Weak curvature conditions on metric graphs

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Abstract

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.

Keywords

metric graphs / curvature conditions / gradient flows / entropic inequalities

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