Stream lines, quasilines and holomorphic motions

Gaven Martin

doi:10.1186/s40627-015-0003-5

Abstract

We give a new application of the theory of holomorphic motions to study the distortion of level lines of harmonic functions and stream lines of ideal planar fluid flow. In various settings, we show they are in fact quasilines—the quasiconformal images of the real line. These methods also provide quite explicit global estimates on the geometry of these curves.

Highlights

The theory of holomorphic motions, introduced by Mané-Sad-Sullivan [1] and advanced by Slodkowski [2], has had a significant impact on the theory of quasiconformal mappings
We consider the geometry of stream lines for ideal fluid flow in a domain and establish bounds on their distortion in terms of a reference line
These bounds come from an analysis of the geometry of the level lines of the hyperbolic metric and seem to be of independent interest

Summary

Introduction

The theory of holomorphic motions, introduced by Mané-Sad-Sullivan [1] and advanced by Slodkowski [2], has had a significant impact on the theory of quasiconformal mappings. We consider the geometry of stream lines for ideal fluid flow in a domain and establish bounds on their distortion in terms of a reference line. Our first theorem states that the level lines are the images of the line γ under a self mapping of the domain with bounded distortion.

Results

Conclusion