Combinatorial Ricci and Calabi flows on discrete conformal structures over surfaces with boundary are proven to converge exponentially to prescribed boundary lengths, globally for two structure classes and locally for the others.
Discrete conformal structures on surfaces with boundary (II) -- Rigidity and Existence
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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.
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Discrete conformal structures on surfaces with boundary (III) -- Deformation
Combinatorial Ricci and Calabi flows on discrete conformal structures over surfaces with boundary are proven to converge exponentially to prescribed boundary lengths, globally for two structure classes and locally for the others.