Potential theory on Sierpinski carpets with applications to uniformization

TL;DR

This paper develops potential theory on Sierpinski carpets to solve uniformization problems, establishing harmonic functions and maps that transform carpets into canonical forms while preserving geometric properties.

Contribution

It introduces a discrete Sobolev space framework for harmonic functions on carpets and proves uniformization results with quasisymmetric maps under specific geometric conditions.

Findings

Existence and uniqueness of harmonic functions on carpets.

Uniformization of carpets to square carpets via quasisymmetric maps.

Construction of harmonic conjugates using new potential theory methods.

Abstract

This research is motivated by the study of the geometry of fractal sets and is focused on uniformization problems: transformation of sets to canonical sets, using maps that preserve the geometry in some sense. More specifically, the main question addressed is the uniformization of planar Sierpinski carpets by square Sierpinski carpets, using methods of potential theory on carpets. We first develop a potential theory and study harmonic functions on planar Sierpinski carpets. We introduce a discrete notion of Sobolev spaces on Sierpinski carpets and use this to define harmonic functions. Our approach differs from the classical approach of potential theory in metric spaces because it takes the ambient space that contains the carpet into account. We prove basic properties such as the existence and uniqueness of the solution to the Dirichlet problem, Liouville's theorem, Harnack's inequality, strong maximum principle, and equicontinuity of harmonic functions. Then we utilize this notion of harmonic functions to prove a uniformization result for Sierpinski carpets. Namely, it is proved that every planar Sierpinski carpet whose peripheral disks are uniformly fat, uniform quasiballs can be mapped to a square Sierpinski carpet with a map that preserves carpet modulus. If the assumptions on the peripheral circles are strengthened to uniformly relatively separated, uniform quasicircles, then the map is a quasisymmetry. The real part of the uniformizing map is the solution of a certain Dirichlet-type problem. Then a harmonic conjugate of that map is constructed using the methods developed by Rajala arXiv:1412.3348.