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Stability of Complete Minimal Lagrangian Submanifold and L2 Harmonic 1-Forms

  • Reiko Miyaoka
  • Satoshi Ueki
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

Abstract

We show that a non-compact complete stable minimal Lagrangian submanifold L in a Kähler manifold with positive Ricci curvature has no non-trivial L 2 harmonic 1-forms, which gives a topological and conformal constraint on L.

Keywords

Riemannian Manifold Complete Riemannian Manifold Minimal Hypersurface Normal Deformation Hodge Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Castro, I., Urbano, F.: Minimal Lagrangian surfaces in S 2 × S 2. Commun. Anal. Geom. 15, 217–248 (2007)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Dodziuk, J.: L 2 harmonic forms on complete manifolds. Ann. Math. Stud. 102, 291–302 (1982)Google Scholar
  3. 3.
    Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Pure Appl. Math. 33, 199–211 (1980)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Lawson, B., Simons, J.: On stable currents and their applications to global problems in real and complex geometry. Ann. Math. 98, 427–450 (1973)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Le, H.V.: Minimal Φ-Lagrangian surfaces in almost Hermitian manifolds. Mat. Sb. 180, 924–936, 991 (1989); translation in Math USSR-Sb. 67, 379–391 (1990)Google Scholar
  6. 6.
    Li, P., Tam, F.: Harmonic functions and the structure of complete manifolds. J. Differ. Geom. 32, 359–383 (1992)MathSciNetGoogle Scholar
  7. 7.
    Ma, H., Ohnita, Y.: Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. J. Differ. Geom. 97(2), 275–348 (2014)MATHMathSciNetGoogle Scholar
  8. 8.
    Miyaoka, R.: L 2 harmonic 1-forms in a complete stable minimal hypersurface. Geometry and Global Analysis, Tohoku University, pp. 289–293 (1993)Google Scholar
  9. 9.
    Oh, Y.G.: Second variation and stabilities of minimal lagrangian submanifolds in Kaähler manifolds. Invent. Math. 101, 501–519 (1990)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Palmer, B.: Stability of minimal hypersurfaces. Comment. Math. Helvetici 66, 185–188 (1991)CrossRefMATHGoogle Scholar
  11. 11.
    Palmer, B.: Buckling eigenvalues, Gauss maps and Lagrangian submanifolds. Differ. Geom. Appl. 4, 391–403 (1994)CrossRefMATHGoogle Scholar
  12. 12.
    Palmer, B.: Biharmonic capacity and the stability of minimal Lagrangian submanifolds. Tohoku Math. J. 55, 529–541 (2003)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    de Rham, G.: Differentiable Manifolds, Forms, Currents, Harmonic Forms. Springer, Berlin (1984)CrossRefMATHGoogle Scholar
  14. 14.
    Simons, J.: Minimal varieties in riemannian manifolds. Ann. Math. 88, 62–105 (1968)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    S. Ueki, Stability of Lagrangian and isotropic submanifolds. Dr. thesis, Tohoku University (2013)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Tohoku UniversityAoba-ku, SendaiJapan
  2. 2.Sado Senior High SchoolSadoJapan

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