Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T06:12:53.076Z Has data issue: false hasContentIssue false
  • English
  • Français

On manifolds with quadratic curvature decay

Part of: Global differential geometry

Published online by Cambridge University Press:  01 March 2009

Nader Yeganefar
Nader Yeganefar*
Affiliation:
CMI, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadratic curvature decay and some properties of the asymptotic cones of such a manifold.

Keywords

asymptotic geometryasymptotic cones

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 528 - 540
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Abresch, U., Lower curvature bounds, Toponogov’s theorem, and bounded topology, Ann. Sci. École Norm. Sup. 18 (1985), 651–670.CrossRefGoogle Scholar
[2]Abresch, U. and Gromoll, D., On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), 355–374.CrossRefGoogle Scholar
[3]Cheeger, J., Critical points of distance functions and applications to geometry, in Geometric topology: recent developments (Montecatini Terme, 1990), Lecture Notes in Mathematics, vol. 1504  (Springer, Berlin, 1991), 1–38.CrossRefGoogle Scholar
[4]Cheeger, J. and Colding, T., On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997), 406–480.Google Scholar
[5]Cheeger, J. and Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443.CrossRefGoogle Scholar
[6]Cheeger, J., Gromov, M. and Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15–53.Google Scholar
[7]Croke, C. B. and Karcher, H., Volumes of small balls on open manifolds: lower bounds and examples, Trans. Amer. Math. Soc. 309 (1988), 753–762.CrossRefGoogle Scholar
[8]Carmo, M. do and Xia, C., Ricci curvature and the topology of open manifolds, Math. Ann. 316 (2000), 391–400.CrossRefGoogle Scholar
[9]Greene, R. E., Petersen, P. and Zhu, S., Riemannian manifolds of faster-than-quadratic curvature decay, Internat. Math. Res. Notices 9 (1994), 363–377.Google Scholar
[10]Greene, R. E. and Wu, H., Gap theorems for noncompact Riemannian manifolds, Duke Math. J. 49 (1982), 731–756.CrossRefGoogle Scholar
[11]Gromov, M., Metric structures for Riemannian and non-Riemannian spaces, in Progress in Mathematics, vol. 152 (Birkhäuser, Boston, 1999).Google Scholar
[12]Grove, K., Critical point theory for distance functions, in Differential geometry: Riemannian geometry, (Los Angeles, CA, 1990, Proceedings of Symposia in Pure Mathematics, vol. 54, Part 3 (American Mathematical Society, Providence, RI, 1993), 357–385.CrossRefGoogle Scholar
[13]Kapovich, M., Hyperbolic manifolds and discrete groups, in Progress in Mathematics, vol. 183 (Birkhäuser, Boston, 2001).Google Scholar
[14]Kleiner, B. and Leeb, B., Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 115–197.CrossRefGoogle Scholar
[15]Klingenberg, W., Contributions to Riemannian geometry in the large, Ann. of Math. (2) 69 (1959), 654–666.CrossRefGoogle Scholar
[16]Lott, J., Manifolds with quadratic curvature decay and fast volume growth, Math. Ann. 325 (2003), 525–541.CrossRefGoogle Scholar
[17]Lott, J. and Shen, Z., Manifolds with quadratic curvature decay and slow volume growth, Ann. Sci. École Norm. Sup. (4) 33 (2000), 275–290.CrossRefGoogle Scholar
[18]Menguy, X., Noncollapsing examples with positive Ricci curvature and infinite topological type, Geom. Funct. Anal. 10 (2000), 600–627.CrossRefGoogle Scholar
[19]Menguy, X., Examples of nonpolar limit spaces, Amer. J. Math. 122 (2000), 927–937.Google Scholar
[20]Petersen, P., Convergence theorems in Riemannian geometry, in Comparison geometry, Berkeley, CA, 1993–94, Mathematical Sciences Research Institute Publications, vol. 30 (Cambridge University Press, Cambridge, 1997), 167–202.Google Scholar
[21]Petrunin, A. and Tuschmann, W., Asymptotical flatness and cone structure at infinity, Math. Ann. 321 (2001), 775–788.Google Scholar
[22]Sha, J. and Shen, Z., Complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity, Amer. J. Math. 119 (1997), 1399–1404.CrossRefGoogle Scholar
[23]Shen, Z., Complete manifolds with nonnegative Ricci curvature and large volume growth, Invent. Math. 125 (1996), 393–404.CrossRefGoogle Scholar
[24]Sormani, C., The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth, Comm. Anal. Geom. 8 (2000), 159–212.CrossRefGoogle Scholar
[25]Sormani, C., Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups, J. Differential Geom. 54 (2000), 547–559.Google Scholar
[26]Tian, G., On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), 101–172.CrossRefGoogle Scholar
[27]Xia, C., Open manifolds with nonnegative Ricci curvature and large volume growth, Comment. Math. Helv. 74 (1999), 456–466.Google Scholar
[28]Yau, S.-T., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.CrossRefGoogle Scholar