We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian, [Formula presented] 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–Émery 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.

Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries / Rizzi, Luca; Stefani, Giorgio. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - 285:9(2023), pp. 1-25. [10.1016/j.jfa.2023.110099]

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

Luca Rizzi
;
Giorgio Stefani
2023-01-01

Abstract

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian, [Formula presented] 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–Émery 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.
2023
285
9
1
25
110099
https://arxiv.org/abs/2301.00735
Rizzi, Luca; Stefani, Giorgio
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/134990
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