In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Carathéodory's theorem is a theorem in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, named after
Constantin Carathéodory, which extends the
Riemann mapping theorem. The theorem, first proved in 1913, states that the
conformal mapping sending the
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
to the region in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
bounded by a
Jordan curve extends continuously to a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on
prime end In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding the boundary circle in an appropriate way.
Historical notes
The concept of prime ends was introd ...
s and the boundary behaviour of univalent holomorphic functions.
Proofs of Carathéodory's theorem
The first proof of Carathéodory's theorem presented here is a summary of the short self-contained account in ; there are related proofs in and .
Clearly if ''f'' admits an extension to a homeomorphism, then ∂''U'' must be a Jordan curve.
Conversely if ∂''U'' is a Jordan curve, the first step is to prove ''f'' extends continuously to the closure of ''D''. In fact this will hold if and only if ''f'' is uniformly continuous on ''D'': for this is true if it has a continuous extension to the closure of ''D''; and, if ''f'' is uniformly continuous, it is easy to check ''f'' has limits on the unit circle and the same inequalities for uniform continuity hold on the closure of ''D''.
Suppose that ''f'' is not uniformly continuous. In this case there must be an ε > 0 and a point ζ on the unit circle and sequences ''z''
''n'', ''w''
''n'' tending to ζ with , ''f''(''z''
''n'') − ''f''(''w''
''n''), ≥ 2ε. This is shown below to lead to a contradiction, so that ''f'' must be uniformly continuous and hence has a continuous extension to the closure of ''D''.
For 0 < ''r'' < 1, let γ
''r'' be the curve given by the arc of the circle , ''z'' − ζ , = ''r'' lying within ''D''. Then ''f'' ∘ γ
''r'' is a Jordan curve. Its length can be estimated using the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
:
:
Hence there is a "length-area estimate":
:
The finiteness of the integral on the left hand side implies that there is a sequence ''r''
''n'' decreasing to 0 with
tending to 0. But the
length of a curve
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
''g''(''t'') for ''t'' in (''a'', ''b'') is given by
:
The finiteness of
therefore implies that the curve has limiting points ''a''
''n'', ''b''
''n'' at its two ends with , ''a''
''n'' – ''b''
''n'', ≤
, so this distance, as well as diameter of the curve, tends to 0. These two limit points must lie on ∂''U'', because ''f'' is a homeomorphism between ''D'' and ''U'' and thus a sequence converging in ''U'' has to be the image under ''f'' of a sequence converging in ''D''. By assumption there exist a homeomorphism β between the circle ∂''D'' and ∂''U''. Since β
−1 is uniformly continuous, the distance between the two points ξ
''n'' and η
''n'' corresponding to ''a''
''n'' and ''b''
''n'' in ∂''U'' must tend to 0. So eventually the smallest circular arc in ∂''D'' joining ξ
''n'' and η
''n'' is defined. Denote τ
''n'' image of this arc under β. By uniform continuity of β, diameter of τ
''n'' in ∂''U'' tends to 0. Together τ
''n'' and ''f'' ∘ γ
''r''''n'' form a simple Jordan curve. Its interior ''U''
''n'' is contained in ''U'' by the Jordan curve theorem for ∂''U'' and ∂''U''
''n'': to see this, notice that ''U'' is the interior of ∂''U'', as it is bounded, connected and it is both open and closed in the complement of ∂''U''; so the exterior region of ∂''U'' is unbounded, connected and does not intersect ∂''U''
''n'', hence its closure is contained in the closure of the exterior of ∂''U''
''n''; taking complements, we get the desired inclusion. The diameter of ∂''U''
''n'' tends to 0 because the diameters of τ
''n'' and ''f'' ∘ γ
''r''''n'' tend to 0. Hence the diameter of ''U''
''n'' tend to 0. (For
is compact set, hence
contains two points ''u'' and ''v'' such that distance between them is maximal. It is easy to see that ''u'' and ''v'' must lie in ∂''U'' and diameters of both ''U'' and ∂''U'' equal
.)
Now if ''V''
''n'' denotes the intersection of ''D'' with the disk , ''z'' − ζ, < ''r''
''n'', then for all sufficiently large ''n'' ''f''(''V''
''n'') = ''U''
''n''. Indeed, the arc γ
''r''''n'' divides ''D'' into ''V''
''n'' and complementary region
, so under the conformal homeomorphism ''f'' the curve ''f'' ∘ γ
''r''''n'' divides ''U'' into
and
a complementary region
; ''U''
''n'' is a connected component of ''U'' \ ''f'' ∘ γ
''r''''n'', as it is connected and is both open and closed in this set, hence
equals either
or
. Diameter of
does not decrease with increasing ''n'', for