Curvature-dimension condition of sub-Riemannian $\alpha$-Grushin half-spaces

Abstract

We provide new examples of sub-Riemannian manifolds with boundary equipped with a smooth measure that satisfy the $\mathsf{RCD}(K,N)$ condition. They are constructed by equipping the half-plane, the hemisphere and the hyperbolic half-plane with a two-dimensional almost-Riemannian structure and a measure that vanishes on their boundary. The construction of these spaces is inspired from the geometry of the $\alpha$-Grushin plane.

16 pages. v2: 19 pages. Some significant clarifications added, corrections to a computation. Final version to appear in the Transactions of the London Mathematical Society