Prior estimates of the gradient for solution of a certain Monge-Amp`ere equation

The resolution of the issue of existence and uniqueness of surfaces with given geometric characteristics in various spaces is associated with finding prior estimates for solution of a nonlinear Monge-Ampère differential equation in the corresponding metrics. Such geometric characteristics include Gaussian curvature, average curvature, the sum of principal radii of curvature, etc. The article describes surfaces homeomorphic to the sphere of a unit radius from the class of regular convex in threedimensional space of constant negative curvature with a given function of intrinsic (Gaussian) curvature. Intrinsic curvature is considered as a function of the point of three-dimensional Lobachevsky space. The solution for a Monge-Ampère differential equation is assumed to be a function explicitly given in spherical coordinates. The article describes the procedure for constructing prior estimates of the first derivatives of equation solution. It is assumed the availability of estimates of the solution itself.