Generalized maximal surfaces in Lorentz–Minkowski space L³

Abstract

In this paper we carry out a systematic study of generalized maximal surfaces in Lorentz–Minkowski space L 3 , with emphasis on their branch points. Roughly speaking, such a surface is given by a conformal mapping from a Riemann surface S in L 3 . In the last years, several authors [1, 2, 5, 6] have dealt with regular maximal surfaces in L 3 , i.e. with isometric immersions, with zero mean curvature, of Riemannian 2-manifolds M in L 3 . So, the term ‘regular’ means free of branch points. As in the minimal case, a conformal structure is naturally induced on M , which becomes a Riemann surface S . The corresponding isometric immersion is then conformal on S , and it does not have any singular points on S (i.e. points on which the differential of the mapping is not one-to-one). This is the way in which generalized maximal surfaces include regular ones. Moreover, branch points are the singular points of the conformal mapping on S . Whereas branch points of generalized minimal surfaces are isolated, we shall show in Section 2 that, in addition to isolated branch points, a generalized maximal surface in L 3 . may have non-isolated ones, in fact they constitute a 1-dimensional submanifold in a certain open subset of S (see Section 2). So our purpose is two-fold, firstly to study and explain in detail the branch points, and secondly to state several geometric results involving prescribed behaviour of those points on the surface.