Equilibrium measures on curves

In this talk I will discuss one of the most classical objects of study in potential theory: equilibrium measures for the logarithmic energy. Given a compact set /E/, the (logarithmic) equilibrium measure on /E/ is the unique (if it exists) minimizer of the logarithmic energy among all measures supported on /E/. In the case of planar sets, the equilibrium measure coincides with the harmonic measure, and is rather well-understood. However, almost Euclidean spaces. In a recent paper with Tuomas Orponen we show that these measures are absolutely continuous with respect to the arc-length measure on C^1, ^α curves in arbitrary dimension. I will describe some ideas of our proof, which involves fractional Laplacians and some singular integral operators.