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

Abstract The Lott–Sturm–Villani curvature-dimension condition $$\textsf{CD}(K,N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>CD</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\textsf{CD}(K,N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>CD</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$C^{1,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Finally, we show the failure of the (weaker) measure contraction property $$\textsf{MCP}(K,N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>MCP</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\textsf{MCP}(0,5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>MCP</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .