On the Hamiltonian Minimality of Normal Bundles

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


A Hamiltonian minimal (shortly, H-minimal) Lagrangian submanifold in a Kähler manifold is a critical point of the volume functional under all compactly supported Hamiltonian deformations. We show that any normal bundle of a principal orbit of the adjoint representation of a compact simple Lie group G in the Lie algebra \(\mathfrak{g}\) of G is an H-minimal Lagrangian submanifold in the tangent bundle \(T\mathfrak{g}\) which is naturally regarded as \(\mathbb{C}^{m}\). Moreover, we specify these orbits with this property in the class of full irreducible isoparametric submanifolds in the Euclidean space.


Symmetric Space Normal Bundle Lagrangian Submanifold Isoparametric Hypersurface Principal Orbit 
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.



The author would like to express his sincere thanks to Profs. Young Jin Suh, Jürgen Berndt, Yoshihiro Ohnita and Byung Hak Kim for inviting me to “Conference on real and complex submanifolds”. He was partially supported by Grant-in-Aid for JSPS Fellows.


  1. 1.
    Amarzaya, A., Ohnita, Y.: Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces. Tohoku Math. J. 55, 583–610 (2003)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Amarzaya, A., Ohnita, Y.: Hamiltonian stability of parallel Lagrangian submanifolds in complex space forms (preprint)Google Scholar
  3. 3.
    Anciaux, H., Castro, I.: Construction of Hamiltonian-minimal Lagrangian submanifolds in complex Euclidean space. Results Math. 60(1-4), 325–349 (2011)Google Scholar
  4. 4.
    Bedulli, L., Gori, A.: Homogeneous Lagrangian submanifolds. Comm. Anal. Geom. 16(3), 591–615 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Berndt, J., Console, S., Olmos, C.: Submanifolds and Holonomy. Chapman & Hall/CRC Research notes in Mathmatics, 434Google Scholar
  6. 6.
    Cecil, T.E.: Isoparametric and Dupin hypersurfaces. Symmetry Integr. Geom. Meth. Appl. 4(062), 1–28 (2008)MathSciNetGoogle Scholar
  7. 7.
    Dadok, J.: Polar coordinates induced by actions of compact Lie groups. Trans. Am. Math. Soc. 288, 125–137 (1985)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dong, Y.: Hamiltonian-minimal Lagrangian submanifolds in Kähler manifolds with symmetries. Nonlinear Anal. 67, 865–882 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Dazord, P.: Sur la géométrie des sous-fibrés et des feuilletages lagrangiens. Ann. Sci. École Norm. Sup. 14(4), 465–480 (1981/1982)Google Scholar
  10. 10.
    Harvey, R., Lawson, H.B.: Calibrated geometry. Acta Math. 148, 47–157 (1982)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Haskins, M., Kapouleas, N.: Closed twisted products and SO(p) × SO(q)-invariant special Lagrangian cones. Comm. Anal. Geom. 20(1), 95–162 (2012)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hélein, F., Romon, P.: Hamiltonian stationary Lagrangian surfaces in \(\mathbb{C}^{2}\). Comm. Anal. Geom. 10, 79–126 (2002)MATHMathSciNetGoogle Scholar
  13. 13.
    Helgason, S.: Differential geometry and symmetric spaces. Academic Press, New York (1962)MATHGoogle Scholar
  14. 14.
    Hsiang, W.Y., Lawson, H.B.: Minimal submanifolds of low cohomogeneity. J. Differ. Geom. 5, 1–38 (1971)MATHMathSciNetGoogle Scholar
  15. 15.
    Ikawa, O., Sakai, T., Tasaki, H.: Weakly reflective submanifolds and austere submanifolds. J. Math. Soc. Jpn. 61(2), 437–481 (2009)Google Scholar
  16. 16.
    Iriyeh, H., Ono H., Sakai, T.: Integral geometry and Hamiltonian volume minimizing property of a totally geodesic Lagrangian torus in S 2 × S 2. Proc. Japan Acad. Ser. A Math. Sci. 79(10), 167–170 (2003)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Iriyeh, H., Sakai, T., Tasaki H.: Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type. J. Math. Soc. Jpn. 65(4), 1135–1151 (2013)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kajigaya, T.: Hamiltonian minimality of normal bundles over the isoparametric submanifolds, to appear in Differ. Geom. Appl.Google Scholar
  19. 19.
    Loos, O.: Symmetric Spaces. II: Compact Spaces and Classification. W. A. Benjamin, New York/Amsterdam (1969)Google Scholar
  20. 20.
    Ma, H., Ohnita, Y.: On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres. Math. Z. 261, 749–785 (2009)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Ma, H., Ohnita, Y.: Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. arXiv:1207.0338v1 [math. DG] (2012)Google Scholar
  22. 22.
    Miyaoka, R.: Isoparametric hypersurfaces with (g, m) = (6, 2). Ann. Math. 177(1), 53–110 (2013)Google Scholar
  23. 23.
    Münzner, H.F.: Isoparametrische Hyperflächen in Sphären I. Math. Ann. 251, 57–71 (1980)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Münzner, H.F.: Isoparametrische Hyperflächen in Sphären II. Math. Ann. 256, 215–232 (1981)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Oh, Y.-G.: Second variation and stability of minimal Lagrangian submanifolds. Invent. Math. 101, 501–519 (1990)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Oh, Y.-G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Zeit. 212, 175–192 (1993)CrossRefMATHGoogle Scholar
  27. 27.
    Oh, Y.-G.: Mean curvature vector and symplectic topology of Lagrangian submanifolds in Einstein-Kähler manifolds. Math. Zeit. 216, 471–482 (1994)CrossRefMATHGoogle Scholar
  28. 28.
    Ohnita, Y.: Certain Lagrangian submanifolds in Hermitian symmetric spaces and Hamiltonian stability problems. In: Proceedings of the Fifteenth International Workshop on Differential Geometry, vol. 15, pp. 1–26 (2011)MathSciNetGoogle Scholar
  29. 29.
    Olmos, C.: Isoparametric submanifolds and their homogeneous structures. J. Differ. Geom. 38, 225–234 (1993)MATHMathSciNetGoogle Scholar
  30. 30.
    Ono, H.: Integral formula of Maslov index and its applications. Jpn. J. Math. 30(2), 413–421 (2004)MATHGoogle Scholar
  31. 31.
    Ozeki, H., Takeuchi, M.: On some types of isoparametric hypersurfaces in spheres. I. Tohoku Math. J. 27, 515–559 (1975)CrossRefMATHGoogle Scholar
  32. 32.
    Palais, R.S., Terng, C.-L.: A general theory of canonical forms. Trans. Am. Math. Soc. 300, 771–789 (1987)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Palmer, B.: Hamiltonian minimality and Hamiltonian stability of Gauss maps. Differ. Geom. Appl. 7, 51–58 (1997)CrossRefMATHGoogle Scholar
  34. 34.
    Segre, B.: Famiglie di ipersuperficie isoparametriche negli spazi euclideo ad un qualinque numero di dimensioni. Atti Accad. Naz. Lincei Rend. VI. Ser. 27, 203–207 (1938)Google Scholar
  35. 35.
    Somigliana, C.: Sulle relazioni fra il principio di Huygenes e l’ottica geometrica. (Italian), Atti Acc. Sc. Torino 54, 974–979 (1918/1919)Google Scholar
  36. 36.
    Takagi R., Takahashi, T.: On the principal curvatures of homogeneous hypersurfaces in a sphere, Differential geometry (in honor of Kentaro Yano), pp. 469–481. Kinokuniya, Tokyo (1972)Google Scholar
  37. 37.
    Terng, C.-L.: Isoparametric submanifolds and their Coxeter groups. J. Differ. Geom. 21, 79–107 (1985)MathSciNetGoogle Scholar
  38. 38.
    Thorbergsson, G.: Isoparametric foliations and their buildings. Ann. Math. 133(2), 429–446 (1991)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Thorbergsson, G.: A survey of isoparametric hypersurfaces and their generalizations. In: Hand Book of Differential Geometry, vol. I, pp. 963–995. North-Holland, Amsterdam (2000)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Mathematical Institute, Graduate School of SciencesTohoku UniversityAoba-ku, SendaiJapan

Personalised recommendations