522 research outputs found
The Green's function and the Ahlfors map
The classical Green's function associated to a simply connected domain in the
complex plane is easily expressed in terms of a Riemann mapping function. The
purpose of this paper is to express the Green's function of a finitely
connected domain in the plane in terms of a single Ahlfors mapping of the
domain, which is a proper holomorphic mapping of the domain onto the unit disc
that is the analogue of the Riemann map in the multiply connected setting.Comment: 14 page
Quadrature domains and kernel function zipping
It is proved that quadrature domains are ubiquitous in a very strong sense in
the realm of smoothly bounded multiply connected domains in the plane. In fact,
they are so dense that one might as well assume that any given smooth domain
one is dealing with is a quadrature domain, and this allows access to a host of
strong conditions on the classical kernel functions associated to the domain.
Following this string of ideas leads to the discovery that the Bergman kernel
can be zipped down to a strikingly small data set. It is also proved that the
kernel functions associated to a quadrature domain must be algebraic.Comment: 13 pages, to appear in Arkiv for matemati
The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc
Given a bounded n-connected domain in the plane bounded by non-intersecting
Jordan curves, and given one point on each boundary curve, L. Bieberbach proved
that there exists a proper holomorphic mapping of the domain onto the unit disc
that is an n-to-one branched covering with the properties that it extends
continuously to the boundary and maps each boundary curve one-to-one onto the
unit circle, and it maps each given point on the boundary to the point 1 in the
unit circle. We modify a proof by H. Grunsky of Bieberbach's result to show
that there is a rational function of 2n+2 complex variables that generates all
of these maps. We also show how to generate all the proper holomorphic mappings
to the unit disc via the rational function.Comment: 17 page
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 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.Comment: 23 page
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