$\widetilde{S}$ onto $\mathbb{P}^{3}=\mathbb{H}^{2}$. That is the coordinate
$x_{4}$ converges to a zero function. $\rho_{t}$ then converges to $\rho_{0}$
whose image lies in the subgroup $PO(2,1)\cong SL(2,\mathbb{R}).$

Apply now the rescaling map to obtain the developing map
$g_{t}\mathcal{D}_{t}:\widetilde{S}\rightarrow\mathbb{X}_{t}$, so that the
holonomy representation is given by $g_{t}\mathcal{D}_{t}g_{t}^{-1}$. Suppose
that $t\rightarrow 0$ then $g_{t}\mathcal{D}_{t}$ converges to a local
diffeomorphism $\mathcal{D}:\widetilde{S}\rightarrow\mathbb{X}_{0}$ and if
$\rho_{\mathcal{D}}:\pi_{1}(S)\rightarrow PGL_{4}(\mathbb{R})$ is the limit of
the holonomy $\rho_{t}$ as $t\rightarrow 0$ then $\mathcal{D}$ is equivariant
with respect to $\rho_{\mathcal{D}}$ . To be precise, for
$\gamma\in\pi_{1}(S)$ if $\rho_{t}$ is of the form

(9) $\displaystyle\rho_{t}=\begin{pmatrix}A(t)&w(t)\\\ v(t)&a(t)\end{pmatrix}$

where $A\in PO(2,1)\cong PSL(2,\mathbb{R})$ and
$w(t),v(t)^{T}\in\mathbb{R}^{3}$, then we have

(10) $\displaystyle\lim_{t\to 0}g_{t}\rho_{t}(\gamma)g_{t}^{-1}=\lim_{t\to
0}\begin{pmatrix}A(t)&tw(t)\\\
v(t)/t&a(t)\end{pmatrix}=\begin{pmatrix}A(0)&0\\\
v^{\prime}(0)&1\end{pmatrix}=\rho_{\mathcal{D}}$

So we have the following:

######  Definition 7.1.

A half-pipe structure on $S\times\mathbb{R}$ is a
$(G_{HP^{3}},\mathbb{HP}^{3})$ structure where
$\mathbb{HP}^{3}=\mathbb{X}_{0}$ and $G_{HP}$ is the subgroup of
$PGL_{4}(\mathbb{R})$ of matrices with the form $\begin{pmatrix}A&0\\\ v&\pm
1\end{pmatrix}$ where $A\in O(2,1)$ and $v^{T}\in\mathbb{R}^{3}$.

we also define:

######  Definition 7.2.

Any path $\rho_{t}$ of representations into $PSL_{2}(\mathbb{C})$ satisfying
Equation (10) is said to be compatible at first order at $t=0$ with
$\rho_{\mathcal{D}}$

.
As observed in[10], $G_{HP^{3}}\cong\mathbb{R}^{2,1}\rtimes O(2,1)$, where an
element of the form $\begin{pmatrix}A&0\\\ v&\pm 1\end{pmatrix}$ can be
interpreted as an infinitesimal deformations of the the hyperbolic structure
given by $A\in PO(2,1)$ and along the direction $v$ normal to $PO(2,1)$ into
$PO(3,1)$. Passing onto quotients, we see that quasi-Fuchsian half-pipe
$3$-manifolds are precisely obtained by infinitesimal deformations in
$\mathcal{QF}(S)$ starting from the point $[c]\in\mathcal{F}(S)$ along a
direction $v_{([c],q)}\in N_{[c]}\mathcal{F}(S)$. So we define:

######  Definition 7.3.

$M^{HP}_{c,q}$ is the half-pipe quasi-Fuchsian structures whose holonomy
representation into $PGL_{4}(\mathbb{R})$ is compatible at first order at
$t=0$ with the holonomy $\rho_{t}$ associated to quasi-Fuchsian metrics in
$\beta_{([c],q)}(t)$ in the sense of Definition 7.2.

###  7.1. Half-pipe Schwarzians and their Foliations

Recall again that $v_{([c],q)}$ is realised by the the path
$\beta_{([c],q)}(t)\in\mathcal{QF}(S)$ associated to which we have unique
minimal immersions of $S$ for each $t<\epsilon$ with immersion data
$I_{t}\in[c]$ and $II_{t}=t\mathfrak{R}(q)$. So for each $t$ we have a
$\mathcal{D}_{t}$ and $\rho_{t}$ in the sense above and a half-pipe structure
as the limit when $t$ goes to $0$. There is also an analogous notion for half-
pipe for the second fundamental form and shape operator in half-pipe geometry
that follows from [15].

So we want to study the limit of these immersion data as $t\rightarrow 0$ and
use the following lemma:

######  Lemma 7.4 ([15]).

Let $\sigma_{t}$ be a $C^{2}$ family of minimal immersions of $\mathbb{H}^{2}$
into $\mathbb{H}^{3}$, such that $\sigma_{0}$ is an embedding of
$\mathbb{H}^{2}$. Let $\sigma=\lim_{t\to 0}G_{t}\circ\sigma_{t}$ be the
rescaled immersion in $\mathbb{HP}^{3}$. Then:

  * •

The first fundamental form of $\sigma$ coincides with the first fundamental
form of $\sigma_{0}$:

$\displaystyle I(v,w)=\lim_{t\to 0}I_{t}(v,w)$

  * •

The second fundamental form of $\sigma$ is the first derivative of the second
fundamental form of $\sigma_{t}$:

$\displaystyle II(v,w)=\lim_{t\to 0}\frac{II_{t}(v,w)}{t}$

  * •

The shape operator $B$ of $\sigma$ is the first derivative of the shape
operator $B_{t}$ of $\sigma_{t}$:

$\displaystyle B(v)=\lim_{t\to 0}\frac{B_{t}}{t}$

We immediately have the following for $M^{HP}_{c,q}$.

######  Proposition 7.5.

The half-pipe manifold $M^{HP}_{c,q}$ contains a smooth minimal surface with
immersion data uniquely given by $I\in[c]$ and $II=\mathfrak{R}(q)$.

###### Proof.

Since $M^{HP}_{c,q}$ appears as a limit of the quasi-Fuchsian structures
defines by the path $\beta_{([c],q)}(t)$ we apply the lemma directly. The
induced metric of the minimal immersion of $S$ into $M^{HP}_{c,q}$ is given by
$I=\lim_{t\to 0}I_{t}\in[c]$. So we see that $II=\lim_{t\to
0}\frac{II_{t}}{t}=\mathfrak{R}(q)$.

So we will introduce an analogous notion for Schwarzian at infinity for half-
pipe manifolds that is quite natural with the tools we have developed so far
and with our definition of $M^{HP}_{cq}$.

######  Definition 7.6.

The positive (resp. negative) half pipe Schwarzian at infinity associated to
$M^{HP}_{cq}$ is defined as the derivative at $\mathcal{F}(S)$ of the
Schwarzian derivatives at the positive (resp. negative) end at infinity for
quasi-Fuchsian metrics in the path $\beta_{([c],q)}(t).$

From Lemma 4.15 So we have

######  Proposition 7.7.

The positive and negative half-pipe Schwarzians at infinity for $M^{HP}_{c,q}$
are $q$ and $-q$.

We can now again consider the horizontal measured foliation $\pm\mathsf{F}$
associated to $\pm q$ on $[c]$ and obtain our Theorem 1.6 by an application of
Theorem 3.19:

######  Theorem 7.8.

Any pair $(\mathsf{F_{+}},\mathsf{F_{-}})\in\mathcal{FMF}(S)$ can be uniquely
realised as the horizontal foliations of the positive and negative half pipe
Schwarzians at infinity associated to quasi-Fuchsian half-pipe manifold.
Moreover, $M^{HP}_{cq}$ defined before is the unique one realising
$(\mathsf{F_{+}},\mathsf{F_{-}})$, where $([c],q)\in T^{*}\mathcal{T}(S)$ is
the unique point realising $(\mathsf{F_{+}},\mathsf{F_{-}})$.

This can be seen as a first-order interpretation of Theorem of Gardiner-Masur
as half-pipe quasi-Fuchsian manifolds correspond to points in
$T^{*}\mathcal{T}(S)$ by Proposition 7.5 via the minimal surface they contain.

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