Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces

TL;DR

This paper introduces a parameterized combinatorial curvature and flow for discrete conformal structures on polyhedral surfaces, proving rigidity, convergence, and providing algorithms for prescribed curvature problems.

Contribution

It generalizes classical combinatorial curvature and Ricci flow to a parameterized setting, confirming conjectures and extending flow methods to handle singularities.

Findings

Proves local and global rigidity of parameterized combinatorial curvature.

Shows exponential convergence of the extended Ricci flow under certain conditions.

Provides an effective algorithm for prescribing combinatorial urvature.

Abstract

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 $\alpha$-curvature for discrete conformal structures on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. Then we prove the local and global rigidity of combinatorial $\alpha$-curvature with respect to discrete conformal structures on polyhedral surfaces, which confirms parameterized Glickenstein rigidity conjecture. To study the Yamabe problem for combinatorial $\alpha$-curvature, we introduce combinatorial $\alpha$-Ricci flow for discrete conformal structures on polyhedral surfaces, which is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings and Luo's combinatorial Yamabe flow for vertex scaling on polyhedral surfaces. To handle the potential singularities of the combinatorial $\alpha$-Ricci flow, we extend the flow through the singularities by extending the inner angles in triangles by constants. Under the existence of a discrete conformal structure with prescribed combinatorial curvature, the solution of extended combinatorial $\alpha$-Ricci flow is proved to exist for all time and converge exponentially fast for any initial value. This confirms a parameterized generalization of another conjecture of Glickenstein on the convergence of combinatorial Ricci flow, gives an almost equivalent characterization of the solvability of Yamabe problem for combinatorial $\alpha$-curvature in terms of combinatorial $\alpha$-Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial $\alpha$-curvatures.