Singularities of maximal surfaces

Abstract

Abstract. We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

“…Furthermore, the condition of holding the duality and that the singular set is {(0, t)} are the same between X h and X h (resp. between Like as the remark, a duality between swallowtail and cuspidal cross cap have been pointed out in many researches, for example, [33,13,22]. In this section, we give an interpretation for this duality.…”

“…Simple criteria for cuspidal edge and swallowtail were given by Kokubu, Rossman, Saji, Umehara and Yamada [28]. Other criteria for singularities of frontals are investigated in [13,32,22]. Here, we briefly review the criteria of frontals.…”

A family of maximal surfaces in Lorentz-Minkowski three-space

“…The above shows that the singularities are in F D ∩ {0 ≤ arg α ≤ θ 0 } (Figure 4) [8,17]. They cover all the singularities of X(M k ).…”

Anti de Sitter horospherical flat timelike surfaces

“…The criteria for cuspidal cross cap is the following: F at (s, t) is a cuspidal cross cap if and only if F is not a front, η T λ T ̸ = 0, ϕ = 0 and ϕ ′ ̸ = 0. Applying this criteria, we have (6). 2 We also give the classification of singularities of the flat tangent TAdS-horocyclic surfaces.…”

“…a 2 ,a 3 ) , by the criteria for singularities of fronts and frontals [6,21], we have the following. Proof.…”