# Conformally invariant operators

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- | On a general compact manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by | + | On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by |

\[ | \[ | ||

A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M), | A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M), | ||

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This is a fourth order operator with leading term $(-\Delta_g)^2$. | This is a fourth order operator with leading term $(-\Delta_g)^2$. | ||

- | * GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ | + | * GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. The formula are not explicit except they are on the standard sphere. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even. |

*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$. | *Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$. |

## Revision as of 02:55, 24 September 2013

On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by \[ A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M), \] where $a, b$ are constant.

Examples of conformally invariant operators include:

- The conformal Laplacian:

\[ L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g, \] where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.

- The Paneitz operator
^{[1]}^{[2]}:

\[ P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q, \] where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor, \[ Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR \] and \[ a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}. \] This is a fourth order operator with leading term $(-\Delta_g)^2$.

- GJMS operators
^{[3]}: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. The formula are not explicit except they are on the standard sphere. A nonexistence result can be found in^{[4]}for $k>\frac n2$ and $n\ge 4$ even.

- Scattering operators
^{[5]}, or the conformally invariant fractional powers of the Laplacian^{[6]}: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors^{[6]}reconciled the way of defining $P_\sigma$ in^{[5]}and the localization method of Caffarelli-Silvestre^{[7]}for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.

Special cases:

- On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.

- On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator
^{[8]}of explicit formula:

\[ P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2}, \] where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$

* is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,

* has the eigenfunctions of spherical harmonics, and

* is the inverse of a spherical Riesz potential.

## References

- ↑ Paneitz, S (1983), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds",
*preprint* - ↑ Paneitz, S (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)",
*SIGMA Symmetry Integrability Geom. Methods Appl.*(4), http://dx.doi.org/10.3842/SIGMA.2008.036 - ↑ Graham, C Robin; Jenne, Ralph; Mason, Lionel J; Sparling, George AJ (1992), "Conformally invariant powers of the Laplacian, I: Existence",
*Journal of the London Mathematical Society***2**: 557--565 - ↑ Gover, A; Hirachi, Kengo (2004), "Conformally invariant powers of the Laplacian—a complete nonexistence theorem",
*Journal of the American Mathematical Society***17**: 389--405 - ↑
^{5.0}^{5.1}Graham, C Robin; Zworski, Maciej (2003), "Scattering matrix in conformal geometry",*Inventiones mathematicae***152**: 89--118 - ↑
^{6.0}^{6.1}Chang, Sun-Yung Alice; González, Maria del Mar (2011), "Fractional Laplacian in conformal geometry",*Advances in Mathematics***226**: 1410--1432 - ↑ Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian",
*Communications in Partial Differential Equations***32**: 1245--1260 - ↑ Branson, Thomas P (1987), "Group representations arising from Lorentz conformal geometry",
*Journal of functional analysis***74**: 199--291