Locally strongly convex hypersurfaces with constant affine mean curvature

Let x : M -> A(n+1) be a locally strongly convex hypersurface, given by a strictly convex function x(n+1) = f(x(1), ... , x(n)) defined in a convex domain Omega subset of A(n). We consider the Riemannian metric G(#) on M, defined by G(#) = Sigma partial derivative(2)f / partial derivative x(i)partial derivative x(j) dx(i) dx(i). 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. (c) 2004 Elsevier B.V. All rights reserved.