Skip to main content
SpringerLink
Log in

The maximum diam theorem on Finsler manifolds

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first eigenvalue of the Finsler-Laplacian attains its lower bound if and only if the Finsler manifold is isometric to a standard Finsler sphere, and moreover, we obtain an explicit 1-st eigenfunction on the sphere.

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. Antonelli, P., Lackey, B.: The Theory of Finslerian Laplacian and Application. Kluwer Acad. Publ., Dordrecht (1998)

  2. Bácsó, S., Cheng, X.Y., Shen, Z.M.: Curvature properties of\((\alpha ,\beta )\)-metrics, In Finsler Geometry, Sapporo 2005-In Memory of M. Matsumoto, ed. S. Sabau and H. Shimada. Adv. Stud. Pure Math., Math. Soc. Jpn. 48, 73–110 (2007)

  3. Bakry, D., Ledoux, M.: Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator. Duke Math. J. 85(1), 253–270 (1996)

    Article  MathSciNet  Google Scholar 

  4. Bao, D.W., Lackey, B.: A Hodge decomposition theorem for Finsler spaces, C. R. Acad. Sci. Paris Sér. I Math. 323, 51–56 (1996)

  5. Bao, D.W., Shen, Z.M.: Finsler metrics of constant cuevature on the Lie group \(\mathbb{S}^3\). J. Lond. Math. Soc. 66, 453–467 (2002)

    Article  Google Scholar 

  6. Bao, D.W., Robles, C., Shen, Z.M.: Zermelo navigation on Riemannian manifolds. J. Diff. Geom. 66, 391–449 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Barthelmé, T.: A natural Finsler-Laplace operator. Israel J. Math. 196(1), 375–412 (2013)

    Article  MathSciNet  Google Scholar 

  8. Centore, P.: Finsler Laplacians and minimal-energy map. Internat. J. Math. 11, 1–13 (2000)

    Article  MathSciNet  Google Scholar 

  9. Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143(3), 289–297 (1975)

    Article  MathSciNet  Google Scholar 

  10. Cheng, X.Y., Shen, Z.M.: Randers metrics with special curvature properties. Osaka J. Math. 40(1), 87–101 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Cheng, X.Y., Shen, Z.M., Tian, Y.F.: A class of Einstein \((\alpha,\beta )\)-metrics. Israel J. Math. 192, 221–249 (2012)

    Article  MathSciNet  Google Scholar 

  12. Ge, Y.X., Shen, Z.M.: Eigenvalues and eigenfunctions of metric measure manifolds. Proc. Lond. Math. Soc. 82, 725–746 (2001)

    Article  MathSciNet  Google Scholar 

  13. Kim, C.W., Yim, J.W.: Finsler manifolds with positive constant flag curvature. Geom. Dedicata. 98, 47–56 (2003)

    Article  MathSciNet  Google Scholar 

  14. Myers, S.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)

    Article  MathSciNet  Google Scholar 

  15. Ohta, S.: Finsler interpolation inequalities. Calc. Var. Partial Differ. Equ. 36, 211–249 (2009)

    Article  MathSciNet  Google Scholar 

  16. Ohta, S., Sturm, K.T.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62, 1386–1433 (2009)

    Article  MathSciNet  Google Scholar 

  17. Qian, Z.M.: Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2), 48, 235–242 (1997)

  18. Ruan, Q.H.: Two rigidity theorems on manifolds with Bakry–Emery Ricci curvature. Proc. Jpn. Acad. Ser. A 85, 71–74 (2009)

    Article  MathSciNet  Google Scholar 

  19. Shen, Z.M.: Finsler spaces of constant positive curvature, (Joint Summer Research conference on Finsler Geometry in July 16-20, 1995), Contem. Math. AMS, 196, 83–92 (1996)

  20. Shen, Z.M.: Volume compasion and its applications in Riemann–Finaler geometry. Adv. Math. 128, 306–328 (1997)

    Article  MathSciNet  Google Scholar 

  21. Shen, Z.M.: Lectures on Finsler geometry , World Sci. 2001, Singapore

  22. Wu, B.Y., Xin, Y.L.: Comparison theorems in Finsler geometry and their applications. Math. Ann. 337, 177–196 (2007)

    Article  MathSciNet  Google Scholar 

  23. Yin, S.T.: Some rigidity theorems on Finsler manifolds. AIMS Math. 6(3), 3025–3036 (2021)

    Article  MathSciNet  Google Scholar 

  24. Yin, S.T., He, Q., Shen, Y.B.: On lower bounds of the first eigenvalue of Finsler–Laplacian. Publ. Math. (Debr.) 83(3), 385–405 (2013)

    Article  MathSciNet  Google Scholar 

  25. Zheng, D.X., He, Q.: \((\alpha,\beta )\)-metric with almost isotropic flag curvature \(K = \frac{3c_x^i y^i}{F} +\sigma \). Adv. Math. (China) 44, 599–606 (2015)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Tongling University, 244000, Tongling, Anhui, China

    Songting Yin

  2. School of Mathematical Sciences, Tongji University, 200092, Shanghai, China

    Qun He

Authors
  1. Songting Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qun He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Songting Yin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This project is supported by NNSFC (Grant Nos. 11971253,11471246), AHNSF (Grant No. 2108085MAxx) and AHGXBJ (Grant No. gxbjZD20210xx)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yin, S., He, Q. The maximum diam theorem on Finsler manifolds. J Geom Anal 31, 12231–12249 (2021). https://doi.org/10.1007/s12220-021-00715-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-021-00715-z

Keywords

Mathematics Subject Classification

Search

Navigation