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 classical curvature. A discrete uniformization theorem established, generalizes Gu-Guo-Luo-Sun-Wu’s curvature [J. Differential Geom. 109 (2018), pp. 431–466]. We further introduce $\alpha$-Yamabe flow and $\alpha$-Calabi to find with prescribed $\alpha$-curvatures. To handle potential singularities along flows, we do surgery flows by edge flipping. Using conformal theory established Gu-Guo-Luo-Sun-Wu 431–466], prove longtime existence convergence surgery, provide effective algorithms finding