## Illinois Journal of Mathematics

### Calabi–Yau structures and special Lagrangian submanifolds of complexified symmetric spaces

Naoyuki Koike

#### Abstract

It is known that there exist Calabi–Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we first construct explicit complete Ricci-flat Kaehler metrics (which give Calabi–Yau structures) for complexified symmetric spaces of arbitrary rank in terms of the Schwarz’s theorem. We consider the case where the Calabi–Yau structure arises from the generalized Stenzel metric. In the complexified symmetric spaces equipped with such a Calabi–Yau structure, we give constructions of special Lagrangian submanifolds of any given phase which are invariant under the actions of symmetric subgroups of the isometry group of the original symmetric space of compact type.

#### Article information

Source
Illinois J. Math., Volume 63, Number 4 (2019), 575-600.

Dates
Revised: 9 September 2019
First available in Project Euclid: 19 November 2019

https://projecteuclid.org/euclid.ijm/1574154084

Digital Object Identifier
doi:10.1215/00192082-8018607

Mathematical Reviews number (MathSciNet)
MR4032815

Zentralblatt MATH identifier
07136347

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index

#### Citation

Koike, Naoyuki. Calabi–Yau structures and special Lagrangian submanifolds of complexified symmetric spaces. Illinois J. Math. 63 (2019), no. 4, 575--600. doi:10.1215/00192082-8018607. https://projecteuclid.org/euclid.ijm/1574154084

#### References

• [1] H. Anciaux, Special Lagrangian submanifolds in the complex sphere, Ann. Fac. Sci. Toulouse Math. Ser. 6, 16 (2007), 215–227.
• [2] M. Arai and K. Baba, Special Lagrangian submanifolds and cohomogeneity one actions on the complex projective space, Tokyo J. Math. 42 (2019), 255–284.
• [3] H. Azad and J.J. Loeb, Plurisubharmonic functions and Kälerian metrics on complexification of symmetric spaces, Indag. Math. (N.S.) 3 (1992), 365–375.
• [4] R. Bielawski, Entire invariant solutions to Monge–Ampère equations, Proc. Amer. Math. Soc. 132 (2004), 2679–2682.
• [5] R. Bielawski, Prescribing Ricci curvature on complexified symmetric spaces, Math. Res. Lett. 11 (2004), 435–441.
• [6] A. Borowiec, M. Francaviglia, and I. Volovich, Anti-Kählerian manifolds, Differential Geom. Appl. 12 (2000) 281–289.
• [7] R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958) 964–1029.
• [8] V. Guillemin and M.B. Stenzel, Grauret tubes and the homogeneous Monge–Ampére equation, J. Differential Geom. 34 (1991), 561–570.
• [9] K. Hashimoto and K. Mashimo, Special Lagrangian submanifolds invariant under the isotropy action of symmetric spaces of rank two, J. Math. Soc. Japan 68 (2016) 839–862.
• [10] K. Hashimoto and T. Sakai, Cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere, Tohoku Math. J. 64 (2012) 141–169.
• [11] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, New York, 1978.
• [12] R. Hermann, Variationally completeness for compact symmetric spaces, Proc. Amer. Math. Soc. 11 (1960) 544–546.
• [13] 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.
• [14] N. Koike, Hermann type actions on a pseudo-Riemannian symmetric space, Tsukuba J. Math. 34 (2010), 137–172.
• [15] N. Koike, Collapse of the mean curvature flow for equifocal submanifolds, Asian J. Math. 15 (2011), 101–128.
• [16] N. Koike, Research of submanifolds in symmetric spaces by using the complexification and the infinite dimensional geometry, SUT J. Math. 50 (2014), 103–129.
• [17] N. Koike, The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry, SUT J. Math. 50 (2014), 271–295.
• [18] A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2002), 571–612.
• [19] G. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68.
• [20] M. B. Stenzel, Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math. 80 (1993), 151–163.
• [21] A. Strominger, S.-T. Yau, and E. Zaslow, Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), 243–259.