Variation of quasiconformal mappings on lines

TL;DR

This paper investigates the regularity of solutions to the reduced Beltrami equation, showing that their restrictions to lines have finite p-variation for all p>1, which advances understanding of their geometric properties.

Contribution

It provides new regularity results for reduced Beltrami solutions, focusing on their variation along lines, which was not previously established.

Findings

Restrictions to lines have finite p-variation for all p>1

Improved regularity results compared to standard Beltrami equation

Not necessarily finite p-variation for p=1

Abstract

We obtain improved regularity of homeomorphic solutions of the reduced Beltrami equation, as compared to the standard Beltrami equation. Such an improvement is not possible in terms of Holder or Sobolev regularity; instead, our results concern the generalized variation of restrictions to lines. Specifically, we prove that the restriction to any line segment has finite p-variation for all p>1 but not necessarily for p=1.