Analytic capacity and holomorphic motions

TL;DR

This paper investigates how the analytic capacity of a compact set changes under holomorphic deformations, demonstrating that its logarithm varies harmonically and establishing the sharpness of the conditions involved.

Contribution

It proves that the logarithm of the analytic capacity varies harmonically under holomorphic conformal deformations, and shows that these conditions are essentially optimal.

Findings

Logarithm of analytic capacity varies harmonically under deformations

Harmonic variation holds under specific holomorphic conformal maps

Conditions for harmonic variation are shown to be sharp

Abstract

We study the behavior of the analytic capacity of a compact set under deformations obtained by families of conformal maps depending holomorphically on the complex parameter. We show that, under those deformations, the logarithm of the analytic capacity varies harmonically. We also show that the hypotheses in this result cannot be substantially weakened.