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Article Dans Une Revue Journal of Functional Analysis Année : 2023

Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries

Résumé

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-\'Emery inequality for the corresponding sub-Laplacian implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry-\'Emery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0,N) condition, yielding a counterexample to gluing theorems in the RCD setting.

Dates et versions

hal-03921243 , version 1 (03-01-2023)

Identifiants

Citer

Luca Rizzi, Giorgio Stefani. Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries. Journal of Functional Analysis, 2023, 285 (9), pp.110099. ⟨10.1016/j.jfa.2023.110099⟩. ⟨hal-03921243⟩
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