[Paper Review] Characterizations of Forman curvature

Jürgen Jost, Florentin Münch|arXiv (Cornell University)|Oct 9, 2021

Geometric Analysis and Curvature Flows27 references4 citations

TL;DR

This paper establishes a semigroup characterization of Forman curvature via the Hodge Laplacian, proving that Forman and Ollivier curvatures coincide when maximizing Forman curvature over 2-cell weights. It introduces generalized curvature notions compatible with arbitrary path distances and shows that the proposed Forman curvature generalizes Forman’s original definition through specific weight choices.

ABSTRACT

We characterize Forman curvature lower bounds via contractivity of the Hodge Laplacian semigroup. We prove that Ollivier and Forman curvature coincide on edges when maximizing the Forman curvature over the choice of 2-cells. To this end, we translate between 2-cells and transport plans. Moreover, we give improved diameter bounds. We explicitly warn the reader that our Forman curvature notion does not coincide with Forman's original definition, but can be seen as generalization of the latter one.

Motivation & Objective

• To provide a semigroup-theoretic characterization of lower bounds for Forman curvature using the Hodge Laplacian on cell complexes.
• To prove that Forman curvature and Ollivier curvature coincide when Forman curvature is maximized over 2-cell weights.
• To generalize curvature notions to arbitrary path distances via a positive weight function ω.
• To reconcile the proposed Forman curvature with Forman’s original definition through specific weight transformations.
• To derive improved diameter bounds under generalized curvature lower bounds.

Proposed method

• Uses the Hodge Laplacian $ H = \delta\delta^* + \delta^*\delta $ on cell complexes with cell weights $ m $, defining Forman curvature as $ F(x) = Hx(x) - \sum_{y} |Hy(x)| $.
• Establishes equivalence between $ F(x) \geq R $ and $ \|e^{-tH}f\|_p \leq e^{-Rt}\|f\|_p $ for all $ f \in C(X_k) $, $ p \in [1,\infty] $, via semigroup estimates and maximum principles.
• Translates between transport plans in Ollivier curvature and cycle weights in Forman curvature using duality in linear programming.
• Introduces generalized Forman curvature $ F_\omega(x) = Hx(x) - \sum_{y \neq x} \frac{\omega(y)}{\omega(x)} |Hy(x)| $ and generalized Ollivier curvature $ \omega(x)\kappa_\omega(x) = \inf \delta\delta^*\delta f(x) $ under path distance $ \omega $.
• Applies duality to show that maximizing minimal Forman curvature over 2-cell weights equals the Ollivier curvature, using dual linear programs.
• Reconciles the generalized Forman curvature with Forman’s original definition via $ m = 1/w $, $ \omega = \sqrt{w} $, showing $ F_\omega(x) = F_{\text{original}}(x)/w(x) $.

Experimental results

Research questions

• RQ1 Can lower bounds on Forman curvature be characterized via the Hodge Laplacian semigroup?
• RQ2 Under what conditions do Forman curvature and Ollivier curvature coincide when Forman curvature is maximized over 2-cell weights?
• RQ3 How can Forman and Ollivier curvature be generalized to arbitrary path distances using a weight function $ \omega $?
• RQ4 What is the relationship between the proposed Forman curvature and Forman’s original definition?
• RQ5 What diameter bounds can be derived under generalized Forman curvature lower bounds?

Key findings

• Theorem 1.1 (Theorem 4.1) establishes that $ F(x) \geq R $ for all $ k $-cells $ x $ if and only if $ \|e^{-tH}f\|_p \leq e^{-Rt}\|f\|_p $ for all $ f \in C(X_k) $, $ p \in [1,\infty] $, providing the first semigroup characterization of Forman curvature.
• Corollary 6.1 (Theorem 6.1) proves that $ \kappa(x) = \max_K F_K(x) $, showing exact coincidence of Ollivier and Forman curvature when Forman curvature is maximized over 2-cell weights.
• Theorem 6.3 (Theorem 7.8) derives a diameter bound: if $ F_\omega(x) \geq R > 0 $ for all $ x \in X_1 $, then $ \text{diam}_\omega \leq \frac{2\min(D_1, D_0)}{R} $, improving prior bounds.
• The generalized Forman curvature $ F_\omega $ and Ollivier curvature $ \kappa_\omega $ are shown to be dual linear programs, with the maximum minimal Forman curvature equaling the Ollivier curvature.
• The paper shows $ F_\omega(x) \leq \kappa_\omega(x) $ for all cells $ x $, establishing a general inequality between the two curvatures.
• With $ m = 1/w $ and $ \omega = \sqrt{w} $, the generalized Forman curvature recovers Forman’s original definition, confirming it as a generalization.