Open Access
Translator Disclaimer
January, 2007 Flat fronts in hyperbolic 3-space and their caustics
Masatoshi KOKUBU, Wayne ROSSMAN, Masaaki UMEHARA
J. Math. Soc. Japan 59(1): 265-299 (January, 2007). DOI: 10.2969/jmsj/1180135510


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.


Download Citation

Masatoshi KOKUBU. Wayne ROSSMAN. Masaaki UMEHARA. "Flat fronts in hyperbolic 3-space and their caustics." J. Math. Soc. Japan 59 (1) 265 - 299, January, 2007.


Published: January, 2007
First available in Project Euclid: 25 May 2007

zbMATH: 1120.53036
MathSciNet: MR2302672
Digital Object Identifier: 10.2969/jmsj/1180135510

Primary: 53C42
Secondary: 53A35

Keywords: caustics , flat fronts

Rights: Copyright © 2007 Mathematical Society of Japan


Vol.59 • No. 1 • January, 2007
Back to Top