Annales Academię Scientiarum Fennicę
Mathematica
Volumen 34, 2009, 303-314
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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