You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 18.208.0.0/16 has been blocked (disabled) by ‪ST47‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider. To prevent abuse, web hosts may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a web host's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: ((unblock|reason=Caught by a web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~)). You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and they should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 01:01, 24 April 2026.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
Your IP address is in a range that has been blocked on all Wikimedia Foundation wikis. The block was made by ‪AntiCompositeNumber‬. The reason given is Open proxy/Webhost: See the help page if you are affected .
- Start of block: 15:08, 30 August 2023
- Expiry of block: 15:08, 30 August 2025
Your current IP address is 18.208.109.86. The blocked range is 18.208.0.0/16. Please include all above details in any queries you make. If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

You can view and copy the source of this page:

In the [[mathematics|mathematical]] field of [[differential geometry]], the '''Paneitz operator''' is a fourth-order [[differential operator]] defined on a [[Riemannian manifold]] of dimension ''n''.  It is named after [[Stephen Paneitz]], who discovered it in 1983, and whose preprint was later published posthumously in ((harvnb|Paneitz|2008)). 
In fact, the same operator was found earlier  in  the context of [[conformal supergravity]] by [[E. Fradkin]] and [[A. Tseytlin]] in 1982
(Phys Lett B 110 (1982) 117   and Nucl Phys B 1982 (1982) 157 ).
It is given by the formula

:<math>P = \Delta^2 - \delta \left\{(n-2)J - 4V\cdot\right\}d + (n-4)Q</math>

where Δ is the [[Laplace&ndash;Beltrami operator]], ''d'' is the [[exterior derivative]], δ is its formal adjoint, ''V'' is the [[Schouten tensor]], ''J'' is the trace of the Schouten tensor, and the dot denotes tensor contraction on either index.  Here ''Q'' is the scalar invariant
:<math>(-4|V|^2+nJ^2+2\Delta J)/4,</math>
where Δ is the positive Laplacian. In four dimensions this yields the [[Q-curvature]].

The operator is especially important in [[conformal geometry]], because in a suitable sense it depends only on the [[conformal structure]]. Another operator of this kind is the [[Laplace operators in differential geometry#Conformal Laplacian|conformal Laplacian]].  But, whereas the conformal Laplacian is second-order, with [[symbol of a differential operator|leading symbol]] a multiple of the Laplace&ndash;Beltrami operator, the Paneitz operator is fourth-order, with leading symbol the ''square'' of the Laplace&ndash;Beltrami operator.  The Paneitz operator is conformally invariant in the sense that it sends [[density on a manifold|conformal densities]] of weight ((nowrap|2 &minus; ''n''/2)) to conformal densities of weight ((nowrap|&minus;2 &minus; ''n''/2)).  Concretely, using the canonical trivialization of the density bundles in the presence of a metric, the  Paneitz operator ''P'' can be represented in terms of a representative the Riemannian metric ''g'' as an ordinary operator on functions that transforms according under a conformal change ((nowrap|''g'' ((mapsto)) &Omega;<sup>2</sup>''g'')) according to the rule
:<math>\Omega^{n/2+2}P(g)\phi = P(\Omega^2g)\Omega^{n/2-2}\phi.\,</math>
The operator was originally derived by working out specifically the lower-order correction terms in order to ensure conformal invariance.  Subsequent investigations have situated the Paneitz operator into a hierarchy of analogous conformally invariant operators on densities: the [[GJMS operator]]s.

The Paneitz operator has been most thoroughly studied in dimension four where it appears naturally in connection with extremal problems for the [[functional determinant]] of the Laplacian (via the [[Polyakov formula]]; see ((harvnb|Branson|Ørsted|1991))).  In dimension four only, the Paneitz operator is the "critical" GJMS operator, meaning that there is a residual scalar piece (the [[Q curvature]]) that can only be recovered by asymptotic analysis.  The Paneitz operator appears in extremal problems for the [[Moser&ndash;Trudinger inequality]] in dimension four as well ((harv|Chang|1999))

==CR Paneitz operator==
There is a close connection between 4 dimensional Conformal Geometry and 3 dimensional CR geometry associated with the study of [[CR manifold]]s. There is a naturally defined fourth order operator on CR manifolds introduced by [[C. Robin Graham]] and [[John M. Lee|John Lee]] that has many properties similar to the classical Paneitz operator defined on 4 dimensional Riemannian manifolds.<ref>((cite journal
| last1=Graham | first1=C. Robin
| last2=Lee | first2=John M.
| title=Smooth Solutions of Degenerate Laplacians on Strictly Pseudo-convex Domains
| journal=Duke Mathematical Journal
| volume=57
| year=1988
| issue=3
| pages=697–720
| doi=10.1215/S0012-7094-88-05731-6 | doi-access=))</ref> This operator in CR geometry is called the '''CR Paneitz operator'''.  The operator defined by Graham and Lee though defined on all odd dimensional CR manifolds, is not known to be conformally covariant in real dimension 5 and higher. The conformal covariance of this operator has been established in real dimension 3 by [[Kengo Hirachi]]. It is always a non-negative operator in real dimension 5 and higher.
Here unlike changing the metric by a conformal factor as in the Riemannian case discussed above, one changes the contact form on the CR 3 manifold by a conformal factor. Non-negativity of the CR Paneitz operator in dimension 3 is a CR invariant condition as proved below. This follows by the conformal covariant properties of the CR Paneitz operator first observed by [[Kengo Hirachi]].<ref>((cite book
| last1=Hirachi | first1=Kengo
| chapter=Scalar Pseudo-hermitian Invariants and the Szegő kernel on three dimensional CR manifolds
| year=1993
| title=Complex Geometry (Osaka 1990)
| series=Lecture Notes in Pure and Applied Mathematics
| volume=143
| pages=67–76
| publisher=Marcel Dekker
| location=New York))</ref> Furthermore, the CR Paneitz operator plays an important role in obtaining the sharp eigenvalue lower bound for Kohn's Laplacian. This is a result of [[Sagun Chanillo]], Hung-Lin Chiu and [[Paul C. Yang]].<ref>((cite journal
| last1=Chanillo | first1=Sagun
| last2=Chiu | first2=Hung-Lin
| last3=Yang | first3=Paul C.
| s2cid=304301
| title=Embeddability for 3-dimensional CR manifolds and CR Yamabe Invariants
| journal=Duke Mathematical Journal
| volume=161
| issue=15
| year=2012
| pages=2909–2921
| doi=10.1215/00127094-1902154
| arxiv=1007.5020))</ref> This sharp eigenvalue lower bound is the exact analog in CR Geometry of the famous [[André Lichnerowicz]] lower bound for the [[Laplace–Beltrami operator]] on compact Riemannian manifolds. It allows one to globally embed, compact, strictly pseudoconvex, abstract CR manifolds into <math> C^n</math>. More precisely, the conditions in [3] to embed a CR manifold into <math> C^n</math> are phrased CR invariantly and non-perturbatively.  There is also a partial converse of the above result where the authors, J. S. Case, S. Chanillo, P. Yang, obtain conditions that guarantee when embedded, compact CR manifolds have non-negative CR Paneitz operators.<ref>((cite journal
| last1=Case | first1=Jeffrey S.
| last2=Chanillo | first2=Sagun
| last3=Yang | first3=Paul C.
| title= The CR Paneitz operator and the Stability of CR Pluriharmonic functions
| journal=[[Advances in Mathematics]]
| volume=287
| year=2016
| pages=109–122
| doi=10.1016/j.aim.2015.10.002 | doi-access=free
| arxiv=1502.01994))</ref> The formal definition of the CR Paneitz operator <math>P_4</math> on CR manifolds of real dimension three is as follows( the subscript <math>4</math> is to remind the reader that this is a fourth order operator)

::<math> P_4\phi=\frac{1}{8}((\Box_b\overline{\Box_b}+\overline{\Box_b}\Box_b)\phi+8Im(A^{11}\phi_1)_1)</math>
<math>\Box_b</math> denotes the Kohn Laplacian which plays a fundamental role in CR Geometry and several complex variables 
and was introduced by [[Joseph J. Kohn]]. One may consult [[CR manifold#The tangential Cauchy–Riemann complex (Kohn Laplacian, Kohn–Rossi complex)|The tangential Cauchy–Riemann complex (Kohn Laplacian, Kohn–Rossi complex)]] for the definition of the Kohn Laplacian.
Further, <math> A^{11}</math> denotes the Webster-Tanaka Torsion tensor and <math> \phi_1</math> the covariant derivative of the function <math>\phi</math> with respect to the Webster-Tanaka connection. Accounts of the Webster-Tanaka, connection, Torsion and curvature tensor may be found in articles by John M. Lee and Sidney M. Webster.<ref name=Lee1988>((cite journal
| last1=Lee | first1=John M.
| title=Pseudo-Einstein Structures on CR manifolds
| journal=American Journal of Mathematics
| volume=110
| issue=1
| year=1988
| pages=157–178
| doi=10.2307/2374543 | jstor=2374543))</ref><ref>((cite journal
| last1=Webster | first1=Sidney M.
| title=Pseudo-hermitian Structures on a Real Hypersurface
| journal=Journal of Differential Geometry
| volume=13
| year=1978
| pages=25–41
| doi=10.4310/jdg/1214434345 | doi-access=free))</ref> There is yet another way to view the CR Paneitz operator in dimension 3. John M. Lee constructed a third order operator <math>P_3</math> which has the property that the kernel of <math> P_3</math> consists of exactly the CR pluriharmonic functions (real parts of CR holomorphic functions).<ref name=Lee1988/> The Paneitz operator displayed above is exactly the divergence of this third order operator <math> P_3</math>. The third order operator <math> P_3</math> is defined as follows:

:<math> P_3\phi=(((\phi_{\bar 1))^{\bar 1))_{1}+\sqrt{-1}A_{11}\phi^1)\theta^1</math>
Here <math> A_{11}</math> is the Webster-Tanaka torsion tensor. The derivatives are taken using the Webster-Tanaka connection and <math>\theta^1</math> is the dual 1-form to the CR-holomorphic tangent vector that defines the CR structure on the compact manifold. Thus <math>P_3</math> sends functions to <math>(1,0)</math> forms. The divergence of such an operator thus will take functions to functions. The third order operator constructed by J. Lee only characterizes CR pluriharmonic functions on CR manifolds of real dimension three.

Hirachi's covariant transformation formula for <math> P_4</math> on three dimensional CR manifolds is as follows. Let the CR manifold be
<math>(M,\theta,J)</math>, where <math>\theta</math> is the contact form and <math> J</math> the CR structure on the kernel of <math>\theta</math> that is on the contact planes. Let us transform the background contact form <math>\theta</math> by a conformal transformation to <math>\tilde{\theta}=e^{2f}\theta</math>. Note this new contact form obtained by a conformal change of the old contact form or background contact form, has not changed the kernel of <math>\theta</math>. That is <math>\tilde{\theta}</math> and <math>\theta</math> have the same kernel, i.e. the contact planes have remained unchanged. The CR structure <math> J</math> has been kept unchanged. The CR Paneitz operator <math> \tilde{P}_4</math> for the new contact form <math> \tilde{\theta}</math> is now seen to be related to the CR Paneitz operator for the contact form <math>\theta</math> by the formula of Hirachi:
:<math>\tilde{P}_4=e^{-4f}P_4</math> 
Next note the volume forms on the manifold <math> M</math> satisfy

:<math> d\tilde{V}=\tilde{\theta}\wedge d\tilde{\theta}= e^{4f}\theta\wedge d\theta=e^{4f}dV</math>
Using the transformation formula of Hirachi, it follows that,
:<math> \int_M \tilde{P}_4\phi\phi d\tilde{V}=\int_M P_4\phi\phi dV</math>
Thus we easily conclude that:
:<math>\int_M P_4\phi\phi dV</math>
is a CR invariant. That is the integral displayed above has the same value for different contact forms describing the '''same''' CR structure <math> J</math>.

The operator <math> P_4</math> is a real self-adjoint operator. On CR manifolds like <math>S^3</math> where the Webster-Tanaka torsion tensor is zero, it is seen from the formula displayed above that only the leading terms involving the Kohn Laplacian survives. Next from the tensor commutation formulae given in [5], one can easily check that the operators <math> \Box_b, \overline{\Box_b}</math> commute when the Webster-Tanaka torsion tensor <math> A_{11}</math> vanishes. More precisely one has
:<math> [\Box_b,\overline{\Box_b}]=4\sqrt{-1}Im Q</math>
where
:<math> Q\phi=2\sqrt{-1}(A_{11}\phi_1)_1</math>

Thus <math> \Box_b,\overline{\Box_b}</math> are simultaneously diagonalizable under the zero torsion assumption. Next note that <math>\Box_b</math> and <math> \overline{\Box_b}</math> have the same sequence of eigenvalues that are also perforce real. Thus we conclude from the formula for <math>P_4</math> that CR structures having zero torsion have CR Paneitz operators that are non-negative. The article [4] among other things shows that real ellipsoids in <math>C^2</math> carry a CR structure inherited from the complex structure of <math>C^2</math> whose CR Paneitz operator is non-negative. This CR structure on ellipsoids has non-vanishing Webster-Tanaka torsion. 
Thus [4] provides the first examples of CR manifolds where the CR Paneitz operator is non-negative and the Torsion tensor too does not vanish. Since we have observed above that the CR Paneitz is the divergence of an operator whose kernel is the pluriharmonic functions, it also follows that the kernel of the CR Paneitz operator contains all CR Pluriharmonic functions. So the kernel of the CR Paneitz operator in sharp contrast to the Riemannian case, has an infinite dimensional kernel. Results on when the kernel is exactly the pluriharmonic functions, the nature and role of the supplementary space in the kernel etc.,  may be found in the article cited as [4] below.

One of the principal applications of the CR Paneitz operator and the results in [3] are to the CR analog of the Positive Mass theorem due to Jih-Hsin Cheng, [[Andrea Malchiodi]] and [[Paul C. Yang]].<ref>((cite journal
| last1=Cheng | first1=Jih-Hsin
| last2=Malchiodi | first2=Andrea
| last3=Yang | first3=Paul
| title=A Positive Mass theorem in three dimensional Cauchy-Riemann Geometry
| journal=[[Advances in Mathematics]]
| volume=308
| date=21 February 2017
| pages=276-347
| arxiv=1312.7764 | bibcode=2013arXiv1312.7764C
| doi=10.1016/j.aim.2016.12.012 | doi-access=free))</ref> This allows one to obtain results on the CR [[Yamabe problem]].

More facts related to the role of the CR Paneitz operator in CR geometry can be obtained from the article [[CR manifold]].

==See also==
*[[Calabi conjecture]]
*[[Monge-Ampere equation]]s
*[[Positive energy theorem|Positive mass conjecture]]
*[[Yamabe conjecture]]

==References==
((Reflist))
*((Citation | last1=Branson | first1=Thomas P. | last2=Ørsted | first2=Bent | title=Explicit functional determinants in four dimensions | doi=10.2307/2048601 | mr=1050018 | year=1991 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=113 | issue=3 | pages=669–682| jstor=2048601 | doi-access=free )).
*((citation|first=Sun-Yung A.|last=Chang|authorlink=Alice Chang|contribution=A fourth order differential operator in conformal geometry|title=Harmonic Analysis and Partial Differential Equations; Essays in honor of Alberto P. Calderón|editor1=M. Christ |editor2=C. Kenig |editor3=C. Sadorsky|series=Chicago Lectures in Mathematics|year=1999|pages=127–150)).
* ((Citation | last1=Paneitz | first1=Stephen M. | s2cid=115155901 | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary) | doi=10.3842/SIGMA.2008.036 | mr=2393291 | year=2008 | journal= Symmetry, Integrability and Geometry: Methods and Applications| issn=1815-0659 | volume=4 | pages=Paper 036, 3| arxiv=0803.4331 | bibcode=2008SIGMA...4..036P )).

[[Category:Conformal geometry]]
[[Category:Differential geometry]]
[[Category:Differential operators]]

Pages transcluded onto the current version of this page (help):

Paneitz operator (edit)
Template:Citation (view source) (protected)
Template:Cite book (view source) (protected)
Template:Cite journal (view source) (protected)
Template:Harv (view source) (template editor protected)
Template:Harvard citation (view source) (template editor protected)
Template:Harvard citation no brackets (view source) (template editor protected)
Template:Harvnb (view source) (template editor protected)
Template:Main other (view source) (protected)
Template:Mapsto (edit)
Template:Nowrap (view source) (protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Module:Arguments (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:Footnotes (view source) (template editor protected)
Module:Footnotes/anchor id list (view source) (template editor protected)
Module:Footnotes/anchor id list/data (view source) (template editor protected)
Module:Footnotes/whitelist (view source) (template editor protected)

Return to Paneitz operator.

Retrieved from "https://en.wikipedia.org/wiki/Paneitz_operator"