[Paper Review] On the properties of the combinatorial Ricci flow for surfaces

Emil Saucan|arXiv (Cornell University)|Apr 11, 2011

Geometric Analysis and Curvature Flows22 references3 citations

TL;DR

This paper establishes the existence, uniqueness, and singularity formation of the combinatorial Ricci flow on polyhedral surfaces using a metric curvature approach, extending results from smooth Ricci flow to discrete settings. It proves that the flow is realizable in ℝ³ and generalizes to CW complexes via a broader notion of metric curvature, ensuring convergence and stability under approximation by smooth surfaces.

ABSTRACT

We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for the combinatorial one and that, moreover, the same results hold for a more general, metric notion of curvature. Furthermore, using the metric curvature approach, we show the existence of the Ricci flow for polyhedral manifolds of piecewise constant curvature. We also study the problem of the realizability of the said flow in $\mathbb{R}^3$.

Motivation & Objective

• To extend the theory of smooth Ricci flow to discrete, polyhedral surfaces via combinatorial Ricci flow.
• To establish existence and uniqueness of the combinatorial Ricci flow, both forward and backward in time.
• To analyze the formation of singularities in the combinatorial flow and its relation to curvature.
• To investigate the realizability of the flow as an embedding in ℝ³ for polyhedral and piecewise constant curvature manifolds.
• To generalize the framework to a broader class of geometric objects, including non-regular CW complexes, via a metric curvature notion.

Proposed method

• Uses Brehm and Kühnel’s approximation theorem to embed a polyhedral surface into a sequence of smooth surfaces converging in Hausdorff and curvature measures.
• Applies classical Ricci flow theory to the approximating smooth surfaces to infer properties for the discrete flow.
• Defines combinatorial curvature via angular defect: $ K(p) = 2\pi - \sum_{i=1}^{m_p} \alpha_i(p) $, where $ \alpha_i(p) $ are face angles at vertex $ p $.
• Models edge lengths via circle packing: $ l_{ij} = \sqrt{r_i^2 + r_j^2 + 2r_i r_j \cos(\Phi(e_{ij}))} $, with $ r_i $ as radii.
• Introduces a generalized metric curvature notion to extend results beyond standard triangulations to broader classes like CW complexes.
• Employs $ \delta $-approximation and $ \alpha $-approximation techniques from differential topology to ensure smooth approximation of piecewise linear maps.

Experimental results

Research questions

• RQ1 Does the combinatorial Ricci flow exist and is it unique for compact polyhedral surfaces?
• RQ2 Can the reverse (backward) Ricci flow be defined and does it preserve geometric properties?
• RQ3 Under what conditions do singularities form in the combinatorial Ricci flow?
• RQ4 Is the combinatorial Ricci flow realizable as a smooth embedding in $\mathbb{R}^3$?
• RQ5 Can the theory be extended beyond regular triangulations to more general geometric objects like CW complexes?

Key findings

• The combinatorial Ricci flow exists and is unique for compact polyhedral surfaces without boundary, extending results from smooth Ricci flow.
• The backward Ricci flow also exists, ensuring time-reversibility of the flow under the same curvature and metric conditions.
• Singularities in the flow correspond to curvature concentration and are characterized by vanishing edge lengths or degenerate configurations.
• The flow is realizable in $\mathbb{R}^3$ for polyhedral manifolds of piecewise constant curvature, provided local embeddings exist.
• A generalized metric curvature approach allows extension of the flow to non-regular CW complexes, broadening applicability beyond standard triangulations.
• Smooth approximations of polyhedral surfaces exist in both metric and curvature senses, ensuring convergence of curvature measures and enabling transfer of smooth flow results to the discrete case.