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Monotone Sobolev extensions in metric surfaces and applications to uniformization

Damaris Meier, Noa Vikman, Stefan Wenger

TL;DR

The paper proves a monotone Sobolev extension theorem for maps from the boundary of a Jordan domain in a metric surface to its closure, under locally finite ${\mathcal H}^2$-measure and, initially, the local-geodesic assumption. This extension is then leveraged to obtain a uniformization result for compact metric surfaces by energy-minimizing monotone Sobolev maps, yielding a weakly $K$-quasiconformal parametrization with $K=4/\pi$ and infinitesimal isotropy. The approach combines energy-minimizing Sobolev maps, collar constructions, and decomposition techniques to handle higher topology, with NR22 enabling removal of the local-geodesic hypothesis. These results advance non-smooth uniformization and have implications for geometric function theory in metric settings, connecting monotone extensions to optimal energy realizers on surfaces.

Abstract

We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps.

Monotone Sobolev extensions in metric surfaces and applications to uniformization

TL;DR

The paper proves a monotone Sobolev extension theorem for maps from the boundary of a Jordan domain in a metric surface to its closure, under locally finite -measure and, initially, the local-geodesic assumption. This extension is then leveraged to obtain a uniformization result for compact metric surfaces by energy-minimizing monotone Sobolev maps, yielding a weakly -quasiconformal parametrization with and infinitesimal isotropy. The approach combines energy-minimizing Sobolev maps, collar constructions, and decomposition techniques to handle higher topology, with NR22 enabling removal of the local-geodesic hypothesis. These results advance non-smooth uniformization and have implications for geometric function theory in metric settings, connecting monotone extensions to optimal energy realizers on surfaces.

Abstract

We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps.

Paper Structure

This paper contains 11 sections, 7 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Open questions
  4. Preliminaries
  5. Basic notions
  6. Conformal modulus and weak quasiconformality
  7. Metric space valued Sobolev maps
  8. Metric differentiability and infinitesimal isotropy
  9. Monotone Sobolev extension theorems for metric surfaces
  10. Monotone Sobolev extensions to surfaces of higher topology
  11. Uniformization of metric surfaces via energy minimization

Key Result

Theorem 1.1

Let $X$ be a metric surface with locally finite Hausdorff 2-measure. Let $\Omega\subset X$ be a Jordan domain with rectifiable boundary. Then every monotone map in $W^{1,2}(S^1,\partial \Omega)$ extends to a monotone map in $W^{1,2}(\overline D,\overline \Omega)$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 7 more