Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature

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.

Polynomial growth harmonic functions on groups of polynomial volume growth

“…Instead of using the volume doubling property (1.2), as in the Riemannian manifold case (see [11,12,34]), Hua-Jost-Liu [27] and Hua-Jost [26] applied a more delicate volume growth property called the relative volume comparison to obtain the optimal dimension estimate on planar graphs with nonnegative combinatorial curvature.…”

Polynomial growth harmonic functions on finitely generated abelian groups

“…In the Cayley graph case, Kleiner [29] obtained the Poincaré inequality, but for non-abelian groups, Bochner's formula is unavailable (consider the free group case). For some special graphs which can be embedded into a surface with nonnegative sectional curvature in the sense of Alexandrov, Hua-Jost-Liu [27] proved the volume doubling property and the Poincaré inequality but Bochner's formula. It seems that Bochner's formula is sensitive to the local structure, but the volume growth property and the Poincaré inequality are not, c.f.…”

“…The simplified argument by the mean value inequality can be found in [35,13] where the dimension estimate is asymptotically optimal. This inspired many generalizations on manifolds [51,50,48,38,39,7,28,34] and on singular spaces [16,29,24,25,27]. In this paper, we give the precise dimension calculation of polynomial growth harmonic functions on finitely generated abelian groups.…”

The first gap for total curvatures of planar graphs with nonnegative curvature

“…Let be a semiplanar graph embedded into a surface (without boundary) with the set of vertices the set of edges , and the set of faces (see ). (It is called planar if is either the 2‐sphere or the plane.)…”

“…In this paper, we study total curvatures of semiplanar graphs with nonnegative combinatorial curvature. Note that a semiplanar graph has nonnegative combinatorial curvature if and only if the polygonal surface is a generalized convex surface (see ). We denote by the set of infinite semiplanar graphs with nonnegative combinatorial curvature.…”