# Conformally invariant operators

### From Mwiki

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

## 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