Conformal welding

This method was first proposed by Pfluger (1960).

If f is a diffeomorphism of the circle, the Alexander extension gives a way of extending f to a diffeomorphism of the unit disk D:

${\displaystyle \displaystyle {F(r,\theta )=r\exp[i\psi (r)g(\theta )+i(1-\psi (r))\theta ],}}$

where ψ is a smooth function with values in [0,1], equal to 0 near 0 and 1 near 1, and

${\displaystyle \displaystyle {f(e^{i\theta })=e^{ig(\theta )},}}$

with g(θ + 2π) = g(θ) + 2π.

The extension F can be continued to any larger disk |z| < R with R > 1. Accordingly in the unit disc

${\displaystyle \displaystyle {\|\mu \|_{\infty }<1,\,\,\,\mu (z)=F_{\overline {z}}/F_{z}.}}$

Now extend μ to a Beltrami coefficient on the whole of C by setting it equal to 0 for |z| ≥ 1. Let G be the corresponding solution of the Beltrami equation:

${\displaystyle \displaystyle {G_{\overline {z}}=\mu G_{z}.}}$

Let F₁(z) = G ∘ F⁻¹(z) for |z| ≤ 1 and F₂(z) = G (z) for |z| ≥ 1. Thus F₁ and F₂ are univalent holomorphic maps of |z| < 1 and |z| > 1 onto the inside and outside of a Jordan curve. They extend continuously to homeomorphisms f_i of the unit circle onto the Jordan curve on the boundary. By construction they satisfy the conformal welding condition:

${\displaystyle \displaystyle {f=f_{1}^{-1}\circ f_{2}.}}$

The use of the Hilbert transform to establish conformal welding was first suggested by the Georgian mathematicians D.G. Mandzhavidze and B.V. Khvedelidze in 1958. A detailed account was given at the same time by F.D. Gakhov and presented in his classic monograph (Gakhov (1990)).

Let e_n(θ) = e^inθ be the standard orthonormal basis of L²(T). Let H²(T) be Hardy space, the closed subspace spanned by the e_n with n ≥ 0. Let P be the orthogonal projection onto Hardy space and set T = 2P - I. The operator H = iT is the Hilbert transform on the circle and can be written as a singular integral operator.

Given a diffeomorphism f of the unit circle, the task is to determine two univalent holomorphic functions

${\displaystyle \displaystyle {f_{-}(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots ,\,\,\,\,\,f_{+}(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots ,}}$

defined in |z| < 1 and |z| > 1 and both extending smoothly to the unit circle, mapping onto a Jordan domain and its complement, such that

${\displaystyle \displaystyle {f_{-}(e^{i\theta })=f_{+}(f(e^{i\theta })).}}$

Let F be the restriction of f₊ to the unit circle. Then

${\displaystyle \displaystyle {TF=-F+2e^{i\theta }}}$

and

${\displaystyle \displaystyle {TF\circ f=F\circ f.}}$

Hence

${\displaystyle \displaystyle {T(F\circ f)\circ f^{-1}-T(F)=2F-2e^{i\theta }.}}$

If V(f) denotes the bounded invertible operator on L² induced by the diffeomorphism f, then the operator

${\displaystyle \displaystyle {K_{f}=V(f)PV(f)^{-1}-P}}$

is compact, indeed it is given by an operator with smooth kernel because P and T are given by singular integral operators. The equation above then reduces to

${\displaystyle \displaystyle {(I-K_{f})F=e^{i\theta }.}}$

The operator I − K_f is a Fredholm operator of index zero. It has zero kernel and is therefore invertible. In fact an element in the kernel would consist of a pair of holomorphic functions on D and D^c which have smooth boundary values on the circle related by f. Since the holomorphic function on D^c vanishes at ∞, the positive powers of this pair also provide solutions, which are linearly independent, contradicting the fact that I − K_f is a Fredholm operator. The above equation therefore has a unique solution F which is smooth and from which f_± can be reconstructed by reversing the steps above. Indeed, by looking at the equation satisfied by the logarithm of the derivative of F, it follows that F has nowhere vanishing derivative on the unit circle. Moreover F is one-to-one on the circle since if it assumes the value a at different points z₁ and z₂ then the logarithm of R(z) = (F(z) − a)/(z - z₁)(z − z₂) would satisfy an integral equation known to have no non-zero solutions. Given these properties on the unit circle, the required properties of f_± then follow from the argument principle.[4]

• Pfluger, A. (1960), "Ueber die Konstruktion Riemannscher Flächen durch Verheftung", J. Indian Math. Soc., 24: 401–412
• Lehto, O.; Virtanen, K.I. (1973), Quasiconformal mappings in the plane, Springer-Verlag, p. 92
• Lehto, O. (1987), Univalent functions and Teichmüller spaces, Springer-Verlag, pp. 100–101, ISBN 0-387-96310-3
• Sharon, E.; Mumford, D. (2006), "2-D analysis using conformal mapping" (PDF), International Journal of Computer Vision, 70: 55–75, doi:10.1007/s11263-006-6121-z, archived from the original (PDF) on 2012-08-03, retrieved 2012-07-01
• Gakhov, F. D. (1990), Boundary value problems. Reprint of the 1966 translation, Dover Publications, ISBN 0-486-66275-6
• Titchmarsh, E. C. (1939), The Theory of Functions (2nd ed.), Oxford University Press, ISBN 0198533497