On the Geometry of Metric Measure Spaces with Variable Curvature Bounds

Abstract

Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound $k$ and upper dimension bound $N\ge 1$. In the case of non-zero constant lower curvature, our approach coincides with the celebrated condition that was proposed by Sturm (Acta Math 196(1):133–177, 2006). We prove several geometric properties as sharp Bishop–Gromov volume growth comparison or a sharp generalized Bonnet–Myers theorem (Schneider’s Theorem). In addition, the curvature-dimension condition is stable with respect to measured Gromov–Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed. We also briefly describe possible extensions for variable dimension bounds.

1 Introduction

Metric measure spaces with generalized lower Ricci curvature bounds have become objects of interest in various fields of mathematics. Since Lott, Sturm, and Villani introduced the so-called curvature-dimension condition CD(K, N) for $K\in {\mathbb {R}}$ and $N\in [1,\infty ]$ via displacement convexity of the Shanon and Reny entropy on the $L^2$-Wasserstein space [24, 33, 34], a rather complete picture of the geometric and analytic properties of these spaces has been developed (e.g. [1, 2, 14, 16, 20–22, 30]). Their approach is based on and inspired by recent fundamental breakthroughs in the theory of optimal transport (e.g. [7, 9, 25, 27]).

However, the condition of lower-bounded Ricci curvature is also very restrictive. Neither non-compact smooth Riemannian manifolds do admit global lower curvature bounds in general, nor does Hamilton’s Ricci flow in general. Moreover, one cannot exceed the information that is encoded by the constant curvature bound. Therefore, the regime of results is limited. However, in the context of smooth Riemannian manifolds, variable lower Ricci curvature plays an important role. One can deduce refined statements for the geometry of the space, e.g. [5, 18, 28, 29, 31, 36]. Therefore, it seems natural to ask for an extension of the theory of Lott, Sturm, and Villani. For dimension-independent situations, a definition is proposed by Sturm in [35]. However, to deduce refined geometric statements, one also must bring a dimension bound into play.

In this article, we will focus on the finite dimensional case and introduce a curvature-dimension condition $CD(k,N)$ for metric measure spaces $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$, where the lower curvature bound $k:X\rightarrow {\mathbb {R}}$ is a lower semi-continuous function and $N=const\ge 1$ (variable dimension bounds will be discussed in Sect. 9). Before we describe our approach, let us recall that Lott, Sturm, and Villani define the curvature-dimension condition CD(0, N) of an arbitrary metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ via displacement convexity for the N-Rény entropy functional

$$\begin{aligned} S_N(\varrho {{\mathrm{m}}}_{{X}})=-\int _X\varrho ^{1-\frac{1}{N}}d{{\mathrm{m}}}_{{X}}. \end{aligned}$$

(The definitions in [24] and in [34] slightly differ.) In [34], Sturm gave a definition of CD(K, N) for general $K\in {\mathbb {R}}$ via the so-called distorted displacement convexity (see also [37]). This approach involves the concept of modified volume distortion coefficients $\tau _{{k},{N}}^{{(t)}}(\theta )$ that do not come from a linear ODE but are motivated by the geometry of Riemannian manifolds. They capture the geometric fact that Ricci curvature of a tangent vector v is the mean value of sectional curvatures of planes intersecting in v. Roughly speaking, non-zero curvature only happens perpendicular to v. Our idea is to introduce generalized volume distortion coefficients as follows. We define

$$\begin{aligned} \tau _{{k}_{\gamma },{N}}^{{(t)}}(\theta )=t^{\frac{1}{N}}\left[ \sigma _{{k}_{\gamma },{N}-1}^{{(t)}}(\theta )\right] ^{\frac{N-1}{N}} \end{aligned}$$

where $k_{\gamma }(t\theta )=k\circ \gamma (t)$, $\gamma :[0,1]\rightarrow X$ is a constant-speed geodesic and $\sigma _{{k}_{\gamma },{N}}^{{(t)}}(\theta )$ is the solution of

$$\begin{aligned}&u''(t)+\frac{k(\gamma (t))}{N}\theta ^2u=0&\end{aligned}$$

(1)

with $u(0)=0 \text{ and } u(1)=1$ where $\theta =|\dot{\gamma }|$. We remark, that in the case of constant curvature $k=K$ this yields $\sigma _{{K},{N}}^{{(t)}}(|\dot{\gamma }|)={\sin _{{K}/{N}}(t|\dot{\gamma }|)}/{\sin _{{K}/{N}}(|\dot{\gamma }|)}$ that is precisely the definition of Sturm in [34].

A key property of the distortion coefficients is their monotonicity w.r.t. $k$ which is a consequence of a classical comparison result of J. C. F. Sturm for one-dimensional Sturm–Liouville-type operators.

Theorem 1.1

(Sturm’s comparison theorem) Let $k,k':[a,b]\rightarrow {\mathbb {R}}$ be continuous function such that ${k}'\ge {k} \text{ on } [a,b]$ and ${{\mathrm{\mathfrak {s}}}}_{k'}>0$ on (a, b]. Then, $\mathfrak {s}_{k}\ge \mathfrak {s}_{{k}'}$ on [a, b].

${{\mathrm{\mathfrak {s}}}}_{k}$ is a solution of (1) with $\theta k/N=k$ and $\gamma (t)=t$, an initial condition $u(0)=0$ and $u'(0)=1$. The theorem is well known in the context of Riemannian manifolds and smooth Jacobi field calculus. Its geometric counterpart is the celebrated Rauch comparison theorem.

In particular, from generalized distortion coefficients, we also obtain a new characterization of the differential inequality $u''\le -ku$ (see Proposition 3.8) that appears naturally in connection with lower curvature bounds on smooth Riemannian manifolds.

Then our curvature-dimension condition takes the following form. Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a metric measure space as in Definition 2.1 and assume for simplicity that for ${{\mathrm{m}}}_{{X}}^2$-a.e. pair (x, y) there exists a unique geodesic. Then $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies the condition $CD(k,N)$ for $N\ge 1$ and a lower semi-continuous function $k:X\rightarrow {\mathbb {R}}$, if for any pair of absolutely continuous probability measures $\mu _0$ and $\mu _1$ on X with bounded support, there exists a dynamic optimal coupling $\Pi \in {\mathcal {P}}({\mathcal {G}}(X))$ such that $(e_t)_{\star }\Pi =\mu _t$ is an $L^2$-Wasserstein geodesic in ${\mathcal {P}}^2({{\mathrm{m}}}_{{X}})$ and

$$\begin{aligned} \varrho _t(\gamma _{t})^{-\frac{1}{N}}\ge \tau _{{k}^-_{\gamma },{N}'}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0(\gamma _{0})^{-\frac{1}{N}}+\tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _1(\gamma _{0})^{-\frac{1}{N}}. \end{aligned}$$

for all $t\in [0,1]$ and $\Pi$-a.e. geodesic $\gamma$. Here $k^+_{\gamma }=k_{\gamma }$ and $k^-_{\gamma }=k_{\gamma ^-}$ where $\gamma ^-$ is the time reverse reparametrization of $\gamma$. $\varrho _t$ is the density of the push-forward of $\Pi$ under the map $\gamma \mapsto \gamma _{t}$. If we replace $\tau _{{k},{N}}$ by $\sigma _{{k}/{N}}$, we say X satisfies the reduced curvature-dimension condition $CD^*(k,N)$. Let us emphasize that we do not assume any non-branching assumption for the metric measure space in general, and we also do not assume a quadratic Cheeger energy as in [1] or an a priori lower curvature bound as in [35].

This is the first part of two articles where we investigate the geometric and and analytic consequences of our curvature-dimension condition. The main results in this article are

• The condition $CD(k,N)$ for $N\in [1,\infty )$ implies $CD(k,\infty )$ in the sense of [35] (Proposition 4.11).

• For Riemannian manifolds, the curvature-dimension condition $CD(k,N)$ is equivalent to a lower bound $k$ for the Ricci tensor and an upper bound N for the dimension (Theorem 4.12).

• A generalized Brunn–Minkowski theorem and a generalized Bishop–Gromov comparison theorem hold (Theorems 5.1, 5.3, 5.9). The latter result in particular yields a local volume doubling property and finite Hausdorff dimension for the support of ${{\mathrm{m}}}_{{X}}$.

• A generalized Bonnet–Myers theorem (Theorem 5.10). This is a non-smooth version of a result by Schneider [31] (see also [3, 15]). It states that if the curvature does not decrease too quickly for large distances from a point in the support of the measure, then the support is compact with explicit bound for the diameter. There are also similar statements in the context of smooth Finsler manifolds and for the Bakry–Emery–Ricci tensor in a smooth context [4, 38].

• The curvature-dimension condition is stable with respect to measured Gromov convergence (Theorem 6.6). In particular, it implies that any family of compact Riemannian manifolds with uniform upper bound for the dimension, uniform upper bound for the diameter, and Ricci curvature uniformly bounded from below admit a converging subsequence such that the liminf function of the variable lower Ricci curvature bounds (that is the function that assigns to each point the smallest eigenvalue of the Ricci tensor) is a lower Ricci curvature bound for the limit space (Corollary 6.8).

• The curvature-dimension condition is stable under tensorization of finitely many metric measure spaces provided a non-branching assumption is satisfied (Theorem 7.4).

• The reduced curvature-dimension condition admits a globalization property (Theorem 8.3).

In the forthcoming addendum to this article [19], we also investigate variants of the condition $CD(k,N)$. Namely, following [14, 26], we introduce an entropic curvature-dimension condition and a measure contraction property as well as an $EVI_{k,N}$-condition for gradient flows on metric spaces where $k$ is a lower semi-continuous function. We will investigate their relation to each other and also to the reduced curvature-dimension condition presented in this paper. Given stronger regularity assumptions, we establish various equivalences and consequences.

In addition, considering the recent approach of Cavalletti and Mondino in [11] to prove isoperimetric inequalities and various other functional inequalities in the context of non-branching CD-spaces with constant curvature bound, our approach seems very well adapted for transforming their ideas to a non-constant curvature setting.

In the second section of this paper, we will present necessary preliminaries of optimal transport, Wasserstein calculus and geometry of metric spaces. In Sect. 3, we will introduce generalized distortion coefficients and we will present a new characterization of $ku$-convexity of a function u. In Sect. 4 we give the definition of $CD(k,N)$ in the general context of metric measure spaces, and in particular we will prove that is consistent with Sturm’s definition in [35]. The topic of Sect. 5 will be geometric consequences of the curvature-dimension condition. In Sects. 6, 7, and 8, we will prove the stability property, the tensorization property under a branching assumption, and the globalization property of the reduced curvature-dimension condition, respectively.

In Sect. 9, we briefly discuss extensions of our approach that also capture a variable dimension bound. This is not obvious since the condition $CD(k,N)$ is defined via N-Reny entropy functionals where $N>0$ has to be a constant parameter.

2 Preliminaries

Definition 2.1

(Metric measure space) Let $(X,{{\mathrm{d}}}_{{X}})$ be a complete and separable metric space, and let ${{\mathrm{m}}}_{{X}}$ be a locally finite Borel measure on $(X,{{\mathrm{d}}}_{{X}})$. That is, for all $x\in X$ there exists $r>0$ such that ${{\mathrm{m}}}_{{X}}(B_r(x))\in (0,\infty )$. Let ${\mathcal {O}}_{{X}}$ and ${\mathcal {B}}_{{X}}$ be the topology of open sets and the family of Borel sets, respectively. A triple $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ will be called metric measure space. We assume that ${{\mathrm{m}}}_{{X}}(X)\ne 0$. If ${{\mathrm{m}}}_{{X}}(X)=1$ we say $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ is normalized.

$(X,{{\mathrm{d}}}_{{X}})$ is called length space if ${{\mathrm{d}}}_{{X}}(x,y)=\inf \text{ L } (\gamma )$ for all $x,y\in X$, where the infimum runs over all rectifiable curves $\gamma$ in X connecting x and y. $(X,{{\mathrm{d}}}_{{X}})$ is called geodesic space if every two points $x,y\in X$ are connected by a curve $\gamma$ such that ${{\mathrm{d}}}_{{X}}(x,y)=\text{ L }(\gamma )$. Distance minimizing curves of constant speed are called geodesics. A length space, which is complete and locally compact, is a geodesic space and proper [6, Theorem 2.5.23 ]. Rectifiable curves always admit a reparametrization proportional to arc length, and therefore become Lipschitz curves. In general, we assume that a geodesic $\gamma :[0,1]\rightarrow X$ is parametrized proportional to its length, and the set of all such geodesics $\gamma :[0,1]\rightarrow X$ is denoted with ${\mathcal {G}}(X)$. The set of all Lipschitz curves $\gamma :[0,1]\rightarrow X$ parametrized proportional to arc-length is denoted with ${\mathcal {LC}}(X)$. $(X,{{\mathrm{d}}}_{{X}})$ is called non-branching if for every quadruple $(z,x_0,x_1,x_2)$ of points in X for which z is a midpoint of $x_0$ and $x_1$ as well as of $x_0$ and $x_2$, it follows that $x_1=x_2$.

${\mathcal {P}}(X)$ denotes the space of probability measures on $(X,{\mathcal {B}}_{{X}})$, and ${\mathcal {P}}_2(X,{{\mathrm{d}}}_{{X}})=:{\mathcal {P}}_2(X)$ denotes the $L^2$-Wasserstein space of probability measures $\mu$ on $(X,{\mathcal {B}}_{{X}})$ with finite second moments, which means that $\int _X{{\mathrm{d}}}_{{X}}^2(x_0,x)d\mu (x)<\infty$ for some (hence all) $x_0\in X$. The $L^2$-Wasserstein distance ${{\mathrm{d}}}_{W}(\mu _0,\mu _1)$ between two probability measures $\mu _0,\mu _1\in {\mathcal {P}}_2(X)$ is defined as

$$\begin{aligned} {{\mathrm{d}}}_{{W}}(\mu _0,\mu _1):={{\mathrm{d}}}_{{X},{W}}(\mu _0,\mu _1):=\sqrt{\inf _{\pi }\int _{X\times X}{{\mathrm{d}}}_{{X}}^2(x,y)\,d\pi (x,y)}. \end{aligned}$$

(2)

Here the infimum ranges over all couplings of $\mu _0$ and $\mu _1$, i.e. over all probability measures on $X\times X$ with marginals $\mu _0$ and $\mu _1$. $({\mathcal {P}}_2(X),{{\mathrm{d}}}_{W})$ is a complete separable metric space. The subspace of ${{\mathrm{m}}}_{{X}}$-absolutely continuous measures is denoted by ${\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})=:{\mathcal {P}}_2({{\mathrm{m}}}_{{X}})$. A minimizer of (2) always exists and is called optimal coupling between $\mu _0$ and $\mu _1$.

A probability measure $\Pi$ on ${\mathcal {G}}(X)$ is called dynamic optimal transference plan if and only if the probability measure $(e_0,e_1)_*\Pi$ on $X\times X$ is an optimal coupling of the probability measures $(e_0)_*\Pi$ and $(e_1)_*\Pi$ on $X$. Here and in the sequel $e_t:\Gamma (X)\rightarrow X$ for $t\in [0,1]$ denotes the evaluation map $\gamma \mapsto \gamma _t$. An absolutely continuous curve $\mu _t$ in ${\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ is a geodesic if and only if there is a dynamic optimal transference plan $\Pi$ such that $(e_t)_*\Pi =\mu _t$. We write ${{\mathrm{DyCpl}}}(\mu _0,\mu _1)$ for the set of dynamic optimal transference plans between $\mu _0$ and $\mu _1$.

Let us recall the notion of Markov kernel. Let $(Y,{{\mathrm{d}}}_{{Y}})$ be a separable and complete metric space. A Markov kernel is a map $Q:Y\times {\mathcal {B}}_{{Y}}\rightarrow [0,1]$ with the following properties. $Q(y,\cdot )$ is a probability measure for each $y\in Y$. The function $Q(\cdot ,A)$ is measurable for each $A\in {\mathcal {B}}_{{X}}$.

Lemma 2.2

For each pair $\mu _0, \mu _1\in {\mathcal {P}}_2(X)$, there exists a dynamic optimal coupling $\Pi$ such that

$$\begin{aligned} {{\mathrm{d}}}_{W}(\mu _0,\mu _1)^2=\int {{\mathrm{d}}}_{{X}}(\gamma (0),\gamma (1))d\Pi (\gamma ). \end{aligned}$$

and there exist Markov kernels $\Pi _{x_0,x_1}$, $\Pi _{x_0}$ and $\Pi _{x_1}$ such that

$$\begin{aligned} d\Pi (\gamma )=d\Pi _{x_0,x_1}(\gamma )d\pi (x_0,x_1)=d\Pi _{x_0}(\gamma )d\mu _{0}(x_0)=d\Pi _{x_1}(\gamma )d\mu _{1}(x_1) \end{aligned}$$

where $(e_0,e_1)_{\star }\Pi =:\pi$.

Proof

For the existence of a dynamic optimal coupling, see [37]. The existence of the corresponding Markov kernels comes from the existence of regular conditional probability measures. $\square$

3 $ku$-Convexity

Let $k:[a,b]\rightarrow {\mathbb {R}}$ be a continuous function. We study solutions of

$$\begin{aligned} v''+kv=0. \end{aligned}$$

(3)

The generalized $\sin$-functions ${{\mathrm{\mathfrak {s}}}}_{k}:[a,b]\rightarrow {\mathbb {R}}$ is the unique solution of (3) such that $\mathfrak {s}_{k}(a)=0$ and $\mathfrak {s}_{k}'(a)=1$. The generalized $\cos$-function is $\mathfrak {c}_{k}={{\mathrm{\mathfrak {s}}}}_{k}'$. Solutions of (3) depend continuously on the coefficient $k$. More precisely, for each $\epsilon >0$ there exists $\delta >0$ such that $|k-k'|_{\infty }<\delta$ implies $|{{\mathrm{\mathfrak {s}}}}_{k}-{{\mathrm{\mathfrak {s}}}}_{k'}|_{\infty }<\epsilon$ where $k,k':[a,b]\rightarrow {\mathbb {R}}$ are continuous. If $\gamma (t)=(1-t)a+tb$ and $v:[a,b]\rightarrow {\mathbb {R}}$ is any solution of (3), then $v\circ \gamma =u:[0,1]\rightarrow {\mathbb {R}}$ solves

$$\begin{aligned} u''+{k}\circ \gamma |\dot{\gamma }|^2u=0. \end{aligned}$$

(4)

In particular, ${{\mathrm{\mathfrak {s}}}}_{k}(\gamma _t)$ solves (4) with ${{\mathrm{\mathfrak {s}}}}_{k}(\gamma _0)=0$ and $\frac{d}{dt}|_{t=0}{{\mathrm{\mathfrak {s}}}}_{k}(\gamma _t)=|\dot{\gamma }(0)|=b-a$.

The next theorems are well-known.

Theorem 3.1

(J. C. F. Sturm’s comparison theorem) Let $k,k':[a,b]\rightarrow {\mathbb {R}}$ be continuous function such that ${k}'\ge {k} \text{ on } [a,b]$ and ${{\mathrm{\mathfrak {s}}}}_{k'}>0$ on (a, b]. Then $\mathfrak {s}_{k}\ge \mathfrak {s}_{{k}'}$.

Theorem 3.2

(Sturm–Picone oscillation theorem) Let $k,k':[a,b]\rightarrow {\mathbb {R}}$ be continuous such that ${k}'\ge {k} \text{ on } [a,b]$. Let u and v be solutions of (3) with respect to $k$ and $k'$ respectively. If $u(a)=u(b)=0$ and $u>0$ on (a, b), then either $u=\lambda v$ for some $\lambda >0$ or there exists $x_1\in (a,b]$ such that $v(x_1)=0$.

Definition 3.3

(generalized distortion coefficients) Consider $k:[0,{L}]\rightarrow {\mathbb {R}}$ that is continuous and $\theta \in (0,L]$. Then

$$\begin{aligned} \sigma _{k}^{{(t)}}(\theta )={\left\{ \begin{array}{ll} \frac{{{\mathrm{\mathfrak {s}}}}_{{k}}(t\theta )}{{{\mathrm{\mathfrak {s}}}}_{{k}}(\theta )}&{} \text{ if } {{\mathrm{\mathfrak {s}}}}_{k}|_{(0,\theta ]}>c>0,\\ \infty &{} \text{ otherwise } . \end{array}\right. } \end{aligned}$$

We also define $\pi _{k}=\sup \left\{ t\in [0,L]:{{\mathrm{\mathfrak {s}}}}_{k}(s)>0 \text{ for } \text{ all } s\le t\right\} .$ If $\sigma _{k}^{{(t)}}(\theta )<\infty$, $t\mapsto \sigma _{k}^{{(t)}}(\theta )$ is a solution of

$$\begin{aligned} u''(t)+k(t\theta )\theta ^2u(t)=0 \end{aligned}$$

(5)

satisfying $u(0)=0$ and $u(1)=1$.

Proposition 3.4

$\sigma ^{{(t)}}_{k}(\theta )$ is non-decreasing with respect to $k:[0,\theta ]\rightarrow {\mathbb {R}}$. More precisely $k(x)\ge k'(x) \ \forall x\in [0,\theta ]\ \ \text{ implies } \ \ \sigma ^{{(t)}}_{k}(\theta )\ge \sigma ^{{(t)}}_{k'}(\theta )\ \forall t\in [0,1].$

Proof

Consider $\sigma ^{{(t)}}_{k}(\theta )$ and $\sigma ^{{(t)}}_{{k}'}(\theta )$ for $k$ and ${k}'$ such that $k(t)\ge {k}'(t)$ for all $t\in [0,1]$. By Sturm–Picone oscillation theorem, $\sigma ^{{(t)}}_{k}(\theta )=\infty$ implies $\sigma ^{{(t)}}_{{k}'}(\theta )=\infty$. Hence, we only need to check the case when $\sigma ^{{(t)}}_{k}(\theta )<\infty$ and $\sigma ^{{(t)}}_{{k}'}(\theta )<\infty$.

We use the idea of the proof of Theorem 14.28 in [37]. We know that $\sigma _{k}^{(0)}(\theta )=\sigma _{{k}'}^{(0)}(\theta )=0$ and $\sigma _{k}^{(1)}(\theta )=\sigma _{{k}'}^{(1)}(\theta )=1$. Consider ${\sigma _{{k}'}^{(t)}(\theta )}/{\sigma _{k}^{(t)}(\theta )}=:h(t)$ for $t\in (0,1]$. We know that $h(1)=1$ and L’Hospital’s rule yields

$$\begin{aligned} \lim _{t\downarrow 0}h(t)=\frac{{{\mathrm{\mathfrak {s}}}}_{k}(\theta )}{{{\mathrm{\mathfrak {s}}}}_{k'}(\theta )}\lim _{t\downarrow 0}\frac{{{\mathrm{\mathfrak {c}}}}_{k'}(t\theta )}{{{\mathrm{\mathfrak {c}}}}_{k}(t\theta )}=\frac{{{\mathrm{\mathfrak {s}}}}_{k}(\theta )}{{{\mathrm{\mathfrak {s}}}}_{k'}(\theta )}\le 1. \end{aligned}$$

Hence, it is sufficient to check that h(t) has no local maximum in (0, 1). For this reason, first we assume that $k>{k}'$. Set $\sigma _{{k}'}^{{(t)}}(\theta )=f$ and $\sigma _{k}^{{(t)}}(\theta )=g$. Assume there is a maximum in $t_0\in (0,1)$. Hence, $(f/g)'(t_0)=0$ and $(f/g)''(t_0)\le 0$. We compute the second derivative of f / g.

$$\begin{aligned} \left( \frac{f}{g}\right) ''&=\frac{f''g^3-g''fg^2}{g^4}+\frac{2gg'fg'-2g'f'g^2}{g^4}\\&=-{k'}\theta ^2\frac{f}{g}+k\theta ^2\frac{f}{g}-\frac{2gg'}{g^2}\frac{f'g-fg'}{g^2} \end{aligned}$$

and therefore

$$\begin{aligned} \left( \frac{f}{g}\right) ''(t_0)=({k'}(t_0\ \theta )-k(t_0\theta ))\theta ^2\frac{f(t_0)}{g(t_0)}>0. \end{aligned}$$

The case where $k\ge {k}'$ follows from that if we replace $k$ by $k+\epsilon$. Then $\sigma _{k+\epsilon }^{{(t)}}(\theta )$ converges uniformly to $\sigma _{k}^{{(t)}}(\theta )$ if $\epsilon \rightarrow 0$. $\square$

Proposition 3.5

For $\theta \in (0,L]$ and $t\in (0,1)$, the map $k\in (C([0,L]),|\cdot |_{\infty })\mapsto \sigma _{k}^{{(t)}}(\theta )\in {\mathbb {R}}_{\ge 0}\cup \left\{ \infty \right\}$ is continuous where ${\mathbb {R}}_{\ge 0}\cup \left\{ \infty \right\}$ is equipped with the usual topology.

Proof

If all the distortion coefficients are finite, this follows from the stability of (3) under uniform changes of $k$. We only have to check the following. If $k_n\rightarrow k$ with respect to $|\cdot |_{\infty }$, and if $\sigma _{k}^{{(t)}}(\theta )=\infty$, then $\sigma _{k_n}^{{(t)}}(\theta )\uparrow \infty$. If $\sigma _{k}^{{(t)}}(\theta )=\infty$, then there exists $r\le \theta$ such that ${{\mathrm{\mathfrak {s}}}}_{k}(r)=0$. If $r<\theta$, then by the stability property, ${{\mathrm{\mathfrak {s}}}}_{k_n}(r_n)=0$ for some $r_n<\theta$ and $n\in {\mathbb {N}}$ sufficiently large. Hence, $\sigma _{k_n}^{{(t)}}(\theta )=\infty$ for n sufficiently large. Otherwise, $r=\theta$ and ${{\mathrm{\mathfrak {s}}}}_{k}>0$ on $(0,\theta )$. Again by stability, it follows that ${{\mathrm{\mathfrak {s}}}}_{k_n}(\theta )\rightarrow 0$ and ${{\mathrm{\mathfrak {s}}}}_{k_n}\rightarrow {{\mathrm{\mathfrak {s}}}}_{k}$ w.r.t. $|\cdot |_{\infty }$ if $n\rightarrow \infty$. Therefore, for any compact $J\subset (0,1)$, there exists $n_0$ such that for each $n\ge n_0$, we have ${{\mathrm{\mathfrak {s}}}}_{k_n}(\cdot \theta )|_J>c>0$ for some $c>0$. Hence, $\sigma _{k_n}^{{(t)}}(\theta )\uparrow \infty$ for each $t\in (0,1)$. $\square$

Lemma 3.6

Let $a,b\in {\mathbb {R}}_{\ge 0}$ and $k:[0,\theta ]\rightarrow {\mathbb {R}}$ as before. If $\sigma _{k}^{{(t)}}(\theta )<\infty$, then

$$\begin{aligned} v(t)=\sigma _{{k}^{{-}}}^{{(1-t)}}(\theta )a+\sigma _{{k}^{{+}}}^{{(t)}}(\theta )b \end{aligned}$$

(6)

solves (5) in the distributional sense satisfying $u(0)=a$ and $u(1)=b$.

Remark 3.7

Given $k$ as above we set ${k}^{{-}}=k\circ \phi$ where $\phi (t)=b+a-t$. We also write $k=:k^{{+}}$. $\sigma _{k}^{{(t)}}(\theta )<\infty$ if and only if $\sigma _{k^{{-}}}^{{(t)}}(\theta )<\infty$. This follows from Sturm’s oscillation theorem.

To see this, we assume $\sigma _{k}^{{(t)}}(\theta )=\infty$ and $\sigma _{k^-}^{{(t)}}(\theta )$ is finite. Then ${{\mathrm{\mathfrak {s}}}}_{k}$ has a zero in $[0,\theta ]$ and ${{\mathrm{\mathfrak {s}}}}_{k^-}$ has no zero in $[0,\theta ]$. However, ${{\mathrm{\mathfrak {s}}}}_{k}(t)$ and ${{\mathrm{\mathfrak {s}}}}_{k}(\theta -t)$ are solutions of $u''+ku=0$, and therefore Sturm’s oscillation theorem yields a contradiction.

Proof

We have

$$\begin{aligned} v''(t)=-{k}^{{-}}((1-t)\theta )\theta ^2\sigma _{{{k}}^{-}}^{{(1-t)}}(\theta )a-k(t\theta )\theta ^2\sigma _{{k}^{{+}}}^{{(t)}}(\theta )b \end{aligned}$$

and

$$\begin{aligned} {k}^{-}((1-t)\theta )=k^{{+}}\circ \phi ((1-t)\theta )=k^{{+}}(\theta -(1-t)\theta ))=k^{{+}}(t\theta ). \end{aligned}$$

Hence (6) solves (5) in the classical sense with the right boundary condition. $\square$

Proposition 3.8

Let $k:[a,b]\rightarrow {\mathbb {R}}$ be continuous and $u:[a,b]\rightarrow {\mathbb {R}}_{\ge 0}$ be an upper semi-continuous. Then the following 4 statements are equivalent:

• (i) $u''+ku\le 0$ in the distributional sense, that is

  $$\begin{aligned} \int _a^b\varphi ''(t)u(t)dt\le -\int _a^b\varphi (t)k(t)u(t)dt \end{aligned}$$

  (7)

  for any $\varphi \in C_0^{\infty }\left( (a,b)\right)$ with $\varphi \ge 0$.

• (ii) It holds

  $$\begin{aligned} u(\gamma (t))\ge (1-t)u(\gamma (0))+tu(\gamma (1)+\int _0^1g(t,s)k(\gamma (s))\theta ^2 u(\gamma (s)) ds \end{aligned}$$

  (8)

  for any constant-speed geodesic $\gamma :[0,1]\rightarrow [a,b]$ where $\theta =|\dot{\gamma }|=\text{ L }(\gamma )$ with g(s, t) being the Green function of [0, 1].

• (iii) There is a constant $0<L\le b-a$ such that

  $$\begin{aligned} u(\gamma (t))\ge \sigma ^{{(1-t)}}_{k^{{-}}_{\gamma }}(\theta )u(\gamma (0))+\sigma ^{{(t)}}_{k^{{+}}_{\gamma }}(\theta )u(\gamma (1)) \end{aligned}$$

  (9)

  for any constant-speed geodesic $\gamma :[0,1]\rightarrow [a,b]$ with $\theta =|\dot{\gamma }|=\text{ L }(\gamma )\le L$. We set $k_{\gamma }=k\circ \bar{\gamma }:[0,\theta ]\rightarrow {\mathbb {R}}$. $\bar{\gamma }:[0,\theta ]\rightarrow [a,b]$ denotes the unit-speed reparametrization of $\gamma$. We use the convention $\infty \cdot 0=0$.

• (iv) The statement in (iii) holds for any geodesic $\gamma :[0,1]\rightarrow [a,b]$.

Proof

1. First, we prove that (iii) implies (i). Since u is upper semi-continuous, it is bounded from above. Hence, $\sigma _{k_{\gamma }}^{{(t)}}(\theta )=\infty$ implies $u\circ \gamma (1)=0$ for any geodesic $\gamma$. Therefore, one can find $L>L'>0$ such that that ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}>0$ on $(0,\theta ]$ for any constant-speed geodesic $\gamma :[0,1]\rightarrow [a,b]$ with $\theta =|\dot{\gamma }|\le L'$. Otherwise $u=const=0$. ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}>0$ implies $\sigma ^{{(t)}}_{k_{\gamma }}(\theta ')<\infty$ for any $\theta '\in (0,\theta ]$. $\square$

Claim

For $k$ and t, fixed $f:h\mapsto \sigma _{k}^{{(t)}}(h)$ is twice differentiable at $h=0$, and we have

$$\begin{aligned} h\in [0,{L}]\mapsto \sigma _{k}^{(t)}(h)=t\left[ 1+\frac{1}{6}(1-t^2)k(0)h^2\right] +o(h^2)_{k}^{t}. \end{aligned}$$

(10)

$\square$

Proof of the claim

We can compute the first and second derivative of f at 0 explicitly by application of L’Hospital’s rule. Then we apply the Taylor expansion formula and the claim follows.

If $\overline{k}\ge k\ge \underline{k}$, then

$$\begin{aligned} o(h^2)_{{k}}^{t}&=\sigma _{{k}}^{(t)}(h)-\frac{1}{3}t(1-t^2){k}(0)h^2-t\\&\le \sigma _{\overline{k}}^{(t)}(h)-t\left[ \frac{1}{3}(1-t^2)\underline{k}h^2+1\right] =t\frac{1}{3}(1-t^2)(\overline{k}-\underline{k})h^2+o(h^2)^t_{\overline{k}} \end{aligned}$$

and similarly,

$$\begin{aligned} o(h^2)_{{k}}^{t}\ge t\frac{1}{3}(1-t^2)(\underline{k}-\overline{k})h^2+o(h^2)^t_{\underline{k}}. \end{aligned}$$

Since $k$ is uniformly continuous on [a, b], we can choose $\overline{h}>0$ and $(r_i)_{i=1,\ldots ,N}$ such that

$$\begin{aligned} \max k|_{[r_i-h,r_i+h]}-\min k|_{[r_i-h,r_i+h]}<\epsilon \end{aligned}$$

for each $i=1,\ldots N$ and each $h\in [0,\overline{h}]$.

Upper semi-continuity of u together with the condition (9) yields continuity of u on [a, b]. We consider $s\in [a,b]$, $h>0$ and a geodesic $\gamma :[0,1]\rightarrow [a,b]$ such that $\gamma _0=s-h$, $\gamma _1=s+h$ and $\gamma _{1/2}=s$ and $s\pm h\in [r_i-\underline{k},r_i+\overline{k}]$ for some $i=1,\ldots N$. Then, from (10) and (9), it follows that

$$\begin{aligned} \frac{2u(s)-u(s-h)-u(s+h)}{h^2}&\ge \underbrace{\frac{k(s-h)u(s-h)+k(s+h)u(s+h)}{2}}_{\rightarrow k(s)u(s)}-\epsilon \\&\quad +\underbrace{\frac{\min _{i=1,\ldots ,N}o(h^2)^t_{\min k|_{[r_i-h,r_i+h]}}}{h^2}}_{\rightarrow 0}. \end{aligned}$$

Multiplication with $\phi \in C^{\infty }_0((a,b))$ such that $\phi \ge 0$, integration with respect to s, a change of variables and taking the limit $h\rightarrow 0$ yields

$$\begin{aligned} \int u(s)\phi ''(s)ds\le -\int k(s)u(s)\phi (s)ds + \epsilon \int \phi (s)ds. \end{aligned}$$

Since $\epsilon >0$ can be chosen arbitrarily small, we obtain the result.

2. We prove the equivalence between (i) and (ii). We assume (i) holds. Consider $v(t)=\int _0^1g(t,s)k(\gamma (s))\theta ^2 u(\gamma (s)) ds$. Then v solves

$$\begin{aligned} v''(t)=-k(\gamma (s))\theta ^2 u(\gamma (s)) \end{aligned}$$

in distributional sense by definition of the Green function. Hence, $u\circ \gamma -v$ has non-positive derivative in the distributional sense, and it follows that $u\circ \gamma -v$ is concave (see Theorem 1.29 in [32]). This implies (ii). The backwards direction is straightforward and works like in the previous step.

3. We prove that (i) implies (iv). The implication (iv) $\Rightarrow$ (iii) is obvious. First, we assume that $u\in C([a,b])\cap C^2((a,b))$. We consider the case when ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}>0$ for any constant-speed geodesic $\gamma :[0,1]\rightarrow (a,b)$. The right-hand-side of (9) is denoted by v(t) where $t\in [0,1]$. It is positive for any t and solves $v''+k_{\gamma }\circ \gamma \ \theta ^2v=0$ with boundary condition $v(0)=u(\gamma (0))$ and $v(1)=u(\gamma (1))$. Hence, it suffices to check that $\frac{u\circ {\gamma }}{v}$ has no local minimum in (0, 1). Otherwise, there is $\tau \in (0,1)$ such that $(\frac{u\circ {\gamma }}{v})'(\tau )=0$ and $(\frac{u\circ {\gamma }}{v})''(\tau )\ge 0$. We can deduce a contradiction exactly like in the proof of Proposition 3.4. $\square$

Next, we consider when there is a a constant-speed geodesic $\gamma :[0,1]\rightarrow (a,b)$ such that ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}(t_0)=0$ for some $t_0\in (0,\theta ]$. Again we adapt parts of the proof of Theorem 14.28 in [37]. We show that $u=0$. Let $v(t)={{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}(\gamma (t))$ and $w(t)=u\circ \gamma (t)$. v satisfies $v''+k_{\gamma }\circ \gamma \theta ^2v=0$ and w satisfies $w''+k_{\gamma }\circ \gamma \theta ^2\le 0$. Consider $\frac{w}{v}=:h$. Then

$$\begin{aligned} (h'v^2)'&=h''v^2+2vv'h'=\left( \frac{w'v-v'w}{v^2}\right) 'v^2+2vv'h'\\&=\frac{w''v-v''w -v'w'+2(v')^2uw}{v^2}+\frac{2v'w'v^2-2(v')^2vw}{v^2}\\&\le -k\theta ^2\frac{w}{v}+k\theta ^2\frac{w}{v}=0 \end{aligned}$$

Hence, $h'v^2$ is non-increasing. Suppose there is $\tau \in [0,1]$ such that $h'(\tau )>0$ then we also have that $h'v^2(\tau )>0$ and $h'v^2\ge C >0$ on $[\tau ,1]$. for some constant $C>0$. Hence $h'\ge C\frac{1}{v^2}$. $v={{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}\circ \gamma$ is in $C^2([0,1])$. Especially, it follows that $v(\delta )=\delta + o(\delta ^2)$. Thus, $h'(h)\ge C \frac{1}{\delta ^2}$. It follows

$$\begin{aligned} \int _{\delta }^{\epsilon }h'(\tau )d\tau =h(\epsilon )-h(\delta )\ge {C}\int _{\delta }^{\epsilon }\frac{1}{\tau ^2}d\tau \rightarrow \infty \ \ \text{ if } \ \delta \rightarrow 0. \end{aligned}$$

Hence $h(\delta )\rightarrow -\infty$ if $\delta \rightarrow 0$ which contradicts $h\ge 0$. On the other hand, if there is $\tau \in [0,1]$ such that $h'(\tau )<0$, the same argument yields $h(\epsilon )\rightarrow -\infty$ if $\delta \rightarrow 0$. It follows that $h'=0$ and $w(t)=c\cdot {{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}(\gamma (t))$. Especially u is differentiable at $\gamma (1)\in (a,b)$ with $u|_{(\gamma (0),\gamma (1))}>0$, $u(\gamma (1))=0$ and $u'(\gamma (1))\ne 0$ if $u\ne 0$ since $u'(\gamma (1))=0$ would contradict the uniqueness of the solution of (3). However, $u(\gamma (1))=0$ and $u'(\gamma (1))\ne 0$ yields $u(x)<0$ for $x\ge \gamma (1)$ which is not possible. Hence, $u=0$ and (9) holds.

Now, let u be just upper semi-continuous. The equivalence between (i) and (ii) yields that u is continuous. Consider $\phi \in C^{\infty }_0((0,1))$ with $\int _{0}^1\phi (t)dt=1$ and $\phi _{\epsilon }(t)=\frac{1}{\epsilon }\phi (\frac{t}{\epsilon })$. $\phi _{\epsilon }\in C^{\infty }_0((0,\epsilon ))$. We set

$$\begin{aligned} \tilde{u}(s)=u\star \phi _{\epsilon }(s)=\int _{-\epsilon }^0\phi _{\epsilon }(-r)u(s-r)dr=\int _{a}^b\phi _{\epsilon }(t-s)u(t)dr \end{aligned}$$

for $s\in [a,c]$ with $c<b$ such that $c+\epsilon \ge b$ and $\epsilon >0$ sufficiently small. $k$ is uniformly continuous on [a, b]. Hence, for $\delta >0$, we can find $\bar{\epsilon }>0$ such that for all $\epsilon <\bar{\epsilon }$ we have $k(s-r)\le k(s)+\delta$. Then

$$\begin{aligned} \tilde{u}''(s)=u\star \phi _{\epsilon }''(s)&=\int _a^b(\phi _{\epsilon }(t-s))''u(t)dt=\int _a^b\phi _{\epsilon }''(t-s)u(t)dt\\&\le -\int _{a}^b\phi _{\epsilon }(-r)k(s-r)u(r-s)dr\le -(k(s)+\delta ) \tilde{u}(s). \end{aligned}$$

Since $\tilde{u}\in C^2((a,c))\cap C^0([a,c])$, the previous conclusion holds for $\tilde{u}$ and $\tilde{k}=k+\delta$. Now, since u is continuous, $\tilde{u}\rightarrow u$ with respect to uniform convergence on [a, c]. And since solutions of (3) change uniformly continuous if the coefficient $k$ changes uniformly continuous on [a, c], we obtain that ${{\mathrm{\mathfrak {s}}}}_{\tilde{k}_{\gamma }}\rightarrow {{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}$ where $\gamma$ is a geodesic in (c, b). Hence, in the case that where ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}>0$ for any constant-speed geodesic $\gamma :[0,1]\rightarrow (a,b)$, we obtain that ${{\mathrm{\mathfrak {s}}}}_{\tilde{k}_{\gamma }}>0$ for any constant-speed geodesic $\gamma$ in (a, c) by Sturm’s comparison theorem. It follows that (3) holds for $\tilde{u}:[a,c]\rightarrow [0,\infty )$ and by uniform convergence, it also holds for $u|_{[a,c]}$ if $\epsilon \rightarrow 0$. Then, it holds for u since c can be chosen arbitrarily close to b.

Finally, consider the case when there is a geodesic $\gamma$ in (a, b) such that ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}(\gamma (1))=0$. Then we can choose c sufficiently close to b and $\epsilon >0$ sufficiently small such that there is a geodesic $\tilde{\gamma }$ in (a, c) with ${{\mathrm{\mathfrak {s}}}}_{\tilde{k}_{\gamma }}(\gamma (1))=0$. By the previous steps, it follows that $\tilde{u}=\phi _{\epsilon }\star u=0$ that implies $u=0$.

3.1 $ku$-Concavity in Metric Spaces

We consider a metric space $(X,{{\mathrm{d}}}_{{X}})$ and a lower semi-continuous function $k:X\rightarrow {\mathbb {R}}\cup \left\{ \infty \right\} =\bar{{\mathbb {R}}}$. $\bar{{\mathbb {R}}}$ is equipped with the usual topology. We define continuous functions $k_n:X\rightarrow {\mathbb {R}}$ that are bounded from above in the following way:

$$\begin{aligned} k_n(x)=\inf _{y\in X}\left\{ \min (k(y)+n{{\mathrm{d}}}_{{X}}(x,y),n)\right\} \le k(x). \end{aligned}$$

We retain this notation for the rest of the article. $k_n$ is monotone non-decreasing and converges pointwise to $k$ as $n\rightarrow \infty$. For each $k_n$ and for each Lipschitz curve $\gamma \in {\mathcal {LC}}(X)$, we can consider ${{\mathrm{\mathfrak {s}}}}_{k_{n,\gamma }}$ where $k_{n,\gamma }=k_n\circ \bar{\gamma }$ and $\bar{\gamma }:[0,\text{ L }(\gamma )]\rightarrow X$ is the 1-speed reparametrization of $\gamma$. If ${{\mathrm{\mathfrak {s}}}}_{k_{n,\gamma }}>0$ for all n, the generalized $\sin$-function ${{\mathrm{\mathfrak {s}}}}_{k_{n,\gamma }}\ge 0$ is monotone non-increasing with respect to n. Hence, the limit exists pointwise in $[0,\text{ L }(\gamma )]$. It is again denoted with ${{{\mathrm{\mathfrak {s}}}}}_{k_{\gamma }}$. ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}$ is upper semi-continuous and if $k$ is continuous, ${{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}$ coincides with the previous definition. This follows since $k_{n,\gamma }$ converges uniformly to $k_{\gamma }$ by Dini’s theorem. Therefore, the stability of solutions of (3) under uniform changes of the coefficient $k_{\gamma }$ implies that ${{\mathrm{\mathfrak {s}}}}_{k_{n,\gamma }}$ converges uniformly to the solution of (3) with coefficient $k_{\gamma }$. We can see that $\mathfrak {s}_{k_{\gamma }}\ge \mathfrak {s}_{{k}_{\gamma }'}$ if $k,k':X\rightarrow {\mathbb {R}}$ are lower semi-continuous and ${k}'\ge {k}$. In particular, we can consider $X=[a,b]\subset {\mathbb {R}}$.

Definition 3.9

Let $k:X\rightarrow \bar{{\mathbb {R}}}$ be lower semi-continuous and let $\gamma :[0,1]\rightarrow X$ be in ${\mathcal {LC}}(X)$ with $|\dot{\gamma }|=\theta$. Consider the sequence $k_n$ defined as above. Then $\sigma _{k_{n,\gamma }}^{{(t)}}(\theta )$ is monotone non-decreasing in ${\mathbb {R}}\cup \left\{ \infty \right\}$. We define the distortion coefficient with respect to $k:X\rightarrow {\mathbb {R}}$ along $\gamma$ as

$$\begin{aligned} \sigma _{k_{\gamma }}^{{(t)}}(\theta ):=\lim _{n\rightarrow \infty }\sigma _{k_{n,\gamma }}^{{(t)}}(\theta )\in {\mathbb {R}}\cup \left\{ \infty \right\} \ \text{ for } t\in [0,1]. \end{aligned}$$

If $k$ is continuous, the definition is consistent with the previous one. That is $\sigma _{k_{\gamma }}^{{(t)}}(\theta )$ equals $\sigma _{k\circ {\bar{\gamma }}}^{{(t)}}(\theta )$ as in Definition 3.3.

Lemma 3.10

Let $k:X\rightarrow \bar{{\mathbb {R}}}$ be lower semi-continuous, and let $\gamma \in {\mathcal {LC}}(X)$ with $|\dot{\gamma }|=\theta$. If $\sigma _{k_{\gamma }}^{{(t_0)}}(\theta )=\infty$ for some $t_0\in (0,1)$ then $\sigma _{k_{\gamma }}^{{(t)}}(\theta )=\infty$ for any $t\in (0,1)$.

In particular, either one has $\sigma _{k_{\gamma }}^{{(t)}}(\theta )<\infty$ for any $t\in (0,1)$ and

$$\begin{aligned} \sigma _{k_{\gamma }}^{{(t)}}(\theta )={{{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}(t\theta )}/{{{\mathrm{\mathfrak {s}}}}_{k_\gamma }(\theta )}\ \ \text{ where } {{\mathrm{\mathfrak {s}}}}_{k_{\gamma }}(\theta )\ne 0, \end{aligned}$$

or $\sigma _{k_{\gamma }}^{{(\cdot )}}(\theta )\equiv \infty$.

Proof

For the proof, we write $k_{n,\gamma }=k_{n}$ and $k_{\gamma }=k$. Assume $\lim _{n\rightarrow \infty }\sigma _{k_n}^{{(t)}}(\theta )<\infty$. We must have that ${{\mathrm{\mathfrak {s}}}}_{k_n}(t_0\theta )/{{\mathrm{\mathfrak {s}}}}_{k_n}(\theta )\rightarrow \infty$. Since $k_n\uparrow$, let $\underline{k}=const\le k_n$ for all n. Hence, ${{\mathrm{\mathfrak {s}}}}_{k_n}(t\theta )$ satisfies $u''+\underline{k}u\le 0$ for every n. By proposition 3.8, we have for every $s\in [0,1]$ that

$$\begin{aligned} {{\mathrm{\mathfrak {s}}}}_{{k}_n}(st_0\theta )\ge \sigma _{\underline{k}}^{{(s)}}(t_0\theta ){{\mathrm{\mathfrak {s}}}}_{{k}_n}(t_0\theta )+\sigma _{\underline{k}}^{{(1-s)}}(t_0\theta ){{\mathrm{\mathfrak {s}}}}_{{k}_n}(0)= \sigma _{\underline{k}}^{{(s)}}(t_0\theta ){{\mathrm{\mathfrak {s}}}}_{{k}_n}(t_0\theta ) \end{aligned}$$

and

$$\begin{aligned} {{\mathrm{\mathfrak {s}}}}_{{k}_n}(((1-s)t_0+s)\theta )\ge \sigma _{\underline{k}}^{{(1-s)}}(t_0\theta ){{\mathrm{\mathfrak {s}}}}_{{k}_n}(t_0\theta )+\sigma _{\underline{k}}^{{(s)}}(t_0\theta ){{\mathrm{\mathfrak {s}}}}_{{k}_n}(\theta )\ge \sigma _{\underline{k}}^{{(1-s)}}(t_0\theta ){{\mathrm{\mathfrak {s}}}}_{{k}_n}(t_0\theta ) \end{aligned}$$

Hence, if we pick $t\in (0,1)$, we can write $t=st_0$ or $t=(1-s)t_0+s$. If $t=st_0$, it follows:

$$\begin{aligned} {{\mathrm{\mathfrak {s}}}}_{k_n}(t\theta )/{{\mathrm{\mathfrak {s}}}}_{k_n}(\theta )\ge \sigma _{\underline{k}}^{{(s)}}(t_0\theta )\underbrace{{{{\mathrm{\mathfrak {s}}}}_{{k}_n}(t_0\theta )}/{{\mathrm{\mathfrak {s}}}}_{k_n}(\theta )}_{\rightarrow \infty }. \end{aligned}$$

and similarly, for $t=(1-s)t_0+s$. Thus, $\sigma _{k_n}^{(t)}(\theta )\rightarrow \infty$ for each $t\in (0,1)$ if $n\rightarrow \infty$. $\square$

Corollary 3.11

Let $k:X\rightarrow \bar{{\mathbb {R}}}$ be lower semi-continuous, $\gamma$ is a geodesic in X. Then $k\mapsto \sigma _{k_{\gamma }}^{(t)}(\theta )$ is monotone non-decreasing in the sense of Proposition 3.4.

Proof

If $k'\ge k$, let $k'_n$ and $k_n$ be the corresponding approximations. It is clear from the definition that $k_{n,\gamma }'\ge k_{n,\gamma }$. Hence, $\sigma _{k_{n,\gamma }'}^{(t)}(\theta )\ge \sigma _{k_{n,\gamma }}^{(t)}(\theta )$. Taking the limit $n\rightarrow \infty$ yields the result.

Remark 3.12

If $\gamma \in {\mathcal {LC}}(X)$, we define $\gamma ^-(t)=\gamma (1-t)$, and we set

$$\begin{aligned} \sigma _{k_{\gamma }^-}^{{(t)}}(\theta )=\sigma _{k_{\gamma ^-}}^{{(t)}}(\theta ). \end{aligned}$$

Therefore, one can see again that $\sigma _{k}^{{(t)}}(\theta )=\infty$ if and only if $\sigma _{k^-}^{{(t)}}(\theta )=\infty$.

Corollary 3.13

Let $k:X\rightarrow \bar{{\mathbb {R}}}$ be lower semi-continuous, and let $u:X\rightarrow {\mathbb {R}}_{\ge 0}$ be upper semi-continuous. Then the following statements are equivalent:

• (i) $(u\circ \bar{\gamma })''+k_\gamma u\circ \bar{\gamma }\le 0$ in the distributional sense for any constant-speed geodesic $\gamma :[0,1]\rightarrow X$.

• (ii) There is a constant $0<L\le b-a$ such that

  $$\begin{aligned} u(\gamma (t))\ge \sigma ^{{(1-t)}}_{k^{{-}}_{\gamma }}(\theta )u(\gamma (0))+\sigma ^{{(t)}}_{k^{{+}}_{\gamma }}(\theta )u(\gamma (1)) \end{aligned}$$

  for any constant-speed geodesic $\gamma :[0,1]\rightarrow X$ with $\theta =|\dot{\gamma }|=\text{ L }(\gamma )\le L$.

• (iii) The statement in (ii) holds for any geodesic $\gamma :[0,1]\rightarrow X$.

Proof

If $k$ is continuous, the result follows from Proposition 3.8. If $k$ is lower semi-continuous, we consider $k_n$ for $n\in {\mathbb {N}}$ as before. We set $u_{\gamma }(t)=u\circ \gamma (t)$.

(ii) $\Rightarrow$ (i): Since $k_n\uparrow k$, we have $\sigma _{k_{n,\gamma }}^{{(t)}}(\theta )\uparrow \sigma _{k_{\gamma }}^{{(t)}}(\theta )$ for $t\in (0,1)$. Then we can apply part 1. of the proof of Proposition 3.8 to obtain (7) for u with $k$ replaced by $k_n$. That is

$$\begin{aligned} -\int \phi ''(t)u_{\gamma }(t)dt&\ge \int \phi (t)k_{n,\gamma }(t)u_{\gamma }(t)dt\\&=\int \underbrace{[\phi (t)k_{n,\gamma }(t)u_{\gamma }(t)]_+}_{\nearrow }dt-\int \underbrace{[\phi (t)k_{n,\gamma }(t)u_{\gamma }(t)]_-}_{\le C<\infty }dt. \end{aligned}$$

for any $\phi \in C^{\infty }_c((0,|\dot{\gamma }|))$ where the left-hand side and C are independent of n. Recall that $u_{\gamma }$ is non-negative and upper semi-continuous. Hence, by the monotone and dominated convergence theorem, the right-hand side converges to the integral of $\phi k_{\gamma }u_{\gamma }$.

(i) $\Rightarrow$ (iii): We can apply part 3. from the proof of Proposition 3.8, and obtain (9) with $k$ replaced by $k_n$. By the definition of distortion coefficients for general $k$, the result follows. $\square$

Lemma 3.14

Consider $\lambda \in [0,1]$, $\theta >0$, a curve $\gamma \in {\mathcal {LC}}(X)$ with $\text{ L }(\gamma )=\theta$ and $k,k':X\rightarrow \bar{{\mathbb {R}}}$ lower semi-continuous. Then

$$\begin{aligned} \sigma ^{(t)}_{k_{\gamma }}(\theta )^{1-\lambda }\cdot \sigma ^{(t)}_{k_{\gamma }'}(\theta )^{\lambda }\ge \sigma ^{(t)}_{(1-\lambda )k_{\gamma }+\lambda k_{\gamma }'}(\theta ). \end{aligned}$$

Especially, $k\mapsto \log \sigma _{k_{\gamma }}$ is convex.

Proof

For the proof, we write $k_{n,\gamma }=k_{n}$ and $k_{\gamma }=k$. Assume $\sigma _{k}^{{(t)}}(\theta )<\infty$ and $\sigma _{k'}^{{(t)}}(\theta )<\infty$ for each $t\in (0,1)$, since otherwise there is nothing to prove. We assume first that $k$ and $k'$ are continuous. $l:\ t\mapsto \log \left[ \sigma ^{(t)}_{k}(\theta )^{1-\lambda }\cdot \sigma ^{(t)}_{k'}(\theta )^{\lambda }\right]$ solves

$$\begin{aligned} l''\le -(1-\lambda )k-\lambda {k'}-(l')^2. \end{aligned}$$

Hence $\sigma ^{(t)}_{k}(\theta )^{1-\lambda }\cdot \sigma ^{(t)}_{k'}(\theta )^{\lambda }$ solves $v''+\big ((1-\lambda )k+\lambda {k}'\big )v\le 0$ with boundary condition $v(0)=0$ and $v(1)=1$. The result follows by corollary 3.13.

If $k$ and $k'$ are lower semi-continuous, we consider again their approximations by $k_n$ and $k'_n$. We easily obtain that

$$\begin{aligned} \sigma ^{(t)}_{k}(\theta )^{1-\lambda }\cdot \sigma ^{(t)}_{k'}(\theta )^{\lambda }\ge \sigma ^{(t)}_{k_n}(\theta )^{1-\lambda }\cdot \sigma ^{(t)}_{k'_n}(\theta )^{\lambda }\ge \sigma ^{(t)}_{(1-\lambda )k_n+\lambda k'_n}(\theta ). \end{aligned}$$

We show that $\sigma ^{(t)}_{(1-\lambda )k_n+\lambda k'_n}(\theta )\rightarrow \sigma ^{(t)}_{(1-\lambda )k+\lambda k'}(\theta )$. One can check that $(1-\lambda )k_n+\lambda k'_n\le ((1-\lambda )k+\lambda k')_n$. On the other hand, by continuity of the approximating sequence for all $n\in {\mathbb {R}}$ and for all $x\in [0,\theta ]$, there exists $m_{x}\ge 2^n$ and $\delta _x>0$ such that $(1-\lambda )k_{\bar{m}}+\lambda k'_{\bar{m}}\ge ((1-\lambda )k+\lambda k')_n$ on $B_{\delta _x}(x)$ for all $\bar{m}\ge m_x$. Hence, by compactness of $[0,\theta ]$, we can choose $x_1,\ldots ,x_n$ such that $[0,\theta ]\subset \bigcup _{i=1,\ldots ,n}B_{\delta _{x_i}}(x_i)$. Then $(1-\lambda )k_{m_n}+\lambda k'_{m_n}\ge ((1-\lambda )k+\lambda k')_n$ for $m_n:=\max _{i}m_{x_i}$. Hence,

$$\begin{aligned} \underbrace{\sigma _{((1-\lambda )k+\lambda k')_{m_n}}^{{(t)}}(\theta )}_{\rightarrow \sigma _{(1-\lambda )k+\lambda k'}^{{(t)}}(\theta )}\le \sigma _{(1-\lambda )k_{m_n}+\lambda k'_{m_n}}^{{(t)}}(\theta )\le \underbrace{ \sigma _{((1-\lambda )k+\lambda k')_n}^{{(t)}}(\theta )}_{\rightarrow \sigma ^{{(t)}}_{(1-\lambda )k+\lambda k'}(\theta )}. \end{aligned}$$

Hence, $\sigma ^{(t)}_{(1-\lambda )k_n+\lambda k'_n}(\theta )\rightarrow \sigma ^{(t)}_{(1-\lambda )k+\lambda k'}(\theta )$. $\square$

Proposition 3.15

Let $k:X\rightarrow \bar{{\mathbb {R}}}$ be continuous (lower semi-continuous). Let $t\in (0,1)$. Then the map

$$\begin{aligned} \gamma \in ({\mathcal {LC}}(X),{{\mathrm{d}}}_{\infty }) \mapsto \sigma _{k^{+/-}_{\gamma }}^{{(t)}}(|\dot{\gamma }|)\in {\mathbb {R}}\cup \left\{ \infty \right\} \end{aligned}$$

is continuous (lower semi-continuous).

Proof

If $k$ is continuous, the result follows from Proposition 3.5. For $k$ lower semi-continuous, we consider its continuous approximation $k_n$. Then, by definition for any Lipschitz curve $\gamma \in {\mathcal {LC}}(X)$,

$$\begin{aligned} \sigma _{k^{+/-}_{n,\gamma }}^{{(t)}}(|\dot{\gamma }|)\uparrow \sigma _{k^{+/-}_{\gamma }}^{{(t)}}(|\dot{\gamma }|). \end{aligned}$$

In particular, $\gamma \mapsto \sigma _{k^{+/-}_{\gamma }}^{{(t)}}(|\dot{\gamma }|)$ is lower semi-continuous $\square$

Definition 3.16

Consider a metric space $(Y,{{\mathrm{d}}}_{{Y}})$ and a lower semi-continuous function $k:Y\rightarrow \bar{{\mathbb {R}}}$. We say an upper semi-continuous function $u:Y\rightarrow [0,\infty )$ is $ku$ -convex if $u<\infty$ and for all geodesics $\gamma :[0,1]\rightarrow Y$

$$\begin{aligned} u(\gamma (t))\ge \sigma ^{{(1-t)}}_{k^{{-}}_{\gamma }}(\text{ L }(\gamma ))u(\gamma (0))+\sigma ^{{(t)}}_{k^{{+}}_{\gamma }}(\text{ L }(\gamma ))u(\gamma (1)) \end{aligned}$$

(11)

where $k_{\gamma }=k\circ \bar{\gamma }:[0,\text{ L }(\gamma )]\rightarrow Y$ and $\bar{\gamma }$ is the unit-speed reparametrization of $\gamma$.

We say u is weakly $ku$-convex if $u<\infty$ and for all $x,y\in Y$ there exists a geodesic $\gamma :[0,1]\rightarrow Y$ between x and y such that (11) holds.

We say a function $f:Y\rightarrow {\mathbb {R}}\cup \left\{ \pm \infty \right\}$ is (weakly) $(k,N)$-convex if $e^{-\frac{f}{N}}=u$ is (weakly) $\frac{k}{N} u$-concave. We use the convention $e^{\infty }=\infty$, $e^{-\infty }=0$.

4 Curvature-Dimension Condition

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a metric measure space. Given a number $N\in {\mathbb {R}}$ with $N\ge 1$, we define the N-Rényi entropy functional

$$\begin{aligned} S_N(\ \cdot \ |{{\mathrm{m}}}_{{X}}):{\mathcal {P}}_2(X)\rightarrow {\mathbb {R}} \end{aligned}$$

with respect to ${{\mathrm{m}}}_{{X}}$ by

$$\begin{aligned} \nu =\varrho {{\mathrm{m}}}_{{X}}+\nu ^s\mapsto S_N(\nu ):=S_N(\nu |{{\mathrm{m}}}_{{X}}):=-\int _X\varrho ^{-\frac{1}{N}}d\nu \end{aligned}$$

where $\varrho {{\mathrm{m}}}_{{X}}+\nu ^s$ is the Lebesgue decomposition of $\nu$. $S_{N}$ is lower semi-continuous for $N>1$. If ${{\mathrm{m}}}_{{X}}$ is a finite measure for each $\nu \in {\mathcal {P}}_2(X)$, we have

$$\begin{aligned} \text{ Ent }(\nu |{{\mathrm{m}}}_{{X}})=\lim _{N\rightarrow \infty }N(1+S_{N}(\nu )), \end{aligned}$$

where $\text{ Ent }$ is the Boltzmann–Shanon entropy functional, and this case ${{\mathrm{Ent}}}$ is lower semi-continuous.

We consider $k={k}/{N}$ where $k:X\rightarrow {\mathbb {R}}\cup \left\{ \infty \right\} =:\bar{{\mathbb {R}}}$ is a lower semi-continuous function and locally bounded from below, and we set $\sigma _{k_{\gamma }/N}^{(t)}(\theta )=\sigma _{\theta ^2k_{\gamma }(\cdot \theta )/N}^{(t)}(1)=\sigma _{k_{\gamma },N}^{(t)}(\theta )$ where $\gamma \in {\mathcal {LC}}(X)$ and $\theta =|\dot{\gamma }|$.

Definition 4.1

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$, $k$ and $\gamma$ as before. We define generalized distortion coefficients with respect to $k$ and N along $\gamma$ as

$$\begin{aligned} \tau _{k_{\gamma },N}^{(t)}(\theta )={\left\{ \begin{array}{ll}\theta \cdot \infty &{} \text{ if } k>0 \text{ and } N=1\\ t^{\frac{1}{{N}}}\big [\sigma _{k_{\gamma },N-1}^{(t)}(\theta )\big ]^{\frac{{N}-1}{{N}}}&{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

We use the conventions $r\cdot \infty =\infty$ for $r>0$, $0\cdot \infty =0$ and $(\infty )^{\alpha }=\infty$ for $\alpha > 0$. If $k>0$, we have $\tau ^{(t)}_{{k}_{\gamma },1}(\theta )<\infty$ if and only if $\theta =0$, and $\tau _{{k}_{\gamma },1}^{(t)}(\theta )=t$ if $k\le 0$.

Corollary 4.2

For $k,k':[0,1]\rightarrow \bar{{\mathbb {R}}}$, $N,N'>0$, $t\in [0,1]$, and $\theta >0$,

$$\begin{aligned} \sigma _{k,N}^{(t)}(\theta )^{N}\sigma _{k',N'}^{(t)}(\theta )^{N'}\ge \sigma _{k+k',N+N'}^{(t)}(\theta )^{N+N'} \end{aligned}$$

and, if $N\ge 1$,

$$\begin{aligned} \tau _{k,N}^{(t)}(\theta )^{N}\sigma _{k',N'}^{(t)}(\theta )^{N'}\ge \tau _{k+k',N+N'}^{(t)}(\theta )^{N+N'}, \end{aligned}$$

and in particular

$$\begin{aligned} \tau _{k,N}^{(t)}(\theta )^{N}\tau _{k',N'}^{(t)}(\theta )^{N'}\ge \tau _{k+k',N+N'}^{(t)}(\theta )^{N+N'}. \end{aligned}$$

Proof

The result follows directly from Lemma 3.14. $\square$

Remark 4.3

For the rest of the article, we always assume that $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ is a metric measure space and $k:X\rightarrow \bar{{\mathbb {R}}}$ is lower semi-continuous and locally bounded from below. In this case, we say that $k$ is an admissible function. It follows from Proposition 3.15 that if $k$ is continuous (lower semi-continuous), the map

$$\begin{aligned} \gamma \in {\mathcal {G}}(X)\mapsto \tau _{{k}^{+/-}_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\in {\mathbb {R}}_{\ge 0}\cup \left\{ \infty \right\} \end{aligned}$$

is continuous (lower semi-continuous) for $t\in [0,1]$. In particular, it is measurable and we can integrate it with respect to probability measures on ${\mathcal {G}}(X)$.

Definition 4.4

Consider an admissible function $k$, and $N\in [1,\infty )$. A metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies the curvature-dimension condition $CD(k,N)$, if for each pair $\nu _0,\nu _1\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ with bounded support there exists a geodesic $(\nu _t)_{t\in [0,1]}\subset {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ and a dynamic optimal coupling $\Pi \in {\mathcal {P}}(X)$ such that $(e_t)_{\star }\Pi =\nu _t$ and

$$\begin{aligned} S_{N'}(\nu _t)\le {\textstyle -}\int \Big [\tau _{k^{{-}}_{\gamma },N'}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0\left( e_0(\gamma )\right) ^{-\frac{1}{N'}}+\tau _{k^{{+}}_{\gamma },N'}^{{(t)}}(|\dot{\gamma }|)\varrho _1\left( e_1(\gamma )\right) ^{-\frac{1}{N'}}\Big ]d\Pi (\gamma ) \end{aligned}$$

(12)

for all $t\in [0,1]$ and all $N'\ge N$. $k_{\gamma }=k\circ \bar{\gamma }$ where $\gamma :[0,1]\rightarrow X$ is a geodesic and $\bar{\gamma }$ its 1-speed reparametrization.

Remark 4.5

The right-hand side in (12) is also denoted with $T^{{(t)}}_{k,N'}(\Pi |{{\mathrm{m}}}_{{X}})$. If $\Pi$ is the optimal dynamic coupling from the previous definition, let $\Pi '(x_0,x_1)(d\gamma )=:\Pi '_{x_0,x_1}(d\gamma )$ be its disintegration with respect to $(e_0,e_1)_{\star }\Pi =\pi$. One can reformulate (12) in the following way

$$\begin{aligned} S_{N'}(\nu _t)\le {\textstyle -}\int \Big [{\mathcal {T}}_{k^{{-}},N'}^{{(1-t)}}(\Pi '_{x_0,x_1})\varrho _0\left( x_0\right) ^{-\frac{1}{N'}}+{\mathcal {T}}_{k^+,N'}^{{(t)}}(\Pi '_{x_0,x_1})\varrho _1\left( x_1\right) ^{-\frac{1}{N'}}\Big ]d\pi (x_0,x_1) \end{aligned}$$

(13)

where ${\mathcal {T}}_{k^-,N'}^{{(1-t)}}(\Pi ')=\int \tau _{k^{{-}}_{\gamma },N'}^{{(1-t)}}(|\dot{\gamma }|)d\Pi '(d\gamma ).$

Conversely, if there is a kernel $\Pi _{x_0,x_1}'(d\gamma )$ such that for $\mu _0$ and $\mu _1$ there exists a geodesic $\mu _t$ and an optimal coupling $\pi$ with (13), then X satisfies $CD(k,N)$.

Remark 4.6

In the case where $k$ is constant, the previous definition is equivalent to a variant of Sturm’s curvature-dimension condition in [34] that is mostly used by other authors (for instance, in [30]). On the right-hand side, Sturm requires integration with respect to an optimal coupling $\pi$ between $\nu _0$ and $\nu _1$ that is not necessarily related to $\mu _t$. However, most authors assume $\pi$ is induced by a dynamic coupling that also induces $\mu _t$. In any case, our definition yields Sturm’s definition for constant lower curvature bounds. On the other hand, if we consider a space that satisfies this variant of the curvature-dimension condition for constant lower curvature bound, it is exactly the condition that we propose.

Definition 4.7

Two metric measure spaces $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ and $(X',{{\mathrm{d}}}_{{X}'},{{\mathrm{m}}}_{{X}'})$ are called isomorphic if there exists an isometry $\psi :{{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}\rightarrow {{\mathrm{supp}}}{{\mathrm{m}}}_{{X}'}$ such that

$$\begin{aligned} \psi _{\star }{{\mathrm{m}}}_{{X}}={{\mathrm{m}}}_{{X}'}. \end{aligned}$$

Remark 4.8

As Sturm states in [34], one might not require that the geodesic $\nu _t$ is the projection of $\Pi$ w.r.t. $e_t$. Consequently, $(e_t)_{\star }\Pi$ might not be necessarily ${{\mathrm{m}}}_{{X}}$-absolutely continuous, or supported in ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$. In this case, we say $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies a modified curvature-dimension condition. However, if $(e_t)_{\star }\Pi$ is not supported in ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$, the condition would not be a property of the isomorphism class of the metric measure space. In this case, the right notion of isomorphism would be to say $\psi :(X,{{\mathrm{d}}}_{{X}})\rightarrow (X',{{\mathrm{d}}}_{{X}'})$ is an isometry and $\psi _{\star }{{\mathrm{m}}}_{{X}}={{\mathrm{m}}}_{{X}'}$. Then, a modified curvature-dimension condition is stable w.r.t. $\psi$ what is apparent from the proof of (i) in the next Proposition.

Proposition 4.9

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a metric measure space which satisfies the condition $CD(k,N)$ for a continuous function $k:X\rightarrow {\mathbb {R}}$ and $N\ge 1$.

• (i) If there is a strong isomorphism $\psi :(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})\rightarrow (X',{{\mathrm{d}}}_{{X}'},{{\mathrm{m}}}_{{X}'})$ onto a metric measure space $(X',{{\mathrm{d}}}_{{X}'},{{\mathrm{m}}}_{{X}'})$ then $(X',{{\mathrm{d}}}_{{X}'},{{\mathrm{m}}}_{{X}'})$ satisfies the condition $CD(\psi ^{\star }k,N)$ with $\psi ^{\star }k=k\circ \psi$.

• (ii) For $\alpha ,\beta >0$ the rescaled metric measure space $(X',\alpha {{\mathrm{d}}}_{{X}'},\beta {{\mathrm{m}}}_{{X}'})$ satisfies $CD(\alpha ^{-2}k,N)$.

• (iii) For each geodesically convex subset $U\subset X$, the metric measure space $(U,{{\mathrm{d}}}_{{X}}|_{U\times U},{{\mathrm{m}}}_{{X}}|_{U})$ satisfies $CD(k|_{U},N)$.

Proof

(i) First, we observe that $\psi ^{\star }k$ is still lower semi-continuous and locally bounded from below. $\psi$ induces an isometry from ${\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ to ${\mathcal {P}}_2(X',{{\mathrm{m}}}_{{X}'})$, and the image of a geodesic in X is a geodesic in $X'$. Moreover,

$$\begin{aligned} \int _X\varrho _t^{-\frac{1}{N}+1}d{{\mathrm{m}}}_{{X}}=\int _{X'}(\varrho _t\circ \psi )^{-\frac{1}{N}+1}d{{\mathrm{m}}}_{{X}'}. \end{aligned}$$

$\psi _{\star }\Pi$ is an optimal dynamic plan, if $\Pi$ is so. Then the result follows.

(ii), (iii) The results follow easily. One can easily adapt the proofs of similar statements in [34].

Definition 4.10

[35] Let $k$ be admissible. We say a metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ with ${{\mathrm{m}}}_{{X}}(X)=1$ satisfies the condition $CD(k,\infty )$ if for $\mu _0,\mu _1\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ there exists a $W_2$-geodesic $\mu _t\in {\mathcal {P}}^2(X,{{\mathrm{m}}}_{{X}})$ and an optimal dynamic plan $\Pi \in {\mathcal {P}}({\mathcal {G}}(X)$ such that $(e_t)_{\star }\Pi =\mu _t$ and

$$\begin{aligned} {{\mathrm{Ent}}}(\mu _t)\le (1-t){{\mathrm{Ent}}}(\mu _0)+t{{\mathrm{Ent}}}(\mu _1)-\int _0^1\int _{}g(s,t) k(\gamma (s))|\dot{\gamma }(s)|^2d\Pi (\gamma )ds \end{aligned}$$

(14)

for all $t\in [0,1]$. $g(s,t)=\min \left\{ s(1-t),t(1-s)\right\}$ is the Green function of [0, 1].

Note that ${{\mathrm{m}}}_{{X}}(X)=1$ guarantees that ${{\mathrm{Ent}}}:{\mathcal {P}}^2(X)\rightarrow {\mathbb {R}}\cup \left\{ \infty \right\}$ is a well-defined, lower semi-continuous function. Otherwise, an exponential growth condition for ${{\mathrm{m}}}_{{X}}$ has to be assumed (for instance, see [17]).

Proposition 4.11

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a metric measure space which satisfies the condition $CD(k,N)$ for a continuous function $k:X\rightarrow {\mathbb {R}}$ and $N\ge 1$.

• (i) If $k':X\rightarrow {\mathbb {R}}$ is a continuous function such that $k'\le k$, and if $N'\ge N$, then $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ also satisfies the condition $CD(k',N')$.

• (ii) If $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ has finite mass then it satisfies the condition $CD(k,\infty )$ in the sense of Sturm.

Proof

(i) is an immediate consequence of the monotonicity of $\sigma _{k}^{{(t)}}(\theta )$ w.r.t. $k$.

For (ii), it suffices to consider $\nu _0,\nu _1\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ with ${{\mathrm{Ent}}}(\nu _0|m_{{X}})<\infty$ and ${{\mathrm{Ent}}}(\nu _1|m_{{X}})<\infty$. In any other case, the right-hand side in (14) is $\infty$. By assumption, $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies $CD(k,N)$. Hence, there exists a dynamic optimal transference plan $\Gamma$ between $\nu _0$ and $\nu _1$ such that (12) is satisfied for $\forall N'\ge N$.

The assumption ${{\mathrm{m}}}_{{X}}(X)<\infty$ implies that ${{\mathrm{Ent}}}((e_t)_*\Gamma |{{\mathrm{m}}}_{{X}})=\lim _{N'\rightarrow \infty }(1+S_{N'}((e_t)_*\Gamma |{{\mathrm{m}}}_{{X}})$ for $t\in [0,1]$. It follows

$$\begin{aligned}&N'(1+S_{N'}((e_t)_*\Gamma |{{\mathrm{m}}}_{{X}}))\\&\quad \le -N'\int \left[ -(1-t)+\tau _{k_{\gamma }^{{-}},N'}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0(e_0(\gamma ))^{-\frac{1}{N'}}-t+\tau _{k_{\gamma }^{{+}},N'}^{{(t)}}(|\dot{\gamma }|)\varrho _1(e_1(\gamma ))^{-\frac{1}{N'}}\right] \Gamma (\gamma )\\&\quad \le (1-t)N'(1+S_{N'}((e_0)_*\Gamma |{{\mathrm{m}}}_{{X}}))+tN'(1+S_{N'}((e_1)_*\Gamma |{{\mathrm{m}}}_{{X}}))\\&\quad \quad -N'\int \left[ \left[ (1-t)+\sigma _{k_{\gamma }^{{-}},N'}^{{(1-t)}}(|\dot{\gamma }|)\right] \varrho _0(e_0(\gamma ))^{\frac{-1}{N'}}+\left[ t+\sigma _{k_{\gamma }^{{+}},N'}^{{(t)}}(|\dot{\gamma }|)\right] \varrho _1(e_1(\gamma ))^{\frac{-1}{N'}}\right] \Gamma (\gamma )\\&\quad \le (1-t)N'(1+S_{N'}((e_0)_*\Gamma |{{\mathrm{m}}}_{{X}}))+tN'(1+S_{N'}((e_1)_*\Gamma |{{\mathrm{m}}}_{{X}}))\\&\quad \quad -\int \underbrace{N'\left[ (1-\sigma _{k_{\gamma }^{{-}},N'}^{{(1-t)}}(|\dot{\gamma }|)-\sigma _{k_{\gamma }^{{+}},N'}^{{(t)}}(|\dot{\gamma }|)\right] }_{=w(t)}\Gamma (\gamma ) \end{aligned}$$

w solves $w''=-k_{\gamma }|\dot{\gamma }|^2(\sigma _{k_{\gamma }^{{-}},N}^{{(1-t)}}(|\dot{\gamma }|)+\sigma _{k_{\gamma }^{{+}},N}^{{(t)}}(|\dot{\gamma }|))$ with $w(0)=w(1)=0$. Hence

$$\begin{aligned} w= \int _0^1\left[ g(s,t){k}_{\gamma }|\dot{\gamma }|^2(\sigma _{k_{\gamma }^{{-}},N}^{{(1-s)}}(|\dot{\gamma }|)+\sigma _{k_{\gamma }^{{+}},N}^{{(s)}}(|\dot{\gamma }|))\right] ds. \end{aligned}$$

Since $\sigma _{k_{\gamma }^{{-}},N}^{{(1-t)}}(|\dot{\gamma }|)+\sigma _{k_{\gamma }^{{+}},N}^{{(t)}}(|\dot{\gamma }|)\rightarrow 1$ if $N'\rightarrow \infty$ uniformly in $\gamma \in {\mathcal {G}}(X)$ for fixed t, it follows

$$\begin{aligned}&N'(1+S_{N'}((e_t)_*\Gamma |{{\mathrm{m}}}_{{X}}))\\&\le (1-t)N'(1+S_{N'}((e_0)_*\Gamma |{{\mathrm{m}}}_{{X}}))+tN'(1+S_{N'}((e_1)_*\Gamma |{{\mathrm{m}}}_{{X}}))\\&-\int \int _0^1\left[ g(s,t){k}_{\gamma }|\dot{\gamma }|^2(\sigma _{k_{\gamma }^{{-}},N}^{{(1-t)}}(|\dot{\gamma }|)+\sigma _{k_{\gamma }^{{+}},N}^{{(t)}}(|\dot{\gamma }|))\right] ds\Gamma (\gamma )\\&\left[ \rightarrow -\int \int _0^1g(s,t){k}_{\gamma }|\dot{\gamma }|^2ds\Gamma (\gamma ) \text{ if } N'\rightarrow \infty \right] \end{aligned}$$

and this implies the result. $\square$

Theorem 4.12

Let $(M,g_{{M}},Vd{{\mathrm{vol}}}_{{M}})$ be a weighted Riemannian manifold for a smooth function $V:M\rightarrow (0,\infty )$. Let $k:M\rightarrow {\mathbb {R}}$ be a lower semi-continuous function and $N\ge 1$.

The metric measure space $(M,{{\mathrm{d}}}_{{M}},Vd{{\mathrm{vol}}}_{{M}})$ satisfies the curvature-dimension condition $CD(k,N)$ if and only if $(M,g_{{M}},Vd{{\mathrm{vol}}}_{{M}}))$ has N-Ricci curvature bounded from below by $k$.

Remark 4.13

For each real number $N>n$, the N-Ricci tensor is defined as

$$\begin{aligned}{{\mathrm{ric}}}^{N,V}(v)&={{\mathrm{ric}}}(v)-(N-n)\frac{\nabla ^2 V^{\frac{1}{N-n}}(v)}{V^{\frac{1}{N-n}}(p)} \end{aligned}$$

where $v\in TM_p$. For $N=n$, we define

$$\begin{aligned}{{\mathrm{ric}}}^{N,V}(v):= {\left\{ \begin{array}{ll} {{\mathrm{ric}}}(v) +\nabla ^2\log V(v) &{}\nabla \log V (v)=0\\ -\infty &{} \text{ else }. \end{array}\right. }\end{aligned}$$

For $1\le N<n$, we define ${{\mathrm{ric}}}^{N,\Psi }(v):=-\infty$ for all $v\ne 0$ and 0 otherwise.

Example 4.14

Let $\overline{(\alpha ,\beta )}=I\subset {\mathbb {R}}$ be some interval where $\alpha ,\beta \in {\mathbb {R}}\cup \left\{ \pm \infty \right\}$. Let $k:I\rightarrow {\mathbb {R}}$ be a lower semi-continuous function and let $u:I\rightarrow [0,\infty )$ be a non-negative solution of $u''+ {\textstyle \frac{k}{N-1}} u=0$ for $N>1$. Then, the metric measure space $(I,|\cdot |_2,u^{N-1}d{\mathcal {L}}^1)$ satisfies the curvature-dimension $CD(k,N)$.

Proof

“$\Leftarrow$”: Pick a point $p\in M$ and $\epsilon >0$ such that $k|_{B_{\epsilon }(p)}\ge k_{\epsilon }$. There exists geodesically convex ball $B_{\delta }(p)$ for $0<\delta <\epsilon$ around p. Hence,

$$\begin{aligned} (B_{\delta }(p),d_{{M}}|_{B_{\delta }(p)},Vd{{\mathrm{vol}}}_{{M}}|_{B_{\delta }}) \end{aligned}$$

satisfies the condition $CD(k_{\epsilon },N)$. It follows that the N-Ricci tensor is bounded from below by $k_{\epsilon }$ (for instance see [34]). If $\epsilon$ goes to 0, we see that $k_{\epsilon }\rightarrow k(p)$ and the result follows.

“$\Rightarrow$”: The proof goes exactly as the proof of the corresponding result in [24, 34] or [9]. $\square$

5 Geometric Consequences

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be metric measure space. All the results of this section stay true if we replace the curvature-dimension condition by the modified curvature-dimension condition of Remark 4.8.

Theorem 5.1

(Brunn–Minkowski inequality) Assume that the metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies $CD(k,N)$ for $k$ admissible and $N\ge 1$. Let $A_0,A_1\subset X$ be bounded Borel sets such that ${{\mathrm{m}}}_{{X}}(A_0){{\mathrm{m}}}_{{X}}(A_1)>0$. We set ${\mathcal {G}}(A_0,A_1)=\left\{ \gamma \in {\mathcal {G}}(X):\gamma (i)\in A_i, i=0,1\right\}$. Then

$$\begin{aligned} {{\mathrm{m}}}_{{X}}(A_t)^{\frac{1}{N}}\ge \inf _{\gamma \in {\mathcal {G}}(A_0,A_1)}\tau _{{{k}}^-_{\gamma },{N}}^{{(1-t)}}(|\dot{\gamma }|){{\mathrm{m}}}_{{X}}(A_0)^{\frac{1}{N}}+ \inf _{\gamma \in {\mathcal {G}}(A_0,A_1)}\tau _{{{k}}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|){{\mathrm{m}}}_{{X}}(A_1)^{\frac{1}{N}}. \end{aligned}$$

(15)

where $\inf _{\gamma \in {\mathcal {G}}(A_0,A_1)}\tau _{{{k}}^{-/+}_{\gamma },{N}}^{{(1-t/t)}}(|\dot{\gamma }|)\ge 0$.

Proof

First, assume ${{\mathrm{m}}}(A_0),{{\mathrm{m}}}(A_1)<\infty$ and set $\mu _i={{\mathrm{m}}}(A_i)^{-1}{{\mathrm{m}}}|_{A_i}$ for $i=0,1$. The curvature-dimension yields

$$\begin{aligned} \int _{A_t}\varrho _t^{\frac{1}{N'}}d\mu _t\ge \int \tau _{{{k}}^-_{\gamma },{N}}^{{(1-t)}}(|\dot{\gamma }|){{\mathrm{m}}}_{{X}}(A_0)^{1/N}+ \tau _{{{k}}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|){{\mathrm{m}}}_{{X}}(A_1)^{1/N} \end{aligned}$$

where $(\mu _t=\varrho _td{{\mathrm{m}}}_{{X}})_t$ denotes the absolutely continuous geodesic that connects $\mu _0$ and $\mu _1$, and $\Pi$ is an optimal dynamic plan. By Jensen’s inequality, the left-hand side of the previous inequality is smaller than ${{\mathrm{m}}}_{{X}}(A_t)^{\frac{1}{N'}}$. The general case follows by approximation of $A_i$ by sets of finite measure. $\square$

Definition 5.2

(Minkowski content) Consider $x_0\in {{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$ and $B_r(x_0)\subset X$. Set $v(r)={{\mathrm{m}}}_{{X}}(\bar{B}_r(x_0))$. The Minkowski content of $\partial B_r(x_0)$ (the r-sphere around $x_0$) is defined as

$$\begin{aligned} s(r):=\limsup _{\delta \rightarrow 0}\frac{1}{\delta }{{\mathrm{m}}}_{{X}}(\bar{B}_{r+\delta }(x_0)\backslash B_r(x_0)). \end{aligned}$$

Theorem 5.3

Assume $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies $CD(k,N)$ for an admissible function $k$ and $N\in [1,\infty )$. Then, $(X,{{\mathrm{d}}}_{{X}})$ is a proper metric space, bounded sets have finite measure and satisfy a doubling property, and either ${{\mathrm{m}}}_{{X}}$ is supported by one point or all points and all sphere have mass 0.

In particular, if $N>1$ then for each $x_0\in {{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$, for all $0<r<R$ and $\underline{k}\in {\mathbb {R}}$ such that $k|_{B_R(x_0)}\ge \underline{k}$ and $R\le \pi \sqrt{(N-1)/\underline{k}\vee 0}$, we have

$$ \begin{aligned} \frac{s(r)}{s(R)}\ge \frac{\sin _{\underline{{k}}/({N}-1)}^{{N}-1}r}{\sin _{\underline{{k}}/({N}-1)}^{{N}-1}R} \ \ \& \ \ \frac{v(r)}{v(R)}\ge \frac{\int _0^r\sin _{\underline{{k}}/({N}-1)}^{{N}-1}tdt}{\int _0^R\sin _{\underline{{k}}/({N}-1)}^{{N}-1}tdt}. \end{aligned}$$

(16)

If $N=1$ and $k\le 0$, then

$$\begin{aligned} \frac{s(r)}{s(R)}\ge 1,\ \ \ \ \frac{v(r)}{v(R)}\ge \frac{r}{R}. \end{aligned}$$

Proof

1. Let us fix a point $x_0\in X$ such that ${{\mathrm{m}}}_{{X}}(\left\{ x_0\right\} )=0$, and let $R>0$ be sufficiently small such that $k|_{B_{2R}(x_0)}\ge \underline{k}$ for some $\underline{k}\in {\mathbb {R}}$. Let $r\in (0,R)$ and put $t=r/R$. We choose $\epsilon >0$ and $\delta >0$ and define $A_0=B_{\epsilon }(x_0)$ and $A_1=\bar{B}_{R+\delta R}(x_0)\backslash B_R(x_0)$. By triangle inequality, one easily verifies that

$$\begin{aligned} A_t\subset \bar{B}_{r+\delta r+\epsilon r/R}(x_0)\backslash B_{r-\epsilon r/R}(x_0)\subset B_{2R}(x_0). \end{aligned}$$

Hence, if we consider measures $\mu _i={{\mathrm{m}}}_{{X}}(A_i)^{-1}{{\mathrm{m}}}_{{X}}|_{A_i}$ for $i=0,1$, the curvature-dimension condition, ${{\mathrm{m}}}_{{X}}(\left\{ x_0\right\} )=\emptyset$, local finiteness of the reference measure, and the monotonicity of the distortion coefficients imply that

$$\begin{aligned} {{\mathrm{m}}}_{{X}}(\bar{B}_{(1+\delta )r}(x_0)\backslash B_r(x_0))^{1/N}\ge \tau _{\underline{k},N}^{{(r/R)}}((1\pm \delta )R){{\mathrm{m}}}_{{X}}(\bar{B}_{(1+\delta )R(x_0)}\backslash B_R(x_0))^{1/N}. \end{aligned}$$

Since ${{\mathrm{m}}}_{{X}}$ is locally finite, we can assume that the right-hand side is finite.

2. Now, we can follow precisely the proof of Theorem 2.3 in [34] to obtain that ${{\mathrm{m}}}_{{X}}(\partial B_r(x_0))=0$ for $r\in (0,R)$, ${{\mathrm{m}}}_{{X}}(\left\{ x\right\} )=0$ for $x\in B_R(x_0)\backslash \left\{ x_0\right\}$ and (16) for $r\in (0,R)$ and $R>0$ as chosen like in the first step. If ${{\mathrm{m}}}_{{X}}(\left\{ x_0\right\} )\ne 0$, we can choose a point x close to $x_0$ such that ${{\mathrm{m}}}_{{X}}(\left\{ x\right\} )=0$ and $B_R(x)\subset B_{2R}(x_0)$. This is implied by the local finiteness of ${{\mathrm{m}}}_{{X}}$ and the existence of $\epsilon$-geodesics. If there is no such point x then necessarily ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}=\left\{ x_0\right\}$. We repeat the previous steps for x instead of $x_0$ and obtain that ${{\mathrm{m}}}_{{X}}(\left\{ x_0\right\} )=0$ unless ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}=\left\{ x_0\right\}$.

3. Hence, for any $x_0\in X$, there is $R>0$ (sufficiently small) such that ${{\mathrm{d}}}_{{X}}$ and ${{\mathrm{m}}}_{{X}}$ restricted to $\bar{B}_R(x_0)$ satisfy a doubling property provided the radius of balls is sufficiently small, and therefore $\bar{B}_r(x_0)$ is compact for $r\in (0,R)$. In particular, X is locally compact. Then, since $(X,{{\mathrm{d}}}_{{X}})$ is also a complete length space, the generalized Hopf–Rinow theorem (for instance, see Theorem 2.5.28 in [6]) implies $(X,{{\mathrm{d}}}_{{X}})$ is a proper metric space. Therefore, any closed ball $\bar{B}_R(x_0)$ is compact, and we can repeat the previous step for any $0<r<R$. In particular, it follows that (16) holds, and any bounded set has finite measure. $\square$

Corollary 5.4

(Doubling) For each metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfying the condition $CD(k,N)$ for an admissible $k$ and $N\ge 1$, the doubling property holds on each bounded set $X'\subset {{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$, and in the case of $k\ge 0$, the doubling constant is $\le 2^N$, and otherwise it can be estimated in terms of $\underline{k}$, N and L as follows

$$\begin{aligned} C\le 2^N{{\mathrm{\mathfrak {c}}}}_{{k}/({N}-1)}^{{N}-1}L \end{aligned}$$

where L is the diameter of the bounded set $X'$, and $\underline{k}=\min _{X'}k$.

Proof

The result follows from the previous theorem (see also [34]).

Corollary 5.5

(Hausdorff dimension) For each metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfying a curvature-dimension condition $CD(k,N)$ for some admissible $k$ and $N\ge 1$, the Hausdorff dimension of ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$ is $\le N$.

Definition 5.6

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be any metric measure space, let $N\ge 1$ and let $k:X\rightarrow {\mathbb {R}}$ be admissible. We define the effective diameter of $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ with respect to $k$ and N as

$$\begin{aligned} \pi _{{k}/(N-1)}=\sup \left\{ {{\mathrm{d}}}_W(\mu _0,\mu _1):\exists \Pi \in {{\mathrm{DyCpl}}}^{abs}(\mu _0,\mu _1) \ \text{ with } \ {\mathcal {T}}_{{k}^{+/-},{N}}^{(t)}(\Pi )<\infty \right\} . \end{aligned}$$

where $\Pi \in {{\mathrm{DyCpl}}}^{abs}(\mu _0,\mu _1)$ if $\Pi \in {{\mathrm{DyCpl}}}(\mu _0,\mu _1)$ with $(e_t)_{\star }\Pi$ ${{\mathrm{m}}}_{{X}}$-absolutely continuous. By definition, we have $\pi _{{k}/({N}-1)}\le {{\mathrm{diam}}}{{{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}}$.

Proposition 5.7

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfy $CD(k,N)$ for an admissible function $k$ and $N\ge 1$. Then $\pi _{{k}/({N}-1)}={{\mathrm{diam}}}{{{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}}$.

Proof

Assume $\pi _{{k}/({N}-1)}<{{\mathrm{diam}}}{{{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}}$. Then, there are points $x,y\in {{{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}}$ such that ${{\mathrm{d}}}_{{X}}(x,y)>c+\pi _{{k}/({N}-1)}$ for some $c>0$. Therefore, we can consider $\epsilon$-balls $B_{\epsilon }(x)=A_0$ and $B_{\epsilon }(y)=A_1$ such that

$$\begin{aligned} {{\mathrm{d}}}_{{X}}(A_0,A_1):=\inf _{x_0\in A_0,x_1\in A_1}{{\mathrm{d}}}_{{X}}(x_0,x_1)>\pi _{{k}/({N}-1)}. \end{aligned}$$

If we define $\mu _{0/1}={{\mathrm{m}}}_{{X}}(A_{0/1})^{-1}{{\mathrm{m}}}_{{X}}|_{A_{0/1}}$, we see that ${{\mathrm{d}}}_W(\mu _0,\mu _1)>\pi _{{k}/({N}-1)}$. Hence, for each dynamical optimal plan $\Pi \in {{\mathrm{DyCpl}}}^{abs}(\mu _0,\mu _1)$

$$\begin{aligned} \infty \le \int \tau _{{k}^-,{N}}^{(1-t)}(|\dot{\gamma }|)d\Pi (\gamma ){{\mathrm{m}}}_{{X}}(A_0)^{\frac{1}{N}}+ \int \tau _{{k}^+,{N}}^{(t)}(|\dot{\gamma }|)d\Pi (\gamma ){{\mathrm{m}}}_{{X}}(A_1)^{\frac{1}{N}}. \end{aligned}$$

However, by the curvature-dimension condition, the right-hand side is smaller than

$$\begin{aligned} -S_{{N}}(\mu _t|{{\mathrm{m}}}_{{X}})\le {{\mathrm{m}}}_{{X}}(A_t)^{\frac{1}{N}}\le {{\mathrm{m}}}_{{X}}(B_R(o))^{\frac{1}{N}} \end{aligned}$$

for some $o\in X$ and $R>0$ sufficiently large such that $A_t\subset B_R(o)$. $A_t$ is the set of all t-midpoints between $A_0$ and $A_1$. However, the Bishop–Gromov comparison tells us that balls have always finite measure. This results in a contradiction. $\square$

Definition 5.8

Fix a point $x\in X$. Since $\partial B_r(x)$ is compact, we can consider $\min _{\partial B_r(x)}k=k_x(r)$ for $r<R_x$ where $R_x=\sup \left\{ r>0:\partial B_r(x)\ne \emptyset \right\}$. Let $\underline{k}_x$ be the lower semi-continuous envelope of $k_x$. It is clear that $\underline{k}_x\le k$ and $\underline{k}_x$ induces a lower semi-continuous function on X - also denoted by $\underline{k}_x$-via

$$\begin{aligned} y\mapsto \underline{k}_x(y):=\underline{k}_x({{\mathrm{d}}}_{{X}}(x,y)). \end{aligned}$$

Theorem 5.9

Let X be a metric measure space satisfying $CD(k,N)$. If $N>1$ then for each $x_0\in X$, for all $0<r<R$ such that $R\le \pi _{\underline{{k}}_x/(N-1)}$, we have

$$ \begin{aligned} \frac{s(r)}{s(R)}\ge \frac{\sin _{\underline{{k}}_x/({N}-1)}^{{N}-1}r}{\sin _{\underline{{k}}_x/({N}-1)}^{{N}-1}R}\ \ \& \ \ \frac{v(r)}{v(R)}\ge \frac{\int _0^r\sin _{\underline{{k}}_x/({N}-1)}^{{N}-1}tdt}{\int _0^R\sin _{\underline{{k}}_x/({N}-1)}^{{N}-1}tdt}. \end{aligned}$$

(17)

Proof

First

$$\begin{aligned} \inf _{{{\mathrm{d}}}_{{X}}(x,z)}\left\{ \underline{k}_x({{\mathrm{d}}}_{{X}}(x,z))+n|{{\mathrm{d}}}_{{X}}(x,z)-{{\mathrm{d}}}_{{X}}(x,y)|\right\} =\underline{k}_{x,n}({{\mathrm{d}}}_{{X}}(x,y)) \end{aligned}$$

and since $\underline{k}_{x,n}(r)\uparrow \underline{k}_x(r)$ we have $\underline{k}_{x,n}({{\mathrm{d}}}_{{X}}(x,y))=:\underline{k}_{x,n}'(y)\uparrow \underline{k}_{x}(y)$. By monotonicity with respect to the curvature function, X satisfies $CD(\underline{k}'_{x,n},N)$. Hence, if we consider $0<r<R<R_x$, and $A_i$ with $\mu _i$ for $i=0,1$ as in Theorem 5.3 (replace $x_0$ by x), then we obtain

$$\begin{aligned}&{{\mathrm{m}}}_{{X}}({\textstyle \bar{B}_{(1+\delta +\epsilon )r}(x)\backslash B_{(1-\epsilon )r}(x)})^{\scriptscriptstyle {\frac{1}{N}}}\\&\quad \ge \int \tau _{\underline{k}_{x,n,\gamma },N}^{{(r/R)}}(|\dot{\gamma }|)d\Pi _{n,\epsilon ,\delta }(\gamma ){{\mathrm{m}}}_{{X}}({\textstyle \bar{B}_{(1+\delta )R}(x)\backslash B_R(x)})^{\scriptscriptstyle {\frac{1}{N}}} \\&\quad \quad + \int \tau _{\underline{k}_{x,n,\gamma },N}^{{(1-r/R)}}(|\dot{\gamma }|)d\Pi _{n,\epsilon ,\delta }(\gamma ){{\mathrm{m}}}_{{X}}(\bar{B}_{\epsilon }(x))^{\scriptscriptstyle {\frac{1}{N}}} \end{aligned}$$

where $\Pi _{n,\epsilon ,\delta }$ is an optimal dynamic plan between $\mu _0$ and $\mu _1$. Since the left-hand side is finite, the right-hand side is uniformly bounded, and the distortion coefficients are finite almost everywhere. If $\epsilon \rightarrow 0$, compactness of closed balls implies that we can find a subsequence of $\Pi _{n,\epsilon ,\delta }$ that converges to $\Pi _{n,\delta }$ for $n\rightarrow \infty$ and with $(e_0)_*\Pi _{n,\delta }=\delta _x$. The previous inequality becomes

$$\begin{aligned} {{\mathrm{m}}}_{{X}}(\bar{B}_{(1+\delta )r}(x)\backslash B_{r}(x))\ge \left( \int \tau _{\underline{k}_{x,n,\gamma },N}^{{(r/R)}}(|\dot{\gamma }|)d\Pi _{n,\epsilon ,\delta }(\gamma )\right) ^N{{\mathrm{m}}}_{{X}}(\bar{B}_{(1+\delta )R}(x)\backslash B_R(x)) \end{aligned}$$

We remark that $\gamma \mapsto \tau _{\underline{k}_{x,n,\gamma },N}^{{(r/R)}}(|\dot{\gamma }|)$ is bounded and continuous for geodesics $\gamma$ in a sufficiently large ball. Similarly, if $\delta$ goes to 0, we can take another sub-sequence of $\Pi _{n,\delta }$ that converges to $\Pi _n$. If we divide both sides by $\delta r$ and take $\delta \rightarrow 0$, the previous inequality becomes

$$\begin{aligned} s_x(r)\ge \left( \int \sigma _{\underline{k}_{x,n,\gamma }/(N-1)}^{{(r/R)}}(|\dot{\gamma }|)d\Pi _{n}(\gamma )\right) ^Ns_x(R). \end{aligned}$$

$(e_0)_*\Pi _n=\delta _x$ and $(e_1)_*\Pi _n$ is a probability measure with $(e_1)_*\Pi _n(\partial B_R(x))=1$. Hence $\Pi _n$ is supported on geodesics with $\gamma (0)=x$ and $|\dot{\gamma }|=R$, and by definition of $\underline{k}_{x,n}'$ we have that $\underline{k}_{x,n}'\circ \bar{\gamma }=\underline{k}_{x,n}'(\cdot R)$ is independent of $\gamma$. Therefore

$$\begin{aligned} \frac{s_x(r)}{s_x(R)}\ge \sigma _{\underline{k}_{x,n}/(N-1)}^{{(r/R)}}(R)^{N-1}. \end{aligned}$$

Now, take $n\rightarrow \infty$. Since $\underline{k}_{x,n}\uparrow \underline{k}_x$, one can check as in Lemma 3.14 that - after choosing another subsequence - ${{\mathrm{\mathfrak {s}}}}_{\underline{k}_{x,n}}\downarrow {{\mathrm{\mathfrak {s}}}}_{\underline{k}_x}$. This is the first claim. The second one follows as in Theorem 5.3. $\square$

Theorem 5.10

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a metric measure space satisfying $CD(k,N)$ for $N>1$. Assume there is point $p\in {{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$, $\alpha >0$ and $R>0$ such that

$$\begin{aligned} k(x)\ge \left( {\textstyle \frac{1}{4}(N-1)}+\alpha ^2\right) {{\mathrm{d}}}_{{X}}(p,x)^{-2}\ \ \text{ for } \text{ all } x\in {{\mathrm{supp}}}{{\mathrm{m}}}_{{X}} \text{ with } {{\mathrm{d}}}_{{X}}(p,x)>R. \end{aligned}$$

Then ${{\mathrm{diam}}}{{{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}}\le R e^{\frac{\pi }{\alpha }}$, and ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$ is compact.

Proof

Assume the contrary. Then we can find a point $q\in X$ such that ${{\mathrm{d}}}_{{X}}(p,q)>(R+\delta )e^{\frac{\pi }{\alpha }}$ for some $0<\delta <R$. We choose $\epsilon >0$ such that $2\epsilon (2-e^{-\frac{\pi }{\alpha }})<\delta$ and

$$\begin{aligned} \min _{x\in B_{\epsilon }(p),y\in B_{\epsilon }(q)}{{\mathrm{d}}}_{{X}}(x,y)=:{{\mathrm{d}}}_{{X}}(\bar{B}_{\epsilon }(q),\bar{B}_{\epsilon }(p))>(R+\delta )e^{\frac{\pi }{\alpha }}. \end{aligned}$$

We set $\bar{B}_{\epsilon }(q)=:A_0$ and $\bar{B}_{\epsilon }(p)=:A_1$ and define probability measures

$$\begin{aligned} \mu _{i}={{{\mathrm{m}}}_{{X}}(A_i)^{-1}}\mu _{{X}}|_{A_i} \end{aligned}$$

where $i=0,1$. Let $q'\in \bar{B}_{\epsilon }(q)$ and $p'\in \bar{B}_{\epsilon }(p)$. We consider a geodesic $\gamma :[0,1]\rightarrow X$ between $q'$ and $p'$ and estimate the curvature along $\gamma$ as follows. Let $\bar{\gamma }$ be the unit speed reparametrization of $\gamma$. For $0<t<[{{\mathrm{d}}}_{{X}}({q'},{p'})+2\epsilon ](1-e^{-\frac{\pi }{\alpha }})$ we have

Therefore

$$\begin{aligned} k(\bar{\gamma }(t))&\ge \frac{c}{{{\mathrm{d}}}_{{X}}(p,\bar{\gamma }(t))^{2}}\ge \frac{c}{({{\mathrm{d}}}_{{X}}({p},{p'})+{{\mathrm{d}}}_{{X}}(p',\bar{\gamma }(t))^{2}}\\&\ge \left( \alpha ^2+{\frac{1}{4}}\right) (N-1)\frac{1}{(\epsilon +{{\mathrm{d}}}_{{X}}(q',p')-t)^2}=:k(t)(N-1) \end{aligned}$$

We obtain a lower estimate for the modified distortion coefficient along $\gamma$. The generalized $\sin$-function ${{\mathrm{\mathfrak {s}}}}_{k\circ \bar{\gamma }/(N-1)}$ is bounded from below by ${{\mathrm{\mathfrak {s}}}}_{k}$ which is given explicitly by

$$\begin{aligned} {{\mathrm{\mathfrak {s}}}}_{k}(t)=C\sqrt{\epsilon +{{\mathrm{d}}}_{{X}}({p'},{q'})-t)}\sin \left[ \alpha \log \left( {\textstyle \frac{\epsilon +{{\mathrm{d}}}_{{X}}(q',p')-t}{\left( \epsilon +{{\mathrm{d}}}_{{X}}(q',p')\right) e^{-\pi /\alpha }}}\right) \right] . \end{aligned}$$

where C is a normalization constant. We see that the second zero of ${{\mathrm{\mathfrak {s}}}}_{k}$ appears at

$$\begin{aligned} (\epsilon +{{\mathrm{d}}}_{{X}}({q'},{p'}))(1-e^{-\frac{\pi }{\alpha }})<{{\mathrm{d}}}_{{X}}({q'},{p'})-R+\epsilon (1-e^{-\frac{\pi }{\alpha }})<{{\mathrm{d}}}_{{X}}({q'},{p'}). \end{aligned}$$

Therefore, the second zero of ${{\mathrm{\mathfrak {s}}}}_{k\circ \bar{\gamma }}$ appears strictly before $t={{\mathrm{d}}}_{{X}}({q},{p})$. Consequently

$$\begin{aligned} \sigma _{k\circ \gamma ,N-1}^{(t)}(\theta )\ge \sigma _{k}^{(t)}(\theta )=\infty . \end{aligned}$$

We conclude that

$$\begin{aligned} {{\mathrm{m}}}_{{X}}(A_t)^{\frac{1}{N}}&\ge {\int \tau _{k^{{-}}_{\gamma },N'}^{{(1-t)}}(|\dot{\gamma }|)d\Pi (\gamma )}{{\mathrm{m}}}_{{X}}\left( A_0\right) ^{\frac{1}{N'}} +{\int \tau _{k^{{+}}_{\gamma },N'}^{{(t)}}(|\dot{\gamma }|)d\Pi (\gamma )}{{\mathrm{m}}}_{{X}}\left( A_1\right) ^{\frac{1}{N'}}\\&=\infty . \end{aligned}$$

$A_t$ is again the set of all t-midpoints between $A_0$ and $A_1$, and $\Pi$ is an optimal dynamic transference for $\mu _0$ and $\mu _1$. As in the previous Proposition, this yields a contradiction.

Example 5.11

The previous theorem is sharp in the sense that one cannot improve the result by replacing the lower bound $\frac{1}{4}(N-1)$ for c by a smaller lower bound. For instance, consider

$$\begin{aligned} ([0,\infty ),|\cdot |_2,\left( \sqrt{r}\right) ^{{N}-1}dr). \end{aligned}$$

Using Theorem 4.12 and Proposition 7.3 one can check that it satisfies the curvature-dimension $CD(k,N)$ for

$$\begin{aligned} k(r)= \frac{1}{4}(N-1)r^{-2} \end{aligned}$$

$k$ satisfies the assumption of the theorem for $c=\frac{1}{4}(N-1)$ and any $p\in [0,\infty )$ since $k(r)\sim \frac{1}{4}(N-1)|r-p|_2^{-2}$ for $r>0$ sufficiently large but one cannot find a point $p\in [0,\infty )$, $c>\frac{1}{4}(N-1)$ and $R>0$ such that $k(r)r^2\ge c$ for $r>0$ with $|r-p|_2\ge R$. A Riemannian manifold of geometric dimension N satisfying this property can be constructed via warped products.

6 Stability

6.1 Measured Gromov–Hausdorff Convergence

A rectifiable curve $\gamma :[0,1]\rightarrow X$ with constant-speed parametrization is called $\epsilon$-geodesic if $\text{ L }(\gamma )-\epsilon \le {{\mathrm{d}}}_{{X}}(\gamma (0),\gamma (1))$. The family of all $\epsilon$-geodesics is denoted with ${\mathcal {G}}^{\epsilon }(X)$, and it is equipped with the topology that comes from ${{\mathrm{d}}}_{\infty }(\gamma ,\tilde{\gamma })=\sup _{t}{{\mathrm{d}}}_{{X}}(\gamma (t),\tilde{\gamma }(t))$. Measurability is understood in the sense of this topology. Obviously, we have ${\mathcal {G}}^{\epsilon }(X)\subseteq {\mathcal {G}}^{\eta }(X)$ if $\epsilon \le \eta$ and ${\mathcal {G}}^{0}(X)={\mathcal {G}}(X)$. If X is compact, then ${\mathcal {G}}^{\epsilon }(X)$ is compact with respect to ${{\mathrm{d}}}_{\infty }$. Since $\text{ L }(\gamma )\le \epsilon +{{\mathrm{diam}}}_{{X}_i}$ for every $\gamma \in {\mathcal {G}}^{\epsilon }(X)$, this follows from Theorem 2.5.14 in [6].

A probability measure $\Pi$ on ${\mathcal {G}}^{\epsilon }(X)$ is called dynamic transference plan between $(e_0)_{\star }\Pi$ and $(e_1)_{\star }\Pi$. If $k:X\rightarrow {\mathbb {R}}\cup \left\{ \infty \right\}$ is an admissible function, we can consider $k_{\gamma }$ for $\gamma \in {\mathcal {G}}^{\epsilon }(X)$ and the corresponding generalized $\sin$-function and the modified distortion coefficient. One can check that $\gamma \mapsto \tau _{{k}_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)$ is a measurable function on ${\mathcal {G}}^{\epsilon }(X)$. The evaluation map $e_t:\gamma \mapsto \gamma (t)$ is continuous and hence measurable.

Definition 6.1

A sequence $(X_i,{{\mathrm{d}}}_{{X}_i})_{i\in {\mathbb {N}}}$ of compact metric spaces converges in Gromov–Hausdorff sense to a compact metric space $(X,{{\mathrm{d}}}_{{X}})$ if there is a compact metric space $(Z,{{\mathrm{d}}}_{{Z}})$ and isometric embeddings $\iota _i:X_i\rightarrow Z$, $\iota :X\rightarrow Z$ such that ${{\mathrm{d}}}_{{Z},H}(\iota _i(X_i),\iota (X))\rightarrow 0$ where ${{\mathrm{d}}}_{{Z},H}$ is the Haudorff distance w.r.t. to ${{\mathrm{d}}}_{{Z}}$.

A sequence of compact metric measure spaces $(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})$ converges in measured Gromov–Hausdorff sense to a compact metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ if there exists a compact metric space $(Z,{{\mathrm{d}}}_{{Z}})$ and isometric embeddings $\iota _i,\iota$ as before such that the corresponding metric spaces converge in Gromov–Hausdorff sense and $(\iota _i)_{\star }{{\mathrm{m}}}_{{X}_i}\rightarrow (\iota )_{\star }{{\mathrm{m}}}_{{X}}$ with respect to weak convergence in Z.

A sequence of isomorphism classes $[(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})]$ of metric measures spaces converges in measured Gromov sense to an isomorphism class of a normalized metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ if there exists a metric space $(Z,{{\mathrm{d}}}_{{Z}})$ and isometric embeddings $\iota _i:X_i\rightarrow Z$ and $\iota :X\rightarrow Z$ such that $(\iota _i)_{\star }{{\mathrm{m}}}_{{X}_i}\rightarrow (\iota )_{\star }{{\mathrm{m}}}_{{X}}$ with respect to weak convergence of finite measures in Z.

Remark 6.2

If a sequence $(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})$ of compact metric measure spaces converges in measured Gromov–Hausdorff sense to $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ then this yields measured Gromov convergence of the corresponding isomorphism classes [33, Lemma 3.18], [17, Theorem 3.30]. The converse is not true in general. However, if the family $(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})_{i\in {\mathbb {N}}}$ satisfies an uniform doubling property and the corresponding isomorphism classes converge in measured Gromov sense to $[(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})]$, then by Gromov’s compactness theorem, one can extract a subsequence that converges in measured Gromov–Hausdorff sense to a limit space $(X',{{\mathrm{d}}}_{{X}'},{{\mathrm{m}}}_{{X}'})$, and consequently $[(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})]=[(X',{{\mathrm{d}}}_{{X}'},{{\mathrm{m}}}_{{X}'})]$. Therefore, if ${{\mathrm{supp}}}{{{\mathrm{m}}}_{{X}}}=X$, then $(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})$ converges in measured Gromov–Hausdorff sense to $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ [33, Lemma 3.18], [17, Theorem 3.33].

Lemma 6.3

Let $k:X\rightarrow {\mathbb {R}}\cup \left\{ \infty \right\}$ be admissible and $N>1$. For dynamic couplings $(\Pi _n)_{n\in {\mathbb {N}}}$ supported on ${\mathcal {G}}^{\eta }(X)$ for some $\eta >0$ with the same marginals $\mu _0, \mu _1\in {\mathcal {P}}(X,{{\mathrm{m}}}_{{X}})$ which converge to a dynamic coupling $\Pi _{\infty }$, it follows

$$\begin{aligned} \limsup _{n\rightarrow \infty }T^{{(t)}}_{k,N}(\Pi _n|{{\mathrm{m}}}_{{X}})\le T^{{(t)}}_{k,N}(\Pi _{\infty }|{{\mathrm{m}}}_{{X}}). \end{aligned}$$

Proof

First, we assume that $k$ is continuous. We will show that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\int \tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _0(\gamma _0)^{-\frac{1}{N}}\Pi _n(d\gamma )\ge \int \tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _0(\gamma _0)^{-\frac{1}{N}}\Pi _{\infty }(d\gamma ). \end{aligned}$$

Let $\Pi _{n,x_0}(d\gamma )$ be a disintegration of $\Pi _n$ with respect to $\mu _0$ for $n\in {\mathbb {N}}\cup \left\{ \infty \right\}$, and let $C\in (0,\infty )$. We put

$$\begin{aligned} v_{0,n}^{C}(x_0):=\int \left[ \tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\wedge C\right] \Pi _{n,x_0}(d\gamma ). \end{aligned}$$

where $n\in {\mathbb {N}}\cup \left\{ \infty \right\}$. Since $C_b(X)$ is dense in $L^1({{\mathrm{m}}}_{{X}})$, and since $v_{0,n}^C$ is bounded by definition, for each $\epsilon >0$, there is $\psi \in C_b(X)$ such that

$$\begin{aligned} \int v_{0,n}^C|\varrho _0^{-\frac{1}{N}}\wedge C-\psi |d\mu _0<\epsilon \ \ \text{ for } \text{ all } \ \ n\in {\mathbb {N}}\cup \left\{ \infty \right\} \end{aligned}$$

(18)

if $C<\infty$. Weak convergence of $\Pi _n\rightarrow \Pi _{\infty }$ on ${\mathcal {G}}^{\eta }(X)$ implies that one can find $n_{\epsilon }$ such that for each $n\ge n_{\epsilon }$, one has

$$\begin{aligned} \int v_{0,\infty }^C\psi d\mu _0\le \int v_{0,n}^C\psi d\mu _0+\epsilon \end{aligned}$$

(19)

Putting together (18) and (19) one gets

$$\begin{aligned} \int v_{0,\infty }^C[\varrho _0^{-\frac{1}{N}}\wedge C]d\mu _0\le \int v_{0,n}^C[\varrho _0^{-\frac{1}{N}}\wedge C]d\mu _0+3\epsilon \le \int v_{0,n}^{\infty }\varrho _0^{-\frac{1}{N}}d\mu _0+3\epsilon . \end{aligned}$$

It follows that for each $C>0$,

$$\begin{aligned} \int v_{0,\infty }^C\varrho _0^{-\frac{1}{N}}\wedge C d\mu _0\le \liminf _{n\rightarrow \infty }\int v_{0,n}^{\infty }\varrho _0^{-\frac{1}{N}}d\mu _0. \end{aligned}$$

(20)

Finally, let $C\rightarrow \infty$

$$\begin{aligned} \int \tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _0(\gamma _0)^{-\frac{1}{N}}\Pi (d\gamma )\le \liminf _{n\rightarrow \infty }\int v_{0,n}^{\infty }\varrho _0^{-\frac{1}{N}}d\mu _0. \end{aligned}$$

The same statement holds with $\varrho _0$ replaced by $\varrho _1$ and $\tau ^{{(t)}}_{{k}^+_{\gamma },{N}}$ replaced by $\tau _{{k}^-_{\gamma },{N}}^{{(1-t)}}$.

Now, let $k$ be lower semi-continuous, and let $k_i$ be a sequence of continuous functions that converge pointwise monotone from below to $k$. By monotonicity of the distortion coefficients, we observe that

$$\begin{aligned} \tau _{k_{i,\gamma }^+,{N}}^{(t)}(|\dot{\gamma }|)\uparrow \tau _{k_{\gamma }^+,{N}}^{(t)}(|\dot{\gamma }|) \end{aligned}$$

for any $\gamma \in {\mathcal {G}}^{\epsilon }$. Therefore, $v_{0,\infty ,i}^C\uparrow v_{0,\infty }^C \text{ and } v_{0,n,i}^{\infty }\uparrow v_{0,n}^{\infty } \text{ if } i\rightarrow \infty .$ In particular, by the monotone convergence theorem for $\epsilon >0$, we can choose $i_{\epsilon }\in {\mathbb {N}}$ such that

$$\begin{aligned} \int \left[ v_{0,\infty }^C-v_{0,\infty ,i}^C\right] \varrho _0^{-\frac{1}{N}}\wedge C d\mu _0<\epsilon \ \ \text{ for } i\ge i_{\epsilon }\text{. } \end{aligned}$$

Hence, together with (20) it follows that

$$\begin{aligned} \int v_{0,\infty }^C\varrho _0^{-\frac{1}{N}}\wedge C d\mu _0-\epsilon \le \liminf _{n\rightarrow \infty }\int v_{0,n,i}^{\infty }\varrho _0^{-\frac{1}{N}}d\mu _0\le \liminf _{n\rightarrow \infty }\int v_{0,n}^{\infty }\varrho _0^{-\frac{1}{N}}d\mu _0 \end{aligned}$$

and finally we let $C\rightarrow \infty$ and $\epsilon \rightarrow 0$, and the result follows as before. $\square$

Proposition 6.4

Let $(X_i,{{\mathrm{d}}}_{{X}_i})$ be a sequence of compact length spaces converging in Gromov–Hausdorff sense to a length space $(X,{{\mathrm{d}}}_{{X}})$. Then for all $\epsilon >0$, there exists $i_{\epsilon }\in {\mathbb {N}}$ with the following property. If $i\ge i_{\epsilon }$, and $\gamma :[0,1]\rightarrow X_i$ is a geodesic, then there exists a geodesic $\gamma ':[0,1]\rightarrow X$ such that ${{\mathrm{d}}}_{{Z},\infty }(\gamma ',\gamma )<\epsilon$. Moreover, we can choose $\Phi _i:\gamma \mapsto \gamma '$ to be a measurable map from ${\mathcal {G}}(Y)$ to ${\mathcal {G}}(X)$.

Proof

We can use exactly the same argument as in the proof of a similar statement in the appendix of [23]. $\square$

Definition 6.5

Let $(X_i,{{\mathrm{d}}}_{{X}_i})$ be a sequence of compact metric spaces that converge to a compact metric space $(X,{{\mathrm{d}}}_{{X}})$ in Gromov–Hausdorff sense. Let Z be a compact metric space where Gromov–Hausdorff convergence is realized. Let $k_i,k:X_i,X\rightarrow {\mathbb {R}}$ be admissible functions. We say $\liminf _{i\rightarrow \infty }k_i\ge k$ if for each $\eta >0$ there exists $i_{\eta }\in {\mathbb {N}}$ and $\delta >0$ such that $k_i(x)\ge k(y)-\eta$ if $i\ge i_{\eta }$, $x\in {X_i}$, $y\in {X}$ and ${{\mathrm{d}}}_{{Z}}(x,y)<\delta$. The definition does not depend on the choice of Z.

Theorem 6.6

Let $(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})_{i\in {\mathbb {N}}}$ be compact metric measure spaces satisfying $CD(k_i,N_i)$ respectively for admissible functions $k_i$ and $N_i\in [1,\infty )$. Assume $(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})$ converges in measured Gromov–Hausdorff sense to a compact metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$. Let $k:X\rightarrow \bar{{\mathbb {R}}}$ be an admissible function and $N\in [1,\infty )$ such that

$$ \begin{aligned} \liminf _{i\rightarrow \infty }k_i\ge k\ \ \& \ \ \limsup _{i\rightarrow \infty }N_i\le N\ \ \& \ \ {{\mathrm{diam}}}_{{X}_i}\le L. \end{aligned}$$

Then $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies $CD(k,N)$.

Remark 6.7

The previous stability theorem uses the notion of measured Gromov–Hausdorff convergence though it is not a property of isomorphism classes of metric measure spaces. The reason for that is that we will apply Proposition 6.4 that guarantees convergence of sequences of geodesics in $(X_i,{{\mathrm{d}}}_{{X}_i})$ where the limit might not be in the support of ${{\mathrm{m}}}_{{X}}$. If we assume that ${{\mathrm{m}}}_{{X}}$ has full support, then a suitable reformulation of the theorem holds for measured Gromov convergence as well by Remark 6.2. We want to emphasize that Gromov’s precompactness theorem—that is a major source of geometric applications in finite dimensional context—yields measured Gromov–Hausdorff convergence in the first place anyway. See also the discussion at the beginning of Sect. 4.2 in [17].

Proof of Theorem

1. Without loss of generality, we assume that ${{\mathrm{m}}}_{{X}_i}$ has full support. Gromov–Hausdorff convergence yields that $(X,{{\mathrm{d}}}_{{X}})$ is a geodesic space, and ${{\mathrm{diam}}}_{{X}}\le L$. The curvature-dimension condition yields a uniform doubling property for ${{\mathrm{supp}}}{{{\mathrm{m}}}_{{X}_i}}$. Therefore, by compactness of $X_i$, the measure ${{\mathrm{m}}}_{{X}_i}$ is finite. Moreover, ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$ satisfies a doubling property as well, and therefore ${{\mathrm{m}}}_{{X}}$ is finite as well. Hence, without loss of generality we normalize any reference measure. By Remark 6.2, measured Gromov–Hausdorff convergence yields measured Gromov convergence of the corresponding isomorphism classes. Let us fix a metric space $(Z,{{\mathrm{d}}}_{{Z}})$ and isometric embeddings $\iota _i:X_i\rightarrow Z$ and $\iota :X\rightarrow Z$ where the measured Gromov and the measured Gromov–Hausdorff convergence are realized according to Definition 6.1. First, assume that $k$ is continuous.

2. ${{\mathrm{m}}}_{{X}_i}$ converges weakly to ${{\mathrm{m}}}_{{X}}$ in Z as probability measures. Hence, let ${q}_i$ be optimal couplings between ${{\mathrm{m}}}_{{X}_i}$ and ${{\mathrm{m}}}_{{X}}$ such that $\int {{\mathrm{d}}}^2_{{Z}}dq_i={{\mathrm{d}}}_{{Z},{W}}({{\mathrm{m}}}_{{X}_i},{{\mathrm{m}}}_{{X}})^2=:d_i^2\rightarrow 0$. Therefore, we can choose $i_{\epsilon }$ such that for $i\ge i_{\epsilon }$ we have that ${{\mathrm{d}}}_i\le \epsilon$. Following [33], for fixed $i\in {\mathbb {N}}$, one can define a map ${Q}_i:{\mathcal {P}}_2({{\mathrm{m}}}_{{X}})\rightarrow {\mathcal {P}}_2({{\mathrm{m}}}_{{X}_i})$ with

$$ \begin{aligned} S_N(Q_i(\mu )|{{\mathrm{m}}}_{{X}_i})\le S_N(\mu |{{\mathrm{m}}}_{{X}})\ \ \& \ \ {{\mathrm{d}}}_{{Z},{W}}^2(\mu ,Q_i(\mu ))<\delta _i \end{aligned}$$

(21)

where ${{\mathrm{d}}}_{{Z},{W}}$ denotes the Wassertstein distance in $(Z,{{\mathrm{d}}}_{{Z}})$ and $\delta _i\rightarrow 0$ for $i\rightarrow \infty$. $Q_i$ is constructed by disintegration ${q}_i$ with respect to ${{\mathrm{m}}}_{{X}_i}$. More precisely, for $\mu =\varrho {{\mathrm{m}}}_{{X}}$, we set $\mu _{i}=Q_i(\mu )=\varrho _{i}d{{\mathrm{m}}}_{{X}_i}$ where

$$\begin{aligned} \varrho _{i}(y)=\int _X\varrho (x)Q_i(y,dx) \end{aligned}$$

and $Q_i$ is a disintegration of $q_i$ w.r.t. ${{\mathrm{m}}}_{{X}}$. Similarly, we define $Q^i:{\mathcal {P}}_2({{\mathrm{m}}}_{{X}_{i}})\rightarrow {\mathcal {P}}_2({{\mathrm{m}}}_{{X}})$ by $\mu ^{i}=Q^i(\mu )=\varrho ^{i}d{{\mathrm{m}}}_{{X}_i}$ where

$$\begin{aligned} \varrho ^{i}(x)=\int _X\varrho (y)Q^i(x,dy) \end{aligned}$$

and $Q^i$ is a disintegration of $q_i$ w.r.t. ${{\mathrm{m}}}_{{X}_i}$. Again we have

$$ \begin{aligned} S_N(Q^i(\mu )|{{\mathrm{m}}}_{{X}})\le S_N(\mu |{{\mathrm{m}}}_{{X}_i})\ \ \& \ \ {{\mathrm{d}}}_{{Z},{W}}^2(\mu ,Q^i(\mu ))<\delta ^i \end{aligned}$$

(22)

with $\delta ^i\rightarrow 0$ if $i\rightarrow \infty$. Hence, we can choose $i_{\epsilon }$ such that $\delta ^i,\delta _i\le \epsilon ^2$ for $i\ge i_{\epsilon }$.

3. Pick measures $\mu _0=\varrho _0{{\mathrm{m}}}_{{X}}$ and $\mu _1=\varrho _1{{\mathrm{m}}}_{{X}}$ in ${\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ with bounded densities, and define $\mu _{j,i}=Q_i(\mu _j)$ for $j=0,1$. Due to the curvature-dimension condition for $X_i$, there exists a geodesic $(\mu _{t,i})_{t\in [0,1]}$ between $\mu _{0,i}$ and $\mu _{1,i}$ and a dynamic optimal plan $\Pi _i$ such that $(e_t)_{\star }\Pi _i=\mu _{t,i}$ and

$$\begin{aligned} S_{N'}(\mu _{t,i}|{{\mathrm{m}}}_{{X}_i})\le T_{{k}_i,{N}'}^{(t)}(\Pi _i|{{\mathrm{m}}}_{{X}_i}). \end{aligned}$$

By (22), we know that $Q^i$ decreases $S_{{N}'}$. Note that $Q^i(\mu _{t,i})=\hat{\mu }_{t,i}$ satisfies

$$\begin{aligned} |{{\mathrm{d}}}_{{X},{W}}(\mu _0,\hat{\mu }_{t,i})-t{{\mathrm{d}}}_{{X},{W}}(\mu _0,\mu _1)|\le 2\epsilon . \end{aligned}$$

By compactness, $\hat{\mu }_{t,i}$ converges weakly in X for each rational $t\in [0,1]$ for $i\rightarrow \infty$ after extracting a subsequence, and by Lipschitz continuity, the limit extends to a geodesic $(\mu _t)_{t\in [0,1]}$ in ${\mathcal {P}}^2(X)$. For instance, see (vi) in the proof of Theorem 3.1 in [34]. For simplicity, we write N for $N'$.

4. In this step, we will generalize the map $\Phi _i:{\mathcal {G}}(X_i)\rightarrow {\mathcal {G}}(X)$. Increase $i_{\epsilon }$ such that ${{\mathrm{d}}}_{{Z},\infty }(\gamma ,\Phi _i(\gamma ))\le \epsilon$ for $i\ge i_{\epsilon }$ and each geodesic $\gamma \in {\mathcal {G}}(X)$. Since X is a compact geodesic space, one can choose a measurable map $\Psi :X^2\rightarrow {\mathcal {G}}(X)$ such that $\Psi (x,y)$ is a geodesic between x and y. For instance, this follows from a measurable selection theorem. Pick a geodesic $\gamma \in X_i$ and consider the geodesic $\Phi _i(\gamma )$ in X. Consider the map $\Xi _i:{\mathcal {G}}(X_i)\times X^2\rightarrow {\mathcal {G}}^{{L}}(X)$ that is defined as follows

$$\begin{aligned} (\gamma ,x_0,x_1)\mapsto \Psi (x_0,\Phi _i(\gamma )(0))*\Phi (\gamma )* \Psi (\Phi _i(\gamma )(1),x_1)\in {\mathcal {G}}^{{L}}(X). \end{aligned}$$

Here, the operator $*$ denotes the cancatenation of curves with constant-speed reparametrization on [0, 1]. It is clear from the construction that $\Xi _i$ maps to ${\mathcal {G}}^{{L}}(X)$ and $\Xi _i$ is measurable. We also set $\Xi _{i,\gamma }(\cdot ):=\Xi _i(\gamma ,\cdot )$ with $\Xi _{i,\gamma }:X^2\rightarrow {\mathcal {G}}^{{L}}(X)$. Then we define ${\mathcal {Q}}(\gamma ,d\sigma )=[(\Xi _{i,\gamma })_{\star }P_{\gamma }](d\sigma )$ where

$$\begin{aligned} P_{\gamma }(dx_0,dx_1):=\frac{\varrho _j(x_0)}{\varrho _{j,i}(\gamma (0))}{Q}_i(\gamma (0),dx_0)\otimes \frac{\varrho _{j}(x_1)}{\varrho _{j,i}(\gamma (1))}{Q}_i(\gamma (1),dx_1). \end{aligned}$$

${\mathcal {Q}}:{\mathcal {G}}(X)\times {\mathcal {P}}({\mathcal {G}}^{{L}}(X))\rightarrow {\mathbb {R}}$ is a Markov kernel. We define a dynamic plan $\hat{\Pi }_i$ via

$$\begin{aligned} \int _{{\mathcal {G}}(X_i)}{\mathcal {Q}}(\gamma ,d\sigma )\Pi _i(d\gamma )=\hat{\Pi }_i(d\sigma ) \in {\mathcal {P}}({\mathcal {G}}^{{L}}(X)). \end{aligned}$$

Set $(e_0,e_1)_{\star }\hat{\Pi }_i=\hat{\pi }_i$. If $f:X^2\rightarrow {\mathbb {R}}$ is continuous and bounded, then we compute

$$\begin{aligned}&\int _{X^2}f(x_0,x_1)\hat{\pi }_i(dx_0,dx_1)=\int _{{\mathcal {G}}(X_i)}\int _{{\mathcal {G}}^{{L}}(X)} f(e_0(\sigma ),e_1(\sigma )){\mathcal {Q}}(\gamma ,d\sigma )\Pi _i(d\gamma )\nonumber \\&\quad =\int _{{\mathcal {G}}(X_i)}\int _{X^2}f( x_0,x_1) \frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{0,i}(\gamma _0)\varrho _{1,i}(\gamma _1)}Q_i(\gamma _0,dx_0)Q_i(\gamma _1,dx_1)\Pi _i(d\gamma )\nonumber \\&\quad =\int _{X_i^2}\int _{X^2} f(x_0,x_1)\frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{0,i}(y_0)\varrho _{1,i}(y_1)}Q_i(y_0,dx_0)Q_i(y_1,dx_1)\pi _i(dy_0,dy_1)\nonumber \\&\quad =\int _{X_i^2}\int _{X^2} f(x_0,x_1)\frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{1,i}(y_1)}Q_i(y_1,dx_1)\pi _i(y_0,dy_1)Q^i(x_0,dy_0){{\mathrm{m}}}_{{X}}(dx_0) \end{aligned}$$

(23)

Here $\pi _i(y_0,dy_1)$ is a disintegration of $\pi _i$ w.r.t. $\mu _{0,i}$. The last equality follows from

$$\begin{aligned} Q_i(y_0,dx_0)\pi _i(dy_0,dy_1)&= Q_i(y_0,dx_0)\pi _i(y_0,dy_1)\rho _{0,i}(y_0){{\mathrm{m}}}_{{X}_i}(dy_0)\nonumber \\&=\rho _{0,i}(y_0)\pi _i(y_0,dy_1)Q_i(y_0,dx_0){{\mathrm{m}}}_{{X}_i}(dy_0)\nonumber \\&=\rho _{0,i}(y_0)\pi _i(y_0,dy_1)Q^i(x_0,dy_0){{\mathrm{m}}}_{{X}}(dx_0). \end{aligned}$$

(24)

Since (23) holds for any f, we obtain an explicit formula for $\hat{\pi }_i$. If one chooses $f(x_0,x_1)=f_0(x_0)$ or $f(x_0,x_1)=f_1(x_1)$, one can see that that the first and the final marginal of $\hat{\Pi }_i$ are $\mu _0$ and $\mu _1$ respectively. Let $\hat{\Pi }_{i,x_0,x_1}(d\sigma )$ be a disintegration of $\hat{\Pi }_i$ with respect to $\hat{\pi }_i$.

Let $C>0$ be a constant. For $\gamma \in {\mathcal {G}}(X_i)$ and for $\sigma \in {\mathcal {G}}^{{L}}(X)$, we define

$$\begin{aligned} \tau _{{k}_{i,\gamma }^{-/+},{N}}^{{(1-t)/(t)}}(|\dot{\gamma }|)=:b^{-/+}(\gamma )\in [0,\infty ]\ \ \text{ and } \ \ \tau _{{k}_{\sigma }^{-/+}-\eta ,{N}}^{{(1-t)/(t)}}(|\dot{\sigma }|)\wedge C=:a^{-/+}(\sigma ). \end{aligned}$$

$\sigma \in {\mathcal {G}}^{{L}}\mapsto a^{-/+}(\sigma )$ is continuous function with respect to ${{\mathrm{d}}}_{\infty }$. The dependence of $a^{-/+}$ and $b^{-/+}$ on $k$, $\eta$, N and C is suppressed in our notation but we wil also write $a^{-/+}_{k}$ if necessary.

5. We consider $(e_0,e_1)_{\star }\Pi _i=\pi _i$, and $(e_0,e_1):\Gamma _i\rightarrow {{\mathrm{supp}}}\pi _i\subset X\times X$. Let $\Pi _{i,y_0,y_1}(d\gamma )$ be the disintegration of $\Pi _i$ with respect to $\pi _i$, and let $\pi _{j,i}(y',dy)$ be a disintegration of $\pi _i$ with respect to $\mu _{j,i}$ for $j=0,1$. We put

$$\begin{aligned} v_0(y_0):=\int _{X_i} {\int _{{\mathcal {G}}(X_i^2)}\tau _{k_{i,\gamma }^-,N'}^{{(1-t)}}(|\dot{\gamma }|)\Pi _{i,y_0,y_1}(d\gamma )} \pi _{0,i}(y_0,dy_1)= {\int _{}\tau _{k_{i,\gamma }^-,N'}^{{(1-t)}}(|\dot{\gamma }|)\Pi _{i}(d\gamma )} \end{aligned}$$

and similarly, we define $v_1(y_1)$ replacing $\tau _{{k}_{i,\gamma }^-,{N}'}^{{(1-t)}}(|\dot{\gamma }|)$ by $\tau _{{k}_{i,\gamma }^+,{N}'}^{{(t)}}(|\dot{\gamma }|)$, and $\pi _{0,j}$ by $\pi _{1,j}$. We compute

$$\begin{aligned} T^{{(t)}}_{k,N'}(\Pi _i|{{\mathrm{m}}}_{{X}_{i}})&=\sum _{j=0,1}\int _{X_i}\left[ \int _X{\varrho }_{j}(x_j){Q}_i(y_j,dx_j)\right] ^{1-\frac{1}{N}}v_j(y_j){{\mathrm{m}}}_{{X}_i}(dy_j)\\&\ge \sum _{j=0,1}\int _{X_i}\int _X{\varrho }_{j}(x_j)^{1-\frac{1}{N}}{Q}_i(y_j,dx_j)v_j(y_j){{\mathrm{m}}}_{{X}_i}(dy_j)\\&=\sum _{j=0,1} \int _{X_i}\int _X{\varrho }_{j}(x_j)^{-\frac{1}{N}}\frac{\varrho _{j}(x_j)}{\varrho _{j,i}(y_j)}{Q}_i(y_j,dx_j)v_j(y_j)\mu _{i}(dy_j)\\&=:\sum _{i=0,1}(\dagger )_j \end{aligned}$$

One has the following identity:

$$\begin{aligned} (\dagger )_{\scriptscriptstyle {0}}&= \int _{X_i}\int _X\varrho _0(x_0)^{-\frac{1}{N}}\frac{\varrho _j(x_0)}{\varrho _{0,i}(y_0)}Q'_i(y_0,dx_0)v_0(y_0)\mu _{i}(dy_0)\\&=\int _{X_i^2}\int \left[ \int _{X^2}\varrho _0(x_0)^{-\frac{1}{N}}{\frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{0,i}(y_0)\varrho _{1,i}(y_1)}{Q}_i(y_1,dx_1)Q_i(y_0,dx_0)}b^-(\gamma )\right] \Pi _{i}(d\gamma )\\&=\int _{X_i^2}\int \left[ \int _{X^2}\varrho _0(x_0)^{-\frac{1}{N}}{\frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{0,i}(\gamma _0)\varrho _{1,i}(\gamma _1)}{Q}_i(\gamma _1,dx_1)Q_i(\gamma _0,dx_0)}b^-(\gamma )\right] \Pi _{i}(d\gamma )\\&=\int _{X_i^2}\int {\int _{X^2}\varrho _0(x_0)^{-\frac{1}{N}}P_{\gamma }(d(x_0,x_1))}b^-(\gamma )\Pi _{i,y_0,y_1}(d\gamma )\pi _i(d(y_0,y_1))=(\#)_{\scriptscriptstyle {0}} \end{aligned}$$

In the third equality, we used that $(e_0,e_1)(\gamma )=(y_0,y_1)$ is constant on the support of $\Pi _{i,y_0,y_1}(d\gamma )$.

$$\begin{aligned} (\#)_{\scriptscriptstyle {0}}&=\int _{X_i}\int \int _{X^2}\varrho _0(x_0)^{-\frac{1}{N}}\left[ a^-\left( (\Xi _{i,\gamma }(x_0,x_1)\right) +\left( b^-(\gamma )-a^-(\Xi _{i,\gamma }(x_0,x_1))\right) \right] \\&\quad \times P_{\gamma }(d(x_0,x_1))\Pi _{i,y_0,y_1}(d\gamma )\pi _i(d(y_0,y_1))\\&={\left\{ \begin{array}{ll} {\displaystyle \int \int _{X^2}\varrho _0(e_0(\Xi _{i,\gamma }(x_0,x_1))^{-\frac{1}{N}}a^-(\Xi _{i,\gamma }(x_0,x_1))P_{\gamma }(d(x_0,x_1))\Pi _{i}(d\gamma )}\end{array}\right. }\\&=(**)_{\scriptscriptstyle {0}}\\&\quad {\left\{ \begin{array}{ll}+{\displaystyle \int _{X_i^2}\int \int _{X^2}\varrho _0(x_0)^{-\frac{1}{N}}\left( b^-(\gamma )-a^-(\Xi _{i,\gamma }(x_0,x_1))\right) } P_{\gamma }(d(x_0,x_1))\Pi _{i}(d\gamma )\end{array}\right. }\\&=(*)_{\scriptscriptstyle {0}} \end{aligned}$$

and similarly for $(\dagger )_{{1}}$.

6. Consider the sequence of optimal couplings $(q_i)_{i\in {\mathbb {N}}}$ between ${{\mathrm{m}}}_{{X}}$ and ${{\mathrm{m}}}_{{X}_i}$. We fix $\lambda >0$. By Markov’s inequality,

$$\begin{aligned} q_i({{\mathrm{d}}}_{{Z}}(x,y)>\lambda )\le \frac{1}{\lambda ^2}\int {{\mathrm{d}}}_{{Z}}^2(x,y)dq_i=\frac{d_i^2}{\lambda ^2}\le \frac{\epsilon ^2}{\lambda ^2}\ \ \text{ for } i\ge i_{\epsilon }. \end{aligned}$$

Hence, if we define ${\mathcal {X}}^{\lambda }=\left\{ (x,y)\in X\times X_i:{{\mathrm{d}}}_{{Z}}(x,y)\le \lambda \right\}$, we have $q_i(({\mathcal {X}}^{\lambda })^c)\rightarrow 0$ for $i\rightarrow \infty$. Recall (24), consider $(*)_{\scriptscriptstyle {0}}$, and rewrite it as

$$\begin{aligned}&\int _{X_i^2}\int \int _{X^2}\varrho _0(x_0)^{-\frac{1}{N}}\left( b^-(\gamma )-a^-(\Xi _{i,\gamma }(x_0,x_1))\right) P_{\gamma }(d(x_0,x_1))\Pi _{i}(d\gamma )\\&\quad =\int _{X\times X_i}\int _{X\times X_i}\int \left( b^-(\gamma )-a^-(\Xi _{i,\gamma }(x_0,x_1))\right) \Pi _{i,y_0,y_1}(d\gamma )\\&\quad \quad \times \frac{\varrho _0(x_0)^{1-\frac{1}{N}}\varrho _1(x_1)}{\varrho _{0,i}(y_0)\varrho _{1,i}(y_1)}{Q}_i(y_1,dx_1)Q_i(y_0,dx_0)\pi _i(dy_0,dy_1)\\&\quad =\int _{X\times X_i}\int _{X\times X_i}\int \left( b^-(\gamma )-a^-(\Xi _{i,\gamma }(x_0,x_1))\right) \Pi _{i,y_0,y_1}(d\gamma )\\&\quad \quad \times \frac{\varrho _0(x_0)^{1-\frac{1}{N}}\varrho _1(x_1)}{\varrho _{1,i}(y_1)}{Q}_i(y_1,dx_1)\pi _i(y_0,dy_1)q_i(dx_0,dy_0)\\&\quad =\left[ \int _{{\mathcal {X}}^{\lambda }}\int _{{\mathcal {X}}^{\lambda }}\int \dots \ \right] =:(II)\ \ + \ \ \left[ \int _{{\mathcal {X}}^{\lambda ,c}}\int _{{\mathcal {X}}^{\lambda ,c}}\int \dots \ \right] =:(I)\ \end{aligned}$$

First, $a^-$ and $\varrho _0$ are bounded independent of i or $\lambda$, and $b^-$ is non-negative. Hence, there is $M>0$ such that

$$\begin{aligned} (I)&\ge -\int _{{\mathcal {X}}^{\lambda , c}}\int _{{\mathcal {X}}^{\lambda , c}}\int M \frac{\varrho _1(x_1)}{\varrho _{1,i}(y_1)}\Pi _{i,y_0,y_1}(d\gamma ){Q}_i(y_1,dx_1)\pi _i(y_0,dy_1)q_i(dx_0,dy_0)\nonumber \\&= -\int _{{\mathcal {X}}^{\lambda , c}}\int _{{\mathcal {X}}^{\lambda , c}} M \frac{\varrho _1(x_1)}{\varrho _{1,i}(y_1)}{Q}_i(y_1,dx_1)\pi _i(y_0,dy_1)q_i(dx_0,dy_0)\nonumber \\&\ge -\int _{{\mathcal {X}}^{\lambda , c}}\int _{X^i} M \pi _i(y_0,dy_1)q_i(dx_0,dy_0)\nonumber \\&= -M\int _{{\mathcal {X}}^{\lambda , c}} q_i(dx_0,dy_0)\ge - \frac{M}{\lambda } {{\mathrm{d}}}_{{Z},{W}}({{\mathrm{m}}}_{{X}_i},{{\mathrm{m}}}_{{X}})^2\ge -M\frac{\epsilon ^2}{\lambda ^2}\ \ \text{ for } i\ge i_{\epsilon }. \end{aligned}$$

(25)

Second, we wan to estimate (II). Observe that ${{\mathrm{d}}}_{{Z},\infty }(\gamma ,\Xi _{i,\gamma }(x_0,x_1))\le 2\lambda +2\epsilon$ and $\Xi _{i,\gamma }(x_0,x_1)\in {\mathcal {G}}^{2\lambda +2\epsilon }(X)$ provided we have $(x_0,\gamma _0),(x_1,\gamma _1)\in {\mathcal {X}}^{\lambda }$ and $i\ge i_{\epsilon }$. Hence, we fix $\eta >0$ and since $\liminf k_i\ge k$, we choose $\lambda >0$, $\epsilon >0$ sufficiently small and $i_{\eta }\ge i_{\epsilon }$ sufficiently large such that $k_{i,\gamma }\ge k_{\sigma }-\eta$ whenever ${{\mathrm{d}}}_{{Z}}(\gamma ,\sigma )\le 2\lambda +2\epsilon$ for every $\gamma \in X_i$ and every $\sigma \in X$ and for $i\ge i_{\eta }$. Hence, by monotonicity of distortion coefficients, we have $\tau _{k_{\gamma },{N}}^{{\scriptscriptstyle {(t)}}}(|\dot{\gamma }|)\ge \tau _{k_{\sigma }-\eta ,{N}}^{{\scriptscriptstyle {(t)}}}(|\dot{\sigma }|)$, and therefore $b^-(\gamma )-a^-(\Xi _{i,\gamma }(x_0,x_1))\ge 0$ provided $(x_0,\gamma _0),(x_1,\gamma _1)\in {\mathcal {X}}^{\lambda }$. Hence, $(II)\ge 0$. Together with (25), we obtain for $i\ge i_{\eta }$

$$\begin{aligned}&-T^{{(t)}}_{k,N'}(\Pi _i|{{\mathrm{m}}}_{{X}_{i}})\ge \int \left[ a^-(\sigma )\varrho _0(\sigma _0)^{-\frac{1}{N}}+ a^+(\sigma )\varrho _1(\sigma _1)^{-\frac{1}{N}}\right] \hat{\Pi }_i(d\sigma )-M\frac{\epsilon ^2}{\lambda ^2}. \end{aligned}$$

(26)

7. Now, choose $\lambda =\sqrt{\epsilon }$, a sequence $\epsilon _i\downarrow 0$ for $i\rightarrow 0$, and pick for every $i\in {\mathbb {N}}$ a measure $\hat{\Pi }_i$ as in (26). Since ${\mathcal {G}}^{{L}}(X)$ is compact with respect to ${{\mathrm{d}}}_{\infty }$, Prohorov’s theorem yields that there is a subsequence of $\hat{\Pi }_i$ that converges to a dynamic transference plan $\Pi$ that is supported on ${\mathcal {G}}^{{L}}(X)$. Recall that $(e_j)_{\star }\hat{\Pi }_i=\mu _j$ for $j=0,1$ and all i. By a modification of Lemma 6.3 (replacing $\tau _{k^{-/+},{N}}$ by $a^{-/+}$), it follows that

$$\begin{aligned} \text{ RHS } \text{ in } \text{(26) }\rightarrow \int \left[ a^{-}({\sigma })\varrho _0(\sigma _0)^{-\frac{1}{N}}+ a^{+}({\sigma })\varrho _1(\sigma _1)^{-\frac{1}{N}}\right] {\Pi }(d\sigma )-{M}\epsilon \ \ \text{ if } i\rightarrow \infty . \end{aligned}$$

We show that $(e_0,e_1)_{\star }\Pi =:\pi$ is an optimal coupling, and $\Pi$ is supported on ${\mathcal {G}}^{}(X)$.

The first claim follows by construction of $\hat{\Pi }_i$. We have an explicit representation for the coupling $\hat{\pi }_i$ by (23) that is the same coupling as constructed by Sturm in [34] (more precisely, this is $\bar{q}^r$ on page 154). It is an almost optimal coupling between $\mu _0$ and $\mu _1$, and the error becomes small if i is large. Therefore, since $\hat{\pi }_i\rightarrow \pi$ weakly and since the total cost of couplings is lower semi-continuous with respect to weak convergence of couplings, $\pi$ is optimal for $\mu _0$ and $\mu _1$.

For the second claim, we decompose $\hat{\Pi }_i$ with respect to $X^{{\scriptscriptstyle \sqrt{\epsilon _i}}}$. Recall

$$\begin{aligned} \int f(\sigma )\hat{\Pi }_i(d\sigma )&=\int _{X_i^2}\int _{X^2}\int _{{\mathcal {G}}(X_i)} f((\Xi _{i,\gamma })_{\star }(x_0,x_1))\frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{0,i}(y_0)\varrho _{1,i}(y_1)}\nonumber \\&\quad \times Q_i(y_1,dx_1)Q_i(y_0,dx_0)\Pi _{i,y_0,y_1}(d\gamma )\pi _i(dy_0,dy_1). \end{aligned}$$

(27)

We set for $f\in C_b({\mathcal {G}}^{{L}}(X))$

$$\begin{aligned} L_{i,{\scriptscriptstyle \sqrt{\epsilon _i}}}f&:=\int _{X_i^2}\int _{X^2}\int _{{\mathcal {G}}(X_i)} f((\Xi _{i,\gamma })_{\star }(x_0,x_1)) 1_{({\mathcal {X}}^{{\scriptscriptstyle \sqrt{\epsilon _i}}})^2}(x_0,y_0,x_1,y_1)\frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{0,i}(y_0)\varrho _{1,i}(y_1)}\\&\quad \times {Q}_i(y_1,dx_1)Q_i(y_0,dx_0)\Pi _{i,y_0,y_1}(d\gamma )\pi _i(dy_0,dy_1). \end{aligned}$$

By Riesz’ theorem, $L_{i,{\scriptscriptstyle \sqrt{\epsilon _i}}}$ yields a measure $\tilde{\Pi }_i$ such that $L_{i,{\scriptscriptstyle \sqrt{\epsilon _i}}}f=\int f(\sigma )d\tilde{\Pi }_i(\sigma )$ and that is supported on ${\mathcal {G}}^{2\epsilon _i+2{\scriptscriptstyle \sqrt{\epsilon _i}}}(X)$ by construction of $X^{\sqrt{\epsilon _i}}$.

Note that ${\mathcal {G}}^{2\epsilon _i+2{\scriptscriptstyle \sqrt{\epsilon _i}}}(X)\subset {\mathcal {G}}^{2\epsilon _j+2{\scriptscriptstyle \sqrt{\epsilon _j}}}(X)$ for $i\ge j$, and since ${\mathcal {G}}^{2\epsilon +2{\scriptscriptstyle \sqrt{\epsilon }}}(X)$ is compact, by diagonal argument, we find a subsequence such that $\tilde{\Pi }_i$ converges to a measure $\tilde{\Pi }$ supported on ${\mathcal {G}}^{2\epsilon _i+2{\scriptscriptstyle \sqrt{\epsilon _i}}}(X)$ for every $i\in {\mathbb {N}}$. Since ${\mathcal {G}}(X)=\bigcap _{i\in {\mathbb {N}}}{\mathcal {G}}^{2\epsilon _i+2{\scriptscriptstyle \sqrt{\epsilon _i}}}(X)$ by the Arzela–Ascoli theorem, $\tilde{\Pi }$ is supported on ${\mathcal {G}}(X)$. For every i, we consider the decomposition $\hat{\Pi }_i-\tilde{\Pi }_i=:\bar{\Pi }_i$ and as in (25) we see that $0\le \bar{\Pi }_i({\mathcal {G}}^{{L}}(X))\le {M}\epsilon _i$ for all $i\ge i_{\eta }$. Hence, $(\Pi -\tilde{\Pi })({\mathcal {G}}^{{L}}(X))=0$, and therefore ${\mathcal {G}}(X)$ has full $\Pi$-measure.

Together with the convergence of $Q^i(\mu _{t,i})$ to $\mu _t$ (step 3.) the curvature-dimension condition on $X_i$ and lower semi-continuity of $S_N$, we get

$$\begin{aligned} S_N(\mu _{t}|{{\mathrm{m}}}_{{X}})\le -\int \left[ a^{-}(\sigma )\varrho _0(\sigma _0)^{-\frac{1}{N}}+ a^{+}(\sigma )\varrho _1(\sigma _1)^{-\frac{1}{N}}\right] {\Pi }(d\sigma ). \end{aligned}$$

(28)

Since $\eta$ was arbitrary, application of another compactness argument on X yields the inequality for $k$ instead $k-\eta$.

8. Now, we will check that $(e_t)_{\star }\Pi =\mu _t$ where $(\mu _t)_{t\in [0,1]}$ is the geodesic that we found in step 3.

Consider again $\tilde{\Pi }_i$. It is a finite measure on ${\mathcal {G}}^{2\epsilon +2{\scriptscriptstyle \sqrt{\epsilon }}}(X)$ and by normalization, we can make it a probability measure. Recall again (27). If $g\in C_b(X_i\times X)$, then

$$\begin{aligned}&\alpha _i\int g(e_t(\gamma ),e_t(\Xi _{i,\gamma }(x_0,x_1)))1_{({\mathcal {X}}^{{\scriptscriptstyle \sqrt{\epsilon }}})^2}(x_0,y_0,x_1,y_1)\\&\quad \times {\textstyle \frac{\varrho _0(x_0)\varrho _1(x_1)}{\varrho _{0,i}(y_0)\varrho _{1,i}(y_1)}} {Q}_i(y_1,dx_1)Q_i(y_0,dx_0)\Pi _{i}(d\gamma ) \end{aligned}$$

defines a coupling between $(e_t)_{\star }\Pi _i=\mu _{t,i}$ from step 3 and $(e_t)_{\star }(\alpha _i\tilde{\Pi }_i)$ where $\alpha _i>0$ is a normalization constant with $\alpha _i\rightarrow 1$ if $i\rightarrow \infty$. Choosing $g={{\mathrm{d}}}^2_{{Z}}|_{X_i\sqcup X}$, we obtain by construction of $\Xi _{i}$ and $X^{\sqrt{\epsilon _i}}$ that ${{\mathrm{d}}}_{{Z},{W}}(\mu _{t,i},(e_t)_{\star }\tilde{\Pi }_i)\le 2\sqrt{\epsilon _i}+2\epsilon _i.$ Recall $\mu _{t,i}\rightarrow \mu _t$ weakly. Moreover, weak convergence of $\alpha _i\tilde{\Pi }_i$ to $\tilde{\Pi }$ implies weak convergence of $(e_t)_{\star }\alpha _i\tilde{\Pi }_i$ to $(e_t)_{\star }\tilde{\Pi }$. Consequently, $(e_t)_{\star }\tilde{\Pi }=\mu _t$ since ${{\mathrm{d}}}_{{Z},{W}}(\mu _{t,i},(e_t)_{\star }\tilde{\Pi }_i)\le 2\sqrt{\epsilon _i}+2\epsilon _i\rightarrow 0$. Moreover, since $((e_t)_{\star }\Pi -(e_t)_{\star }\tilde{\Pi })(X)=0$, we have $\mu _t=(e_t)_{\star }\Pi$.

9. In the last step, we want to remove the remaining assumptions, namely continuity of $k$ and boundedness of $\varrho _j$ and $a^{-/+}$.

Consider general probability measures $\mu _0,\mu _1\in {\mathcal {P}}^2({{\mathrm{m}}}_{{X}})$ with densities $\rho _j$ for $j=0,1$. Fix an arbitrary optimal coupling $\tilde{\pi }$ between $\mu _0$ and $\mu _1$, and set for $r\in (0,\infty )$

$$\begin{aligned} E_r:=\left\{ (x_0,x_1))\in X^2: \rho _0(x_0)\le r, \rho _1(x_1)\le r\right\} , \ \alpha _r=\tilde{\pi }(E_r),\ \tilde{\pi }^r:=\frac{1}{\alpha _r}\tilde{\pi }|_{E_r}. \end{aligned}$$

The coupling $\tilde{\pi }^r$ is an optimal coupling between its marginals $\mu _0^r$ and $\mu ^r_1$ such that

$$\begin{aligned} {{\mathrm{d}}}_{{X},{W}}(\mu _j,\mu _j^r)\le \epsilon \ \text{ for } j=0,1 \hbox { if }r>0 \text{ sufficiently } \text{ large }. \end{aligned}$$

(29)

Depending on $r>0$ we can construct $\mu _t^r$ and $\Pi ^r$ as before. After successively choosing subsequences - that $\mu _t^r$ converges weakly to a probability $\mu _t$ for $t\in [0,1]\cap {\mathbb {Q}}$. Then, again as in step (vi) of the proof of Theorem 3.1 in [34] $\mu _t$ extends to geodesic between $\mu _0$ and $\mu _1$ and $\liminf _{i\rightarrow \infty } S_{{N}}(\mu ^r_{t}|{{\mathrm{m}}}_{{X}_{\infty }})\ge S_{{N}}(\mu _{t}|{{\mathrm{m}}}_{{X}_{\infty }})$ for $t\in [0,1]$.

Set $a^{-}(\sigma )\varrho _0(\sigma _0)^{-\frac{1}{N}}+ a^{+}(\sigma )\varrho _1(\sigma _1)^{-\frac{1}{N}}=\psi (\gamma )$. $\psi$ is integrable w.r.t. $\Pi$, since the coefficients $a^{+/-}$ are bounded, since $\rho _0$ and $\rho _1$ are probability densities for $\mu _0$ and $\mu _1$, respectively, and since $\Pi$ is a coupling between $\mu _0$ and $\mu _1$. Therefore, if we set $\Pi ^{\epsilon }=\alpha _r\Pi ^r+\Psi _{\star }\tilde{\pi }|_{X^2\backslash E_r}$, it follows that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\left| \int \psi (\gamma )d\Pi ^{\epsilon }(\gamma )-\int \psi (\gamma ) d\Pi ^r(\gamma )\right| =0. \end{aligned}$$

(see also step (v) in the proof of Theorem 3.1 in [34] for similar argument) Now, by compactness, we can choose subsequence $\epsilon _i$ such that $\Pi ^{\epsilon _i}$ converges weakly to an optimal coupling $\Pi$ between $\mu _0$ and $\mu _1$. Since $\Pi ^{\epsilon }$ is a coupling between $\mu _0$ and $\mu _1$ for every $\epsilon >0$, we can apply again Lemma 6.3. Hence,

$$\begin{aligned} S_{{N}}(\mu _{t}|{{\mathrm{m}}}_{{X}_{\infty }})&\le \liminf _{i\rightarrow \infty } S_{{N}}(\mu ^{r(\epsilon _i)}_{t}|{{\mathrm{m}}}_{{X}_{\infty }})\le \limsup _{i\rightarrow \infty }-\int \psi (\gamma )d\Pi ^{\epsilon _i}(\gamma )\\&\le -\int \psi (\gamma ) d\Pi (\gamma ). \end{aligned}$$

If $k$ is lower semi-continuous, we take monotone sequence of continuous functions $k_n$ that approximates $k$ from below. Since we can repeat all the previous steps, for any n, we obtain an optimal dynamic coupling $\Pi ^{n}$ and a Wasserstein geodesic $\mu _t^{n}$ such that (28) holds with $k$ replaced by $k_n$. The right-hand side of (28) is monotone with respect to $k_n$. Therefore, we obtain

$$\begin{aligned} S_N(\mu _{t}^{n}|{{\mathrm{m}}}_{{X}})&\le -\int \left[ a^{-}_{k_{n}}(\gamma )\varrho _0(\gamma _0)^{-\frac{1}{N}}+ a^{+}_{k_{n}}(\gamma )\varrho _1(\gamma _1)^{-\frac{1}{N}}\right] {\Pi }^{n}(d\gamma ) \\&\le -\int \left[ a^{-}_{k_{\hat{n}}}(\gamma )\varrho _0(\gamma _0)^{-\frac{1}{N}}+ a^{+}_{k_{\hat{n}}}(\gamma )\varrho _1(\gamma _1)^{-\frac{1}{N}}\right] {\Pi }^{{n}}(d\gamma ). \end{aligned}$$

for $n\ge \hat{n}$. Compactness yields a subsequence such that $\Pi _{n_i}$ and $\mu _t^{n_i}$ converge if $i\rightarrow \infty$ and by another application of Lemma 6.3 the limits of $\Pi$ and $\mu _t$ satisfy

$$\begin{aligned} S_N(\mu _{t}|{{\mathrm{m}}}_{{X}}) \le -\int \left[ a^{-}_{k_{\hat{n}}}(\gamma )\varrho _0(\gamma _0)^{-\frac{1}{N}}+ a^{+}_{k_{\hat{n}}}(\gamma )\varrho _1(\gamma _1)^{-\frac{1}{N}}\right] {\Pi }(d\gamma ). \end{aligned}$$

(30)

We let $\hat{n}\rightarrow \infty$. Then the theorem of monotone convergence yields the estimate

$$\begin{aligned} S_N(\mu _{t}|{{\mathrm{m}}}_{{X}})\le -\int \left[ a^{-}_{k}(\sigma )\varrho _0(\sigma _0)^{-\frac{1}{N}}+ a^{+}_{k}(\sigma )\varrho _1(\sigma _1)^{-\frac{1}{N}}\right] {\Pi }(d\sigma ). \end{aligned}$$

(31)

Finally, by a similar reasoning as before, $C\nearrow \infty$ yields $a^{-/+}(\gamma )\nearrow \tau _{{k}_{\gamma }^{-/+},{N}}^{{(t)}}(|\dot{\gamma }|)\in {\mathbb {R}}\cup \left\{ \infty \right\}$ for any $\gamma \in {\mathcal {G}}(X)$, and again by monotone convergence, the left-hand side in (31) converges to

$$\begin{aligned} S_N(\mu _{t}|{{\mathrm{m}}}_{{X}})\le -\int \left[ \tau _{{k}_{\gamma }^{-},{N}}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0(\gamma _0)^{-\frac{1}{N}}+ \tau _{{k}_{\gamma }^{+},{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _1(\gamma _1)^{-\frac{1}{N}}\right] {\Pi }(d\gamma ). \end{aligned}$$

This finishes the proof. $\square$

Corollary 6.8

Let $(M_i,g_{{M}_i})_{i\in {\mathbb {N}}}$ be a family of compact Riemannian manifolds with

$$ \begin{aligned}{{\mathrm{ric}}}_{{M}_i}\ge k_i\ \ \& \ \ \dim _{{M}_i}\le N\ \ \& \ \ {{\mathrm{diam}}}_{{M}_i}\le L\end{aligned}$$

where $k_i:M_i\rightarrow {\mathbb {R}}$ are lower semi-continuous functions such that $k_i\ge -C$ for some $C>0$. Let $\bar{{{\mathrm{vol}}}}_{{M}_i}$ be the normalized Riemannian volume. Then, there exists subsequence of $(M_i,{{\mathrm{d}}}_{{M}_i},\bar{{{\mathrm{vol}}}}_{{M}_i})$ that converges in measured Gromov–Hausdorff sense to a normalized metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$, there exists a subsequence $(i_n)_{n\in {\mathbb {N}}}$ and a lower semi-continuous function $k:X\rightarrow {\mathbb {R}}$ such that $\liminf \kappa _{i_n}\ge \kappa$ and $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies the condition $CD(k,N)$.

Proof

By Gromov’s precompactness theorem, there is a converging subsequence $(M_i,{{\mathrm{d}}}_{{M}_i},\bar{{{\mathrm{vol}}}}_{{M}_i})$ in the measured Gromov–Hausdorff sense, and a normalized limit metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$. $\bar{{{\mathrm{vol}}}}$ is the normalized Riemannian volume. Consider a compact metric space $(Z,{{\mathrm{d}}}_{{Z}})$ where the convergence is realized, and define $\hat{\kappa }:Z\rightarrow {\mathbb {R}}$ with $\hat{\kappa }(x_i)=k_i(x_i)$ if $x_i\in X_i$, and $+\infty$ otherwise. Let $\kappa :Z\rightarrow {\mathbb {R}}$ be the lower semi-continuous envelope of $\hat{\kappa }$ (see, for instance, [12]). We have $\hat{\kappa }>-\infty$ since $k_i\ge -C>-\infty$. We define $\kappa |_{X}=k:X\rightarrow {\mathbb {R}}$. By compactness, $\kappa$ is uniformly lower semi-continuous. More precisely, for $\epsilon >0$, there is $\delta >0$ such that for all $x,y\in Z$ with ${{\mathrm{d}}}_{{Z}}(x,y)\le \delta$ we have $f(y)\ge f(x)-\epsilon$. This implies $\liminf \kappa _i\ge \kappa$ in the sense of our definition. Hence, applying the previous theorem yields the statement.

Remark 6.9

Recall the notion of pointed measured Gromov–Hausdorff (GH) and pointed measured Gromov convergence from [17]. These notions generalize the previous concepts of convergence of metric measure spaces to the context of non-compact spaces and measures with infinite mass. From Remark 3.29 in [17], we see that if a sequence of pointed metric measure spaces converges in pointed measured GH sense to a metric measure space that is a length space, then pointed measured GH convergence is equivalent to measured GH convergence of closed R-balls around the center point to closed R-balls around the center point in the limit space. Hence, it is possible to extend the previous stability statement to pointed measured GH convergence.

7 Non-branching Spaces and Tensorization Property

Lemma 7.1

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a non-branching metric measure space that satisfies $CD(k,N)$. Then, for every $x\in {{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}$, there exists a unique geodesic between x and ${{\mathrm{m}}}_{{X}}$-a.e. $y\in X$. Consequently, there exists a measurable map $\Psi : X^2\rightarrow {\mathcal {G}}(X)$ such that $\Psi (x,y)$ is the unique geodesic between x and y ${{\mathrm{m}}}_{{X}}\otimes {{\mathrm{m}}}_{{X}}$-a.e. .

Proof

Since $k$ is bounded from below on any ball $B_R(x)$ by Theorem 5.3, one can adapt the proof of Lemma 4.1 in [34]. $\square$

Proposition 7.2

Let $k:X\rightarrow {\mathbb {R}}$ be admissible, $N\ge 1$ and $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a metric measure space that is non-branching. Then the following statements are equivalent

• (i) $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies the curvature-dimension condition $CD(k,N)$.

• (ii) For each pair, $\mu _0,\mu _1\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$, there exists an optimal dynamic transference plan $\Pi$ such that

  $$\begin{aligned} \varrho _t(\gamma _t)^{-\frac{1}{N}}\ge \tau _{{k}^-_{\gamma },{N}'}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0(\gamma _0)^{-\frac{1}{N}}+\tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _1(\gamma _1)^{-\frac{1}{N}}. \end{aligned}$$

  (32)

  for all $t\in [0,1]$ and $\Pi$-a.e. $\gamma \in {\mathcal {G}}(X)$. Here, $\varrho _t$ is the density of the push-forward of $\Pi$ under the map $\gamma \mapsto \gamma _t$. That is determined by

  $$\begin{aligned} \int _Xu(y)\varrho _t(y)d{{\mathrm{m}}}_{{X}}(y)=\int u(\gamma _t)d\Pi (\gamma ). \end{aligned}$$

  for all bounded measurable functions $u:X\rightarrow {\mathbb {R}}$.

Proof

“$\Leftarrow$”: Let $N'>N$ and $\varrho _id{{\mathrm{m}}}_{{X}}=\mu _i\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ for $i=0,1$. Hölder’s inequality yields

$$\begin{aligned} \varrho _t(\gamma _t)^{-\frac{1}{N'}}&\ge \left( \tau _{{k}^-_{\gamma },{N}'}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0(\gamma _0)^{-\frac{1}{N}}+\tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _1(\gamma _1)^{-\frac{1}{N}}\right) ^{\frac{N}{N'}}\\&\ge \tau _{{k}^-_{\gamma },{N}'}^{{(1-t)}}(|\dot{\gamma }|)^{\frac{N}{N'}}(1-t)^{(1-\frac{N}{N'})}\varrho _0(\gamma _0)^{\frac{-1}{N'}}+\tau _{{k}^+_{\gamma },{N}'}^{{(t)}}(|\dot{\gamma }|)^{\frac{N}{N'}}t^{(1-\frac{N}{N'})}\varrho _1(\gamma _1)^{\frac{-1}{N'}} \end{aligned}$$

In addition, Lemma 3.14 yields the estimate

$$\begin{aligned} \tau _{{k}^{-}_{\gamma },{N}}^{{(1-t)}}(|\dot{\gamma }|)^{\frac{N}{N'}}(1-t)^{1-\frac{N}{N'}}\ge \tau _{{k}^-_{\gamma },{N}'}^{{(1-t)}}(|\dot{\gamma }|) \end{aligned}$$

and similarly for the term involving $k^+_{\gamma }$. Finally, integrating the previous inequality with respect to $\Pi$ yields the condition $CD(k,N)$.

“$\Rightarrow$”: Consider probability measures $\mu _i=\varrho _id{{\mathrm{m}}}_{{X}}$ for $i=0,1$. Let $\Pi$ be an optimal dynamic coupling. Since for ${{\mathrm{m}}}_{{X}}\otimes {{\mathrm{m}}}_{{X}}$-a.e. pair (x, y), there exists a unique geodesic $\gamma _{x,y}$, there exist an optimal coupling $\pi$ such that $\Pi$ can be written in the form: $\delta _{\gamma _{x,y}}d\pi (x,y)$. Therefore, the curvature-dimension condition for $\mu _0$ and $\mu _1$ becomes

$$\begin{aligned}&\int _X\int \varrho _t(\gamma _t)^{-\frac{1}{N}}\delta _{\gamma _{x,y}}(d\gamma )d\pi (x,y)\\&\ge \int _{X^2}\int \left[ \tau _{{k}^-_{\gamma },{N}}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0(\gamma _0)^{-\frac{1}{N}}+\tau _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|)\varrho _1(\gamma _1)^{-\frac{1}{N}}\right] \delta _{\gamma _{x,y}}(d\gamma )d\pi (x,y). \end{aligned}$$

Now, we can follow exactly the proof of the corresponding result in [34]. $\square$

Proposition 7.3

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a non-branching metric measure space that satisfies $CD(k,N)$, let $k':X\rightarrow {\mathbb {R}}$ be lower semi-continuous and let $V:X\rightarrow [0,\infty )$ be strongly $k'V$-convex in the sense of Definition 3.16. Then $(X,{{\mathrm{d}}}_{{X}},V^{N'}{{\mathrm{m}}}_{{X}})$ satisfies the condition $CD(k+k',N+N')$.

Proof

The proof is a straightforward calculation using the characterization of $CD(k,N)$ for non-branching spaces, Corollary 4.2 and Hölder’s inequality. $\square$

Theorem 7.4

Let $(X_i,{{\mathrm{d}}}_{{X}_i},{{\mathrm{m}}}_{{X}_i})$ be non-branching metric measure spaces for $i=1,\ldots ,k$ satisfying the condition $CD(k_i,N_i)$ for admissible functions $k_i:X_i\rightarrow {\mathbb {R}}$ and $N_i\ge 1$. Then the metric measure space

$$\begin{aligned} \left( \Pi _{i=1}^kX_i,\ \sqrt{\sum \nolimits _{i=1}^k{{\mathrm{d}}}_{{X}_i}^2},\ \bigotimes \nolimits _{i=1}^k{{\mathrm{m}}}_{{X}_i}\right) =(Y,{{\mathrm{d}}}_{{Y}},{{\mathrm{m}}}_{{Y}}) \end{aligned}$$

satisfies the condition

$$\begin{aligned} CD\left( \min _{i=1,\ldots ,k}k_i,\max _{i=1,\ldots ,k}N\right) \end{aligned}$$

where $(\min _{i=1,\ldots , k}k_i)(x_1,\ldots ,x_k)=\min \left\{ k_i(x_i):i=1,\ldots ,k\right\}$.

Proof

It is enough to consider $k=2$ and measures of $\mu _0$ and $\mu _1$ in ${\mathcal {P}}_2(Y,{{\mathrm{m}}}_{{Y}})$ of the form $\mu _0=\mu _0^{{(1)}}\otimes \mu _0^{{(2)}} \text{ and } \mu _1=\mu _1^{{(1)}}\otimes \mu _1^{{(2)}}.$ Then general case follows in the same way as in [8], for instance. Consider dynamic optimal couplings $\Pi ^{{(i)}}$ for $\mu _0^{{(i)}}$ and $\mu _0^{{(i)}}$ such that (32) holds according to our curvature assumption. Let $(e_0,e_1)_{\star }\Pi ^{{(i)}}=\pi ^{{(i)}}$. The pushforward of $\pi ^{{(1)}}\otimes \pi ^{{(2)}}$ with respect to

$$\begin{aligned} (x_0^{{(1)}}, x_1^{{(1)}},x_0^{{(2)}}, x_1^{{(2)}})\mapsto (x_0^{{(1)}},x_0^{{(2)}}, x_1^{{(1)}}, x_1^{{(2)}}) \end{aligned}$$

becomes an optimal coupling $\pi$ between $\mu _0$ and $\mu _1$. There is also a measurable map $(\gamma ^{{(1)}},\gamma ^{{(2)}})\in {\mathcal {G}}(X_1)\times {\mathcal {G}}(X_2)\mapsto (\gamma ^{{(1)}},\gamma ^{{(2)}})\in {\mathcal {G}}(Z)$. Therefore, we can consider the pushforward $\Pi$ of $\Pi ^{{(1)}}\times \Pi ^{{(2)}}$ with respect to this map. Since $(e_0,e_1)_{\star }\Pi =\pi$, $\Pi$ is an optimal dynamic plan for $\mu _0$ and $\mu _1$.

Claim: For geodesics $\gamma ^{{(1)}}\in {\mathcal {G}}(X_1)$ and $\gamma ^{{(2)}}\in {\mathcal {G}}(X_2)$ consider $\gamma =(\gamma ^{{(1)}},\gamma ^{{(2)}})\in {\mathcal {G}}(Y)$, then we have

$$\begin{aligned} \tau ^{{(t)}}_{{k}_{1,\gamma },{N}_1}(|\dot{\gamma }^{{(1)}}|)^{{N}_1}\cdot \tau ^{{(t)}}_{{k}_{2,\gamma },{N}_2}(|\dot{\gamma }^{{(2)}}|)^{{N}_2}\ge \tau ^{{(t)}}_{k_{\gamma },{N}_1+{N}_2}\left( |\dot{\gamma }|\right) ^{{N}_1+{N}_2} \end{aligned}$$

The claim follows immediately from Corollary 4.2 combined with the observations that $\tau ^{{(t)}}_{{k}_{\gamma },{N}}(|\dot{\gamma }|)=\tau ^{{(t)}}_{{k}_{\gamma }|\dot{\gamma }|^2,{N}}(1)$, that $|\dot{\gamma }|^2=|\dot{\gamma }^{{(1)}}|^2+|\dot{\gamma }^{{(2)}}|^2$, and that

$$\begin{aligned} k_i\circ \bar{\gamma }^{{(i)}}(t|\dot{\gamma }^{{(i)}}|) =k_i\circ \gamma ^{{(i)}}(t)\ge \min _{i=1,2}\left\{ k_i\circ \gamma (t)\right\} =\left( \min \nolimits _{i=1,2}k_i\circ \bar{\gamma }\right) (t|\dot{\gamma }|). \end{aligned}$$

for $i=1,2$. The rest of the proof works exactly like the proof of the corresponding result in [13]. $\square$

8 Globalization of the Reduced Curvature-Dimension Condition

Definition 8.1

If we replace in Definition 4.4

$$\begin{aligned} \tau _{k^{{-/+}}_{\gamma },N'}^{{(1-t)/(t)}}(|\dot{\gamma }|) \text{ by } \sigma _{k^{{-/+}}_{\gamma },N'}^{{(1-t)/(t)}}(|\dot{\gamma }|). \end{aligned}$$

we say $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies the reduced curvature-dimension condition $CD^*(k,N)$. Obviously, we always have that $CD(k,N)$ implies $CD^*(k,N)$.

We say that $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies the the curvature-dimension condition locally - denoted by $CD_{loc}(k,N)$ - if for any point x there exists a neighborhood $U_x$ such that for each pair $\mu _0,\mu _1\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ with bounded support in $U_x$, then one can find a geodesic $\mu _t\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ and an optimal dynamic coupling $\Pi \in {\mathcal {P}}({\mathcal {G}}(X))$ such that (12) holds. Similarly, we define $CD^*_{loc}(k,N)$.

Remark 8.2

All the previous results of this article also hold for the condition $CD^*(k,N)$ although constants and estimates are in general not sharp.

Theorem 8.3

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a non-branching and geodesic metric measure space with ${{\mathrm{supp}}}{{\mathrm{m}}}_{{X}}=X$. Let $k:X\rightarrow {\mathbb {R}}$ be admissible. Then the curvature-dimension condition $CD^*(k,N)$ holds if and only if it holds locally.

Proof

We only have to show the implication $CD^*_{loc}(k,N)$ implies $CD^*(k,N)$. Let us assume the curvature-dimension condition holds locally. Therefore, a Bishop–Gromov volume growth result holds locally, and it implies the space is locally compact. Then the metric Hopf-Rinow theorem implies that X is proper. Hence, we can assume that X is compact. Otherwise, we choose an exhaustion of X with compact balls $\overline{B_R(o)}$ such that the optimal transport between measures supported in $B_R(o)$ does not leave $\overline{B_{2R}(o)}$. For instance, compare with the proof of Theorem 5.1 in [8]. Similar to the proof of Proposition 7.2, one can also see that a measure contraction property holds locally. Then, the result of [10] implies uniqueness of $L^2$-Wasserstein geodesics.

By compactness of X, there is $\lambda \in (0,{{\mathrm{diam}}}_{{X}})$, finitely many disjoints sets $L_1,\ldots , L_k$ that cover X and have non-zero measure, and finitely many open sets $M_1,\ldots , M_k$ such that $B_{\lambda }(L_i)\subset M_i$ for $i\in \left\{ 1,\ldots k\right\}$ and such that (12) holds in $M_i$ for each i (for instance, see the proof of Theorem 5.1 in [8]).

Let $\mu _0,\mu _1\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ be arbitrary and let $\mu _t$ be the $L^2$-Wasserstein geodesic between $\mu _0$ and $\mu _1$. Consider $\mu _{\bar{t}}$ and $\mu _{\bar{s}}$ such that $\bar{s}-\bar{t}\le \lambda /{{\mathrm{diam}}}_{{X}}$. We define $\nu _{\tau }=\mu _{(1-\tau )\bar{t}+\tau \bar{s}}$ is a geodesic between $\mu _{\bar{t}}$ and $\mu _{\bar{s}}$, and any transport geodesic has length less than $\lambda$. $\Pi$ denotes the optimal dynamic transference plan that corresponds to $\nu _t$. We decompose $\nu _0$ with respect to $(L_i)_{i=1,\ldots ,k}$ as follows

$$\begin{aligned} \nu _0=\sum _{i=1}^{k}\frac{1}{\nu _0(L_i)}\nu _0|_{L_i}=\sum _{i=1}^{k}\nu _0^i. \end{aligned}$$

Define ${\mathcal {L}}_i=\left\{ \gamma \in {\mathcal {G}}(X):\gamma (0)\in L_i\right\}$ with $\nu _0(L_i)=\Pi ({\mathcal {L}}_i)$. The restriction property of optimal transport yields that $\Pi ^i=\Pi ({\mathcal {L}}_i)^{-1}\Pi |_{{\mathcal {L}}_i}$ are optimal dynamic couplings between $\nu _0^i$ and $\nu _1^i=(e_1)_*\Pi ^i$ and $\Pi ^i$ induces a geodesic $\nu _{\tau }^i$ between $\nu ^i_0$ and $\nu ^i_1=(e_1)_*\Pi ^i$. By construction, $\nu _1^i$ is supported in $M_i$. Hence, the condition $CD(k,N)$ implies

$$\begin{aligned} {\varrho }_t^i(\gamma (t))^{-\frac{1}{N}}\ge \sigma _{{k}^-_{\gamma },{N}}^{{(1-t)}}(|\dot{\gamma }|){\varrho }_0^i(\gamma (0))^{-\frac{1}{N}}+\sigma _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|){\varrho }_1^i(\gamma (1))^{-\frac{1}{N}} \end{aligned}$$

for $\Pi ^i$-a.e. $\gamma \in {\mathcal {G}}(X)$ where $\varrho _t^id{{\mathrm{m}}}_{{X}}=d\nu _t^i$. In particular, $\nu _t$ is absolutely continuous with density $\varrho _t=\sum _{i=1}^k\varrho _t^i$.

The measures $\nu _0^i$ are disjoint. Therefore, the measures $\nu _t^i$ for $i=1,\ldots ,k$ are disjoint for any $t\in [0,1)$ (see for instance, Lemma 2.6 in [8]). Since any optimal transport between absolutely continuous probability measures is induced by an optimal map, we can conclude that also $\nu _1^i$ are disjoint. Therefore, for any $t\in [0,1]$,

$$\begin{aligned} \varrho _t(x)^{-\frac{1}{N}}= \sum _{i=1}^k \frac{1}{\Pi ({\mathcal {L}}_i)}\varrho _t^i(x)^{-\frac{1}{N}} \end{aligned}$$

(33)

where $\varrho _t^i d{{\mathrm{m}}}_{{X}}=d\nu _t^i$. Hence

$$\begin{aligned}&{\varrho }_t(\gamma (t))^{-\frac{1}{N}}\ge \sigma _{{k}^-_{\gamma },{N}}^{{(1-t)}}(|\dot{\gamma }|){\varrho }_0(\gamma (0))^{-\frac{1}{N}}+\sigma _{{k}^+_{\gamma },{N}}^{{(t)}}(|\dot{\gamma }|){\varrho }_1(\gamma (1))^{-\frac{1}{N}} \\&\quad \text{ for } \Pi -\hbox {a.e.} \gamma \in {\mathcal {G}}(X). \end{aligned}$$

In particular, the previous argument holds for each $\bar{s},\bar{t}\in [0,1]\cap {\mathbb {Q}}$. Thus, if $\mu _t$ is the unique geodesic between $\mu _0,\mu _1$ and $\Pi$ is the corresponding optimal dynamic plan, we showed that

$$\begin{aligned}&{\rho }_{\tau (t)}(\gamma (\tau (t)))^{-\frac{1}{N}}\ge \sigma _{{k}^-_{\gamma },{N}}^{{(1-\tau (t))}}((s-t)|\dot{\gamma }|){\rho }_t(\gamma (t))^{-\frac{1}{N}}\\&\quad +\sigma _{{k}^+_{\gamma },{N}}^{{(\tau (t))}}((s-t)|\dot{\gamma }|){\rho }_s(\gamma (s))^{-\frac{1}{N}} \end{aligned}$$

for $\Pi$-a.e. geodesic $\gamma$ and each $\bar{t},\bar{s}\in [0,1]\cap {\mathbb {Q}}$ where $\tau (t)=(1-t)\bar{t}+t\bar{s}$. If we pick such a geodesic $\gamma$, the inequality holds also globally along $\gamma$ for $\rho _t$ by Corollary 3.13. Then the result follows. $\square$

9 Curvature-Dimension Condition with Variable Dimension Bound

We briefly discuss two possibilities to define a curvature-dimension condition $CD(k,{\mathcal {N}})$ with variable dimension bound ${\mathcal {N}}:X\rightarrow (0,\infty )$. Following the previous approach for variable lower curvature bounds, it is not obvious how to pose a reasonable definition when $N={\mathcal {N}}$ is variable as well. This is because for our definition, we use the N-Reny entropy functional where N is a constant parameter.

However, the problem can be resolved in the following way. Consider the non-branching situation of Sect. 7, and the reduced curvature-dimension condition $CD^*(\kappa ,N)$ that is introduced in Sect. 8. A metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies $CD^{*}(k,N)$ if and only if for each pair $\mu _0,\mu _1\in {\mathcal {P}}^2(X,{{\mathrm{m}}}_{{X}})$ there exists a geodesic $\mu _t\in {\mathcal {P}}^2(X,{{\mathrm{m}}}_{{X}})$ and a dynamic optimal plan $\Pi$ with $(e_t)_{\star }\Pi =\mu _t$ and

$$\begin{aligned} \rho _t(\gamma _t)^{-\frac{1}{{N}}}\ge \sigma _{\kappa ^-_{\gamma }/{N}}^{(1-t)}(|\dot{\gamma }|)\rho _0(\gamma _0)^{-\frac{1}{{N}}} +\sigma _{\kappa ^+_{\gamma }/{N}}^{(t)}(|\dot{\gamma }|)\rho _0(\gamma _0)^{-\frac{1}{{N}}} \end{aligned}$$

(34)

for $\Pi$-a.e. $\gamma \in {\mathcal {G}}(X)$, where $\mu _t=\rho _td{{\mathrm{m}}}_{{X}}$. (34) is equivalent to the following differential inequality in the distributional sense:

$$\begin{aligned} \frac{d^2}{dt^2}\rho _t(\gamma _t)^{-\frac{1}{{N}}}\le -\frac{\kappa _{\gamma }(t)|\dot{\gamma }|^2}{N} \rho _t(\gamma _t)^{-\frac{1}{{N}}} \ \ \text{ on } \ [0,1]. \end{aligned}$$

(35)

Note (35) implies that $t\mapsto \rho _t(\gamma _t)$ is absolutely continuous. If $N={\mathcal {N}}$ is variable along $\gamma$, (35) would involve second derivatives of ${\mathcal {N}}$ along $\gamma$. However, one can fix this problem by another reformulation. If we assume that $\rho _t(\gamma _t)$ is $C^2$ in t for $\Pi$-a.e. $\gamma$ (this is true, for instance, on Riemannian manifolds), then (35) is equivalent to the Ricatti equation

$$\begin{aligned} \frac{d^2}{dt^2}\log \rho _t(\gamma _t)+{\textstyle \frac{1}{N}}\left( {\textstyle \frac{d}{dt}\log \rho _t(\gamma _t)}\right) ^2+\kappa _{\gamma }(t)\left| \dot{\gamma }\right| ^2\le 0. \end{aligned}$$

(36)

Here, it is no problem to replace N by an upper semi-continuous function ${\mathcal {N}}\circ \gamma$. Then, the following definition is meaningful.

Definition 9.1

Let $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ be a non-branching metric measure space, and let $\kappa :X\rightarrow {\mathbb {R}}$ be lower semi-continuous and let ${\mathcal {N}}:X\rightarrow (0,\infty )$ be upper semi-continuous. We say $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies $CD^*(\kappa ,{\mathcal {N}})$ if for each pair $\mu _0,\mu _1\in {\mathcal {P}}^2(X,{{\mathrm{m}}}_{{X}})$ there exists a geodesic $\mu _t\in {\mathcal {P}}^2(X,{{\mathrm{m}}}_{{X}})$ and a dynamic optimal plan $\Pi$ such that $(e_t)_{\star }\Pi =\rho _t{{\mathrm{m}}}_{{X}}\in {\mathcal {P}}^2({{\mathrm{m}}}_{{X}})$, $\rho _t(\gamma _t)$ is absolutely continuous in $t\in [0,1]$ for $\Pi$-a.e. $\gamma$ and (36) holds in the distributional sense with N replaced by ${\mathcal {N}}\circ \gamma$.

Similarly, one can define an entropic curvature-dimension condition $CD^e(\kappa ,{\mathcal {N}})$ for $\kappa$ lower semi-continuous and ${\mathcal {N}}$ upper semi-continuous following ideas of [14] and [19] where no non-branching assumption is required. However, Definition 9.1 has an important disadvantage. It is not clear to the author how to formulate an equivalent, integrated version that is desirable for proving geometric consequences and stability properties. Moreover, the full curvature-dimension condition $CD(\kappa ,{\mathcal {N}})$ does not make sense since the coefficients $\tau ^{\scriptscriptstyle {(t)}}_{\kappa _{\gamma }}(|\dot{\gamma }|)$ are not derived from an ODE.

Following a clever suggestion of the referee another possibility to extend our definition to variable upper dimension bounds would be as follows.

Definition 9.2

Consider an admissible function $k:X\rightarrow {\mathbb {R}}$, and a upper semi-continuous function ${\mathcal {N}}:X\rightarrow {\mathbb {R}}$ bounded from above. A metric measure space $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies the curvature-dimension condition $CD(k,{\mathcal {N}})$ if for each pair $\nu _0,\nu _1\in {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ with bounded support there exists a geodesic $(\nu _t)_{t\in [0,1]}\subset {\mathcal {P}}_2(X,{{\mathrm{m}}}_{{X}})$ and a dynamic optimal coupling $\Pi \in {\mathcal {P}}(X)$ such that $(e_t)_{\star }\Pi =\nu _t$ and

$$\begin{aligned} S_{N'}(\nu _t)\le {\textstyle -}\int \Big [\tau _{k^{{-}}_{\gamma },N'}^{{(1-t)}}(|\dot{\gamma }|)\varrho _0\left( e_0(\gamma )\right) ^{-\frac{1}{N'}}+\tau _{k^{{+}}_{\gamma },N'}^{{(t)}}(|\dot{\gamma }|)\varrho _1\left( e_1(\gamma )\right) ^{-\frac{1}{N'}}\Big ]d\Pi (\gamma ) \end{aligned}$$

(37)

for all $t\in [0,1]$ and all $N'\ge \overline{N}$ where $\overline{N}=\overline{N}(\Pi )=\sup _{\gamma {{\mathrm{supp}}}\Pi ,t\in [0,1]}{\mathcal {N}}(\gamma (t))$.

The upper bound for ${\mathcal {N}}$ guarantees that $(X,{{\mathrm{d}}}_{{X}},{{\mathrm{m}}}_{{X}})$ satisfies $CD(\kappa ,N)$ for some N, and in particular it is locally compact. Therefore $\overline{N}(\Pi )$ is well defined. This definition seems to be better for proving geometric consequences and stability of the condition, and it might be possible that one can extend the results of this article.

Example 9.3

First, it is clear that any $CD(\kappa ,N)$-space satisfies a condition $CD(\kappa ,{\mathcal {N}})$ for N constant and ${\mathcal {N}}$ upper semi-continuous if ${\mathcal {N}}\ge N$.

Now, we construct a more interesting example. Let $N,N'\in [1,\infty )$ with $N> N'$, and $K\in (0,\infty )$. Consider

$$\begin{aligned} I=\left[ -\pi \sqrt{\frac{K}{N-1}},0]\cup (0,\pi \sqrt{\frac{K}{N'-1}}\right] \end{aligned}$$

and define

$$\begin{aligned} f(x)={\left\{ \begin{array}{ll} \cos _{{K}/({N}-1)}(x)^{N-1}&{} \ \text{ if } x\in [-\pi \sqrt{\frac{K}{N-1}},0],\\ \cos _{{K}/({N}'-1)}(x)^{N'-1}&{} \ \text{ if } x\in (0,\pi \sqrt{\frac{K}{N'-1}}]. \end{array}\right. } \end{aligned}$$

Then, $(I,|\cdot |_2, fd{\mathcal {L}}^1)$ satisfies the curvature-dimension condition $CD(K,{\mathcal {N}})$ in the sense of Definition 9.2 and the condition $CD^*(K,{\mathcal {N}})$ in the sense of Definition 9.1 where

$$\begin{aligned} {\mathcal {N}}(x)={\left\{ \begin{array}{ll} N\ \text{ if } x\in [-\pi \sqrt{\frac{K}{N-1}},0],\\ N'\ \text{ if } x\in (0,\pi \sqrt{\frac{K}{N'-1}}]. \end{array}\right. } \end{aligned}$$

Note that the space satisfies the condition CD(K, N) already.