Bochner-type Formulas for the Weyl Tensor on Four-dimensional Einstein Manifolds

Catino, Giovanni; Mastrolia, Paolo

doi:10.1093/imrn/rny127

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…the critical k-th order Lovelock term (4.1) has no explicit dependence on the coframe, implying that the equation (4.10) is trivially satisfied. A similar proof for critical Lovelock terms for arbitrary order in the metric formalism can be found in [37]. Consequently the only remaining equation is the one for the connection.…”

Section: Exploring Non-trivial Solutions Of the Critical Case Of Arbisupporting

confidence: 53%

On the topological character of metric-affine Lovelock Lagrangians in critical dimensions

Janssen

Jiménez-Cano

2019

Physics Letters B

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In this paper we prove that the k-th order metric-affine Lovelock Lagrangian is not a total derivative in the critical dimension n = 2k in the presence of non-trivial non-metricity. We use a bottom-up approach, starting with the study of the simplest cases, Einstein-Palatini in two dimensions and Gauss-Bonnet-Palatini in four dimensions, and focus then on the critical Lovelock Lagrangian of arbitrary order. The two-dimensional Einstein-Palatini case is solved completely and the most general solution is provided. For the Gauss-Bonnet case, we first give a particular configuration that violates at least one of the equations of motion and then show explicitly that the theory is not a pure boundary term. Finally, we make a similar analysis for the k-th order critical Lovelock Lagrangian, proving that the equation of the coframe is identically satisfied, while the one of the connection only holds for some configurations. In addition to this, we provide some families of non-trivial solutions.

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Section: Exploring Non-trivial Solutions Of the Critical Case Of Arbisupporting

confidence: 53%

On the topological character of metric-affine Lovelock Lagrangians in critical dimensions

Janssen

Jiménez-Cano

2019

Physics Letters B

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“…Finally, the following identity holds (see [6]) Lemma 2.1. On every n-dimensional, n ≥ 4, Riemannian manifold one has…”

Section: Dimension Fourmentioning

confidence: 99%

Four dimensional closed manifolds admit a weak harmonic Weyl metric

Catino¹,

Mastrolia²,

Monticelli³

et al. 2018

Preprint

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On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.

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“…There are many gap results for Einstein manifolds, Ricci solitons and closed manifolds; such as Catino and Mastrolia [11], Hebey and Vaugon [32], Li and Wang [38,39], Munteanu and Wang [43], Petersen and Wylie [47], Singer [51], Tran [52], Zhang [63] and their references. In this paper we provide a different gap criterion, which depends on the constant C(n) of Sobolev inequality.…”

Section: Introductionmentioning

confidence: 99%

Geometric inequalities and rigidity of gradient shrinking Ricci solitons

2020

Preprint

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Bochner-type Formulas for the Weyl Tensor on Four-dimensional Einstein Manifolds

Cited by 5 publications

References 12 publications

On the topological character of metric-affine Lovelock Lagrangians in critical dimensions

On the topological character of metric-affine Lovelock Lagrangians in critical dimensions

Four dimensional closed manifolds admit a weak harmonic Weyl metric

Geometric inequalities and rigidity of gradient shrinking Ricci solitons

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