Metrics with prescribed Ricci curvature on homogeneous spaces

Pulemotov, Artem

doi:10.1016/j.geomphys.2016.04.003

Cited by 27 publications

(56 citation statements)

References 30 publications

(66 reference statements)

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“…We will elaborate on this after we provide an overview of our main results. According to [22,Theorem 1.1], if H is a maximal connected Lie subgroup of G, the functional S attains its greatest value on M T at some g ∈ M T . It is easy to show that this g satisfies (1.1) with c > 0.…”

Section: Introductionmentioning

confidence: 99%

“…To illustrate the applicability of this result, we explore cases where M is a generalised Wallach space and a generalised flag manifold.Keywords: Prescribed Ricci curvature, homogeneous space, generalised Wallach space, generalised flag manifold for some c > 0, where T is a given G-invariant (0, 2)-tensor field. The study of (1.1) in the framework of homogeneous spaces was initiated in [22] and continued in [16]; see also [17,15,11]. It is on the basis of [22] that the first results about the Ricci iteration in the non-Kähler setting were obtained in [23].…”

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confidence: 99%

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Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces

Pulemotov

2019

J Geom Anal

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Consider a compact Lie group G and a closed Lie subgroup H < G. Let M be the set of G-invariant Riemannian metrics on the homogeneous space M = G/H. By studying variational properties of the scalar curvature functional on M, we obtain an existence theorem for solutions to the prescribed Ricci curvature problem on M . To illustrate the applicability of this result, we explore cases where M is a generalised Wallach space and a generalised flag manifold.Keywords: Prescribed Ricci curvature, homogeneous space, generalised Wallach space, generalised flag manifold for some c > 0, where T is a given G-invariant (0, 2)-tensor field. The study of (1.1) in the framework of homogeneous spaces was initiated in [22] and continued in [16]; see also [17,15,11]. It is on the basis of [22] that the first results about the Ricci iteration in the non-Kähler setting were obtained in [23]. These results provided a new approach to uniformisation on homogeneous spaces. Moreover, they led to the discovery of several dynamical properties of the Ricci curvature.Assume the (0, 2)-tensor field T lies in M. As [22, Lemma 2.1] demonstrates, a metric g ∈ M satisfies (1.1) for some c ∈ R if and only if it is (up to scaling) a critical point of the functional S onThis parallels the well-known variational interpretation of the Eisntein equation. Indeed, a metric g ∈ M satisfies Ric g = λg

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces

Pulemotov

2019

J Geom Anal

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“…We invite the reader to see [7,4] for the history of the subject. The list of recent references not mentioned in [7,4] includes but is not limited to [16,15,27,28,22,26].…”

Section: Introductionmentioning

confidence: 99%

The Dirichlet problem for the prescribed Ricci curvature equation on cohomogeneity one manifolds

Pulemotov

2015

Annali di Matematica

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Let M be a domain enclosed between two principal orbits on a cohomogeneity one manifold M 1 . Suppose that T and R are symmetric invariant (0, 2)-tensor fields on M and ∂ M, respectively. The paper studies the prescribed Ricci curvature equation Ric(G) = T for a Riemannian metric G on M subject to the boundary condition G ∂ M = R (the notation G ∂ M here stands for the metric induced by G on ∂ M). Imposing a standard assumption on M 1 , we describe a set of requirements on T and R that guarantee global and local solvability.

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“…Turning to the main result of this Ph.D. thesis, global existence and uniqueness for the Ricci curvature equation and for boundary value problems thereof are well studied topics but results are difficult to come by: not mentioning the work on Kähler-Einstein metrics and the Calabi conjecture, on manifolds without boundary see for instance [7,8,13,15,22,34,43,47], and on manifolds with nonempty boundary see for instance [44,46,50,51], and the references therein. Indeed, strong obstructions to global existence are presented in for instance [3,12,24,34].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The Bianchi identity and the Ricci curvature equation

Smith¹

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The main result of this Ph.D. thesis is to present conditions on the curvature and boundary of an orientable, compact manifold under which there is a unique global solution to the Dirichlet boundary value problem (BVP) for the prescribed Ricci curvature equation. This Dirichlet BVP is a determined, non-elliptic system of 2nd-order, quasilinear partial differential equations, which is supplemented with a constraint equation in the form of the so-called Bianchi identity. Indeed, in order to prove the main result of this Ph.D. thesis, we are also required to prove that the kernel of the Bianchi operator is a smooth tame Fréchet submanifold of the space of Riemannian metrics on a compact Riemannian manifold with boundary. After presenting an overview of the literature in Chapter 1 and the required notation and background in Riemannian geometry and geometric analysis in Chapter 2, the main results of this thesis are organised into the following two chapters.In Chapter 3 we present conditions under which the kernel of the Bianchi operator is Furthermore, the material in Chapter 3 of this thesis sits more generally within the literature on linearisation stability of nonlinear PDE. Indeed, the method by which we prove that the kernel of the Bianchi operator is a global submanifold of the space of metrics on a compact manifold is an application of the more general technique of proving the linearisation stability of a system of nonlinear PDE, and yields that the Bianchi operator is itself linearisation stable in a sense made precise in Chapter 3.In Chapter 4 we then use the fact that the kernel of the Bianchi operator is globally a smooth tame Fréchet manifold, and the Nash-Moser implicit function theorem (IFT) in the smooth tame category, to find and conditions under which globally there is a unique

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Metrics with prescribed Ricci curvature on homogeneous spaces

Cited by 27 publications

References 30 publications

Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces

Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces

The Dirichlet problem for the prescribed Ricci curvature equation on cohomogeneity one manifolds

The Bianchi identity and the Ricci curvature equation

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