Inmathematics,conformal welding (sewing orgluing) is a process ingeometric function theory for producing aRiemann surface by joining together two Riemann surfaces, each with a disk removed, along their boundary circles. This problem can be reduced to that of finding univalent holomorphic mapsf,g of theunit disk and its complement into the extended complex plane, both admitting continuous extensions to the closure of their domains, such that the images are complementary Jordan domains and such that on the unit circle they differ by a givenquasisymmetric homeomorphism. Several proofs are known using a variety of techniques, including theBeltrami equation,^[1] theHilbert transform on the circle^[2] and elementary approximation techniques.^[3]Sharon & Mumford (2006) describe the first two methods of conformal welding as well as providing numerical computations and applications to the analysis of shapes in the plane.

This method was first proposed byPfluger (1960).

Iff is a diffeomorphism of the circle, theAlexander extension gives a way of extendingf to a diffeomorphism of the unit diskD:

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

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

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

withg(θ + 2π) =g(θ) + 2π.

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

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

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

LetF₁(z) =G${\displaystyle \circ }$F⁻¹(z) for |z| ≤ 1 andF₂(z) =G (z) for |z| ≥ 1. ThusF₁ andF₂ are univalent holomorphic maps of |z| < 1 and |z| > 1 onto the inside and outside of a Jordan curve. They extend continuously to homeomorphismsf_i of the unit circle onto the Jordan curve on the boundary. By construction they satisfy theconformal 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)).

Lete_n(θ) =e^inθ be the standardorthonormal basis of L²(T). Let H²(T) beHardy space, the closed subspace spanned by thee_n withn ≥ 0. LetP be the orthogonal projection onto Hardy space and setT = 2P -I. The operatorH =iT is theHilbert transform on the circle and can be written as asingular integral operator.

Given a diffeomorphismf 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

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

LetF be the restriction off₊ to the unit circle. Then

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

and

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

Hence

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

IfV(f) denotes the bounded invertible operator on L² induced by the diffeomorphismf, 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 becauseP andT are given by singular integral operators. The equation above then reduces to

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

The operatorI −K_f is aFredholm 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 onD andD^c which have smooth boundary values on the circle related byf. Since theholomorphic function onD^c vanishes at ∞, the positive powers of this pair also provide solutions, which are linearly independent, contradicting the fact thatI −K_f is a Fredholm operator. The above equation therefore has a unique solutionF which is smooth and from whichf_± can be reconstructed by reversing the steps above. Indeed, by looking at the equation satisfied by the logarithm of the derivative ofF, it follows thatF has nowhere vanishing derivative on the unit circle. MoreoverF is one-to-one on the circle since if it assumes the valuea at different pointsz₁ andz₂ then the logarithm ofR(z) = (F(z) −a)/(z -z₁)(z −z₂) would satisfy anintegral equation known to have no non-zero solutions. Given these properties on the unit circle, the required properties off_± then follow from theargument 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 fromthe original(PDF) on 2012-08-03, retrieved2012-07-01
• 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{{citation}}:ISBN / Date incompatibility (help)

• Theorems in mathematical analysis
• Complex analysis
• Riemann surfaces

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