In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, published by Carathéodory in 1913, states that any conformal mapping sending the unit disk to some region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.
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 zn, wn tending to ζ with |f(zn) − f(wn)| ≥ 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 γr be the curve given by the arc of the circle lying within D. Then f ∘ γr is a Jordan curve. Its length can be estimated using the Cauchy–Schwarz inequality:
\displaystyle{\ell(f\circ\gammar)=
\int | |
\gammar |
|f\prime(z)||dz|\le
\left(\int | |
\gammar |
|dz|\right)1/2 ⋅
\left(\int | |
\gammar |
|f\prime(z)|2|dz|\right)1/2\le(2\pir)1/2
⋅ \left(\int | |
\{\theta:|\zeta+rei\theta|<1\ |
Hence there is a "length-area estimate":
1 | |
\displaystyle{\int | |
0 |
\ell(f\circ
2 | |
\gamma | |
r) |
{dr\overr}\le2\pi\int|z|<1|f\prime(z)|2dxdy=2\pi ⋅ {\rmArea}f(D)<infty.}
The finiteness of the integral on the left hand side implies that there is a sequence rn decreasing to 0 with
\ell(f\circ
\gamma | |
rn |
)
\displaystyle{\ell(g)
=\sup | |
a<t1<t2< … <tk<b |
k-1 | |
\sum | |
i=1 |
|g(ti+1)-g(ti)|.}
The finiteness of
\ell(f\circ
\gamma | |
rn |
)
|an-bn|\le\ell(f\circ
\gamma | |
rn |
)
\barUn x \barUn
\barUn
|u-v|
Now if Vn denotes the intersection of D with the disk |z − ζ| < rn, then for all sufficiently large n f(Vn) = Un. Indeed, the arc γrn divides D into Vn and complementary region
Vn'
f(Vn)
f(Vn')
Un
f(Vn)
f(Vn')
f(Vn')
n<n'
Vn'\subsetVn''
f(Vn')
So the diameter of f(Vn) tends to 0. On the other hand, passing to subsequences of (zn) and (wn) if necessary, it may be assumed that zn and wn both lie in Vn. But this gives a contradiction since |f(zn) − f(wn)| ≥ ε. So f must be uniformly continuous on U.
Thus f extends continuously to the closure of D. Since f(D) = U, by compactness f carries the closure of D onto the closure of U and hence ∂D onto ∂U. If f is not one-one, there are points u, v on ∂D with u ≠ v and f(u) = f(v). Let X and Y be the radial lines from 0 to u and v. Then is a Jordan curve. Arguing as before, its interior V is contained in U and is a connected component of . On the other hand, is the disjoint union of two open sectorsW1 and W2. Hence, for one of them, W1 say, f(W1) = V. Let Z be the portion of ∂W1 on the unit circle, so that Z is a closed arc and f(Z) is a subset of both ∂U and the closure of V. But their intersection is a single point and hence f is constant on Z. By the Schwarz reflection principle, f can be analytically continued by conformal reflection across the circular arc. Since non-constant holomorphic functions have isolated zeros, this forces f to be constant, a contradiction. So f is one-one and hence a homeomorphism on the closure of D.
Two different proofs of Carathéodory's theorem are described in and . The first proof follows Carathéodory's original method of proof from 1913 using properties of Lebesgue measure on the circle: the continuous extension of the inverse function g of f to ∂U is justified by Fatou's theorem on the boundary behaviour of bounded harmonic functions on the unit disk. The second proof is based on the method of, where a sharpening of the maximum modulus inequality was established for bounded holomorphic functions h defined on a bounded domain V: if a lies in V, then
|h(a)| ≤ mt ⋅ M1 − t,
where 0 ≤ t ≤ 1, M is maximum modulus of h for sequential limits on ∂U and m is the maximum modulus of h for sequential limits on ∂U lying in a sector centred on a subtending an angle 2πt at a.
An extension of the theorem states that a conformal isomorphism
g\colonD\toU
where
U
U
This result is often also attributed to Carathéodory, but was first stated and proved by Marie Torhorst in her 1918 thesis, under the supervision of Hans Hahn, using Carathéodory's theory of prime ends. More precisely, Torhorst proved that local connectivity is equivalent to the domain having only prime ends of the first kind. By the theory of prime ends, the latter property, in turn, is equivalent to
g