Vol. 315, No. 1, 2021

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Variational principles and combinatorial $p$-th Yamabe flows on surfaces

Chunyan Li, Aijin Lin and Chang Yang

Vol. 315 (2021), No. 1, 129–150

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 p = 2 to any p > 1 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 p = 2 to any p > 1 by a new discrete conformal theory and discrete uniformization theorems established by Gu et al. It is shown that for our generalized p-th flows (p > 1,p2), there exists only curvature convergence but no exponential convergence as in the case of p = 2.

combinatorial Yamabe flow, combinatorial Ricci potential, variational principles, surgery
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 52C26
Received: 6 November 2019
Revised: 24 October 2020
Accepted: 5 August 2021
Published: 13 December 2021
Chunyan Li
Department of Physics
National University of Defense Technology
Changsha 410073
Aijin Lin
Department of Mathematics
National University of Defense Technology
Changsha 410073
Chang Yang
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP)
College of Mathematics and Statistics
Hunan Normal University
Changsha 410081