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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-Wus discrete uniformization theorem for classical combinatorial curvature. 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, 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.
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hy
Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth conformal structure on surfaces that assigns discrete metrics by scalar functions defined on vertices. In this paper, we introduce combinatorial $alpha$-curvatur
The paper proves a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin-Sullivans theorem on convergence of circle packing mappings to the Riemann mapping in the new setting of
The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations of surfaces
In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth func