On the singularities of spaces with bounded Ricci curvature

Geometric and Functional Analysis

Volumn 12, Issue 5, 2002, Pages 873-914

  Cheeger, J ^a   Colding, T H ^a   Tian, G ^b  

^a NEW YORK UNIVERSITY   (United States)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036433827     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00012649     Document Type: Article

References (38)

1. [AbG1] U. Abresch, D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. AMS 3 (1990), 355-374.
2. [An1] M. Anderson, Ricci curvature bounds and Einstein metric on compact manifolds, J. Amer. Math. Soc. V. 2 (1989), 455-490.
3. [An2] M.T. Anderson, Convergence and rigidity of metrics under Ricci curvature bounds, Invent. Math. 102 (1990), 429-445.
4. 2 structure of moduli spaces of Einstein metrics on 4-manifolds, GAFA, Geom. funct, anal. 2 (1992), 29-89.
5. [An4] M.T. Anderson, Einstein metrics and metrics with bounds on Ricci curvature, Proc. International Congress of Mathematicians, Zürich (1994), vol. 1, Birkhäuser, Basel (1995), 443-452.
6. [BaKaNa] S. Bando, A. Kasue, H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), 313-349.
7. [Be] A.L. Besse, Einstein Manifolds, Springer Verlag, New York, 1987.
8. [Ch1] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, GAFA, Geom. funct, anal. 9 (1999), 428-517.
9. p-bounds on curvature, elliptic estimates and rectifiability of singular sets, C.R. Acad. Sci. Paris, série I, 334:1 (2002), 195-198.
10. [Ch3] J. Cheeger, Integral bounds on curvature, elliptic estimates and rectifiability of singular sets, preprint.
11. [Ch4] J. Cheeger, Degeneration of Riemannian Metrics under Ricci Curvature Bounds, Lezione Fermiane (2001), Accademia Nazionale del Lincei, Scuola Normale Superiore, Pisa, 2002.
12. [ChCo1] J. Cheeger, T.H. Colding, Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below-, C.R. Acad. Sci. Paris, t. 320, Serie I (1995), 353-357.
13. [ChCo2] J. Cheeger, T.H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Annals of Math. 144:1 (1996), 189-237.
14. [ChCo3] J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below; I, Jour. of Diff. Geom. 46 (1997), 406-480.
15. [ChCo4] J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below; II, Jour. of Diff. Geom. 52 (1999), 13-35.
16. [ChCo5] J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below; III, Jour. of Diff. Geom. 52 (1999), 37-74.
17. [ChCoTi1] J. Cheeger, T.H. Colding, G. TIAN, Constraints on singularities under Ricci curvature bounds, C.R. Acad. Sci. Paris, Serie I, t. 324 (1997), 645-649.
18. [ChEb] J. Cheeger, D. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland, New York, 1975.
19. [ChSm] J. Cheeger, J. Simons, Differential characters and geometric invariants, Geometry and Topology (College Park, Md., 1983/84), Springer Lecture Notes in Math. 1167 (1985), 50-80.
20. [ChTi1] J. Cheeger, G. TIAN, On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math. 118 (1994), 493-571.
21. [ChTi2] J. Cheeger, G. Tian, Anti-self-duality of curvature and riemannian metrics with special holonomy, in preparation.
22. [CgY] S.Y. Cheng, S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure and Appl. Math. 28 (1975), 333-354.
23. [Co1] T.H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 Fasc. 1-3 (1996), 175-191.
24. [Co2] T.H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 Fasc. 1-3 (1996), 193-214.
25. [Co3] T.H. Colding, Ricci curvature and volume convergence, Annals of Math. 145:3 (1997), 477-501.
26. [Co4] T.H. Colding, Ricci curvature and stability, C.R. Acad. Sci. Paris, t. 320, Serie I (1995), 1343-1347.
27. [Fe] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.
28. [Fu] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math. 87 (1987), 517-547.
29. [GiTr] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.
30. [GvLP] M. Gromov, J. Lafontaine, P. Pansu, Structures métriques pour les varieties riemanniennces, Cedid/Fernand Nathan, Paris, 1981.
31. [Jo] D. Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
32. [Mi] J. Millson, Examples of nonvanishing Chen-Simons invariants, J. Diff. Geom. 10:153 (1975), 589-600.
33. [Sim] L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag Basel, 1996.
34. [Ti1] G. Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101:1 (1990), 101-172.
35. [Ti2] G. Tian, On one of Calabi's problems, in several complex variables and complex geometry, Proc. of Symp. in Pure Math. 52:2 (Santa Cruz, CA, 1989), Amer. Math. Soc., Providence, RI, (1991), 543-556.
36. [Ti3] G. Tian, Kähler-Einstein metrics on algebraic manifolds, Proc. of ICM 1,2 (Kyoto, 1990), Math. Soc. Japan, Tokyo (1990), 587-598.
37. [Ti4] G. Tian, Compactness theorems for Kähler-Einstein manifolds of dimension 3 and up, Jour. of Diff. Geom. 35 (1992), 535-558.
38. [Wo] J. Wolf, Spaces of constant curvature, University of California, 1972.