The Weyl problem with nonnegative Gauss curvature
The Weyl problem with nonnegative Gauss curvature
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DOI：
10.4310/jdg/1214454874
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发表时间：
1994
影响因子：
2.5
通讯作者：
Pengfei Guan;Yanyan Li
中科院分区：
文献类型：
--
作者：
Pengfei Guan;Yanyan Li
Weyl在1916年提出了以下问题：考虑2球S，假设g°是S上的黎曼度规，其高斯曲率处处为正。是否存在一个全局的二氧化碳等距嵌入X:（S, g°）-> (i？， δ)其中δ是欧几里德三维空间i中的标准平面度规？？ 解决这个问题的第一次尝试是由Weyl自己提出的。他提出了连续性方法，并获得了直到二阶导数的先验估计。后来，Lewy[13]在g°是解析的情况下解决了这个问题。1953年，Nirenberg在一篇漂亮的论文中给出了完整的解，在非常温和的假设下，度规g°有连续的四阶导数。他的结果依赖于他对二维均匀椭圆方程的强先验估计。1962年Heinz[9]将结果推广到度规连续三阶导数的情况。Alexandroff[1]用一种完全不同的方法得到了WeyFs问题作为多面体极限的广义解。Pogorelov[18]，[19]证明了该广义解的正则性。
Weyl posed the following problem in 1916 [21]: consider the 2-sphere S and suppose g° is a Riemannian metric on S whose Gauss curvature is everywhere positive. Does there exist a global C 2 isometric embedding X: (S, g°) -> (i?, δ) where δ is the standard flat metric in a Euclidean 3-space i? ? The first attempt to solve the problem was made by Weyl himself. He suggested the continuity method and obtained a priori estimates up to the second derivatives. Later Lewy [13] solved the problem in the case of g° being analytic. The complete solution was given in 1953 by Nirenberg in a beautiful paper [16] under very mild hypothesis that the metric g° has continuous fourth derivatives. His result depends on the strong a priori estimates he had derived for uniformly elliptic equations in dimension two [17]. The result was extended to the case of continuous third derivatives of the metric by Heinz [9] in 1962. In a completely different approach to the problem, Alexandroff [1] obtained a generalized solution of WeyFs problem as a limit of polyhedra. The regularity of this generalized solution was proved by Pogorelov [18], [19].