Hostname: page-component-6bb9c88b65-9rk55 Total loading time: 0 Render date: 2025-07-22T22:45:49.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 from minimal surfaces in 𝕊3

Part of: Global differential geometry Local differential geometry Symplectic geometry, contact geometry

Published online by Cambridge University Press:  27 December 2018

Burcu Bektaş ,
Marilena Moruz ,
Joeri Van der Veken  and
Luc Vrancken
Burcu Bektaş
Affiliation:
Faculty of Science and Letters, Department of Mathematics, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey (bektasbu@itu.edu.tr)
Marilena Moruz
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B - Box 2400, BE-3001 Leuven, Belgium (marilena.moruz@kuleuven.be)
Joeri Van der Veken
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B – Box 2400, BE-3001 Leuven, Belgium (joeri.vanderveken@wis.kuleuven.be)
Luc Vrancken
Affiliation:
LAMAV, ISTV2 Université de Valenciennes Campus du Mont Houy 59313 Valenciennes Cedex 9, France and KU Leuven Department of Mathematics, Celestijnenlaan 200B – Box 2400 BE-3001 Leuven, Belgium (luc.vrancken@univ-valenciennes.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

Keywords

Lagrangian submanifoldsNearly Kaehler 𝕊3 × 𝕊3minimal surfaces, 𝕊3

MSC classification

Primary: 53B25: Local submanifolds
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53D12: Lagrangian submanifolds; Maslov index

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 655 - 689
Copyright
Copyright © Royal Society of Edinburgh 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Bektaş, B., Moruz, M., Van der Veken, J. and Vrancken, L.. Lagrangian submanifolds with constant angle functions of the nearly Kähler 𝕊3 × 𝕊3. J. Geom. Phys. 127 (2018), 1–13.Google Scholar
2Bolton, J., Dillen, F., Dioos, B. and Vrancken, L.. Almost complex surfaces in the nearly Kähler 𝕊3 × 𝕊3. Tohoku Math. J. 67 (2015), 1–17.Google Scholar
3Boothby, W. M.. An introduction to differentiable manifolds and Riemannian geometry, Revised 2nd edn (Singapore: Elsevier, 2007).Google Scholar
4Butruille, J.-B.. Homogeneous nearly Kähler manifolds. In Handbook of Pseudo-Riemannian geometry and supersymmetry, RMA Lect. Math. Theor. Phys., vol. 16, pp. 399–423 (Zürich: Eur. Math. Soc., 2010).Google Scholar
5Cortés, V. and Vásquez, J. J.. Locally homogeneous nearly Kähler manifolds. Ann. Global Anal. Geom. 48 (2015), no. 3, 269–294.Google Scholar
6Dillen, F. and Vrancken, L.. Totally real submanifolds in 𝕊6(1) satisfying Chen's equality. Trans. Amer. Math. Soc. 348( 4) (1996), 1633–1646.Google Scholar
7Dillen, F., Verstraelen, L. and Vrancken, L.. Classification of totally real 3-dimensional submanifolds of 𝕊6(1) with K ⩾ 1/16. J. Math. Soc. Japan 42(1990), 565–584.Google Scholar
8Dioos, B., Vrancken, L. and Wang, X.. Lagrangian submanifolds in the homogeneous nearly Kähler 𝕊3 × 𝕊3. Ann. Global Anal. Geom. 53(2018), no. 1, 39–66.Google Scholar
9Ejiri, N.. Totally real submanifolds in a 6-sphere. Proc. Amer. Math. Soc. 83(1981), 759–763.Google Scholar
10Ejiri, N.. Equivariant minimal immersions of 𝕊2 into 𝕊2m(1). Trans. Amer. Math. Soc. 297(1986), 105–124.Google Scholar
11Foscolo, L. and Haskins, M.. New G2-holonomy cones and exotic nearly Kähler structures on 𝕊6 and 𝕊3 × 𝕊3. Ann. of Math. (2) 185(2017), no. 1,59–130.Google Scholar
12Gray, A.. Nearly Kähler manifolds. J. Diff. Geom. 4 (1970), 283–309.Google Scholar
13Gutowski, J., Ivanov, S. and Papadopoulos, G.. Deformations of generalized calibrations and compact non-Kähler manifolds with vanishing first Chern class. Asian J. Math. 7(2003), 039–080.Google Scholar
14John, F.. Partial differential equations, vol. 1 (US: Springer, 1978).Google Scholar
15Lotay, J. D.. Ruled Lagrangian submanifolds of the 6-sphere. Trans. Amer. Math. Soc. 363(2011), 2305–2339.Google Scholar
16Moroianu, A. and Semmelmann, U.. Generalized Killing spinors and Lagrangian graphs. Diff. Geom. Appl. 37 (2014), 141–151.Google Scholar
17Nagy, P.-A.. Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6(2002a), 481–504.Google Scholar
18Nagy, P.-A.. On nearly- Kähler geometry. Ann. Global Anal. Geom. 22(2002b), 167–178.Google Scholar
19Palmer, B.. Calibrations and Lagrangian submanifolds in the six sphere (English summary). Tohoku Math. J. (2) 50(1998), 303–315.Google Scholar
20Schäfer, L. and Smoczyk, K.. Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds. Ann. Global Anal. Geom. 37(2010), 221–240.Google Scholar
21Szabo, Z. I.. Structure theorems on riemannian spaces satisfying R(X, Y) · R = 0. I. the local version. J. Differ. Geom. 17(1982), 531–582.Google Scholar
22Vrancken, L.. Killing vector fields and Lagrangian submanifolds of the nearly Kaehler 𝕊6. J. Math. Pures Appl. (9) 77(1998), 631–645.Google Scholar
23Vrancken, L.. Special Lagrangian submanifolds of the nearly Kaehler 6-sphere. Glasg. Math. J. 45(2003), 415–426.Google Scholar
24Zhang, Y., Dioos, B., Hu, Z., Vrancken, L. and Wang, X.. Lagrangian submanifolds in the 6-dimensional nearly Kähler manifolds with parallel second fundamental form. J. Geom. Phys. 108 (2016), 21–37.Google Scholar