Vol. 264, No. 1, 2013

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Helicoidal flat surfaces in hyperbolic 3-space

Antonio Martínez, João Paulo dos Santos and Keti Tenenblat

Vol. 264 (2013), No. 1, 195–211
Abstract

A flat surface in hyperbolic space 3 is determined by a harmonic function as well as by its meromorphic data. In this paper, helicoidal flat surfaces in 3 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 3 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
Authors
Antonio Martínez
Departamento de Geometría y Topología
Universidad de Granada
E-18071 Granada
Spain
João Paulo dos Santos
Departamento de Matemática
Universidade de Brasília
70910-900 Brasília
Brazil
Keti Tenenblat
Departamento de Matemática
Universidade de Brasília
70910-900 Brasília
Brazil