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Quasiregular values and Rickman's Picard theorem

Ilmari Kangasniemi, Jani Onninen

TL;DR

This work proves a broad Picard-type theorem for maps with quasiregular values, showing that for each $K\, ext{and}\,n$ there exists a finite bound $q=q(n,K)$ such that no nonconstant $W^{1,n}_{loc}(\,\b R^n,\bb R^n)$ map can have $(K,\Sigma)$-quasiregular values at more than $q$ boundary points when $\Sigma$ lies in $L^{1+\varepsilon}$ and $L^{1-\varepsilon}$ on $\mathbb{R}^n$. The authors develop a robust framework based on a logarithmic- singularity approach, Caccioppoli-type inequalities, and a pseudosupremum notion to control bounded components of level sets, then adapt Bonk–Poggi-Corradini’s strategy to higher dimensions without relying on conformance structures. The paper also establishes spherical-QR-value variants and delivers sharp planar cases, along with counterexamples illustrating the sharpness of the integrability conditions. Overall, the results extend Rickman’s Picard theorem to a wide class of Sobolev maps by linking global boundary behavior to quantitative integrability, with significant implications for geometric function theory and nonlinear analysis.

Abstract

We prove a far-reaching generalization of Rickman's Picard theorem for a surprisingly large class of mappings, based on the recently developed theory of quasiregular values. Our results are new even in the planar case.

Quasiregular values and Rickman's Picard theorem

TL;DR

This work proves a broad Picard-type theorem for maps with quasiregular values, showing that for each there exists a finite bound such that no nonconstant map can have -quasiregular values at more than boundary points when lies in and on . The authors develop a robust framework based on a logarithmic- singularity approach, Caccioppoli-type inequalities, and a pseudosupremum notion to control bounded components of level sets, then adapt Bonk–Poggi-Corradini’s strategy to higher dimensions without relying on conformance structures. The paper also establishes spherical-QR-value variants and delivers sharp planar cases, along with counterexamples illustrating the sharpness of the integrability conditions. Overall, the results extend Rickman’s Picard theorem to a wide class of Sobolev maps by linking global boundary behavior to quantitative integrability, with significant implications for geometric function theory and nonlinear analysis.

Abstract

We prove a far-reaching generalization of Rickman's Picard theorem for a surprisingly large class of mappings, based on the recently developed theory of quasiregular values. Our results are new even in the planar case.
Paper Structure (24 sections, 30 theorems, 155 equations, 2 figures)

This paper contains 24 sections, 30 theorems, 155 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background on quasiregular maps and the Picard theorem
  3. The theory of quasiregular values
  4. Other versions of Theorem \ref{['thm:Picard_for_QR_values']}
  5. The planar case
  6. Main ideas of the proof
  7. The structure of this paper
  8. Acknowledgments
  9. Preliminaries on Sobolev differential forms
  10. Quasiregular values and maps between spheres
  11. Maps into $\mathbb{S}^n$
  12. Logarithmic singularity and Caccioppoli inequalities
  13. The logarithmic singularity function
  14. Quasiregular values and superlevel sets
  15. Measure estimates and Caccioppoli-type inequalities
  16. ...and 9 more sections

Key Result

Theorem 1.1

For every $K \geq 1$ and $n \geq 2$, there exists a positive integer $q = q(n, K) \in \mathbb{Z}_{> 0}$ such that if $f \colon \mathbb{R}^n \to \mathbb{R}^n$ is $K$-quasiregular and $\mathbb{R}^n \setminus f(\mathbb{R}^n)$ contains $q$ different points, then $f$ is constant.

Figures (2)

  • Figure 1: The chain of balls $B_i$ from $0$ to $x_0$.
  • Figure 2: Rough illustration of the map $f$ of Example \ref{['ex:plus_epsilon_counterexample']} in the case $n = 2$. The map $f$ takes each of the infinitely many shaded annuli on the domain side to one of the open-ended stalks on the target side, stopping partway through. In the lighter shaded part of $\mathbb{R}^2$ the map $f$ is locally constant, with the unbounded component mapped to the center of the stalks. The tips of the stalks are quasiregular values of $f$ and are contained in $\partial f(\mathbb{R}^2)$.

Theorems & Definitions (51)

  • Theorem 1.1: Rickman's Picard Theorem
  • Theorem 1.2
  • Theorem 1.3: Kangasniemi-Onninen_Heterogeneous and Kangasniemi-Onninen_1ptReshetnyak
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 3.1
  • proof
  • ...and 41 more