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Convergence of an Iterative Algorithm for Teichmüller Maps via Harmonic Energy Optimization

Ronald Lok Ming Lui The Chinese University of Hong Kong Xianfeng Gu Stony Brook University Shing-Tung Yau Harvard University

Computational Geometry mathscidoc:1609.09012

Mathematics of Computation, 84, 2823-2842, 2015.3
Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichmüller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichmüller map between Riemann surfaces of finite type are theoretically guaranteed (see Fletcher and Markovic, Quasiconformal maps and Teichmüller theory, Oxford Graduate Texts in Math., vol. 11, Oxford University Press, Oxford, 2007). Recently, a simple iterative algorithm for computing the Teichmüller maps between connected Riemann surfaces with given boundary value was proposed by Lui, Lam, Yau, and Gu in Teichmüller extremal mapping and its applications to landmark matching registration, arXiv:1211.2569. Numerical results were reported in the paper to show the effectiveness of the algorithm. The method was successfully applied to landmarkmatching registration. The purpose of this paper is to prove the iterative algorithm proposed in loc. cit., indeed converges.
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@inproceedings{ronald2015convergence,
  title={Convergence of an Iterative Algorithm for Teichmüller Maps via Harmonic Energy Optimization},
  author={Ronald Lok Ming Lui, Xianfeng Gu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160905170438982140624},
  booktitle={Mathematics of Computation},
  volume={84},
  pages={2823-2842},
  year={2015},
}
Ronald Lok Ming Lui, Xianfeng Gu, and Shing-Tung Yau. Convergence of an Iterative Algorithm for Teichmüller Maps via Harmonic Energy Optimization. 2015. Vol. 84. In Mathematics of Computation. pp.2823-2842. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160905170438982140624.
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