Remarks on GJMS operator of order six

@article{Chen2016RemarksOG,
  title={Remarks on GJMS operator of order six},
  author={Xuezhang Chen and Fei Hou},
  journal={arXiv: Differential Geometry},
  year={2016}
}
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… Expand
Existence result for the higher-order $Q$-curvature equation
We obtain an existence result for the $Q$-curvature equation of arbitrary order $2k$ on a closed Riemannian manifold of dimension $n\ge 2k+4$, where $k\ge1$ is an integer. We obtain this result underExpand

References

SHOWING 1-10 OF 23 REFERENCES
Compactness of conformal metrics with constant Q-curvature. I
We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, weExpand
Laplacian Operators and Q-curvature on Conformally Einstein Manifolds
A new definition of canonical conformal differential operators Pk (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. TheseExpand
Conformally invariant powers of the Laplacian — A complete nonexistence theorem
Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observationExpand
A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature
In this paper we consider Riemannian manifolds (M, g) of dimension n ≥ 5, with semi-positive Q-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operatorExpand
Explicit Formulas for GJMS-Operators and Q-Curvatures
We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré–Einstein metrics and renormalized volumeExpand
Paneitz-type operators and applications
To the memory of André Lichnerowicz Given (M, g) a smooth 4-dimensional Riemannian manifold, let S g be the scalar curvature of g, and let Rc g be the Ricci curvature of g. The Paneitz operator,Expand
Q-Curvature on a Class of Manifolds with Dimension at Least 5
For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q-curvature, and dimension at least 5, we prove the existence of a conformal metric with constant Q-curvature. OurExpand
Mountain pass critical points for Paneitz-Branson operators
Abstract. Given $(M,g)$ a smooth compact Riemannian manifold of dimension $n \ge 5$, we study fourth order equations involving Paneitz-Branson type operators and the critical Sobolev exponent.
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
A maximizing function, f, is shown to exist for the HLS inequality on R': 11 IXI - * fIq < Np f A , Iif IIwith Nbeing the sharp constant and i/p + X/n = 1 + 1/q, 1 <p, q, n/X < x. When p =q' or p = 2Expand
The Concentration-Compactness Principle in the Calculus of Variations. (The limit case, Part I.)
After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where theExpand
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