Uniqueness of solutions of the Monge-Ampere a class at a sphere as two-dimensional manifolds

Restoring surfaces for given geometric characteristics is one of the most important and difficult tasks of modern differential geometry. The article presents the proof of uniqueness of solutions adversely elliptical differential equation of the Monge-Ampere equation on the sphere as a two-dimensional manifold linearization method. On the basis of the theorem proved considered a consequence of the uniqueness of a convex surface homeomorphic to a sphere in Euclidean space with a predetermined function of the Gaussian curvature. The conditions for the uniqueness of the surface in the hyperbolic space and elliptic space.