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

Abstract

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian,12Δ(‖∇u‖2)≥g(∇u,∇Δu)+K‖∇u‖2,K∈R, 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.