Flat fronts in hyperbolic 3-space and their caustics

Abstract

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.

Extrinsic diameter of immersed flat tori in S 3

“…A C ∞ -map f : M n → N n+1 is called a wave front or a front if f lifts to a Legendrian immersion L f : M n → P(T * N n+1 ), namely, d L f (T M n ) lies in the contact hyperplane field on P(T * N n+1 ). Moreover if a front L f can be lifted to a C ∞ -map into T * N n+1 , f is called co-orientable, and otherwise it is called non-co-orientable (In [13], non-co-orientable fronts are called p-fronts, but in this paper, a front is not assumed to be co-orientable). Fronts generalize immersions, as they allow for singularities.…”

“…1, we give a formulation of flat fronts in S 3 . Global behaviour of flat surfaces in H 3 and R 3 regarded as wave fronts is given in [13] and [15]. For the case of flat surfaces in S 3 , Gálvez and Mira [4] and Okada [16] gave a representation formula for flat surfaces that are wave fronts (i.e.…”

Weakly Complete Wave Fronts With One Principal Curvature Constant

“…is a complete metric, f is said to be weakly complete. The notion of weak completeness was introduced in [14] for constant negative curvature surfaces in R 3 and in [9] for flat fronts in the hyperbolic 3-space. On the other hand, f is said to be complete if there exists a symmetric covariant tensor T on M 2 with compact support such that ds 2 + T gives a complete metric on M 2 .…”

“…Such a phenomenon also holds in the case of the hyperbolic 3-space [17] and the 3-sphere [7] (cf. [6,9]). …”

Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space

“…We also introduce in this section the basic invariants for the flat lightlike hypersurfaces. It should be noticed that the singularities of certain classes of surfaces (i.e., kinds of "flat"surfaces) have been recently investigated by several authors ( [6,8,14,15,20]) from a differential geometry viewpoint. One of the main purposes of the submanifold theory in differential geometry is to study some special classes of submanifolds in different ambient spaces such as "flat"surfaces.…”