We consider the equilibrium problem for an external background potential in weighted potential theory, and show that for a large class of background potentials there is a complementarity relationship between the measure solving the weighted equilibrium problem-the weighted equilibrium measure-and a certain partial balayage measure.

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, and it is closely related to (construction of) quadrature domains for subharmonic functions, growth processes such as Laplacian growth and to weighted equilibrium distributions.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.

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