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

Petersen, Peter; Wei, Guofang

doi:10.1090/S0002-9947-00-02621-0

Analysis and geometry on manifolds with integral Ricci curvature bounds. II
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by Peter Petersen and Guofang Wei PDF

Trans. Amer. Math. Soc. 353 (2001), 457-478 Request permission

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

Uwe Abresch and Detlef Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), no. 2, 355–374. MR 1030656, DOI 10.1090/S0894-0347-1990-1030656-6
M. T. Anderson, Scalar curvature, metric degeneration and the static vacuum Einstein equations on 3-manifolds, I. preprint from SUNY Stony Brook.
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
Jeff Cheeger and Tobias H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189–237. MR 1405949, DOI 10.2307/2118589
Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. MR 1484888
Tobias H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996), no. 1-3, 193–214. MR 1369415, DOI 10.1007/s002220050050
Tobias H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996), no. 1-3, 175–191. MR 1369414, DOI 10.1007/s002220050049
Tobias H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), no. 3, 477–501. MR 1454700, DOI 10.2307/2951841
Michel Le Couturier and Gilles F. Robert, $L^p$-pinching and the geometry of compact Riemannian manifolds, Comment. Math. Helv. 69 (1994), no. 2, 249–271. MR 1282370, DOI 10.1007/BF02564485
Sylvestre Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque 157-158 (1988), 191–216. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976219
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
G. Perelman, Manifolds of positive Ricci curvature with almost maximal volume, J. Amer. Math. Soc. 7 (1994), no. 2, 299–305. MR 1231690, DOI 10.1090/S0894-0347-1994-1231690-7
P. Petersen, On eigenvalue pinching in positive Ricci curvature, Invt. Math. (1999) DOI 10.1007/s002229900931.
P. Petersen and C. Sprouse, Integral curvature bounds, distance estimates and applications, J. Diff. Geo. 50 (1998) 269-298.
P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1031–1045. MR 1487753, DOI 10.1007/s000390050036
Deane Yang, Convergence of Riemannian manifolds with integral bounds on curvature. I, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 1, 77–105. MR 1152614, DOI 10.24033/asens.1644
Deane Yang, Convergence of Riemannian manifolds with integral bounds on curvature. II, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 2, 179–199. MR 1169351, DOI 10.24033/asens.1647
Deane Yang, Riemannian manifolds with small integral norm of curvature, Duke Math. J. 65 (1992), no. 3, 501–510. MR 1154180, DOI 10.1215/S0012-7094-92-06519-7