Gradient shrinking solitons with vanishing Weyl tensor

Zhang, Zhuhong

doi:10.2140/pjm.2009.242.189

Cited by 95 publications

(70 citation statements)

References 13 publications

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“…In the shrinking case, the analysis of Kotschwar [23] of rotationally invariant shrinking, gradient Ricci solitons gives the following classification where the Gaussian soliton is defined as the flat R n with a potential function f = α|x| 2 /2n, for a constant α ∈ R. This classification of shrinking, gradient, LCF Ricci solitons was obtained in the paper of P. Petersen and W. Wylie [29] (using also results of Z.-H. Zhang [33]). Many other authors contributed to the subject, including X. Cao, B. Wang and Z. Zhang [7], B.-L. Chen [8], M. Fernández-López and E. García-Río [14], L. Ni and N. Wallach [27], O. Munteanu and N. Sesum [25] and again P. Petersen and W. Wilye [30].…”

Section: The Classification Of Steady and Shrinking Gradient Lcf Riccmentioning

confidence: 83%

“…The class of solitons with nonnegative Ricci tensor is particularly interesting as it includes all the shrinking and steady Ricci solitons by the results TOME 61 (2011), FASCICULE 4 in [8] and [33], where it is proved that a complete ancient solution g(t) to the Ricci flow with zero Weyl tensor has a nonnegative curvature operator for every time t. In particular, this holds for any steady or shrinking Ricci soliton (even if not gradient) being them special ancient solutions.…”

Section: The Classification Of Steady and Shrinking Gradient Lcf Riccmentioning

confidence: 99%

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The Evolution of the Weyl Tensor under the Ricci Flow

Catino¹,

Mantegazza²

2011

Ann. inst. Fourier

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Section: The Classification Of Steady and Shrinking Gradient Lcf Riccmentioning

confidence: 83%

Section: The Classification Of Steady and Shrinking Gradient Lcf Riccmentioning

confidence: 99%

The Evolution of the Weyl Tensor under the Ricci Flow

Catino¹,

Mantegazza²

2011

Ann. inst. Fourier

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“…In particular, it is known (cf. [23,21,7]) that any complete 3-dimensional gradient shrinking Ricci soliton is a finite quotient of either the round sphere S 3 , or the Gaussian shrinking soliton R 3 , or the round cylinder S 2 × R. For higher dimensions, it has been proven that complete locally conformally flat gradient shrinking Ricci solitons are finite quotients of either the round sphere S n , or the Gaussian shrinking soliton R n , or the round cylinder S n−1 × R (first due to Z. H. Zhang [26] based on the work of Ni-Wallach [21], see also 0 2000 Mathematics Subject Classification. Primary 53C21, 53C25.…”

Section: The Resultsmentioning

confidence: 99%

On Bach-flat gradient shrinking Ricci solitons

Cao¹,

Chen²

2013

Duke Math. J.

138

173

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Abstract. In this paper, we classify n-dimensional (n ≥ 4) complete Bachflat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton R 4 or the round cylinder S 3 ×R. More generally, for n ≥ 5, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton R n or the product N n−1 × R, where N n−1 is Einstein. The resultsA 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 ij satisfies the equation R ij + ∇ i ∇ j f = ρg ij for some constant ρ. For ρ = 0 the Ricci soliton is steady, for ρ > 0 it is shrinking and for ρ < 0 expanding. The function f is called a potential function of the gradient Ricci soliton. Clearly, when f is a constant the gradient Ricci soliton is simply an Einstein manifold. Thus Ricci solitons are natural extensions of Einstein metrics. Gradient Ricci solitons play an important role in Hamilton's Ricci flow as they correspond to self-similar solutions, and often arise as singularity models. Therefore it is important to classify gradient Ricci solitons or understand their geometry.In this paper we shall focus our attention on gradient shrinking Ricci solitons, which are possible Type I singularity models in the Ricci flow. We normalize the constant ρ = 1/2 so that the shrinking soliton equation is given byIn recent years, inspired by Perelman's work [22,23], much efforts have been devoted to study the geometry and classifications of gradient shrinking Ricci solitons. We refer the reader to the survey papers [4,5] by the first author and the references therein for recent progress on the subject. In particular, it is known (cf. [23,21,7]) that any complete 3-dimensional gradient shrinking Ricci soliton is a finite quotient of either the round sphere S 3 , or the Gaussian shrinking soliton R 3 , or the round cylinder S 2 × R. For higher dimensions, it has been proven that complete locally conformally flat gradient shrinking Ricci solitons are finite quotients of either the round sphere S n , or the Gaussian shrinking soliton R n , or the round cylinder S n−1 × R (first due to Z. H. Zhang that n-dimensional complete gradient shrinking solitons with harmonic Weyl tensor are rigid in the sense that they are finite quotients of the product of an Einstein manifold N k with the Gaussian shrinking soliton R n−k . Our aim in this paper is to investigate an interesting class of complete gradient shrinking Ricci solitons: those with vanishing Bach tensor. This well-known tensor was introduced by R. Bach [1] in early 1920s' to study conformal relativity. On any n-dimensional manifold (M n , g ij ) (n ≥ 4), the Bach tensor is defined byHere W ikjl is the Weyl tensor. It is easy to see that if (M n , g ij ) is either locally conformally flat (i.e., W ikjl = 0) or...

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“…Zhang [19] has shown that complete gradient shrinking Ricci solitons with vanishing Weyl tensor have nonnegative Ricci tensor and that the Riemann curvature has at most exponential growth (see also [12]). As an application of Theorem 2.3 we have the following (which has already been obtained by Petersen and Wylie [18] under the assumption M |Ric| 2 e − f < ∞, which holds true for any complete shrinking Ricci soliton as shown in [12]).…”

Section: Theorem 23 Let (M N G) Be An N-dimensional Complete Noncomentioning

confidence: 99%