Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space
Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space
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DOI：
10.1007/bf01195136
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发表时间：
1995-06
影响因子：
0.6
通讯作者：
An-Min Li
中科院分区：
文献类型：
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作者：
An-Min Li
1. 简介。众所周知，双曲空间 H z (_ 1) 可以规范地嵌入到 Minkowski 空间 R 3 中，因为表面 x~+ x~-x~=-l, x3> 0，这是 R 3 中的类空间、完全脐形表面。在 [6] 中，Hano 和 Nomizu 描述了 H2 (-1) 的其他非全等等距嵌入。下面的问题自然出现：对 R~ 中具有恒定内禀高斯曲率-1 的完整类空间曲面进行分类。此类问题向更高维度的推广是对具有恒定高斯-克罗内克曲率 K= i 的完整、类空间、凸超曲面进行分类。在本文中，我们将自己限制在具有有界主曲率的超曲面上。我们将在第 3 节中展示，具有恒定高斯-克罗内克曲率 K= 1 且具有有界主曲率的完整、类空间、凸超曲面的研究可简化为以下方程
1. Introduction. It is well-known that the hyperbolic space H z (_ 1) can be canonically embedded into the Minkowski space R 3 as the surface x~+ x~-x~=-l, x3> 0 which is a spacelike, totally umbilic surface in R 3. In [6] Hano and Nomizu described other noncongruent isometric embeddings of H2 (-1). The following problem arises naturally: classify the complete spacelike surfaces in R~ with constant intrinsic Gauss curvature-1. A generalization to higher dimensions of such a problem is to classify the complete, spacelike, convex hypersurfaces with constant Gauss-Kronecker curvature K= i. In this paper we restrict ourselves to hypersurfaces with bounded principal curvatures. We will show in Section 3 that the study of complete, spacelike, convex hypersurfaces with constant Gauss-Kronecker curvature K= 1 and with bounded principal curvatures is reduced to the following equation