Generalized circle patterns on surfaces with boundary

In [Rigidity of polyhedral surfaces, II, Geom. Topol. 13(3) (2009) 1265–1312], Guo and Luo generalized Bobenko–Springborn’s hyperbolic circle patterns on closed surfaces to surfaces with boundary, and proved their rigidity. In this paper, we prove the existence of Guo–Luo’s generalized circle patterns with prescribed combinatorial curvatures on surfaces with boundary by the continuity method. This result provides a partial answer to a question posed by Guo and Luo in the above-mentioned paper. Furthermore, we introduce two combinatorial curvature flows associated with these generalized circle patterns on surfaces with boundary, and prove the long-time existence and convergence of the solutions to these combinatorial curvature flows.