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Transactions of the American Mathematical Society

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Combinatorial Calabi flows on surfaces


Author: Huabin Ge
Journal: Trans. Amer. Math. Soc. 370 (2018), 1377-1391
MSC (2010): Primary 53C44, 52C26
DOI: https://doi.org/10.1090/tran/7196
Published electronically: October 16, 2017
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Abstract: For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of the smooth Calabi flow. We prove that the solution to the combinatorial Calabi flow exists for all time and converges if and only if the Thurston's circle packing exists. As a consequence, the combinatorial Calabi flow provides a new algorithm to find circle packings with prescribed curvatures. The proofs rely on careful analysis of the combinatorial Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.


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Additional Information

Huabin Ge
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Email: hbge@bjtu.edu.cn

DOI: https://doi.org/10.1090/tran/7196
Keywords: Circle packing, combinatorial Calabi flow, combinatorial Calabi energy, combinatorial Ricci potential
Received by editor(s): March 10, 2015
Received by editor(s) in revised form: July 5, 2016, and December 23, 2016
Published electronically: October 16, 2017
Additional Notes: This research was supported by the National Natural Science Foundation of China under grant No.11501027, and Fundamental Research Funds for the Central Universities (Nos. 2015JBM103, 2014RC028 and 2016JBM071).
Article copyright: © Copyright 2017 American Mathematical Society

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