Prescribing singularities of maximal surfaces via a singular Björling representation formula

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

“…It turns out that generalized spacelike maximal surfaces admit singular Bjölring representation formula. Theorem 2.2 (Singular Bjölring representation formula for generalized spacelike maximal surfaces [12]). Given an analytic null curve γ : (a, b) → L 3 and an analytic null vector field L : (a, b) → L 3 such that γ (u) ⊥ L(u) and that at least one of γ and L does not vanish identically,…”

“…Following [12], we divide A into three disjoint sets as in the following definition, where D p ( ) is an open disk of radius centered at p. Definition 2.5. We call p ∈ A a shrinking singular point if there is some D p ( ) ⊂ U and a regular embedded curve γ :…”

“…[12], generalized timelike minimal surfaces also satisfy reflection principles with respect to a folded singularity or with respect to a collapsed singularity. …”

“…Note that regular and singular Björling formulae for spacelike maximal surfaces have been already studied in several articles [3,12,16].…”

Björling Formula for Mean Curvature One Surfaces in Hyperbolic Three-Space and in De Sitter Three-Space

“…It should be remarked that, unlike the Riemannian counterparts, the maximal surfaces in L 3 and the CMC 1 surfaces in S representation formula [19,31]. It roughly says that given an analytic null curve and null directions on it perpendicular to the curve, there exists a unique maximal or CMC 1 surface which contains the given null curve as a singular curve and the null directions as the normal directions of the surface.…”