Weak curvature conditions on metric graphs

Starting from pointwise gradient estimates for the heat semigroup, we study three characterizations of weak lower curvature bounds on metric graphs. More precisely, we prove the equivalence between a weak notion of the Bakry-Émery curvature condition, a weak Evolutionary Variational Inequality and a weak form of geodesic convexity. The proof is based on a careful regularization of absolutely continuous curves together with an explicit representation of the Cheeger energy. We conclude with a brief discussion on possible applications to the Schrödinger bridge problem on metric graphs.

💡 Research Summary

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This paper investigates weak lower curvature bounds on metric graphs by exploiting pointwise gradient estimates for the heat semigroup. The author introduces three equivalent formulations of a weak curvature condition: a weak Bakry‑Émery (BE) inequality, a weak Evolutionary Variational Inequality (EVI), and a weak geodesic convexity condition (RCD(_w)).

The setting is a compact metric graph (G=(V,E,\ell)) equipped with the natural length metric (d) and the Lebesgue measure (L). On each edge the standard one‑dimensional heat equation is considered, and Kirchhoff-type coupling conditions are imposed at vertices, yielding a global heat semigroup ((P_t)_{t\ge0}). The author first identifies the Cheeger energy (\mathrm{Ch}) on the graph with the classical Dirichlet energy, proving that (W^{1,\infty}= \mathrm{Lip}) holds on metric graphs. This identification allows the heat flow to be interpreted simultaneously as the (L^2)-gradient flow of (\mathrm{Ch}) and as the Wasserstein gradient flow of the relative entropy (\mathrm{Ent}_m).

A central technical contribution is the construction of a regularization procedure for absolutely continuous curves ((\mu_s)_{s\in