Ngô Quốc Anh

December 21, 2014

Conformal Changes of the Green function for the conformal Laplacian

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 11:00

Long time ago, I talked about conformal changes for various geometric quantities on a given Riemannian manifold (M,g) of dimension n, 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

\displaystyle L_g(u) = - \frac{n-1}{4(n-2)}\Delta_g u + \text{Scal}_g u

where \text{Scal}_g is the scalar curvature of the metric g. The operator L_g is conformal in the sense that any change of metric \widehat g = \varphi ^\frac{4}{n-2}g would give the following magic identity

\displaystyle L_{\widehat g} (u) =\varphi^{-\frac{n+2}{n-2}} L_g (\varphi u).

Associated to the conformal Laplacian operator L_g is the Green function, if exists, \mathbb G_{L,g}. Mathematically, the Green function \mathbb G_{L,g} is defined to be a continuous function

\mathbb G_{L,g} : M \times M \backslash \{(x,x) : x \in M\} \to \mathbb R

such that for any x\in M, \mathbb G_{L,g} (x, \cdot) \in L^1(M) and for any u \in C^2(M) and any x \in M, we have the following representation

\displaystyle u(x) = \int_M \mathbb G_{L,g}(x,y) L_g(u)(y) dv_g (y).

In this note, I will show that under the conformal change \widehat g = \varphi^\frac{4}{n-2} g, the Green function \mathbb G_{L,g} follows the following magic rule

\displaystyle \mathbb G_{L,\widehat g}(x,y) = \varphi (x)^{-1} \varphi (y)^{-1} \mathbb G_{L,g} (x,y).

Obviously, as long as \mathbb G_{L,g} is the Green function for L_g, the new function \mathbb G_{L,\widehat g} given by the above formula is well-defined and continuous. Since \varphi \in C^\infty (M), there also holds \mathbb G_{L,\widehat g} (x, \cdot) \in L^1(M). The only thing we need to check is the representation

\displaystyle u(x) = \int_M \mathbb G_{L,\widehat g}(x,y) L_{\widehat g}(u)(y) dv_{\widehat g} (y).

for any u \in C^2(M) and any x \in M. To see this, by well-known facts, we obtain:

  • First the conformal change for L_{\widehat g}(u) as shown above

    \displaystyle L_g (u) =\varphi^{\frac{n+2}{n-2}} L_{\widehat g} (\varphi^{-1} u).

  • Then we have the conformal change for the volume

    \displaystyle dv_g =\varphi^{-\frac{2n}{n-2}} dv_{\widehat g}.

Hence, using the formula for \mathbb G_{L,\widehat g} given above, we obtain

 \begin{array}{lcl} u(x) \varphi(x) &=& \displaystyle \int_M \mathbb G_{L,g}(x,y) L_g(\varphi u)(y) dv_g (y) \\&=& \displaystyle \int_M \mathbb G_{L,g}(x,y) \varphi (y) ^{\frac{n+2}{n-2}} L_{\widehat g}( u)(y) \varphi (y) ^{-\frac{2n}{n-2}} dv_{\widehat g} (y) \\&=& \displaystyle \int_M \mathbb G_{L,g}(x,y) \varphi (y) ^{-1} L_{\widehat g}( u)(y) dv_{\widehat g} (y).\end{array}

Hence, we have just proved that

 \begin{array}{lcl} u(x) &=& \displaystyle \int_M \mathbb G_{L,g}(x,y) \varphi(x) ^{-1} \varphi (y)^{-1} L_{\widehat g} (u)(y) dv_{\widehat g} (y) \\&=& \displaystyle \int_M \mathbb G_{L,\widehat g}(x,y) L_{\widehat g}(u)(y) dv_{\widehat g} (y).\end{array}

Thus, there must hold

\displaystyle \mathbb G_{L,\widehat g}(x,y) = \varphi (x)^{-1} \varphi (y)^{-1} \mathbb G_{L,g} (x,y)

as claimed.

It is interesting to note that this type of transformation also holds for the 4th order Paneitz operator P associated with Q-curvature for any dimension 4 \ne n \geqslant 3. More precisely, there holds

\displaystyle \mathbb G_{P,\widehat g}(x,y) = \varphi (x)^{-1} \varphi (y)^{-1} \mathbb G_{P,g} (x,y)

where \widehat g =\varphi^\frac{4}{n-4} g. For a precise formulas and definition for 4th order Paneitz operator P 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 L 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 P comes from a joint paper by Hang and Yang recently appeared in IMRN.

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