The problem of recovering a surface by the given external curvature and solutions of the Monge–Ampère equation

In this paper, we generalize the concept of the spherical mapping of a surface in Euclidean space. The normal mapping of a surface introduced by I. Ya. Bakelman is a special case of the generalized curvature. We prove general properties of the generalized curvature and special properties of the generalized curvature extended to a hyperbolic cylinder. Using these properties, we prove the existence and uniqueness of a solution of the Monge–Ampère equation in a multiply connected domain.