Weak curvature conditions on metric graphs

Juliane Krautz; Juliane Krautz

doi:10.3934/nhm.2026031

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 725-754. doi: 10.3934/nhm.2026031

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Research article

Weak curvature conditions on metric graphs

Juliane Krautz ^,

Universität Augsburg, Institut für Mathematik, Universitätsstraße 12a, 86159 Augsburg, Germany

Received: 17 December 2025 Revised: 05 March 2026 Accepted: 17 March 2026 Published: 27 April 2026

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.
- metric graphs,
- curvature conditions,
- gradient flows,
- entropic inequalities
Citation: Juliane Krautz. Weak curvature conditions on metric graphs[J]. Networks and Heterogeneous Media, 2026, 21(3): 725-754. doi: 10.3934/nhm.2026031

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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.

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