Journal of Differential Geometry

On the structure of spaces with Ricci curvature bounded below. III

Jeff Cheeger and Tobias H. Colding

Full-text: Open access

Article information

Source
J. Differential Geom., Volume 54, Number 1 (2000), 37-74.

Dates
First available in Project Euclid: 24 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1214342146

Digital Object Identifier
doi:10.4310/jdg/1214342146

Mathematical Reviews number (MathSciNet)
MR1815411

Zentralblatt MATH identifier
1027.53043

Subjects
Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citation

Cheeger, Jeff; Colding, Tobias H. On the structure of spaces with Ricci curvature bounded below. III. J. Differential Geom. 54 (2000), no. 1, 37--74. doi:10.4310/jdg/1214342146. https://projecteuclid.org/euclid.jdg/1214342146


Export citation

References

  • [1] M.T. Anderson and J. Cheeger, Ca -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 (1992) 265-281.
  • [2] P. Buser, A note on the isoperimetric constant, Ann. Sci. Ecole Norm. Sup., Paris 15 (1982) 213-230.
  • [3] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces G AFA, Geom. Funct. Anal. 9 (1999) 428-517.
  • [4] J. Cheeger and T. H. Colding, Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below, C.R. Acad. Sci. Paris t. 320, Ser. 1 (1995) 353-357.
  • [5] J. Cheeger and T. H. Colding, Lower bounds on the Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189-237.
  • [6] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997) 406-480.
  • [7] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. II, Preprint.
  • [8] S. Y. Cheng and S.-T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975) 333-354.
  • [9] R. Coifman and G. Weiss, Extensions of Hardy Spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569-645.
  • [10] T. H. Colding, Stability and Ricci curvature, C.R. Acad. Sci. Paris, t. 320, Ser. 1 (1995) 1343-1347.
  • [11] T. H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996) 175-191.
  • [12] T. H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996) 193-214.
  • [13] T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997) 477-501.
  • [14] T. H. Colding and W. Minnicozzi II, Harmonic functions on manifolds, Ann. of Math. 146 (1997) 725-747.
  • [15] G. David and S. Semmes, Analysis on and of uniformly rectifiable sets, Math. Surv. and Monographs 38 Amer. Math. Soc. (Providence) 1993.
  • [16] L. C. Evans and R. Gariepy, Measure theory and fine properties of functions, CRC Press (Boca Raton) 1992.
  • [17] E. Fabes, C. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. in PDE, 7 (1) (1982) 77-116.
  • [18] H. Fédérer, Geometric measure theory, Springer, Berlin-New York, 1969.
  • [19] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math. 87 (1987) 517-547.
  • [20] M. Fukushima, Dirichlet forms and Markoff processes, North Holland (Amsterdam) 1980.
  • [21] M. Gromov, J. Lafontaine and P. Pansu, Structures métriques pour les varieties riemanniennes, Cedic/Fernand Nathan, Paris, 1981.
  • [22] M. Gromov, Dimension, non-linear spectra and width, Springer Lecture Notes 1317 (1988) 132-184.
  • [23] P. Hajlasz, Sobolev spaces on an arbitrary metric space, J. Potential Anal. 5 (1995) 1211-1215.
  • [24] P. Hajlasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris 320 (1995) 1211-1215.
  • [25] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory for degenerate elliptic equations, Clarendon Press (Oxford, Tokyo, New York) 1993.
  • [26] J. Heinonen and P. Koskela, Weighted Sobolev and Poincaré inequalities and quasiregular mappings of polynomial type, Math. Scand. 77 (1995) 251-271.
  • [27] J. Heinonen and P. Koskela, Quasiconformai maps in metric spaces with controlled geometry, Acta Math. 181 (1998) 1-61.
  • [28] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1985) 309-338.
  • [29] J. Kinnunen and O. Martio, The Sobolev capacity on metric spaces, Ann. Sci. Acad. Fenn. Math. 21 (1996) 367-382.
  • [30] T. Kilpeläinen, Smooth approximation in weighted Sobolev spaces, Comment. Math. Univ. Carolinae 38 (1995) 1-8.
  • [31] X. Menguy, Noncollapsing examples with positive Ricci curvature and infinite topological type, Preprint.
  • [32] C. B. Morrey Jr., Multiple integrals in the calculus of variations, Springer, New York, 1966.
  • [33] G. Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers, Comparision geometry, MSRI (1994) 157-163.
  • [34] S. Rickman, Quasiregular mappings, Springer, Berlin, 1993.
  • [35] S.-T. Yau, Open problems in geometry, Chern - a great Geometer of the Twentieth Century, (S.-T. Yau ed.), Internat. Press (Hong Kong) 1992.

See also

  • Part I: Jeff Cheeger, Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom., Volume 46, Number 3, (1997), 406--480.
  • Part II: Jeff Cheeger, Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom., Volume 54, Number 1, (2000), 13--35.