Boundary correspondence of two conformal maps onto a Jordan domain #
Let f and g be holomorphic bijections of the upper half-plane onto the same bounded domain U
whose frontier is a Jordan curve. Pair a real point x with a real point y when f at x and
g at y have the same boundary limit. Then g⁻¹ ∘ f is an automorphism of the upper
half-plane. Its boundary action is naturally defined on the extended real line, where a real
Möbius transformation may send a finite point to infinity. In the Cayley coordinate
z ↦ (z - i) / (z + i) it is a standard disc automorphism w ↦ u * (w - c) / (1 - conj c * w).
Two consequences are recorded. Paired points have equal cross-ratios
(x₁ - x₃) * (x₂ - x₄) / ((x₁ - x₄) * (x₂ - x₃)). Three pairs x ↦ x force the Möbius map
to be the identity, so if f and g have the same boundary limits at three distinct real points
then f = g. This is the three-point normalization of conformal maps onto a Jordan domain.
Distinct real points also have distinct boundary limits. For Schwarz--Christoffel maps these
facts say that the prevertices of a polygon are determined up to a real Möbius transformation.
Main results #
TauCeti.exists_sub_I_div_add_I_eq_unitDiscStandardAutomorphismFormula_of_tendsto-- in Cayley coordinates, the boundary correspondence is a disc automorphism.TauCeti.crossRatio_eq_of_tendsto_of_bijOn_upperHalfPlaneSet-- the boundary correspondence preserves cross-ratios.TauCeti.eqOn_upperHalfPlaneSet_of_tendsto_of_bijOn-- two conformal maps onto a Jordan domain with the same boundary limits at three distinct real points coincide.TauCeti.eq_of_tendsto_of_bijOn_upperHalfPlaneSet-- a conformal map onto a Jordan domain has distinct boundary limits at distinct real points.
References #
- L. Ahlfors, Complex Analysis, Ch. 6, Sections 1--2.
- C. Carathéodory, Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung, Math. Ann. 73 (1913).
Transport to the disc #
The boundary correspondence #
The boundary correspondence of two conformal maps is a Möbius transformation. Let f and
g be holomorphic bijections of the upper half-plane onto a bounded domain U whose frontier is a
Jordan curve. There are u on the unit circle and c in the unit disc such that whenever f at
the real point x and g at the real point y have the same limit along the upper half-plane,
the Cayley transforms of x and y are related by the standard disc automorphism
w ↦ u * (w - c) / (1 - conj c * w).
The boundary correspondence of two conformal maps preserves cross-ratios. Let f and g
be holomorphic bijections of the upper half-plane onto a bounded domain U whose frontier is a
Jordan curve. If f at x i and g at y i have the same limit w i along the upper
half-plane for every index i, then the families x and y of real points have the same
cross-ratios.
Three-point normalization of conformal maps onto a Jordan domain. Two holomorphic
bijections f and g of the upper half-plane onto a bounded domain whose frontier is a Jordan
curve coincide on the upper half-plane as soon as, at each of three distinct real points, they have
the same limit along the upper half-plane.
Distinct real points have distinct boundary limits. If a holomorphic bijection of the upper half-plane onto a bounded domain whose frontier is a Jordan curve has the same limit along the upper half-plane at two real points, then these points are equal.