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Summary: Boundary maps in the Riemann mapping theorem
Posted:
Mar 19, 1992 12:49 PM
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Thanks to everyone responding to my request on this topic.
Below I have exerpted some of the most useful information.
Executive summary:
Pommerenke, _Univalent Functions_, chapters 9 and 10 are on boundary
behavior. Whether the conformal map (on the closure) satisfies a Lipschitz
condition is related to whether the boundary has any cusps. I did not find any
quantitative information on what happens at cusps that is
fine enough to explain the behavior near the z=1/4 cusp of the
Mandelbrot set.
-------------------
original question:
Boundary maps in the Riemann mapping theorem
I have some questions related to the boundary behavior in the
Riemann mapping theorem. References dealing with this
would be welcome. I am interested in the RATE OF APPROACH
to the boundary.
Suppose we have a conformal map f of the open unit disk U
one-to-one onto a domain S in the Riemann sphere. Suppose
f extends continuously to the boundary of U. [I guess there
is a theorem to the effect that this always happens?
Of course the extended map may no longer be one-to-one.]
Thus, as a point z in U approaches the boundary of U, the
corresponding point w = f(z) in S approaches the boundary of S.
Here is the question: what is the rate at which w approaches
the boundary point? This depends, of course, on the nature of the
region S near that point. If S is smooth there, and f is differentiable
there, then when z approaches 1 along the x-axis at speed 1,
w = f(z) approaches f(1) at speed given by the absolute value of
the derivative of f at 1.
But what happens at more interesting boundary points?
-------------------------------------------------------
from Marius Overholt
This is in reply to your recent posting on math.sci.research about
the boundary behavior of the Riemann mapping. A standard reference on
boundary behavior is Pommerenke s Univalent Functions. It contains among
other things a proof of Caratheodory s Theorem that the Riemann mapping
extends to a homeomorphism between the closures if the domain
is a Jordan domain. There is also a theorem that asserts that the mapping
extends to a continuous function if the boundary of the domain is locally
connected (this can be interpreted in terms of prime ends, also covered
by Pommerenke). Pommerenke has been writing a book
entirely devoted to boundary behavior, and if it has now appeared, it
should be very useful.
There is also a theorem asserting smoothness of the Riemann mapping
onto a smoothly bounded Jordan domain (Kellogg s Theorem). See for example
Steve Bell and Laszlo Lempert: A C-infinity Schwarz Reflection... in
J. Differential Geometry 32(1990) for a modern proof of a related result.
-------------------------------------------------
from Joe Christy
The Mandelbrot set IS known to be locally connected at the cusp
of the main cardioid, so convergence does make sense there. The question
could probably be answered using Douady & Hubbard's construction of
the R-map of the complement and their theory of external rays. I can't
seem to put my hands on the reference, but I do seem to recall that it
was in the Annales de l'Ecole Norm. Sup., about 3-7 years ago.
-----------------
from F. David Lesly
If the boundary of S is locally connected then the Riemann mapping extends
continuously to U closure. If Boundary S is a Jordan curve it extends as
a homeomorphism. In the general cases on refers to the theory of "prime
ends" All of this is in "Univalent Functions" by Christian Pommerenke.
More specially, if boundary S is a "quasicircle" then the boundary
mapping is Holder continuous(umlaut over the o), with exponent depending on the way one quantifies being a quasicircle. Quasicircles are cusp free, but many
fractals are quasicircles(e.g. Julia sets for c in the main cardioid of the
Mandelbrotset). They are also called quasiconformal curves and quasiarcs.
Two different approaches to this are in: 1. Holder continuity of conformal map-
pings at the boundary via the strip method. F.D. Lesley Indiana U. Math J., 31,
(1982) P 341-354. 2. Quasiconformal circles and Lipschitz classes. R. Nakki
and B Palka, Comment Math Helvet. 55 (1980) P 485-?. A more concrete case is
studied in Conformal mappings of domains satisfying a wedge condition. Proc.
AMS 93 (1985) P 483-488. That one is also by me. Cusps are considered by Nakki
and Palka in Bounary angles, cusps and conformal mappings. Complex Variables 5
(1986) P 165-180. They also have some other papers on these topics, some in
higher dimensions for quasiconformal mappings.
All of these papers use the method of extremal length, or modulus of curve
families, which is nicely laid out in a little van Nostrand paperback by
Wolfgang Fuchs. The SOURCE is the books of Ahlfors on Conformal Invariants
and Quasiconformal Mappings. Indeed the method is related to harmonic measure.
--
Gerald A. Edgar Internet: edgar@mps.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
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