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

TL;DR

This paper extends discrete uniformization theorems and introduces curvature flows with surgery for polyhedral surfaces, enabling the computation of hyperbolic metrics with prescribed combinatorial curvatures.

Contribution

It generalizes classical uniformization results to a parameterized setting and develops new curvature flows with surgery for polyhedral surfaces.

Findings

Established a discrete uniformization theorem for combinatorial α-curvature.

Proved long-time existence and convergence of α-Yamabe and α-Calabi flows with surgery.

Provided effective algorithms for computing hyperbolic metrics with prescribed curvatures.

Abstract

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. 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.