Annales Academię Scientiarum Fennicę

Mathematica

Volumen 34, 2009, 303-314

# MATCHING UNIVALENT FUNCTIONS AND CONFORMAL WELDING

## Erlend Grong, Pavel Gumenyuk and Alexander Vasil'ev

University of Bergen, Department of Mathematics

Johannes Brunsgate 12, Bergen 5008, Norway; Erlend.Grong 'at' math.uib.no

University of Bergen, Department of Mathematics

Johannes Brunsgate 12, Bergen 5008, Norway; Pavel.Gumenyuk 'at' math.uib.no

University of Bergen, Department of Mathematics

Johannes Brunsgate 12, Bergen 5008, Norway; Alexander.Vasiliev 'at' math.uib.no

**Abstract.**
Given a conformal mapping *f* of the unit disk **D** onto a
simply connected domain *D* in the complex plane bounded by a closed Jordan curve,
we consider the problem of constructing a matching conformal mapping, i.e., the
mapping of the exterior of the unit disk **D**^{*} onto the exterior
domain *D*^{*} regarding to *D*. The answer is expressed in
terms of a linear differential equation with a driving term given as the kernel
of an operator dependent on the original mapping *f*. Examples are provided.
This study is related to the problem of conformal welding and to representation of
the Virasoro algebra in the space of univalent functions.

**2000 Mathematics Subject Classification:**
Primary 30C35; Secondary 17B68.

**Key words:**
Matching univalent functions, conformal welding,
Kirillov's manifold, Virasoro algebra.

**Reference to this article:** E. Grong, P. Gumenyuk and A. Vasil'ev:
Matching univalent functions and conformal welding.
Ann. Acad. Sci. Fenn. Math. 34 (2009), 303-314.

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