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Failure of the Curvature-Dimension Condition in Sub-Finsler Manifolds

Year: 2026
Type: article
Abstract: The Lott–Sturm–Villani curvature-dimension condition $$\textsf{CD}(K,N)$$ CD ( K , N ) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N . It has been recently proved that this condition does not hold in Sub-Riemannian geometry for every choice of the parameters K and N . In this paper, we extend this result to the context Sub-Finsler geometry, showing that the $$\textsf{CD}(K,N)$$ CD ( K , N ) condition is not well-suited to characterize curvature in this setting. Firstly, we show that this condition fails in (strict) Sub-Finsler manifolds equipped with an analytic strongly convex norm and with a positive smooth measure. Secondly, we focus on the Sub-Finsler Heisenberg group, proving that curvature-dimension bounds cannot hold also when the reference norm is less regular, in particular when it is of class $$C^{1,1}$$ C 1 , 1 . Finally, we show the failure of the (weaker) measure contraction property $$\textsf{MCP}(K,N)$$ MCP ( K , N ) in the Sub-Finsler Heisenberg group, equipped with a singular strictly convex norm and with a positive smooth measure. This result contrasts with what happens in the Sub-Riemannian Heisenberg group, which instead satisfies $$\textsf{MCP}(0,5)$$ MCP ( 0 , 5 ) .
DOI: 10.1007/s12220-026-02511-z
Institution: University of Bonn
Language: English
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4383469275
FWCI:
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Cited by: 3
Related to: 10