A Bernstein property of solutions to a class of prescribed affine mean curvature equations

Abstract

Let $x: M \rightarrow A^{n+1}$ be a locally strongly convex hypersurface, given as the graph of a locally strongly convex function x _n+1 = z(x ₁, ..., x _n ). In this paper we prove a Bernstein property for hypersurfaces which are complete with respect to the metric $G^{\sharp} = \sum \left( \frac{\partial^{2}z}{\partial x_{i} \partial x_{j}} \right) dx_{i} dx_{j}$ and which satisfy a certain Monge–Ampère type equation. This generalises in some sense the earlier result of Li and Jia for affine maximal hypersurfaces of dimension n = 2 and n = 3 (Li, A.-M., Jia, F.: A Bernstein property of affine maximal hypersurfaces. Ann. Glob. Anal. Geom. 23, 359–372 (2003)), related results (Li, A.-M., Jia, F.: Locally strongly convex hypersurfaces with constant affine mean curvature. Diff. Geom. Appl. 22(2), 199–214 (2005)) and results for n = 2 of Trudinger and Wang (Trudinger, N.S., Wang, X.-J.: Bernstein-Jörgens theorem for a fourth order partial differential equation. J. Partial Diff. Equ. 15(2), 78–88 (2002)).