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Transactions of the American Mathematical Society

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3-dimensional combinatorial Yamabe flow in hyperbolic background geometry


Authors: Huabin Ge and Bobo Hua
Journal: Trans. Amer. Math. Soc. 373 (2020), 5111-5140
MSC (2010): Primary 51M10, 52C17, 05E45
DOI: https://doi.org/10.1090/tran/8062
Published electronically: April 28, 2020
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Abstract: We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e., the total (extended) solid angle at each vertex is equal to $ 4\pi $. In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (extended) combinatorial scalar curvature if the number of tetrahedra incident to each vertex is at most 22.


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Additional Information

Huabin Ge
Affiliation: School of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China
Email: hbge@ruc.edu.cn

Bobo Hua
Affiliation: School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China; and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China
Email: bobohua@fudan.edu.cn

DOI: https://doi.org/10.1090/tran/8062
Received by editor(s): April 29, 2019
Received by editor(s) in revised form: November 25, 2019
Published electronically: April 28, 2020
Additional Notes: The first author was supported by the NSFC of China (No. 11871094).
The second author was supported by the NSFC of China (No. 11831004 and No. 11826031).
Article copyright: © Copyright 2020 American Mathematical Society