Abstract
Luo (Commun Contemp Math 6:765–780, 2004) conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution for all time. Moreover, we consider the convergence of such solution. We show that the extended solution converges exponentially fast to the constant curvature PL-metric if it exists. In addition, we investigate the geometric meaning of the limit of the extended solution.
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Acknowledgments
We would like to thank Professor Gang Tian for his encouragement and support. We are very grateful to the referee for carefully reading the original manuscript and providing many helpful suggestions that greatly improve the article. The first author would also like to thank Professor Feng Luo for helpful discussions. The research is supported by National Natural Science Foundation of China under Grant (No.11501027), and Fundamental Research Funds for the Central Universities (Nos. 2015JBM103, 2014RC028, 2016JBM071 and 2016JBZ012).
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Communicated by A. Chang.
