Rapport asymptotique de courbure, courbure positive et non effondrement
Séminaire de théorie spectrale et géométrie, Tome 30 (2011-2012), pp. 47-75.

On s’intéresse ici à un invariant géométrique associé à toute variété riemannienne non compacte : le rapport asymptotique de courbure. On étudie son influence sur la topologie de la variété sous-jacente en présence d’autres contraintes géométrico-topologiques portant sur le volume asymptotique, la positivité de la courbure (de Ricci) et/ou la finitude du groupe fondamental (à l’infini).

We focus on a geometric invariant associated to any noncompact Riemannian manifold : the asymptotic curvature ratio introduced by Gromov. We study how it interacts with the topology of the underlying manifold with other geometric constraints such as positive asymptotic volume ratio, nonnegative (Ricci) curvature and finiteness of the fundamental group (at infinity).

DOI : 10.5802/tsg.290
Classification : 53, 58
Mot clés : géométrie riemannienne, courbure positive, cône asymptotique, effondrement à l’infini, topologie des variétés riemanniennes non compactes.
Keywords: Riemannian geometry, nonnegative curvature, asymptotic cone, collapsing at infinity, topology of noncompact Riemannian manifolds.
Deruelle, Alix 1

1 Institut Fourier, Université de Grenoble I 38402 Saint-Martin d’Hères, France
@article{TSG_2011-2012__30__47_0,
     author = {Deruelle, Alix},
     title = {Rapport asymptotique de courbure, courbure positive et non effondrement},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {47--75},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     year = {2011-2012},
     doi = {10.5802/tsg.290},
     language = {fr},
     url = {http://www.numdam.org/articles/10.5802/tsg.290/}
}
TY  - JOUR
AU  - Deruelle, Alix
TI  - Rapport asymptotique de courbure, courbure positive et non effondrement
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2011-2012
SP  - 47
EP  - 75
VL  - 30
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/tsg.290/
DO  - 10.5802/tsg.290
LA  - fr
ID  - TSG_2011-2012__30__47_0
ER  - 
%0 Journal Article
%A Deruelle, Alix
%T Rapport asymptotique de courbure, courbure positive et non effondrement
%J Séminaire de théorie spectrale et géométrie
%D 2011-2012
%P 47-75
%V 30
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/tsg.290/
%R 10.5802/tsg.290
%G fr
%F TSG_2011-2012__30__47_0
Deruelle, Alix. Rapport asymptotique de courbure, courbure positive et non effondrement. Séminaire de théorie spectrale et géométrie, Tome 30 (2011-2012), pp. 47-75. doi : 10.5802/tsg.290. http://www.numdam.org/articles/10.5802/tsg.290/

[1] Anderson, Michael T. Short geodesics and gravitational instantons, J. Differential Geom., Volume 31 (1990) no. 1, pp. 265-275 http://projecteuclid.org/getRecord?id=euclid.jdg/1214444097 | MR | Zbl

[2] Bando, Shigetoshi; Kasue, Atsushi; Nakajima, Hiraku On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math., Volume 97 (1989) no. 2, pp. 313-349 | DOI | MR | Zbl

[3] Belegradek, I.; Wei, G. Metrics of positive Ricci curvature on vector bundles over nilmanifolds, Geom. Funct. Anal., Volume 12 (2002) no. 1, pp. 56-72 | DOI | MR | Zbl

[4] Bérard-Bergery, Lionel Sur de nouvelles variétés riemanniennes d’Einstein, Institut Élie Cartan, 6 (Inst. Élie Cartan), Volume 6, Univ. Nancy, Nancy, 1982, pp. 1-60 | Zbl

[5] Brendle, Simon; Schoen, Richard Sphere theorems in geometry, Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry : forty years of the Journal of Differential Geometry (Surv. Differ. Geom.), Volume 13, Int. Press, Somerville, MA, 2009, pp. 49-84 | MR | Zbl

[6] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001 | MR

[7] Burago, Yu.; Gromov, M.; Perel’man, G. A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk, Volume 47 (1992) no. 2(284), p. 3-51, 222 | DOI | MR | Zbl

[8] Cao, Huai-Dong Limits of solutions to the Kähler-Ricci flow, J. Differential Geom., Volume 45 (1997) no. 2, pp. 257-272 http://projecteuclid.org/getRecord?id=euclid.jdg/1214459797 | MR | Zbl

[9] Carron, G. Some old and new results about rigidity of critical metric, ArXiv e-prints (2010)

[10] Cheeger, Jeff; Colding, Tobias H. Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), Volume 144 (1996) no. 1, pp. 189-237 | DOI | MR | Zbl

[11] Cheeger, Jeff; Colding, Tobias H. On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., Volume 46 (1997) no. 3, pp. 406-480 http://projecteuclid.org/getRecord?id=euclid.jdg/1214459974 | MR | Zbl

[12] Cheeger, Jeff; Fukaya, Kenji; Gromov, Mikhael Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc., Volume 5 (1992) no. 2, pp. 327-372 | DOI | MR | Zbl

[13] Cheeger, Jeff; Gromoll, Detlef The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry, Volume 6 (1971/72), pp. 119-128 | MR | Zbl

[14] Cheeger, Jeff; Gromoll, Detlef On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2), Volume 96 (1972), pp. 413-443 | MR | Zbl

[15] Cheeger, Jeff; Tian, Gang Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc., Volume 19 (2006) no. 2, p. 487-525 (electronic) | DOI | MR | Zbl

[16] Chen, C.-W.; Deruelle, A. Structure at infinity of expanding gradient Ricci soliton, ArXiv e-prints (2011)

[17] Chow, Bennett; Lu, Peng; Ni, Lei Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI, 2006 | MR | Zbl

[18] Deruelle, Alix Géométrie à l’infini de certaines variétés riemanniennes non compactes, Université de Grenoble I (2012) (Masters thesis)

[19] Drees, Günter Asymptotically flat manifolds of nonnegative curvature, Differential Geom. Appl., Volume 4 (1994) no. 1, pp. 77-90 | DOI | MR | Zbl

[20] Eschenburg, J.-H. Comparison theorems and hypersurfaces, Manuscripta Math., Volume 59 (1987) no. 3, pp. 295-323 | DOI | MR | Zbl

[21] Eschenburg, J.-H.; Schroeder, V.; Strake, M. Curvature at infinity of open nonnegatively curved manifolds, J. Differential Geom., Volume 30 (1989) no. 1, pp. 155-166 http://projecteuclid.org/getRecord?id=euclid.jdg/1214443288 | MR | Zbl

[22] Fukaya, Kenji A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom., Volume 28 (1988) no. 1, pp. 1-21 http://projecteuclid.org/getRecord?id=euclid.jdg/1214442157 | MR | Zbl

[23] Greene, R. E.; Shiohama, K. Convex functions on complete noncompact manifolds : topological structure, Invent. Math., Volume 63 (1981) no. 1, pp. 129-157 | DOI | MR | Zbl

[24] Greene, R. E.; Wu, H. Lipschitz convergence of Riemannian manifolds, Pacific J. Math., Volume 131 (1988) no. 1, pp. 119-141 http://projecteuclid.org/getRecord?id=euclid.pjm/1102690072 | MR | Zbl

[25] Gromov, Misha Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2007 (Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates) | MR | Zbl

[26] Guijarro, Luis; Kapovitch, Vitali Restrictions on the geometry at infinity of nonnegatively curved manifolds, Duke Math. J., Volume 78 (1995) no. 2, pp. 257-276 | DOI | MR | Zbl

[27] Guijarro, Luis; Petersen, Peter Rigidity in non-negative curvature, Ann. Sci. École Norm. Sup. (4), Volume 30 (1997) no. 5, pp. 595-603 | DOI | MR | Zbl

[28] Holck Colding, T.; Naber, A. Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications, ArXiv e-prints (2011) | MR | Zbl

[29] Kasue, Atsushi A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. (4), Volume 21 (1988) no. 4, pp. 593-622 | Numdam | MR | Zbl

[30] Lott, John Manifolds with quadratic curvature decay and fast volume growth, Math. Ann., Volume 325 (2003) no. 3, pp. 525-541 | DOI | MR | Zbl

[31] Lott, John; Shen, Zhongmin Manifolds with quadratic curvature decay and slow volume growth, Ann. Sci. École Norm. Sup. (4), Volume 33 (2000) no. 2, pp. 275-290 | DOI | Numdam | MR | Zbl

[32] Minerbe, Vincent On the asymptotic geometry of gravitational instantons, Ann. Sci. Éc. Norm. Supér. (4), Volume 43 (2010) no. 6, pp. 883-924 | Numdam | MR | Zbl

[33] Perelman, G. A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone, Comparison geometry (Berkeley, CA, 1993–94) (Math. Sci. Res. Inst. Publ.), Volume 30, Cambridge Univ. Press, Cambridge, 1997, pp. 165-166 | MR | Zbl

[34] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, New York, 2006 | MR | Zbl

[35] Petrunin, Anton; Tuschmann, Wilderich Asymptotical flatness and cone structure at infinity, Math. Ann., Volume 321 (2001) no. 4, pp. 775-788 | DOI | MR | Zbl

[36] Rong, Xiaochun On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. (2), Volume 143 (1996) no. 2, pp. 397-411 | DOI | MR | Zbl

[37] Šarafutdinov, V. A. The Pogorelov-Klingenberg theorem for manifolds that are homeomorphic to R n , Sibirsk. Mat. Ž., Volume 18 (1977) no. 4, p. 915-925, 958 | MR | Zbl

[38] Sha, Jiping; Shen, Zhongmin Complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity, Amer. J. Math., Volume 119 (1997) no. 6, pp. 1399-1404 http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.6sha.pdf | MR | Zbl

[39] Shen, Zhongmin; Sormani, Christina The topology of open manifolds with nonnegative Ricci curvature, Commun. Math. Anal. (2008) no. Conference 1, pp. 20-34 | MR | Zbl

[40] Shi, Wan-Xiong Complete noncompact three-manifolds with nonnegative Ricci curvature, J. Differential Geom., Volume 29 (1989) no. 2, pp. 353-360 http://projecteuclid.org/getRecord?id=euclid.jdg/1214442879 | MR | Zbl

[41] Sormani, Christina The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth, Comm. Anal. Geom., Volume 8 (2000) no. 1, pp. 159-212 | MR | Zbl

[42] Unnebrink, Stefan Asymptotically flat 4-manifolds, Differential Geom. Appl., Volume 6 (1996) no. 3, pp. 271-274 | DOI | MR | Zbl

Cité par Sources :