# Conformally invariant operators

(Difference between revisions)
 Revision as of 19:15, 23 September 2013 (view source)Tianling (Talk | contribs) (Created page with "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 $g_w=e^{2w}g$...")← Older edit Revision as of 19:48, 23 September 2013 (view source)Tianling (Talk | contribs) Newer edit → Line 4: Line 4: \] \] where $a, b$ are constant. 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 . + + + + + + + == References == + {{reflist|refs= + {{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds |year=1983 | journal=preprint}} + + {{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary) | url=http://dx.doi.org/10.3842/SIGMA.2008.036 | doi:10.3842/SIGMA.2008.036 | year=2008 | journal=SIGMA Symmetry Integrability Geom. Methods Appl. | issue=4 | Paper=036}} + + }}

## Revision as of 19:48, 23 September 2013

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

1. Paneitz, S (1983), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds", preprint
2. Paneitz, S (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)", SIGMA Symmetry Integrability Geom. Methods Appl. (4)