Comptes Rendus
no. 23-24
Differential Geometry
Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons
[Minorer des la courbures scalaires de solitons de Ricci gradient non compact]

Présenté par : Jean-Pierre Demailly

Bennett Chow1 ; Peng Lu2 ; Bo Yang1
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
Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1265-1267.

Nous montrons que les travaux récents de Ni et Wilking (in preparation) [11] donne le résultat dʼun non plate soliton contractant de type gradient non compact a tout au plus sa courbure scalaire avec décroissance quadratique. Les exemples de solitons de Kähler–Ricci contractant de type non compact par Feldman, Ilmanen, et Knopf (2003) [7] montre que ce résultat est optimales. Nous prouvons aussi un résultat similaire pour certains solitons de Ricci stable de type gradient non compact.

We show that recent work of Ni and Wilking (in preparation) [11] yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler–Ricci shrinkers by Feldman, Ilmanen, and Knopf (2003) [7] exhibit that this result is sharp. We also prove a similar result for certain noncompact steady gradient Ricci solitons.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.11.004
Bennett Chow 1 ; Peng Lu 2 ; Bo Yang 1

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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Bennett Chow; Peng Lu; Bo Yang. Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1265-1267. doi : 10.1016/j.crma.2011.11.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.004/

[1] Simon Brendle Uniqueness of gradient Ricci solitons, Mathematical Research Letters, Volume 18 (2011), pp. 531-538

[2] Huai-Dong Cao, Qiang Chen, On locally conformally flat gradient steady Ricci solitons, Transactions of the American Mathematical Society, in press.

[3] Huai-Dong Cao; De-Tang Zhou On complete gradient shrinking Ricci solitons, Journal of Differential Geometry, Volume 85 (2010), pp. 175-185

[4] Bing-Long Chen Strong uniqueness of the Ricci flow, Journal of Differential Geometry, Volume 82 (2009), pp. 363-382

[5] Manolo Eminenti; Gabriele La Nave; Carlo Mantegazza Ricci solitons: the equation point of view, Manuscripta Mathematica, Volume 127 (2008), pp. 345-367

[6] Fu-Quan Fang; Jian-Wen Man; Zhen-Lei Zhang Complete gradient shrinking Ricci solitons have finite topological type, Comptes Rendus Mathematique Academie des Sciences Paris, Volume 346 (2008), pp. 653-656

[7] Mikhail Feldman; Tom Ilmanen; Dan Knopf Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, Journal of Differential Geometry, Volume 65 (2003), pp. 169-209

[8] Hongxin Guo Area growth rate of the level surface of the potential function on the 3-dimensional steady Ricci soliton, Proceedings of the American Mathematical Society, Volume 137 (2009), pp. 2093-2097

[9] Richard S. Hamilton The formation of singularities in the Ricci flow, Surveys in Differential Geometry, vol. II, International Press, Cambridge, MA, 1995, pp. 7-136

[10] Robert Haslhofer; Reto Müller A compactness theorem for complete Ricci shrinkers, Geometric and Functional Analysis, Volume 21 (2011) no. 5, pp. 1091-1116

[11] Lei Ni, Burkhard Wilking, in preparation.

[12] Ovidiu Munteanu, Natasa Sesum, On gradient Ricci solitons, , Journal of Geometric Analysis, in press. | arXiv

[13] Stefano Pigola; Michele Rimoldi; Alberto G. Setti Remarks on non-compact gradient Ricci solitons, Mathematische Zeitschrift, Volume 268 (2011) no. 3–4, pp. 777-790 | DOI

[14] Peng Wu Remarks on gradient steady Ricci solitons | arXiv

[15] Shijin Zhang On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below, Acta Mathematica Sinica, Volume 27 (2011) no. 5, pp. 871-882

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