Arkiv för Matematik

  • Ark. Mat.
  • Volume 51, Number 2 (2013), 223-249.

An improved Riemann mapping theorem and complexity in potential theory

Steven R. Bell

Full-text: Open access

Abstract

We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another “double quadrature domain,” i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann mapping theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain is close to the original domain. We explore some of the parallels between this new theorem and the classic theorem, and some of the similarities between the unit disc and the double quadrature domains that arise here. The new results shed light on the complexity of many of the objects of potential theory in multiply connected domains.

Note

Research supported by the NSF Analysis and Cyber-enabled Discovery and Innovation programs, grant DMS 1001701.

Article information

Source
Ark. Mat., Volume 51, Number 2 (2013), 223-249.

Dates
Received: 7 October 2011
First available in Project Euclid: 1 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907216

Digital Object Identifier
doi:10.1007/s11512-012-0168-6

Mathematical Reviews number (MathSciNet)
MR3090195

Zentralblatt MATH identifier
1291.30047

Rights
2012 © Institut Mittag-Leffler

Citation

Bell, Steven R. An improved Riemann mapping theorem and complexity in potential theory. Ark. Mat. 51 (2013), no. 2, 223--249. doi:10.1007/s11512-012-0168-6. https://projecteuclid.org/euclid.afm/1485907216


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