Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature

Hua, Bobo; Jost, Jürgen; Liu, Shiping

Abstract

We apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature. We obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point. Moreover, we show the volume doubling property and the Poincaré inequality on such graphs. The quadratic volume growth of these graphs implies the parabolicity. Finally, we prove the polynomial growth harmonic function theorem analogous to the case of Riemannian manifolds.

Citation

Hua, B., Jost, J., & Liu, S. (2013). Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature. Journal für die reine und angewandte Mathematik, 2015(700), 1-36. https://doi.org/10.1515/crelle-2013-0015

• Journal Article Type: Article
• Acceptance Date: Jan 21, 2013
• Publication Date: Apr 11, 2013
• Deposit Date: Dec 16, 2014
• Publicly Available Date: Sep 1, 2015
• Journal: Journal für die reine und angewandte Mathematik
• Print ISSN: 0075-4102
• Electronic ISSN: 1435-5345
• Publisher: De Gruyter
• Peer Reviewed: Peer Reviewed
• Volume: 2015
• Issue: 700
• Pages: 1-36
• DOI: https://doi.org/10.1515/crelle-2013-0015
• Public URL: https://durham-repository.worktribe.com/output/1418018