Electrostatics on closed two-dimensional Riemannian manifolds

Abstract This article gives a self-contained geometric formulation of electrostatics on closed Riemannian surfaces. On a compact surface without boundary, Gauss’s law imposes a global compatibility condition: the total charge must vanish. Hence an isolated nonzero point charge is incompatible with the closed geometry unless it is neutralized by a compensating background. The appropriate replacement for the Euclidean Coulomb kernel is the normalized Green function of the Laplace–Beltrami operator, obtained by removing the constant zero mode. Within this framework, we organize the analytic structures needed for closed-surface electrostatics. These include solvability of the Poisson equation for mean-zero sources, the Hodge decomposition of smooth 1 -forms, the resulting classification of smooth electrostatic fields, spectral and heat-kernel representations of normalized Green functions, and the Robin function as the diagonal regular part of the Green kernel. In two dimensions, the Robin function gives the renormalized self-energy of a neutralized point charge. We also derive the conformal transformation laws for the normalized Green function and the Robin function, together with the corresponding conformal-response formula for compact-surface Coulomb Hamiltonians. The general framework is illustrated on several representative closed surfaces. For the round sphere and for flat tori, we give explicit formulas for the normalized Green kernel; on flat tori, the area-normalized Fourier representation is compared with the Jacobi theta-function representation. For compact Riemann surfaces of higher genus, we recall the construction of the canonical Arakelov Green function from the prime form, including the Abel–Jacobi quadratic correction. As an application, we formulate the two-dimensional one-component plasma on a compact surface in terms of the normalized Green kernel and the Robin self-energy. We prove a leading N 2 -scale mean-field equilibrium theorem for this model: the normalized Riemannian area measure is the unique leading-order equilibrium measure, and asymptotically minimizing configurations equidistribute with respect to it. The paper thus provides a unified and explicit reference framework for Coulomb-type interactions on closed curved surfaces.