关于一类四阶偏微分方程解的一个Bernstein 性质

设x∶M→An+1是由定义在凸域Ω(∪)An上的某局部严格凸函数 xn+1=f(x1,...,xn)给出的超曲面. 记ρx=det((e)2f)/((e)xi(e)xj)(x)-1/n+2.假设M,ɡ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△ɡρ=β(‖▽ρ‖2ɡ)/ρ(β≠1)则M一定是椭圆抛物面.

A Bernstein property of solutions to a class fourth order PDE

Let x:M→An+1 be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x1,...,xn) defined in a convex domain Ω(＜) An. The Hessian metric ɡ on M is considered, defined by ɡ=∑((e)2f)/((e)xi(e)xj)dxidxj. Denote ρ(x)=(det((e)2f)/((e)xi(e)xj)(x))-1/(n+2).Suppose (M,ɡ) is a complete 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.