GJMS operator
Encyclopedia
In the 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 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 operator between the line bundle
of conformal densities
of weight for k a positive integer
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
of Charles Fefferman
and 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 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 ΔkF, 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 Ln/2 that take a true function on the manifold and produce a multiple of the volume form
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of differential geometry, the GJMS operators are a family of differential operator
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, accepting a function and returning another .This article considers only linear operators,...
s, that are defined on a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
. In an appropriate sense, they depend only on the conformal structure of the manifold. The GJMS operators generalize the Paneitz operator
Paneitz operator
In the 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...
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 operator between the line bundle
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 organising these...
of conformal densities
Density on a manifold
In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold which can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain trivial line bundle, called the density bundle...
of weight for k a positive integer
The operators have leading symbol
Symbol of a differential operator
In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this...
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
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 as the boundary of a certain Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian...
of Charles Fefferman
Charles Fefferman
Charles Louis Fefferman is an American mathematician at Princeton University. His primary field of research is mathematical analysis....
and 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 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 ΔkF, 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 Ln/2 that take a true function on the manifold and produce a multiple of the volume form
Volume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...
.