×

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.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.