Discrete conformal structures on surfaces with boundary (II) -- Rigidity and Existence

TL;DR

This paper investigates the rigidity and existence of discrete conformal structures on surfaces with boundary, extending previous results to a more general setting within an axiomatic framework.

Contribution

It extends prior work by providing generalized rigidity and existence results for discrete conformal structures on bordered surfaces.

Findings

Extended Guo-Luo and Guo results to broader contexts

Established conditions for rigidity of discrete conformal structures

Proved existence theorems for these structures on surfaces with boundary

Abstract

In \cite{X-Z DCS1}, we introduced discrete conformal structures on surfaces with boundary via an axiomatic framework, and provided a classification of such discrete conformal structures. The present work focuses on the rigidity and existence of these discrete conformal structures on surfaces with boundary. As a direct consequence, the results by Guo-Luo in \cite{GL2} and Guo in \cite{Guo}, which deal with the rigidity and existence of discrete conformal structures on surfaces with boundary, are extended to a very general context.