A new class of discrete conformal structures on surfaces with boundary

Abstract

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 of the new discrete conformal structures using variational principles, which is a complement of Guo–Luo’s rigidity of the discrete conformal structures in Guo and Luo (Geom Topol 13(3):1265–1312, 2009) and Guo’s rigidity of vertex scaling in Guo (Commun Contemp Math 13(5):827–842, 2011) on surfaces with boundary. As a byproduct, new results on the convexity of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases 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 these 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.