Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II

TL;DR

This paper refines the understanding of harmonic functions on infinite semiplanar graphs with nonnegative curvature by using Alexandrov space methods to achieve a linear growth rate estimate, improving previous quadratic bounds.

Contribution

It introduces a direct Alexandrov space approach to obtain the optimal linear dimension estimate of polynomial growth harmonic functions on these graphs.

Findings

Achieved a linear growth rate estimate for harmonic functions.

Connected graph harmonic functions to functions on Alexandrov surfaces.

Improved previous quadratic growth bounds to linear.

Abstract

In a previous paper Hua-Jost-Liu, we have applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincar\'e inequality on these graphs to obtain an dimension estimate of polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on graphs, we directly argue on Alexandrov spaces to obtain the optimal dimension estimate of polynomial growth harmonic functions on graphs which is actually linear in the growth rate. From a harmonic function on the graph, we construct a function on the corresponing Alexandrov surface that is not necessarily harmonic, but satisfies crucial estimates.