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The degree of strong nondegeneracy of the bisectional curvature of exceptional bounded symmetric domains

  • Zhong Jia-Qing

Abstract

In [1], [2] Siu discovered the complex-analyticity of harmonic maps between two Kähler manifolds under some conditions and prove the strong rigidity of compact quotients of irreducible bounded symmetric domains of dimension at least two. Furthermore, he proposed the following conjecture:

Suppose f: N → M is a harmonic map between compact Kähler manifolds and M is a compact quotient of an irreducible symmetric domain. Let r be the maximal rank of df (over IR). Then f is either holomorphic or antiholomorphic provided that r is appropriately large. In [3] Siu confirmed this conjecture for the four classical domains and indicated that the confirmation for the two exceptional domains depends on the computation of the degree of the strong nondegeneracy of the bisectional curvature in the two exceptional cases.

Keywords

Symmetric Space Curvature Tensor Symmetric Domain Root Vector Bisectional Curvature 
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References

  1. 1.
    Yum-Tong Siu: The complex-analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds. Ann. of Math. 112 (1980), 73–111.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Yum-Tong Siu: Strong rigidity of compact quotients of exceptional domains. Duke Math. J. 48 (1981), 857–871.MATHGoogle Scholar
  3. 3.
    Yum-Tong Siu: Complex-analyticity of harmonic maps, Vanishing and Lefschetz Theorems. J. Diff. Geom. 17(1982), 55–138.MATHGoogle Scholar
  4. 4.
    Cheeger, J. and Ebin, D.: Comparison theorems in Riemannian geometry. North-Holland, Amsterdam 1975.MATHGoogle Scholar
  5. 5.
    Hergason, S.: Differential geometry and symmetric spaces. Academic Press, New York, 1962.Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1984

Authors and Affiliations

  • Zhong Jia-Qing
    • 1
  1. 1.Institute of MathematicsAcademia SinicaChina

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