Generalized quasidisks and conformality: progress and challenges

Abstract

In this note, we survey the recent developments on theory of generalized quasidisks. Based on standard techniques used earlier, we also provide some minor improvements on the recorded results. A few natural questions are posed.

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Acknowledgements

This survey is dedicated to our former Ph.D supervisor Prof. Pekka Koskela for his excellent guidance during our postgraduate studies and for bringing us to the world of quasiconformal analysis.

Funding

C.-Y.Guo is supported by the Qilu funding of Shandong University (No. 62550089963197). H. Xu is supported by the Academy of Finland (No. 21000046081).

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Correspondence to Chang-Yu Guo.

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Dedicated to Professor Pekka Koskela on the occasion of his 60th birthday.

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Guo, CY., Xu, H. Generalized quasidisks and conformality: progress and challenges. Complex Anal Synerg 7, 2 (2021). https://doi.org/10.1007/s40627-021-00065-0

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Keywords

  • Homeomorphism of finite distortion
  • Generalized quasidisk
  • Local connectivity
  • Three point property
  • Cusps

Mathematics Subject Classification

  • 30C62
  • 30C65