Asymptotic behaviors of hyperbolic hypersurfaces with constant affine Gauss-Kronecker curvature (Q2430803)

It is well-known that, by a Legendre transformation, a hypersurface \(x:M\to A^{n+1}\) with constant affine Gauss-Kronecker curvature 1 of hyperbolic type corresponds to the Monge-Ampère equation \(\text{det}({\partial^2 u\over\partial\xi_i \partial\xi_j})= (-\widetilde u)^{-n-2}\) in \(\Omega\) with \(u=\varphi\) on \(\partial\Omega\) \((\varphi\in{\mathcal C}^\infty(\partial\Omega))\) which is linked with the solution \(\widetilde u\) of the second Monge-Ampère equation \(\text{det}({\partial^2\tilde u\over \partial\xi_i \partial\xi_j})= (-\widetilde u)^{-n-2}\) in \(\Omega\) and \(\widetilde u= 0\) on \(\partial\Omega\). Hereby, the immersion \(x:M\to A^{n+1}\) is considered as the graph of a convex function \(x_{n+1}=f(x_1,\dots,x_n)\), and \(\Omega\) is the Legendre domain of \(f\), a bounded convex domain. Next, the author considers the case where \(\partial\Omega\) is smooth and strictly convex. Then the main theorem states that, for hypersurfaces \(M\) of constant affine Gauss-Kronecker curvature 1 with bounded affine mean curvature, there exist interesting consequences with respect to their asymptotic behaviour.