COMPARISON THEOREMS IN FINSLER GEOMETRY WITH WEIGHTED CURVATURE BOUNDS AND RELATED RESULTS

- Journal title : Journal of the Korean Mathematical Society
- Volume 52, Issue 3, 2015, pp.603-624
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2015.52.3.603

Title & Authors

COMPARISON THEOREMS IN FINSLER GEOMETRY WITH WEIGHTED CURVATURE BOUNDS AND RELATED RESULTS

Wu, Bing-Ye;

Wu, Bing-Ye;

Abstract

We first extend the notions of weighted curvatures, including the weighted flag curvature and the weighted Ricci curvature, for a Finsler manifold with given volume form. Then we establish some basic comparison theorems for Finsler manifolds with various weighted curvature bounds. As applications, we obtain some McKean type theorems for the first eigenvalue of Finsler manifolds, some results on weighted curvature and fundamental group for Finsler manifolds, as well as an estimation of Gromov simplicial norms for reversible Finsler manifolds.

Keywords

Finsler manifold;weighted flag curvature;weighted Ricci curvature;first eigenvalue;fundamental group;

Language

English

References

1.

M. Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology 29 (1990), no. 1, 41-55.

2.

D. Bao, S. S. Chern, and Z. Shen, An introduction to Riemann-Finsler Geometry, GTM 200, Springer-Verlag, 2000.

3.

E. Calabi, On manifolds with nonnegative Ricci curvature II, Notices Amer. Math. Soc. 22 (1975), A-205, Abstract No. 720-53-6.

4.

I. Chavel, Riemannian Geometry: A Modern Introduction, Camb. Univ. Press, 1993.

5.

Q. Ding, A new Laplacian comparison theorem and the estimate of eigenvalues, Chin. Ann. Math. Ser. 15 (1994), no. 1, 35-42.

6.

M. Gromov, Volume and bounded cohomology, IHES Publ. Math. 56 (1983), 213-307.

7.

H. P. McKean, An upper bound for the spectrum of ${\Delta}$ on a manifold of negative curvature, J. Differential Geom. 4 (1970), 359-366.

8.

J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1-7.

9.

S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 2, 211-249.

10.

S. Ohta and K. T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math. 62 (2009), no. 10, 1386-1433.

11.

H. B. Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Ann. 328 (2004), no. 3, 373-387.

12.

Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128 (1997), no. 2, 306-328.

13.

Z. Shen, Lectures on Finsler Geometry, World Sci., Singapore, 2001.

14.

Y. B. Shen and W. Zhao, On fundamental groups of Finsler manifolds, Sci. China Math. 54 (2011), no. 9, 1951-1964.

15.

B. Y. Wu, Volume form and its applications in Finsler geometry, Publ. Math. Debrecen 78 (2011), no. 3-4, 723-741.

16.

B. Y. Wu, Some results on curvature and topology of Finsler manifolds, Ann. Polon. Math. 107 (2013), ni. 3, 309-320.

17.

B. Y. Wu and Y. L. Xin, Comparison theorems in Finsler geometry and their applications, Math. Ann. 337 (2007), no. 1, 177-196.