## Osaka Journal of Mathematics

### Stability and rigidity of special Lagrangian cones over certain minimal Legendrian orbits

Yoshihiro Ohnita

#### Abstract

Special Lagrangian cones in complex Euclidean spaces are obtained as cones over compact minimal Legendrian submanifolds in the odd dimenisonal standard hypersphere. The notion of the stability, the Legendrian stability and the rigidity of special Lagrangian cones were recently introduced and investigated by D. Joyce, M. Haskins etc. In this paper we determine explicitly the stability-index, the Legendrian-index, and the rigidity of special Lagrangian cones over compact irreducible symmeric spaces of type $A$ obtained as minimal Legendrian orbits and over a minimal Legendrian $\mathit{SU}(2)$-orbit. We obtain the examples of stable and rigid special Lagrangian cones in higher dimensions. Moreover we discuss a relationship of these properties with the Hamiltonian stability of minimal Lagrangian submanifolds in complex projective spaces.

#### Article information

Source
Osaka J. Math., Volume 44, Number 2 (2007), 305-334.

Dates
First available in Project Euclid: 5 July 2007

https://projecteuclid.org/euclid.ojm/1183667983

Mathematical Reviews number (MathSciNet)
MR2351004

Zentralblatt MATH identifier
1141.53048

#### Citation

Ohnita, Yoshihiro. Stability and rigidity of special Lagrangian cones over certain minimal Legendrian orbits. Osaka J. Math. 44 (2007), no. 2, 305--334. https://projecteuclid.org/euclid.ojm/1183667983

#### References

• A. Amarzaya and Y. Ohnita: Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces, Tohoku Math. J. (2) 55 (2003), 583--610.
• A. Amarzaya and Y. Ohnita: Hamiltonian stability of certain H-minimal Lagrangian sub-manifolds and related problems; in General Study on Riemannian submanifolds (Japanese) (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku 1292, RIMS, Kyoto University, Kyoto, 2002, 72--93.
• A. Amarzaya and Y. Ohnita: Hamiltonian stability of certain symmetric $R$-spaces embedded in complex Euclidean spaces, preprint, Tokyo Metropolitan University, 2002.
• N. Bourbaki: Éléments de Mathématique, Fasc. XXXIV, Groupes et algebres de Lie, Actualites Scientifiques et Industrielles 1337, Hermann, Paris, 1968.
• R. Chiang: New Lagrangian submanifolds of $\mathbf{CP}^{n}$, Int. Math. Res. Not. 45 (2004), 2437--2441.
• M. Haskins: Special Lagrangian cones, Amer. J. Math. 126 (2004), 845--871.
• M. Haskins: The geometric complexity of special Lagrangian $T^{2}$-cones, Invent. Math. 157 (2004), 11--70.
• R. Harvey and H.B. Lawson: Calibrated geometries, Acta Math. 148 (1982), 47--157.
• D. Joyce: Special Lagrangian $m$-folds in $\mathbf{C}^{m}$ with symmetries, Duke Math. J. 115 (2002), 1--51, math.DG/0008021.
• D. Joyce: Special Lagrangian submanifolds with isolated conical singularities. I. Regularity, Ann. Global Anal. Geom. 25 (2004), 201--251, math.DG/0211294.
• D. Joyce: Special Lagrangian submanifolds with isolated conical singularities. II. Moduli spaces, Ann. Global Anal. Geom. 25 (2004), 301--352, math.DG/0211295.
• D. Joyce: Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications, J. Differential Geom. 63 (2003), 279--347, math.DG/0303272.
• W.G. McKay and J. Patera: Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras, Lecture Notes in Pure and Applied Mathematics 69, Marcel Dekker, Inc., New York, 1981.
• R.C. McLean: Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705--747.
• H. Muto, Y. Ohnita and H. Urakawa: Homogeneous minimal hypersurfaces in the unit spheres and the first eigenvalues of their Laplacian, Tohoku Math. J. (2) 36 (1984), 243--267.
• Y. Ohnita: The first standard minimal immersions of compact irreducible symmetric spaces; in Differential Geometry of Submanifolds (Kyoto, 1984), edited by Katsuei Kenmotsu, Lecture Notes in Math. 1090, Springer-Verlag, 1984, 37--49.
• Y. Ohnita: Satbility and Rigidity of Certain Special Lagrangian Cones; in Differential Geometry and Submanifold Theory (Kyoto, 2005), Sūrikaisekikenkyūsho Kōkyūroku 1460 (2005), RIMS, Kyoto University, Kyoto, 2005, 43--52.
• T. Takahashi: Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380--385.
• M. Takeuchi: Modern Theory of Spherical Functions, Iwanami, Tokyo, 1975, (in Japanese). Modern Spherical Functions, Translations of Mathematical Monographs 135, Amer. Math. Soc., Providence, RI, 1994.