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The Heinonen-Semmes problems after thirty years

Guy C. David, Pekka Pankka, Jeremy T. Tyson

TL;DR

This survey reexamines the 33 Heinonen–Semmes problems thirty years after their inception, charting which questions have been resolved and which remain open across geometric measure theory, metric geometry, and quasiconformal geometry. It highlights core achievements—such as Bonk–Kleiner’s quasisymmetric uniformization of S$^2$ under LLC and Ahlfors regularity, Laakso-type non-embeddability constructions, and Romney’s negative results on absolute continuity—while detailing persistent barriers in higher dimensions, Heisenberg geometry, and topological uniformization. The work situates these results within the broader evolution of analysis in metric spaces, Cheeger differentiation, uniform rectifiability, and the interaction with geometric group theory, suggesting that the field has matured into a broadly interconnected discipline with many open avenues. Overall, the article emphasizes both the depth of progress since 1997 and the enduring challenge of formulating intrinsic, dimensionally robust criteria for uniformization and parametrization in non-smooth settings.

Abstract

We survey the current status of the questions posed by Juha Heinonen and Stephen Semmes in `Thirty-three yes or no questions about mappings, measures, and metrics' (Conformal Geometry and Dynamics, 1997).

The Heinonen-Semmes problems after thirty years

TL;DR

This survey reexamines the 33 Heinonen–Semmes problems thirty years after their inception, charting which questions have been resolved and which remain open across geometric measure theory, metric geometry, and quasiconformal geometry. It highlights core achievements—such as Bonk–Kleiner’s quasisymmetric uniformization of S under LLC and Ahlfors regularity, Laakso-type non-embeddability constructions, and Romney’s negative results on absolute continuity—while detailing persistent barriers in higher dimensions, Heisenberg geometry, and topological uniformization. The work situates these results within the broader evolution of analysis in metric spaces, Cheeger differentiation, uniform rectifiability, and the interaction with geometric group theory, suggesting that the field has matured into a broadly interconnected discipline with many open avenues. Overall, the article emphasizes both the depth of progress since 1997 and the enduring challenge of formulating intrinsic, dimensionally robust criteria for uniformization and parametrization in non-smooth settings.

Abstract

We survey the current status of the questions posed by Juha Heinonen and Stephen Semmes in `Thirty-three yes or no questions about mappings, measures, and metrics' (Conformal Geometry and Dynamics, 1997).
Paper Structure (26 sections, 16 theorems, 22 equations, 1 figure)

This paper contains 26 sections, 16 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quasiconformal uniformization and parametrization
  3. Definitions and notation
  4. Commentary
  5. Bi-Lipschitz parametrizations and embeddings
  6. Definitions and notation
  7. Commentary
  8. Geometric topology
  9. Definitions and notation
  10. Commentary
  11. Absolute continuity of quasisymmetric and quasiconformal mappings
  12. Definitions and notation
  13. Commentary
  14. Analysis on metric measure spaces
  15. Definitions and notation
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $G$ be a metric graph of the above type, and let $E \subset \mathbb{R}^2$ be the corresponding fractal realization. For suitable powers $s>0$, the weight $w(x) = \mathop{\mathrm{dist}}\nolimits(x,E)^s$ defines a strong $A_\infty$ weight whose deformed metric geometry $(\mathbb{R}^2,d_w')$ does n

Figures (1)

  • Figure 1: An approximation to the slit carpet $M$.

Theorems & Definitions (17)

  • Theorem 2.1: Laakso
  • Theorem 2.2: Laakso
  • Theorem 2.3: Bishop
  • Conjecture 2.4: Cannon
  • Theorem 2.5: Bonk--Kleiner
  • Theorem 3.1: Reifenberg, 1960
  • Theorem 3.2: Schul
  • Theorem 4.1: Heinonen--Wu, Pankka--Wu
  • Theorem 5.1: David--Toro
  • Theorem 5.2: Romney
  • ...and 7 more