Long time ago, I talked about conformal changes for various geometric quantities on a given Riemannian manifold of dimension , see this post.
Frequently used in conformal geometry in general, or when solving the prescribed scalar curvature equation in particular, is the conformal Laplacian, defined as follows
where is the scalar curvature of the metric . The operator is conformal in the sense that any change of metric would give the following magic identity
Associated to the conformal Laplacian operator is the Green function, if exists, . Mathematically, the Green function is defined to be a continuous function
such that for any , and for any and any , we have the following representation
In this note, I will show that under the conformal change , the Green function follows the following magic rule
Obviously, as long as is the Green function for , the new function given by the above formula is well-defined and continuous. Since , there also holds . The only thing we need to check is the representation
for any and any . To see this, by well-known facts, we obtain:
- First the conformal change for as shown above
- Then we have the conformal change for the volume
Hence, using the formula for given above, we obtain
Hence, we have just proved that
Thus, there must hold
It is interesting to note that this type of transformation also holds for the 4th order Paneitz operator associated with Q-curvature for any dimension . More precisely, there holds
where . For a precise formulas and definition for 4th order Paneitz operator as well as its Q-curvature, I refer to this topic.
Finally, the idea of using conformal change for the Green function for the conformal Laplacian goes back to the work by Lee and Parker in their famous paper about the Yamabe problem published in 1987. The similar identity for the 4th order Paneitz operator comes from a joint paper by Hang and Yang recently appeared in IMRN.