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Abstract
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The combinatorial Yamabe flow was introduced by Luo (2004) to study the combinatorial
Yamabe problem. To handle the possible singularities along the combinatorial
Yamabe flow, Ge and Jiang initiated the “extended Yamabe flow algorithm”, while
Gu, Luo, Sun and Wu initiated the “doing surgery by flipping algorithm”. We
generalize Ge and Jiang’s results on the extended combinatorial Yamabe flow from
to
any
based on the work of Bobenko, Pinkall and Springborn on the explicit
formula of the Ricci potential functional. On the other hand, we generalize
the work of Gu et al. on the combinatorial Yamabe flow with surgery from
to
any
by a new discrete conformal theory and discrete uniformization
theorems established by Gu et al. It is shown that for our generalized
-th flows
(),
there exists only curvature convergence but no exponential convergence as in the case
of
.
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Keywords
combinatorial Yamabe flow, combinatorial Ricci potential,
variational principles, surgery
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Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 52C26
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Milestones
Received: 6 November 2019
Revised: 24 October 2020
Accepted: 5 August 2021
Published: 13 December 2021
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