Cheeger isoperimetric constants of Gromov-hyperbolic spaces with quasi-poles

Communications in Contemporary Mathematics

Volumn 2, Issue 4, 2000, Pages 511-533

  Cao, Jianguo ^a  

^a UNIVERSITY OF NOTRE DAME   (United States)

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EID: 23044523610     PISSN: 02191997     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. A. Ancona, Negatively curved manifolds, elliptic operators, and Martin boundary, Annals of Math. 125 (1987) 495-536.
2. _, Positive harmonic functions and hyperbolicity, in Potential Theory, Surveys and Problems, eds. J. Král et al., Lecture Notes in Math., No. 1344, Springer-Verlag (1988), pp. 1-24.
3. _, Theorie du potentiel sur les graphes et les varieties, in Ecolé d'Eté de Probabilités de Saint-Flour XVII-1988, eds. A. Ancona et al., Lecture Notes in Math., No. 1427, Springer-Verlag (1990).
4. M. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Diff. Geom. 18 (1983) 701-721.
5. α-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Diff. Geom. 35 (1992) 265-281.
6. M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Annals Math. 121 (1985) 429-461.
7. W. Ballmann, On the Dirichlet problem at infinity for manifolds of non-positive curvature, Forum Math. 1 (1989) 201-213.
8. W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Non-positive Curvature, Progress in Mathematics, Birkhäuser, Boston 61 (1985).
9. W. Ballmann and F. Ledrappier, The Poisson boundary for rank one manifolds and their co-compact lattices, Forum Math. 6 (1994) 301-313.
10. M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000) 266-306.
11. M. Brin and Y. Kifer, Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes, to appear in Trans. AMS.
12. K. Burns, The flat strip theorem fails for surfaces without conjugate points, Proc. Amer. Math. Soc. 115 (1992) 199-206.
13. P. Buser, A note on the isoperimetric constant, Ann. Scient. Éc. Norm. Sup. 15 (1982) 213-230.
14. J. W. Cannon, Theory of negatively curved spaces and groups, in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, eds, T. Bedford, M. Keene and C. Series, Oxford University Press (1991), pp. 315-369.
15. J. Chao, An isoperimetric estimate and its application to the Martin problem, Cornell University, preprint (1993).
16. _, Rigidity for non-compact surfaces of finite area and certain Kähler manifolds, Ergod. Th. & Dynam. Syst. 15 (1995) 475-516.
17. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, ed. R. Gunning, Princeton University Press (1970), pp. 195-199.
18. J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. Math. 96 (1972) 414-443.
19. R. Chung, A. Grigor'yan and S. T. Yau, Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs, preprint 1998, to appear in Comm. Geom. Analysis.
20. M. Coornaert, T. Delzant and A. Papadopoulas, Géometrie et Théorie des Groupes, Lecture Notes in Math. No. 1441, Springer-Verlag, Berlin (1990).
21. P. Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167 (1972) 151-170.
22. _, B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973) 45-109.
23. E. Ghys and P. de la Harpe (eds.), Sur les Groupes Hyperboliques d'après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser, Boston (1990).
24. R. Greene and H. Wu, Function Theory on Manifolds which Possess a Pole (1979), Lecture Notes in Mathematics, No. 699, Springer-Verlag.
25. _, Hyperbolic groups in Essays in Group Theory, ed. S. Gersten, MSRI Pub., Springer-Verlag 8 (1987) 75-263.
26. M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory, eds. G. Niblo and M. Roller, London Math. Soc. Lecture Notes Series, Cambridge Univ. Press 181 (1993).
27. R. Gulliver, On the variety of manifolds without conjugate points, Trans. Amer. Math. Soc. 210 (1975) 185-201.
28. W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton (1948).
29. M. Kanai, Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds, J. Math. Soc. Japan 37 (1985) 391-413.
30. _, Analytic inequality and rough isometries between non-compact Riemannian manifolds, Lecture Notes in Math., No. 1201, Springer-Verlag (1987), pp. 122-137.
31. Y. Kifer, Spectrum, harmonic functions and hyperbolic metric spaces, Israel J. Math. 89 (1995) 377-428.
32. H. P. McKean, An upper bound to the spectrum of Δ on α manifold of negative curvature, J. Differential Geometry 4 (1970) 359-366.
33. R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press Co., Ltd., Hong Kong, Boston (1994) 65-74.
34. H. Short et al., Notes on word hyperbolic groups, in Group Theory from a Geometric Viewpoint, eds. E. Ghys, A. Haefliger and A. Verjovsky, World Scientific, Singapore, New Jersey (1991), pp. 3-63.
35. D. Sullivan, The Dirichlet problem, at infinity for a negatively curved manifold, J. Diff. Geom. 18 (1983) 723-732.
36. S. T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. 8(4) (1975) 487-507.