The Cauchy Integral Formula, Quadrature Domains, and Riemann Mapping Theorems
- PDF / 435,478 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 113 Downloads / 191 Views
The Cauchy Integral Formula, Quadrature Domains, and Riemann Mapping Theorems Steven R. Bell1
Received: 6 December 2017 / Revised: 21 March 2018 / Accepted: 22 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract It is well known that a domain in the plane is a quadrature domain with respect to area measure if and only if the function z extends meromorphically to the double, and it is a quadrature domain with respect to boundary arc length measure if and only if the complex unit tangent vector function T (z) extends meromorphically to the double. By applying the Cauchy integral formula to z¯ , we will shed light on the density of area quadrature domains among smooth domains with real analytic boundary. By extending z¯ and T (z) and applying the Cauchy integral formula to the Szeg˝o kernel, we will obtain conformal mappings to nearby arc length quadrature domains and even domains that are like the unit disc in that they are simultaneously area and arc length quadrature domains. These “double quadrature domains” can be thought of as analogs of the unit disc in the multiply connected setting and the mappings so obtained as generalized Riemann mappings. The main theorems of this paper are not new, but the methods used in their proofs are new and more constructive than previous methods. The new computational methods give rise to numerical methods for computing generalized Riemann maps to nearby quadrature domains. Keywords Bergman kernel · Szeg˝o kernel · Double quadrature domains Mathematics Subject Classification 30C20 · 30C40 · 31A35
Communicated by Vladimir Andrievskii.
B 1
Steven R. Bell [email protected] Mathematics Department, Purdue University, West Lafayette, IN 47907, USA
123
S. R. Bell
1 Introduction The Riemann mapping theorem states that any simply connected domain in the plane that is not equal to the whole plane is biholomorphic to the unit disc, which is a wellknown quadrature domain with respect to both area measure and boundary arc length measure, i.e., a double quadrature domain. This classic theorem was generalized in [6] to state that any finitely connected domain in the plane, = C, is biholomorphic to a double quadrature domain. Furthermore, if the domain is a bounded domain bounded by finitely many non-intersecting Jordan curves, the mapping function can be taken to be arbitrarily close to the identity in the uniform topology up to the boundary. If the boundary curves are C ∞ smooth, the mapping can be taken to be arbitrarily close to the identity in C ∞ (). If the boundary curves are further assumed to be smooth real analytic curves, the mapping extends holomorphically past the boundary and can be taken to be close to the identity on a neighborhood of the closure of the domain. The proofs of these theorems given in [6] are rather long and use ideas from Riemann surface theory as well as regularity properties of the Bergman and Szeg˝o projections. The double quadrature domains that arise in the proofs are one-point double quadrature domains like the un
Data Loading...
