A new class of discrete conformal structures on surfaces with boundary

TL;DR

This paper introduces a novel class of discrete conformal structures on surfaces with boundary, proves their global rigidity, and develops curvature flows to compute hyperbolic metrics with prescribed boundary lengths.

Contribution

It presents a new discrete conformal structure with proven global rigidity and introduces curvature flows for hyperbolic surface metrics with boundary.

Findings

Proved global rigidity of the new conformal structures.

Established properties of combinatorial Ricci and Calabi flows.

Provided algorithms for hyperbolic metrics with prescribed boundary lengths.

Abstract

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo-Luo's rigidity of the discrete conformal structures and Guo's rigidity of vertex scaling on surface with boundary. As a result, new convexities of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow-Luo's combinatorial Ricci flow and Luo's combinatorial Yamabe flow on closed surfaces, we further introduce combinatorial Ricci flow and combinatorial Calabi flows to deform the new discrete conformal structures on surfaces with boundary. The basic properties of these combinatorial curvature flows are established. These combinatorial curvature flows provide effective algorithms for constructing hyperbolic metrics on surfaces with totally geodesic boundary components of prescribed lengths.