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The method of images and the Green's functions for spherical domains
Eugene Gutkin and Paul K. Newton
Motivated by problems in electrostatics and vortex dynamics, we develop two general methods for constructing the Green's function for simply connected domains on the surface of the unit sphere. We prove a Riemann mapping theorem showing that such domains can be conformally mapped to the upper hemisphere. We then categorize all domains on the sphere for which the Green's function can be constructed by an extension of the classical method of images. We illustrate our methods by several examples, such as the upper-hemisphere, geodesic triangles, and latitudinal rectangles. We describe the point vortex motion in these domains, which is governed by a Hamiltonian determined by the Dirichlet Green's function.