On complete manifolds of nonnegative 𝑘th-Ricci curvature

Shen, Zhong Min

doi:10.1090/S0002-9947-1993-1112548-6

On complete manifolds of nonnegative $k$th-Ricci curvature
HTML articles powered by AMS MathViewer

by Zhong Min Shen PDF

Trans. Amer. Math. Soc. 338 (1993), 289-310 Request permission

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.

References

U. Abresch, Lower curvature bounds, Toponogov’s theorem, and bounded topology. II, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 475–502. MR 925724, DOI 10.24033/asens.1541
Michael T. Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology 29 (1990), no. 1, 41–55. MR 1046624, DOI 10.1016/0040-9383(90)90024-E
Michael T. Anderson, Peter B. Kronheimer, and Claude LeBrun, Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys. 125 (1989), no. 4, 637–642. MR 1024931, DOI 10.1007/BF01228345
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
K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505
Jeff Cheeger, Critical points of distance functions and applications to geometry, Geometric topology: recent developments (Montecatini Terme, 1990) Lecture Notes in Math., vol. 1504, Springer, Berlin, 1991, pp. 1–38. MR 1168042, DOI 10.1007/BFb0094288
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 309010, DOI 10.2307/1970819
Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128. MR 303460
Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208
M. Gromov, Manifolds of negative curvature, J. Differential Geometry 13 (1978), no. 2, 223–230. MR 540941, DOI 10.4310/jdg/1214434487
Detlef Gromoll and Wolfgang Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75–90. MR 247590, DOI 10.2307/1970682

On the excess of metric spaces and manifolds

Karsten Grove and Katsuhiro Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), no. 2, 201–211. MR 500705, DOI 10.2307/1971164
R. E. Greene and H. Wu, $C^{\infty }$ approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 47–84. MR 532376, DOI 10.24033/asens.1361
R. E. Greene and H. Wu, Integrals of subharmonic functions on manifolds of nonnegative curvature, Invent. Math. 27 (1974), 265–298. MR 382723, DOI 10.1007/BF01425500
Peter Li and Luen-Fai Tam, Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set, Ann. of Math. (2) 125 (1987), no. 1, 171–207. MR 873381, DOI 10.2307/1971292
John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942, DOI 10.1515/9781400878055

Morse theory

Zhong Min Shen, Finiteness and vanishing theorems for complete open Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 2, 241–244. MR 998628, DOI 10.1090/S0273-0979-1989-15817-5

Finite topological type and vanishing theorems for riemannian manifolds

Richard Schoen and Shing Tung Yau, Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 209–228. MR 645740
Ji-Ping Sha and DaGang Yang, Examples of manifolds of positive Ricci curvature, J. Differential Geom. 29 (1989), no. 1, 95–103. MR 978078
Ji-Ping Sha and DaGang Yang, Positive Ricci curvature on the connected sums of $S^n\times S^m$, J. Differential Geom. 33 (1991), no. 1, 127–137. MR 1085137

convexity of manifolds with boundary

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, DOI 10.1016/0020-7683(93)90194-c
H. Wu, An elementary method in the study of nonnegative curvature, Acta Math. 142 (1979), no. 1-2, 57–78. MR 512212, DOI 10.1007/BF02395057
H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525–548. MR 905609, DOI 10.1512/iumj.1987.36.36029
Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051