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On Bach-flat gradient shrinking Ricci solitons. (English) Zbl 1277.53036
A complete Riemannian manifold \((M^n,g_{ij})\) is called a gradient Ricci soliton if there exists a smooth function \(f\) on \(M^n\) such that the Ricci tensor \(R_{ij}\) of the metric \(g\) satisfies the equation \[ R_{ij}+\nabla_i\nabla_j f=\rho g_{ij} \] for some constant \(\rho\). For \(\rho=0\) the Ricci soliton is steady, for \(\rho>0\) it is shrinking, and for \(\rho<0\) expanding. In this paper, the authors investigate an interesting class of complete gradient shrinking Ricci solitons: those with vanishing Bach tensor. On any manifold \((M^n,g_{ij})\), \(n\geqslant 4\), the Bach tensor is defined by \[ B_{ij}=\frac{1}{n-3}\nabla^{k}\nabla^{l} W_{ikjl}+\frac{1}{n-2}R_{kl}W_{i_j}{}^{k l}, \] \(W_{ikjl}\) being the Weyl tensor. It is easy to see that if \((M^n,g_{ij})\) is either locally conformally flat or Einstein, then \((M^n,g_{ij})\) is Bach-flat: \(B_{ij}=0\). The authors show that Bach-flat 4-dimensional gradient shrinking Ricci solitons are either Einstein or locally conformally flat. More generally, for \(n\geqslant 5\), they prove that a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton \(\mathbb R^n\) or the product \(N^{n-1}\times\mathbb R\), where \(N^{n-1}\) is an Einstein manifold of positive scalar curvature.

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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[1] R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs , Math. Z. 9 (1921), 110-135. · JFM 48.1035.01
[2] A. L. Besse, Einstein Manifolds , Ergeb. Math. Grenzgeb. (3) 10 , Springer, Berlin, 1987. · Zbl 0613.53001
[3] S. Brendle, Uniqueness of gradient Ricci solitons , Math. Res. Lett. 18 (2011), 531-538. · Zbl 1246.53066
[4] H.-D. Cao, “Recent progress on Ricci solitons” in Recent Advances in Geometric Analysis (Taipei, 2007) , Adv. Lect. Math. (ALM) 11 , International Press, Somerville, Mass., 2010, 1-38.
[5] H.-D. Cao, “Geometry of complete gradient shrinking Ricci solitons” in Geometry and Analysis, No. 1 (Cambridge, Mass., 2008) , Adv. Lect. Math. (ALM) 17 , International Press, Somerville, Mass., 2011, 227-246. · Zbl 1268.53047
[6] H.-D. Cao, G. Catino, Q. Chen, C. Mantegazza, and L. Mazzieri, Bach-flat gradient steady Ricci solitons , to appear in Calc. Var., preprint, [math.DG] 1107.4591v2
[7] H.-D. Cao, B.-L. Chen, and X.-P. Zhu, “Recent developments on Hamilton’s Ricci flow” in Surveys in Differential Geometry, Vol. XII , Surv. Differ. Geom. 12 , International Press, Somerville, Mass., 2008, 47-112. · Zbl 1157.53002
[8] H.-D. Cao and Q. Chen, On locally conformally flat gradient steady Ricci solitons , Trans. Amer. Math. Soc. 364 , no. 5 (2012), 2377-2391. · Zbl 1245.53038
[9] H.-D. Cao and D. Zhou, On complete gradient shrinking Ricci solitons , J. Differential Geom. 85 (2010), 175-185. · Zbl 1246.53051
[10] X. Cao, B. Wang, and Z. Zhang, On locally conformally flat gradient shrinking Ricci solitons , Commun. Contemp. Math. 13 (2011), 269-282. · Zbl 1215.53061
[11] G. Catino and C. Mantegazza, The evolution of the Weyl tensor under the Ricci flow , Ann. Inst. Fourier (Grenoble) 61 (2011), 1407-1435. · Zbl 1255.53034
[12] B.-L. Chen, Strong uniqueness of the Ricci flow , J. Differential Geom. 82 (2009), 363-382. · Zbl 1177.53036
[13] X. X. Chen and Y. Wang, On four-dimensional anti-self-dual gradient Ricci solitons , preprint, [math.DG] 1102.0358v2
[14] A. Derdzinski, Self-dual Kähler manifolds and Einstein manifolds of dimension four , Compos. Math. 49 (1983), 405-433. · Zbl 0527.53030
[15] M. Eminenti, G. La Nave, and C. Mantegazza, Ricci solitons: The equation point of view , Manuscripta Math. 127 (2008), 345-367. · Zbl 1160.53031
[16] M. Fernández-López and E. García-Río, Rigidity of shrinking Ricci solitons , Math. Z. 269 (2011), 461-466. · Zbl 1226.53047
[17] R. S. Hamilton, Three-manifolds with positive Ricci curvature , J. Differential Geom. 17 (1982), 255-306. · Zbl 0504.53034
[18] Hamilton, R. S., “The formation of singularities in the Ricci flow” in Surveys in Differential Geometry (Cambridge, Mass., 1993) , International Press, Cambridge, Mass., 1995, 7-136. · Zbl 0867.53030
[19] O. Munteanu and N. Sesum, On gradient Ricci solitons , to appear in J. Geom. Anal., preprint, [pdf, ps, other] 0910.1105 · Zbl 1275.53061
[20] L. Ni and N. Wallach, On a classification of gradient shrinking solitons , Math. Res. Lett. 15 (2008), 941-955. · Zbl 1158.53052
[21] G. Perelman, The entropy formula for the Ricci flow and its geometric applications , preprint, [math.DG]. · Zbl 1130.53001
[22] Perelman, G., Ricci flow with surgery on three-manifolds , preprint, [math.DG] · Zbl 1130.53002
[23] P. Petersen and W. Wylie, On the classification of gradient Ricci solitons , Geom. Topol. 14 (2010), 2277-2300. · Zbl 1202.53049
[24] S. Pigola, M. Rimoldi, and A. G. Setti, Remarks on non-compact gradient Ricci solitons , Math. Z. 268 (2011), 777-790. · Zbl 1223.53034
[25] Z.-H. Zhang, Gradient shrinking solitons with vanishing Weyl tensor , Pacific J. Math. 242 (2009), 189-200. · Zbl 1171.53332
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