Positively curved cubic plane graphs are finite

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 + Σx∈F 1 |F | > 0, where the summation is taken over all facial cycles F containing x. Note that if G is positively curved then the maximum degree of G is at most 5. As a discrete analogue 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. Partially supported by a grant from China Scholarship Council Partially supported by NSF grant DMS-9970527