De Sitter空间中的类空Weingarten超曲面

The Space-like Weingarten' s Hypersurfaces in De Sitter Space

在超曲面几何学中,对主曲率的研究是至关重要的。特别,当主曲率之间满足某种关系式时,这种超曲面存在性的研究是极其有意义的。一般地说,这种问题可归结为解相应地偏微分方程。由于解某些偏分微方程十分困难,目前,许多几何学家设法将偏微分方程转化为常微分方程。本文就是利用这一方法,去确定De Sitter空间S_1^(n+1)中的主曲率k_1,…,K_n满足某一关系的超曲面M。具体地说,有:给定R^(n-1)内开集(0,∞)^(n-1)上一个C^1函数k_n=f(k_1,…,k_(n-1))(n≥2),一定存在S_1^(n+1)内n维类空旋转超曲面M,使得M的n个主曲率k_1,…,k_n恰有上述函数关系。

It is essential to study the principal curvature of hypersurface in the hypersurface geometry. Particularly, the existence of the hypersurface whose principal curvature satisfies some equations is very interesting. In general, the problem can be dealt with by the way of solving some partial differential equations. Today, many geometrists reduce the partial differential equation to the ordinary differential equation. Using this method, the paper obtains: 'Given a c1-functional K=f(K1,…K-l) ( n≥2 ) , on open set(0, ∞ )^(n-1)in R(4-1), there exists a n-dimensional space-like rotatory hypersurface M in De Sitter S_λ^(n+1) so that the principal curvature KI,…, K. of M satisfies exactly the above equation' .