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WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (8853)

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In the [[mathematics|mathematical]] field of [[differential geometry]], the '''GJMS operators''' are a family of [[differential operator]]s, that are defined on a [[Riemannian manifold]]. In an appropriate sense, they depend only on the [[conformal structure]] of the manifold. The GJMS operators generalize the [[Paneitz operator]] and the [[Laplace operators in differential geometry#Conformal Laplacian|conformal Laplacian]]. The initials GJMS are for its discoverers [[#CITEREFGrahamJenneMasonSparling1992|Graham, Jenne, Mason & Sparling (1992)]].

Properly, the GJMS operator on a conformal manifold of dimension ''n'' is a conformally invariant operator between the [[line bundle]] of [[density on a manifold|conformal densities]] of weight {{nowrap|''k'' − ''n''/2}} for ''k'' a positive integer
:<math>L_k : E[k-n/2] \to E[-k-n/2].</math>
The operators have [[symbol of a differential operator|leading symbol]] given by a power of the [[Laplace–Beltrami operator]], and have lower order correction terms that ensure conformal invariance.

The original construction of the GJMS operators used the [[ambient construction]] of [[Charles Fefferman]] and [[Robin Graham (mathematician)|Robin Graham]]. A conformal density defines, in a natural way, a function on the [[null cone]] in the ambient space. The GJMS operator is defined by taking density ''&fnof;'' of the appropriate weight {{nowrap|''k'' − ''n''/2}} and extending it arbitrarily to a function ''F'' off the null cone so that it still retains the same homogeneity. The function Δ<sup>''k''</sup>''F'', where Δ is the ambient [[Laplace–Beltrami operator]], is then homogeneous of degree {{nowrap|−''k'' − ''n''/2}}, and its restriction to the null cone does not depend on how the original function ''&fnof;'' was extended to begin with, and so is independent of choices. The GJMS operator also represents the obstruction term to a formal asymptotic solution of the Cauchy problem for extending a weight {{nowrap|''k'' − ''n''/2}} function off the null cone in the ambient space to a harmonic function in the full ambient space.

The most important GJMS operators are the ''critical'' GJMS operators. In even dimension ''n'', these are the operators ''L''<sub>''n''/2</sup> that take a true function on the manifold and produce a multiple of the [[volume form]].

==References==
* {{Citation | last1=Graham | first1=C. Robin | last2=Jenne | first2=Ralph | last3=Mason | first3=Lionel J. | last4=Sparling | first4=George A. J. | title=Conformally invariant powers of the Laplacian. I. Existence | doi=10.1112/jlms/s2-46.3.557 | mr=1190438 | year=1992 | journal=Journal of the London Mathematical Society. Second Series | issn=0024-6107 | volume=46 | issue=3 | pages=557–565}}.

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

PPA (complexity) - Revision history

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (8853)