Riemannian Manifolds with Positive Yamabe Invariant and Paneitz Operator

Gursky, Matthew J.; Hang, Fengbo; Lin, Yueh Ju

doi:10.1093/imrn/rnv176

Cited by 33 publications

(19 citation statements)

References 10 publications

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“…and the constant Q-curvature problem for Paneitz-Branson operator (cf. [3,4,8,10], etc.). Especially, F. Hang and P. Yang [10] set up a dual variational method of the minimization for the Paneitz-Branson functional to seek a positive maximizer of the dual functional, such a scheme heavily relies on the positivity and expansion of its Green's function.…”

Section: Expansion Of Green's Function Of P Gmentioning

confidence: 99%

“…One key ingredient in such works is that a strong maximum principle for the fourth order Paneitz-Branson operator is discovered under a hypothesis on the positivity of some conformal invariants or Q-curvature of the background metric. The readers are referred to [8,9,10,13] and the references therein. This naturally stimulates us to study GJMS operator of order six and its associated Q-curvature problem, the analogue to the Yamabe problem and Q-curvature problem for Paneitz-Branson operator.…”

Section: Introductionmentioning

confidence: 99%

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Remarks on GJMS operator of order six

Chen

Hou

2017

Pacific J. Math.

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We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Green's function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n \geq 10$, let $(M^n,g)$ be a closed Einstein manifold of positive scalar curvature and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a maximum point of $f$ and $f$ satisfies a vanishing order condition at this maximum point, then there exists a conformal metric $\tilde g$ of $g$ such that its $Q$-curvature $Q_{\tilde g}^6$ equals $f$.Comment: to appear Pacific Journal of Mathematic

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Section: Expansion Of Green's Function Of P Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Remarks on GJMS operator of order six

Chen

Hou

2017

Pacific J. Math.

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“…There are several criteria for the positivity of P (see [3, theorem 1.6] and [12,35]). On the other hand, in a recent preprint [11], it is proved that if .M; g/ is a smooth, compact Riemannian manifold with dimension n 6, and Y .g/ > 0, Y 4 .g/ > 0, then we can find a conformal metric z g with z R > 0 and Q > 0. In particular, it follows from [12] that any conformal metric with constant Q-curvature must have positive scalar curvature.…”

Section: Introductionmentioning

confidence: 99%

Q‐Curvature on a Class of Manifolds with Dimension at Least 5

Hang

Yang

2015

Comm Pure Appl Math

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“…, and these invariants coincide in some special cases. For instance, if n = dim M ≥ 6 and there exists g ∈ [g 0 ], with scal g > 0 and Q g > 0, 1 then Y 4 (M, g 0 ) = Y + 4 (M, g 0 ), and the (positive) infimum in (2.7) is attained by a positive function u such that u 4 n−4 g 0 has positive constant Qcurvature, and everywhere positive scalar curvature [GHL16].…”

Section: Variational Setupmentioning

confidence: 99%

“…1 If dim M ≥ 6, the existence of g ∈ [g 0 ] with with scalg > 0 and Qg > 0 is proved in [GHL16] to be equivalent to Y (M, g 0 ) > 0 and Pg 0 > 0.…”

Section: Variational Setupmentioning

confidence: 99%

Nonuniqueness of Conformal Metrics With Constant Q-curvature

Bettiol

Piccione

Sire

2019

International Mathematics Research Notices

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We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) Q-curvature on compact and noncompact manifolds of dimension ≥ 5. Infinitely many branches of metrics with constant Q-curvature, but without constant scalar curvature, are found to bifurcate from Berger metrics on spheres and complex projective spaces. These provide examples of nonisometric metrics with the same constant negative Q-curvature in a conformal class with negative Yamabe invariant, echoing the absence of a Maximum Principle. We also discover infinitely many complete metrics with constant Q-curvature conformal to S m × R d , m ≥ 4, d ≥ 1, and S m × H d , 2 ≤ d ≤ m − 3; which give infinitely many solutions to the singular constant Q-curvature problem on round spheres S n blowing up along a round subsphere S k , for all 0 ≤ k < (n − 4)/2.

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Riemannian Manifolds with Positive Yamabe Invariant and Paneitz Operator

Abstract: For a Riemannian manifold with dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.

Cited by 33 publications

References 10 publications

Remarks on GJMS operator of order six

Remarks on GJMS operator of order six

Q‐Curvature on a Class of Manifolds with Dimension at Least 5

Nonuniqueness of Conformal Metrics With Constant Q-curvature

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