Bernstein Property Of Solution To A Class Forth Order PDE

Let x: M→Aⁿ⁺¹ be a locally strongly convex hypersurface,given by the graph of a convex function x_n+1=f(x₁,…, x_n) defined in a convex domainΩ(?) Aⁿ. We consider the Hessian metric g on M, defined byg=Σ((?)²f)/((?)x_i(?)x_j) dx_idx_j. Denote p(x)=(det(((?)²f)/((?)x_i(?)x_j)(x)))^-1/(n+2). Suppose (M,g)is a complete Hessian manifold with non-negative Ricci curvature. Ifρsatisfiesâ–³_gρ=β(‖▽ρ‖_g²)/ρ(β≠1), then x(M) must be an elliptic paraboloid.