Supports of Weighted Equilibrium Measures: Complete Characterization

TL;DR

This paper provides a complete characterization of the supports of weighted equilibrium measures in complex spaces, extending previous results to higher dimensions and offering a new proof even in one dimension.

Contribution

It offers a complete characterization of supports of weighted equilibrium measures in several complex variables, generalizing prior one-dimensional results with a novel proof.

Findings

A compact set is the support of a weighted equilibrium measure iff it is not pluripolar at each point.

The characterization extends Saff and Totik's result to higher dimensions.

Provides a new proof technique applicable in one dimension.

Abstract

In this paper, we prove that a compact set $K\subset \mathbb{C}^n$ is the support of a weighted equilibrium measure if and only it is not pluripolar at each of its points extending a result of Saff and Totik to higher dimensions. Thus, we characterize the supports weighted equilibrium measures completely. Our proof is a new proof even in one dimension.