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Let be a two-dimensional Brownian excursion with darning on a finitely connected domain. Using Koebe's theorem and conformal invariance of Brownian motion with darning we derive that a two-sided restriction measure exists for . Based on the existence of the two-sided restriction measure, we show that possesses conformal restriction property, extending the conformal restriction of the Brownian excursion for a simply connected domain. [ABSTRACT FROM AUTHOR]

For a fixed integer, letbe anm-connected region in the Riemann spherewhose complementis a union ofmdisjoint closed disksand letbe quasisymmetric mappings defined onfor. We construct discrete conformal welding forbased 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. [ABSTRACT FROM AUTHOR]

Möbius invariants of general circle packings are defined in terms of cross ratios. The necessary and sufficient conditions of existence of circle packings are established by the techniques of Möbius invariants. It is shown that circle packings are uniquely determined, up to Möbius transformations, by their Möbius invariants. The rigidity of infinite circle packings with bounded degree is proved using the approach of Möbius invariants. [ABSTRACT FROM AUTHOR]

It is known that hexagonal circle packings can be used to approximate the Riemann mappings. Here, the convergence of the hexagonal circle packings is generalized to that of circle packings with bounded degree. We shall show that the first two derivatives of the bounded degree circle packings converge to the Riemann mapping function's first two derivatives, respectively; moreover we give the explicit estimates for the rates of convergence. [ABSTRACT FROM AUTHOR]