An analogue of the Descartes-Euler formula for infinite graphs and Higuchi's conjecture

Let $\mathcal{R}$ be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let $V$ be the set of vertices, and for every $v \in V$, let $\kappa(v)$ denote the (Gaussian) curvature of $v$: $2 \pi$ minus the sum of incident polygon angles. Descartes showed that $\sum_{v \in V} \kappa(v) = 4 \pi$ whenever $\mathcal{R}$ may be realized as the surface of a convex polytope in $\mathbb{R}^3$. More generally, if $\mathcal{R}$ is made of finitely many polygons, Euler's formula is equivalent to the equation $\sum_{v \in V} \kappa(v) = 2 \pi \chi(\mathcal{R})$ where $\chi(\mathcal{R})$ is the Euler characteristic of $\mathcal{R}$. Our main theorem shows that whenever $\sum_{v \in V : \kappa(v) < 0} \kappa(v)$ converges and there is a positive lower bound on the distance between any pair of vertices in $\mathcal{R}$, there exists a compact closed 2-manifold $\mathcal{S}$ and an integer $t$ so that $\mathcal{R}$ is homeomorphic to $\mathcal{S}$ minus $t$ points, and further $\sum_{v \in V} \kappa(v) \le 2 \pi \chi(\mathcal{S}) - 2 \pi t$.

In the special case when every polygon is regular of side length one and $\kappa(v) > 0$ for every vertex $v$, we apply our main theorem to deduce that $\mathcal{R}$ is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless $\mathcal{R}$ is a prism, antiprism, or the projective planar analogue of one of these that $\vert V\vert \le 3444$. This resolves a recent conjecture of Higuchi.