Stability of the graph of a convex function (Q1428273)

Let \(x:M\to \mathbb R^{n+1}\) be the graph of some strongly convex function \(x_{n+1}= f(x_1,\dots,x_n)\) defined on a domain \(\Omega \subset \mathbb R^n\). There exist a positive Riemannian metric \(G\), defined by \(G=\sum\frac{\partial^2 f}{\partial x_i\partial x_j}\,dx_idx_j\), and a volume element \(dV=\sqrt{\det(G_{ij})}dx_1 \wedge\cdots\wedge dx_n\). In this paper, the author calculates the first and the second variation of the volume, and shows that the parabolic affine hyperspheres are stable.