具有更高的亏格和有限的总曲率的极小曲面。 Costa[Co]在R.Hoffman和Meeks[HM2]中发现了属一的完整最小曲面，并证明了其嵌入性，并且他们发现了与Costa曲面类似的更高属嵌入曲面，但具有高阶旋转对称性。 Wohlgemuth [W] 严格证明了几个具有嵌入末端的高等属极小曲面的存在，其中包括 Hotfman、Meeks 和 Callahan 仅在数值上发现的曲面。 1982 年，Chen-Gackstatter [CG] 发现了具有一端且属于第一属和第二属的曲面。属一 C-G 面由 Karcher [K] 推广，属二 C-G 面由 Thayer [T] 推广。这些概括与 C-G 表面类似，但最终具有更高的缠绕阶数。由于 R 中的最小曲面是由 Weierstrass 数据和路径积分给出的，因此我们始终必须检查曲面的明确性，这称为周期问题。 Chen-Gackstatter [CG] 也给出了亏格三曲面的 Weierstrass 数据，但他们没有解决其周期问题。 Thayer [T] 猜想可以解决任意属的周期问题，并给出了数值证据来支持这一点。 Espίrito-Santo [E] 用数值论证解决了属三情况。罗斯曼向我建议使用同伦论证（可以将其视为多个变量的中间值定理）来解决周期问题。 Wohlgemuth [W] 使用同伦论证来解决具有四个嵌入端的最小曲面的周期问题。在本文中，我们利用多变量中间值定理解决了任意亏格（2.1）的广义C-G曲面的周期问题。 1991年数学学科分类。初级53A10；次级 53C42。 230 K.佐藤主要定理。属 jk 的 R 3 中存在单端完全极小曲面 X(Mjtk)，对于所有 j,k=l,2,... ，总曲率为 4(y + \)kπ 。在第一节中，我们总结了有关极小曲面理论的一些基本事实。在第 2 节中，我们介绍广义 C-G 曲面 X(Mjtk) 并讨论它们的对称性。在第 3 节中，我们证明 /? 中的每个 X(Mjtk) 都是完备且正则的。 ，我们看到第 2 节中的对称性可用于减少 yfc toy 的周期数。此外，我们写下了求解周期的具体条件。在第 4 节中，我们重写了 Chen-Gackstatter [CG] 和 Thayer [T] 的一些结果，为第 5 节做准备。在第 5 节中，我们通过假设几个不等式来归纳解决周期问题。最后在第 6 节中，我们证明了这些不等式。评论。一旦我们证明了参数 a2 存在一定的值，. 。 。 , aj9 ceR 使得最小浸入度 (2.1) 得到明确定义，我们说周期问题已解决。尽管我们无法证明 a2 值的唯一性，. 。 。 , aj9 c，我们的数值计算表明这些表面是唯一的。 （最近，在 j= 2 和 fc=1 的情况下，Lopez、Martin 和 Rodriguez [LMR] 已经证明了值 a2 和 c 的唯一性。）有关数值结果和几个猜想的更多信息写在 [T] 中。也感谢裁判和Rossman博士提出的宝贵建议。 1、基本属性。以下定理 1.1-1.3 由 Osserman [Osl][Os2] 给出（另见 [HM1]）。定理 1.1。设 M 为黎曼曲面，g: M->Cu {00} 为亚纯函数，dh 为 M 上的全纯ίc \-形式。我们通过 (1.1) * = (Φι, Φ2, Φι) = (~-gdh, i— + ̂ \dh, 2dh 定义向量值 \-形式 Φ。然后实部 (1.2) PQ 定义到 /? 的最小映射，这是明确定义的对 M 当且仅当
We prove that there exist complete minimal surfaces in the Euclidean 3-space with one Enneper-type end and finite total curvature which have two parameters 7, k and are of genus jk, where j and k are positive integers. Our main problem is the period problem', each surface has j periods to be killed. We prove that these periods can be killed simultaneously. Introduction. Recently, many minimal surfaces with higher genus and finite total curvature have been found. Costa [Co] found a complete minimal surface of genus one in R. Hoffman and Meeks [HM2] proved its embeddedness, and they found higher genus embedded surfaces which are similar to Costa's surface, but with higher-order rotational symmetry. Wohlgemuth [W] proved rigorously the existence of several higher genus minimal surfaces which have embedded ends, including a surface which was found by Hoίfman, Meeks and Callahan only numerically. In 1982, Chen-Gackstatter [CG] found surfaces which have one end and are of genera one and two. The genus one C-G surface was generalized by Karcher [K] and the genus two C-G surface was generalized by Thayer [T]. These generalizations are similar to C-G surfaces, but with higher winding order at the end. Since minimal surfaces in R are given by Weierstrass data and path integrals, we always must check well-definedness of the surfaces, and this is called the period problem. Chen-Gackstatter [CG] also gave Weierstrass data for a genus three surface, but they did not solve its period problem. Thayer [T] conjectured that the period problem can be solved for arbitrary genus and gave numerical evidence to support this. Espίrito-Santo [E] solved the genus three case with a numerical argument. Rossman suggested to me a homotopy argument (which can be thought of as an intermediate value theorem of several variables) for solving the period problem. Wohlgemuth [W] used the homotopy argument to solve period problems for minimal surfaces with four embedded ends. In this paper, we solve the period problem of the generalized C-G surfaces for arbitrary genus (2.1) by using the intermediate value theorem of several variables. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 53C42. 230 K. SATO MAIN THEOREM. There exist one-ended complete minimal surfaces X(Mjtk) in R 3 of genus jk with total curvature — 4(y + \)kπ for all j,k=l,2,... . In Section 1, we summarize some basic fact about minimal surface theory. In Section 2, we introduce the generalized C-G surfaces X(Mjtk) and discuss their symmetries. In Section 3, we show that each X(Mjtk) is complete and regular in /? , and we see that the symmetries in Section 2 can be used to reduce the number of periods fromyfc toy. Furthermore, we write down a concrete condition for solving the periods. In Section 4, we rewrite some of the results of Chen-Gackstatter [CG] and Thayer [T] in preparation for Section 5. In Section 5, we solve the period problem inductively, by assuming several inequalities. Finally in Section 6, we prove these inequalities. REMARK. Once we prove the existence of certain value for the parameters a2, . . . , aj9 ceR so that the minimal immersion (2.1) is well-defined, we say that the period problem is solved. Though we could not show uniqueness for the value of a2, . . . , aj9 c, our numerical computations suggest that the surfaces are unique. (Recently, in the case j= 2 and fc=l, Lopez, Martin and Rodriguez [LMR] have shown the uniqueness for the value a2 and c.) More information about the numerical results and several conjectures are written in [T]. Thanks are also due to the referee and Dr. Rossman for valuable suggestions. 1. Basic properties. The following Theorems 1.1-1.3 were given by Osserman [Osl][Os2] (see also [HM1]). THEOREM 1.1. Let M be a Riemann surface, g: M->Cu {00} a meromorphie function and dh a holomorphίc \-form on M. We define a vector-valued \-form Φ by (1.1) * = (Φι, Φ2, Φι) = (~-gdh, i— + ̂ \dh, 2dh. Then the real part (1.2) PQ defines a minimal mapping into /?, which is well-defined on M if and only if