关于一类四阶偏微分方程解的一个Bernstein性质(英文)

A Bernstein property of solutions to a class fourth order PDE

设x:M→A^(n+1)是由定义在凸域Ω(?)A^n上的某局部严格凸函数x_(n+1)=f(x_1,…,x_n)给出的超曲面.记ρ(x)=det(((?)~2f)/((?)x_i(?)x_j)(x)))^(-1/(n+2)).假设(M,g)是一完备的Hessian流形且具有非负的李奇曲率,作者证明了如果ρ满足△_(9ρ)=β(‖▽ρ‖_g^2)/ρ(β≠1)则M一定是椭圆抛物面.

Let x ： M→An＋1 be a locally strongly convex hypersurface, given by the graph of a convexfunction xn＋1 =f（x1 ,…,xn ） defined in a convex domain Ω（∪）An.The Hessian metric g on M is consid-ered, defined by g=∑a2f/axiaxjdxidxj. Denote ρx=det（（e）2f）/（（e）xi（e）xj）（x）-1/n＋2.Suppose （M,9） is acomplete Hessian manifold with nonnegative Ricci curvature. It is proved that if ρ satisfies△ɡρ=β（‖▽ρ‖2ɡ）/ρ（β≠1）, then x（M） must be an elliptic paraboloid.