Metric Curvatures and their Applications 2: Metric Ricci Curvature and Flow

In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and we largely review our previous studies on the subject, based upon Wald’s curvature. In addition to our previous metric approaches to the discretization of Ricci curvature, we consider yet another one, based on the Haantjes curvature, interpreted as a geodesic curvature. We also try to understand the mathematical reasons behind the recent proliferation of discretizations of Ricci curvature. Furthermore, we propose another approach to the metrization of Ricci curvature, based on Forman’s discretization, and in particular we propose on that uses our graph version of Forman’s Ricci curvature.

💡 Research Summary

The paper “Metric Curvatures and their Applications 2: Metric Ricci Curvature and Flow” presents a comprehensive study of discrete Ricci curvature and Ricci flow from a purely metric perspective. The authors begin by motivating the need for a metric analogue of Ricci curvature, citing its central role in volume growth, the classical Ricci flow, and recent curvature‑dimension conditions (CD(K,N)) derived from optimal transport (Lott‑Villani‑Sturm) and Bakry‑Émery’s Bochner‑Weitzenböck formula. They review several existing discretizations—Ollivier’s transport‑based curvature, Bakry‑Émery, Lott‑Villani‑Sturm, Morgan’s approach, and graph‑based definitions by Stone and Forman—highlighting that each captures a different facet of the smooth Ricci tensor.

The core contribution is a metric Ricci curvature for piecewise‑linear (PL) surfaces based on Wald’s curvature, and a corresponding metric Ricci flow. Using smoothing techniques, the authors construct a sequence of smooth surfaces (S^2_m) that converge to a given PL surface (S^2_{Pol}) both in Hausdorff distance and in angle approximation. They prove that the Wald curvature (K_W) of the PL surface converges weakly to the classical Gaussian curvature (K) of the smooth approximants, establishing a bridge between combinatorial (angle‑defect) curvature and metric curvature.

From this foundation they define a metric Ricci flow for PL surfaces. The basic evolution law for an edge length (l_{ij}) incident to vertex (v_i) is \