# The harmonic maps compactification of Teichmüller spaces for punctured
Riemann surfaces

Kento Sakai  Graduate School of Science, Osaka University
<EMAIL_ADDRESS>
###### Abstract.

Wolf gave a homeomorphism from the Teichmüller space to the space of quadratic
differentials on a closed Riemann surface by using harmonic maps. Moreover,
using harmonic maps rays, he gave a compactification of the Teichmüller space
and show that it coincides with the Thurston compactification. In this paper,
we extend the harmonic maps compactification to the Teichmüller spaces of
punctured Riemann surfaces, and show that it still coincides with the Thurston
compactification.

##  1\. Introduction

Let $S$ be a finite type surface of genus $g$ with $n$ punctures such that
$\chi(S)=2-2g-n<0$. The Teichmüller space of $S$ is defined as the set of
complete finite-area hyperbolic metrics on $S$ modulo isotopy. Let
$\mathcal{T}_{g,n}$ denote the Teichmüller space of $S$.

For $\sigma,\rho\in\mathcal{T}_{g,n}$, there exists a unique harmonic
diffeomorphism $h(\sigma,\rho)$ from $(S,\sigma)$ to $(S,\rho)$ isotopic to
the identity with finite energy [Loh91]. Then the $(2,0)$-part of the pull-
buck metric $h(\sigma,\rho)^{\ast}\rho$ is a holomorphic quadratic
differential on $(S,\sigma)$, which is called the Hopf differential of
$h(\sigma,\rho)$. Therefore fixing the domain metric $\sigma$, we obtain a map
$\Phi\colon\mathcal{T}_{g,n}\to\operatorname{\mathrm{QD}}(\sigma)$ given by
$\Phi(\rho)=(h(\sigma,\rho)^{\ast}\rho)^{2,0}$, where
$\operatorname{\mathrm{QD}}(\sigma)$ denotes the vector space of holomorphic
quadratic differentials with finite $L^{1}$-norm on $(S,\sigma)$. It is shown
that $\Phi$ is a homeomorphism by Wolf for $n=0$, by Lohkamp for $n>0$ [Wol89,
Loh91].

Using the map $\Phi$, we can obtain the compactification of
$\mathcal{T}_{g,n}$ since the vector space
$\operatorname{\mathrm{QD}}(\sigma)$ has the natural compactification which is
obtained by adding an infinity point to each ray from the origin. This
compactification is called the harmonic maps compactification and denote by
$\overline{\mathcal{T}_{g,n}^{\textrm{harm}}}$. The boundary of the harmonic
maps compactification is identified with the unit sphere
$\operatorname{\mathrm{SQD}}(\sigma)$ in $\operatorname{\mathrm{QD}}(\sigma)$.
On the other hand, Thurston introduced the Thurston compactification
$\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$, which is given by embedding
$\mathcal{T}_{g,n}$ to the projective space of functionals on the set
$\mathcal{C}=\mathcal{C}(S)$ of isotopy classes of essential simple closed
curves on $S$. The boundary
$\partial\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$ of the Thurston
compactification is the projective space $\operatorname{\mathcal{PMF}}$ of
measured foliations on $S$. The action of the mapping class group on
$\mathcal{T}_{g,n}$ extends contiunously to
$\partial\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$.

Wolf showed that, if $S$ is closed, the homeomorphism $\Phi$ continuously
extends to the homeomorphism $\overline{\Phi}$ from
$\overline{\mathcal{T}_{g,0}^{\mathrm{Th}}}$ to
$\overline{\mathcal{T}_{g,0}^{\mathrm{harm}}}$. This implies that the
restriction of $\overline{\Phi}$ to the Thurston boundary coincides with the
canonical identification $\overline{F_{v}}$ of
$\operatorname{\mathrm{SQD}}(\sigma)$ and $\operatorname{\mathcal{PMF}}$,
where $\overline{F_{v}}$ is induced by the map
$F_{v}\colon\operatorname{\mathrm{QD}}(\sigma)\to\operatorname{\mathcal{MF}}$
given by the vertical measured foliation $F_{v}(\Psi)$ for
$\Psi\in\operatorname{\mathrm{QD}}(\sigma)$. Therefore, the harmonic maps
compactification is independent on the choice of the domain metric $\sigma$.
In addition, the extension $\overline{\Phi}$ provides the Thurston
compactification with a global parametrization by the closed unit ball in
$\operatorname{\mathrm{QD}}(\sigma)$.

In this paper, we extend the compatibility of the harmonic maps
compactification and the Thurston compactification for the Teichmüller spaces
of punctured surfaces.

######  Main Theorem.

The homeomorphism
$\Phi\colon\mathcal{T}_{g,n}\to\operatorname{\mathrm{QD}}(\sigma)$
continuously extends to a homeomorphism from
$\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$ to
$\overline{\mathcal{T}_{g,n}^{\mathrm{harm}}}$. Moreover, this extension on
the boundary coincides with the inverse of the canonical identification
$\overline{F_{v}}$.

Our strategy is based on Wolf’s proof for a closed Riemann surface. However,
the compactness of the surface is essentially used in his proof. In our
setting, a holomorphic quadratic differential may have a simple pole at a
puncture, so we have to substaintially change his proof.

We outline our proof below, comparing with Wolf’s proof. Fixing
$\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$, we set
$\rho_{t}\coloneqq\Phi^{-1}(t\Phi_{0})$ for $t>0$. The one-parameter family
$\\{\rho_{t}\\}$ is called the harmonic maps ray in the direction of
$\Phi_{0}$. We first show that, for the harmonic diffeomorphsim
$h(t)\colon(S,\sigma)\to(S,\rho_{t})$, the norm of the Beltrami differential
of $h(t)$ converges monotonically to $1$ as $t\to\infty$ (3.7). For the proof
of this convergence, [Wol89] uses the compactness of the closed surface.
Therefore, using some results on holomorphic energy functions for punctured
surfaces in [Loh91], we prove 3.7 without the compactness. From 3.7, we obtain
the asymptotic length of arcs along the leaves of the horizontal or vertical
measured foliation of $\Phi_{0}$ (3.8).

Let $\mathcal{C}=\mathcal{C}(S)$ be the set of isotopy classes of essential
simple closed curves on $S$ and $\beta$ be the identification of
$\operatorname{\mathrm{QD}}(\sigma)$ and $\operatorname{\mathcal{MF}}$ given
by $\beta\Phi=F_{v}(4\Phi)$. We next recall the fundamental lemma, which is an
inequality of the intersection number function $i(\beta\Phi(\rho),\cdot)$ and
the hyperbolic length function $\ell_{\rho}$ on $\mathcal{C}$ for a metric
$\rho\in\mathcal{T}_{g,0}$.

######  Theorem ([Wol89, Lemma 4.1]).

Fix an isotopy class $[\gamma]\in\mathcal{C}$. Then, for every
$\rho\in\mathcal{T}_{g,0}$, there exist positive constants
$k_{0}=k_{0}(\|\Phi(\rho)\|)$ and $\eta(\|\Phi(\rho)\|,[\gamma])$ such that

$i(\beta\Phi(\rho),[\gamma])\leq\ell_{\rho}([\gamma])\leq
k_{0}i(\beta\Phi(\rho),[\gamma])+\eta$

where $k_{0}\searrow 1$ and $\eta\|\Phi(\rho)\|^{-1/2}\to 0$ as
$\|\Phi(\rho)\|\to\infty$.

The key to show the lemma is constructing a “staircase” representative
$\gamma_{\Phi}$ for $\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$. The
representative $\gamma_{\Phi}$ satisfies some conditions. For example, it
consists of curves along leaves of horizontal or vertical foliation of $\Phi$
and the intersection number $i(\beta\Phi(\rho),\gamma_{\Phi})$ is equal to
$i(\beta\Phi(\rho),[\gamma])$. However, for punctured surfaces, we cannot
construct $\gamma_{\Phi}$ in the same manner as [Wol89], since a quadratic
differential may have a simple pole at each puncture. Hence, we show the
following instead for punctured surfaces.

######  Theorem 4.1.

Fix an isotopy class $[\gamma]\in\mathcal{C}$ and $\varepsilon>0$. Then, there
exists a nonnegative number $c_{0}=c_{0}([\gamma],\varepsilon)<\varepsilon$
satisfying the following: for every $\rho\in\mathcal{T}_{g,n}$, there exist
positive constants $k_{0}=k_{0}(\|\Phi(\rho)\|,\varepsilon)$ and
$\eta=\eta(\|\Phi(\rho)\|,[\gamma],\varepsilon)$ such that

$i(\beta\Phi(\rho),[\gamma])\leq\ell_{\rho}([\gamma])\leq
k_{0}i(\beta\Phi(\rho),[\gamma])+\eta$

where $k_{0}\searrow 1$ and $\eta\|\Phi(\rho)\|^{-1/2}\to c_{0}$ as
$\|\Phi(\rho)\|\to\infty$.

This propositon is different from Wolf’s lemma in that the limit of
$\eta\|\Phi(\rho)\|^{-1/2}$ as $\|\Phi(\rho)\|\to\infty$ may not be $0$.
Setting the allowance for arbitraly small $\varepsilon>0$, we can construct
such a “staircase” representative $\gamma_{\Phi}$. However, unlike the above
lemma, we need to increase the horizontal measure of the representative
$\gamma_{\Phi}$ from $i(\beta\Phi(\rho),[\gamma])$ by a little. Therefore, in
order to estimate the length of the additional horizontal arcs, we use some
results on quadratic differential metrics in [Min92, Wol89]. Theorem 4.1 is a
week version of [Wol89, Lemma 4.1], but still leads to the main theorem.

### Acknowlagement

The author would like to thank my supervisor, Shinpei Baba, for many
discussions and helpful advice.

##  2\. Background

###  2.1. Harmonic maps

Let $\overline{S}$ be a closed, oriented, connected surface of genus $g$, and
$S\coloneqq\overline{S}-\\{p_{1},\ldots,p_{n}\\}$. If $\chi(S)=2-2g-n<0$, the
surface admits (complete, finite area) hyperbolic metrics. Throughout this
paper, we fix $S$ such that $\chi(S)<0$ and $S$ is not a thrice punctured
surface. Let $\sigma$ be a hyperbolic metric on $S$. The hyperbolic surface
$(S,\sigma)$ can be regarded as a (punctured) Riemann surface by taking the
isothermal coordinates system for $\sigma$.

Let $\sigma|dz|^{2},\rho|dw|^{2}$ be hyperbolic metrics on $S$, where $z=x+iy$
and $w=u+iv$ denote the conformal structures of $(S,\sigma)$ and $(S,\rho)$,
respectively. For a $C^{2}$ map
$f\colon(S,\sigma|dz|^{2})\to(S,\rho|dw|^{2})$, we define the energy density
of $f$ by

$e(f)\coloneqq\frac{\rho(f(z))}{\sigma(z)}(|f_{z}|^{2}+|f_{\bar{z}}|^{2}),$

the total energy of $f$ by

$\mathcal{E}(f)\coloneqq\int_{S}\rho(f(z))(|f_{z}|^{2}+|f_{\bar{z}}|^{2})\,\frac{i}{2}dzd\bar{z}$

and, the holomorphic energy $H(f)$ and anti-holomorphic energy $L(f)$ by

$\displaystyle H(f)\coloneqq\frac{\rho(f(z))}{\sigma(z)}|f_{z}|^{2},\
L(f)\coloneqq\frac{\rho(f(z))}{\sigma(z)}|f_{\bar{z}}|^{2}.$

Let $J(f)$ be the Jacobian of $f$, then $J(f)=H(f)-L(f)$. The Beltrami
differential of $f$ is defined by

$\nu(f)\coloneqq\frac{f_{\bar{z}}{d\bar{z}}}{f_{z}dz}.$

The norm of the Beltrami differential of $f$ is the well-defined function on
$(S,\sigma)$. Clearly, we have $|\nu(f)|^{2}=L(f)/H(f)$.

A $C^{2}$ map $f\colon(S,\sigma)\to(S,\rho)$ is said to be harmonic if $f$ is
a critical point of the energy functional $\mathcal{E}$. By considering the
variation problem for energy, $f$ is a harmonic map if and only if $f$
satisfies the Euler-Lagrange equation

$f_{z\bar{z}}+\frac{\rho_{w}}{\rho}f_{z}f_{\bar{z}}=0.$

It is a well-known fact that if $S$ is closed, there exists a unique harmonic
diffeomorphism to the identity map from $(S,\sigma)$ to $(S,\rho)$. (Eells and
Sampson proved the existence of a harmonic map in each homotopy class [ES64].
Hartman proved its uniquness [Har67]. Shoen-Yau and Sampson indepndently
proved that the harmonic map homotopic to the identity is a diffeomorphism
[SY78, Sam78].)

In the case of punctured surfaces, Lohkamp showed the following.

######  Theorem 2.1 ([Loh91]).

Let $S$ be a surface of finite type, and $\sigma,\rho$ be complete hyperbolic
metrics of finite area on $S$. Then, there exists a unique harmonic
diffeomorphism $h(\sigma,\rho)\colon(S,\sigma)\to(S,\rho)$ such that
$h(\sigma,\rho)$ is homotopic to the identity and
$\mathcal{E}(h(\sigma,\rho))<+\infty$.

###  2.2. Compactifications of the Teichmüller space

The Teichmüller space of the surface $S$ is the quotient space of hyperbolic
metrics on $S$ by the pull-buck action of diffeomorphisms isotopic to the
identity. Let $\mathcal{T}_{g,n}$ denote the Teichmüller space of $S$. We
often write $\rho$ simply for $[\rho]\in\mathcal{T}_{g,n}$.

A holomorphic quadratic differential $\Phi$ on $(S,\sigma)$ is said to be
integrable, if the $|\Phi|$-area of $(S,\sigma)$ is finite, namely

$\|\Phi\|=\int_{S}|\Phi|dxdy<+\infty.$

Let $\operatorname{\mathrm{QD}}(\sigma)$ denote the space of integrable
holomorphic quadratic differentials on $(S,\sigma)$. For a holomorphic
quadratic differential $\Phi$ on $(S,\sigma)$, $\Phi$ is integrable if and
only if $\Phi$ has poles of at most one degree at punctures.

Fixing the metric $\sigma$ of the domain surface, we set $h(\rho)\coloneqq
h(\sigma,\rho)$ for $\rho$. Since $(h(\rho)^{\ast}\rho)^{2,0}$ is a
holomorphic quadratic differenttial on $(S,\sigma)$ with finite norm ([Loh91]
Lemma 7), we can define the map $\Phi$ as

$\Phi:\mathcal{T}_{g,n}\to\operatorname{\mathrm{QD}}(\sigma);[\rho]\mapsto(h(\rho)^{\ast}\rho)^{2,0}.$

The uniqueness of the harmonic diffeomorphism shows that the map $\Phi$ is
well-defined. Here, we list very useful well-known formulae.

######  Proposition 2.2.

Let $J(\rho)=J(h(\rho)),H(\rho)=H(h(\rho)),L(\rho)=L(h(\rho))$, and
$\nu(\rho)=\nu(h(\rho))$. Then the following hold:

  1. (I)

$J(\rho)=H(\rho)-L(\rho)$

  2. (II)

$|\Phi(\rho)|/\sigma^{2}=H(\rho)L(\rho)$

  3. (III)

$|\nu(\rho)|H(\rho)\sigma=|\Phi(\rho)|$

  4. (IV)

$\nu(\rho)=L(\rho)/H(\rho)$

  5. (V)

$\Delta_{\sigma}H(\rho)=2H(\rho)-2L(\rho)-2$

  6. (VI)

$\Delta_{\sigma}L(\rho)=2L(\rho)-2H(\rho)-2$ on $S-\\{\Phi(\rho)=0\\}$

Here,

$\Delta_{\sigma}\coloneqq\frac{4}{\sigma}\frac{\partial^{2}}{\partial
z\partial\bar{z}}$

is the Laplace-Beltrami operator on $(S,\sigma)$.

For the proof of (V) and (VI), see [Jos13, Lemma 3.10.1].

The following theorem is shown by Wolf for $n=0$ and by Lohkamp for $n>0$.

######  Theorem 2.3 ([Wol89], [Loh91]).

$\Phi$ is a homeomorphism.

Using the Riemann-Roch theorem, we see that
$\operatorname{\mathrm{QD}}(\sigma)$ is a vector space of real dimension
$6g-6+2n$. Since a vector space can be compactified by adding an infinite
point to the endpoint of every ray from the origin, we can obtain the
compactification of $\mathcal{T}_{g,n}$ through the homeomorphism $\Phi$. The
compactification of $\mathcal{T}_{g,n}$ is called the harmonic maps
compactification and denoted by
$\overline{\mathcal{T}^{\mathrm{harm}}_{g,n}}$. In other words,

$\overline{\mathcal{T}^{\mathrm{harm}}_{g,n}}=\operatorname{\mathrm{BQD}}(\sigma)\cup\operatorname{\mathrm{SQD}}(\sigma),$

where
$\operatorname{\mathrm{BQD}}(\sigma)\coloneqq\\{\Phi\in\operatorname{\mathrm{QD}}(\sigma)\mid\|\Phi\|<1\\}$
and
$\operatorname{\mathrm{SQD}}(\sigma)\coloneqq\\{\Phi\in\operatorname{\mathrm{QD}}(\sigma)\mid\|\Phi\|=1\\}$.

On the other hand, Thurston introduced a compactification of
$\mathcal{T}_{g,n}$, which is called the Thruston compactification and denoted
by $\overline{\mathcal{T}^{\mathrm{Th}}_{g,n}}$. Here, we put a brief
description of $\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$ with reference to
[FLP12].

A simple closed curve $\gamma$ on $S$ is peripheral if $\gamma$ bounds a
punctured disk. If a simple closed curve $\gamma$ is neither null-homotopic
nor peripheral, $\gamma$ is said to be essential. Let
$\mathcal{C}=\mathcal{C}(S)$ be the set of homotopy classes of essential
simple closed curves in $S$. Given functionals $f,g\in\mathbb{R}_{\geq
0}^{\mathcal{C}}$, $f$ and $g$ are said to be equivalent, if there exists a
positive real number $\lambda>0$ such that $f=\lambda g$. The quotient space
of the functionals by the equivalence relation is denoted by
$P(\mathbb{R}_{\geq 0}^{\mathcal{C}})$, and let $\pi:\mathbb{R}_{\geq
0}^{\mathcal{C}}-\\{0\\}\to P(\mathbb{R}_{\geq 0}^{\mathcal{C}})$ be the
projection. Let $\operatorname{\mathcal{MF}}^{\ast}$ be the set of nontrivial
measured foliations on $S$ which may have possibly one pronged singularities
at the punctures. Then, $\operatorname{\mathcal{MF}}^{\ast}$ can be embedded
into $\mathbb{R}_{\geq 0}^{\mathcal{C}}-\\{0\\}$ by the map $I_{\ast}$ which
is defined by

$I_{\ast}(F)\coloneqq(i(F,\cdot):\mathcal{C}\to\mathbb{R}_{\geq
0})\qquad(F\in\operatorname{\mathcal{MF}}^{\ast}),$

where $i(F,[\gamma])$ is the infimum of the transeverse measure of
representatives of $[\gamma]\in\mathcal{C}$. Thus we identify
$\operatorname{\mathcal{MF}}^{\ast}$ with its image in $\mathbb{R}_{\geq
0}^{\mathcal{C}}$. Thurston showed that $\pi\circ
I_{\ast}(\operatorname{\mathcal{MF}}^{\ast})$ is homeomorphic to a sphere of
dimension $6g-7+2n$.

For $\rho\in\mathcal{T}_{g,n}$, the length functional
$\ell_{\ast}(\rho)\in\mathbb{R}_{\geq 0}^{\mathcal{C}}$ is given by

$\ell_{\ast}(\rho)([\gamma])\coloneqq\ell_{\rho}([\gamma])=\inf_{\gamma\in[\gamma]}\ell_{\rho}(\gamma)\qquad([\gamma]\in\mathcal{C}).$

It is known that $\ell_{\ast}$ is an embedding of $\mathcal{T}_{g,n}$ into
$\mathbb{R}_{\geq 0}^{\mathcal{C}}-\\{0\\}$ and $\pi\circ\ell_{\ast}$ is still
an embedding of $\mathcal{T}_{g,n}$ into $P(\mathbb{R}_{\geq
0}^{\mathcal{C}})$.

Let $\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$ be the subset
$\pi\circ\ell_{\ast}(\mathcal{T}_{g,n})\cup\pi\circ
I_{\ast}(\operatorname{\mathcal{MF}}^{\ast})$ in $P(\mathbb{R}_{\geq
0}^{\mathcal{C}})$. We call $\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$ the
Thuston compactification of $\mathcal{T}_{g,n}$. In fact, it is a manifold
with boundary and homeomorphic to a closed ball of dimension $6g-6+2n$. The
boundary $\pi\circ I_{\ast}(\operatorname{\mathcal{MF}}^{\ast})$ of the
Thurston compactification is denoted by $\operatorname{\mathcal{PMF}}$ and
called the Thrston boundary. By the construction, the mapping class group
action on $\mathcal{T}_{g,n}$ extends continuously to the Thurston
compactification.

##  3\. Deformation along harmonic maps rays

###  3.1. Norm functions of Beltrami differentials

We denote the inverse homeomorphism
$\Phi^{-1}:\operatorname{\mathrm{QD}}(\sigma)\to\mathcal{T}_{g,n}$ by $\rho$ .

######  Definition 3.1.

Let $\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$. The harmonic maps ray in
the direction of $\Phi_{0}$ is the ray defined by
$\\{\rho_{t}\coloneqq\rho(t\Phi_{0})\\}_{t>0}$ in $\mathcal{T}_{g,n}$.

Let $h(t)$ denote the unique harmonic diffeomorphism $h(\rho_{t})$ homotopic
to the identity with $\mathcal{E}(h(\rho_{t}))<+\infty$. We denote the
holomorphic energy by $H(t)$, anti-holomorphic energy by $L(t)$, and the
Beltrami differentials of $h(t)$ by $\nu(t)$. Clearly
$\Phi(\rho_{t})=t\Phi_{0}$ by the definition. The main purpose of this
subsection is to prove the following propositon.

######  Proposition 3.2.

For any $\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$, let
$\\{\rho_{t}\\}_{t>0}$ be the harmonic maps ray in the direction of
$\Phi_{0}$. Then, for every nonzero point $p$ of $\Phi_{0}$, we have

$|\nu(t)(p)|^{2}\nearrow 1\qquad(\text{as}\ t\to\infty).$

######  Remark 3.3.

Let $M$ be the domain surface $(S,\sigma)$. In the case of closed surfaces,
Wolf [Wol89] proved this propositon in the following three steps.

  1. Step 1:

Show that $|\nu(t)(p)|^{2}$ converges to $1$ at alomost everywhere on $M$.

  2. Step 2:

Show that $(|\nu(t)(p)|^{2})^{\prime}>0$ on $M-\\{\Phi_{0}(p)=0\\}$.

  3. Step 3:

Exclude the possibility that $|\nu(t)(p)|^{2}\to\delta\neq 1$ as $t\to\infty$
for a nonzero point $p$ of $\Phi_{0}$.

If the surface $S$ has punctures, we can proceed Step 1 and Step 3 in
essentially the same way as the paper [Wol89]. Therefore we give the proof of
Step 2 here. Since the compactness of the surface is critical for Step 2 in
[Wol89], We prove this inequality for punctured surfaces.

######  Proposition 3.4.

For every nonzero point $p\in M-\\{\Phi_{0}(q)=0\\}$, the derivative
$(|\nu(t)(p)|^{2})^{\prime}$ with respect to $t$ is positive.

###### Proof.

By the formula (II) in 2.2, we have

$(H(t)L(t))^{\prime}=H^{\prime}(t)L(t)+L^{\prime}(t)H(t)=\frac{2t|\Phi_{0}|^{2}}{\sigma^{2}}=\frac{2}{t}H(t)L(t).$

Therefore, for every $p\in M-\\{\Phi_{0}(p)=0\\}$,

(3.1)
$\frac{H^{\prime}(t)}{H(t)}(p)+\frac{L^{\prime}(t)}{L(t)}(p)=\frac{2}{t}.$

Next, we introduce some results on the holomorphic energy functions $H(t)$.
The following is shown by Lohkamp.

######  Lemma 3.5 ([Loh91] Lemma 14).

For $t_{1},t_{2}>0$,

$\min\left\\{1,\frac{t_{2}}{t_{1}}\right\\}\cdot H(t_{1})\leq
H(t_{2})\leq\max\left\\{1,\frac{t_{2}}{t_{1}}\right\\}\cdot H(t_{1})\ \
\text{on}\ M.$

Using Lemma 3.5, we show the following.

######  Lemma 3.6.

For every $t>0$ and $p\in M$ , we have

$0\leq\frac{H^{\prime}(t)}{H(t)}(p)<\frac{1}{t},$

where $H^{\prime}$ is a derivative with respect to $t$.

###### Proof.

Since the holomorphic energy $H(t)$ is positive, by Lemma 3.5,

$\min\left\\{0,\log\frac{t_{2}}{t_{1}}\right\\}\leq\log\frac{H(t_{2})}{H(t_{1})}\leq\max\left\\{0,\log\frac{t_{2}}{t_{1}}\right\\}.$

Therefore, we obtain

$0\leq\frac{\log H(t_{2})-\log H(t_{1})}{t_{2}-t_{1}}\leq\frac{\log t_{2}-\log
t_{1}}{t_{2}-t_{1}}.$

Setting $t_{1}=t,t_{2}=t+h$ and tending $h$ to $0$, we find

(3.2) $0\leq\frac{H^{\prime}(t)}{H(t)}\leq\frac{1}{t}.$

Next, we show that the the second inequality of 3.2 is strict. If there exists
a point $p_{0}\in M$ such that $H^{\prime}(t)/H(t)(p_{0})=1/t$, since $p_{0}$
maximizes $H^{\prime}(t)/H(t)$, we have

$0\geq\Delta_{\sigma}\frac{H^{\prime}(t)}{H(t)}(p_{0})=\Delta_{\sigma}(\log
H(t))^{\prime}=2(H^{\prime}(t)-L^{\prime}(t)).$

Therefore, we find $H^{\prime}(t)(p_{0})\leq L^{\prime}(t)(p_{0})$.

This implies that $p_{0}$ is not a zero of $\Phi_{0}$. If $p_{0}$ is a zero,
by $H(t)>0$ and $H(t)L(t)=t^{2}|\Phi_{0}(p_{0})|^{2}/\sigma^{2}=0$,

$L(t)(p_{0})=0$

holds for any $t>0$. Hence we have $H^{\prime}(t)(p_{0})\leq 0$, however this
is impossible since $H(t)(p_{0})>0$ and $H^{\prime}(t)/H(t)(p_{0})=1/t$. Thus,
we find that $\Phi_{0}(p_{0})\neq 0$.

By $H(t)-L(t)>0$, we have

$\frac{L^{\prime}(t)}{L(t)}(p_{0})\geq\frac{H^{\prime}(t)}{L(t)}(p_{0})>\frac{H^{\prime}(t)}{H(t)}(p_{0}).$

However, this contradicts $H^{\prime}(t)/H(t)(p_{0})=1/t$ and 3.1. ∎

By Lemma 3.6 and 3.1, we have

$\frac{L^{\prime}(t)}{L(t)}>\frac{H^{\prime}(t)}{H(t)}\ \ \text{on}\
M-\\{\Phi_{0}(p)=0\\}.$

Thus, we obtain

$(|\nu(t)|^{2})^{\prime}=\left(\frac{L(t)}{H(t)}\right)^{\prime}=\frac{L^{\prime}(t)H(t)-L(t)H^{\prime}(t)}{H(t)^{2}}>0\
\ \text{on}\ M-\\{\Phi_{0}(p)=0\\}.$

This complets the proof. ∎

###  3.2. Asymptotic length of horizontal and vertical arcs

Given hyperbolic structures $\sigma,\rho$ on $S$, we can isotope $\rho$ so
that the identity map $\operatorname{id}\colon(S,\sigma)\to(S,\rho)$ is
harmonic. For the rest of this paper, we always take such a representative in
its isotopy class.

A holomorphic quadratic differential $\Phi$ defines two measured foliations on
$M$ which are orhthogonal to each other away from the zeros. By changing a
conformal coordinate $z$ on $M$ to the natural coordinate $\zeta=\xi+i\eta$ of
$\Phi$, the representation of $\Phi$ in terms of $\zeta$ is identically $1$
away from zero, that is

$\Phi\,dz^{2}=d\zeta^{2}.$

Then lines parallel to $\xi$-axis (resp. $\eta$-axis) define a singular
foliation on $M$ such that its singular points are zeros of $\Phi_{0}$, and
also $|d\eta|$ (resp. $|d\xi|$) defines a transverse measure for the singular
foliation. We call the measured foliation the horizontal (resp.vertical)
foliation of $\Phi_{0}$, and it is denoted by $F_{h}(\Phi)$ (resp.
$F_{v}(\Phi)$). If $\Phi_{0}$ has a pole of digree one at a puncture on $S$,
the foliation is one-pronged at the puncture. An arc along a leaf of
horizontal (resp.vertical) measured foliation of $\Phi$ is called a horizontal
(resp.vertical) arc of $\Phi_{0}$.

Let $\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$ and $\\{\rho_{t}\\}$ be
the harmonic maps ray in the direction of $\Phi_{0}$. In this subsection, we
consider the asymptotic $\rho_{t}$-length of horizontal and vertical arcs of
$\Phi_{0}$. Here, we recall Wolf’s setting in [Wol89]. For the ray
$\\{\rho_{t}\\}$, we define conformal coordinates $z=x+iy$ on $M$ such that
$\partialderivative{x}$ and $\partialderivative{y}$ give an orthonormal frame
field on $M$ and also they are respectively maximum and minimum streching
directions of the differential map $dh(t)$. Then, they are tangent to the
horizontal and vertical foliations of $\Phi_{0}$, respectively. The conformal
coordinates $z$ are defined from away zeros of $\Phi_{0}$. By the definitions
of $\partialderivative{x}$ and $\partialderivative{y}$, they are orthogonal to
each other in the $\rho_{t}$-metric. Thus, we have

$t\Phi_{0}\,dz^{2}=\frac{1}{4}\left(\left\|\partialderivative{x}\right\|^{2}_{\rho_{t}}-\left\|\partialderivative{y}\right\|^{2}_{\rho_{t}}\right)\,dz^{2}.$

Therefore, the $|\Phi(\rho_{t})|$ metric length of a tangent vector
$\partialderivative{x}$ is

(3.3)
$\left\|\partialderivative{x}\right\|_{\Phi(\rho_{t})}^{2}=\frac{1}{4}\left(\left\|\partialderivative{x}\right\|^{2}_{\rho_{t}}-\left\|\partialderivative{y}\right\|^{2}_{\rho_{t}}\right).$

Moreover, by computation, we find

(3.4)
$|\nu(t)|=\frac{1-\|\partialderivative{y}\|_{\rho_{t}}/\|\partialderivative{x}\|_{\rho_{t}}}{1+\|\partialderivative{y}\|_{\rho_{t}}/\|\partialderivative{x}\|_{\rho_{t}}}.$

Hence, by 3.2, the following holds.

######  Proposition 3.7.

Let $\\{\rho_{t}\\}_{t>0}$ be the harmonic maps ray in the direction
$\Phi_{0}$. Then

  1. (1)

for every $p$ with $\Phi_{0}(p)\neq 0$,
$\|\partialderivative{y}\|_{\rho_{t}}/\|\partialderivative{x}\|_{\rho_{t}}\searrow
0$ as $t\to\infty$, and

  2. (2)

for every $p$ with $\Phi_{0}(p)\neq 0$,
$\|\partialderivative{x}\|_{|4\Phi(\rho_{t})|}/\|\partialderivative{x}\|_{\rho_{t}}\nearrow
1$ as $t\to\infty$.

######  Proposition 3.8.

Let $\\{\rho_{t}\\}_{t>0}$ be a harmonic maps ray in the direction $\Phi_{0}$.
For an arc $\gamma$ on $M$, let $\ell_{\rho_{t}}(\gamma)$ denote the
$\rho_{t}$-length of $\gamma$.

  1. (1)

Let $\gamma$ be a compact horizontal arc of $\Phi_{0}$ containing no zeros of
$\Phi_{0}$. Then, there exist constants $c_{0},c_{1}>0$ depending only on
$\gamma$ such that for every $t>1$

$0<c_{0}<\ell_{\rho_{t}}(\gamma)t^{-1/2}<c_{1}<\infty.$

  2. (2)

Let $\gamma$ be a compact vertical arc of $\Phi_{0}$ (that may contain zeros
of $\Phi_{0}$). Then

$\ell_{\rho_{t}}(\gamma)t^{-1/2}\to 0\ \ \text{as}\ t\to\infty.$

###### Proof.

This proposition can be shown similarly to [Wol89], using the fact that the
holomorphic energy function is bounded [Loh91, Corollary 3 and Lemma 9]
instead of the compactness for a closed surface. ∎

Following lemma for closed surfaces is proved by Minsky in [Min92, Lemma 3.3].
Since Minsky’s proof works for Lemma 3.9, we omit the proof.

######  Lemma 3.9.

Let $\Phi\in\operatorname{\mathrm{QD}}(\sigma)$. Suppose that $p\in M$ is at
least distance $d$ away from poles and zeros of $\Phi$, and

$\log|\nu(\Phi)|^{-1}<b\ \ \ \text{on}\ B_{|\Phi|}(p,d),$

where $B_{|\Phi|}(p,d)$ denotes the $|\Phi|$-radius $d$ disk centred at $p$,
and $\nu(\Phi)=\nu(\rho(\Phi))$. Then

$\log|\nu(\Phi)(p)|^{-1}<\frac{b}{\cosh d}\,.$

By using Lemma 3.9 and some estimates in [Wol91], we can bound the asymptotic
$\rho_{t}$-length of horizontal arcs of $\Phi_{0}$ from above by the
$|\Phi_{0}|$-length.

######  Proposition 3.10.

Let $\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$, and let
$\\{\rho_{t}\\}_{t>0}$ be the harmonic maps ray in the direction of
$\Phi_{0}$. If $\gamma$ be a compact horizontal arc of $\Phi_{0}$ containing
no zeros, then there exist $C,D>0$ depending only on $\gamma$ such that

$\ell_{\rho_{t}}(\gamma)<t^{1/2}\ell_{|4\Phi_{0}|}(\gamma)(1+Ce^{-D\sqrt{t}})\
\text{for all}\ t>0.$

###### Proof.

Let $\Sigma(\Phi_{0})$ denote the set consisting of punctures and zeros of
$\Phi_{0}$. We set

$d(t)=\inf\\{d_{t|\Phi_{0}|}(p,q)\mid p\in\gamma,q\in\Sigma(\Phi_{0})\\}.$

Then we find that $d(t)=t^{1/2}d(1)$ by the definition of quadratic
differential metrics. We define a constant $C_{0}$ as

$C_{0}=\sup\\{\log|\nu(1)(q)|^{-1}\mid q\in N_{|\Phi_{0}|}(\gamma,d(1)/2)\\},$

where $N_{|\Phi_{0}|}(\gamma,d(1)/2)$ denote the $(d(1)/2)$-neighborhood of
$\gamma$ in the $|\Phi_{0}|$ metric. Since $|\nu(t)|$ increases monotonically
in $t$, we see that for every $q\in N_{|\Phi_{0}|}(\gamma,d(1)/2)$ and every
$t>1$,

$\log|\nu(t)(q)|^{-1}<\log|\nu(1)(q)|^{-1}\leq C_{0}.$

Therefore, by Lemma 3.9,

$\log|\nu(t)|^{-1}<\frac{C_{0}}{\cosh(d(t)/2)}=\frac{C_{0}}{\cosh(t^{1/2}d(1)/2)}<C_{0}e^{-\sqrt{t}d(1)/2}\
\ $

holds on $\gamma$. Then, we have

$\displaystyle|\nu(t)|^{-1/2}-1$
$\displaystyle<\frac{1}{|\nu(1)|^{1/2}}(1-|\nu(t)|^{2})$
$\displaystyle\leq\frac{1}{\min_{\gamma}|\nu(1)|^{1/2}}\log|\nu(t)|^{-2}$
$\displaystyle\leq\frac{2}{\min_{\gamma}|\nu(1)|^{1/2}}C_{0}e^{-\sqrt{t}d(1)/2}.$

Setting

$C=\frac{2}{\min_{\gamma}|\nu(1)|^{1/2}}C_{0}\ \ \text{and}\ \ D=d(1)/2,$

we have

$\displaystyle\ell_{\rho_{t}}(\gamma)$
$\displaystyle=\int_{\gamma}\left\|\partialderivative{x}\right\|_{\rho_{t}}ds$
$\displaystyle=\int_{\gamma}\\{H(t)^{1/2}+L(t)^{1/2}\\}\,ds_{\sigma}$
$\displaystyle=\int_{\gamma}H(t)^{1/2}(1+|\nu(t)|)\,ds_{\sigma}$
$\displaystyle=\int_{\gamma}\frac{t^{1/2}|\Phi_{0}|^{1/2}}{|\nu(t)|^{1/2}}(1+|\nu(t)|)\frac{ds_{\sigma}}{\sigma^{1/2}}$
$\displaystyle=t^{1/2}\int_{\gamma}(1+(|\nu(t)|^{-1/2}-1))(2-(1-|\nu(t)|))\,ds_{|\Phi_{0}|}$
$\displaystyle<2t^{1/2}\int_{\gamma}(1+(|\nu(t)|^{-1/2}-1))\,ds_{|\Phi_{0}|}$
$\displaystyle<t^{1/2}\ell_{|4\Phi_{0}|}(\gamma)(1+Ce^{-D\sqrt{t}}),$

as $2\ell_{|\Phi_{0}|}(\gamma)=\ell_{|4\Phi_{0}|}(\gamma)$. We obtain the
desired inequality. ∎

######  Corollary 3.11.

Under the assumptions of 3.10,

$\lim_{t\to\infty}t^{-1/2}\ell_{\rho_{t}}(\gamma)\leq\ell_{|4\Phi_{0}|}(\gamma)$

holds.

The following proposition will be used in the proof of 4.3.

######  Lemma 3.12.

Let $q\in\overline{S}-S$, a puncture of $S$. Then for every
$\Phi\in\operatorname{\mathrm{QD}}(\sigma)$,

$\ell_{|\Phi|}(\partial B_{|\Phi|}(q,R))\leq L_{1}R,$

where $L_{1}$ is a constant which depends only on the topological type of $S$.

######  Remark 3.13.

Note for closed Riemann surfaces that Lemma 3.12 is a special case of Lemma
4.1 in [Min92]. For every point $p\in\overline{S}$, let $\deg(p)$ denote the
degree of $\Phi$ at $p$. We know that a singularity $p$ has a cone angle of
$(n+2)\pi$, or concentrated curvature $-n\pi$, where $n=\deg(p)$. Note that,
if $p$ is a pole of $\Phi$, then $\deg(p)=-1$.

###### Proof.

For a number $r$ with $0\leq r\leq R$, we set $\gamma_{r}=\partial
B_{|\Phi|}(q,R)$. Then, we have

$\frac{d}{dr}\ell_{|\Phi|}(\gamma_{r})=\kappa(\gamma_{r}),$

where $\kappa(\gamma_{r})$ is a total curvature of $\gamma_{r}$ in the
$|\Phi|$ metiric. By the Gauss-Bonnet theorem, we have

$\displaystyle\kappa(\gamma_{r})$
$\displaystyle=2\pi\chi(B_{|\Phi|}(q,r))-\sum_{p\in
B_{|\Phi|}(q,R)}(-\deg(p)\pi)$ $\displaystyle\leq 2\pi+(4g-4+n)\pi.$

Therefore,
$\ell_{|\Phi|}(\gamma_{R})=\int_{0}^{R}\kappa(\gamma_{r})\,dr\leq(4g-2+n)\pi
R$ follows. ∎

##  4\. The identification of the compactifications

###  4.1. The Fundamental Lemma

As described at the begginning of the previous section, a holomorphic
quadratic differential $\Phi$ on $M$ defines the vertical measured foliation
on $M$. In fact, by the Hubbard-Masur theorem, the map $\Phi\mapsto
F_{v}(\Phi)$ is a homeomorphism from $\operatorname{\mathrm{QD}}(\sigma)$ to
$\operatorname{\mathcal{MF}}$ ([HM79], also see [Gar87, p.206]), where
$\operatorname{\mathcal{MF}}$ denotes the set of measured foliations on $S$
(here $\operatorname{\mathcal{MF}}$ contains the empty measured foliation). We
define a homeomorphism
$\beta\colon\operatorname{\mathrm{QD}}(\sigma)\to\operatorname{\mathcal{MF}}$
by $\beta\Phi=F_{v}(4\Phi)$. Then, by the definition of the transverse measure
of $\beta\Phi$, we find that, for every $[\gamma]\in\mathcal{C}$

$i(\beta\Phi,[\gamma])=\|\Phi\|^{1/2}i(\beta(\Phi/\|\Phi\|),[\gamma]).$

The main purpose of this subsection is to prove the following proposition
analogous to [Wol89, Lemma 4.6].

######  Theorem 4.1.

Fix $[\gamma]\in\mathcal{C}$ and $\varepsilon>0$. Then, for every
$\rho\in\mathcal{T}_{g,n}$, there exist nonnegative constants
$c_{0}=c_{0}([\gamma],\varepsilon)$, $k_{0}=k_{0}(\varepsilon,\|\Phi(\rho)\|)$
and $\eta=\eta([\gamma],\varepsilon,\|\Phi(\rho)\|)$ such that

$i(\beta\Phi(\rho),[\gamma])\leq\ell_{\rho}([\gamma])\leq
k_{0}i(\beta\Phi(\rho),[\gamma])+\eta,$

and

$k_{0}\searrow 1,\ \ \eta\|\Phi(\rho)\|^{-1/2}\to c_{0}<\varepsilon\ \
\text{as}\ \|\Phi(\rho)\|\to\infty.$

######  Remark 4.2.

In the statement of Lemma 4.6 in [Wol89], it is written that the constant
$k_{0}$ depends on $[\gamma]$. However, from his proof, one can observe it is
actually independent.

###### Proof.

The lower bound $i(\beta\Phi(\rho),[\gamma])\leq\ell_{\rho}([\gamma])$ is
shown in the same manner with [Wol89], so we omit the proof here.

In order to show the upper bound, we first show the following lemma.

######  Proposition 4.3.

For each $\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$, we can construct a
representative $\gamma_{\Phi}\in[\gamma]$ so that there exist domains
$R_{0}=R_{0}([\gamma],\varepsilon)$ and $R_{1}=R_{1}(\varepsilon)$ containing
all punctures of $M$, constants
$\delta=\delta(\varepsilon)>0,K=K([\gamma])>0$, and positive integers
$L=L([\gamma]),m=m([\gamma],\varepsilon)$ satisfying the following conditions:

  1. (1)

The curve $\gamma_{\Phi}$ consists of horizontal and vertical arcs of $\Phi$
and does not intersect with the neighborhood
$R_{0}=R_{0}([\gamma],\varepsilon)$ of the punctures of $M$.

  2. (2)

The horizontal arcs of $\gamma_{\Phi}$ are divided into the main part
$\gamma_{\Phi}^{h}$ and the additional part $\widetilde{\gamma}_{\Phi}^{h}$
such that

     * •

the main part $\gamma_{\Phi}^{h}$ is disjoint from

$\left(\bigcup_{p}B_{\sigma}(p,\delta)\right)\cup R_{1},$

where the union is over all zeros $p$ of $\Phi$.

     * •

the number of the connected segments constituting the additional part
$\widetilde{\gamma}_{\Phi}^{h}$ is at most $m=m([\gamma],\varepsilon)$, and
they are disjoint from some neighborhood of zeros of $\Phi$.

  3. (3)

$i(\beta\Phi,\gamma_{\Phi}^{h})=i(\beta\Phi,[\gamma])$ and
$i(\beta\Phi,\widetilde{\gamma}_{\Phi}^{h})<\varepsilon$.

  4. (4)

The total $|\Phi|$-length of the vertical arcs of $\gamma_{\Phi}$ is uniformly
bounded by $K=K([\gamma])$, i.e. $i(\beta(-\Phi),\gamma_{\Phi}^{v})<K$, where
$\gamma_{\Phi}^{v}$ is the union of the vertical arcs. Moreover, the number of
the connected segments of the vertical arcs which contain a zero of $\Phi$ is
at most $L=L([\gamma])$.

###### Proof.

We fix $\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$. By the compactness of
$\operatorname{\mathrm{SQD}}(\sigma)$, it is enough to show that the claim
holds on a neighborhood of $\Phi_{0}$.

For a quadratic differential $\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$, we
let

$\displaystyle Z(\Phi)$ $\displaystyle=\\{\text{zeros of }\Phi\text{ in
}M\\},$ $\displaystyle P_{\mathrm{reg}}(\Phi)$
$\displaystyle=\\{\text{punctures which are regular points or zeros of
}\Phi\\},$ $\displaystyle P_{\mathrm{pole}}(\Phi)$
$\displaystyle=\\{\text{punctures which are poles of }\Phi\\},$
$\displaystyle\Sigma(\Phi)$ $\displaystyle=Z(\Phi)\cup
P_{\mathrm{reg}}(\Phi)\cup P_{\mathrm{pole}}(\Phi),\text{the singular set of
}\Phi.$

Furthermore, we pick a positive constant
$\delta^{\prime}=\delta^{\prime}(\Phi_{0})$ such that, for all distinct
$p_{i},p_{j}\in\Sigma(\Phi_{0})$, the $|\Phi_{0}|$-distance between
$B_{|\Phi_{0}|}(p_{i},2\delta^{\prime})$ and
$B_{|\Phi_{0}|}(p_{j},2\delta^{\prime})$ is at least $2\delta^{\prime}$, and
also

$\delta^{\prime}(\Phi_{0})<\Bigl{(}\min_{\Phi\in\operatorname{\mathrm{SQD}}(\sigma)}\operatorname{\mathrm{inj}}|\Phi|\Bigr{)}/(8g-5+2n),$

where we set
$\operatorname{\mathrm{inj}}|\Phi|\coloneqq\inf_{[\gamma]\in\mathcal{C}}\ell_{|\Phi|}([\gamma])/2$.
We may assume $\varepsilon<\delta^{\prime}$, since $\varepsilon>0$ is
arbitrary. Then, we pick a sufficiently small neighborhood $\mathcal{N}$ of
$\Phi_{0}$ in $\operatorname{\mathrm{SQD}}(\sigma)$ so that every
$\Phi\in\mathcal{N}$ satisfies the following conditions:

  1. (1)

The connected components of

$N(\Sigma(\Phi_{0}),\delta^{\prime})\coloneqq\bigcup_{p\in\Sigma(\Phi_{0})}B_{|\Phi_{0}|}(p,\delta^{\prime})$

bijectively correspond to the connected components of

$N(\Sigma(\Phi),\delta^{\prime})\coloneqq\bigcup_{p\in\Sigma(\Phi)}B_{|\Phi|}(p,\delta^{\prime})$

by the correspondence between $\Sigma(\Phi)$ and $\Sigma(\Phi_{0})$.

  2. (2)

If we fill the punctures, every connected component $C$ of
$N(\Sigma(\Phi),\delta^{\prime})$ is a topologically disk, and the
$|\Phi|$-distance between $C$ and the other connected components is at least
$\delta^{\prime}$.

  3. (3)

The total length of critical vertical leaves of $\Phi$ contained in
$N(\Sigma(\Phi),\delta^{\prime})$ is uniformly bounded from above by a
constant $K_{1}=K_{1}(\Phi_{0})$.

  4. (4)

The zeros of $\Phi$ splitting off (see Figure 1) from $q\in
P_{\mathrm{reg}}(\Phi_{0})$ is contained in $B_{|\Phi|}(q,r/2)$, where $r$ is
a sufficiently small constant which depends only on $[\gamma]$, $\varepsilon$
and $\Phi_{0}$ and is defined later in 4.3.

Figure 1. The zeros splits off from a puncture.

Under the preparation, we describe the construction of the representative
$\gamma_{\Phi}$ for each $\Phi\in\mathcal{N}$. First, we beggin with the
$|\Phi|$-geodesic representative $\Gamma_{\Phi}$ of $[\gamma]$. Note that
$\Gamma_{\Phi}$ may touch some punctures and cannot be realized on $M$ in a
strict sense (see Figure 2). (If all of punctures of $S$ are simple poles of
$\Phi$, there exists a strict $|\Phi|$-geodesic representative of $[\gamma]$
in $M$, see [Sya96].)

Figure 2. Left and middle: we illustrate examples of the geodesic
representative in a slightly broad sense around a puncture. Right: there is no
such a representative in the right of the figures.

Each segment of the geodesic representative $\Gamma_{\Phi}$ outside of
$N(\Sigma(\Phi),\delta^{\prime})$ is a Euclidean straight line segment, so we
replace such a straight segment of $\Gamma_{\Phi}$ with a $\Phi$-staircase
curve which is union of horizontal and vertical arcs of $\Phi$. Let
$\Gamma_{\Phi}^{\prime}$ be the resulting curve. Then

$i(\beta\Phi,\Gamma^{\prime}_{\Phi})=i(\beta\Phi,[\gamma])\ \text{and}\
i(\beta(-\Phi),\Gamma^{\prime}_{\Phi})<\max_{\Phi\in\operatorname{\mathrm{SQD}}(\sigma)}\ell_{|4\Phi|}([\gamma]).$

Next, let $C$ be a connected component of $N(\Sigma(\Phi),\delta^{\prime})$.
Then, the number of punctures contained in $C$ is at most one.

######  Claim.

The number of connected components of $\Gamma_{\Phi}^{\prime}\cap C$ is, at
most, $\alpha=\alpha([\gamma],\delta^{\prime})$.

###### Proof.

Case I. We suppose that $C$ contains a puncture $q$. Then, $C$ contains, at
most, $4g-4+n$ zeros of $\Phi$. Therefore, $C$ is contained in the disk

(4.1) $D\coloneqq B_{|\Phi|}(q,(8g-7+2n)\delta^{\prime}).$

The disk $D$ does not contain the other puncture and $D$ is embedded into $M$,
since $(8g-7+2n)\delta^{\prime}<\operatorname{\mathrm{inj}}|\Phi|$. Let $p$ be
a zero of $\Phi$ contained in $C$. Let $\Gamma_{\Phi}^{\prime\prime}$ denote a
connected component of $\Gamma_{\Phi}\cap D$. Notice that
$B_{|\Phi|}(p,\delta^{\prime})$ may not be a convex set, since it can contain
a pole of $\Phi$ at the puncture $q$. However, considering a double branched
covering of $D$, we find that the number of the connected components of
$\Gamma_{\Phi}^{\prime\prime}\cap B_{|\Phi|}(p,\delta^{\prime})$ is at most
two (see Figure 3).

\begin{overpic}[scale={1.3}]{DoubleBranchedCover.pdf} \put(14.0,26.0){$q$}
\put(22.0,26.0){$p$} \put(68.0,26.0){$\tilde{q}$} \put(61.5,17.0){$\tilde{p}$}
\put(79.0,32.0){$\tilde{p}$} \end{overpic} Figure 3. Left: it is difficult to
directly understand the $\delta^{\prime}$-neighborhood of a zero near a pole
of $\Phi$. Right: the lift of the double cover branched at the puncture $q$.

Thus, the number of connected components of $\Gamma_{\Phi}^{\prime\prime}\cap
C$ is at most $2(4g-3+n)$, where $4g-3+n$ is the maximum of the number of the
disks constituting $C$. Let $\gamma^{\prime}$ denote a subarc of
$\Gamma_{\Phi}$ which leaves $D$ and comes back to $D$. The endpoints of
$\gamma^{\prime}$ is on $\partial D$. Then, since $D$ is convex and
$\Gamma_{\Phi}$ is geodesic, $\gamma^{\prime}$ is not isotopic to $\partial D$
rel the endpoints. Thus, we see that the $|\Phi|$-length of $\gamma^{\prime}$
is at least $\delta^{\prime}$. By the above discussion, we find that, every
time $\Gamma_{\Phi}$ intersects $D$, the number of the connected components of
$C\cap\Gamma_{\Phi}$ increases by at most $8g-6+2n$ and also the number of the
connected components of $\Gamma_{\Phi}\cap D$ is at most

(4.2)
$\biggl{\lceil}\max_{\Phi\in\operatorname{\mathrm{SQD}}(\sigma)}\\{\ell_{|4\Phi|}([\gamma])\\}/\delta^{\prime}(\Phi_{0})\biggr{\rceil}.$

Case II. Suppose next that $C$ contains no punctures. Then, $C$ contains, at
most $4g-4+n$, zeros of $\Phi$. Therefore, the $|\Phi|$-diamiter of $C$ is, at
most, $(8g-8+2n)\delta^{\prime}$. Hence, $C$ is contained in a ball
$B=B_{|\Phi|}(p_{0},(4g-4+n)\delta^{\prime})$ for a point $p_{0}\in M$. Then
the ball contains at most one puncture, since
$(8g-8+2n)\delta^{\prime}<(8g-5+2n)\delta^{\prime}<\operatorname{\mathrm{inj}}|\Phi|$.

Then we have two cases. First, suppose that $B$ contains a puncture $q$. Then

$D=B_{|\Phi|}(q,(8g-7+2n)\delta^{\prime})$

contains $B$, and $D$ contains no other punctures. Therefore $D$ is convex,
and thus we can apply the argument of Case I to $D$. However,
$B_{|\Phi|}(p,\delta^{\prime})$ is now convex for a zero $p$ of $\Phi$
contained in $C$, so we do not need to take a double branched covering. Thus
we obtain the desired upper bounds. Second, suppose that $B$ contains no
punctures. If the neighborhood $N_{|\Phi|}(B,\delta^{\prime})$ of $B$ contains
a puncture $q$, $B$ is contained in the disk

$D=B_{|\Phi|}(q,(8g-7+2n)\delta^{\prime}).$

Therefore we can show the desired claim as in the first case that $C$ contains
a puncture $q$. If the neighborhood $N_{|\Phi|}(B,\delta^{\prime})$ of $B$
does not contain a puncture, each connected component of
$\Gamma_{\Phi}\setminus B$ has length at least $\delta^{\prime}$. Then,
setting $D=B$, we again apply the argument of Case I to $D$. ∎

Let $\overline{\gamma}$ denote a connected component of
$\Gamma_{\Phi}^{\prime}\cap C$. If a vertical arc of $\Phi$ contains a
singular point of $\Phi$ as its end point, then it is called a critical
vertical arc. Then, either (i) $\overline{\gamma}$ has at most one
intersection with each critical vertical arc contained in $C$, or (ii)
$\overline{\gamma}$ is a union of critical vertical arcs. We leave the curves
in the case (ii), and deform the curves in the case (i). The number of the
critical vertical arcs contained in $C$ is, at most, $3(4g-4+n)+1$, so the
number of the intersections of $\overline{\gamma}$ and the critical vertical
arcs is clearly finite. Then, we divide $\overline{\gamma}$ at these
intersection points into finitely many segments. For each such segment of
$\overline{\gamma}$, we replace it with a curve consisting of horizontal arcs
and vertical arcs of $\Phi$. Then, we drag the horizontal measure out of $C$
by adding vertical arcs (Figure 4).

Figure 4. The dotted lines denote vertical leaves of $\Phi$. The black points
denote intersections of critical vertical arcs and $\overline{\gamma}$. The
upper right figure illustrates the lift of the foliation in the upper left
figure.

Then we set

$R_{1}=\bigcap_{\Phi\in\mathcal{N}}\left(\bigcup_{q\in\overline{S}-S}B_{|\Phi|}(q,\delta^{\prime})\right).$

Let $\widehat{\gamma}$ denote the resulting curve from $\overline{\gamma}$.

Next, we add the following operation, which is denoted by $(\ast)$, for each
the connected component $\alpha$ of $\widehat{\gamma}\cap B_{|\Phi|}(q,r)$:

  1. (1)

isotope $\alpha$ to $\partial B_{|\Phi|}(q,r)$ rel the endpoints, and

  2. (2)

then further isotope $\alpha$ on $\partial B_{|\Phi|}(q,r)$ to a union of
vertical and horizontal arcs tangent to $\partial B_{|\Phi|}(q,r)$ (see Figure
5).

By this operaton ($\ast$), $\gamma_{\Phi}$ does not enter the $r$-neighborhood
of the puncture $q$. Since, by Lemma 3.12, the $|\Phi|$-length of $\partial
B_{|\Phi|}(q,r)$ is at most $L_{1}r$, this operation $(\ast)$ increases the
horizontal measure by at most $L_{1}r$. Thus, if we take $r$ so that

(4.3) $m_{1}\alpha L_{1}r<\varepsilon,$

then the increase due to this operation ($\ast$) is less than $\varepsilon$,
where $m_{1}$ is the upper bounds on the total number of the operation
$(\ast)$ all over each $\widehat{\gamma}$ and $\alpha$ is as in the above
claim. Moreover, $m_{1}$ depends only on the topology of $M$. Thus, we set a
domain $R_{0}$ by

$R_{0}=\bigcap_{\Phi\in\mathcal{N}}\left(\bigcup_{q\in\overline{S}-S}B_{|\Phi|}(q,r)\right)$

and define $\gamma_{\Phi}$ as the resulting representative of $[\gamma]$. The
constant $\delta$ and the domain $R_{1}$ in 4.3 is determined by
$\delta^{\prime}$.

Figure 5. This is an example of a vertical arc in $B_{|\Phi|}(q,r)$. By the
operation $(\ast)$, we let the curve not enter $B_{|\Phi|}(q,r)$.

∎

Next, we describe the constant $k_{0}$. For $\rho\in\mathcal{T}_{g,n}$, we set

$M^{\prime}=M-R_{1},\ M_{\delta}(\rho)=M^{\prime}-\bigcup_{p\in
Z(\Phi)}B_{\sigma}(p,\delta).$

Furthermore, we define a function $k_{2}$ by

$k_{2}(\rho)=\max_{p\in
M_{\delta(\rho)}}\frac{\|(\partialderivative{x})_{p}\|_{\rho}}{\|(\partialderivative{x})_{p}\|_{|4\Phi(\rho)|}}.$

######  Claim.

The function $k_{2}$ is upper semicontinuous in $\mathcal{T}_{g,n}$.

###### Proof.

This proof is similar to that in [Wol89]. [Wol89, p.464]. ∎

Hence, defining a function
$\kappa_{A}:\operatorname{\mathrm{SQD}}(\sigma)\to\mathbb{R}_{\geq 0}$ for
$A>0$ by

$\kappa_{A}(\Phi_{0})\coloneqq k_{2}(\rho(A\Phi_{0})),$

we find that $\kappa_{A}$ is the upper semicontinuous function on
$\operatorname{\mathrm{SQD}}(\sigma)$. Since a upper semicontinuous function
on a compact set has a maximum, we set

$k_{0}(A,\varepsilon)=\max_{\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)}\kappa_{A}(\Phi_{0}).$

Then, by 3.7, for every $p\in M_{\delta}(\Phi_{0})$,

$\frac{\|(\partialderivative{x})_{p}\|_{\rho(A\Phi_{0})}}{\|(\partialderivative{x})_{p}\|_{|4A\Phi_{0}|}}\searrow
1\ \ \text{as }A\to\infty.$

Therefore $\kappa_{A}(\Phi_{0})\searrow 1$ as $A\to\infty$. Since
$\\{\kappa_{A}\\}$ converges uniformly on
$\operatorname{\mathrm{SQD}}(\sigma)$ by Dini’s theorem, we find that
$k_{0}\searrow 1$ as $A\to\infty$. Thus we have

$\displaystyle\ell_{\rho}(\gamma^{h}_{\Phi_{0}(\rho)})$
$\displaystyle=\sum_{\gamma\subset\gamma^{h}_{\Phi_{0}}}\int_{\gamma}\left\|\partialderivative{x}\right\|_{\rho}ds$
$\displaystyle\leq\sup_{\gamma_{\Phi_{0}}^{h}}\frac{\|(\partialderivative{x})_{p}\|_{\rho}}{\|(\partialderivative{x})_{p}\|_{|4\Phi(\rho)|}}\sum\int_{\gamma}\left\|\partialderivative{x}\right\|_{|4\Phi(\rho)|}ds$
$\displaystyle\leq k_{2}(\rho)i(\beta\Phi(\rho),\gamma_{\Phi_{0}}^{h})$
$\displaystyle\leq
k_{0}(A,\varepsilon)i(\beta\Phi(\rho),\gamma_{\Phi_{0}}^{h}),$

where $\Phi_{0}(\rho)\coloneqq\Phi(\rho)/\|\Phi(\rho)\|$.

Finally, we describe the way of setting the constant
$\eta=\eta([\gamma],\varepsilon,A)$. By 3.8, for any
$\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$

$\ell_{\rho(A\Phi_{0})}(\gamma_{\Phi_{0}}^{v})A^{-1/2}\to 0\ \ \text{as
}A\to\infty.$

Since $\gamma_{\Phi_{0}}$ continuously changes with respect to $\Phi_{0}$ and
$\gamma_{\Phi_{0}}^{v}$ contains at most finite zeros of $\Phi_{0}$, the total
$\sigma$-length of $\gamma_{\Phi_{0}}^{v}$ is bounded from above. Therefore,
we conclude that there exists a constant $\eta_{1}([\gamma],A)$ such that

$\sum_{\gamma\subset\gamma_{\Phi_{0}(\rho)}^{v}}\ell_{\rho}(\gamma)<\eta_{1}\
\text{and\ \ }\eta_{1}A^{-1/2}\to 0\ \ \ \text{as }A\to\infty.$

By 3.10 and the construction of the representative $\gamma_{\Phi}\in[\gamma]$
for $\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$, we can take constants $C$
and $D$ (see Remark 4.4) such that, for every
$\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$,

(4.4)
$\ell_{\rho(t\Phi)}(\widetilde{\gamma}_{\Phi}^{h})<t^{1/2}i(\beta\Phi,\widetilde{\gamma}_{\Phi}^{h})(1+Ce^{-D\sqrt{t}}).$

Therefore, if we set

$c_{0}=c_{0}([\gamma],\varepsilon)=\sup_{\Phi\in\operatorname{\mathrm{SQD}}(\sigma)}i(\beta\Phi,\widetilde{\gamma}_{\Phi}^{h})\
$

and

$\eta_{2}=\eta_{2}([\gamma],\varepsilon,A)=A^{1/2}c_{0}(1+Ce^{-D\sqrt{t}}),$

then, for every $\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$,

$\ell_{\rho(t\Phi)}(\widetilde{\gamma}_{\Phi}^{h})<\eta_{2}\ \ \text{and}\ \
\eta_{2}A^{-1/2}\to c_{0}<\varepsilon$

as $A\to\infty$. Hence, setting $\eta\coloneqq\eta_{1}+\eta_{2}$, we have
$\eta A^{-1/2}\to c_{0}$ as $A\to\infty$. Thus, for every
$\rho\in\rho(A\cdot\operatorname{\mathrm{SQD}}(\sigma))$,

$\displaystyle\ell_{\rho}([\gamma])$
$\displaystyle\leq\ell_{\rho}(\gamma_{\Phi_{0}(\rho)})$
$\displaystyle=\ell_{\rho}(\gamma_{\Phi_{0}(\rho)}^{h})+\ell_{\rho}(\widetilde{\gamma}_{\Phi_{0}(\rho)}^{h})+\ell_{\rho}(\gamma_{\Phi_{0}(\rho)}^{v})$
$\displaystyle<k_{0}i(\beta\Phi(\rho),[\gamma])+\eta,$

and the proof is complete. ∎

######  Remark 4.4.

Here, we describe the setting of $\eta_{2}$, namely the constants $C$ and $D$
in 4.4. In the proof of 3.10, $D$ is determined by the $|\Phi|$-distance from
a horizontal arc to zeros of $\Phi$. Now, for each
$\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$, $\widetilde{\gamma}_{\Phi}^{h}$
is at least $r/2$ away from zeros of $\Phi$ in the $|\Phi|$ metric, so the
constant $D$ can be taken uniformly on $\operatorname{\mathrm{SQD}}(\sigma)$.
Moreover, we take the constant $C$ as follows. We set $r^{\prime}>0$ so that,
for every $\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$ and every $p\in
Z(\Phi)$,

$B_{\sigma}(p,r^{\prime})\subset B_{|\Phi|}(p,r/2).$

By the same manner as the upper semicontinuity of $k_{2}$, we conclude that
the map

$\Phi\mapsto\min_{p\in M_{r^{\prime}}(\rho(\Phi))}|\nu(\Phi)|$

is lower semicontinuous on $\operatorname{\mathrm{QD}}(\sigma)$. Therefore,
there exists a lower bound of $\displaystyle\min_{p\in
M_{r^{\prime}}(\rho(\Phi))}|\nu(\Phi)(p)|$ for
$\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$.

Thus we find that $|\nu(\Phi)|\geq C_{1}>0$ and $\log|\nu(\Phi)|^{-1}\leq
C_{2}<+\infty$ on $\widetilde{\gamma}_{\Phi}^{h}\ (\subset
M_{r^{\prime}}(\rho(\Phi)))$ for every
$\Phi\in\operatorname{\mathrm{SQD}}(\sigma)$. Therefore we obtain the desired
constant $C$, since it is given by $C_{1}$ and $C_{2}$ as in the proof of
3.10.

###  4.2. The extended homeomorphism

The following lemma is an extension of [Wol89, Lemma 4.7] for punctured
surfaces.

######  Proposition 4.5.

Let $\\{\rho_{i}\\}\subset\mathcal{T}_{g,n}$ be a sequence diverging to
$\infty$ (i.e. it leaves every compact in $\mathcal{T}_{g,n}$). Then
$\\{\pi\circ\ell(\rho_{i})\\}$ converges in $\operatorname{\mathcal{PMF}}$ if
and only if $\\{\pi\circ\beta\Phi(\rho_{i})\\}$ converges in
$\operatorname{\mathcal{PMF}}$. Moreover, their limits in
$\operatorname{\mathcal{PMF}}$ coincide when they converges.

###### Proof.

We can take finitely many essential simple closed curves
$[\gamma_{1}],\ldots,[\gamma_{k}]\in\mathcal{C}$ and a positive number
$\delta>0$ so that

$\sum_{j=1}^{k}i(\beta\Phi_{0},[\gamma_{j}])>\delta>0$

holds for every $\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)$.

Suppose that $\pi\circ\ell(\rho_{n})$ converges. Then, there exists a sequence
$\\{\lambda_{n}\\}\subset\mathbb{R}_{>0}$ such that
$\lambda_{n}\ell(\rho_{n})$ converges in $\mathbb{R}_{>0}^{\mathcal{C}}$.
Therefore, there exists $B>0$ such that, for each $n\in\mathbb{N}$,

$\displaystyle B$
$\displaystyle>\sum_{j=1}^{k}\lambda_{n}\ell_{\rho_{n}}([\gamma_{j}])$ (4.5)
$\displaystyle>\lambda_{n}\sum_{j=1}^{k}i(\beta\Phi(\rho_{n}),[\gamma_{j}])\ \
\ (\text{By \lx@cref{creftypecap~refnum}{prop:FundamentalInequality}})$
$\displaystyle=\lambda_{n}\|\Phi(\rho_{n})\|^{1/2}\sum_{j=1}^{k}i(\beta(\Phi(\rho_{n})/\|\Phi(\rho_{n})\|),[\gamma_{j}])$
$\displaystyle>(\lambda_{n}\|\Phi(\rho_{n})\|^{1/2})\delta.$

Hence, we see that

$\lambda_{n}\cdot\eta([\gamma],\varepsilon,\|\Phi(\rho_{n})\|)<(B/\delta)\cdot\eta\|\Phi(\rho_{n})\|^{-1/2}.$

By Theorem 4.1,

$\lim_{n\to\infty}\eta\|\Phi(\rho_{n})\|^{-1/2}=c_{0}([\gamma],\varepsilon)<\varepsilon.$

Morever, for any $\varepsilon>0$ and $[\gamma]\in\mathcal{C}$,

$\displaystyle\lambda_{n}i(\beta\Phi(\rho_{n}),[\gamma])$
$\displaystyle=\lambda_{n}\|\Phi(\rho_{n})\|^{1/2}i(\beta(\Phi(\rho_{n})/\|\Phi(\rho_{n})\|),[\gamma])$
$\displaystyle<(B/\delta)\max_{\Phi_{0}\in\operatorname{\mathrm{SQD}}(\sigma)}i(\beta\Phi_{0},[\gamma]).$

Thus by Theorem 4.1, for any $\varepsilon>0$ and $[\gamma]\in\mathcal{C}$,

$|\lambda_{n}\ell_{\rho_{n}}([\gamma])-\lambda_{n}i(\beta\Phi(\rho_{n}),[\gamma])|<(k_{0}-1)\lambda_{n}i(\beta\Phi(\rho_{n}),[\gamma])+\lambda_{n}\eta$

and

$(k_{0}-1)\lambda_{n}i(\beta\Phi(\rho_{n}),[\gamma])\to 0$

as $n\to\infty$. Therefore, for any $\varepsilon>0$ and
$[\gamma]\in\mathcal{C}$,

$\lim_{n\to\infty}(\lambda_{n}\ell_{\rho_{n}}([\gamma])-\lambda_{n}i(\beta\Phi(\rho_{n}),[\gamma]))<\varepsilon.$

Since we can take arbitraly small $\varepsilon>0$, we have

$\lim_{n\to\infty}\lambda_{n}\ell(\rho_{n})=\lim_{n\to\infty}\lambda_{n}\beta\Phi(\rho_{n})\
\ \text{in}\ \mathbb{R}_{\geq 0}^{\mathcal{C}}.$

Thus we conclude that

$\lim_{n\to\infty}\pi\circ\beta\Phi(\rho_{n})=\lim_{n\to\infty}\pi\circ\ell(\rho_{n}).$

We can show the converse in the same manner by starting with 4.5. ∎

Define a map
$\psi:\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}\to\overline{\mathcal{T}_{g,n}^{\mathrm{harm}}}$
by

$\psi(x)=\begin{cases}\left(\displaystyle\lim_{n\to\infty}\frac{\Phi(x_{n})}{\|\Phi(x_{n})\|},1\right)&(x\in\partial_{\mathrm{Th}}\mathcal{T}_{g,n},\
\\{x_{n}\\}\subset\mathcal{T}_{g,n}\ \text{with}\ x_{n}\to x)\\\\[15.0pt]
\left(\displaystyle\frac{\Phi(x)}{\|\Phi(x)\|},\frac{4\|\Phi(x)\|}{1+4\|\Phi(x)\|}\right)&(x\in\mathcal{T}_{g,n}),\end{cases}$

where we use the polar coordinates in
$\overline{\operatorname{\mathrm{BQD}}(\sigma)}$ (i.e. for
$(r,\theta)\in\overline{\operatorname{\mathrm{BQD}}(\sigma)}$,
$\theta\in\operatorname{\mathrm{SQD}}(\sigma)$ and $r\in[0,1]$).

Our main theorem, Theorem 4.6, is shown from 4.5. The proof is essentially
same as the proof of [Wol89, Theorem 4.1]. However we write the proof here for
the sake of completeness.

######  Theorem 4.6.

The map $\psi$ is a homeomorphism.

###### Proof.

We first show that $\psi$ is well-defined on $\operatorname{\mathcal{PMF}}$.
For sequences $\\{x_{n}\\}$, $\\{x_{n}^{\prime}\\}$ with
$x_{n},x_{n}^{\prime}\to x\in\operatorname{\mathcal{PMF}}$, there exist

$\lim_{n\to\infty}\pi\circ\beta\Phi(x_{n})\ \text{and}\
\lim_{n\to\infty}\pi\circ\beta\Phi(x_{n}^{\prime}),$

and they coincide by 4.5. Therefore there exist sequences $\\{\lambda_{n}\\}$,
$\\{\lambda_{n}^{\prime}\\}$ such that

$\lim_{n\to\infty}\lambda_{n}\beta\Phi(x_{n})=\lim_{n\to\infty}\lambda_{n}^{\prime}\beta\Phi(x_{n}^{\prime}).$

Since $\beta$ is a homeomorphism,

$\lim_{n\to\infty}\lambda_{n}^{2}\Phi(x_{n})=\lim_{n\to\infty}{\lambda^{\prime}_{n}}^{2}\Phi(x_{n}^{\prime}).$

Hence, we have

$\lim_{n\to\infty}\frac{\Phi(x_{n})}{\|\Phi(x_{n})\|}=\lim_{n\to\infty}\frac{\lambda_{n}^{2}\Phi(x_{n})}{{\lambda_{n}}^{2}\|\Phi(x_{n})\|}=\lim_{n\to\infty}\frac{{\lambda_{n}^{\prime}}^{2}\Phi(x_{n}^{\prime})}{{\lambda_{n}^{\prime}}^{2}\|\Phi(x_{n}^{\prime})\|}=\lim_{n\to\infty}\frac{\Phi(x_{n}^{\prime})}{\|\Phi(x_{n}^{\prime})\|}.$

Thus, $\psi$ is well-defined.

We secondly show that $\psi$ is continuous. In particular, we need to show
$\psi$ is continuous at a point $x\in\operatorname{\mathcal{PMF}}$. Let
$\\{x_{n}\\}$ be a sequence in $\mathcal{T}_{g,n}$ with $x_{n}\to x$. Then,
since $x_{n}$ diverges to $\infty$,

$\frac{4\|\Phi(x_{n})\|}{1+4\|\Phi(x_{n})\|}\to 1\ \ \text{as}\ \ n\to\infty.$

Thus, we find that the second componet of $\psi$ is continuous. The first
componet is continuous by the definition of $\psi$.

We thirdly show that $\psi$ is injective. The injectivity on
$\mathcal{T}_{g,n}$ follows from the injectivity of $\Phi$. Therefore, we need
to show $\psi$ is injective on $\operatorname{\mathcal{PMF}}$. Suppose that
$\psi(x)=\psi(x^{\prime})$ for $x,x^{\prime}\in\operatorname{\mathcal{PMF}}$.
Let $x_{n}$ and $x_{n}^{\prime}$ be sequences in $\mathcal{T}_{g,n}$ such that
$x_{n}\to x\in\operatorname{\mathcal{PMF}}$ and $x_{n}^{\prime}\to
x^{\prime}\in\operatorname{\mathcal{PMF}}$ as $n\to\infty$. Since
$\psi(x)=\psi(x^{\prime})$, we have

$\lim_{n\to\infty}\frac{\Phi(x_{n})}{\|\Phi(x_{n})\|}=\lim_{n\to\infty}\frac{\Phi(x_{n}^{\prime})}{\|\Phi(x_{n}^{\prime})\|}.$

Therefore, we have

$\lim_{n\to\infty}\pi\circ\beta\Phi(x_{n})=\lim_{n\to\infty}\pi\circ\beta\Phi(x_{n}^{\prime}).$

By 4.5, we have

$\lim_{n\to\infty}\pi\circ\ell(x_{n})=\lim_{n\to\infty}\pi\circ\ell(x_{n}^{\prime}).$

This imples $x=x^{\prime}$. Thus we find that $\psi$ is injective on
$\operatorname{\mathcal{PMF}}$.

We forthly show that $\psi$ is surjective. The restriction
$\psi|_{\mathcal{T}_{g,n}}$ is surjective, since $\Phi$ is a homeomorphism, so
we need to show the surjectivity for $\operatorname{\mathrm{SQD}}(\sigma)$.
Taking each $\theta\in\operatorname{\mathrm{SQD}}(\sigma)$, we set
$x_{n}=\Phi^{-1}(n\theta)$. Then
$\pi\circ\beta\Phi(x_{n})=\pi\circ\beta(n\theta)=[F_{v}(\theta)]\in\operatorname{\mathcal{PMF}}$
for every $n$. Therefore, we have

$\lim_{n\to\infty}\pi\circ\ell(x_{n})=[F_{v}(\theta)]$

by 4.5. Thus,

$\psi([F_{v}(\theta)])=\left(\lim_{n\to\infty}\frac{\Phi(x_{n})}{\|\Phi(x_{n})\|},1\right)=(\theta,1).$

Thus, $\psi$ is surjective.

We finally show that $\psi^{-1}$ is continuous. Let
$(\theta_{n},r_{n})\to(\theta,1)$ as $n\to\infty$ for each
$\theta\in\operatorname{\mathrm{SQD}}(\sigma)$, where
$\theta_{n}\in\operatorname{\mathrm{SQD}}(\sigma)$ and $r_{n}\in(0,1)$. Then

$\psi^{-1}(\theta_{n},r_{n})=\Phi^{-1}\left(\frac{r_{n}\theta_{n}}{4(1-r_{n})}\right)\text{
and }\psi^{-1}(\theta,1)=[F_{v}(\theta)].$

Therefore, setting $x_{n}=\psi^{-1}(\theta_{n},r_{n})$, we have

$\pi\circ\beta\Phi(x_{n})=[F_{v}(\theta_{n})].$

Thus,

$\psi^{-1}(\theta,1)=[F_{v}(\theta)]=\lim_{n\to\infty}[F_{v}(\theta_{n})]=\lim_{n\to\infty}\pi\circ\beta\Phi(x_{n})=\lim_{n\to\infty}\psi^{-1}(\theta_{n},r_{n}),$

and this completes the proof. ∎

######  Remark 4.7.

This is another proof that the Thurston compactification
$\overline{\mathcal{T}_{g,n}^{\mathrm{Th}}}$ is a closed ball of dimension
$6g-6+2n$.

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