FLAT SURFACES WITH SINGULARITIES IN EUCLIDEAN 3-SPACE

It is classically known that complete flat (that is, zero Gaussian curvature) surfaces in Euclidean 3-space R-3 are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface f admits singularities but its Gauss map v is globally defined on the surface and can be smoothly extended across the singular set, f is called a frontal. In addition, if the pair (f, v) defines an immersion into R-3 x S-2, f is called a front. A front f is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the caustic (i.e. focal surface) of a flat front f are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fronts, and give a representation formula for a flat front which has a non-empty compact singular set and whose ends are all immersed and complete. As an application, we show that such a flat front has properly embedded ends if and only if its Gauss map image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.