Areas of spherical polyhedral surfaces with regular faces

TL;DR

This paper classifies spherical polyhedral surfaces with regular faces and curvature constraints, establishing bounds on their area and identifying conditions for spherical tilings.

Contribution

It provides a complete classification of spherical tilings with regular polygons and proves an area gap for non-tiling surfaces under curvature bounds.

Findings

Classified all spherical tilings with regular polygons.

Proved an area upper bound for non-tiling surfaces with curvature ≥ 1.

Established a positive gap between surface area and that of the unit sphere.

Abstract

For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of planar graphs which admit spherical polyhedral surfaces with the curvature bounded below by 1 in the sense of Alexandrov, i.e. the total angle at each vertex is at most $2\pi$. We classify all spherical tilings with regular spherical polygons, i.e. total angles at vertices are exactly $2\pi$. We prove that for any graph in this class which does not admit a spherical tiling, the area of the associated spherical polyhedral surface with the curvature bounded below by 1 is at most $4\pi - \epsilon_0$ for some $\epsilon_0 > 0$. That is, we obtain a definite gap between the area of such a surface and that of the unit sphere.