in the Minkowski problem are a priori for the solution of the Monge-Ampère equation with the function At that time there was no approach to studying this completely nonlinear equation. A. V. Pogorelov has created the theory of the Monge-Ampère equation by using the geometric methods. First, going from polyhedra, he proved the existence of generalized solutions under natural conditions on the right-hand side. After that he has found the a priori estimates for the derivatives up to the third order inclusively for the regular solutions. Using the a priori estimates, he has proved the regularity of strictly convex solutions, the existence of solutions of the Dirichlet problem and their regularity. The Monge-Ampère equation is an essential component of the Monge-Kantorovich transport problem; it is used in conformal, affine, Kähler geometries, in meteorology and in financial mathematics. A.V. Pogorelov once said about the Monge-Ampère equation: this is a great equation with which I had the honor to work. One of the most conceptual works of A. V. Pogorelov refers to the cycle of works about smooth surfaces of bounded external curvature. A.D. Aleksandrov created a theory of general metric manifolds that naturally generalize Riemannian manifolds. In particular, he introduced the class of two-dimensional manifolds of bounded curvature. They exhaust the class of all metrized two-dimensional manifolds that admit, in a neighborhood of each point, a uniform approximation by Riemannian metrics with absolute integral curvature (i.e., the integral of the module of Gaussian curvature) bounded in aggregate. Naturally, the question arose about the class of surfaces in three-dimensional Euclidean space carrying such a metric with preservation of connections between the metric and the extrinsic geometry of the surface. Partially answering this question, A.V. Pogorelov introduced the class of С1-smooth surfaces with the requirement on the area of a spherical image to be bounded, taking into account the multiplicity of the covering in some neighborhood of each point of the surface. Such surfaces are called surfaces of bounded extrinsic curvature. For such surfaces there is also a very close connection between the intrinsic geometry of the surface and its extrinsic shape: a complete surface with a bounded extrinsic curvature and a nonnegative intrinsic curvature (not equal to zero) is either a closed convex surface or an unbounded convex surface; a complete surface with zero intrinsic curvature and bounded extrinsic curvature is a cylinder. The first work of A. V. Pogorelov on surfaces