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 L3, with emphasis on their branch points. Roughly speaking, such a surface is given by a conformal mapping from a Riemann surface S in L3. In the last years, several authors [1, 2, 5, 6] have dealt with regular maximal surfaces in L3, i.e. with isometric immersions, with zero mean curvature, of Riemannian 2-manifolds M in L3. 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 L3. 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.

Spacelike Maximal Surfaces, Timelike Minimal Surfaces, and Björling Representation Formulae

“…Definition 2.1 ( [7]). Let S be a Riemann surface and y = (x, y, t) : S → L 3 be nonconstant and harmonic.…”

“…In [7] it is introduced as the unique, up to homothety and translation, spacelike maximal surface in L 3 which is the graph of a function of the form t = f (x) + g(y). Being spacelike, the graph is considered only over the region {(x, y) : tanh 2 x + tanh 2 y < 1}.…”

Nonexistence results and convex hull property for maximal surfaces in Minkowski three-space

“…If D is an open disc and X : D → ‫ތ‬ 3 is a maximal immersion with a spacelike singular point in q ∈ D, the local behavior at the singularity is similar to the case of minimal surfaces in ‫ޒ‬ 3 (see [Dierkes et al 1992;Estudillo and Romero 1992;Fernández et al 2005]): X is not a topological embedding, D−{q} with the induced metric is conformally equivalent to {z ∈ ‫ރ‬ | 0 < |z| < 1}, the Weierstrass data (g, 3 )…”

“…In this line, Bartnik and Simon [1982/83] obtained results on the existence and regularity of spacelike solutions to the boundary value problem for the mean curvature operator in ‫ތ‬ n+1 , and Kobayashi [1984] investigated surfaces with conelike singularities. Estudillo and Romero [1992] defined a class of maximal surfaces with singularities of other types and studied criteria for such a surface to be a plane. On the other hand, Klyachin and Mikyukov [1993] have tackled the problem of existence of solutions to the maximal hypersurface equation in ‫ތ‬ n+1 with prescribed boundary conditions and a finite number of singularities.…”

Causal Characters of Zero Mean Curvature Surfaces of Riemann Type in the Lorentz-Minkowski 3-Space

“…On the other hand, in contrast to the case of minimal surfaces in E 3 , motivated by Calabi's work [1], Cheng and Yau [2] showed that the only complete maximal surfaces are spacelike planes. From this fact, more general classes of maximal surfaces with singularities such as maxfaces in [22] or generalized maximal surfaces in [3] have been frequently studied. Similarly, timelike minimal surfaces with singularities have been studied and many examples constructed.…”