Helicoidal flat surfaces in hyperbolic 3-space

Antonio Martínez, João Paulo dos Santos and Keti Tenenblat

Vol. 264 (2013), No. 1, 195–211

DOI: 10.2140/pjm.2013.264.195

Abstract

A flat surface in hyperbolic space ℍ³ is determined by a harmonic function as well as by its meromorphic data. In this paper, helicoidal flat surfaces in ℍ³ are considered. A complete classification of the helicoidal flat fronts is given in terms of their hyperbolic Gauss maps as well as by means of linear harmonic functions. A family of examples that provides the classification of the helicoidal flat fronts is included. Moreover, it is shown that a flat surface in ℍ³ that corresponds to a linear harmonic function is locally congruent to a helicoidal flat front or to a peach front.

Keywords

helicoidal surfaces, flat surfaces, flat fronts, hyperbolic space, conformal representation

Mathematical Subject Classification 2010

Primary: 53A35, 53C42

Milestones

Received: 29 February 2012

Revised: 20 April 2012

Accepted: 15 May 2012

Published: 5 July 2013