Differential geometry
Uniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decay
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1007-1009.

We show that any complete noncompact shrinking gradient Ricci soliton with (1) |Rc|0 at infinity or (2) R0 at infinity, |Rm| bounded, and κ-noncollapsed has a unique asymptotic cone.

Nous montrons que tout soliton gradient de Ricci contractant complet non compact vérifiant la propriété (1) |Rc|0 à l'infini ou (2) R0 à l'infini, avec |Rm| bornée et κ-non-effrondée, possède un cône asymptotique unique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.009
Chow, Bennett 1; Lu, Peng 2

1 Department of Mathematics, University of California San Diego, La Jolla, CA 92093, United States
2 Department of Mathematics, University of Oregon, Eugene, OR 97403, United States
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Chow, Bennett; Lu, Peng. Uniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decay. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1007-1009. doi : 10.1016/j.crma.2015.09.009. http://www.numdam.org/articles/10.1016/j.crma.2015.09.009/

[1] Cao, H.-D.; Zhou, D.-T. On complete gradient shrinking Ricci solitons, J. Differ. Geom., Volume 85 (2010), pp. 175-186

[2] Cao, H.-D.; Chen, B.-L.; Zhu, X.-P. Recent developments on Hamilton's Ricci flow, Geometric Flows, Surveys in Differential Geometry, vol. XII, Int. Press, Somerville, MA, USA, 2008, pp. 47-112

[3] Cheeger, J.; Colding, T.H. Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), Volume 144 (1996), pp. 189-237

[4] Cheeger, J.; Tian, G. On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math., Volume 118 (1994), pp. 493-571

[5] Chen, B.-L. Strong uniqueness of the Ricci flow, J. Differ. Geom., Volume 82 (2009), pp. 363-382

[6] Colding, T.H.; Minicozzi, W.P. II On uniqueness of tangent cones for Einstein manifolds, Invent. Math., Volume 196 (2014), pp. 515-588

[7] Gromov, M. Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, USA, 1999 (with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates)

[8] Hamilton, R.S. The Ricci flow on surfaces, Santa Cruz, CA, 1986 (Contemp. Math.), Volume vol. 71 (1988), pp. 237-262

[9] Hamilton, R.S. The formation of singularities in the Ricci flow, Cambridge, MA, 1993, International Press, Cambridge, MA, USA (1995), pp. 7-136

[10] Kotschwar, B. Backwards uniqueness for the Ricci flow, Int. Math. Res. Not., Volume 21 (2010), pp. 4064-4097

[11] Kotschwar, B.; Wang, L. Rigidity of asymptotically conical shrinking gradient Ricci solitons, J. Differ. Geom., Volume 100 (2015), pp. 55-108

[12] Munteanu, O.; Wang, J. Geometry of shrinking Ricci solitons | arXiv

[13] Munteanu, O.; Wang, J. Conical structure for shrinking Ricci solitons | arXiv

[14] Munteanu, O.; Wang, J. Positively curved shrinking Ricci solitons are compact | arXiv

[15] Naber, A. Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math., Volume 645 (2010), pp. 125-153

[16] Ni, L.; Wallach, N. On a classification of gradient shrinking solitons, Math. Res. Lett., Volume 15 (2008), pp. 941-955

[17] Perelman, G. A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone, Berkeley, CA, 1993–1994 (Math. Sci. Res. Inst. Publ.), Volume vol. 30, Cambridge University Press, Cambridge, UK (1997), pp. 165-166

[18] Perelman, G. The entropy formula for the Ricci flow and its geometric applications | arXiv

[19] Petersen, P.; Wylie, W. On the classification of gradient Ricci solitons, Geom. Topol., Volume 14 (2010), pp. 2277-2300

[20] Wang, L. Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc., Volume 27 (2014), pp. 613-638

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