[Paper Review] Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I

Xu Xu|arXiv (Cornell University)|Jun 11, 2018

Geometric Analysis and Curvature Flows44 references4 citations

TL;DR

This paper introduces a parameterized discrete curvature ($\alpha$-curvature) for polyhedral surfaces and establishes a discrete uniformization theorem generalizing prior results. It proves global rigidity of $\alpha$-curvature under vertex scaling and demonstrates convergence of both combinatorial $\alpha$-Yamabe and $\alpha$-Calabi flows with surgery to metrics of constant $\alpha$-curvature, confirming generalized conjectures by Luo.

ABSTRACT

In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uniformization theorem obtained by Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete curvature with respect to the discrete conformal factors, which confirms a generalized Luo conjecture on rigidity of discrete curvatures. We further introduce a parameterized discrete Yamabe flow for piecewise linear metrics on surfaces. To handle the possible singularities along the flow, we do surgery on the flow by flipping. Then we prove that the flow with surgery converges to a piecewise linear metric with constant discrete $α$-curvature, which confirms another generalized Luo conjecture on convergence of discrete Yamabe flow with surgery. We also introduce a parameterized discrete Calabi flow and prove the convergence of the flow with surgery, which generalizes the convergence proved by Zhu and the author.

Motivation & Objective

• To generalize classical discrete curvature to a parameterized $\alpha$-curvature for piecewise linear metrics on triangulated surfaces.
• To establish a discrete uniformization theorem for $\alpha$-curvature using a variational principle.
• To prove global rigidity of $\alpha$-curvature with respect to vertex scaling, confirming a generalized Luo conjecture.
• To define and analyze combinatorial $\alpha$-Yamabe and $\alpha$-Calabi flows with surgery for convergence to constant $\alpha$-curvature metrics.
• To extend convergence results for discrete curvature flows beyond the $\alpha=0$ case, generalizing prior work on Calabi flow.

Proposed method

• Introduce $\alpha$-curvature as $R_{\alpha,i} = K_i / w_i^\alpha$, where $K_i$ is the classical combinatorial curvature and $w_i$ is the vertex scaling factor.
• Define the admissible space $\Omega^\mathcal{T}(d)$ of conformal factors ensuring triangle inequalities are preserved under vertex scaling.
• Construct a convex functional $W_\alpha(u)$ on the logarithmic conformal factor space $\mathcal{U}^\mathcal{T}(d)$, which is proper under the condition $\alpha\chi(S) \leq 0$.
• Define the combinatorial $\alpha$-Yamabe flow via $du_i/dt = (\Delta_\alpha \mathbf{F}_\alpha)_i$, with surgery to handle singularities from degenerating triangles.
• Define the combinatorial $\alpha$-Calabi flow as $du_i/dt = (\Delta_\alpha \mathbf{F}_\alpha)_i$, preserving the total $\sum w_i^\alpha$ under the flow.
• Use Lyapunov function techniques and convexity of $W_\alpha(u)$ to prove long-time existence and convergence of both flows with surgery.

Experimental results

Research questions

• RQ1 Can a discrete uniformization theorem be established for the generalized $\alpha$-curvature on polyhedral surfaces?
• RQ2 Is the $\alpha$-curvature globally rigid under vertex scaling, as conjectured by Luo?
• RQ3 Does the combinatorial $\alpha$-Yamabe flow with surgery converge to a metric of constant $\alpha$-curvature?
• RQ4 Does the combinatorial $\alpha$-Calabi flow with surgery converge to a constant $\alpha$-curvature metric when $\alpha\chi(S) \leq 0$?
• RQ5 Can the convergence results for the $\alpha=0$ case be generalized to arbitrary real $\alpha$?

Key findings

• The discrete uniformization theorem for $\alpha$-curvature is established, generalizing the result of Gu-Luo-Sun-Wu for $\alpha=0$.
• Global rigidity of $\alpha$-curvature with respect to vertex scaling is proven, confirming a generalized version of Luo’s conjecture.
• The combinatorial $\alpha$-Yamabe flow with surgery exists for all time and converges to a piecewise linear metric with constant $\alpha$-curvature.
• The combinatorial $\alpha$-Calabi flow with surgery exists for all time and converges to a constant $\alpha$-curvature metric when $\alpha\chi(S) \leq 0$, generalizing the result in [70].
• The functional $W_\alpha(u)$ is shown to be strictly convex and proper on the constraint set $\sum w_i^\alpha = N$, ensuring convergence of the flows.
• The convergence of both flows is proven via the decay of a Lyapunov functional $W_\alpha(u(t))$, which decreases monotonically and has a unique minimum at the desired constant $\alpha$-curvature metric.