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Convergence of an iterative algorithm for Teichmüller maps via harmonic energy optimization


Authors: Lok Ming Lui, Xianfeng Gu and Shing-Tung Yau
Journal: Math. Comp. 84 (2015), 2823-2842
MSC (2010): Primary 52C26, 65D18, 65E05; Secondary 52B20, 52C99
DOI: https://doi.org/10.1090/S0025-5718-2015-02962-7
Published electronically: March 24, 2015
MathSciNet review: 3378849
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Abstract: 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 landmark-matching registration. The purpose of this paper is to prove the iterative algorithm proposed in loc. cit., indeed converges.


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Additional Information

Lok Ming Lui
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: lmlui@math.cuhk.edu.hk

Xianfeng Gu
Affiliation: Department of Computer Sciences, State University of New York at Stony Brook, Stony Brook, New York
Email: gu@cs.sunysb.edu

Shing-Tung Yau
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts
Email: yau@math.harvard.edu

DOI: https://doi.org/10.1090/S0025-5718-2015-02962-7
Received by editor(s): September 18, 2013
Published electronically: March 24, 2015
Additional Notes: The first author was supported by RGC GRF (Project ID: 401811), CUHK Direct Grant (Project ID: 2060413), and CUHK FIS Grant (Project ID: 1902036)
Article copyright: © Copyright 2015 American Mathematical Society