Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space

Abstract

Consider a surface S immersed in the Lorentz-Minkowski 3-space R 3 1 . A complete light-like line in R 3 1 is called an entire null line on the surface S in R 3 1 if it lies on S and consists of only null points with respect to the induced metric. In this paper, we show the existence of embedded space-like maximal graphs containing entire null lines. If such a graph is defined on a convex domain in R 2 , then it must be a light-like plane (cf. Remark 3.3). Our example is critical in the sense that it is defined on a certain non-convex domain.

Reflection principle for lightlike line segments on maximal surfaces

“…See Figure 14, for example. Typical examples of Case 2 are the following hyperbolic catenoid H and the parabolic catenoid P, which are invariant under rotations in L 3 with respect to spacelike and lightlike axes, respectively (see Figure 14 and also [13,14] Also, examples of Case 1 were given in [4], and it is known that such a maximal From the viewpoint of the function theory, in the present article we discussed bounded harmonic mappings. On the other hand, Cases 1 and 2 lead us to considering unbounded harmonic mappings for which the Poisson integrals do not work.…”

Bernstein-Type Theorem for Zero Mean Curvature Hypersurfaces Without Time-like Points in Lorentz–Minkowski Space

“…(As remarked in [5], it is due to Bartnik): IfΩ is not convex, the assertion fails. Such an example is given in [2].…”

Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

“…We first recall the definition of real analytic ZMC DC-immersion from [4] as follows: Let U be a domain in the uv-plane R 2 , and let f : U → R 3 1 be a C 2differentiable map into the Lorentz-Minkowski 3-space R 3 1 of signature (− + +). We set…”

“…which is a singly-periodic surface with a countably infinite number of cone-like singular points, giving an analytic extension of f H (see Examples 2.3 and 2.15). The figures of P and H are found in [9], and they can be expressed as the zero sets of the single analytic functions F P (x, y, t) := 12(x 2 − t 2 ) − (x + t) 4 + 12y 2 , F H (x, y, t) := sin 2 x + y 2 − t 2 respectively, and by this fact, we can show that they are analytically complete (cf. Proposition 1.5).…”

“…In the remainder of this paper, we shall demonstrate that this criterion is actually applicable to concrete examples. In fact, in the last two sections of this paper, it is applied to show the analytic completeness of analytic extensions of constant mean curvature one catenoids in de Sitter 3-space, as follows: Let R 4 1 be the Lorentz-Minkowski 4-space with the metric , of signature (− + + +). Then…”