Advertisement

Convergence

  • Peter Petersen
Chapter
  • 7.6k Downloads
Part of the Graduate Texts in Mathematics book series (GTM, volume 171)

Abstract

In this chapter we offer an introduction to several of the convergence ideas for Riemannian manifolds.

Keywords

Gromov Hausdorff Distance Cheeger Harmonic Coordinates Curvature Pinching Injectivity Radius 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. 4.
    M.T. Anderson, J. Cheeger, C α-compactness for manifolds with Ricci curvature and injectivity radius bounded below. J. Differ. Geom. 35(2), 265–281 (1992)MathSciNetzbMATHGoogle Scholar
  2. 16.
    C. Böhm, B. Wilking, Manifolds with positive curvature operators are space forms. Ann. Math. (2) 167(3), 1079–1097 (2008)Google Scholar
  3. 20.
    S. Brendle, R. Schoen, Manifolds with 1/4-pinched curvature are space forms. J. Am. Math. Soc. 22(1), 287–307 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 22.
    E. Calabi, P. Hartman, On the smoothness of isometries. Duke Math. J. 37, 741–750 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 25.
    J. Cheeger, Comparison and Finiteness Theorems for Riemannian Manifolds, Ph. D. thesis, Princeton University, 1967Google Scholar
  6. 26.
    J. Cheeger, Finiteness theorems for Riemannian manifolds. Am. J. Math. 92, 61–75 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 27.
    J. Cheeger, Pinching theorems for a certain class of Riemannian manifolds. Am. J. Math. 92, 807–834 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 29.
    J. Cheeger, T.H. Colding, On the structure of space with Ricci curvature bounded below. J. Differ. Geom. 46, 406–480 (1997)MathSciNetzbMATHGoogle Scholar
  9. 34.
    T.H. Colding, Ricci curvature and volume convergence. Ann. Math. 145, 477–501 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 35.
    R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II (Wiley Interscience, New York, 1962)zbMATHGoogle Scholar
  11. 36.
    X. Dai, P. Petersen, G. Wei, Integral pinching theorems. Manuscripta Math. 101(2), 143–152 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 37.
    D. DeTurk, J. Kazdan, Some regularity theorems in Riemannian geometry. Ann. sci. Éc. Norm. Sup. 14, 249–260 (1981)MathSciNetzbMATHGoogle Scholar
  13. 39.
    R. Edwards, R. Kirby, Deformations of spaces of embeddings. Ann. Math. 93, 63–88 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 43.
    T. Farrell, L. Jones, Negatively curved manifolds with exotic smooth structures. J. AMS 2, 899–908 (1989)MathSciNetzbMATHGoogle Scholar
  15. 44.
    K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, Advanced Studies in Pure Math. 18-I (1990). Recent topics in Differential and Analytic Geometry, pp. 143–238Google Scholar
  16. 50.
    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second-Order, 2nd edn. (Springer, Berlin-Heidelberg, 1983)CrossRefzbMATHGoogle Scholar
  17. 51.
    R. Greene, S.T. Yau (eds.), Proceedings of Symposia in Pure Mathematics, vol. 54, 3 (1994)Google Scholar
  18. 52.
    M. Gromov, Manifolds of negative curvature. J. Differ. Geom. 12, 223–230 (1978)MathSciNetzbMATHGoogle Scholar
  19. 53.
    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces (Birkhäuser, Boston 1999)zbMATHGoogle Scholar
  20. 54.
    K. Grove, P. Petersen (eds.), Comparison Geometry, vol. 30 (MSRI publications, New York; Cambridge University Press, Cambridge, 1997)Google Scholar
  21. 55.
    M. Gromov, W. Thurston, Pinching constants for hyperbolic manifolds. Invest. Math. 89, 1–12 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 58.
    K. Grove, H. Karcher, E. Ruh, Group actions and curvature. Invest. Math. 23, 31–48 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 66.
    J. Jost, H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen. Manuf. Math. 19, 27–77 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 86.
    S. Peters, Cheeger’s finiteness theorem for diffeomorphism classes of manifolds. J. Reine Angew. Math. 349, 77–82 (1984)MathSciNetzbMATHGoogle Scholar
  25. 88.
    P. Petersen, G. Wei, Relative volume comparison with integral Ricci curvature bounds. Geom. Funct. Anal. 7, 1031–1045 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 91.
    P. Petersen, S. Shteingold, G. Wei, Comparison geometry with integral curvature bounds. Geom. Funct. Anal. 7, 1011–1030 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 94.
    E. A. Ruh, Almost flat manifolds. J. Differ. Geom. 17(1), 1–14 (1982)MathSciNetzbMATHGoogle Scholar
  28. 99.
    M.E. Taylor, Partial Differential Equations, vol. I-III (Springer, New York, 1996)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Peter Petersen
    • 1
  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

Personalised recommendations