#### 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
##### Abstract

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 ($p>1,p\ne 2$), there exists only curvature convergence but no exponential convergence as in the case of $p=2$.

##### Keywords
combinatorial Yamabe flow, combinatorial Ricci potential, variational principles, surgery
Primary: 53C44
Secondary: 52C26