Arkiv för Matematik
- Ark. Mat.
- Volume 51, Number 2 (2013), 223-249.
An improved Riemann mapping theorem and complexity in potential theory
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.
Research supported by the NSF Analysis and Cyber-enabled Discovery and Innovation programs, grant DMS 1001701.
Ark. Mat., Volume 51, Number 2 (2013), 223-249.
Received: 7 October 2011
First available in Project Euclid: 1 February 2017
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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