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

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.