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A combinatorial Yamabe problem on two and three dimensional manifolds

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Abstract

In this paper, we define a new discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study the corresponding constant curvature problem, which is called the combinatorial Yamabe problem, by the corresponding combinatorial versions of Ricci flow and Calabi flow for surfaces and Yamabe flow for 3-dimensional manifolds. The basic tools are the discrete maximal principle and variational principle.

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Acknowledgements

The first author would like to thank BICMR, Professor Gang Tian and Professor Yanxun Chang for constant support and encouragement. The research of the first author is supported by National Natural Science Foundation of China under grant no. 11871094. The research of the second author is partially supported by Fundamental Research Funds for the Central Universities and National Natural Science Foundation of China under grant no. 11301402 and 11301399. He would also like to thank Professor Guofang Wang for the invitation to the Institute of Mathematics of the University of Freiburg and for his encouragement and many useful conversations during the work. Both authors would also like to thank Dr. Dongfang Li for many helpful conversations.

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China

    Huabin Ge

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Xu Xu

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  1. Huabin Ge
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  2. Xu Xu
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Correspondence to Xu Xu.

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Communicated by J. Jost.

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Ge, H., Xu, X. A combinatorial Yamabe problem on two and three dimensional manifolds. Calc. Var. 60, 20 (2021). https://doi.org/10.1007/s00526-020-01900-8

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