Abstract
In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Burstall, F.E., Kilian, M.: Equivariant harmonic cylinders. Q. J. Math. 57, 449–468 (2006)
Castro, I., Urbano, F.: New examples of minimal Lagrangian tori in the complex projective plane. Manuscirta Math. 85(3–4), 265–281 (1994)
Dorfmeister, J., Pedit, F., Wu, H.: Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6, 633–668 (1998)
Farkas, H.M., Kra, I.: Riemann Surfaces. Springer, Berlin (1991)
Haskins, M.: Special Lagrangian cones. Amer. J. Math. 126(4), 845–871 (2004)
Haskins, M.: The geometric complexity of special Lagranian \(T^2\)-cones. Invent. Math. 157(1), 11–70 (2004)
Joyce, D.: Special Lagrangian 3-folds and integrable systems. In: Surveys on geometry and integrable systems, vol. 51, pp. 189–233, Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo (2008)
Ludden, G.D., Okumura, M., Yano, K.: A totally real surface in \({\cal C}P^{2}\) that is not totally geodesic. Proc. Amer. Math. Soc. 53, 186–190 (1975)
Ma, H., Ma, Y.: Totally real minimal tori in \(\mathbb{C}P^2\). Math. Z. 249, 241–267 (2005)
McIntosh, I.: Special Lagrangian cones in \(\mathbb{C}^3\) and primitive harmonic maps. J. London Math. Soc. 67(2), 769–789 (2003)
Naitoh, H., Takeuchi, M.: Totally real submanifolds and symmetric bounded domains. Osaka Math. J. 19, 717–731 (1982)
Pressley, A., Segal, G.: Loop groups. Oxford Science Publications, Oxford Science Monographs (1998)
Sharipov, R.A.: Minimal tori in the five-dimensional sphere in \(\mathbb{C}^3\). Theor. Math. Phys. 87(1), 363–369 (1991)
Yau, S.T.: Submanifolds with constant mean curvature. I. Amer. J. Math. 96, 346–366 (1974)
Acknowledgments
This work has been done during the first author’s visits at Tsinghua University and the second author’s visit at TU-München. The authors would like to thank both institutions for their generous support. Most of the results of this paper were reported by the second author during the 10th geometry conference for the friendship between China and Japan in 2014. She would like to thank the organizers for the invitation to the conference. This work is partially supported by NSFC grant No. 11271213.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Japan
About this paper
Cite this paper
Dorfmeister, J.F., Ma, H. (2016). A New Look at Equivariant Minimal Lagrangian Surfaces in \({\mathbb {C}} P^2\) . 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_5
Download citation
DOI: https://doi.org/10.1007/978-4-431-56021-0_5
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-56019-7
Online ISBN: 978-4-431-56021-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)