Carathéodory's theorem (conformal mapping)


In mathematics, Carathéodory's theorem is a theorem in complex analysis, 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 to the 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.

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 zn, wn tending to ζ with |f − f| ≥ 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 | z − ζ | = r lying within D. Then f ∘ γr is a Jordan curve. Its length can be estimated using the Cauchy–Schwarz inequality:
Hence there is a "length-area estimate":
The finiteness of the integral on the left hand side implies that there is a sequence rn decreasing to 0 with tending to 0. But the length of a curve g for t in is given by
The finiteness of therefore implies that the curve has limiting points an, bn at its two ends with |an – bn| ≤ , so this difference 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. Since ∂U is a homeomorphic image of the circle ∂D, the distance between the two corresponding parameters ξn and ηn in ∂U must tend to 0. So eventually the smallest circular arc in ∂D joining ξn and ηn is defined and, by uniform continuity, the diameter of its image τn tends to 0. Together τn and f ∘ γrn form a simple Jordan curve. Its interior Un is contained in U by the Jordan curve theorem for ∂U and ∂Un: 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 ∂Un, hence its closure is contained in the closure of the exterior of ∂Un; taking complements, we get the desired inclusion. The diameter of ∂Un tends to 0 because the diameters of τn and f ∘ γrn tend to 0. Hence the diameter and the area of Un tend to 0.
Now if Vn denotes the intersection of D with the disk |z − ζ| < rn, then f = Un. Indeed, the arc γrn divides D into Vn and a complementary region; Un is a connected component of U \ f ∘ γrn, as it is connected and is both open and closed in this set, so under the conformal homeomorphism f the curve f ∘ γrn divides U into Un and
a complementary region Un′, one of which equals f. Since the areas of f and Un tend to 0, while the sum of the areas of Un and Un′ is fixed, it follows that f = Un.
So the diameter of f tends to 0. On the other hand, passing to subsequences of and if necessary, it may be assumed that zn and wn both lie in Vn. But this gives a contradiction since |f − f| ≥ ε. So f must be uniformly continuous on U.
Thus f extends continuously to the closure of D. Since f = 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 = f. Let X and Y be the radial lines from 0 to u and v. Then f is a Jordan curve. Arguing as before, its interior V is contained in U and is a connected component of U \ f. On the other hand, D \ is the disjoint union of two open sectors
W1 and W2. Hence, for one of them, W1 say, f = V. Let Z be the portion of ∂W1 on the unit circle, so that Z is a closed arc and f 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
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.

Discussion

Intuitively, Carathéodory's theorem says that compared to general simply connected open sets in the complex plane C, those bounded by Jordan curves are particularly well-behaved.
Carathéodory's theorem is a basic result in the study of boundary behavior of conformal maps, a classical part of complex analysis. In general it is very difficult to decide whether or not the Riemann map from an open set U to the unit disk D extends continuously to the boundary, and how and why it may fail to do so at certain points.
While having a Jordan curve boundary is sufficient for such an extension to exist, it is by no means necessary . For example, the map
from the upper half-plane H to the open set G that is the complement of the positive real axis is holomorphic and conformal, and it extends to a continuous map from the real line R to the positive real axis R+; however, the set G is not bounded by a Jordan curve.