Osaka Journal of Mathematics

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

Yoshihiro Ohnita

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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

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1183667983

Mathematical Reviews number (MathSciNet)
MR2351004

Zentralblatt MATH identifier
1141.53048

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C38: Calibrations and calibrated geometries 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

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


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