[Paper Review] Electrical Networks and Stephenson's Conjecture

Sa’ar Hersonsky|arXiv (Cornell University)|Jun 27, 2016

Analytic and geometric function theory44 references3 citations

TL;DR

This paper constructs a sequence of electrical networks on triangulated planar annuli that converge uniformly to a conformal map onto a concentric Euclidean annulus, thereby affirming Ken Stephenson's 1990s conjecture that the Riemann mapping can be approximated via discrete electrical networks on increasingly fine triangulations.

ABSTRACT

In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.

Motivation & Objective

• To investigate whether the Riemann mapping of a planar annulus can be approximated by discrete electrical networks on triangulations.
• To establish uniform convergence of a sequence of discrete harmonic maps to a conformal homeomorphism onto a Euclidean annulus.
• To resolve Ken Stephenson's conjecture from the 1990s regarding the approximation of conformal mappings using electrical networks.
• To provide a constructive, geometrically grounded method for approximating conformal maps using discrete potential theory on planar domains.

Proposed method

• Construct a sequence of triangulations that exhaust a planar annulus, ensuring increasingly fine mesh refinement.
• Define electrical networks on each triangulation using conductances derived from the geometry of the triangulation.
• Construct discrete harmonic maps from the triangulated domain to the unit disk, using Dirichlet boundary conditions.
• Apply a normalization procedure to align the discrete maps with the conformal structure of the annulus.
• Prove uniform convergence on compact subsets of the domain to a conformal homeomorphism onto a concentric Euclidean annulus.
• Use properties of discrete harmonic functions and convergence theorems in discrete complex analysis to establish the limit behavior.

Experimental results

Research questions

• RQ1 Can the Riemann mapping of a planar annulus be approximated by a sequence of electrical networks on increasingly fine triangulations?
• RQ2 Does the sequence of discrete harmonic maps on triangulated annuli converge uniformly to a conformal homeomorphism onto a Euclidean annulus?
• RQ3 Is the limit map conformal and onto a concentric Euclidean annulus, as predicted by Stephenson's conjecture?
• RQ4 What geometric and analytic conditions ensure convergence of electrical network maps to the continuous Riemann map?

Key findings

• The sequence of discrete harmonic maps constructed from electrical networks on triangulated annuli converges uniformly on compact subsets to a conformal homeomorphism.
• The limit map is a conformal homeomorphism onto a Euclidean annulus bounded by two concentric circles.
• The convergence is robust under mesh refinement, confirming the stability of the approximation scheme.
• The result affirms Ken Stephenson's conjecture that the Riemann mapping can be realized as the limit of electrical network maps on planar domains.