Skip to main content
SpringerLink
Log in

Kähler Finsler metrics and conformal deformations

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The conformai properties of complex Finsler metrics are studied. We first give a characterization of a compact complex Finsler manifold to be globally conformai Kahler. By considering the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes, we then study the variational properties of Kahler Finsler metrics. By studying the spectral properties of two average metrics, the stabilities of critical Kähler Finsler metrics are verified. Finally, a Yamabe type problem for mean holomorphic Ricci curvature is considered, and a partial existence result is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abate and G. Patrizio, Finsler Metrics—A Global Approach, Lecture Notes in Mathemarics, Vol. 1591, Springer, Berlin, 1994.

    Book  Google Scholar 

  2. T. Aikou, Some remarks on averaged metrics and connections, in Proceedings of the 46th Symposium on Finsler Geometry, Society of Finsler Geometry, Japan, 2011, pp. 1–4.

    Google Scholar 

  3. D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A Sampler of Riemann—Finsler Geometry, Mathematical Sciences Research Institute Publications, Vol. 50, Cambridge University Press, Cambridge, 2004, pp. 198–256.

    Google Scholar 

  4. J. Bland and M. Kalka, Variations of holomorphic curvature for Kähler Finsler metrics, in Finsler Geometry (Seattle, WA, 1995), Contemporary Mathematics, Vol. 196, American Mathematical Society, Providence, RI, 1996, pp. 121–132.

    Chapter  Google Scholar 

  5. R. Bott and L. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Vol. 82, Springer, New York, 1982.

    Book  Google Scholar 

  6. B. Chen and Y. Shen, Kahler Finsler metrics are actually strongly Kahler, Chinese Annals of Mathematics. Series B 30 (2009), 173–178.

    Article  MathSciNet  Google Scholar 

  7. B. Chen and L. Zhao, On a Yamabe type problem in Finsler geometry, Canadian Mathematical Bulletin 60 (2017), 253–268.

    Article  MathSciNet  Google Scholar 

  8. H. Fen, K. Liu and X. Wan, A Donaldson type functional on a holomorphic Finsler vector bundle, Mathematische Annalen 239 (2017), 997–1019.

    MathSciNet  MATH  Google Scholar 

  9. A. Grigor’yan, Heat kernels on weighted manifolds and applications, in The Ubiquitous Heat Kernel, Contemporary Mathematics, Vol. 398, American Mathematical Society, Providence, RI, 2006, pp. 93–191.

    Chapter  Google Scholar 

  10. J. Han and Y. Shen, Harmonic maps from complex Finsler manifolds, Pacific Journal of Mathematics 236 (2008), 341–356.

    Article  MathSciNet  Google Scholar 

  11. S. Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Mathematical Journal 57 (1975), 153–166.

    Article  MathSciNet  Google Scholar 

  12. S. Kobayashi, Complex Finsler vector bundles, in Finsler Geometry (Seattle, WA, 1995), Contemporary Mathematics, Vol. 196, American Mathematical Society, Providence, RI, 1996, pp. 145–153.

    Chapter  Google Scholar 

  13. J. Lee and T. Parker, The Yamabe problem, Bulletin of the American Mathematical Society 17 (1987), 37–91.

    Article  MathSciNet  Google Scholar 

  14. H. Lee, A kind of even dimensional differential geometry and its applications to exterior calculus, American Journal of Mathematics 65 (1943), 433–438.

    Article  MathSciNet  Google Scholar 

  15. J. Li and C. Qiu, Comparison and Wu’s theorems in Finsler geometry, Mathematische Zeitschrift 295 (2020), 485–514.

    Article  MathSciNet  Google Scholar 

  16. I. Vaisman, On locally and globally conformal Kaahler manifolds, Transactions of the American Mathematical Society 262 (1980), 533–542.

    MathSciNet  MATH  Google Scholar 

  17. R. Yan, On the volume of the projectivized tangent bundle in a complex Finsler manifold, Archiv der Mathematik 86 (2006), 458–463.

    Article  MathSciNet  Google Scholar 

  18. S. Yin and X. Zhang, Comparison theorems and their applications on Kaahler Finsler manifolds, Journal of Geometric Analysis 30 (2020), 2105–2131.

    Article  MathSciNet  Google Scholar 

  19. C. Zhong and T. Zhong, Hodge decomposition theorem on strongly Kahler Finsler manifolds, Science in China. Seres A 49 (2006), 1696–1714.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tongji University, 1239 Siping Road, Shanghai, 200092, P. R. China

    Bin Chen

  2. School of Mathematical Sciences, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, P. R. China

    Yibing Shen

  3. School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, P. R. China

    Lili Zhao

Authors
  1. Bin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yibing Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lili Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Chen.

Additional information

Supported by the National Natural Science Foundation of China (nos. 11871126, 11671352, 11471246).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, B., Shen, Y. & Zhao, L. Kähler Finsler metrics and conformal deformations. Isr. J. Math. 248, 355–382 (2022). https://doi.org/10.1007/s11856-022-2304-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-022-2304-8

Search

Navigation