The Effect of Planar Harmonic Mappings on the Lebesgue Measure of Sets

TL;DR

This paper establishes sharp area distortion inequalities for harmonic mappings of the unit disk, demonstrating how such maps affect the Lebesgue measure of sets and identifying conditions for equality.

Contribution

It provides a complete solution to a longstanding problem by deriving optimal bounds on area distortion under harmonic mappings, including rigidity and extremal cases.

Findings

Global area contraction for disks and star-shaped sets

Sharp bounds with equality only for conformal automorphisms

Examples illustrating the sharpness of the estimates

Abstract

We investigate the effect of planar univalent harmonic mappings on the Lebesgue measure of measurable sets in the complex plane. Motivated by Problem 3.25 of Koh and Kovalev (HQM2010), we establish sharp quantitative area distortion inequalities for disks and for arbitrary measurable sets under sense-preserving harmonic self-maps of the unit disk. Using the area formula and the canonical decomposition of harmonic mappings, we derive bounds in terms of the Jacobian and the dilatation, and we identify rigidity phenomena characterizing equality. In particular, we prove global area contraction for disks, star-shaped sets, and sufficiently small sets, and we refine the results using Hardy space methods to obtain sharp bounds with equality only for conformal automorphisms. Extremal affine and non-affine examples illustrate the sharpness of our estimates. Our results provide a complete, rigorous, and strengthened solution to Problem 3.25 and highlight several natural conjectures on global area contraction, extremal distortion, and rigidity for harmonic mappings.