Cuspidal Cross Caps and Singularities of Maximal Surfaces

We shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap. As an application, 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. Introduction Let U be a domain in R and f : U → (N, g) a C∞-map from U into a Riemannian 3-manifold (N, g). The map f is called a frontal map if there exists a unit vector field ν on N along f such that ν is perpendicular to f∗(TU). Identifying the unit tangent bundle T1N 3 with the unit cotangent bundle T ∗ 1N , the map ν is identified with the map L = g(ν, ∗) : U −→ T ∗ 1N. The unit cotangent bundle T ∗ 1N 3 has a canonical contact form μ and L is an isotropic map, that is, the pull back of μ by L vanishes. Namely, a frontal map is the projection of an isotropic map. We call L the Legendrian lift (or isotropic lift) of f . If L is an immersion, the projection f is called a front. Whitney [W] proved that the generic singularities of C∞-maps of 2-manifolds into 3-manifolds can only be cross caps. (For example, fCR(u, v) = (u , v, uv) gives a cross cap.) On the other hand, a cross cap is not a frontal map, and it is also well-known that cuspidal edges and swallowtails are generic singularities of fronts (see, for example, [AGV], Section 21.6, page 336). The typical examples of a cuspidal edge fC and a swallowtail fS are given by fC(u, v) := (u , u, v), fS(u, v) := (3u 4 + uv, 4u + 2uv, v). A cuspidal cross cap is a singular point which is A-equivalent to the C∞-map (see Figure 1) (1) fCCR(u, v) := (u, v , uv), which is not a front but a frontal map with unit normal vector field νCCR := 1 √ 4 + 9u2v2 + 4v6 (−2v,−3uv, 2). Here, two C∞-maps f : (U, p) → N and g : (V, q) → N are A-equivalent (or rightleft equivalent) at the points p ∈ U and q ∈ V if there exists a local diffeomorphism φ of R with φ(p) = q and a local diffeomorphism Φ of N with Ψ(f(p)) = g(q) such that g = Ψ ◦ f ◦ φ−1. In this paper, we shall give a simple criterion for a given singular point on the surface to be a cuspidal cross cap. Let (N, g) be a Riemannian 3-manifold and Ω Date: October 18, 2005. The second author was supported by JSPS Research Fellowships for Young Scientists. 1 2 S. FUJIMORI, K. SAJI, M. UMEHARA, AND K. YAMADA Figure 1. The cuspidal cross cap the Riemannian volume element on N. Let f : U → N be a frontal map defined on a domain U on R. Then we can take the unit normal vector field ν : U → T1N of f as mentioned above. The smooth function λ : U → R defined by (2) λ(u, v) := Ω(fu, fv, ν) is called the signed area density function, where (u, v) is a local coordinate system of U . The singular points of f are the zeros of λ. A singular point p ∈ U is called non-degenerate if the exterior derivative dλ does not vanish at p. When p is a non-degenerate singular point, the singular set {λ = 0} consists of a regular curve near p, called the singular curve, and we can express it as a parametrized curve γ(t) : (−ε, ε) → U such that γ(0) = p and λ ( γ(t) ) = 0 (t ∈ (−ε, ε)). We call the tangential direction γ′(t) the singular direction. Since dλ 6= 0, fu and fv do not vanish simultaneously. So the kernel of df is 1-dimensional at each singular point p. A nonzero tangential vector η ∈ TpU belonging to the kernel is called the null direction. There exists a smooth vector field η(t) along the singular curve γ(t) such that η(t) is the null direction at γ(t) for each t. We call it the vector field of the null direction. In [KRSUY], the following criteria for cuspidal edges and swallowtails are given: Fact. Let f : U → N be a front and p ∈ U a non-degenerate singular point. Take a singular curve γ(t) with γ(0) = 0 and a vector field of null directions η(t). Then (1) The germ of f at p = γ(0) is A-equivalent to a cuspidal edge if and only if the null direction η(0) is transversal to the singular direction γ′(0). (2) The germ of f at p = γ(0) is A-equivalent to a swallowtail if and only if the null direction η(0) is proportional to the singular direction γ′(0) and d dt ∣