A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial $α$-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures, we introduce the combinatorial $α$-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial $α$-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial $α$-Ricci flow with surgery. As an application of the combinatorial $α$-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. We further introduce the combinatorial $α$-Calabi flow with surgery and study its longtime behavior.