The Evolution of the Weyl Tensor under the Ricci Flow
[L’évolution du tenseur de Weyl d’une variété par le flot de Ricci]
Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1407-1435.

Nous calculons l’équation d’évolution du tenseur de Weyl d’une variété riemannienne par le flot de Ricci et nous discutons des conséquences pour la classification des solitons de Ricci localement conformément plats.

We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons.

DOI : 10.5802/aif.2644
Classification : 53C21, 53C25
Keywords: Ricci solitons, singularity of Ricci flow
Mot clés : solitons de Ricci, singularités du flot de Ricci
Catino, Giovanni 1 ; Mantegazza, Carlo 2

1 SISSA International School for Advanced Studies Via Bonomea 265 Trieste 34136 (Italy)
2 Scuola Normale Superiore di Pisa P.za Cavalieri 7 Pisa 56126 (Italy)
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Catino, Giovanni; Mantegazza, Carlo. The Evolution of the Weyl Tensor under the Ricci Flow. Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1407-1435. doi : 10.5802/aif.2644. http://www.numdam.org/articles/10.5802/aif.2644/

[1] Baird, P.; Danielo, L. Three–dimensional Ricci solitons which project to surfaces, J. Reine Angew. Math., Volume 608 (2007), pp. 65-91 | DOI | MR | Zbl

[2] Besse, A. L. Einstein manifolds, Springer–Verlag, Berlin, 2008 | MR | Zbl

[3] Böhm, C.; Wilking, B. Manifolds with positive curvature operators are space forms, Ann. of Math. (2), Volume 167 (2008) no. 3, pp. 1079-1097 | DOI | MR | Zbl

[4] Brozos-Vázquez, M.; García-Río, E.; Vázquez-Lorenzo, R. Some remarks on locally conformally flat static space-times, J. Math. Phys., Volume 46 (2005) no. 2, pp. 022501, 11 | DOI | MR | Zbl

[5] Bryant, R. L. Local existence of gradient Ricci solitons (1987) (Unpublished work)

[6] Cao, H.-D.; Chen, Q. On locally conformally flat gradient steady Ricci solitons (2009) (ArXiv Preprint Server – http://arxiv.org)

[7] Cao, X.; Wang, B.; Zhang, Z. On locally conformally flat gradient shrinking Ricci solitons (2008) (ArXiv Preprint Server – http://arxiv.org)

[8] Chen, B.-L. Strong uniqueness of the Ricci flow, J. Diff. Geom., Volume 82 (2009), pp. 363-382 | MR | Zbl

[9] Chow, B.; Chu, S.-C.; Glickenstein, D.; Guenther, C.; Isenberg, J.; Ivey, T.; Knopf, D.; Lu, P.; Luo, F.; Ni, L. The Ricci flow: techniques and applications. Part I. Geometric aspects, Mathematical Surveys and Monographs, 135, American Mathematical Society, Providence, RI, 2007 | MR | Zbl

[10] Chow, B.; Chu, S.-C.; Glickenstein, D.; Guenther, C.; Isenberg, J.; Ivey, T.; Knopf, D.; Lu, P.; Luo, F.; Ni, L. The Ricci flow: techniques and applications. Part II. Analytic aspects, Mathematical Surveys and Monographs, 144, American Mathematical Society, Providence, RI, 2008 | MR | Zbl

[11] Chow, B.; Lu, P.; Ni, L. Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI, 2006 | Zbl

[12] Derdzinski, A. Some remarks on the local structure of Codazzi tensors, Global differential geometry and global analysis (Berlin, 1979) (Lect. Notes in Math.), Volume 838, Springer–Verlag, Berlin, 1981, pp. 243-299 | Zbl

[13] Eminenti, M.; Nave, G. La; Mantegazza, C. Ricci solitons: the equation point of view, Manuscripta Math., Volume 127 (2008) no. 3, pp. 345-367 | DOI | MR | Zbl

[14] Fernández–López, M.; García–Río, E. Rigidity of shrinking Ricci solitons (2009) (preprint) | Zbl

[15] Gallot, S.; Hulin, D.; Lafontaine, J. Riemannian geometry, Springer–Verlag, 1990 | MR | Zbl

[16] Hamilton, R. S. Three–manifolds with positive Ricci curvature, J. Diff. Geom., Volume 17 (1982) no. 2, pp. 255-306 | MR | Zbl

[17] Hamilton, R. S. Four–manifolds with positive curvature operator, J. Diff. Geom., Volume 24 (1986) no. 2, pp. 153-179 | MR | Zbl

[18] Hamilton, R. S. Eternal solutions to the Ricci flow, J. Diff. Geom., Volume 38 (1993) no. 1, pp. 1-11 | MR | Zbl

[19] Hamilton, R. S. A compactness property for solutions of the Ricci flow, Amer. J. Math., Volume 117 (1995) no. 3, pp. 545-572 | DOI | MR | Zbl

[20] Hamilton, R. S. The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995, pp. 7-136 | MR | Zbl

[21] Hiepko, S. Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann., Volume 241 (1979) no. 3, pp. 209-215 | DOI | EuDML | MR | Zbl

[22] Hiepko, S.; Reckziegel, H. Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter, Manuscripta Math., Volume 31 (1980) no. 1–3, pp. 269-283 | DOI | EuDML | MR | Zbl

[23] Kotschwar, B. On rotationally invariant shrinking Ricci solitons, Pacific J. Math., Volume 236 (2008) no. 1, pp. 73-88 | DOI | MR | Zbl

[24] Ma, L.; Cheng, L. On the conditions to control curvature tensors or Ricci flow, Ann. Global Anal. Geom., Volume 37 (2010) no. 4, pp. 403-411 | DOI | MR | Zbl

[25] Munteanu, O.; Sesum, N. On gradient Ricci solitons (2009) (ArXiv Preprint Server – http://arxiv.org) | Zbl

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

[27] Ni, L.; Wallach, N. On a classification of gradient shrinking solitons, Math. Res. Lett., Volume 15 (2008) no. 5, pp. 941-955 | MR | Zbl

[28] Perelman, G. The entropy formula for the Ricci flow and its geometric applications (2002) (ArXiv Preprint Server – http://arxiv.org) | Zbl

[29] Petersen, P.; Wylie, W. On the classification of gradient Ricci solitons (2007) (ArXiv Preprint Server – http://arxiv.org) | Zbl

[30] Petersen, P.; Wylie, W. Rigidity of gradient Ricci solitons, Pacific J. Math., Volume 241 (2009) no. 2, pp. 329-345 | DOI | MR | Zbl

[31] Sesum, N. Convergence of the Ricci flow toward a soliton, Comm. Anal. Geom., Volume 14 (2006) no. 2, pp. 283-343 | MR | Zbl

[32] Tojeiro, R. Conformal de Rham decomposition of Riemannian manifolds, Houston J. Math., Volume 32 (2006) no. 3, p. 725-743 (electronic) | MR | Zbl

[33] Zhang, Z.-H. Gradient shrinking solitons with vanishing Weyl tensor, Pacific J. Math., Volume 242 (2009) no. 1, pp. 189-200 | DOI | MR | Zbl

[34] Zhang, Z.-H. On the completeness of gradient Ricci solitons, Proc. Amer. Math. Soc., Volume 137 (2009) no. 8, pp. 2755-2759 | DOI | MR | Zbl

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