Flat Surfaces with singularities in Euclidean 3-space
  It is classically known that complete flat surfaces in Euclidean 3-space 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 $\nu$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,\nu)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fonts, and give a representation formula for complete flat fronts. As an application, we show that a complete flat front has properly embedded ends if and only if its Gauss 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. 
Satoko Murata and Masaaki Umehara