On the structure of spaces with Ricci curvature bounded below. I
Jeff Cheeger and Tobias H. Colding
Source: J. Differential Geom. Volume 46, Number 3
(1997), 406-480.
First Page:
Show
Hide
Related Works:
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jdg/1214459974
Mathematical Reviews number (MathSciNet): MR1484888
Zentralblatt MATH identifier: 0902.53034
References
[1] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci survature, J. Amer. Math. Soc. 3 (1990) 355-374.
Zentralblatt MATH: 0704.53032
Mathematical Reviews (MathSciNet): MR1030656
Digital Object Identifier: doi:10.2307/1990957
JSTOR: links.jstor.org
[2] A. D. Alexandrov, A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov 38 (1951) 5-23.
[3] M.T.Anderson, Metrics of positive Ricci curvaturewith large diameter, Manuscripta Math. 68 (1990) 405-415.
Zentralblatt MATH: 0711.53036
Mathematical Reviews (MathSciNet): MR1068264
Digital Object Identifier: doi:10.1007/BF02568774
[4] M.T.Anderson, Convergence and rigidity of metrics under Ricci curvature bounds, Invent. Math. 102 (1990) 429-445.
Zentralblatt MATH: 0711.53038
Mathematical Reviews (MathSciNet): MR1074481
Digital Object Identifier: doi:10.1007/BF01233434
[5] M.T.Anderson, Short geodesics and gravitational instantons, J. Differential Geom. 31 (1990) 265-275.
Zentralblatt MATH: 0696.53029
Mathematical Reviews (MathSciNet): MR1030673
Project Euclid: euclid.jdg/1214444097
[6] M.T.Anderson, Degenerations of metrics with bounded curvature and applicationsto critical metrics of Riemannian functionals, Proc. Sympos. Pure Math. Amer. Math. Soc. 54 (1993) 53-79.
Zentralblatt MATH: 0791.58027
Mathematical Reviews (MathSciNet): MR1216611
[7] M. T. Anderson and J. Cheeger, C -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 (1992) 265-281.
Zentralblatt MATH: 0774.53021
Mathematical Reviews (MathSciNet): MR1158336
Project Euclid: euclid.jdg/1214448075
[8] R. Bishop, A relation between volume, mean curvature and diameter, Notices Amer. Math. Soc. 10 (1963) 364.
[9] Y. Burago, M. Gromov and G. Perelman, A. D. Alexandrov spaces with curvature bounded below, Uspekhi Mat. Nauk. 47: 2 (1992) 3-51.
Zentralblatt MATH: 0802.53018
Mathematical Reviews (MathSciNet): MR1185284
[10] E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958) 45-56.
Zentralblatt MATH: 0079.11801
Mathematical Reviews (MathSciNet): MR92069
Digital Object Identifier: doi:10.1215/S0012-7094-58-02505-5
Project Euclid: euclid.dmj/1077467776
[11] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970) 61-74.
Zentralblatt MATH: 0194.52902
Mathematical Reviews (MathSciNet): MR263092
Digital Object Identifier: doi:10.2307/2373498
JSTOR: links.jstor.org
[12] J.Cheeger and T. H.Colding, On the structure of spaces with Ricci curvature bounded below. II, to appear.
Zentralblatt MATH: 1027.53042
Mathematical Reviews (MathSciNet): MR1815410
Project Euclid: euclid.jdg/1214342145
[13] J.Cheeger and T. H.Colding, On the structure of spaces with Ricci curvature bounded below. III, to appear.
Zentralblatt MATH: 1027.53043
Mathematical Reviews (MathSciNet): MR1815410
Project Euclid: euclid.jdg/1214342145
[14] 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, Serie I, 320 (1995) 353-357.
Zentralblatt MATH: 0840.53024
Mathematical Reviews (MathSciNet): MR1320384
[15] J.Cheeger and T. H.Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189-237.
Zentralblatt MATH: 0865.53037
Mathematical Reviews (MathSciNet): MR1405949
Digital Object Identifier: doi:10.2307/2118589
JSTOR: links.jstor.org
[16] J. Cheeger, T. H. Colding and G. Tian, Constraints on singularities under Ricci curvature bounds, C.R. Acad. Sci. Paris, Serie I, 324 (1997) 645-649.
Zentralblatt MATH: 0876.53018
Mathematical Reviews (MathSciNet): MR1447035
Digital Object Identifier: doi:10.1016/S0764-4442(97)86982-0
[17] J. Cheeger, T. H. Colding and G. Tian, On the singularities of spaces with bounded Ricci curvature, to appear.
Zentralblatt MATH: 1030.53046
Mathematical Reviews (MathSciNet): MR1937830
Digital Object Identifier: doi:10.1007/PL00012649
[18] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North Holland, Amsterdam, 1975.
Zentralblatt MATH: 0309.53035
Mathematical Reviews (MathSciNet): MR458335
[19] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative sectional curvature, Ann. of Math. 96 (1972) 413-443.
Zentralblatt MATH: 0246.53049
Mathematical Reviews (MathSciNet): MR309010
Digital Object Identifier: doi:10.2307/1970819
JSTOR: links.jstor.org
[20] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971) 119-128.
Zentralblatt MATH: 0223.53033
Mathematical Reviews (MathSciNet): MR303460
Project Euclid: euclid.jdg/1214430220
[21] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 31 (1990) 269-298.
Zentralblatt MATH: 0727.53043
Mathematical Reviews (MathSciNet): MR1064875
Project Euclid: euclid.jdg/1214445047
[22] J. Cheeger and G. Tian, On the cone structure at in nity of Ricci at manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math. 118 (1994) 493-571.
Zentralblatt MATH: 0814.53034
Mathematical Reviews (MathSciNet): MR1296356
Digital Object Identifier: doi:10.1007/BF01231543
[23] S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975) 333-354.
Zentralblatt MATH: 0312.53031
Mathematical Reviews (MathSciNet): MR385749
Digital Object Identifier: doi:10.1002/cpa.3160280303
[24] T. H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996) Fasc 1-3, 175-191.
Zentralblatt MATH: 0871.53027
Mathematical Reviews (MathSciNet): MR1369414
Digital Object Identifier: doi:10.1007/s002220050049
[25] T. H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996) Fasc 1-3, 193-214.
Zentralblatt MATH: 0871.53028
Mathematical Reviews (MathSciNet): MR1369415
Digital Object Identifier: doi:10.1007/s002220050050
[26] T. H. Colding, Ricci curvature and volume convergence, Ann. of Math., to appear.
Zentralblatt MATH: 0879.53030
Mathematical Reviews (MathSciNet): MR1454700
Digital Object Identifier: doi:10.2307/2951841
JSTOR: links.jstor.org
[27] T. H. Colding, Stability and Ricci curvature, C.R. Acad. Sci. Paris, Serie I, 320 (1995) 1343-1347.
Zentralblatt MATH: 0830.53035
Mathematical Reviews (MathSciNet): MR1338284
[28] G. David and S. Semmes, Analysis of and on uniformly recti able sets, Math. Surveys Monographs, Amer. Math. Soc. 38 (1993).
Zentralblatt MATH: 0832.42008
Mathematical Reviews (MathSciNet): MR1251061
[29] H. Federer, Geometric measure theory, Springer, New York, 1969.
Zentralblatt MATH: 0176.00801
Mathematical Reviews (MathSciNet): MR257325
[30] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math. 87 (1987) 517-547.
Zentralblatt MATH: 0589.58034
Mathematical Reviews (MathSciNet): MR874035
Digital Object Identifier: doi:10.1007/BF01389241
[31] K. Fukaya and T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Ann. of Math. 136 (1992) 253-333.
Zentralblatt MATH: 0770.53028
Mathematical Reviews (MathSciNet): MR1185120
Digital Object Identifier: doi:10.2307/2946606
JSTOR: links.jstor.org
[32] K. Fukaya and T. Yamaguchi, Isometry group of singular spaces, Math. Z. 216 (1994) 31-44.
Zentralblatt MATH: 0797.53033
Mathematical Reviews (MathSciNet): MR1273464
Digital Object Identifier: doi:10.1007/BF02572307
[33] E. Giusti, Minimal surfaces and functions of bounded variation, Birkhauser, Basel, 1984.
Zentralblatt MATH: 0545.49018
Mathematical Reviews (MathSciNet): MR775682
[34] M. Gromov, Synthetic geometry of Riemannian manifolds, Proc. ICM-1978, Vol. 1, 415-419.
Zentralblatt MATH: 0427.53018
Mathematical Reviews (MathSciNet): MR562635
[35] M. Gromov, Dimension, non-linear spectra and width, Lecture Notes, Springer, New York, Vol. 1317, 1988, 132-184.
Zentralblatt MATH: 0664.41019
Mathematical Reviews (MathSciNet): MR950979
Digital Object Identifier: doi:10.1007/BFb0081739
[36] M. Gromov, Sign and geometric meaning of curvature, Rondi. Sem. Mat. Fis. Milano 61 (1991) 9-123.
Zentralblatt MATH: 0820.53035
Mathematical Reviews (MathSciNet): MR1297501
Digital Object Identifier: doi:10.1007/BF02925201
[37] M. Gromov, J. Lafontaine and P. Pansu, Structures metriques pour les varieties Riemanniennes, Cedic-Fernand Nathan, Paris, 1981.
Zentralblatt MATH: 0509.53034
Mathematical Reviews (MathSciNet): MR682063
[38] K. Grove and P. Petersen, Bounding homotopy types by geometry, Ann. of Math. 128 (1988) 195-206.
Zentralblatt MATH: 0655.53032
Mathematical Reviews (MathSciNet): MR951512
Digital Object Identifier: doi:10.2307/1971439
JSTOR: links.jstor.org
[39] K. Grove and P. Petersen, Excess and rigidity of inner metric spaces, Preprint.
[40] K. Grove, P. Petersen and J.-Y. Wu, Geometric niteness theorems via controlled topology, Invent. Math. 99 (1990) 205-213.
Zentralblatt MATH: 0747.53033
Mathematical Reviews (MathSciNet): MR1029396
Digital Object Identifier: doi:10.1007/BF01234417
[41] K. Grove, P. Petersen and J.-Y. Wu, Geometric niteness theorems via controlled topology, Invent. Math. 104 (1991) 221-222.
Zentralblatt MATH: 0778.53034
Mathematical Reviews (MathSciNet): MR1094053
Digital Object Identifier: doi:10.1007/BF01245073
[42] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. 106 (1977) 201-211.
Zentralblatt MATH: 0341.53029
Mathematical Reviews (MathSciNet): MR500705
Digital Object Identifier: doi:10.2307/1971164
[43] C. B. Morrey, Multiple integrals in the calculus of variations, Grundlehren Math. Wiss., Band 130, Springer, New York, 1966.
Zentralblatt MATH: 0142.38701
Mathematical Reviews (MathSciNet): MR202511
[44] M. E. Munroe, Introduction to measure and integration, Addison Wesley, Reading, Massachusetts, 1959.
Zentralblatt MATH: 0050.05603
Mathematical Reviews (MathSciNet): MR53186
[45] G. Perelman, (private communication).
[46] M. E. Munroe, Manifolds of positive Ricci curvature with almost maximal volume, J. Amer. Math. Soc. 7 (1994) 299-305.
Zentralblatt MATH: 0799.53050
Mathematical Reviews (MathSciNet): MR1231690
Digital Object Identifier: doi:10.2307/2152760
JSTOR: links.jstor.org
[47] M. E. Munroe, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers, Preprint.
[48] M. E. Munroe, Alexandrov's spaces with curvatures bounded below, II, Preprint.
[49] P. Petersen, F. Wilhelm and S. Zhu, Spaces on and beyond the boundary of existence, J. Geom. Anal. 5 (1995) 419-426.
Zentralblatt MATH: 0839.53032
Mathematical Reviews (MathSciNet): MR1360829
[50] E. R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1962) 1-92.
Zentralblatt MATH: 0099.08503
Mathematical Reviews (MathSciNet): MR114145
Digital Object Identifier: doi:10.1007/BF02547186
[51] L. Schwartz, Theorie des distributions, Hermann, Paris, 1966.
Zentralblatt MATH: 0149.09501
Mathematical Reviews (MathSciNet): MR209834
[52] S. Semmes, Chord-arc surfaces with small constant II: Good parameterizations, Adv. Math. 88 (1991) 170-199.
Zentralblatt MATH: 0733.42016
Mathematical Reviews (MathSciNet): MR1120612
Digital Object Identifier: doi:10.1016/0001-8708(91)90007-T
[53] S. Semmes, Finding structure in sets with little smoothness, Proc. ICM Zurich, Vol. 2 Birkhauser, Basel, 1994, 875-885.
Zentralblatt MATH: 0897.28003
Mathematical Reviews (MathSciNet): MR1403987
[54] Z. Shen, Volume comparison and its applications in Riemannian-Finsler geometry, Adv. Math., to appear.
Zentralblatt MATH: 0919.53021
Mathematical Reviews (MathSciNet): MR1454401
Digital Object Identifier: doi:10.1006/aima.1997.1630
[55] Z. Shen, to appear.
Mathematical Reviews (MathSciNet): MR2244601
Zentralblatt MATH: 05049011
Digital Object Identifier: doi:10.1512/iumj.2006.55.2558
[56] L. Simon, Lectures on geometric measure theory, Center for Math. Anal., Austral. Nat. Univ., Vol.3 (1983).
Zentralblatt MATH: 0546.49019
Mathematical Reviews (MathSciNet): MR756417
[57] S.L. Sobolev, Sur un theorem d'analyse fonctionalle, Math. USSR-Sb. (N.S) 46 (1938) 471-496.
[58] T. Toro, Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J. 77 (1995) 193-227.
Zentralblatt MATH: 0847.42011
Mathematical Reviews (MathSciNet): MR1317632
Digital Object Identifier: doi:10.1215/S0012-7094-95-07708-4
Project Euclid: euclid.dmj/1077286151
[59] M. Wang and W. Ziller, Existence and non-existence of homogeneous Einstein metrics, Invent. Math. 84 (1986) 177-194.
Zentralblatt MATH: 0596.53040
Mathematical Reviews (MathSciNet): MR830044
Digital Object Identifier: doi:10.1007/BF01388738
[60] M. Wang and W. Ziller, Einstein metrics on principle torus bundles, J. Differential Geom. 31 (1990) 215-248.
Zentralblatt MATH: 0691.53036
Mathematical Reviews (MathSciNet): MR1030671
Project Euclid: euclid.jdg/1214444095
[61] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991) 317-357.
Zentralblatt MATH: 0737.53041
Mathematical Reviews (MathSciNet): MR1097241
Digital Object Identifier: doi:10.2307/2944340
JSTOR: links.jstor.org
[62] S. T. Yau, Some function theoretic properties of complete riemannian manifolds and their applications in geometry, Indiana Univ. Math. J. 2 (1976) 659-670.
Zentralblatt MATH: 0335.53041
Mathematical Reviews (MathSciNet): MR417452
Digital Object Identifier: doi:10.1512/iumj.1976.25.25051
Journal of Differential Geometry
