We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature κ satisfy the inequality K 2 − κ 2 > 0. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that any minimal Lorentz surface of general type is determined (up to a rigid motion) by two invariant functions satisfying a system of two natural partial differential equations. Using a concrete solution to this system we construct an example of a minimal Lorentz surface of general type.2010 Mathematics Subject Classification. Primary 53B30, Secondary 53A35, 53B25.

Abstract. We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.

Abstract. We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classification of minimal general rotational surfaces of elliptic and hyperbolic type, general rotational surfaces with parallel normalized mean curvature vector field, flat general rotational surfaces, and general rotational surfaces with flat normal connection.

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