A Bernstein Property for F Relative Parabolic Affine Hyperspheres

Abstract

Let \(\mathbf f : M\rightarrow R^{n+1}\) be a locally strongly convex hypersurface, locally given as graph of a strongly convex function \(\xi _{n+1}=u(\xi _1,\xi _2,\ldots ,\xi _n)\) defined on a convex domain in n dimensional affine space \(R^{n}\). We study a relative affine differential geometry with the conormal field U given by \(U=F^{-1}(-u_1,-u_2,\ldots ,-u_n,1)\), where \(F>0\) is a function of \(\rho \), called F normaliztion. We derive the differential equations satisfied by the F relative affine hyperspheres and prove a Bernstein property of these hypersurfaces.