In the mathematical field of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the GJMS operators are a family of

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

s, that are defined on a

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

. In an appropriate sense, they depend only on the

conformal structure In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...

of the manifold. The GJMS operators generalize the

Paneitz operator 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 who ...

and the conformal Laplacian. The initials GJMS are for its discoverers Graham, Jenne, Mason & Sparling (1992). Properly, the GJMS operator on a conformal manifold of dimension ''n'' is a

conformally invariant In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...

operator between the

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

of conformal densities of weight for ''k'' a positive integer :

L_k : E -n/2 \to E k-n/2

The operators have 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 In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham for which a conformal manifold of dimension ''n'' is realized (''ambiently'') as the boundary of a certain Poincaré manifold, or alte ...

of Charles Fefferman and Robin Graham. A conformal density defines, in a natural way, a function on the

null cone In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms and ...

in the ambient space. The GJMS operator is defined by taking density ''ƒ'' of the appropriate weight and extending it arbitrarily to a function ''F'' off the null cone so that it still retains the same homogeneity. The function Δ^''k''''F'', where Δ is the ambient Laplace–Beltrami operator, is then homogeneous of degree , and its restriction to the null cone does not depend on how the original function ''ƒ'' 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 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''_''n''/2 that take a true function on the manifold and produce a multiple of the

volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...

References

* {{Citation , last1=Graham , first1=C. Robin, author1-link= C. Robin Graham , 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 , series=Second Series , issn=0024-6107 , volume=46 , issue=3 , pages=557–565. Conformal geometry Differential operators