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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II
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by Bobo Hua and Jürgen Jost PDF
Trans. Amer. Math. Soc. 367 (2015), 2509-2526 Request permission

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
  • MR Author ID: 865783
  • 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
  • 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
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3301872