It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

We investigate flat surfaces in hyperbolic 3-space with admissible singularities, called flat fronts. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded, and give new examples for which equality holds.

After Gálvez, Martínez and Milán discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space H 3 , the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H 3 . In the present work we broaden the notion of completeness to weak completeness, and of front to p-front. As a front is a p-front and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces.This more general class contains the caustics of flat fronts -shown also to be flat by Roitman (who gave a holomorphic representation formula for them) -which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only p-fronts. Using the new framework, we obtain characterizations for caustics.

Dedicated to Professor Osamu Kobayashi for his sixtieth birthday.Abstract. It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space R 3 1 is called of mixed type if it changes causal type from space-like to time-like. In R 3 1 , Osamu Kobayashi found two zero mean curvature entire graphs of mixed type that are not planes. As far as the authors know, these two examples were the only known examples of entire zero mean curvature graphs of mixed type without singularities. In this paper, we construct several families of real analytic zero mean curvature entire graphs of mixed type in Lorentz-Minkowski 3-space. The entire graphs mentioned above lie in one of these classes.

We give a representation formula for minimal surfaces in hyperbolic space. It is a natural generalization of the Weierstrass-Enneper formula for minimal surfaces in Euclidean space. Furthermore, we define the normal Gauss map and discuss some of its properties.Introduction. Weierstrass-Enneper formula, which explicitly describes a minimal immersion of a surface into Euclidean space, plays an important role in minimal surface theory.More generally, Kenmotsu [K] gave a representation formula for surfaces of prescribed mean curvature in Euclidean 3-space and, as a special case, for surfaces of constant mean curvature. By virtue of the formula, if a harmonic map φ from a Riemann surface Σ into S 2 is given, then one can construct an immersion of a constant mean curvature surface whose Gauss map is φ. It is remarkable that the harmonic map equation for φ is the complete integrability condition for a system of partial differential equations of first order which should be satisfied by the corresponding constant mean curvature immersion.On the other hand, Bryant [B] obtained an explicit representation formula for surfaces of constant mean curvature one (CMC-1 surfaces) in hyperbolic 3-space H 3 ( -1) of sectional curvature -1: Any CMC-1 surface in H 3 (-1) can be constructed from an sl(2, C)-valued holomorphic 1-form satisfying some conditions (or equivalently a pair of a meromorphic function and a holomorphic 1-form) on a Riemann surface. The study of CMC-1 surfaces in H 3 (-1) is making steady progress thanks to Bryant's formula (cf. [U-Y]).The purpose of this paper is to provide a Weierstrass type representation formula for minimal surfaces in hyperbolic «-space. In Section 2, we consider hyperbolic w-space to be a Lie group equipped with a left invariant metric, which is also obtained by deforming Euclidean space under certain change of the Riemannian metric. We use it as a model of hyperbolic space mainly for the following three reasons: (1) Since it is R n as a differentiable manifold, an immersion is written in terms of an «-tuple of real-valued functions. (2) We can see that the formula obtained in this paper is a generalization of the Weierstrass formula in the case ofR n .