Priori estimates of solutions in the metric $C^0 (S^2_{1})$ equations of Monge-Ampere on the sphere as a two-dimensional manifolds in spaces of constant curvature

The article provides a solution to the problem of finding sufficient conditions for the unique solvability of a differential equation of the Monge-Ampere equation on the sphere as a two-dimensional manifold in spaces of constant curvature, in particular three-dimensional Lobachevskii space. This problem is connected with the restoration of the surfaces, homeomorphic to a sphere with a predetermined function of the Gaussian curvature. In the course of the proof, a priori estimates of solutions of equations of the Monge-Ampere equation in the metric $C^0 (S^2_{1})$. Results investigation for particular types of Monge-Ampere equations in three-dimensional Lobachevskii space, in three-dimensional Euclidean space.