LAUSR

This paper investigates the combinatorial α-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial α-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu’s discrete uniformization theorem for classical combinatorial curvature [16]. We further introduce combinatorial α-Yamabe flow and combinatorial α-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial α-curvatures. To handle the potential singularities along the combinatorial curvature flows, we do surgery along the flows by edge flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu [16], we prove the longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial α-curvatures.