Tohoku Mathematical Journal

Construction of homogeneous Lagrangian submanifolds in $\boldsymbol{CP}^n$ and Hamiltonian stability

David Petrecca and Fabio Podestà

Full-text: Open access

Abstract

We apply the concept of castling transform of prehomogeneous vector spaces to produce new examples of minimal homogeneous Lagrangian submanifolds in the complex projective space. Furthermore we verify the Hamiltonian stability of a low dimensional example that can be obtained in this way.

Article information

Source
Tohoku Math. J. (2), Volume 64, Number 2 (2012), 261-268.

Dates
First available in Project Euclid: 2 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1341249374

Digital Object Identifier
doi:10.2748/tmj/1341249374

Mathematical Reviews number (MathSciNet)
MR2948822

Zentralblatt MATH identifier
1252.53091

Subjects
Primary: Primary
Secondary: Secondary 32J27: Compact Kähler manifolds: generalizations, classification 57S25: Groups acting on specific manifolds

Keywords
Homogeneous spaces Lagrangian submanifolds Hamiltonian stability

Citation

Petrecca, David; Podestà, Fabio. Construction of homogeneous Lagrangian submanifolds in $\boldsymbol{CP}^n$ and Hamiltonian stability. Tohoku Math. J. (2) 64 (2012), no. 2, 261--268. doi:10.2748/tmj/1341249374. https://projecteuclid.org/euclid.tmj/1341249374


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References

  • A. Amarzaya and Y. Ohnita, Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces, Tohoku Math. J. 55 (2003), 583–610.
  • L. Bedulli and A. Gori, Homogeneous Lagrangian submanifolds, Comm. Anal. Geom. 16 (2008), 591–615.
  • L. Bedulli and A. Gori, A Hamiltonian stable minimal Lagrangian submanifold of projective space with non-parallel second fundamental form, Transform. Groups 12 (2007), 611–617.
  • M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
  • V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984.
  • E. Heintze, Extrinsic upper bounds for $\Lambda_1$, Math. Ann. 280 (1988), 389–402.
  • S. Helgason, Differential geometry and symmetric spaces, Pure Appl. Math. 12, Academic Press, New York-London, 1962.
  • T. Kimura, A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications, J. Algebra 83 (1983), 72–100.
  • S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience Tracts in Pure and Applied Mathematics, No.15, Vol I, Vol II, Interscience Publishers, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1963, 1969.
  • Y. Matsushima, Espaces homogenes de Stein des groupes de Lie complexes, Nagoya Math. J. 16 (1960), 205–218.
  • H. Ma and Y. Ohnita, On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres, Math. Z. 261 (2009), 749–785.
  • H. Mutô and H. Urakawa, On the least positive eigenvalue of Laplacian for compact homogeneous spaces, Osaka J. Math. 17 (1980), 471–484.
  • H. Naitoh and M. Takeuchi, Totally real submanifolds and symmetric bounded domains, Osaka J. Math. 19 (1982), 717–731.
  • H. Ono, Minimal Lagrangian submanifolds in adjoint orbits and upper bounds on the first eigenvalue of the Laplacian, J. Math. Soc. Japan 55 (2003), 243–254.
  • Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), 501–519.
  • Y. Ohnita, Stability and rigidity of special Lagrangian cones over certain minimal Legendrian orbits, Osaka J. Math. 44 (2007), 305–334.
  • A. Ros, Spectral geometry of CR-minimal submanifolds in the complex projective space, Kodai Math. J. 6 (1983), 88–99.
  • M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.