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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
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:<math>(-4|V|^2+nJ^2+2\Delta J)/4\,</math>
which in four dimensions 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}}
 
==See also==
*[[Calabi conjecture]]
*[[Positive energy theorem|Positive mass conjecture]]
*[[Yamabe conjecture]]
*[[Monge-Ampere equation]]s
 
==References==
*{{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}}.
*{{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|editors=M. Christ, C. Kenig and C. Sadorsky|series=Chicago Lectures in Mathematics|year=1999|pages=127–150}}.
* {{Citation | last1=Paneitz | first1=Stephen M. | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary) | doi=10.3842/SIGMA.2008.036 | mr=2393291 | year=2008 | journal=SIGMA. Symmetry, Integrability and Geometry. Methods and Applications | issn=1815-0659 | volume=4 | pages=Paper 036, 3}}.
 
[[Category:Conformal geometry]]
[[Category:Differential geometry]]
[[Category:Differential operators]]
[[Category:Conformal supergravity]]
 
{{mathanalysis-stub}}

Latest revision as of 15:31, 24 September 2014


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