It is sufficient to prove the result for the case $\mathsf{G}=\mathsf{D}$. If
$(M,\mathsf{d})$ is ideal, there exists a unique $\pi\in{\rm
Opt}(\mu_{0},\mu_{1})$, and it is induced by a transport map $T:M\to M$.
Furthermore, for $\mu_{0}$-a.e. $x\in M$, either $T(x)=x$, or
$T(x)\notin\mathrm{Cut}(x)$, cf. [24, Cor. 38]. Since
$\operatorname{\mathrm{supp}}\mu_{0}$ and
$\operatorname{\mathrm{supp}}\mu_{1}$ are disjoint, the first case never
occurs. It follows that $\mathsf{D}$ is continuous at $(x,y)$ for $\pi$-a.e.
$(x,y)\in M\times M$. ∎

######  Remark 7.21.

One can appreciate in the above proof the importance of the restriction to
measures $\mu_{0},\mu_{1}$ with disjoint support. In general (and more
precisely, whenever the sub-Riemannian structure is not Riemannian), the
diagonal $\Delta\subset M\times M$ is always part of the cut locus, and
$\mathsf{D}$ is not continuous there. Thus, the regularity condition of
Definition 5.12 would not be verified without that restriction.

######  Remark 7.22.

The ideal assumption cannot be removed. In fact, in [20, Fig. 2], the authors
built an explicit example, on corank 1 Carnot groups, and measures
$\mu_{i}\in\mathcal{P}^{*}_{bs}(X,\mathsf{d},\mathfrak{m})$ with disjoint
support where any point $x\in\operatorname{\mathrm{supp}}\mu_{0}$ is sent to
$T(x)\in\mathrm{Cut}(x)$, and more precisely $T(x)$ is the endpoint of an
abnormal geodesic from $x$. Here, it turns out that $\mathsf{D}$ is not
continuous at every $(x,y)\in\operatorname{\mathrm{supp}}\pi$.

####  7.3.3 G-boundedness condition

We discuss here the boundedness properties of natural gauge functions. We
recall the following definition, see [2, Sec. 3.1] for further details.

######  Definition 7.23 (Step).

A sub-Riemannian metric space has step $k\in\mathbb{N}$ at $x\in M$ if Lie
brackets of length $k$ are sufficient to satisfy the bracket-generating
condition at $x$, and $k=k(x)$ is the smallest number with this property. The
step of a sub-Riemannian metric space is the supremum of the step at all its
points.

The next result is a reformulation of well-known facts about the regularity of
the sub-Riemannian distance.

######  Theorem 7.24 (Boundedness properties of natural gauge functions).

Let $(M,\mathsf{d},\mathfrak{m})$ be a sub-Riemannian m.m.s. Let $\mathsf{G}$
be a natural gauge function. Then the following hold:

  1. (i)

The function $\mathsf{G}$ is locally bounded on an open and dense set of
$M\times M$. In particular for any non-empty open set $\mathcal{O}$ it holds

$\mathfrak{m}(\\{y\in
M\;:\;|\mathsf{G}(x,y)|<+\infty\\}\cap\mathcal{O})>0,\qquad\forall\,x\in M.$
(7.25)

  2. (ii)

If $(M,\mathsf{d})$ satisfies the minimizing Sard property, then for any $x\in
M$ the function $\mathsf{G}(x,\cdot)$ is locally bounded almost everywhere. In
particular it holds

$\mathfrak{m}(\\{y\in
M\;:\;|\mathsf{G}(x,y)|=+\infty\\})=0,\qquad\forall\,x\in M.$ (7.26)

  3. (iii)

If $(M,\mathsf{d})$ has step $\leq 2$, then $\mathsf{G}$ is locally bounded,
that is for every bounded subset $\mathcal{O}$ there exists $C>0$ such that

$|\mathsf{G}(x,y)|\leq C,\qquad\forall\,x,y\in\mathcal{O}.$ (7.27)

  4. (iv)

If $\mathsf{d}$ is locally Lipschitz in charts outside of the diagonal
$\Delta\subset M\times M$, then $\mathsf{G}$ is locally bounded away from the
diagonal, that is for all $\varepsilon>0$ and bounded subset $\mathcal{O}$
there exists $C>0$ such that

$|\mathsf{G}(x,y)|\leq
C,\qquad\forall\,x,y\in\mathcal{O},\;\mathsf{d}(x,y)\geq\varepsilon.$ (7.28)

###### Proof.

It is sufficient to prove the statements for $\mathsf{G}=\mathsf{D}$.

Proof of (i). Fix $x\in M$ and $y\notin\mathrm{Cut}(x)$. Then $\mathsf{d}$ is
smooth in a neighborhood of $(x,y)$, and
$\mathsf{D}(x,y)=\|\nabla^{R}_{x}\mathsf{d}(\cdot,y)\|_{R}<+\infty$. The set
$M\setminus\mathrm{Cut}(x)$ is open and dense by Theorem 7.12, and since
$\mathfrak{m}$ is smooth the statement follows.

Proof of (ii). By the previous argument, if $\mathsf{D}(x,y)=+\infty$ we must
have $y\in\mathrm{Cut}(x)$. If the Sard property holds, then $\mathrm{Cut}(x)$
has zero measure.

Proof of (iii). The step of $(M,\mathsf{d})$ is $\leq 2$ if and only if
$\mathsf{d}$ is locally Lipschitz in charts [4, Cor. 6.2]. Since $\mathsf{D}$
is the asymptotic Lipschitz constant of $\mathsf{d}$ with respect to a
Riemannian metric $\mathsf{d}_{R}$, then $\mathsf{D}$ must be locally bounded.

Proof of (iv). Obvious. ∎

######  Remark 7.25.

Item (i) yields the non-triviality condition (4.41) for natural gauge
functions. Item (ii) yields the finiteness condition (4.54) required in the
generalized Bishop-Gromov Theorem 4.24.

######  Remark 7.26.

Item (iv) holds for example when $(M,\mathsf{d})$ is ideal, and more generally
if there are no non-trivial Goh paths, see [4, Thm. 5.5] for details. This is
true, for example, if the underlying distribution is medium-fat [46, Sec.
4.5], [12].

######  Remark 7.27.

The step $2$ assumption in (iii) cannot be removed. Notice that, by the Ball-
Box theorem [52, 31], $\mathsf{d}(o,\cdot)$ cannot be Lipschitz in charts in a
neighborhood of any point $o$ where the step $s(o)\geq 3$, so that
$\mathsf{D}(o,o)=+\infty$. One can show that, in general, $\mathsf{D}$ may not
be essentially bounded in any neighborhood of the diagonal around such points.

To illustrate this fact, let us discuss the case of Carnot groups
$(\mathbb{G},\mathsf{d})$. These are left-invariant structures on a nilpotent,
connected and simply connected Lie group $\mathbb{G}\simeq\mathbb{R}^{n}$. We
denote points as $(x_{1},\dots,x_{n})$ in exponential coordinates, with
$o=(0,\dots,0)$ the identity of the group. Carnot groups are equipped with
metric dilations $\delta_{\lambda}$, $\lambda>0$, that in these coordinates
read $\delta_{\lambda}(x)=(\lambda^{w_{1}}x_{1},\dots,\lambda^{w_{n}}x_{n})$,
where $w_{i}$ denotes the weight of the $i$-th coordinate. Furthermore, we may
choose a Riemannian reference $\mathsf{d}_{R}$, left-invariant by the group
action of $\mathbb{G}$. Notice that, for any bounded neighborhood
$\mathcal{O}\subset\mathbb{R}^{n}$, there exists a constant
$C=C_{\mathcal{O}}>0$ such that it holds

$\frac{1}{C}\sum_{i=1}^{n}|x_{i}-x_{i}^{\prime}|\leq\mathsf{d}_{R}(x,x^{\prime})\leq
C\sum_{i=1}^{n}|x_{i}-x_{i}^{\prime}|,\qquad\forall\,x,x^{\prime}\in\mathcal{O}.$
(7.29)

Furthermore, using dilations, one checks that there exists a constant
$C^{\prime}>0$ such that

$\frac{1}{C^{\prime}}\sum_{i=1}^{n}|x_{i}-x^{\prime}_{i}|^{1/w_{i}}\leq\mathsf{d}(x,x^{\prime})\leq
C^{\prime}\sum_{i=1}^{n}|x_{i}-x^{\prime}_{i}|^{1/w_{i}},\qquad\forall\,x,x^{\prime}\in\mathbb{R}^{n}.$
(7.30)

Let now $\mathcal{O}$ be a bounded neighborhood of the origin
$o\in\mathbb{G}$. Up to restricting it, we may assume that
$p\in\mathbb{O}\Leftrightarrow p^{-1}\in\mathbb{O}$ (recall that, in
exponential coordinates, group inversion corresponds to the map $x\mapsto-x$).
We have, for all $x\in\mathcal{O}$

$\displaystyle\mathsf{D}(o,x^{-1})=\mathsf{D}(x,o)$
$\displaystyle=\frac{1}{2}\limsup_{z,w\to
x}\frac{\mathsf{d}(z,o)^{2}-\mathsf{d}(w,o)^{2}}{\mathsf{d}_{R}(z,w)}$ (7.31)
$\displaystyle\geq\frac{1}{2}\limsup_{\lambda,\lambda^{\prime}\to
1}\frac{\mathsf{d}(\delta_{\lambda}(x),o)^{2}-\mathsf{d}(\delta_{\lambda^{\prime}}(x),o)^{2}}{\mathsf{d}_{R}(\delta_{\lambda}(x),\delta_{\lambda^{\prime}}(x))}$
(7.32) $\displaystyle\geq\frac{1}{2}\limsup_{\lambda,\lambda^{\prime}\to
1}\frac{|\lambda^{2}-\lambda^{{}^{\prime}2}|\mathsf{d}(x,o)^{2}}{|\lambda-\lambda^{\prime}|C^{2}\mathsf{d}_{R}(x,o)}$
(7.33)
$\displaystyle\geq\frac{1}{C^{2}}\frac{\mathsf{d}(x,o)^{2}}{\mathsf{d}_{R}(x,o)}\geq
C^{\prime\prime}\frac{\sum_{i=1}^{n}|x_{i}|^{2/w_{i}}}{\sum_{i=1}^{n}|x_{i}|},$
(7.34)

where in the first equality we used the left-invariance of $\mathsf{D}$ (which
follows from the left-invariance of $\mathsf{d}$ and $\mathsf{d}_{R}$), and we
note that the constant $C^{\prime\prime}>0$ depends only on $\mathcal{O}$. It
follows that, if there is one $w_{i}\geq 3$ (that is, precisely when the step
is $\geq 3$), then it holds

$\mathfrak{m}\shortminus\operatorname*{ess\,sup}_{x\in\mathcal{O},x\neq
o}\mathsf{G}(o,x)=+\infty,$ (7.35)

where $\mathfrak{m}$ is any smooth measure on $\mathbb{R}^{n}$ (this includes
the Haar measures of $\mathbb{G}$). Notice that in the last estimate,
$\mathcal{O}$ can be any bounded neighborhood of $o$. It follows by left-
invariance that for any neighborhood $\mathcal{O}$ in $\mathbb{G}$ we have

$\operatorname{\mathrm{diam}}_{\mathsf{G}}(\mathcal{O})=+\infty.$ (7.36)

Thus, for sub-Riemannian structures of step $3$, natural gauge functions do
not satisfy, in general, the $\mathsf{G}$-boundedness assumptions of the
stability Theorems 5.6 and 5.14.

The diagram in Figure 2 illustrates most of the properties for natural gauge
functions we proved in this section, on sub-Riemannian structures.

step $\leq 2$ (Def. 7.23)$\mathsf{G}$ is locally bounded minimizing Sard
---
(Def. 7.13)
$\mathsf{G}$ is locally bounded
---
almost everywhere
$\mathsf{G}$ satisfies the
---
regularity for stability
of $\mathsf{MCP}$ (Def. 5.3)
$*$-minimizing Sard
---
(Def. 7.13)
$\mathsf{G}$ is meek (Def. 4.22)ideal (Def. 7.19) $\mathsf{G}$ satisfies the
---
regularity for stability
of $\mathsf{CD}$ (Def. 5.12)
no non-trivial Goh
---
geodesics
$\mathsf{G}$ is locally bounded
---
away from diagonal
Theorem 7.24Theorem 7.18Theorem 7.24real-analyticTheorem 7.16Theorem
7.20Remark 7.26 Figure 2: Sub-Riemannian properties (left), and of natural
gauge functions $\mathsf{G}$ (right). When $\mathsf{G}=\mathsf{d}$ all
properties are trivially verified with no assumptions.

###  7.4 Expression of $\mathsf{D}$ out of the cut locus

In this final section we show how $\mathsf{D}$ is related with more familiar
objects in sub-Riemannian geometry. It will also serve the purpose to better
motivate the expression $\sqrt{\mathsf{D}^{2}-\mathsf{d}^{2}}$ that appears in
the list of natural gauge functions in Section 7.2.

######  Proposition 7.28.

Let $(M,\mathsf{d})$ be a sub-Riemannian metric space, equipped with a
reference Riemannian metric $\mathsf{d}_{R}$. Let $\mathsf{D}:M\times
M\to[0,+\infty]$ be as in Definition 7.4. For all $x\in M$, let
$T_{x}M=\mathscr{D}_{x}\oplus\mathscr{V}_{x}$, where $\mathscr{D}_{x}$ is the
horizontal subspace at $x$ and $\mathscr{D}_{x}\perp\mathscr{V}_{x}$, and
denote with $\pi_{x}^{\mathscr{V}},\pi_{x}^{\mathscr{D}}$ the corresponding
projection. Let also $\nabla^{R}$ and $\|\cdot\|_{R}$ denote the gradient and
the norm of the Riemannian metric. Then it holds:

$\mathsf{D}(x,y)=\|\nabla_{x}^{R}c(\cdot,y)\|_{R},\qquad\forall\,(x,y)\notin\mathrm{Cut}(M).$
(7.37)

If, in addition, the reference $\mathsf{d}_{R}$ is an extension of
$\mathsf{d}$ (cf. Definition 7.1), then it holds

$\mathsf{D}(x,y)^{2}=\mathsf{d}(x,y)^{2}+\|\pi^{\mathscr{V}}_{x}\nabla_{x}^{R}c(\cdot,y)\|_{R}^{2},\qquad\forall\,(x,y)\notin\mathrm{Cut}(M).$
(7.38)

Equivalently, identifying $T_{x}M\simeq T_{x}^{*}M$ by means of the Riemannian
structure, it holds

$\|\pi_{x}^{\mathscr{V}}\lambda^{x,y}\|^{2}=\mathsf{D}(x,y)^{2}-\mathsf{d}(x,y)^{2},\qquad(x,y)\notin\mathrm{Cut}(M),$
(7.39)

where $\lambda^{x,y}\in T_{x}^{*}M$ is the initial covector of the unique
geodesic joining $x$ with $y$.

###### Proof.

By Theorem 7.12, on $M\setminus\mathrm{Cut}(y)$ the function
$c(\cdot,y)=\tfrac{1}{2}\mathsf{d}(\cdot,y)^{2}$ is smooth. Thus, out of
$\mathrm{Cut}(y)$, the asymptotic Lipschitz constant with respect to
$\mathsf{d}_{R}$ coincides with the Riemannian norm of the Riemannian
gradient, thus proving (7.37).

We then prove that, if $\mathsf{d}_{R}$ is a Riemannian extension induced by
the Riemannian metric $g_{R}$, and $g$ denotes the sub-Riemannian metric, for
all $x\in X$ and $v_{x}\in\mathscr{D}_{x}$ it holds
$\|v_{x}\|_{g_{R}}=\|v_{x}\|_{g}$. In fact, let $X_{1},\dots,X_{L}$ be a
generating family for the sub-Riemannian structure. There exist
$c_{i}\in\mathbb{R}$ such that $v_{x}=\sum_{i=1}^{N}c_{i}X_{i}(x)$, then let
$V:=\sum_{i=1}^{N}c_{i}X_{i}$. Let also $\gamma_{t}=e^{tV}(x)$ be the flow of
$V$, for $t\in[0,\varepsilon]$. This is by construction a horizontal curve
with $\dot{\gamma}_{t}=V(\gamma_{t})$. Furthermore, it holds (cf. [2, Ex.
3.51, 3.52]):

$L_{\mathsf{d}}(\gamma|_{[0,t]})=\int_{0}^{t}\|V(\gamma_{s})\|_{g}\,\mathrm{d}s,\qquad\Longrightarrow\qquad\|v_{x}\|_{g}=\left.\frac{d}{dt}\right|_{t=0}L_{\mathsf{d}}(\gamma|_{[0,t]}).$
(7.40)

Since $\mathsf{d}_{R}$ is an extension, $\gamma_{t}$ is rectifiable also for
$\mathsf{d}_{R}$, and it holds
$L_{\mathsf{d}}(\gamma|_{[0,t]})=L_{\mathsf{d}_{R}}(\gamma|_{[0,t]})$. We
deduce that $\|v_{x}\|_{g_{R}}=\|v_{x}\|_{g}$, which proves (7.38).

As a consequence, letting $k(x)=\dim\mathscr{D}_{x}$, we can find a
$g_{R}$-orthonormal basis $v_{1},\dots,v_{n}\in T_{x}M$ such that
$v_{1},\dots,v_{k(x)}$ is an orthonormal basis for $\mathscr{D}_{x}$ with
respect to the sub-Riemannian metric at the point $x$. Therefore, if
$(x,y)\notin\mathrm{Cut}(M)$, it holds

$\mathsf{D}(x,y)^{2}=\|\nabla^{R}_{x}c(\cdot,y)\|_{g_{R}}^{2}=\sum_{i=1}^{k(x)}\langle\mathrm{d}_{x}c(\cdot,y),v_{i}\rangle^{2}+\sum_{i=k(x)+1}^{n}\langle\mathrm{d}_{x}c(\cdot,y),v_{i}\rangle^{2},$
(7.41)

where $\langle\cdot,\cdot\rangle:T_{x}^{*}M\times T_{x}M\to\mathbb{R}$ denotes
the dual action of covectors on vectors. Notice that
$\mathrm{d}_{x}c(\cdot,y)=\lambda^{x,y}$ is the initial covector of the unique
geodesic $\gamma:[0,1]\to M$ joining $x$ with $y$: this follows by the
characterization of sub-Riemannian geodesics via the Lagrange multipliers
rule, cf. for example [3, Prop. 4.3, Lemma 2.20]. The first term in the r.h.s.
of (7.41) is $2H(\lambda^{x,y})=\mathsf{d}(x,y)^{2}$, concluding the proof of
(7.39). ∎

##  8 Sub-Riemannian comparison theory

We recall from [25] a general comparison theory for lower Ricci curvature
bounds in the sub-Riemannian setting. We remind first what comparison models
are in this setting.

###  8.1 LQ optimal control problems

Comparison models are a special class of dynamical systems, called _linear
quadratic optimal control problems_ (LQ in the following), and are a classical
topic in optimal control theory. They arise as variational problems in
$\mathbb{R}^{n}$ with a quadratic cost and linear dynamics. We recall their
general features, and we refer to [7, Ch. 16], [43, Ch. 1] and [55, Ch. 7] for
further details.

Let $A,B$ be $\ell\times\ell$ matrices, with $B\geq 0$ and symmetric. Letting
$k\leq\ell$ be the rank of $B$, there exist
$b_{1},\dots,b_{k}\in\mathbb{R}^{\ell}$, unique up to orthogonal
transformations, such that $B=\sum_{i=1}^{k}b_{i}b_{i}^{*}$. Let also $Q$ be a
symmetric $n\times n$ matrix, and $T>0$. We are interested in _admissible
trajectories_ , namely curves $q:[0,T]\to\mathbb{R}^{\ell}$ for which there
exists a _control_ $u\in L^{2}([0,T],\mathbb{R}^{k})$ such that

$\dot{q}=Aq+\sum_{i=1}^{k}u_{i}b_{i}.$ (8.1)

Thus, we look for admissible trajectories with fixed endpoints $q(0)=x$,
$q(T)=y$, that minimize the quadratic functional
$C_{T}:L^{2}([0,T],\mathbb{R}^{k})\to\mathbb{R}$

$C_{T}(u)=\frac{1}{2}\int_{0}^{T}\left(u^{*}u-q^{*}Qq\right)dt.$ (8.2)

Admissible trajectories minimizing (8.2) are called _minimizers_. The vector
$Aq$ represents the _drift_ , while $b_{1},\dots,b_{k}$ are the _controllable
directions_. The matrix $Q$ is the _potential_ of the LQ problem.

We only deal with _controllable_ systems, i.e. for which there exists $s>0$
such that

$\operatorname{\mathrm{rank}}(B,AB,\ldots,A^{s-1}B)=\ell.$ (8.3)

Condition (8.3) is known as _Kalman condition_ in control theory. It is
equivalent to the fact that, for any choice of $x,y\in\mathbb{R}^{\ell}$ and
$T>0$, there is a non-empty set of admissible trajectories
$q:[0,T]\to\mathbb{R}^{\ell}$ joining $x$ with $y$.

It is well-known that the admissible trajectories minimizing (8.2) are
projections $(p,q)\mapsto q$ of the solutions of the Hamilton equations

$\dot{p}=-\partial_{q}H,\qquad\dot{q}=\partial_{p}H,\qquad(p,q)\in
T^{*}\mathbb{R}^{\ell}=\mathbb{R}^{2\ell},$ (8.4)

where the Hamiltonian function $H:\mathbb{R}^{2\ell}\to\mathbb{R}$ is defined
by

$H(p,q)=\frac{1}{2}\left(p^{*}Bp+2p^{*}Aq+q^{*}Qq\right).$ (8.5)

We remark that the Hamiltonian flow is defined for all times, since $H$ is
quadratic.

######  Definition 8.1 (Conjugate time).

We say that $t_{*}>0$ is a _conjugate time_ if there exists a non-trivial
solution of the Hamilton equations (8.4) such that $q(0)=q(t_{*})=0$.

The first conjugate time $t_{c}=\inf\\{t_{*}>0\mid t_{*}\text{ is a conjugate
time}\\}\in(0,+\infty]$ determines existence and the uniqueness of solutions
of minimizers, as specified by the following proposition (see [7, Sec. 16.4]).

######  Proposition 8.2 (Conjugate times and existence of minimizers).

Let $t_{c}$ be the first conjugate time of the LQ problem (8.1)-(8.2). Then,
for any $x,y\in\mathbb{R}^{\ell}$,

  * •

if $T<t_{c}$ there exists a unique minimizer between $x$ with $y$;

  * •

if $T>t_{c}$ there exist no minimizers between $x$ with $y$;

  * •

if $T=t_{c}$ existence of minimizers depends on $x$, $y$.

The minimization of the functional (8.2) with fixed endpoints and $T>0$ does
not define a metric on $\mathbb{R}^{\ell}$, in general. Nevertheless, one can
still define a distortion coefficient.

######  Definition 8.3 (LQ distortion coefficient).

Consider the LQ problem (8.1)-(8.2), with $T=1$. For $x,y\in\mathbb{R}^{\ell}$
and $t\in[0,1]$, define

$Z_{t}(x,y)=\\{q(t)\mid q:[0,1]\to\mathbb{R}^{\ell}\text{ is a minimizer s.t.
}q(0)=x,\,q(1)=y\\}.$ (8.6)

The _distortion coefficient_ of the LQ problem is

$\beta_{t}^{A,B,Q}(x,y)=\limsup_{r\to
0}\frac{|Z_{t}(x,B_{r}(y))|}{|B_{r}(y)|},\qquad t\in[0,1],$ (8.7)

where $x,y\in\mathbb{R}^{\ell}$, $B_{r}(y)$ denotes the Euclidean ball with
center $y$ and radius $r>0$, and $|\cdot|$ denotes the Lebesgue measure of
$\mathbb{R}^{\ell}$.

The next proposition, proven in [25, Prop. 27], links the distortion
coefficient with the Hamilton equations of the LQ problem.

######  Proposition 8.4.

Consider the LQ problem (8.1)-(8.2), with $T=1$, and assume that $t_{c}>1$.
Its distortion coefficient does not depend on the choice of $x,y$, and
satisfies

$\beta_{t}^{A,B,Q}=\frac{\det N(t)}{\det N(1)}>0,\qquad\forall\,t\in[0,1],$
(8.8)

where $M(t),N(t):[0,+\infty)\to\mathrm{Mat}(\ell\times\ell)$ are the solutions
of the Hamiltonian system

$\frac{d}{dt}\begin{pmatrix}M\\\ N\end{pmatrix}=\begin{pmatrix}-A^{*}&-Q\\\
B&A\end{pmatrix}\begin{pmatrix}M\\\
N\end{pmatrix},\qquad\begin{pmatrix}M(0)\\\
N(0)\end{pmatrix}=\begin{pmatrix}\mathbb{1}\\\ \mathbb{0}\end{pmatrix}.$ (8.9)

Equivalently, we have

$\beta_{t}^{A,B,Q}=\exp\left(-\int_{t}^{1}\operatorname{\mathrm{tr}}(BV(s)+A)\mathrm{d}s\right)>0,\qquad\forall\,t\in(0,1],$
(8.10)

where $V:(0,t_{c})\to\mathrm{Sym}(\ell\times\ell)$ is the maximal solution of
the matrix Riccati equation

$\dot{V}+A^{*}V+VA+VBV+Q=\mathbb{0},\qquad\lim_{t\to
0^{+}}V^{-1}(t)=\mathbb{0}.$ (8.11)

Notice that, by the Kalman condition (8.3), the Cauchy problem with limit
initial datum (8.11) is well-posed, its solution is well-defined on the
maximal interval $(0,t_{c})$, where $t_{c}$ is the first conjugate time of the
corresponding LQ problem. In particular, the standing assumption $t_{c}>1$
makes (8.10) well-defined (see [22, Appendix A]).

###  8.2 Constant curvature models

A LQ problem depends on data $A,B,Q$. In this section we explain how we choose
these, yielding a class of constant curvature models. The matrices $A$ and $B$
are determined by a Young diagram. The general case can be reduced to diagrams
with one row, which can be seen as basic building blocks. Consider thus the
following Young diagram:

$\ytableausetup{centertableaux}Y=\ytableau 1&2\none[\cdots]\ell\,,$ (8.12)

of length $\ell\in\mathbb{N}$. The case $\ell=1$ must be thought as the
“Riemannian case”. We associate the matrices $A,B$ as follows:

$A=\Gamma_{1}^{*}(Y),\qquad B=\Gamma_{2}(Y),$ (8.13)

(the transpose is a convention to agree with the one of [22]), where

$\Gamma_{1}(Y):=\begin{pmatrix}\mathbb{0}&\mathbb{1}_{\ell-1}\\\
0&\mathbb{0}\end{pmatrix},\qquad\Gamma_{2}(Y):=\begin{pmatrix}1&\mathbb{0}\\\
\mathbb{0}&\mathbb{0}_{\ell-1}\end{pmatrix}.$ (8.14)

The matrices $A=\Gamma_{1}^{*}(Y)$ and $B=\Gamma_{2}(Y)$ satisfy the Kalman
condition (8.3). The potential $Q$ is a diagonal matrix, whose entries
represent the Ricci curvature bounds:

$Q=\mathrm{diag}(\kappa_{1},\dots,\kappa_{\ell}),\qquad\kappa\in\mathbb{R}^{\ell}.$
(8.15)

##### Notation.

Following the above prescriptions, a basic LQ model is uniquely specified by
the choice of $\ell\in\mathbb{N}$, $\kappa\in\mathbb{R}^{\ell}$. We denote the
corresponding distortion coefficient by

$\beta_{t}^{\kappa}:=\beta_{t}^{A,B,Q},\qquad\text{$A$, $B$, $Q$ as in
\eqref{eq:AandB}-\eqref{eq:QQ}}.$ (8.16)

These LQ problems have constant curvature in the sense discussed in Appendix
B. More precisely, one can show that any Jacobi curve of the corresponding
Hamiltonian flow has Young diagram $Y$ of one row of length $\ell$, constant
canonical curvature $\mathfrak{R}_{\lambda_{t}}=Q$, and constant canonical
Ricci curvatures $\mathfrak{Ric}_{\lambda_{t}}^{i}=\kappa_{i}$, where
$i=1,\dots,\ell$. This can be proved by observing that the restrictions to any
extremal of $E_{i}=\partial_{p_{i}}$ and $F_{i}=\partial_{q_{i}}$, for
$i=1,\dots,\ell$ is a canonical frame along that extremal (see Section B.2.2).

######  Proposition 8.5 (Basic models).

Let $\ell\in\mathbb{N}$, and $\kappa\in\mathbb{R}^{\ell}$. Consider the LQ
problem (8.1)-(8.2) on $\mathbb{R}^{\ell}$, with $A$, $B$ as in (8.13) and $Q$
as in (8.15). Then there exist a computable real-analytic function
$\mathsf{s}_{\kappa}:[0,+\infty)\to\mathbb{R}$, and
$t_{\kappa}\in(0,+\infty]$, such that:

  1. (i)

$\mathsf{s}_{\kappa}$ is strictly positive on $(0,t_{\kappa})$, and vanishes
at the endpoints;

  2. (ii)

the first conjugate time of the LQ problem is $t_{\kappa}$;

  3. (iii)

there exists $c>0$ depending only on $\ell$ (and not on $\kappa$) and
$N=\ell^{2}$ such that

$\mathsf{s}_{\kappa}(t)\sim ct^{N},\qquad t\to 0;$

  4. (iv)

assuming that $t_{\kappa}>1$, then the distortion coefficient of the LQ
problem is

$\beta_{t}^{\kappa}=\frac{\mathsf{s}_{\kappa}(t)}{\mathsf{s}_{\kappa}(1)},\qquad\forall\,t\in[0,1];$
(8.17)

  5. (v)

for $\lambda\geq 0$, let
$\lambda\odot\kappa:=(\kappa_{1}\lambda^{2},\dots,\kappa_{\ell}\lambda^{2\ell})$.
Then it holds:

$t_{\lambda\odot\kappa}=\frac{t_{\kappa}}{\lambda},\qquad\text{and}\qquad\mathsf{s}_{\lambda\odot\kappa}(t)=\frac{\mathsf{s}_{\kappa}(t\lambda)}{\lambda^{\ell^{2}}},\qquad\forall\,t\in\mathbb{R},$
(8.18)

with the convention that, if $\lambda=0$, then $t_{0}=+\infty$ and
$\mathsf{s}_{0}(t)=t^{\ell^{2}}$ for all $t\in[0,1]$;

  6. (vi)

let $m\in\mathbb{N}$, and assume that
$\bar{\kappa}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{\ell}$ where each
$\bar{\kappa}_{i}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ is a homogeneous function
of degree $2i$, for $i=1,\dots,\ell$. Let
$\mathsf{s}_{\bar{\kappa}}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ be the function
defined by

$\mathsf{s}_{\bar{\kappa}}(\theta):=\mathsf{s}_{\bar{\kappa}(\theta/|\theta|)}(|\theta|),\qquad\qquad\forall\,\theta\in\mathbb{R}^{m}_{+}$
(8.19)

with the convention $\mathsf{s}_{\bar{\kappa}}(0)=0$.

The function $\mathsf{s}_{\bar{\kappa}}$ is radially real-analytic, satisfies
$\mathsf{s}_{\bar{\kappa}}(\theta)=c|\theta|^{N}+o(|\theta|^{N})$ for some
$c>0$ and $N=\ell^{2}$. Its positivity domain, that is the largest open star-
shaped subset ${\rm{DOM}}_{\bar{\kappa}}\subseteq\mathbb{R}^{m}_{+}$ such that
$\mathsf{s}_{\bar{\kappa}}>0$ on ${\rm{DOM}}_{\bar{\kappa}}\setminus\\{0\\}$,
is characterized as

${\rm{DOM}}_{\bar{\kappa}}=\\{\theta\in\mathbb{R}^{m}_{+}\mid
1<t_{\bar{\kappa}(\theta)}\\}=\\{\theta\in\mathbb{R}^{m}_{+}\mid|\theta|<t_{\bar{\kappa}(\theta/|\theta|)}\\},$
(8.20)

and for all $t\in[0,1]$ and $\theta\in{\rm{DOM}}_{\bar{\kappa}}$ it holds

$\beta_{t}^{\bar{\kappa}(\theta)}=\begin{cases}t^{N}&|\theta|=0,\\\
\frac{\mathsf{s}_{\bar{\kappa}}(t\theta)}{\mathsf{s}_{\bar{\kappa}}(\theta)}&|\theta|\neq
0\text{ and }\theta\in{\rm{DOM}}_{\bar{\kappa}},\end{cases}$ (8.21)

with $N=\ell^{2}$.

######  Remark 8.6 (Link with CD theory).

Proposition 8.5, and in particular item vi, is the bridge between the
curvature-dimension theory of Section 3 and the comparison theory of Section
8. It establishes that model distortion coefficients of LQ problems coincide
with model distortion coefficients in the sense of (3.14) (for the scalar-
valued case) and (6.4) (for the vector-valued case).

######  Remark 8.7 (Scalar case).

Item vi is stated in the general case $m\in\mathbb{N}$, in order to describe
model coefficients for vector-valued gauge functions. The scalar case
corresponds to $m=1$. In this case, we identify
$\bar{\kappa}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{\ell}$ with
$\kappa\in\mathbb{R}^{\ell}$ by the identity
$\bar{\kappa}_{i}(\theta)=\kappa_{i}\theta^{2i}$ for all $i=1,\dots,\ell$. Out
notation is consistent since it holds

$\mathsf{s}_{\bar{\kappa}}(t\theta)=\mathsf{s}_{\kappa}(t\theta)\qquad\text{and}\qquad{\rm{DOM}}_{\bar{\kappa}}=[0,t_{\kappa}).$
(8.22)

In particular, there is no need to introduce the $\mathsf{s}_{\bar{\kappa}}$
functions, and the positivity domain is just a segment. For general
$m\in\mathbb{N}$, however

$\mathsf{s}_{\kappa}:[0,+\infty)\to\mathbb{R}\qquad\text{and}\qquad\mathsf{s}_{\bar{\kappa}}:\mathbb{R}^{m}_{+}\to\mathbb{R}$
(8.23)

act on different domains. The link between them is provided by (8.19).

######  Remark 8.8 (Convexity).

If $\ell=m=1$ then
$(0,t_{\kappa})\ni\theta\mapsto\log\beta_{t}^{\kappa}(\sqrt{\theta})$ is
convex and monotone for fixed $t\in[0,1]$ (the case of Riemannian
coefficients, see Section 8.2.1). Convexity, in particular, is important for
the deduction of the standard form of the $\mathsf{CD}(K,N)$ inequality. Both
properties fail, in general, if $\ell>1$. Take for example $\kappa=(2,-1)$,
and compute the solution of the Hamiltonian system (8.9) (the values are
chosen so that the system has a simple Jordan form, with eigenvalues $\pm i$).

###### Proof.

Using the notation of Proposition 8.4, we set

$\mathsf{s}_{\kappa}(t):=\det N(t).$ (8.24)

By definition, the the first conjugate time of the LQ problem is

$t_{\kappa}:=\inf\\{t>0\mid\mathsf{s}_{\kappa}(t)=0\\}.$ (8.25)

Kalman condition implies that $\det N(t)\neq 0$ for small $t>0$ so that
$t_{\kappa}>0$. Furthermore one can check, by studying the asymptotic
behaviour of the solutions of (8.9), that $\mathsf{s}_{\kappa}(t)>0$ for small
$t$. Thus i, ii and iv follow immediately.

To prove iii, we first observe that $\mathsf{s}_{\kappa}(\cdot)$ is real-
analytic, and thus there exist $N\in\mathbb{N}$ and $c\neq 0$ such that
$\mathsf{s}_{\kappa}(t)\sim ct^{N}$ as $t\to 0$. To prove that $N=\ell^{2}$,
we claim that the following asymptotic formula holds:

$\lim_{t\to 0}\left(\frac{\mathrm{d}}{\mathrm{d}t}\log\det
N(t)-\frac{\ell^{2}}{t}\right)=0.$ (8.26)

To prove it, observe that the first term on the left hand side is

$\frac{\mathrm{d}}{\mathrm{d}t}\log\det
N(t)=\operatorname{\mathrm{tr}}(\dot{N}(t)N(t)^{-1})=\operatorname{\mathrm{tr}}(\Gamma_{2}M(t)N(t)^{-1}+\Gamma_{1}^{*}).$
(8.27)

The asymptotics of $M(t)N(t)^{-1}$ as $t\to 0$ is obtained in [3, Thm. 7.4 and
Cor. 7.5], yielding (8.26). We do not repeat the details here. We remark that
(8.26) can also be proved by deducing asymptotic formulas for the solutions
$M(t),N(t)$ of the Hamiltonian system in Proposition 8.4, using the properties
of $A,B$ in (8.13). Such a proof also shows that the constant $c$ depends only
on $\ell$ and not on $Q$ (i.e. not on $\kappa\in\mathbb{R}^{\ell}$), and
$c>0$.

To prove v, for all $\lambda>0$ let

$Q=\mathrm{diag}(\kappa_{1},\dots,\kappa_{\ell}),\qquad
Q_{\lambda}=\mathrm{diag}(\kappa_{1}\lambda^{2},\dots,\kappa_{\ell}\lambda^{2\ell}),$
(8.28)

be the potentials corresponding to $\kappa$ and $\lambda\odot\kappa$,
respectively. We remark that

$Q_{\lambda}=\Omega_{\lambda}Q\Omega_{\lambda},\qquad\Omega_{\lambda}:=\mathrm{diag}(\lambda,\dots,\lambda^{\ell}).$
(8.29)

For the specific choice $A=\Gamma_{1}^{*}(Y)$ and $B=\Gamma_{2}(Y)$ of the
statement, see (8.13), we denote by $M(t),N(t)$ (resp.
$M_{\lambda}(t),N_{\lambda}(t)$) the solution of the Hamiltonian system (8.9)
for potential $Q$ (resp. $Q_{\lambda}$). We observe from the explicit form
(8.14) that

$\Omega_{\lambda}A=\lambda A\Omega_{\lambda},\qquad
B\Omega_{\lambda}=\Omega_{\lambda}B=\lambda B.$ (8.30)

Using these properties and (8.9) we immediately see that

$M_{\lambda}(t)=\Omega_{\lambda}M(t\lambda)\Omega_{\lambda}^{-1},\qquad
N_{\lambda}(t)=\lambda\Omega_{\lambda}^{-1}N(t\lambda)\Omega_{\lambda}^{-1},\qquad\forall\,t\in\mathbb{R}.$
(8.31)

It follows that, for conjugate times
$t_{\lambda\odot\kappa}=t_{\kappa}/\lambda$, and furthermore

$\mathsf{s}_{\lambda\odot\kappa}(t)=\det
N_{\lambda}(t)=\lambda^{-\ell^{2}}\mathsf{s}_{\kappa}(t\lambda),\qquad\forall\,\lambda>0.$
(8.32)

The above rescaling formula yields v in the case $\lambda>0$. The case
$\lambda=0$ can be seen as a limit of the previous one, or also proved
directly by solving (8.9); in this case $Q=0$ and thus
$M(t)_{ij}=(-1)^{j-i}t^{j-i}/(j-i)!$ for $j\geq i$, and zero otherwise, while
$N(t)_{ij}=(-1)^{j+1}t^{i+j-1}/(i+j-1)!$ for all $i,j=1,\dots,\ell$.

The proof of vi is a straightforward consequence of all previous items,
unraveling the notation. The fact that $\mathsf{s}_{\bar{\kappa}}$ is radially
real-analytic follows from the fact that $\mathsf{s}_{\kappa}$ is real-
analytic for $\kappa\in\mathbb{R}^{\ell}$. Notice that if the
$\bar{\kappa}_{i}$ are homogeneous of degree $2i$, then for $\theta\neq 0$ we
have $\bar{\kappa}(\theta)=|\theta|\odot\bar{\kappa}(\theta/|\theta|)$.
Therefore it holds

$\mathsf{s}_{\bar{\kappa}}(\theta)=\mathsf{s}_{\bar{\kappa}(\theta/|\theta|)}(|\theta|)\sim
c|\theta|^{\ell^{2}},\qquad|\theta|\to 0,$ (8.33)

where we used iii, and the fact that the constant $c>0$ in iii does not depend
on $\kappa\in\mathbb{R}^{\ell}$. The characterization of the positivity domain
follows from the homogeneity property v. Finally, by definition, and using v,
it holds

$\beta_{t}^{\bar{\kappa}(\theta)}=\frac{\mathsf{s}_{\bar{\kappa}(\theta)}(t)}{\mathsf{s}_{\bar{\kappa}(\theta)}(1)}=\frac{\mathsf{s}_{|\theta|\odot\bar{\kappa}(\theta/|\theta|)}(t)}{\mathsf{s}_{|\theta|\odot\bar{\kappa}(\theta/|\theta|)}(1)}=\frac{\mathsf{s}_{\bar{\kappa}(\theta/|\theta|)}(t|\theta|)}{\mathsf{s}_{\bar{\kappa}(\theta/|\theta|)}(|\theta|)}=\frac{\mathsf{s}_{\bar{\kappa}}(t\theta)}{\mathsf{s}_{\bar{\kappa}}(\theta)},$
(8.34)

for all $\theta\in{\rm{DOM}}_{\bar{\kappa}}\setminus\\{0\\}$ and all
$t\in(0,1)$. Finally if $|\theta|=0$, we have $\bar{\kappa}(0)=0$ and we
proceed in a similar fashion reminding that $\mathsf{s}_{0}(t)=t^{\ell^{2}}$
by item v. The cases $t=0$ and $t=1$ are trivially verified. ∎

The proof of Proposition 8.5 is constructive and the distortion coefficients
are obtained by solving the Hamiltonian system (8.9) associated with
corresponding LQ problem. In the next three sections we give explicit
examples, recovering all typical comparison functions that have arisen in the
recent literature in sub-Riemannian geometry.

####  8.2.1 The Riemannian case

Let $\ell=1$, so that $Y=\ytableausetup{smalltableaux}\ytableaushort{}$.
According to (8.13), we have $A=0$, $B=1$, and we set $Q=\kappa$ for some
$\kappa\in\mathbb{R}$. The Hamiltonian of the LQ problem is

$H(p,q)=\frac{1}{2}\left(p^{2}+\kappa q^{2}\right),$ (8.35)

which is the Hamiltonian of a one-dimensional harmonic oscillator. From
Proposition 8.5 we obtain easily that

$\mathsf{s}_{\kappa}(t)=\begin{cases}\sin(\sqrt{\kappa}t)&\kappa>0,\\\
\sinh(\sqrt{-\kappa}t)&\kappa\leq 0,\end{cases}\qquad
t_{\kappa}=\begin{cases}\frac{\pi}{\sqrt{\kappa}}&\kappa>0,\\\
+\infty&\kappa\leq 0.\end{cases}$ (8.36)

The typical right hand side of Ricci curvature lower bounds has the form
$\bar{\kappa}(\mathsf{d})=\kappa\mathsf{d}^{2}$. Thus, according to
Proposition 8.5, we obtain that for $\theta\in[0,+\infty)$

$\mathsf{s}_{\bar{\kappa}}(\theta)=\mathsf{s}_{\kappa}(\theta),\qquad{\rm{DOM}}_{\bar{\kappa}}=[0,t_{\kappa}).$
(8.37)

Here we stress that $m=1$. The corresponding distortion coefficient is:

$\beta_{t}^{\bar{\kappa}(\theta)}=\beta_{t}^{\kappa\theta^{2}}=\frac{\sin(\sqrt{\kappa}\theta
t)}{\sin(\sqrt{\kappa}\theta)},\qquad\forall\,t\in[0,1],\quad\theta\in{\rm{DOM}}_{\bar{\kappa}}=[0,t_{\kappa}),$
(8.38)

with the standard conventions if $\kappa\leq 0$. The condition
$\theta\in{\rm{DOM}}_{\bar{\kappa}}$ corresponds to
$\bar{\kappa}(\theta)=\kappa\theta^{2}<\pi^{2}$, which is reminiscent of the
Bonnet-Myers bound.

##### Relation with classical coefficients.

This is the building block for the construction of Riemannian distortion
coefficients. Consider a space form of dimension $N$ and (sectional) curvature
equal to $K/(N-1)$, and thus Ricci curvature equal to $K$. The potential
“felt” in the directions orthogonal to the one of the motion, along a geodesic
of length $\mathsf{d}$, is $K\mathsf{d}^{2}/(N-1)$. There are $N-1$ orthogonal
directions, and one direction where there is no curvature (the direction of
the motion). Therefore, taking the product of all these factors we obtain

$\beta_{t}^{0}\prod_{i=1}^{N-1}\beta_{t}^{K\mathsf{d}^{2}/(N-1)}=t\left(\frac{\sin(t\mathsf{d}\sqrt{K/(N-1)})}{\sin(\mathsf{d}\sqrt{K/(N-1)})}\right)^{N-1},\qquad\forall\,t\in[0,1],\quad\mathsf{d}<t_{K/(N-1)}.$
(8.39)

This is the usual distortion coefficient for a space form of Ricci curvature
$K$ and dimension $N$. Compare with Section 3.3.1.

####  8.2.2 The Sasakian case

In this section we describe the typical coefficients arising in the
description of three-dimensional Sasakian space forms, which were the first
sub-Riemannian structures where an effective comparison theory was established
[51, 5, 61].

Let $\ell=2$, so that $Y=\ytableausetup{smalltableaux}\ytableaushort{}$. We
have $A=\left(\begin{smallmatrix}0&0\\\ 1&0\end{smallmatrix}\right)$,
$B=\left(\begin{smallmatrix}1&0\\\ 0&0\end{smallmatrix}\right)$, and we set
$Q=\mathrm{diag}(\kappa_{1},0)$ for some $\kappa_{1}\in\mathbb{R}$ (formally,
$\kappa=(\kappa_{1},\kappa_{2})$ with $\kappa_{2}=0$, which greatly simplifies
this case). Proposition 8.4 can be used to obtain

$\mathsf{\mathsf{s}}_{\kappa_{1},0}(t)=\frac{2-2\cos\left(\sqrt{\kappa_{1}}t\right)-\sqrt{\kappa_{1}}t\sin\left(\sqrt{\kappa_{1}}t\right)}{\kappa_{1}^{2}},\qquad
t_{\kappa_{1},0}=\begin{cases}\tfrac{2\pi}{\sqrt{\kappa_{1}}}&\kappa_{1}>0,\\\
+\infty&\kappa_{1}\leq 0,\end{cases}$ (8.40)

with usual interpretation when $\kappa_{1}\leq 0$. Furthermore we obtain

$\beta_{t}^{\kappa_{1},0}=\frac{2-2\cos\left(\sqrt{\kappa_{1}}t\right)-\sqrt{\kappa_{1}}t\sin\left(\sqrt{\kappa_{1}}t\right)}{2-2\cos\left(\sqrt{\kappa_{1}}\right)-\sqrt{\kappa_{1}}\sin\left(\sqrt{\kappa_{1}}\right)}.$
(8.41)

On a three-dimensional Sasakian manifold with Tanaka-Webster curvature equal
to $K\in\mathbb{R}$, the sharp distortion coefficient along a geodesic
$\gamma$ is the product of two factors. One flat factor $t$, arising from a
one-dimensional subspace (the direction of the motion), and one factor
$\beta_{t}^{\kappa_{1},0}$ as the one we just described, with

$\kappa_{1}=\bar{\kappa}_{1}(|h_{0}|,\mathsf{d}):=|h_{0}|^{2}+K\mathsf{d}^{2},$
(8.42)

where $\mathsf{d}$ is the length of the geodesic and $h_{0}$ is the Reeb
component of its initial covector. In particular
$\bar{\kappa}_{1}:\mathbb{R}^{2}_{+}\to\mathbb{R}$ is homogeneous of degree
$2$ and the distortion coefficient thus obtained is suitable for a theory with
vector-valued gauge function (the gauge function being
$\mathsf{G}=(\mathsf{G}_{1},\mathsf{G}_{2})$ with $\mathsf{G}_{1}=|h_{0}|$ and
$\mathsf{G}_{2}=\mathsf{d}$).

We will describe more precisely this case in Section 10.4.

####  8.2.3 The two columns case

This example generalizes the previous one. Let again $\ell=2$ so that
$Y=\ytableausetup{smalltableaux}\ytableaushort{}$, and let
$Q=\mathrm{diag}(\kappa_{1},\kappa_{2})$, with
$\kappa_{1},\kappa_{2}\in\mathbb{R}$. The difference with respect to Section
8.2.2 is that now $\kappa_{2}$ may be non-zero. The Hamiltonian of the
corresponding LQ problem is

$H(p,q)=\frac{1}{2}\left(p_{1}^{2}+2p_{2}q_{1}+\kappa_{1}q_{1}^{2}+\kappa_{2}q_{2}^{2}\right).$
(8.43)

We can compute the distortion coefficient using Proposition 8.4. By reduction
to Jordan normal form of the Hamiltonian system (8.9) (see details in [78,
Prop. 28]), one obtains

$\mathsf{s}_{\kappa_{1},\kappa_{2}}(t)=\frac{\xi_{-}^{2}\sin^{2}\left(\xi_{+}t\right)-\xi_{+}^{2}\sin^{2}\left(\xi_{-}t\right)}{4\xi_{-}^{2}\xi_{+}^{2}(\xi_{-}^{2}-\xi_{+}^{2})},$
(8.44)

where we have set $\kappa_{1}=2(\xi_{+}^{2}+\xi_{-}^{2})$ and
$\kappa_{2}=-(\xi_{+}^{2}-\xi_{2}^{2})^{2}$. In other words, choosing the
principal branch of the square root:

$\xi_{\pm}=\frac{1}{2}(\sqrt{x+y}\pm\sqrt{x-y}),\qquad\text{with}\qquad
x=\frac{\kappa_{1}}{2},\quad y=\frac{\sqrt{4\kappa_{2}+\kappa_{1}^{2}}}{2}.$
(8.45)

It seems not trivial to deduce an explicit expression for
$t_{\kappa_{1},\kappa_{2}}$ from the above expression. Abstract necessary and
sufficient conditions in terms of $\kappa_{1},\kappa_{2}$ for the finiteness
of $t_{\kappa_{1},\kappa_{2}}$ are obtained via the results in [6]. Estimates
for $t_{\kappa_{1},\kappa_{2}}$ are found in [78, Prop. 1].

Thus, assuming $t_{\kappa_{1},\kappa_{2}}>1$, the distortion coefficient is

$\beta_{t}^{\kappa_{1},\kappa_{2}}=\frac{\xi_{-}^{2}\sin^{2}\left(\xi_{+}t\right)-\xi_{+}^{2}\sin^{2}\left(\xi_{-}t\right)}{\xi_{-}^{2}\sin^{2}\left(\xi_{+}\right)-\xi_{+}^{2}\sin^{2}\left(\xi_{-}\right)},\qquad\forall\,t\in[0,1],$
(8.46)

understood with the usual conventions as a real-analytic function of
$\xi_{\pm}\in\mathbb{C}$.

###  8.3 Main comparison result

Recall from Definition 4.6 the distortion coefficient of a metric measure
space. For a sub-Riemannian m.m.s $(M,\mathsf{d},\mathfrak{m})$, it can always
be written as

$\beta_{t}^{(M,\mathsf{d},\mathfrak{m})}(x,y)=t\beta_{t}^{\perp}(x,y),\qquad\forall\,(x,y)\notin\mathrm{Cut}(M).$
(8.47)

The factor $t$ corresponds to the distortion felt in the direction of the
geodesic between $x$ and $y$, while the factor $\beta_{t}^{\perp}$ represents
the distortion felt in the perpendicular directions.

The following result is a reformulation of [25, Thm. 16], with $N=n$ (cf.
Rmks. 17 and 12 there), with an explicit dependence on a possibly vector-
valued gauge function $\mathsf{G}:M\times M\to\mathbb{RP}^{m}_{+}$, combined
with Proposition 8.5vi. We will state directly the general vector-valued case,
the scalar-valued one is obtained by setting $m=1$, in which case the
statement greatly simplifies, see in particular Remark 8.7.

######  Theorem 8.9.

Let $(M,\mathsf{d},\mathfrak{m})$ be a sub-Riemannian metric measure space,
with $n=\dim M$, and let $\mathsf{G}:M\times M\to\mathbb{R}^{m}_{+}$ be a
finite Borel function, $m\in\mathbb{N}$. Let $(x,y)\notin\mathrm{Cut}(M)$ and
$\gamma:[0,1]\to M$ be a geodesic joining them, with initial covector
$\lambda$, and reduced Young diagram $Y$. Assume that there exists a function
$\bar{C}:\mathbb{R}^{m}_{+}\to\mathbb{R}$, homogeneous of degree $1$, such
that

$\rho_{\mathfrak{m},\lambda}\leq\bar{C}(\mathsf{G}(x,y)).$ (8.48)

Denote with $\Upsilon$ the set of levels of $Y$. Assume that for any level
$\alpha\in\Upsilon$, with size $r_{\alpha}$ and length $\ell_{\alpha}$, there
exist $\bar{\kappa}_{\alpha}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{\ell_{\alpha}}$
such that each component
$\bar{\kappa}_{\alpha_{i}}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ is homogeneous of
degree $2i$, for $i=1,\dots,\ell_{\alpha}$, and such that for any superbox
$\alpha_{i}$ of that level it holds

$\frac{1}{r_{\alpha}}\mathfrak{Ric}^{\alpha_{i}}_{\lambda}\geq\bar{\kappa}_{\alpha_{i}}(\mathsf{G}(x,y)),\qquad
i=1,\dots,\ell_{\alpha},$ (8.49)

with the convention that if $\alpha$ is the level of length $\ell_{\alpha}=1$
then $r_{\alpha}$ is replaced by $r_{\alpha}-1$; and if $r_{\alpha}=0$ after
this replacement, then that level is omitted. Then:

  * •

denoting with ${\rm{DOM}}_{\bar{\kappa}_{\alpha}}\subseteq\mathbb{R}^{m}_{+}$
the positivity domain of
$\mathsf{s}_{\bar{\kappa}_{\alpha}}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ as in
Proposition 8.5, it holds

$\mathsf{G}(x,y)\in\bigcap_{\alpha\in\Upsilon}{\rm{DOM}}_{\bar{\kappa}_{\alpha}},$
(8.50)

so that the distortion coefficients
$\beta_{t}^{\bar{\kappa}_{\alpha}(\mathsf{G}(x,y))}$ are well-defined for all
$\alpha\in\Upsilon$ as in Proposition 8.5;

  * •

with the same convention as above, we have that

$\frac{\beta_{t}^{\perp}(x,y)}{\displaystyle\prod_{\alpha\in\Upsilon}\left(\beta_{t}^{\bar{\kappa}_{\alpha}(\mathsf{G}(x,y))}\right)^{r_{\alpha}}}e^{-t\bar{C}(\mathsf{G}(x,y))}\quad\text{is
a non-increasing function of $t\in(0,1]$},$ (8.51)

and in particular, the following holds

$\beta_{t}^{(M,\mathsf{d},\mathfrak{m})}(x,y)\geq
te^{(t-1)\bar{C}(\mathsf{G}(x,y))}\prod_{\alpha\in\Upsilon}\left(\beta_{t}^{\bar{\kappa}_{\alpha}(\mathsf{G}(x,y))}\right)^{r_{\alpha}},\qquad\forall\,t\in[0,1].$
(8.52)

######  Remark 8.10.

We recall from Proposition 8.5 that, for
$\bar{\kappa}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{\ell}$, and $t\in[0,1]$

$\beta_{t}^{\bar{\kappa}(\mathsf{G})}=\begin{cases}t^{N}&|\mathsf{G}|=0,\\\
\frac{\mathsf{s}_{\bar{\kappa}}(t\mathsf{G})}{\mathsf{s}_{\bar{\kappa}}(\mathsf{G})}&|\mathsf{G}|\neq
0\text{ and }\mathsf{G}\in{\rm{DOM}}_{\bar{\kappa}},\end{cases}$ (8.53)

with $N=\ell^{2}$, is the distortion coefficient of the LQ model with a Young
diagram of one row and $\ell$ columns, with potential
$Q=\mathrm{diag}(\bar{\kappa}_{1}(\mathsf{G}),\dots,\bar{\kappa}_{\ell}(\mathsf{G}))$.
In particular the functions
$\mathsf{s}_{\bar{\kappa}}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ depend only on the
$\ell$ homogeneous functions $\bar{\kappa}_{1},\dots,\bar{\kappa}_{\ell}$.

Theorem 8.9 yields a comparison result for distortion along a single geodesic.
If its assumptions hold globally, then we obtain the following result,
establishing the compatibility of our $\mathsf{CD}(\beta,n)$ theory with the
one of sub-Riemannian Ricci bounds. Once again, in the scalar-valued case
$m=1$ what follows greatly simplifies, see Remark 8.7.

######  Definition 8.11 (Sub-Riemannian m.m.s. with Ricci curvature bounded
from below).

Let $(M,\mathsf{d},\mathfrak{m})$ be a sub-Riemannian metric measure space,
with $n=\dim M$, equipped with a finite gauge function $\mathsf{G}:M\times
M\to\mathbb{R}^{m}_{+}$, $m\in\mathbb{N}$. We say that
$(M,\mathsf{d},\mathfrak{m},\mathsf{G})$ has _Ricci curvatures bounded from
below_ if there exist:

  * •

a Young diagram $Y$,

  * •

functions $\bar{C}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ homogeneous of degree $1$,

  * •

functions
$\bar{\kappa}_{\alpha}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{\ell_{\alpha}}$ such
that each component
$\bar{\kappa}_{\alpha_{i}}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ is homogeneous of
degree $2i$, for $i=1,\dots,\ell_{\alpha}$ for any level $\alpha$ of $Y$, with
length $\ell_{\alpha}$,

such that for all $x\in M$ and $\mathfrak{m}$-a.e. $y$ there exists a unique
geodesic $\gamma\in\mathrm{Geo}(M)$ joining $x$ with $y$, with Young diagram
$Y$, initial covector $\lambda\in T_{x}^{*}M$, for which the following hold:

  * •

bound on the geodesic volume derivative:

$\rho_{\mathfrak{m},\lambda}\leq\bar{C}(\mathsf{G}(x,y)),$ (8.54)

  * •

bound on the Ricci curvatures: for all levels $\alpha$ of the Young diagram
$Y$

$\frac{1}{r_{\alpha}}\mathfrak{Ric}^{\alpha_{i}}_{\lambda}\geq\bar{\kappa}_{\alpha_{i}}(\mathsf{G}(x,y)),\qquad
i=1,\dots,\ell_{\alpha},$ (8.55)

with the convention that if $\alpha$ is the level of length $\ell_{\alpha}=1$
then $r_{\alpha}$ is replaced by $r_{\alpha}-1$; and if $r_{\alpha}=0$ after
this replacement, then that level is omitted.

The above data determine a function
$\mathsf{s}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ defined by

$\mathsf{s}(\theta):=|\theta|\cdot
e^{\bar{C}(\theta)}\cdot\prod_{\alpha\in\Upsilon}\mathsf{s}_{\bar{\kappa}_{\alpha}}(\theta)^{r_{\alpha}},\qquad\forall\,\theta\in\mathbb{R}^{m}_{+},$
(8.56)

where $\mathsf{s}_{\bar{\kappa}_{\alpha}}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ are
as in Proposition 8.5vi, $\Upsilon$ denotes the sets of levels of $Y$, and
with the convention that if $\alpha$ is the level of length $\ell_{\alpha}=1$
then $r_{\alpha}$ is replaced by $r_{\alpha}-1$; and if $r_{\alpha}=0$ after
this replacement, then that level is omitted from the product in (8.56).
Finally, let $\beta$ be the corresponding distortion coefficient defined as in
(6.4).

We note that the positivity domain ${\rm{DOM}}$ of $\mathsf{s}$ (see Section
6) is

${\rm{DOM}}=\bigcap_{\alpha\in\Upsilon}{\rm{DOM}}_{\bar{\kappa}_{\alpha}},$
(8.57)

and that $\mathsf{s}$ satisfies the asymptotic relation
$\mathsf{s}(\theta)=c|\theta|^{N}+o(|\theta|^{N})$ for some $c>0$ and

$N=\sum_{\alpha\in\Upsilon}r_{\alpha}\ell_{\alpha}^{2}.$ (8.58)

######  Theorem 8.12 (Ideal sub-Riemannian structures with Ricci bounded below
are $\mathsf{CD}$).

Let $(M,\mathsf{d},\mathfrak{m})$ be an ideal sub-Riemannian metric measure
space, with $n=\dim M$, equipped with a finite gauge function
$\mathsf{G}:M\times M\to\mathbb{R}^{m}_{+}$, $m\in\mathbb{N}$, with Ricci
curvatures bounded from below in the sense of Definition 8.11, and let $\beta$
be the corresponding distortion coefficient. Then
$(M,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies the $\mathsf{CD}(\beta,n)$
condition.

###### Proof of Theorem 9.23.

Step 1: comparison out of the cut locus. We use the Ricci curvature bounds
assumption of Definition 8.11, paired with Theorem 8.9, to deduce that for all
$x\in M$ and $\mathfrak{m}$-a.e. $y\in M$

$\mathsf{G}(x,y)\in{\rm{DOM}}=\bigcap_{\alpha\in\Upsilon}{\rm{DOM}}_{\bar{\kappa}_{\alpha}},$
(8.59)

where ${\rm{DOM}}$ is the positivity domain of
$\mathsf{s}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ in (8.56), and furthermore

$\beta_{t}^{(M,\mathsf{d},\mathfrak{m})}(x,y)\geq\beta_{t}^{\mathrm{mod}}(\mathsf{G}(x,y)),\qquad\forall\,t\in[0,1],$
(8.60)

where $\beta^{\mathrm{mod}}$ in the right hand side is the distortion
coefficient built out of $\mathsf{s}$ according to the construction in Section
3 (Section 6 for the vector case). See Remark 8.10.

Step 2: interpolation inequalities for densities. By [24, Thm. 4], for any
$\mu_{0}\in\mathcal{P}_{bs}(M,\mathsf{d},\mathfrak{m})$,
$\mu_{1}\in\mathcal{P}_{bs}^{*}(M,\mathsf{d},\mathfrak{m})$ there exists a
unique $W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$, associated with $\nu\in{\rm
OptGeo}(\mu_{0},\mu_{1})$ such that $\mu_{t}\ll\mathfrak{m}$ for all
$t\in(0,1]$, and letting
$\rho_{t}:=\tfrac{\mathrm{d}\mu_{t}}{\mathrm{d}\mathfrak{m}}$ be the
corresponding density, it holds for all $t\in(0,1)$:

$\frac{1}{\rho_{t}(\gamma_{t})^{1/n}}\geq\frac{\beta_{1-t}^{(M,\mathsf{d},\mathfrak{m})}(\gamma_{1},\gamma_{0})^{1/n}}{\rho_{0}(\gamma_{0})^{1/n}}+\frac{\beta_{t}^{(M,\mathsf{d},\mathfrak{m})}(\gamma_{0},\gamma_{1})^{1/n}}{\rho_{1}(\gamma_{1})^{1/n}},\quad\nu\text{-a.e.}\
\gamma\in\mathrm{Geo}(M),$ (8.61)

with the understanding that the first term in the right hand side of (4.11) is
omitted if $\mu_{0}\notin\mathcal{P}_{ac}(M,\mathfrak{m})$. We also remark
that $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$, the corresponding optimal plan
$\pi$, and the corresponding $W_{2}$-geodesic $\mu_{t}$ are unique, and they
are induced by an optimal transport map $T:M\to M$ such that
$T_{\sharp}\mu_{1}=\mu_{0}$.

Furthermore, by [24, Cor. 3.8], optimal transport on ideal structures almost
surely is either static or avoids the cut locus. More precisely:

$\mu_{1}\left(M\setminus\\{x\in M\mid T(x)=x\\}\cup\\{x\in M\mid
T(x)\notin\mathrm{Cut}(x)\\}\right)=0.$ (8.62)

Recall that the $\mathsf{CD}(\beta,n)$ inequality (3.15) must be tested only
for measures such that
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$.
In this case, $\nu$ is concentrated on a set of geodesics with endpoints out
of the cut locus.

Thus, using (8.60), we have that (8.61) holds for $\nu$-a.e. geodesic
$\gamma\in\mathrm{Geo}(M)$, with the replacements:

$\beta_{1-t}^{(M,\mathsf{d},\mathfrak{m})}(\gamma_{1},\gamma_{0})\mapsto\beta_{1-t}^{\mathrm{mod}}(\mathsf{G}(\gamma_{1},\gamma_{0})),\qquad\beta_{t}^{(M,\mathsf{d},\mathfrak{m})}(\gamma_{0},\gamma_{1})\mapsto\beta_{t}^{\mathrm{mod}}(\mathsf{G}(\gamma_{0},\gamma_{1})).$
(8.63)

Integrating the aforementioned inequality with respect to $\nu$, and using
Jensen’s inequality, we obtain the $\mathsf{CD}(\beta,n)$ condition (3.15) for
$\beta=\beta^{\mathrm{mod}}$. ∎

##  9 Curvature estimates for fat sub-Riemannian structures

In this section, we establish new Ricci curvature lower bounds for fat sub-
Riemannian structures. We address a non-expert reader to Appendix B for a
self-contained summary on the canonical curvature in sub-Riemannian geometry
that we use extensively.

We provide here an informal explanation of the basic concepts of Young diagram
and Ricci curvatures. To the generic222More precisely, for an open and dense
set of points $x\in M$ there exists a non-empty Zariski open (and thus dense)
set $A_{x}\subseteq T_{x}^{*}M$ of initial covectors for which the associated
geodesic has a well-defined Young diagram. See see [89, Sec. 5] and [3, Sec.
5.2]. geodesic of a sub-Riemannian manifold we can associate a Young diagram.
Each Young diagram must be thought as an ordered collection of $n=\dim M$
boxes, each one corresponding to a one-dimensional subspace of the tangent
space along the given geodesic. The position of the box in the diagram is
determined by the number of derivations (aka Lie brackets in the direction of
the geodesic) that one must perform to access to that direction starting with
the original distribution. According to the general theory, a sub-Riemannian
canonical curvature operator can be defined on the tangent space along the
geodesic. It behaves in a different way when restricted to each one of these
one-dimensional subspaces. It is natural then to group up the one-dimensional
subspaces having homogeneous behaviour. We obtain in this way a so-called
_reduced_ Young diagram, whose boxes are called superboxes. Each superbox of a
reduced Young diagram represents a subspace of the tangent space along the
geodesic where the curvature behaves in a homogeneous way, and thus it makes
sense to trace the curvature operator in these subspaces. This gives rise to
Ricci curvatures, one for each superbox. Note that, on Riemannian manifolds,
the reduced Young diagram is the same for any geodesic, it is formed by only
one superbox, and a large part of the canonical curvature theory explained in
Appendix B trivializes.

###  9.1 Fat curvature estimates

A distribution $\mathscr{D}$ on a smooth manifold $M$ of dimension $n\geq 3$
is called _fat_ if it is _strong bracket-generating_ [80, 70]: for any smooth
vector field $X$ on $M$ it holds

$X|_{x}\neq
0\quad\Longrightarrow\quad\mathscr{D}_{x}+[X,\mathscr{D}]_{x}=T_{x}M,\qquad\forall\,x\in
M.$ (9.1)

Let $g$ be a sub-Riemannian metric on $\mathscr{D}$. We call $(\mathscr{D},g)$
a _fat sub-Riemannian structure_ on $M$. Letting $\mathsf{d}$ be the
corresponding Carnot-Carathéodory distance.

######  Remark 9.1.

We recall that the following popular classes sub-Riemannian structures are
fat: the Heisenberg groups, contact structures, all H-type groups in the sense
of Kaplan [56], and more generally H-type foliations [30].

For a fat structure, any non-trivial geodesic has the same Young diagram with
two columns. We label the superboxes of the reduced Young diagram as follows:

$\ytableausetup{centertableaux,nosmalltableaux}\ytableau b&a\\\ c\\\ $

The sizes of the superboxes are $\mathrm{size}(a)=\mathrm{size}(b)=n-k$ and
$\mathrm{size}(c)=2k-n$. To each superbox, we associate the canonical Ricci
curvatures $\mathfrak{Ric}^{a}$, $\mathfrak{Ric}^{b}$, $\mathfrak{Ric}^{c}$
respectively.

Without loss of generality, we assume that $g$ is the restriction to
$\mathscr{D}$ of a Riemannian metric $g_{R}$. Recall from Definition 7.4 the
function $\mathsf{D}:M\times M\to[0,+\infty]$ associated with the Riemannian
extension $\mathsf{d}_{R}=\mathsf{d}_{g_{R}}$. By Proposition 7.28, it holds

$\mathsf{D}(x,y)=\|\nabla^{R}_{x}c(\cdot,y)\|_{R},\qquad\forall\,(x,y)\notin\mathrm{Cut}(M),$
(9.2)

where $c=\tfrac{1}{2}\mathsf{d}^{2}$. To fix the ideas, in the Heisenberg
group, and choosing the canonical Riemannian extension, for
$(x,y)\notin\mathrm{Cut}(M)$ it holds

$\mathsf{D}(x,y)^{2}=\mathsf{d}(x,y)^{2}+h_{0}(\lambda^{x,y})^{2},$ (9.3)

where $h_{0}(\lambda^{x,y})$ is the vertical component of the unique initial
covector $\lambda^{x,y}$ of the geodesic between $x$ and $y$.

The next result will be pivotal to show that fat sub-Riemannian structures on
a compact $n$-dimensional manifold are $\mathsf{CD}(\beta,n)$ spaces, for
suitable $\beta$ (cf. Theorem 9.23).

######  Theorem 9.2 (Fat curvature estimates).

Let $(M,\mathsf{d})$ be a sub-Riemannian metric space with dimension $n$ and
with fat distribution of rank $k<n$. Then, for any compact set $K\subset M$
there exists a constant $\kappa>0$ such that for any $x,y\in
K\setminus\mathrm{Cut}(M)$ and for the unique geodesic $\gamma:[0,1]\to M$
joining them, with initial covector $\lambda\in T_{x}^{*}M$, it holds

$\displaystyle\mathfrak{Ric}^{a}_{\lambda}$
$\displaystyle\geq-\kappa\mathsf{D}(x,y)^{4},$ (9.4)
$\displaystyle\mathfrak{Ric}^{b}_{\lambda}$
$\displaystyle\geq-\kappa\mathsf{D}(x,y)^{2},$ (9.5)
$\displaystyle\mathfrak{Ric}^{c}_{\lambda}$
$\displaystyle\geq-\kappa\mathsf{D}(x,y)^{2}.$ (9.6)

Furthermore, if $\mathfrak{m}$ is a smooth measure on $M$, there exists $C>0$
such that for any $x,y\in K\setminus\mathrm{Cut}(M)$ and for the unique
geodesic $\gamma:[0,1]\to M$ joining them, with initial covector $\lambda\in
T_{x}^{*}M$, it holds:

$|\rho_{\mathfrak{m},\lambda}|\leq C\mathsf{D}(x,y),$ (9.7)

where $\rho_{\mathfrak{m},\lambda}$ is the geodesic volume derivative (see
Section B.3).

######  Remark 9.3.

If the last block is one-dimensional, i.e. $\mathrm{size}(c)=2k-n=1$, then the
lower bound for $\mathfrak{Ric}^{c}$ is redundant as
$\mathfrak{Ric}^{c}_{\lambda}=0$.

######  Remark 9.4.

The constants in Theorem 9.2 do depend on the choice of $g_{R}$.

Recall that the canonical frame and curvature, seen as functions of the
initial covector $\lambda\in T^{*}M$, are not defined when the associated
geodesic is the trivial (abnormal) one, corresponding to
$\lambda\in\mathscr{D}^{0}$ (the annihilator bundle of $\mathscr{D}$). From an
analytic viewpoint, this translates into a singularity as
$\lambda\to\mathscr{D}^{0}$ of the canonical frame. This does not happen in
Riemannian geometry, where the canonical frame is defined even for the zero
covector.

In this section we will prove that even if the canonical frame is singular,
the canonical curvature is not. More precisely, we will establish a stronger
result on the structure of the canonical frame in a neighborhood of the
annihilator bundle, which is the main result of this section (see Theorem
9.22). The latter yields Theorem 9.2 as a consequence.

###  9.2 Preliminary setup

We start by setting up some notation. Let $X_{1},\dots,X_{n}$ be a local
Riemannian orthonormal frame defined on a neighborhood $\mathcal{O}\subset M$,
such that $X_{1},\dots,X_{k}$ are horizontal (i.e., smooth sections of
$\mathscr{D}$). Define linear-on-fibers functions
$h_{i}:T^{*}\mathcal{O}\to\mathbb{R}^{n}$ via
$h_{i}(\lambda)=\langle\lambda,X_{i}\rangle$, for $i=1,\dots,n$. These
functions define a local trivialization
$T^{*}\mathcal{O}\simeq\mathcal{O}\times\mathbb{R}^{n}$. Let also
$c_{ij}^{k}:\mathcal{O}\to\mathbb{R}$ be the smooth functions defined by

$[X_{i},X_{j}]=\sum_{k=1}^{n}c_{ij}^{k}X_{k},\qquad i,j=1,\dots,n.$ (9.8)

We routinely use symplectic calculus (cf. [2, Ch. 4]). We recall some basic
concepts here. To any $a\in C^{\infty}(T^{*}\mathcal{O})$ we associate the
vector field $\vec{a}$ on $T^{*}\mathcal{O}$

$\vec{a}(f)=\\{a,f\\},\qquad\forall\,f\in C^{\infty}(T^{*}\mathcal{O}),$ (9.9)

where $\\{\cdot,\cdot\\}$ denotes the Poisson bracket induced by the
symplectic structure on $T^{*}M$. Thus, on $T^{*}\mathcal{O}$, we define a
frame for $T(T^{*}\mathcal{O})$ given by

$\vec{h}_{1},\dots,\vec{h}_{n},\partial_{h_{1}},\dots,\partial_{h_{n}}.$
(9.10)

Notice that it holds $\pi_{*}\vec{h}_{i}=X_{i}$, where $\pi:T^{*}M\to M$ is
the bundle projection. Let also $\nu_{1},\dots,\nu_{n}$ be the dual frame of
one-forms on $\mathcal{O}$ such that $\langle\nu_{i},X_{j}\rangle=\delta_{ij}$
for $i,j=1,\dots,n$. In terms of this frame, the tautological one-form is
$\tau=\sum_{\ell=1}^{n}h_{\ell}\pi^{*}\nu_{\ell}$ on $T^{*}\mathcal{O}$ and
the symplectic form $\sigma=d\tau$ is

$\sigma=\sum_{\ell=1}^{n}dh_{\ell}\wedge\pi^{*}\nu_{\ell}+h_{\ell}\pi^{*}d\nu_{\ell}.$
(9.11)

In terms of the frame (9.10) we have, on $T^{*}\mathcal{O}$ and for
$i,j=1,\dots,n$:

$\sigma(\partial_{h_{i}},\partial_{h_{j}})=0,\qquad\sigma(\partial_{h_{i}},\vec{h}_{j})=\delta_{ij},\qquad\sigma(\vec{h}_{i},\vec{h}_{j})=\sum_{\mu=1}^{n}h_{\mu}c_{ij}^{\mu}.$
(9.12)

Let $H:T^{*}M\to\mathbb{R}$ be the sub-Riemannian Hamiltonian and denote with
$\vec{H}$ the corresponding Hamiltonian vector field, which we assume to be
complete, and denote with $e^{t\vec{H}}$ its flow. The curves
$\lambda:[0,1]\to T^{*}M$ given by

$\lambda_{t}=e^{t\vec{H}}(\lambda),\qquad\forall\,t\in[0,1],\qquad\forall\,\lambda\in
T^{*}M,$ (9.13)

are called normal extremals, and the projections $\gamma=\pi\circ\gamma$ are
locally length-minimizing curves parametrized with constant speed
$\|\dot{\gamma}_{t}\|^{2}=2H(\lambda)$, for all $t\in[0,1]$. In particular if
$\gamma$ is a geodesic it holds
$\mathsf{d}(\gamma_{0},\gamma_{1})^{2}=2H(\lambda)$.

For any tensor field on $T^{*}M$, and without risk of confusion, the dot
denotes the derivation with respect to $\vec{H}$. For example if
$V\in\Gamma(T(T^{*}M))$:

$\dot{V}|_{\lambda}=\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}e^{-\varepsilon\vec{H}}_{*}V|_{e^{\varepsilon\vec{H}}(\lambda)}=[\vec{H},V]|_{\lambda},\qquad\forall\,\lambda\in
T^{*}M,$ (9.14)

and for functions $a:T^{*}M\to\mathbb{R}$, we have $\dot{a}=\vec{H}(a)$.

The frame (9.10) will serve as a reference for the computation of the
canonical frame. It is convenient to adopt the following notation to
distinguish between “horizontal” and “vertical” coordinates. We let
$u_{i}=h_{i}$ for $i=1,\dots,k$ and $v_{j}=h_{j}$ for $j=k+1,\dots,n$. In
particular $u,v$ denote tuples $u=(u_{1},\dots,u_{k})$,
$v=(u_{k+1},\dots,u_{n})$, and thus $h=(h_{1},\dots,h_{n})=(u,v)$. In this
notation we have the following trivialization of $T^{*}\mathcal{O}$:

$T^{*}\mathcal{O}=\mathcal{O}\times\mathbb{R}^{k}\times\mathbb{R}^{n-k},$
(9.15)

and we denote a point $\lambda\in T^{*}\mathcal{O}$ as
$\lambda=(x;h)=(x;u,v)$. On $T^{*}\mathcal{O}$, the sub-Riemannian and
Riemannian Hamiltonians $H,H_{R}:T^{*}M\to\mathbb{R}$, respectively, read:

$H=\frac{1}{2}|u|^{2}\qquad\text{and}\qquad
H_{R}=\frac{1}{2}\left(|u|^{2}+|v|^{2}\right).$ (9.16)

For any set $\Lambda\subset T^{*}M$, we adopt the notation (recall that
$\mathscr{D}^{0}$ is the annihilator of $\mathscr{D}$)

$\Lambda_{\neq 0}:=\Lambda\setminus\mathscr{D}^{0}=\\{\lambda\in\Lambda\mid
H(\lambda)>0\\},$ (9.17)

so that, if $\Lambda\subset T^{*}\mathcal{O}$, in the given trivialization we
have

$\Lambda_{\neq 0}=\\{(x;u,v)\in\Lambda\mid u\neq 0\\}.$ (9.18)

####  9.2.1 Fundamental computations

The following lemma collects computations and notations that will be used
throughout the section. The proofs are routine computation, and are omitted.

######  Lemma 9.5 (Fundamental computations).

The following formulas hold on $T^{*}\mathcal{O}$:

$\displaystyle\dot{\partial}_{v_{\mu}}$ $\displaystyle=-u_{\ell}c_{\ell
A}^{\mu}\partial_{h_{A}},$ (9.19) $\displaystyle\dot{\partial}_{u_{i}}$
$\displaystyle=-\vec{u}_{i}-u_{\ell}c_{\ell A}^{i}\partial_{h_{A}},$ (9.20)
$\displaystyle\dot{{\vec{h}}}_{A}$
$\displaystyle=-h_{B}c_{A\ell}^{B}\vec{u}_{\ell}+u_{\ell}c_{\ell
A}^{B}\vec{h}_{B}+u_{\ell}h_{B}f_{\ell AC}^{B}\partial_{h_{C}},$ (9.21)

where for lower-case latin indices $i,j,\ell=1,\dots,k$, for greek ones
$\mu,\nu,\sigma=k+1,\dots,n$, for upper-case latin ones $A,B,C=1,\dots,n$,
repeated indices are understood as summed over their range, and we defined the
following smooth functions on $\mathcal{O}$:

$f_{\ell AC}^{B}\in C^{\infty}(\mathcal{O}),\qquad f_{\ell
AC}^{B}:=X_{\ell}(c_{AC}^{B})-X_{A}(c_{\ell C}^{B})-c_{\ell
A}^{D}c_{DC}^{B}+c_{\ell D}^{B}c_{AC}^{D}-c_{AD}^{B}c_{\ell C}^{D}.$ (9.22)

We rewrite these equations in a compact notation as follows:

$\displaystyle\dot{\partial}_{v}$
$\displaystyle=\mathbb{A}\cdot\partial_{u}+\mathbb{B}\cdot\partial_{v},$
(9.23) $\displaystyle\dot{\partial}_{u}$
$\displaystyle=-\vec{u}+\mathbb{C}\cdot\partial_{u}+\mathbb{D}\cdot\partial_{v},$
(9.24) $\displaystyle\dot{\vec{u}}$
$\displaystyle=\mathbb{E}\cdot\vec{u}-\mathbb{A}^{*}\cdot\vec{v}+\mathbb{F}\cdot\partial_{u}+\mathbb{G}\cdot\partial_{v},$
(9.25) $\displaystyle\dot{\vec{v}}$
$\displaystyle=\mathbb{H}\cdot\vec{u}-\mathbb{B}^{*}\cdot\vec{v}+\mathbb{I}\cdot\partial_{u}+\mathbb{L}\cdot\partial_{v},$
(9.26)

where the dot denotes multiplication of tuples333Here and in the following,
the notation $V=O\cdot W$, where $O$ is a $n\times m$ matrix,
$V=(V_{1},\dots,V_{n})$ is a $n$-tuple of vector fields, and similarly
$W=(W_{1},\dots,W_{m})$, with $n,m\in\mathbb{N}$, means that
$V_{i}=\sum_{j=1}^{m}O_{ij}W_{j}$., and where we defined the following smooth
matrix-valued maps:

$\displaystyle\mathbb{A}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(n-k,k),$
$\displaystyle\mathbb{A}_{\mu i}$ $\displaystyle:=-u_{\ell}c_{\ell i}^{\mu},$
(9.27) $\displaystyle\mathbb{B}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(n-k,n-k),$
$\displaystyle\mathbb{B}_{\mu\nu}$
$\displaystyle:=-u_{\ell}c_{\ell\nu}^{\mu},$ (9.28) $\displaystyle\mathbb{C}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(k,k),$
$\displaystyle\mathbb{C}_{ij}$ $\displaystyle:=-u_{\ell}c_{\ell j}^{i},$
(9.29) $\displaystyle\mathbb{D}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(k,n-k),$
$\displaystyle\mathbb{D}_{i\mu}$ $\displaystyle:=-u_{\ell}c_{\ell\mu}^{i},$
(9.30) $\displaystyle\mathbb{E}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(k,k),$
$\displaystyle\mathbb{E}_{ij}$
$\displaystyle:=-h_{B}c_{ij}^{B}+u_{\ell}c_{\ell i}^{j},$ (9.31)
$\displaystyle\mathbb{F}$ $\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(k,k),$
$\displaystyle\mathbb{F}_{ij}$ $\displaystyle:=u_{\ell}h_{B}f_{\ell ij}^{B},$
(9.32) $\displaystyle\mathbb{G}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(k,n-k),$
$\displaystyle\mathbb{G}_{i\mu}$ $\displaystyle:=u_{\ell}h_{B}f_{\ell
i\mu}^{B},$ (9.33) $\displaystyle\mathbb{H}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(n-k,k),$
$\displaystyle\mathbb{H}_{\mu i}$ $\displaystyle:=-h_{B}c^{B}_{\mu
i}+u_{\ell}c_{\ell\mu}^{i},$ (9.34) $\displaystyle\mathbb{I}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(n-k,k),$
$\displaystyle\mathbb{I}_{\mu i}$ $\displaystyle:=u_{\ell}h_{B}f_{\ell\mu
i}^{B},$ (9.35) $\displaystyle\mathbb{L}$
$\displaystyle:T^{*}\mathcal{O}\to\mathrm{M}(n-k,n-k),$
$\displaystyle\mathbb{L}_{\mu\nu}$
$\displaystyle:=u_{\ell}h_{B}f_{\ell\mu\nu}^{B}.$ (9.36)

####  9.2.2 Partial dilations

Recall that we have fixed a Riemannian extension so that
$TM=\mathscr{D}\oplus\mathscr{V}$, with $\mathscr{V}\perp\mathscr{D}$, and
thus $T^{*}M=\mathscr{V}^{0}\oplus\mathscr{D}^{0}$. This induces partial
dilations on the fibers of the cotangent bundle. For simplicity we only
provide local definitions in the trivialization $T^{*}\mathcal{O}$, even
though these maps are defined on the whole $T^{*}M$.

######  Definition 9.6 ($u$-dilation).

For $a>0$, we define the $u$-dilation $\delta_{a}^{hor}:T^{*}\mathcal{O}\to
T^{*}\mathcal{O}$ by

$\delta_{a}^{hor}(x;u,v):=(x;au,v),\qquad(x;u,v)\in T^{*}\mathcal{O}.$ (9.37)

######  Definition 9.7 ($u$-homogeneous functions).

Let $V$ be a vector space. A map $f:T^{*}\mathcal{O}\to V$ is _$u$
-homogeneous of degree $d$_ if

$f(\delta_{a}^{hor}(\lambda))=a^{d}f(\lambda),\qquad\forall\,a>0,\;\lambda\in
T^{*}\mathcal{O}.$ (9.38)

######  Definition 9.8 ($u$-star shaped set).

A set $\Lambda\subset T^{*}\mathcal{O}$ is _$u$ -star-shaped_ if

$\Lambda=\bigcup_{a\in[0,1]}\delta_{a}^{hor}(\Lambda).$ (9.39)

Similar definitions can be given for $v$-dilations, $v$-homogeneous functions,
$v$-star-shaped sets but they will not be used, so they are omitted.

####  9.2.3 Pseudo-homogeneous maps

The class of $u$-homogeneous maps is not closed with respect to
$\vec{H}$-derivations. Thus, we introduce a larger class of maps with this
property. We only give a local definition.

######  Definition 9.9 ($u$-pseudo-homogeneous maps).

Let $V$ be a vector space and let $\Lambda\subset T^{*}\mathcal{O}$ be an open
set. We say that $f:\Lambda\to V$ is _$u$ -pseudo-homogeneous_ of degree
$d\in\mathbb{N}$ if there exist

  * •

a finite set of indices $I$;

  * •

smooth functions $g_{i},c_{i}:T^{*}\mathcal{O}\to\mathbb{R}$, $i\in I$
(defined on the whole $T^{*}\mathcal{O}$);

such that it holds

$f(x;u,v)=|u|^{d}\sum_{i\in
I}c_{i}(x;u,v)g_{i}(x;u/|u|,v),\qquad\forall\,(x;u,v)\in\Lambda_{\neq 0}.$
(9.40)

The following simple observation will be crucial.

######  Lemma 9.10.

If $f:\Lambda\to V$ is $u$-pseudo-homogeneous of degree $d$, then $\vec{H}(f)$
is also $u$-pseudo-homogeneous of degree $d$. In particular, if
$\Lambda\subset T^{*}\mathcal{O}$ is relatively compact domain,
$j\in\mathbb{N}$, there exists $C=C_{j,\Lambda}$ such that

$\|\vec{H}^{(j)}(f)(\lambda)\|\leq
CH(\lambda)^{d/2},\qquad\forall\,\lambda\in\Lambda_{\neq 0},$ (9.41)

where $\|\cdot\|$ denotes a fixed arbitrary norm on $V$.

###### Proof.

Notice that $H=\tfrac{1}{2}|u|^{2}$ so that
$\vec{H}=\sum_{i=1}^{k}u_{i}\vec{u}_{i}$, and it holds $\vec{H}(|u|)=0$ on
$T^{*}\mathcal{O}_{\neq 0}$. In particular $\vec{H}$ is a derivation that acts
only on $x,v$ and the spherical part of $u$. The statement follows easily from
(9.40). ∎

######  Remark 9.11.

All $u$-homogeneous functions of degree $d$ are $u$-pseudo-homogeneous. In
this case (9.40) is satisfied for $I=\\{1\\}$, $c_{1}=1$, and $g_{1}$ is any
everywhere-smooth function such that $g_{1}(x;u,v)=f(x;u,v)$ for
$u\in\mathbb{S}^{k-1}$.

###  9.3 Regularity of the canonical frame

In the next series of lemmas we prove that one can choose in a smooth and
locally bounded fashion a canonical frame as a function of the initial
covector. The key feature is that the sets $\Lambda$ in the statements are
$u$-star-shaped so that the estimates hold arbitrarily close to the
singularity at $\mathscr{D}^{0}$, corresponding to $2H=0$ or, equivalently,
$u=0$.

Recall that, for a fat structure, all non-trivial normal extremals have the
same Young diagram, with two columns. The elements of the frame are labeled
according to the boxes of the (reduced) Young diagram:

$\ytableausetup{centertableaux,nosmalltableaux}Y=\ytableau b&a\\\ c\\\
\qquad\qquad\begin{aligned} \mathrm{size}(a)&=\mathrm{size}(b)=n-k,\\\
\mathrm{size}(c)&=2k-n.\end{aligned}$ (9.42)

We organize a canonical frame along an extremal
$\lambda_{t}=e^{t\vec{H}}(\lambda)$ as a collection of tuples

$\\{E_{a},E_{b},E_{c},F_{a},F_{b},F_{c}\\},$ (9.43)

where, for $\boxdot=a,b,c$, each $E_{\boxdot}$, $F_{\boxdot}$ in this list is
a tuple of smooth vector fields along $\lambda_{t}$, of size equal to the one
of the corresponding superbox $\mathrm{size}(\boxdot)$.

######  Remark 9.12.

In Lemmas 9.13, 9.14, $\Lambda$ is a neighborhood of any $\bar{\lambda}\in
T^{*}M$. Furthermore, the $u$-star-shaped property of $\Lambda$ is a
triviality since, as it is clear from the proof, $\Lambda$ can actually be any
relatively bounded domain contained in a local trivialization
$T^{*}\mathcal{O}$. Starting from Lemma 9.15 onward, however,
$\bar{\lambda}\in T^{*}M_{\neq 0}$, and furthermore the $u$-star-shaped
property is not trivial and must be proved in the construction.

######  Lemma 9.13.

For any $\bar{\lambda}\in T^{*}M$ there exists a $u$-star-shaped neighborhood
$\Lambda\subset T^{*}M$ of $\bar{\lambda}$, $T>0$, and a smooth map
$\mathbb{P}:[0,T]\times\Lambda_{\neq 0}\to\mathrm{GL}(\mathbb{R}^{n-k})$, such
that for all $\lambda\in\Lambda_{\neq 0}$ the tuple

$E_{a}|_{\lambda_{t}}=\frac{1}{\sqrt{2H(\lambda)}}\mathbb{P}(t,\lambda)\cdot\partial_{v}|_{\lambda_{t}},\qquad
t\in[0,T],$ (9.44)

is (a choice of) the $E_{a}$-component of the canonical frame along
$\lambda_{t}=e^{t\vec{H}}(\lambda)$.

Furthermore, the map $\mathbb{P}$ and all its time-derivatives are uniformly
bounded, that is for all $j\in\mathbb{N}$ there exists a constant $C=C_{j}>0$
such that

$\|\partial_{t}^{j}\mathbb{P}(t,\lambda)\|\leq
C,\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0},$ (9.45)

where $\|\cdot\|$ denotes a fixed matrix norm.

###### Proof.

Let $\bar{x}=\pi(\bar{\lambda})$ and let $\mathcal{O}\subseteq M$ be an open
neighborhood of $\bar{x}$ equipped with a local orthonormal frame, as
explained in Section 9.2, so that we have the trivialization

$T^{*}\mathcal{O}=\mathcal{O}\times\mathbb{R}^{k}\times\mathbb{R}^{n-k},\qquad\lambda=(x;u,v).$
(9.46)

Let $\Lambda\subset T^{*}\mathcal{O}$ be any $u$-star-shaped and relatively
compact neighborhood of $\bar{\lambda}$. Let
$\Lambda_{t}=e^{t\vec{H}}(\Lambda)$. Indeed, the union of all $\Lambda_{t}$,
for $t\in[0,T]$, is relatively compact. Furthermore, $\pi(\Lambda_{t})$ is
contained in the union of all the sub-Riemannian balls with centers in the
bounded set $\pi(\Lambda)$ and radius $t\rho$, where
$\rho=\sup\\{|u|\mid(x;u,v)\in\Lambda\\}$. If $T$ is sufficiently small, then
$\Lambda_{t}\subset T^{*}\mathcal{O}$ for all $t\leq T$. Hence, in the
following, all extremals $\lambda_{t}$ with $\lambda_{0}=\lambda\in\Lambda$,
are contained in a common compact subset of $T^{*}\mathcal{O}$.

Now fix $\lambda\in\Lambda_{\neq 0}$ and $t\in[0,T]$. The tuple $E_{a}$ is
determined (up to a constant orthogonal transformation) by the following
conditions along $\lambda_{t}$:

  * (i)

$E_{a}$ is vertical (i.e. $\pi_{*}E_{a}=0$);

  * (ii)

$\dot{E}_{a}$ is vertical;

  * (iii)

normalization condition: $\mathbb{1}=\sigma(\ddot{E}_{a},\dot{E}_{a})$;

  * (iv)

Darboux frame condition: $\mathbb{0}=\sigma(\ddot{E}_{a},\ddot{E}_{a})$.

By Lemma 9.5, points (i) and (ii), $E_{a}$ must have the form

$E_{a}|_{\lambda_{t}}=\theta(t)\cdot\partial_{v}|_{\lambda_{t}},\qquad\forall\,t\in[0,T],$
(9.47)

for some smooth $t\mapsto\theta(t)\in\mathrm{GL}(\mathbb{R}^{n-k})$.

Using Lemma 9.5, we obtain

$\ddot{\partial}_{v}=(\dot{\mathbb{A}}+\mathbb{A}\mathbb{C}+\mathbb{B}\mathbb{A})\cdot\partial_{u}+(\dot{\mathbb{B}}+\mathbb{A}\mathbb{D}+\mathbb{B}^{2})\cdot\partial_{v}-\mathbb{A}\cdot\vec{u}.$
(9.48)

Here $\dot{\mathbb{A}}:T^{*}\mathcal{O}\to\mathrm{M}(n-k\times k)$ is the
smooth map given by $\dot{\mathbb{A}}(\lambda)=(\vec{H}\mathbb{A})(\lambda)$,
where the action of $\vec{H}$ is meant component-wise. The notation is
consistent as
$\dot{\mathbb{A}}(\lambda_{t})=\frac{d}{dt}\mathbb{A}(\lambda_{t})$. Similarly
for $\dot{\mathbb{B}}$.

Using (9.48) and (9.23) in the ansatz (9.47), we obtain:

$\displaystyle\dot{E}_{a}$
$\displaystyle=\theta\mathbb{A}\cdot\partial_{u}+(\dot{\theta}+\theta\mathbb{B})\cdot\partial_{v},$
(9.49) $\displaystyle\ddot{E}_{a}$
$\displaystyle=[2\dot{\theta}\mathbb{A}+\theta(\dot{\mathbb{A}}+\mathbb{A}\mathbb{C}+\mathbb{B}\mathbb{A})]\cdot\partial_{u}+[\ddot{\theta}+2\dot{\theta}\mathbb{B}+\theta(\dot{\mathbb{B}}+\mathbb{A}\mathbb{D}+\mathbb{B}^{2})]\cdot\partial_{v}-\theta\mathbb{A}\cdot\vec{u},$
(9.50)

where we omitted the explicit evaluation along $\lambda_{t}$.

We now impose (iii). It yields:

$\mathbb{1}=-\sigma(\dot{E}_{a},\ddot{E}_{a})=\sigma(\theta\mathbb{A}\cdot\partial_{u},\theta\mathbb{A}\cdot\vec{u})=(\theta\mathbb{A})\sigma(\partial_{u},\vec{u})(\theta\mathbb{A})^{*}=\theta\mathbb{A}\mathbb{A}^{*}\theta^{*},$
(9.51)

where we used (9.12) for $\sigma(\partial_{u},\vec{u})=\mathbb{1}$, and the
star denotes the transpose.

Notice that $\mathbb{A}\mathbb{A}^{*}:T^{*}\mathcal{O}\to\mathrm{M}(n-k,n-k)$
is given by

$\mathbb{A}\mathbb{A}^{*}(x;u,v)_{\mu\nu}=-\sum_{i,j,\ell=1}^{k}u_{\ell}u_{j}c_{\ell
i}^{\mu}(x)c_{ji}^{\nu}(x),\qquad\mu,\nu=k+1,\dots,n.$ (9.52)

In particular $\mathbb{A}\mathbb{A}^{*}$ is $u$-homogeneous of degree $2$. The
fat assumption means that $\mathbb{A}=\mathbb{A}(x;u,v)$ has non-trivial
kernel if and only if $u=0$, or equivalently that $\mathbb{A}\mathbb{A}^{*}>0$
on $T^{*}\mathcal{O}_{\neq 0}$, and thus the restriction

$\mathbb{A}\mathbb{A}^{*}|_{T^{*}\mathcal{O}_{\neq 0}}:T^{*}\mathcal{O}_{\neq
0}\to\mathrm{GL}(\mathbb{R}^{n-k}),$ (9.53)

is a smooth map taking values in the space of scalar products.

By the Gram-Schmidt process applied to the columns of
$\mathbb{A}\mathbb{A}^{*}|_{T^{*}\mathcal{O}_{\neq 0}}$, and the fact that the
latter is smooth, we deduce the existence of a smooth map

$\mathbb{S}:T^{*}\mathcal{O}_{\neq 0}\to\mathrm{GL}(\mathbb{R}^{n-k}),$ (9.54)

which is $u$-homogeneous of degree $1$, and such that

$\mathbb{A}\mathbb{A}^{*}(x;u,v)=\mathbb{S}\mathbb{S}^{*}(x;u,v),\qquad\forall\,(x;u,v)\in
T^{*}\mathcal{O}_{\neq 0}.$ (9.55)

Notice that $\mathbb{A}$ is a rectangular matrix, while $\mathbb{S}$ is an
invertible square matrix.

We define then the smooth map
$t\mapsto\phi(t)\in\mathrm{GL}(\mathbb{R}^{n-k})$ by

$\phi(t):=\theta(t)\mathbb{S}(\lambda_{t}),\qquad t\in[0,T].$ (9.56)

Condition (iii) in (9.51), expressed in terms of $\phi$, is equivalent to

$\phi\phi^{*}=\theta\mathbb{S}\mathbb{S}^{*}\theta^{*}=\theta\mathbb{A}\mathbb{A}^{*}\theta^{*}=\mathbb{1},$
(9.57)

omitting evaluation along $\lambda_{t}$. Since $\phi(t)$ is invertible we
obtain that

$\text{condition
(iii)}\qquad\Leftrightarrow\qquad\phi(t)\in\mathrm{O}(\mathbb{R}^{n-k}),\qquad\forall\,t\in[0,T].$
(9.58)

So that we can refine (9.47) as:

$E_{a}|_{\lambda_{t}}=\phi(t)\mathbb{S}(\lambda_{t})^{-1}\cdot\partial_{v}|_{\lambda_{t}},\qquad\forall\,t\in[0,T],$
(9.59)

for a smooth map $t\mapsto\phi(t)$ taking values now in
$\mathrm{O}(\mathbb{R}^{n-k})$.

We now impose condition (iv), namely
$\mathbb{0}=\sigma(\ddot{E}_{a},\ddot{E}_{a})$. This yields an ODE which will
characterize $t\mapsto\phi(t)$. Using (9.50) and (9.12), we obtain

$\mathbb{0}=[2\dot{\theta}\mathbb{A}+\theta(\dot{\mathbb{A}}+\mathbb{A}\mathbb{C}+\mathbb{B}\mathbb{A})](\theta\mathbb{A})^{*}-\theta\mathbb{A}[2\dot{\theta}\mathbb{A}+\theta(\dot{\mathbb{A}}+\mathbb{A}\mathbb{C}+\mathbb{B}\mathbb{A})]^{*}.$
(9.60)

We re-express this condition in terms of $\phi$. To do this, we use again the
shorthand $\dot{\mathbb{S}}=\vec{H}(\mathbb{S}):T^{*}\mathcal{O}_{\neq
0}\to\mathrm{GL}(\mathbb{R}^{n-k})$, so that
$\dot{\mathbb{S}}(\lambda_{t})=\frac{d}{dt}\mathbb{S}(\lambda_{t})$,
consistently with the previous notation. Omitting $t$ and evaluation along
$\lambda_{t}$, and denoting with $\mathcal{A}(M)=\tfrac{1}{2}(M-M^{*})$ the
skew-symmetric part of a matrix $M$, we obtain the following ODE for $\phi$:

$\displaystyle 2\dot{\phi}$
$\displaystyle=\phi\mathbb{S}^{-1}\mathcal{A}\left(2\dot{\mathbb{S}}\mathbb{S}-(\dot{\mathbb{A}}+\mathbb{A}\mathbb{C}+\mathbb{B}\mathbb{A})\mathbb{A}^{*}+\tfrac{1}{2}\mathbb{A}\sigma(\vec{u},\vec{u})\mathbb{A}^{*}\right)\mathbb{S}^{-1},$
(9.61)

where we used
$\phi\in\mathrm{O}(\mathbb{R}^{n-k})\Rightarrow\dot{\phi}\phi^{*}=-\phi\dot{\phi}^{*}$.
We can rewrite the above ODE as $\dot{\phi}(t)=\phi(t)\Xi(\lambda_{t})$, where
the smooth map

$\Xi:T^{*}\mathcal{O}_{\neq 0}\to\mathrm{M}(n-k\times n-k),$ (9.62)

takes values in skew-symmetric matrices and is defined by

$\Xi:=\frac{1}{2}\mathbb{S}^{-1}\mathcal{A}\left[2\dot{\mathbb{S}}\mathbb{S}-(\dot{\mathbb{A}}+\mathbb{A}\mathbb{C}+\mathbb{B}\mathbb{A})\mathbb{A}^{*}+\tfrac{1}{2}\mathbb{A}\sigma(\vec{u},\vec{u})\mathbb{A}^{*}\right]\mathbb{S}^{-1}.$
(9.63)

The terms
$\mathbb{A},\dot{\mathbb{A}},\mathbb{B},\mathbb{C},\mathbb{D},\sigma(\vec{u},\vec{u})$
are smoothly defined on $T^{*}\mathcal{O}$, on the other hand $\mathbb{S}$ and
$\mathbb{S}^{-1}$ are well-defined and smooth only on $T^{*}\mathcal{O}_{\neq
0}$, and they blow-up as $H(\lambda)\to 0$.

We claim that $\Xi$ and all its $\vec{H}$-derivatives $\vec{H}^{(j)}(\Xi)$
remain uniformly bounded on $\Lambda_{\neq 0}$. To prove this claim, observe
that $\mathbb{A}$, $\mathbb{B}$, $\mathbb{C}$, $\sigma(\vec{u},\vec{u})$,
$\mathbb{S}$ appearing in (9.63) are $u$-homogeneous of degree one, in
particular they are $u$-pseudo-homogeneous of degree $1$. By Lemma 9.10, the
$\vec{H}$-derivatives of these maps are $u$-pseudo-homogeneous of degree $1$.
In particular this is the case for $\dot{\mathbb{A}}$ and $\dot{\mathbb{S}}$
appearing in (9.63). Taking into account the form (9.40) of $u$-pseudo-
homogeneous maps all possible singularities at $u=0$ in (9.63) cancel out, and
$\Xi$ and its $\vec{H}$-derivatives are $u$-pseudo-homogeneous of degree
$d=0$. Another application of Lemma 9.10 proves the claim.

We choose $t\mapsto\phi(t)$ in (9.59) as the solution of the Cauchy problem

$\dot{\phi}(t)=\phi(t)\Xi(\lambda_{t}),\qquad\phi(0)=\mathbb{1}.$ (9.64)

Of course one might choose as the initial condition $\phi(0)$ any orthogonal
matrix, and this freedom in the choice of the initial condition in expected as
the canonical frame is uniquely defined only up to a constant orthogonal
transformation.

By the construction of $\Lambda$ and $T$ all extremals
$\lambda_{t}=e^{t\vec{H}}(\lambda)$, for $t\leq T$ and $\lambda\in\Lambda$,
are contained in a common relatively compact subset of $T^{*}\mathcal{O}_{\neq
0}$. Furthermore, $\Xi$ and its $\vec{H}$-derivatives are uniformly bounded
for all $t\in[0,T]$, $\lambda\in\Lambda_{\neq 0}$. In particular the solution
to the ODE (9.64) is well-defined on $[0,T]$. Define the map

$\Phi:[0,T]\times\Lambda_{\neq 0}\to\mathrm{O}(\mathbb{R}^{n-k}),$ (9.65)

such that $\Phi(\cdot,\lambda)$ is the solution of (9.64) with
$\lambda_{t}=e^{t\vec{H}}(\lambda)$. Since $\Xi$ is skew-symmetric then $\Phi$
is orthogonal for $t\in[0,T]$.

Thanks to the aforementioned claim of uniform boundedness property of $\Xi$,
we can apply Gronwall’s lemma to (9.64), and we obtain that for any
$j\in\mathbb{N}$ there exists $C=C(j,\Lambda,T)>0$ such that

$\|\partial_{t}^{j}\Phi(t,\lambda)\|\leq
C,\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0}.$ (9.66)

In order to recover the statement of the lemma, we set

$\mathbb{P}(t,\lambda):=\Phi(t,\lambda)\sqrt{2H(\lambda)}\mathbb{S}(\lambda_{t})^{-1},\qquad\lambda_{t}=e^{t\vec{H}}(\lambda),\qquad\forall\,(t,\lambda)\in[0,1]\times\Lambda_{\neq
0}.$ (9.67)

By construction, the map

$(t,\lambda)\mapsto
E_{a}|_{\lambda_{t}}=\frac{1}{\sqrt{2H(\lambda)}}\mathbb{P}(t,\lambda)\cdot\partial_{v}|_{\lambda_{t}},$
(9.68)

is smooth for $(t,\lambda)\in[0,T]\times\Lambda_{\neq 0}$, and
$E_{a}|_{\lambda_{t}}$ is the $E_{a}$-component of a canonical frame along the
extremal $\lambda_{t}$ with initial covector $\lambda$.

It remains to prove that the map $\mathbb{P}$ and its time-derivatives
$\partial_{t}^{j}\mathbb{P}$ are uniformly bounded as required on
$[0,T]\times\Lambda_{\neq 0}$. Notice that $\Phi(t,\lambda)$ and its time-
derivatives already satisfy the required property by (9.66), while
$\lambda\mapsto\sqrt{2H(\lambda)}\mathbb{S}(\lambda)^{-1}$ is $u$-homogeneous
of degree $0$ and smooth on $T^{*}\mathcal{O}_{\neq 0}$, so that we can apply
Lemma 9.10 for $d=0$, so that also
$\sqrt{2H(\lambda_{t})}\mathbb{S}(\lambda_{t})^{-1}$ and all its time-
derivatives are uniformly bounded as required. ∎

Lemma 9.13 for the $E_{a}$-component of the frame is the starting point for
analogous statement for all the other tuples.

######  Lemma 9.14.

For any $\bar{\lambda}\in T^{*}M$ there exists an $u$-star-shaped neighborhood
$\Lambda\subset T^{*}M$ of $\bar{\lambda}$, $T>0$, and smooth maps
$\mathbb{Q}_{i}:[0,T]\times\Lambda_{\neq 0}\to\mathrm{M}(n_{i},m_{i})$, with
$n_{i},m_{i}\in\mathbb{N}$, $i=1,\dots,5$, such that for all
$\lambda\in\Lambda_{\neq 0}$ the tuples

$\displaystyle E_{b}|_{\lambda_{t}}$
$\displaystyle=\mathbb{Q}_{1}(t,\lambda)\cdot\partial_{u}|_{\lambda_{t}}+\frac{1}{\sqrt{2H(\lambda)}}\mathbb{Q}_{2}(t,\lambda)\cdot\partial_{v}|_{\lambda_{t}},$
$\displaystyle t\in[0,T],$ (9.69) $\displaystyle F_{b}|_{\lambda_{t}}$
$\displaystyle=\mathbb{Q}_{3}(t,\lambda)\cdot\vec{u}|_{\lambda_{t}}+\mathbb{Q}_{4}(t,\lambda)\cdot\partial_{u}|_{\lambda_{t}}+\frac{1}{\sqrt{2H(\lambda)}}\mathbb{Q}_{5}(t,\lambda)\cdot\partial_{v}|_{\lambda_{t}},$
$\displaystyle t\in[0,T],$ (9.70)

are (a choice of) the $E_{b}$ and $F_{b}$-components of the canonical frame
along $\lambda_{t}=e^{t\vec{H}}(\lambda)$.

Furthermore, the maps $\mathbb{Q}_{i}$ and all their time-derivatives are
bounded, that is for all $j\in\mathbb{N}$ there exists a constant $C=C_{j}>0$
such that

$\|\partial_{t}^{j}\mathbb{Q}_{i}(t,\lambda)\|\leq
C,\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0},\;i=1,\dots,5,$ (9.71)

where $\|\cdot\|$ denotes a fixed matrix norm.

###### Proof.

Let $\Lambda$ and $T>0$ as in Lemma 9.13 (cf. first paragraph of its proof).
Let $\lambda\in\Lambda_{\neq 0}$. By the structural equations
$E_{b}|_{\lambda_{t}}=\dot{E}_{a}|_{\lambda_{t}}$. Using Lemma 9.5, we obtain

$E_{b}|_{\lambda_{t}}=\frac{1}{\sqrt{2H}}\left[\mathbb{P}(t,\lambda)\mathbb{A}(\lambda_{t})\cdot\partial_{u}|_{\lambda_{t}}+(\dot{\mathbb{P}}(t,\lambda)+\mathbb{P}(t,\lambda)\mathbb{B}(\lambda_{t}))\cdot\partial_{v}|_{\lambda_{t}}\right],$
(9.72)

where $\dot{\mathbb{P}}(t,\lambda)=\partial_{t}\mathbb{P}(t,\lambda)$. We set
hence:

$\mathbb{Q}_{1}(t,\lambda):=\frac{1}{\sqrt{2H(\lambda)}}\mathbb{P}(t,\lambda)\mathbb{A}(\lambda_{t}),\qquad\mathbb{Q}_{2}(t,\lambda):=\dot{\mathbb{P}}(t,\lambda)+\mathbb{P}(t,\lambda)\mathbb{B}(\lambda_{t}).$
(9.73)

Notice that $\mathbb{P}$ and all its time-derivatives are uniformly bounded by
the estimate in Lemma 9.13. Since $\mathbb{A}$ and $\mathbb{B}$ are
$u$-pseudo-homogeneous of degree $1$, an application of Lemma 9.10 yields the
required bounds on $\mathbb{Q}_{1}$ and $\mathbb{Q}_{2}$ and their time-
derivatives.

Likewise, $F_{b}|_{\lambda_{t}}=-\dot{E}_{b}|_{\lambda_{t}}$, so that, using
Lemma 9.5, we obtain:

$F_{b}=\frac{1}{\sqrt{2H}}\left[\mathbb{P}\mathbb{A}\cdot\vec{u}-(\dot{\mathbb{P}}\mathbb{A}+\mathbb{P}\dot{\mathbb{A}}+\mathbb{P}\mathbb{A}\mathbb{C}+\mathbb{P}\mathbb{B}\mathbb{A})\cdot\partial_{u}\right.\\\
\left.-(\ddot{\mathbb{P}}+2\dot{\mathbb{P}}\mathbb{B}+\mathbb{P}\dot{\mathbb{B}}+\mathbb{P}\mathbb{A}\mathbb{D}+\mathbb{P}\mathbb{B}^{2})\cdot\partial_{v}\right],$
(9.74)

omitting evaluation along $\lambda_{t}$ from $\mathbb{A}$, $\mathbb{B}$,
$\mathbb{D}$, and evaluation at $(t,\lambda)$ for $\mathbb{P}$. To obtain the
statement for $F_{b}$ we set

$\displaystyle\mathbb{Q}_{3}$
$\displaystyle:=\frac{1}{\sqrt{2H}}\mathbb{P}\mathbb{A},$ (9.75)
$\displaystyle\mathbb{Q}_{4}$
$\displaystyle:=-\frac{1}{\sqrt{2H}}(\dot{\mathbb{P}}\mathbb{A}+\mathbb{P}\dot{\mathbb{A}}+\mathbb{P}\mathbb{A}\mathbb{C}+\mathbb{P}\mathbb{B}\mathbb{A}),$
(9.76) $\displaystyle\mathbb{Q}_{5}$
$\displaystyle:=-(\ddot{\mathbb{P}}+2\dot{\mathbb{P}}\mathbb{B}+\mathbb{P}\dot{\mathbb{B}}+\mathbb{P}\mathbb{A}\mathbb{D}+\mathbb{P}\mathbb{B}^{2}),$
(9.77)

and argue as above using Lemma 9.10 to prove the required estimates on the
$\mathbb{Q}_{i}$’s. ∎

######  Lemma 9.15.

For any $\bar{\lambda}\in T^{*}M_{\neq 0}$ there exists an $u$-star-shaped
neighborhood $\Lambda\subset T^{*}M$ of $\bar{\lambda}$, $T>0$, and smooth
maps $\mathbb{Q}_{i}:[0,T]\times\Lambda_{\neq 0}\to\mathrm{M}(n_{i},m_{i})$,
with $n_{i},m_{i}\in\mathbb{N}$, $i=6,\dots,10$, such that for all
$\lambda\in\Lambda_{\neq 0}$ the tuples

$\displaystyle E_{c}|_{\lambda_{t}}$
$\displaystyle=\mathbb{Q}_{6}(t,\lambda)\cdot\partial_{u}|_{\lambda_{t}}+\frac{1}{\sqrt{2H(\lambda)}}\mathbb{Q}_{7}(t,\lambda)\cdot\partial_{v}|_{\lambda_{t}},$
$\displaystyle t\in[0,T],$ (9.78) $\displaystyle F_{c}|_{\lambda_{t}}$
$\displaystyle=\mathbb{Q}_{8}(t,\lambda)\cdot\vec{u}|_{\lambda_{t}}+\mathbb{Q}_{9}(t,\lambda)\cdot\partial_{u}|_{\lambda_{t}}+\frac{1}{\sqrt{2H(\lambda)}}\mathbb{Q}_{10}(t,\lambda)\cdot\partial_{v}|_{\lambda_{t}},$
$\displaystyle t\in[0,T],$ (9.79)

are (a choice of) the $E_{c}$ and $F_{c}$-components of the canonical frame
along $\lambda_{t}=e^{t\vec{H}}(\lambda)$.

Furthermore, the maps $\mathbb{Q}_{i}$ and all their time-derivatives are
bounded, that is for all $j\in\mathbb{N}$ there exists a constant $C=C_{j}>0$
such that

$\|\partial_{t}^{j}\mathbb{Q}_{i}(t,\lambda)\|\leq
C,\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0},\;i=6,\dots,10,$
(9.80)

where $\|\cdot\|$ denotes a fixed matrix norm.

###### Proof.

Let $\Lambda$ and $T>0$ as in Lemma 9.13 (cf. first paragraph of its proof).
Let $\lambda_{t}=e^{t\vec{H}}(\lambda)$, for $\lambda\in\Lambda_{\neq 0}$ and
$t\in[0,T]$. The $(2k-n)$-tuple $E_{c}|_{\lambda_{t}}$ is determined, up to a
constant orthogonal transformation, by the following conditions along
$\lambda_{t}$:

  * (i)

$E_{c}$ is vertical (i.e. $\pi_{*}E_{c}=0$);

  * (ii)

Darboux frame conditions: $\mathbb{0}=\sigma(\dot{E}_{b},E_{c})$ and
$\mathbb{0}=\sigma(\ddot{E}_{b},E_{c})$;

  * (iii)

normalization condition: $\mathbb{1}=\sigma(\dot{E}_{c},E_{c})$;

  * (iv)

isotropic condition: $\mathbb{0}=\sigma(\dot{E}_{c},\dot{E}_{c})$.

We observe that if $\bar{E}_{c}$ is a tuple which verifies (i), (ii) and (iii)
along $\lambda_{t}$, then all these conditions will be also verified by

$E_{c}|_{\lambda_{t}}:=\psi(t)\cdot\bar{E}_{c}|_{\lambda_{t}},\qquad
t\in[0,T],$ (9.81)

for any smooth map $t\mapsto\psi(t)\in\mathrm{O}(\mathbb{R}^{2k-n})$. Then, we
find first smooth tuple $\bar{E}_{c}$ such that (i), (ii) and (iii) are
verified, and subsequently we will fix $\psi$ via (iv).

By (i), we must have

$\bar{E}_{c}|_{\lambda_{t}}=U(t)\cdot\partial_{u}|_{\lambda_{t}}+V(t)\cdot\partial_{v}|_{\lambda_{t}},\qquad
t\in[0,T],$ (9.82)

for some smooth $t\mapsto U(t)\in\mathrm{M}(2k-n,k)$ and $t\mapsto
V(t)\in\mathrm{M}(2k-n,n-k)$. From this, and Lemma 9.5, we obtain:

$\displaystyle\dot{\bar{E}}_{c}$
$\displaystyle=-U\cdot\vec{u}+(\dot{U}+U\mathbb{C}+V\mathbb{A})\cdot\partial_{u}+(\dot{V}+U\mathbb{D}+V\mathbb{B})\cdot\partial_{v},$
(9.83) $\displaystyle\ddot{\bar{E}}_{c}$
$\displaystyle=-(2\dot{U}+U\mathbb{E}+U\mathbb{C}+V\mathbb{A})\cdot\vec{u}+U\mathbb{A}^{*}\cdot\vec{v}+\mathrm{span}\\{\partial_{u},\partial_{v}\\},$
(9.84)

Similarly from (9.72) in the proof of Lemma 9.14, and Lemma 9.5 we obtain:

$\displaystyle\dot{E}_{b}$
$\displaystyle=\frac{1}{\sqrt{2H}}\mathbb{P}\mathbb{A}\cdot\vec{u}+\mathrm{span}\\{\partial_{u},\partial_{v}\\},$
(9.85) $\displaystyle\ddot{E}_{b}$
$\displaystyle=\mathbb{T}\cdot\vec{u}-\frac{1}{\sqrt{2H}}\mathbb{P}\mathbb{A}\mathbb{A}^{*}\cdot\vec{v}+\mathrm{span}\\{\partial_{u},\partial_{v}\\},$
(9.86)

where, for brevity of notation, we set $\mathbb{T}=\mathbb{T}(t,\lambda)$, as

$\mathbb{T}(t,\lambda):=-\frac{1}{\sqrt{2H}}\Big{[}2(\partial_{t}\mathbb{P})\mathbb{A}+3\mathbb{P}\dot{\mathbb{A}}+\mathbb{P}(\mathbb{A}\mathbb{E}+\mathbb{A}\mathbb{C}+\mathbb{B}\mathbb{A})\Big{]},$
(9.87)

where the matrix maps $\mathbb{A}$, $\mathbb{B}$, $\mathbb{C}$, $\mathbb{E}$
in the r.h.s. is evaluated along $\lambda_{t}=e^{t\vec{H}}(\lambda)$,
$\mathbb{P}=\mathbb{P}(t,\lambda)$, for $\lambda\in\Lambda_{\neq 0}$ and
$t\in[0,T]$. Notice that $\mathbb{A}$ is $u$-pseudo-homogeneous of degree $1$,
together with $\dot{\mathbb{A}}$, by Lemma 9.10. Therefore we obtain that
$\mathbb{T}$ is uniformly bounded with all its time-derivatives, that is for
all $j\in\mathbb{N}$ there exists $C=C_{j}$ such that

$\|\partial_{t}^{j}\mathbb{T}(t,\lambda)\|\leq
C_{j},\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0}.$ (9.88)

Using relations (9.12), and the fact that $\mathbb{P}$ and
$\mathbb{A}\mathbb{A}^{*}$ are invertible on $\Lambda_{\neq 0}$ (cf. proof of
Lemma 9.13), conditions (ii) are equivalent to:

$\mathbb{A}U^{*}=0,\qquad\text{and}\qquad
V^{*}=\sqrt{2H}(\mathbb{P}\mathbb{A}\mathbb{A}^{*})^{-1}\mathbb{T}U^{*},$
(9.89)

along $\lambda_{t}$ and where $\mathbb{P}=\mathbb{P}(t,\lambda)$ and
$\mathbb{T}=\mathbb{T}(t,\lambda)$. The second equation of (9.89) determines
$V(t)$ in terms of $U(t)$. The first one is an orthogonality relation between
the rows of $\mathbb{A}(\lambda_{t})$ and those of $U(t)$. Recall from Lemma
9.5 that $\mathbb{A}:T^{*}\mathcal{O}\to\mathrm{M}(n-k,k)$ has constant rank
equal to $n-k$ on $T^{*}\mathcal{O}_{\neq 0}$, while
$U:[0,T]\to\mathrm{M}(2k-n,k)$ must have rank $2k-n$. By applying the Gram-
Schmidt process to a local orthogonal complement to the rows of
$\mathbb{A}(\bar{\lambda})$ we can find a neighborhood
$\Lambda^{\prime}\subset T^{*}\mathcal{O}$ of $\bar{\lambda}$ and a smooth map

$\mathbb{A}^{\perp}:\Lambda^{\prime}_{\neq 0}\to\mathrm{M}(2k-n,k).$ (9.90)

such that

$\mathbb{A}^{\perp}\mathbb{A}^{*}=\mathbb{0},\qquad\mathbb{A}^{\perp}\mathbb{A}^{\perp*}=|u|^{2}\mathbb{1}.$
(9.91)

Furthermore, since $\mathbb{A}$ is $u$-homogeneous of degree one, we can take
$\mathbb{A}^{\perp}$ to be also $u$-homogeneous of degree $1$ and
$\Lambda^{\prime}$ to be $u$-star-shaped. The $u$-star-shaped neighborhood
$\Lambda$ of $\bar{\lambda}$ built at the beginning of the proof of Lemma 9.13
was arbitrary. Up to restriction of $\Lambda$ to a smaller $u$-star-shaped
set, and up to taking a smaller $T$ (and since the flow of $\vec{H}$ is smooth
and preserves $|u|$), we can assume that for all $\lambda\in\Lambda_{\neq 0}$
and $t\in[0,T]$, the extremals $\lambda_{t}=e^{t\vec{H}}(\lambda)$ take value
in a common bounded neighborhood contained in $\Lambda^{\prime}_{\neq 0}$, so
that $\mathbb{A}^{\perp}(\lambda_{t})$ is well-defined. We then set:

$\displaystyle U(t):=\frac{1}{\sqrt{2H}}\mathbb{A}^{\perp},\qquad
V(t):=-\mathbb{A}^{\perp}\mathbb{T}^{*}(\mathbb{P}\mathbb{A}\mathbb{A}^{*})^{-1*},$
(9.92)

where the matrix maps $\mathbb{A}$, $\mathbb{A}^{\perp}$ in the r.h.s. are
evaluated along $\lambda_{t}=e^{t\vec{H}}(\lambda)$ and
$\mathbb{P}=\mathbb{P}(t,\lambda)$, $\mathbb{T}=\mathbb{T}(t,\lambda)$ for
$\lambda\in\Lambda_{\neq 0}$ and $t\in[0,T]$. Since $\mathbb{A}$ and
$\mathbb{A}^{\perp}$ are $u$-homogeneous of degree $1$,
$\mathbb{P}(t,\lambda)$ and $\mathbb{T}(t,\lambda)$ are uniformly bounded with
all time-derivatives and $\lambda\in\Lambda_{\neq 0}$, and using Lemma 9.10,
we observe that for all $j\in\mathbb{N}$ there exists a $C=C_{j}>0$
independent on $\lambda$ such that

$\|\partial_{t}^{j}U(t)\|\leq C,\qquad\|\partial_{t}V(t)\|\leq
CH(\lambda)^{-1/2},\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0}.$
(9.93)

Estimates (9.93) will be used later.

With the definition (9.92), conditions (i) and (ii) are fully verified, while
(iii) corresponds to the following normalization:

$\mathbb{1}=UU^{*}=\frac{1}{2H}\mathbb{A}^{\perp}\mathbb{A}^{\perp*},$ (9.94)

along $\lambda_{t}$, which is verified by our choice in the normalization of
$\mathbb{A}^{\perp}$ in (9.91). This concludes the construction of the tuple
$\bar{E}_{c}$.

Set hence $E_{c}|_{\lambda_{t}}=\psi(t)\cdot\bar{E}_{c}|_{\lambda_{t}}$, for
$\psi(t)\in\mathrm{O}(\mathbb{R}^{2k-n})$. Condition (iv) yields an ODE that
determines $\psi$, once the initial condition is fixed, for example
$\phi(0)=\mathbb{1}$. We obtain

$\dot{\psi}(t)=\frac{1}{2}\psi(t)\sigma(\dot{\bar{E}}_{c}|_{\lambda_{t}},\dot{\bar{E}}_{c}|_{\lambda_{t}}),\qquad\psi(0)=\mathbb{1}.$
(9.95)

Since
$(t,\lambda)\mapsto\sigma(\bar{E}_{c}|_{\lambda_{t}},\bar{E}_{c}|_{\lambda_{t}})$
is skew-symmetric and smooth, and $\mathrm{O}(\mathbb{R}^{2k-n})$ is compact,
(9.95) will have a unique solution in $\mathrm{O}(\mathbb{R}^{2k-n})$ for all
$t\in[0,T]$, and for any $\lambda\in\Lambda_{\neq 0}$. We define then the
smooth map

$\Psi:[0,T]\times\Lambda_{\neq 0}\to\mathrm{O}(\mathbb{R}^{2k-n}),$ (9.96)

such that $t\mapsto\Psi(t,\lambda)$ is the solution of (9.95) on $[0,T]$
corresponding to $\lambda_{t}=e^{t\vec{H}}(\lambda)$.

The matrix in the r.h.s. of (9.95) is given more explicitly by

$\displaystyle\sigma(\dot{\bar{E}}_{c}|_{\lambda_{t}},\dot{\bar{E}}_{c}|_{\lambda_{t}})$
$\displaystyle=U\sigma(\vec{u},\vec{u})U^{*}-2\mathcal{A}\left[(\dot{U}+U\mathbb{C}+V\mathbb{A})U^{*}\right].$
(9.97)

where $\mathcal{A}(M)=\tfrac{1}{2}(M-M^{*})$ denotes the skew-symmetric part
of a matrix $M$, and $U$, $V$ are defined in (9.92). By the uniform estimates
for $U(t)$ and $V(t)$ in (9.93), and since $\mathbb{A}$ is $u$-homogeneous of
degree $1$, we see that
$\sigma(\dot{\bar{E}}_{c}|_{\lambda_{t}},\dot{\bar{E}}_{c}|_{\lambda_{t}})$
and all its time-derivatives remain uniformly bounded for all extremals
$\lambda_{t}=e^{t\vec{H}}(\lambda)$ for $t\in[0,T]$ and
$\lambda\in\Lambda_{\neq 0}$. By Gronwall’s lemma we deduce an analogous
uniform boundedness property of $\Psi$ and its time-derivatives, that is for
all $j\in\mathbb{N}$ there exists $C=C_{j}>0$ such that

$\|\partial_{t}^{j}\Psi(t,\lambda)\|\leq
C,\qquad\forall\,t\in[0,T]\;\lambda\in\Lambda_{\neq 0}.$ (9.98)

Finally, to obtain the statement of the lemma, we set then

$\displaystyle\mathbb{Q}_{6}(t,\lambda)$
$\displaystyle:=\frac{1}{\sqrt{2H(\lambda)}}\Psi(t,\lambda)\mathbb{A}^{\perp}(\lambda_{t}),$
(9.99) $\displaystyle\mathbb{Q}_{7}(t,\lambda)$
$\displaystyle:=-\sqrt{2H(\lambda)}\Psi(t,\lambda)\mathbb{A}^{\perp}(\lambda_{t})\mathbb{T}^{*}(t,\lambda)\left[\mathbb{P}(t,\lambda)\mathbb{A}(\lambda_{t})\mathbb{A}(\lambda_{t})^{*}\right]^{-1*}.$
(9.100)

Using the fact that $\mathbb{A}^{\perp}$ is $u$-homogeneous of degree one,
that $\Psi,\mathbb{T},\mathbb{P}$ are uniformly bounded on
$[0,T]\times\Lambda_{\neq 0}$ with all their time-derivatives, and applying
Lemma 9.10, we obtain the desired uniform bounds on
$\mathbb{Q}_{6},\mathbb{Q}_{7}$.

To prove the analogous statement for the tuple $F_{c}$, we recall that by the
structural equations $F_{c}|_{\lambda_{t}}=-\dot{E}_{c}|_{\lambda_{t}}$. Then,
making use of the Leibniz rule, we obtain the corresponding statement of the
lemma, for suitably defined matrix-valued maps
$\mathbb{Q}_{8},\mathbb{Q}_{9},\mathbb{Q}_{10}$ defined on
$[0,T]\times\Lambda_{\neq 0}$, satisfying the required uniform bounds. ∎

We recall first, that for a general reduced Young diagram $Y$ and for two
superboxes $\boxtimes,\boxplus$ of $Y$, the corresponding curvature map can be
computed as

$R_{t}(\boxtimes,\boxplus)=\sigma(\dot{F}_{\boxtimes}|_{\lambda_{t}},F_{\boxplus}|_{\lambda_{t}}).$
(9.101)

Hence a first series of estimates for some of the curvature maps comes as a
consequence of the previous ones for the canonical frame.

######  Remark 9.16.

Notice that $R_{t}(\boxtimes,\boxplus)$ is a one-parameter family of
$\mathrm{size}(\boxtimes)\times\mathrm{size}(\boxplus)$ matrices representing
the canonical curvature operator for the given choice of the canonical frame
along $\lambda_{t}$. By the uniqueness part of Theorem B.2, and since the
Hilbert-Schmidt norm $\|M\|^{2}=\operatorname{\mathrm{tr}}(MM^{*})$ is
invariant under this action of the orthogonal group, the quantities
$\|\partial_{t}^{j}R_{t}(\boxtimes,\boxplus)\|$ do not depend on choice of the
canonical frame and are thus well-defined estimates for the corresponding
canonical curvature operator $\mathfrak{R}_{\lambda_{t}}^{\boxtimes\boxplus}$
and their time-derivatives.

######  Lemma 9.17 (Curvature estimates I).

For any $\bar{\lambda}\in T^{*}M_{\neq 0}$ there exists an $u$-star-shaped
neighborhood $\Lambda\subset T^{*}M$ of $\bar{\lambda}$ and $T>0$ such that,
for all $j\in\mathbb{N}$, there exists a constant $C=C_{j}>0$ such that for
the canonical curvature map $R_{t}$ along the extremal
$\lambda_{t}=e^{t\vec{H}}(\lambda)$ it holds:

$\|\partial_{t}^{j}R_{t}(\boxtimes,\boxplus)\|\leq
C,\qquad\forall\,t\in[0,T],\,\lambda\in\Lambda_{\neq 0},$ (9.102)

where $\|\cdot\|$ denotes the Hilbert-Schmidt norm, and
$\boxtimes,\boxplus=a,b,c$, with the exclusion of the pairs
$(\boxtimes,\boxplus)=(a,a)$, $(a,c)$ and $(c,a)$.

###### Proof.

Let $T>0$ and $\Lambda$ as in all previous lemmas (9.13, 9.14, 9.15). Take
$\lambda\in\Lambda_{\neq 0}$, $t\in[0,T]$. We routinely omit the evaluation
along the extremal $\lambda_{t}$ for vector fields, and on $(t,\lambda)$ for
maps $\mathbb{Q}_{i}$. Recall from the previous lemma that

$\displaystyle F_{b}$
$\displaystyle=\mathbb{Q}_{3}\cdot\vec{u}+\mathbb{Q}_{4}\cdot\partial_{u}+\frac{1}{\sqrt{2H}}\mathbb{Q}_{5}\cdot\partial_{v},$
(9.103) $\displaystyle F_{c}$
$\displaystyle=\mathbb{Q}_{8}\cdot\vec{u}+\mathbb{Q}_{9}\cdot\partial_{u}+\frac{1}{\sqrt{2H}}\mathbb{Q}_{10}\cdot\partial_{v},$
(9.104)

where $\mathbb{Q}_{i}=\mathbb{Q}_{i}(t,\lambda)$ are uniformly bounded with
all their time-derivatives for $(t,\lambda)\in[0,T]\times\Lambda_{\neq 0}$.
Thanks to these estimates, the only possible singularities as $H\to 0$ can
arise from the $\partial_{v}$-term in the above expressions. Recalling the
symplectic product in the static basis
$\partial_{u},\partial_{v},\vec{u},\vec{v}$ in (9.12), and using Leibniz rule
we see that, for example

$\displaystyle R(b,b)=\sigma(\dot{F}_{b},F_{b})=\text{regular
part}+\frac{1}{\sqrt{2H}}\left(\mathbb{Q}_{3}\sigma(\dot{\vec{u}},\partial_{v})\mathbb{Q}_{5}^{*}+\mathbb{Q}_{5}\sigma(\dot{\partial_{v}},\vec{u})\mathbb{Q}_{3}^{*}\right),$
(9.105)

where the regular part is the multiplication of a finite number of matrix maps
$\mathbb{Q}_{i}$ and their time-derivatives, evaluated at $(t,\lambda)$, and
the smooth matrix maps $\mathbb{A},\mathbb{B},\dots$ from Lemma 9.5 evaluated
along $\lambda_{t}$. In particular the regular part will be bounded with all
its time-derivatives, uniformly for $(t,\lambda)\in[0,T]\times\Lambda_{\neq
0}$.

For what concerns the possibly singular part, recall from Lemma 9.5 that
$\sigma(\dot{\vec{u}},\partial_{v})=-\mathbb{A}^{*}$, and
$\sigma(\dot{\partial_{v}},\vec{u})=\mathbb{A}$. Since
$\mathbb{A}:T^{*}\mathcal{O}_{\neq 0}\to\mathrm{M}(n-k,n-k)$ is
$u$-homogeneous of degree $1$, by Lemma 9.10 the term

$\frac{1}{\sqrt{2H}}\left(\mathbb{Q}_{3}\sigma(\dot{\vec{u}},\partial_{v})\mathbb{Q}_{5}^{*}+\mathbb{Q}_{5}\sigma(\dot{\partial_{v}},\vec{u})\mathbb{Q}_{3}^{*}\right),$
(9.106)

when evaluated along the extremal $\lambda_{t}$, is bounded with all its time-
derivatives, uniformly for $(t,\lambda)\in[0,T]\times\Lambda_{\neq 0}$.

Therefore $R(b,b)$ is uniformly bounded with all its time-derivatives as
required. The same argument proves the similar estimates for $R(c,c)$,
$R(c,b)$ and $R(b,c)=R(c,b)^{*}$.

The estimate of $R(a,b)$ is more delicate. A direct estimates requires the
knowledge of $F_{a}$, which are not in position to estimate yet. However,
using the normal condition in the construction of the canonical frame, one can
determine $R(a,b)$ only using the estimates for $F_{b}$ appearing in Lemma
9.14. Indeed, recall that for a Young diagram with two columns, the normal
condition is (cf. Appendix B):

$R(b,a)=-R(b,a)^{*}.$ (9.107)

Therefore, using (9.115), and the fact that the canonical frame is Darboux, we
obtain

$\displaystyle R(b,a)$
$\displaystyle=\sigma(\dot{F}_{b},F_{a})=-\sigma(\dot{F}_{b},\dot{F}_{b})+\sum_{\mu=a,b,c}\sigma(\dot{F}_{b},E_{\mu})R(b,\mu)^{*}$
(9.108)
$\displaystyle=-\sigma(\dot{F}_{b},\dot{F}_{b})+R(b,a)^{*}=-\sigma(\dot{F}_{b},\dot{F}_{b})-R(b,a),$
(9.109)

from which it follows that

$R(b,a)=-\frac{1}{2}\sigma(\dot{F}_{b},\dot{F}_{b}).$ (9.110)

Using the explicit estimates for $F_{b}$ and its derivative, one can now
estimate $R(b,a)$. Using Leibniz rule and arguing as above we obtain that

$R(b,a)=\text{regular
part}+\frac{1}{\sqrt{2H}}\left[\mathbb{Q}_{3}\sigma(\dot{\vec{u}},\partial_{v})\dot{\mathbb{Q}}_{5}^{*}+\mathbb{Q}_{3}\sigma(\dot{\vec{u}},\dot{\partial_{v}})\mathbb{Q}_{5}^{*}+\dot{\mathbb{Q}}_{3}\sigma(\vec{u},\dot{\partial_{v}})\mathbb{Q}_{5}^{*}\right.\\\
\left.+\dot{\mathbb{Q}}_{5}\sigma(\partial_{v},\dot{\vec{u}})\mathbb{Q}_{3}^{*}+\mathbb{Q}_{5}\sigma(\dot{\partial_{v}},\dot{\vec{u}})\mathbb{Q}_{3}^{*}+\mathbb{Q}_{5}\sigma(\dot{\partial_{v}},\vec{u})\dot{\mathbb{Q}}_{3}^{*}\right],$
(9.111)

where $\dot{\mathbb{Q}}_{i}=\partial_{t}\mathbb{Q}_{i}(t,\lambda)$. Using
Lemma 9.5 we observe that

$\displaystyle\sigma(\dot{\vec{u}},\partial_{v})=-\sigma(\partial_{v},\dot{\vec{u}})^{*}$
$\displaystyle=-\mathbb{A}^{*},$ (9.112)
$\displaystyle\sigma(\dot{\vec{u}},\dot{\partial_{v}})=-\sigma(\dot{\partial_{v}},\dot{\vec{u}})^{*}$
$\displaystyle=-\mathbb{E}\mathbb{A}^{*}+\mathbb{A}^{*}\mathbb{B}^{*},$
(9.113)
$\displaystyle\sigma(\vec{u},\dot{\partial_{v}})=-\sigma(\dot{\partial_{v}},\vec{u})^{*}$
$\displaystyle=-\mathbb{A}^{*}.$ (9.114)

Using again the fact that $\mathbb{A}$ is $u$-homogeneous of degree $1$, and
applying Lemma 9.10 to the possibly singular part in (9.111), we see that all
possible singularities cancel out and also $R(b,a)$ remains uniformly bounded
with all derivatives along $\lambda_{t}$, as claimed.

The estimates from the missing pairs of superboxes in the statement of the
lemma follow by the global symmetry of the curvature maps
$R(\boxtimes,\boxplus)=R(\boxplus,\boxtimes)^{*}$. ∎

We can now prove the estimate for the last element of the canonical frame,
which is the $n-k$-tuple $F_{a}$. Using the structural equations, we have the
general formula:

$F_{a}|_{\lambda_{t}}=-\dot{F}_{b}|_{\lambda_{t}}+\sum_{\boxtimes=a,b,c}R_{t}(b,\boxtimes)\cdot
E_{\boxtimes}|_{\lambda_{t}}.$ (9.115)

The curvatures $R_{t}(b,\boxtimes)$ in (9.115) are precisely those estimated
in Lemma 9.17.

######  Lemma 9.18.

For any $\bar{\lambda}\in T^{*}M_{\neq 0}$ there exists an $u$-star-shaped
neighborhood $\Lambda\subset T^{*}M$ of $\bar{\lambda}$, $T>0$, and smooth
maps $\mathbb{Q}_{i}:[0,T]\times\Lambda_{\neq 0}\to\mathrm{M}(n_{i},m_{i})$,
with $n_{i},m_{i}\in\mathbb{N}$, $i=11,\dots,14$, such that for all
$\lambda\in\Lambda_{\neq 0}$ the tuple

$F_{a}|_{\lambda_{t}}=\mathbb{Q}_{11}(t,\lambda)\cdot\vec{u}|_{\lambda_{t}}+\sqrt{2H(\lambda)}\mathbb{Q}_{12}(t,\lambda)\cdot\vec{v}|_{\lambda_{t}}\\\
+\mathbb{Q}_{13}(t,\lambda)\cdot\partial_{u}|_{\lambda_{t}}+\frac{1}{\sqrt{2H(\lambda)}}\mathbb{Q}_{14}(t,\lambda)\cdot\partial_{v}|_{\lambda_{t}},$
(9.116)

for $t\in[0,T]$, is (a choice of) the $F_{a}$-component of the canonical frame
along $\lambda_{t}=e^{t\vec{H}}(\lambda)$.

Furthermore, the maps $\mathbb{Q}_{i}$ and all their time-derivatives are
bounded, that is for all $j\in\mathbb{N}$ there exists a constant $C=C_{j}>0$
such that

$\|\partial_{t}^{j}\mathbb{Q}_{i}(t,\lambda)\|\leq
C,\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0},\;i=11,\dots,14,$
(9.117)

where $\|\cdot\|$ denotes a fixed matrix norm.

###### Proof.

By the structural equations we have, along any given extremal
$\lambda_{t}=e^{t\vec{H}}(\lambda)$:

$F_{a}=-\dot{F}_{b}+R(b,b)\cdot E_{b}+R(b,a)\cdot E_{a}+R(b,c)\cdot E_{c}.$
(9.118)

The claim is now a consequence of lemmas 9.13, 9.14, 9.15 for the previous
parts of the canonical frame and Lemma 9.17 for estimate on the curvature maps
appearing in (9.118). Using Leibniz rule and omitting as usual evaluation
along $\lambda_{t}$ and $(t,\lambda)$ we obtain

$F_{a}=\left(\dot{\mathbb{Q}}_{3}+\mathbb{Q}_{3}\mathbb{E}-\mathbb{Q}_{4}\right)\cdot\vec{u}-\mathbb{Q}_{3}\mathbb{A}^{*}\cdot\vec{v}+\bigg{(}\mathbb{Q}_{3}\mathbb{F}+\dot{\mathbb{Q}}_{4}+\mathbb{Q}_{4}\mathbb{C}+\frac{1}{\sqrt{2H}}\mathbb{Q}_{5}\mathbb{A}\\\
+R(b,b)\mathbb{Q}_{1}+R(b,c)\mathbb{Q}_{6}\bigg{)}\cdot\partial_{u}+\bigg{(}\mathbb{Q}_{3}\mathbb{G}+\mathbb{Q}_{4}\mathbb{D}+\frac{1}{\sqrt{2H}}\dot{\mathbb{Q}}_{5}+\frac{1}{\sqrt{2H}}\mathbb{Q}_{5}\mathbb{B}\\\
+\frac{1}{\sqrt{2H}}R(b,b)\mathbb{Q}_{2}+\frac{1}{\sqrt{2H}}R(b,a)\mathbb{P}+\frac{1}{\sqrt{2H}}R(b,c)\mathbb{Q}_{7}\bigg{)}\cdot\partial_{v}.$
(9.119)

Using the $u$-homogeneity of $\mathbb{A}$ coupled with Lemma 9.10, we obtain
the required uniform estimates, for $(t,\lambda)\in[0,T]\times\Lambda_{\neq
0}$, where $T$ and $\Lambda$ are as in the previous lemmas. ∎

We are ready to complete the estimates on the canonical curvature maps.

######  Lemma 9.19 (Curvature estimates II).

For any $\bar{\lambda}\in T^{*}M$ there exists an $u$-star-shaped neighborhood
$\Lambda\subset T^{*}M_{\neq 0}$ of $\bar{\lambda}$, and $T>0$ such that, for
all $j\in\mathbb{N}$ there exists a constant $C=C_{j}>0$ such that for the
canonical curvature maps $R_{t}$ along the extremal
$\lambda_{t}=e^{t\vec{H}}(\lambda)$ it holds:

$\|\partial_{t}^{j}R_{t}(\boxtimes,\boxplus)\|\leq
C,\qquad\forall\,t\in[0,T],\;\lambda\in\Lambda_{\neq 0},$ (9.120)

where $\|\cdot\|$ denotes the Hilbert-Schmidt norm, and
$\boxtimes,\boxplus=a,b,c$.

###### Proof.

We only need to prove the statement for the pairs $\boxtimes,\boxplus$
excluded in Lemma 9.17, that is the pairs $(a,a)$, $(a,c)$ and $(c,a)$. We
sketch the proof since the argument is analogous to the one in Lemma 9.17 for
the first curvature estimates, coupled with the new piece of information on
$F_{a}$ coming from Lemma 9.18.

Let $T>0$ and $\Lambda$ so that all previous lemmas and estimates about the
canonical frame hold. Then we decomposing, for example

$R(a,a)=\sigma(\dot{F}_{a},F_{a})=\text{regular part}+\text{possibly singular
part},$ (9.121)

where the possibly singular part contains all terms with a factor containing a
positive power of $H^{-1/2}$. Then, using the estimates for $F_{a}$, Lemma 9.5
and Lemma 9.10, one can prove that all singularities cancel out. We stress in
particular that the presence of the factor $\sqrt{2H}$ in the
$\vec{v}$-component of $F_{a}$ of Lemma 9.18 is necessary for these
cancellations. One then argues similarly for
$R(c,a)=\sigma(\dot{F}_{c},F_{a})$ and $R(a,c)=\sigma(\dot{F}_{a},F_{c})$. ∎

We now estimate the geodesic volume derivative. Recall that the fat sub-
Riemannian manifold $M$ comes equipped with a Riemannian extension. We denote
with $\mathrm{vol}$ the associated Riemannian density, while $M$ is equipped
with a smooth reference volume $\mathfrak{m}=e^{-V}\mathrm{vol}$, where
$V:M\to\mathbb{R}$ is a smooth function. The geodesic volume derivative with
respect to $\mathfrak{m}$ is the smooth function
$\rho_{\mathfrak{m}}:T^{*}M_{\neq 0}\to\mathbb{R}$, defined by

$\rho_{\mathfrak{m},\lambda}:=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\log\mathfrak{m}(\pi_{*}F_{a}|_{\lambda_{t}},\pi_{*}F_{b}|_{\lambda_{t}},\pi_{*}F_{c}|_{\lambda_{t}}),\qquad\forall\,\lambda\in
T^{*}M_{\neq 0},$ (9.122)

for any choice of canonical frame along $\lambda_{t}=e^{t\vec{H}}(\lambda)$,
see Section B.3.

######  Lemma 9.20 (Volume derivative estimates).

For any $\bar{\lambda}\in T^{*}M_{\neq 0}$ there exists an $u$-star-shaped
neighborhood $\Lambda\subset T^{*}M_{\neq 0}$ of $\bar{\lambda}$ such that,
for all $j\in N$ there exists a constant $C=C_{j}>0$ such that for the
geodesic volume derivative $\rho_{\mathfrak{m},\lambda}$ it holds:

$\vec{H}^{(j)}\rho_{\mathfrak{m},\lambda}\leq
C,\qquad\forall\,\lambda\in\Lambda_{\neq 0}.$ (9.123)

###### Proof.

Let $\Lambda$ and $T>0$ as in the previous lemmas. By construction, for all
$\lambda\in\Lambda_{\neq 0}$ the extremal $\lambda_{t}$ remains for all
$t\in[0,T]$ in a common bounded subset of a trivial neighborhood
$T^{*}\mathcal{O}$, and the corresponding geodesic
$\gamma_{t}=\pi(\lambda_{t})$ remains in $\mathcal{O}$. Recall that from
Lemmas 9.14, 9.15 and 9.18 we have

$\displaystyle\pi_{*}F_{a}|_{\lambda_{t}}$
$\displaystyle=\mathbb{Q}_{11}(t,\lambda)X|_{\gamma_{t}}+\sqrt{2H(\lambda)}\mathbb{Q}_{12}(t,\lambda)Z_{\gamma_{t}},$
(9.124) $\displaystyle\pi_{*}F_{b}|_{\lambda_{t}}$
$\displaystyle=\mathbb{Q}_{3}(t,\lambda)X|_{\gamma_{t}},$ (9.125)
$\displaystyle\pi_{*}F_{c}|_{\lambda_{t}}$
$\displaystyle=\mathbb{Q}_{8}(t,\lambda)X|_{\gamma_{t}},$ (9.126)

for all $t\in[0,T]$, and where $X=(X_{1},\dots,X_{k})$,
$Z=(X_{k+1},\dots,X_{n})$ are tuples forming an adapted orthonormal frame for
$g$ on $\mathcal{O}$. In particular since $\mathfrak{m}(X,Z)=1$, we have

$\displaystyle\rho_{\mathfrak{m},\lambda_{t}}=-\vec{H}(V)(\lambda_{t})+\frac{\mathrm{d}}{\mathrm{d}t}\log\left\lvert\det\mathbb{Q}_{12}(t,\lambda)\det\begin{matrix}\mathbb{Q}_{3}(t,\lambda)\\\
\mathbb{Q}_{8}(t,\lambda)\end{matrix}\right\rvert,$ (9.127)

for all $\lambda\in\Lambda_{\neq 0}$ and $t\in[0,T]$. Notice that by
construction $\mathbb{Q}_{12}$ is a $n-k\times n-k$ matrix of full rank, while
$\left(\begin{smallmatrix}\mathbb{Q}_{3}\\\
\mathbb{Q}_{8}\end{smallmatrix}\right)$ is a $k\times k$ matrix of full rank.
To prove the statement, observe that
$\vec{H}^{(j)}\rho_{\mathfrak{m},\lambda}=\partial_{t}^{j}\rho_{\mathfrak{m},\lambda_{t}}|_{t=0}$,
so that one has only to bound the time-derivatives of (9.127) at $t=0$ and
$\lambda\in\Lambda_{\neq 0}$. The first term $\vec{H}(V)(\lambda_{t})$ is
uniformly bounded with all time-derivatives since all curves $\lambda_{t}$
remain in a common bounded subset of the trivial neighborhood
$T^{*}\mathcal{O}$ by construction, and $V$ is smooth. The second term in
(9.127) is also uniformly bounded with all time-derivatives, by the properties
of the $\mathbb{Q}_{i}$’s. ∎

###  9.4 Regularity of the canonical curvature maps

We collect the local estimates from Lemmas 9.17–9.19 in a unified statement
for the canonical curvature operators defined as in Section B.2.

######  Remark 9.21.

The fact that the neighborhoods $\Lambda$ of Lemmas from 9.13 to 9.19 are
$u$-star-shaped will be used now.

######  Theorem 9.22 (Fat curvature estimates).

Let $(\mathscr{D},g)$ be a fat sub-Riemannian structure on $M$. Then for any
compact set $B\subset M$ and $j\in\mathbb{N}$ there exists a constant
$C=C_{j}(B)>0$ such that, for all superboxes $\boxtimes,\boxplus=a,b,c$ of the
Young diagram $Y$ it holds

$\|\vec{H}^{(j)}\mathfrak{R}_{\lambda}^{\boxtimes\boxplus}\|\leq
CH_{R}(\lambda)^{\tfrac{j+\mathrm{col}(\boxtimes)+\mathrm{col}(\boxplus)}{2}},\qquad\forall\,\lambda\in
T^{*}B_{\neq 0},$ (9.128)

where $\|\cdot\|$ denotes the Hilbert-Schmidt norm, and
$\mathrm{col}(\boxtimes)$ is the column index of the superbox. As a
consequence for some $C^{\prime}=C^{\prime}_{j}(B)>0$ it also holds

$|\vec{H}^{(j)}\mathfrak{Ric}_{\lambda}^{\boxtimes}|\leq
C^{\prime}H_{R}(\lambda)^{\tfrac{j}{2}+\mathrm{col}(\boxtimes)},\qquad\forall\,\lambda\in
T^{*}B_{\neq 0}.$ (9.129)

Furthermore, if $\mathfrak{m}=e^{-V}\mathrm{vol}$, for the geodesic volume
derivative it holds

$|\vec{H}^{(j)}\rho_{\mathfrak{m},\lambda}|\leq
CH_{R}(\lambda)^{\tfrac{j+1}{2}},\qquad\forall\,\lambda\in T^{*}B_{\neq 0}.$
(9.130)

###### Proof.

With no loss of generality we assume that $B$ is properly contained in a
trivial neighborhood $\mathcal{O}$ such that

$T^{*}\mathcal{O}=\mathcal{O}\times\mathbb{R}^{k}\times\mathbb{R}^{n-k},\qquad\lambda=(x;u,v),$
(9.131)

and for the sub-Riemannian and Riemannian Hamiltonians it holds:

$H(x;u,v)=\frac{1}{2}|u|^{2},\qquad
H_{R}(x;u,v)=\frac{1}{2}(|u|^{2}+|v|^{2}).$ (9.132)

Take the following compact set $K\subseteq T^{*}B$:

$K=\\{(x;u,v)\in
B\times\mathbb{R}^{k}\times\mathbb{R}^{n-k}\mid|u|^{2}+|v|^{2}=1\\}.$ (9.133)

We can find a finite number of neighborhoods $\Lambda^{i}$, $i=1,\dots,N$ as
in Lemmas 9.17–9.19, that cover $K_{\neq 0}$. This is possible, even if
$K_{\neq 0}$ is non-compact, since the neighborhoods $\Lambda$ obtained in the
previous lemmas are $u$-star-shaped (otherwise, their size in the
$u$-coordinate might get smaller and smaller as $u\to 0$ if they were not
$u$-star-shaped). We deduce that for all $j\in\mathbb{N}$ there exists
$C=C_{j}>0$ such that

$\|\partial_{t}^{j}\mathfrak{R}_{\lambda_{t}}^{\boxtimes\boxplus}|_{t=0}\|\leq
C,\qquad\forall\,t\in[0,T],\;\lambda\in K_{\neq 0},$ (9.134)

where $\lambda_{t}=e^{t\vec{H}}(\lambda)$. Recall the homogeneity properties
of the curvature from [23, Thm. 4.7]: for any $c>0$, and up to constant
congruence (cf. Remark 9.16), it holds

$\mathfrak{R}_{\lambda^{c}_{t}}^{\boxtimes\boxplus}\simeq
c^{\mathrm{col}(\boxtimes)+\mathrm{col}(\boxplus)}\mathfrak{R}_{\lambda_{ct}}^{\boxtimes\boxplus},\qquad\forall\,\lambda\in
T^{*}M_{\neq 0},$ (9.135)

where $\lambda^{c}_{t}=e^{t\vec{H}}(c\lambda)$. In particular taking
derivatives with respect to $t$, and evaluating at $t=0$, we have

$\partial_{t}\mathfrak{R}_{\lambda_{t}}^{\boxtimes\boxplus}|_{t=0}\simeq
c^{\mathrm{col}(\boxtimes)+\mathrm{col}(\boxplus)+j}\partial_{t}\mathfrak{R}_{\eta_{t}}^{\boxtimes\boxplus}|_{t=0},\qquad\forall\,\lambda\in
T^{*}M_{\neq 0},$ (9.136)

where $\eta=\lambda/c$. The statement then follows by reduction to $K_{\neq
0}$, since any point $\lambda\in T^{*}\mathcal{O}_{\neq 0}$ has the form
$c\eta$ with $\eta\in K_{\neq 0}$ and $c=\sqrt{2H_{R}(\lambda)}$.

The statement concerning the canonical Ricci curvatures follows immediately
as, for any superbox $\boxtimes\in Y$ and $\lambda\in T^{*}M_{\neq 0}$ it
holds by definition:
$\mathfrak{Ric}_{\lambda}^{\boxtimes}=\operatorname{\mathrm{tr}}(\mathfrak{R}_{\lambda}^{\boxtimes\boxtimes})$.

The final statement about the geodesic volume derivative follows in a similar
fashion, starting from Lemma 9.20 and using the fact that the following
property holds:

$\rho_{\mathfrak{m},\lambda^{c}_{t}}=c\,\rho_{\mathfrak{m},\lambda_{ct}},\qquad\forall\,\lambda\in
T^{*}M_{\neq 0},$ (9.137)

which is immediately obtained by using [23, Prop. 4.9] in (9.122). ∎

###  9.5 Proof of Theorem 9.2

We can now prove Theorem 9.2. In fact, it is a corollary of Theorem 9.22.

###### Proof of Theorem 9.2.

If $x,y\notin\mathrm{Cut}(M)$, we have by Proposition 7.28 that

$\mathsf{D}(x,y)^{2}=\|\nabla^{R}_{x}c(\cdot,y)\|_{R}^{2}=2H_{R}(\mathrm{d}_{x}c(\cdot,y))=2H_{R}(\lambda^{x,y}),$
(9.138)

where $\lambda^{x,y}=\mathrm{d}_{x}c(\cdot,y)\in T_{x}^{*}M_{\neq 0}$ is the
initial covector of the unique strictly normal geodesic $\gamma:[0,1]\to M$
joining $x$ with $y$. Notice that the first equality follows by the very
definition of $\mathsf{D}$ in the sub-Riemannian case and by the fact that $c$
is smooth outside of $\mathrm{Cut}(M)$. The second equality is obvious by the
usual duality between norm and Hamiltonian. In the third equality we used the
fact that the initial covector can be recovered by the differential of the
squared distance (cf. for example [3, Lemma 2.20(ii)]).

Let $K\subset M$ be a compact set as in the statement of Theorem 9.2. Let $B$
be a larger compact set such that for any $x,y\in K$ all geodesics joining
them belong to $B$. On $B$ we apply Theorem 9.22. In particular if $\lambda\in
T_{x}^{*}M$ is the initial covector of the unique geodesic $\gamma$ joining
$x,y\in K\setminus\mathrm{Cut}(M)$ there exists $C^{\prime}>0$ such that

$|\mathfrak{Ric}_{\lambda}^{\boxdot}|\leq
C^{\prime}H_{R}(\lambda)^{\mathrm{col}(\boxdot)}=\frac{C^{\prime}}{2^{\mathrm{col}(\boxdot)}}\mathsf{D}(x,y)^{2\mathrm{col}(\boxdot)},$
(9.139)

and where $\boxdot=a,b,c$. This (stronger) inequality in particular implies
the desired lower bounds on $\mathfrak{Ric}$ by recalling that
$\mathrm{col}(a)=\mathrm{col}(b)=2$ and $\mathrm{col}(c)=1$.

In an analogous way one proves the statement about the geodesic volume
derivative starting from the corresponding estimate in Theorem 9.22. ∎

###  9.6 Compact fat structures satisfy the $\mathsf{CD}(\beta,n)$ condition

We combine the curvature estimates of Theorem 9.2 for fat structures, and
Theorem 8.12, to obtain the main result of this section.

Recall that $\mathsf{s}_{\kappa}:[0,+\infty)\to\mathbb{R}$, for
$\kappa\in\mathbb{R}^{\ell}$, $\ell\in\mathbb{N}$, are the basic comparison
functions of Proposition 8.5, associated with a Young diagram of length
$\ell$. In particular:

  * •

for $\ell=1$, $\mathsf{s}_{\kappa}$ is a Riemannian-type function (see Section
8.2.1);

  * •

for $\ell=2$, $\mathsf{s}_{\kappa_{1},\kappa_{2}}$ is a two-columns-type
function (see Section 8.2.3).

In all cases, the $\mathsf{s}_{\kappa}$ are real-analytic and have order
$N=\ell^{2}$ at zero.

######  Theorem 9.23.

Let $(M,\mathsf{d},\mathfrak{m})$ be a compact $n$-dimensional sub-Riemannian
metric measure space, with fat distribution of rank $k<n$. Let $\mathsf{D}$ be
the natural gauge function associated with a Riemannian extension. There exist
constants $C,\kappa_{a},\kappa_{b},\kappa_{c}\in\mathbb{R}$ with the following
property: let $\mathsf{s}:[0,+\infty)\to\mathbb{R}$ be the real-analytic
function

$\mathsf{s}(\theta):=\theta\cdot
e^{C\theta}\cdot\mathsf{s}_{\kappa_{b},\kappa_{a}}(\theta)^{n-k}\cdot\mathsf{s}_{\kappa_{c}}(\theta)^{2k-n-1},$
(9.140)

and let $\beta$ be defined accordingly as in (3.14). Then
$(M,\mathsf{d},\mathfrak{m},\mathsf{D})$ satisfies the $\mathsf{CD}(\beta,n)$.

Since $\mathsf{s}$ is real-analytic, using Proposition 3.4(vii), we obtain the
following.

######  Corollary 9.24.

Let $(M,\mathsf{d},\mathfrak{m})$ be a compact $n$-dimensional sub-Riemannian
metric measure space, with fat distribution of rank $k<n$. Then there exists
$N^{\prime}\geq 3n-2k$ such that $(M,\mathsf{d},\mathfrak{m})$ satisfies the
classical $\mathsf{MCP}(0,N^{\prime})$.

######  Remark 9.25.

Both these results can be seen as a sub-Riemannian counterpart of the fact
that any compact Riemannian manifold has Ricci curvature bounded from below.

######  Remark 9.26.

Corollary 9.24 removes the real-analytic assumption of the analogous result
proved in [18, Thm. 1.3], in the case when the underlying distribution is fat.

######  Remark 9.27.

Fat sub-Riemannian structures are ideal and their step is $2$. In particular
$\mathsf{D}$ is locally bounded, $\mathsf{D}$ is meek, and the regularity
conditions for the stability of the $\mathsf{CD}$ and $\mathsf{MCP}$ hold. See
Figure 2.

###### Proof of Theorem 9.23.

For a fat structure, all normal extremals corresponding to non-trivial
geodesics have the same Young diagram, with two columns. It has two levels,
denoted by $I$ and $II$, the first one with length $\ell_{I}=2$ and size
$r_{I}=n-k$, while the second one with length $\ell_{II}=1$ and size
$r_{II}=2k-n$. We label the superboxes as follows:

$\ytableausetup{nosmalltableaux}\ytableausetup{centertableaux}Y=\ytableau
b&a\\\ c\\\ .$ (9.141)

By Theorem 9.2, and since $M$ is compact, we find
$\kappa_{a},\kappa_{b},\kappa_{c},C\in\mathbb{R}$ such that, for any
$(x,y)\notin\mathrm{Cut}(M)$ and the corresponding geodesic $\gamma:[0,1]\to
M$ joining them with initial covector $\lambda\in T^{*}_{x}M$, it holds

$\displaystyle\frac{1}{n-k}\mathfrak{Ric}_{\lambda}^{a}\geq\kappa_{a}\mathsf{D}(x,y)^{4},\qquad\frac{1}{n-k}\mathfrak{Ric}^{b}_{\lambda}\geq\kappa_{b}\mathsf{D}(x,y)^{2},$
(9.142)
$\displaystyle\frac{1}{2k-n-1}\mathfrak{Ric}_{\lambda}^{c}\geq\kappa_{c}\mathsf{D}(x,y)^{2},\qquad\rho_{\mathfrak{m},\lambda}\leq
C\mathsf{D}(x,y),$ (9.143)

the bound on $\mathfrak{Ric}_{\lambda}^{c}$ being omitted if $2k-n=1$. Thus it
follows that $(M,\mathsf{d},\mathfrak{m})$, equipped with the gauge function
$\mathsf{D}$, has Ricci curvatures bounded from below according to Definition
8.11, with the above Young diagram and

$\displaystyle\bar{\kappa}_{I}$ $\displaystyle:[0,+\infty)\to\mathbb{R}^{2},$
$\displaystyle\bar{\kappa}_{I}(\mathsf{D})$
$\displaystyle:=(\kappa_{b}\mathsf{D}^{2},\kappa_{a}\mathsf{D}^{4}),$ (9.144)
$\displaystyle\bar{\kappa}_{II}$ $\displaystyle:[0,+\infty)\to\mathbb{R},$
$\displaystyle\bar{\kappa}_{II}(\mathsf{D})$
$\displaystyle:=\kappa_{c}\,\mathsf{D}^{2},$ (9.145) $\displaystyle\bar{C}$
$\displaystyle:[0,+\infty)\to\mathbb{R},$ $\displaystyle\bar{C}(\mathsf{D})$
$\displaystyle:=C\mathsf{D}.$ (9.146)

We conclude by applying Theorem 8.12 obtaining that
$(M,\mathsf{d},\mathfrak{m},\mathsf{D})$ is a $\mathsf{CD}(\beta,n)$ space.
The distortion coefficients $\beta$ is induced by the function
$\mathsf{s}:\mathbb{R}^{m}_{+}\to\mathbb{R}$ of (8.56). To describe the latter
explicitly, we recall that we are in the scalar case $m=1$, and thus by Remark
8.7 the function $\mathsf{s}:[0,+\infty)\to\mathbb{R}$ is given by

$\mathsf{s}(\theta)=\theta\cdot
e^{C\theta}\cdot\mathsf{s}_{\kappa_{b},\kappa_{a}}(\theta)^{n-k}\cdot\mathsf{s}_{\kappa_{c}}(\theta)^{2k-n-1},\qquad\theta\in[0,+\infty).$
(9.147)

The positivity domain of $\mathsf{s}$ is given by

${\rm{DOM}}={\rm{DOM}}_{\bar{\kappa}_{I}}\cap{\rm{DOM}}_{\bar{\kappa}_{II}}=[0,\min\\{t_{\kappa_{c}},t_{\kappa_{b},\kappa_{a}}\\}],$
(9.148)

and $\mathsf{s}$ has order $N=1+(n-k)\ell_{I}^{2}+(2k-n-1)\ell_{II}^{2}=3n-2k$
as $\theta\to 0$. ∎

##  10 Examples and applications

###  10.1 Heisenberg group

In this section we describe the results already presented in Section 3.3.2, in
the special three-dimensional case, in order to make a bridge with the
comparison theory developed in the previous sections.

The first Heisenberg group $\mathbb{H}^{1}$ is the non-commutative group
structure on $\mathbb{R}^{3}$ given by the law

$(x,y,z)\star(x^{\prime},y^{\prime},z^{\prime})=\left(x+x^{\prime},y+y^{\prime},z+z^{\prime}+\sum_{i=1}^{d}\frac{1}{2}(xy^{\prime}-yx^{\prime})\right).$
(10.1)

Consider the left-invariant vector fields

$X=\frac{\partial}{\partial x}-\frac{y}{2}\frac{\partial}{\partial z},\qquad
Y=\frac{\partial}{\partial y}+\frac{x}{2}\frac{\partial}{\partial z},$ (10.2)

and the left-invariant metric making $\\{X,Y\\}$ a global orthonormal frame.
We equip $\mathbb{H}^{1}$ with the associated sub-Riemannian (Carnot-
Carathéodory) distance, denoted by $\mathsf{d}_{0}$ (this notation is slightly
different from the one used in Section 3.3.2, but it is more convenient in
what follows). We also equip $\mathbb{H}^{1}$ with the Lebesgue measure
$\mathscr{L}^{3}$, which a Haar measure. We set

$\mathsf{s}(\theta)=\theta\sin\left(\frac{\theta}{2}\right)\left[\sin\left(\frac{\theta}{2}\right)-\frac{\theta}{2}\cos\left(\frac{\theta}{2}\right)\right],\qquad\theta\in[0,+\infty).$
(10.3)

Notice that $\mathsf{s}(\theta)$ has finite order equal to $N=5$ as $\theta\to
0$. Moreover, it is clear that

$\mathcal{D}=\inf\\{\theta>0\mid\mathsf{s}(\theta)=0\\}=2\pi.$ (10.4)

The corresponding model distortion coefficient defined as in (3.14) is then

$[0,1]\times[0,\infty]\ni(t,\theta)\mapsto\beta^{\mathbb{H}^{1}}_{t}(\theta)=\begin{cases}t^{5}\quad&\theta=0,\\\
t\dfrac{\sin\left(\frac{t\theta}{2}\right)\left(\sin\left(\frac{t\theta}{2}\right)-\frac{t\theta}{2}\cos\left(\frac{t\theta}{2}\right)\right)}{\sin\left(\frac{\theta}{2}\right)\left(\sin\left(\frac{\theta}{2}\right)-\frac{\theta}{2}\cos\left(\frac{\theta}{2}\right)\right)}&0<\theta<2\pi,\\\
+\infty&\theta\geq 2\pi,t\neq 0,\\\ 0&\theta\geq 2\pi,t=0.\\\ \end{cases}$
(10.5)

The main result in [19, Thm. 1.1] is a Jacobian determinant inequality which
yields by standard manipulations an interpolation inequality for optimal
transport, see [19, Eq. (3.17), p. 61]. More precisely, for all
$\mu_{0},\mu_{1}\in\mathcal{P}_{bs}^{*}(\mathbb{H}^{1},\mathsf{d}_{0},\mathscr{L}^{3})$,
there exists $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ associated with a
$W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ such that $\mu_{t}\ll\mathscr{L}^{3}$
for all $t\in(0,1]$ and for $\nu$-a.e. $\gamma$ it holds

$\frac{1}{\rho_{t}(\gamma_{t})^{1/3}}\geq\frac{\beta^{\mathbb{H}^{1}}_{1-t}(\theta^{\gamma_{1},\gamma_{0}})^{1/3}}{\rho_{0}(\gamma_{0})^{1/3}}+\frac{\beta^{\mathbb{H}^{1}}_{t}(\theta^{\gamma_{0},\gamma_{1}})^{1/3}}{\rho_{1}(\gamma_{1})^{1/3}},\qquad\forall\,t\in[0,1],$
(10.6)

where $\rho_{t}=\frac{\mathrm{d}\mu_{t}}{\mathrm{d}\mathscr{L}^{3}}$, and
$\theta^{x,y}$ is the vertical norm of the covector associated with the
geodesic joining $x,y$. In (10.6) we use the convention that if $\mu_{0}$ is
not absolutely continuous then the first term in the right hand side is
omitted.

Notice that the argument in the r.h.s. of (10.6) does not depend on the
distance. It is then natural to set as a gauge function the map
$\mathsf{G}:\mathbb{H}^{1}\times\mathbb{H}^{1}\to[0,+\infty]$ by

$\mathsf{G}(x,y):=\theta^{x,y}.$ (10.7)

By standard arguments (cf. again Section 3.3.2), the results of [19] can be
restated within our framework as the following $\mathsf{CD}$ condition.

######  Theorem 10.1.

The gauge m.m.s. $(\mathbb{H}^{1},\mathsf{d}_{0},\mathscr{L}^{3},\mathsf{G})$
satisfies the $\mathsf{CD}(\beta^{\mathbb{H}^{1}},3)$ condition.

One can check that $\beta^{\mathbb{H}^{1}}_{t}(\theta)\geq t^{5}$ for all
$t\in[0,1]$ and $\theta\in[0,2\pi]$. Thus, by Proposition 3.10(iii),
$\mathsf{CD}(\beta^{\mathbb{H}^{1}},3)\Rightarrow\mathsf{CD}(t^{5},3)$. In
turn, the latter implies the classical $\mathsf{MCP}(0,5)$.

####  10.1.1 Application of the comparison theory to the Heisenberg group

To make the link with the curvature computations of Sections 8-9, notice that
for every pair $x,y\in\mathbb{H}^{1}\setminus\mathrm{Cut}(\mathbb{H}^{1})$ and
for the geodesic $\gamma:[0,1]\to\mathbb{H}^{1}$ joining them, the (reduced)
Young diagram $Y$ along $\gamma$ has two columns which can be labeled as
follows

$Y=\ytableausetup{centertableaux}\ytableau b&a\\\ c\\\ $

with all sizes of the superboxes equal to 1. The sub-Riemannian Ricci
curvatures have been computed in [5], and are:

$\displaystyle\mathfrak{Ric}^{a}_{\lambda}=\mathfrak{Ric}^{c}_{\lambda}=0,\qquad\mathfrak{Ric}^{b}_{\lambda}$
$\displaystyle=\theta^{2},$ (10.8)

where, letting

$\theta=\theta^{x,y}=|\langle\lambda^{x,y},\partial_{z}\rangle|,$ (10.9)

is the absolute value of the $z$-component of the unique initial covector
$\lambda^{x,y}$ of the geodesic $\gamma$ joining $x$ and $y$ with constant
speed on the interval $[0,1]$. We stress that, for such geodesics,
$\theta<2\pi$. We have (cf. Section 8.2.2)

$\mathsf{\mathsf{s}}_{\kappa_{1},0}(t)=\frac{2-2\cos\left(\sqrt{\kappa_{1}}t\right)-\sqrt{\kappa_{1}}t\sin\left(\sqrt{\kappa_{1}}t\right)}{\kappa_{1}^{2}},\qquad
t_{\kappa_{1},0}=\begin{cases}\tfrac{2\pi}{\sqrt{\kappa_{1}}}&\kappa_{1}>0,\\\
+\infty&\kappa_{1}\leq 0.\end{cases}$ (10.10)

From Theorem 8.9, one obtains the following comparison theorem

$\displaystyle\beta_{t}^{(\mathbb{H}^{1},\mathsf{d},\mathfrak{m})}(x,y)$
$\displaystyle\geq
t\cdot\frac{2-2\cos\left(t\theta\right)-t\theta\sin\left(t\theta\right)}{2-2\cos\left(\theta\right)-\theta\sin\left(\theta\right)}$
(10.11)
$\displaystyle=t\cdot\frac{\sin\left(\frac{t\theta}{2}\right)\left[\sin\left(\frac{t\theta}{2}\right)-\frac{t\theta}{2}\cos\left(\frac{t\theta}{2}\right)\right]}{\sin\left(\frac{\theta}{2}\right)\left[\sin\left(\frac{\theta}{2}\right)-\frac{\theta}{2}\cos\left(\frac{\theta}{2}\right)\right]}$
(10.12) $\displaystyle=\beta^{\mathbb{H}^{1}}_{t}(\theta),$ (10.13)

where $\beta^{\mathbb{H}^{1}}$ is the expression given in Section 3.3.2.

###  10.2 Canonical variation

In this section we consider what is called the _canonical variation_ of the
Heisenberg group, namely the one-parameter family of left-invariant Riemannian
distances $\mathsf{d}_{\varepsilon}$ on $\mathbb{H}^{1}$, which are those
associated with a left-invariant Riemannian metric for which

$\\{X,Y,\varepsilon Z\\},\qquad\varepsilon>0,$ (10.14)

is orthonormal, where

$X=\partial_{x}-\frac{y}{2}\partial_{z},\qquad
Y=\partial_{y}+\frac{x}{2}\partial_{z},\qquad Z=\partial_{z}.$ (10.15)

The terminology canonical variation is used in the context of Riemannian
submersions [32]. Only for this subsection, it will be handy to denote by
$\mathsf{d}_{0}$ the Carnot-Carathéodory metric, so that we study the family
of metric measure spaces
$(\mathbb{H}^{1},\mathsf{d}_{\varepsilon},\mathscr{L}^{3})$ for all
$\varepsilon\geq 0$.

We remark that the Riemannian Ricci curvature of
$(\mathbb{H}^{1},\mathsf{d}_{\varepsilon})$ is unbounded as $\varepsilon\to
0$. Therefore it is convenient to study the distortion coefficients of the
corresponding metric measure spaces by direct methods. To this purpose, let
$\beta_{t}^{\varepsilon}:\mathbb{H}^{1}\times\mathbb{H}^{1}\to[0,+\infty]$ be
the distortion coefficient of