Stability of the Graph of a Convex Function

Abstract

Let x: M → A n+1 be the graph of some strongly convex function x n+1= ƒ( x1,…,xn) defined on a domain Ω ⊂ A n in a real affine space. We consider the relative metric G, defined by \( G=\sum{\partial^{2}f\over\partial x_{i}\partial x_{j}}dx_{i}dx_{j}\).In this paper, we calculate the second variation of the area integral with respect to the relative metric G. We prove that the parabolic affine hyperspheres are stable.