# Comparison of the Hitchin metric and the semi-flat metric
in the rank two case

Takuro Mochizuki Research Institute for Mathematical Sciences, Kyoto
University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>
###### Abstract

Let $(E,\theta)$ be a Higgs bundle of rank $2$ and degree $0$ on a compact
Riemann surface $X$ whose spectral curve is smooth. The tangent space of the
moduli space of Higgs bundles at $(E,\theta)$ is equipped with two natural
metrics called the Hitchin metric and the semi-flat metric. It is known that
the difference between two metrics along the curve $(E,t\theta)$ $(t\geq 1)$
decays in an exponential way. In this paper, we shall study how the
exponential rate is improved.

MSC: 53C07, 58E15, 14D21, 81T13.
Keywords: harmonic bundle, moduli space of Higgs bundles, Hitchin metric,
semi-flat metric

##  1 Introduction

###  1.1 Hitchin metric and semi-flat metric of the moduli space of Higgs
bundles

Let $X$ be a compact Riemann surface. Let $\mathcal{M}_{H}$ denote the moduli
space of stable Higgs bundles of rank $2$ and degree $0$ on $X$. According to
Hitchin [10], $\mathcal{M}_{H}$ is equipped with the hyperkähler metric
$g_{H,{\mathbb{R}}}$ called the Hitchin metric.

There also exists the Hitchin fibration
$\Phi_{H}:\mathcal{M}_{H}\to\mathcal{A}_{H}=H^{0}(X,K_{X}^{\otimes 2})$, which
admits a section $\mathcal{A}_{H}\to\mathcal{M}_{H}$ called a Hitchin section
depending only on the choice of $K_{X}^{1/2}$. Let
$\mathcal{A}_{H}^{\prime}\subset\mathcal{A}_{H}$ denote the set of regular
values. We set
$\mathcal{M}_{H}^{\prime}=\Phi_{H}^{-1}(\mathcal{A}_{H}^{\prime})$. The
restriction of the Hitchin fibration
$\mathcal{M}_{H}^{\prime}\to\mathcal{A}_{H}^{\prime}$ is an algebraic
integrable system, and in this situation $\mathcal{M}_{H}^{\prime}$ is
equipped with the canonically defined hyperkähler metric $g_{\mathop{\rm
sf}\nolimits,{\mathbb{R}}}$ [7], called the semi-flat metric.

Let $(E,\theta)\in\mathcal{M}_{H}^{\prime}$. We obtain the curve $(E,t\theta)$
$(t\geq 1)$ in $\mathcal{M}_{H}^{\prime}$. Let $g_{H}$ and $g_{\mathop{\rm
sf}\nolimits}$ denote the Kähler metrics of $\mathcal{M}_{H}$ with the natural
complex structure induced by $g_{H,{\mathbb{R}}}$ and $g_{\mathop{\rm
sf}\nolimits,{\mathbb{R}}}$. It was predicted in [8, 9] and proved in [4] that
there exists $\epsilon>0$ such that

$g_{H|(E,t\theta)}-g_{\mathop{\rm sf}\nolimits|(E,t\theta)}=O(e^{-\epsilon
t})$ (1)

as $t\to\infty$. Such an estimate was obtained in [4] under a mild assumption
on $(E,\theta)$, and in [18] without the additional assumption by a different
method. However, the original prediction in [8, 9] suggests an explicit range
of $\epsilon$. In this paper, we shall partially verify it in the rank two
case by using the method in [18].

Before explaining our main result more explicitly, let us recall more detail
about the related objects.

####  1.1.1 Hitchin metric

Let $(E,\theta)$ be a Higgs bundle on a compact Riemann surface $X$ of degree
$0$ and $\mathop{\rm rank}\nolimits 2$. Assume that $\mathop{\rm
tr}\nolimits\theta=0$ and $\det(E)=\mathcal{O}_{X}$, for simplicity. We also
assume $(E,\theta)\in\mathcal{M}^{\prime}_{H}$, which is equivalent to the
condition that the spectral curve $\Sigma_{\theta}$ of $(E,\theta)$ is smooth.

According to the fundamental theorem of Hitchin and Simpson, under the
condition, $(E,\theta)$ has a unique harmonic metric $h$ such that
$\det(h)=1$, i.e., the Hitchin equation $R(h)+[\theta,\theta^{\dagger}_{h}]=0$
holds, where $R(h)$ denotes the curvature of the Chern connection of $(E,h)$,
and $\theta^{\dagger}_{h}$ denotes the adjoint of $\theta$ with respect to
$h$. We remark that the induced harmonic metric on $(\mathop{\rm
End}\nolimits(E),\mathop{\rm ad}\nolimits\theta)$ is also denoted by $h$.

We consider the Deformation complex $\mathop{\rm Def}\nolimits(E,\theta)$
given by

$\begin{CD}\mathop{\rm End}\nolimits(E)@>{\mathop{\rm
ad}\nolimits\theta}>{}>\mathop{\rm End}\nolimits(E)\otimes K_{X},\end{CD}$

where the first term sits at the degree $0$. It is well known that

$T_{(E,\theta)}\mathcal{M}^{\prime}_{H}\simeq H^{1}(X,\mathop{\rm
Def}\nolimits(E,\theta)).$ (2)

We recall that an $\mathop{\rm End}\nolimits(E)$-valued $1$-form $\sigma$ is
called harmonic if

$(\overline{\partial}_{E}+\mathop{\rm
ad}\nolimits\theta)\sigma=(\overline{\partial}_{E}+\mathop{\rm
ad}\nolimits\theta)^{\ast}_{h,g_{X}}\sigma=0$

where $\overline{\partial}_{E}$ denotes the holomorphic structure of $E$, and
$g_{X}$ denotes a compact Kähler metric $g_{X}$ of $X$. It is equivalent to
the condition that

$(\overline{\partial}_{E}+\mathop{\rm
ad}\nolimits\theta)\sigma=(\partial_{E,h}+\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h})\sigma=0,$

where $\partial_{E,h}$ denotes the $(1,0)$-part of the Chern connection of
$(E,h)$. Then, (2) is also isomorphic to the space
$\mathcal{H}^{1}(\mathop{\rm End}\nolimits(E),\mathop{\rm
ad}\nolimits\theta,h)$ of harmonic $1$-forms of $(\mathop{\rm
End}\nolimits(E),\mathop{\rm ad}\nolimits\theta,h)$,

We consider the standard $L^{2}$-Hermitian product of $\mathop{\rm
End}\nolimits(E)$-valued $1$-forms $\sigma_{i}$ $(i=1,2)$:

$(\sigma_{1},\sigma_{2})_{L^{2},h}=2\sqrt{-1}\int_{X}\mathop{\rm
Tr}\nolimits\Bigl{(}\sigma_{1}^{1,0}\cdot(\sigma_{2}^{1,0})^{\dagger}_{h}-\sigma_{1}^{0,1}\cdot(\sigma_{2}^{0,1})^{\dagger}_{h}\Bigr{)}.$

Then, the Hitchin metric $g_{H|(E,\theta)}$ is obtained as the restriction of
$(\cdot,\cdot)_{L^{2},h}$ to $\mathcal{H}^{1}(\mathop{\rm
End}\nolimits(E),\mathop{\rm ad}\nolimits\theta,h)\simeq
T_{(E,\theta)}\mathcal{M}_{H}^{\prime}$.

####  1.1.2 Semi-flat metric

Note that $\Phi_{H}:\mathcal{M}_{H}^{\prime}\to\mathcal{A}_{H}^{\prime}$ is a
locally principal torus bundle. It is a Lagrangian fibration, and the fiber
$\Phi_{H}^{-1}(\Phi_{H}(E,\theta))$ is isomorphic to $\mathop{\rm
Pic}\nolimits_{2g(X)-2}(\Sigma_{\theta})$. There exists a Hitchin section
$\mathcal{A}_{H}^{\prime}\to\mathcal{M}_{H}^{\prime}$. There uniquely exists
an integrable connection of $\Phi_{H}$ such that any Hitchin section is
horizontal. We obtain the decomposition of
$T_{(E,\theta)}\mathcal{M}_{H}^{\prime}$ into the vertical part
$(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm ver}\nolimits}$ and the
horizontal part $(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
hor}\nolimits}$. Because the fiber is identified with a component of the
Picard variety of $\Sigma_{\theta}$, there exists an isomorphism

$(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm ver}\nolimits}\simeq
H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}}).$

Because the vertical part is a Lagrangian subspace,
$(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm hor}\nolimits}$ is the
dual space of $(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
ver}\nolimits}$, and hence

$(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm hor}\nolimits}\simeq
H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}}).$

Therefore, we obtain

$T_{(E,\theta)}\mathcal{M}_{H}^{\prime}\simeq
H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})\oplus
H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}}).$

It is also identified with the space $\mathcal{H}^{1}(\Sigma_{\theta})$ of the
harmonic $1$-forms of $\Sigma_{\theta}$.

We consider the standard $L^{2}$-Hermitian product of $1$-forms $\mu_{i}$
$(i=1,2)$ on $\Sigma_{\theta}$:

$(\mu_{1},\mu_{2})_{L^{2},\Sigma_{\theta}}=2\sqrt{-1}\int_{\Sigma_{\theta}}\Bigl{(}\mu_{1}^{1,0}\wedge\overline{\mu_{2}^{1,0}}-\mu_{1}^{0,1}\wedge\overline{\mu_{2}^{0,1}}\Bigr{)}.$

Then, the semi-flat metric $g_{\mathop{\rm sf}\nolimits|(E,\theta)}$ is
obtained as the restriction of $(\cdot,\cdot)_{L^{2},\Sigma_{\theta}}$ to
$\mathcal{H}^{1}(\Sigma)\simeq T_{(E,\theta)}\mathcal{M}^{\prime}_{H}$.

###  1.2 An auxiliary Hermitian product

####  1.2.1 Decompositions

Let $D_{\theta}$ denote the set of the critical values of
$\pi:\Sigma_{\theta}\to X$. Let $Q\in X\setminus D_{\theta}$. There exists a
neighbourhood $X_{Q}$ of $Q$ and a decomposition

$(E,\theta)_{|X_{Q}}=(E_{Q,1},\theta_{Q,1})\oplus(E_{Q,2},\theta_{Q,2}).$

By setting $\mathop{\rm End}\nolimits(E)_{|X_{Q}}^{\circ}=\mathop{\rm
End}\nolimits(E_{Q,1})\oplus\mathop{\rm End}\nolimits(E_{Q,2})$ and
$\mathop{\rm End}\nolimits(E)_{|X_{Q}}^{\bot}=\mathop{\rm
Hom}\nolimits(E_{Q,1},E_{Q,2})\oplus\mathop{\rm
Hom}\nolimits(E_{Q,2},E_{Q,1})$, we obtain the decomposition

$\mathop{\rm End}\nolimits(E)_{|X_{Q}}=\mathop{\rm
End}\nolimits(E)_{|X_{Q}}^{\circ}\oplus\mathop{\rm
End}\nolimits(E)_{|X_{Q}}^{\bot}.$

By varying $Q\in X\setminus D_{\theta}$, we obtain the decomposition

$\mathop{\rm End}\nolimits(E)_{|X\setminus D_{\theta}}=\mathop{\rm
End}\nolimits(E)^{\circ}\oplus\mathop{\rm End}\nolimits(E)^{\bot}.$

For any local section $s$ of $\mathop{\rm End}\nolimits(E)$ on an open set in
$X\setminus D_{\theta}$, we obtain the decomposition $s=s^{\circ}+s^{\bot}$.
We have similar decompositions for any $\mathop{\rm End}\nolimits(E)$-valued
$1$-forms on an open subset of $X\setminus D_{\theta}$.

####  1.2.2 Functions and $1$-forms on the spectral curves

We recall there exists a holomorphic line bundle $L$ on $\Sigma_{\theta}$ such
that $\pi_{\ast}(L)\simeq(E,\theta)$. Here, $\pi_{\ast}(L)$ is equipped with
the Higgs field induced by the tautological $1$-form of $T^{\ast}X$, and the
isomorphism preserves the Higgs fields.

Let $U\subset X\setminus D_{\theta}$ be an open subset. Let $\alpha$ be a
function on $\Sigma_{\theta|U}=\pi^{-1}(U)\subset\Sigma_{\theta}$, then the
multiplication of $\alpha$ on $L$ induces an endomorphism $F_{\alpha}$ of
$E_{|U}$ such that $[\theta,F_{\alpha}]=0$. Similarly, a $1$-form $\tau$ on
$\Sigma_{\theta|U}$ induces an $\mathop{\rm End}\nolimits(E)$-valued $1$-form
$F_{\tau}$ on $U$. If $\alpha$ (resp. $\tau$) is holomorphic, then
$F_{\alpha}$ (resp. $F_{\tau}$) is also holomorphic. Note that even if $U\cap
D_{\theta}\neq\emptyset$, a holomorphic function $\alpha$ on
$\Sigma_{\theta|U}$ induces a holomorphic endomorphism $F_{\alpha}$ of
$E_{|U}$ such that $[\theta,F_{\alpha}]=0$. A holomorphic $1$-form $\nu$ on
$\Sigma_{\theta|U}$ induces a holomorphic section of $\mathop{\rm
End}\nolimits(E)\otimes K_{X}(D_{\theta})$ on $U$.

####  1.2.3 Holomorphic $1$-forms on the spectral curve induced by harmonic
metrics

Let $h_{\infty}$ be the limiting configuration of $(E,\theta)$ in the sense of
[13] (see §3.3.1). We consider the following $\mathop{\rm
End}\nolimits(E)$-valued $1$-form on $X\setminus D_{\theta}$:

$\Psi_{h}=(\partial_{E,h_{\infty}}+\theta^{\dagger}_{h_{\infty}})-(\partial_{E,h}+\theta^{\dagger}_{h}).$

Because both $\partial_{E,h_{\infty}}+\theta^{\dagger}_{h_{\infty}}$ and
$\partial_{E,h}+\theta^{\dagger}_{h}$ commute with
$\overline{\partial}_{E}+\theta$, we obtain
$(\overline{\partial}_{E}+\mathop{\rm ad}\nolimits\theta)\Psi_{h}=0$. There
exists a holomorphic $1$-form $\eta_{(E,\theta)}$ on
$\Sigma_{\theta|X\setminus D_{\theta}}$ such that

$(\Psi_{h}^{1,0})^{\circ}=F_{\eta_{(E,\theta)}}.$

######  Lemma 1.1 (Lemma 3.29)

$\eta_{(E,\theta)}$ is holomorphic on $\Sigma_{\theta}$.

####  1.2.4 An auxiliary Hermitian product

Let $\phi^{1/2}$ denote the holomorphic $1$-form obtained as the canonical
root of $\pi^{\ast}(-\det\theta)$ on $\Sigma_{\theta}$. Let
$\widetilde{D}_{\theta}$ be the set-theoretic inverse image
$\pi^{-1}(D_{\theta})$. For $\nu_{i}\in
H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$, we obtain

$\nu_{1}\cdot\nu_{2}\cdot\eta_{(E,\theta)}\cdot(\phi^{1/2})^{-1}\in
H^{0}\Bigl{(}\Sigma_{\theta},(K_{\Sigma_{\theta}}(\widetilde{D}_{\theta}))^{2}\Bigr{)}.$

Note that for each $P\in D_{\theta}$, there exists a natural map

$\mathop{\rm
Res}\nolimits^{(2)}_{\pi^{-1}(P)}:H^{0}\bigl{(}\Sigma_{\theta},(K_{\Sigma_{\theta}}(\widetilde{D}))^{2}\bigr{)}\longrightarrow{\mathbb{C}}$

obtained as the residue. We define

$\langle\nu_{1},\nu_{2}\rangle^{\mathop{\rm
aux}\nolimits}_{(E,\theta)}=-4\pi\sum_{P\in D_{\theta}}\mathop{\rm
Res}\nolimits^{(2)}_{\pi^{-1}(P)}\Bigl{(}\nu_{1}\cdot\nu_{2}\cdot\eta_{(E,\theta)}\cdot(\phi^{1/2})^{-1}\Bigr{)}.$

By using the isomorphisms $\iota^{\mathop{\rm
hor}\nolimits}:H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})\simeq(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
hor}\nolimits}$ and $\iota^{\mathop{\rm
ver}\nolimits}:H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})\simeq(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
ver}\nolimits}$, we define the Hermitian product $g^{\mathop{\rm
aux}\nolimits}_{(E,\theta)}$ on $T_{(E,\theta)}\mathcal{M}_{H}^{\prime}$ by
the following conditions.

  * •

$g^{\mathop{\rm aux}\nolimits}_{(E,\theta)}(u_{1},u_{2})=0$ if
$u_{i}\in(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm ver}\nolimits}$
$(i=1,2)$ or $u_{i}\in(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
hor}\nolimits}$ $(i=1,2)$.

  * •

For $\nu\in H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$ and $\tau\in
H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})=H^{0}(\Sigma_{\theta}^{\dagger},K_{\Sigma_{\theta}^{\dagger}})$,
we set

$g^{\mathop{\rm aux}\nolimits}_{|(E,\theta)}(\iota^{\mathop{\rm
hor}\nolimits}(\nu),\iota^{\mathop{\rm
ver}\nolimits}(\tau))=\langle\nu,\overline{\tau}\rangle^{\mathop{\rm
aux}\nolimits}_{(E,\theta)}.$

###  1.3 Main result

We set $\phi=-\det(\theta)\in H^{0}(X,K_{X}^{\otimes 2})$, which induces the
distance $d_{\phi}$ on $X$. According to [3, §3.2], a saddle connection of
$\phi$ is a geodesic segment on $(X,d_{\phi})$ such that (i) the starting
point and the ending point are contained in $D_{\phi}$, (ii) any other point
is contained in $X\setminus D_{\phi}$. Note that the starting point and the
ending point are not necessarily distinct. Let $M(\phi)$ denote the minimum
length of the saddle connections of $\phi$, which is called the threshold.

######  Proposition 1.2 (Corollary 3.34)

$g^{\mathop{\rm aux}\nolimits}_{|(E,t\theta)}=O(e^{-2at})$ with respect to
$g_{\mathop{\rm sf}\nolimits|(E,t\theta)}$ for any $0<a<M(\phi)$. As a result,
there exists $t_{0}>0$ such that $g_{\mathop{\rm
sf}\nolimits|(E,t\theta)}+g^{\mathop{\rm aux}\nolimits}_{|(E,t\theta)}$ are
positive definite Hermitian products on
$T_{(E,t\theta)}\mathcal{M}_{H}^{\prime}$ for any $t\geq t_{0}$.

The following theorem is the main result of this paper.

######  Theorem 1.3 (Theorem 3.35)

For any $(E,\theta)\in\mathcal{M}_{H}^{\prime}$, we obtain

$g_{H|(E,t\theta)}-\bigl{(}g_{\mathop{\rm
sf}\nolimits|(E,t\theta)}+g^{\mathop{\rm
aux}\nolimits}_{|(E,t\theta)}\bigr{)}=O\bigl{(}e^{-4at}\bigr{)}\quad(0<a<M(\phi)).$
(3)

####  1.3.1 Symmetric case

A non-degenerate symmetric product of $(E,\theta)$ is a non-degenerate
symmetric product $C:E\otimes E\to\mathcal{O}_{X}$ such that $\theta$ is self-
adjoint with respect to $C$. Note that if $(E,\theta)$ is contained in a
Hitchin section, then $(E,\theta)$ has a non-degenerate symmetric product.

######  Corollary 1.4 (Lemma 3.31)

If $(E,\theta)$ is equipped with a non-degenerate symmetric product, then we
obtain $\eta_{(E,\theta)}=0$. As a result, we obtain

$g_{H|(E,t\theta)}-g_{\mathop{\rm
sf}\nolimits|(E,t\theta)}=O(e^{-4at}),\quad(0<a<M(\phi)).$ (4)

####  1.3.2 Previous works

Originally, Gaiotto-Moore-Neitzke [8, 9] predicted (4). (See also [3, 4, 6,
20].) In [3], Dumas and Neitzke studied the issues in the case where
$(E,\theta)$ is contained in a Hitchin section, and they proved

$g_{H|(E,t\theta)}-g_{\mathop{\rm
sf}\nolimits|(E,t\theta)}=O(e^{-4at}),\quad(0<a<M(\phi)/2)$ (5)

on the subspace $T_{t^{2}\phi}\mathcal{A}_{H}^{\prime}\subset
T_{(E,t\theta)}\mathcal{M}_{H}^{\prime}$ induced by the Hitchin section. As we
mentioned, Fredrickson obtained the estimate (1) on the basis of [5, 13, 14].
It was further generalized to the parabolic case by Fredrickson, Mazzeo,
Swoboda and Weiss in [6], where they suggested that they could also obtain the
estimate (5). In [11], Holdt studied the original approach of [8, 9] in the
case that a Higgs bundle has a non-trivial parabolic structure. More recently,
Chen and Li studied more singular case in [2].

In [18], we obtained (1) for higher rank case without any assumption on the
ramification of $\Sigma_{\theta}\to X$ in a way different from [4]. A naive
hope was to obtain (4) by the method in [18]. Corollary 1.4 is a satisfactory
solution in the symmetric case. In the general case, Theorem 1.3 implies (5),
but it is not a complete answer to the prediction (4) of Gaiotto-Moore-Neitzke
because it is not clear whether $g^{\mathop{\rm
aux}\nolimits}_{(E,t\theta)}=O(e^{-4at})$ $(0<a<M(\phi))$ or not with respect
to $g_{\mathop{\rm sf}\nolimits|(E,\theta)}$.

###  1.4 Outline in the symmetric case

It would be instructive to explain an outline to obtain (4) directly in the
case where $(E,\theta)$ is equipped with a non-degenerate symmetric pairing
$C$ such that $\det(C)=1$, the arguments can be simplified because some issues
disappear. It is close to the case of [3]. Note that if $(E,\theta)$ is
contained in a Hitchin section, then $(E,\theta)$ is equipped with a non-
degenerate symmetric pairing.

Let $D_{\theta}$ denote the zero set of $\phi=-\det(\theta)$. Let
$\Sigma_{\theta}\subset T^{\ast}X$ denote the spectral curve. Because
$(E,\theta)\in\mathcal{M}_{H}^{\prime}$, $\Sigma_{\theta}$ is smooth, and the
projection $\pi:\Sigma_{\theta}\to X$ is a ramified covering of degree $2$. By
[1, 10], there exists a line bundle $L$ on $\Sigma_{\theta}$ such that
$\pi_{\ast}(L)\simeq(E,\theta)$, where the Higgs field of $\pi_{\ast}(L)$ is
induced by the tautological $1$-form on $T^{\ast}X$.

####  1.4.1 The deformation complexes

Let $\mathop{\rm End}\nolimits(E)^{\mathop{\rm sym}\nolimits}$ denote the
sheaf of endomorphisms of $E$ which are symmetric with respect to $C$.
Similarly, let $\mathop{\rm End}\nolimits(E)^{\mathop{\rm asym}\nolimits}$
denote the sheaf of endomorphisms of $E$ which are anti-symmetric with respect
to $C$. We obtain the following subcomplex of $\mathop{\rm
Def}\nolimits(E,\theta)$:

$\mathop{\rm Def}\nolimits(E,\theta,C)=\Bigl{(}\mathop{\rm
End}\nolimits(E)^{\mathop{\rm asym}\nolimits}\to\mathop{\rm
End}\nolimits(E)^{\mathop{\rm sym}\nolimits}\otimes K_{X}\Bigr{)}.$

We may also consider the following subcomplex of $\mathop{\rm
Def}\nolimits(E,\theta)$:

$\mathop{\rm Def}\nolimits^{\prime}(E,\theta,C)=\Bigl{(}\mathop{\rm
End}\nolimits(E)^{\mathop{\rm sym}\nolimits}\to\mathop{\rm
End}\nolimits(E)^{\mathop{\rm asym}\nolimits}\otimes K_{X}\Bigr{)}.$

We have the decomposition of the complexes:

$\mathop{\rm Def}\nolimits(E,\theta)=\mathop{\rm
Def}\nolimits(E,\theta,C)\oplus\mathop{\rm
Def}\nolimits^{\prime}(E,\theta,C).$ (6)

The natural action $\mathcal{O}_{\Sigma}$ on $L$ induces
$\pi_{\ast}(\mathcal{O}_{\Sigma})\to\mathop{\rm End}\nolimits(E)$. It induces
a morphism of complexes
$\pi_{\ast}(\mathcal{O}_{\Sigma})\longrightarrow\mathop{\rm
Def}\nolimits^{\prime}(E,\theta,C)\subset\mathop{\rm Def}\nolimits(E,\theta)$,
and the isomorphism of sheaves

$\pi_{\ast}(\mathcal{O}_{\Sigma_{\theta}})\simeq\mathcal{H}^{0}(\mathop{\rm
Def}\nolimits^{\prime}(E,\theta,C))\simeq\mathcal{H}^{0}(\mathop{\rm
Def}\nolimits(E,\theta)).$

We also have $\mathcal{H}^{1}(\mathop{\rm
Def}\nolimits^{\prime}(E,\theta,C))=0$. We obtain the following injection:

$H^{0}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})\simeq
H^{1}(X,\mathop{\rm Def}\nolimits^{\prime}(E,\theta,C))\longrightarrow
H^{1}(X,\mathop{\rm Def}\nolimits(E,\theta)).$ (7)

As explained in [18], there exist natural isomorphisms

$\mathcal{H}^{1}(\mathop{\rm
Def}\nolimits(E,\theta))\simeq\mathcal{H}^{1}(\mathop{\rm
Def}\nolimits(E,\theta,C))\simeq\pi_{\ast}(K_{\Sigma_{\theta}}).$

We also have $\mathcal{H}^{0}(\mathop{\rm Def}\nolimits(E,\theta,C))=0$. We
obtain the following injection:

$H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})\simeq H^{1}(X,\mathop{\rm
Def}\nolimits(E,\theta,C))\longrightarrow H^{1}(X,\mathop{\rm
Def}\nolimits(E,\theta)).$ (8)

####  1.4.2 The tangent space, the vertical direction and the horizontal
direction

The restriction of the Hitchin fibration
$\mathcal{M}_{H}^{\prime}\to\mathcal{A}_{H}^{\prime}$ is locally a principal
torus bundle equipped with an integrable connection. There exists the
decomposition into the horizontal direction and the vertical direction:

$T_{(E,\theta)}\mathcal{M}_{H}=(T_{(E,\theta)}\mathcal{M}_{H})^{\mathop{\rm
hor}\nolimits}\oplus(T_{(E,\theta)}\mathcal{M}_{H})^{\mathop{\rm
ver}\nolimits}.$ (9)

We identify $T_{(E,\theta)}\mathcal{M}_{H}\simeq H^{1}(X,\mathop{\rm
Def}\nolimits(E,\theta))$. The vertical direction
$(T_{(E,\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm ver}\nolimits}$ equals
$T_{(E,\theta)}\Phi_{H}^{-1}(\Phi_{H}(E,\theta))$, which is identified with
the image of (7). The horizontal direction equals the image of (8) as
explained in [18]. In other words, the decomposition (9) is induced by the
decomposition (6) in the symmetric case.

####  1.4.3 The Hitchin metric and the semi-flat metric

Let $\tau\in H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})$ and $\nu\in
H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$. We regard $\tau$ as an anti-
holomorphic $1$-form on $\Sigma_{\theta}$, i.e., $\tau\in
H^{0}(\Sigma_{\theta}^{\dagger},K_{\Sigma_{\theta}^{\dagger}})$. Because
$\Sigma_{\theta}=\Sigma_{t\theta}$, there exist the corresponding elements

$\iota_{t}^{\mathop{\rm ver}\nolimits}(\tau),\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu)\in H^{1}(X,\mathop{\rm
Def}\nolimits(E,t\theta))=T_{(E,t\theta)}\mathcal{M}_{H}.$

We have the following:

$g_{\mathop{\rm sf}\nolimits}(\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau),\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau))=-2\sqrt{-1}\int_{\Sigma_{\theta}}\tau\wedge\overline{\tau}=\|\tau\|_{L^{2}}^{2},$
(10) $g_{\mathop{\rm sf}\nolimits}(\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu))=2\sqrt{-1}\int_{\Sigma_{\theta}}\nu\wedge\overline{\nu}=\|\nu\|_{L^{2}}^{2},$
(11) $g_{\mathop{\rm sf}\nolimits}(\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm ver}\nolimits}(\tau))=0.$ (12)

Let $h_{t}$ be the harmonic metric of $(E,t\theta)$, which are compatible with
$C$. The Hitchin metrics $g_{H}$ at $(E,t\theta)$ are induced by the natural
$L^{2}$-metric on the space of $\mathop{\rm End}\nolimits(E)$-valued
$1$-forms. Namely, for any $L^{2}$-sections
$\rho_{i}=\rho_{i}^{1,0}+\rho_{i}^{0,1}$ $(i=1,2)$ of $\mathop{\rm
End}\nolimits(E)\otimes\Omega^{1,0}\oplus\mathop{\rm
End}\nolimits(E)\otimes\Omega^{0,1}$ on $W\subset X$, we set

$(\rho_{1},\rho_{2})_{L^{2},h_{t},W}=2\sqrt{-1}\int_{W}\mathop{\rm
Tr}\nolimits\Bigl{(}\rho_{1}^{1,0}\cdot(\rho_{2}^{1,0})^{\dagger}_{h_{t}}-\rho_{1}^{0,1}\cdot(\rho_{2}^{0,1})^{\dagger}_{h_{t}}\Bigr{)}.$

If $W=X$, $(\rho_{1},\rho_{2})_{L^{2},h_{t},X}$ is denoted by
$(\rho_{1},\rho_{2})_{L^{2},h_{t}}$. Let ${\tt H}_{t}(\nu)$ and ${\tt
V}_{t}(\tau)$ $(t\geq 1)$ denote the harmonic $1$-forms of $(\mathop{\rm
End}\nolimits(E),\mathop{\rm ad}\nolimits\theta,h_{t})$ corresponding to
$\iota^{\mathop{\rm hor}\nolimits}_{t}(\nu)$ and $\iota^{\mathop{\rm
ver}\nolimits}_{t}(\tau)$, respectively. We have the following:

$g_{H}(\iota_{t}^{\mathop{\rm ver}\nolimits}(\tau),\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau))=({\tt V}_{t}(\tau),{\tt V}_{t}(\tau))_{L^{2},h_{t}},$
(13) $g_{H}(\iota_{t}^{\mathop{\rm hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu))=({\tt H}_{t}(\nu),{\tt H}_{t}(\nu))_{L^{2},h_{t}},$ (14)
$g_{H}(\iota_{t}^{\mathop{\rm hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau))=({\tt H}_{t}(\nu),{\tt V}_{t}(\tau))_{L^{2},h_{t}}.$
(15)

Because $C$ is compatible with $h_{t}$, the decomposition $\mathop{\rm
End}\nolimits(E)=\mathop{\rm End}\nolimits(E)^{\mathop{\rm
sym}\nolimits}\oplus\mathop{\rm End}\nolimits(E)^{\mathop{\rm asym}\nolimits}$
is orthogonal with respect to $h$. We can observe that ${\tt H}_{t}(\nu)$ is a
section of the Dolbeault resolution of $\mathop{\rm
Def}\nolimits(E,\theta,C)$, and ${\tt V}_{t}(\nu)$ is a section of the
Dolbeault resolution of $\mathop{\rm Def}\nolimits^{\prime}(E,\theta,C)$.
Hence, we obtain

$g_{H}(\iota_{t}^{\mathop{\rm hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau))=0=g_{\mathop{\rm sf}\nolimits}(\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm ver}\nolimits}(\tau)).$

We would like to show the following estimates for any $0<\kappa<M(\phi)/2$:

$g_{H}(\iota_{t}^{\mathop{\rm hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu))=g_{\mathop{\rm sf}\nolimits}(\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu),\iota_{t}^{\mathop{\rm
hor}\nolimits}(\nu))+O\bigl{(}e^{-8\kappa t}\|\nu\|_{L^{2}}^{2}\bigr{)}.$ (16)
$g_{H}(\iota_{t}^{\mathop{\rm ver}\nolimits}(\tau),\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau))=g_{\mathop{\rm sf}\nolimits}(\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau),\iota_{t}^{\mathop{\rm
ver}\nolimits}(\tau))+O\bigl{(}e^{-8\kappa t}\|\tau\|_{L^{2}}^{2}\bigr{)}.$
(17)

####  1.4.4 Canonically decoupled harmonic metric

Let $Q\in X\setminus D_{\theta}$. Let $(X_{Q},z_{Q})$ denote a holomorphic
coordinate neighbourhood around $Q$ such that $\phi=dz_{Q}\,dz_{Q}$. There
exists the decomposition, which is orthogonal with respect to $C$:

$(E,\theta)_{|X_{Q}}=(E_{Q,1},dz_{Q})\oplus(E_{Q,2},-dz_{Q}).$

Let $v_{i}$ $(i=1,2)$ be frames of $E_{Q,i}$ such that $C(v_{i},v_{i})=1$. We
define $h_{\infty,Q}$ by $h_{\infty,Q}(v_{i},v_{i})=1$ and
$h_{\infty,Q}(v_{1},v_{2})=0$, which is independent of $v_{i}$. By varying
$Q\in X\setminus D_{\theta}$, we obtain the canonical decoupled harmonic
metric $h_{\infty}$ of $(E,\theta)_{|X\setminus D_{\theta}}$ compatible with
$C$. It equals the limiting configuration in [13].

Let $s_{t}$ be the automorphism of $E_{|X\setminus D_{\theta}}$ determined by
$h_{t}=h_{\infty}s_{t}$. We have $\det(s_{t})=1$.

We set $g_{\phi}=|\phi|$ as the Kähler metric of $X\setminus D_{\theta}$,
which induces the distance $d_{\phi}$ on $X$. We obtain the connection
$\nabla_{\infty}$ of $\mathop{\rm End}\nolimits(E)\otimes T^{\ast}(X\setminus
D_{\theta})^{\otimes m}$ for any $m\in{\mathbb{Z}}_{\geq 0}$ induced by
$h_{\infty}$ and $g_{\phi}$.

For any $r>0$ and $P\in X$, we set $X_{P}(r)=\\{Q\in
X\,|\,d_{\phi}(Q,P)<r\\}$. We set $\kappa_{0}=M(\phi)/2$. We have
$X_{P}(\kappa_{0})\cap X_{P^{\prime}}(\kappa_{0})=\emptyset$ if $P,P^{\prime}$
are two distinct points in $D_{\theta}$.

For $0<\kappa<\kappa_{0}$, we set
$\delta(\kappa):=\min\\{(\kappa-\kappa_{0})/10,\kappa/10\\}$. For each $P\in
D_{\theta}$, we set

$X^{(1)}_{P,\kappa}=X_{P}(\kappa_{0}-\delta(\kappa)),\quad
X^{(2)}_{P,\kappa}=X_{P}(\kappa_{0}-2\delta(\kappa)).$

For any non-negative integer $j$, there exist $C_{j}>0$ such that the
following holds on $X\setminus\bigcup X^{(2)}_{P,\kappa}$ for any $t\geq 1$
(see Corollary 2.5):

$\bigl{|}\nabla_{\infty}^{j}(s_{t}-\mathop{\rm
id}\nolimits)\bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+7\delta(\kappa))t}.$ (18)

This is essentially contained in [3].

The multiplication of $\nu\in H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$ on
$L$ induces a section $F_{\nu}$ of $(\mathop{\rm
End}\nolimits(E)\otimes\Omega^{1,0})_{|X\setminus D_{\theta}}$ such that
$(\overline{\partial}_{E}+t\mathop{\rm ad}\nolimits\theta)F_{\nu}=0$.
Similarly, $\tau$ induces a section $F_{\tau}$ of $(\mathop{\rm
End}\nolimits(E)\otimes\Omega^{0,1})_{|X\setminus D_{\theta}}$ such that
$(\overline{\partial}_{E}+t\mathop{\rm ad}\nolimits\theta)F_{\tau}=0$. We can
regard $F_{\nu}$ and $F_{\tau}$ as approximations of ${\tt H}_{t}(\nu)$ and
${\tt V}_{t}(\tau)$ on $X\setminus\bigcup_{P}X_{P,\kappa}^{(2)}$.

####  1.4.5 Approximation of ${\tt H}_{t}(\nu)$ around $P\in D_{\theta}$

For any $W\subset X$, we set $\Sigma_{\theta|W}=\Sigma_{\theta}\times_{X}W$.
There exists a holomorphic function $\alpha_{P}$ on
$\Sigma_{\theta|X_{P}(\kappa_{0})}$ such that $\nu=d\alpha_{P}$ and that
$\alpha_{P}=0$ at $\pi^{-1}(P)$. The multiplication of $\alpha_{P}$ on $L$
induces a holomorphic endomorphism $F_{\alpha_{P}}$ on
$E_{|X_{P}(\kappa_{0})}$ such that $[\theta,F_{\alpha_{P}}]=0$. We obtain the
following $\mathop{\rm End}\nolimits(E)$-valued $C^{\infty}$ $1$-forms for
$t\geq 1$:

${\tt H}_{P,t}(\nu)=(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}})F_{\alpha_{P}}.$

They are harmonic $1$-forms of $(\mathop{\rm End}\nolimits(E),t\mathop{\rm
ad}\nolimits\theta,h_{t})_{|X_{P}}$, i.e.,

$(\overline{\partial}_{E}+t\mathop{\rm ad}\nolimits\theta){\tt
H}_{P,t}(\nu)=0,\quad(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}}){\tt H}_{P,t}(\nu)=0.$

We have ${\tt H}_{P,t}(\nu)^{\circ}=F_{\nu}$ on
$X_{P}(\kappa_{0})\setminus\\{P\\}$. There exists a $C^{\infty}$-section
$\rho_{P}$ of $\mathop{\rm End}\nolimits(E)^{\mathop{\rm
asym}\nolimits}_{|X_{P}(\kappa_{0})\setminus\\{P\\}}$ such that

$(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\rho_{P,t}=F_{\nu}-{\tt H}_{P,t}(\nu).$

####  1.4.6 Approximation of ${\tt H}_{t}(\nu)$ on $X$

By patching $F_{\nu}$ and ${\tt H}_{P,t}(\nu)$ $(P\in D_{\theta})$, we
construct ${\tt H}^{\prime}_{\kappa,t}(\nu)$ as follows.

  * •

On $X\setminus\bigcup X_{P,\kappa}^{(1)}$, we set ${\tt
H}^{\prime}_{\kappa,t}(\nu)=F_{\nu}$.

  * •

For $P\in D_{\theta}$, let $\chi_{P,\kappa}:X\to[0,1]$ be a
$C^{\infty}$-function such that $\chi_{P,\kappa}=1$ on $X_{P,\kappa}^{(2)}$
and $\chi_{P,\kappa}=0$ on $X\setminus X_{P,\kappa}^{(1)}$. Then, on
$X^{(1)}_{P,\kappa}\setminus X^{(2)}_{P,\kappa}$, we set ${\tt
H}^{\prime}_{\kappa,t}(\nu)=F_{\nu}-(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)$ $(\chi_{P,\kappa}\rho_{P})$.

  * •

On $X^{(2)}_{P,\kappa}$, we set ${\tt H}^{\prime}_{\kappa,t}(\nu)={\tt
H}_{P,t}(\nu)$.

By the construction, we obtain $(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta){\tt H}_{\kappa,t}^{\prime}(\nu)=0$. We can observe that
${\tt H}^{\prime}_{\kappa,t}(\nu)$ is a section of the Dolbeault resolution of
$\mathop{\rm Def}\nolimits(E,\theta,C)$, and that ${\tt
H}_{\kappa,t}^{\prime}(\nu)^{\circ}=F_{\nu}$. As explained in [18], the
cohomology class of ${\tt H}_{\kappa,t}^{\prime}(\nu)$ is
$\iota_{t}^{\mathop{\rm hor}\nolimits}(\nu)$.

There exist $C_{j}>0$ $(j\geq 0)$ such that the following holds on
$X^{(1)}_{P,\kappa}\setminus X^{(2)}_{P,\kappa}$ by (18):

$\Bigl{|}\nabla_{\infty}^{j}({\tt
H}_{P,t}(\nu)-F_{\nu})\Bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+6\delta(\kappa))t}\|\nu\|_{L^{2}}.$ (19)

There exist $C_{j}>0$ $(j\geq 0)$ such that the following holds on
$X^{(1)}_{P,\kappa}\setminus X^{(2)}_{P,\kappa}$:

$\bigl{|}\nabla^{j}_{\infty}\rho_{P}\bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+6\delta(\kappa))t}\|\nu\|_{L^{2}}.$

Hence, there exist $C_{j}>0$ $(j\geq 0)$ such that on
$X^{(1)}_{P,\kappa}\setminus X^{(2)}_{P,\kappa}$:

$\bigl{|}\nabla^{j}_{\infty}({\tt
H}^{\prime}_{\kappa,t}(\nu)-F_{\nu})\bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+6\delta(\kappa))t}\|\nu\|_{L^{2}}.$

Let $g_{X}$ be a Kähler metric. Note that $(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}}){\tt H}^{\prime}_{\kappa,t}(\nu)=0$ on
$X\setminus\bigcup_{P}X^{(1)}_{P,\kappa}$ and $\bigcup_{P}X_{P,\kappa}^{(2)}$.
There exists $C>0$ such that the following holds on $X$:

$\bigl{|}(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}}){\tt
H}^{\prime}_{\kappa,t}(\nu)\bigr{|}_{h_{t},g_{X}}\leq
Ce^{-4(\kappa+6\delta(\kappa))t}\|\nu\|_{L^{2}}.$

There exist unique $C^{\infty}$-sections $\gamma_{\kappa,t}(\nu)$ of
$\mathop{\rm End}\nolimits(E)^{\mathop{\rm asym}\nolimits}$ such that

$(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}})(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\gamma_{\kappa,t}(\nu)=(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}}){\tt H}^{\prime}_{\kappa,t}(\nu).$

By [18, Proposition 2.39], there exists $C>0$ such that

$\|\gamma_{\kappa,t}(\nu)\|_{L^{2},h_{t},g_{X}}+\|(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\gamma_{\kappa,t}(\nu)\|_{L^{2},h_{t},g_{X}}\leq
Ce^{-4(\kappa+5\delta(\kappa))t}\|\nu\|_{L^{2}}.$

We have

${\tt H}_{t}(\nu)={\tt H}^{\prime}_{\kappa,t}(\nu)-\gamma_{\kappa,t}(\nu).$

####  1.4.7 Estimate of the pairings of ${\tt H}_{t}(\nu)$

First, we observe

$({\tt H}_{t}(\nu),{\tt H}_{t}(\nu))_{L^{2},h_{t}}=({\tt
H}^{\prime}_{\kappa,t}(\nu),{\tt
H}^{\prime}_{\kappa,t}(\nu))_{L^{2},h_{t}}+O\bigl{(}e^{-8\kappa
t}\|\nu\|_{L^{2}}^{2}\bigr{)}.$ (20)

Indeed, because $(\overline{\partial}+t\mathop{\rm
ad}\nolimits\theta)^{\ast}_{h_{t},g_{X}}{\tt H}_{t}(\nu)=0$, we obtain

$({\tt H}_{t}(\nu),{\tt H}_{t}(\nu))_{L^{2},h_{t}}=({\tt H}_{t}(\nu),{\tt
H}^{\prime}_{\kappa,t}(\nu))_{L^{2},h_{t}}+\bigl{(}{\tt
H}_{t}(\nu),(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\gamma_{\kappa,t}(\nu)\bigr{)}_{L^{2},h_{t}}=\\\ ({\tt
H}_{t}(\nu),{\tt
H}^{\prime}_{\kappa,t}(\nu))_{L^{2},h_{t}}+\bigl{(}(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)^{\ast}_{h_{t},g_{X}}{\tt
H}_{t}(\nu),\gamma_{\kappa,t}(\nu)\bigr{)}_{L^{2},h_{t},g_{X}}=({\tt
H}_{t}(\nu),{\tt H}^{\prime}_{\kappa,t}(\nu))_{L^{2},h_{t}}.$ (21)

Because $(\overline{\partial}+t\mathop{\rm
ad}\nolimits\theta)^{\ast}_{h_{t},g_{X}}{\tt
H}^{\prime}_{\kappa,t}(\nu)=O\bigl{(}e^{-4\kappa t}\|\nu\|_{L^{2}}\bigr{)}$
and $\gamma_{\kappa,t}(\nu)=O(e^{-4\kappa t}\|\nu\|_{L^{2}})$, we obtain

$\bigl{(}{\tt
H}^{\prime}_{\kappa,t}(\nu),(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\gamma_{\kappa,t}(\nu)\bigr{)}_{L^{2},h_{t}}=\bigl{(}(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)^{\ast}_{h_{t},g_{X}}{\tt
H}^{\prime}_{\kappa,t}(\nu),\gamma_{\kappa,t}(\nu)\bigr{)}_{L^{2},h_{t},g_{X}}=O\bigl{(}e^{-8\kappa
t}\|\nu\|_{L^{2}}^{2}\bigr{)}.$

Hence, we obtain (20).

We have the following estimates on $W_{\kappa}=X\setminus\bigcup
X_{P,\kappa}^{(2)}$:

${\tt
H}_{\kappa,t}^{\prime}(\nu)^{\circ}=F_{\nu},\quad\quad\Bigl{|}\bigl{(}{\tt
H}^{\prime}_{\kappa,t}(\nu)\bigr{)}^{\bot}\Bigr{|}_{h_{t},g_{X}}=O\bigl{(}e^{-4\kappa
t}\|\nu\|_{L^{2}}\bigr{)},\quad\quad\bigl{|}s_{t}-\mathop{\rm
id}\nolimits\bigr{|}_{h_{t}}=O(e^{-4\kappa t}).$ (22)

We can prove the following by using (22) with an elementary consideration as
in §3.6.1:

$({\tt H}^{\prime}_{\kappa,t}(\nu),{\tt
H}^{\prime}_{\kappa,t}(\nu))_{L^{2},h_{t},W_{\kappa}}=2\sqrt{-1}\int_{W_{\kappa}}\mathop{\rm
Tr}\nolimits\bigl{(}F_{\nu}\cdot(F_{\nu})^{\dagger}_{h_{\infty}}\bigr{)}+O\bigl{(}e^{-8\kappa
t}\|\nu\|_{L^{2}}^{2}\bigr{)}\\\
=2\sqrt{-1}\int_{\Sigma_{\theta|W_{\kappa}}}\nu\wedge\overline{\nu}+O\bigl{(}e^{-8\kappa
t}\|\nu\|_{L^{2}}^{2}\bigr{)}.$ (23)

On $X^{(2)}_{P,\kappa}$, we have ${\tt H}^{\prime}_{\kappa,t}(\nu)={\tt
H}_{P,t}(\nu)$. Because
$(\partial_{E,h_{t}}+t\theta^{\dagger}_{h_{t}})^{\ast}_{h_{t},g_{X}}{\tt
H}_{P,t}(\nu)=0$, we obtain the following by using the Stokes formula (Lemma
3.55):

$\bigl{(}{\tt H}^{\prime}_{\kappa,t}(\nu),{\tt
H}^{\prime}_{\kappa,t}(\nu)\bigr{)}_{L^{2},h_{t},X^{(2)}_{P,\kappa}}=2\sqrt{-1}\int_{\partial
X_{P,\kappa}^{(2)}}\mathop{\rm Tr}\nolimits\Bigl{(}F_{\alpha_{P}}\cdot({\tt
H}_{P,t}(\nu)^{1,0})^{\dagger}_{h_{t}}\Bigr{)}.$

By using the estimates (18) and (19) with the elementary consideration in
§3.6.1, we obtain

$2\sqrt{-1}\int_{\partial X_{P,\kappa}^{(2)}}\mathop{\rm
Tr}\nolimits\Bigl{(}F_{\alpha_{P}}\cdot({\tt
H}_{P,t}(\nu)^{1,0})^{\dagger}_{h_{t}}\Bigr{)}=2\sqrt{-1}\int_{\partial
X_{P,\kappa}^{(2)}}\mathop{\rm
Tr}\nolimits\Bigl{(}F_{\alpha_{P}}\cdot(F_{\nu})^{\dagger}_{h_{\infty}}\Bigr{)}+O\bigl{(}e^{-8\kappa
t}\|\nu\|^{2}_{L^{2}}\bigr{)}\\\
=2\sqrt{-1}\int_{\partial\Sigma_{\theta|X_{P,\kappa}^{(2)}}}\alpha_{P}\cdot\overline{\nu}+O\bigl{(}e^{-8\kappa
t}\|\nu\|^{2}_{L^{2}}\bigr{)}=2\sqrt{-1}\int_{\Sigma_{\theta|X_{P,\kappa}^{(2)}}}\nu\overline{\nu}+O\bigl{(}e^{-8\kappa
t}\|\nu\|^{2}_{L^{2}}\bigr{)}.$ (24)

In all, we obtain

$({\tt H}_{t}(\nu),{\tt
H}_{t}(\nu))_{L^{2},h_{t}}=(\nu,\nu)_{L^{2}}+O\bigl{(}e^{-8\kappa
t}\|\nu\|_{L^{2}}^{2}\bigr{)}\quad(0<\kappa<M(\phi)/2).$

This implies (16).

####  1.4.8 Approximation of ${\tt V}_{t}(\tau)$ on $X$

For each $P\in D_{\theta}$, there exists an anti-holomorphic function
$\beta_{P}$ on $\Sigma_{\theta|X_{P}(\kappa_{0})}$ such that
$\tau=d\beta_{P}$. Let $\chi_{P,\kappa}:X\to[0,1]$ be as before. We set

$\tau^{\circ}_{\kappa}=\tau-\sum_{P\in
D_{\theta}}\overline{\partial}(\pi^{\ast}(\chi_{P,\kappa})\cdot\beta_{P}).$

The multiplication of $\tau^{\circ}_{\kappa}$ on $L$ induces a
$C^{\infty}$-section $F_{\tau^{\circ}_{\kappa}}$ of $\mathop{\rm
End}\nolimits(E)\otimes\Omega^{0,1}$ such that
$[\theta,F_{\tau^{\circ}_{\kappa}}]=0$. We obtain
$(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)F_{\tau^{\circ}_{\kappa}}=0$. The cohomology class of
$F_{\tau^{\circ}_{\kappa}}$ is $\iota^{\mathop{\rm ver}\nolimits}_{t}(\tau)$.

We may regard $\pi^{\dagger}:\Sigma_{\theta}^{\dagger}\to X^{\dagger}$ as the
spectral curve of the Higgs bundle
$(E,\partial_{E,h_{t}},\theta^{\dagger}_{h_{t}})$ on $X^{\dagger}$. We may
regard $\beta_{P}$ as a holomorphic function on
$\Sigma^{\dagger}_{\theta|X_{P}(\kappa_{0})}$. There exists $L_{t}$ on
$\Sigma^{\dagger}_{\theta}$ with an isomorphism
$\pi^{\dagger}_{\ast}(L_{t})\simeq(E,\partial_{E,h_{t}},\theta^{\dagger}_{h_{t}})$.
The multiplication of $\beta_{P}$ induces a holomorphic endomorphism
$F^{\dagger}_{\beta_{P}}$ of $(E,\partial_{E,h_{t}})$. We have
$F^{\dagger}_{\beta_{P}}=(F_{\overline{\beta_{P}}})^{\dagger}_{h_{t}}$. We
note that $(\overline{\partial}_{E}+t\theta)F^{\dagger}_{\beta_{P}}$ is a
harmonic $1$-form of $(\mathop{\rm End}\nolimits(E),t\mathop{\rm
ad}\nolimits\theta,h_{t})$.

We set

${\tt V}^{\prime}_{\kappa,t}(\tau)=F_{\tau^{\circ}_{\kappa}}+\sum_{P\in
D_{\theta}}(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)(\chi_{P,\kappa}F^{\dagger}_{\beta_{P}}).$

By the construction, we have $(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta){\tt V}^{\prime}_{\kappa,t}(\tau)=0$, and its cohomology
class equals $\iota_{t}^{\mathop{\rm ver}\nolimits}(\tau)$.

By (18), there exists $C>0$ such that

$\Bigl{|}(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}}){\tt
V}^{\prime}_{\kappa,t}(\tau)\Bigr{|}_{h_{t},g_{X}}\leq
Ce^{-4(\kappa+7\delta(\kappa))t}\|\tau\|_{L^{2}}.$

We also have $\partial_{X}\mathop{\rm tr}\nolimits({\tt
V}^{\prime}_{\kappa,t}(\tau))=0$. There exist unique sections
$\gamma_{\kappa,t}(\tau)$ of $\mathop{\rm End}\nolimits(E)$ such that

$(\partial_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}})(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\gamma_{\kappa,t}(\tau)=(\overline{\partial}_{E,h_{t}}+t\mathop{\rm
ad}\nolimits\theta^{\dagger}_{h_{t}}){\tt V}^{\prime}_{\kappa,t}(\tau).$

By [18, Proposition 2.39], there exists $C>0$ such that

$\|\gamma_{\kappa,t}(\tau)\|_{L^{2},h_{t},g_{X}}+\|(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\gamma_{\kappa,t}(\tau)\|_{L^{2},h_{t},g_{X}}\leq
Ce^{-4(\kappa+5\delta(\kappa))t}\|\tau\|_{L^{2}}.$

We have

${\tt V}_{t}(\tau)={\tt
V}^{\prime}_{\kappa,t}(\tau)-(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\gamma_{\kappa,t}(\tau).$

####  1.4.9 Estimate of the pairings of ${\tt V}_{t}(\tau)$

As in the case of (20), we obtain

$\bigl{(}{\tt V}_{t}(\tau),{\tt
V}_{t}(\tau)\bigr{)}_{L^{2},h_{t}}=\bigl{(}{\tt
V}^{\prime}_{\kappa,t}(\tau),{\tt
V}^{\prime}_{\kappa,t}(\tau)\bigr{)}_{L^{2},h_{t}}+O\bigl{(}e^{-8\kappa
t}\|\tau\|_{L^{2}}^{2}\bigr{)}.$

By using (18) with the elementary consideration in §3.6.2, we obtain

$\bigl{(}{\tt V}^{\prime}_{\kappa,t}(\tau),{\tt
V}^{\prime}_{\kappa,t}(\tau)\bigr{)}_{L^{2},h_{t},W_{\kappa}}=-2\sqrt{-1}\int_{W_{\kappa}}\mathop{\rm
Tr}\nolimits\bigl{(}F_{\tau}\cdot(F_{\tau})^{\dagger}_{h_{\infty}}\bigr{)}+O\bigl{(}e^{-8\kappa
t}\|\tau\|_{L^{2}}^{2}\bigr{)}\\\
=-2\sqrt{-1}\int_{\Sigma_{\theta|W_{\kappa}}}\tau\overline{\tau}+O\bigl{(}e^{-8\kappa
t}\|\tau\|_{L^{2}}^{2}\bigr{)}.$ (25)

By using the Stokes formula (Lemma 3.55), and by using (18) with the
elementary consideration in §3.6.2, we obtain

$\bigl{(}{\tt V}^{\prime}_{\kappa,t}(\tau),{\tt
V}^{\prime}_{\kappa,t}(\tau)\bigr{)}_{L^{2},h_{t},X^{(2)}_{P,\kappa}}=-2\sqrt{-1}\int_{\partial
X^{(2)}_{P,\kappa}}\mathop{\rm
Tr}\nolimits\Bigl{(}F^{\dagger}_{\beta_{P}}\cdot\bigl{(}{\tt
V}^{\prime}_{t}(\tau)^{0,1}\bigr{)}^{\dagger}_{h_{t}}\Bigr{)}\\\
=-2\sqrt{-1}\int_{\partial\Sigma_{\theta|X^{(2)}_{P,\kappa}}}\beta_{P}\cdot\overline{\tau}+O\bigl{(}e^{-8\kappa
t}\|\tau\|_{L^{2}}^{2}\bigr{)}=-2\sqrt{-1}\int_{\Sigma_{\theta|X^{(2)}_{P,\kappa}}}\tau\overline{\tau}+O\bigl{(}e^{-8\kappa
t}\|\tau\|_{L^{2}}^{2}\bigr{)}.$ (26)

In all, we obtain

$\bigl{(}{\tt V}_{t}(\tau),{\tt
V}_{t}(\tau)\bigr{)}_{L^{2},h_{t}}=(\tau,\tau)_{L^{2}}+O\bigl{(}e^{-8\kappa
t}\|\tau\|_{L^{2}}^{2}\bigr{)}\quad\quad(0<\kappa<M(\phi)/2).$

This implies (17).

##### Acknowledgement

This study grew out of my attempt to understand the work of David Dumas and
Andrew Neitzke [3]. I thank Laura Fredrickson who asked whether the method in
[18] is useful to improve the exponential rate. I thank Rafe Mazzeo, Motohico
Mulase and Olivia Dumitrescu for some discussions. I thank Qiongling Li and
Szilard Szabo for discussions and collaborations. This study is partially
based on the joint works [12, 19]. I thank Laura Fredrickson and Andy Neitzke
for encouraging comments. I am grateful to Gao Chen and Nianzi Li for
discussions. I thank Yoshifumi Tsuchimoto and Akira Ishii for their constant
encouragements.

I prepared the original version of this manuscript for the workshops “Complex
Lagrangians, Mirror Symmetry, and Quantization” at Banff International
Research Station in October 2023, and “Higgs bundles, non-Abelian Hodge
correspondence and related topics 2023” at Chern Institute of Mathematics in
November 2023. I revised it for the workshop “New developments in Kobayashi-
Hitchin correspondence and Higgs bundles” at Osaka Metropolitan University in
August 2024. I thank the organizers for the opportunities of the talks.

I am partially supported by the Grant-in-Aid for Scientific Research (A) (No.
21H04429), the Grant-in-Aid for Scientific Research (A) (No. 22H00094), the
Grant-in-Aid for Scientific Research (A) (No. 23H00083), and the Grant-in-Aid
for Scientific Research (C) (No. 20K03609), Japan Society for the Promotion of
Science. I am also partially supported by the Research Institute for
Mathematical Sciences, an International Joint Usage/Research Center located in
Kyoto University.

##  2 Local estimates

###  2.1 Variations of Simpson’s main estimate

We explain a variant of Simpson’s main estimate in the rank $2$ case. We also
use the functions used in [3] to improve the estimate.

####  2.1.1 A general estimate in the rank $2$ case

For any $R>0$, we set
$U(R)=\bigl{\\{}z\in{\mathbb{C}}\,\big{|}\,|z|<R\bigr{\\}}$.

Let $R_{0}>0$. Let $(E,\overline{\partial}_{E},\theta)$ be a Higgs bundle of
rank $2$ on $U(R)$ for some $R>2R_{0}$. We fix an isomorphism
$\kappa:\det(E)\simeq\mathcal{O}_{U(R)}$. Let $f$ be the endomorphism of $E$
determined by $\theta=f\,dz$. We assume that there exits a decomposition

$(E,f)=(E_{1},1)\oplus(E_{-1},-1).$

Let $\mathcal{H}(E,\theta;\kappa)$ denote the set of harmonic metrics $h$ of
$(E,\theta)$ such that $\det(h)=1$ under the isomorphism
$\kappa:\det(E)\simeq\mathcal{O}_{X}$.

For any $h\in\mathcal{H}(E,\theta;\kappa)$, let $\pi_{1}$ denote the
projection onto $E_{1}$. We obtain the orthogonal projection $\pi_{1}^{h}$ of
$E$ onto $E_{1}$. We regard $\pi_{1}$ and $\pi_{1}^{h}$ as endomorphisms of
$E$ in natural ways. We set $\rho:=\pi_{1}-\pi_{1}^{h}$. We have
$|\pi_{1}|_{h}^{2}=|\pi_{1}^{h}|_{h}^{2}+|\rho|_{h}^{2}$. We shall prove the
following proposition in §2.1.6.

###### Proposition 2.1

For any $0<\gamma<2\sqrt{2}$, there exists $C(R_{0},\gamma)>0$ depending only
on $R_{0}$ and $\gamma$, such that the following holds on $U(R-R_{0})$ for any
$h\in\mathcal{H}(E,\theta;\kappa)$:

$|\rho|_{h}\leq C(R_{0},\gamma)e^{-\gamma(R-|z|)}.$

####  2.1.2 Refinement under an additional assumption

Let $v_{i}$ $(i=1,2)$ be frames of $E_{i}$ such that $\kappa(v_{1}\wedge
v_{2})=1$. We shall prove the following proposition in §2.1.7.

###### Proposition 2.2

Let $h\in\mathcal{H}(E,\theta;\kappa)$. Suppose that there exist
$\gamma_{0}>2$ and $B(h)>0$ such that the following holds on $U(R)$:

$h(v_{1},v_{1})+h(v_{2},v_{2})+|\partial_{z}h(v_{1},v_{1})|+|\partial_{z}h(v_{2},v_{2})|+|\partial_{z}h(v_{1},v_{2})|+|\partial_{\overline{z}}h(v_{1},v_{2})|\leq
B(h)e^{-\gamma_{0}(R-|z|)}.$ (27)

Then, for any $0<\gamma<4$, there exist $C(R_{0},\gamma,\gamma_{0},B(h))>0$
depending only on $R_{0},\gamma,\gamma_{0},B(h)$, such that the following
holds on $U(R-R_{0})$:

$|\rho|_{h}\leq
C\bigl{(}R_{0},\gamma,\gamma_{0},B(h)\bigr{)}e^{-\gamma(R-|z|)}.$

####  2.1.3 Refinement in the symmetric case

Let $C$ be a non-degenerate symmetric pairing of $(E,\theta)$ such that
$\det(C)=1$ under the isomorphism $\kappa$. Let $\mathcal{H}(E,\theta;C)$
denote the set of harmonic metrics of $(E,\theta)$ compatible with $C$. For
any $h\in\mathcal{H}(E,\theta;C)$, we obtain an estimate as in Proposition 2.2
without the assumption (27). We shall prove the following proposition in
§2.1.8.

###### Proposition 2.3

For any $0<\gamma<4$, there exists $C(R_{0},\gamma)>0$ such that the following
holds on $U(R-R_{0})$ for any $h\in\mathcal{H}(E,\theta;C)$:

$|\rho|_{h}\leq C(R_{0},\gamma)e^{-\gamma(R-|z|)}.$

Let $v_{i}$ be frames of $E_{i}$ such that $C(v_{i},v_{i})=1$. The following
proposition is a refinement of Proposition 2.3.

###### Proposition 2.4

For any $0<\gamma<4$ and $(j,k)\in{\mathbb{Z}}^{2}_{\geq 0}$, there exists
$C_{j,k}(R_{0},\gamma)>0$ such that the following holds for any
$h\in\mathcal{H}(E,\theta;C)$ on $U(R-R_{0})$:

$|\partial_{z}^{j}\partial_{\overline{z}}^{k}(h(v_{i},v_{i})-1)|\leq
C_{j,k}(R_{0},\gamma)e^{-2\gamma(R-|z|)},$ (28)
$|\partial_{z}^{j}\partial_{\overline{z}}^{k}h(v_{1},v_{2})|\leq
C_{j,k}(R_{0},\gamma)e^{-\gamma(R-|z|)}.$ (29)

Let $h^{C}$ denote the unique decoupled harmonic metric of $(E,\theta)$
compatible with $C$ (see [12]). We have $h^{C}(v_{i},v_{i})=1$ and
$h^{C}(v_{1},v_{2})=0$. Let $\nabla$ denote the Chern connection of $E$
determined by $h^{C}$. For any $h\in\mathcal{H}(E,\theta;C)$, let $s(h^{C},h)$
be the automorphism of $E$ determined by $h=h^{C}\cdot s$. We set $\mathop{\rm
End}\nolimits(E)^{\circ}=\mathop{\rm End}\nolimits(E_{1})\oplus\mathop{\rm
End}\nolimits(E_{2})$ and $\mathop{\rm End}\nolimits(E)^{\bot}=\mathop{\rm
Hom}\nolimits(E_{1},E_{2})\oplus\mathop{\rm Hom}\nolimits(E_{2},E_{1})$. We
obtain the decomposition

$\mathop{\rm End}\nolimits(E)=\mathop{\rm
End}\nolimits(E)^{\circ}\oplus\mathop{\rm End}\nolimits(E)^{\bot}.$

We have the corresponding decomposition
$s(h^{C},h)=s(h^{C},h)^{\circ}+s(h^{C},h)^{\bot}$.

###### Corollary 2.5

For any $0<\gamma<4$ and $(j,k)\in{\mathbb{Z}}^{2}_{\geq 0}$, there exists
$C^{\prime}_{j,k}(R_{0},\gamma)>0$ such that the following holds for any
$h\in\mathcal{H}(E,\theta;C)$ on $U(R-R_{0})$:

$|\nabla_{z}^{j}\nabla_{\overline{z}}^{k}(s(h^{C},h)^{\circ}-\mathop{\rm
id}\nolimits)|\leq C^{\prime}_{j,k}(R_{0},\gamma)e^{-2\gamma(R-|z|)},$
$|\nabla_{z}^{j}\nabla_{\overline{z}}^{k}(s(h^{C},h)^{\bot})|\leq
C^{\prime}_{j,k}(R_{0},\gamma)e^{-\gamma(R-|z|)}.$

###### Remark 2.6

Proposition 2.3 and Proposition 2.4 are essentially contained in [3]. They can
be proved independently from Proposition 2.1 and Proposition 2.2.

####  2.1.4 A general inequality

Let $h\in\mathcal{H}(E,\theta;\kappa)$. We have the following equality:

$-\partial_{z}\partial_{\overline{z}}|\pi_{1}|_{h}^{2}=-|\partial_{E,h}\pi_{1}|^{2}_{h}-\bigl{|}[f^{\dagger}_{h},\pi_{1}]\bigr{|}_{h}^{2}.$
(30)

Let $v_{i}$ $(i=1,2)$ be frames of $E_{i}$ such that $v_{1}\wedge v_{2}=1$.
Let $H$ be the Hermitian matrix valued function determined by
$H_{i,j}=h(v_{i},v_{j})$. To simplify, we set $a=h(v_{1},v_{1})$,
$b=h(v_{1},v_{2})$ and $c=h(v_{2},v_{2})$, i.e.,

$H=\left(\begin{array}[]{cc}a&b\\\ \overline{b}&c\end{array}\right).$

Because $\det(h)=1$, we have $\det(H)=ac-|b|^{2}=1$. We also have
$|\rho|_{h}^{2}=|b|^{2}$.

We shall use the following lemma implicitly.

###### Lemma 2.7

Let $g$ be an endomorphism of $E$. Let $g^{\dagger}_{h}$ be the adjoint of $g$
with respect to $h$. Let $G$ and $G^{\dagger}_{H}$ denote the matrix valued
functions determined by $g{\boldsymbol{v}}={\boldsymbol{v}}G$ and
$g^{\dagger}_{h}{\boldsymbol{v}}={\boldsymbol{v}}G^{\dagger}_{H}$. Then, we
have
$G^{\dagger}_{H}=\overline{H}^{-1}\cdot({}^{t}\\!\overline{G})\cdot\overline{H}$.

###### Lemma 2.8

We have
$\bigl{|}[f^{\dagger}_{h},\pi_{1}]\bigr{|}_{h}^{2}=8(1+|b|^{2})|b|^{2}$.

Proof Let $F$ and $F^{\dagger}_{H}$ be the matrix valued functions determined
by $f{\boldsymbol{v}}={\boldsymbol{v}}F$ and
$f^{\dagger}_{h}{\boldsymbol{v}}={\boldsymbol{v}}\cdot F^{\dagger}_{H}$. More
explicitly,

$F=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\quad
F^{\dagger}_{H}=\left(\begin{array}[]{cc}c&-\overline{b}\\\
-b&a\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\left(\begin{array}[]{cc}a&\overline{b}\\\
b&c\end{array}\right)=\left(\begin{array}[]{cc}1+2|b|^{2}&2\overline{b}c\\\
-2ab&-1-2|b|^{2}\end{array}\right).$

Let $\Pi_{1}$ be the matrix valued function determined by
$\pi_{1}{\boldsymbol{v}}={\boldsymbol{v}}\cdot\Pi_{1}$, i.e.,

$\Pi_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right).$

We can check the following:

$[F^{\dagger}_{H},\Pi_{1}]=\left(\begin{array}[]{cc}0&-2\overline{b}c\\\
-2ab&0\end{array}\right)=[F^{\dagger}_{H},\Pi_{1}]^{\dagger}_{H}.$

We obtain

$[F^{\dagger}_{H},\Pi_{1}]\cdot[F^{\dagger}_{H},\Pi_{1}]^{\dagger}_{H}=\left(\begin{array}[]{cc}4(1+|b|^{2})|b|^{2}&0\\\
0&4(1+|b|^{2})|b|^{2}\end{array}\right).$

Hence, we obtain

$\bigl{|}[f^{\dagger}_{h},\pi_{1}]\bigr{|}_{h}^{2}=\mathop{\rm
Tr}\nolimits\bigl{(}[F^{\dagger}_{H},\Pi_{1}]^{\dagger}_{H}\cdot[F^{\dagger}_{H},\Pi_{1}]\bigr{)}=8(1+|b|^{2})|b|^{2}.$

Thus, we obtain Lemma 2.8.

###### Corollary 2.9

We obtain $-\partial_{z}\partial_{\overline{z}}|b|_{h}^{2}\leq-8|b|_{h}^{2}$.

####  2.1.5 Bessel function of the first kind

We recall some functions by following [3, §5]. Let
$I_{0}(z):{\mathbb{C}}\to{\mathbb{R}}_{>0}$ be the function as in [3, §5],
i.e., $I_{0}$ denotes the unique positive even $C^{\infty}$-function such that
(i) $I_{0}(z)$ depends only on $|z|$, (ii)
$(\partial_{x}^{2}+\partial_{y}^{2})I_{0}=I_{0}$, (iii) $I_{0}(0)=1$. We have
the convergence $I_{0}(z)\cdot(2\pi|z|)^{1/2}e^{-|z|}\to 1$ as $|z|\to\infty$.
In particular, we have the following lemma.

###### Lemma 2.10

For any $0<u<1$, there exists $C_{0}(u)>0$ such that the following holds on
${\mathbb{C}}$:

$I_{0}(z)^{-1}\leq C_{0}(u)e^{-u|z|}.$

######  Lemma 2.11 ([3])

For any $0<\gamma_{1}<\gamma_{2}$, there exists $A(\gamma_{1},\gamma_{2})>0$
such that the following holds for any $0<b<a$:

$e^{-\gamma_{2}(a-b)}<A(\gamma_{1},\gamma_{2})\frac{I_{0}(\gamma_{1}b)}{I_{0}(\gamma_{1}a)}$

####  2.1.6 Proof of Proposition 2.1

For any $r>0$ and $z_{0}\in{\mathbb{C}}$, we set
$U(z_{0},r)=\bigl{\\{}z\in{\mathbb{C}}\,\big{|}\,|z-z_{0}|<r\bigr{\\}}$. We
set $r_{0}=R_{0}/10$. We recall the following general lemma.

###### Lemma 2.12

There exists $C_{1}(r_{0})$ depending only on $r_{0}$ such that the following
holds:

  * •

Let $h$ be any harmonic metric of $(E,\theta)_{|U(z_{0},r_{0})}$. Then,
$|f|_{h}\leq C_{1}(r_{0})$ holds on $U(z_{0},r_{0}/2)$.

###### Corollary 2.13

$|f|_{h}\leq C_{1}(r_{0})$ holds on $U(R-r_{0}/2)$ for any
$h\in\mathcal{H}(E,\theta)$.

Note that $|f|_{h}^{2}=2(1+2|b|^{2})$.

###### Corollary 2.14

There exists $C_{2}(r_{0})>0$ such that $|b|\leq C_{2}(r_{0})$ on
$U(R-r_{0}/2)$.

###### Lemma 2.15

The following hols:

$|b(0)|^{2}\leq 2C_{2}(r_{0})^{2}I_{0}(4\sqrt{2}(R-r_{0}/2))^{-1}.$

Proof We set
$J(z)=2C_{2}(r_{0})^{2}I_{0}(4\sqrt{2}z)I_{0}(4\sqrt{2}(R-r_{0}/2))^{-1}$. We
have

$-\partial_{z}\partial_{\overline{z}}J(z)=-8J(z),\quad
J(R-r_{0}/2)=2C_{2}(r_{0})^{2}.$

Let $Z=\\{z\in U(R-r_{0}/2)\,|\,|b(z)|^{2}>J(z)\\}$. We deduce a contradiction
by assuming $Z\neq\emptyset$. It is relatively compact in $U(R-r_{0}/2)$. We
have $|b(z)|^{2}=J(z)$ on $\partial Z$. On $Z$, we have

$-\partial_{z}\partial_{\overline{z}}(|b|^{2}-J)\leq-8(|b|^{2}-J)<0.$

Hence, $|b|^{2}-J$ is subharmonic on $Z$. By the maximum principle, we obtain
$|b|^{2}-J\leq 0$, which contradicts with the definition of $Z$. Hence, we
obtain $Z=\emptyset$. In particular, $|b(0)|^{2}\leq J(0)$.

###### Corollary 2.16

For any $0<u<1$,

$|b(0)|^{2}\leq 2C_{2}(r_{0})^{2}C_{0}(u)e^{-4\sqrt{2}(R-r_{0}/2)u}\leq
2C_{2}(r_{0})^{2}C_{0}(u)e^{2\sqrt{2}r_{0}}e^{-4\sqrt{2}uR}.$

For $z_{0}\in U(R-R_{0})$, we have $U(z_{0},R-|z_{0}|)\subset U(R)$. By
applying Corollary 2.16, we obtain

$|b(z_{0})|^{2}\leq
2C_{2}(r_{0})^{2}C_{0}(u)e^{-4\sqrt{2}(R-|z_{0}|-r_{0}/2)u}\leq
2C_{2}(r_{0})^{2}C_{0}(u)e^{2\sqrt{2}r_{0}}e^{-4\sqrt{2}u(R-|z_{0}|)}.$

It implies the claim of Proposition 2.1.

####  2.1.7 Proof of Proposition 2.2

###### Lemma 2.17

We obtain

$\bigl{|}\partial_{E,h,z}\pi\bigr{|}_{h}^{2}=(1+|b|^{2})^{2}\bigl{(}|\partial_{\overline{z}}b|^{2}+|\partial_{z}b|^{2}\bigr{)}+|b|^{2}(a^{2}|\partial_{z}c|^{2}+c^{2}|\partial_{z}a|^{2})\\\
-2\mathop{\rm
Re}\nolimits\Bigl{(}ba^{2}c\partial_{z}\overline{b}\partial_{\overline{z}}c+\overline{b}ac^{2}\partial_{z}b\partial_{\overline{z}}a+b^{2}\bigl{(}-c\partial_{z}\overline{b}+\overline{b}\partial_{z}c\bigr{)}\bigl{(}-\overline{b}\partial_{\overline{z}}a+a\partial_{\overline{z}}\overline{b}\bigr{)}\Bigr{)}.$
(31)

Proof We have

$\overline{H}^{-1}\partial_{z}\overline{H}=\left(\begin{array}[]{cc}c&-\overline{b}\\\
-b&a\end{array}\right)\left(\begin{array}[]{cc}\partial_{z}a&\partial_{z}\overline{b}\\\
\partial_{z}b&\partial_{z}c\end{array}\right)=\left(\begin{array}[]{cc}c\partial_{z}a-\overline{b}\partial_{z}b&c\partial_{z}\overline{b}-\overline{b}\partial_{z}c\\\
-b\partial_{z}a+a\partial_{z}b&-b\partial_{z}\overline{b}+a\partial_{z}c\end{array}\right).$

Let $A$ be determined by
$(\partial_{E,h,z}\pi_{1}){\boldsymbol{v}}={\boldsymbol{v}}\cdot A$. We set
$\beta=-c\partial_{z}\overline{b}+\overline{b}\partial_{z}c$ and
$\gamma=-b\partial_{z}a+a\partial_{z}b$. We obtain

$A=\partial_{z}\Pi_{1}+[\overline{H}^{-1}\partial_{z}\overline{H},\Pi_{1}]=\left(\begin{array}[]{cc}0&\beta\\\
\gamma&0\end{array}\right).$

We have

$A\cdot\overline{H}^{-1}{}^{t}\\!\overline{A}\cdot\overline{H}=\left(\begin{array}[]{cc}-b^{2}\beta\overline{\gamma}+a^{2}|\beta|^{2}&-bc\beta\overline{\gamma}+a\overline{b}|\beta|^{2}\\\
bc|\gamma|^{2}-a\overline{b}\gamma\overline{\beta}&|\gamma|^{2}c^{2}-\gamma\overline{\beta}\overline{b}^{2}\end{array}\right).$

We have

$\bigl{|}\partial_{E,h,z}\pi_{1}\bigr{|}_{h}^{2}=\mathop{\rm
Tr}\nolimits\bigl{(}A\cdot\overline{H}^{-1}{}^{t}\\!\overline{A}\cdot\overline{H}\bigr{)}=a^{2}|\beta|^{2}+c^{2}|\gamma|^{2}-2\mathop{\rm
Re}\nolimits(b^{2}\beta\overline{\gamma}).$

It is equal to

$a^{2}\Bigl{(}c^{2}|\partial_{z}\overline{b}|^{2}+|b|^{2}|\partial_{z}c|^{2}-2\mathop{\rm
Re}\nolimits(bc\partial_{z}\overline{b}\partial_{\overline{z}}c)\Bigr{)}+c^{2}\Bigl{(}|b|^{2}|\partial_{z}a|^{2}+|a|^{2}|\partial_{z}b|^{2}-2\mathop{\rm
Re}\nolimits(\overline{b}a\partial_{z}b\partial_{\overline{z}}a)\Bigr{)}\\\
-2\mathop{\rm
Re}\nolimits\Bigl{(}b^{2}(-c\partial_{z}\overline{b}+\overline{b}\partial_{z}c)(-\overline{b}\partial_{\overline{z}}a+a\partial_{\overline{z}}\overline{b})\Bigr{)}.$
(32)

Then, by using $ac=1+|b|^{2}$, we obtain (31).

We set $W=\\{z\in U(R)\,|\,b(z)\neq 0\\}$. We take
$4<\gamma_{1}<\gamma_{0}+\min\\{2\sqrt{2},\gamma_{0}\\}$.

###### Lemma 2.18

There exists $C_{20}(B(h),\gamma_{1})>0$, depending only on $B(h)$ and
$\gamma_{1}$ such that the following holds on $W$:

$-\partial_{z}\partial_{\overline{z}}|b|\leq-4|b|+C_{20}(B(h),\gamma_{1})e^{-\gamma_{1}(R-|z|)}.$
(33)

Proof On $W$, we have

$\partial_{z}|b|=\partial_{\overline{z}}|b|\leq\frac{1}{2}\bigl{(}|\partial_{z}b|+|\partial_{\overline{z}}b|\bigr{)}.$

We obtain

$2\partial_{z}|b|\partial_{\overline{z}}|b|\leq\frac{1}{2}\bigl{(}|\partial_{z}b|+|\partial_{\overline{z}}b|\bigr{)}^{2}\leq|\partial_{z}b|^{2}+|\partial_{\overline{z}}b|^{2}.$

Then, we obtain (33) from (30) and (31).

Let $0<\gamma<4$. We set $\eta=(4+\gamma)/2$. We have

$-\partial_{z}\partial_{\overline{z}}I_{0}(\eta
z)=-\frac{1}{4}\eta^{2}I_{0}(\eta z).$

There exists $C_{21}(B(h),\gamma_{1},\gamma)$ such that
$|b|<C_{21}(B(h),\gamma_{1},\gamma)$ on $|z|=R-r_{0}$, and that the following
holds for $g=C_{21}(B(h),\gamma_{1},\gamma)I_{0}(\eta
z)I_{0}(\eta(R-r_{0}))^{-1}$ on $U(R-r_{0})$:

$-\partial_{z}\partial_{\overline{z}}g\geq-4g+C_{20}(B(h),\gamma_{1})e^{-\gamma_{1}(R-|z|)}.$

###### Lemma 2.19

We have $|b|\leq g$ on $U(R-r_{0})$. In particular, $|b(0)|\leq
C_{21}(B(h),\gamma_{1},\gamma)I_{0}(\eta(R-r_{0}))^{-1}$.

Proof Let $Z=\\{z\in U(R-r_{0})\,|\,|b(z)|>g(z)\\}$. We shall deduce a
contradiction by assuming $Z\neq\emptyset$. We note that $Z\cap\partial
U(R-r_{0})=\emptyset$. We also have $Z\subset W$. Hence, we have $|b|=g$ on
$\partial Z$, and the following inequality on $Z$:

$-\partial_{z}\partial_{\overline{z}}(|b|-g)\leq-4(|b|-g).$

Then, we obtain $|b|\leq g$ on $Z$ which is a contradiction.

We obtain

$|b(0)|\leq C_{21}(B(h),\gamma,\gamma_{1})C_{0}(\gamma/\eta)e^{\gamma
r_{0}}e^{-\gamma R}.$

By applying the argument to the restriction of $(E,\theta,h)$ to
$U(z_{0},R-|z_{0}|)$ for $z_{0}\in U(R_{1})$, we obtain

$|b(z_{0})|\leq C_{21}(B(h),\gamma,\gamma_{1})C_{0}(\gamma/\eta)e^{\gamma
r_{0}}e^{-\gamma(R-|z_{0}|)}.$

Thus, we obtain Proposition 2.2.

####  2.1.8 Proof of Proposition 2.3 and the $C^{0}$-part of Proposition 2.4

Let $v_{1},v_{2}$ be a frame such that $C(v_{i},v_{i})=1$ and
$C(v_{1},v_{2})=0$. Let $h\in\mathcal{H}(E,\theta;C)$. Let $H$ be the
Hermitian matrix valued function determined by $H_{i,j}=h(v_{i},v_{j})$. We
can describe $H$ as

$H=\left(\begin{array}[]{cc}a&\sqrt{-1}\widetilde{b}\\\
-\sqrt{-1}\widetilde{b}&a\end{array}\right),$

where $a$ is an ${\mathbb{R}}_{>0}$-valued function, and $\widetilde{b}$ is an
${\mathbb{R}}$-valued function such that $a^{2}-\widetilde{b}^{2}=1$.

###### Lemma 2.20

The following holds:

$|\partial_{E,h,z}\pi_{1}|^{2}_{h}=\frac{2(1+2\widetilde{b}^{2})}{1+\widetilde{b}^{2}}|\partial_{z}\widetilde{b}|^{2}.$

Proof We set

$\beta=-\sqrt{-1}\widetilde{b}\partial_{z}a+\sqrt{-1}a\partial_{z}\widetilde{b}=\sqrt{-1}a^{-1}\partial_{z}\widetilde{b}.$

We have

$\overline{H}^{-1}\partial_{z}\overline{H}=\left(\begin{array}[]{cc}0&-\beta\\\
\beta&0\end{array}\right).$

Then, we obtain

$\bigl{[}\partial_{z}+\overline{H}^{-1}\partial_{z}\overline{H},\Pi_{1}\bigr{]}=\left(\begin{array}[]{cc}0&\beta\\\
\beta&0\end{array}\right).$

We obtain

$\bigl{|}\partial_{E,h,z}\pi_{1}\bigr{|}^{2}_{h}=\mathop{\rm
tr}\nolimits\left(\left(\begin{array}[]{cc}0&\beta\\\
\beta&0\end{array}\right)\left(\begin{array}[]{cc}a&\sqrt{-1}\widetilde{b}\\\
-\sqrt{-1}\widetilde{b}&a\end{array}\right)\left(\begin{array}[]{cc}0&\overline{\beta}\\\
\overline{\beta}&0\end{array}\right)\left(\begin{array}[]{cc}a&-\sqrt{-1}\widetilde{b}\\\
\sqrt{-1}\widetilde{b}&a\end{array}\right)\right)\\\
=2(1+2\widetilde{b}^{2})|\beta|^{2}=\frac{2(1+2\widetilde{b}^{2})}{1+\widetilde{b}^{2}}|\partial_{z}\widetilde{b}|^{2}.$
(34)

We set $W=\\{z\in U(R)\,|\,\widetilde{b}(z)\neq 0\\}$.

###### Lemma 2.21

On $W$, the following holds:

$-\partial_{z}\partial_{\overline{z}}|\widetilde{b}|\leq-4|\widetilde{b}|.$

Proof We have

$-\partial_{z}\partial_{\overline{z}}\widetilde{b}^{2}=-2|\widetilde{b}|\partial_{z}\partial_{\overline{z}}|\widetilde{b}|-2(\partial_{z}|\widetilde{b}|)\cdot(\partial_{\overline{z}}|\widetilde{b}|).$

Because $\widetilde{b}$ is real valued, we obtain

$\partial_{z}|\widetilde{b}|=\partial_{\overline{z}}|\widetilde{b}|\leq|\partial_{z}\widetilde{b}|.$

We have

$-2\frac{1+2\widetilde{b}^{2}}{1+\widetilde{b}^{2}}|\partial_{z}\widetilde{b}|^{2}+2\partial_{z}|\widetilde{b}|\partial_{\overline{z}}|\widetilde{b}|\leq
0.$

Then, we obtain
$-2|\widetilde{b}|\partial_{z}\partial_{\overline{z}}|\widetilde{b}|\leq-8|\widetilde{b}|^{2}$,
which implies the claim of the lemma.

###### Proposition 2.22

For any $0<\gamma<4$, there exists $C^{\prime}(R_{0},\gamma)>0$ such that the
following holds for any $h\in\mathcal{H}(E,\theta;C)$ on $U(R-R_{0})$:

$|\widetilde{b}(z)|\leq C^{\prime}(R_{0},\gamma)e^{-\gamma(R-|z|)}.$

It implies $|a-1|\leq C^{\prime}(R_{0},\gamma)e^{-2\gamma(R-|z|)}$.

Proof We use the notation in §2.1.6.

###### Lemma 2.23

The following holds:

$|\widetilde{b}(0)|\leq 2C_{2}(r_{0})I_{0}(4(R-r_{0}/2))^{-1}.$

Proof We set $J(z)=2C_{2}(r_{0})I_{0}(4z)I_{0}(4(R-r_{0}/2))^{-1}$.

$-\partial_{z}\partial_{\overline{z}}J(z)=-4J(z),\quad
J(z)_{|z|=R-r_{0}/2}=2C_{2}(r_{0}).$

Let $Z=\\{z\in U(R-r_{0}/2)\,|\,|\widetilde{b}(z)|>J(z)\\}$. We deduce a
contradiction by assuming $Z\neq\emptyset$. Clearly, $Z$ is contained in $W$,
and relatively compact in $U(R-r_{0}/2)$. We have $|\widetilde{b}(z)|=J(z)$ on
$\partial Z$, and

$-\partial_{z}\partial_{\overline{z}}(|\widetilde{b}|-J)\leq-4(|\widetilde{b}|^{2}-J)<0$

on $Z$. Hence, $|\widetilde{b}|-J$ is subharmonic on $Z$. By the maximum
principle, we obtain $|\widetilde{b}|-J\leq 0$, which contradicts with the
definition of $Z$. Hence, we obtain $Z=\emptyset$. In particular,
$|\widetilde{b}(0)|\leq J(0)$.

###### Corollary 2.24

For any $0<u<1$, we have

$|\widetilde{b}(0)|\leq 2C_{2}(r_{0})C_{0}(u)e^{-4(R-r_{0}/2)u}\leq
2C_{2}(r_{0})C_{0}(u)e^{2r_{0}}e^{-4uR}.$

By applying Corollary 2.24 to the restriction to $U(z_{0},R-|z_{0}|)$, we
obtain

$|\widetilde{b}(z_{0})|\leq 2C_{2}(r_{0})C_{0}(u)e^{-4(R-r_{0}/2)u}\leq
2C_{2}(r_{0})C_{0}(u)e^{2r_{0}}e^{-4u(R-|z_{0}|)}.$

It implies the claim of Proposition 2.22.

Proposition 2.22 implies the claim of Proposition 2.4 in the case
$(j,k)=(0,0)$. It implies Proposition 2.3.

####  2.1.9 Proof of Proposition 2.4

We continue to use the notation in §2.1.8. We have

$\partial_{\overline{z}}\Bigl{(}\overline{H}^{-1}\partial_{z}\overline{H}\Bigr{)}=\left(\begin{array}[]{cc}0&-\sqrt{-1}\partial_{\overline{z}}(a^{-1}\partial_{z}\widetilde{b})\\\
\sqrt{-1}\partial_{\overline{z}}(a^{-1}\partial_{z}\widetilde{b})&0\end{array}\right).$

We have

$F^{\dagger}_{H}=\overline{H}^{-1}F\overline{H}=\left(\begin{array}[]{cc}1+\widetilde{b}^{2}&-2\sqrt{-1}a\widetilde{b}\\\
-2\sqrt{-1}a\widetilde{b}&-1-2\widetilde{b}^{2}\end{array}\right).$

We obtain

$[F,F^{\dagger}_{H}]=\left(\begin{array}[]{cc}0&-4\sqrt{-1}a\widetilde{b}\\\
4\sqrt{-1}a\widetilde{b}&0\end{array}\right).$

Because
$\partial_{\overline{z}}\left(\overline{H}^{-1}\partial_{z}\overline{H}\right)-[F,F^{\dagger}_{H}]=0$,
we obtain

$\partial_{\overline{z}}\bigl{(}a^{-1}\partial_{z}\widetilde{b}\bigr{)}-4a\widetilde{b}=0.$
(35)

By using $a^{2}-\widetilde{b}^{2}=1$, we obtain

$\partial_{\overline{z}}\partial_{z}\widetilde{b}=\frac{1}{1+\widetilde{b}^{2}}\widetilde{b}|\partial_{z}\widetilde{b}|^{2}+4\widetilde{b}.$
(36)

For $z_{0}\in U(R_{1})$, let $\chi_{z_{0}}:{\mathbb{C}}\to[0,1]$ be a
$C^{\infty}$-function such that $\chi_{z_{0}}(z)=1$ $(|z-z_{0}|<r_{0}/2)$ and
$\chi_{z_{0}}(z)=0$ $(|z-z_{0}|\geq 2r_{0}/3)$. We also assume that
$\chi_{z_{0}}^{-1/2}d\chi_{z_{0}}$ induces a $C^{\infty}$-function. By (36),
we obtain

$0\leq\int\chi_{z_{0}}\widetilde{b}\partial_{\overline{z}}\partial_{z}\widetilde{b}=-\int\partial_{\overline{z}}(\chi_{z_{0}})\widetilde{b}\partial_{z}\widetilde{b}-\int\chi_{z_{0}}|\partial_{z}\widetilde{b}|^{2}.$

We obtain

$\int\chi_{z_{0}}|\partial_{z}\widetilde{b}|^{2}\leq\left(\int|\chi_{z_{0}}^{-1/2}d\chi_{z_{0}}|\cdot|\widetilde{b}|^{2}\right)^{1/2}\left(\int\chi_{z_{0}}|\partial_{z}\widetilde{b}|^{2}\right)^{1/2}.$

Hence,

$\int\chi_{z_{0}}|\partial_{z}\widetilde{b}|^{2}\leq\int|\chi_{z_{0}}^{-1/2}d\chi_{z_{0}}|\cdot|\widetilde{b}|^{2}.$

For any $0<\gamma<4$, we obtain

$\left(\int_{|z-z_{0}|<r_{0}/2}|\partial_{z}\widetilde{b}|^{2}\right)^{1/2}\leq\pi^{1/2}(r_{0}/2)C^{\prime}(R,R_{1}+r_{0},\gamma)e^{\gamma
r_{0}}e^{-\gamma(R-|z_{0}|)}.$

By using (35), we obtain the estimate for the sup norms of
$\partial_{z}\widetilde{b}$ on $\\{|z-z_{0}|<r_{0}/3\\}$. By a standard
bootstrapping argument, we obtain the estimate (29). Because
$a^{2}=1+\widetilde{b}^{2}$, it implies the estimate (28).

###  2.2 An estimate for derivatives

Fix $R_{0}>0$ and $C_{0}>0$. Let $(E,\theta)$ be a Higgs bundle of rank $r$ on
$U(R)$ for $R>2R_{0}$. Let $f$ be the endomorphism determined by
$\theta=f\,dz$. Assume that $|\mathop{\rm tr}\nolimits(f^{j})|<C_{0}$ for
$j=1,\ldots,r$. Let $h_{0}$ be a harmonic metric. Let $g_{0}$ be the Euclidean
metric $dz\,d\overline{z}$. Let $\nabla$ denote the Chern connection of
$(E,h_{0})$. Let $\mathfrak{R}$ denote the endomorphism of $E$ such that
$\mathfrak{R}\,dz\,d\overline{z}$ is the curvature of $\nabla$.

###### Lemma 2.25

There exist $C(j,k)>0$ depending only on $R_{0}$, $C_{0}$, $r$ and $(j,k)$
such that the following holds on $U(R-R_{0})$ for any harmonic metric $h$ of
$(E,\theta)$:

$|\nabla_{z}^{j}f|_{h_{0}}+|\nabla_{\overline{z}}^{k}f^{\dagger}_{h_{0}}|_{h_{0}}+|\nabla_{z}^{j}\nabla_{\overline{z}}^{k}\mathfrak{R}|_{h_{0}}\leq
C(j,k).$

Proof For $z_{0}\in U(R-R_{0})$, we have $U(z_{0},R_{0})\subset U(R)$. By
applying [19, Proposition 4.1] to $(E,\theta)_{|U(z_{0},R_{0})}$, we obtain
the estimate on $U(z_{0},R_{0}/2)$. Hence, we obtain the desired estimate on
$U(R-R_{0})$.

For positive constants $C_{1},\epsilon_{0},\epsilon_{1}$, let
$\mathcal{H}(E,\theta;h_{0},C_{1},\epsilon_{0},\epsilon_{1})$ denote the set
of harmonic metrics $h$ of $(E,\theta)$ such that the following holds on
$U(R)$.

  * •

$|s(h_{0},h)|_{h_{0}}+|s(h_{0},h)^{-1}|_{h_{0}}\leq C_{1}$.

  * •

$\int\bigl{|}\overline{\partial}s(h_{0},h)\bigr{|}_{h_{0}}^{2}+\int\bigl{|}[\theta,s(h_{0},h)]\bigr{|}_{h_{0}}^{2}\leq\epsilon_{1}^{2}$.

  * •

$\Bigl{|}\overline{\partial}\bigl{(}s(h_{0},h)^{-1}\partial_{h_{0}}s(h_{0},h)\bigr{)}\Bigr{|}_{h_{0},g_{0}}\leq\epsilon_{0}$.

We set $\epsilon_{2}=\max\\{\epsilon_{0},\epsilon_{1}\\}$.

###### Lemma 2.26

There exists $C_{2}$ depending only on
$R_{0},C_{0},C_{1},r,\epsilon_{0},\epsilon_{1}$, such that the following holds
on $U(R-R_{0})$ for any
$h\in\mathcal{H}(E,\theta;h_{0},C_{1},\epsilon_{0},\epsilon_{1})$:

$\bigl{|}\nabla
s(h_{0},h)\bigr{|}_{h_{0},g_{0}}+\bigl{|}[\theta,s(h_{0},h)]\bigr{|}_{h_{0},g_{0}}\leq
C_{2}\cdot\epsilon_{2}.$

Proof To simplify the description, we denote $s(h_{0},h)$ by $s$. We obtain
the estimate for
$\sup_{U(R-R_{0})}\bigl{|}\overline{\partial}s\bigr{|}_{h_{0},g_{0}}$ by using
the Sobolev embedding theorem and Uhlenbeck’s theorem on the existence of
Coulomb gauge. (See the proof of [19, Proposition 4.1], for example.) Because
$s$ is self-adjoint with respect to $h_{0}$, we obtain the estimate for
$\sup_{U(R-R_{0})}\bigl{|}\nabla s\bigr{|}_{h_{0},g_{0}}$. Because
$\overline{\partial}[\theta,s]=-[\theta,\overline{\partial}s]$, we also obtain
the estimate for $\sup_{U(R-R_{0})}\bigl{|}[\theta,s]\bigr{|}_{h_{0},g_{0}}$.

###### Lemma 2.27

For any $(j,k)\in{\mathbb{Z}}_{\geq 0}^{2}\setminus\\{(0,0)\\}$, there exist
positive constants $C_{3}(j,k)$ depending only on $R_{0}$, $C_{0}$, $C_{1}$,
$r$, $\epsilon_{0}$ and $\epsilon_{1},(j,k)$, such that the following holds on
$U(R-R_{0})$:

$\bigl{|}\nabla_{z}^{j}\nabla_{\overline{z}}^{k}s(h_{0},h)\bigr{|}_{h_{0}}\leq
C_{3}(j,k)\cdot\epsilon_{2}.$

Proof We have

$\overline{\partial}\bigl{(}s(h_{0},h)^{-1}\partial_{E,h_{0}}s(h_{0},h)\bigr{)}+\bigl{[}\theta,s^{-1}[\theta^{\dagger}_{h_{0}},s]\bigr{]}=0.$

Then, we obtain the desired estimate by using a standard bootstrapping
argument, the Sobolev embedding theorem and Uhlenbeck’s theorem on the
existence of Coulomb gauge.

###### Corollary 2.28

For any $(j,k)\in{\mathbb{Z}}_{\geq 0}^{2}$, there exist positive constants
$C_{4}(j,k)$ depending only on $R_{0}$, $C_{0}$, $C_{1}$, $r$, $\epsilon_{0}$
and $\epsilon_{1},(j,k)$, such that the following holds on $U(R-R_{0})$:

$\bigl{|}\nabla_{z}^{j}\nabla_{\overline{z}}^{k}[\theta,s(h_{0},h)]\bigr{|}_{h_{0}}+\bigl{|}\nabla_{z}^{j}\nabla_{\overline{z}}^{k}[\theta^{\dagger}_{h_{0}},s(h_{0},h)]\bigr{|}_{h_{0}}\leq
C_{4}(j,k)\cdot\epsilon_{2}.$

##  3 Global estimates

###  3.1 Flat metrics induced by quadratic differentials

####  3.1.1 Holomorphic local coordinates

Let $X$ be a compact Riemann surface. Let $\phi$ be a non-zero quadratic
differential on $X$ such that each zero of $\phi$ is simple. Let $D_{\phi}$
denote the set of zeroes of $\phi$. Let $g_{\phi}$ be the Kähler metric of
$X\setminus D_{\phi}$ obtained as $g_{\phi}=|\phi|$. Let $d_{\phi}$ denote the
distance of $X$ induced by $g_{\phi}$. For any $r>0$ and $P\in X$, we set
$X_{P}(r)=\\{Q\in X\,|\,d_{\phi}(P,Q)<r\\}$.

For each $P\in X\setminus D_{\phi}$, there exists a coordinate neighbourhood
$(X_{P},z_{P})$ around $P$ such that $\phi_{|X_{P}}=dz_{P}\,dz_{P}$ and
$z_{P}(P)=0$. We obtain $g_{\phi|X_{P}}=dz_{P}\,d\overline{z}_{P}$. Such a
coordinate $z_{P}$ is well defined up to the multiplication of $\pm 1$. There
exists $r_{0}>0$ such that the coordinate $z_{P}$ induces the following
holomorphic isomorphism for any $r<r_{0}$:

$(X_{P}(r),\phi)\simeq(U(r),(dz_{P})^{2}).$

For each $P\in D_{\phi}$, there exists a coordinate neighbourhood
$(X_{P},z_{P})$ such that
$\phi=\left(\frac{3}{2}\right)^{2}z_{P}\,dz_{P}\,dz_{P}$. We have
$g_{\phi|X_{P}}=\left(\frac{3}{2}\right)^{2}|z_{P}|\,dz_{P}\,d\overline{z}_{P}$.
Such a coordinate $z_{P}$ is well defined up to the multiplication of cubic
roots of $1$. We consider the quadratic differential
$\phi_{0}=\left(\frac{3}{2}\right)^{2}z\,(dz)^{2}$ on ${\mathbb{C}}$. It
induces the distance $d_{\phi_{0}}$ on ${\mathbb{C}}$. We set

$\mathcal{U}_{P}(r)=\bigl{\\{}z\in{\mathbb{C}}\,\big{|}\,d_{\phi_{0}}(0,z)<r\bigr{\\}}=\bigl{\\{}z\in{\mathbb{C}}\,\big{|}\,|z|<r^{2/3}\bigr{\\}}.$

There exists $r_{0}>0$ such that the coordinate $z_{P}$ induces the following
holomorphic isomorphism for any $r<r_{0}$:

$(X_{P}(r),\phi)\simeq\bigl{(}\mathcal{U}_{P}(r),\phi_{0}\bigr{)}.$

####  3.1.2 Threshold

According to [3, §3.2], a saddle connection of $\phi$ is a geodesic segment on
$(X,d_{\phi})$ such that (i) the starting point and the ending point are
contained in $D_{\phi}$, (ii) any other point is contained in $X\setminus
D_{\phi}$. Note that the starting point and the ending point are not
necessarily distinct. Let $M(\phi)$ denote the minimum length of the saddle
connections of $\phi$, which is called the threshold. The following lemma is
obvious.

###### Lemma 3.1

For $P\in D_{\phi}$, we have $X_{P}(M(\phi))\cap D_{\phi}=\\{P\\}$.

We shall prove the following proposition in §3.1.4.

###### Proposition 3.2

Let $P\in D_{\phi}$.

  * •

A local holomorphic coordinate $z_{P}$ around $P$ as in §3.1.1 is analytically
continued to a holomorphic coordinate on $X_{P}(M(\phi)/2)$, and it induces a
holomorphic isomorphism

$(X_{P}(\kappa),\phi)\simeq(\mathcal{U}_{P}(\kappa),\phi_{0})$

for any $0<\kappa\leq M(\phi)/2$. In particular,
$\phi_{|X_{P}(\kappa)}=(\frac{3}{2})^{2}z_{P}(dz_{P})^{2}$.

  * •

The inverse function $z_{P}^{-1}:\mathcal{U}_{P}(M(\phi)/2)\simeq
X_{P}(M(\phi)/2)$ is analytically continued to a locally bi-holomorphic
function

$\psi_{P}:\mathcal{U}_{P}(M(\phi))\longrightarrow X.$

We have $\psi_{P}^{\ast}(\phi)=\phi_{0}$, $\psi_{P}^{-1}(D_{\theta})=\\{0\\}$,
and $\psi_{P}(\mathcal{U}_{P}(\kappa))\subset X_{P}(\kappa)$ for any
$0<\kappa\leq M(\phi)$.

The following lemma is easier which is also proved in §3.1.4.

###### Lemma 3.3

Let $P\in X\setminus D_{\phi}$. Let $z_{P}$ be a holomorphic local coordinate
around $P$ as in §3.1.1. Then, $z_{P}^{-1}$ is analytically continued to a
locally bi-holomorphic function

$\psi_{P}:U\bigl{(}d_{\phi}(P,D_{\phi})\bigr{)}\longrightarrow X.$

We have $\psi_{P}^{\ast}(\phi)=(dz)^{2}$, $\psi_{P}^{-1}(D_{\phi})=\emptyset$,
and $\psi_{P}(\mathcal{U}_{P}(\kappa))\subset X_{P}(\kappa)$ for any
$0<\kappa\leq d_{\phi}(P,D_{\phi})$.

####  3.1.3 Metrics induced by holomorphic $1$-forms

Let $Y$ be a simply connected Riemann surface isomorphic to a disc. Let $a$ be
a holomorphic $1$-form on $Y$. We set $Z_{a}=\\{Q\in Y\,|\,a(Q)=0\\}$.

We obtain the Kähler metric $|a|^{2}$ on $Y\setminus Z_{a}$ whose curvature is
flat. As explained in [23], it induces the distance $d_{a}$ on $Y$. For a
curve $\gamma$, the length of $\gamma$ is denoted by $L_{a}(\gamma)$. In this
subsection, any geodesic is assumed to be parameterized by the arc length. We
assume the following.

###### Assumption 3.4

$(Y,d_{a})$ is a complete metric space.

Under the assumption, the following holds. See [23, Theorem 14.2.1, Theorem
16.1, Corollary 18.2].

###### Proposition 3.5

For any two points $P_{1},P_{2}\in Y$, there is a unique geodesic
$\gamma_{P_{1},P_{2}}$ starting from $P_{1}$ and ending at $P_{2}$. We have
$L_{a}(\gamma_{P_{1},P_{2}})=d_{a}(P_{1},P_{2})$. If the length of a curve
connecting $P_{1}$ and $P_{2}$ is $d_{a}(P_{1},P_{2})$, the curve equals to
the geodesic $\gamma_{P_{1},P_{2}}$ after a re-parametrization of the
parameter.

We fix a base point $P\in Y$. We set $M_{a,P}:=\inf_{Q\in
Z_{a}\setminus\\{P\\}}d_{a}(P,Q)$. Let $Z_{a,P}$ denote the set of $Q\in
Z_{a}\setminus\\{P\\}$ such that $\gamma_{P,Q}(s)\not\in Z_{a}$ for any
$0<s<d_{a}(P,Q)$. It is easy to see $M_{a,P}=\inf_{Q\in Z_{a,P}}d_{a}(P,Q)$.

Let $n\in{\mathbb{Z}}_{>0}$ denote the order of zero of $a$ at $P$. Let
$f_{a,P}$ be the holomorphic function on $Y$ such that $df_{a,P}=a$ and
$f_{a,P}(P)=0$. There exists a holomorphic function $f_{a,P}^{\frac{1}{n+1}}$
defined on a neighbourhood $\mathcal{U}$ of $P$ such that
$(f_{a,P}^{\frac{1}{n+1}})^{n+1}=f_{a,P}$ and that $f_{a,P}^{\frac{1}{n+1}}$
has simple zero at $P$.

For any $r>0$, let $Y_{P}(r):=\\{Q\in Y\,|\,d_{a}(P,Q)<r\\}$ and
$Y_{P}(r)^{\ast}=Y_{P}(r)\setminus\\{P\\}$. We also set
$U(r):=\\{w\in{\mathbb{C}}\,|\,|w|<r\\}$ and
$U(r)^{\ast}=U(r)\setminus\\{0\\}$.

###### Proposition 3.6

Let $r\leq M_{a,P}$.

  * •

$f_{a,P}$ induces a proper map $Y_{P}(r)\to U(r)$, and the induced map
$Y_{P}(r)^{\ast}\to U(r)^{\ast}$ is a covering map of degree $n+1$.

  * •

$f_{a,P}^{\frac{1}{n+1}}$ is analytically continued to a holomorphic function
on $Y_{P}(r)$, and it induces a bi-holomorphic map $Y_{P}(r)\simeq
U(r^{\frac{1}{n+1}})$. We have $(f_{a,P}^{\frac{1}{n+1}})^{n+1}=f_{a,P}$ on
$Y_{P}(r)$.

Proof We shall give some lemmas.

###### Lemma 3.7

For any $Q\in Y_{P}(M_{a,P})^{\ast}$, $\gamma_{P,Q}(s)$ $(0<s\leq d_{a}(P,Q))$
are not contained in $Z_{a}$. In particular, $Y_{P}(M_{a,P})^{\ast}\cap
Z_{a}=\emptyset$.

Proof Let $S$ denote the set of $0<s\leq d_{a}(P,Q)$ such that
$\gamma_{P,Q}(s)\in Z_{a}$. Assume that $S$ is not empty. Let $s_{0}=\min S$.
Then, $\gamma_{P,Q}(s_{0})\in Z_{a,P}$, and $d_{a}(P,\gamma_{P,Q}(s_{0}))\leq
d_{a}(P,Q)<M_{a,P}$, which contradicts the definition of $M_{a,P}$.

For $Q\in Y_{P}(M_{a,P})^{\ast}$, by Lemma 3.7, we obtain

$f_{a,P}\circ\gamma_{P,Q}(s)=s\cdot|f_{a,P}(Q)|^{-1}f_{a,P}(Q),$

and $d_{a}(P,Q)=|f_{a,P}(Q)|$. In particular, we obtain
$f_{a,P}(Y_{P}(r))\subset U(r)$ for any $0<r<M_{a,P}$.

Let $w\in{\mathbb{C}}\setminus\\{0\\}$. There exist $(n+1)$-distinct points
$w_{n+1}^{(i)}\in{\mathbb{C}}$ $(i=1,\ldots,n+1)$ such that
$(w_{n+1}^{(i)})^{n+1}=w$. There exist $\epsilon>0$ and $C^{\infty}$-maps
$\eta^{(i)}_{w,\epsilon}:[0,\epsilon]\to\mathcal{U}\subset Y$
$(i=1,\ldots,n+1)$ such that $\eta^{(i)}_{w,\epsilon}(0)=P$ and
$f_{a,P}^{\frac{1}{n+1}}\circ\eta^{(i)}_{w,\epsilon}(s)=s^{\frac{1}{n+1}}w_{n+1}^{(i)}$
for $0\leq s\leq\epsilon$.

###### Lemma 3.8

Suppose $|w|<M_{a,P}$.

  * •

There exist $C^{\infty}$-maps $\eta^{(i)}_{w}:[0,1]\to Y$ such that (i)
$\eta^{(i)}_{w}(s)=\eta^{(i)}_{w,\epsilon}(s)$ $(0\leq s\leq\epsilon)$, (ii)
$f_{a,P}\circ\eta^{(i)}_{w}(s)=sw$ $(0\leq s\leq 1)$.

  * •

For $i\neq j$, we have $\eta^{(i)}_{w}(s)\neq\eta^{(j)}_{w}(s)$ for any
$0<s\leq 1$.

Proof Fix $i\in\\{1,\ldots,n+1\\}$. Let $I$ be the set of $\epsilon\leq\nu\leq
1$ such that there exists $\eta^{(i)}_{w,\nu}:[0,\nu]\to Y$ such that (i)
$\eta^{(i)}_{w,\nu}(s)=\eta^{(i)}_{w,\epsilon}(s)$ $(0\leq s\leq\epsilon)$,
(ii) $f_{a,P}\circ\eta^{(i)}_{w,\nu}(s)=sw$ $(0\leq s\leq\nu)$. Note that
$\epsilon\in I$, and hence $I\neq\emptyset$. We set $\nu_{0}=\sup I$. Suppose
$\nu_{0}<1$. For any $\epsilon\leq\nu<\nu_{0}$, we have
$\eta^{(i)}_{w,\nu}:[0,\nu]\to Y$ satisfying the above conditions. Note that

$d_{a}(P,\eta^{(i)}_{w,\nu}(s))\leq\bigl{|}\eta^{(i)}_{w,\nu}(s)\bigr{|}=s|w|<M_{a,P}.$

Because $\eta^{(i)}_{w,\nu}(s)\not\in Z_{a}$ for any $\epsilon\leq\nu<\nu_{0}$
and $0<s\leq\nu$, we have
$\eta^{(i)}_{w,\nu_{1}}(s)=\eta^{(i)}_{w,\nu_{2}}(s)$ for any
$\epsilon\leq\nu_{1}\leq\nu_{2}$ and $0\leq s\leq\nu_{1}$. Let us consider the
sequence $\eta^{(i)}_{w,\nu}(\nu)$ $(\nu\in I)$. We note that
$d_{a}(\eta^{(i)}_{w,\nu_{1}}(\nu_{1}),\eta^{(i)}_{w,\nu_{2}}(\nu_{2}))=|\nu_{1}-\nu_{2}|\cdot|w|$.
Because $d_{a}$ is complete, there exists the limit
$z_{\infty}=\lim_{\nu\to\nu_{0}}\eta^{(i)}_{w,\nu}(\nu)\in Y$. By the
continuity of $f_{a}$, we have $f_{a}(z_{\infty})=\nu_{0}w$. Because $f_{a}$
is bi-holomorphic around $z_{\infty}$ and $\nu_{0}w$, we can construct
$\eta^{(i)}_{w,\nu_{0}+\delta}$ satisfying the above conditions for some
$\delta>0$. It contradicts the definition of $\nu_{0}$. Thus, we obtain the
first claim.

Fix a pair $1\leq i,j\leq n+1$ with $i\neq j$. Let $J$ be the set of $0<s\leq
1$ such that $\eta^{(i)}_{w}(s)=\eta^{(j)}_{w}(s)$. Because the condition is
closed, $J$ is a closed subset of $]0,1]$. Because $f_{a}$ is locally bi-
holomorphic, $J$ is an open subset of $]0,1]$. Because
$J\subset\\{\epsilon\leq s\leq 1\\}$, we obtain $J\neq]0,1]$, and hence
$J=\emptyset$. Thus, we obtain the second claim.

###### Lemma 3.9

For any $w\in U(r)^{\ast}$, we have

$\bigl{\\{}Q\in
Y_{P}(M_{a,P})\setminus\\{P\\}\,\big{|}\,f_{a,P}(Q)=w\bigr{\\}}=\bigl{\\{}\eta^{(i)}_{w}(1)\,\big{|}\,i=1,\ldots,n+1\bigr{\\}}.$

Proof The right hand side is contained in the left hand side. Let $Q\in
Y_{P}(M_{a,P})$ such that $f_{a,P}(Q)=P$. Because
$f_{a,P}\circ\gamma_{P,Q}(|w|s)=s\cdot w$ for $0\leq s\leq 1$,
$\gamma_{P,Q}(|w|\cdot s)$ equals one of $\eta^{(i)}_{w}$. Hence, the left
hand side is contained in the right hand side.

Let $0<r\leq M_{a,P}$. Let us consider the induced map
$f_{a|Y_{P}(r)^{\ast}}:Y_{P}(r)^{\ast}\to U(r)^{\ast}$. It is locally bi-
holomorphic. By Lemma 3.9, it is a covering map. Then, $f_{a}^{\frac{1}{n+1}}$
is analytically continued on $Y_{P}(r)$. The other claims of Proposition 3.6
also follow.

####  3.1.4 Proof of Proposition 3.2 and Lemma 3.3

Let $\Sigma\subset T^{\ast}X$ be the spectral curve of $\phi$, which is also a
compact Riemann surface. Let $\pi:\Sigma\to X$ denote the projection. There
exists a holomorphic $1$-form $a$ on $\Sigma$ such that
$\pi^{\ast}(\phi)=a^{2}$.

Let $\varphi:\widetilde{\Sigma}\to\Sigma$ be a universal covering. We obtain
$\widetilde{a}=\varphi^{\ast}(a)$. The Riemann surface $\widetilde{\Sigma}$ is
bi-holomorphic to a disc. According to [23, Lemma 18.2], the metric space
$(\widetilde{\Sigma},d_{\widetilde{a}})$ is complete.

Let $P\in D_{\phi}$. There exists a unique point $\widetilde{P}_{1}\in\Sigma$
such that $\pi(\widetilde{P}_{1})=P$. Let
$\widetilde{P}_{2}\in\widetilde{\Sigma}$ be a point such that
$\varphi(\widetilde{P}_{2})=\widetilde{P}_{1}$. By definitions, we obtain

$M(\phi)\leq M_{\widetilde{a},\widetilde{P}_{2}}.$

We obtain
$f_{\widetilde{a},\widetilde{P}_{2}}:\widetilde{\Sigma}\to{\mathbb{C}}$ such
that $df_{\widetilde{a},\widetilde{P}_{2}}=\widetilde{a}$ and
$f_{\widetilde{a},\widetilde{P}_{2}}(\widetilde{P}_{2})=0$. As explained in
Proposition 3.6, there exists a holomorphic function
$f_{\widetilde{a},\widetilde{P}_{2}}^{\frac{1}{3}}$ on
$\widetilde{\Sigma}_{\widetilde{P}_{2}}(M_{\widetilde{a},\widetilde{P}_{2}})$
such that
$(f_{\widetilde{a},\widetilde{P}_{2}}^{\frac{1}{3}})^{3}=f_{\widetilde{a},\widetilde{P}_{2}}$,
and $f_{\widetilde{a},\widetilde{P}_{2}}^{\frac{1}{3}}$ induces a bi-
holomorphic isomorphism

$\widetilde{\Sigma}_{\widetilde{P}_{2}}(r)\simeq\\{|\zeta|<r^{1/3}\\}$

for any $r<M_{\widetilde{a},\widetilde{P}_{2}}$. We have
$\widetilde{a}=d\zeta^{3}$.

Let $d_{a}$ be the distance on $\Sigma$ induced by $|a|^{2}$. For $r>0$, let
$\Sigma_{\widetilde{P}_{1}}(r)=\\{Q\in\Sigma\,|\,d_{a}(\widetilde{P}_{1},Q)<r\\}$
and
$\Sigma_{\widetilde{P}_{1}}(r)^{\ast}=\Sigma_{\widetilde{P}_{1}}(r)\setminus\\{\widetilde{P}_{1}\\}$.

###### Lemma 3.10

Let $r\leq M_{\widetilde{a},\widetilde{P}_{2}}/2$.

  * •

$\varphi$ induces an isomorphism
$\widetilde{\Sigma}_{\widetilde{P}_{2}}(r)\simeq\Sigma_{\widetilde{P}_{1}}(r)$.

  * •

$\pi$ induces a ramified covering map $\Sigma_{\widetilde{P}_{1}}(r)\to
X_{P}(r)$ of degree $2$. It induces a covering map
$\Sigma_{\widetilde{P}_{1}}(r)^{\ast}\to X_{P}(r)^{\ast}$.

Proof We have
$\varphi(\widetilde{\Sigma}_{\widetilde{P}_{2}}(r))\subset\Sigma_{\widetilde{P}_{1}}(r)$
for any $r>0$. Suppose $r\leq M_{\widetilde{a},\widetilde{P}_{2}}/2$. Let
$Q_{1},Q_{2}\in\widetilde{\Sigma}_{\widetilde{P}_{2}}(r)$ such that
$\varphi(Q_{1})=\varphi(Q_{2})$ and $Q_{1}\neq Q_{2}$. Then, there exists
$\widetilde{P}^{\prime}_{2}\in\varphi^{-1}(\widetilde{P}_{1})\setminus\\{\widetilde{P}_{2}\\}$
such that $d_{\widetilde{a}}(\widetilde{P}_{2}^{\prime},Q_{1})<r$. We obtain
$d_{\widetilde{a}}(\widetilde{P}_{2},\widetilde{P}_{2}^{\prime})<2r\leq
M_{\widetilde{a},\widetilde{P}_{2}}$, which contradicts the condition of
$M_{\widetilde{a},\widetilde{P}_{2}}$. Hence, the restriction of $\varphi$
induces a bi-holomorphic isomorphism between
$\widetilde{\Sigma}_{\widetilde{P}_{2}}(r)$ and its image
$\varphi(\widetilde{\Sigma}_{\widetilde{P}_{2}}(r))$.

Let $Q\in\Sigma_{\widetilde{P}_{1}}(r)$. There exists a curve $\gamma^{Q}_{0}$
starting from $\widetilde{P}_{1}$ and ending at $Q$ whose length is strictly
smaller than $r$. According to [23, Theorem 18.2.2], there exists a unique
geodesic $\gamma^{Q}$ from $\widetilde{P}_{1}$ to $Q$ such that $\gamma^{Q}$
is homotopic to $\gamma^{Q}_{0}$. The length $L$ of $\gamma^{Q}$ is strictly
less than $r$. Let $\widetilde{\gamma}^{Q}$ be the lift of $\gamma^{Q}$ to
$\widetilde{\Sigma}$ starting from $\widetilde{P}_{2}$. Then,
$\widetilde{\gamma}^{Q}(L)$ is contained in
$\widetilde{\Sigma}_{\widetilde{P}_{2}}(r)$. We obtain
$\Sigma_{\widetilde{P}_{1}}(r)=\varphi(\widetilde{\Sigma}_{\widetilde{P}_{2}}(r))$.
Thus, we obtain the first claim.

We have $\pi(\Sigma_{\widetilde{P}_{1}}(r))\subset X_{P}(r)$ for any $r>0$.
Suppose that $r<M_{\widetilde{a},\widetilde{P}_{2}}/2$. The induced map
$\Sigma_{\widetilde{P}_{1}}(r)^{\ast}\to X_{P}(r)^{\ast}$ is locally bi-
holomorphic. Let $Q\in X_{P}(r)$. There exists a curve $\gamma^{Q}$ starting
from $P$ and ending at $Q$ whose length is strictly smaller than $r$.
According to [23, Theorem 18.2.2], there exists a unique geodesic $\gamma^{Q}$
from $P$ to $Q$ such that $\gamma^{Q}$ is homotopic to $\gamma^{Q}_{0}$. The
length $L$ of $\gamma^{Q}$ is strictly less than $r$. Because
$L<M_{\widetilde{a},\widetilde{P}_{2}}/2$, $\gamma^{Q}(s)$ is not contained in
$D_{\phi}$ for any $0<s\leq L$. Hence, there exists exactly two lifts
$\gamma^{Q(i)}$ $(i=1,2)$ of $\gamma^{Q}$ to $\Sigma_{\widetilde{P}_{1}}(r)$.
Hence, $\Sigma_{\widetilde{P}_{1}}(r)^{\ast}\to X_{P}(r)^{\ast}$ is a covering
map of degree $2$. Thus, we obtain the second claim.

Let $\sigma$ be the involution of $\Sigma$ induced by the multiplication of
$-1$ on $T^{\ast}X$. It induces an involution $\sigma$ on
$\Sigma_{\widetilde{P}_{1}}(r)$ for any $r$. By Lemma 3.10, we obtain the
involution $\sigma$ on $\widetilde{\Sigma}_{\widetilde{P}_{2}}(r)$ for
$r<M_{\widetilde{a},\widetilde{P}_{2}}/2$.

###### Lemma 3.11

The induced involution $\sigma$ on
$\widetilde{\Sigma}_{\widetilde{P}_{2}}(M_{\widetilde{a},\widetilde{P}_{2}})$
is described as $\sigma(\zeta)=-\zeta$.

Proof Because $\widetilde{a}=d(\zeta^{3})$,
$\sigma^{\ast}\widetilde{a}=-\widetilde{a}$,
$\sigma(\widetilde{P}_{2})=\widetilde{P}_{2}$ and $\sigma^{2}=\mathop{\rm
id}\nolimits$, we obtain $\sigma^{\ast}(\zeta)=-\zeta$.

There exists a bi-holomorphic map

$\Phi_{1}:U\bigl{(}(M_{\widetilde{a},\widetilde{P}_{2}}/2)^{2/3}\bigr{)}\longrightarrow
X_{P}(M_{\widetilde{a},\widetilde{P}_{2}}/2)$

such that $\pi\circ\varphi=\Phi_{1}\circ
f^{\frac{1}{3}}_{\widetilde{a},\widetilde{P}_{2}}$ on
$\widetilde{\Sigma}_{\widetilde{P}_{2}}(M_{\widetilde{a},\widetilde{P}_{2}}/2)$.
For each $0<r<M_{\widetilde{a},\widetilde{P}_{2}}/2$, $\Phi_{1}$ induces an
isomorphism $U(r^{2/3})\simeq X_{P}(r)$. Because $M(\phi)\leq
M_{\widetilde{a},\widetilde{P}_{2}}$, this implies the first claim of
Proposition 3.2. We obtain the second claim of Proposition 3.2 from the
following lemma.

###### Lemma 3.12

$\Phi_{1}$ is analytically continued to a locally bi-holomorphic map

$\Phi_{2}:U\bigl{(}M_{\widetilde{a},\widetilde{P}_{2}}^{2/3}\bigr{)}\longrightarrow
X_{P}(M_{\widetilde{a},\widetilde{P}_{2}})$

such that $\pi\circ\varphi=\Phi_{2}\circ
f^{\frac{1}{3}}_{\widetilde{a},\widetilde{P}_{2}}$ on
$\widetilde{\Sigma}_{\widetilde{P}_{2}}(M_{\widetilde{a},\widetilde{P}_{2}})$.

Proof For $\theta_{1}<\theta_{2}$ such that $\theta_{2}-\theta_{1}<2\pi$, we
consider the following regions:

$S_{0}(\theta_{1},\theta_{2})=\bigl{\\{}\xi\in{\mathbb{C}}\,\big{|}\,(M_{\widetilde{a},\widetilde{P}_{2}}/3)^{2/3}<|\xi|<(M_{\widetilde{a},\widetilde{P}_{2}}/2)^{2/3},\,\,\theta_{1}<\arg(\xi)<\theta_{2}\bigr{\\}},$
$S_{1}(\theta_{1},\theta_{2})=\bigl{\\{}\xi\in{\mathbb{C}}\,\big{|}\,(M_{\widetilde{a},\widetilde{P}_{2}}/3)^{2/3}<|\xi|<M_{\widetilde{a},\widetilde{P}_{2}}^{2/3},\,\,\theta_{1}<\arg(\xi)<\theta_{2}\bigr{\\}}.$

Let $\rho:{\mathbb{C}}\to{\mathbb{C}}$ be the map defined by
$\rho(\zeta)=\zeta^{2}$. For $i=0,1$, we have the decomposition into the
connected components

$\rho^{-1}(S_{i}(\theta_{1},\theta_{2}))=\rho^{-1}(S_{i}(\theta_{1},\theta_{2}))_{1}\sqcup\rho^{-1}(S_{i}(\theta_{1},\theta_{2}))_{2}.$

We obtain

$(\rho^{-1})_{j}:S_{1}(\theta_{1},\theta_{2})\simeq\rho^{-1}(S_{1}(\theta_{1},\theta_{2}))_{j}$

obtained as the inverse of $\rho$. We have the equalities

$\pi\circ\varphi\circ(f_{\widetilde{a},\widetilde{P}_{2}}^{\frac{1}{3}})^{-1}\circ(\rho^{-1})_{1}=\pi\circ\varphi\circ(f_{\widetilde{a},\widetilde{P}_{2}}^{\frac{1}{3}})^{-1}\circ(\rho^{-1})_{2}$

on $S_{1}(\theta_{1},\theta_{2})$, because their restriction to
$S_{0}(\theta_{1},\theta_{2})$ are equal. Then, we obtain the claim of Lemma
3.12, which implies the second claim of Proposition 3.2.

Let $P\in X\setminus D_{\phi}$. Let
$\widetilde{P}\in(\pi\circ\varphi)^{-1}(P)$. By definitions, we have

$M_{\widetilde{a},\widetilde{P}}=d_{\phi}(P,D_{\phi}).$

We obtain the locally bi-holomorphic map
$U(M_{\widetilde{a},\widetilde{P}})\to X$ as the composition of

$\begin{CD}U(M_{\widetilde{a},\widetilde{P}}<EMAIL_ADDRESS>
This gives a desired analytic continuation of $z_{P}^{-1}$. Thus, we obtain
Lemma 3.3.

###  3.2 Higgs bundles of rank $2$ with smooth spectral curve

Let $X$ be a compact Riemann surface. Let $(E,\theta)$ be a Higgs bundle of
degree $0$ and rank $2$ on $X$. We assume that $\det(E)=\mathcal{O}_{X}$ and
$\mathop{\rm tr}\nolimits\theta=0$. We assume that the spectral curve
$\Sigma_{\theta}$ is smooth, reduced and irreducible. Let $D_{\theta}\subset
X$ denote the set of the critical values of the projection
$\pi:\Sigma_{\theta}\to X$. At each point of $D_{\theta}$,
$\phi:=-\det(\theta)$ has a simple zero. For any $W\subset X$, we set
$\Sigma_{\theta|W}=\Sigma_{\theta}\times_{X}W$.

We obtain the flat Kähler metric $g_{\phi}=|\phi|$ of $X\setminus D_{\theta}$.
Let $d_{\phi}$ denote the distance of $X$ induced by $g_{\phi}$. For any $P\in
X$ and $r>0$, we set $X_{P}(r)=\\{Q\in X\,|\,d_{\phi}(P,Q)<r\\}$. We set

$\kappa_{0}=M(\phi)/2.$

####  3.2.1 Decomposition into the diagonal part and the off-diagonal part

Let $P\in X\setminus D_{\theta}$. There exists a holomorphic coordinate
neighbourhood $(X_{P},z_{P})$ such that $\phi=(dz_{P})^{2}$ and $z_{P}(P)=0$.
We set $E_{P}=E_{|X_{P}}$. Let $f_{P}$ be the endomorphism of $E_{P}$
determined by $\theta_{|X_{P}}=f_{P}\,dz_{P}$.

We have the decomposition

$(E,\theta)_{|X_{P}}=(E_{P,1},dz_{P})\oplus(E_{P,2},-dz_{P}).$

We set

$\mathop{\rm End}\nolimits(E)^{\circ}_{P}=\bigoplus\mathop{\rm
End}\nolimits(E_{P,i}),\quad\quad\mathop{\rm
End}\nolimits(E)^{\bot}_{P}=\mathop{\rm
Hom}\nolimits(E_{P,1},E_{P,2})\oplus\mathop{\rm
Hom}\nolimits(E_{P,2},E_{P,1}).$

The subbundles $\mathop{\rm End}\nolimits(E)^{\circ}_{P}$ and $\mathop{\rm
End}\nolimits(E^{\bot})_{P}$ of $\mathop{\rm End}\nolimits(E)_{|X_{P}}$ are
well defined. By varying $P\in X\setminus D_{\theta}$, we obtain the
subbundles $\mathop{\rm End}\nolimits(E)^{\circ}$ and $\mathop{\rm
End}\nolimits(E)^{\bot}$ of $\mathop{\rm End}\nolimits(E)_{|X\setminus
D_{\theta}}$, and the decomposition

$\mathop{\rm End}\nolimits(E)_{|X\setminus D_{\theta}}=\mathop{\rm
End}\nolimits(E)^{\circ}\oplus\mathop{\rm End}\nolimits(E)^{\bot}.$ (37)

###  3.3 Approximation of large solutions of Hitchin equation

####  3.3.1 Limiting configuration

Let us recall the notion of the limiting configuration in [13]. There exists a
line bundle $L$ on $\Sigma_{\theta}$ with an isomorphism
$\pi_{\ast}\Sigma\simeq(E,\theta)$, where the Higgs field of $\pi_{\ast}L$ is
induced by the tautological $1$-form of $T^{\ast}X$. There exists a singular
flat Hermitian metric $h_{L}$ of $L$ with the following property.

  * •

Let $Q\in\Sigma_{\theta}$ be any point of $\pi^{-1}(D_{\theta})$. Then, there
exists a holomorphic local coordinate $\zeta_{Q}$ of $\Sigma_{\theta}$ and a
frame $v_{Q}$ of $L$ around $Q$ such that $h_{L}(v_{Q},v_{Q})=|\zeta_{Q}|$.

We obtain the Hermitian metric $h_{\infty}=\pi_{\ast}h_{L}$ of $E_{|X\setminus
D_{\theta}}$. Because $\det(h_{\infty})$ induces a $C^{\infty}$ Hermitian
metric of $\det(E)$, we may adjust $h_{\infty}$ so that $\det(h_{\infty})=1$.

Let $\nabla_{\infty}$ denote the flat unitary connection of $(E_{|X\setminus
D_{\theta}},h_{\infty})$ obtained as the Chern connection of $h_{\infty}$. We
consider the flat connection of $T^{\ast}(X\setminus D_{\theta})$ obtained as
the Levi-Civita connection of $g_{\phi}$. For any $m\in{\mathbb{Z}}_{\geq 0}$,
the induced connections on $\mathop{\rm
End}\nolimits(E)\otimes(T^{\ast}(X\setminus D_{\theta}))^{\otimes m}$ are also
denoted by $\nabla_{\infty}$. We obtain $\nabla_{\infty}^{j}:\mathop{\rm
End}\nolimits(E)\otimes(T^{\ast}X)^{\otimes k}\to\mathop{\rm
End}\nolimits(E)\otimes(T^{\ast}X)^{\otimes(k+j)}$ for any $k,j\geq 0$.

####  3.3.2 Model Higgs bundle

We set
$E_{0}=\mathcal{O}_{{\mathbb{C}}}u_{1}\oplus\mathcal{O}_{{\mathbb{C}}}u_{2}$.
We consider the Higgs field $\theta_{0}$ of $E_{0}$ defined by

$\theta_{0}(u_{1})=\frac{3}{2}u_{2}\,dz,\quad\theta_{0}(u_{2})=\frac{3}{2}u_{1}\,z\,dz.$

Let $C_{0}$ be the non-degenerate symmetric pairing such that
$C_{0}(u_{i},u_{i})=0$ $(i=1,2)$ and $C_{0}(u_{1},u_{2})=1$. There exists a
unique harmonic metric $h_{0}$ of $(E_{0},\theta_{0})$ compatible with $C_{0}$
(see [12]). Let $\rho_{t}:{\mathbb{C}}\to{\mathbb{C}}$ be given by
$\rho_{t}(z)=t^{2/3}z$. By setting

$\rho_{t}^{\ast}(u_{1})=t^{1/6}u_{1},\quad\rho_{t}^{\ast}(u_{2})=t^{-1/6}u_{2},$

we obtain the isomorphism
$\rho_{t}^{\ast}(E_{0},\theta_{0},C_{0})\simeq(E_{0},t\theta_{0},C_{0})$. We
obtain harmonic metrics $h_{0,t}=\rho_{t}^{\ast}h_{0}$ of
$(E_{0},t\theta_{0})$ which are compatible with $C_{0}$.

####  3.3.3 Harmonic metrics around $P\in D_{\theta}$

Let $P\in D_{\theta}$. We obtain the map
$\psi_{P,2\kappa_{0}}:\mathcal{U}_{P}(2\kappa_{0})\to X$ as in §3.1. We obtain
the Higgs bundle

$(E_{P},\theta_{P})=\psi_{P,2\kappa_{0}}^{\ast}(E,\theta)$

on $\mathcal{U}_{P}(2\kappa_{0})$. There exists the decomposition

$\mathop{\rm
End}\nolimits(E_{P})_{|\mathcal{U}_{P}(2\kappa_{0})\setminus\\{0\\}}=\mathop{\rm
End}\nolimits(E_{P})^{\circ}\oplus\mathop{\rm End}\nolimits(E_{P})^{\bot}$
(38)

as in §3.2.1.

We set $h_{P,\infty}:=\psi_{P,2\kappa_{0}}^{\ast}(h_{\infty})$ on
$\mathcal{U}_{P}(2\kappa_{0})\setminus\\{0\\}$. There exists a non-degenerate
symmetric pairing $C_{P}$ of $(E_{P},\theta_{P})$ such that $C_{P}$ is
compatible with $h_{P,\infty}$ on
$\mathcal{U}_{P}(2\kappa_{0})\setminus\\{0\\}$ (see [19]). For any
$m\in{\mathbb{Z}}_{\geq 0}$, we obtain the connections $\nabla_{P,\infty}$ of
$\mathop{\rm End}\nolimits(E_{P})\otimes
T^{\ast}(\mathcal{U}_{P}(2\kappa_{0})\setminus\\{0\\})^{\otimes m}$ induced by
the Chern connection of $h_{P,\infty}$ and the Levi-Civita connection of
$|\phi_{0}|$.

There exists a natural isomorphism

$(E_{P},\theta_{P},C_{P})\simeq(E_{0},\theta_{0},C_{0})_{|\mathcal{U}_{P}(2\kappa_{0})}.$
(39)

For any $t>0$, let $h_{P,t}$ be the harmonic metric of $(E_{P},t\theta_{P})$
obtained as the restriction of $h_{0,t}$ under the isomorphism (39). It is
compatible with $C_{P}$. We define the endomorphism $s_{P,t}$ of
$E_{P|\mathcal{U}_{P}(2\kappa_{0})\setminus\\{P\\}}$ by
$h_{P,t}=h_{P,\infty}\cdot s_{P,t}$. We obtain the decomposition

$s_{P,t}=s_{P,t}^{\circ}+s_{P,t}^{\bot}$

on $\mathcal{U}_{P}(2\kappa_{0})\setminus\\{0\\}$ corresponding to the
decomposition (38).

Let $0<\kappa<\kappa_{0}$. We set
$\delta(\kappa):=\min\\{(\kappa_{0}-\kappa)/10,\kappa/20\\}$. We set

$\mathcal{U}^{(1)}_{P,\kappa}=\mathcal{U}_{P}(\kappa_{0}-\delta(\kappa)),\quad\mathcal{U}^{(2)}_{P,\kappa}=\mathcal{U}_{P}(\kappa_{0}-2\delta(\kappa)).$

###### Lemma 3.13

For any $j\in{\mathbb{Z}}_{\geq 0}$, there exists $C_{j}>0$ such that the
following holds on
$\mathcal{U}^{(1)}_{P,\kappa}\setminus\mathcal{U}^{(2)}_{P,\kappa}$:

$|\nabla_{\infty}^{j}(s_{P,t}^{\circ}-\mathop{\rm
id}\nolimits)|_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-8(\kappa+7\delta(\kappa))t},\quad\quad\quad|\nabla_{\infty}^{j}(s_{P,t}^{\bot})|_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+7\delta(\kappa))t}.$ (40)

Proof Let
$Q\in\mathcal{U}^{(1)}_{P,\kappa}\setminus\mathcal{U}^{(2)}_{P,\kappa}$. There
exists a holomorphic embedding
$\psi_{Q}:U(\kappa_{0}-2\delta(\kappa))\to\mathcal{U}^{(0)}_{P,\kappa}$ such
that $\psi_{Q}(0)=Q$, $\psi_{Q}^{\ast}(\phi_{0})=(dz)^{2}$, and
$0\not\in\mathop{\rm Im}\nolimits(\psi_{Q})$. Take
$\kappa+7\delta(\kappa)<\kappa^{\prime}<\kappa+8\delta(\kappa)=\kappa_{0}-2\delta(\kappa)$.

For any $t>0$, let $\rho_{t}:{\mathbb{C}}\to{\mathbb{C}}$ be the map defined
by $\rho_{t}(z)=t^{-1}z$. We put $\psi_{Q,t}=\psi_{Q}\circ\rho_{t}$. Let
$g_{0}$ be the standard Euclidean metric on ${\mathbb{C}}$. We apply Corollary
2.5 to $\psi_{Q,t}^{\ast}(E_{P},t\theta_{P})$ on
$U(t(\kappa_{0}-2\delta(\kappa)))$. For any $j\geq 0$, there exists
$C_{j}^{\prime}>0$ such that the following holds around $0\in
U\bigl{(}t(\kappa_{0}-2\delta(\kappa))\bigr{)}$:

$\Bigl{|}\psi_{Q,t}^{\ast}\bigl{(}\nabla_{\infty}^{j}(s_{P,t}^{\circ}-\mathop{\rm
id}\nolimits)\bigr{)}\Bigr{|}_{\psi_{Q,t}^{\ast}h_{\infty},g_{0}}\leq
C^{\prime}_{j}e^{-8\kappa^{\prime}t},\quad\quad\quad\Bigl{|}\psi_{Q,t}^{\ast}\bigl{(}\nabla_{\infty}^{j}(s_{P,t}^{\bot})\bigr{)}\Bigr{|}_{\psi_{Q,t}^{\ast}h_{\infty},g_{0}}\leq
C_{j}^{\prime}e^{-4\kappa^{\prime}t}.$

Hence, the following holds around $0\in
U\bigl{(}\kappa_{0}-2\delta(\kappa)\bigr{)}$:

$\Bigl{|}\psi_{Q}^{\ast}\bigl{(}\nabla_{\infty}^{j}(s_{P}^{\circ}-\mathop{\rm
id}\nolimits)\bigr{)}\Bigr{|}_{\psi_{Q}^{\ast}h_{\infty},g_{0}}\leq
C^{\prime}_{j}t^{j}e^{-8\kappa^{\prime}t},\quad\quad\quad\Bigl{|}\psi_{Q}^{\ast}\bigl{(}\nabla_{\infty}^{j}(s_{P}^{\bot})\bigr{)}\Bigr{|}_{\psi_{Q}^{\ast}h_{\infty},g_{0}}\leq
C_{j}^{\prime}t^{j}e^{-4\kappa^{\prime}t}.$

Then, we obtain (40) around $Q$.

For any $\kappa^{\prime}<\kappa_{0}$, we identify
$\mathcal{U}_{P}(\kappa^{\prime})$ and $X_{P}(\kappa^{\prime})$. We also
identify $(E_{P},\theta_{P})_{|\mathcal{U}_{P}(\kappa)}$ and
$(E,\theta)_{|X_{P}(\kappa)}$. We obtain $X^{(i)}_{P,\kappa}\subset
X_{P}(\kappa_{0})$ $(i=1,2)$ corresponding to $\mathcal{U}^{(i)}_{P,\kappa}$.

####  3.3.4 Approximation of harmonic metrics

For $0<\kappa<\kappa_{0}$, by patching $h_{\infty}$ and $h_{P,t}$ $(P\in
D_{\theta})$, we construct Hermitian metrics $\widetilde{h}_{\kappa,t}$ such
that the following holds.

  * •

$\widetilde{h}_{\kappa,t}=h_{P,t}$ on $X^{(2)}_{P,\kappa}$.

  * •

$\widetilde{h}_{\kappa,t}=h_{\infty}$ on $X\setminus\bigcup
X^{(1)}_{P,\kappa}$.

  * •

Let $s(h_{\infty},\widetilde{h}_{\kappa,t})$ be the automorphism of
$E_{|X\setminus D_{\theta}}$ determined by
$\widetilde{h}_{\kappa,t}=h_{\infty}\cdot
s(h_{\infty},\widetilde{h}_{\kappa,t})$. For any $j\in{\mathbb{Z}}_{\geq 0}$,
there exists $C_{j}>0$ such that the following holds on
$\bigcup(X_{P,\kappa}^{(1)}\setminus X^{(2)}_{P,\kappa})$:

$\Bigl{|}\nabla_{\infty}^{j}\bigl{(}s(h_{\infty},\widetilde{h}_{\kappa,t})^{\circ}-\mathop{\rm
id}\nolimits\bigr{)}\Bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-8(\kappa+7\delta(\kappa))t},$
$\bigl{|}\nabla_{\infty}^{j}(s(h_{\infty},\widetilde{h}_{\kappa,t})^{\bot})\bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+7\delta(\kappa))t}.$

  * •

$\det(\widetilde{h}_{\kappa,t})=1$.

Let $s(\widetilde{h}_{\kappa,t},h_{t})$ be the automorphism of $E$ determined
by $h_{t}=\widetilde{h}_{\kappa,t}\cdot s(\widetilde{h}_{\kappa,t},h_{t})$.

Let $\nabla_{h_{t}}$ denote the Chern connection of $h_{t}$. Let $g_{X}$ be a
Kähler metric of $X$. The Levi-Civita connection and $\nabla_{h_{t}}$ induce
connections on $\mathop{\rm End}\nolimits(E)\otimes(T^{\ast}X)^{\otimes m}$,
which are denoted by $\nabla_{h_{t},g_{X}}$. We obtain the differential
operators $\nabla_{h_{t},g_{X}}^{j}:\mathop{\rm
End}\nolimits(E)\otimes(T^{\ast}X)^{\otimes m}\to\mathop{\rm
End}\nolimits(E)\otimes(T^{\ast}X)^{\otimes m+j}$. We shall prove the
following theorem in §3.3.5–3.3.10.

###### Theorem 3.14

For any $j\geq 0$, there exists $C_{j}>0$ such that the following holds on $X$
for any $t\geq 1$:

$\Bigl{|}\nabla_{h_{t},g_{X}}^{j}\bigl{(}s(\widetilde{h}_{\kappa,t},h_{t})-\mathop{\rm
id}\nolimits\bigr{)}\Bigr{|}_{h_{t},g_{X}}+\Bigl{|}\nabla_{h_{t},g_{X}}^{j}\bigl{(}[\theta,s(\widetilde{h}_{\kappa,t},h_{t})]\bigr{)}\Bigr{|}_{h_{t},g_{X}}+\Bigl{|}\nabla_{h_{t},g_{X}}^{j}\bigl{(}[\theta^{\dagger}_{h_{t}},s(\widetilde{h}_{\kappa,t},h_{t})]\bigr{)}\Bigr{|}_{h_{t},g_{X}}\leq
C_{j}e^{-4\kappa t}.$

To simplify the description, we set
$s_{\kappa,t}=s(\widetilde{h}_{\kappa,t},h_{t})$. To simplify the argument, we
use the following result already proved in [13]. (See also [19].)

######  Proposition 3.15 ([13])

There exists $C>0$ and $\epsilon>0$ such that $|s_{\kappa,t}-\mathop{\rm
id}\nolimits|_{\widetilde{h}_{\kappa,t}}\leq Ce^{-\epsilon t}$. Higher
derivatives are also dominated.

It is our purpose to improve the rate of the convergence.

####  3.3.5 Preliminary

Let $R(\widetilde{h}_{\kappa,t})$ denote the curvature of the Chern connection
of $E$ with $\widetilde{h}_{\kappa,t}$. We set

$F(\widetilde{h}_{\kappa,t})=R(\widetilde{h}_{\kappa,t})+t^{2}[\theta,\theta^{\dagger}_{\widetilde{h}_{\kappa,t}}].$

We obtain the following lemma by the construction of
$\widetilde{h}_{\kappa,t}$.

###### Lemma 3.16

The support of $F(\widetilde{h}_{\kappa,t})$ is contained in
$\bigcup_{P}\bigl{(}X^{(1)}_{P,\kappa}\setminus X^{(2)}_{P,\kappa}\bigr{)}$.
Moreover, there exists $C>0$ such that

$|F(\widetilde{h}_{\kappa,t})^{\bot}|_{\widetilde{h}_{\kappa,t},g_{X}}\leq
Ce^{-4(\kappa+6\delta(\kappa))t},\quad\quad|F(\widetilde{h}_{\kappa,t})^{\circ}|_{\widetilde{h}_{\kappa,t},g_{X}}\leq
Ce^{-8(\kappa+6\delta(\kappa))t}.$

####  3.3.6 Weaker estimate of the $L^{2}$-norm of the first derivatives

Let $s(h_{\infty},h_{t})$ be the automorphism of $E_{|X\setminus D_{\theta}}$
determined by $h_{t}=h_{\infty}s(h_{\infty},h_{t})$.

###### Lemma 3.17

There exists $C>0$ such that $|s(h_{\infty},h_{t})^{\bot}|\leq
Ce^{-2\sqrt{2}(\kappa+7\delta(\kappa))t}$ on $X\setminus\bigcup
X^{(2)}_{P,\kappa}$.

Proof Let $Q\in X\setminus\bigcup X^{(2)}_{P,\kappa}$. By Lemma 3.3, there
exists a locally bi-holomorphic map
$\psi_{Q,t}:U\bigl{(}t(\kappa_{0}-2\delta(\kappa))\bigr{)}\to X\setminus
D_{\theta}$ such that $\psi_{Q}(0)=Q$, $\psi_{Q}^{\ast}(t^{2}\phi)=(dz)^{2}$.
By applying Proposition 2.1, we obtain the claim of Lemma 3.17.

###### Corollary 3.18

There exists $C>0$ such that $|s_{\kappa,t}^{\bot}|\leq
Ce^{-2\sqrt{2}(\kappa+7\delta(\kappa))t}$ on $X\setminus\bigcup
X^{(2)}_{P,\kappa}$.

###### Corollary 3.19

There exists $C>0$ such that the following holds on $X\setminus D_{\theta}$:

$\Bigl{|}\mathop{\rm
Tr}\nolimits\bigl{(}s_{\kappa,t}F(\widetilde{h}_{\kappa,t})\bigr{)}\Bigr{|}\leq
Ce^{-(4+2\sqrt{2})(\kappa+6\delta(\kappa))t}.$ (41)

Proof Because $\mathop{\rm
Tr}\nolimits\bigl{(}s_{\kappa,t}F(\widetilde{h}_{\kappa,t})\bigr{)}=\mathop{\rm
Tr}\nolimits\bigl{(}s^{\circ}_{\kappa,t}F(\widetilde{h}_{\kappa,t})^{\circ}\bigr{)}+\mathop{\rm
Tr}\nolimits\bigl{(}s^{\bot}_{\kappa,t}F(\widetilde{h}_{\kappa,t})^{\bot}\bigr{)}$,
we obtain (41) from Proposition 3.15 Lemma 3.16 and Lemma 3.17.

According to [21, Lemma 3.1], we have

$\Delta_{g_{X}}\mathop{\rm Tr}\nolimits(s_{\kappa,t})=-\mathop{\rm
Tr}\nolimits\bigl{(}F(\widetilde{h}_{\kappa,t})s_{\kappa,t}\bigr{)}-\bigl{|}\overline{\partial}_{E}(s_{t})s_{t}^{-1/2}\bigr{|}^{2}_{\widetilde{h}_{\kappa,t},g_{X}}-\bigl{|}[\theta,s_{\kappa,t}]s_{\kappa,t}^{-1/2}\bigr{|}^{2}_{\widetilde{h}_{\kappa,t},g_{X}}$

Because $\int\Delta_{g_{X}}\mathop{\rm Tr}\nolimits(s_{\kappa,t})=0$, there
exists $C^{\prime}>0$ such that

$\|\overline{\partial}_{E}(s_{\kappa,t})s_{\kappa,t}^{-1/2}\|_{L^{2},\widetilde{h}_{\kappa,t},g_{X}}\leq
C^{\prime}e^{-(2+\sqrt{2})(\kappa+6\delta(\kappa))t},\quad\quad\|[\theta,s_{\kappa,t}]s_{\kappa,t}^{-1/2}\|_{L^{2},\widetilde{h}_{\kappa,t},g_{X}}\leq
C^{\prime}e^{-(2+\sqrt{2})(\kappa+6\delta(\kappa))t}.$

By Proposition 3.15, there exists $C^{\prime\prime}>0$ such that

$\|\overline{\partial}_{E}(s_{\kappa,t})\|_{L^{2},\widetilde{h}_{\kappa,t},g_{X}}\leq
C^{\prime\prime}e^{-(2+\sqrt{2})(\kappa+6\delta(\kappa))t},\quad\quad\|[\theta,s_{\kappa,t}]\|_{L^{2},\widetilde{h}_{\kappa,t},g_{X}}\leq
C^{\prime\prime}e^{-(2+\sqrt{2})(\kappa+6\delta(\kappa))t}.$

Note that it is equivalent to

$\|\overline{\partial}_{E}(s_{\kappa,t})\|_{L^{2},\widetilde{h}_{\kappa,t},g_{\phi}}\leq
C^{\prime\prime}e^{-(2+\sqrt{2})(\kappa+6\delta(\kappa))t},\quad\quad\|[\theta,s_{\kappa,t}]\|_{L^{2},\widetilde{h}_{\kappa,t},g_{\phi}}\leq
C^{\prime\prime}e^{-(2+\sqrt{2})(\kappa+6\delta(\kappa))t}.$ (42)

####  3.3.7 Weaker estimate of higher derivatives

###### Lemma 3.20

For any $j>0$, there exists $C_{j}>0$ such that the following holds on
$X\setminus\bigcup X^{(2)}_{P,\kappa}$:

$|\nabla_{\infty}^{j}(s_{\kappa,t})|_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-2\sqrt{2}(\kappa+6\delta(\kappa))t}.$ (43)

Proof By using Lemma 3.17, there exists $C^{\prime}>0$ such that

$\Bigl{|}t^{2}[\theta,\theta^{\dagger}_{h_{t}}]\Bigr{|}_{h_{\infty},g_{\phi}}\leq
C^{\prime}t^{2}e^{-2\sqrt{2}(\kappa+7\delta(\kappa))t}$

on $X\setminus\bigcup X^{(2)}_{P,\kappa}$. Let $R(h_{t})$ denote the curvature
of the Chern connection of $h_{t}$. Because
$R(h_{t})+t^{2}[\theta,\theta^{\dagger}_{h_{t}}]=0$, we obtain

$\Bigl{|}R(h_{t})\Bigr{|}_{h_{\infty},g_{\phi}}\leq
C^{\prime}t^{2}e^{-2\sqrt{2}(\kappa+7\delta(\kappa))t}$

on $X\setminus\bigcup X^{(2)}_{P,\kappa}$. By using Lemma 2.26 and Lemma 2.27
with the estimate (42), we obtain the desired estimate (43).

We consider the connection $\nabla_{\widetilde{h}_{\kappa,t},g_{\phi}}$ of
$\mathop{\rm End}\nolimits(E)\otimes T^{\ast}(X\setminus D_{\phi})^{\otimes
m}$ induced by the Chern connection of $\widetilde{h}_{\kappa,t}$ and the
Levi-Civita connection of $g_{\phi}$. Let
$\rho_{P,\kappa}:X_{P,\kappa}^{(2)}\to{\mathbb{R}}_{\geq 0}$ be defined by
$\rho_{P,\kappa}(Q)=d_{\phi}(Q,X_{P}(2\delta(\kappa)))$.

###### Lemma 3.21

Let $P\in D_{\theta}$. For any $j>0$ and $0<\gamma<2\sqrt{2}$, there exists
$C_{j,\gamma}>0$ such that the following holds on $X^{(2)}_{P,\kappa}\setminus
X_{P}(2\delta(\kappa))$:

$\Bigl{|}\nabla_{\widetilde{h}_{\kappa,t},g_{\phi}}^{j}(s_{\kappa,t})\Bigr{|}_{\widetilde{h}_{\kappa,t},g_{\phi}}\leq
C_{j,\gamma}e^{-\gamma\cdot\rho_{P,\kappa}t}.$

Proof Let $Q\in X^{(2)}_{P,\kappa}\setminus X_{P}(2\delta(\kappa))$. There
exists a locally bi-holomorphic map

$\psi_{Q,t}:U\bigl{(}t\rho_{P}(Q)\bigr{)}\longrightarrow X$

such that $\psi_{Q}(0)=Q$ and $\psi_{Q}^{\ast}(t^{2}\phi)=(dz)^{2}$. We apply
Proposition 2.1 to the Higgs bundle
$(E_{Q,t},\theta_{Q,t})=\psi_{Q,t}^{\ast}(E,t\theta)$ and harmonic metrics
$h_{t,Q}=\psi_{Q,t}^{\ast}(h_{t})$ and $h_{P,t,Q}=\psi_{Q,t}^{\ast}(h_{P,t})$.
Let $\gamma<\gamma^{\prime}<2\sqrt{2}$ and $t\geq 10\delta(\kappa)^{-1}$.
There exists $C>0$ such that the following holds on
$U(t\rho_{P,\kappa}(Q)-1)$:

$\Bigl{|}\overline{\partial}_{E_{Q,t}}\Bigl{(}\psi_{Q,t}^{\ast}(s_{\kappa,t})^{-1}\partial_{E_{Q,t},h_{P,t,Q}}\psi_{Q,t}^{\ast}(s_{\kappa,t})\Bigr{)}\Bigr{|}_{h_{P,t,Q},g_{0}}\leq
Ce^{-\gamma^{\prime}(t\rho_{P,\kappa}(Q)-|z|)}.$

We have $U(2)\subset U(t\rho_{P,\kappa}(Q))$. By (42), we obtain

$\int_{U(1)}\bigl{|}\partial_{E_{Q,t},h_{P,t,Q}}\psi_{Q,t}^{\ast}(s_{\kappa,t})\bigr{|}_{h_{P,t,Q},g_{0}}^{2}\leq
Ce^{-(2+\sqrt{2})(\kappa+6\delta(\kappa))t}.$

Note that $\rho_{P,\kappa}(Q)<\kappa+6\delta(\kappa)$, and hence
$2\sqrt{2}\rho_{P,\kappa}(Q)<(2+\sqrt{2})\bigl{(}\kappa+6\delta(\kappa)\bigr{)}$.
By Lemma 2.25 and Lemma 2.26, we obtain the following on $U(1)$:

$\bigl{|}\psi_{Q,t}^{\ast}\nabla^{j}_{h_{P,t},g_{\phi}}(s_{\kappa,t})\bigr{|}_{\psi_{Q,t}^{\ast}(h_{P,t}),g_{0}}\leq
Ce^{-\gamma^{\prime}\rho_{P,\kappa}(Q)t}.$

We obtain the following around $Q$:

$\bigl{|}\nabla^{j}_{h_{P,t},g_{\phi}}(s_{\kappa,t})\bigr{|}_{h_{P,t},g_{\phi}}\leq
Ct^{j}e^{-\gamma^{\prime}\rho_{P,\kappa}(Q)t}.$

Then, we obtain the claim of the lemma.

###### Lemma 3.22

Let $P\in D_{\theta}$. For any $j\geq 0$ and $0<\gamma<2\sqrt{2}$, there
exists $C_{j,\gamma}>0$ such that the following holds on
$X^{(2)}_{P,\kappa}\setminus X_{P}(2\delta(\kappa))$:

$\Bigl{|}\nabla_{\infty}^{j}(s(h_{\infty},h_{t})-\mathop{\rm
id}\nolimits)\Bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j,\gamma}e^{-\gamma\rho_{P,\kappa}t}.$

Proof There exists $C_{j,\gamma}^{\prime}>0$ such that the following holds on
$X^{(2)}_{P,\kappa}\setminus X_{P}(2\delta(\kappa))$:

$\Bigl{|}\nabla_{\infty}^{j}(s(h_{\infty},\widetilde{h}_{\kappa,t})-\mathop{\rm
id}\nolimits)\Bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j,\gamma}^{\prime}e^{-\gamma\rho_{P,\kappa}t}.$

Together with the estimate in Lemma 3.21, we obtain the claim of Lemma 3.22.

Let $\rho_{\kappa}:X\to{\mathbb{R}}_{\geq 0}$ be defined by
$\rho_{\kappa}(Q)=\min d_{\phi}(Q,X_{P}(2\delta(\kappa)))$.

###### Corollary 3.23

For any $j>0$ and $0<\gamma<2\sqrt{2}$, there exists $C_{j}>0$ such that the
following holds on $X\setminus\bigcup_{P}X_{P}(2\delta(\kappa))$:

$\Bigl{|}\nabla_{\infty}^{j}(s(h_{\infty},h_{t}))\Bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j,\gamma}e^{-\gamma\rho_{\kappa}t}.$

####  3.3.8 Refinement

We obtain the following.

###### Lemma 3.24

There exists $C>0$ such that the following holds on
$X\setminus\bigcup_{P}X^{(2)}_{P,\kappa}$:

$|s(h_{\infty},h_{t})^{\bot}|\leq Ce^{-4(\kappa+5\delta(\kappa))t}.$

Proof It follows from Proposition 2.2 and Corollary 3.23.

###### Corollary 3.25

There exists $C>0$ such that the following holds on
$X\setminus\bigcup_{P}X^{(2)}_{P,\kappa}$:

$|s_{\kappa,t}^{\bot}|\leq Ce^{-4(\kappa+5\delta(\kappa))t}.$ (44)

There exists $C^{\prime}>0$ such that

$\Bigl{|}\mathop{\rm
Tr}\nolimits\bigl{(}s_{\kappa,t}F(\widetilde{h}_{\kappa,t})\bigr{)}\Bigr{|}\leq
C^{\prime}e^{-8(\kappa+4\delta(\kappa))t}.$ (45)

There exists $C^{\prime\prime}>0$ such that

$\|\overline{\partial}_{E}(s_{\kappa,t})\|_{L^{2},\widetilde{h}_{t},g_{X}}\leq
Ce^{-4(\kappa+4\delta(\kappa))t},\quad\|[\theta,s_{\kappa,t}]\|_{L^{2},\widetilde{h}_{t},g_{X}}\leq
Ce^{-4(\kappa+4\delta(\kappa))t}.$ (46)

Proof We obtain (44) from Lemma 3.24. We obtain the others by repeating the
argument in §3.3.6.

We refine Lemma 3.20.

###### Lemma 3.26

For any $j>0$, there exists $C_{j}>0$ such that the following holds on
$X\setminus\bigcup X^{(2)}_{P,\kappa}$:

$|\nabla_{\infty}^{j}(s_{\kappa,t})|_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+4\delta(\kappa))t}.$ (47)

For any $j\geq 0$, there exists $C^{\prime}_{j}\geq 0$ such that the following
holds on $X\setminus\bigcup X^{(2)}_{P,\kappa}$:

$\bigl{|}\nabla_{\infty}^{j}([\theta,s_{\kappa,t}])\bigr{|}_{h_{\infty},g_{\phi}}+\bigl{|}\nabla_{\infty}^{j}([\theta^{\dagger}_{h_{t}},s_{\kappa,t}])\bigr{|}_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+4\delta(\kappa))t}.$ (48)

Proof By using Corollary 3.25, we obtain that there exists $C>0$ such that

$\Bigl{|}R(h_{t})\Bigr{|}_{h_{\infty},g_{\phi}}\leq
Ct^{2}e^{-4(\kappa+5\delta(\kappa))t}$

on $X\setminus\bigcup X^{(2)}_{P,\kappa}$. Then, we obtain the estimate (47)
by using Lemma 2.26 and Lemma 2.27. We also obtain (48) from Corollary 2.28.

####  3.3.9 $C^{0}$-estimate

###### Proposition 3.27

There exists $C>0$ such that the following holds on $X$:

$|s_{\kappa,t}-1|_{\widetilde{h}_{\kappa,t}}\leq
Ce^{-4(\kappa+4\delta(\kappa))t}.$ (49)

Proof We already have the following estimates on $X\setminus\bigcup
X_{P,\kappa}^{(2)}$:

$|s_{\kappa,t}^{\bot}|_{h_{\infty}}\leq
Ce^{-4(\kappa+5\delta(\kappa))t},\quad\quad|\nabla_{\infty}(s_{\kappa,t})|_{h_{\infty}}\leq
Ce^{-4(\kappa+4\delta(\kappa))t}.$

Let $Q\in X\setminus\bigcup X_{P,\kappa}^{(2)}$. Let $v_{i}$ $(i=1,2)$ be a
holomorphic frame of $E_{Q,i}$ around $Q$ such that
$h_{\infty}(v_{i},v_{i})=1$. We have

$\bigl{|}h_{t}(v_{1},v_{2})\bigr{|}\leq Ce^{-4(\kappa+4\delta(\kappa))t}.$

We also have

$\Bigl{|}h_{t}(v_{1},v_{1})\cdot h_{t}(v_{2},v_{2})-1\Bigr{|}\leq
Ce^{-4(\kappa+4\delta(\kappa))t},\quad\quad|dh_{t}(v_{i},v_{i})|\leq
Ce^{-4(\kappa+4\delta(\kappa))t}.$

Note that the inclusion $X\setminus\bigcup X_{P,\kappa}^{(2)}\longrightarrow
X$ induces the surjection of the fundamental groups. Hence, there exists a
path $\gamma:[0,1]\to X\setminus\bigcup X_{P,\kappa}^{(2)}$ such that (i)
$\gamma(0)=\gamma(1)=Q$, (ii) the two points of $\pi^{-1}(Q)$ are exchanged by
the monodromy along $\gamma$. By considering the above estimates along the
path, we obtain

$\Bigl{|}h_{t}(v_{1},v_{1})-h_{t}(v_{2},v_{2})\Bigr{|}\leq
Ce^{-4(\kappa+4\delta(\kappa))t}.$

Then, we obtain

$\Bigl{|}h_{t}(v_{i},v_{i})-1\Bigr{|}\leq Ce^{-4(\kappa+4\delta(\kappa))t}.$

Hence, (49) holds for some $C>0$ on $X\setminus\bigcup_{P}X_{P,\kappa}^{(2)}$.

On $X_{P,\kappa}^{(2)}$, the function $\mathop{\rm
Tr}\nolimits(s_{\kappa,t}-\mathop{\rm id}\nolimits)\geq 0$ is subharmonic.
Hence, (49) also holds on $\bigcup_{P}X_{P,\kappa}^{(2)}$.

####  3.3.10 Estimates for higher derivatives

We consider the connection $\nabla_{\widetilde{h}_{\kappa,t},g_{X}}$ of
$\mathop{\rm End}\nolimits(E)\otimes T^{\ast}X$ induced by the Chern
connection of $\widetilde{h}_{\kappa,t}$ and the Levi-Civita connection of
$g_{X}$.

###### Lemma 3.28

For any $j>0$, there exists $C_{j}>0$ such that

$\Bigl{|}\nabla_{\widetilde{h}_{\kappa,t},g_{X}}^{j}\bigl{(}s_{\kappa,t}\bigr{)}\Bigr{|}_{\widetilde{h}_{\kappa,t},g_{X}}\leq
C_{j}t^{j}e^{-4(\kappa+3\delta(\kappa))t}.$ (50)

For any $j\geq 0$, there exists $C^{\prime}_{j}>0$ such that

$\Bigl{|}\nabla_{\widetilde{h}_{\kappa,t},g_{X}}^{j}\bigl{(}[\theta,s_{\kappa,t}]\bigr{)}\Bigr{|}_{\widetilde{h}_{\kappa,t},g_{X}}+\Bigl{|}\nabla_{\widetilde{h}_{\kappa,t},g_{X}}^{j}\bigl{(}[\theta^{\dagger}_{h},s_{\kappa,t}]\bigr{)}\Bigr{|}_{\widetilde{h}_{\kappa,t},g_{X}}\leq
C_{j}t^{j}e^{-4(\kappa+3\delta(\kappa))t}.$ (51)

Proof By Proposition 3.27, there exists $C>0$ such that

$\left|t^{2}[\theta,\theta^{\dagger}_{\widetilde{h}_{\kappa,t}}]-t^{2}[\theta,\theta^{\dagger}_{h_{t}}]\right|_{\widetilde{h}_{\kappa,t},g_{X}}\leq
Ct^{2}e^{-4(\kappa+4\delta(\kappa)t)}.$

We obtain

$\Bigl{|}\overline{\partial}_{E}\bigl{(}s_{\kappa,t}^{-1}\partial_{E,\widetilde{h}_{\kappa,t}}s_{\kappa,t}\bigr{)}\Bigr{|}_{\widetilde{h}_{\kappa,t},g_{X}}\leq
Ct^{2}e^{-4(\kappa+4\delta(\kappa))t}.$

Then, we obtain (50) from Lemma 2.26, Lemma 2.27 and Corollary 3.25. We also
obtain (51) by using Corollary 2.28.

We obtain Theorem 3.14 from Proposition 3.27 and Lemma 3.28.

###  3.4 An auxiliary Hermitian product

####  3.4.1 Induced $1$-forms on the spectral curve

We obtain the following $\mathop{\rm End}\nolimits(E)$-valued $1$-form on
$X\setminus D_{\theta}$:

$\Psi_{t}=(\partial_{E,h_{\infty}}+t\theta^{\dagger}_{h_{\infty}})-(\partial_{E,h_{t}}+t\theta^{\dagger}_{h_{t}}).$

It satisfies $(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\Psi_{t}=0$. Hence, there exist holomorphic $1$-forms
$\eta_{t}$ on $\Sigma_{\theta|X\setminus D_{\theta}}$ such that
$F_{\eta_{t}}=(\Psi_{t}^{\circ})^{1,0}$.

###### Lemma 3.29

$\eta_{t}$ are holomorphic $1$-forms on $\Sigma_{\theta}$.

Proof Let $P\in D_{\theta}$. We consider the following $\mathop{\rm
End}\nolimits(E)$-valued $1$-form on $X(\kappa_{0})\setminus\\{P\\}$:

$\Psi^{\prime}_{P,t}=(\partial_{E,h_{\infty}}+t\theta^{\dagger}_{h_{\infty}})-(\partial_{E,h_{P,t}}+t\theta^{\dagger}_{h_{P,t}}).$

It is a section of

$\mathop{\rm End}\nolimits(E_{|X_{P}(\kappa_{0})})^{\mathop{\rm
asym}\nolimits}\otimes\Omega^{1,0}\oplus\mathop{\rm
End}\nolimits(E_{|X_{P}}(\kappa_{0}))^{\mathop{\rm
sym}\nolimits}\otimes\Omega^{0,1}.$

Hence, we obtain $((\Psi^{\prime}_{P,t})^{\circ})^{1,0}=0$. We consider the
following $\mathop{\rm End}\nolimits(E)$-valued $1$-form on
$X_{P}(\kappa_{0})$:

$\Psi_{P,t}=(\partial_{E,h_{P,t}}+t\theta^{\dagger}_{h_{P,t}})-(\partial_{E,h_{t}}+t\theta^{\dagger}_{h_{t}}).$

It satisfies $(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\Psi_{P,t}=0$. Because $\Psi_{P,t}$ is given on
$X_{P}(\kappa_{0})$, there exist holomorphic $1$-forms $\eta_{P,t}$ on
$\Sigma_{\theta|X_{P}(\kappa)}$ such that
$F_{\eta_{P,t}}=((\Psi_{P,t})^{1,0})^{\circ}$ on
$X_{P}(\kappa_{0})\setminus\\{P\\}$. (See [18].) Because $\eta_{t}=\eta_{P,t}$
on $\Sigma_{\theta|X_{P}(\kappa_{0})\setminus\\{P\\}}$, we obtain the claim of
Lemma 3.29.

###### Lemma 3.30

$\eta_{t}=O(e^{-4\kappa t})$ for any $\kappa<\kappa_{0}$.

Proof Let $W_{\kappa}=X\setminus\bigcup_{P}X^{(2)}_{P,\kappa}$. Because

$\Psi_{t}=s(h_{\infty},h_{t})^{-1}\partial_{h_{\infty}}s(h_{\infty},h_{t})+s(h_{\infty},h_{t})^{-1}\bigl{[}\theta^{\dagger}_{h_{\infty}},s(h_{\infty},h_{t})\bigr{]},$

we obtain $\eta_{t}=O(e^{-4\kappa t})$ on $\Sigma_{\theta|W_{\kappa}}$.
Because $\eta_{t}$ are holomorphic, it implies the estimate on
$\Sigma_{\theta}$.

###### Lemma 3.31

If $(E,\theta)$ has a globally defined non-degenerate symmetric pairing $C$,
then $\eta_{t}=0$.

Proof Because $\Psi_{t}$ is a section of $\mathop{\rm
End}\nolimits(E)^{\mathop{\rm
asym}\nolimits}\otimes\Omega^{1,0}\oplus\mathop{\rm
End}\nolimits(E)^{\mathop{\rm sym}\nolimits}\otimes\Omega^{0,1}$, we obtain
$(\Psi_{t}^{\circ})^{1,0}=0$. It implies the claim of the lemma.

####  3.4.2 Canonical choice of a root of $\phi$ and a root of the coordinate
$z_{P}$

We obtain the quadratic differential $\pi^{\ast}\phi$ on $\Sigma_{\theta}$.
Let $a_{T^{\ast}X}$ denote the tautological $1$-form on $T^{\ast}X$. Let
$\phi^{1/2}$ denote the pull back of $a_{T^{\ast}X}$ by the inclusion
$\Sigma_{\theta}\to T^{\ast}X$.

###### Lemma 3.32

We obtain $(\phi^{1/2})^{2}=\pi^{\ast}(\phi)$.

For each $P\in D_{\theta}$, let $z_{P}$ be a holomorphic coordinate on
$X_{P}(\kappa_{0})$ such that
$\phi_{|X_{P}(\kappa_{0})}=(\frac{3}{2})^{2}z_{P}(dz_{P})^{2}$. It induces a
holomorphic coordinate system $(z_{P},\xi_{P})$ on $T^{\ast}X_{P}$ such that
$\xi_{P}\,dz_{P}$ is the tautological $1$-form. Because

$\Sigma_{\theta|X_{P}(\kappa_{0})}=\left\\{(z_{P},\xi_{P})\,\left|\,\xi_{P}^{2}-\left(\frac{3}{2}\right)^{2}z_{P}=0\right.\right\\},$

$\xi_{P}$ is a holomorphic local coordinate on
$\Sigma_{\theta|X_{P}(\kappa_{0})}$. We have

$\phi^{1/2}=\frac{2^{3}}{3^{2}}\xi_{P}^{2}d\xi_{P}.$

####  3.4.3 Induced pairings on the cohomology group of $\Sigma_{\theta}$

For each $P\in D_{\theta}$, let $\widetilde{P}\in\Sigma_{\theta}$ denote the
pre-image of $P$. We set $\widetilde{D}_{\theta}=\\{\widetilde{P}\,|\,P\in
D_{\theta}\\}$, which can be regarded as the divisor on $\Sigma_{\theta}$. For
each $\widetilde{P}\in\widetilde{D}_{\theta}$, there exists the residue
morphism

$\mathop{\rm
Res}\nolimits^{(2)}_{\widetilde{P}}:H^{0}\bigl{(}\Sigma_{\theta},K_{\Sigma_{\theta}}^{2}(2\widetilde{D})\bigr{)}\longrightarrow{\mathbb{C}}.$

Let $\nu_{1},\nu_{2}\in H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$. We obtain

$\nu_{1}\cdot\nu_{2}\cdot\eta_{t}\cdot(\phi^{1/2})^{-1}\in
H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}}^{2}(2D)).$

We define the pairing $\langle\cdot,\cdot\rangle^{\mathop{\rm
aux}\nolimits}_{P,h_{t}}$ on $H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$ as
follows:

$\langle\nu_{1},\nu_{2}\rangle^{\mathop{\rm
aux}\nolimits}_{P,h_{t}}=-4\pi\mathop{\rm
Res}\nolimits^{(2)}_{\widetilde{P}}\Bigl{(}\nu_{1}\cdot\nu_{2}\cdot\eta_{t}\cdot(\phi^{1/2})^{-1}\Bigr{)}.$

We set

$\langle\cdot,\cdot\rangle_{h_{t}}^{\mathop{\rm aux}\nolimits}=\sum_{P\in
D_{\theta}}\langle\cdot,\cdot\rangle^{\mathop{\rm aux}\nolimits}_{P,h_{t}}.$

####  3.4.4 Induced Hermitian pairings on
$T_{(E,t\theta)}\mathcal{M}_{H}^{\prime}$

We identify $T_{(E,t\theta)}\mathcal{M}_{H}^{\prime}=H^{1}(X,\mathop{\rm
End}\nolimits(E,t\theta))$. As recalled in §1.1.2, we have the decomposition

$T_{(E,t\theta)}\mathcal{M}_{H}^{\prime}=(T_{(E,t\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
ver}\nolimits}\oplus(T_{(E,t\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
hor}\nolimits},$

and there exist the natural isomorphisms

$\iota^{\mathop{\rm
ver}\nolimits}_{t}:H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})\simeq(T_{(E,t\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
ver}\nolimits},\quad\iota^{\mathop{\rm
hor}\nolimits}_{t}:H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})\simeq(T_{(E,t\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
hor}\nolimits}.$

(See [18] for more details.) We define $g^{\mathop{\rm
aux}\nolimits}_{|(E,t\theta)}$ by using
$\langle\cdot,\cdot\rangle^{\mathop{\rm aux}\nolimits}_{h_{t}}$ in §3.4.3 as
follows.

  * •

$g^{\mathop{\rm aux}\nolimits}_{|(E,t\theta)}(v_{1},v_{2})=0$ if
$v_{i}\in(T_{(E,t\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
ver}\nolimits}$ $(i=1,2)$ or
$v_{i}\in(T_{(E,t\theta)}\mathcal{M}_{H}^{\prime})^{\mathop{\rm
hor}\nolimits}$ $(i=1,2)$.

  * •

Let $\nu\in H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$ and $\tau\in
H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})=H^{0}(\Sigma_{\theta}^{\dagger},K_{\Sigma_{\theta}^{\dagger}})$.
Then,

$g^{\mathop{\rm aux}\nolimits}_{|(E,t\theta)}(\iota^{\mathop{\rm
hor}\nolimits}_{t}(\nu),\iota^{\mathop{\rm
ver}\nolimits}_{t}(\tau))=\langle\nu,\overline{\tau}\rangle^{\mathop{\rm
aux}\nolimits}_{h_{t}}$

We obtain the following lemma from Lemma 3.30.

###### Lemma 3.33

$g^{\mathop{\rm aux}\nolimits}_{|(E,t\theta)}(\nu,\tau)=O(e^{-4\kappa
t})\|\nu\|_{L^{2}}\cdot\|\tau\|_{L^{2}}$ for any $0<\kappa<\kappa_{0}$.

###### Corollary 3.34

$g^{\mathop{\rm aux}\nolimits}_{|(E,t\theta)}=O(e^{-4\kappa t})$ with respect
to $g_{\mathop{\rm sf}\nolimits|(E,t\theta)}$ for any $0<\kappa<\kappa_{0}$.
As a result, there exists $t_{0}>0$ such that $g_{\mathop{\rm
sf}\nolimits|(E,t\theta)}+g^{\mathop{\rm aux}\nolimits}_{|(E,t\theta)}$ are
positive definite Hermitian products on
$T_{(E,t\theta)}\mathcal{M}_{H}^{\prime}$ for any $t\geq t_{0}$.

###  3.5 Comparison of the Hitchin metric and the semi-flat metric

####  3.5.1 Main theorem

Let $t_{0}$ be as in Corollary 3.34. For any $t\geq t_{0}$, let $\sigma_{t}$
be the automorphism of $T_{(E,t\theta)}\mathcal{M}_{H}$ determined by
$g_{H|(E,t\theta)}=\bigl{(}g_{\mathop{\rm
sf}\nolimits|(E,t\theta)}+g^{\mathop{\rm
aux}\nolimits}_{|(E,t\theta)}\bigr{)}\cdot\sigma_{t}$.

###### Theorem 3.35

For any $0<\kappa<\kappa_{0}$, there exists $C(\kappa)>0$ such that

$|\sigma_{t}-\mathop{\rm id}\nolimits|_{g_{\mathop{\rm sf}\nolimits}}\leq
C(\kappa)e^{-8\kappa t}.$

Let $\nu\in H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$. It determines the
horizontal class

$\iota_{t}^{\mathop{\rm hor}\nolimits}(\nu)\in H^{1}(X,\mathop{\rm
Def}\nolimits(E,t\theta))^{\mathop{\rm hor}\nolimits}\subset
H^{1}(X,\mathop{\rm Def}\nolimits(E,t\theta)).$

(See [18].) Let ${\tt H}_{t}(\nu)$ denote the harmonic representative of
$\iota_{t}^{\mathop{\rm hor}\nolimits}(\nu)$ with respect to $h_{t}$. We shall
prove the following proposition in §3.5.5.

###### Proposition 3.36

For any $0<\kappa<\kappa_{0}$,

$({\tt H}_{t}(\nu),{\tt
H}_{t}(\nu))_{L^{2},h_{t}}=\|\nu\|_{L^{2}}^{2}+O\bigl{(}e^{-8\kappa
t}\|\nu\|_{L^{2}}^{2}\bigr{)}.$

Let $\tau\in
H^{0}(\Sigma_{\theta}^{\dagger},K_{\Sigma_{\theta}^{\dagger}})\simeq
H^{1}(\Sigma_{\theta},\mathcal{O}_{\Sigma_{\theta}})$. We also regard $\tau$
as a harmonic $(0,1)$-form on $\Sigma_{\theta}$. We obtain $\iota^{\mathop{\rm
ver}\nolimits}_{t}(\tau)\in H^{1}(X,(\mathop{\rm End}\nolimits(E),t\theta))$.
There exists a harmonic representative ${\tt V}_{t}(\tau)$ of $(\mathop{\rm
End}\nolimits(E),t\theta,h_{t})$. We shall prove the following proposition in
§3.5.7.

###### Proposition 3.37

For any $0<\kappa<\kappa_{0}$,

$({\tt V}_{t}(\tau),{\tt
V}_{t}(\tau))_{L^{2},h_{t}}=(\tau,\tau)_{L^{2}}+O\bigl{(}e^{-8\kappa
t}\|\tau\|_{L^{2}}^{2}\bigr{)}.$

We shall prove the following proposition in §3.5.8.

###### Proposition 3.38

$({\tt H}_{t}(\nu),{\tt
V}_{t}(\tau))_{L^{2},h_{t}}=\langle\nu,\overline{\tau}\rangle^{\mathop{\rm
aux}\nolimits}_{h_{t}}+O\bigl{(}e^{-8\kappa
t}\cdot\|\nu\|_{L^{2}}\cdot\|\tau\|_{L^{2}}\bigr{)}$.

We obtain Theorem 3.35 from Proposition 3.36, Proposition 3.37 and Proposition
3.38.

####  3.5.2 Preliminary

Let $P\in D_{\theta}$. Let $s_{P,t}$ be determined by $h_{P,t}=h_{t}\cdot
s_{P,t}$ on $X_{P}(\kappa_{0})$. (Note that we use a different notation from
§3.3.3.) We set

$\Psi_{P,t}:=\bigl{(}\partial_{h_{P,t}}+t\theta^{\dagger}_{h_{P,t}}\bigr{)}-\bigl{(}\partial_{h_{t}}+t\theta^{\dagger}_{h_{t}}\bigr{)}=s_{P,t}^{-1}\partial_{h_{t}}s_{P,t}+s_{P,t}^{-1}[t\theta^{\dagger}_{h_{t}},s_{P,t}].$

Because
$[\overline{\partial}_{E}+t\theta,\partial_{h_{P,t}}+t\theta^{\dagger}_{h_{P,t}}]=0$
and
$[\overline{\partial}_{E}+t\theta,\partial_{h_{t}}+t\theta^{\dagger}_{h_{t}}]=0$,
we obtain $(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\Psi_{P,t}=0$.

Let $f_{P}$ be the endomorphism determined by $\theta=f_{P}\,dz_{P}$. Let
$M_{P,i,t}$ be the endomorphisms of $E_{|X_{P}(\kappa_{0})}$ defined by

$\Psi_{P,t}=M_{P,1,t}\,dz_{P}+M_{P,2,t}\,d\overline{z}_{P}.$

Because $(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\Psi_{P,t}=0$, we obtain

$\overline{\partial}M_{P,1,t}-[tf_{P},M_{P,2,t}\,d\overline{z}_{P}]=0.$ (52)

###### Lemma 3.39

We have $\overline{\partial}M_{P,1,t}^{\circ}=0$. There exist $C_{j}>0$ such
that the following holds on $X^{(1)}_{P,\kappa}\setminus X^{(2)}_{P,\kappa}$:

$|\nabla_{\infty}^{j}M_{P,1,t}|_{h_{\infty},g_{\phi}}\leq
C_{j}e^{-4(\kappa+\delta(\kappa))t}.$

Proof The first claim follows from (52). The second follows from Theorem 3.14.

Let $\eta_{t}$ be the $1$-forms on $\Sigma_{\theta}$ as in §3.4.1. Let
$(z_{P},\xi_{P})$ be the coordinate as in §3.4.2. Let $\upsilon_{P,t}$ be the
holomorphic function on $\Sigma_{\theta|X_{P}(\kappa_{0})\setminus\\{P\\}}$
determined by

$\eta_{t}=\upsilon_{P,t}\cdot\pi^{\ast}(dz_{P}).$

We note that $\upsilon_{P,t}$ has at most pole of order $1$ at $\pi^{-1}(P)$.
By the construction, we have the following lemma.

###### Lemma 3.40

$M_{P,1,t}^{\circ}=F_{\upsilon_{P,t}}$.

####  3.5.3 Some $1$-forms around $P\in D_{\theta}$

Let $\nu\in H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$. We obtain the
holomorphic section $F_{\nu}$ of $\mathop{\rm End}\nolimits(E)\otimes K_{X}$
on $X\setminus D_{\theta}$.

Let $P\in D_{\theta}$. There exists a holomorphic function $\alpha_{P}$ on
$\Sigma_{\theta|X_{P}(\kappa)}$ such that $d\alpha_{P}=\nu$ and that
$\alpha_{P}=0$ at $\pi^{-1}(P)$. We obtain the induced section
$F_{\alpha_{P}}$ of $\mathop{\rm End}\nolimits(E_{P})$ on $X_{P}(\kappa_{0})$.
Let $\alpha_{0}$ and $\alpha_{1}$ be the holomorphic functions on
$X_{P}(\kappa_{0})$ determined by

$\alpha_{P}(\xi_{P})=\alpha_{0}(z_{P})+\alpha_{1}(z_{P})\xi_{P},$

where $(z_{P},\xi_{P})$ be the holomorphic coordinate system as in §3.4.2.
Then, the following holds:

$F_{\alpha_{P}}=\alpha_{0}\mathop{\rm id}\nolimits+\alpha_{1}f_{P}.$

###### Lemma 3.41

There exists $C>0$ independently from $\nu\in
H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$ and $t\geq 1$ such that the
following holds:

$\|F_{\alpha_{P}}\|^{2}_{h_{P,t},X_{P}(\kappa_{0})}\leq C\|\nu\|_{L^{2}}^{2}.$

Proof We have $|\alpha_{P}|\leq C_{0}\|\nu\|_{L^{2}}$. By Theorem 3.14, there
exists $C_{1}>0$ such that $|F_{\alpha_{P}}|_{h_{P,t}}^{2}\leq
C_{1}\|\nu\|_{L^{2}}^{2}$ on $\partial X_{P}^{(2)}$ for any $t\geq 1$. Because
$(\overline{\partial}_{E}+t\mathop{\rm ad}\nolimits\theta)F_{\alpha_{P}}=0$,
we obtain
$-\partial_{z}\partial_{\overline{z}}\bigl{|}F_{\alpha_{P}}\bigr{|}_{h_{t}}^{2}\leq
0$. Then, we obtain the claim of the lemma.

We obtain the following harmonic $1$-forms of $(\mathop{\rm
End}\nolimits(E_{P}),t\theta_{P},h_{P,t})$ on $X_{P}(\kappa_{0})$:

${\tt H}_{P,t}(\nu):=\bigl{(}\partial_{h_{P,t}}+\mathop{\rm
ad}\nolimits(t\theta^{\dagger}_{h_{P,t}})\bigr{)}F_{\alpha_{P}}.$

According to [18, Lemma 4.14], there exists $C>0$ such that the following
holds on $X^{(1)}_{P,\kappa}$ for any $t\geq 1$:

$|{\tt H}_{P,t}(\nu)|_{h_{P,t},g_{X}}\leq C(1+t)\|\nu\|_{L^{2}}.$

We remark the following.

###### Lemma 3.42

The $(1,0)$-part ${\tt H}_{P,t}(\nu)^{1,0}$ is symmetric, and the $(0,1)$-part
${\tt H}_{P,t}(\nu)^{0,1}$ is anti-symmetric with respect to $C_{P}$. We have
${\tt H}_{P,t}(\nu)^{\circ}=F_{\nu}$ on $X_{P}(\kappa_{0})$.

On $X_{P}(\kappa_{0})\setminus\\{P\\}$, there exists a $C^{\infty}$-section
$\rho_{P,t}(\nu)$ of $\mathop{\rm End}\nolimits(E)$, which is anti-symmetric
with respect to $C_{P}$, such that

${\tt H}_{P,t}(\nu)-F_{\nu}=(\overline{\partial}_{E}+t\mathop{\rm
ad}\nolimits\theta)\rho_{P,t}(\nu).$

On $X_{P,\kappa}^{(1)}\setminus X_{P,\kappa}^{(2)}$, we have

$\|\rho_{P,t}(\nu)\|_{L_{\ell}^{p},h_{\infty},g_{\phi}}=O\bigl{(}e^{-4(\kappa+7\delta(\kappa))t}\|\nu\|_{L^{2}}\bigr{)}.$
(53)

We also note that $\rho_{P,t}(\nu)^{\circ}=0$.

We also obtain the following harmonic $1$-forms of $(\mathop{\rm
End}\nolimits(E),t\mathop{\rm ad}\nolimits\theta,h_{t})$ on
$X_{P}(\kappa_{0})$:

$\widetilde{{\tt H}}_{P,t}(\nu):=(\partial_{E,h_{t}}+\mathop{\rm ad}\nolimits
t\theta^{\dagger}_{h_{t}})F_{\alpha_{P}}.$

We set ${\tt I}_{P,t}(\nu)={\tt H}_{P,t}(\nu)-\widetilde{{\tt H}}_{P,t}(\nu)$.

###### Lemma 3.43

We have

${\tt I}_{P,t}(\nu)=\bigl{(}\overline{\partial}+t\mathop{\rm
ad}\nolimits\theta\bigr{)}(-t^{-1}\alpha_{1}M_{P,1,t}).$ (54)

Proof We have

${\tt
I}_{P,t}(\nu)=[\Psi_{t},F_{\alpha_{P}}]=[M_{P,1,t}dz_{P},\alpha_{1}f_{P}]+[M_{P,2,t}\,d\overline{z}_{P},\alpha_{1}f_{P}]=-[tf_{P}\,dz_{P},t^{-1}\alpha_{1}M_{P,1,t}]-[f_{P},\alpha_{1}M_{P,2,t}d\overline{z}_{P}].$

We also have

$\overline{\partial}(t^{-1}\alpha_{1}M_{P,1,t})=t^{-1}\alpha_{1}\overline{\partial}M_{P,1,t}=t^{-1}\alpha_{1}[tf_{P},M_{P,2,t}d\overline{z}_{P}]=[f_{P},\alpha_{1}M_{P,2,t}d\overline{z}_{P}].$

Then, we obtain (54).

###### Lemma 3.44

We have ${\tt I}_{P,t}(\nu)^{\circ}=0$. There exists $C>0$ such that $|{\tt
I}_{P,t}(\nu)|_{h_{\infty},g_{\phi}}\leq Ce^{-4\kappa t}\|\nu\|_{L^{2}}$ on
$X_{P,\kappa}^{(1)}\setminus X_{P,\kappa}^{(2)}$.

Proof The first claim is clear by the construction. The second follows from
Lemma 3.39.

####  3.5.4 Approximations of harmonic $1$-forms in the horizontal direction

Let $\nu\in H^{0}(\Sigma_{\theta},K_{\Sigma_{\theta}})$. We recall a
construction in [18]. Let $\chi_{P,\kappa}:X\to[0,1]$ be a
$C^{\infty}$-function such that $\chi_{P,\kappa}=1$ on $X^{(2)}_{P,\kappa}$