On complete manifolds of nonnegative thRicci curvature
Author:
Zhong Min Shen
Journal:
Trans. Amer. Math. Soc. 338 (1993), 289310
MSC:
Primary 53C21; Secondary 31C12, 57R70
MathSciNet review:
1112548
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Abstract: In this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature. Key tools are comparison techniques and Morse Theory of Busemann and distance functions.
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Math. Soc. (N.S.) 21 (1989), no. 2, 241–244. MR 998628
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, Finite topological type and vanishing theorems for riemannian manifolds, Ph.D. thesis, SUNY at Stony Brook, 1990.
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Tung Yau, Complete threedimensional manifolds with positive Ricci
curvature and scalar curvature, Seminar on Differential Geometry,
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1982, pp. 209–228. MR 645740
(83k:53060)
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JiPing
Sha and DaGang
Yang, Examples of manifolds of positive Ricci curvature, J.
Differential Geom. 29 (1989), no. 1, 95–103. MR 978078
(90c:53110)
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JiPing
Sha and DaGang
Yang, Positive Ricci curvature on the connected sums of
𝑆ⁿ×𝑆^{𝑚}, J. Differential Geom.
33 (1991), no. 1, 127–137. MR 1085137
(92f:53048)
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J. Sha, convexity of manifolds with boundary, Ph.D. Dissertation, SUNY at Stony Brook, 1986.
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Zhong
Min Shen and Guofang
Wei, Volume growth and finite topological type, Differential
geometry: Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure
Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993,
pp. 539–549. MR 1216645
(94e:53036)
 [W1]
H.
Wu, An elementary method in the study of nonnegative
curvature, Acta Math. 142 (1979), no. 12,
57–78. MR
512212 (80c:53054), http://dx.doi.org/10.1007/BF02395057
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H.
Wu, Manifolds of partially positive curvature, Indiana Univ.
Math. J. 36 (1987), no. 3, 525–548. MR 905609
(88k:53068), http://dx.doi.org/10.1512/iumj.1987.36.36029
 [Y]
Shing
Tung Yau, Some functiontheoretic properties of complete Riemannian
manifold and their applications to geometry, Indiana Univ. Math. J.
25 (1976), no. 7, 659–670. MR 0417452
(54 #5502)
 [A]
 U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology. II, Ann. Sci. École Norm. Sup. 20 (1987), 475502. MR 925724 (89d:53080)
 [An]
 M. Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology 29 (1990), 4155. MR 1046624 (91b:53041)
 [AKL]
 M. Anderson, P. Kronheimer and C. LeBrun, Complete Ricciflat Kähler manifolds of infinite topological type, preprint. MR 1024931 (91a:53073)
 [AG]
 U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), 355374. MR 1030656 (91a:53071)
 [BY]
 S. Bochner and K. Yano, Curvature and Betti numbers, Ann. of Math. Studies, no. 32, Princeton Univ. Press, Princeton, N.J., 1953. MR 0062505 (15:989f)
 [C]
 J. Cheeger, Critical points of distance functions and applications to geometry, preprint 1990. MR 1168042 (94a:53075)
 [CE]
 J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, American Elsevier, New York, 1975. MR 0458335 (56:16538)
 [CG1]
 J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1974), 413443. MR 0309010 (46:8121)
 [CG2]
 , The splitting theorems for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971), 119128. MR 0303460 (46:2597)
 [CGT]
 J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete riemannian manifolds, J. Differential Geom. 17 (1982), 1553. MR 658471 (84b:58109)
 [G1]
 M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), 179195. MR 630949 (82k:53062)
 [G2]
 , Manifolds of negative curvature, J. Differential Geom. 13 (1978), 223230. MR 540941 (80h:53040)
 [GM]
 D. Gromoll and W. T. Meyer, Complete open manifolds of positive curvature, Ann. of Math. (2) 96 (1969), 7590. MR 0247590 (40:854)
 [GP]
 K. Grove and P. Petersen V, On the excess of metric spaces and manifolds, preprint.
 [GS]
 K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), 201211. MR 0500705 (58:18268)
 [GW1]
 R. E. Greene and H. Wu, approximation of convex, subharmonic and plurisubharmonic functions, Ann. Sci. École. Norm. Sup. 12 (1979), 4784. MR 532376 (80m:53055)
 [GW2]
 , Integrals of subharmonic function on manifolds of nonnegative curvature, Invent. Math. 27 (1974), 265298. MR 0382723 (52:3605)
 [LT]
 P. Li and L. F. Tam, Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set, Ann. of Math. (2) 125 (1987), 171207. MR 873381 (88m:58039)
 [M1]
 J. Milnor, Lectures on the cobordism theorem, Princeton Univ. Press, Princeton, N.J., 1965. MR 0190942 (32:8352)
 [M2]
 , Morse theory, Princeton Univ. Press, Princeton, N.J., 1975.
 [S1]
 Z. Shen, Finiteness and vanishing theorems for complete open riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 21 (1989), 241244. MR 998628 (90c:53111)
 [S2]
 , Finite topological type and vanishing theorems for riemannian manifolds, Ph.D. thesis, SUNY at Stony Brook, 1990.
 [SHY]
 R. Schoen and S. T. Yau, Complete three dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry (S. T. Yau, ed.), Ann. of Math. Studies, no. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 209228. MR 645740 (83k:53060)
 [SY1]
 J. Sha and D. Yang, Examples of manifolds of positive Ricci curvature, J. Differential Geom. 29 (1989), 95103. MR 978078 (90c:53110)
 [SY2]
 , Positive Ricci curvature on the connected sums of , J. Differential Geom. 33 (1991), 127138. MR 1085137 (92f:53048)
 [SH]
 J. Sha, convexity of manifolds with boundary, Ph.D. Dissertation, SUNY at Stony Brook, 1986.
 [SW]
 Z. Shen and G. Wei, Volume growth and finite topological type, MSRI preprint, 1990. MR 1216645 (94e:53036)
 [W1]
 H. Wu, An elementary method in the study of nonnegative curvature, Acta Math. 142 (1979), 5778. MR 512212 (80c:53054)
 [W2]
 , Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), 525548. MR 905609 (88k:53068)
 [Y]
 S. T. Yau, Some functiontheoretic properties of complete riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659670. MR 0417452 (54:5502)
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DOI:
http://dx.doi.org/10.1090/S00029947199311125486
PII:
S 00029947(1993)11125486
Article copyright:
© Copyright 1993
American Mathematical Society
