Partial Balayage on Riemannian Manifolds

TL;DR

This paper develops a comprehensive theory of partial balayage on Riemannian manifolds, connecting it to quadrature domains, growth processes, and equilibrium measures, with specific results in two-dimensional cases.

Contribution

It introduces a general framework for partial balayage on Riemannian manifolds, including new results relating harmonic and geodesic balls to Gaussian curvature.

Findings

Harmonic and geodesic balls coincide iff Gaussian curvature is constant in 2D.

Provides examples illustrating the theory.

Establishes connections between balayage, quadrature domains, and growth processes.

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, 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.