On the logarithmic equilibrium measure on curves

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 $\mathrm{\Omega }={\mathbb{R}}^2\setminus \gamma$ with pole at $\mathrm{\infty }$. For $d\ge 3$, however, our result is new even for $C^{\mathrm{\infty }}$-graphs. In fact, up to now it was not even known if the support of $\mu$ has positive dimension.