Boundary estimates of solutions for a class of Monge-Ampère equations (Q2402689)

It is known that the classification of Euclidean-complete hyperbolic affine hyperspheres is reduced to the study of the boundary value problem: \[ \begin{cases} \det\left(\frac{\partial^2u}{\partial x_i\partial x_j}\right)=(-u)^{-n-2}&\text{ in }\Omega,\\ u=0&\text{ on }\partial\Omega,\end{cases} \] where \(\Omega\) is a bounded convex domain in the Euclidean \(n\)-space \(\mathbb R^n\). In this paper, the author first establishes some boundary estimates of the solution to the above equation when \(\Omega\) is a strictly convex bounded domain with smooth boundary in \(\mathbb R^2\). On the unit ball in \(\mathbb R^n\), he also gives some sharp estimates of the solutions of a class of Monge-Ampère equations which is an extension of the above equation and occurs in the study of the hyperbolic hypersurfaces with prescribed Gauss-Kronecker curvature. As an application, he considers the two linked Monge-Ampère equations and proves that a relative hyperbolic hypersurface with Li-normalization constructed from the solution has constant Gauss-Kronecker curvature and bounded principal curvatures.