Curvature‐dimension condition of sub‐Riemannian α$\alpha$‐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.

INTRODUCTION

In the past few decades, the curvature-dimension condition $$\mathsf {CD}(K, N)$$ was specifically developed to generalize to the nonsmooth setting of metric measure spaces the concept of a lower bound on the $$N$$-Bakry–Émery Ricci tensor $$\mathrm{Ric}_{N,V} \geqslant K$$ on weighted Riemannian manifolds, using the theory of optimal transport (see [24, 38, 39]). The $$\mathsf {RCD}(K,N)$$ condition is a restriction of the $$\mathsf {CD}(K,N)$$ condition by requiring in addition that the space is infinitesimally Hilbertian, which excludes Finsler manifolds from being $$\mathsf {RCD}$$-spaces (see [2, 16, 18] for example). The conditions $$\mathsf {CD}(K,N)$$ and $$\mathsf {RCD}(K,N)$$ have not only established known meaningful geometric inequalities to nonsmooth spaces, but also helped obtain new results, even in the smooth setting.

However, it has been found that the large class of sub-Riemannian manifolds equipped with a smooth positive measure does not satisfy any of the $$\mathsf {CD}$$ conditions. In the Heisenberg group, this was first proven in [21] and then for any Carnot groups in [5]. The same author then showed in [22] that the same result holds for any sub-Riemannian manifold whose distribution has constant rank strictly smaller than its topological dimension. A no-$$\mathsf {CD}$$ theorem for two-dimensional rank-varying structures was then found in [27]. Finally, the most general result to date, established in [36], states that no curvature-dimension condition can hold for any sub-Riemannian manifold equipped with a positive smooth measure. As a side note, we refer the reader interested in the study of curvature-dimension in the sub-Finsler setting to the works [8, 9, 28] and [10].

Surprisingly, it was discovered in [34, 35] that it suffices to consider a sub-Riemannian structure on a manifold with boundary and equip it with a smooth measure that vanishes on the boundary points to construct an example of a $$\mathsf {CD}$$-space in sub-Riemannian geometry. This example, which we revisit here, is also discussed in [36]. For $$\alpha \geqslant 0$$, the $$\alpha$$-Grushin plane $$\mathbb {G}_\alpha$$ is the sub-Riemannian structure on $$\mathbb {R}^2$$ induced by the vector fields $$$\begin{equation*} X={\partial}_{x},\hspace*{0.202em}\hspace*{0.202em}\hspace*{0.202em}\hspace*{0.202em}{Y}^{\alpha}=|x{|}^{\alpha}{\partial}_{y}. \end{equation*}$$$This defines a metric space $$(\mathbb {G}_\alpha, \mathop {}\!\mathrm{d}_{\mathbb {G}_\alpha })$$ that admits the metric tensor $$$\begin{equation*} g_{\mathbb {G}_\alpha }=\mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x+\frac{1}{|x|^{2\alpha }}\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y, \end{equation*}$$$at nonsingular points. In [35, 36], the authors equip this metric space with the $$\beta$$-weighted measure $${\mathfrak{m}}_{\beta}=|x{|}^{\beta -\alpha} \mathrm{d}x\mathrm{d}y$$. Note that this weighted measure vanishes on the singular set $$\lbrace x = 0\rbrace$$ if $$\beta >\alpha$$, it coincides with the Lebesgue measure if $$\beta =\alpha$$, and it fails to be locally finite if $$\beta \leqslant \alpha -1$$. Although [35] and [36] examined the same space, their techniques are different. In [35], Pan–Wei constructed a Riemannian manifold $$M:= [0,+\infty) \times _{f} \mathbb {S}^{n-1} \times _{g} \mathbb {S}^1$$ with a doubly warped product metric such that $$f(r)\sim \sqrt {r}$$ and $$g(r)\sim r^{-2\alpha }$$ as the first factor $$r$$ goes to $$+\infty$$. It can be verified that $$M$$ has positive Ricci curvature. By taking the asymptotic cone of the universal cover based at a specific point, it can be argued that its (collapsed) limit is a Ricci limit space satisfying $$\mathsf {RCD}(0,n+1)$$. Montgomery pointed out that this limit space is actually the $$\alpha$$-Grushin half-plane.

In [36], Rizzi–Stefani revisited this example by showing that the half-plane $$\overline{\mathbb {G}}_\alpha ^+:= \lbrace x \geqslant 0\rbrace$$ and the open half-plane $$\mathbb {G}_\alpha ^+:= \lbrace x > 0\rbrace$$ are both geodesically convex subsets of $$\mathbb {G}_\alpha$$ and that the weighted incomplete Riemannian manifold $$(\mathbb {G}_\alpha ^+, \mathop {}\!\mathrm{d}_{\mathbb {G}_\alpha }, \mathfrak {m}_{\beta })$$ satisfies the condition $$\mathrm{Ric}_{N,V} \geqslant K$$ if and only if 1 $$$\begin{equation} K\leqslant 0,\nobreakspace \nobreakspace \text{and}\nobreakspace \nobreakspace -\alpha ^2-\alpha +\beta \cdot \min {\left(\alpha,1-\frac{\beta }{N-2}\right)}\geqslant 0, \end{equation}$$$where $$V(x,y)=-\beta \log |x|$$ is a singular potential. Actually, [36] only considered the case $$\alpha = 1$$ but the general case $$\alpha \geqslant 0$$ is obtained in the same way. They showed that the $$\alpha$$-Grushin half-plane $$\overline{\mathbb {G}}_\alpha ^+$$ verifies $$\mathsf {CD}(K,N)$$ for the values of $$K \in \mathbb {R}$$ and $$N \in (2, +\infty]$$ such that (1) holds, using the geodesic convexity and the weighted Ricci curvature lower bound on the interior $$\mathbb {G}_\alpha ^+$$. They also proved that $$\overline{\mathbb {G}}_\alpha ^+$$ is infinitesimally Hilbertian, and thus the $$\mathsf {RCD}(K, N)$$ is also satisfied.

The goal of this paper is to provide a few more examples of this phenomenon. We introduce a sub-Riemannian structure on the hemisphere and on the hyperbolic half-plane that depends, on a parameter $$\alpha \geqslant 0$$, and by tweaking their unbounded “Riemannian” volume measure with a parameter $$\beta \geqslant \alpha$$, we obtain metric measure spaces that are $$\mathsf {RCD}$$-spaces. One motivation for studying these kinds of singular spaces is the singular Weyl's laws studied in [11, 14, 15] and the references therein.

In Subsection 3.1, we define the $$\alpha$$-Grushin hemisphere and its $$\beta$$-weighted measure. This metric measure space is an example of sub-Riemannian manifold where the $$\mathsf {RCD}(K, N)$$ condition holds for some $$K > 0$$. $1.1$

Theorem

The metric measure space consisting of the sub-Riemannian $$\alpha$$-Grushin hemisphere equipped with its $$\beta$$-weighted measure satisfies the following properties.

• (i)For any $$\alpha \geqslant 0$$ and $$K>0$$, there is $$\beta \geqslant \alpha$$ such that the $$\mathsf {RCD}(K,\infty)$$ condition is satisfied.
• (ii)For any $$\alpha \geqslant 1$$ and $$N\geqslant 2+4\alpha (\alpha +1)$$, there is $$\beta >\alpha$$ such that $$\mathsf {RCD}(0,N)$$ holds.
• (iii)When $$\mathsf {RCD}(0,N)$$ is satisfied for some $$N \in \left[2, +\infty \right)$$, there is $$K>0$$ such that $$\mathsf {RCD}(K,N)$$ holds.

Similarly, we introduce in Subsection 3.2 the $$\alpha$$-Grushin hyperbolic half-plane and its $$\beta$$-weighted measure. This metric measure space is an example of sub-Riemannian manifold where the $$\mathsf {RCD}(K, N)$$ condition holds for some $$K < 0$$. $1.2$

Theorem

The metric measure space consisting of the sub-Riemannian $$\alpha$$-Grushin hyperbolic half-plane equipped with its $$\beta$$-weighted measure satisfies the following properties.

• (i)For any $$\alpha \geqslant 1$$ and $$N\geqslant 2+4\alpha (\alpha +1)$$, there are $$\beta >\alpha$$ and $$K<0$$ such that $$\mathsf {RCD}(K,N)$$ holds.
• (ii)For any $$\alpha,\beta \geqslant 0$$ and $$N\in (2,+\infty)$$, it does not satisfy $$\mathsf {RCD}(0,N)$$. Furthermore, it satisfies $$\mathsf {RCD}(0,+\infty)$$ if and only if $$\alpha =1$$ and $$\beta \geqslant 2$$.
• (iii)For any $$\alpha \geqslant 1$$, there are $$\beta >\alpha$$ and $$N\in (-\infty,0)$$ such that $$\mathsf {CD}(0,N)$$ holds.

The negativity of $$K$$ is essential in the sense that the $$\alpha$$-Grushin hyperbolic half-plane satisfies the $$\mathsf {CD}(0,N)$$ condition only if $$N$$ is negative or $$+\infty$$. An odd phenomenon occurs at $$\alpha =1$$, where the 1-Grushin hyperbolic half-plane satisfies $$\mathsf {RCD}(0,+\infty)$$. It is unclear if this phenomenon is relevant to the validity of hyperbolicity or $$\mathsf {CAT}(0)$$ property in the 1-Grushin hyperbolic plane. $1.3$

Remark

In [34], Pan constructed a sequence of doubly warped metric spaces which collapses to the 1-Grushin hemisphere. We do not know if a similar construction holds for other $$\alpha$$-Grushin spaces, nor if the limit measure in such a construction would be equal to our $$\beta$$-weighted measure.

As another generalization of the $$\alpha$$-Grushin plane, we introduce a new sub-Riemannian manifold with infinite Hausdorff dimension, which we call the $$\infty$$-Grushin plane. The description of this space is found in Subsection 3.3. By tweaking the Riemannian volume measure by multiplicative factors that depend on two parameters $$\beta$$ and $$\gamma$$, the $$\infty$$-Grushin half-plane is shown to satisfy the $$\mathsf {RCD}(0,+\infty)$$. $1.4$

Theorem

The metric measure space consisting of the sub-Riemannian $$\infty$$-Grushin half-plane equipped with its $$(\beta, \gamma)$$-weighted measure satisfies the following properties.

• (i)There are $$\beta,\gamma \geqslant 0$$ such that $$\mathsf {RCD}(0,+\infty)$$ holds.
• (ii)For any $$\beta,\gamma \geqslant 0$$ and $$K\in \mathbb {R}$$, there is no $$N\in (2,+\infty)$$ such that $$\mathsf {RCD}(K,N)$$ holds.

$1.5$

Remark

The following consequence is to be noted. Theorem 1.4 implies that, for any $$N\geqslant 2$$, Gromov–Hausdorff limits of a sequence of metric measure spaces in the class$$$\begin{equation*} \mathcal {X}_N:={\left\lbrace (X,\mathsf {d},\mathfrak {m})\mid \text{proper, }\mathsf {RCD}(0,\infty) \text{ and } \dim _H(X,\mathsf {d})\leqslant N\right\rbrace} \end{equation*}$$$can have infinite Hausdorff dimension, that is, $$\overline{\mathcal {X}_N}\not\subset \bigcup _{n\in \mathbb {N}}\mathcal {X}_n$$. Indeed, for $$\varepsilon >0$$, the subsets $$\mathbb {R}_{\geqslant \varepsilon }\times \mathbb {R}$$ of the $$\infty$$-Grushin half-plane are weighted Riemannian manifolds with boundary contained in $$\mathcal {X}_2$$, which converges to the $$\infty$$-Grushin half-plane as $$\varepsilon \rightarrow 0$$, which has infinite Hausdorff dimension (see Lemma 3.11).

The strategy of the proof of the previous results is fairly simple. After studying these spaces in detail in Section 3, we show in Section 4 that, under the right conditions of smoothness and geodesic convexity (see Theorem 4.1), the validity of the $$\mathsf {CD}(K, N)$$ condition for a metric measure space whose interior is Riemannian is equivalent to the bound $$\mathrm{Ric}_{N,V} \geqslant K$$ on its interior. The values of $$(K, N)$$ for which the $$\mathsf {CD}(K, N)$$ and $$\mathsf {RCD}(K, N)$$ conditions holds in the $$\alpha$$-Grushin half-spaces are then just a matter of Ricci curvature computations, which we provide in Section 5.

PRELIMINARIES

The $$\mathsf {CD}(K,N)$$ and $$\mathsf {RCD}(K,N)$$ conditions

First, we start by recalling the curvature-dimension condition $$\mathsf {CD}(K,N)$$ and Riemannian curvature-dimension condition $$\mathsf {RCD}(K,N)$$. A metric measure space is a triple $$(\mathsf {X},\mathop {}\!\mathrm{d},\mathfrak {m})$$ where $$(\mathsf {X},\mathop {}\!\mathrm{d})$$ is a complete, separable, locally compact and geodesic metric space, and $$\mathfrak {m}$$ is a nonnegative Radon measure on it. We emphasize that the metric measure spaces will always be assumed to be essentially nonbranching. We denote by $$C([0, 1], \mathsf {X})$$ the space of continuous curves from $$[0, 1]$$ to $$\mathsf {X}$$, and for $$s \in [0, 1]$$, we let $$e_s: C([0, 1], \mathsf {X})\ni \gamma \mapsto \gamma (s) \in \mathsf {X}$$ be the evaluation map. A geodesic $$\gamma:[0,1]\rightarrow \mathsf {X}$$ is a curve such that $$\mathop {}\!\mathrm{d}(\gamma (s), \gamma (t)) = |t - s| \mathop {}\!\mathrm{d}(\gamma (0), \gamma (1))$$ for all $$s, t \in [0, 1]$$, and we denote by $$\operatorname{Geo}(\mathsf {X})$$ the space of all geodesics on $$(\mathsf {X},\mathop {}\!\mathrm{d})$$. Furthermore, the set of Borel probability measures on $$\mathsf {X}$$ is denoted by $$\mathcal {P}(\mathsf {X})$$ and the set of those having finite second momentum by $$\mathcal {P}_2(\mathsf {X}) \subseteq \mathcal {P}(\mathsf {X})$$. We endow the space $$\mathcal {P}_2(\mathsf {X})$$ with the Wasserstein distance $$W_2$$, defined by 2 $$$\begin{equation} {W}_{2}^{2}({\mu}_{0},{\mu}_{1}):=\underset{\pi \in \mathrm{Adm}({\mu}_{0},{\mu}_{1})}{\rm inf}\int \mathrm{d}{(x,y)}^{2}\, \mathrm{d}\pi (x,y), \end{equation}$$$where $$\mathsf {Adm}(\mu _0, \mu _1):=\lbrace \pi \in \mathsf {X}\times \mathsf {X}\mid (\mathtt {p}_i)_\sharp \pi = \mu _i,\ \mathtt {p}_i: \text{the projection to the $i$th factor}\nobreakspace (i=1,2)\rbrace$$. The metric space $$(\mathcal {P}_2(\mathsf {X}),W_2)$$ is itself complete, separable and geodesic. A probability measure $$\pi \in \mathcal P(\mathsf {X}\times \mathsf {X})$$ which attains the minimum values in (2) is called an optimal transport plan.

The $$\mathsf {CD}(K,N)$$ condition for positive $$N$$

In this context, the curvature-dimension condition is defined as follows. For every $$K \in \mathbb {R}$$, $$N\in [1,\infty)$$ and $$t\in [0,1]$$, the distortion coefficients are the functions $$$\begin{equation*} \tau _{K,N}^{(t)}(\theta):= {\begin{cases} \displaystyle +\infty & \textrm {if}\ (N-1)\pi ^{2}\leqslant K\theta ^{2} \nobreakspace \text{ and }\nobreakspace K\theta ^2>0,\\ \displaystyle t^{1/N}{\left(\frac{\sin (t\theta \sqrt {K/N})}{\sin (\theta \sqrt {K/N})}\right)}^{1-1/N} & \textrm {if}\ 0 < K\theta ^{2} < (N-1)\pi ^{2},\\ t & \textrm {if}\ K\theta ^2 =0, \nobreakspace \nobreakspace \text{or}\nobreakspace \nobreakspace K\theta ^2<0\nobreakspace \text{ and }\nobreakspace N=1, \\ \displaystyle t^{1/N}{\left(\frac{\sinh (t\theta \sqrt {-K/N})}{\sinh (\theta \sqrt {-K/N})}\right)}^{1-1/N} & \textrm {if}\ K\theta ^2 < 0\nobreakspace \text{ and }\nobreakspace N>1. \end{cases}} \end{equation*}$$$$2.1$

Definition

$$\mathsf {CD}(K,N)$$

For $$K\in \mathbb {R}$$ and $$N\in [1,+\infty]$$, a metric measure space $$(\mathsf {X},\mathsf {d},\mathfrak {m})$$ is said to satisfy the $$\mathsf {CD}(K,N)$$ condition if for every pair of absolutely continuous measures $$\mu _0=\rho _0\mathfrak {m},\mu _1= \rho _1 \mathfrak {m}\in \mathcal {P}_2(\mathsf {X})$$, there exists an optimal transport plan $$\pi \in \mathcal P(\mathsf {X}\times \mathsf {X})$$ and an absolutely continuous $$W_2$$-geodesic $$(\rho _t\mathfrak {m})_{t\in [0,1]}$$ connecting them such that the following inequality holds for every $$N^{\prime } \geqslant N$$ and every $$t \in [0,1]$$:$$$\begin{equation*} \int _\mathsf {X}\rho _t^{1-\frac{1}{N^{\prime }}} \mathop {}\!\mathrm{d}\mathfrak {m}\geqslant \int _{\mathsf {X}\times \mathsf {X}} {\left[ \tau ^{(1-t)}_{K,N^{\prime }} {\left(\mathop {}\!\mathrm{d}(x,y) \right)} \rho _{0}(x)^{-\frac{1}{N^{\prime }}} + \tau ^{(t)}_{K,N^{\prime }} {\left(\mathop {}\!\mathrm{d}(x,y) \right)} \rho _{1}(y)^{-\frac{1}{N^{\prime }}} \right]} \mathop {}\!\mathrm{d}\pi (x,y), \end{equation*}$$$if $$N < +\infty$$, and, if $$N = +\infty$$,$$$\begin{equation*} \int _\mathsf {X}\rho _t\log \rho _t \mathop {}\!\mathrm{d}\mathfrak {m}\leqslant (1 - t)\int _\mathsf {X}\rho _0\log \rho _0 \mathop {}\!\mathrm{d}\mathfrak {m}+ t \int _\mathsf {X}\rho _1\log \rho _1 \mathop {}\!\mathrm{d}\mathfrak {m}- \frac{K}{2} t(1 - t) W_2^2(\mu _0, \mu _1). \end{equation*}$$$

It is not difficult to see that if a metric measure space satisfies the $$\mathsf {CD}(K, N)$$ condition for some $$K \in \mathbb {R}$$ and $$N \in [1, +\infty]$$, then it also satisfies the $$\mathsf {CD}(K^{\prime }, N^{\prime })$$ for any $$K^{\prime } \leqslant K$$ and $$N^{\prime } \in [N,+\infty]$$, see [38, Proposition 1.6].

The $$\mathsf {CD}(K,N)$$ condition for negative $$N$$

The curvature-dimension condition $$\mathsf {CD}(K,N)$$ for negative $$N$$ is defined in a similar way. For $$K\in \mathbb {R}$$ and $$N<0$$, we set the distortion coefficients $$$\begin{equation*} \tilde{\tau }_{K,N}^{(t)}(\theta):= {\begin{cases} \displaystyle +\infty & \textrm {if}\ (N-1)\pi ^{2}\geqslant K\theta ^{2},\\ \displaystyle t^{1/N}{\left(\frac{\sin (t\theta \sqrt {K/N})}{\sin (\theta \sqrt {K/N})}\right)}^{1-1/N} & \textrm {if}\ 0>K\theta ^{2} > (N-1)\pi ^{2},\\ t & \textrm {if}\ K\theta ^2 =0, \\ \displaystyle t^{1/N}{\left(\frac{\sinh (t\theta \sqrt {-K/N})}{\sinh (\theta \sqrt {-K/N})}\right)}^{1-1/N} & \textrm {if}\ K\theta ^2 > 0. \end{cases}} \end{equation*}$$$

The following definition was first introduced in [33]. $2.2$

Definition

$$\mathsf {CD}(K,N)$$

For $$K\in \mathbb {R}$$ and $$N\in (-\infty,0)$$, a metric measure space $$(\mathsf {X},\mathsf {d},\mathfrak {m})$$ is said to satisfy the $$\mathsf {CD}(K,N)$$ condition if for every pair of absolutely continuous measures $$\mu _0=\rho _0\mathfrak {m},\mu _1= \rho _1 \mathfrak {m}\in \mathcal {P}_2(\mathsf {X})$$, there are an optimal transport plan $$\pi \in \mathcal P(\mathsf {X}\times \mathsf {X})$$ and an absolutely continuous $$W_2$$-geodesic $$(\rho _t\mathfrak {m})_{t\in [0,1]}$$ connecting them such that the following inequality holds for every $$N^{\prime }\in [N,0)$$ and every $$t \in [0,1]$$:$$$\begin{equation*} \int _\mathsf {X}\rho _t^{1-\frac{1}{N^{\prime }}} \mathop {}\!\mathrm{d}\mathfrak {m}\leqslant \int _{\mathsf {X}\times \mathsf {X}} {\left[ \tilde{\tau }^{(1-t)}_{K,N^{\prime }} {\left(\mathop {}\!\mathrm{d}(x,y) \right)} \rho _{0}(x)^{-\frac{1}{N^{\prime }}} + \tilde{\tau }^{(t)}_{K,N^{\prime }} {\left(\mathop {}\!\mathrm{d}(x,y) \right)} \rho _{1}(y)^{-\frac{1}{N^{\prime }}} \right]} \mathop {}\!\mathrm{d}\pi (x,y). \end{equation*}$$$

The $$\mathsf {CD}(K,N)$$ condition for negative $$N$$ has been found to appear naturally when studying harmonic measures on the sphere in [32], where the author uses previous results from [31] to obtain new isoperimetric inequalities for these harmonic measures. This condition is further studied in works such as [25, 26, 37]. We also have the following consistency property: if a metric measure space satisfies the $$\mathsf {CD}(K, N)$$ condition for some $$K \in \mathbb {R}$$ and $$N < 0$$, then it also satisfies the $$\mathsf {CD}(K^{\prime }, N^{\prime })$$ condition for every $$K^{\prime } \leqslant K$$ and $$N^{\prime } \in [N, 0)$$. We also have that the $$\mathsf {CD}(K, +\infty)$$ condition implies the $$\mathsf {CD}(K, N)$$ condition for any $$N < 0$$, see [33, Lemma 2.9].

The $$\mathsf {RCD}(K,N)$$ condition

In addition to the $$\mathsf {CD}(K,N)$$ condition, we need to introduce infinitesimal Hilbertianity to define $$\mathsf {RCD}(K,N)$$ spaces, see [18]. On a proper metric measure space $$(\mathsf {X},\mathop {}\!\mathrm{d},\mathfrak {m})$$, the Cheeger energy $$\mathrm{Ch}:L^2(\mathsf {X},\mathfrak {m})\rightarrow \mathbb {R}$$ is defined by $$$\begin{equation*} \mathrm{Ch}(f):=\inf {\left\lbrace \liminf _{n\rightarrow \infty }\frac{1}{2}\int _\mathsf {X}\mathsf {Lip}(f_n)\mathop {}\!\mathrm{d}\mathfrak {m}\mid f_n\in \mathsf {Lip}_b(\mathsf {X},\mathop {}\!\mathrm{d})\cap L^2(\mathsf {X},\mathfrak {m}),\Vert f_n-f\Vert _{L^2}\rightarrow 0\right\rbrace}, \end{equation*}$$$where $$\mathsf {Lip}_b(\mathsf {X},\mathop {}\!\mathrm{d})$$ is the space of bounded Lipschitz functions and for $$f\in \mathsf {Lip}_b(\mathsf {X},\mathop {}\!\mathrm{d})$$ and $$x\in \mathsf {X}$$, $$$\begin{equation*} \mathsf {Lip}(f)(x):=\limsup _{y\rightarrow x}\frac{|f(y)-f(x)|}{\mathop {}\!\mathrm{d}(x,y)}. \end{equation*}$$$Then the space of functions $$H^{1,2}(\mathsf {X},\mathop {}\!\mathrm{d},\mathfrak {m}):=\lbrace f\in L^2(\mathsf {X},\mathfrak {m})\mid \mathrm{Ch}(f)<+\infty \rbrace$$ defines a Banach space with the norm $$\Vert f\Vert _{H^{1,2}}:=\left(\Vert f\Vert _{L^2}^2+2\mathrm{Ch}(f)^2\right)^{1/2}$$.

There are different equivalent definitions of infinitesimal Hilbertianity, and the following is the one we choose here, see [18] and [3] for more details. $2.3$

Definition

$$\mathsf {RCD}(K,N)$$

A metric measure space $$(\mathsf {X},\mathop {}\!\mathrm{d},\mathfrak {m})$$ is said to be infinitesimally Hilbertian if $$(H^{1,2}(\mathsf {X},\mathop {}\!\mathrm{d},\mathfrak {m}),\Vert \cdot \Vert _{H^{1,2}})$$ is a Hilbert space.

Furthermore, given $$K\in \mathbb {R}$$ and $$N\in [1,+\infty]$$, the metric measure space $$(\mathsf {X},\mathop {}\!\mathrm{d},\mathfrak {m})$$ satisfies the $$\mathsf {RCD}(K,N)$$ condition if it verifies the $$\mathsf {CD}(K,N)$$ condition and is infinitesimally Hilbertian.

$2.4$

Remark

We do not speak about the $$\mathsf {RCD}$$ condition for $$N < 0$$ for a few reasons. First, the $$\mathsf {CD}$$ condition for negative $$N$$, contrary to when $$N \geqslant 1$$, allows for measures $$\mathfrak {m}$$ that are not locally finite. If that is the case, it is not known if the equivalences of the weak gradients in [3, sections 7 and 8] still hold. Second, one of the reasons the $$\mathsf {RCD}(K, N)$$ condition is particularly successful is because it is equivalent to the (synthetic) Bakry–Émery condition $$\mathsf {BE}(K, N)$$ under the Sobolev-to-Lipschitz property, see [16] and [4].

In our specific setting, the measures are always nonnegative, and we only use the equivalence between $$\mathsf {CD}(K,N)$$ and the smooth Ricci lower bound $$\mathrm{Ric}_{N,V} \geqslant K$$. Once the gaps in the $$\mathsf {RCD}$$ theory for negative effective dimension will have been filled, one should be able to replace $$\mathsf {CD}(K, N)$$ with $$\mathsf {RCD}(K, N)$$ when $$N < 0$$ in this paper, for example, in (iii) of Theorem 1.2

Sub-Riemannian geometry

Before we introduce the $$\alpha$$-Grushin spaces, we recall some basic facts about sub-Riemannian and metric geometry. For a more comprehensive account of these topics, we refer the reader to [1] and [12], for example. The $$\alpha$$-Grushin spaces considered in this work will all be two-dimensional almost-Riemannian manifolds and we recommend especially [1, chapter 9].

A sub-Riemannian structure on an $$n$$-dimensional manifold $$M$$ is given by a set of $$m$$ globally defined vector field $$\mathcal {F}:= \lbrace X_1, \dots, X_m\rbrace$$, also called the generating frame. The associated distribution is defined as the family, indexed with $$x \in M$$, of the vector subspaces $$$\begin{equation*} \mathcal {D}_x:= \mathrm{span} \lbrace X_1(x), \dots, X_m(x)\rbrace \subseteq T_xM. \end{equation*}$$$From these data, it is possible to introduce an inner product $$g_x$$ on $$\mathcal {D}_x$$ by applying the polarization formula to 3 $$$\begin{equation} g_x(v, v):= \inf {\left\lbrace \sum _{i = 1}^m u_i^2 \mid \sum _{i = 1}^m u_i X_i(x) = v \right\rbrace} . \end{equation}$$$

The rank of the sub-Riemannian structure at $$x \in M$$ is defined by $$r(x):= \dim (\mathcal {D}_x)$$. A two-dimensional almost-Riemannian manifold is a sub-Riemannian structure on a two-dimensional manifold $$M$$ such that the cardinal of $$\mathcal {F}$$ is two.

In a two-dimensional sub-Riemannian structure, a point $$x \in M$$ such that $$r(x) = 2$$ (resp., $$r(x) = 1$$) is called a Riemannian point (resp., singular point). The singular set, that is, the set of singular points, must necessarily be small (see [1, section 9.1.1]). In a neighborhood of Riemannian points, the inner product Equation 3 is a well-defined metric tensor and we can introduce, at those points, a Riemannian volume density (using the same formula as in Riemannian geometry) which diverges when approaching a singular point.

An admissible (or horizontal) curve $$\gamma :[0,1]\to M$$ is an absolutely continuous path such that there exists a control $$u \in \mathrm{L}^2(\left[0, 1\right], \mathbb {R}^m)$$ satisfying$$$\begin{equation*} \dot{\gamma }(t) = \sum _{i = 1}^m u_i(t) X_i(\gamma (t)), \ \text{ for almost every } t \in {\left[0, 1\right]}. \end{equation*}$$$The sub-Riemannian length of an admissible curve $$\gamma :[0,1]\to M$$ is then defined by$$$\begin{equation*} \mathrm{Length}(\gamma) := \int _0^1 \sqrt {g_{\gamma (t)}(\dot{\gamma }(t), \dot{\gamma }(t))} \mathrm{d}t, \end{equation*}$$$and the sub-Riemannian distance between two points $$x, y \in M$$ is4 $$$\begin{equation} \mathrm{d}(x, y):= \inf {\left\lbrace \mathrm{Length}(\gamma) \mid \gamma \text{ admissible and joins } x \text{ and } y \right\rbrace} . \end{equation}$$$It is not always given that $$\mathrm{d}$$ really defines a distance function. If $$\mathcal {F}$$ satisfies the bracket-generating condition (see [1, Definition 3.1]), for instance, then Rashevskii–Chow theorem [1, section 3.2] implies that there exists an admissible curve between every two points of $$M$$, and that $$(M, \mathop {}\!\mathrm{d})$$ is a metric space with the metric and manifold topology coinciding.

Denoting by $$\pi: T^\ast M \rightarrow M$$ is the canonical bundle projection, the Hamiltonian is the map $$H: T^*M \rightarrow \mathbb {R}$$ defined by $$$\begin{equation*} H(\lambda) := \frac{1}{2}\sum _{k = 1}^m \langle \lambda, X_k(\pi (\lambda)) \rangle ^2 \nobreakspace \nobreakspace \nobreakspace \nobreakspace \forall \lambda \in T_xM. \end{equation*}$$$Pontryagin's Maximum Principle is very helpful in the search for geodesics. $2.5$

Theorem

$Pontryagin\text{'}s\; Maximum\; Principle$

If $$\gamma$$ is length minimiser parameterized by constant speed, then there exists a Lipschitz curve $$\lambda (t) \in T^*_{\gamma (t)}M$$ such that one and only one of the following is satisfied:

• (i)$$\dot{\lambda } = \overrightarrow{H}(\lambda)$$, where $$\overrightarrow{H}$$ is the unique vector field in $$T^*M$$ such that $$\sigma (\cdot, \overrightarrow{H}(\lambda)) = \mathrm{d}_\lambda H$$ for all $$\lambda \in T^*M$$;
• (ii)$$\langle \lambda (t), X_i(\gamma (t)) \rangle = 0$$ for all $$i = 1, \dots, m$$, and $$\lambda (t) \ne 0$$ for all $$t\in [0,1]$$.

A curve $$\lambda: \left[0, 1\right] \rightarrow T^*M$$ satisfying (i) (resp., (ii)) in the theorem above is called a normal (resp., abnormal) extremal. Theorem 2.5 is thus stating that a (constant speed) minimizing geodesic has a cotangent lift that is a normal or an abnormal extremal. Note that an extremal in a two-dimensional almost-Riemannian manifold is abnormal if and only if its projection is a constant curve that lies on the singular set (see [1, Theorem 9.2]).

For the remainder of this work, we will adopt the notation $$(x)^{2 \alpha }:= (x^2)^\alpha \in \mathbb {R}_{\geqslant 0}$$ for every $$\alpha \geqslant 0$$ and $$x\in \mathbb {R}$$.

GEOMETRY OF $$\alpha$$-GRUSHIN HALF-SPACES

The $$\alpha$$-Grushin sphere and hemisphere

On the two-dimensional Riemannian sphere $$(\mathbb {S}^2, g_{\mathbb {S}^2})$$, we introduce the following coordinate chart, the validity of which can be found in [11]. Fix a large circle $$\gamma:\mathbb {R}/2\pi \mathbb {Z}\rightarrow \mathbb {S}^2$$, and let $$N$$ and $$S$$ be the north pole and south pole of $$\mathbb {S}^2$$, respectively, with respect to $$\gamma$$. For $$p\in \mathbb {S}^2\setminus \lbrace N,S\rbrace$$, we define $$x:=x(p)\in (-\pi/2,\pi/2)$$ as the signed (spherical) distance $$\mathop {}\!\mathrm{d}_{\mathbb {S}^2}(p,\mathrm{Im}(\gamma))$$, where the sign is positive (resp., negative) if $$p$$ belongs to the hemisphere containing $$N$$ (resp., $$S$$). Furthermore, define the number $$y=y(p)\in \mathbb {R}/2\pi \mathbb {Z}$$ so that $$\gamma (y)$$ is the perpendicular foot from $$p$$ to $$\mathrm{Im}(\gamma)$$. The map $${\mathbb{S}}^{2}\to (-\pi/2,\pi/2)\ensuremath{\times{}}\mathbb{R}/2 \pi \mathbb{Z}:p\mapsto (x(p),y(p))$$ is a well-defined coordinate chart which can be naturally extended to $$N$$ and $$S$$ under the identification $$(\frac{\pi }{2},y_1) \sim (\frac{\pi }{2},y_2)$$ and $$(-\frac{\pi }{2},y_1) \sim (-\frac{\pi }{2},y_2)$$ for any $$y_1,y_2\in \mathbb {R}/2\pi \mathbb {Z}$$. Note that $$N$$ (resp., $$S$$) corresponds to the equivalence class of $$(\frac{\pi }{2},y)$$ (resp., $$(-\frac{\pi }{2},y)$$). It is easily shown that in these coordinates, the spherical Riemannian metric tensor $$g_{\mathbb {S}^2}$$ possesses the warped product structure$$$\begin{equation*} g_{\mathbb {S}^2} = \mathop {}\!\mathrm{d}x \otimes \mathop {}\!\mathrm{d}x + \cos ^2(x) \mathop {}\!\mathrm{d}y \otimes \mathop {}\!\mathrm{d}y. \end{equation*}$$$$3.1$

Remark

The coordinate system $$(x, y)$$ is, up to rotations, the same as the spherical coordinates $$(\varphi,\theta) \in (-\pi /2, \pi /2)\times \mathbb {R}/ 2\pi \mathbb {Z}$$ which parameterizes the standard sphere $$\mathbb {S}^2 \setminus \lbrace N, S\rbrace \subseteq \mathbb {R}^3$$ by$$$\begin{equation*} (\theta,\varphi)\mapsto (\cos (\theta)\cos (\varphi), \sin (\theta)\cos (\varphi),\sin (\varphi)). \end{equation*}$$$

With this in mind, we can introduce the $$\alpha$$-Grushin sphere and hemisphere. $3.2$

Definition

For $$\alpha \geqslant 0$$, the $$\alpha$$-Grushin sphere $$\mathbb {S}_\alpha$$ is the sub-Riemannian structure on $$\mathbb {S}^2$$ induced from the vector field $$X$$ and $$Y^\alpha$$ given by$$$\begin{equation*} X:={\partial}_{x},\hspace*{0.202em}\hspace*{0.202em}{Y}^{\alpha}:=\frac{|\sin (x){|}^{\alpha}}{\cos (x)}{\partial}_{y}. \end{equation*}$$$The $$\alpha$$-Grushin hemisphere $$\overline{\mathbb {S}}^+_\alpha$$ (resp., open hemisphere $$\mathbb {S}^+_\alpha$$) is the subset of $$\mathbb {S}_\alpha$$ defined by$$$\begin{equation*} \overline{\mathbb {S}}^+_\alpha:= \lbrace p \in \mathbb {S}^2 \mid x \in [0,\pi /2] \rbrace \text{ (resp., $\mathbb {S}^+_\alpha:= \lbrace p \in \mathbb {S}^2 \mid x \in (0,\pi /2] \rbrace)$}. \end{equation*}$$$

$3.3$

Remark

When $$\alpha$$ is noninteger, the vector fields are not smooth and they are not bracket-generating. However, any pair of points can still be joined with a horizontal curve and Pontryagin's Maximum Principle stated in Theorem 2.5 can be applied. Therefore, the Carnot–Carathéodory metric Equation (4) can be constructed as in smooth sub-Riemannian geometry. This remark remains valid for the other model spaces introduced in this work.

$3.4$

Remark

Note that, strictly speaking, the structure introduced in Definition 3.2 does not fall into the definition of sub-Riemannian structure laid out in Subsection 2.2. Indeed, the vector fields $$X$$ and $$Y^\alpha$$ are not global vector fields: they are not defined at the poles, that is, at $$x=\pm \pi /2$$. To be completely rigorous, one should therefore check that it is still a sub-Riemannian manifold but according to the general definition of [1, Definition 3.2]. There, a sub-Riemannian manifold is given by the pair $$(E, f)$$ where $$E$$ is a Euclidean vector bundle and $$f: E \rightarrow T M$$ is a morphism of vector bundles. In our setting, $$E = T\mathbb {S}^2$$ and $$f$$ is a morphism that we now construct explicitly.

Denote by $$p:T\mathbb {S}^2\rightarrow \mathbb {S}^2$$ the bundle projection. Letting $$\mathcal {U}_1:= \mathbb {S}^2 \setminus \lbrace N, S\rbrace$$, the coordinate chart $$\varphi _1:= (x, y): \mathcal {U}_1 \rightarrow \mathbb {R}^2$$ induces a chart on $$T\mathbb {S}^2$$, and we define $$f_{1}: p^{-1}(\mathcal {U}_1) \rightarrow T \mathbb {S}^2$$ as the map, linear on fibers, that satisfies$$$\begin{equation*} {f}_{1}({\partial}_{x})={\partial}_{x},\hspace*{2em}{f}_{1}({\partial}_{y})=|\sin (x){|}^{\alpha}{\partial}_{y}. \end{equation*}$$$Letting $$\mathcal {U}_2:= \lbrace (s, t, z) \in \mathbb {S}^2\subset \mathbb {R}^3 \mid z > 0 \rbrace$$, the map$$$\begin{equation*} \varphi _2: \mathcal {U}_2 \rightarrow \mathbb {R}^2: (s, t, \sqrt {1 - s^2 - t^2}) \mapsto (s, t) \end{equation*}$$$is another coordinate chart on $$\mathbb {S}^2$$, which also induces a chart on $$T \mathbb {S}^2$$. We set $$f_{2}: p^{-1}(\mathcal {U}_2) \rightarrow T \mathbb {S}^2$$ the map, linear on fibers, satisfying$$$\begin{equation*} f_2(\partial _s) = \frac{1}{s^2+t^2} {\left[ {\left(s^2+(1-s^2-t^2)^{\alpha /2}t^2\right)} \partial _s + st(1-(1-s^2-t^2)^{\alpha /2}) \partial _t \right]} \end{equation*}$$$and$$$\begin{equation*} f_2(\partial _t) = \frac{1}{s^2+t^2} {\left[ st(1-(1-s^2-t^2)^{\alpha /2}) \partial _s + {\left((1-s^2-t^2)^{\alpha /2}s^2 + t^2 \right)} \partial _t \right]}. \end{equation*}$$$This ensures $$f_1=f_2$$ on $$p^{-1}(\mathcal {U}_1\cap \mathcal {U}_2)$$ via the coordinate transformation $$s=\cos (x)\cos (y)$$, $$t=\cos (x)\sin (y)$$. The apparent singularity at $$(s,t)=(0,0)$$ is removable because$$$\begin{equation*} f_2(\partial _s)\rightarrow \partial _s, \quad f_2(\partial _t)\rightarrow \partial _t \quad \text{as }(s,t)\rightarrow (0,0), \end{equation*}$$$so $$f_2$$ extends smoothly over the origin, giving the identity on the fiber at $$N$$. The maps $$f_1$$ and $$f_2$$ are patched together to define a smooth, globally defined bundle morphism $$f\colon E\rightarrow T\mathbb {S}^2$$, and the pair $$(E,f)$$ defines a sub-Riemannian manifold in the sense of [1, Definition 3.2].

The 0-Grushin sphere is simply the two-dimensional Riemannian sphere $$\mathbb {S}^2$$. When $$\alpha > 0$$, the $$\alpha$$-Grushin sphere is a two-dimensional almost-Riemannian structure with $$\lbrace x = 0\rbrace$$ being its set of singular points. The Grushin sphere studied in [11, 34] corresponds to the 1-Grushin sphere. At nonsingular points, this sub-Riemannian structure admits the Riemannian metric 5 $$$\begin{equation} g_{\mathbb {S}_{\alpha }} = \mathop {}\!\mathrm{d}x \otimes \mathop {}\!\mathrm{d}x + \frac{\cos ^2(x)}{\sin ^{2\alpha }(x)} \mathop {}\!\mathrm{d}y \otimes \mathop {}\!\mathrm{d}y. \end{equation}$$$A simple computation shows that the Riemannian volume induced from Equation 5, is given by $$$\begin{equation*} \text{dvol}_{{\mathbb{S}}_{\alpha}}=\cos (x)|\sin (x){|}^{-\alpha} \mathrm{d}x \mathrm{d}y. \end{equation*}$$$We introduce the following weighted measure. $3.5$

Definition

For $$\beta \geqslant \alpha$$, we consider the weighted measure given by$$$\begin{equation*} {\mathfrak{m}}_{{\mathbb{S}}_{\alpha}}^{\beta}:=|\sin (x){|}^{\beta}\text{dvol}_{{\mathbb{S}}_{\alpha}}=\cos (x)|\sin (x){|}^{\beta -\alpha} \mathrm{d}x \mathrm{d}y={e}^{-{V}_{{\mathbb{S}}_{\alpha}}}\text{dvol}_{{\mathbb{S}}_{\alpha}}, \end{equation*}$$$where $${V}_{{\mathbb{S}}_{\alpha}}(x,y):=-\beta \log |\sin (x)|$$.

The $$\alpha$$-Grushin hemisphere is a geodesically convex subset of $$\mathbb {S}_\alpha$$ and a geodesic space when seen as a length subspace of $$\mathbb {S}_\alpha$$. This is made clear by the next result. $3.6$

Proposition

There is a minimizing geodesic contained within $$\overline{\mathbb {S}}^+_\alpha$$ that joins any two given points in $$\overline{\mathbb {S}}^+_\alpha$$. Furthermore, the $$\alpha$$-Grushin open hemisphere $$\mathbb {S}_\alpha ^+$$ is a geodesically convex subset of $$\overline{\mathbb {S}}^+_\alpha$$ and has the structure of an incomplete weighted Riemannian manifold when equipped with the restriction of the measure $$\mathfrak {m}^\beta _{\mathbb {S}_\alpha }$$.

$$

Proof

We start by noting that between every two points on the $$\alpha$$-Grushin sphere, there is indeed a horizontal path controlled by the vector fields $$X$$ and $$Y^{\alpha }$$ joining them. This means that the induced sub-Riemannian distance $$\mathop {}\!\mathrm{d}_{\mathbb {S}_\alpha }$$ is well-defined and that $$(\mathbb {S}_\alpha, \mathop {}\!\mathrm{d}_{\mathbb {S}_\alpha })$$ is a locally compact metric space. Even though the bracket generating condition is not verified when $$\alpha \notin \mathbb {N}$$, it is easy to see that the metric topology still coincides with the original topology of $$\mathbb {S}^2$$ by the monotonicity property of $$\mathrm{d}_{\mathbb {S}_\alpha }$$ with respect to $$\alpha \geqslant 0$$. Furthermore, any metric ball is compact and thus the metric space $$(\mathbb {S}_\alpha, \mathop {}\!\mathrm{d}_{\mathbb {S}_\alpha })$$ is complete.

The sub-Riemannian Hamiltonian $$H: T^*(\mathbb {S}_\alpha) \rightarrow \mathbb {R}$$ can be written in the canonical coordinates $$(x, y, u,v)$$ induced from $$(x, y)$$ as$$$\begin{equation*} H(\lambda):=\frac{1}{2}{\left[\langle \lambda,X\rangle ^2+\langle \lambda,Y^\alpha \rangle ^2\right]}=\frac{1}{2}{\left[u^2+\frac{\sin ^{2\alpha }(x)}{\cos ^2(x)}v^2\right]}. \end{equation*}$$$A normal extremal $$\lambda: \left[0, T\right] \rightarrow T^*(\mathbb {S}_\alpha) : t \mapsto (x(t), y(t), u(t), v(t))$$ satisfies the following Hamiltonian system of equations6 $$$\begin{equation} {\begin{cases} \dot{x}=u,\\ \displaystyle \dot{y}=\frac{\sin ^{2\alpha }(x)}{\cos ^2(x)}v,\\ \displaystyle \dot{u}= - v^2 \ \sin ^{2(\alpha - 1)}(x) \tan (x) (\alpha + \tan ^2(x)),\\ \dot{v}=0. \end{cases}} \end{equation}$$$

Here note that the extremal reaches to the undefined point $$x=\pm \pi /2$$ only if $$v\equiv 0$$ (this follows from the nonintegrability of $$\tan (x)$$ near $$x=\pm \pi /2$$). In this case, a (Euclidean) large circle passing through the north pole becomes a length minimizing geodesic. By completeness, there is a sub-Riemannian geodesic between every two points of $$\mathbb {S}_\alpha$$ by [12, Theorem 2.5.23]. These are obtained from Hamilton's equation Equation 6 because there are no nontrivial abnormal geodesics.

If a horizontal path of $$\mathbb {S}_\alpha$$ is contained in both $$\overline{\mathbb {S}}^+_\alpha$$ and $$\overline{\mathbb {S}}^-_\alpha:= \lbrace p \in \mathbb {S}^2 \mid x \in [-\pi /2,0] \rbrace$$, then a reflection $$(x, y) \mapsto (-x,y)$$ of the part of path that is in $$\overline{\mathbb {S}}^-_\alpha$$ produces a curve contained in $$\overline{\mathbb {S}}^+_\alpha$$ with the same length. This shows that a geodesic between points in the $$\alpha$$-Grushin hemisphere $$\overline{\mathbb {S}}^+_\alpha$$ is contained within $$\overline{\mathbb {S}}^+_\alpha$$. Length-minimizers are smooth because they satisfy Hamilton's equation Equation 6. Thus, a constant-speed minimizing geodesic $$\gamma (t) = (x(t), y(t))$$ that touches the singular equator at a point other than its endpoints must do so tangentially, and Equation 6 implies that $$x(t)$$ vanishes for all $$t$$. Consequently, $$y(t)$$ also vanishes, and $$\gamma$$ becomes a constant curve. In particular, a minimizing geodesic between points of $$\mathbb {S}_\alpha ^+$$ is also contained within $$\mathbb {S}_\alpha ^+$$.

The fact that $$\mathbb {S}_\alpha ^+$$ is also an incomplete Riemannian manifold follows easily because it does not contain any singular points of $$\mathbb {S}_\alpha$$.$$\Box$$

The $$\alpha$$-Grushin hyperbolic plane and half-plane

On the two-dimensional hyperbolic plane $$(\mathbb {H}^2, g_{\mathbb {H}^2})$$, we consider the following coordinate chart, called Lobachevsky's coordinates (see [29, section 33.1], for example). We fix an infinite minimizing geodesic ray $$\gamma: \mathbb {R}\rightarrow \mathbb {H}^2$$, and for $$p \in \mathbb {H}^2$$, we let $$x:= x(p) \in \mathbb {R}$$ be signed hyperbolic distance $$\mathop {}\!\mathrm{d}_{\mathbb {H}^2}(p, \mathrm{Im}(\gamma))$$, where the signature is positive (resp., negative) if $$p$$ belongs to the left-hand side (resp., right-hand side) of $$\gamma$$. Furthermore, let $$y=y(p)\in \mathbb {R}$$ be the unique number such that $$\gamma (y)$$ is the perpendicular foot from $$p$$ to $$\mathrm{Im}(\gamma)$$. The map $$\mathbb {H}^2 \rightarrow \mathbb {R}\times \mathbb {R}: p \mapsto (x(p), y(p))$$ defines a global coordinate chart, and a short computation shows that the hyperbolic Riemannian metric $$g_{\mathbb {H}^2}$$ has the warped product structure $$$\begin{equation*} g_{\mathbb {H}^2} = \mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x+\cosh ^2(x)\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y. \end{equation*}$$$$3.7$

Definition

For $$\alpha \geqslant 0$$, the $$\alpha$$-Grushin hyperbolic plane $$\mathbb {H}_\alpha$$ is the sub-Riemannian structure on $$\mathbb {H}^2$$ induced from the vector field $$X$$ and $$Y^\alpha$$ given by$$$\begin{equation*} X:={\partial}_{x},\hspace*{0.202em}\hspace*{0.202em}{Y}^{\alpha}:=\frac{|\mathrm{\sinh}(x){|}^{\alpha}}{\cosh (x)}{\partial}_{y}. \end{equation*}$$$The $$\alpha$$-Grushin hyperbolic half-plane $$\overline{\mathbb {H}}^+_\alpha$$ (resp., open half-plane $$\mathbb {H}^+_\alpha$$) is the subset of $$\mathbb {H}_\alpha$$ defined by$$$\begin{equation*} \overline{\mathbb {H}}^+_\alpha:= \lbrace p \in \mathbb {H}^2 \mid x \geqslant 0 \rbrace \text{ (resp., $\mathbb {H}^+_\alpha:= \lbrace p \in \mathbb {H}^2 \mid x > 0 \rbrace $)}. \end{equation*}$$$

The 0-Grushin hyperbolic plane is simply the two-dimensional Riemannian hyperbolic plane $$\mathbb {H}^2$$. When $$\alpha > 0$$, the $$\alpha$$-Grushin hyperbolic plane is a two-dimensional almost-Riemannian structure with $$\lbrace x = 0\rbrace$$ being its set of singular points. To the best of our knowledge, this definition, although very natural, is new. At nonsingular points, this sub-Riemannian structure admits the Riemannian metric 7 $$$\begin{equation} g_{\mathbb {H}_{\alpha }} = \mathop {}\!\mathrm{d}x \otimes \mathop {}\!\mathrm{d}x + \frac{\cosh ^2(x)}{\sinh ^{2\alpha }(x)} \mathop {}\!\mathrm{d}y \otimes \mathop {}\!\mathrm{d}y. \end{equation}$$$A simple computation shows that the Riemannian volume induced from Equation 7 is given by $$$\begin{equation*} \text{dvol}_{{\mathbb{H}}_{\alpha}}=\cosh (x)|\mathrm{\sinh}(x){|}^{-\alpha} \mathrm{d}x \mathrm{d}y. \end{equation*}$$$We introduce the following weighted measure. $3.8$

Definition

For $$\beta \geqslant \alpha$$, we consider the weighted measure given by $$$\begin{equation*} {\mathfrak{m}}_{{\mathbb{H}}_{\alpha}}^{\beta}:=|\mathrm{\sinh}(x){|}^{\beta}\text{dvol}_{{\mathbb{H}}_{\alpha}}=\cosh (x)|\mathrm{\sinh}(x){|}^{\beta -\alpha} \mathrm{d}x \mathrm{d}y={e}^{-{V}_{{\mathbb{H}}_{\alpha}}}\text{dvol}_{{\mathbb{H}}_{\alpha}}, \end{equation*}$$$where $${V}_{{\mathbb{H}}_{\alpha}}(x,y):=-\beta \log |\mathrm{\sinh}(x)|$$.

As for the previous section, the $$\alpha$$-Grushin hyperbolic half-plane is a geodesically convex subset of $$\mathbb {H}_\alpha$$ and is a geodesic space when seen as a length subspace of $$\mathbb {H}_\alpha$$. The open half-plane $$\mathbb {H}_\alpha ^+$$ is also a geodesically convex subset and it is an incomplete Riemannian manifold because it does not contain any singular points of $$\mathbb {H}_\alpha$$. $3.9$

Proposition

There is a minimizing geodesic contained within $$\overline{\mathbb {H}}^+_\alpha$$ that joins any two given points in $$\overline{\mathbb {H}}^+_\alpha$$. Furthermore, The $$\alpha$$-Grushin hyperbolic open half-plane $$\mathbb {H}_\alpha ^+$$ is a geodesically convex subset of $$\overline{\mathbb {H}}_\alpha ^+$$ and has the structure of an incomplete weighted Riemannian manifold when equipped with the restriction of the measure $$\mathfrak {m}^\beta _{\mathbb {H}_\alpha }$$.

$$

Proof

Here, the Hamiltonian and the corresponding Hamilton's equation are given in Lobachevsky's coordinates, by$$$\begin{equation*} H(\lambda)=\frac{1}{2}{\left[u^2+\frac{\sinh ^{2\alpha }(x)}{\cosh ^2(x)}v^2\right]}, \text{ and } {\begin{cases} \dot{x}=u,\\ \displaystyle \dot{y}=\frac{\sinh ^{2\alpha }(x)}{\cosh ^2(x)}v,\\ \displaystyle \dot{u}= - 2 v^2 \ \sinh ^{2(\alpha - 1)}(x) \tanh (x) (\alpha - \tanh ^2(x)),\\ \dot{v}=0. \end{cases}} \end{equation*}$$$The rest of the proof follows exactly the arguments of Proposition 3.6.$$\Box$$

The $$\infty$$-Grushin plane and half-plane

The geometry of the so-called $$\alpha$$-Grushin plane, where $$\alpha \geqslant 0$$, has been studied in [6, 13], and [7]. The $$\alpha$$-Grushin half-plane and the validity of the $$\mathsf {CD}$$ condition in this space is studied in [36], and we recalled some details in Section 1. Instead, we introduce a model of a Grushin plane with infinite Hausdorff dimension. The global chart $$(x, y)$$ simply denotes the cartesian coordinates in this section. $3.10$

Definition

The $$\infty$$-Grushin plane $$\mathbb {G}_\infty$$ is the sub-Riemannian structure on $$\mathbb {R}^2$$ induced from the vector field $$X$$ and $$Y$$ given by$$$\begin{equation*} X:={\partial}_{x},\hspace*{0.202em}\hspace*{0.202em}Y:={e}^{-1/|x|}{\partial}_{y}. \end{equation*}$$$The $$\infty$$-Grushin half-plane $$\overline{\mathbb {G}}^+_\infty$$ (resp., open half-plane $$\mathbb {G}^+_\infty$$) is the subset of $$\mathbb {G}_\infty$$ defined by$$$\begin{equation*} \overline{\mathbb {G}}^+_\infty:= \lbrace p \in \mathbb {R}^2 \mid x \geqslant 0 \rbrace \text{ (resp., $\mathbb {G}^+_\infty:= \lbrace p \in \mathbb {R}^2 \mid x > 0 \rbrace $)}. \end{equation*}$$$

$3.11$

Lemma

The $$\infty$$-Grushin plane and half-plane have infinite Hausdorff dimension.

$$

Proof

We will show that the Hausdorff dimension of $$\mathcal {S}:=\lbrace (0,y)\mid y\in \mathbb {R}\rbrace \subseteq \mathbb {G}_\infty$$ is $$+\infty$$. Let us denote by $$\mathsf {d}_\alpha$$ (resp., $$\mathsf {d}_\infty$$) the induced distance on $$\mathbb {G}_\alpha$$ (resp., $$\mathbb {G}_\infty$$). It is well-known that the Hausdorff dimension of $$\mathcal {S}\subseteq \mathbb {G}_\alpha$$ is $$\alpha +1$$, see, for example, [17]. As the inequality $$|x|^\alpha \geqslant e^{-1/|x|}$$ holds for sufficiently small $$|x|$$, we have the inequality $$\mathsf {d}_\alpha \geqslant \mathsf {d}_\infty$$ in a small neighbourhood of an arbitrary point in $$\mathcal {S}$$. This implies that $$\dim _H(\mathcal {S},\mathsf {d}_\alpha)\leqslant \dim _H(\mathcal {S},\mathsf {d}_\infty)$$ and concludes the lemma.$$\Box$$

The $$\infty$$-Grushin plane is a two-dimensional almost-Riemannian structure with $$\lbrace x = 0\rbrace$$ being its set of singular points. To the best of our knowledge, this definition of $$\infty$$-Grushin plane is also new. At nonsingular points, this sub-Riemannian structure admits the Riemannian metric 8 $$$\begin{equation} {g}_{{\mathbb{G}}_{\infty}}= \mathrm{d}x\otimes \mathrm{d}x+{e}^{2/|x|} \mathrm{d}y\otimes \mathrm{d}y. \end{equation}$$$A simple computation shows that the Riemannian volume induced from Equation 8 is given by $$$\begin{equation*} \text{dvol}_{{\mathbb{G}}_{\infty}}={e}^{1/|x|} \mathrm{d}x \mathrm{d}y. \end{equation*}$$$We introduce the following weighted measure. $3.12$

Definition

For $$\beta \geqslant 0$$ and $$\gamma >0$$, we consider the weighted (Radon) measure given by$$$\begin{equation*} {\mathfrak{m}}_{{\mathbb{G}}_{\infty}}^{\beta ,\gamma}:=|x{|}^{\beta}{e}^{-\gamma /{x}^{2}}\text{dvol}_{{\mathbb{G}}_{\infty}}=|x{|}^{\beta}{e}^{-\gamma /{x}^{2}+1/|x|} \mathrm{d}x \mathrm{d}y={e}^{-V}\text{dvol}_{{\mathbb{G}}_{\infty}}, \end{equation*}$$$where $${V}_{{\mathbb{G}}_{\infty}}(x,y):=\frac{\gamma}{{x}^{2}}-\beta \log |x|$$.

The next result is analogous to the corresponding one in the previous two sections. The $$\infty$$-Grushin half-plane is a geodesically convex subset of $$\mathbb {G}_\infty$$ and a geodesic space when seen as a length subspace of $$\mathbb {G}_\infty$$. Similarly, the open half-plane $$\mathbb {G}_\infty ^+$$ is also a geodesically convex subset and it is an incomplete Riemannian manifold because it does not contain any singular points of $$\mathbb {G}_\infty$$. $3.13$

Proposition

There is a minimizing geodesic contained within $$\overline{\mathbb {G}}^+_\infty$$ that joins any two given points in $$\overline{\mathbb {G}}^+_\infty$$. Furthermore, the $$\infty$$-Grushin open half-plane $$\mathbb {G}_\infty ^+$$ is a geodesically convex subset of $$\overline{\mathbb {G}}_\infty ^+$$ and has the structure of an incomplete weighted Riemannian manifold when equipped with the restriction of the measure $$\mathfrak {m}^{\beta, \gamma }_{\mathbb {G}_\infty }$$.

$$

Proof

The argument is again analogous to the proofs of Proposition 3.6 and Proposition 3.9, with the Hamiltonian and Hamilton's equation given in cartesian coordinates by $$$\begin{equation*} H(\lambda)=\frac{1}{2}{\left[u+e^{-2/\vert x \vert}v^2\right]}, \text{ and } {\begin{cases} \dot{x}=u,\\ \displaystyle \dot{y}=e^{-2/\vert x \vert }v,\\ \displaystyle \dot{u}= - v^2 \frac{e^{-2/\vert x \vert }}{{\vert x \vert}^2} x,\\ \dot{v}=0. \end{cases}} \end{equation*}$$$$$\Box$$

Hereafter, we will collectively refer to the $$\alpha$$-Grushin plane (resp., half-plane), the $$\infty$$-Grushin plane (resp., half-plane), the $$\alpha$$-Grushin sphere (resp., hemisphere), and the $$\alpha$$-Grushin hyperbolic plane (resp., half-plane) as the $$\alpha$$-Grushin spaces (resp., half-spaces).

EQUIVALENCE BETWEEN $$\mathsf {CD}(K,N)$$ AND $$\mathrm{Ric}_N\geqslant K$$ FOR SOME ALMOST RIEMANNIAN MANIFOLDS

The $$\alpha$$-Grushin spaces and half-spaces introduced in Section 3 are metric measure spaces $$(\mathsf {X}, \mathsf {d}, \mathfrak {m})$$ with a weighted Riemannian manifold $$(M, \mathop {}\!\mathrm{d}_g, e^{-V}\mathrm{dvol}_g)$$ as their interior. For $$N \in (-\infty,0)\cup [n,+\infty]$$, we recall that the $$N$$-Ricci tensor of an $$n$$-dimensional weighted Riemannian $$(M, \mathop {}\!\mathrm{d}_g, e^{-V} \mathrm{dvol}_g)$$ is defined by 9 $$$\begin{equation} \mathrm{Ric}_{N,V}:= {\begin{cases} \mathrm{Ric}& \text{ if } N=n, \\ \mathrm{Ric}+ \mathrm{Hess}(V) & \text{ if } N=+\infty,\\ \mathrm{Ric}+\mathrm{Hess}(V)-\dfrac{\mathop {}\!\mathrm{d}V\otimes \mathop {}\!\mathrm{d}V}{N-n} & \text{ otherwise}, \end{cases}} \end{equation}$$$with the convention that $$V$$ must be constant when $$N=n$$.

The following theorem provides sufficient conditions under which the differential condition $$\mathrm{Ric}_{N,V} \geqslant K$$ on $$M$$ is equivalent to $$(\mathsf {X}, \mathsf {d}, \mathfrak {m})$$ satisfying the $$\mathsf {CD}(K, N)$$ condition. The proof generalizes the sketch found in [36, section 3.5], which is specific to the $$\alpha$$-Grushin half-plane. This theorem is, to some extent, related to the conjecture stated in [20]. $4.1$

Theorem

Let $$K \in \mathbb {R}$$, $$N\in (-\infty ,0)\cup [n,+\infty ]$$, $$(\mathsf {X}, \mathsf {d}, \mathfrak {m})$$ be a metric measure space and $$M$$ be an open subset of $$\mathsf {X}$$ such that

• (i)$$M$$ is a geodesically convex subset of $$(\mathsf {X}, \mathsf {d})$$, that is, for every $$x,y\in M$$, there is a geodesic joining $$x$$ and $$y$$ and any such curve is contained in $$M$$,
• (ii)$$(M, \mathsf {d}|_M, \mathfrak {m}|_M)$$ possesses a weighted $$n$$-dimensional Riemannian manifold structure,
• (iii)$$\mathfrak {m}(\mathsf {X}\setminus M) = 0$$.

Then, the metric measure space $$(\mathsf {X}, \mathsf {d}, \mathfrak {m})$$ satisfies the $$\mathsf {CD}(K,N)$$ condition if and only if $$\mathrm{Ric}_{N,V} \geqslant K$$ on $$M$$.

$$

Proof

We provide the proof only for $$N \in \left[n, +\infty \right]$$. The details for $$N \in \left(-\infty, 0\right)$$ are exactly the same, but one needs to replace the relevant key results used in the proof by the analogous ones when $$N$$ is negative. In particular, the Gromov–Hausdorff convergence which we use below must be replaced by the pointed $$\mathsf {iKRW}$$ convergence described in [26].

First, we note that the $$\mathsf {CD}(K,N)$$ condition on $$({\rm X},{\rm d},{\rm m})$$ directly implies that $$\mathrm{Ric}_{N,V} \geqslant K$$ on $$M$$ by [39, Theorem 17.36] (see also [38, Part (e) of the proof of Theorem 1.7]). We therefore focus now on the other implication.

For a closed metric subspace $$A\subseteq \mathsf {X}$$, we shall denote by $$W_2^A$$ the Wasserstein distance on $$A$$. Consider $$\mu _0, \mu _1 \in \mathcal {P}_2^{\mathrm{ac}}(\mathsf {X}, \mathsf {d}, \mathfrak {m})$$ with continuous densities $$\rho _0$$ and $$\rho _1$$, respectively. We need to argue that there exists a $$W_2^\mathsf {X}$$-geodesic $$(\mu _s)_{s \in \left[0, 1\right]}$$ joining $$\mu _0$$ to $$\mu _1$$ with $$\mu _s = (e_s)_\sharp \nu$$ for some $$\nu \in \mathcal {P}(\mathrm{Geo}(\mathsf {X}, \mathsf {d}))$$ such that the inequality in Definition 2.1 is satisfied. For all $$k \in \mathbb {N}$$ and given an arbitrary $$x_0 \in \mathsf {X}$$, we define the sets$$$\begin{equation*} M_k:=\mathrm{cl}(\mathrm{geo}(N_k)), \text{ and } \ \ N_k:= \lbrace x\in M\mid k \geqslant \mathsf {d}(x_0,x), \text{ and }\mathsf {d}(x,\mathsf {X}\setminus M)\geqslant 1/k\rbrace, \end{equation*}$$$where $$\mathrm{geo}(A)$$ is the geodesic hull of a subset $$A \subseteq \mathsf {X}$$, that is, the union of all geodesics starting at $$x \in A$$ and ending at $$y \in A$$. The set $$N_k$$ is clearly closed and bounded. As $$(\mathsf {X}, \mathop {}\!\mathrm{d}, \mathfrak {m})$$ is a complete metric measure space, the Heine–Borel property (see [12, Theorem 2.5.28]) implies that $$N_k$$ is also compact in $$\mathsf {X}$$.

Step 1. The closure of the geodesic hull of a compact subset of $$M$$ is in $$M$$

Let $$K$$ be a compact set of $$\mathsf {X}$$ contained in $$M$$. We claim that $$\mathrm{cl}(\mathrm{geo}(K))$$ is contained within $$M$$. As $$M$$ is assumed to be geodesically convex in $$\mathsf {X}$$, we note that their closure in $$\mathsf {X}$$ satisfies $$\mathrm{cl}(\mathrm{geo}(K)) \subseteq \mathrm{cl}(M)$$. Because $$M$$ is open in $$\mathsf {X}$$, we also have that $$\mathop {}\!\mathrm{d}(x, \partial M) \geqslant \mathop {}\!\mathrm{d}(K, \partial M) > 0$$ for all $$x \in K$$. Suppose by contradiction that there exists $$p \in \mathrm{cl}(\mathrm{geo}(K))$$ such that $$p \in \partial M$$. This means that for all $$n \in \mathbb {N}$$, there exist $$x_n, y_n \in K$$, a geodesic $$\gamma _n$$ joining $$x_n$$ to $$y_n$$ contained in $$M$$, and a point $$p_n \in M$$ lying on $$\gamma _n$$ such that $$p_n \rightarrow p$$ as $$n \rightarrow +\infty$$. By compactness, we have, after extracting a converging subsequence, that $$x_n \rightarrow x$$ and $$y_n \rightarrow y$$ for some $$x, y \in K$$. In particular, the geodesics $$\gamma _n$$ have uniformly bounded lengths. As $$(\mathsf {X}, \mathop {}\!\mathrm{d})$$ is a complete locally compact length space, the Arzela–Ascoli theorem from [12, Theorem 2.5.14] implies that the sequence of curves $$\gamma _n$$ contains a uniformly converging subsequence. This limit, which we denote by $$\gamma$$, is a curve of $$\mathsf {X}$$ with endpoints $$x$$ and $$y$$ and passing through $$p$$. Actually, the curve $$\gamma$$ is a minimizing geodesic between $$x$$ and $$y$$ by [12, Proposition 2.5.17]. By geodesic convexity of $$M$$ assumed at (ii), the curve $$\gamma$$ is contained in $$M$$ while the point $$p \notin M$$, leading to a contradiction.

Step 2. The support of $$\mu _0$$ and $$\mu _1$$ are compact and contained in $$N_k$$

Assume that $$\operatorname{supp}(\mu _0)$$ and $$\operatorname{supp}(\mu _1)$$ are compact and that $$\operatorname{supp}(\mu _0) \cup \operatorname{supp}(\mu _1) \subseteq N_k$$ for some $$k \in \mathbb {N}$$. The previous step applied to the compact $$N_k$$ implies that each pair of points $$x \in \operatorname{supp}(\mu _0)$$ and $$y \in \operatorname{supp}(\mu _1)$$ are connected by a geodesic contained in $$M$$. By [30, Theorem 13], there exists a unique $$W_2^{M_k}$$-geodesic $$(\mu _s)_{s \in \left[0, 1\right]}$$ joining $$\mu _0$$ to $$\mu _1$$. By [24, Proposition 2.10], we also know that there exists $$\nu \in \mathcal {P}(\mathrm{Geo}(\mathsf {X}, \mathsf {d}))$$ such that $$\mu _s = (e_s)_\sharp \nu$$ for all $$s \in \left[0, 1\right]$$ and $$\mathrm{supp}(\nu) \subseteq \Gamma:= (e_0 \times e_1)^{-1}(\mathrm{supp}(\mu _0) \times \mathrm{supp}(\mu _1))$$. A geodesic in $$\gamma \in \Gamma$$ joins points in $$\mathrm{supp}(\mu _0)$$ with points in $$\mathrm{supp}(\mu _1)$$ and is contained in $$M$$ thanks to Step 1. As $$\mathrm{dim}(M) \leqslant N$$ and $$\mathrm{Ric}_{V}^N \geqslant K$$ on $$(M, g)$$, [39, Theorem  17.36] (or also [38, Parts (a), (b), (c), and (d) of the proof of Theorem 1.7]) implies that the inequality in Definition 2.1 for all $$N^{\prime } \geqslant N$$.

Step 3. The support of $$\mu _0$$ and $$\mu _1$$ are compact and contained in $$\mathsf {X}$$

We can assume without loss of generality that $$\mathsf {X}\setminus M \subseteq \partial M$$. Indeed, when it comes to the $$\mathsf {CD}(K, N)$$ condition or the Gromov–Hausdorff convergence that we are going to discuss in this step, only the support of the measure matters (see [19, Remark 3.1]). The definition of $$N_k$$ and assumption (iii) imply that $$N_k \rightarrow \mathsf {X}$$ as $$k \rightarrow +\infty$$ in the pointed measured Gromov–Hausdorff convergence (see [19, Definition 3.9]), when taking the inclusion map $$\iota _k:N_k\rightarrow \mathsf {X}$$ as approximation maps. Although $$N_k$$ is not a geodesically convex subspace, we have shown in Step 1 that $$M_k:= \mathrm{cl}(\mathrm{geo}(N_k)) \subseteq M$$. Then, an argument similar to the proof of [39, Theorem 28.13] shows that the metric space $$(\mathcal P(N_k),W_2^{N_k})$$ converges to $$(\mathcal P(\mathsf {X}),W_2^\mathsf {X})$$ in the geodesic local Gromov–Hausdorff topology, via the inclusion map $$(\iota _k)_\#$$. Hence, there are sequences of probability measures $$(\mu _i^k)_{k\in \mathbb {N}}$$ $$(i=0,1)$$, supported on $$N_k$$, such that their pushforwards $$(\iota _k)_\#\mu _i^k$$ converges to $$\mu _i$$ in $$(\mathcal P_2(\mathsf {X}),W_2^\mathsf {X})$$.

Let $$(\mu _s^k)_{s \in \left[0, 1\right]}$$ be the Wasserstein geodesic joining $$\mu _0^k$$ and $$\mu _1^k$$ and $$\nu _k$$ be the optimal transport plan joining $$\mu _0^k$$ to $$\mu _1^k$$ in $$M_k$$. By Step 3, the inequality in Definition 2.1 holds along $$(\mu _s^k)_{s \in \left[0, 1\right]}$$ for all $$N^{\prime } \geqslant N$$. Furthermore, by [39, Theorem 28.9 and Exercise 28.15], $$\nu _k$$ weakly converges to an optimal transport plan $$\nu$$ in $$\mathsf {X}$$ joining $$\mu _0$$ to $$\mu _1$$, and the $$W_2^{M_k}$$-geodesics $$(\mu _s^k)_{s \in \left[0, 1\right]}$$ uniformly converges to a $$W_2^{\mathsf {X}}$$-geodesic $$(\mu _s)_{s \in \left[0, 1\right]}$$ in $$\mathsf {X}$$ joining $$\mu _0$$ to $$\mu _1$$. By the same argument as in [39, Theorem  29.24 and 29.21], we conclude that the inequality in Definition 2.1 remains valid along $$(\mu _s)_{s \in \left[0, 1\right]}$$ for all $$N^{\prime } \geqslant N$$, by lower semicontinuity.

Step 4. The general case

The case where the support of $$\mu _0$$ and $$\mu _1$$ are not necessarily compact is obtained by the previous step and by using an exhaustion by compact sets, see [24, appendix E].$$\Box$$

As for the $$\mathsf {RCD}(K, N)$$ condition (for $$K \in \mathbb {R}$$ and $$N \in \left[1, +\infty \right]$$), it is enough for our purposes to use the result of [23, Theorem 1.2], which shows that sub-Riemannian manifolds equipped with a nonnegative Radon measure are infinitesimally Hilbertian. Although the results in [23] assume the bracket-generating condition and our $$\alpha$$-Grushin half-spaces do not when $$\alpha \notin \mathbb {N}$$, their Finsler approximation techniques can still be applied. It is sufficient to regard the $$\alpha$$-Grushin half-spaces as $$\alpha$$-Grushin (full) spaces equipped with a measure supported on the half-space, for the following theorem to follow directly. $4.2$

Theorem

For $$K \in \mathbb {R}$$ and $$N\in [1,+\infty ]$$, the $$\mathsf {CD}(K,N)$$ condition is equivalent to the $$\mathsf {RCD}(K,N)$$ condition for the $$\alpha$$-Grushin half-spaces.

GENERALIZED RICCI CURVATURE OF $$\alpha$$-GRUSHIN HALF-SPACES

The previous section shows that establishing the validity of the $$\mathsf {CD}(K, N)$$ condition in the $$\alpha$$-Grushin half-spaces introduced in Section 3 is equivalent to a computation of the Bakry–Émery Ricci curvature. The following simple computation will be handy. $5.1$

Lemma

Let $$(M, g)$$ be a two-dimensional Riemannian manifold, $$\mathcal {U} \subseteq M$$ an open set, and $$N \in (-\infty, 0) \cup (2, +\infty]$$. Assume that $$(x, y): \mathcal {U} \rightarrow \mathbb {R}^2$$ is a chart and that we are given two smooth functions $$f: \mathcal {U} \rightarrow \mathbb {R}\setminus \lbrace 0\rbrace$$ and $$V: \mathcal {U} \rightarrow \mathbb {R}$$ that only depend on $$x$$. If, in this coordinate,$$$\begin{equation*} g = \mathop {}\!\mathrm{d}x \otimes \mathop {}\!\mathrm{d}x + \frac{1}{f(x)^2} \mathop {}\!\mathrm{d}y \otimes \mathop {}\!\mathrm{d}y, \end{equation*}$$$then it holds that $$$\begin{equation*} \begin{aligned} \mathrm{Ric}_{N,V} \,{=}\, {\left[ {\left(\frac{f^{\prime }}{f}\right)}^{\prime } \,{-}\, {\left(\frac{f^{\prime }}{f}\right)}^2 \,{+}\, V^{\prime \prime } \,{-}\, \frac{(V^{\prime })^2}{N \,{-}\, 2}\right]} \mathop {}\!\mathrm{d}x \otimes \mathop {}\!\mathrm{d}x \,{+}\, \frac{1}{f^2}{\left[{\left(\frac{f^{\prime }}{f}\right)}^{\prime } \,{-}\, {\left(\frac{f^{\prime }}{f}\right)}^2 \,{-}\, \frac{f^{\prime }}{f} V^{\prime }\right]} \mathop {}\!\mathrm{d}y \otimes \mathop {}\!\mathrm{d}y, \end{aligned} \end{equation*}$$$where all the derivatives are understood with respect to the variable $$x$$.

$$

Proof

The two vector fields $$X = \partial _x$$ and $$Y = f(x) \partial _y$$ defined on $$\mathcal {U}$$ form a family of $$g$$-orthonormal fields. The only nonzero bracket relation is$$$\begin{equation*} [X, Y] = \frac{f^{\prime }}{f} Y. \end{equation*}$$$Using the Koszul formula, we easily obtain$$$\begin{equation*} \nabla _X X = \nabla _X Y = 0, \qquad \nabla _Y X = - \frac{f^{\prime }}{f} Y, \qquad \nabla _Y Y = \frac{f^{\prime }}{f} X. \end{equation*}$$$It follows that the only nonzero entry of the Riemann curvature tensor is$$$\begin{equation*} \mathrm{R}(X, Y, X, Y) = {\left(\frac{f^{\prime }}{f}\right)}^{\prime } - {\left(\frac{f^{\prime }}{f}\right)}^2. \end{equation*}$$$Using the symmetries, we obtain $$$\begin{equation*} \mathrm{Ric} = {\left[{\left(\frac{f^{\prime }}{f}\right)}^{\prime } - {\left(\frac{f^{\prime }}{f}\right)}^2\right]} \mathop {}\!\mathrm{d}x \otimes \mathop {}\!\mathrm{d}x + \frac{1}{f^2}{\left[{\left(\frac{f^{\prime }}{f}\right)}^{\prime } - {\left(\frac{f^{\prime }}{f}\right)}^2\right]} \mathop {}\!\mathrm{d}y \otimes \mathop {}\!\mathrm{d}y. \end{equation*}$$$We also find that $$\nabla V =X(V)X$$, and that the only nonzero entries of the Hessian are $$$\begin{equation*} \mathrm{Hess}(V)(X,X) = V^{\prime \prime },\nobreakspace \nobreakspace \mathrm{Hess}(V)(Y,Y)=-\frac{f^{\prime }}{f}V^{\prime }. \end{equation*}$$$Finally, one has the nonzero entry $$$\begin{equation*} \mathop {}\!\mathrm{d}V\otimes \mathop {}\!\mathrm{d}V(X,X)=(V^{\prime })^2, \end{equation*}$$$and the proof is complete with Equation 9.$$\Box$$

The $$\alpha$$-Grushin hemisphere

We make use of the coordinate chart and the notation described in Subsection 3.1. $5.2$

Proposition

Given $$K \in \mathbb {R}$$, $$N \in \left(-\infty, 0\right) \cup \left[2, +\infty \right)$$, $$\alpha \geqslant 0$$ and $$\beta \geqslant \alpha$$, the $$N$$-Ricci curvature of the $$\alpha$$-Grushin open hemisphere $$(\mathbb {S}_\alpha ^+, \mathop {}\!\mathrm{d}_{\mathbb {S}_\alpha }, \mathfrak {m}^\beta _{\mathbb {S}_\alpha })$$ is $$$\begin{align*} \mathrm{Ric}_{N,V}=&-\frac{1}{\sin ^2(x)}{\left[3\alpha -1+(\alpha -1)^2\cos ^2(x) - \beta + \frac{\beta ^2}{N-2}\cos ^2(x) \right]}\mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x \\ &-\frac{\cos ^2(x)}{\sin ^{2\alpha +2}(x)}{\left[3\alpha -1+(\alpha -1)^2\cos ^2(x) - \beta {\left(1+(\alpha -1)\cos ^2(x)\right)}\right]}\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y. \end{align*}$$$Furthermore, it holds $$\mathrm{Ric}_{N,V} \geqslant K$$ if and only if $$$\begin{equation*} \beta -\alpha ^2-\alpha +\min {\left(-K+(\alpha -1)^2,\nobreakspace -\frac{\beta ^2}{N-2},\nobreakspace \beta (\alpha -1)\right)}\geqslant 0. \end{equation*}$$$

$$

Proof

The first part is done by Lemma 5.1 with $$f(x)=\frac{|\sin (x){|}^{\alpha}}{\cos (x)}$$ and $$V(x)=-\beta \log |\sin (x)|$$. For the second part, we see by comparing the coefficients of the tensors $$\mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x$$ and $$\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y$$, respectively, that $$\mathrm{Ric}_{N,V}\geqslant K$$ holds if and only if $$$\begin{equation*} -3\alpha +1+\beta -K+{\left[K-(\alpha -1)^2-\frac{\beta ^2}{N-2}\right]}\cos ^2(x)\geqslant 0 \end{equation*}$$$and $$$\begin{equation*} -3\alpha +1+\beta -K+{\left[K-(\alpha -1)^2+\beta (\alpha -1)\right]}\cos ^2(x)\geqslant 0. \end{equation*}$$$As $$\cos (x)$$ takes its value in [0,1) on $$\mathbb {S}_\alpha ^+$$, the above two inequalities are equivalent to the following inequality: $$$\begin{equation*} -3\alpha +1+\beta -K+\min {\left(0,K-(\alpha -1)^2-\frac{\beta ^2}{N-2},K-(\alpha -1)^2+\beta (\alpha -1)\right)}\geqslant 0, \end{equation*}$$$which is trivially equivalent to the inequality in the statement.$$\Box$$

The $$\alpha$$-Grushin hyperbolic half-plane

We make use of the coordinate chart and the notation described in Subsection 3.2. $5.3$

Proposition

Given $$K \in \mathbb {R}$$, $$N\in (-\infty ,0)\cup [2,+\infty )$$, $$\alpha \geqslant 0$$ and $$\beta \geqslant \alpha$$, the $$N$$-Ricci curvature of the $$\alpha$$-Grushin open hyperbolic half-plane $$(\mathbb {H}_\alpha ^+, \mathop {}\!\mathrm{d}_{\mathbb {H}_\alpha }, \mathfrak {m}^\beta _{\mathbb {H}_\alpha })$$ is$$$\begin{align*} \mathrm{Ric}_{N,V}=&-\frac{1}{\sinh ^2(x)}{\left[3\alpha -1+(\alpha -1)^2\cosh ^2(x) - \beta + \frac{\beta ^2}{N-2}\cosh ^2(x) \right]}\mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x \\ &-\frac{\cosh ^2(x)}{\sin ^{2\alpha +2}(x)}{\left[3\alpha -1+(\alpha -1)^2\cosh ^2(x) - \beta {\left(1+(\alpha -1)\cosh ^2(x)\right)}\right]}\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y. \end{align*}$$$Furthermore, it holds $$\mathrm{Ric}_{N,V} \geqslant K$$ if and only if$$$\begin{equation*} \min {\left(-K-(\alpha -1)^2,\beta -\alpha ^2-\alpha \right)}+\min {\left(-\frac{\beta ^2}{N-2},\nobreakspace \beta (\alpha -1)\right)}\geqslant 0. \end{equation*}$$$

$$

Proof

The first part is done by Lemma 5.1 with $$f(x)=\frac{|\mathrm{\sinh}(x){|}^{\alpha}}{\cosh (x)}$$ and $$V(x)=-\beta \log |\mathrm{\sinh}(x)|$$. For the second part, we see by comparing the coefficients of the tensors $$\mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x$$ and $$\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y$$, respectively, that $$\mathrm{Ric}_{N,V}\geqslant K$$ holds if and only if$$$\begin{equation*} -3\alpha +1+\beta +K+{\left[-K-(\alpha -1)^2-\frac{\beta ^2}{N-2}\right]}\cosh ^2(x)\geqslant 0 \end{equation*}$$$and$$$\begin{equation*} -3\alpha +1+\beta +K+{\left[-K-(\alpha -1)^2+\beta (\alpha -1)\right]}\cosh ^2(x)\geqslant 0. \end{equation*}$$$As $$\cosh (x)\in [1,+\infty)$$, the two inequalities hold for all $$x\geqslant 0$$ if and only if$$$\begin{equation*} \min {\left(0,-3\alpha +1+\beta +K\right)}+\min {\left(-K-(\alpha -1)^2-\frac{\beta ^2}{N-2},-K-(\alpha -1)^2+\beta (\alpha -1)\right)}\geqslant 0 \end{equation*}$$$holds, which is equivalent to the inequality in the statement.$$\Box$$

The $$\infty$$-Grushin half-plane

We make use of the coordinate chart and the notation described in Subsection 3.3. $5.4$

Proposition

Given $$K \in \mathbb {R}$$, $$N\in (-\infty ,0)\cup [2,+\infty )$$ and $$\beta, \gamma \geqslant 0$$, the $$N$$-Ricci curvature of the $$\infty$$-Grushin open half-plane $$(\mathbb {G}_\infty ^+, \mathop {}\!\mathrm{d}_{\mathbb {G}_\infty }, \mathfrak {m}^{\beta,\gamma }_{\mathbb {G}_\infty })$$ is$$$\begin{align*} \mathrm{Ric}_{N,V}=&{\left[\frac{6\gamma -1}{|x|^4}-\frac{2}{|x|^3}+\frac{\beta }{|x|^2}-\frac{1}{(N-2)x^6}{\left(2\gamma +\beta x^2\right)}^2\right]}\mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x\\ &+{\left[\frac{2\gamma }{|x|^5}-\frac{1}{|x|^4}+\frac{\beta -2}{|x|^3}\right]}e^{\frac{2}{|x|}}\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y. \end{align*}$$$It holds $$\mathrm{Ric}_{N,V} \geqslant K$$ if and only if the following inequalities hold for any $$x\geqslant 0$$:$$$\begin{equation*} (6\gamma -1)x^2-2x^3+\beta x^4-\frac{1}{N-2}(2\gamma +\beta x^2)^2\geqslant Kx^6\nobreakspace \nobreakspace \text{and}\nobreakspace \nobreakspace 2\gamma -x+(\beta -2)x^2\geqslant Kx^5. \end{equation*}$$$In particular, $$\mathrm{Ric}_{\infty,V}\geqslant 0$$ holds if and only if$$$\begin{equation*} \min \lbrace \beta (6\gamma -1),8(\beta -2)\gamma \rbrace \geqslant 1. \end{equation*}$$$

$$

Proof

The proof is reached by using Lemma 5.1 with $$f(x)=e^{-1/|x|}$$ and $$V(x)=\frac{\gamma }{|x|^2}-\beta \log |x|$$, and by comparing the coefficients of $$\mathop {}\!\mathrm{d}x\otimes \mathop {}\!\mathrm{d}x$$ and $$\mathop {}\!\mathrm{d}y\otimes \mathop {}\!\mathrm{d}y$$, respectively.$$\Box$$

Theorems 1.1, 1.2, and 1.4 follow from the Ricci computations this section, noting that the validity of the assumptions in Theorem 4.1 are verified because of the results of Section 3.

ACKNOWLEDGMENTS

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Program (Grant Agreement No. 945655). This research was funded in part by the Austrian Science Fund (FWF) [Grant DOI 10.55776/EFP6]. The authors thank Shouhei Honda for leading the authors’ attention to these problems, and telling us about the consequences in Remark 1.5.

JOURNAL INFORMATION

The Transactions of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.

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