Approximation of conformal welding for finitely connected regions

Abstract

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.

Keywords:

• Circle packing
• conformal welding
• quasisymmetric mapping
• quasicircle

AMS Subject Classifications:

• 65N10
• 65N30
• 52C15
• 30G62

Acknowledgements

The authors would like to express their gratitude to Prof. Wei Lin for useful discussions and excellent suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is partially supported by the NSF of China [grant number 11161004], [grant number 11171354], [grant number 11661011]; NSF of Guangxi [grant number 2016GXNSFAA380099].