Stream lines, quasilines and holomorphic motions

TL;DR

This paper applies holomorphic motion theory to analyze the distortion of level lines of harmonic functions and stream lines in fluid flow, demonstrating they are quasilines and providing explicit geometric estimates.

Contribution

It introduces a novel application of holomorphic motions to characterize level and stream lines as quasilines with explicit geometric bounds.

Findings

Level lines of harmonic functions are quasilines.

Stream lines of ideal planar fluid flow are quasilines.

Explicit global estimates on the geometry of these curves are provided.

Abstract

We give a new application of the theory of holomorphic motions to the 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.