Prescribed diagonal Ricci tensor in locally conformally flat manifolds

Pina, Romildo; Adriano, Levi; Pieterzack, Mauricio

doi:10.1016/j.jmaa.2014.07.058

Cited by 8 publications

(6 citation statements)

References 10 publications

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“…Clearly, the first obstacle in the general Riemannian setting is to understand the prescribed Ricci curvature equation. While this equation has been the subject of active research (e.g., [6,Chapter 5], [26,20,21,19,18,34,33,40])-when compared to the Kähler case-it is understood rather poorly on general Riemannian manifolds. Recent results of the first-named author [35] provide a replacement for the Calabi-Yau theorem on certain classes of homogeneous spaces and are crucial in the present article.…”

Section: Introductionmentioning

confidence: 99%

Ricci iteration on homogeneous spaces

Pulemotov¹,

Rubinstein²

2016

Preprint

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The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not Kähler. The Ricci iteration in the non-Kähler setting exhibits new phenomena. Among them is the existence of so-called ancient Ricci iterations. As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times. In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov-Hausdorff topology) from a collapsed Einstein metric and converge smoothly to a second Einstein metric. In the case of compact homogeneous spaces with maximal isotropy, we prove a relative compactness result that excludes collapsing. Our work can also be viewed as proposing a dynamical criterion for detecting whether an ancient Ricci flow exists on a given Riemannian manifold as well as a method for predicting its limit.

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Section: Introductionmentioning

confidence: 99%

Ricci iteration on homogeneous spaces

Pulemotov¹,

Rubinstein²

2016

Preprint

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“…Many mathematicians have investigated the global existence of Riemannian metrics with prescribed Ricci curvature. The papers [10,22,21,23] provide a snapshot of the recent progress on this topic. We refer to [2,Chapter 5] and [1,Section 9.2] for surveys of older work and to [6] for a sample of the research done in the Lorentzian setting.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Metrics with prescribed Ricci curvature on homogeneous spaces

Pulemotov

2016

Journal of Geometry and Physics

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Let G be a compact connected Lie group and H a closed subgroup of G. Suppose the homogeneous space G/H is effective and has dimension 3 or higher. Consider a G-invariant, symmetric, positivesemidefinite, nonzero (0,2)-tensor field T on G/H. Assume that H is a maximal connected Lie subgroup of G. We prove the existence of a G-invariant Riemannian metric g and a positive number c such that the Ricci curvature of g coincides with cT on G/H. Afterwards, we examine what happens when the maximality hypothesis fails to hold.

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“…When λ = 0 and λ = − 1 2 , the problem (1.1)is known in the literature as the prescribed Ricci and Einstein tensor, respectively. This problem has been studied for a particular family of tensors (see [16][17][18][19][20]). Recently, Pulemotov studied the problem (1.1) for λ = 0 in homogeneous manifolds (see [21]).…”

Section: Introductionmentioning

confidence: 99%

Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

2019

Self Cite

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We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.

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Prescribed diagonal Ricci tensor in locally conformally flat manifolds

Cited by 8 publications

References 10 publications

Ricci iteration on homogeneous spaces

Ricci iteration on homogeneous spaces

Metrics with prescribed Ricci curvature on homogeneous spaces

Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

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