Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Cover of the book

2016 | OriginalPaper | Chapter

Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group

Authors : Reiko Aiyama, Kazuo Akutagawa

Published in: Geometry and Topology of Manifolds

Publisher: Springer Japan

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we give a representation formula for Legendrian surfaces in the 5-dimensional Heisenberg group \(\mathfrak {H}^5\), in terms of spinors. For minimal Legendrian surfaces especially, such data are holomorphic. We can regard this formula as an analogue (in Contact Riemannian Geometry) of Weierstrass representation for minimal surfaces in \(\mathbb {R}^3\). Hence for minimal ones in \(\mathfrak {H}^5\), there are many similar results to those for minimal surfaces in \(\mathbb {R}^3\). In particular, we prove a Halfspace Theorem for properly immersed minimal Legendrian surfaces in \(\mathfrak {H}^5\).

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
next chapter Gluing Principle for Orbifold Stratified Spaces
Literature
1.
Aiyama, R.: Totally real surfaces in the complex 2-space. In: Steps in Differential Geometry, pp. 15–22. Inst. Math. Inform., Debrecen (2001)
2.
3.
Blair, D.C.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Birkhäuser (2010)
4.
Boyer, C.P.: The Sasakian geometry of the Heisenberg group. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52, 251–262 (2009)
5.
Castro, I., Li, H.Z., Urbano, F.: Hamiltonian-minimal Lagrangian submanifolds in complex space forms. Pac. J. Math. 227, 43–63 (2006)MathSciNetCrossRefMATH
6.
Castro, I., Urbano, F.: On a minimal Lagrangian submanifold of \(\mathbb{C}^n\) foliated by spheres. Michigan Math. J. 46, 71–82 (1999)MathSciNetCrossRefMATH
7.
Chen, B.-Y., Morvan, J.-M.: Geometrie des surfaces Lagrangiennes de \(\mathbb{C}^2\). J. Math. Pures Appl. 66, 321–325 (1987)MathSciNetMATH
8.
Collin, P., Kusner, R., Meeks III, W.H., Rosenberg, H.: The topology, geometry and conformal structure of properly embedded minimal surfaces. J. Differ. Geom. 67, 377–393 (2004)MathSciNetMATH
9.
Dragomir, S., Tomassini, G.: Differential geometry and analysis on CR manifolds. In: Progress in Mathematics, vol. 246. Birkhäuser (2006)
10.
Ekholm, T., Etnyre, J., Sullivan, M.: Non-isotopic Legendrian submanifolds in \(\mathbb{R}^{2n+1}\). J. Differ. Geom. 71, 85–128 (2005)MathSciNetMATH
11.
12.
Hélein, F., Romon, P.: Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions. Comment. Math. Helv. 75, 668–680 (2000)MathSciNetCrossRefMATH
13.
14.
15.
Hoffman, D., Osserman, R.: The geometry of the generalized Gauss map. Mem. Am. Math. Soc. 28(236) (1980)
16.
Hoffman, D., Osserman, R.: The Gauss map of surfaces in \(\mathbb{R}^3\) and \(\mathbb{R}^4\). Proc. Lond. Math. Soc. 50, 27–56 (1985)MathSciNetCrossRefMATH
17.
18.
19.
Meeks, W.H.: Minimal surfces in flat three-dimensional spaces. Lect. Note Math. 1775, 1–14 (2002)MathSciNetMATH
20.
21.
22.
Ono, H.: Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds. Differ. Geom. Appl. 22, 327–340 (2005)MathSciNetCrossRefMATH
23.
Osserman, R.: A Survey of Minimal Surfaces, 2nd edn. Dover Publ. Inc, New York (1986)
24.
Reckziegel, H.: A correspondence between horizontal submanifolds of Sasakian manifolds and totally real submanifolds of Kähler manifolds. In: Szenthe, J., Tamássy, L. (Eds.) Topics in Differential Geometry, vol. II, pp. 1063–1081. Debrecen, North-Holland (1988)
25.
Schoen, R., Wolfson, J.: Minimizing area among Lagrangian surfaces: the mapping problem. J. Differ. Geom. 38, 1–86 (2001)
Metadata
Title
Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group
Authors
Reiko Aiyama
Kazuo Akutagawa
Copyright Year
2016
Publisher
Springer Japan
Book
Geometry and Topology of Manifolds

Print ISBN: 978-4-431-56019-7

Electronic ISBN: 978-4-431-56021-0

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-4-431-56021-0_1

Premium Partner

    Image Credits