Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II.

This paper investigates the combinatorial \alpha-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 \alpha-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature [J. Differential Geom. 109 (2018), pp. 431–466]. We further introduce combinatorial \alpha-Yamabe flow and combinatorial \alpha-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial \alpha-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 [J. Differential Geom. 109 (2018), pp. 431–466], we prove the longtime existence and convergence of combinatorial \alpha-Yamabe flow and combinatorial \alpha-Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial \alpha-curvatures. [ABSTRACT FROM AUTHOR]