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A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results

Published online by Cambridge University Press:  20 November 2018

Wei Zhao  and
Yibing Shen
Wei Zhao
Affiliation:
Center of Mathematical Sciences, Zhejiang University, Hangzhou, China, e-mail: zhaowei008@yahoo.cn, yibingshen@zju.edu.cn
Yibing Shen
Affiliation:
Center of Mathematical Sciences, Zhejiang University, Hangzhou, China, e-mail: zhaowei008@yahoo.cn, yibingshen@zju.edu.cn
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Abstract

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In this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger–Kazdan inequality and Santalá's formula in Finsler geometry. Based on these, we derive a Berger–Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

Keywords

53B4053C6552A38Finsler manifoldBerger–Kazdan inequalityBerger–Kazdan comparison theoremSantalá's formulaCroke's isoperimetric inequality
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 6 , 01 December 2013 , pp. 1401 - 1435
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Alvarez Paiva, J. C. and Berck, G., What is wrong with the Hausdorff measure in Finsler spaces. Adv. in Math. 204(2006), no. 2, 647–663. http://dx.doi.org/10.1016/j.aim.2005.06.007 Google Scholar
[2] Alvarez-Paiva, J. and Thompson, A. C., Volumes in normed and Finsler spaces. In: A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., 50, Cambridge University Press, Cambridge, 2004, pp. 1-48.Google Scholar
[3] Auslander, L., On curvature in Finsler geometry. Trans. Amer. Math. Soc. 79(1955), 378-388. http://dx.doi.org/10.1090/S0002-9947-1955-0071833-6 Google Scholar
[4] Bao and, D. Chern, S. S., A note on the Gauss-Bonnet theorem for Finsler spaces. Ann. of Math. 143(1996), no. 2, 233–252. http://dx.doi.org/10.2307/2118643 Google Scholar
[5] Bao, D., Chern, S. S., and Shen, Z., An introduction to Riemannian-Finsler geometry. Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000.Google Scholar
[6] Berger, M., Lectures on geodesics in Riemannian geometry. Tata Institute of Fundamental Research Lectures on Mathematics, 33, Tata Institute of Fundamental Research, Bombay, 1965.Google Scholar
[7] Berger, M., Une borne inférieure pour le volume d’une variété riemannienes en fonction du rayon d’injectivité. Ann. Inst. Fourier 30(1980), no. 3, 259–265. http://dx.doi.org/10.5802/aif.802 Google Scholar
[8] Berger, M. and Kazdan, J. L., A Sturm-Liouville inequality with applications to an isoperimetric inequality for volume in terms of injectivity radius, and to Wiedersehen manifolds. In: General inequalities, 2, (Proc. Second Internat. Conf., Oberwolfach, 1978), Birkhäuser, Basel-Boston, MA, 1980, pp. 367–377.Google Scholar
[9] Besse, L., Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete, 93, Springer-Verlag, Berlin-New York, 1978.Google Scholar
[10] Bishop, R. L. and Crittenden, R. J., Geometry of manifolds. Pure and Applied Mathematics, 15, Academic Press, New York-London, 1964.Google Scholar
[11] Burago, D., Burago, Y., and Ivanov, S., A course in metric geometry. Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001.Google Scholar
[12] H. Busemann, , Intrinsic area. Ann. of Math. 48(1947), 234–267. http://dx.doi.org/10.2307/1969168 Google Scholar
[13] Chavel, I., Riemannian geometry—a modern introduction. Cambridge Tracts in Mathematics, 108, Cambridge University Press, Cambridge, 1993.Google Scholar
[14] Chen, B., Some geometric and analysis problems in Finsler geometry.(Chinese), Doctoral thesis, Zhejiang University, 2010.Google Scholar
[15] Chow, B., Lu, P., and Ni, L., Hamilton's Ricci flow. Graduate Studeis in Mathematics, 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006.Google Scholar
[16] Croke, C. B., Curvature free volume estimates. Invent. Math. 76(1984), no. 3, 515–521. http://dx.doi.org/10.1007/BF01388471 Google Scholar
[17] Croke, C. B., Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. École Norm. Sup. 13(1980), no. 4, 419–435.Google Scholar
[18] Dazord, P., Propriétés globales des géodesiques des espaces de Finsler. Thèse, Universitéde Lyon, 1969.Google Scholar
[19] Durán, C. E., A volume comparison theorem for Finsler manifolds. Proc. Amer. Math. Soc. 126(1998), no. 10, 3079–3082.http://dx.doi.org/10.1090/S0002-9939-98-04629-2 Google Scholar
[20] Egloff, D., Uniform Finsler Hadamard manifolds. Ann. Ins. H. Poincaré. Phys. Théor. 66(1997), no. 3, 323–357.Google Scholar
[21] Gray, A., Tubes. Progress in Mathematics, 221, Birkhäuser Verlag, Basel, 2004.Google Scholar
[22] Gromov, M., Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics, 152, Birkhäuser Boston, Boston, MA, 1999.Google Scholar
[23] He, Q. and Shen, Y.-B., On Bernstein type theorems in Finsler spaces with the volume form induced from the projective sphere bundle. Proc. Amer. Math. Soc. 134(2006), no. 3, 871–880. http://dx.doi.org/10.1090/S0002-9939-05-08017-2 Google Scholar
[24] Holmes, R. D. and Thompson, A. C., n-dimensional area and content in Minkowski spaces. Pacific J. Math. 85(1979), no. 1, 77–110.Google Scholar
[25] Kim, C.-W. and Yim, J.-W., Finsler manifolds with positive constant flag curvature. Geom. Dedicata. 98(2003), 47–56.http://dx.doi.org/10.1023/A:1024034012734 Google Scholar
[26] Matsumoto, M., Theory of curves in tangent planes of two-dimensional Finsler spaces. Tensor (N.S.) 37(1982), no. 1, 35–42.Google Scholar
[27] O'Neill, B., Semi-Riemannian geometry. Pure and Applied Mathematics, 103, Academic Press, New York, 1983.Google Scholar
[28] Petersen, P. V., Gromov-Hausdorff convergence of metric spaces. In: Riemannian geometry (Los Angeles,CA, 1990), Proc. Sympos. Pure Math., 54, Part 3, American Mathematical Society, Providence, RI, 1993, pp. 489–504.Google Scholar
[29] Petersen, P. V., Riemannian geometry. Graduate Texts in Mathematics, 171, Springer-Verlag, New York, 1998.Google Scholar
[30] Rademacher, H.-B., A sphere theorem for non-reversible Finsler metrics. Math. Ann. 328(2004), no. 3, 373–387. http://dx.doi.org/10.1007/s00208-003-0485-y Google Scholar
[31] Santaló, L., Integral geometry and geometric probability. Encyclopedia of Mathematics and its Applications, 1, Addison-Wesley, Reading, MA, 1976.Google Scholar
[32] Shen, Y.-B. and Zhao, W., Gromov pre-compactness theorems for nonreversible Finsler manifolds. Differential Geom. Appl. 28(2010), no. 5, 565–581.http://dx.doi.org/10.1016/j.difgeo.2010.04.006 Google Scholar
[33] Shen, Z., Differential geometry of spray and Finsler spaces. Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
[34] Shen, Z., Lectures on Finsler geometry. World Scientific, Singapore, 2001.Google Scholar
[35] Shen, Z., On Finsler geometry of submanifolds. Math. Ann. 311(1998), no. 3, 549–576.http://dx.doi.org/10.1007/s002080050200 Google Scholar
[36] Shen, Z., Volume comparison and its applications in Riemannian-Finsler geometry. Adv. Math. 128(1997), no. 2, 306–328. http://dx.doi.org/10.1006/aima.1997.1630 Google Scholar
[37] Shen, Z., Finsler manifolds of constant positive curvature. In: Finsler geometry (Seattle,WA, 1995), Contemp. Math., 196, American Mathematical Society, Providence, RI, 1996, pp. 83–93.Google Scholar
[38] Y.Wu, B., Volume form and its applications in Finsler geometry. Publ. Math. Debrecen 78(2011), no. 3–4, 723–741. http://dx.doi.org/10.5486/PMD.2011.4998 Google Scholar
[39] Y.Wu, B. and Xin, Y. L., Comparison theorems in Finsler geometry and their applications. Math. Ann. 337(2007), no. 1, 177–196.Google Scholar
[40] Yau, S. T., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25(1976), no. 7, 659–670. http://dx.doi.org/10.1512/iumj.1976.25.25051 Google Scholar