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.