The Ricci–Bourguignon flow

Catino, Giovanni; Cremaschi, Laura; Djadli, Zindine; Mantegazza, Carlo; Mazzieri, Lorenzo

doi:10.2140/pjm.2017.287.337

Cited by 103 publications

(73 citation statements)

References 28 publications

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“…Another strong motivation for our map is the fact that, although the Ricci and Yamabe flows are identical in dimension two, they are essentially different in higher dimensions. Let us remark that an interpolation flow between Ricci and Yamabe flows is considered in [7] under the name Ricci-Bourguignon flow but it depends on a single scalar, see also the pages 79-80 of the book [6].…”

Section: Introductionmentioning

confidence: 99%

Ricci–Yamabe maps for Riemannian flows and their volume variation and volume entropy

Güler¹,

Crâşmăreanu²

2019

Turk J Math

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The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow g(t). The considered flow in covariant symmetric 2-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of g(t). Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most commonly addressed situation we express the Ricci flow equation in all four orthogonal separable coordinate systems of the plane.

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Section: Introductionmentioning

confidence: 99%

Ricci–Yamabe maps for Riemannian flows and their volume variation and volume entropy

Güler¹,

Crâşmăreanu²

2019

Turk J Math

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“…The first one of them corresponds to self-similar solutions of Ricci flow and often arises as limits of dilations of singularities in the Ricci flow, see Hamilton [18]. A special family of the second one of them arises from the Ricci-Bourguignon flow, see Catino et al [11] or Catino and Mazzieri [12]. The third one of them is originated from the study of Einstein warped product manifolds, see Besse [4].…”

Section: Introductionmentioning

confidence: 99%

Gradient Ricci almost soliton warped product

Feitosa

Filho

Gomes

et al. 2019

Journal of Geometry and Physics

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We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an example of particular solutions of the PDEs that arise from our construction. We also prove a rigidity result for a gradient Ricci soliton Riemannian product in the class of gradient Ricci almost soliton warped products under some natural geometric assumptions on the warping function.2010 Mathematics Subject Classification. Primary 53C15, 53C25; Secondary 53C21.

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“…Although gradient Ricci solitons are a special case of quasi-Einstein metrics, they exhibit quite different properties (see [18]). We emphasize that the gradient Ricci almost soliton equation is not just a formal generalization of the Ricci soliton equation, but includes families of self-similar solutions of other geometric flows such as the Ricci-Bourguignon flow [21]. This flow is defined for a κ ∈ R by the evolution equation ∂ t g(t) = −2(ρ(t) − κτ (t) g(t)).…”

Section: Introductionmentioning

confidence: 99%