Geometric problems related to the Monge–Ampere equation

Many questions in the theory of convex surfaces in their analytical interpretation lead to questions of the existence and uniqueness of solutions to partial differential equations. Often these are Monge–Ampere equations of elliptic type.
First, we give definitions of some concepts of geometric characteristics of a convex surface related to this problem.
Then we talk about A.V.Pogorelov’s theorem on the existence and uniqueness of convex polyhedra with a back monotonic function of polyhedral angles at the vertices. The transition to regular surfaces is carried out by A.D.Aleksandrov’s method.