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

Chunyan Li, Aijin Lin and Chang Yang

Vol. 315 (2021), No. 1, 129–150

DOI: 10.2140/pjm.2021.315.129

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

Mathematical Subject Classification 2010

Primary: 53C44

Secondary: 52C26

Milestones

Received: 6 November 2019

Revised: 24 October 2020

Accepted: 5 August 2021

Published: 13 December 2021