# Higgs bundles in the Hitchin section over non-compact hyperbolic surfaces

Qiongling Li Chern Institute of Mathematics and LPMC, Nankai University,
Tianjin 300071, China<EMAIL_ADDRESS>Takuro Mochizuki Research
Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8512, Japan,
<EMAIL_ADDRESS>
###### Abstract

Let $X$ be an arbitrary non-compact hyperbolic Riemann surface, that is, not
$\mathbb{C}$ or $\mathbb{C}^{*}$. Given a tuple of holomorphic differentials
$\boldsymbol{q}=(q_{2},\cdots,q_{n})$ on $X$, one can define a Higgs bundle
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ in the Hitchin section. We show
there exists a harmonic metric $h$ on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ satisfying (i) $h$ weakly
dominates $h_{X}$; (ii) $h$ is compatible with the real structure. Here
$h_{X}$ is the Hermitian metric on $\mathbb{K}_{X,n}$ induced by the conformal
complete hyperbolic metric $g_{X}$ on $X.$ Moreover, when
$q_{i}(i=2,\cdots,n)$ are bounded with respect to $g_{X}$, we show such a
harmonic metric on $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ satisfying
(i)(ii) uniquely exists. With similar techniques, we show the existence of
harmonic metrics for $SO(n,n+1)$-Higgs bundles in Collier’s component and
$Sp(4,\mathbb{R})$-Higgs bundles in Gothen’s component over $X$, under some
mild assumptions.

MSC: 53C07, 58E15, 14D21, 81T13.
Keywords: higgs bundles, harmonic metric, Hitchin section

###### Contents

  1. 1 Introduction
    1. 1.1 Harmonic metrics for Higgs bundles in the Hitchin section
    2. 1.2 Harmonic metrics for Higgs bundles which admit a full filtration
    3. 1.3 $SO(n,n+1)$-Higgs bundles and $Sp(4,\mathbb{R})$-Higgs bundles
    4. 1.4 Further questions
  2. 2 Preliminaries on existence of harmonic metrics
    1. 2.1 Dirichlet problem
    2. 2.2 Convergence
    3. 2.3 Appendix
  3. 3 Domination property and the existence of harmonic metrics
    1. 3.1 Full flags and Hermitian metrics
    2. 3.2 Set-up
      1. 3.2.1 The graded case
      2. 3.2.2 Symmetric pairings
      3. 3.2.3 Symmetric pairing and graded bundles
    3. 3.3 Domination property and the Dirichlet problem
    4. 3.4 Domination property and the existence of harmonic metrics
      1. 3.4.1 Preliminary from linear algebra
      2. 3.4.2 Notation
      3. 3.4.3 Local estimate in the nowhere vanishing case
      4. 3.4.4 Local estimate in the general case
      5. 3.4.5 Proof of Theorem 3.10 and Theorem 3.12
  4. 4 Uniqueness in a bounded case
    1. 4.1 Statement
      1. 4.1.1 A characterization of the mutual boundedness with $h_{X}$
    2. 4.2 Preliminary from Linear algebra
      1. 4.2.1 Cyclic vectors
      2. 4.2.2 Real structure and self-adjoint endomorphisms
    3. 4.3 An estimate
    4. 4.4 Proof of Theorem 4.1
  5. 5 Hitchin section for $SL(n,\mathbb{R})$
    1. 5.1 Existence of weakly dominant harmonic metric in the general case
    2. 5.2 Uniqueness in the case of bounded differentials
      1. 5.2.1 Compact case
      2. 5.2.2 Pull back
  6. 6 Existence with bounded condition on the unit disk
    1. 6.1 Some function spaces
    2. 6.2 General existence with bounded condition
    3. 6.3 Existence for holomorphic chains
    4. 6.4 Relation to prescribed curvature equation
    5. 6.5 Holomorphic chains of type $(1,1,\cdots,1)$
  7. 7 $SO(n,n+1)$-Higgs bundles
    1. 7.1 Dirichlet problem
    2. 7.2 The generically regular semisimple case
      1. 7.2.1 Appendix: Preliminary from linear algebra
    3. 7.3 Collier section
      1. 7.3.1 Existence for the case $\mu\neq 0$
      2. 7.3.2 The generically regular semisimple case
  8. 8 $Sp(4,\mathbb{R})$-Higgs bundles
    1. 8.1 Dirichlet problem
    2. 8.2 The generically regular semisimple case
    3. 8.3 Gothen section
      1. 8.3.1 The generically regular semisimple case
      2. 8.3.2 The case $(\mu,\nu)=(0,0)$
      3. 8.3.3 The case $\mu\neq 0$
  9. A Discussions on Green functions
  10. B Various expressions of Higgs bundles in the Hitchin section

##  1 Introduction

Let $X$ be a Riemann surface and $(E,\overline{\partial}_{E},\theta)$ be a
Higgs bundle on $X$. Let $h$ be a Hermitian metric of $E$. We obtain the Chern
connection $\nabla_{h}=\overline{\partial}_{E}+\partial_{E,h}$ and the adjoint
$\theta^{*h}$ of $\theta$. The metric $h$ is called a harmonic metric of the
Higgs bundle $(E,\overline{\partial}_{E},\theta)$ if
$\nabla_{h}+\theta+\theta^{*h}$ is flat, i.e.,
$\nabla_{h}\circ\nabla_{h}+[\theta,\theta^{*h}]=0$. It was introduced by
Hitchin [Hit87], and it has been one of the most important and interesting
mathematical objects. A starting point is the study of the existence and the
classification of harmonic metrics. If $X$ is compact, the results of Hitchin
[Hit87] and Simpson [Sim88] show that a Higgs bundle is polystable of degree
$0$ if and only if it admits a harmonic metric. Together with the work of
Corlette [Cor88] and Donaldson [Don87], one obtains the non-Abelian Hodge
correspondence which says the moduli space of polystable
$SL(n,\mathbb{C})$-Higgs bundles is isomorphic to the representation variety
of the surface group $\pi_{1}(S)$ into $SL(n,\mathbb{C})$. The study of
harmonic metrics for Higgs bundles in the non-compact case was pioneered by
Simpson [Sim88, Sim92], and pursued by Biquard-Boalch [BB04] and the second
author [Moc21].

Let $\boldsymbol{q}=(q_{2},\cdots,q_{n})$, where $q_{j}$ is a holomorphic
$j$-differential on $X$. One can naturally construct a Higgs bundle
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ as follows. Let $K_{X}$ be the
canonical line bundle of $X$. The multiplication of $q_{j}$ induces the
following morphisms:

$K_{X}^{(n-2i+1)/2}\to K_{X}^{(n-2i+2(j-1)+1)/2}\otimes K_{X}\quad(j\leq i\leq
n).$

We also have the identity map for $i=1,\ldots,n-1$:

$K_{X}^{(n-2i+1)/2}\to K_{X}^{(n-2(i+1)+1)/2}\otimes K_{X}.$

They define a Higgs field $\theta(\boldsymbol{q})$ of
$\mathbb{K}_{X,n}=\oplus_{i=1}^{n}K_{X}^{(n+1-2i)/2}$. The natural pairings
$K_{X}^{(n-2i+1)/2}\otimes K_{X}^{-(n-2i+1)/2}\to\mathcal{O}_{X}$ induce a
non-degenerate symmetric bilinear form $C_{\mathbb{K},X,n}$ of
$\mathbb{K}_{X,n}$. There exists a basis of $SL(n,\mathbb{C})$-invariant
homogeneous polynomials $p_{i}$ of deg $i(i=2,\cdots,n)$ on $sl(n,\mathbb{C})$
such that $p_{i}(\theta(\boldsymbol{q}))=q_{i}$. The Hitchin fibration is from
the moduli space of polystable $SL(n,\mathbb{C})$-Higgs bundles to the vector
space $\oplus_{i=2}^{n}H^{0}(X,K_{X}^{i})$ given by

$[(E,\theta)]\longmapsto(p_{2}(\theta),\cdots,p_{n}(\theta)).$

Such Higgs bundles $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ were introduced
by Hitchin in [Hit92] for compact hyperbolic Riemann surfaces. They form a
section of the Hitchin fibration. For this reason, for arbitrary (not
necessarily compact) Riemann surfaces, we call
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ Higgs bundles in the Hitchin
section.

For the compact hyperbolic surface case, Hitchin in [Hit92] showed that
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ are always stable and the Hitchin
section corresponds to Hitchin component, a connected component in the
representation variety of $\pi_{1}(X)$ into $SL(n,\mathbb{R})$ which contains
embedded Fuchsian representations. In particular, when $n=2$, the Hitchin
section parametrize the Teichmüller space. Hitchin component has been the
central object in the field of higher Teichmüller theory. For the case when
$X=\bar{X}-D$ where $\bar{X}$ is a compact Riemann surface and $D$ is a finite
set of points, let $q_{j}(j=2,\cdots,n)$ be meromorphic differentilas on
$\bar{X}$ with possible poles at $D$ of pole order at most $j-1$. Using the
work of Simpson [Sim90] on parabolic Higgs bundles, Biswas-Arés-Gastesi-
Govindarajan in [BAGG97] showed $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$
can be prolonged to a stable parabolic Higgs bundle of degree $0$ over
$\bar{X}$ and thus admits a harmonic metric. Moreover, the Hitchin section
corresponds to a connected component of the representation variety of
$\pi_{1}(X)$ into $SL(n,\mathbb{R})$ such that the holonomy of loops around
punctures are of certain parabolic holonomy.

We want to study Higgs bundles in the Hitchin section in general case: tuples
of holomorphic differentials on an arbitrary non-compact Riemann surfaces,
e.g., unit disk, of infinite topology, etc. We focus on the following natural
question.

###### Question 1.1

Given a tuple of holomorphic differentials
$\boldsymbol{q}=(q_{2},\cdots,q_{n})$ on a non-compact Riemann surface $X$,
(1) does there exist a harmonic metric on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ compatible with
$C_{\mathbb{K},X,n}$?
(2) If so, can one find a notion of “best” harmonic metric such that it
uniquely exists?

###### Remark 1.2

1\. When $X$ is parabolic, that is, $\mathbb{C}$ or $\mathbb{C}^{*}$, there
exists no harmonic metric on $(\mathbb{K}_{X,n},\theta(\mathbf{0}))$. When $X$
is hyperbolic, each hyperbolic Kähler metric over $X$ induces a harmonic
metric on $(\mathbb{K}_{X,n},\theta(\mathbf{0}))$.
2\. Suppose $n=2$, $q_{2}\neq 0$ and $X$ is an arbitrary non-compact Riemann
surface. The work of [Wan92] [WA94] [Li18] together show there uniquely exists
a harmonic metric $h$ of unit determinant of
$(\mathbb{K}_{X,n},\theta(q_{2}))$ satisfying $(h|_{K_{X}^{-1/2}})^{2}$
defines a complete metric on $X$.
3\. Suppose $\boldsymbol{q}=(0,\cdots,0,q_{n})$, $q_{n}\neq 0$ and $X$ is an
arbitrary non-compact Riemann surface, the authors in [LM20a] introduce the
notion of a complete metric $h$ on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$, that is $h$ is diagonal of unit
determinant and satisfies
$(h|_{K_{X}^{(n+1-2i)/2}})^{-1}\otimes(h|_{K_{X}^{(n+1-2(i+1))/2}})(i=1,\cdots,n-1)$
defines a complete metric on $X$. And we show the existence and uniqueness for
a complete metric of $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$. Sagman [Sag]
later extends the existence of complete metric to the subcyclic case
$(0,\cdots,0,q_{n-1},0)$. For such two cases of lower ranks, there are rich
related geometry including hyperbolic affine spheres in $\mathbb{R}^{3}$
[Lab07, Lof01], maximal surfaces in $\mathbb{H}^{2,2}$ [CTT19], $J$-complex
curves in $\mathbb{H}^{4,2}$[Bar10, Nie22, CT23]. There are extensive studies
on the harmonic metrics for such two cases over non-compact surfaces, see e.g.
[BH13, BH14, DW15, Nie23, TW20, Eva22, GL14, GIL15, Moc, Moc14].
4\. In [LM22], the authors consider generically regular semisimple Higgs
bundles which admit a non-degenerate symmetric pairing $C$. Here the condition
“generically regular semisimple” means there exists a point such that the
Higgs field has $n$ distinct eigen $1$-forms. For such Higgs bundles, the
authors show the existence of a harmonic metric compatible with $C$. Note that
this result is not restricted to Higgs bundles in the Hitchin section.

A harmonic metric on $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ compatible
with $C_{\mathbb{K},X,n}$ gives rise to a representation
$\rho:\pi_{1}(X)\rightarrow SL(n,\mathbb{R})$ and a $\rho$-equivariant
harmonic map to the symmetric space $SL(n,\mathbb{R})/SO(n)$. Here
$SL(n,\mathbb{R})/SO(n)$ is equipped with the $SL(n,\mathbb{R})$-invariant
Riemannian metric induced by the Killing form $B(X,Y)=2n\cdot\mathop{\rm
tr}\nolimits(XY)$ on $sl(n,\mathbb{R})$. A closely related question is as
follows.

###### Question 1.3

Given a tuple of holomorphic differentials
$\boldsymbol{q}=(q_{2},\cdots,q_{n})$ on a non-compact Riemann surface $X$,
does there exist an equivariant harmonic map $f:\widetilde{X}\rightarrow
SL(n,\mathbb{R})/SO(n)$ such that $q_{i}=p_{i}(-\frac{1}{2}f^{-1}\partial f)$
for $i=2,\cdots,n$?

Here we used the explicit relation $-\frac{1}{2}f^{-1}\partial
f=\theta(\boldsymbol{q}),$ see e.g. [Li19b, Section 5.1]. If Question 1.1(1)
holds for some $\boldsymbol{q}_{0}$ on $X$, then Question 1.3 automatically
holds for $\boldsymbol{q}_{0}$ on $X$.

###### Remark 1.4

When $X=\mathbb{C}$ and $\boldsymbol{q}$ are polynomial differentials,
Question 1.3 reduces to the question of Tamburelli-Wolf in [TW20, Question A].

###  1.1 Harmonic metrics for Higgs bundles in the Hitchin section

Suppose $X$ is a non-compact hyperbolic Riemann surface, equivalently, it is
not $\mathbb{C}$ nor $\mathbb{C}^{*}$ . Let $g_{X}$ be the unique complete
hyperbolic Kähler metric on $X$. Let $h_{X}=\oplus_{k=1}^{n}a_{k}\cdot
g_{X}^{-\frac{n+1-2k}{2}}$, where $a_{k}$ are some fixed constants. Such
$a_{k}$’s are chosen so that $h_{X}$ is a harmonic metric for the Higgs bundle
$(\mathbb{K}_{X,n},\theta(\mathbf{0}))$.

Let $F_{k}=\oplus_{l\leq k}K_{X}^{\frac{n+1-2l}{2}}.$ Then $\\{0\subset
F_{1}\subset F_{2}\subset\cdots\subset F_{n}\\}$ forms an increasing
filtration of $\mathbb{K}_{X,n}$. We call a Hermitian metric $h$ on
$\mathbb{K}_{X,n}$ weakly dominates $h_{X}$ if
$\det(h|_{F_{k}})\leq\det(h_{X}|_{F_{k}})$ for $1\leq k\leq n-1.$ Our main
result in this paper is the following two theorems, as an answer to Question
1.1.

###### Theorem 1.5

(Theorem 5.1) On a non-compact hyperbolic surface $X$, there exists a harmonic
metric $h$ on $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ satisfying (i) $h$
weakly dominates $h_{X}$; (ii) $h$ is compatible with $C_{\mathbb{K},X,n}.$

As a result, the associated harmonic map
$f:(\widetilde{X},\widetilde{g_{X}})\rightarrow SL(n,\mathbb{R})/SO(n)$
satisfies the energy density $e(f)\geq\frac{n^{2}(n^{2}-1)}{6}.$ The equality
holds if $\boldsymbol{q}=0.$

###### Theorem 1.6

(Theorem 5.2) On a non-compact hyperbolic surface $X$, suppose
$q_{i}(i=2,\cdots,n)$ are bounded with respect to $g_{X}$. Then there uniquely
exists a harmonic metric $h$ on $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$
satisfying (i) $h$ weakly dominates $h_{X}$; (ii) $h$ is compatible with
$C_{\mathbb{K},X,n}.$

Moreover, $h$ is mutually bounded with $h_{X}.$

As an application of Theorem 1.6, we reprove the existence and uniqueness of a
harmonic metric on $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ over a compact
hyperbolic Riemann surface. Note that our proof here does not invoke the
Hitchin-Kobayashi correspondence by using the stability of Higgs bundle.

###### Theorem 1.7

(Theorem 5.4) Given a tuple of holomorphic differentials
$\boldsymbol{q}=(q_{2},\cdots,q_{n})$ on a compact hyperbolic surface $X$,
there uniquely exists a harmonic metric $h$ on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ satisfying $h$ is compatible with
$C_{\mathbb{K},X,n}.$

Moreover, $h$ weakly dominates $h_{X}$.

###  1.2 Harmonic metrics for Higgs bundles which admit a full filtration

In fact, we prove the existence of harmonic metrics for a more general family
of Higgs bundles than Higgs bundles in the Hitchin section. Consider a Higgs
bundle $(E,\theta)$ over a Riemann surface $X$ which admits a full holomorphic
filtration $\mathbf{F}=\\{0\subset F_{1}\subset F_{2}\subset\cdots\subset
F_{n}\\}$. We require that the induced map $\theta$ on each
$Gr_{k}(E):=F_{k}/F_{k-1}$ is not a zero map, $k=1,\cdots,n-1$. Let
$E_{0}=\oplus_{k=1}^{n}Gr_{k}(E)$ and $\theta_{0}$ are induced by $\theta$ on
the graded bundles $Gr_{k}(E)$. Then $(E_{0},\theta_{0})$ is a holomorphic
chain of type $(1,1,\cdots,1).$ Take $F_{k}^{0}=\oplus_{l\leq k}Gr_{l}(E)$.
There is a canonical way identifying $\det(F_{k})$ and $\det(F_{k}^{0})$, for
$1\leq k\leq n$. So a metric on $\det(F_{k})$ can be viewed as a metric on
$\det(F_{k}^{0})$.

###### Definition 1.8

Let $h,h_{1}$ be Hermitian metrics on $E,E_{0}$ respectively. Call $h$ weakly
dominates $h_{1}$ if

$\det(h|_{F_{k}})\leq\det(h_{1}|_{F_{k}^{0}}),\quad 1\leq k\leq n-1.$

We prove the following existence result.

###### Theorem 1.9

(Theorem 3.10) Suppose there exists a diagonal harmonic metric $h_{1}$ on
$(E_{0},\theta_{0})$, then there exists a harmonic metric $h$ on $(E,\theta)$
satisfying (i) $\det(h)=\det(h_{1})$; (ii) $h$ weakly dominates $h_{1}$.

Because of Theorem 1.9, we are interested in the existence of a diagonal
harmonic metric on a holomorphic chain of type $(1,\cdots,1).$ However, we
find that such metric does not always exist, see Proposition 6.8 and
Proposition 6.10. In Theorem 6.3, we provide a sufficient condition of the
existence of a harmonic metric on holomorphic chains.

###  1.3 $SO(n,n+1)$-Higgs bundles and $Sp(4,\mathbb{R})$-Higgs bundles

The Higgs bundles we consider in Theorem 1.9 also appear in $SO(n,n+1)$-Higgs
bundles in Collier section and $Sp(4,\mathbb{R})$-Higgs bundles in Gothen
section. As applications of Theorem 1.9 and the existence result for diagonal
harmonic metric on holomorphic chains, we show in §7 the existence of harmonic
metric on $SO(n,n+1)$-Higgs bundles in Collier section. In §8, we show the
existence of harmonic metric on $Sp(4,\mathbb{R})$-Higgs bundles in Gothen
section.

###  1.4 Further questions

1\. Our techniques here only apply to hyperbolic Riemann surfaces since it
relies on the existence of harmonic metric on the graded Higgs bundle. Since
the graded Higgs bundles are nilpotent, the existence of a harmonic metric
forces the Riemann surface to be hyperbolic. Therefore,
$\boldsymbol{q}\neq\mathbf{0}$ is a necessary condition for the existence of
harmonic metric on $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ over a
parabolic Riemann surface. So it would be interesting to ask if
$\boldsymbol{q}\neq\mathbf{0}$ is a sufficient condition. So far, the best
answer we can provide are Higgs bundles satisfying generically regular
semisimple condition.

2\. We would like to see the uniqueness result in Theorem 1.6 extends to all
$\boldsymbol{q}$ without the boundedness condition.

3\. For holomorphic chains, we find a sufficient condition for the existence
of a harmonic metric in Theorem 6.3. We would like to find a sufficient and
necessary condition for the existence of a diagonal harmonic metric for
holomorphic chains of type $(1,\cdots,1)$.

4\. There is a natural $\mathbb{C}^{*}$-action on the space of gauge
equivalent classes of Higgs bundles as follows:
$t\cdot[(E,\theta)]=[(E,t\theta)].$ We want to ask if the
$\mathbb{C}^{*}$-action preserve the property admitting a harmonic metric.
More precisely, suppose a Higgs bundle $(E,\theta)$ admits a harmonic metric,
does there exist a harmonic metric on $(E,t\cdot\theta)$ for
$t\in\mathbb{C}^{*}$? This is true if the base Riemann surface is compact
hyperbolic since the stability is preserved by the $\mathbb{C}^{*}$-action.
For non-compact Riemann surfaces, the answer is unclear. The evidence for this
conjecture is that the properties in the two cases we can prove the existence
of harmonic metrics are preserved by the $\mathbb{C}^{*}$-action: (1) Higgs
bundles being in the Hitchin section; (2) generically regular semisimple and
admits a non-degenerate symmetric pairing.

### Organization

In §2, we give some results on the existence of harmonic metric using
exhaustion family of harmonic metrics of Dirichlet problem. In §3, we study
the existence of harmonic metric for the Higgs bundles which admit a full
holomorphic filtration. In §4, we study the uniqueness of real harmonic
metrics of some Higgs bundles which are mutually bounded with a canonically
constructed metric. In §5, we apply the existence result to Higgs bundles in
the Hitchin section and show the uniqueness result for the case of bounded
differentials. In §6, we show the existence of harmonic metric under
boundedness condition on the Higgs bundle and apply it to holomorphic chains.
In the last two sections, we show the existence of harmonic metric on
$SO(n,n+1)$-Higgs bundles and $Sp(4,\mathbb{R})$-Higgs bundles.

### Acknowledgement

The first author is partially supported by the National Key R&D Program of
China No. 2022YFA1006600, the Fundamental Research Funds for the Central
Universities and Nankai Zhide foundation. The second author is 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. He is
also partially supported by the Research Institute for Mathematical Sciences,
an International Joint Usage/Research Center located in Kyoto University.

##  2 Preliminaries on existence of harmonic metrics

In this section, we give some results on the existence of harmonic metric
using exhaustion family of harmonic metrics of Dirichlet problem. A variant
version also appears in [LM20b, Section 2].

###  2.1 Dirichlet problem

Let $X$ be any Riemann surface. Let $(E,\overline{\partial}_{E},\theta)$ be a
Higgs bundle on $X$. For a Hermitian metric $h$ of $E$, we obtain the Chern
connection $\nabla_{h}=\overline{\partial}_{E}+\partial_{E}^{h}$ of
$(E,\overline{\partial}_{E},h)$. The curvature of $\nabla_{h}$ is denoted by
$F(\nabla_{h})$ or $F(h)$. We also obtain the adjoint $\theta^{*h}$ of
$\theta$ with respect to $h$. The curvature of $\nabla_{h}+\theta+\theta^{*h}$
is denoted by $F(E,\overline{\partial}_{E},\theta)$, i.e.,
$F(E,\overline{\partial}_{E},\theta)=F(\nabla_{h})+[\theta,\theta^{*h}]$.

Let $Y\subset X$ be a relatively compact connected open subset with smooth
boundary $\partial Y$. Assume that $\partial Y$ is non-empty. Let $h_{\partial
Y}$ be any Hermitian metric of $E_{|\partial Y}$.

######  Proposition 2.1 (Donaldson)

There exists a unique harmonic metric $h$ of
$(E,\overline{\partial}_{E},\theta)$ such that $h_{|\partial Y}=h_{\partial
Y}$.

Proof This was proved by Donaldson [Don92, Theorem 2] in the case $Y$ is a
disc. The general case is essentially the same. We explain an outline of the
proof for the convenience of the reader.

We may assume that $X$ is an open Riemann surface. According to [GN67], there
exists a nowhere vanishing holomorphic $1$-form $\tau$ on $X$. Let $f$ be the
automorphism of $E$ determined by $\theta=f\,\tau$. We consider the Kähler
metric $g_{X}=\tau\,\overline{\tau}$ of $X$.

Let $\Gamma$ be a lattice of ${\mathbb{C}}$ and let $T$ be a real
$2$-dimensional torus obtained as ${\mathbb{C}}/\Gamma$. We set
$g_{T}=dz\,d\overline{z}$. We set $\widetilde{X}:=X\times T$ with the
projection $p:\widetilde{X}\longrightarrow X$. It is equipped with the flat
Kähler metric $g_{\widetilde{X}}$ induced by $g_{T}$ and $g_{X}$. We set
$\widetilde{Y}:=p^{-1}(Y)$.

Let $\widetilde{E}$ be the pull back of $E$ with the holomorphic structure
$p^{\ast}(\overline{\partial}_{E})+p^{\ast}(f)\,d\overline{z}$. According to
the dimensional reduction of Hitchin, a Hermitian metric $h$ of $E_{|Y}$ is a
harmonic metric of $(E,\overline{\partial}_{E},\theta)_{|Y}$ if and only if
$\Lambda_{\widetilde{Y}}F(p^{\ast}h)=0$. According to a theorem of Donaldson
[Don92, Theorem 1], there exists a unique Hermitian metric $\widetilde{h}$ of
$\widetilde{E}$ such that $\Lambda_{\widetilde{Y}}F(\widetilde{h})=0$ and that
$\widetilde{h}_{|p^{-1}(\partial Y)}=p^{\ast}(h_{\partial Y})$. By the
uniqueness, $\widetilde{h}$ is $T$-invariant. Hence, there uniquely exists a
harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)_{|Y}$ which induces
$\widetilde{h}$. It satisfies $h_{|\partial Y}=h_{\partial Y}$.

Let $h_{0}$ be a Hermitian metric of $E$. Assume that $\det(h_{0})$ is flat.

###### Corollary 2.2

There exists a unique harmonic metric $h$ of $E_{|Y}$ such that $h_{|\partial
Y}=h_{0|\partial Y}$ and that $\det(h)=\det(h_{0})_{|Y}$.

Proof There exists a unique harmonic metric $h$ such that $h_{|\partial
Y}=h_{0|\partial Y}$. We obtain $\det(h)_{|\partial Y}=\det(h_{0})_{|\partial
Y}$. Note that both $\det(h)$ and $\det(h_{0})_{|Y}$ are flat. By the
uniqueness in Proposition 2.1, we obtain $\det(h)=\det(h_{0})_{|Y}$.

###  2.2 Convergence

Let $X$ be an open Riemann surface. Let $h_{0}$ be a Hermitian metric of $E$.

###### Definition 2.3

An exhaustive family $\\{X_{i}\\}$ of a Riemann surface $X$ means an
increasing sequence of relatively compact open subsets $X_{1}\subset
X_{2}\subset\cdots$ of $X$ such that $X=\bigcup X_{i}$. The family is called
smooth if $\partial X_{i}$ are smooth.

Let $\\{X_{i}\\}$ be a smooth exhaustive family of $X$. The restriction
$h_{0|X_{i}}$ is denoted by $h_{0,i}$. Let $h_{i}$ $(i=1,2,\ldots)$ be
harmonic metrics of $(E,\overline{\partial}_{E},\theta)_{|X_{i}}$. Let $s_{i}$
be the automorphism of $E_{|X_{i}}$ determined by $h_{i}=h_{0,i}\cdot s_{i}$.
Let $f$ be an ${\mathbb{R}}_{>0}$-valued function on $X$ such that each
$f_{|X_{i}}$ is bounded. Though the following proposition is proved in
[LM20b], we include the proof for the convenience of the readers.

###### Proposition 2.4

Assume that $|s_{i}|_{h_{0,i}}+|s^{-1}_{i}|_{h_{0,i}}\leq f_{|X_{i}}$ for any
$i$. Then, there exists a subsequence $s_{i(j)}$ which is convergent to an
automorphism $s_{\infty}$ of $E$ on any relatively compact subset of $X$ in
the $C^{\infty}$-sense. As a result, we obtain a harmonic metric
$h_{\infty}=h_{0}s_{\infty}$ of $(E,\overline{\partial}_{E},\theta)$ as the
limit of the subsequence $h_{i(j)}$. Moreover, we obtain
$|s_{\infty}|_{h_{0}}+|s_{\infty}^{-1}|_{h_{0}}\leq f$. In particular, if $f$
is bounded, $h_{0}$ and $h_{\infty}$ are mutually bounded.

Proof We explain an outline of the proof. Let $g_{X}$ be a Kähler metric of
$X$. According to a general formula (5) below, the following holds on any
$X_{i}$:

$\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm
tr}\nolimits(s_{i})=-\sqrt{-1}\mathop{\rm tr}\nolimits\bigl{(}s_{i}\Lambda
F(h_{0,i})\bigr{)}-\bigl{|}(\overline{\partial}+\theta)(s_{i})\cdot
s_{i}^{-1/2}\bigr{|}^{2}_{h_{0,i},g}.$ (1)

Let $K$ be any compact subset of $X$. Let $N$ be a relatively compact
neighbourhood of $K$ in $X$. Let $\chi:X\longrightarrow{\mathbb{R}}_{\geq 0}$
be a $C^{\infty}$-function such that (i) $\chi_{|K}=1$, (ii)
$\chi_{|X\setminus N}=0$, (iii) $\chi^{-1/2}\partial\chi$ and
$\chi^{-1/2}\overline{\partial}\chi$ on $\\{P\in X\,|\,\chi(P)>0\\}$ induces a
$C^{\infty}$-function on $X$.

There exist $i_{0}$ such that $N$ is a relatively compact open subset of
$X_{i}$ for any $i\geq i_{0}$. We obtain the following:

$\sqrt{-1}\Lambda\overline{\partial}\partial(\chi\mathop{\rm
tr}\nolimits(s_{i}))=\chi\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm
tr}\nolimits(s_{i})+(\sqrt{-1}\Lambda\overline{\partial}\partial\chi)\cdot\mathop{\rm
tr}\nolimits(s_{i})+\sqrt{-1}\Lambda(\overline{\partial}\chi\partial\mathop{\rm
tr}\nolimits(s_{i}))-\sqrt{-1}\Lambda(\partial\chi\overline{\partial}\mathop{\rm
tr}\nolimits(s_{i})).$ (2)

Note that
$|\overline{\partial}_{E}s_{i}|_{h_{i},g_{X}}=|\partial_{E,h_{i}}s_{i}|_{h_{i},g_{X}}$,
and that

$\left|\int_{X}\sqrt{-1}\Lambda(\overline{\partial}\chi\partial\mathop{\rm
tr}\nolimits(s_{i}))\right|\leq\left(\int_{X}|\chi^{-1/2}\overline{\partial}\chi|^{2}\right)^{1/2}\cdot\left(\int_{X}\chi|\partial_{E,h_{i}}s_{i}|^{2}_{h_{0},g_{X}}\right)^{1/2}.$
(3)

Note that there exists $C_{0}>0$ such that
$|s_{i}|_{h_{0}}+|s_{i}^{-1}|_{h_{0}}\leq C_{0}$ on $N$. By (1), (2) and (3),
there exist $C_{j}>0$ $(j=1,2)$ such that the following holds for any
sufficiently large $i$:

$\int\chi\bigl{|}\overline{\partial}_{E}s_{i}\bigr{|}^{2}_{h_{0},g_{X}}+\int\chi\bigl{|}[\theta,s_{i}]\bigr{|}^{2}_{h_{0},g_{X}}\leq
C_{1}+C_{2}\left(\int\chi\bigl{|}\overline{\partial}_{E}s_{i}\bigr{|}^{2}_{h_{0},g_{X}}+\int\chi\bigl{|}[\theta,s_{i}]\bigr{|}^{2}_{h_{0},g_{X}}\right)^{1/2}$

Therefore, there exists $C_{3}>0$ such that the following holds for any
sufficiently large $i$:

$\int\chi\bigl{|}\overline{\partial}_{E}s_{i}\bigr{|}^{2}_{h_{0},g_{X}}+\int\chi\bigl{|}[\theta,s_{i}]\bigr{|}^{2}_{h_{0},g_{X}}\leq
C_{3}.$

We obtain the boundedness of the $L^{2}$-norms of
$\overline{\partial}_{E}s_{i}$ and $\partial_{E,h_{i}}s_{i}$ $(i\geq i_{0})$
on $K$ with respect to $h_{0}$ and $g_{X}$. By a variant of Simpson’s main
estimate (see [Moc16, Proposition 2.1]), we obtain the boundedness of the sup
norms of $\theta$ on $N$ with respect to $h_{i}$ and $g_{X}$. By the Hitchin
equation, we obtain the boundedness of the sup norms of
$\overline{\partial}_{E}(s_{i}^{-1}\partial_{E,h_{i}}s_{i})$ on $N$ with
respect to $h_{i}$ and $g_{X}$. By using the elliptic regularity, we obtain
that the $L_{1}^{p}$-norms of $s_{i}^{-1}\partial_{E,h_{i}}(s_{i})$ on a
relatively compact neighbourhood of $K$ are bounded for any $p>1$. It follows
that $L_{2}^{p}$-norms of $s_{i}$ on a relatively compact neighbourhood of $K$
are bounded for any $p$. Hence, a subsequence of $s_{i}$ is weakly convergent
in $L_{2}^{p}$ on a relatively compact neighbourhood of $K$. By the
bootstrapping argument using a general formula (4) below, we obtain that the
sequence is convergent on a relatively compact neighbourhood of $K$ in the
$C^{\infty}$-sense. By using the diagonal argument, we obtain that a
subsequence of $s_{i}$ is weakly convergent in $C^{\infty}$-sense on any
compact subset.

###  2.3 Appendix

We recall some fundamental formulas due to Simpson [Sim88, Lemma 3.1] for the
convenience of the readers.

Let $h_{i}$ $(i=1,2)$ be Hermitian metrics of $E$. We obtain the automorphism
$s$ of $E$ determined by $h_{2}=h_{1}\cdot s$. Let $g$ be a Kähler metric of
$X$, let $\Lambda$ denote the adjoint of the multiplication of the associated
Kähler form. Then, according to [Sim88, Lemma 3.1 (a)], we obtain the
following on $X$:

$\sqrt{-1}\Lambda\bigl{(}\overline{\partial}_{E}+\theta\bigr{)}\circ\bigl{(}\partial_{E,h_{1}}+\theta^{*h_{1}}\bigr{)}s=s\sqrt{-1}\Lambda\bigl{(}F(h_{2})-F(h_{1})\bigr{)}+\sqrt{-1}\Lambda\Bigl{(}\bigl{(}\overline{\partial}_{E}+\theta\bigr{)}(s)s^{-1}\bigl{(}\partial_{E,h_{1}}+\theta^{*h_{1}}\bigr{)}(s)\Bigr{)}.$
(4)

By taking the trace, and by using [Sim88, Lemma 3.1 (b)], we obtain

$\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm
tr}\nolimits(s)=\sqrt{-1}\mathop{\rm
tr}\nolimits\Bigl{(}s\Lambda\bigl{(}F(h_{2})-F(h_{1})\bigr{)}\Bigr{)}-\Bigl{|}\bigl{(}\overline{\partial}_{E}+\theta\bigr{)}(s)s^{-1/2}\Bigr{|}^{2}_{h_{1},g}.$
(5)

Note that
$(\overline{\partial}_{E}+\theta)(s)=\overline{\partial}_{E}(s)+[\theta,s]$.
Moreover, $\overline{\partial}_{E}(s)$ is a $(0,1)$-form, and $[\theta,s]$ is
a $(1,0)$-form. Hence, (5) is also rewritten as follows:

$\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm
tr}\nolimits(s)=\sqrt{-1}\mathop{\rm
tr}\nolimits\Bigl{(}s\Lambda\bigl{(}F(h_{2})-F(h_{1})\bigr{)}\Bigr{)}-\bigl{|}[\theta,s]s^{-1/2}\bigr{|}^{2}_{h_{1},g}-\bigl{|}\overline{\partial}_{E}(s)s^{-1/2}\bigr{|}_{h_{1},g}.$
(6)

We also recall the following inequality [Sim88, Lemma 3.1 (d)]:

$\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm
tr}\nolimits(s)\leq\bigl{|}\Lambda F(h_{1})\bigr{|}_{h_{1}}+\bigl{|}\Lambda
F(h_{2})\bigr{|}_{h_{2}}.$ (7)

In particular, if both $h_{i}$ are harmonic, the functions $\mathop{\rm
tr}\nolimits(s)$ and $\log\mathop{\rm tr}\nolimits(s)$ are subharmonic:

$\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm
tr}\nolimits(s)=-\bigl{|}(\overline{\partial}_{E}+\theta)(s)s^{-1/2}\bigr{|}^{2}_{h,g}\leq
0,\quad\quad\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm
tr}\nolimits(s)\leq 0.$ (8)

##  3 Domination property and the existence of harmonic metrics

###  3.1 Full flags and Hermitian metrics

Let $V$ be a complex vector space equipped with a base
$\mathbf{e}=(e_{1},\cdots,e_{n})$. For $k=1,\cdots,n$, let $F_{k}(V)$ denote
the subspace generated by $e_{1},\cdots,e_{k}.$ We set $F_{0}(V)=0$. We set
$Gr_{k}^{F}(V)=F_{k}(V)/F_{k-1}(V)$. There exists a natural isomorphism

$\rho_{k}:Gr_{k}^{F}(V)\otimes\det(F_{k-1})\cong\det(F_{k}).$

Let $h$ be a Hermitian metric of $V$. Let $F_{k}(h)$ denote the induced metric
of $F_{k}(V).$ It induces a Hermitian metric $\det(F_{k}(h))$ of
$\det(F_{k}(V)).$ Let $G_{k}(V,h)$ be the orthogonal complement of
$F_{k-1}(V)$ in $F_{k}(V).$ The projection $F_{k}(V)\rightarrow Gr_{k}^{F}(V)$
induces an isomorphism $G_{k}(V,h)\cong Gr_{k}^{F}(V).$ We obtain the metric
$Gr_{k}^{F}(h)$ of $Gr_{k}^{F}(V)$ which is induced by $h|_{Gr_{k}(V,h)}$ and
the isomorphism $G_{k}(V,h)\cong Gr_{k}^{F}(V).$

###### Lemma 3.1

$\rho_{k}$ is isometric with respect to $\det(F_{k}(h))$ and
$Gr_{k}^{F}(h)\otimes\det(F_{k-1}(h)).$

Proof There exists the orthogonal decomposition
$F_{k}(V)=\oplus_{j=1}^{k}G_{j}(V,h).$ We choose $v_{j}\in G_{j}(V,h)$ such
that $h(v_{j},v_{j})=1.$ The norm of $v_{1}\wedge\cdots\wedge v_{k}$ with
respect to $\det(F_{k}(h))$ is $1$. Let $[v_{k}]\in Gr_{k}^{F}(V)$ denote the
element induced by $v_{k}$. The norm of $[v_{k}]$ with respect to
$Gr_{k}^{F}(h)$ is $1$. Then we obtain the claim of the lemma.

Denote $F_{k}^{0}(V)=\oplus_{l\leq k}Gr_{k}^{F}(V).$ It has the induced metric
$F_{k}^{0}(h)$ from $h$ on $V$. From $\rho_{k}$’s, one naturally has an
isomorphism between $F_{k}(V)$ with $F_{k}^{0}(V)$, which is an isometry with
respect to $\det(F_{k}(h))$ and $\det(F_{k}^{0}(h)).$

###  3.2 Set-up

Let $X$ be a hyperbolic Riemann surface and $K$ be its canonical line bundle.

Consider a Higgs bundle $(E,\theta)$ over $X$ which admits a full holomorphic
filtration

$\mathbf{F}=\\{0=F_{0}\subset F_{1}\subset F_{2}\subset\cdots\subset
F_{n}=E\\}$

and $\theta:F_{k}\rightarrow F_{k+1}\otimes K$. We require that the induced
map $\theta$ on each $F_{k}/F_{k-1}$ is not a zero map, denoted by $\phi_{k}$,
for $1\leq k\leq n-1$.

Denote by $Gr_{k}^{F}(E)$ the quotient line bundle $F_{k}/F_{k-1}$, equipped
with the quotient holomorphic structure. Consider the holomorphic vector
bundle $E_{0}=Gr^{F}(E):=\oplus_{k=1}^{n}Gr_{k}^{F}(E)$. Let $\theta_{0}$ be
formed by $\phi_{k}:Gr_{k}^{F}(E)\rightarrow Gr_{k+1}^{F}(E)\otimes K,$ for
$1\leq k\leq n-1$. Therefore, $(E_{0},\theta_{0})$ is a holomorphic chain of
type $(1,1,\cdots,1).$ Let $F_{k}^{(0)}=\oplus_{l\leq k}Gr_{l}^{F}(E).$

Let $h$ be a Hermitian metric on $E$. Let $F_{k}(h)$ denote the induced metric
of $h$ on $F_{k}$. The metric $h$ induces a metric $Gr_{k}^{F}(h)$ on each
$Gr_{k}^{F}(E)$, a diagonal metric $F_{k}^{0}(h)$ on $F_{k}^{0}$, and a
diagonal metric on $E_{0}$.

###### Definition 3.2

Suppose $h$ is a Hermitian metric on $E$, and $h_{1}$ is a diagonal Hermitian
metrics on $E_{0}$. Call $h$ weakly dominates $h_{1}$ if

$\det(F_{k}^{0}(h))\leq\det(F_{k}^{0}(h_{1})),\quad 1\leq k\leq n-1.$ (9)

Under the natural identification between $\det(F_{k}^{0})$ and $\det(F_{k})$
in §3.1, we can write the condition (9) as follows

$\det(F_{k}(h))\leq\det(F_{k}^{0}(h_{1})),\quad 1\leq k\leq n-1.$ (10)

####  3.2.1 The graded case

If $E=\oplus_{i=1}^{n}L_{i}$ and $F_{k}=\oplus_{l\leq k}L_{l}$ for some
holomorphic line bundles $L_{i}$ over $X$, there is a canonical isomorphism
between $E$ and $E_{0}$ by mapping $L_{k}$ to $Gr_{k}^{F}(E)$. In this case,
we can identify $E$ and $E_{0}$ and view the metric $h_{1}$ as a metric on $E$
too. Then we call $h$ weakly dominates $h_{1}$ if

$\det(F_{k}(h))\leq\det(F_{k}(h_{1})),\quad 1\leq k\leq n-1.$ (11)

Because of the following lemma, we may assume the existence of such a grading
if the Riemann surface is non-compact.

###### Lemma 3.3

Let $E$ be a holomorphic vector bundle of rank $r$ on a non-compact Riemann
surface $X$ equipped with an increasing filtration $F_{j}(j=1,\cdots,n)$ such
that $\mathop{\rm rank}\nolimits\mathop{\rm Gr}\nolimits^{F}_{j}(E)=1$ for
$j=1,\cdots,n$. Then, there exists a frame $e_{1},\cdots,e_{n}$ of $E$ such
that $F_{j}=\oplus_{i\leq j}\mathcal{O}_{X}e_{i}$.

Proof It is well known that $H^{1}(X,\mathcal{O}_{X})=0$. Because any
holomorphic vector bundle $E^{\prime}$ on $X$ is isomorphic to
$\mathcal{O}_{X}^{\mathop{\rm rank}\nolimits(E^{\prime})}$, we have
$H^{1}(X,E^{\prime})=0$. Let $\pi:E_{1}\rightarrow E_{2}$ be an epimorphism of
holomorphic vector bundles on $X$. Let $K$ be the kernel. Because

$H^{0}(X,\mathop{\rm Hom}\nolimits(E_{2},E_{1}))\rightarrow
H^{0}(X,\mathop{\rm Hom}\nolimits(E_{2},E_{2}))\rightarrow H^{1}(X,\mathop{\rm
Hom}\nolimits(E_{2},K))=0$

is exact, there exists a splitting $s:E_{2}\rightarrow E_{1}$ such that
$\pi\circ s=id_{E_{2}}$. Then, the claim of the lemma follows.

####  3.2.2 Symmetric pairings

Let us recall the notion of compatibility of a non-degenerate symmetric
pairing and a Hermitian metric on a complex vector space $V$. (See [LM22,
§2.1] for more details.) Let $V^{\lor}$ denote the dual space of $V$. Let
$\langle\cdot,\cdot\rangle:V^{\lor}\times V\to{\mathbb{C}}$ denote the
canonical pairing.

Let $C:V\times V\to{\mathbb{C}}$ be a non-degenerate symmetric bilinear form.
We obtain the linear isomorphism $\Psi_{C}:V\simeq V^{\lor}$ by
$\langle\Psi_{C}(u),v\rangle=C(u,v)$. We obtain the symmetric bilinear form
$C^{\lor}:V^{\lor}\times V^{\lor}\to{\mathbb{C}}$ by

$C^{\lor}(u^{\lor},v^{\lor})=C(\Psi_{C}^{-1}(u^{\lor}),\Psi_{C}^{-1}(v^{\lor})).$

We have $\Psi_{C^{\lor}}\circ\Psi_{C}=\mathop{\rm id}\nolimits_{V}$.

Let $h$ be a Hermitian metric of $V$. We obtain the sesqui-linear isomorphism
$\Psi_{h}:V\simeq V^{\lor}$ by $\langle\Psi_{h}(u),v\rangle=h(v,u)$. We obtain
the Hermitian metric $h^{\lor}$ of $V^{\lor}$ by

$h^{\lor}(u^{\lor},v^{\lor})=h\bigl{(}\Psi_{h}^{-1}(v^{\lor}),\Psi_{h}^{-1}(u^{\lor})\bigr{)}.$

It is easy to see that $\Psi_{h^{\lor}}\circ\Psi_{h}=\mathop{\rm
id}\nolimits_{V}$.

###### Definition 3.4

We say that $h$ is compatible with $C$ if $\Psi_{C}$ is isometric with respect
to $h$ and $h^{\lor}$.

###### Lemma 3.5

The following conditions are equivalent.

  * •

$h$ is compatible with $C$.

  * •

$C(u,v)=\overline{C^{\lor}(\Psi_{h}(u),\Psi_{h}(v))}$ holds for any $u,v\in
V$.

  * •

$\Psi_{C^{\lor}}\circ\Psi_{h}=\Psi_{h^{\lor}}\circ\Psi_{C}$ holds. It is also
equivalent to $\Psi_{C}\circ\Psi_{h^{\lor}}=\Psi_{h}\circ\Psi_{C^{\lor}}$.

Note that $h$ and $C$ induce a Hermitian metric $\det(h)$ and a non-degenerate
symmetric pairing $\det(C)$ of $\wedge^{n}V$ respectively. The following lemma
is clear.

###### Lemma 3.6

If $h$ is compatible with $C$, then $|\det(h)|=|\det(C)|$.

####  3.2.3 Symmetric pairing and graded bundles

Consider the graded case $E=\oplus_{i=1}^{n}L_{i}$ and $F_{k}=\oplus_{l\leq
k}L_{l}$ for some holomorphic line bundles $L_{i}$ over $X$. Suppose in
addition $L_{i}=L_{n+1-i}^{-1}$. Then it induces a natural symmetric pairing
induced by $L_{i}\otimes L_{n+1-i}\rightarrow\mathcal{O}$, denoted by $C$. In
this case $\det(E)\cong\mathcal{O}$ and $|\det(C)|=1$. If $h$ is compatible
with $C$, then $\det(h)=1$.

With respect to the decomposition $\mathop{\rm
End}\nolimits(E)=\oplus_{ij}\mathop{\rm Hom}\nolimits(L_{i},L_{j})$, the Higgs
field $\theta=\sum_{ij}\theta_{ij}$ .

Suppose moreover $\theta$ is symmetric with respect to $C$. That is,
$\theta_{ij}=\theta_{n+1-j,n+1-i}$ under the identification between
$\mathop{\rm Hom}\nolimits(L_{i},L_{j})\cong\mathop{\rm
Hom}\nolimits(L_{n+1-j},L_{n+1-i})=\mathop{\rm
Hom}\nolimits(L_{j}^{-1},L_{i}^{-1})$.

The graded bundle $E_{0}$ has an induced pairing that
$C_{0}(Gr_{k}^{F}E,Gr_{l}^{F}E)=\delta_{kl}$. The canonical isomorphism
between $E$ and $E_{0}$ takes $C$ to $C_{0}$. So we identify $(E_{0},C_{0})$
with $(E,C)$. Since $\theta_{0}=\sum_{i=1}^{n-1}\theta_{i,i+1}$, it is again
symmetric with respect to $C$.

###  3.3 Domination property and the Dirichlet problem

Let $X$, $(E,\theta)$ and $(E_{0},\theta_{0})$ be as in §3.2. Let $h_{1}$ be a
harmonic metric on $(E_{0},\theta_{0})$ orthogonal to the decomposition
$E_{0}=\oplus_{k=1}^{n}Gr_{k}^{F}(E)$. The following proposition is motivated
by the result for Higgs bundles in the Hitchin section over compact Riemann
surfaces in [Li19a]. Here we extend the result to surfaces with boundaries and
more general Higgs bundles. This domination property turns out to be the key
property in showing the convergence of harmonic metrics in the exhaustion
process.

###### Proposition 3.7

On a Riemann surface $X$ with boundary $\partial X$, suppose $(E,\theta)$ has
a harmonic metric $h$ satisfying $h=h_{1}$ on $\partial X.$ Then $h$ weakly
dominates $h_{1}.$

Proof For a holomorphic subbundle $F$ of $E$, we would like to deduce the
Hitchin equation which respects $F$. Denote by $F^{\perp}$ the subbundle of
$E$ perpendicular to $F$ with respect to the harmonic metric $H$. $F^{\perp}$
can be equipped with the quotient holomorphic structure from $E/F$. With
respect to the $C^{\infty}$ orthogonal decomposition

$E=F\oplus F^{\perp},$

we have the expression of the holomorphic structure $\overline{\partial}_{E}$
and the Higgs field $\phi$ as follows:

$\overline{\partial}_{E}=\begin{pmatrix}\bar{\partial}_{F}&\beta\\\
0&\bar{\partial}_{F^{\perp}}\end{pmatrix},\quad\theta=\begin{pmatrix}\phi_{1}&\alpha\\\
B&\phi_{2}\end{pmatrix},\quad H=\begin{pmatrix}H_{1}&0\\\
0&H_{2}\end{pmatrix},$

where the term $B\in\Omega^{1,0}(X,\mathop{\rm Hom}\nolimits(F,F^{\perp}))$,
$\alpha\in\Omega^{1,0}(X,\mathop{\rm Hom}\nolimits(F^{\perp},F))$, and
$\beta\in\Omega^{0,1}(X,\mathop{\rm Hom}\nolimits(F^{\perp},F))$.

The Chern connection $\nabla_{H}$ and the adjoint $\theta^{*_{H}}$ of the
Higgs field are

$\nabla_{H}=\begin{pmatrix}\nabla_{H_{1}}&\beta\\\
-\beta^{*_{H}}&\nabla_{H_{2}}\end{pmatrix},\quad\theta^{*_{H}}=\begin{pmatrix}\phi_{1}^{*_{H_{1}}}&B^{*_{H}}\\\
\alpha^{*_{H}}&\phi_{2}^{*_{H_{2}}}\end{pmatrix}.$

We calculate the Hitchin equation with respect to the decomposition $E=F\oplus
F^{\perp}$ and by restricting to $\mathop{\rm Hom}\nolimits(F,F)$, we obtain

$F(\nabla_{H_{1}})-\beta\wedge\beta^{*_{H}}+\alpha\wedge\alpha^{*_{H}}+B^{*_{H}}\wedge
B+[\phi_{1},\phi_{1}^{*_{H_{1}}}]=0.$

By taking trace and noting that $\mathop{\rm
tr}\nolimits([\phi_{1},\phi_{1}^{*_{H_{1}}}])=0$, we obtain

$\mathop{\rm tr}\nolimits(F(\nabla_{H_{1}}))-\mathop{\rm
tr}\nolimits(\beta\wedge\beta^{*_{H}})+\mathop{\rm
tr}\nolimits(\alpha\wedge\alpha^{*_{H}})+\mathop{\rm
tr}\nolimits(B^{*_{H}}\wedge B)=0.$

Let $g_{X}=g(z)(dx^{2}+dy^{2})$ be a conformal Riemannian metric on $X$. The
associated Kähler form associated to $g_{X}$ is

$\omega=\frac{\sqrt{-1}}{2}g(z)dz\wedge d\bar{z}.$

Note that

$|\partial/\partial_{z}|_{g_{X}}^{2}=\frac{g(z)}{2},\quad|dz|_{g_{X}}^{2}=\frac{2}{g(z)}.$

Thus the induced Hermitian metric on $K_{X}^{-1}$ can be written as
$\frac{g(z)}{2}dz\otimes d\bar{z}$, still denoted as $g_{X}$. Denote by
$\Lambda_{g_{X}}$ the contraction with respect to the Kähler form $\omega$.
Therefore,

$\displaystyle-\sqrt{-1}\Lambda_{g_{X}}\mathop{\rm
tr}\nolimits(F(\nabla_{H_{1}}))-\sqrt{-1}\Lambda_{g_{X}}\mathop{\rm
tr}\nolimits(B^{*_{H}}\wedge B)$ $\displaystyle=$
$\displaystyle-\sqrt{-1}\Lambda_{g_{X}}\mathop{\rm
tr}\nolimits(\beta\wedge\beta^{*_{H}})+\sqrt{-1}\Lambda_{g_{X}}\mathop{\rm
tr}\nolimits(\alpha\wedge\alpha^{*_{H}})$ (12) $\displaystyle=$
$\displaystyle||\beta||_{H,g_{X}}^{2}+||\alpha||_{H,g_{X}}^{2}\geq 0.$

We will apply the above procedure to $F=F_{k}$ for each $k=1,2,\cdots,n-1$. We
take $L_{i}$ to be the perpendicular line bundle of $F_{i-1}$ inside $F_{i}$
with respect to he harmonic metric $h.$ Then we have a smooth decomposition of

$E=L_{1}\oplus L_{2}\oplus\cdots\oplus L_{n}.$

With respect to the decomposition, we have the following:

I. the Hermitian metric $H$ solving the Hitchin equation is given by

$H=\begin{pmatrix}h|_{L_{1}}&&&\\\ &h|_{L_{2}}&&\\\ &&\ddots&\\\
&&&h|_{L_{n}}\end{pmatrix}$ (13)

where $h|_{L_{i}}$ is the induced Hermitian metric on $L_{i}$ and
$h|_{L_{i}}=\det(h|_{F_{i}})/\det(h|_{F_{i-1}})$;

II. the holomorphic structure on $E$ is given by the $\bar{\partial}$-operator

$\displaystyle\overline{\partial}_{E}=\begin{pmatrix}\bar{\partial}_{1}&\beta_{12}&\beta_{13}&\cdots&\beta_{1n}\\\
&\bar{\partial}_{2}&\beta_{23}&\cdots&\beta_{2n}\\\
&&\bar{\partial}_{3}&\cdots&\beta_{3n}\\\ &&&\ddots&\vdots\\\
&&&&\bar{\partial}_{n}\end{pmatrix}$ (14)

where $\bar{\partial}_{k}$ are $\bar{\partial}$-operators defining the
holomorphic structures on $L_{k}$, and
$\beta_{ij}\in\Omega^{0,1}(X,\mathop{\rm Hom}\nolimits(L_{j},L_{i}))$;

III. the Higgs field is of the form

$\displaystyle\theta=\begin{pmatrix}a_{11}&a_{12}&a_{13}&\cdots&a_{1n}\\\
\gamma_{1}&a_{22}&a_{23}&\cdots&a_{2n}\\\ &\gamma_{2}&a_{33}&\cdots&a_{3n}\\\
&&\ddots&\ddots&\vdots\\\ &&&\gamma_{n-1}&a_{nn}\end{pmatrix}$ (15)

where $a_{ij}\in\Omega^{1,0}(X,\mathop{\rm Hom}\nolimits(L_{j},L_{i}))$ and
$\gamma_{k}:L_{k}\rightarrow L_{k+1}\otimes K$ is holomorphic.

We then consider the subbundle $F=F_{k}$ for $k=1,\cdots,n-1$. Then the
associated factor $B$ is

$B=\begin{pmatrix}0&0&\cdots&0&\gamma_{k}\\\ 0&0&\cdots&0&0\\\
\vdots&\vdots&\cdots&\vdots&\vdots\\\
0&0&\cdots&0&0\end{pmatrix}:F_{k}\rightarrow(L_{k+1}\oplus\cdots\oplus
L_{n})\otimes K$

then

$\sqrt{-1}\Lambda_{g_{X}}\mathop{\rm tr}\nolimits(B^{*_{H}}\wedge
B)=-|\gamma_{k}|^{2}(h|_{L_{k}})^{-1}h|_{L_{k+1}}/g_{X}=-|\gamma_{k}|^{2}\frac{\det(h|_{F_{k-1}})\det(h|_{F_{k+1}})}{\det(h|_{F_{k}})^{2}}/g_{X}.$

Therefore the Hitchin equation for $(E,\theta,h)$ and
$F=F_{k}(k=1,\cdots,n-1)$ becomes

$\displaystyle-\sqrt{-1}\Lambda_{g_{X}}\mathop{\rm
tr}\nolimits(F(h|_{F_{k}}))\geq-|\gamma_{k}|^{2}\frac{\det(h|_{F_{k-1}})\det(h|_{F_{k+1}})}{\det(h|_{F_{k}})^{2}}/g_{X},\quad
k=1,\cdots,n-1.$ (16)

Note that the Hitchin equation for $(E_{0},\theta_{0},h_{1})$ and $F=F_{k}$

$\displaystyle-\sqrt{-1}\Lambda_{g_{X}}\mathop{\rm
tr}\nolimits(F(h_{1}|_{F_{k}}))=-|\gamma_{k}|^{2}\frac{\det(h_{1}|_{F_{k-1}})\det(h_{1}|_{F_{k+1}})}{\det(h_{1}|_{F_{k}})^{2}}/g_{X},\quad
k=1,\cdots,n-1.$ (17)

Set $\displaystyle v_{k}=\log\frac{\det(h|_{F_{k}})}{\det(h_{1}|_{F_{k}})}$
for $1\leq k\leq n$ and $v_{0}=0$. The Laplacian with respect to $g_{X}$ is
$2\sqrt{-1}\Lambda_{g_{X}}\partial\bar{\partial}$, denoted by
$\triangle_{g_{X}}$. We obtain

$\frac{1}{2}\triangle_{g_{X}}v_{k}+(e^{v_{k-1}+v_{k+1}-2v_{k}}-1)\cdot|\gamma_{k}|^{2}\frac{\det(h_{1}|_{F_{k-1}})\det(h_{1}|_{F_{k+1}})}{\det(h_{1}|_{F_{k}})^{2}}/g_{X}\geq
0,\quad k=1,\cdots,n-1.$ (18)

Let

$c_{k}=|\gamma_{k}|^{2}\frac{\det(h_{1}|_{F_{k-1}})\det(h_{1}|_{F_{k+1}})}{\det(h_{1}|_{F_{k}})^{2}}/g_{X}\int_{0}^{1}e^{(1-t)(v_{k-1}+v_{k+1}-2v_{k})}dt,\quad
k=1,\cdots,n-1.$

Then $v_{k}$’s satisfy

$\displaystyle\frac{1}{2}\triangle_{g_{X}}v_{k}+c_{k}(v_{k-1}-2v_{k}+v_{k+1})$
$\displaystyle\geq$ $\displaystyle 0,\quad k=1,\cdots,n-1.$ (19)

By the assumption on the boundary $\partial X$, $v_{k}=0,k=1,\cdots,n-1.$ It
is easy to check that the above system of equations satisfies the assumptions
in Lemma 3.8. Moreover, $(1,1,\cdots,1)$ is indeed a supersolution of the
system (19). Then one can apply Lemma 3.8 and obtain $v_{k}\leq
0,k=1,\cdots,n-1$.

###### Lemma 3.8

([Sir09, Theorem 1]) Let $(X,g)$ be a Riemannian manifold with boundary. For
each $1\leq i\leq n$, let $u_{i}$ be a $C^{2}$ real-valued function on $X$
satisfying

$\displaystyle\triangle_{g}u_{i}+\sum_{j=1}^{n}c_{ij}u_{j}\geq 0,\quad 1\leq
i\leq n,\quad\text{in $X$},$

where $c_{ij}$ are continuous functions on $X$, $1\leq i,j\leq n$, satisfying
$(a)$ cooperative: $c_{ij}\geq 0,~{}i\neq j$,
$(b)$ fully coupled: the index set $\\{1,\cdots,n\\}$ cannot be split up in
two disjoint nonempty sets $\alpha,\beta$ such that $c_{ij}\equiv 0$ for
$i\in\alpha,j\in\beta.$

Suppose that there exists a supersolution
$(\psi_{1},\psi_{2},\cdots,\psi_{n})$ satisfying $\psi_{i}\geq 1$ of the above
system, i.e.,

$\displaystyle\triangle_{g}\psi_{i}+\sum_{j=1}^{n}c_{ij}\psi_{j}\leq 0,\quad
1\leq i\leq n.$

Then

$\sup_{X}u_{i}\leq\sup_{\partial X}u_{i},\quad 1\leq i\leq n.$

###  3.4 Domination property and the existence of harmonic metrics

We assume that $X$ is non-compact.

Let $(E,\theta)$, $(E_{0},\theta_{0})$ be as in §3.2. Moreover, we assume the
following.

###### Condition 3.9

There exists a harmonic metric $h_{0}$ of $(E_{0},\theta_{0})$ such that the
decomposition $E_{0}=\oplus_{i=1}^{n}\mathop{\rm Gr}\nolimits^{F}_{i}(E)$ is
orthogonal with respect to $h_{0}$. Note that $X$ has to be hyperbolic, see
[LM20a, Lemma 3.13].

Let $\mathop{\rm Harm}\nolimits^{dom}(E,\theta:h_{0})$ denote the set of
harmonic metrics $h$ of $(E,\theta)$ such that (i) $h$ weakly dominates
$h_{0}$, (ii) $\det(h)=\det(h_{0})$. We shall prove the following theorem in
§3.4.5 after the preliminaries in §3.4.1–§3.4.4.

###### Theorem 3.10

  * •

$\mathop{\rm Harm}\nolimits^{dom}(E,\theta:h_{0})$ is not empty.

  * •

$\mathop{\rm Harm}\nolimits^{dom}(E,\theta:h_{0})$ is compact in the following
sense: any sequence $h_{i}$ in $\mathop{\rm
Harm}\nolimits^{dom}(E,\theta:h_{0})$ contains a subsequence $h_{i}^{\prime}$
such that the sequence $h_{i}^{\prime}$ and their derivatives are convergent
on any relatively compact open subset $K$ of $X$.

Let $(E,\theta,C),(E_{0},\theta_{0},C)$ be defined in §3.2.3. In addition to
Condition 3.9, we assume the following.

###### Condition 3.11

$h_{0}$ is compatible with $C$.

Let $\mathop{\rm Harm}\nolimits^{dom}(E,\theta,C:h_{0})$ denote the set of
harmonic metrics $h$ of $(E,\theta)$ such that (i) $h$ weakly dominates
$h_{0}$, (ii) $h$ is compatible with $C$. We shall also prove the following
theorem in §3.4.5.

###### Theorem 3.12

$\mathop{\rm Harm}\nolimits^{dom}(E,\theta,C:h_{0})$ is non-empty and compact.

####  3.4.1 Preliminary from linear algebra

Let $P$ be an upper triangular $n\times n$ matrix with non-vanishing diagonal
terms.

Let $A$ be an $n\times n$ matrix with

$A_{j,k}=0(j>k+1),\quad A_{k+1,k}\neq 0.$

Set

$|A|:=\max_{j,k}|A_{j,k}|,\quad\widetilde{|A|}=\max_{1\leq k\leq
n-1}|(A_{k+1,k})^{-1}|.$

In this section, our goal is to show the following.

###### Proposition 3.13

Suppose $|P^{-1}AP|\leq c,|P_{1,1}|\geq d,\det(P)\leq e$, then there exists a
constant $C=C(|A|,\widetilde{|A|},c,d,e)$ such that

$|(P^{-1})_{i,j}|+|P_{i,j}|\leq C.$

The proof of Proposition 3.13 follows from Proposition 3.16 and 3.17.

First we investigate the properties of $P^{-1}$ in terms of $P$.

###### Lemma 3.14

  * •

$(P^{-1})_{i,j}=0$ for $i>j$.

  * •

$(P^{-1})_{j,j}=(P_{j,j})^{-1}$ for $j=1,\cdots,n.$

  * •

For $1\leq i<j\leq n$ and $m\in\mathbb{Z}_{\geq 1}$, let
$\mathcal{S}_{m}(i,j)$ denote the set of
$\mathbf{i}=(i_{0},i_{1},\cdots,i_{m})\in\mathbb{Z}_{\geq 1}^{m}$ such that
$i_{0}=i<i_{1}<\cdots<i_{m}=j$. Then,

$(P^{-1})_{i,j}=\sum_{m\geq
1}\sum_{\mathbf{i}\in\mathcal{S}_{m}(i,j)}(-1)^{m}\prod_{p=0}^{m}(P_{i_{p},i_{p}})^{-1}\prod_{p=0}^{m-1}P_{i_{p},i_{p+1}}.$

Proof Let $Q$ be the diagonal matrix such that $Q_{i,i}=P_{i,i}$. We set
$R=P-Q$, which is strictly upper triangular matrix. Let $I$ denote the
identity matrix. Because $P=Q(I+Q^{-1}R)$, we obtain

$P^{-1}=Q^{-1}+\sum_{m\geq 1}(-1)^{m}(Q^{-1}R)^{m}Q^{-1}.$

Then, the claims of the lemma are obvious.

###### Lemma 3.15

Assume $|(P_{i,i})^{-1}|\leq B_{1}$. Suppose $|P_{i,i+t}|\leq B_{2}$ for all
$0\leq t\leq t_{0}$, then

$|(P^{-1})_{i,i+t}|\leq c\sum_{m=0}^{t}B_{1}^{m+1}B_{2}^{m}$

is bounded by a constant $c=c(n)$ for all $0\leq t\leq t_{0}$.

Proof Note that each term of the formula of $P^{-1}_{i,i+t}$ involves terms of
products of $(P_{i_{p},i_{p}})^{-1}$ and $P_{i_{p},i_{p+1}}$ for $i\leq
i_{p}<i_{p+1}\leq j.$ By assumption, all such terms are bounded by $B$ since
$i_{p+1}\leq i_{p}+t_{0}$.

###### Proposition 3.16

Assume $B_{1}^{-1}\leq|P_{i,i}|\leq B_{1}$ and suppose $|P^{-1}AP|\leq B_{2}.$
Then we have

$|(P^{-1})_{i,j}|+|P_{i,j}|\leq C,\quad 1\leq i,j\leq n$

for some constant $C=C(n,B_{1},B_{2},|A|,\widetilde{|A|}).$

Proof It is enough to estimate $P_{i,j}$ with $i\leq j.$ We prove by induction
on $j-i$. First of all, $P_{i,i}$ satisfies the estimates by assumption.
Assume that

$|P_{i,i+t}|\leq C(t_{0}),\quad 1\leq i\leq n,0\leq t\leq t_{0}.$ (20)

We are going to show $|P_{i,i+t_{0}+1}|\leq C.$

By Assumption (20) and Lemma 3.15, $|(P^{-1})_{i,i+t}|\leq C(t_{0})$ for all
$i,0\leq t\leq t_{0}$. We set

$\mathcal{T}(i,t_{0})=\bigl{\\{}(\ell,k)\,\big{|}\,i-1\leq\ell-1\leq k\leq
i+t_{0}\bigr{\\}},\quad\mathcal{T}^{\prime}(i,t_{0})=\mathcal{T}(i,t_{0})\setminus\\{(i,i-1),(i+t_{0}+1,i+t_{0})\\}.$

For any $(k,\ell)\in\mathcal{T}^{\prime}(i,t_{0})$, we have $\ell-i\leq t_{0}$
and $i+t_{0}-k\leq t_{0}$. We obtain

$\displaystyle(P^{-1}AP)_{i,i+t_{0}}$ $\displaystyle=$
$\displaystyle\sum_{(l,k)\in\mathcal{T}(i,t_{0})}(P^{-1})_{i,l}A_{l,k}P_{k,i+t_{0}}$
$\displaystyle=$
$\displaystyle\sum_{(l,k)\in\mathcal{T}^{\prime}(i,t_{0})}(P^{-1})_{i,l}A_{l,k}P_{k,i+t_{0}}+(P^{-1})_{i,i+t_{0}+1}A_{i+t_{0}+1,i+t_{0}}P_{i+t_{0},i+t_{0}}+(P^{-1})_{i,i}A_{i,i-1}P_{i-1,i+t_{0}}$
$\displaystyle=$
$\displaystyle\sum_{(l,k)\in\mathcal{T}^{\prime}(i,t_{0})}(P^{-1})_{i,l}A_{l,k}P_{k,i+t_{0}}+\sum_{m\geq
2}\sum_{\mathbf{i}\in\mathcal{S}_{m}(i,i+t_{0}+1)}(-1)^{m}A_{i+t_{0}+1,i+t_{0}}P_{i+t_{0},i+t_{0}}\prod_{p=0}^{m}(P_{i_{0},i_{0}})^{-1}\prod_{p=0}^{m-1}P_{i_{p},i_{p+1}}$
$\displaystyle-
P_{i,i+t_{0}+1}P_{i,i}^{-1}P_{i+t_{0}+1,i+t_{0}+1}^{-1}A_{i+t_{0}+1,i+t_{0}}P_{i+t_{0},i+t_{0}}+(P^{-1})_{i,i}A_{i,i-1}P_{i-1,i+t_{0}}.$

Here, we formally put $A_{1,0}=A_{n+1,n}=0$,
$(P^{-1})_{i,n+1}=P_{i,n+1}=P_{0,1+t_{0}}=0$ and $P_{n+1,n+1}=1$. The first
and second terms of the right hand side in the formula of
$(P^{-1}AP)_{i,i+t_{0}}$ only involve $P_{\mu\nu}$ where $\nu\leq\mu+t_{0}$
and $A_{lk}$. By the formula in the case $i=1$, we obtain an estimate for
$P_{1,t_{0}+2}$. Inductively, we obtain an estimate for $P_{i,i+t_{0}+1}$
$(i=2,\ldots,n-t_{0}-1)$ by using the formula in the case $i$.

###### Proposition 3.17

Suppose $|(P^{-1}AP)_{k+1,k}|\leq c$, $P_{1,1}\geq d$ and $\det(P)\leq e$.
Then

$d\bigl{(}c\widetilde{|A|}\bigr{)}^{1-i}\leq|P_{i,i}|\leq(ed^{-i+1})^{\frac{1}{n+1-i}}(c\widetilde{|A|})^{-\frac{(i-1)(i-2)}{2(n+1-i)}}(c\widetilde{|A|})^{-\frac{1}{2}(n-i)}$
(21)

Proof We set $\widetilde{c}=c\widetilde{|A|}$ to simplify the notation. Recall
that $(P^{-1}AP)_{k+1,k}=(P^{-1})_{k+1,k+1}A_{k+1,k}P_{k,k}.$ Thus

$|(P^{-1})_{k+1,k+1}P_{k,k}|\leq\widetilde{c}.$

By Lemma 3.14, $(P^{-1})_{k+1,k+1}=P_{k+1,k+1}^{-1}$. Thus

$|P_{k,k}|\leq\widetilde{c}|P_{k+1,k+1}|.$

So

$|P_{j,j}|\geq\widetilde{c}^{j-1}|P_{1,1}|\geq\widetilde{c}^{j-1}d.$

Because $|P_{j,j}|\geq\widetilde{c}^{j-i}|P_{i,i}|$ for $i\leq j$, we obtain

$\prod_{j=1}^{i-1}(\widetilde{c}^{j-1}d)\cdot\prod_{j=i}^{n}(\widetilde{c}^{j-i}|P_{i,i}|)\leq\prod_{j=1}^{n}|P_{j,j}|=e.$

It implies

$|P_{i,i}|^{n+1-i}\leq
ed^{-i+1}\widetilde{c}^{-\frac{1}{2}(i-1)(i-2)}\cdot\widetilde{c}^{-\frac{1}{2}(n-i)(n+1-i)}.$

Thus, we obtain the right inequality in (21).

####  3.4.2 Notation

Let $V$ be a complex vector space equipped with a base
$\mathbf{e}=(e_{1},\cdots,e_{n})$. Let $h$ be any Hermitian metric of $V$. By
applying the Gram-Schmidt process to the base $\mathbf{e}$, we obtain a base
$\mathbf{v}(h)=(v_{1}(h),\cdots,v_{n}(h)).$ Let $P(h)=(P(h)_{j,k})$ be the
matrix determined by $\mathbf{v}=\mathbf{e}P(h)$. Then $P(h)_{j,k}=0(j>k)$,
i.e.,

$v_{k}(h)=\sum_{j\leq k}P(h)_{j,k}e_{j}.$

Let $P^{-1}(h)=(P^{-1}(h)_{j,k})$ be the inverse matrix of $P(h)$. In terms of
the frame ${\boldsymbol{e}}$, the metric $h$ is represented by the matrix
$h({\boldsymbol{e}})=(P(h)^{-1})^{t}\cdot\overline{P(h)^{-1}}$.

We use a similar notation for a vector bundle equipped with a frame and a
Hermitian metric.

####  3.4.3 Local estimate in the nowhere vanishing case

We set $U(R)=\\{z\in\mathbb{C}||z|<R\\}$ and
$\overline{U}(R)=\\{z\in\mathbb{C}||z|\leq R\\}$ for any $R>0$.

Let $R_{1}<R_{2}$. Let $E=\oplus_{i=1}^{n}\mathcal{O}_{U(R_{2})}e_{i}$. Let
$f$ be an endomorphism of $E$. Let $A$ be the matrix determined by
$f(e_{j})=\sum_{i=1}^{n}A_{ij}e_{i}.$ That is $A$ is the matrix representation
of $f$ in terms of ${\boldsymbol{e}}.$ Note that $|f|_{h}=|P(h)^{-1}AP(h)|.$
We assume the following.

###### Condition 3.18

  * •

$A_{ij}=0(i>j+1)$.

  * •

$A_{j+1,j}(j=1,\cdots,n-1)$ are nowhere vanishing on $\overline{U}(R_{1}).$

  * •

$|\mathop{\rm tr}\nolimits(f^{l})|(l=1,\cdots,n)$ are bounded on $U(R_{2}).$

We set

$B_{1}(f)=\max_{1\leq l\leq n}\sup_{U(R_{2})}|\mathop{\rm
tr}\nolimits(f^{l})|$ $B_{2}(f)=\min_{1\leq j\leq
n-1}\min_{\overline{U}(R_{1})}|A_{j+1,j}|>0.$

We obtain the Higgs field $\theta=fdz$ of $E$. We recall the following lemma.

###### Lemma 3.19

([LM20a, Proposition 3.12]) There exists $C_{1}>0$ depending only on
$R_{1},R_{2},n$ and $B_{1}(f)$ such that $|f|_{h}\leq C_{1}$ on $U(R_{1})$ for
any harmonic metric $h$ of $(E,\theta)$ on $U(R_{2}).$

Let $f_{0}$ be the endomorphism of $E$ determined by
$f_{0}(e_{j})=A_{j+1,j}e_{j+1}$ for $j=1,\cdots,n-1$ and $f_{0}(e_{n})=0.$ We
obtain the Higgs field $\theta_{0}=f_{0}dz$ of $E$. Assume that there exists a
harmonic metric $h_{0}$ of $(E,\theta_{0})$ such that the decomposition
$E=\oplus_{i=1}^{n}\mathcal{O}_{U(R_{2})}e_{i}$ is orthogonal.

Let $\mathop{\rm Harm}\nolimits^{dom}(E,\theta:h_{0})$ denote the set of
harmonic metrics $h$ of $(E,\theta)$ such that (i) $h$ weakly dominates
$h_{0}$, (ii) $\det(h)=\det(h_{0})$. For two Hermitian metrics $h_{1},h_{2}$
on $E$, let $s(h_{1},h_{2})$ be the automorphism of $E$ such that

$h_{2}(u,v)=h_{1}(s(h_{1},h_{2})u,v),$

for any two sections $u,v$ of $E$. In terms of the frame ${\boldsymbol{e}}$,
$s(h_{1},h_{2})$ is represented by a matrix $J(h_{1},h_{2})$. Then
$J(h_{1},h_{2})$ satisfies $h_{2}({\boldsymbol{e}})=J(h_{1},h_{2})^{t}\cdot
h_{1}({\boldsymbol{e}}).$ So

$J(h_{1},h_{2})=P(h_{1})\cdot\overline{P(h_{1})^{t}}\cdot\overline{(P(h_{2})^{-1})^{t}}\cdot
P(h_{2})^{-1}.$

We obtain the following proposition.

###### Proposition 3.20

There exists $C_{2}>0$ depending only on $n,R_{i}(i=1,2)$ and
$B_{k}(f)(k=1,2)$ such that

$|s(h_{0},h)|_{h_{0}}+|s(h_{0},h)^{-1}|_{h_{0}}\leq C_{2}$

for any $h\in\mathop{\rm Harm}\nolimits^{dom}(E,\theta:h_{0}).$

Proof From Lemma 3.19, we have

$|f|_{h}=|P(h)^{-1}AP(h)|\leq C_{1}.$

Since $h$ weakly dominates $h_{0},$ we obtain that
$|P(h)_{1,1})\geq|P(h_{0})_{1,1}|\geq C_{1}^{\prime}$ for some positive
constant $C_{1}^{\prime}$. And $\det(P(h))=\det(P(h_{0}))\leq C_{2}^{\prime},$
for some positive constant $C_{2}^{\prime}$.

From Proposition 3.13, we have $|P(h)|+|P(h)^{-1}|\leq C_{3}^{\prime}$, for
some positive constant $C_{3}^{\prime}$.

The rest follows from the matrix expression $J(h_{0},h)$ of $s(h_{0},h)$ is

$J(h_{0},h)=P(h_{0})\cdot\overline{P(h_{0})^{t}}\cdot\overline{(P(h)^{-1})^{t}}\cdot
P(h)^{-1},$ $|s(h_{0},h)|_{h_{0}}=|P(h_{0})^{-1}J(h_{0},h)P(h_{0})|$

and $P(h_{0}),P(h_{0})^{-1}$ are bounded.

####  3.4.4 Local estimate in the general case

Let $X$, $(E,\theta)$, $(E_{0},\theta_{0})$ and $h_{0}$ be as in §3.4. We fix
an isomorphism $E\simeq E_{0}$ as in §3.2.1, and we regard $h_{0}$ as a
Hermitian metric of $E$.

Let $K_{1}\subset X$ be a relatively compact open subset. Let $K_{2}$ be a
relatively compact open neighbourhood of $\overline{K}_{1}$ in $X$.

###### Proposition 3.21

There exists $C_{3}>0$ such that the following holds on $K_{1}$ for any
$h\in\mathop{\rm Harm}\nolimits^{dom}((E,\theta:h_{0})|_{K_{2}})$:

$|s(h_{0},h)|_{h_{0}}+|s(h_{0},h)^{-1}|_{h_{0}}\leq C_{3}.$ (22)

Proof By making $K_{1}$ larger if necessary, we may assume that
$A_{j+1,j}(j=1,\cdots,r-1)$ are nowhere vanishing on a neighbourhood $N$ of
$\partial K_{1}$. Let $N^{\prime}$ be a relatively compact neighbourhood of
$\partial K_{1}$ in $N$. By using Proposition 3.20, we can prove that there
exists $C_{4}>0$ such that the following holds on $N^{\prime}$ for any
$h\in\mathop{\rm Harm}\nolimits^{dom}((E,\theta:h_{0})|_{N})$:

$|s(h_{0},h)|_{h_{0}}+|s(h_{0},h)^{-1}|_{h_{0}}\leq C_{4}.$ (23)

Let $h_{1}$ be a harmonic metric of $(E,\theta)|_{\overline{K}_{2}}$ such that
$\det(h_{1})=\det(h_{0})$. There exists $C_{5}>0$ such that the following
holds on $\overline{K}_{2}$:

$|s(h_{1},h_{0})|_{h_{1}}+|s(h_{1},h_{0})^{-1}|_{h_{1}}\leq C_{5}.$ (24)

By Equation (23) and (24), there exists $C_{6}>0$ such that the following
holds for any $h\in\mathop{\rm
Harm}\nolimits^{dom}((E,\theta:h_{0})|_{K_{2}})$ on $N^{\prime}:$

$|s(h_{1},h)|_{h_{1}}+|s(h_{1},h)^{-1}|_{h_{1}}\leq C_{6}.$

Because $\log\mathop{\rm tr}\nolimits(s(h_{1},h))$ are subharmonic, the
following holds on $K_{1}:$

$|s(h_{1},h)|_{h_{1}}+|s(h_{1},h)^{-1}|_{h_{1}}\leq C_{6}.$

Therefore, together with Equation (24), we obtain Equation (22).

####  3.4.5 Proof of Theorem 3.10 and Theorem 3.12

Let $X_{i}(i=1,2,\cdots)$ be a smooth exhaustion family of $X$. Let $h^{(i)}$
be the harmonic metrics of $(E,\theta)|_{X_{i}}$ such that $h^{(i)}|_{\partial
X_{i}}=h_{0}|_{\partial X_{i}}$.

###### Theorem 3.22

$h^{(i)}$ contains a convergent subsequence.

Proof By Proposition 3.7, $h^{(i)}$ weakly dominates $h_{0}$. By Proposition
2.4 and Proposition 3.21, $h^{(i)}$ contains a convergent subsequence.

Hence, we obtain the first claim of Theorem 3.10. We also obtain the second
claim of Theorem 3.10 from Proposition 2.4 and the argument in the proof of
Proposition 2.4.

Suppose moreover, $E,\theta,C,E_{0},\theta_{0},h_{0}$ are in the setting of
Theorem 3.12. By the uniqueness of solutions to the Dirichlet problem and
$h_{0}$ is compatible with $C$, $h^{(i)}$ is also compatible with $C$. So the
limit metric is again compatible with $C$. So we obtain the first claim of
Theorem 3.12. The second claim of Theorem 3.12 follows from the second claim
of Theorem 3.10 and that compatibility with $C$ is preserved under limit.

##  4 Uniqueness in a bounded case

###  4.1 Statement

Let $X$ be a Riemann surface. Let $g_{X}$ be a complete Kähler metric whose
Gauss curvature is bounded below.

We fix a line bundle $K_{X}^{1/2}$ and an isomorphism $K_{X}^{1/2}\otimes
K_{X}^{1/2}\simeq K_{X}$. We set

$\mathbb{K}_{X,n}=\bigoplus_{i=1}^{n}K_{X}^{(n+1-2i)/2}.$

We set $F_{j}\mathbb{K}_{X,n}=\bigoplus_{i\leq j}K_{X}^{(n+1-2i)/2}$. We
obtain the Hermitian metric $h_{X}=\bigoplus g_{X}^{-(n+1-2i)/2}$ of
$\mathbb{K}_{X,n}$. Let $C$ be a holomorphic non-degenerate symmetric pairing
of $\mathbb{K}_{X,n}$ which is compatible with $h_{X}$.

Let $\theta$ be a Higgs field of $\mathbb{K}_{X,n}$. We assume the following.

  * •

$\theta(F_{j}\mathbb{K}_{X,n})\subset F_{j+1}\mathbb{K}_{X,n}\otimes K_{X}$.
Moreover, the induced morphisms $\phi_{j}\colon\mathop{\rm
Gr}\nolimits^{F}_{j}\mathbb{K}_{X,n}\to\mathop{\rm
Gr}\nolimits^{F}_{j+1}\mathbb{K}_{X,n}\otimes K_{X}$ are the identity
morphisms under the natural isomorphisms $\mathop{\rm
Gr}\nolimits^{F}_{j}\mathbb{K}_{X,n}=K_{X}^{(n+1-2j)/2}=\mathop{\rm
Gr}\nolimits^{F}_{j+1}\mathbb{K}_{X,n}\otimes K_{X}$.

  * •

$\theta$ is bounded with respect to $h_{X}$ and $g_{X}$.

  * •

$\theta$ is self-adjoint with respect to $C$.

We shall prove the uniqueness of harmonic metrics which are compatible with
$C$ and mutually bounded with $h_{X}$.

###### Theorem 4.1

Let $h_{1}$ and $h_{2}$ be harmonic metrics of $(\mathbb{K}_{X,n},\theta)$.
Suppose that both $h_{i}$ are compatible with $C$, and that both $h_{i}$ are
mutually bounded with $h_{X}$. Then, $h_{1}=h_{2}$ holds.

####  4.1.1 A characterization of the mutual boundedness with $h_{X}$

We also have the following characterization for a harmonic metric to be
mutually bounded with $h_{X}$.

###### Proposition 4.2

Let $h$ be a harmonic metric of $(\mathbb{K}_{X,n},\theta)$ such that
$\det(h)=1$. Then, $h$ is mutually bounded with $h_{X}$ if and only there
exists $b>0$ such that $h_{|F_{1}\mathbb{K}_{X,n}}\leq
bh_{X|F_{1}\mathbb{K}_{X,n}}$.

Proof The “only if” part of Proposition 4.2 is clear. Let us prove the “if”
part. Let $h$ be a harmonic metric of $(\mathbb{K}_{X,n},\theta)$ such that
$\det(h)=1$. Because the spectral curve of the Higgs bundle
$(\mathbb{K}_{X,n},\theta)$ is bounded with respect to $g_{X}$, we obtain the
following lemma from [LM20a, Proposition 3.12].

###### Lemma 4.3

$|\theta|_{h,g_{X}}$ is bounded on $X$.

Let $x$ be any point of $X$. Let $\tau$ be a base of $K_{X|x}$ such that
$|\tau|_{g_{X|x}}=1$. By setting $e_{i}=\tau^{(n+1-2i)/2}$ $(i=1,\ldots,n)$,
we obtain an orthonormal frame ${\boldsymbol{e}}=(e_{1},\ldots,e_{n})$ of
$\mathbb{K}_{X,n|x}$ with respect to $h_{X|x}$. Let $A$ be the matrix
determined by $\theta_{|x}({\boldsymbol{e}})={\boldsymbol{e}}\cdot A\,\tau$.
Because $\theta$ is bounded with respect to $h_{X}$ and $g_{X}$, there exists
$B_{1}>0$ which is independent of $x$ such that $|A|\leq B_{1}$. Moreover,
$A_{k+1,k}=1$ for $k=1,\ldots,n-1$.

By applying the Gram-Schmidt process to the frame ${\boldsymbol{e}}$ and the
metric $h_{|x}$, we obtain the base ${\boldsymbol{v}}$ of $\mathbb{K}_{X,n|x}$
which is orthonormal with respect to $h_{|x}$. Let $P$ be the matrix
determined by ${\boldsymbol{v}}={\boldsymbol{e}}\cdot P$. Because $\theta$ is
bounded with respect to $h$ and $g_{X}$, there exists $B_{2}>0$, which is
independent of $x$, such that $|P^{-1}AP|\leq B_{2}$. Because
$h_{|F_{1}\mathbb{K}_{X,n}}\leq bh_{X|F_{1}\mathbb{K}_{X,n}}$, we obtain
$P_{1,1}\geq b^{-1}$. Because $\det(h)=1$, we have $\det(P)=1$. By Proposition
3.13, there exists $B_{3}>0$ which is independent of $x$ such that
$|P|+|P^{-1}|\leq B_{3}$. Therefore, there exists $B_{4}$ such that the
following holds on $X$:

$\bigl{|}s(h,h_{X})\bigr{|}_{h_{X}}+\bigl{|}s(h,h_{X})^{-1}\bigr{|}_{h_{X}}\leq
B_{4}$

Thus, we obtain Proposition 4.2.

###  4.2 Preliminary from Linear algebra

####  4.2.1 Cyclic vectors

Let $V$ be an $n$-dimensional complex vector space equipped with an
endomorphism $f$. A vector $v\in V$ is called an $f$-cyclic vector if
$v,f(v),\ldots,f^{n-1}(v)$ generate $V$. The following proposition is well
known. (For example, see [Rom08, §6,§7].)

###### Proposition 4.4

There exists an $f$-cyclic vector if and only if the characteristic polynomial
of $f$ equals the minimal polynomial of $f$.

###### Corollary 4.5

Suppose that there exists an $f$-cyclic vector. For any eigenvalue $\alpha$ of
$f$, the space of eigen vectors associated with $\alpha$ is one dimensional.

Let $h$ be a Hermitian metric of $V$. For any $v\in V$, we set
$\omega(f,v)=v\wedge f(v)\wedge\cdots\wedge f^{n-1}(v)$. Then, $v$ is an
$f$-cyclic vector of $V$ if and only if $\omega(f,v)\neq 0$. We always have
$|\omega(f,v)|_{h}\leq|f|_{h}^{n(n-1)/2}|v|^{n}_{h}$.

###### Lemma 4.6

Let $A>0$ and $\rho>0$. There exists $\epsilon_{0}(n,A,\rho)>0$ depending only
on $n$, $A$ and $\rho$, such that the following holds.

  * •

Suppose that $|f|_{h}\leq A$ and that $\rho|v|_{h}^{n}\leq|\omega(f,v)|_{h}$
for a non-zero element $v\in V$. Let $f_{1}$ be an endomorphism of $V$ such
that $|f-f_{1}|_{h}\leq\epsilon_{0}(n,A,\rho)$. Then,
$\frac{1}{2}\rho|v|_{h}^{n}<|\omega(f_{1},v)|_{h}$. In particular, $f_{1}$
also has a cyclic vector.

Proof If $|f-f_{1}|_{h}<1$, we obtain

$\bigl{|}f_{1}^{j}(v)-f^{j}(v)\bigr{|}_{h}\leq\sum_{k=1}^{j}C(j,k)|f|^{j-k}_{h}|f-f_{1}|^{k}_{h}|v|_{h}\leq|f-f_{1}|_{h}(1+|f|_{h})^{j}|v|_{h}.$

Here, $C(j,k)$ denote the binomial coefficients. We obtain

$\Bigl{|}v\wedge f_{1}(v)\wedge\cdots\wedge
f_{1}^{j-1}(v)\wedge(f_{1}^{j}(v)-f^{j}(v))\wedge f^{j+1}(v)\wedge\cdots
f^{n-1}(v)\Bigr{|}_{h}\leq|v|_{h}^{n}\cdot|f_{1}|_{h}^{j(j-1)/2}\cdot|f^{j}-f_{1}^{j}|_{h}\cdot|f|_{h}^{n(n-1)/2-j(j+1)/2}\\\
\leq|v|_{h}^{n}\cdot|f-f_{1}|_{h}\cdot(1+|f|_{h})^{n(n-1)/2}.$ (25)

We obtain

$\bigl{|}\omega(f,v)-\omega(f_{1},v)\bigr{|}_{h}\leq
n(1+|f|_{h})^{n(n-1)/2}|f-f_{1}|_{h}\cdot|v|_{h}^{n}.$

Then, the claim of the lemma is clear.

####  4.2.2 Real structure and self-adjoint endomorphisms

Let $C$ be a non-degenerate symmetric pairing of a finite dimensional complex
vector space $V$. Let $f$ be an endomorphism of $V$ such that $f$ is self-
adjoint with respect to $C$, i.e., $C(fu,v)=C(u,fv)$ for any $u,v\in V$. There
exists the generalized eigen decomposition
$V=\bigoplus_{\alpha\in{\mathbb{C}}}V_{\alpha}$, where $V_{\alpha}$ denote the
space of generalized eigen vectors of $f$ associated with $\alpha$. The
following lemma is well known.

###### Lemma 4.7

If $\alpha\neq\beta$, then $V_{\alpha}$ and $V_{\beta}$ are orthogonal with
respect to $C$.

Proof We explain a proof just for the convenience of readers. For
$j\in{\mathbb{Z}}_{\geq 1}$, we set

$\mathcal{F}_{j}V_{\alpha}=\bigl{\\{}u_{\alpha}\in
V_{\alpha}\,\big{|}\,(f-\alpha\mathop{\rm
id}\nolimits_{V})^{j}u_{\alpha}=0\bigr{\\}}.$

Let us prove that $\mathcal{F}_{i}V_{\alpha}$ and $\mathcal{F}_{j}V_{\beta}$
$(i+j=\ell)$ are orthogonal by an induction on $\ell$. Let us consider the
case $\ell=2$. For $u_{\alpha}\in\mathcal{F}_{1}V_{\alpha}$ and
$v_{\beta}\in\mathcal{F}_{1}V_{\beta}$, we obtain $\alpha
C(u_{\alpha},v_{\beta})=C(f(u_{\alpha}),v_{\beta})=C(u_{\alpha},f(v_{\beta}))=\beta
C(u_{\alpha},v_{\beta})$. It implies $C(u_{\alpha},v_{\beta})=0$. Suppose that
we have already proved the claim in the case $i+j=\ell$, and let us consider
the case $i+j=\ell+1$. For $u_{\alpha}\in\mathcal{F}_{i}V_{\alpha}$ and
$v_{\beta}\in\mathcal{F}_{j}V_{\beta}$, we have $f(u_{\alpha})-\alpha
u_{\alpha}\in\mathcal{F}_{i-1}V_{\alpha}$ and $f(u_{\beta})-\beta
u_{\beta}\in\mathcal{F}_{j-1}V_{\beta}$. By the assumption of the induction,
we obtain $C(f(u_{\alpha})-\alpha u_{\alpha},v_{\beta})=0$ and
$C(u_{\alpha},f(v_{\beta})-\beta v_{\beta})=0$. Because
$C(f(u_{\alpha}),v_{\beta})=C(u_{\alpha},f(v_{\beta}))$, we obtain
$C(u_{\alpha},v_{\beta})=0$.

Let $h$ be a Hermitian metric compatible with $C$. Let $\kappa$ be the real
structure of $V$ induced by $C$ and $h$. Let $W\subset V$ be a vector subspace
such that (i) $f(W)\subset W$ and $f(\kappa(W))\subset\kappa(W)$, (ii)
$W\cap\kappa(W)=0$.

###### Proposition 4.8

We have either (i) $W=0$, or (ii) $f_{|W}$ and $f_{|\kappa(W)}$ have a common
eigenvalue.

Proof Suppose that there is no common eigenvalue of $f_{|W}$ and
$f_{|\kappa(W)}$. By Lemma 4.7, $W$ and $\kappa(W)$ are orthogonal with
respect to $C$. For any $u\in W$, we obtain $h(u,u)=C(u,\kappa(u))=0$, and
hence $W=0$.

###### Corollary 4.9

Suppose moreover that there exists an $f$-cyclic vector. Then, we obtain
$W=0$.

Proof If $W\neq 0$, then $f_{|W}$ and $f_{|\kappa(W)}$ have a common
eigenvalue $\alpha$. Hence, the dimension of the eigen space associated with
$\alpha$ is larger than $2$, which contradicts Corollary 4.5.

Let us explain how to use Corollary 4.9 in a simple case.

###### Proposition 4.10

Let $s$ be an automorphism of $V$ such that (i) $s$ is Hermitian and positive
definite with respect to $h$, (ii) $s$ is self-adjoint with respect to $C$,
(iii) $[f,s]=0$. Suppose that there exists an $f$-cyclic vector. Then, we
obtain $s=\mathop{\rm id}\nolimits_{V}$.

Proof There exists the eigen decomposition $V=\bigoplus_{a>0}V_{a}$ of $s$.
Because $[s,f]=0$, we obtain $f(V_{a})\subset V_{a}$ for any $a$. Recall
$\kappa(V_{a})=V_{a^{-1}}$ as in [LM22, Lemma 2.10]. Hence, we obtain
$V_{a}=0$ for any $a\neq 1$ by Corollary 4.9.

###### Remark 4.11

Theorem 4.12 below is a quantified version of Proposition 4.10.

###  4.3 An estimate

Let $V$ be a complex vector space equipped with a base
${\boldsymbol{e}}=(e_{1},\ldots,e_{n})$. Let $C$ be a non-degenerate symmetric
pairing of $V$. For $\rho>0$, let $\mathcal{H}(V,{\boldsymbol{e}},C;\rho)$ be
the space of Hermitian metrics $h$ of $V$ such that (i) $h$ are compatible
with $C$, (ii) $|e_{1}\wedge\cdots\wedge e_{n}|_{h}\geq\rho|e_{1}|^{n}$.

Let $f$ be an endomorphism of $V$ which is self-adjoint with respect to $C$.
Let $\mathcal{A}(f)=(\mathcal{A}(f)_{i,j})$ be the matrix representing $f$
with respect to ${\boldsymbol{e}}$, i.e.,
$f(e_{k})=\sum_{j=1}^{n}\mathcal{A}(f)_{j,k}e_{j}$. We assume that
$\mathcal{A}(f)_{j,k}=0$ $(j>k+1)$ and $\mathcal{A}(f)_{k+1,k}=1$, i.e.,

$f(e_{k})=e_{k+1}+\sum_{j\leq
k}\mathcal{A}(f)_{j,k}e_{j}\quad(k=1,\ldots,n-1),\quad f(e_{n})=\sum_{j\leq
n}\mathcal{A}(f)_{j,n}e_{j}.$

###### Theorem 4.12

Let $A>0$ and $\rho>0$. There exist $\epsilon_{1}(n,A,\rho)>0$ and
$C_{1}(n,A,\rho)>0$ depending only on $n$, $A$ and $\rho$ such that the
following holds for any $0<\epsilon<\epsilon_{1}(n,A,\rho)$:

  * •

Suppose $|f|_{h}\leq A$. Then, for any
$h,h^{\prime}\in\mathcal{H}(V,{\boldsymbol{e}},C;\rho)$ such that
$\bigl{|}[s(h,h^{\prime}),f]\bigr{|}_{h}\leq\epsilon$, we obtain

$|s(h,h^{\prime})-\mathop{\rm id}\nolimits_{V}|_{h}\leq
C_{1}(n,A,\rho)\epsilon.$

Proof Let $h,h^{\prime}\in\mathcal{H}(V,{\boldsymbol{e}},C;\rho)$. We obtain
the automorphism $s(h,h^{\prime})$ of $V$ determined by
$h^{\prime}(u,v)=h(s(h,h^{\prime})u,v)$ for any $u,v\in V$, which is self-
adjoint with respect to both $h$ and $h^{\prime}$. There exists the eigen
decomposition $V=\bigoplus_{a>0}V_{a}$ of $s(h,h^{\prime})$.

Let $\kappa$ be the real structure induced by $C$ and $h$. Note that
$\kappa(V_{a})=V_{a^{-1}}$. We set
$\mathcal{S}(h,h^{\prime}):=\\{a>1\,|\,V_{a}\neq 0\\}$. If
$\mathcal{S}(h,h^{\prime})=\emptyset$, we obtain $s(h,h^{\prime})=\mathop{\rm
id}\nolimits_{V}$. Let us consider the case where
$\mathcal{S}(h,h^{\prime})\neq\emptyset$.

Let $\nu_{1}$ be any positive number such that
$\nu_{1}\leq\min\\{1,\max\mathcal{S}(h,h^{\prime})-1\\}$. Let
$c_{1}<c_{2}<\cdots<c_{m}$ denote the elements of $\mathcal{S}(h,h^{\prime})$.
We set $c_{0}=1$. Because $|\mathcal{S}(h,h^{\prime})|\leq n/2$, there exists
$1\leq m(0)\leq m$ such that the following holds.

  * •

$c_{i}-c_{i-1}\leq\frac{1}{2}n^{-1}\nu_{1}$ for any $i<m(0)$.

  * •

$c_{m(0)}-c_{m(0)-1}>\frac{1}{2}n^{-1}\nu_{1}$.

We set $\mathcal{S}(h,h^{\prime};\nu_{1})_{0}=\\{c_{1},\ldots,c_{m(0)-1}\\}$
and $\mathcal{S}(h,h^{\prime};\nu_{1})_{1}=\\{c_{m(0)},\ldots,c_{m}\\}$.

###### Lemma 4.13

The set $\mathcal{S}(h,h^{\prime};\nu_{1})_{0}$ is contained in $\\{1<a\leq
2\\}$. The set $\mathcal{S}(h,h^{\prime};\nu_{1})_{1}$ is non-empty. For any
$a_{0}\in\mathcal{S}(h,h^{\prime};\nu_{1})_{0}\cup\\{1\\}$ and
$a_{1}\in\mathcal{S}(h,h^{\prime};\nu_{1})_{1}$, we obtain
$|a_{0}^{-1}-a_{1}^{-1}|\geq\frac{1}{12}n^{-1}\nu_{1}$.

Proof Because $\nu_{1}\leq 1$, we obtain the first claim. The second claim is
clear. For any $a_{0}\in\mathcal{S}(h,h^{\prime};\nu_{1})_{0}\cup\\{1\\}$ and
$a_{1}\in\mathcal{S}(h,h^{\prime};\nu_{1})_{1}$, we obtain

$|a_{0}^{-1}-a_{1}^{-1}|\geq\bigl{|}a_{0}^{-1}-(a_{0}+n^{-1}\nu_{1}/2)^{-1}\bigr{|}=|a_{0}|^{-1}|a_{0}+n^{-1}\nu_{1}/2|^{-1}\frac{1}{2}n^{-1}\nu_{1}\geq\frac{1}{2}\cdot\frac{1}{3}\cdot\frac{1}{2}n^{-1}\nu_{1}=\frac{1}{12}n^{-1}\nu_{1}.$

Thus, we obtain the third claim of Lemma 4.13.

We set

$W^{(\nu_{1})}=\bigoplus_{a\in\mathcal{S}(h,h^{\prime};\nu_{1})_{1}}V_{a},\quad\quad
V^{(\nu_{1})}=V_{1}\oplus\bigoplus_{a\in\mathcal{S}(h,h^{\prime};\nu_{1})_{0}}V_{a}\oplus\bigoplus_{a^{-1}\in\mathcal{S}(h,h^{\prime}:\nu_{1})_{0}}V_{a}.$

Because $\mathcal{S}(h,h^{\prime};\nu_{1})_{1}\neq\emptyset$, we have
$W^{(\nu_{1})}\neq 0$. We have $W^{(\nu_{1})}\cap\kappa(W^{(\nu_{1})})=0$ and
the decomposition

$V=V^{(\nu_{1})}\oplus W^{(\nu_{1})}\oplus\kappa(W^{(\nu_{1})}).$

We obtain the decomposition

$f=\sum_{U_{1},U_{2}=V^{(\nu_{1})},W^{(\nu_{1})},\kappa(W^{(\nu_{1})})}f_{U_{1},U_{2}},$

where $f_{U_{1},U_{2}}\in\mathop{\rm Hom}\nolimits(U_{2},U_{1})$. We set

$\widetilde{f}^{(\nu_{1})}=f_{V^{(\nu_{1})},V^{(\nu_{1})}}+f_{W^{(\nu_{1})},W^{(\nu_{1})}}+f_{\kappa(W^{(\nu_{1})}),\kappa(W^{(\nu_{1})})}.$

###### Lemma 4.14

$\widetilde{f}^{(\nu_{1})}$ is self-adjoint with respect to $C$.

Proof To simplify the description, we denote $W^{(\nu_{1})}$ by $W$. We set
$\widetilde{W}=W\oplus\kappa(W)$. The decomposition
$V^{(\nu_{1})}\oplus\widetilde{W}$ is orthogonal with respect to $C$. We
obtain the decomposition
$f=\sum_{U_{1},U_{2}=V^{(\nu_{1})},\widetilde{W}}f_{U_{2},U_{1}}$. Because $f$
is self-adjoint with respect to $C$, we obtain that
$f_{V^{(\nu_{1})},V^{(\nu_{1})}}$ and $f_{\widetilde{W},\widetilde{W}}$ are
self-adjoint with respect to $C$.

We have the decompositions $\widetilde{W}=W\oplus\kappa(W)$ and
$f_{\widetilde{W},\widetilde{W}}=\sum_{U_{1},U_{2}=W,\kappa(W)}f_{U_{2},U_{1}}$.
The restrictions of $C$ to $W$ and $\kappa(W)$ are $0$. Then, it is easy to
check that $f_{W,W}+f_{\kappa(W),\kappa(W)}$ is self-adjoint with respect to
$C$. Thus, we obtain Lemma 4.14.

###### Lemma 4.15

We have
$|f-\widetilde{f}^{(\nu_{1})}|_{h}\leq\nu_{1}^{-1}(10n)^{3}\bigl{|}[f,s(h,h^{\prime})]\bigr{|}_{h}$.

Proof We denote $s(h,h^{\prime})$ by $s$ to simplify the description. We have
the decomposition

$[s,f]=\sum_{U_{1},U_{2}=V^{(\nu_{1})},W,\kappa(W)}[s,f_{U_{1},U_{2}}].$

We have
$\bigl{|}[s,f_{U_{1},U_{2}}]\bigr{|}_{h}\leq\bigl{|}[s,f]\bigr{|}_{h}$.

Let $U_{1}\neq U_{2}$. Let $F_{U_{2},U_{1}}:\mathop{\rm
Hom}\nolimits(U_{1},U_{2})\to\mathop{\rm Hom}\nolimits(U_{1},U_{2})$ be
defined by

$F_{U_{2},U_{1}}(g)=[s,g]=s_{|U_{2}}\circ g-g\circ s_{|U_{1}}.$

For any eigenvalues $a_{i}$ $(i=1,2)$ of $s_{|U_{i}}$, we have
$|a_{1}-a_{2}|>(12n)^{-1}\nu_{1}$. Hence, $F_{U_{2},U_{1}}$ is invertible, and
$|F_{U_{2},U_{1}}^{-1}|_{h}\leq\nu_{1}^{-1}(12n)n^{2}$. Thus, we obtain Lemma
4.15.

By using a positive constant $\epsilon_{0}(n,A,\rho)$ in Lemma 4.6, we set

$\epsilon_{1}(n,A,\rho):=\frac{1}{2}(10n)^{-3}\epsilon_{0}(n,A,\rho),\quad
C_{1}(n,A,\rho):=n\epsilon_{1}(n,A,\rho)^{-1}.$

Let $0<\epsilon<\epsilon_{1}(n,A,\rho)$. Suppose
$\bigl{|}[s(h,h^{\prime}),f]\bigr{|}_{h}\leq\epsilon$. We set

$\nu_{2}:=\frac{1}{2}\epsilon_{1}(n,A,\rho)^{-1}\epsilon<\frac{1}{2}.$

If $\nu_{2}\leq\max\mathcal{S}(h,h^{\prime})-1$, we obtain

$|f-\widetilde{f}^{(\nu_{2})}|_{h}\leq\nu_{2}^{-1}(10n)^{3}\bigl{|}[f,s(h,h^{\prime})]\bigr{|}_{h}\leq\epsilon_{1}(n,A,\rho)(10n)^{3}\leq\epsilon_{0}(n,A,\rho).$

By Lemma 4.6, there exists an $\widetilde{f}^{(\nu_{2})}$-cyclic vector. But,
it contradicts $W^{(\nu_{2})}\neq 0$, according to Corollary 4.9. Hence, we
obtain $\max\mathcal{S}(h,h^{\prime})-1<\nu_{2}$. Then, we obtain
$|s-\mathop{\rm id}\nolimits|_{h}\leq n\nu_{2}\leq C_{1}(n,A,\rho)\epsilon$.

###  4.4 Proof of Theorem 4.1

Let $X$, $(\mathbb{K}_{X,n},\theta)$, $C$ and $h_{i}$ $(i=1,2)$ be as in
Theorem 4.1. Let $s$ be the automorphism of $\mathbb{K}_{X,n}$ determined by
$h_{2}=h_{1}\cdot s$. We have

$\sqrt{-1}\Lambda_{g_{X}}\overline{\partial}\partial\mathop{\rm
tr}\nolimits(s)\leq-\bigl{|}\overline{\partial}(s)s^{-1/2}\bigr{|}^{2}_{h_{1},g_{X}}-\bigl{|}\bigl{[}s,\theta\bigr{]}s^{-1/2}\bigr{|}^{2}_{h_{1},g_{X}}.$

By Omori-Yau maximum principle, there exist $m_{0}\in{\mathbb{Z}}_{>0}$ and a
family of points $p_{m}\in X$ $(m\geq m_{0})$ such that

$\mathop{\rm tr}\nolimits(s)(p_{m})\geq\sup\mathop{\rm
tr}\nolimits(s)-\frac{1}{m},\quad\quad\sqrt{-1}\Lambda_{g_{X}}\overline{\partial}\partial\mathop{\rm
tr}\nolimits(s)\geq-\frac{1}{m}.$

Because $h_{1}$ and $h_{2}$ are mutually bounded, there exists $C_{1}>0$ such
that

$\bigl{|}\bigl{[}s,\theta\bigr{]}\bigr{|}^{2}_{h_{1},g_{X}}(p_{m})\leq\frac{C_{1}}{m}.$

Let $\tau_{m}$ be a frame of the cotangent space of $X$ at $p_{m}$ such that
$|\tau_{m}|_{g_{X}}=1$. It induces a frame $e_{m,j}=\tau_{m}^{(n+1-2j)/2}$
$(j=1,\ldots,n)$ of $\mathbb{K}_{X,n|p_{m}}$. Because both $h_{i}$ are
mutually bounded with $h_{X}$, there exists a constant $B>0$ such that
$|e_{m,1}|_{h_{i}}\leq B$ for any $m$ and $i$. Let $f_{m}$ be the endomorphism
of $\mathbb{K}_{X,n|p_{m}}$ determined by $\theta_{|p_{m}}=f_{m}\,\tau_{m}$.
Because $\theta$ is bounded with respect to $h_{i}$ and $g_{X}$, there exists
$C_{2}>0$ independently from $m$ such that $|f_{m}|_{h_{i}}\leq C_{2}$. By
Theorem 4.12, there exists $C_{3}>0$ independently from $m$ such that

$\bigl{|}s-\mathop{\rm
id}\nolimits\bigr{|}_{h_{1}}(p_{m})\leq\frac{C_{3}}{\sqrt{m}}.$

Because both $h_{i}$ are compatible with the non-degenerate pairing $C$, we
have $\det(s)=1$. There exists $C_{4}>0$ independently from $m$ such that

$n\leq\sup_{X}\mathop{\rm tr}\nolimits(s)\leq\mathop{\rm
tr}\nolimits(s)(p_{m})+\frac{1}{m}\leq n+\frac{C_{4}}{\sqrt{m}}+\frac{1}{m}.$

We obtain that $\mathop{\rm tr}\nolimits(s)$ is constantly $n$, i.e.,
$s=\mathop{\rm id}\nolimits$.

##  5 Hitchin section for $SL(n,\mathbb{R})$

###  5.1 Existence of weakly dominant harmonic metric in the general case

Given a tuple of holomorphic differentials
$\boldsymbol{q}=(q_{2},q_{3},\cdots,q_{n})$, one can construct a
$SL(n,\mathbb{R})$-Higgs bundle

$\Big{(}\mathbb{K}_{X,n}=K_{X}^{\frac{n-1}{2}}\oplus
K_{X}^{\frac{n-3}{2}}\oplus\cdots\oplus K_{X}^{\frac{3-n}{2}}\oplus
K_{X}^{\frac{1-n}{2}},\quad\theta(\boldsymbol{q})=\begin{pmatrix}0&q_{2}&q_{3}&q_{4}&\cdots&q_{n}\\\
1&0&q_{2}&q_{3}&\ddots&\vdots\\\ &1&0&q_{2}&\ddots&\vdots\\\
&&\ddots&\ddots&\ddots&q_{3}\\\ &&&\ddots&\ddots&q_{2}\\\
&&&&1&0\end{pmatrix}\Big{)}.$

The natural pairings $K_{X}^{(n-2i+1)/2}\otimes
K_{X}^{-(n-2i+1)/2}\to\mathcal{O}_{X}$ induce a non-degenerate symmetric
bilinear form $C_{\mathbb{K},X,n}$ of $\mathbb{K}_{X,n}$. It is a non-
degenerate symmetric pairing of $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$.
Such Higgs bundles are called Higgs bundles in the Hitchin section. They were
first introduced by Hitchin in [Hit92] for compact hyperbolic Riemann
surfaces. There are various expressions of Higgs bundles in the Hitchin
section and they are equivalent to each other. One may refer Appendix B for
details.

A non-compact Riemann surface $X$ is called hyperbolic if its universal cover
is isomorphic to the unit disk $\mathbb{D}$. A non-compact Riemann surface $X$
is hyperbolic iff it is not $\mathbb{C}$ nor $\mathbb{C}^{*}$. Suppose $X$ is
hyperbolic.

Let $g_{X}$ be the unique complete conformal hyperbolic metric on $X$.
Locally, write $g_{X}=g_{0}(dx^{2}+dy^{2})$. The induced Hermitian metric on
$K_{X}^{-1}$ is $\frac{g_{0}}{2}dz\otimes d\bar{z}$, also denoted by $g_{X}$.
Denote by $F(g_{X})$ the curvature of the Chern connection of the Hermitian
metric $g_{X}$ on $K_{X}^{-1}$. So
$F(g_{X})=\bar{\partial}\partial\log\frac{g_{0}}{2}=-\partial\bar{\partial}\log
g_{0}$. The Gaussian curvature of $g_{X}$ is
$k_{g_{X}}:=\sqrt{-1}\Lambda_{g_{X}}F(g_{X})=-\frac{2}{g_{0}}\partial_{z}\partial_{\bar{z}}\log
g_{0}=-\frac{1}{2}\triangle_{g(X)}\log g_{0}$. Here $g_{X}$ is hyperbolic
means $k_{g_{X}}=-1.$

Let $F_{i}=\oplus_{k=1}^{i}K_{X}^{\frac{n+1-2k}{2}}.$ Thus
$\mathbf{F}=\\{F_{1}\subset F_{2}\subset\cdots\subset F_{n}\\}$ forms a full
holomorphic filtration of $\mathbb{K}_{X,n}$. And $\theta(\boldsymbol{q})$
takes $F_{i}$ to $F_{i+1}\otimes K_{X}$ and induces an isomorphism between
$F_{i}/F_{i-1}\rightarrow F_{i+1}/F_{i}\otimes K_{X}$ for $i=1,\cdots,n-1$.
Then, $(\mathbb{K}_{X,n},\theta(\mathbf{0}))$ is the graded Higgs bundle of
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ with respect to the filtration
$\mathbf{F}.$

Let

$h_{X}=\oplus_{k=1}^{n}a_{k,n}\cdot g_{X}^{-\frac{n+1-2k}{2}},$

where

$a_{k,n}=\prod_{l=1}^{k-1}(\frac{l(n-l)}{2})^{\frac{1}{2}}\cdot\prod_{l=k}^{n-1}(\frac{l(n-l)}{2})^{-\frac{1}{2}}.$
(26)

One may check that $h_{X}$ is a harmonic metric for the Higgs bundle
$(\mathbb{K}_{X,n},\theta(\mathbf{0}))$.

We call a Hermitian metric $h$ on $\mathbb{K}_{X,n}$ weakly dominates $h_{X}$
if $\det(h|_{F_{k}})\leq\det(h_{X}|_{F_{k}})$ for $1\leq k\leq n-1.$

###### Theorem 5.1

On a hyperbolic surface $X$, there exists a harmonic metric $h$ on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ satisfying (i) $h$ weakly
dominates $h_{X}$; (ii) $h$ is compatible with $C_{\mathbb{K},X,n}.$

Moreover, the norm of Higgs field satisfies
$|\theta(\boldsymbol{q})|_{h,g_{X}}^{2}\geq|\theta(\mathbf{0})|_{h_{X},g_{X}}^{2}=\frac{n(n^{2}-1)}{12}.$

As a result, the associated harmonic map
$f:(\widetilde{X},\widetilde{g_{X}})\rightarrow SL(n,\mathbb{R})/SO(n)$
satisfies the energy density $e(f)\geq\frac{n^{2}(n^{2}-1)}{6}.$ The equality
holds if $\boldsymbol{q}=0.$

Proof The existence follows from Part (i) of Theorem 3.12.

The proof of the moreover statement is identical to the one in [Li19a, Theorem
4.2].

From [Li19b, Section 5.2], we know that the energy density is
$e(f)=2n\cdot|\theta(\boldsymbol{q})|_{h,g_{X}}^{2}$. So $e(f)\geq
2n\cdot|\theta(\mathbf{0})|_{h_{X},g_{X}}^{2}=\frac{n^{2}(n^{2}-1)}{6}.$

###  5.2 Uniqueness in the case of bounded differentials

Next, we consider the case when $q_{i}(i=2,\cdots,n)$ are bounded with respect
to $g_{X}$, that is, $(q_{i}\bar{q}_{i})/g_{X}^{i}$ is bounded.

###### Theorem 5.2

On a hyperbolic surface $X$, suppose $q_{i}(i=2,\cdots,n)$ are bounded with
respect to $g_{X}$. Then there uniquely exists a harmonic metric $h$ of
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ over $X$ such that (i) $h$ weakly
dominates $h_{X}$, (ii) $h$ is compatible with $C_{\mathbb{K},X,n}$.

Moreover, $h$ is mutually bounded with $h_{X}$.

Proof The existence follows from Theorem 5.1. Let $h_{i}$ $(i=1,2)$ be
harmonic metrics of $(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ compatible
with $C_{\mathbb{K},X,n}$ which weakly dominate $h_{X}$. By Proposition 4.2,
both $h_{i}$ are mutually bounded with $h_{X}$. By Theorem 4.1, we obtain
$h_{1}=h_{2}$.

###### Remark 5.3

The condition (i) in Theorem 5.2 can be replaced by (i’) there exists a
positive constant $c$ such that $h_{|F_{1}(\mathbb{K}_{X,n})}\leq c\cdot
h_{X|F_{1}(\mathbb{K}_{X,n})}$.

####  5.2.1 Compact case

We reprove the existence and uniqueness of a harmonic metric on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ over a compact hyperbolic Riemann
surface. Note that here our proof does not invoke the Hitchin-Kobayashi
correspondence by using the stability of Higgs bundle.

###### Theorem 5.4

Given a tuple of holomorphic differentials
$\boldsymbol{q}=(q_{2},\cdots,q_{n})$ on a compact hyperbolic surface $X$,
there uniquely exists a harmonic metric $h$ on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ satisfying $h$ is compatible with
$C_{\mathbb{K},X,n}.$

Moreover, $h$ weakly dominates $h_{X}$;

Proof We first show the existence. Let $X$ be covered by $\mathbb{D}$ under
the map $p:\mathbb{D}\rightarrow X$, with the covering transformation group of
$X$ be $\Gamma<Aut(\mathbb{D})=PSL(2,\mathbb{R})$, i.e. $X=\mathbb{D}/\Gamma$.
Lift
$\boldsymbol{q},g_{X},h_{X},\mathbb{K}_{X,n},\theta(\boldsymbol{q}),C_{\mathbb{K},X,n}$
to
$\hat{\boldsymbol{q}},g_{\mathbb{D}},h_{\mathbb{D}},\mathbb{K}_{\mathbb{D},n},\theta(\hat{\boldsymbol{q}}),C_{\mathbb{K},\mathbb{D},n}$
on $\mathbb{D}$, which are invariant under $\Gamma$. By Theorem 5.2, there
exists a harmonic metric $\hat{h}\in\mathop{\rm
Harm}\nolimits^{dom}(\mathbb{K}_{\mathbb{D},n},\theta(\boldsymbol{q}),C_{\mathbb{K},\mathbb{D},n}:h_{\mathbb{D}})$.
From §5.2.2, each $\gamma\in\Gamma$ induces an automorphism on $\mathop{\rm
Harm}\nolimits^{dom}(\mathbb{K}_{\mathbb{D},n},\theta(\boldsymbol{q}),C_{\mathbb{K},\mathbb{D},n}:h_{\mathbb{D}})$.
By the uniqueness in Theorem 5.2, $\gamma^{*}(\hat{h})=\hat{h},$ for
$\gamma\in\Gamma$. Hence $\hat{h}$ descends to a harmonic metric $h$ on
$(\mathbb{K}_{X,n},\theta(\boldsymbol{q}))$ over $\mathbb{D}/\Gamma=X$.

The lifted $\hat{\boldsymbol{q}}$ are bounded with respect to $g_{\mathbb{D}}$
and any lifted harmonic metric $\hat{h}$ satisfies there exists a positive
constant $c$ such that $\hat{h}_{|F_{1}(\mathbb{K}_{\mathbb{D},n})}\leq c\cdot
h_{\mathbb{D}|F_{1}(\mathbb{K}_{\mathbb{D},n})}$. By Theorem 5.2 and Remark
5.3, $\hat{h}$ is unique and weakly dominates $h_{\mathbb{D}}$. Thus the
descended $h$ is unique and weakly dominates $h_{X}$.

####  5.2.2 Pull back

Let $F:X_{1}\longrightarrow X_{2}$ be a holomorphic map of Riemann surfaces
which is locally an isomorphism, i.e., the derivative of $F$ is nowhere
vanishing. Let $\boldsymbol{q}=(q_{2},\cdots,q_{n})$ be a tuple of holomorphic
differentials on $X_{2}$.

Because $F$ is locally an isomorphism, there exists a natural isomorphism

$F^{\ast}(\mathbb{K}_{X_{2},n},\theta(\boldsymbol{q}),C_{\mathbb{K},X_{2},n})\simeq(\mathbb{K}_{X_{1},n},\theta(F^{\ast}\boldsymbol{q}),C_{\mathbb{K},X_{1},n}).$

For any harmonic metric $h$ of
$(\mathbb{K}_{X_{2},n},\theta(F^{\ast}\boldsymbol{q}))$ compatible with
$C_{\mathbb{K},X_{2},n}$, it is well known and easy to check that the induced
metric $F^{\ast}(h)$ of $\mathbb{K}_{X_{1},r}$ is a harmonic metric of
$(\mathbb{K}_{X_{1},r},\theta(F^{\ast}\boldsymbol{q}))$ compatible with
$C_{\mathbb{K},X_{1},n}$.

Let $h_{0}$ be a Hermitian metric on $\mathbb{K}_{X_{2},n}$. If $h$ weakly
dominates $h_{0}$, then $F^{*}(h)$ weakly dominates $F^{*}(h_{0})$. Let
$h_{0}$ be a Hermitian metric on $\mathbb{K}_{X,n}$.

In this way, we obtain the map

$F^{\ast}:\mathop{\rm
Harm}\nolimits^{dom}(\mathbb{K}_{X_{2},n},\theta(\boldsymbol{q}),C_{\mathbb{K},X_{2},n}:h_{0})\longrightarrow\mathop{\rm
Harm}\nolimits^{dom}(\mathbb{K}_{X_{1},n},\theta(F^{*}\boldsymbol{q}),C_{\mathbb{K},X_{1},n}:F^{*}(h_{0})).$

If $X_{1}=X_{2}$ and $F^{\ast}(\boldsymbol{q})=\boldsymbol{q}$,
$F^{*}h_{0}=h_{0}$, then $F$ induces an automorphism on $\mathop{\rm
Harm}\nolimits^{dom}(\mathbb{K}_{X_{1},n},\theta(F^{*}\boldsymbol{q}):h_{0})$.

##  6 Existence with bounded condition on the unit disk

In this section, let $X$ be the unit disk $\\{z\in\mathbb{C}|~{}|z|<1\\}.$

###  6.1 Some function spaces

Let $\mathcal{A}$ be the set consisting of all smooth nonnegative functions
$f$ such that

$\int_{X}f(z)(1-|z|^{2})d\sigma<\infty,$

where $d\sigma$ is the Lebesgue measure on the unit disk $X$.

Let $G(z,\xi)$ denote the Green function in $X$. Equivalently, from Lemma A.1,
$\mathcal{A}$ is the set consisting of all smooth nonnegative functions $f$
such that for some (thus for all) $z$,

$\int_{X}G(z,\xi)f(\xi)d\sigma_{\xi}<\infty.$

Let $\mathcal{A}^{b}$ be the set consisting of all smooth nonnegative
functions $f$ such that

$\sup_{z\in X}\int_{X}G(z,\xi)f(\xi)d\sigma_{\xi}<\infty.$

It is clear that $\mathcal{A}^{b}\subset\mathcal{A}.$ From Lemma A.1, for
$p>-2$, $(1-|z|^{2})^{p}\in\mathcal{A}^{b};$ for $p\leq-2,$
$(1-|z|^{2})^{p}\notin\mathcal{A}.$

###  6.2 General existence with bounded condition

We set $X=\\{z\in\mathbb{C}|~{}|z|<1\\}$ with the Poincaré metric $g_{X}$ and
Euclidean metric $g_{0}(X)$:

$g_{X}=\frac{dx^{2}+dy^{2}}{(1-|z|^{2})^{2}},\quad g_{0}(X)=dx^{2}+dy^{2}.$

###### Proposition 6.1

Suppose
$(E,\overline{\partial}_{E}=\overline{\partial}_{E}^{0}+\xi,\theta=\theta_{0}+\phi)$
is a Higgs bundle over $X$ for $\phi\in A^{1,0}(X,\mathop{\rm
End}\nolimits(E))$ and $\xi\in A^{0,1}(X,\mathop{\rm End}\nolimits(E))$.
Assume $(E,\overline{\partial}_{E}^{0},\theta_{0})$ is a Higgs bundle over $X$
which admits a harmonic metric $h_{1}$.

  * •

Suppose

$|[\phi,(\theta_{0})^{*h_{1}}]|_{h_{1},g_{0}(X)}\in\mathcal{A},\quad|\phi|_{h_{1},g_{0}(X)}^{2}\in\mathcal{A},\quad|\overline{\partial}_{E}^{0}\xi^{*h_{1}}|_{h_{1},g_{0}(X)}\in\mathcal{A},\quad|\xi|_{h_{1},g_{0}(X)}^{2}\in\mathcal{A}.$

Then there exists a harmonic metric $h$ on
$(E,\overline{\partial}_{E},\theta)$.

  * •

Suppose

$|[\phi,(\theta_{0})^{*h_{1}}]|_{h_{1},g_{0}(X)}\in\mathcal{A}^{b},\quad|\phi|_{h_{1},g_{0}(X)}^{2}\in\mathcal{A}^{b},\quad|\overline{\partial}_{E}^{0}\xi^{*h_{1}}|_{h_{1},g_{0}(X)}\in\mathcal{A}^{b},\quad|\xi|_{h_{1},g_{0}(X)}^{2}\in\mathcal{A}^{b}.$

Then there exists a harmonic metric $h$ on
$(E,\overline{\partial}_{E},\theta)$ mutually bounded with $h_{1}$.

Proof Let $\nabla_{h_{1}}=\partial_{E}^{h_{1}}+\overline{\partial}_{E}^{0}$ be
the Chern connection of $E$ determined by $h_{1}$ and
$\overline{\partial}_{E}^{0}$. We have

$\displaystyle F(\overline{\partial}_{E}^{0}+\xi,\theta_{0}+\phi,h_{1})$
$\displaystyle=$ $\displaystyle
F(\nabla_{h_{1}})+\partial_{E}^{h_{1}}\xi-\overline{\partial}_{E}^{0}\xi^{*h_{1}}-[\xi,\xi^{*h_{1}}]+[\theta_{0}+\phi,(\theta_{0}+\phi)^{*h_{1}}]$
$\displaystyle=$
$\displaystyle-[\theta_{0},{\theta_{0}}^{*h_{1}}]+\partial_{E}^{h_{1}}\xi-\overline{\partial}_{E}^{0}\xi^{*h_{1}}-[\xi,\xi^{*h_{1}}]+[\theta_{0}+\phi,(\theta_{0}+\phi)^{*h_{1}}]$
$\displaystyle=$
$\displaystyle\partial_{E}^{h_{1}}\xi-\overline{\partial}_{E}^{0}\xi^{*h_{1}}-[\xi,\xi^{*h_{1}}]-[\phi,{\theta_{0}}^{*h_{1}}]-[\theta_{0},\phi^{*h_{1}}]-[\phi,\phi^{*h_{1}}].$

By assumption,

$\displaystyle\Big{|}\Lambda_{g_{0}(X)}F(\overline{\partial}_{E}^{0}+\xi,\theta_{0}+\phi,h_{1})\Big{|}_{h_{1}}\in\mathcal{A}.$

For $0<r<1,$ we set $X_{r}:=\\{|z|<r\\}.$ Let $h_{X_{r}}$ be the harmonic
metric of $(E,\theta)|_{X_{r}}$ such that $h_{X_{r}}=h_{1}$ on $\partial
X_{r}$. We have $\det(s(h_{1}|_{X_{r}},h_{X_{r}}))=1.$ Recall
$\triangle=\partial_{x}^{2}+\partial_{y}^{2}=4\partial_{z}\partial_{\bar{z}}$.
We have the following inequality on $X_{r}:$

$\frac{1}{2}\triangle\log\mathop{\rm
tr}\nolimits(s(h_{1}|_{X_{r}},h_{X_{r}}))=\sqrt{-1}\Lambda_{g_{0}(X)}\partial\bar{\partial}\log\mathop{\rm
tr}\nolimits(s(h_{1}|_{X_{r}},h_{X_{r}}))\geq-\Big{|}\Lambda_{g_{0}(X)}F(\theta+\phi,h_{1})\Big{|}_{h_{1}}.$

###### Lemma 6.2

Let $f$ be a nonnegative smooth function on $X$. Suppose $f\in\mathcal{A}.$
Let $u_{r}$ be the unique solution satisfying

$\displaystyle\triangle u_{r}=-f,\quad\text{in $X_{r}$}$ $\displaystyle
u_{r}=0\quad\text{on $\partial X_{r}$}$

There exists a smooth nonnegative function $v$ on $X$ such that $0\leq
u_{r}\leq v,$ for any $r\in(0,1)$.

If moreover $f\in\mathcal{A}^{b}$, we can choose a bounded $v$ satisfying the
above property.

Proof Define $v(z)=\frac{1}{2\pi}\int_{X}f(\xi)G(z,\xi)d\sigma_{\xi}.$ By
Lemma A.1, we know $v(z)$ is well-defined and nonnegative. Then we have

$\displaystyle\triangle v=-f,\quad\text{in $X$}$ $\displaystyle v\geq
0\quad\text{in $X$}$

By the maximum principle, $u_{r}\leq v$ holds in $X_{r}$. Also, note that $0$
is a subsolution to the equation. By the maximum principle, $u_{r}\geq 0.$

If $f\in\mathcal{A}^{b},$ then $v$ is bounded by definition.

By Lemma 6.2 and using the maximum principle, there exists a smooth function
$v$ on $X$ such that

$\log(\mathop{\rm
tr}\nolimits(s(h_{1}|_{X_{r}},h_{X_{r}}))/\text{rank}(E))\leq v.$

Then, by Proposition 2.4, there is a convergence subsequence of $h_{X_{r}}$
whose limit is denoted by $h$. Then $h$ is a harmonic metric of $(E,\theta).$

If $v$ is bounded, $h$ is mutually bounded with $h_{1}$.

###  6.3 Existence for holomorphic chains

Given $k$ holomorphic vector bundles $E_{i}$ over $X$ of rank $n_{i}$,
$i=1,\cdots,k$. We can consider a Higgs bundle as following:

$(E=E_{1}\oplus E_{2}\oplus\cdots\oplus
E_{k},\quad\theta=\begin{pmatrix}0&&&&\\\ \theta_{1}&0&&&\\\
&\theta_{2}&0&&\\\ &&\ddots&\ddots&\\\ &&&\theta_{k-1}&0\end{pmatrix}),$

where $\theta_{i}\in H^{0}(X,\mathop{\rm Hom}\nolimits(E_{i},E_{i+1})\otimes
K).$ Such Higgs bundle is called a holomorphic chain of type
$(n_{1},\cdots,n_{k})$. We call a Hermitian metric $h$ on $E$ orthogonal if
$h$ is orthogonal with respect to the decomposition $E=\oplus_{i=1}^{k}E_{i}$.

For such $(E,\theta)$, we also consider a holomorphic chain $(E,\theta_{0})$
as follows:

$(E,\quad\theta_{0}=\begin{pmatrix}0&&&&\\\ \phi_{1}&0&&&\\\ &\phi_{2}&0&&\\\
&&\ddots&\ddots&\\\ &&&\phi_{k-1}&0\end{pmatrix}),$

where there exists a subset $S$ of $\\{1,2,\cdots,k-1\\}$, $\phi_{i}=0$ for
$i\in S$ and $\phi_{i}=\theta_{i}$ for $i\notin S$.

In the following, we will deduce the existence of a harmonic metric on
$(E,\theta)$ from the one on $(E,\theta_{0})$ if it exists.

###### Theorem 6.3

We consider two holomorphic chains $(E,\theta),(E,\theta_{0})$ as above.
Suppose there exists an orthogonal harmonic metric $h_{1}$ on
$(E,\theta_{0})$. Suppose
$|\theta_{i}|_{h_{1},g_{0}(X)}^{2}\in\mathcal{A}(i\in S),$ then there exists
an orthogonal harmonic metric $h$ on $(E,\theta)$.

Moreover, $|\theta_{i}|_{h_{1},g_{0}(X)}^{2}\in\mathcal{A}^{b}(i\in S)$ if and
only if there exists an orthogonal harmonic metric $h$ on $(E,\theta)$
mutually bounded with $h_{1}$.

Proof Note that $[\theta-\theta_{0},(\theta_{0})^{*h_{1}}]=0.$ The existence
part under both assumptions follows from Proposition 6.1.

We only need to prove the inverse direction for a bounded metric. Suppose we
have a diagonal harmonic metric $h$ mutually bounded with $h_{1}$.

The Hitchin equation for $h_{1}$ on $(E_{1}\oplus E_{2}\oplus\cdots\oplus
E_{k},\theta_{0})$ is

$\displaystyle F(h_{1}|_{E_{1}})=-(\phi_{1})^{*h_{1}}\wedge\phi_{1}$
$\displaystyle
F(h_{1}|_{E_{2}})=-(\phi_{2})^{*h_{1}}\wedge\phi_{2}-\phi_{1}\wedge(\phi_{1})^{*h_{1}}$
$\displaystyle\cdots$ $\displaystyle
F(h_{1}|_{E_{k-1}})=-(\phi_{k-1})^{*h_{1}}\wedge\phi_{k-1}-\phi_{k-2}\wedge(\phi_{k-2})^{*h_{1}}$
$\displaystyle F(h_{1}|_{E_{k}})=-\phi_{k-1}\wedge(\phi_{k-1})^{*h_{1}}$

Denote by

$|\phi_{i}|_{h_{1},g_{0}(X)}^{2}=\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(\phi_{i}\wedge(\phi_{i})^{*h_{1}})=-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits((\phi_{i})^{*h_{1}}\wedge\phi_{i}).$

So

$\displaystyle-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h_{1}|_{E_{1}}))=-|\phi_{1}|_{h_{1},g_{0}(X)}^{2}$
$\displaystyle-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h_{1}|_{E_{2}}))=-|\phi_{2}|_{h_{1},g_{0}(X)}^{2}+|\phi_{1}|_{h_{1},g_{0}(X)}^{2}$
$\displaystyle\cdots$ $\displaystyle-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h_{1}|_{E_{k-1}}))=-|\phi_{k-1}|_{h_{1},g_{0}(X)}^{2}+|\phi_{k-2}|_{h_{1},g_{0}(X)}^{2}$
$\displaystyle-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h_{1}|_{E_{k}}))=|\phi_{k-1}|_{h_{1},g_{0}(X)}^{2}$

Therefore,

$-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h_{1}|_{\oplus_{l=1}^{i}E_{l}}))=-|\phi_{i}|_{h_{1},g_{0}(X)}^{2},\quad
i=1,2,\cdots,k-1.$

Similarly,

$-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h|_{\oplus_{l=1}^{i}E_{l}}))=-|\theta_{i}|_{h,g_{0}(X)}^{2},\quad
i=1,2,\cdots,k-1.$

Let
$u_{i}=\log(\det(h|_{\oplus_{l=1}^{i}E_{l}})/\det(h_{1}|_{\oplus_{l=1}^{i}E_{l}})).$
Then

$-\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h|_{\oplus_{l=1}^{i}E_{l}}))+\sqrt{-1}\Lambda_{g_{0}(X)}\mathop{\rm
tr}\nolimits(F(h_{1}|_{\oplus_{l=1}^{i}E_{l}}))=2\partial_{z}\partial_{\bar{z}}u_{i}=\frac{1}{2}\triangle
u_{i}.$

Then we obtain the equation for $u_{i}$’s as follows:

$\frac{1}{2}\triangle
u_{i}=-|\theta_{i}|_{h,g_{0}(X)}^{2}+|\phi_{i}|_{h_{1},g_{0}(X)}^{2},,\quad
i=1,2,\cdots,k-1.$

We only focus on $u_{i}$’s where $i\in S$. For $i\in S$, $\phi_{i}=0$ and the
equation becomes:

$\frac{1}{2}\triangle u_{i}=-|\theta_{i}|_{h,g_{0}(X)}^{2}$

Suppose $|u_{i}|\leq M$ for $i=1,2,\cdots,k$. Applying Lemma 6.6, we obtain
$|\theta_{i}|_{h,g}^{2}\in\mathcal{A}^{b}$ for $i\in S.$ Since $|u_{i}|\leq M$
for $i=1,2,\cdots,k$, $|\theta_{i}|_{h_{1},g}^{2}\in\mathcal{A}^{b}$ for $i\in
S.$

In the following, we will show Lemma 6.6 which was used in proving Theorem
6.3.

Let $u$ be a subharmonic function on a domain $D$. A harmonic majorant of $u$
is a harmonic function $h$ on $D$ such that $h\geq u$ there. If also $h\leq
k$, for every other harmonic majorant $k$ of $u,$ then $h$ is called the least
harmonic majorant of $u$.

###### Lemma 6.4

([RR94, Theorem 3.3]) Let $u$ be a subharmonic function on $X$ with
$u\neq-\infty.$ There exists a harmonic majorant for $u$ if and only if

$\sup\limits_{0<r<1}\frac{1}{2\pi}\int_{0}^{2\pi}u(re^{it})dt<\infty.$

###### Lemma 6.5

[Ran95, Theorem 4.5.4] Let $u$ be a subharmonic function on $X$ such that
$u\neq-\infty.$ If $u$ has a harmonic majorant on $X$, then it has a least
one, $h$, and

$u(z)=h(z)-\frac{1}{2\pi}\int_{X}G(z,\xi)\triangle u(\xi)d\sigma_{\xi}.$

###### Lemma 6.6

Let $f$ be a nonnegative or nonpositive smooth function. If $|u|\leq M$ and
$\triangle u=f$ on $X$, then $|f|\in\mathcal{A}^{b}.$

Proof It is enough to show the case for $f$ being nonnegative. If $f$ is
nonpositive, we can consider $\triangle(-u)=-f.$

By Lemma 6.4, since $|u|\leq M$, there exists a least harmonic majorant $h$
for $u$ and $h\leq M$ since the constant function $M$ is a harmonic majorant
of $u$.

By Lemma 6.5,

$\frac{1}{2\pi}\int_{X}G(z,\xi)\triangle u(\xi)d\sigma_{\xi}=h(z)-u(z)\leq
2M.$

Thus $f=\triangle u\in\mathcal{A}^{b}.$

###  6.4 Relation to prescribed curvature equation

Consider the curvature equation on $X$:

$\frac{1}{4}\triangle u=|\alpha|^{2}e^{2u},$ (27)

where $\alpha$ is a holomorphic function on $X$. That is, we are looking for
the function $u$ such that the metric $e^{2u}(dx^{2}+dy^{2})$ on $X$ has
Gaussian curvature $-4|\alpha|^{2}.$

As a corollary of Theorem 6.3, we can recover the following theorem shown by
Kraus.

###### Proposition 6.7

([Kra13, Theorem 3.1]) (1) $|\alpha|^{2}\in\mathcal{A}^{b}$ if and only if
there exists a solution $u$ bounded from above and below, of Equation (27).

(2) if $|\alpha|^{2}\in\mathcal{A}$, then there exists a solution $u$ of
Equation (27).

Proof Consider the Higgs bundle
$(\mathcal{O}\oplus\mathcal{O},\begin{pmatrix}0&0\\\ \alpha&0\end{pmatrix}dz)$
over the unit disk $X$. Note that the Higgs bundle is symmetric to the non-
degenerate symmetric pairing $C=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$. On
$\mathcal{O}\oplus\mathcal{O}$, there is a flat Hermitian metric
$h_{1}=\mathop{\rm diag}\nolimits(1,1)$ which is symmetric with respect to $C$
and of unit determinant. Then we obtain $|\alpha\cdot
dz|_{h_{1},g_{0}(X)}^{2}=2|\alpha|^{2}.$ Also, a diagonal harmonic metric
which is compatible with $C$ is of unit determinant. It is of the form
$h=\mathop{\rm diag}\nolimits((h^{0})^{-1},h^{0})$. Let $h^{0}=e^{u},$ then
$u$ satisfies Equation (27).

The rest follows from Theorem 6.3.

In fact, Kraus showed the converse direction for the existence of the
curvature equation.

###### Proposition 6.8

([Kra13, Theorem 1.3]) If there exists a solution of Equation (27), then there
exists a non-vanishing holomorphic function $f$ on $X$ such that $|\alpha\cdot
f|^{2}\in\mathcal{A}$.

###### Remark 6.9

Kraus’ proof relies on the Littlewood-Paley identity for holomorphic
functions. It is not clear if such conditions are still necessary for higher
rank nilpotent Higgs bundles.

###  6.5 Holomorphic chains of type $(1,1,\cdots,1)$

The following proposition indicates that only a proper subset of
$\theta_{i}$’s being nice does not imply the existence of a diagonal harmonic
metric.

###### Proposition 6.10

Consider a Higgs bundle
$(\mathcal{O}\oplus\mathcal{O}\oplus\cdots\oplus\mathcal{O},\begin{pmatrix}0&&&&\\\
\gamma_{1}&0&&&\\\ &\gamma_{2}&0&&\\\ &&\ddots&\ddots&\\\
&&&\gamma_{n-1}&0\end{pmatrix}dz)$ over the unit disk $X$ satisfying
$\prod_{i=1}^{n-1}\gamma_{i}^{i(n-i)}=\alpha^{\frac{n(n^{2}-1)}{6}}$ for a
holomorphic function $\alpha$ which is not constantly $0$. A necessary
condition for the existence of a diagonal harmonic metric is that there exists
a non-vanishing holomorphic function $f$ on $X$ such that $|\alpha\cdot
f|^{2}\in\mathcal{A}.$

Proof Suppose there is a harmonic metric $h=\mathop{\rm
diag}\nolimits(e^{-u_{1}},e^{-u_{2}},\cdots,e^{-u_{n}})$. Let
$w_{k}=\sum_{i=1}^{k}u_{i}.$ Then following from the calculations in the proof
of Theorem 6.3 and $|dz|_{g_{0}(X)}^{2}=2$, the Hitchin equation becomes

$\displaystyle\frac{1}{4}\triangle w_{1}=|\gamma_{1}|^{2}e^{2w_{1}-w_{2}}$
$\displaystyle\frac{1}{4}\triangle
w_{2}=|\gamma_{2}|^{2}e^{2w_{2}-w_{1}-w_{3}}$ $\displaystyle\cdots$
$\displaystyle\frac{1}{4}\triangle
w_{n-2}=|\gamma_{n-2}|^{2}e^{2w_{n-2}-w_{n-3}-w_{n-1}}$
$\displaystyle\frac{1}{4}\triangle
w_{n-1}=|\gamma_{n-1}|^{2}e^{2w_{n-1}-w_{n-2}}$

Summing up the above $(n-1)$-equations, we obtain

$\frac{1}{4}\triangle(w_{1}+w_{2}+\cdots+w_{n-1})=|\gamma_{1}|^{2}e^{2w_{1}-w_{2}}+\sum_{i=2}^{n-2}|\gamma_{i}|^{2}e^{2w_{i}-w_{i-1}-w_{i+1}}+|\gamma_{n-1}|^{2}e^{2w_{n-1}-w_{n-2}}$
(28)

Let $r_{i}=\frac{i(n-i)}{2}.$ Then

$2r_{1}-r_{2}=1,2r_{i}-r_{i-1}-r_{i+1}=1(i=2,\cdots,n-2),2r_{n-1}-r_{n-2}=1,\sum_{i=1}^{n-1}r_{i}=\frac{n(n^{2}-1)}{12}.$

Note that the right hand side of Equation (28) satisfies

$\displaystyle|\gamma_{1}|^{2}e^{2w_{1}-w_{2}}+\sum_{i=2}^{n-2}|\gamma_{i}|^{2}e^{2w_{i}-w_{i-1}-w_{i+1}}+|\gamma_{n-1}|^{2}e^{2w_{n-1}-w_{n-2}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{\max_{i=1,\cdots,n-1}r_{i}}\cdot(r_{1}|\gamma_{1}|^{2}e^{2w_{1}-w_{2}}+\sum_{i=2}^{n-2}r_{i}|\gamma_{i}|^{2}e^{2w_{i}-w_{i-1}-w_{i+1}}+r_{n-1}|\gamma_{n-1}|^{2}e^{2w_{n-1}-w_{n-2}})$
$\displaystyle\geq$
$\displaystyle\frac{1}{\max_{i=1,\cdots,n-1}r_{i}}\cdot\sum_{i=1}^{n-1}r_{i}\cdot\big{(}\prod_{i=1}^{n-1}|\gamma_{i}|^{2r_{i}}e^{r_{1}(2w_{1}-w_{2})+\sum_{i=2}^{n-2}r_{i}(2w_{i}-w_{i-1}-w_{i+1})+r_{n-1}(2w_{n-1}-w_{n-2})}\big{)}^{\frac{1}{\sum_{i=1}^{n-1}r_{i}}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{\max_{i=1,\cdots,n-1}r_{i}}\cdot\frac{n(n^{2}-1)}{12}\cdot\big{(}\prod_{i=1}^{n-1}|\gamma_{i}|^{2r_{i}}e^{(2r_{1}-r_{2})w_{1}+\sum_{i=2}^{n-2}(2r_{i}-r_{i-1}-r_{i+1})w_{i}+(2r_{n-1}-r_{n-2})w_{n-1}}\big{)}^{\frac{12}{n(n^{2}-1)}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{\max_{i=1,\cdots,n-1}r_{i}}\cdot\frac{n(n^{2}-1)}{12}\cdot\big{(}\prod_{i=1}^{n-1}|\gamma_{i}|^{2r_{i}}e^{w_{1}+\cdots+w_{n-1}}\big{)}^{\frac{12}{n(n^{2}-1)}}.$

So

$\frac{1}{4}\triangle(w_{1}+\cdots+w_{n-1})\geq\frac{1}{\max_{i=1,\cdots,n-1}r_{i}}\cdot\frac{n(n^{2}-1)}{12}\big{(}\prod_{i=1}^{n-1}|\gamma_{i}|^{i(n-i)}\big{)}^{\frac{12}{n(n^{2}-1)}}\cdot
e^{\frac{12}{n(n^{2}-1)}(w_{1}+\cdots+w_{n-1})}.$ (29)

Consider the equation

$\frac{1}{4}\triangle
u=\frac{1}{\max_{i=1,\cdots,n-1}r_{i}}\big{(}\prod_{i=1}^{n-1}|\gamma_{i}|^{i(n-i)}\big{)}^{\frac{12}{n(n^{2}-1)}}\cdot
e^{u}.$ (30)

Then $\frac{12}{n(n^{2}-1)}\cdot(w_{1}+\cdots+w_{n-1})$ is a subsolution to
the equation (30). Note that $\gamma_{i}$’s are holomorphic functions and not
constantly zero. So the function
$-\big{(}\prod_{i=1}^{n-1}|\gamma_{i}|^{i(n-i)}\big{)}^{\frac{12}{n(n^{2}-1)}}$
satisfies the essential negative property in [KY93, Definition 0.1]. By [KY93,
Theorem 4], the existence of a subsolution implies there exists a $C^{2}$
solution to the equation (30).

The rest follows from Proposition 6.8 and the assumption
$\prod_{i=1}^{n-1}\gamma_{i}^{i(n-i)}=\alpha^{\frac{n(n^{2}-1)}{6}}$ for a
holomorphic function $\alpha$.

###### Remark 6.11

It would be interesting if one could find a necessary and sufficient condition
on $\gamma_{i}(i=1,\cdots,n-1)$ for the existence of a diagonal harmonic
metric.

##  7 $SO(n,n+1)$-Higgs bundles

In this section, we discuss the existence of harmonic metrics on
$SO(n,n+1)$-Higgs bundles over non-compact Riemann surfaces by using the
techniques developed in this paper and our previous paper [LM22].

###### Definition 7.1

  * •

An $SO(n,n+1)$-Higgs bundle over a Riemann surface $X$ is given by
$((V,Q_{V}),(W,Q_{W}),\eta)$, where $(V,Q_{V})$ is an orthogonal bundle of
rank $n$ satisfying $\det V=\mathcal{O}_{X}$, $(W,Q_{W})$ is an orthogonal
bundle of rank $n+1$ satisfying $\det W=\mathcal{O}_{X}$, and
$\eta:V\rightarrow W\otimes K_{X}$ is a holomorphic bundle map.

  * •

The associated $SL(2n+1,\mathbb{C})-$Higgs bundle is

$(E,\theta)=\left(V\oplus W,\begin{pmatrix}0&\eta^{\dagger}\\\
\eta&0\end{pmatrix}\right),$

where $\eta^{\dagger}$ is the adjoint of $\eta$ with respect to $Q_{V},Q_{W}$.

  * •

A harmonic metric $h$ on $(E,\theta)$ is called compatible with
$SO(n,n+1)$-structure if $h=h|_{V}\oplus h|_{W}$ where $h_{V},h_{W}$ are
compatible with $Q_{V},Q_{W}$ respectively.

###  7.1 Dirichlet problem

Let $Y\subset X$ be a relatively compact connected open subset with smooth
boundary $\partial Y$. Assume that $\partial Y$ is non-empty. Let $h_{\partial
Y}$ be any Hermitian metric of $E_{|\partial Y}$.

###### Lemma 7.2

Let $h$ be a harmonic metric of $(E,\theta)$ such that $h|_{\partial
Y}=h_{\partial Y}$. Suppose $h_{\partial Y}$ is compatible with
$SO(n.n+1)$-structure. Then $h$ is compatible with $SO(n.n+1)$-structure.

Proof First we show that $h=\mathop{\rm diag}\nolimits(h_{V},h_{W}).$ There
exists the automorphism $\varphi=1_{V}\oplus(-1_{W})$ on $E=V\oplus W$.
Because $\varphi^{*}\theta=-\theta$, $\varphi^{*}(h)$ is also a harmonic
metric of $(E,\theta)$. Because $\varphi^{*}(h)|_{\partial Y}=h_{\partial Y}$,
we obtain $\varphi^{*}(h)=h$. It means that $h$ is the direct sum of the
Hermitian metrics of $V$ and $W$.

Next we show that $h_{V},h_{W}$ are compatible with $Q_{V},Q_{W}$
respectively. The metric $h$ induces a harmonic metric
$h^{\lor}=(h|_{V})^{\lor}\oplus(h|_{W})^{\lor}$ on $(E^{\lor},\theta^{\lor})$.
Let $\Psi_{Q_{V}}:V\rightarrow V^{\lor}$ and $\Psi_{Q_{W}}:W\rightarrow
W^{\lor}$ be the induced isomorphism by $Q_{V},Q_{W}$, respectively. Then
$(\Psi_{Q_{V}})^{*}((h|_{V})^{\lor})\oplus(\Psi_{Q_{W}})^{*}((h|_{W})^{\lor})$
is again a harmonic metric on $(E,\theta)$. Since
$\big{(}(\Psi_{Q_{V}})^{*}((h|_{V})^{\lor})\oplus(\Psi_{Q_{W}})^{*}((h|_{W})^{\lor})\big{)}|_{\partial
Y}=h_{\partial Y}$, we obtain
$(\Psi_{Q_{V}})^{*}((h|_{V})^{\lor})\oplus(\Psi_{Q_{W}})^{*}((h|_{W})^{\lor})=h|_{V}\oplus
h|_{W}$. It means $h_{V},h_{W}$ are compatible with $Q_{V},Q_{W}$
respectively.

###  7.2 The generically regular semisimple case

Let $((V,Q_{V}),(W,Q_{W}),\eta)$ be an $SO(n,n+1)$-Higgs bundle on $X$. Let
$(E,\theta)$ be the associated $SL(2n+1,\mathbb{C})$-Higgs bundle. We obtain
$\eta^{\dagger}\circ\eta\in\mathop{\rm End}\nolimits(V)\otimes K_{X}^{2}.$ Let
$(T^{*}X)^{\otimes 2}$ denote the total space of the line bundle $K_{X}^{2}$.
Let $Z_{X}\subset(T^{*}X)^{\otimes 2}$ denote the zero-section. The spectral
curve $\Sigma_{\eta^{\dagger}\circ\eta}\subset(T^{*}X)^{\otimes 2}$ of
$\eta^{\dagger}\circ\eta$ is defined as usual. We obtain the finite map
$\pi:\Sigma_{\eta^{\dagger}\circ\eta}\cup Z_{X}\rightarrow X$.

###### Definition 7.3

We say that the tuple $((V,Q_{V}),(W,Q_{W}),\eta)$ is generically regular
semisimple if there exists $P\in X$ such that $|\pi^{-1}(P)|=n+1$.

###### Theorem 7.4

If $((V,Q_{V}),(W,Q_{W}),\eta)$ is generically regular semisimple, then there
exists a harmonic metric $h$ of $(E,\theta)$ compatible with
$SO(n,n+1)$-structure.

Proof The following lemma follows from Corollary 7.7 below.

###### Lemma 7.5

$((V,Q_{V}),(W,Q_{W}),\eta)$ is generically regular semisimple if and only if
the associated $SL(2n+1,\mathbb{C})$-Higgs bundle $(E,\theta)$ is generically
regular semisimple. (See [LM22, Definition] for generically regular
semisimplicity for Higgs bundles.)

Let $h_{0}=h_{0}|_{V}\oplus h_{0}|_{W}$ be a Hermitian metric of $E=V\oplus W$
such that $h_{0}|_{V}$ and $h_{0}|_{W}$ are compatible with $Q_{V}$ and
$Q_{W}$, respectively. Let $X_{i}$ $(i=1,\cdots)$ be an exhaustion family of
$X$. Let $E_{i},V_{i}$ and $W_{i}$ denote the restriction of $E,V$ and $W$ to
$X_{i}$, respectively. Let $h_{0,i}$ denote the restriction of $h_{0}$ to
$X_{i}$.

Let $h_{i}$ be a harmonic metric of $(E_{i},\theta_{i})$ such that
$h_{i}|_{\partial X_{i}}=h_{0}|_{\partial X_{i}}$. By Lemma 7.2, $h_{i}$ is
compatible with $SO(n,n+1)$-structure. It implies that $h_{i}$ is compatible
with the non-degenerate symmetric pairing $Q_{V}\oplus Q_{W}$. Let $s_{i}$ be
the automorphism of $E_{i}$ determined by $h_{i}=h_{0,i}\cdot s_{i}$ as in
§2.2. Note that the Higgs field $\theta$ is self-adjoint with respect to the
non-degenerate symmetric pairing $Q_{V}\oplus Q_{W}$ of $E$. By Lemma 7.5 and
[LM22, Proposition 2.37], there exist positive constants $C_{i}$
$(i=1,2,\ldots)$ such that the following holds on $X_{i}$ for $j\geq i+1$:

$\bigl{|}s_{j}\bigr{|}_{h_{0,i}}+\bigl{|}s_{j}^{-1}\bigr{|}_{h_{0,i}}\leq
C_{i}.$

By Proposition 2.4, there exists a convergent subsequence $h_{i}^{\prime}$. As
the limit, we obtain a harmonic metric $h$ of $(E,\theta)$ compatible with
$SO(n,n+1)$-structure.

####  7.2.1 Appendix: Preliminary from linear algebra

Let $R$ be any field. In this subsection, we consider matrices whose entries
are contained in $R$. For any positive integer $n$, let $I_{n}$ denote the
$(n\times n)$-identity matrix, and let $0_{n}$ denote the $(n\times n)$-zero
matrix.

Let $n\geq m$ be positive integers. Let $A$ be an $(n\times m)$-matrix. Let
$B$ be an $(m\times n)$-matrix. Let $C$ be the $(n+m)$-square matrix given as
follows:

$C=\begin{pmatrix}0_{n}&A\\\ B&0_{m}\end{pmatrix}.$

###### Lemma 7.6

We have $\det(tI_{n+m}-C)=t^{n-m}\det(t^{2}I_{m}-BA)$ in $R[t]$.

Proof It is enough to prove the equality in $R[t,t^{-1}]$. Let $0_{m,n}$
denote the $(m\times n)$-zero matrix. We have

$\det(tI_{n+m}-C)=\det\begin{pmatrix}tI_{n}&-A\\\
-B&tI_{m}\end{pmatrix}=\det\begin{pmatrix}tI_{n}&-A\\\
0_{m,n}&tI_{m}-t^{-1}BA\end{pmatrix}\\\
=t^{n}\det(tI_{m}-t^{-1}BA)=t^{n-m}\det(t^{2}I_{m}-BA).$ (31)

We recall that an $(\ell\times\ell)$-matrix is called regular semisimple if it
has $\ell$-distinct eigen values.

###### Corollary 7.7

If $n\geq m+2$, $C$ cannot be regular semisimple. If $n=m,m+1$, $C$ is regular
semisimple if and only if $BA$ is invertible and regular semisimple.

###  7.3 Collier section

Given a holomorphic line bundle $M$ on $X$, $\mu\in H^{0}(X,M^{-1}\otimes
K_{X}^{n})$, $\nu\in H^{0}(X,M\otimes K_{X}^{n})$, $q_{2i}\in
H^{0}(X,K_{X}^{2i})$ $(i=1,\cdots,n-1)$, one can construct the following
$SO(n,n+1)$-Higgs bundle
$((V,Q_{V}),(W,Q_{W}),\eta_{\mu,\nu}(\boldsymbol{q}))$ given by

$\displaystyle(V,Q_{V})=(K_{X}^{n-1}\oplus K_{X}^{n-3}\oplus\cdots\oplus
K_{X}^{3-n}\oplus K_{X}^{1-n},\begin{pmatrix}&&1\\\ &\iddots&\\\
1&&\end{pmatrix})$ $\displaystyle(W,Q_{W})=(M\oplus K_{X}^{n-2}\oplus
K_{X}^{n-4}\oplus\cdots\oplus K_{X}^{4-n}\oplus K_{X}^{2-n}\oplus
M^{-1},\begin{pmatrix}&&1\\\ &\iddots&\\\ 1&&\end{pmatrix})$
$\displaystyle\eta_{\mu,\nu}(\boldsymbol{q})=\begin{pmatrix}0&0&0&\cdots&\cdots&0&\nu\\\
1&q_{2}&q_{4}&\cdots&\cdots&q_{2n-4}&q_{2n-2}\\\
&1&q_{2}&q_{4}&\cdots&\cdots&q_{2n-4}\\\ &&1&q_{2}&\ddots&\ddots&q_{2n-6}\\\
&&&\ddots&\ddots&\ddots&\vdots\\\ &&&&\ddots&\ddots&\vdots\\\ &&&&&1&q_{2}\\\
&&&&&&\mu\end{pmatrix}:V\rightarrow W\otimes K_{X}.$ (32)

When $X$ is a compact Riemann surface of genus at lease two, for each integer
$d\in(0,n(2g-2)]$, Brian Collier in [Col20, Theorem 4.11] defined a component
$X_{d}$ of the moduli space of $SO(n,n+1)-$Higgs bundles formed by the above
Higgs bundles determined by $(M,\mu,\nu,q_{2},\cdots,q_{2n-2})$ where
$\deg(M)=d$ and $\mu\neq 0$. In particular, when $d=n(2g-2),$ $X_{d}$
coincides with the Hitchin component for $SO(n,n+1).$ Such components are
analogues of Hitchin components. Such Higgs bundles correspond to positive
$SO(n,n+1)$ representations.

We call the above Higgs bundles are in the Collier section. We are going to
discuss the existence of harmonic metrics of Higgs bundles in the Collier
section over non-compact Riemann surfaces.

####  7.3.1 Existence for the case $\mu\neq 0$

Let $(E,\theta)$ be the Higgs bundle associated with the $SO(n,n+1)$-Higgs
bundle $((V,Q_{V}),(W,Q_{W}),\eta_{\mu,\nu}(\boldsymbol{q}))$ in (7.3). We
introduce a holomorphic full filtration $\mathbf{F}(E)=\\{F_{1}(E)\subset
F_{2}(E)\subset\cdots\subset F_{2n+1}(E)\\}$ of $E$ as follows. We define
$F_{2i+1}(W)$ $(i=0,\ldots,n)$ by

$F_{1}(W)=M,\quad F_{2i+1}(W)=M\oplus K_{X}^{n-2}\oplus\cdots\oplus
K_{X}^{n-2j}\,\,\,(i=1,\ldots,n-1),\quad F_{2n+1}(W)=W,$

We also set $F_{2i}(W)=F_{2i-1}(W)$ for $i=1,\ldots,n$ and $F_{0}(W)=0$. We
define $F_{2i}(V)$ $(i=1,\ldots,n)$ by

$F_{2i}(V)=K_{X}^{n-1}\oplus\cdots\oplus K_{X}^{n+1-2i}\,\,\,(i=1,\ldots,n).$

We also set $F_{2i+1}(V)=F_{2i}(V)$ for $i=1,\ldots,n$ and $F_{1}(V)=0$. Then,
$\theta$ takes $F_{j}(W)$ to $F_{j+1}(V)\otimes K_{X}$, and $F_{j}(V)$ to
$F_{j+1}(W)\otimes K_{X}$. We define

$F_{j}(E)=F_{j}(V)\oplus F_{j}(W).$

Then, $\theta$ takes $F_{j}(E)$ to $F_{j+1}(E)\otimes K_{X}$.

With respect to the filtration $\mathbf{F}(E)$, the associated graded Higgs
bundle is

$(E_{0}=M\oplus K_{X}^{n-1}\oplus K_{X}^{n-2}\oplus\cdots\oplus
K_{X}^{2-n}\oplus K_{X}^{1-n}\oplus
M^{-1},\quad\theta_{0}=\begin{pmatrix}0&&&&&\\\ \mu&0&&&&\\\ &1&0&&&\\\
&&\ddots&\ddots&\\\ &&&1&0&\\\ &&&&\mu&0\end{pmatrix}).$ (33)

###### Proposition 7.8

Let $X$ be a non-compact hyperbolic Riemann surface. Suppose $\mu\neq 0$.
Suppose there exists a diagonal harmonic metric $h_{1}$ on
$(E_{0},\theta_{0})$, compatible with $SO(n,n+1)$-structure. Then there exists
a harmonic metric $h$ on $(E,\theta)$ which is compatible with
$SO(n,n+1)$-structure and weakly dominates $h_{1}.$

Proof Let $X_{i}$ $(i=1,2,\cdots)$ be a smooth exhaustion family of $X$. Let
$h^{(i)}$ be the harmonic metrics of $(E,\theta)|_{X_{i}}$ such that
$h^{(i)}|_{\partial X_{i}}=h_{0}|_{\partial X_{i}}$. Note that
$h_{0}=h_{0}|_{V}\oplus h_{0}|_{W}$, where $h_{0}|_{V},h_{0}|_{W}$ are
compatible with $Q_{V},Q_{W}$ respectively. By Lemma 7.2, $h^{(i)}$ is
compatible with $SO(n,n+1)$-structure. By Theorem 3.22, $h^{(i)}$ has a
convergence subsequence and has a smooth limit harmonic metric $h$. As a
result, $h=h|_{V}\oplus h|_{W},$ where $h|_{V},h|_{W}$ are compatible with
$Q_{V},Q_{W}$ respectively.

###### Theorem 7.9

Suppose $X$ is the unit disk. Suppose there exists a flat Hermitian metric
$h_{M}$ on $M$ and $\mu\in H^{0}(X,M^{-1}K_{X}^{n})$ satisfies
$h_{M}^{-1}g_{X}^{-n}(\mu,\mu)\in\mathcal{A}$ and not constantly $0$, then
there exists a harmonic metric $h$ on $(E,\theta)$, compatible with
$SO(n,n+1)$-structure.

Proof Consider

$(E_{1}=M\oplus K_{X}^{n-1}\oplus K_{X}^{n-2}\oplus\cdots\oplus
K_{X}^{2-n}\oplus K_{X}^{1-n}\oplus
M^{-1},\quad\theta_{1}=\begin{pmatrix}0&&&&&\\\ 0&0&&&&\\\ &1&0&&&\\\
&&\ddots&\ddots&\\\ &&&1&0&\\\ &&&&0&0\end{pmatrix}).$ (34)

Let

$h_{X}=\oplus_{k=1}^{2n-1}a_{k,2n-1}g_{X}^{k-n}$

be a diagonal metric on

$K_{X}^{n-1}\oplus K_{X}^{n-2}\oplus\cdots\oplus K_{X}^{2-n}\oplus
K_{X}^{1-n},$

where $a_{k,2n-1}$ is defined in Equation (26).

Then $h_{1}=\mathop{\rm diag}\nolimits(h_{M},h_{X},h_{M}^{-1})$ is a diagonal
harmonic metric on $(E_{1},\theta_{1})$. We compare the Higgs bundle
$(E_{1},\theta_{1})$ with $(E_{0},\theta_{0})$. It follows from Theorem 6.3
and $h_{M}^{-1}g_{X}^{-n}(\mu,\mu)\in\mathcal{A}$ that there exists a diagonal
harmonic metric $h_{0}$ on the Higgs bundle $(E_{0},\theta_{0})$. Similar to
the argument in Proposition 7.8 and the proof of Theorem 6.3 and Proposition
6.1, one can impose that $h_{0}$ is compatible with $SO(n,n+1)$-structure.
Then the statement follows from Proposition 7.8.

####  7.3.2 The generically regular semisimple case

In this subsection, we use the notation $(E,\theta(\boldsymbol{q}))$ to denote
the Higgs bundle associated with $SO(n,n+1)$-Higgs bundle
$((V,Q_{V}),(W,Q_{V}),\eta_{\mu,\nu}(\boldsymbol{q}))$ in (7.3) to emphasize
the dependence on $\boldsymbol{q}$. According to Theorem 7.4, if
$((V,Q_{V}),(W,Q_{V}),\eta_{\mu,\nu}(\boldsymbol{q}))$ is generically regular
semisimple, then $(E,\theta(\boldsymbol{q}))$ has a harmonic metric compatible
with $SO(n,n+1)$-structure. Let us mention some examples.

The following lemma is obvious.

###### Lemma 7.10

If $\boldsymbol{q}=\mathbf{0}=(0,\ldots,0)$, then
$\eta_{\mu,\nu}(\mathbf{0})^{\dagger}\circ\eta_{\mu,\nu}(\mathbf{0})\in\mathop{\rm
End}\nolimits(V)\otimes K_{X}^{2}$ is induced by the identity morphisms
$K_{X}^{n+1-2i}\cong K_{X}^{n-1-2i}\otimes K_{X}^{2}$ $(i=1,\cdots,n-1)$ and
$2\mu\nu:K_{X}^{-n+1}\rightarrow K_{X}^{n-1}\otimes K_{X}^{2}$. Therefore, if
$\mu\nu$ is not constantly $0$,
$((V,Q_{V}),(W,Q_{W}),\eta_{\mu,\nu}(\mathbf{0}))$ is generically regular
semisimple.

We obtain the following corollary from Lemma 7.10 and Theorem 7.4.

###### Corollary 7.11

If $\boldsymbol{q}=0$ and if $\mu\nu$ is not constantly $0$, then there exists
a harmonic metric $h$ of $(E,\theta(\mathbf{0}))$ compatible with
$SO(n,n+1)$-structure.

Let us consider the case $X=\mathbb{C}$. Let $M=\mathcal{O}_{\mathbb{C}}$. Let
$\mu_{0}$ and $\nu_{0}$ be non-zero polynomials. We set $\mu=\mu_{0}dz^{n}$
and $\nu=\nu_{0}dz^{n}$. For a positive integer $N$, we set
$\mathcal{P}_{N}=\\{g(z)\in\mathbb{C}[z]\,|\,\deg g\leq N\\}$. We consider the
following affine space.

$\mathcal{Q}_{N}=\\{(g_{1}(z)dz^{2},g_{2}(z)dz^{4},\cdots,g_{n-1}(z)dz^{2n-2})\,|\,g_{i}\in\mathcal{P}_{N}\\}.$

###### Proposition 7.12

There exists a non-empty Zariski open subset
$\mathcal{U}\subset\mathcal{Q}_{N}$ such that for any
$\boldsymbol{q}\in\mathcal{Q}_{N}$ the associated $SO(n,n+1)$-Higgs bundle
$((V,Q_{V}),(W,Q_{W}),\eta_{\mu,\nu}(\boldsymbol{q}))$ is generically regular
semisimple. As a result, for any $\boldsymbol{q}\in\mathcal{U}$, the Higgs
bundle $(E,\theta(\boldsymbol{q}))$ on $\mathbb{C}$ has a harmonic metric
compatible with $SO(n,n+1)$-structure.

Proof We obtain the first claim from Lemma 7.10 which says
$\mathbf{0}\in\mathcal{U}$ under the assumption $\mu\nu$ is not constantly
$0$. The second claim follows from Theorem 7.4.

##  8 $Sp(4,\mathbb{R})$-Higgs bundles

In this section, we discuss the existence of harmonic metrics on
$Sp(2n,\mathbb{R})$-Higgs bundles over non-compact Riemann surfaces by using
the techniques developed in this paper and our previous paper [LM22]. We are
mainly interested in the case $n=2$.

###### Definition 8.1

  * •

An $Sp(2n,\mathbb{R})$-Higgs bundle over a Riemann surface $X$ is determined
by $(V,\gamma,\beta)$, where $V$ is a rank $n$ vector bundle, $\gamma\in
H^{0}(X,S^{2}V^{\lor}\otimes K_{X})$ and $\beta\in H^{0}(X,S^{2}V\otimes
K_{X})$.

  * •

The associated $SL(2n,\mathbb{C})$-Higgs bundle is $(E=V\oplus
V^{\lor},\theta=\begin{pmatrix}0&\beta\\\ \gamma&0\end{pmatrix}).$

  * •

A harmonic metric $h$ on $(E,\theta)$ is said to be compatible with
$Sp(2n,\mathbb{R})$-structure if $h=h|_{V}\oplus(h|_{V})^{\lor}.$

The natural perfect pairing of $V$ and $V^{\lor}$ induces a non-degenerate
symmetric pairing $Q_{E}$ of $E=V\oplus V^{\lor}$. The Higgs field $\theta$ is
self-adjoint with respect to $Q_{E}$. If a harmonic $h$ of $(E,\theta)$ is
compatible with ${\mathcal{S}p}(2n,{\mathbb{R}})$-structure then $h$ is
compatible with $Q_{E}$.

###  8.1 Dirichlet problem

Let $Y\subset X$ be a relatively compact connected open subset with smooth
boundary $\partial Y$. Assume that $\partial Y$ is non-empty. Let $h_{\partial
Y}$ be any Hermitian metric of $E_{|\partial Y}$.

###### Lemma 8.2

Let $h$ be a harmonic metric of $(E,\theta)$ such that $h|_{\partial
Y}=h_{\partial Y}$. Suppose $h_{\partial Y}$ is compatible with
$Sp(2n,\mathbb{R})$-structure. Then $h$ is compatible with
$Sp(2n,\mathbb{R})$-structure.

Proof First we show that $h=h|_{V}\oplus h|_{V^{\lor}}.$ There exists the
automorphism $\varphi=1_{V}\oplus(-1_{V^{\lor}})$ on $E=V\oplus V^{\lor}$.
Because $\varphi^{*}\theta=-\theta$, $\varphi^{*}(h)$ is also a harmonic
metric of $(E,\theta)$. By the uniqueness of the solution for Dirichlet
problem for harmonic metric, $\varphi^{*}(h)|_{\partial Y}=h_{\partial Y}$, we
obtain $\varphi^{*}(h)=h$. It means that $h$ is the direct sum of the
Hermitian metrics of $V$ and $V^{\lor}$.

Next we show that $h|_{V^{\lor}}=(h|_{V})^{\lor}$. The metric $h$ induces the
harmonic metric $h^{\lor}=(h|_{V})^{\lor}\oplus(h|_{V^{\lor}})^{\lor}$ on
$(E^{\lor}=V^{\lor}\oplus V,\theta^{\lor}=\begin{pmatrix}0&\gamma\\\
\beta&0\end{pmatrix}).$ Re-ordering $V$ and $V^{\lor}$, we have
$(h|_{V^{\lor}})^{\lor}\oplus(h|_{V})^{\lor}$ is a harmonic metric on
$(E,\theta)$. Note that
$\big{(}(h|_{V^{\lor}})^{\lor}\oplus(h|_{V})^{\lor}\big{)}|_{\partial
Y}=h_{\partial Y}$. By the uniqueness of solutions of Dirichlet problem for
harmonic metrics, we obtain
$(h|_{V^{\lor}})^{\lor}\oplus(h|_{V})^{\lor}=h|_{V}\oplus h|_{V^{\lor}}$.
Thus, $h|_{V^{\lor}}=(h|_{V})^{\lor}$.

###  8.2 The generically regular semisimple case

Let $(V,\gamma,\beta)$ be an $Sp(2n,{\mathbb{R}})$-Higgs bundle on $X$. Let
$(E,\theta)$ denote the associated $SL(2n,{\mathbb{C}})$-Higgs bundle. We
obtain $\beta\circ\gamma\in\mathop{\rm End}\nolimits(V)\otimes K_{X}^{2}$. The
spectral curve $\Sigma_{\beta\circ\gamma}\subset(T^{\ast}X)^{\otimes 2}$ of
$\beta\circ\gamma$ is defined as usual. We obtain the finite map
$\pi:\Sigma_{\beta\circ\gamma}\to X$.

###### Definition 8.3

$(V,\gamma,\beta)$ is called generically regular semisimple if there exists
$P\in X$ such that $|\pi^{-1}(P)|=n$ and $0\not\in\pi^{-1}(P)$.

The following theorem says $(E,\theta)$ has a harmonic metric compatible with
$Sp(2n,{\mathbb{R}})$-structure in most cases. See §8.3.1 for examples in the
Gothen section.

###### Theorem 8.4

Suppose $X$ is a general non-compact Riemann surface. If $(V,\gamma,\beta)$ is
generically regular semisimple, there exists a harmonic metric $h$ of
$(E,\theta)$ compatible with $Sp(2n,\mathbb{R})$-structure.

Proof By Corollary 7.7, $(E,\theta)$ is generically regular semisimple. It is
standard to obtain the claim of Theorem 8.4 by using Lemma 8.2, [LM22,
Proposition 2.37] and Proposition 2.4. (See the proof of Theorem 7.4.)

###### Corollary 8.5

Suppose $X$ is a general non-compact Riemann surface. If $\bigl{(}\mathop{\rm
tr}\nolimits(\beta\gamma)\bigr{)}^{2}-4\det\beta\cdot\det\gamma$ and
$\det(\beta)\det(\gamma)$ are not constantly $0$, there exists a harmonic
metric $h$ of $(E,\theta)$ compatible with $Sp(4,\mathbb{R})$-structure.

Proof If $\bigl{(}\mathop{\rm
tr}\nolimits(\beta\gamma)\bigr{)}^{2}-4\det\beta\cdot\det\gamma$ and
$\det(\beta)\det(\gamma)$ are not constantly $0$, $(V,\beta,\gamma)$ is
generically regular semisimple. Hence, the claim follows from Theorem 8.4.

###  8.3 Gothen section

Given a holomorphic line bundle $N$ on $X$ and

$\mu\in H^{0}(X,N^{-2}K_{X}^{3}),\quad\nu\in H^{0}(X,N^{2}K_{X}),\quad
q_{2}\in H^{0}(X,K_{X}^{2}),$

one can construct a $Sp(4,\mathbb{R})$-Higgs bundle $(V,\gamma,\beta)$ as
follows:

$V=N\oplus N^{-1}K_{X},\quad\gamma=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\beta=\begin{pmatrix}\nu&q_{2}\\\
q_{2}&\mu\end{pmatrix}.$

The associated $SL(4,\mathbb{C})$-Higgs bundle is

$(E=N\oplus N^{-1}K_{X}\oplus N^{-1}\oplus
NK_{X}^{-1},\quad\theta=\begin{pmatrix}0&0&\nu&q_{2}\\\ 0&0&q_{2}&\mu\\\
0&1&0&0\\\ 1&0&0&0\end{pmatrix}).$ (35)

When $X$ is a compact Riemann surface of genus at least two, for each integer
$d\in(g-1,3g-3],$ there is a component $X_{d}$ (see [Got01], [BGPG12,
Proposition 3.23]), called Gothen component, of the moduli space of
$Sp(4,\mathbb{R})$-Higgs bundles formed by the above Higgs bundles determined
by $(N,\mu,\nu,q_{2})$ where $\deg(N)=d$ and $\mu\neq 0$. In particular, when
$d=3(g-1),$ it coincides with the Hitchin component for $Sp(4,\mathbb{R}).$
Such Higgs bundles are maximal and correspond to maximal $Sp(4,\mathbb{R})$
representations.

We call the above Higgs bundles are in the Gothen section. We are going to
discuss the existence of harmonic metrics of Higgs bundles in the Gothen
section over non-compact Riemann surfaces.

####  8.3.1 The generically regular semisimple case

###### Proposition 8.6

Suppose $X$ is a non-compact Riemann surface. If $\mu\nu$ and $\mu\nu-
q_{2}^{2}$ are not constantly $0$, there exists a harmonic metric $h$ of
$(E,\theta)$ compatible with $Sp(4,\mathbb{R})$-structure.

Proof Because $\det(\beta\gamma)=q_{2}^{2}-\mu\nu$ and $(\mathop{\rm
tr}\nolimits\beta\gamma)^{2}-4\det(\beta\gamma)=(2q_{2})^{2}-4(q_{2}^{2}-\mu\nu)=4\mu\nu$,
we obtain the claim from Corollary 8.5.

####  8.3.2 The case $(\mu,\nu)=(0,0)$

###### Proposition 8.7

Suppose $X$ is a non-compact Riemann surface. Suppose in addition $q_{2}\neq
0$ when $X$ is parabolic. If $(\mu,\nu)=(0,0)$, then there exists a harmonic
metric $h$ of $(E,\theta)$ compatible with $Sp(4,\mathbb{R})$-structure.

Proof For $(\mu,\nu)=(0,0)$, the Higgs bundle

$(E,\theta)=\big{(}N\oplus NK_{X}^{-1},\begin{pmatrix}0&q_{2}\\\
1&0\end{pmatrix}\big{)}\oplus\big{(}N^{-1}K_{X}\oplus
N^{-1},\begin{pmatrix}0&q_{2}\\\ 1&0\end{pmatrix}\big{)}.$

Fix a square root line bundle $K_{X}^{\frac{1}{2}}$ of $K_{X}$. Let
$L=NK_{X}^{-\frac{1}{2}}$ and let $h_{L}$ be a flat Hermitian metric on $L$.
Let $\mathop{\rm diag}\nolimits(h_{0},h_{0}^{-1})$ be a harmonic metric on
$\big{(}K_{X}^{\frac{1}{2}}\oplus
K_{X}^{-\frac{1}{2}},\begin{pmatrix}0&q_{2}\\\ 1&0\end{pmatrix}\big{)}.$ Then

$\mathop{\rm diag}\nolimits(h_{L}\otimes h_{0},h_{L}\otimes
h_{0}^{-1})\oplus\mathop{\rm diag}\nolimits(h_{L}^{-1}\otimes
h_{0},h_{L}^{-1}\otimes h_{0}^{-1})$

is a harmonic metric of $(E,\theta)$ compatible with
$Sp(4,\mathbb{R})$-structure.

####  8.3.3 The case $\mu\neq 0$

Set

$F_{1}=N,\quad F_{2}=N\oplus NK_{X}^{-1},\quad F_{3}=N\oplus NK_{X}^{-1}\oplus
N^{-1}K_{X},\quad F_{4}=E.$

Then $\mathbf{F}=\\{F_{1}\subset F_{2}\subset F_{3}\subset F_{4}\\}$ is a full
holomorphic filtration of $E$ and $\theta$ takes $F_{i}$ to $F_{i+1}\otimes
K.$ And the graded Higgs bundle is

$(E_{0}=N\oplus NK_{X}^{-1}\oplus N^{-1}K_{X}\oplus
N^{-1},\quad\theta_{0}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\\\ 0&\mu&0&0\\\
0&0&1&0\end{pmatrix}).$ (36)

###### Proposition 8.8

Let $X$ be a non-compact hyperbolic Riemann surface. Suppose $\mu\neq 0$.
Suppose there exists a diagonal harmonic metric $h_{1}$ on