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The Liouville property for geometric graphs in the plane

  • Kuijun Liu
  • Feng Luo
  • Yanwen Luo
  • Zhenghao Rao
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Abstract

We show that every bounded discrete harmonic function with respect to the cotangent Laplacian on a Delaunay triangulation of the Euclidean plane is constant; equivalently, the cotangent-weighted triangulation has the Liouville property. Our main theorem establishes this property for a more general class of connected weighted geometric graphs in the plane. Furthermore, we show that a planar Delaunay triangulation with cotangent weights could be transient but is recurrent if edge lengths are uniformly bounded.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026liouville,
  title = {{The Liouville property for geometric graphs in the plane}},
  author = {Kuijun Liu and Feng Luo and Yanwen Luo and Zhenghao Rao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39242},
  eprint = {2610.39242},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39242},
}
RIS
TY  - UNPB
TI  - The Liouville property for geometric graphs in the plane
AU  - Kuijun Liu
AU  - Feng Luo
AU  - Yanwen Luo
AU  - Zhenghao Rao
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39242
UR  - https://arxiv.org/abs/2610.39242
AB  - We show that every bounded discrete harmonic function with respect to the cotangent Laplacian on a Delaunay triangulation of the Euclidean plane is constant; equivalently, the cotangent-weighted triangulation has the Liouville property. Our main theorem establishes this property for a more general class of connected weighted geometric graphs in the plane. Furthermore, we show that a planar Delaunay triangulation with cotangent weights could be transient but is recurrent if edge lengths are uniformly bounded.
ER  -