New surfaces of constant mean curvature
New surfaces of constant mean curvature
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平均曲率恒定的新曲面
DOI：
10.1007/bf02572424
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发表时间：
1993
影响因子：
0.8
通讯作者：
Karsten Große
中科院分区：
文献类型：
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作者：
Karsten Große
欧氏空间中常平均曲率为1的浸没曲面(以下简称H-曲面)已被许多作者发现。我们想提到以下几点。1841年，Delaunay确定了除球面外的所有旋转曲面[D]，它们都是简单的周期曲面。1970年，Lawson描述了两个双周期曲面[-L]。1984年，Wente发现了浸没的H-Tori，其他人对H-Tori的进一步研究做出了贡献，即Pinkall和Sterling实现了分类[PSI。Karcher在1989年得到了一些三周期H-曲面[KA]。1990年，Kapouleas证明了大量紧致和非紧致H-曲面的存在[KPL，KP2]。Karcher曲面和Kapouleas曲面属于相同拓扑类型的连续族。在这项工作中，我们将Karcher的方法扩展到更广泛的曲面类别，即具有末端的曲面，该方法又基于Lawson的原始思想。我们得到了某些对称曲面族的完整的单参数。这些族的极限情况如下：一方面，如Kapouleas所描述的，其Delaunay形积木具有小颈项且中心是球形的曲面；另一方面，具有同样小颈项但具有新类型中心(渐近为n型)的曲面。在这两者之间，我们找到了颈部尺寸最大的曲面。例如，我们可以将Lawson曲面嵌入到具有相同对称性的连续的单参数曲面族中，并且我们能够证明它们正是具有最大颈长的曲面族中的曲面(定理3.3)。对于端部为n的对称曲面，我们得到了类似的结果，见定理6.1。这项工作安排如下。第一节给出了H-曲面的共轭曲面构造。这基本上是一个几何变换，它将H-曲面的期望基本曲面片的自由边值问题归结为三个球面83中测地多边形的平台问题。如果这个多边形嵌入到H-凸集的边界中，Morrey在$3中对Platform问题的解，然后是共轭，得到了所需的欧几里德H-曲面片。由于Morrey解最小化，只要它们的基本面片足够小，我们就可以得到H曲面，即我们必须假设足够高的对称性。
Immersed surfaces of constant mean curvature 1 in Euclidean three-space (henceforth: H-surfaces) have been found by many authors. We would like to mention the following. In 1841 Delaunay determined all surfaces of revolution [D], which are simply periodic except for the sphere. In 1970 Lawson described two doubly periodic surfaces [-L]. In 1984 Wente discovered immersed H-tori, and others contributed to a further study of H-tori, namely Pinkall and Sterling achieved a classification [PSI. Karcher obtained some triply periodic H-surfaces in 1989 [Ka]. In 1990 Kapouleas proved existence of a wealth of H-surfaces, compact and non-compact [Kpl, Kp2]. Karcher's and Kapouleas' surfaces come in continuous families of the same topological type. In this work we extend Karcher's method which is in turn based on Lawson's original ideas-to a broader class of surfaces, namely to surfaces with ends. We get the full one-parameter family of certain symmetric surfaces. The limiting cases for these families are as follows: surfaces whose Delaunay-shaped building blocks have small necks and the centres are spherical, as described by Kapouleas on the one hand, and surfaces with again small necks but a new type of centre (which asymptotically is n-noid-shaped) on the other hand. In between we find surfaces with maximal neck-size. For example we can embed Lawson's surfaces into a continuous one-parameter family of surfaces with the same symmetry and we are able to prove that they are exactly those in the family having maximal neck-size (Theorem 3.3). We obtain a similar result for the symmetric surfaces with n onduloid ends, see Theorem 6.1. The work is organized as follows. The conjugate surface construction for H-surfaces is given in Sect. 1. Basically this is a geometric transformation which reduces the free bundary value problem for desired fundamental patches of H-surfaces to a Plateau problem for a geodesic polygon in the three-sphere 83. If this polygon is embedded in the boundary of an H-convex set, Morrey's solution to the Plateau problem in $3 followed by conjugation yields the desired Euclidean H-surface patch. Since the Morrey solution minimizes we get H-surfaches provided their fundamental patch is small enough, ie we have to SUppose sufficiently high symmetry.