SHADOW LEMMA ON FINSLER MANIFOLDS OF HYPERBOLIC TYPE

Year 2014, Volume: 2 Issue: 2, 76 - 88, 01.12.2014

CYRIAQUE Atındogbe CARLOSOGOUYANDJOU

Abstract

Let (M, F ) be a compact Finsler manifold of hyperbolic type, ˜MFbe its universal Finslerian covering and αFthe critical exponent of the groupof the deck transformations of ˜F. In this paper we prove the existence of anαF-Busemann quasi-density on the Gromov boundary ˜MG(∞) of ˜. Furthermore, we generalize the Shadow lemma to the compact Finsler manifoldsof hyperbolic type

Keywords

Finsler manifold , Gromov hyperbolic manifold , Busemann quasidensity , Shadow lemma

References

• Ancona A., Th´eorie du potentiel sur les graphes et les vari´et´es, In Ancona A, et Al (eds.) ”Potential Theory, Surveys and Problems”, Lecture Notes in Math.,No. 1344, Springer-Verlag, 19
• D. Bao, S.-S. Chern and Z. Shen., An introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics 200, Springer-Verlag, 1982.
• J. Cao., Cheeger isoperimetric constants of Gromov hyperbolic spaces with quasi-pole. Com- munication in Contemporay Math. no. 4, (2000) 2:511-533.
• Coornaert M., Delzant T., Papadopoulos A., G´eom´etrie et th´eorie des groupes. Lecture notes in Math. No. 1441, Springer-Verlag, Berlin 1990.
• Coornaert M., Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific J. of Math. 159, n. 2 (1993), 241-270.
• Coornert M., Papadoupoulos A., Symbolic Dynamics and Hyperbolic Groups, Lecture Notes in Math., No. 1539, Springer-Verlag, 1993.
• W. Klingenberg., Geod¨atischer Fluss auf Mannigfaltigkeiten vom hyperbolishen Typ. Inven- tiones math. 14 (1971), 63-82.
• G. Knieper., On the asymptotic geometry of nonpositively curved manifolds. GAFA vol. 7, (1997) 755-782.
• G. Knieper., Volume growth, entropy and the geodesic stretch. Mathematical Research Letters 2, (1995) 39-58.
• S. Patterson., The limit set of Fuchsian group. Acta Math. 163 (1976), 241-273.
• Z. Shen., Volume comparison and its applications in Riemann-Finsler geometry. Adv. Math. 128 (1977), 306-328.
• J. P. Schr¨oder., Minimal rays on surfaces of genus greater than one. arXiv :1404.0573v1 [math.DS]
• D. Sullivan., The density at infinity of discrete group of hyperbolic motions. Publ. math. I.H.E.S., 50 (1979), 171-202.
• E. M. Zaustinsky., Extremals on compact E-surfaces. Transactions of the American Mathe- matical Society. 102.3 (1962), 433-445.
• Institut de Math´ematiques et de Sciences Physiques (IMSP),Universit´e d’Abomey- Calavi 01 BP 613 Porto-Novo, Republic of Benin. E-mail address: [e-mail] E-mail address: [e-mail] E-mail address: [e-mail]