A new class of discrete conformal structures on surfaces with boundary

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpretations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity using variational principles, is complement Guo–Luo’s Guo and Luo (Geom Topol 13(3):1265–1312, 2009) Guo’s vertex scaling (Commun Contemp Math 13(5):827–842, 2011) boundary. As byproduct, results convexity volume generalized pyramids right-angled hexagonal bases are obtained. Motivated by Chow–Luo’s combinatorial Ricci flow Luo’s Yamabe closed surfaces, further Calabi flows to deform The basic properties these curvature established. These provide effective algorithms for constructing metrics totally geodesic boundary components prescribed lengths.