A new class of discrete conformal structures on surfaces with boundary

Abstract

We introduce a new type 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 on surfaces with boundary using variational principles, which is dual to Guo-Luo's rigidity of the discrete conformal structures on surfaces with boundary they introduced in [13]. As a result, some new convexities of the volume functions of some generalized hyperbolic tetrahedra 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 the 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.