The Laplace-Beltrami operator is invertible on the space of 1-forms on $ S^2$ (since $ S^2$ has zero first betti number). Therefore it has an inverse, the Green’s function. Now let the radius $ r$ of $ S^2$ go to infinity, or more precisely, we can identify $ S^2\setminus\{\textrm{north pole}\}$ with $ \mathbb{R}^2$ and send the radius $ r$ round metric on the former to the flat metric on the latter. Do the sequence of Green’s operators $ G_r$ (restricted to $ S^2\setminus\{\textrm{north pole}\}$ ) converge to the usual Green’s operator $ G$ on $ \mathbb{R}^2$ (which would be $ G(x,y) = -\frac{1}{2\pi}\log(|x-y|)$ times the identity operator on $ 1$ -forms)?