The Evolution of the Weyl Tensor under the Ricci Flow
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, p. 1407-1435

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.

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.

DOI : https://doi.org/10.5802/aif.2644
Classification:  53C21,  53C25
Keywords: Ricci solitons, singularity of Ricci flow
@article{AIF_2011__61_4_1407_0,
     author = {Catino, Giovanni and Mantegazza, Carlo},
     title = {The Evolution of the Weyl Tensor under the Ricci Flow},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     pages = {1407-1435},
     doi = {10.5802/aif.2644},
     mrnumber = {2951497},
     zbl = {1255.53034},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_4_1407_0}
}
Catino, Giovanni; Mantegazza, Carlo. The Evolution of the Weyl Tensor under the Ricci Flow. Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1407-1435. doi : 10.5802/aif.2644. http://www.numdam.org/item/AIF_2011__61_4_1407_0/

[1] Baird, P.; Danielo, L. Three–dimensional Ricci solitons which project to surfaces, J. Reine Angew. Math., Tome 608 (2007), pp. 65-91 | Article | MR 2339469 | Zbl 1128.53020

[2] Besse, A. L. Einstein manifolds, Springer–Verlag, Berlin (2008) | MR 2371700 | Zbl 0613.53001

[3] Böhm, C.; Wilking, B. Manifolds with positive curvature operators are space forms, Ann. of Math. (2), Tome 167 (2008) no. 3, pp. 1079-1097 | Article | MR 2415394 | Zbl 1185.53073

[4] Brozos-Vázquez, M.; García-Río, E.; Vázquez-Lorenzo, R. Some remarks on locally conformally flat static space-times, J. Math. Phys., Tome 46 (2005) no. 2, pp. 022501, 11 | Article | MR 2121707 | Zbl 1076.53084

[5] Bryant, R. L. Local existence of gradient Ricci solitons (1987) (Unpublished work)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., Tome 82 (2009), pp. 363-382 | MR 2520796 | Zbl 1177.53036

[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, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 135 (2007) | MR 2302600 | Zbl 1216.53057

[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, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 144 (2008) | MR 2365237 | Zbl 1157.53035 | Zbl 1216.53057

[11] Chow, B.; Lu, P.; Ni, L. Hamilton’s Ricci flow, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 77 (2006) | Zbl 1118.53001

[12] Derdzinski, A. Some remarks on the local structure of Codazzi tensors, Global differential geometry and global analysis (Berlin, 1979), Springer–Verlag, Berlin (Lect. Notes in Math.) Tome 838 (1981), pp. 243-299 | Zbl 0437.53012

[13] Eminenti, M.; Nave, G. La; Mantegazza, C. Ricci solitons: the equation point of view, Manuscripta Math., Tome 127 (2008) no. 3, pp. 345-367 | Article | MR 2448435 | Zbl 1160.53031

[14] Fernández–López, M.; García–Río, E. Rigidity of shrinking Ricci solitons (2009) (preprint) | Zbl 1226.53047

[15] Gallot, S.; Hulin, D.; Lafontaine, J. Riemannian geometry, Springer–Verlag (1990) | MR 1083149 | Zbl 0716.53001

[16] Hamilton, R. S. Three–manifolds with positive Ricci curvature, J. Diff. Geom., Tome 17 (1982) no. 2, pp. 255-306 | MR 664497 | Zbl 0504.53034

[17] Hamilton, R. S. Four–manifolds with positive curvature operator, J. Diff. Geom., Tome 24 (1986) no. 2, pp. 153-179 | MR 862046 | Zbl 0628.53042

[18] Hamilton, R. S. Eternal solutions to the Ricci flow, J. Diff. Geom., Tome 38 (1993) no. 1, pp. 1-11 | MR 1231700 | Zbl 0792.53041

[19] Hamilton, R. S. A compactness property for solutions of the Ricci flow, Amer. J. Math., Tome 117 (1995) no. 3, pp. 545-572 | Article | MR 1333936 | Zbl 0840.53029

[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 1375255 | Zbl 0867.53030

[21] Hiepko, S. Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann., Tome 241 (1979) no. 3, pp. 209-215 | Article | MR 535555 | Zbl 0387.53014

[22] Hiepko, S.; Reckziegel, H. Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter, Manuscripta Math., Tome 31 (1980) no. 1–3, pp. 269-283 | Article | MR 576500 | Zbl 0441.53035

[23] Kotschwar, B. On rotationally invariant shrinking Ricci solitons, Pacific J. Math., Tome 236 (2008) no. 1, pp. 73-88 | Article | MR 2398988 | Zbl 1152.53056

[24] Ma, L.; Cheng, L. On the conditions to control curvature tensors or Ricci flow, Ann. Global Anal. Geom., Tome 37 (2010) no. 4, pp. 403-411 | Article | MR 2601499 | Zbl 1188.35033

[25] Munteanu, O.; Sesum, N. On gradient Ricci solitons (2009) (ArXiv Preprint Server – http://arxiv.org) | Zbl 1275.53061

[26] Naber, A. Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math, Tome 645 (2010), pp. 125-153 | Article | MR 2673425 | Zbl 1196.53041

[27] Ni, L.; Wallach, N. On a classification of gradient shrinking solitons, Math. Res. Lett., Tome 15 (2008) no. 5, pp. 941-955 | MR 2443993 | Zbl 1158.53052

[28] Perelman, G. The entropy formula for the Ricci flow and its geometric applications (2002) (ArXiv Preprint Server – http://arxiv.org) | Zbl 1130.53001

[29] Petersen, P.; Wylie, W. On the classification of gradient Ricci solitons (2007) (ArXiv Preprint Server – http://arxiv.org) | Zbl 1202.53049

[30] Petersen, P.; Wylie, W. Rigidity of gradient Ricci solitons, Pacific J. Math., Tome 241 (2009) no. 2, pp. 329-345 | Article | MR 2507581 | Zbl 1176.53048

[31] Sesum, N. Convergence of the Ricci flow toward a soliton, Comm. Anal. Geom., Tome 14 (2006) no. 2, pp. 283-343 | MR 2255013 | Zbl 1106.53045

[32] Tojeiro, R. Conformal de Rham decomposition of Riemannian manifolds, Houston J. Math., Tome 32 (2006) no. 3, p. 725-743 (electronic) | MR 2247906 | Zbl 1116.53044

[33] Zhang, Z.-H. Gradient shrinking solitons with vanishing Weyl tensor, Pacific J. Math., Tome 242 (2009) no. 1, pp. 189-200 | Article | MR 2525510 | Zbl 1171.53332

[34] Zhang, Z.-H. On the completeness of gradient Ricci solitons, Proc. Amer. Math. Soc., Tome 137 (2009) no. 8, pp. 2755-2759 | Article | MR 2497489 | Zbl 1176.53046