Green's Function for Laplacian on $S^1 \times S^2$

Question

As indicated by the title, I am looking to find the Green's function for the Laplacian on $S^1 \times S^2$. Is such a function known? If not, does anyone have an approach to constructing such a function? My first idea isto combine the Green's function on $S^2$ in a nice way with some $1$-periodic function on $\mathbb{R}$, but I haven't had much luck.

Answer 1

Here's a way to get a representation as series. It works in a more general situation. Let $M$ and $N$ be smooth closed Riemannian manifolds. Denote by $\{\varphi_i\}_{i=1}^\infty$, $\{\psi_j\}_{i=j}^\infty$ the eigenfunctions with eigenvalues $\lambda_i$ and $\mu_j$ for the Laplace operators on $M$ and $N$ respectively. Let $L_2$ norms of eigenfunctions be equal to one so delta-functions can be expanded as $$ \delta_M(x-x')=\sum_{i=1}^\infty \varphi_i(x) \varphi_i(x'), $$ $$ \delta_N(y-y')=\sum_{i=1}^\infty \psi_i(y) \psi_i(y'). $$ Then Green's function for $M$ is given by series $$ G_M(x,x')=\sum_{i=1}^\infty \frac{\varphi_i(x) \varphi_i(x')}{\lambda_i} $$ and analogously for $N$. The Green's function for $M\times N$ is $$ G_{M\times N}(x,y,x',y')= \sum_{i,j=1}^\infty \frac{\varphi_i(x)\varphi_i(x') \psi_j(y)\psi_j(y')}{\lambda_i+\mu_j} $$ since $$ \Delta_{x,y} G_{M\times N}(x,y,x',y')= \sum_{i,j=1}^\infty \varphi_i(x)\varphi_i(x') \psi_j(y)\psi_j(y')= $$ $$ \delta_M(x-x')\delta_N(y-y')=\delta_{M\times N}(x-x',y-y'). $$ So one has to combine eigenfunctions rather than Green's functions themselves.

For your case $\varphi_i$ are cosines (up to a constant) and $\psi_j$ are spherical harmonics. A possibility to get a closed form for this series seems rather thin to me.