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Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II


Authors: Bobo Hua and Jürgen Jost
Journal: Trans. Amer. Math. Soc. 367 (2015), 2509-2526
MSC (2010): Primary 31C05, 05C81, 05C63
DOI: https://doi.org/10.1090/S0002-9947-2014-06167-9
Published electronically: November 24, 2014
MathSciNet review: 3301872
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Abstract: In a previous paper, we applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincaré inequality on these graphs to obtain a dimension estimate for polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on the graph, we translate the problem to a polygonal surface by filling polygons into the graph with edge lengths 1. This polygonal surface then is an Alexandrov space of nonnegative curvature. From a harmonic function on the graph, we construct a function on the polygonal surface that is not necessarily harmonic, but satisfies crucial estimates. Using the arguments on the polygonal surface, we obtain the optimal dimension estimate for polynomial growth harmonic functions on the graph which is linear in the growth rate.


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Additional Information

Bobo Hua
Affiliation: Max Planck Institute for Mathematics in the Sciences, Leipzig, 04103, Germany
Address at time of publication: School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, People’s Republic of China; Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Email: bobohua@mis.mpg.de, bobohua@fudan.edu.cn

Jürgen Jost
Affiliation: Max Planck Institute for Mathematics in the Sciences, Leipzig, 04103, Germany
Email: jost@mis.mpg.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06167-9
Received by editor(s): September 28, 2012
Published electronically: November 24, 2014
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 267087
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.