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

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)


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.


Riemannian Manifold Complete Riemannian Manifold Minimal Hypersurface Normal Deformation Hodge Theory 


  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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