LAUSR

For a fixed integer , let be an m-connected region in the Riemann sphere whose complement is a union of m disjoint closed disks and let be quasisymmetric mappings defined on for . We construct discrete conformal welding for based on the circle packing approach. We show that the discrete conformal welding mappings induced by circle packings converge uniformly on compact subsets to their continuous counterparts and that the corresponding discrete conformal welding curves converge uniformly to quasicircles determined by . This gives a constructive proof of the existence and uniqueness theorem for conformal welding of finitely connected regions.