Complete Convex Solutions of Monge–Ampere-type Equations and Their Analogs

In this paper, we study complete convex solutions of certain nonlinear elliptic equations by using geometric methods. We present a proof of the Jörgens–Calabi–Pogorelov theorem about improper convex affine spheres based on the study of complete convex solutions of the simplest Monge–Ampere equation. We consider a similar problem for Monge–Ampere equations of more general type. We prove that, under certain assumptions, solutions of these equations are quadratic polynomials.