A SYMPLECTIC FUNCTIONAL ANALYTIC PROOF OF THE CONFORMAL WELDING THEOREM

We give a new functional-analytic/symplectic geometric proof of the conformal welding theorem. This is accomplished by representing composition by a quasisymmetric map phi as an operator on a suitable Hilbert space and algebraically solving the conformal welding equation for the unknown maps f and g satisfying g o phi = f. The univalence and quasiconformal extendibility of f and g is demonstrated through the use of the Grunsky matrix.