On the Convergence of Analytic Boundary Approximations to Schwarz–Christoffel Mappings

Uniform versus Pointwise Behavior under Contour Degeneration

Erik Goughnour

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Abstract

We investigate boundary degeneration in conformal maps arising from smooth planar domains that converge to polygonal limits. Focusing on a family of Lamé domains that approach a rectangle, we show that although the associated Riemann maps converge uniformly on compact subsets of the unit disk, boundary correspondence becomes increasingly non-uniform near corners.

This phenomenon is quantified using harmonic measure. We demonstrate that boundary neighborhoods of physical thickness O(1/n) near the rectangle corners correspond to preimage arcs of length O(1/n²) on the unit circle, reflecting the quadratic scaling associated with right-angle wedges.

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Key Results

1. Uniform interior convergence: Riemann maps for Lamé curves converge uniformly on compact subsets of the unit disk as the exponent n → ∞

2. Non-uniform boundary behavior: Boundary correspondence fails to extend uniformly to the boundary near corners

3. Quadratic scaling law: Corner harmonic measure scales as O(1/n²), consistent with right-angle wedge theory

4. Numerical validation: Boundary integral methods confirm the scaling law without explicit Riemann map computation

Reproducibility

All numerical results can be reproduced using the companion software packages:

Packages (PyPI)

Package	Description
superellipse	Lamé curve geometry and discretization
panel-bie	Boundary integral equation solvers
harmonic-measure	Harmonic measure computation

Interactive Notebooks

Binder	Purpose
	Numerical verification
	Visual illustrations

Source

• Paper source + CI - LaTeX and build automation
• Numerics glue - Experiment configuration

Quick Start

# Install packages
pip install harmonic-measure superellipse panel-bie

# Run scaling diagnostic
harmonic-measure scaling --ns 8 12 16 24 32 48

# Generate paper figures
pip install matplotlib
python -c "
from harmonic_measure import corner_scaling_diagnostic
ns, arcs, slope = corner_scaling_diagnostic([8, 12, 16, 24, 32, 48])
print(f'Fitted slope: {slope:.4f} (expected: -2.0)')
"

Citation

@article{Goughnour2026,
  author  = {Erik Goughnour},
  title   = {On the Convergence of Analytic Boundary Approximations
             to Schwarz--Christoffel Mappings},
  year    = {2026},
  note    = {Preprint}
}

License

Paper: All rights reserved Code: MIT License