Locally strongly convex hypersurfaces with constant affine mean curvature

Let x:M→An+1 be a locally strongly convex hypersurface, given by a strictly convex function xn+1=f(x1,…,xn) defined in a convex domain Ω⊂An. We consider the Riemannian metric G# on M, defined by G#=∑∂2f∂xi∂xjdxidxj. In this paper we prove that if M is a locally strongly convex surface with constant affine mean curvature and if M is complete with respect to the metric G#, then M must be an elliptic paraboloid.