Holomorphic motions, dimension, area and quasiconformal mappings

TL;DR

This paper investigates how the dimensions and area of sets change under holomorphic motions, offering a unified approach to key theorems in quasiconformal mapping theory, including area and dimension distortion results.

Contribution

It introduces a new method to analyze the variation of dimensions and area under holomorphic motions, unifying several classical theorems in quasiconformal mapping theory.

Findings

Dimensions and area vary holomorphically under holomorphic motions.

Unified approach recovers Astala's area and dimension distortion theorems.

Provides new insights into the behavior of quasicircles and related sets.

Abstract

We describe the variation of the Minkowski, packing and Hausdorff dimensions of a set moving under a holomorphic motion, as well as the variation of its area. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles.