Hyperbolic polyhedral surfaces with regular faces

TL;DR

This paper investigates hyperbolic polyhedral surfaces with regular faces, establishing their combinatorial equivalence to Euclidean polyhedral surfaces with negative curvature, and analyzes the area gap between non-smooth and smooth hyperbolic surfaces.

Contribution

It characterizes hyperbolic polyhedral surfaces with regular faces and demonstrates a quantifiable area gap compared to smooth hyperbolic surfaces.

Findings

Combinatorial equivalence with Euclidean polyhedral surfaces with negative curvature.

Existence of a measurable area gap between non-smooth and smooth hyperbolic surfaces.

Numerical estimation of the area gap for double torus surfaces with cubic graph skeletons.

Abstract

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2\pi.$ The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.