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

Hunduma Legesse Geleta

TL;DR

The paper addresses how planar sense-preserving univalent harmonic self-maps distort the planar Lebesgue measure, focusing on the unit disk. It uses the area formula together with the canonical decomposition $f=h+ar{g}$ and the dilatation $ obreak\omega=g'/h'$ to derive sharp area-distortion inequalities for disks and general measurable sets. Key results include global area contraction for disks, star-shaped sets, and sufficiently small sets, with Hardy-space arguments yielding equality only for conformal automorphisms. These findings provide a rigorous resolution toward Koh–Kovalev Problem 3.25 in the harmonic setting and motivate conjectures on global area contraction and rigidity, supported by explicit affine and non-affine extremals.

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.

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

TL;DR

The paper addresses how planar sense-preserving univalent harmonic self-maps distort the planar Lebesgue measure, focusing on the unit disk. It uses the area formula together with the canonical decomposition and the dilatation to derive sharp area-distortion inequalities for disks and general measurable sets. Key results include global area contraction for disks, star-shaped sets, and sufficiently small sets, with Hardy-space arguments yielding equality only for conformal automorphisms. These findings provide a rigorous resolution toward Koh–Kovalev Problem 3.25 in the harmonic setting and motivate conjectures on global area contraction and rigidity, supported by explicit affine and non-affine extremals.

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.
Paper Structure (8 sections, 10 theorems, 65 equations)

This paper contains 8 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lebesgue measure and the area formula
  4. Complex-valued harmonic mappings
  5. Auxiliary Results
  6. Main results
  7. Sharpness and examples
  8. Concluding remarks

Key Result

Theorem 2.1

Let $\Omega \subset \mathbb{R}^2$ be open and let $f:\Omega \to \mathbb{R}^2$ be an injective $C^1$-mapping. Then for every measurable set $E\subset\Omega$,

Theorems & Definitions (23)

  • Theorem 2.1: Area formula
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 3.1: Exact area formula
  • proof
  • Proposition 3.1: Quantitative area distortion
  • proof
  • Theorem 4.1: Integral or Avereged Area Schwarz-Pick type Inequality
  • ...and 13 more