The Weyl and Minkowski problems in differential geometry in the large
The Weyl and Minkowski problems in differential geometry in the large
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DOI：
10.1002/cpa.3160060303
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发表时间：
1953-08
影响因子：
3
通讯作者：
L. Nirenberg
中科院分区：
文献类型：
--
作者：
L. Nirenberg
Weyl和Minkowski问题是微分几何中两个经典的嵌入问题。这类问题通常减少到有关非线性微分方程的问题，这里处理的问题导致非线性方程的椭圆性质。第一个问题是H。Weyl [29]在1916年提出的，是欧氏空间中凸曲面的一个不同的实现问题。给出了单位球面上正曲率的一个简单几何度量。换句话说，给定一个在单位球面的每一点上定义的正定二次型，其在局部参数（u，v）中采用以下形式：
The problems of Weyl and Minkowski treated in this paper are two classical embedding problems of differential geometry in the large. Such problems usually reduce to questions concerning nonlinear differential equations and those treated here lead to nonlinear equations of elliptic character. Consequently, much of the paper is concerned with questions in the field of elliptic differential equations.The first problem, which was considered by H. Weyl [29] in 1916, is the problem of the realization by a convex surface in Euclidean 3-space of a different. ial geometric metric of positive curvature given on the unit sphere. In other words, one is given a positive definite quadratic form defined at every point of the unit sphere-which in local parameters (u, v) takes, the form