Partial balayage on Riemannian manifolds

Abstract

A general theory of partial balayage on Riemannian manifolds is developed, with emphasis on compact manifolds. Partial balayage is an operation of sweeping measures, or charge distributions, to a prescribed density, which then is attained on an a priori unknown set. It is closely related to (construction of) quadrature domains for subharmonic functions, to weighted equilibrium distributions and, as for methods used, to the obstacle problem.Several examples are given in the paper, as well as some specific results. For instance, it is proved that, in two dimensions, harmonic and geodesic balls are the same if and only if the Gaussian curvature of the manifold is constant.