Positively curved cubic plane graphs are finite

This is the authors' abstract. We don't add key points for this paper.

Let G be an infinite plane graph such that G is locally finite and every face of G is bounded by a cycle. Then G is said to be positively curved if, for every vertex x of G, 1-d(x) 2+∑ x∈ F1 |F|>0 , where the summation is taken over all facial cycles F of G containing x and |F| denotes the number of vertices in F. Note that if G is positively curved then the maximum degree of G is at most 5. As a discrete analog of a result in Riemannian geometry, Higuchi conjectured that if G is positively curved then G is finite. In this paper, we establish this conjecture for cubic graphs.