A Local to Global Principle for the Complexity of Riemann Mappings (Extended Abstract)
Robert Rettinger
TL;DR
This paper establishes that the complexity of computing Riemann mappings can be derived from local boundary computations, providing new upper bounds for Schwarz-Christoffel and piecewise analytic boundary mappings.
Contribution
It introduces a local-to-global principle that bounds Riemann mapping complexity based on local boundary computations, with formal proofs for specific classes of mappings.
Findings
Bounded Riemann mapping complexity using local boundary data
Provided first formal upper bounds for Schwarz-Christoffel mappings
Extended results to mappings of domains with piecewise analytic boundaries
Abstract
We show that the computational complexity of Riemann mappings can be bounded by the complexity needed to compute conformal mappings locally at boundary points. As a consequence we get first formally proven upper bounds for Schwarz-Christoffel mappings and, more generally, Riemann mappings of domains with piecewise analytic boundaries.
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Taxonomy
TopicsAnalytic and geometric function theory · Geometric Analysis and Curvature Flows · Holomorphic and Operator Theory