Deformations of Q-curvature I

In this article, we define a symmetric 2-tensor canonically associated to Qcurvature called J-tensor on any Riemannian manifold with dimension at least three. The relation between J-tensor and Q-curvature is precisely like Ricci tensor and scalar curvature. Thus it can be interpreted as a higher-order analogue of Ricci tensor. This tensor can also be used to understand Chang-Gursky-Yang's theorem on 4-dimensional Q-singular metrics. Moreover, we show an Almost-Schur Lemma holds for Q-curvature, which gives an estimate of Q-curvature on closed manifolds.

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“…[2,13]) such that it satisfies certain conformal invariant properties. For more details, please refer to the appendix of [12].…”

Section: Introductionmentioning

confidence: 99%

A symmetric 2-tensor canonically associated to Q-curvature and its applications

Lin

Yuan

2017

Pacific J. Math.

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“…Recall that the principal symbol of a differential operator is an invariant that captures some very strong properties of the operator, as example, the ellipticity. In our case, it was observed in [28] that the principal symbol of L * g is σ ξ (L * g ) = −a n g|ξ| 2 − ξ ⊗ ξ |ξ| 2 . Notice that L * g has an injective symbol.…”

Section: Local Surjectivitymentioning

confidence: 58%

“…• Rm · h) jk := R ijkl h il . Following notation in [28] we have Proposition 2.1 (Lin-Yuan, [28]). Given an infinitesimal variation h, the linearization of the Q-curvature Q at g, denoted by L g , in the direction of h is given by…”

Section: Local Surjectivitymentioning

confidence: 99%

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On the prescribed Q-curvature problem in Riemannian manifolds

Cruz

Tiarlos

2020

manuscripta math.

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We prove the existence of metrics with prescribed Qcurvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on R n can be realized as the Q-curvature of a Riemannian metric.

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“…We notice that the definition (1.4) of the Einstein operator differ from [31, Definition 1.6] and [30,Definition 1.6] by a signal. Using (2.6) we have the following.…”

Section: Volume Comparisonmentioning

confidence: 99%

On the $σ_2$-curvature and volume of compact manifolds

Andrade¹,

Tiarlos²,

Santos³

2021

Preprint

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In this work we are interested in studying deformations of the σ2-curvature and the volume. For closed manifolds, we relate critical points of the total σ2-curvature functional to the σ2-Einstein metrics and, as a consequence of results of H. J. Gursky and J. A. Viaclovsky [25] and Z. Hu and H. Li [27], we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to σ2-curvature which shows that the volume can be controlled by the σ2-curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant σ2-curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second order properties of the volume functional we show that there is a variation for which geodesic balls are indeed local minimum in a natural direction.

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Deformations of Q-curvature I

Cited by 14 publications

References 33 publications

A symmetric 2-tensor canonically associated to Q-curvature and its applications

A symmetric 2-tensor canonically associated to Q-curvature and its applications

On the prescribed Q-curvature problem in Riemannian manifolds

On the $σ_2$-curvature and volume of compact manifolds

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