Bernstein theorems for complete α-relative extremal hypersurfaces

Abstract

Let \({y : M \rightarrow \mathbb{R}^{n+1}}\) be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space \({\mathbb{R}^{n+1}}\), given as graph of a strictly convex function xn+1 = f(x1, . . . ,xn) defined on a domain \({\Omega \subset \mathbb{R}^{n}}\). Considering the Li-normalization of the graph of the convex function f, we will prove the Bernstein theorems for relative extremal hypersurfaces with complete α-metrics. As one case of main theorem, we obtain that affine complete maximal surface given by graph is an elliptic paraboloid, which is a special case of Calabi conjecture about affine maximal surfaces.