Bernstein Properties for Some Relative Parabolic Affine Hyperspheres

Abstract.

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.