The Evolution of the Weyl Tensor under the Ricci Flow

“…In the appendix we review the proof of the classification of 2-dimensional solitons giving a proof that follows from an Obata-type characterization of warped product manifolds originally due to Brinkmann [5]. [7] and Catino and Mantegazza [10] independently give the complete classification of locally conformally flat steady and expanding gradient solitons. In particular, these results, combined with the classification of rotationally symmetric shrinking gradient Ricci solitons by Kotschwar [22], gives yet another route to the classification of shrinking gradient Ricci solitons on locally conformally flat manifolds.…”

Section: Introductionmentioning

confidence: 99%

On the classification of gradient Ricci solitons

2010

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We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones S n , S n 1 R and R n . This gives a new proof of the Hamilton-Ivey-Perelman classification of 3-dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of53C25

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“…On the other hand, if n ≥ 4, from the conditions (4.2), (M, g, f ) is a locally conformally flat gradient Ricci soliton. Proposition 2.1 now follows from the classifications results in the shrinking ([45,53,48]), steady ([16,21]) and expanding ([21]) cases. To the best of our knowledge, the complete classification of locally conformally flat, gradient expanding Ricci solitons is still open; however it is known that around any regular point of f the manifold (M, g) is locally a warped product with codimension one fibers of constant sectional curvature.LS f and LSE f : proof of Proposition 2.2.…”

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confidence: 81%

A potential generalization of some canonical Riemannian metrics

Catino

Mastrolia

2019

Ann Glob Anal Geom

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The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper we also describe the "nongradient" version of this construction.2010 Mathematics Subject Classification. 53C20, 53C25. 1 2 GIOVANNI CATINO AND PAOLO MASTROLIAOn the other hand, from the analytic point of view, the aim is to simplify the curvature by imposing some differential condition. A quite natural and not too restrictive way to do this is to consider curvature tensors belonging to the kernel of a first order linear differential operator. Some well known conditions of this type can be given by saying that (M, g) belongs to• LS if ∇(Riem) = 0 (locally symmetric metrics); • PR if ∇(Ric) = 0 (metrics with parallel Ricci curvature); • HC if div(Riem) = 0 (harmonic curvature metric).Note that, by Bianchi identities, we can redefine the class Y of Yamabe metrics using the condition div(Ric) = 0 or, equivalently, ∇R = 0. Here and in the rest of the paper div denotes the divergence operator (see Section 3 for the definition). Obviously, SF ⊂ LS ⊂ PR and, by Bianchi identities, PR ⊂ HC, E ⊂ HC ⊂ Y. Thus, we have the inclusionswhere, by definition, LSE := LS ∩ E (locally symmetric Einstein metrics). Note that, in dimension n ≥ 4, all the inclusions are strict.This classes of metrics certainly do not exhaust all the possible canonical metrics on a Riemannian manifold: our choice is essentially made in such a way that Einstein metrics (and Ricci solitons, as we will see) are the "cornerstone" of our construction, and the conditions that we impose are consequently focused on the Ricci tensor. We note that, in principle, one could also consider "higher order" conditions, such as ∇ k Riem = 0 or ∇ k Ric = 0, k ≥ 2, but these relations give rise again to LS and PR, respectively, by the results in [46,50]. However, one can consider other higher order analytic curvature conditions in order to generalize locally symmetric metrics, such as, for instance, the class of semi-symmetric spaces introduced by Cartan in [19].The class SF is the most rigid, since, up to quotients, it contains only S n , R n and H n with their standard metrics. Locally symmetric spaces LS were classified by Cartan [18], while, from the de Rham decomposition theorem ([6]), PR metrics are locally Riemannian products of Einstein metrics. On the other hand, given any compact manifold M, there always exists a Riemannian metric g such that (M, g) ∈ Y (see e.g. [41]). The remaining classes are more flexible. In particular E and HC, in the last decades, have been studied by many researchers, also for their connections with Physics in General Relativity and Yang-Mills Theory. In fact, these metrics ...

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The Evolution of the Weyl Tensor under the Ricci Flow

Abstract: 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.

Cited by 54 publications

References 30 publications

Formal matched asymptotics for degenerate Ricci flow neckpinches

Formal matched asymptotics for degenerate Ricci flow neckpinches

On the classification of gradient Ricci solitons

A potential generalization of some canonical Riemannian metrics

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