Helicoids in 𝕊² × ℝ and ℍ² × ℝ

Young Wook Kim, Sung-Eun Koh, Heayong Shin and Seong-Deog Yang

Vol. 242 (2009), No. 2, 281–297

DOI: 10.2140/pjm.2009.242.281

Abstract

We provide two characterizations of helicoids in 𝕊² × ℝ and in ℍ² × ℝ. First, we show that any nontrivial ruled minimal surface in 𝕊² × ℝ and in ℍ² × ℝ is a part of a helicoid. Second, we also show that these surfaces can be characterized as the only surface with zero mean curvature with respect to both the Riemannian product metric and the Lorentzian product metric on 𝕊² × ℝ or ℍ² × ℝ.

Keywords

helicoid, ruled surface, minimal surface, spacelike maximal surface

Mathematical Subject Classification 2000

Primary: 53A35

Milestones

Received: 14 January 2009

Revised: 8 April 2009

Accepted: 14 April 2009

Published: 1 October 2009