Harmonic Maps in Complex Finsler Geometry

  • Seiki Nishikawa
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 59)


Given a smooth map from a compact Riemann surface to a complex manifold M, equipped with a strongly pseudoconvex Finsler metric F, we define a natural notion of the \(\overline \partial\)-energy of the map. A harmonic map is then defined to be a critical point of the \(\overline \partial\)-energy functional. Under the condition that F, is weakly Kähler, we obtain the second variation formula of the functional, and prove that any \(\overline \partial\)-energy minimizing harmonic map from a Riemann sphere to a weakly Kähler Finsler manifold M, of positive curvature is either holomorphic or antiholomorphic. Significance of complex Finsler metrics in the study of holomorphic vector bundles, in particular its relation with the Hartshorne conjecture in complex algebraic geometry, is represented. A brief overview of complex Finsler geometry is also provided.


Vector Bundle Compact Riemann Surface Holomorphic Vector Bundle Finsler Manifold Finsler Metrics 
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  1. [1]
    M. Abate and G. Patrizio, Finsler Metrics - A Global Approach with applications to geometric function theory, Lecture Notes in Math. Vol. 1591, Springer-Verlag, Berlin-Heidelberg-New York, 1994.Google Scholar
  2. [2]
    M. Abate and G. Patrizio, Holomorphic curvature of Finsler metrics and complex geodesics, J. Geom. Anal. 3 (1996), 341–363.MathSciNetCrossRefGoogle Scholar
  3. [3]
    M. Abate and G. Patrizio, Kähler Finsler manifolds of constant holomorphic curvature, Internat. J. Math. 8 (1997), 169–186.MathSciNetzbMATHGoogle Scholar
  4. [4]
    R.L. Bishop and S.I. Goldberg, On the second cohomology group of a Kähler manifold of positive curvature, Proc. Amer. Math. Soc. 16 (1965), 119–122.MathSciNetzbMATHGoogle Scholar
  5. [5]
    P. Centore, Finsler Laplacians and minimal-energy maps, Internat. J. Math. 11 (2000), 1–13.MathSciNetzbMATHGoogle Scholar
  6. [6]
    J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, C. B. M. S. Regional Conference Series No. 50, Amer. Math. Soc., Providence, Rhode Island, 1983.Google Scholar
  7. [7]
    T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165–174.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S.I. Goldberg and S. Kobayashi, Holomorphic bisectional curvature, J. Differential Geom. 1 (1967), 225–233.MathSciNetzbMATHGoogle Scholar
  9. [9]
    A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphères de Riemann, Amer. J. Math. 79 (1957), 121–138.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Lecture Notes in Math. Vol. 156, Springer-Verlag, Berlin-Heidelberg-New York, 1970.zbMATHCrossRefGoogle Scholar
  11. [11]
    S. Kobayashi, On compact Kähler manifolds with positive Ricci tensor, Ann. of Math. 74 (1961), 570–574.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    S. Kobayashi, Negativity of vector bundles and complex Finsler structures, Nagoya J. Math. 57 (1975), 153–166.zbMATHGoogle Scholar
  13. [13]
    S. Kobayashi, Complex Finsler vector bundles, Contemp. Math. 196 (1996), 145–152.CrossRefGoogle Scholar
  14. [14]
    S. Kobayashi and T. Ochiai, On complex manifolds with positive tangent bundles, J. Math. Soc. Japan 22 (1970), 499–525.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    S. Kobayashi and T. Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47.MathSciNetzbMATHGoogle Scholar
  16. [16]
    T. Mabuchi, C3-actions and algebraic three-folds with ample tangent bundle, Nagoya Math. J. 69 (1978), 33–64.MathSciNetGoogle Scholar
  17. [17]
    M. Marcus and V. J. Mizel, Continuity of certain Nemitsky operators on Sobolev spaces and the chain rule, J. Anal. Math. 28 (1975), 303–334.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    X. Mo, Harmonic maps from Finsler manifolds, Illinois J. Math. 45 (2001), 1331–1345.MathSciNetzbMATHGoogle Scholar
  19. [19]
    S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. 110 (1979), 593–606.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    S. Nishikawa, Harmonic maps into complex Finsler manifolds, in preparation.Google Scholar
  21. [21]
    H.L. Royden, Complex Finsler metrics, Contemp. Math. 49 (1986), 119–124.MathSciNetCrossRefGoogle Scholar
  22. [22]
    H. Rund, Generalized metrics on complex manifolds, Math. Nachr. 34 (1967), 55–77.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981), 1–24.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Y.-T. Siu and S.-T. Yau, Complex Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), 189–204.MathSciNetzbMATHGoogle Scholar
  25. [25]
    E. Vesentini, Complex geodesics, Composito Math. 44 (1981), 375–394.MathSciNetzbMATHGoogle Scholar

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© Springer Basel AG 2004

Authors and Affiliations

  • Seiki Nishikawa
    • 1
  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan

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