Green's function on sphere Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\Delta_{S^n}$ can be written as $$\Delta_{S^n} = \partial^2_r + (n - 1)\text{cot }r\partial_r + \frac{1}{\text{sin}^2\text{ }r}\Delta_\theta.$$
I was wondering, is there a radial Green's function for $\Delta_{S^n}$, and how to find it out. Or how about the Laplacian in the upper hemisphere $\{x = (x_1,...,x_{n + 1}) \in \mathbb{R}^{n + 1} : x_{n + 1} \geq 0\}$, and imposing the Dirichlet boundary conditions on the boundary $\{ x \in S^n : x_{n + 1} = 0\}$? Any help in appreciated.
 A: A good reference for Green's function for the Laplacian on compact manifolds (with or without boundary) is the book "Some nonlinear problems in Riemannian geometry" by Aubin, page 108ff.
The Green's function for the Laplacian on a compact manifold $M$ without boundary is unique up to an additive constant. If we fix $y\in M$, then all Green's functions $G_y$ at $y$ satisfy
$$\Delta G_y=\delta_y-\frac{1}{\mathrm{vol}(M)}$$
in the sense of distributions. The term $-\frac{1}{\mathrm{vol}(M)}$ appears since one has to project to the orthogonal complement of the kernel of the Laplacian. In particular, $G_y$ are smooth on $M\setminus\{y\}$.
In the example of the standard sphere with geodesic polar coordinates at a point $y$ with radial variable $r$ all Green's functions at $y$ are functions of $r$ only (they must be invariant under rotations fixing $y$ since the Laplacian is invariant under such rotations). In order to get an explicit formula for the Green's function in your coordinate system on the sphere you therefore have to solve
$$f''(r)+(n-1)\cot(r)f'(r)=-\frac{1}{\mathrm{vol}(S^n)}$$
for $r>0$ and look for a solution which is singular at $r=0$ and has a limit as $r\to\pi$.
Similar things hold for the Green's function on the upper hemisphere with Dirichlet boundary condition (see e.g. Aubin's book).
A: Pushing further another guest answer, you can obtain semi-explicit formulas such as in Section 2.5 of "Discrete and continuous green energy on compact manifolds" by 
C. Beltran, N. Corral and J. G. Criado Del Rey. 
In particular, for the two-sphere, $G(x,y)=\frac1{2\pi}\log\|x-y\|$ (up to an additive constant), where $\|\cdot\|$ is the Euclidean norm of $\mathbb R^3$. Higher dimensional spheres have more involved Green functions unfortunately...
