On the logarithmic equilibrium measure on curves

TL;DR

This paper proves that the logarithmic equilibrium measure on certain smooth curves in Euclidean space is absolutely continuous with respect to arc length, providing new insights especially in dimensions three and higher where such properties were previously unknown.

Contribution

The paper establishes the absolute continuity of the equilibrium measure on smooth curves in higher dimensions, extending known results from two-dimensional cases.

Findings

The equilibrium measure is absolutely continuous on $C^{1,eta}$-graphs.

New results on the support's positive dimension in higher dimensions.

Extension of classical harmonic measure results to higher dimensions.

Abstract

Let $\mu$ be the logarithmic equilibrium measure on a compact set $\gamma \subset \mathbb{R}^{d}$. We prove that $\mu$ is absolutely continuous with respect to the length measure on the part of $\gamma$ which can be locally expressed as the graph of a $C^{1,\alpha}$-function $\mathbb{R} \to \mathbb{R}^{d - 1}$, $\alpha > 0$. For $d = 2$, at least in the case where $\gamma$ is a compact $C^{1,\alpha}$-graph, our result can also be deduced from the classical fact that $\mu$ coincides with the harmonic measure of $\Omega =\mathbb{R}^{2} \, \setminus \, \gamma$ with pole at $\infty$. For $d \geq 3$, however, our result is new even for $C^{\infty}$-graphs. In fact, up to now it was not even known if the support of $\mu$ has positive dimension.