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A rigidity phenomenon on Riemannian manifolds with reverse volume pinching

  • Peihe WangEmail author
  • Yuliang Wen
Original Paper

Abstract

In the article stated here, we introduce the Hausdorff convergence to discuss the differentiable sphere theorem with some curvature conditions and volume pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds will be derived.

Keywords

Volume Comparison Theorem Hausdorff convergence Differentiable sphere theorem Harmonic coordinate Harmonic radius 

Mathematics Subject Classification (2000)

53C20 53C23 53C24 

References

  1. 1.
    Anderson M. (1990). Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102: 429–445 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Anderson M. and Cheeger J. (1992). C α—compactness for manifolds with Ricci curvature and injective radius bounded below. J. Differ. Geom. 35(2): 265–281 zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cai M. (1992). Rigidity of manifolds with large volume. Math. Z. 213: 17–31 Google Scholar
  4. 4.
    Cheeger, J., Ebin, D.G.: Comparison Theorems in Riemannian Geometry, North-Holland. Amesterdam (1975)Google Scholar
  5. 5.
    Coghlan L. and Itokawa Y. (1991). A sphere theorem for reverse volume pinching on even-dimension manifolds. Proc. Am. Math. Soc. 111(3): 815–819 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd ed. Grundlehoen der mathematischen Wissenschaften 224, Springer-Verlag (1983)Google Scholar
  7. 7.
    Gromov M. (1981). Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. Math. 53: 183–215 CrossRefGoogle Scholar
  8. 8.
    Gromov, M.: Metric Structures for Riemannian and non-Riemannian Spaces. Birkhäuser, Boston (1999)zbMATHGoogle Scholar
  9. 9.
    Grove K. and Petersen V.P. (1991). Manifolds near the boundary of existence. J. Differ. Geom. 33: 379–394 zbMATHMathSciNetGoogle Scholar
  10. 10.
    Grove K. and Shiohama K. (1977). A generalized sphere theorem. Ann. Math. 106: 201–211 CrossRefMathSciNetGoogle Scholar
  11. 11.
    Grove K. and Wilhelm F. (1995). Hard and soft packing radius theorems. Ann Math. 142(2): 213–237 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hartman P. (1979). Oscillation criteria for self-adjoint second-order differential systems and principle sectional curvature. J. Diff. Equations 34: 326–338 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hebey E.: Nonlinear analysisi on manifolds: Sobolev space and inequalities. New York University, New York (1998)Google Scholar
  14. 14.
    Kazdan, J.L.: An isoperimetric inequality and Wiedersehen manifolds. In: Yau, S.T. (ed.) Seminar on differential geometry (Ann. Math. Stud. no 102). Princeton University Press, Princeton, pp. 143–157 (1982)Google Scholar
  15. 15.
    Pawel K. (1992). On the spectral gap for compact manifolds. J. Differ Geom. 36(2): 315–330 zbMATHGoogle Scholar
  16. 16.
    Pawel, L., Andreis, T.: Applications of eigenvalue techniques to geometry. Comtemporary geometry, pp 21–52, Univ. Ser. Math., Plenum, New York (1991)Google Scholar
  17. 17.
    Peters S. (1987). Convergence of Riemannian manifolds. Compositio Math. 62: 3–16 zbMATHMathSciNetGoogle Scholar
  18. 18.
    Petersen, P.: Convergence theorems in Riemannian geometry, vol. 30, pp. 167–202. Comparison Geometry. MSRI Publications (1997)Google Scholar
  19. 19.
    Petersen P. (1998). Riemannian Geometry. Springer, New York zbMATHGoogle Scholar
  20. 20.
    Rong, X.: Introduction to the convergence and collapse theory in Riemannian Geometry. Lectures given by Xiaochun RongGoogle Scholar
  21. 21.
    Sakai T. (1983). On continuity of injectivity radius function. Math. J. Okayama Univ. 25: 91–97 zbMATHMathSciNetGoogle Scholar
  22. 22.
    Sakai T. (1992). Riemannian Geometry. Shokabo Publishing Co. Ltd., Tokyo Google Scholar
  23. 23.
    Shiohama K. (1983). A sphere theorem for manifolds of positive Ricci curvature. Tans. Am. Math. Soc. 27: 811–819 CrossRefMathSciNetGoogle Scholar
  24. 24.
    Tuschmann W. (2001). Smooth diameter and eigenvalue rigidity in positive Ricci curvature. Proc AMS 30: 303–306 Google Scholar
  25. 25.
    Wang P. (2007). A gap phenomenon on Riemannian manifolds with reverse volume pinching. Acta Mathematica Hungarica 115(1–2): 133–144 zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Wang, P.: A differential sphere theorem on manifolds with reverse volume pinching. Acta Mathematica Hungarica, to appearGoogle Scholar
  27. 27.
    Wen Y. (2004). A note on pinching sphere theorem. C. R. Acad. Sci. Paris, Ser. I 338: 229–234 zbMATHGoogle Scholar
  28. 28.
    Wu J.Y. (1990). A diameter pinching sphere theorem. Proc. Am. Math. Soc. 197(3): 796–802 Google Scholar
  29. 29.
    Wu J.-Y. (1993). Hausdorff convergence and sphere theorem. Proc. Sym. Pure Math. 54: 685–692 Google Scholar
  30. 30.
    Xia C.Y. (1994). Rigidity and sphere theorem for manifolds with positive Ricci curvature. Manuscript Math. 85: 79–87 zbMATHCrossRefGoogle Scholar
  31. 31.
    Yang H.C. (1990). Estimates of the first eigenvalue for a compact Riemann manifold. Sci. China Ser. A 33(1): 39–51 zbMATHMathSciNetGoogle Scholar
  32. 32.
    Yang D.G. (1999). Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature. Pacific J. Math. 190(2): 383–398 zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhong J.Q. and Yang H.C. (1984). On the estimates of the first eigenvalue of a compact Riemannian manifold. Sci. Sinica, Ser. A 27(12): 1265–1273 zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesQufu Normal UniversityQufuChina
  2. 2.Department of MathematicsEast China Normal UniversityShanghaiChina

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