## 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

#### 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
First available in Project Euclid: 1 February 2017

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

#### 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

#### References

• Aharonov, D. and Shapiro, H. S., Domains on which analytic functions satisfy quadrature identities, J. Anal. Math. 30 (1976), 39–73.
• Bell, S., Solving the Dirichlet problem in the plane by means of the Cauchy integral, Indiana Univ. Math. J. 39 (1990), 1355–1371.
• Bell, S., The Szegő projection and the classical objects of potential theory in the plane, Duke Math. J. 64 (1991), 1–26.
• Bell, S., The Cauchy Transform, Potential Theory, and Conformal Mapping, CRC Press, Boca Raton, FL, 1992.
• Bell, S., Unique continuation theorems for the $\bar{\partial}$-operator and applications, J. Geom. Anal. 3 (1993), 195–224.
• Bell, S., Complexity of the classical kernel functions of potential theory, Indiana Univ. Math. J. 44 (1995), 1337–1369.
• Bell, S., Ahlfors maps, the double of a domain, and complexity in potential theory and conformal mapping, J. Anal. Math. 78 (1999), 329–344.
• Bell, S., The fundamental role of the Szegő kernel in potential theory and complex analysis, J. Reine Angew. Math. 525 (2000), 1–16.
• Bell, S., Complexity in complex analysis, Adv. Math. 172 (2002), 15–52.
• Bell, S., The Bergman kernel and quadrature domains in the plane, in Quadrature Domains and their Applications (Santa Barbara, CA, 2003 ), Operator Theory: Advances and Applications 156, pp. 35–52, Birkhäuser, Basel, 2005.
• Bell, S., Quadrature domains and kernel function zipping, Ark. Mat. 43 (2005), 271–287.
• Bell, S., Bergman coordinates, Studia Math. 176 (2006), 69–83.
• Bell, S., The Green’s function and the Ahlfors map, Indiana Univ. Math. J. 57 (2008), 3049–3063.
• Bell, S., Density of quadrature domains in one and several complex variables, Complex Var. Elliptic Equ. 54 (2009), 165–171.
• Bell, S., Ebenfelt, P., Khavinson, D. and Shapiro, H. S., On the classical Dirichlet problem in the plane with rational data, J. Anal. Math. 100 (2006), 157–190.
• Bell, S., Gustafsson, B. and Sylvan, Z., Szegő coordinates, quadrature domains, and double quadrature domains, Comput. Methods Funct. Theory 11 (2011), 25–44.
• Crowdy, D., Quadrature domains and fluid dynamics, in Quadrature Domains and their Applications (Santa Barbara, CA, 2003 ), Operator Theory: Advances and Applications 156, pp. 113–129, Birkhäuser, Basel, 2005.
• Ebenfelt, P., Singularities encountered by the analytic continuation of solutions to Dirichlet’s problem, Complex Variables Theory Appl. 20 (1992), 75–91.
• Ebenfelt, P., Gustafsson, B., Khavinson, D. and Putinar, M. (eds.), Quadrature Domains and Their Applications, Operator Theory: Advances and Applications 156, Birkhäuser, Basel, 2005.
• Farkas, H. and Kra, I., Riemann Surfaces, Springer, New York, 1980.
• Gustafsson, B., Quadrature domains and the Schottky double, Acta Appl. Math. 1 (1983), 209–240.
• Gustafsson, B., Applications of half-order differentials on Riemann surfaces to quadrature identities for arc-length, J. Anal. Math. 49 (1987), 54–89.
• Gustafsson, B. and Shapiro, H. S., What is a quadrature domain? in Quadrature Domains and their Applications (Santa Barbara, CA, 2003 ), Operator Theory: Advances and Applications 156, pp. 1–25, Birkhäuser, Basel, 2005.
• Kerzman, N. and Stein, E. M., The Cauchy kernel, the Szegő kernel, and the Riemann mapping function, Math. Ann. 236 (1978), 85–93.
• Kerzman, N. and Trummer, M., Numerical conformal mapping via the Szegő kernel, J. Comput. Appl. Math. 14 (1986), 111–123.
• Putinar, M. and Shapiro, H. S., The Friedrichs operator of a planar domain II, in Recent Advances in Operator Theory and Related Topics (Szeged, 1999 ), Operator Theory: Advances and Applications 127, pp. 519–551, Birkhäuser, Basel, 2001.
• Shapiro, H. S., The Schwarz Function and its Generalization to Higher Dimensions, Univ. of Arkansas Lecture Notes in the Mathematical Sciences, Wiley, New York, 1992.
• Shapiro, H. S. and Ullemar, C., Conformal mappings satisfying certain extremal properties and associated quadrature identities, Preprint, TRITA-MAT-1986-6, Royal Inst. of Technology, Stockholm, 1981.