On minimal lightlike surfaces in Minkowski space–time

Weierstrass Representation of Lightlike Surfaces in Lorentz-Minkowski 4-Space

“…The mean curvature H cannot be defined for lightlike surfaces in the same way as it is defined for spacelike and timelike surfaces. Therefore, to define minimal lightlike surfaces, a different approach is used ( [7]).…”

“…This characterization was used as a definition of a minimal lightlike surface in [7]. It was further proved that locally there are two disjoint classes of lightlike surfaces which satisfy the definition: ruled surfaces with lightlike rulings and the so-called l-minimal surfaces.…”

“…Since in M 4 there exit lightlike surfaces which are not minimal, our aim is to find necessary and sufficient conditions on the functions (ρ, f, g) such that the lightlike surface given by (3.6) is minimal. We require the following lemma, which was proved in [7].…”

“…The first is that the vector fields x v and x vv are collinear, which describes the class of ruled lightlike surfaces. The second possibility is that x v and x vv are linearly independent and x uv is their linear combination, which gives exactly the class of l-minimal surfaces ( [7]).…”

“…is l-minimal. This parametrization is based on a very general example given in [7] on how to construct an lminimal translation surface in M n , n ≥ 4. We will only comment here that such a parametrization in M 4 cannot be conformal because the only way to decompose…”