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

TL;DR

This paper introduces new sub-Riemannian manifolds with boundary, inspired by the $eta$-Grushin plane, that satisfy the $ ext{RCD}(K,N)$ condition, expanding the class of spaces with Ricci curvature bounds.

Contribution

It constructs novel examples of sub-Riemannian manifolds with boundary satisfying the $ ext{RCD}(K,N)$ condition, using almost-Riemannian structures inspired by the $eta$-Grushin plane.

Findings

New sub-Riemannian spaces satisfy $ ext{RCD}(K,N)$ condition.

Construction involves measures vanishing on the boundary.

Examples include half-plane, hemisphere, and hyperbolic half-plane.

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.