Volume doubling, Poincar\'{e} inequality and Gaussian heat kernel estimate for non-negatively curved graphs

Paul Horn Denver University Yong Lin Renmin University of China Shuang Liu Renmin University of China Shing-Tung Yau Harvard University

Combinatorics Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:1608.10074

Journal für die reine und angewandte Mathematik, 2017.12
By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincar\'{e} inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is finite, which original is a conjecture of Yau on Riemannian manifold proved by Colding and Minicozzi. Under the assumption of positive curvature on graphs, we derive the Bonnet-Myers type theorem that the diameter of graphs is finite and bounded above in terms of the positive curvature by proving some Log Sobolev inequalities.
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@inproceedings{paul2017volume,
  title={Volume doubling, Poincar\'{e} inequality and Gaussian heat kernel estimate for non-negatively curved graphs},
  author={Paul Horn, Yong Lin, Shuang Liu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160825085906189304440},
  booktitle={Journal für die reine und angewandte Mathematik},
  year={2017},
}
Paul Horn, Yong Lin, Shuang Liu, and Shing-Tung Yau. Volume doubling, Poincar\'{e} inequality and Gaussian heat kernel estimate for non-negatively curved graphs. 2017. In Journal für die reine und angewandte Mathematik. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160825085906189304440.
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