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Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group

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Geometry and Topology of Manifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 154))

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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 \(\mathfrak {H}^5\).

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Acknowledgments

The both authors would like to thank Katsuei Kenmotsu, Yu Kawakami and Katsutoshi Yamanoi for helpful discussions and continuous encouragements. They would also like to thank Reiko Miyaoka and the anonymous referee for valuable comments. The second author is supported in part by the Grant-in-Aid for Challenging Exploratory Research, Japan Society for the Promotion of Science, No. 24654009.

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Authors and Affiliations

  1. University of Tsukuba, Tsukuba, Japan

    Reiko Aiyama

  2. Tokyo Institute of Technology, Tokyo, Japan

    Kazuo Akutagawa

Authors
  1. Reiko Aiyama
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  2. Kazuo Akutagawa
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Corresponding author

Correspondence to Kazuo Akutagawa .

Editor information

Editors and Affiliations

  1. The University of Tokyo, Meguro-ku, Tokyo, Japan

    Akito Futaki

  2. Mathematical Institute, Tohoku University, Sendai,Aoba-ku, Japan

    Reiko Miyaoka

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, China

    Zizhou Tang

  4. Chern Institute of Mathematics, Nankai University, Tianjin, China

    Weiping Zhang

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Aiyama, R., Akutagawa, K. (2016). Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group. In: Futaki, A., Miyaoka, R., Tang, Z., Zhang, W. (eds) Geometry and Topology of Manifolds. Springer Proceedings in Mathematics & Statistics, vol 154. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56021-0_1

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