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Liouville theorem for bounded harmonic functions on manifolds and graphs satisfying non-negative curvature dimension condition

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Abstract

Brighton (in J Geom Anal 23(2):562–570, 2013) proved the Liouville theorem for bounded harmonic functions on weighted manifolds satisfying non-negative curvature dimension condition, i.e. \(\mathrm {CD}(0,\infty ).\) In this paper, we provide a new proof of this result by using the reverse Poincaré inequality. Moreover, we adopt this approach to prove the Liouville theorem for bounded harmonic functions on graphs satisfying the \(\mathrm {CD}(0,\infty )\) condition.

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Acknowledgements

The author would like to thank Yong Lin and Ariel Yadin for many stimulating discussions on Liouville theorems on discrete harmonic functions. The author is supported by NSFC (China), Grant Nos. 11831004, 11826031 and 11401106.

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Authors and Affiliations

  1. School of Mathematical Sciences, LMNS, Fudan University, Shanghai, 200433, China

    Bobo Hua

  2. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Bobo Hua

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Correspondence to Bobo Hua.

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Hua, B. Liouville theorem for bounded harmonic functions on manifolds and graphs satisfying non-negative curvature dimension condition. Calc. Var. 58, 42 (2019). https://doi.org/10.1007/s00526-019-1485-4

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