# Green's function for the Yamabe problem

I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8.

Theorem 2.8 (Existence of the Green Function).

Suppose $M$ is a compact Riemannian manifold of dimension $n\ge 3$, and $h$ is a strictly positive smooth function on $M$. For each $P\in M$, there exists a unique smooth function $\Gamma_P$ on $M\setminus \{P\}$, called the Green function for $\Delta + h$ at $P$, such that $(\Delta + h)\Gamma_P = \delta_P$ in the distribution sense, where $\delta_P$ is the Dirac measure on $P$.

Does anyone know where I might find out about general existence results for Green's functions (in particular this one)? Thanks!

• Chapter 4 of Aubin's Nonlinear Analysis on Manifolds constructs the Green's function for the Laplacian on a Riemannian manifold (without the $h$). – Henry T. Horton Feb 17 '13 at 19:57
• @HenryT.Horton: Thanks! I'll check it out. – Sam Feb 17 '13 at 23:13

2. As Tom Parker and I commented after the proof of Theorem 6.5 in our paper, that proof can be adapted to prove the existence of the Green function, if you've already proved that $\Delta+h$ is an isomorphism between appropriate Sobolev spaces: First you use the procedure in the proof of Theorem 6.5 to construct an approximation to the Green function of the form $\Gamma_0 = r^{2-n}(1+\bar\psi)$; then let $f_0 = (\Delta+h)\Gamma_0$ and note that $f_0$ is in some Sobolev space on which $(\Delta+h)^{-1}$ exists. The Green function is $\Gamma_P = \Gamma_0 - (\Delta + h)^{-1}(f_0)$.
3. Or you can use the Schwarz kernel theorem to observe that $(\Delta+h)^{-1}$ has an integral kernel $K$, which is a distribution on $M\times M$. Elliptic regularity shows that it is smooth on the complement of the diagonal. The Green function is $\Gamma_P(Q) = K(P,Q)$.