On the structure of spaces with Ricci curvature bounded below. I

Journal of Differential Geometry - Tập 46 Số 3 - 1997
Jeff Cheeger, Tobias Colding1
1New York University

Tóm tắt

Từ khóa


Tài liệu tham khảo

[1] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci survature, J. Amer. Math. Soc. 3 (1990) 355-374.

[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.

[4] M.T.Anderson, Convergence and rigidity of metrics under Ricci curvature bounds, Invent. Math. 102 (1990) 429-445.

[5] M.T.Anderson, Short geodesics and gravitational instantons, J. Differential Geom. 31 (1990) 265-275.

[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.

[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.

[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.

[10] E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958) 45-56.

[11] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970) 61-74.

[12] J.Cheeger and T. H.Colding, On the structure of spaces with Ricci curvature bounded below. II, to appear.

[13] J.Cheeger and T. H.Colding, On the structure of spaces with Ricci curvature bounded below. III, to appear.

[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.

[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.

[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.

[17] J. Cheeger, T. H. Colding and G. Tian, On the singularities of spaces with bounded Ricci curvature, to appear.

[18] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North Holland, Amsterdam, 1975.

[19] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative sectional curvature, Ann. of Math. 96 (1972) 413-443.

[20] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971) 119-128.

[21] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 31 (1990) 269-298.

[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.

[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.

[24] T. H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996) Fasc 1-3, 175-191.

[25] T. H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996) Fasc 1-3, 193-214.

[26] T. H. Colding, Ricci curvature and volume convergence, Ann. of Math., to appear.

[27] T. H. Colding, Stability and Ricci curvature, C.R. Acad. Sci. Paris, Serie I, 320 (1995) 1343-1347.

[28] G. David and S. Semmes, Analysis of and on uniformly recti able sets, Math. Surveys Monographs, Amer. Math. Soc. 38 (1993).

[29] H. Federer, Geometric measure theory, Springer, New York, 1969.

[30] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math. 87 (1987) 517-547.

[31] K. Fukaya and T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Ann. of Math. 136 (1992) 253-333.

[32] K. Fukaya and T. Yamaguchi, Isometry group of singular spaces, Math. Z. 216 (1994) 31-44.

[33] E. Giusti, Minimal surfaces and functions of bounded variation, Birkhauser, Basel, 1984.

[34] M. Gromov, Synthetic geometry of Riemannian manifolds, Proc. ICM-1978, Vol. 1, 415-419.

[35] M. Gromov, Dimension, non-linear spectra and width, Lecture Notes, Springer, New York, Vol. 1317, 1988, 132-184.

[36] M. Gromov, Sign and geometric meaning of curvature, Rondi. Sem. Mat. Fis. Milano 61 (1991) 9-123.

[37] M. Gromov, J. Lafontaine and P. Pansu, Structures metriques pour les varieties Riemanniennes, Cedic-Fernand Nathan, Paris, 1981.

[38] K. Grove and P. Petersen, Bounding homotopy types by geometry, Ann. of Math. 128 (1988) 195-206.

[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.

[41] K. Grove, P. Petersen and J.-Y. Wu, Geometric niteness theorems via controlled topology, Invent. Math. 104 (1991) 221-222.

[42] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. 106 (1977) 201-211.

[43] C. B. Morrey, Multiple integrals in the calculus of variations, Grundlehren Math. Wiss., Band 130, Springer, New York, 1966.

[44] M. E. Munroe, Introduction to measure and integration, Addison Wesley, Reading, Massachusetts, 1959.

[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.

[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.

[50] E. R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1962) 1-92.

[51] L. Schwartz, Theorie des distributions, Hermann, Paris, 1966.

[52] S. Semmes, Chord-arc surfaces with small constant II: Good parameterizations, Adv. Math. 88 (1991) 170-199.

[53] S. Semmes, Finding structure in sets with little smoothness, Proc. ICM Zurich, Vol. 2 Birkhauser, Basel, 1994, 875-885.

[54] Z. Shen, Volume comparison and its applications in Riemannian-Finsler geometry, Adv. Math., to appear.

[55] Z. Shen, to appear.

[56] L. Simon, Lectures on geometric measure theory, Center for Math. Anal., Austral. Nat. Univ., Vol.3 (1983).

[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.

[59] M. Wang and W. Ziller, Existence and non-existence of homogeneous Einstein metrics, Invent. Math. 84 (1986) 177-194.

[60] M. Wang and W. Ziller, Einstein metrics on principle torus bundles, J. Differential Geom. 31 (1990) 215-248.

[61] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991) 317-357.

[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.