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A Mean Value Laplacian for Strongly Kähler-Finsler Manifolds

  • Zhong Chunping
  • Zhong Tongde
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

It is well known that the Laplace operator plays an important role in the theory of harmonic integrals and the Bochner technique both in Riemannian and Kähler manifolds. In recent years, under the initiation of S.S. Chern, the global differential geometry of real and complex Finsler manifolds has gained a great development ([1], [2], [3], [4]). A lot of results about the Laplacian and its applications have been obtained in a real Finsler manifold ([5], [6]). But up to now there are no results for the Laplacian and its applications in a complex Finsler manifold. The key point in the theory of the Bochner technique and harmonic integrals is to define a suitable Laplace operator. In the case of Finsler manifolds the difficulty is that the Finsler metric depends on the fibre coordinates. Using the idea that the Laplacian on Euclidean space or on a Riemannian manifold measures the average value of a function around a point, P. Centore ([7]) generalizes the Laplacian on a Riemannian manifold to a real Finsler manifold. Considering a complex manifold as a real manifold, there is a one-to-one correspondence between the real coordinates and the complex coordinates ([8]). In this paper, we use the mean value idea to define the Laplacian on a strongly Kähler-Finsler manifold, first for functions and then for forms, and we derive some remarkable properties for the Laplacian for functions and extend the Laplacian to arbitrary forms. Indeed our Laplacian on strongly Kähler-Finsler manifolds generalizes the Kählerian Laplacian. And it is worth to remark that using the osculating Kähler metric — which we obtain in the following to define the pointwise and global inner product when we define the Hodge-Laplace operator of (p, q)-forms — is more natural than using the fundamental tensor of the Finsler metric and can avoid many complicated calculations.

Keywords

Strongly Kähler-Finsler manifold mean value Laplacian Hodge—Laplace operator 

Mathematics Subject Classification (2000)

32C10 53B40 53C55 

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References

  1. [1]
    S.S. Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction. AMS Notices 43 (1996), 959–963.MathSciNetzbMATHGoogle Scholar
  2. [2]
    D. Bao, S.S. Chern and Z. Shen (eds.), Finsler geometry (Proceedings of the Joint Summer Research Conference on Finsler Geometry, July 16–20,1995, Seattle, Washington). Cont. Math. Vol. 196 (1996) Amer. Math. Soc. Providence, RI.zbMATHGoogle Scholar
  3. [3]
    D. Bao, S.S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry. New York: Springer-Verlag, New York, 2000.CrossRefGoogle Scholar
  4. [4]
    M. Abate and G. Patrizio, Finsler metrics - A global approach. Lecture Notes in Math, Vol 1591, Springer-Verlag, 1994.zbMATHGoogle Scholar
  5. [5]
    D. Bao and B. Lackey, A Hodge decomposition theorem for Finsler spaces. C.R. Acad. Sci. Paris 323 (1996), 51–56.MathSciNetzbMATHGoogle Scholar
  6. [6]
    P.L. Antonelli and B. Lackey (eds.), The Theory of Finslerian Laplacians and applications. MAIA 459, Kluwer Academic Publishers, 1998.zbMATHGoogle Scholar
  7. [7]
    P. Centore, A mean-value Laplacian for Finsler spaces. The theory of Finslerian Laplacians and applications, MAIA 459,Kluwer Academic Publishers, 1998, 151–186.CrossRefGoogle Scholar
  8. [8]
    K. Yano and S. Bochner, Curvature and Betti Numbers. Princeton Univ. Press, 1953.zbMATHGoogle Scholar
  9. [9]
    Zhong Chunping and Zhong Tongde, Hodge-Laplace operator on complex Finsler manifolds. Proceedings of the ICM 2002 Satellite Conference on Geometric Function Theory in Several Complex Variables, USTC. Hefei, Anhui, P.R. of China, Aug. 30-Sep. 2, 2002. Edited by C.H. Fitzgerald and Sheng Gong. World Scientific. (to appear).Google Scholar
  10. [10]
    J. Morrow, K. Kodaira, Complex manifolds, New York: Holt, Rinehart and Winston, Inc, 1971.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Zhong Chunping
    • 1
  • Zhong Tongde
    • 2
  1. 1.School of Mathematical SciencesXiamen UniversityXiamenP.R. China
  2. 2.Institute of MathematicsXiamen UniversityXiamenP.R. China

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