Volume Doubling, Poincaré Inequality and Gaussian Heat Kernel Estimate for Non-negatively Curved Graphs
Publication Date
10-17-2017
Document Type
Article
Organizational Units
Mathematics
Keywords
Heat equation, Curvature-dimension, Gaussian heat kernel
Abstract
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 CDE0 (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.
Publication Statement
Copyright held by author or publisher. User is responsible for all copyright compliance.
Recommended Citation
Horn, P, Y Lin, S Liu, and S.-T Yau. "Volume Doubling, Poincaré Inequality and Gaussian Heat Kernel Estimate for Non-Negatively Curved Graphs." Journal Fur Die Reine Und Angewandte Mathematik. 2019.757 (2021): 89-130. Print. doi: 10.1515/crelle-2017-0038.