Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T12:00:35.966Z Has data issue: false hasContentIssue false
  • English
  • Français

HELICOIDAL MINIMAL SURFACES IN ℍ2×ℝ

Part of: Classical differential geometry

Published online by Cambridge University Press:  15 December 2011

YOUNG WOOK KIM ,
SUNG-EUN KOH ,
HEAYONG SHIN  and
SEONG-DEOG YANG
YOUNG WOOK KIM
Affiliation:
Dept. of Mathematics, Korea University, Seoul 136-701, Korea (email: ywkim@korea.ac.kr)
SUNG-EUN KOH*
Affiliation:
Dept. of Mathematics, Konkuk University, Seoul 143-701, Korea (email: skoh@konkuk.ac.kr)
HEAYONG SHIN
Affiliation:
Dept. of Mathematics, Chung-Ang University, Seoul 156-756, Korea (email: hshin@cau.ac.kr)
SEONG-DEOG YANG
Affiliation:
Dept. of Mathematics, Korea University, Seoul 136-701, Korea (email: sdyang@korea.ac.kr)
*
✉For correspondence; e-mail: skoh@konkuk.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that a minimal surface in ℍ2×ℝ is invariant under a one-parameter group of screw motions if and only if it lies in the associate family of helicoids. It is also shown that the conjugate surfaces of the parabolic and hyperbolic helicoids in ℍ2×ℝ are certain types of catenoids.

Keywords

helicoidal surfacehelicoidcatenoidassociate minimal surfacesconjugate minimal surfaces

MSC classification

Secondary: 53A35: Non-Euclidean differential geometry
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 135 - 149
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The first named author was supported by NRF 2009-0086794. The second named author was supported by NRF 2009-0086794 and NRF 2009-0086441.

References

[1]Abresch, U. and Rosenberg, H., ‘A Hopf differential for constant mean curvature surfaces in 𝕊2×ℝ and ℍ×ℝ’, Acta Math. 193 (2004), 141–174.CrossRefGoogle Scholar
[2]Daniel, B., ‘Isometric immersions into 𝕊n×ℝ and ℍn×ℝ and applications to minimal surfaces’, Trans. Amer. Math. Soc. 361 (2009), 6255–6282.CrossRefGoogle Scholar
[3]Galvez, D., Martinez, A. and Mira, P., ‘The Bonnet problem for surfaces in homogeneous 3-manifolds’, Comm. Anal. Geom. 16 (2008), 907–935.CrossRefGoogle Scholar
[4]Hauswirth, L., Sa Earp, R. and Toubiana, E., ‘Associate and conjugate minimal immersions in 𝕄×ℝ’, Tohoku Math. J. 60 (2008), 267–286.CrossRefGoogle Scholar
[5]Kim, Y. W., Koh, S.-E., Shin, H. and Yang, S.-D., ‘Helicoids in 𝕊2×ℝ and ℍ2×ℝ’, Pacific J. Math. 242 (2009), 281–297.CrossRefGoogle Scholar
[6]Mira, P. and Pastor, A. S., ‘Helicoidal maximal surfaces in Lorentz–Minkowski space’, Monatsch. Math. 140 (2003), 315–334.CrossRefGoogle Scholar
[7]Meeks, W. and Rosenberg, H., ‘The theory of minimal surfaces in M×ℝ’, Comment. Math. Helv. 80 (2005), 811–858.CrossRefGoogle Scholar
[8]Nitsche, J. C. C., Lectures on Minimal Surfaces, Vol. 1 (Cambridge University Press, Cambridge, 2008).Google Scholar
[9]Rosenberg, H., ‘Minimal surfaces in M×ℝ’, Illinois J. Math. 46 (2002), 1177–1195.Google Scholar
[10]Sa Earp, R., ‘Parabolic and hyperbolic screw motion surfaces in ℍ2×ℝ’, J. Aust. Math. Soc. 85 (2008), 113–143.CrossRefGoogle Scholar
[11]Sa Earp, R. and Toubiana, E., ‘Screw motion surfaces in ℍ2×ℝ and 𝕊2×ℝ’, Illinois J. Math. 49 (2005), 1323–1362.Google Scholar