Applying of the surface theory of non-Euclidean spaces to the solution of the Monge-Ampere equation of elliptic type

Solutions of many geometry problems in the whole are related to solutions of differential equations. A.D. Alexandrov’s problem of recovering a surface from a given extrinsic curvature function leads to the solution of the Dirichlet problem for the Monge-Ampere equation in a convex domain. The paper presents a method for generalizing the problem of recovering a convex surface from extrinsic curvature in non-Euclidean spaces. For this purpose, a cylindrical mapping is defined, which is a generalization of the spherical mapping, and a method is given for determining the extrinsic curvature of a convex surface in non-Euclidean spaces. Formulas for the extrinsic curvature of convex surfaces in some specific non-Euclidean spaces are calculated. It has been proven that in non-Euclidean spaces it is possible to generalize the problem of A.D. Alexandrov, which makes it possible to prove the existence of a solution to the Monge-Ampere equation in non-convex and non-simply connected domains with different boundary conditions. The problem of A.D. Alexandrov will lead to the solution of the Monge-Ampere equation, which is a special case of solutions to the equation proved by I.Ya. Bakelman.