On Gradient Ricci Solitons

Munteanu, Ovidiu; Šešum, Natasa

doi:10.1007/s12220-011-9252-6

Cited by 178 publications

(160 citation statements)

References 31 publications

(64 reference statements)

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“…the works of Eminenti-La Nave-Mantegazza [15], Petersen-Wylie [24], X. Cao, B. Wang and Z. Zhang [10], and Munteanu-Sesum [20]). Moreover, it follows from the works of Fernández-López and García-Río [16], and Munteanu-Sesum [20] 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 .…”

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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Section: The Resultsmentioning

confidence: 99%

On Bach-flat gradient shrinking Ricci solitons

Cao¹,

Chen²

2013

Duke Math. J.

138

173

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“…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: 99%

The Evolution of the Weyl Tensor under the Ricci Flow

Catino¹,

Mantegazza²

2011

Ann. inst. Fourier

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“…Recently, gradient Ricci solitons with harmonic Weyl curvature became an interesting theme to study [2,6]. Now one may need to understand noncompact solitons with λ of any sign.…”

Section: Introductionmentioning

confidence: 99%

Some Doubly-Warped Product Gradient Ricci Solitons

Kim¹

2016

Communications of the Korean Mathematical Society

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Abstract. In this paper, we study certain doubly-warped products which admit gradient Ricci solitons with harmonic Weyl curvature and non-constant soliton function. The metric is of the form g = dx 2 1 + p(x 1 ) 2 dx 2 2 + h(x 1 ) 2g on R 2 × N , where x 1 , x 2 are the local coordinates on R 2 andg is an Einstein metric on the manifold N . We obtained a full description of all the possible local gradient Ricci solitons.

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On Gradient Ricci Solitons

Cited by 178 publications

References 31 publications

On Bach-flat gradient shrinking Ricci solitons

On Bach-flat gradient shrinking Ricci solitons

The Evolution of the Weyl Tensor under the Ricci Flow

Some Doubly-Warped Product Gradient Ricci Solitons

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