Asymptotic curvature ratio, nonnegative curvature and non-collapsing
- [1] Institut Fourier, Université de Grenoble I 38402 Saint-Martin d’Hères, France
Séminaire de théorie spectrale et géométrie (2011-2012)
- Volume: 30, page 47-75
- ISSN: 1624-5458
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topDeruelle, Alix. "Rapport asymptotique de courbure, courbure positive et non effondrement." Séminaire de théorie spectrale et géométrie 30 (2011-2012): 47-75. <http://eudml.org/doc/275728>.
@article{Deruelle2011-2012,
abstract = {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).},
affiliation = {Institut Fourier, Université de Grenoble I 38402 Saint-Martin d’Hères, France},
author = {Deruelle, Alix},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {Riemannian geometry; nonnegative curvature; asymptotic cone; collapsing at infinity; topology of noncompact Riemannian manifolds},
language = {fre},
pages = {47-75},
publisher = {Institut Fourier},
title = {Rapport asymptotique de courbure, courbure positive et non effondrement},
url = {http://eudml.org/doc/275728},
volume = {30},
year = {2011-2012},
}
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
PB - Institut Fourier
VL - 30
SP - 47
EP - 75
AB - 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).
LA - fre
KW - Riemannian geometry; nonnegative curvature; asymptotic cone; collapsing at infinity; topology of noncompact Riemannian manifolds
UR - http://eudml.org/doc/275728
ER -
References
top- Michael T. Anderson, Short geodesics and gravitational instantons, J. Differential Geom. 31 (1990), 265-275 Zbl0696.53029MR1030673
- Shigetoshi Bando, Atsushi Kasue, Hiraku Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), 313-349 Zbl0682.53045MR1001844
- I. Belegradek, G. Wei, Metrics of positive Ricci curvature on vector bundles over nilmanifolds, Geom. Funct. Anal. 12 (2002), 56-72 Zbl1029.53045MR1904556
- Lionel Bérard-Bergery, Sur de nouvelles variétés riemanniennes d’Einstein, Institut Élie Cartan, 6 6 (1982), 1-60, Univ. Nancy, Nancy Zbl0544.53038
- Simon Brendle, Richard Schoen, Sphere theorems in geometry, Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry : forty years of the Journal of Differential Geometry 13 (2009), 49-84, Int. Press, Somerville, MA Zbl1184.53037MR2537082
- Dmitri Burago, Yuri Burago, Sergei Ivanov, A course in metric geometry, 33 (2001), American Mathematical Society, Providence, RI Zbl1232.53037MR1835418
- Yu. Burago, M. Gromov, G. Perel’man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), 3-51, 222 Zbl0802.53018MR1185284
- Huai-Dong Cao, Limits of solutions to the Kähler-Ricci flow, J. Differential Geom. 45 (1997), 257-272 Zbl0889.58067MR1449972
- G. Carron, Some old and new results about rigidity of critical metric, ArXiv e-prints (2010) Zbl1315.53024
- Jeff Cheeger, Tobias H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), 189-237 Zbl0865.53037MR1405949
- Jeff Cheeger, Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406-480 Zbl0902.53034MR1484888
- Jeff Cheeger, Kenji Fukaya, Mikhael Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), 327-372 Zbl0758.53022MR1126118
- Jeff Cheeger, Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119-128 Zbl0223.53033MR303460
- Jeff Cheeger, Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413-443 Zbl0246.53049MR309010
- Jeff Cheeger, Gang Tian, Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc. 19 (2006), 487-525 (electronic) Zbl1092.53034MR2188134
- C.-W. Chen, A. Deruelle, Structure at infinity of expanding gradient Ricci soliton, ArXiv e-prints (2011) Zbl1335.53083
- Bennett Chow, Peng Lu, Lei Ni, Hamilton’s Ricci flow, 77 (2006), American Mathematical Society, Providence, RI Zbl1118.53001MR2274812
- Alix Deruelle, Géométrie à l’infini de certaines variétés riemanniennes non compactes, (2012)
- Günter Drees, Asymptotically flat manifolds of nonnegative curvature, Differential Geom. Appl. 4 (1994), 77-90 Zbl0796.53041MR1264910
- J.-H. Eschenburg, Comparison theorems and hypersurfaces, Manuscripta Math. 59 (1987), 295-323 Zbl0642.53044MR909847
- J.-H. Eschenburg, V. Schroeder, M. Strake, Curvature at infinity of open nonnegatively curved manifolds, J. Differential Geom. 30 (1989), 155-166 Zbl0678.53031MR1001273
- Kenji Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988), 1-21 Zbl0652.53031MR950552
- R. E. Greene, K. Shiohama, Convex functions on complete noncompact manifolds : topological structure, Invent. Math. 63 (1981), 129-157 Zbl0468.53033MR608531
- R. E. Greene, H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), 119-141 Zbl0646.53038MR917868
- Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, (2007), Birkhäuser Boston Inc., Boston, MA Zbl1113.53001MR2307192
- Luis Guijarro, Vitali Kapovitch, Restrictions on the geometry at infinity of nonnegatively curved manifolds, Duke Math. J. 78 (1995), 257-276 Zbl0838.53036MR1333500
- Luis Guijarro, Peter Petersen, Rigidity in non-negative curvature, Ann. Sci. École Norm. Sup. (4) 30 (1997), 595-603 Zbl1008.53042MR1474806
- T. Holck Colding, A. Naber, Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications, ArXiv e-prints (2011) Zbl1271.53042MR3037899
- Atsushi Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. (4) 21 (1988), 593-622 Zbl0662.53032MR982335
- John Lott, Manifolds with quadratic curvature decay and fast volume growth, Math. Ann. 325 (2003), 525-541 Zbl1034.53033MR1968605
- John Lott, Zhongmin Shen, Manifolds with quadratic curvature decay and slow volume growth, Ann. Sci. École Norm. Sup. (4) 33 (2000), 275-290 Zbl0996.53026MR1755117
- Vincent Minerbe, On the asymptotic geometry of gravitational instantons, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 883-924 Zbl1215.53043MR2778451
- G. Perelman, A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone, Comparison geometry (Berkeley, CA, 1993–94) 30 (1997), 165-166, Cambridge Univ. Press, Cambridge Zbl0887.53038MR1452873
- Peter Petersen, Riemannian geometry, 171 (2006), Springer, New York Zbl1220.53002MR2243772
- Anton Petrunin, Wilderich Tuschmann, Asymptotical flatness and cone structure at infinity, Math. Ann. 321 (2001), 775-788 Zbl1004.53026MR1872529
- Xiaochun Rong, On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. (2) 143 (1996), 397-411 Zbl0974.53029MR1381991
- V. A. Šarafutdinov, The Pogorelov-Klingenberg theorem for manifolds that are homeomorphic to , Sibirsk. Mat. Ž. 18 (1977), 915-925, 958 Zbl0374.53018MR487896
- Jiping Sha, Zhongmin Shen, Complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity, Amer. J. Math. 119 (1997), 1399-1404 Zbl0901.53023MR1481819
- Zhongmin Shen, Christina Sormani, The topology of open manifolds with nonnegative Ricci curvature, Commun. Math. Anal. (2008), 20-34 Zbl1167.53309MR2452400
- Wan-Xiong Shi, Complete noncompact three-manifolds with nonnegative Ricci curvature, J. Differential Geom. 29 (1989), 353-360 Zbl0668.53026MR982179
- Christina Sormani, The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth, Comm. Anal. Geom. 8 (2000), 159-212 Zbl0970.53024MR1730892
- Stefan Unnebrink, Asymptotically flat -manifolds, Differential Geom. Appl. 6 (1996), 271-274 Zbl0856.53031MR1408311
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