The Problem of Plateau on a Riemannian Manifold
The Problem of Plateau on a Riemannian Manifold
复制标题
DOI：
10.2307/1969401
复制
发表时间：
1948-10
影响因子：
4.9
通讯作者：
C. B. Morrey
中科院分区：
文献类型：
--
作者：
C. B. Morrey
Plateau问题是变分学中的经典问题之一，它证明了欧氏空间中由给定的若当围线包围的最小面积曲面的存在性。这个问题已经研究了至少从时间的黎曼，但直到1930年，令人满意的证明，一般轮廓是独立的道格拉斯[25]和雷达[15]。然而，在此之前，收集了大量信息，并在许多有趣的特殊情况下解决了这个问题。对于这一时期的文献，我们参考Rad 6 [38]关于高原问题的出色报告。在道格拉斯和拉多的解出现后不久，麦克沙恩提出了一个有趣的解，这个解可能更沿着简单的“变分法的直接方法”（见[30]）。在过去的17年中，通过用几个轮廓r1，*，rk代替r并通过规定所需表面的拓扑结构，已经将该问题推广[2019 - 05 - 19][2019 - 05][或者通过将表面由给定轮廓限定的条件替换为仅规定边界的部分的条件，边界的其余部分仅限于位于沿着某些流形（见[18]，[22]，[23]，[24]）。最后，讨论了不提供甚至相对最小值的极小曲面（参见[20]，[21]，[31]，[32]，[33]，[34]，[35]，[39]，[40]，[42]，[43]）。本文推广的问题，通过替换基本的欧氏空间的黎曼流形相当的一般性。作者只知道一篇在欧氏空间以外的空间中解决了Plateau问题的论文，即A. Lonseth [7]，其中所考虑的空间是“双曲空间”。然而，S. Bochner [2]研究了在一般黎曼度量（充分可微）下的调和函数，但没有证明它们的极小化性质。这项工作当然是非常感兴趣的，因为它仍然是真实的黎曼空间（足够的可微性），如果一个最小的表面是共形映射在一个平面区域，然后代表向量是“谐波”在博赫纳的意义上，也是一个表面的最小面积（如果足够的可微性）是一个最小的表面（在意义上的微分几何）。本文作者并不试图推广前面关于Plateau问题的所有结果，而是把自己限制在由k条给定轮廓线围成的k重连通平面区域类型的曲面的情形。这个空间是一个C '类的“齐次正则的”（见定义3.1）黎曼流形9）。每一个紧致的，正则的C'类黎曼流形被认为是齐次的，
The problem of Plateau or the proof of the existence of a surface of least area bounded by a given Jordan contour in Euclidean space is one of the classical problems of the Calculus of Variations. This problem has been studied at least since the time of Riemann but it was not until 1930 that satisfactory proofs for a general contour were given independently by Douglas [25]' and Rado [15]. However, before that time, considerable information was gathered and the problem was solved in many interesting special cases. For the literature of this period, we refer to the excellent report of Rad6 [38] on the problem of Plateau. Shortly after the appearance of the solutions of Douglas and Rado, McShane presented an interesting solution which was perhaps more along the lines of the straightforward "direct methods of the Calculus of Variations" (see [30]). In the past seventeen years the problem has been generalized by replacing r by several contours r1, * * *, rk and by prescribing the topological structure of the desired surface ([5], [19], [27], [28], 129], [33], 135], [41], [43]), or by replacing the condition that the surface be bounded by a given contour by the condition that only parts of the boundary be prescribed, the remainder of the boundary being merely restricted to lie along certain manifolds (see [18], [22], [23], [24]). Finally minimal surfaces which do not furnish even relative minima have been discussed (see [20], [21], [31], [32], [33], [34], [35], [39], [40], [42], [43]). The present paper generalizes the problem by replacing the underlying Euclidean space by a Riemannian manifold of considerable generality. The writer knows of only one paper in which the Plateau problem has been solved in any space other than Euclidean space, namely the recent paper by A. Lonseth [7] where the space considered is "hyperbolic space". However, S. Bochner [2] has studied the "harmonic" functions in a general Riemannian metric (of sufficient differentiability) but has not demonstrated their minimizing property in general. This work is, of course of great interest as it is still true in a Riemannian space (of sufficient differentiability) that if a minimal surface is mapped conformally on a plane region, then the representing vector is "harmonic" in Bochner's sense and also that a surface of least area (if of sufficient differentiability) is a minimal surface (in the sense of Differential Geometry). The present writer does not attempt to generalize all of the preceding results on the Plateau problem but restricts himself to the case of a surface of the type of a k-fold connected plane region bounded by k given contours. The space is a "homogeneously regular" (see Def. 3.1) Riemannian manifold 9) of class C'. Every compact, regular Riemannian manifold of class C' is seen to be homo-