## Journal of the Mathematical Society of Japan

### Special Lagrangian submanifolds invariant under the isotropy action of symmetric spaces of rank two

#### Abstract

We study special Lagrangian submanifolds of the cotangent bundle $T^*S^n$ of the sphere in the tangent space of Riemannian symmetric space of rank two. We show that the special Lagrangian submanifolds correspond to the solution of a differential equation on $\mathbb{R}^2$ under the assumption that the submanifold is of cohomogeneity one. Our result is the generalization of the former work of Sakai and the first author [5]. We study the qualitative properties of the solution for the special Lagrangian submanifolds and give some examples.

#### Article information

Source
J. Math. Soc. Japan, Volume 68, Number 2 (2016), 839-862.

Dates
First available in Project Euclid: 15 April 2016

https://projecteuclid.org/euclid.jmsj/1460727384

Digital Object Identifier
doi:10.2969/jmsj/06820839

Mathematical Reviews number (MathSciNet)
MR3488149

Zentralblatt MATH identifier
1348.53060

Subjects
Primary: 53C38: Calibrations and calibrated geometries

#### Citation

HASHIMOTO, Kaname; MASHIMO, Katsuya. Special Lagrangian submanifolds invariant under the isotropy action of symmetric spaces of rank two. J. Math. Soc. Japan 68 (2016), no. 2, 839--862. doi:10.2969/jmsj/06820839. https://projecteuclid.org/euclid.jmsj/1460727384

#### References

• H. Anciaux, Special Lagrangian submanifolds in the complex sphere, Ann. Fac. Sci. Toulouse Math. (6), 16 (2007), 215–227.
• I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1–88.
• P. Hartman, Ordinary differential equations, Society for Industrial and Applied Mathematics, Philadelphia, 2002.
• R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math., 148 (1982), 47–157.
• K. Hashimoto and T. Sakai, Cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere, Tohoku Math. J. (2), 64 (2012), 141–169.
• K. Hashimoto, On the construction of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere, Differential Geometry of Submanifolds and Its Related Topics, World Sci. Publ., Hackensack, NJ, 2014, 135–146.
• M. Ionel and M. Min-Oo, Cohomogeneity one special Lagrangian $3$-folds in the deformed conifold and the resolved conifolds, Illinois J. Math., 52 (2008), 839–865.
• D. D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, 12, Oxford University Press, Oxford, 2007.
• S. Karigiannis and M. Min-Oo, Calibrated subbundles in noncompact manifolds of special holonomy, Ann. Global Anal. Geom., 28 (2005), 371–394.
• O. Loos, Symmetric spaces II: Compact Spaces and Classification, W. A. Benjamin, Inc., New York, Amsterdam, 1969.
• M. Stenzel, Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math., 80 (1993), 151–163.
• R. Szöke, Complex structures on tangent bundles of Riemannian manifolds, Math. Ann., 291 (1991), 409–428.
• M. Takeuchi, Modern Spherical Functions, Translations of Mathematical Monographs, 135. American Mathematical Society, Providence, RI, 1994.
• M. Yamaguchi, Mathematics of non-linear phenomena (in Japanese), Asakura shoten, Tokuo, 1972.