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Analysis and geometry on manifolds with integral Ricci curvature bounds. II


Authors: Peter Petersen and Guofang Wei
Journal: Trans. Amer. Math. Soc. 353 (2001), 457-478
MSC (2000): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9947-00-02621-0
Published electronically: September 21, 2000
MathSciNet review: 1709777
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Abstract | References | Similar Articles | Additional Information

Abstract:

We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding's volume convergence results and extend the Cheeger-Colding splitting theorem.


References [Enhancements On Off] (What's this?)

  • 1. U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, JAMS 3 (1990) 355-374. MR 91a:53071
  • 2. M. T. Anderson, Scalar curvature, metric degeneration and the static vacuum Einstein equations on 3-manifolds, I. preprint from SUNY Stony Brook.
  • 3. S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333-354. MR 52:6608
  • 4. J. Cheeger and T.H. Colding, Lower bounds on Ricci curvature and almost rigidity of warped products, Ann. Math. 144 (1996), 189-237. MR 97h:53038
  • 5. J. Cheeger and T.H. Colding, On the structure of spaces with Ricci curvature bounded below I, J. Diff. Geom. 45 (1997), 406-480. MR 98k:53044
  • 6. T. H. Colding, Large manifolds with positive curvature, Invt. Math. 124 (1996), 193-214. MR 96k:53068
  • 7. T. H. Colding, Shape of manifolds with positive curvature, Invt. Math. 124 (1996), 175-191. MR 96k:53067
  • 8. T. H. Colding, Ricci curvature and volume convergence, Ann. Math. 145 (1997), 477-501. MR 98d:53050
  • 9. M. Le Couturier and G. F. Robert, $L^{p}$- pinching and the geometry of compact Riemannian manifolds, Comm. Math. Helv. 69 (1994) 249-271. MR 95c:58179
  • 10. S. Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque, 157-158, (1988), 191-216. MR 90a:58179
  • 11. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Berlin: Springer Verlag, 1983. MR 86c:35035
  • 12. G. Perel$'$man, Manifolds of positive Ricci curvature with almost maximal volume, JAMS 7 (1994) 299-305. MR 94f:53077
  • 13. P. Petersen, On eigenvalue pinching in positive Ricci curvature, Invt. Math. (1999) DOI 10.1007/s002229900931.
  • 14. P. Petersen and C. Sprouse, Integral curvature bounds, distance estimates and applications, J. Diff. Geo. 50 (1998) 269-298. CMP 99:11
  • 15. P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997) 1031-1045. MR 99c:53023
  • 16. D. Yang, Convergence of Riemannian manifolds with integral bounds on curvature I, Ann. Scient. École Norm. Sup. (4) 25 (1992) 77-105. MR 93a:53037
  • 17. D. Yang, Convergence of Riemannian manifolds with integral bounds on curvature II, Ann. Scient. École Norm. Sup. (4) 25 (1992) 179-199. MR 93m:53037
  • 18. D. Yang, Riemannian manifolds with small integral norm of curvature, Duke J. Math. 65 (1992), 501-510. MR 93e:53052

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Additional Information

Peter Petersen
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
Email: petersen@math.ucla.edu

Guofang Wei
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: wei@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02621-0
Keywords: Integral curvature bounds, maximum principle, gradient estimate, excess estimate, volume and Gromov-Hausdorff convergence.
Received by editor(s): November 30, 1998
Received by editor(s) in revised form: July 30, 1999
Published electronically: September 21, 2000
Additional Notes: Both authors were supported by the NSF
Article copyright: © Copyright 2000 American Mathematical Society

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