Bernstein Properties for Some Relative Parabolic Affine Hyperspheres

摘要

Let y : $M rightarrow {mathbb{R}^{n+1}}$ be a locally strongly convex hypersurface immersion of a smooth, connected manifold into real affine space ${mathbb{R}^{n+1}}$ , given as graph of a strictly convex function x n+1 = f(x 1, … , x n ) defined on a convex domain $Omega subset {mathbb{R}}^n$ . Let Y = (0, 0, … , 1) denote the canonical relative normal of the hypersurface, then the associated conormal field U is given by $U = (-frac{{partial{f}}}{partial{x_{1}}},ldots,-frac{{partial f}}{partial{x_{n}}},,1)$ . In this paper, we define another relative normalization in terms of the conormal vector field $tilde{U} = [{rm det}(frac{partial^{2}f}{partial x_i partial x_j})]^{-frac{alpha}{n+2}},U$ , where $alpha in {mathbb{R}}$ is a constant. With this relative normalization, the relative parabolic affine hyperspheres satisfy a system of fourth order nonlinear PDEs (see (1.2) below). We study these PDEs and obtain some Bernstein properties of relative parabolic affine hyperspheres.