The existence of infinite polygons in a two-dimensional metric of variable negative curvature

We prove the existence of certain classes of convex noncompact domains on two-dimensional manifolds of variable negative Gaussian curvature. These domains are defined as the intersection of finitely or countably many half-planes (a half-plane is one of the two parts of the whole manifold bounded by a geodesic) whose boundaries have no common points. The boundary of such convex noncompact domains consists of complete geodesics. In this paper, these domains are called infinite polygons (IP for short). There exist IPs in the Lobachevsky plane.