$F$- complete Parabolic Affine Hyperspheres

Abstract:

Let $x:M\rightarrow R^{n+1}$ be a locally strongly convex hypersurface, given by the graph of a locally strongly convex function $x_{n+1}=f(x_{1},...,x_{n})$ defined in a convex domain $D \subset R^{n}$. Defining the $F$- metric $\tilde{G}=F(\rho)\sum\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j}$ on $M$, we derive the PDEs of the $F$- complete parabolic affine hyperspheres and obtain some Bernstein properties.