相对极值超曲面的Bernstein性质

设x:M→A~(n+1)是一个局部严格凸的超曲面,由Ω(<)A~n上的凸函数x_(n+1)=f(x_1,…,x_n)定义.考虑M上的相对度量G~α=p~(α+1)∑δ~2f/x_ix_jdx_idx_j,其中P=(det(δ~2f/δx_iδx_j))-1/n+2,α为常数.作者对由一个四阶偏微分方程的凸解所给出的局部严格凸超曲面进行了研究,给出了这个非线性偏微分方程凸解的Bernstein性质的证明.

A Bernstein property for relative extremal hypersurfaces

Let x:M→A~(n+1) be a locally strongly convex hypersurface, given by a convex→function x_(n+1)=f(x_1 ,…,x_n) defined in a domain Ω(<)A~n.Consider the relative metric G~α on M, defined by G~α=p~(α+1)∑δ~2f/δx_iδx_jdx_idx_j, where p = (det(δ2~f/δx_iδx_j))-1/(n+2) and a is a constant. In this paper the authors investigate locally strongly convex hypersurface, given by a convex solution of the forth order PDE and prove a Bernstein property of the convex solutions of this nonlinear partial differential equation.