The geometric meaning of the complex dilatation

TL;DR

This paper explores the geometric interpretation of complex dilatation by linking it to hyperbolic geometry and conformal structures, providing a new perspective on distortion measurement in complex analysis.

Contribution

It introduces a geometric approach to complex dilatation using hyperbolic models of conformal structures, connecting it with classical and modern geometric concepts.

Findings

Reveals the hyperbolic geometric nature of complex dilatation.

Connects conformal structure distortion with hyperbolic plane models.

Relates the approach to Arnold's proof of the hyperbolic altitudes theorem.

Abstract

The paper is devoted to an approach to the notion of the complex dilatation based on the following observations. (1) A natural measure of the distortion of the conformal structure by a real linear automorphism of the complex plane is the pull-back of the standard conformal structure of complex plane. (2) The set of all conformal structures on the complex plane carries a canonical structure of a model of the hyperbolic plane and can be naturally identified with the unit disc together with its structure of the Klein model of the hyperbolic plane. (3) The standard isomorphism of the Klein model with the Poincar\'e unit disc model transforms this measure of distortion into the classical complex dilatation. In version 2 this approach is related to Arnold's proof of the hyperbolic altitudes theorem.