(see the paragraph preceding Lemma 4.9), ${\boldsymbol{\alpha}}_{2}$ is the
sum of holomorphic $(1,0)$-forms ${\boldsymbol{\alpha}}_{2}^{+}$ and
${\boldsymbol{\alpha}}_{2}^{-}$ with values in them respectively, as required.

Case $n\geq 3$. In this case, the conformality condition in Definition 4.3
takes effect. In particular, ${\boldsymbol{\alpha}}_{2}(X)$ is conformal for
any $X$. So we may use Lemma 4.9 to conclude that ${\mathcal{L}}_{2}$ carries
a unique orientation under which ${\boldsymbol{\alpha}}_{2}$ is a nonzero
$\mathrm{Hom}_{\mathbb{C}}({\mathcal{L}}_{1},{\mathcal{L}}_{2})$-valued
holomorphic $(1,0)$-form, as required.

Thus, we have shown that ${\mathcal{L}}_{2}$ can be oriented and
${\boldsymbol{\alpha}}_{2}$ has the required description.

Next, we assume $n\geq 3$ and show that ${\mathcal{L}}_{3}$ can be oriented
and ${\boldsymbol{\alpha}}_{3}$ has the required description. As
${\mathcal{L}}_{2}$ has already been oriented and hence is a hermitian
holomorphic line bundle,
$\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{2},{\mathcal{L}}_{3})$ is a rank $2$
hermitian holomorphic vector bundle and we may write
${\boldsymbol{\alpha}}_{3}={\boldsymbol{\alpha}}_{3}^{\prime}+{\boldsymbol{\alpha}}_{3}^{\prime\prime}$
under the decomposition into $(1,0)$\- and $(0,1)$-parts

$\Omega^{1}(\Sigma,\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{2},{\mathcal{L}}_{3}))=\Omega^{1,0}(\Sigma,\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{2},{\mathcal{L}}_{3}))\oplus\Omega^{0,1}(\Sigma,\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{2},{\mathcal{L}}_{3})).$

Since ${\boldsymbol{\alpha}}_{2}$ and ${\boldsymbol{\alpha}}_{3}^{\prime}$ are
$(1,0)$-forms with values in
$\mathrm{Hom}_{\mathbb{C}}({\mathcal{L}}_{1},{\mathcal{L}}_{2})$ and
$\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{2},{\mathcal{L}}_{3})$,
respectively, their wedge-composition
${\boldsymbol{\alpha}}_{3}^{\prime}\wedge{\boldsymbol{\alpha}}_{2}$ is a
$\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{1},{\mathcal{L}}_{3})$-valued
$(2,0)$-form, which must vanish. Therefore, the $k=3$ case of the second
equation in (4.15) gives
${\boldsymbol{\alpha}}_{3}^{\prime\prime}\wedge{\boldsymbol{\alpha}}_{2}={\boldsymbol{\alpha}}_{3}\wedge{\boldsymbol{\alpha}}_{2}=0$.
This forces ${\boldsymbol{\alpha}}_{3}^{\prime\prime}$ to vanish on the set
$\\{{\boldsymbol{\alpha}}_{2}\neq 0\\}\subset\Sigma$. But since
${\boldsymbol{\alpha}}_{2}$ is holomorphic and is not identically zero as
noticed early, this set is dense. So we conclude by continuity that
${\boldsymbol{\alpha}}_{3}^{\prime\prime}$ vanishes everywhere, or in other
words, ${\boldsymbol{\alpha}}_{3}={\boldsymbol{\alpha}}_{3}^{\prime}$ is a
$(1,0)$-form. Now the rest of the argument is the same as the above one for
${\boldsymbol{\alpha}}_{2}$: we first use the second equation in (4.12) to
infer that ${\boldsymbol{\alpha}}_{3}$ is holomorphic, then show in the cases
$n=3$ and $n\geq 4$ separately that ${\mathcal{L}}_{3}$ can be oriented and
${\boldsymbol{\alpha}}_{3}$ has the required description.

This scheme of argument carries on and eventually shows that
${\mathcal{L}}_{i}$ can be oriented and ${\boldsymbol{\alpha}}_{i}$ has the
required holomorphic description for all $i=2,\cdots,n$. The proof of Part (2)
is thus finished.

For Part (1), we have already seen that
${\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1}$ are nonzero. To
show the uniqueness of the Frenet splitting, we note that as a subbundle of
$\mathsf{T}\mathbb{H}^{p,q}|_{\Sigma}$, ${\mathcal{L}}_{1}=\mathsf{T}\Sigma$
is already completely determined, whereas each ${\mathcal{L}}_{i}$ is
determined from ${\mathcal{L}}_{1},\cdots,{\mathcal{L}}_{i-1}$ because

  * •

if $i=n$, then ${\mathcal{L}}_{i}$ is just the orthogonal complement of
${\mathcal{L}}_{1}\oplus\cdots\oplus{\mathcal{L}}_{i-1}$ in
$\mathsf{T}\mathbb{H}^{p,q}|_{\Sigma}$;

  * •

if $i\leq n-1$, ${\mathcal{L}}_{i}$ is also determined because the image of
the $\mathrm{Hom}_{\mathbb{C}}({\mathcal{L}}_{i-1},{\mathcal{L}}_{i})$-valued
holomorphic $(1,0)$-form ${\boldsymbol{\alpha}}_{i}$ is full in
${\mathcal{L}}_{i}$ except at the isolated zeros.

This implies the required uniqueness and completes the proof. ∎

In what follows, given an _oriented_ A-surface $\Sigma\subset\mathbb{H}^{p,q}$
not contained in a codimension $4$ pseudo-hyperbolic subspace, we will always
view $\Sigma$ as a Riemann surface as in Theorem 4.6 (2) and refer to the
collection
$({\mathcal{L}}_{1},\cdots,{\mathcal{L}}_{n},{\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{\pm})$
of hermitian holomorphic line bundles and holomorphic forms as the _structural
data_ of $\Sigma$. Moreover, we let ${\boldsymbol{h}}_{i}$ denote the
hermitian metric on ${\mathcal{L}}_{i}$. Note that ${\mathcal{L}}_{1}$ is the
anti-canonical line bundle ${\mathcal{K}}^{-1}$ and
${\boldsymbol{g}}:=\operatorname{\mathsf{Re}}{\boldsymbol{h}}_{1}=\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle_{1}$
is the first fundamental form.

###  4.3. Affine Toda system from Gauss-Codazzi equations

In the above proof, we wrote the Gauss-Codazzi equations of an A-surface
$\Sigma\subset\mathbb{H}^{p,q}$ as the three sets of equations (4.9), (4.12)
and (4.15), and basically deduced Theorem 4.6 from the latter two. We show
next that the first set of equations (4.9) is an _affine Toda system_ , namely
the same type of equation as the Hitchin equation of cyclic Higgs bundles (see
[5, 6, 20]). This is an expected result, as A-surfaces will be linked directly
with cyclic Higgs bundles in §6 below.

######  Theorem 4.10.

For any oriented A-surface $\Sigma\subset\mathbb{H}^{p,q}$ (with
$n=\frac{p+q}{2}\geq 2$) not contained in a pseudo-hyperbolic subspace of
codimension $4$, the structural data satisfies

$\displaystyle\text{if $n=2$:}\quad\begin{cases}\partial_{z\bar{z}}^{2}\log
h_{1}=\tfrac{1}{2}h_{1}-\tfrac{h_{2}}{h_{1}}|\alpha_{2}^{+}|^{2}-\frac{1}{h_{1}h_{2}}|\alpha_{2}^{-}|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{2}=\tfrac{h_{2}}{h_{1}}|\alpha_{2}^{+}|^{2}-\frac{1}{h_{1}h_{2}}|\alpha_{2}^{-}|^{2};\end{cases}$
$\displaystyle\text{if $n\geq
3$:}\quad\begin{cases}\partial_{z\bar{z}}^{2}\log
h_{1}=\tfrac{1}{2}h_{1}-\tfrac{h_{2}}{h_{1}}|\alpha_{2}|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{k}=\tfrac{h_{k}}{h_{k-1}}|\alpha_{k}|^{2}-\tfrac{h_{k+1}}{h_{k}}|\alpha_{k+1}|^{2}\quad\text{for
$2\leq k\leq n-2$},\\\\[5.69046pt] \partial_{z\bar{z}}^{2}\log
h_{n-1}=\tfrac{h_{n-1}}{h_{n-2}}|\alpha_{n-1}|^{2}-\tfrac{h_{n}}{h_{n-1}}|\alpha_{n}^{+}|^{2}-\tfrac{1}{h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{n}=\tfrac{h_{n}}{h_{n-1}}|\alpha_{n}^{+}|^{2}-\tfrac{1}{h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2};\end{cases}$

where notations are as follows: we view ${\boldsymbol{\alpha}}_{j}$ ($2\leq
j\leq n-1$) and ${\boldsymbol{\alpha}}_{n}^{\pm}$ as holomorphic sections of
${\mathcal{K}}\\!{\mathcal{L}}_{j-1}^{-1}{\mathcal{L}}_{j}$ and
${\mathcal{K}}\\!{\mathcal{L}}_{n-1}^{-1}{\mathcal{L}}_{n}^{\pm 1}$,
respectively, then pick a conformal coordinate $z$ of $\Sigma$ and a
holomorphic local frame $\ell_{i}(z)$ of ${\mathcal{L}}_{i}$ to write locally
$h_{i}={\boldsymbol{h}}_{i}(\ell_{i},\ell_{i})$,
${\boldsymbol{\alpha}}_{j}=\alpha_{j}\ell_{j-1}^{-1}\ell_{j}{\mathsf{d}z}$ and
${\boldsymbol{\alpha}}_{n}^{\pm}=\alpha_{n}^{\pm}\ell_{n-1}^{-1}\ell_{n}^{\pm
1}{\mathsf{d}z}$.

######  Remark 4.11.

Despite having been exhibited in a local form, these equations make sense
globally. In fact, after being divided by $h_{1}$, every term is a globally
defined function on $\Sigma$: the left-hand side
$\tfrac{1}{h_{1}}\partial_{z\bar{z}}^{2}\log h_{i}$ is a constant times the
ratio between the curvature form ${\bar{\partial}}\partial\log h_{i}$ of the
Chern connection of ${\mathcal{L}}_{i}$ and the volume form
$h_{1}{\mathsf{d}x}\wedge{\mathsf{d}y}$ of the first fundamental form
${\boldsymbol{g}}=\operatorname{\mathsf{Re}}{\boldsymbol{h}}_{1}$, whereas
each term on the right-hand side is the (pointwise) squared norm of
${\boldsymbol{\alpha}}_{j}$ or ${\boldsymbol{\alpha}}_{n}^{\pm}$ under the
hermitian metric on the respective line bundle induced by the
${\boldsymbol{h}}_{i}$’s, namely

$\|{\boldsymbol{\alpha}}_{j}\|^{2}=\tfrac{h_{j}}{h_{1}h_{j-1}}|\alpha_{j}|^{2}\quad(2\leq
j\leq
n-1),\quad\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}=\tfrac{h_{n}}{h_{1}h_{n-1}}|\alpha_{n}^{+}|^{2},\quad\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2}=\tfrac{1}{h_{1}h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2}.$

###### Proof.

In the proof of Theorem 4.6, we obtained equations (4.9), (4.12) and (4.15)
after choosing an orthonormal local frame $(u_{i},v_{i})$ for each
${\mathcal{L}}_{i}$. The matrices $\Omega_{i}$ and ${\boldsymbol{\alpha}}_{j}$
therein depend on the choice. Under the special choice
$(u_{i},v_{i})=h_{i}^{-\frac{1}{2}}(\ell_{i},\boldsymbol{i}\ell_{i})$, we have

(4.18)
${\boldsymbol{\alpha}}_{j}=\tfrac{\sqrt{h_{j}}}{\sqrt{h_{j-1}}}\scalebox{0.9}{
$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})&-\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})&\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})\end{pmatrix}$}$

for $2\leq j\leq n-1$ and

(4.19)
${\boldsymbol{\alpha}}_{n}^{+}=\tfrac{\sqrt{h_{n}}}{\sqrt{h_{n-1}}}\scalebox{0.9}{
$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{n}^{+}{\mathsf{d}z})&-\operatorname{\mathsf{Im}}(\alpha_{n}^{+}{\mathsf{d}z})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{n}^{+}{\mathsf{d}z})&\operatorname{\mathsf{Re}}(\alpha_{n}^{+}{\mathsf{d}z})\end{pmatrix}$},\quad{\boldsymbol{\alpha}}_{n}^{-}=\tfrac{1}{\sqrt{h_{n-1}h_{n}}}\scalebox{0.9}{
$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{n}^{-}{\mathsf{d}z})&-\operatorname{\mathsf{Im}}(\alpha_{n}^{-}{\mathsf{d}z})\\\\[2.84544pt]
-\operatorname{\mathsf{Im}}(\alpha_{n}^{-}{\mathsf{d}z})&-\operatorname{\mathsf{Re}}(\alpha_{n}^{-}{\mathsf{d}z})\end{pmatrix}$}.$

We shall show that (4.9) reduces to the required equations. After expanding
the $1$-forms in (4.18) and (4.19) in the real coordinates $(x,y)$ with
$z=x+\boldsymbol{i}y$ (e.g.
$\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})=\operatorname{\mathsf{Re}}(\alpha_{j}){\mathsf{d}x}-\operatorname{\mathsf{Im}}(\alpha_{j}){\mathsf{d}y}$,
etc.), we obtain by computations that

(4.20)
$\displaystyle{\boldsymbol{\alpha}}_{j}\wedge\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{j}$
$\displaystyle=-\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{j}\wedge{\boldsymbol{\alpha}}_{j}=-\tfrac{2\,h_{j}}{h_{j-1}}|\alpha_{j}|^{2}\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}{\mathsf{d}x}\wedge{\mathsf{d}y}\quad(2\leq j\leq n-1),$
(4.21)
$\displaystyle{\boldsymbol{\alpha}}_{n}^{+}\wedge\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{+}$
$\displaystyle=-\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{+}\wedge{\boldsymbol{\alpha}}_{n}^{+}=-\tfrac{2\,h_{n}}{h_{n-1}}|\alpha_{n}^{+}|^{2}\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}{\mathsf{d}x}\wedge{\mathsf{d}y},$ (4.22)
$\displaystyle{\boldsymbol{\alpha}}_{n}^{-}\wedge{\boldsymbol{\alpha}}_{n}^{-}$
$\displaystyle=\tfrac{2}{h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2}\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}{\mathsf{d}x}\wedge{\mathsf{d}y},$ (4.23)
$\displaystyle{\boldsymbol{\alpha}}_{n}^{+}\wedge{\boldsymbol{\alpha}}_{n}^{-}$
$\displaystyle=\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{+}\wedge{\boldsymbol{\alpha}}_{n}^{-}=0.$

Since ${\boldsymbol{\alpha}}_{n}^{-}$ is a symmetric and
${\boldsymbol{\alpha}}_{n}={\boldsymbol{\alpha}}_{n}^{+}+{\boldsymbol{\alpha}}_{n}^{-}$,
we deduce from (4.21)$\sim$(4.23) that

(4.24)
$\displaystyle{\boldsymbol{\alpha}}_{n}\wedge\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}$
$\displaystyle=({\boldsymbol{\alpha}}_{n}^{+}+{\boldsymbol{\alpha}}_{n}^{-})\wedge(\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{+}+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{-})=2\left(-\tfrac{h_{n}}{h_{n-1}}|\alpha_{n}^{+}|^{2}+\tfrac{1}{h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2}\right)\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}{\mathsf{d}x}\wedge{\mathsf{d}y},$ (4.25)
$\displaystyle\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\wedge{\boldsymbol{\alpha}}_{n}$
$\displaystyle=(\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{+}+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{-})\wedge({\boldsymbol{\alpha}}_{n}^{+}+{\boldsymbol{\alpha}}_{n}^{-})=2\left(\tfrac{h_{n}}{h_{n-1}}|\alpha_{n}^{+}|^{2}+\tfrac{1}{h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2}\right)\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}{\mathsf{d}x}\wedge{\mathsf{d}y}.$

Next, since $\Omega_{i}\wedge\Omega_{i}=0$ (see the proof of Theorem 4.6),
$\mathsf{d}\Omega_{i}$ is the matrix of the curvature $2$-form
$F_{\nabla_{i}}\in\Omega^{2}(\Sigma,\operatorname{End}_{\mathbb{C}}({\mathcal{L}}_{i}))$
of $\nabla_{i}$ under the frame $(u_{i},v_{i})$. But since $\nabla_{i}$ is the
Chern connection of ${\mathcal{L}}_{i}$, its curvature is expressed in the
holomorphic frame $\ell_{i}$ as ${\bar{\partial}}\partial\log
h_{i}=2\boldsymbol{i}\partial_{z\bar{z}}^{2}\,\log
h_{i}{\mathsf{d}x}\wedge{\mathsf{d}y}$. Changing to the real frame
$(u_{i},v_{i})$, we get

(4.26) $\mathsf{d}\Omega_{i}=2\,\partial_{z\bar{z}}^{2}\log
h_{i}\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}{\mathsf{d}x}\wedge{\mathsf{d}y}.$

Finally, since $\theta=\leftidx{{}^{\mathsf{t}}}\\!{(}\theta_{1},\theta_{2})$
is the frame of $\mathsf{T}^{*}\Sigma$ dual to the orthonormal frame
$(u_{1},v_{1})$ of ${\mathcal{L}}_{1}=\mathsf{T}\Sigma$,
$\theta_{1}\wedge\theta_{2}$ equals the volume form
$h_{1}{\mathsf{d}x}\wedge{\mathsf{d}y}$ of first fundamental form
${\boldsymbol{g}}=\operatorname{\mathsf{Re}}{\boldsymbol{h}}_{1}$. It follows
that

(4.27)
$\theta\wedge\leftidx{{}^{\mathsf{t}}}\\!{\theta}=-\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}\theta_{1}\wedge\theta_{2}=-h_{1}\scalebox{0.8}{$\begin{pmatrix}&-1\\\
1&\end{pmatrix}$}{\mathsf{d}x}\wedge{\mathsf{d}y}.$

Inserting (4.20) and (4.24)$\sim$(4.27) into (4.9), we get the required
equations. ∎

###  4.4. Gauss map

The classical notion of Gauss map has a nice generalization to spacelike
surfaces $\Sigma$ in $\mathbb{H}^{2,q}$, which plays a key role in [10, 21].
In this case, the Gauss map $f$ of $\Sigma$ takes values in the symmetric
space $X_{\mathrm{SO}_{0}(2,q+1)}$ (see §2.1) and is defined in such a way
that $f(z)$ is the timelike totally geodesic $q$-sphere
$S\subset\mathbb{H}^{2,q}$ passing through $z$ with $\mathsf{T}_{z}S$ being
the orthogonal complement of $\mathsf{T}_{z}\Sigma$ in
$\mathsf{T}_{z}\mathbb{H}^{2,q}$. For A-surfaces in $\mathbb{H}^{p,q}$, we
introduce:

######  Definition 4.12.

Given an A-surface $\Sigma\subset\mathbb{H}^{p,q}$ not contained in any
pseudo-hyperbolic subspace of codimension $4$, we let
${\mathcal{N}}^{-}:={\mathcal{L}}_{2}\oplus{\mathcal{L}}_{4}\oplus\cdots$
denote the timelike part of the normal bundle of $\Sigma$ given by the Frenet
splitting and define the Gauss map

$f:\Sigma\to X_{\mathrm{SO}_{0}(p,q+1)}=\big{\\{}\text{timelike totally
geodesic $q$-spheres in $\mathbb{H}^{p,q}$}\big{\\}}$

in such a way that for every $z\in\Sigma$, $f(z)$ is the totally geodesic
$q$-sphere $S\subset\mathbb{H}^{p,q}$ passing through $z$ with
$\mathsf{T}_{z}S={\mathcal{N}}^{-}_{z}\subset\mathsf{T}_{z}\mathbb{H}^{p,q}$.

Note that since the definition requires the uniqueness of the Frenet splitting
(Theorem 4.6 (1)), the assumption that $\Sigma$ is not in a subspace of
codimension $4$ is indispensable (see Remark 4.8).

######  Theorem 4.13.

In the above setting, $f$ is a conformal minimal immersion, with pullback
metric

(4.28)
$f^{*}{\boldsymbol{g}}_{X}=(8n-4)\big{(}\tfrac{1}{2}+\|{\boldsymbol{\alpha}}_{2}\|^{2}+\cdots+\|{\boldsymbol{\alpha}}_{n-1}\|^{2}+\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}+\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2}\big{)}{\boldsymbol{g}},$

where ${\boldsymbol{g}}_{X}$ is the metric on $X:=X_{\mathrm{SO}_{0}(p,q+1)}$
and ${\boldsymbol{g}}=\operatorname{\mathsf{Re}}{\boldsymbol{h}}_{1}$ is the
first fundamental form of $\Sigma$.

Here, the squared norms in (4.28) are as define in Remark 4.11, while the
metric ${\boldsymbol{g}}_{X}$ is the one induced by the Killing form
$B(x,y)=(2n-1)\operatorname{Tr}(xy)$ of $\mathfrak{so}(n,n+1)$.

###### Proof.

We first describe the metric and connection on $f^{*}\mathsf{T}X$ (see §3.5)
by using Lemma 3.8 and the moving frame
$(\iota,u_{1},v_{1},\cdots,u_{n},v_{n})$ from the proof of Theorem 4.6.
Viewing $X$ as the space of $(q+1)$-dimensional negative definite subspaces of
$\mathbb{R}^{p,q+1}$, by definition of the Gauss map $f$, we can write

$f(z)=\mathrm{Span}\big{\\{}\iota(z),\,u_{2}(z),\,v_{2}(z),\,u_{4}(z),\,v_{4}(z),\,\cdots\big{\\}}.$

In particular, the matrix-valued function

$\widetilde{f}:\Sigma\to
G:=\mathrm{SO}_{0}(p,q+1),\quad\widetilde{f}:=(u_{1},\,v_{1},\,u_{3},\,v_{3},\,\cdots,\,\iota,\,u_{2},\,v_{2},\,u_{4},\,v_{4},\,\cdots)$

is a lift of $f$. Thus, by Lemma 3.8, $f^{*}\mathsf{T}X$ identifies locally
with the trivial $\mathfrak{p}$-bundle endowed with the metric given by the
Killing form $B$ and the connection $\widehat{\nabla}$ given by the
$\operatorname{End}(\mathfrak{p})$-valued $1$-form
$\operatorname{ad}(\widetilde{f}^{*}\omega)^{\mathfrak{k}}$, while
$\mathsf{d}f\in\Omega^{1}(\Sigma,f^{*}\mathsf{T}X)$ is locally the
$\mathfrak{p}$-valued $1$-form $(\widetilde{f}^{*}\omega)^{\mathfrak{p}}$. In
order to get more explicit expressions, we view
$\mathfrak{g}:=\mathfrak{so}(p,q+1)$ and $G$ as matrix algebra and matrix
group contained in $\mathbb{R}^{(p+q+1)\times(p+q+1)}$, so that the Maurer-
Cartan form $\omega$ can be written as $g^{-1}\mathsf{d}g$ (where $g\in G$ is
the matrix coordinate), and hence
$\widetilde{f}^{*}\omega=\widetilde{f}^{-1}\mathsf{d}\widetilde{f}$. By
equation (4.4) in the proof of Theorem 4.6, we obtain

$\widetilde{f}^{*}\omega=\widetilde{f}^{-1}\mathsf{d}\widetilde{f}=\left(\begin{array}[]{c|c}\scalebox{0.85}{$\begin{matrix}\Omega_{1}&&\\\
&\Omega_{3}&\\\\[-2.84544pt]
&&\ddots\end{matrix}$}&\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}\\\
\hline\cr\Gamma&\scalebox{0.85}{$\begin{matrix}0&&&\\\\[-2.84544pt]
&\Omega_{2}&&\\\ &&\Omega_{4}&\\\\[-2.84544pt]
&&&\ddots\end{matrix}$}\end{array}\right),$

where

(4.29)
$\Gamma=\scalebox{0.9}{$\begin{pmatrix}\leftidx{{}^{\mathsf{t}}}\\!{\theta}&&&&\\\
{\boldsymbol{\alpha}}_{2}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}&&&\\\
&{\boldsymbol{\alpha}}_{4}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{5}&&\\\
&&\ddots&\ddots&\\\
&&&{\boldsymbol{\alpha}}_{n-2}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}\\\
&&&&{\boldsymbol{\alpha}}_{n}\end{pmatrix}$}\text{ if $n$ is
even,}\quad\Gamma=\scalebox{0.9}{$\begin{pmatrix}\leftidx{{}^{\mathsf{t}}}\\!{\theta}&&&&\\\
{\boldsymbol{\alpha}}_{2}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}&&&\\\
&{\boldsymbol{\alpha}}_{4}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{5}&&\\\
&&\ddots&\ddots&\\\
&&&{\boldsymbol{\alpha}}_{n-1}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\end{pmatrix}$}\text{
if $n$ is odd}.$

Meanwhile, since the components $\mathfrak{k}$ and $\mathfrak{p}$ of the
Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ are

$\mathfrak{k}=\left\\{\begin{pmatrix}\xi_{1}&\\\ &\xi_{2}\end{pmatrix}\
\Bigg{|}\ \parbox{56.9055pt}{$\xi_{1}\in\mathfrak{so}(p)$,\\\
$\xi_{2}\in\mathfrak{so}(q+1)$}\right\\},\quad\mathfrak{p}=\left\\{\begin{pmatrix}&\leftidx{{}^{\mathsf{t}}}\\!{\eta}\\\
\eta&\end{pmatrix}\ \Bigg{|}\ \eta\in\mathbb{R}^{p\times(q+1)}\right\\},$

we may identify $\mathfrak{p}$ with $\mathbb{R}^{p\times(q+1)}$ in such a way
that the action of $\mathrm{diag}(\xi_{1},\xi_{2})\in\mathfrak{k}$ sends
$\eta\in\mathbb{R}^{p\times(q+1)}$ to $\xi_{2}\eta-\eta\xi_{1}$, while the
Killing form $B$ on $\mathfrak{p}$ is $(4n-2)$-times the standard metric of
$\mathbb{R}^{p\times(q+1)}$. Therefore, we conclude that $f^{*}\mathsf{T}X$
identifies locally with the trivial $\mathbb{R}^{p\times(q+1)}$-bundle endowed
with the metric
$\langle\eta_{1},\eta_{2}\rangle=(4n-2)\operatorname{Tr}(\leftidx{{}^{\mathsf{t}}}\\!{\eta}_{1}\eta_{2})$
and the connection $\widehat{\nabla}$ defined by

$\widehat{\nabla}\eta=\mathsf{d}\eta+\scalebox{0.9}{$\begin{pmatrix}0&&&\\\\[-2.84544pt]
&\Omega_{2}&&\\\ &&\Omega_{4}&\\\\[-2.84544pt]
&&&\ddots\end{pmatrix}$}\eta-\eta\scalebox{0.9}{$\begin{pmatrix}\Omega_{1}&&\\\
&\Omega_{3}&\\\\[-2.84544pt] &&\ddots\end{pmatrix}$},$

while $\mathsf{d}f\in\Omega^{1}(\Sigma,f^{*}\mathsf{T}X)$ is locally the
$\mathbb{R}^{p\times(q+1)}$-valued $1$-form $\Gamma$ in (4.29).

Next, we verify the required expression (4.28) of $f^{*}{\boldsymbol{g}}_{X}$.
Using the above identifications, we obtain by computations that

$f^{*}{\boldsymbol{g}}_{X}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})=B(\mathsf{d}f({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}),\mathsf{d}f({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}))=(4n-2)\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\Gamma({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=(4n-2)\Big{\\{}\leftidx{{}^{\mathsf{t}}}\\!{\theta}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\theta({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})+\sum_{j=2}^{n}\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}\Big{\\}},$

where “$\,\operatorname{Tr}\,$” takes the matrix trace of each matrix-valued
$2$-tensor to produce a scalar $2$-tensor. Since
$\theta=\leftidx{{}^{\mathsf{t}}}\\!{(}\theta_{1},\theta_{2})$ is the frame of
$\mathsf{T}^{*}\Sigma$ dual to the orthonormal frame $(u_{1},v_{1})$ of
$\mathsf{T}\Sigma$, the term
$\leftidx{{}^{\mathsf{t}}}\\!{\theta}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\theta({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})$
is just the first fundamental form ${\boldsymbol{g}}$. To compute the term
$\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}$,
we pick a coordinate $z$ of $\Sigma$ and a holomorphic local frame $\ell_{i}$
of ${\mathcal{L}}_{i}$ as in Theorem 4.10 and let $(u_{i},v_{i})$ be as in the
proof of that theorem. Using the expression of ${\boldsymbol{\alpha}}_{j}$
given in that proof, we get, for $2\leq j\leq n-1$,

$\displaystyle\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}$
$\displaystyle=\tfrac{h_{j}}{h_{j-1}}\operatorname{Tr}\scalebox{0.9}{
$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})&\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})\\\\[2.84544pt]
-\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})&\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})\end{pmatrix}$}\otimes\scalebox{0.9}{
$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})&-\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})&\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})\end{pmatrix}$}$
$\displaystyle=\tfrac{2\,h_{j}}{h_{j-1}}\big{(}\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})\otimes\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})+\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})\otimes\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})\big{)}$
$\displaystyle=\tfrac{2\,h_{j}}{h_{j-1}}|\alpha_{j}|^{2}({\mathsf{d}x}^{2}+{\mathsf{d}y}^{2})=\tfrac{2\,h_{j}}{h_{1}h_{j-1}}|\alpha_{j}|^{2}h_{1}({\mathsf{d}x}^{2}+{\mathsf{d}y}^{2})=2\|{\boldsymbol{\alpha}}_{j}\|^{2}{\boldsymbol{g}}.$

As for
${\boldsymbol{\alpha}}_{n}={\boldsymbol{\alpha}}_{n}^{+}+{\boldsymbol{\alpha}}_{n}^{-}$,
we have

$\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{n}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}+\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{n}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}+\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{n}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}+\operatorname{Tr}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{n}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}.$

By similar computations, the last two terms vanish and the first two terms
equal $2\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}{\boldsymbol{g}}$ and
$2\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2}{\boldsymbol{g}}$, respectively.
Putting the results together, we get (4.28).

This already implies that $f$ is a conformal immersion. We finally show that
$f$ is minimal via Lemma 3.7. Note that the above local description of
$f^{*}\mathsf{T}X$ effectively identifies $f^{*}\mathsf{T}X$ globally as the
homomorphism bundle
$\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{1}\oplus{\mathcal{L}}_{3}\oplus\cdots,\
\underline{\mathbb{R}}\oplus{\mathcal{L}}_{2}\oplus{\mathcal{L}}_{4}\oplus\cdots)$
with metric and connection induced by those on the trivial line bundle
$\underline{\mathbb{R}}$ and each ${\mathcal{L}}_{i}$. In particular,

(4.30) $\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{k},{\mathcal{L}}_{l}),\ \
\mathrm{Hom}({\mathcal{L}}_{k},\underline{\mathbb{R}})\quad(k\in\\{1,3,\cdots\\},\
l\in\\{2,4,\cdots\\})$

are parallel subbundles orthogonal to each other. Meanwhile, by the expression
(4.29), the only nonzero components of $\mathsf{d}f=\Gamma$ are
${\boldsymbol{\alpha}}_{l}\in\Omega^{1}(\Sigma,\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{l-1},{\mathcal{L}}_{l}))$,
${\boldsymbol{\alpha}}_{k}^{*}\in\Omega^{1}(\Sigma,\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{k},{\mathcal{L}}_{k-1}))$
and
${\boldsymbol{g}}\in\Omega^{1}(\Sigma,\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{1},\underline{\mathbb{R}}))=C^{\infty}(\Sigma,\mathsf{T}^{*}\Sigma\otimes\mathsf{T}^{*}\Sigma)$
(the last one corresponds to the term $\leftidx{{}^{\mathsf{t}}}\\!{\theta}$).
To apply Lemma 3.7, we need to give $f^{*}\mathsf{T}X$ an orthogonal,
parallel, complex structure under which $\mathsf{d}f$ is a $(1,0)$-form. Since
each ${\mathcal{L}}_{i}$ is a hermitian holomorphic vector bundle (Theorem
4.6), so is every subbundle in (4.30) (see the paragraph preceding Lemma 4.9),
and the component ${\boldsymbol{\alpha}}_{l}$ is a $(1,0)$-form by Theorem
4.6. Also, the component ${\boldsymbol{g}}$ is a $(1,0)$-form since the metric
${\boldsymbol{g}}$ is conformal. On the other hand, the component
${\boldsymbol{\alpha}}_{k}^{*}$ is a $(0,1)$-form, but we can invert the
complex structure on
$\mathrm{Hom}_{\mathbb{R}}({\mathcal{L}}_{k},{\mathcal{L}}_{k-1})$ to make it
a $(1,0)$-from. Thus, we conclude that $f^{*}\mathsf{T}X$ does admit a complex
structure fulfilling the assumption of Lemma 3.7, so we can apply the lemma to
complete the proof. ∎

###  4.5. ${\boldsymbol{J}}$-holomorphic curves in $\mathbb{H}^{4,2}$ as
A-surfaces

We now restrict to the case $n=3$ and consider the interplay between
A-surfaces $\Sigma\subset\mathbb{H}^{4,2}$ and the almost complex structure
${\boldsymbol{J}}$ on $\mathbb{H}^{4,2}$ (see §2.6). By Lemma 2.12 and the
description of $X_{G_{2}^{\prime}}$ in Corollary 2.14, we immediately deduce
from Theorem 4.13:

######  Corollary 4.14.

For any A-surface $\Sigma\subset\mathbb{H}^{4,2}$ which is not part of a
totally geodesic $\mathbb{H}^{2}\subset\mathbb{H}^{4,2}$, the following
conditions are equivalent to each other:

  * •

the Gauss map $f:\Sigma\to X_{\mathrm{SO}_{0}(3,4)}$ takes values in the
totally geodesic submanifold $X_{G_{2}^{\prime}}\subset
X_{\mathrm{SO}_{0}(3,4)}$ (hence is a conformal minimal immersion into
$X_{G_{2}^{\prime}}\,$);

  * •

the subbundle bundle
${\mathcal{L}}_{2}\subset\mathsf{T}\mathbb{H}^{4,2}|_{\Sigma}$ is preserved by
${\boldsymbol{J}}$.

A particularly interesting subclass of A-surfaces satisfying these conditions
are:

######  Proposition 4.15.

Let $\Sigma\subset\mathbb{H}^{4,2}$ be a spacelike
${\boldsymbol{J}}$-holomorphic curve with nowhere vanishing second fundamental
form. Then $\Sigma$ is an A-surface if and only if the osculation line
$\\{{\boldsymbol{I\\!\\!I}}(X,Y)\mid
X,Y\in\mathsf{T}_{z}\Sigma\\}\subset{\mathcal{N}}_{z}$ (see §3.4) is negative
definite for all $z\in\Sigma$. In this case, the whole Frenet splitting
$\mathsf{T}\mathbb{H}^{4,2}|_{\Sigma}={\mathcal{L}}_{1}\oplus{\mathcal{L}}_{2}\oplus{\mathcal{L}}_{3}$
is preserved by ${\boldsymbol{J}}$. In particular, $\Sigma$ satisfies the
conditions in Corollary 4.14.

###### Proof.

If $\Sigma$ is a ${\boldsymbol{J}}$-holomorphic curve and an A-surface at the
same time, then the fiber of ${\mathcal{L}}_{2}$ at any $z\in\Sigma$ where
${\boldsymbol{I\\!\\!I}}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})$
is nonzero is exactly the osculation line at $z$, so the “only if” part is
trivial. Conversely, assuming that the osculation lines are negative definite,
we let ${\mathcal{L}}_{2}$ be the subbundle of the normal bundle
${\mathcal{N}}$ formed by these complex lines and ${\mathcal{L}}_{3}$ be the
orthogonal complement of ${\mathcal{L}}_{2}$ in ${\mathcal{N}}$. Since the
almost complex structure ${\boldsymbol{J}}$ of $\mathbb{H}^{4,2}$ is
orthogonal, the subbundles ${\mathcal{L}}_{1}=\mathsf{T}\Sigma$,
${\mathcal{L}}_{2}$ and ${\mathcal{L}}_{3}$ of
$\mathsf{T}\mathbb{H}^{4,2}|_{\Sigma}$ form a Frenet splitting preserved by
${\boldsymbol{J}}$. Also,
${\boldsymbol{\alpha}}_{2}(X)={\boldsymbol{I\\!\\!I}}(X,{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}):{\mathcal{L}}_{1}\to{\mathcal{L}}_{2}$
is conformal because
${\boldsymbol{I\\!\\!I}}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})$
is complex bilinear by Proposition 3.4 (2). Therefore, $\Sigma$ is an
A-surface. This proves the “if” part and the “In this case” statement. ∎

######  Remark 4.16.

The A-surfaces that we construct from cyclic $G_{2}^{\prime}$-Higgs bundles in
§6.3 below belong to the above subclass. They also have the following extra
properties by construction:

  * •

${\boldsymbol{\alpha}}_{3}^{+}$ is nowhere zero as well; as a consequence,
${\mathcal{L}}_{1}$, ${\mathcal{L}}_{2}$ and ${\mathcal{L}}_{3}$ identify with
${\mathcal{K}}^{-1}$, ${\mathcal{K}}^{-2}$ and ${\mathcal{K}}^{-3}$,
respectively, as holomorphic line bundles;

  * •

the hermitian metrics ${\boldsymbol{h}}_{1}$, ${\boldsymbol{h}}_{2}$ and
${\boldsymbol{h}}_{3}$ on these line bundles satisfy
${\boldsymbol{h}}_{3}={\boldsymbol{h}}_{1}{\boldsymbol{h}}_{2}/4$.

By using $G_{2}^{\prime}$-moving frames, it can be shown that these properties
are not ad hoc and are local in nature. Namely, they are satisfied by every
A-surface $\Sigma$ as in Proposition 4.15.

######  Remark 4.17.

As the concluding remark for this section, we note that all the results till
now are local in nature. Therefore, Theorems 4.6, 4.10 and Proposition 4.15
still hold if the ambient space $\mathbb{H}^{p,q}$ is replaced by an
orientable pseudo-Riemannian manifold locally modeled on $\mathbb{H}^{p,q}$,
or more generally if $\Sigma$ is replaced by an _A-surface immersion_ from the
universal cover $\widetilde{\Sigma}$ of an abstract surface $\Sigma$ to
$\mathbb{H}^{p,q}$ which is equivariant with respect to some representation
$\rho$ of $\pi_{1}(\Sigma)$ in $\mathrm{SO}_{0}(p,q+1)$ or $G_{2}^{\prime}$.
In particular, we can treat the structural data of such an immersion as
defined on $\Sigma$ rather than $\widetilde{\Sigma}$ by virtue of the
invariance under the $\pi_{1}(\Sigma)$ action. The definition and results
concerning the Gauss map $f$ also hold in this equivariant setting, and $f$ is
$\rho$-equivariant as well.

##  5\. Infinitesimal rigidity and unique determination from Gauss map

In this section, we restrict to A-surfaces $\Sigma$ satisfying an extra
hypothesis related to Dai-Li’s work [23] and prove two result for them: the
first is the core result of this paper that $\Sigma$ is infinitesimally rigid;
the second is that $\Sigma$ is determined by its Gauss map up to the antipodal
map of $\mathbb{H}^{p,q}$.

###  5.1. Hypothesis ($\star$ ‣ 5.1) and a sufficient condition

Fix an oriented A-surface $\Sigma$ in an orientable pseudo-Riemannian manifold
$M$ locally modeled on $\mathbb{H}^{p,q}$ with $n=\frac{p+q}{2}\geq 3$. Also
assume that $\Sigma$ is not locally contained in any pseudo-hyperbolic
subspace of codimension $4$, so that we have a unique Frenet splitting
$\mathsf{T}M|_{\Sigma}={\mathcal{L}}_{1}\oplus\cdots\oplus{\mathcal{L}}_{n}$
by Theorem 4.6 and Remark 4.17, with each ${\mathcal{L}}_{i}$ being a
hermitian holomorphc line bundle. The extra hypothesis that we consider in
this section can be simply stated as _the first $n-1$ line bundles
${\mathcal{L}}_{1},\cdots,{\mathcal{L}}_{n-1}$ have semi-negative Chern
forms_. Here, the _Chern form_ is $\tfrac{\boldsymbol{i}}{2\pi}$-times the
curvature form of the Chern connection, which is an $\mathbb{R}$-valued
$2$-form representing the first Chern class, and we call a $2$-form $\omega$
_semi-negative_ if the open set $\\{\omega\neq 0\\}\subset\Sigma$ is dense and
$\omega$ is opposite to the orientation of $\Sigma$ on this set. Although this
notion of negativity depends on the choice of orientation for $\Sigma$, the
hypothesis as a whole does not, because reversing the orientation of $\Sigma$
will also reverse the one of each ${\mathcal{L}}_{i}$ with $2\leq i\leq n-1$
(see Remark 4.7), hence will reverse the complex structure and Chern form of
${\mathcal{L}}_{i}$ as well.

######  Remark 5.1.

It is reasonable that the semi-negativity assumption is not imposed on
${\mathcal{L}}_{n}$: unlike ${\mathcal{L}}_{i}$ with $2\leq i\leq n-1$, whose
orientation (and hence complex structure) is uniquely determined, we are free
to choose the orientation of ${\mathcal{L}}_{n}$, which just reflects the
orientation of the ambient space $M$ (see Remark 4.7), and such a choice is
irrelevant to what we are doing.

By virtue of Theorem 4.10, we can reformulate the hypothesis as follows. Since
the Chern form of ${\mathcal{L}}_{i}$ is expressed locally as
$\tfrac{\boldsymbol{i}}{2\pi}{\bar{\partial}}\partial\log
h_{i}=-\tfrac{1}{\pi}\partial_{z\bar{z}}^{2}\log
h_{i}{\mathsf{d}x}\wedge{\mathsf{d}y}$, the hypothesis is equivalent to (see
Theorem 4.10 and Remark 4.11 for the notations):

($\star$) $\parbox{82.51282pt}{on an open dense subset of $\Sigma$,
}\begin{cases}\tfrac{1}{h_{1}}\partial_{z\bar{z}}^{2}\log
h_{1}=\tfrac{1}{2}-\|{\boldsymbol{\alpha}}_{2}\|^{2}>0,\\\\[5.69046pt]
\tfrac{1}{h_{1}}\partial_{z\bar{z}}^{2}\log
h_{k}=\|{\boldsymbol{\alpha}}_{k}\|^{2}-\|{\boldsymbol{\alpha}}_{k+1}\|^{2}>0\quad(2\leq
k\leq n-2),\\\\[5.69046pt] \tfrac{1}{h_{1}}\partial_{z\bar{z}}^{2}\log
h_{n-1}=\|{\boldsymbol{\alpha}}_{n-1}\|^{2}-\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}-\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2}>0.\end{cases}$

This clearly implies some restrictions on the zeros of the holomorphic forms
${\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{\pm}$.
For example, the last inequality in ($\star$ ‣ 5.1) forces every zero of
${\boldsymbol{\alpha}}_{n-1}$ to be a zero of ${\boldsymbol{\alpha}}_{n}^{+}$
and ${\boldsymbol{\alpha}}_{n}^{-}$ at the same time. More specifically,
letting $({\boldsymbol{\alpha}})$ denote the divisor333When
${\boldsymbol{\alpha}}\not\equiv 0$, the divisor $({\boldsymbol{\alpha}})$ is
just the $\mathbb{Z}_{\geq 0}$-valued function on $\Sigma$ assigning to every
$z\in\Sigma$ the multiplicity of zero of ${\boldsymbol{\alpha}}$ at $z$. When
${\boldsymbol{\alpha}}\equiv 0$, we let $({\boldsymbol{\alpha}})$ be the
constant function $+\infty$ by convention. of any holomorphic section
${\boldsymbol{\alpha}}$, one may check that a necessary condition for ($\star$
‣ 5.1) is

(5.1)
$({\boldsymbol{\alpha}}_{2})\leq\cdots\leq({\boldsymbol{\alpha}}_{n-1})\leq\min\big{\\{}({\boldsymbol{\alpha}}_{n}^{+}),({\boldsymbol{\alpha}}_{n}^{-})\big{\\}},$

where “$\leq$” and “$\min$” are pointwise inequality and minimum of
$\mathbb{Z}_{\geq 0}\cup\\{+\infty\\}$-valued functions on $\Sigma$.

When $\Sigma$ is closed, we generalize the work of Dai and Li [23] to get the
theorem below on the affine Toda system. It implies that a specific condition
stronger than (5.1) is sufficient for ($\star$ ‣ 5.1), although we do not know
whether (5.1) itself is sufficient. The condition is given by replacing every
“$\leq$” in (5.1) by a stronger relation “$\prec$”, where we write
“$D_{1}\prec D_{2}$” if the value of the divisor $D_{1}$ is strictly less than
that of $D_{2}$ at every point where the former is nonzero (which always holds
when $D_{1}=\emptyset$).

######  Theorem 5.2.

Let $\Sigma$ be a closed Riemann surface of genus $\geq 2$,
${\mathcal{L}}_{1},\cdots,{\mathcal{L}}_{n}$ ($n\geq 2$) be holomorphic line
bundles on $\Sigma$ and ${\boldsymbol{h}}_{i}$ be a hermitian metric on
${\mathcal{L}}_{i}$. Let ${\boldsymbol{\alpha}}_{1}$,
${\boldsymbol{\alpha}}_{j}$ ($2\leq j\leq n-1$) and
${\boldsymbol{\alpha}}_{n}^{\pm}$ be holomorphic sections of the line bundles
${\mathcal{K}}\\!{\mathcal{L}}_{1}$,
${\mathcal{K}}\\!{\mathcal{L}}_{j-1}^{-1}{\mathcal{L}}_{j}$ and
${\mathcal{K}}\\!{\mathcal{L}}_{n-1}^{-1}{\mathcal{L}}_{n}^{\pm 1}$,
respectively, with
${\boldsymbol{\alpha}}_{1},\cdots,{\boldsymbol{\alpha}}_{n-1}\not\equiv 0$.
Suppose the following equations (with the same local notation as in Theorem
4.10) hold on $\Sigma$:

(5.2) $\displaystyle\text{if
$n=2$:}\quad\begin{cases}\partial_{z\bar{z}}^{2}\log
h_{1}=h_{1}|\alpha_{1}|^{2}-\tfrac{h_{2}}{h_{1}}|\alpha_{2}^{+}|^{2}-\frac{1}{h_{1}h_{2}}|\alpha_{2}^{-}|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{2}=\tfrac{h_{2}}{h_{1}}|\alpha_{2}^{+}|^{2}-\frac{1}{h_{1}h_{2}}|\alpha_{2}^{-}|^{2};\end{cases}$
(5.3) $\displaystyle\text{if $n\geq
3$:}\quad\begin{cases}\partial_{z\bar{z}}^{2}\log
h_{1}=h_{1}|\alpha_{1}|^{2}-\tfrac{h_{2}}{h_{1}}|\alpha_{2}|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{k}=\tfrac{h_{k}}{h_{k-1}}|\alpha_{k}|^{2}-\tfrac{h_{k+1}}{h_{k}}|\alpha_{k+1}|^{2}\quad\text{for
$2\leq k\leq n-2$},\\\\[5.69046pt] \partial_{z\bar{z}}^{2}\log
h_{n-1}=\tfrac{h_{n-1}}{h_{n-2}}|\alpha_{n-1}|^{2}-\tfrac{h_{n}}{h_{n-1}}|\alpha_{n}^{+}|^{2}-\tfrac{1}{h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{n}=\tfrac{h_{n}}{h_{n-1}}|\alpha_{n}^{+}|^{2}-\tfrac{1}{h_{n-1}h_{n}}|\alpha_{n}^{-}|^{2}.\end{cases}$

Then the following statements hold.

  1. (1)

If
$({\boldsymbol{\alpha}}_{1})\prec({\boldsymbol{\alpha}}_{2})\cdots\prec({\boldsymbol{\alpha}}_{n-1})\prec\min\big{\\{}({\boldsymbol{\alpha}}_{n}^{+}),({\boldsymbol{\alpha}}_{n}^{-})\big{\\}}$,
then we have $\partial_{z\bar{z}}^{2}\log h_{i}\geq 0$ on $\Sigma$ for
$i=1,\cdots,n-1$, with strict inequality away from the zeros of
${\boldsymbol{\alpha}}_{i}$.

  2. (2)

If
$({\boldsymbol{\alpha}}_{n}^{+})\prec({\boldsymbol{\alpha}}_{n}^{-})\neq\emptyset$,
then $\partial_{z\bar{z}}^{2}\log h_{n}\geq 0$ on $\Sigma$, with strict
inequality away from the zeros of ${\boldsymbol{\alpha}}_{n}^{+}$.

The proof is directly adapted from Dai-Li’s and is postponed to the appendix.

######  Remark 5.3.

For the sake of completeness, we have given in Theorem 5.2 a statement that is
stronger than required in two ways. First, the setting is more general than
the one we are working on: the line bundle ${\mathcal{L}}_{1}$ is not
necessarily ${\mathcal{K}}^{-1}$, so an extra ${\boldsymbol{\alpha}}_{1}\in
H^{0}(\Sigma,{\mathcal{K}}{\mathcal{L}}_{1})$ comes up correspondingly.
Second, Parts (1) and (2) are about the left-hand side of the 1st to $(n-1)$th
and the last equations in (5.3), respectively, but we only need the former.
The original result of Dai-Li (see [23, Lemma 5.4]) is the two parts combined
(for
$({\boldsymbol{\alpha}}_{1})=\cdots=({\boldsymbol{\alpha}}_{n-1})=({\boldsymbol{\alpha}}_{n}^{+})=\emptyset$),
without noticing that they are unrelated.

######  Remark 5.4.

Without affecting the statement, we can add to Theorem 5.2 the extra
assumption that ${\boldsymbol{\alpha}}_{n}^{+}\not\equiv 0$. In fact, if
${\boldsymbol{\alpha}}_{n}^{-}\not\equiv 0$ but
${\boldsymbol{\alpha}}_{n}^{+}\equiv 0$, we can switch the roles of
${\mathcal{L}}_{n}$ and ${\mathcal{L}}_{n}^{-1}$, which interchanges
${\boldsymbol{\alpha}}_{n}^{\pm}$ and does affect the statement. On the other
hand, if ${\boldsymbol{\alpha}}_{n}^{+}={\boldsymbol{\alpha}}_{n}^{-}\equiv
0$, then the statement just reduces from $n$ to the $n-1$ case, with
${\boldsymbol{\alpha}}_{n-1}^{-}\equiv 0$ and
${\boldsymbol{\alpha}}_{n-1}^{+}$ being the original
${\boldsymbol{\alpha}}_{n-1}$, whereas the $n=2$ case reduces to a trivial
statement.

###  5.2. Infinitesimal rigidity

An A-surface in a $4$-dimensional pseudo-Riemannian manifold is just a
spacelike minimal surface with timelike normal bundle (see Remark 4.4). In
this case, by Proposition 3.2, $\Sigma$ locally maximizes the area as long as
$M$ is negatively curved. We show the following generalization to higher
dimensions (see Theorem 4.6, Remark 4.17 and §3.3 for notation and
background):

######  Theorem 5.5.

Let $\Sigma$ be an A-surface in an orientable pseudo-Riemannian manifold $M$
locally modeled on $\mathbb{H}^{p,q}$ with $n=\frac{p+q}{2}\geq 3$. Suppose
$\Sigma$ is not locally contained in any pseudo-hyperbolic subspace of
codimension $4$ and its structural data satisfies ($\star$ ‣ 5.1). Consider
the spacelike part
${\mathcal{N}}^{+}:={\mathcal{L}}_{3}\oplus{\mathcal{L}}_{5}\oplus\cdots$ and
timelike part
${\mathcal{N}}^{-}:={\mathcal{L}}_{2}\oplus{\mathcal{L}}_{4}\oplus\cdots$ of
the normal bundle ${\mathcal{N}}$ of $\Sigma$. Then for any compactly
supported variation of $\Sigma$ whose variational vector field $\xi$ is a
section of ${\mathcal{N}}^{+}$ (resp. ${\mathcal{N}}^{-}$) and is not
identically zero, the associated second variation of volume is positive (resp.
negative).

######  Remark 5.6.

Besides maximal surfaces in $\mathbb{H}^{2,q}$, minimal surfaces in hyperbolic
$3$-manifolds have also been extensively studied (see e.g. [42, 58, 59]). For
these surfaces, we cannot expect such kind of variational property to hold in
general, because the term in the second variation formula given by the second
fundamental form ${\boldsymbol{I\\!\\!I}}$ has different sign from the
Laplacian term and the curvature term, which makes the sign of the whole
formula indefinite. The common feature of maximal surfaces and A-surfaces
which frees us from this sign issue is that all osculation vectors are
timelike, as they belong to ${\mathcal{L}}_{2}$ (see Remark 4.2).

###### Proof.

Recall from Theorem 3.1 (1) that the second variation of volume is given by
the integral

$\int_{\Sigma}\operatorname{Tr}_{{\boldsymbol{g}}}\Big{(}\langle\nabla_{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}^{\mathcal{N}}\xi,\nabla_{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}^{\mathcal{N}}\xi\rangle-\langle{\boldsymbol{S}}_{\xi}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}),{\boldsymbol{S}}_{\xi}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\rangle-\langle
R_{M}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},\xi){\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},\xi\rangle\Big{)}\mathsf{d}\mathit{vol}_{\boldsymbol{g}}$

(where the first fundamental form ${\boldsymbol{g}}$ is the real part of the
hermitian metric ${\boldsymbol{h}}_{1}$ on
${\mathcal{L}}_{1}=\mathsf{T}\Sigma$). To prove the theorem, we shall show
that the integrand satisfies

(5.4)
$\operatorname{Tr}_{{\boldsymbol{g}}}\Big{(}\langle\nabla_{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}^{\mathcal{N}}\xi,\nabla_{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}^{\mathcal{N}}\xi\rangle-\langle{\boldsymbol{S}}_{\xi}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}),{\boldsymbol{S}}_{\xi}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\rangle-\langle
R_{M}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},\xi){\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},\xi\rangle\Big{)}\
\left\\{\begin{array}[]{l}\geq 0\ \text{ if $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{+})$}\\\ \leq 0\ \text{ if $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{-})$}\end{array}\right.$

everywhere on $\Sigma$, with strict inequality on an open dense subset of
$\\{\xi\neq 0\\}$.

To this end, suppose the connection $\nabla^{\mathcal{N}}$ decomposes under
the splitting ${\mathcal{N}}={\mathcal{N}}^{+}\oplus{\mathcal{N}}^{-}$ as

(5.5)
$\nabla^{\mathcal{N}}=\begin{pmatrix}\nabla^{{\mathcal{N}}^{+}}&\Gamma_{0}^{-}\\\
\Gamma_{0}^{+}&\nabla^{{\mathcal{N}}^{-}}\end{pmatrix},$

where $\nabla^{{\mathcal{N}}^{\pm}}$ is a connection on ${\mathcal{N}}^{\pm}$
and $\Gamma_{0}^{\pm}$ is a
$\mathrm{Hom}({\mathcal{N}}^{\pm},{\mathcal{N}}^{\mp})$-valued $1$-form. Then
the left-hand side of (5.4) can be rewritten as

$\underbrace{\operatorname{Tr}_{{\boldsymbol{g}}}\langle\nabla_{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}^{{\mathcal{N}}^{\pm}}\xi,\nabla_{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}^{{\mathcal{N}}^{\pm}}\xi\rangle}_{\raisebox{-0.5pt}{\small{1}}⃝}+\underbrace{\operatorname{Tr}_{{\boldsymbol{g}}}\langle\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle}_{\raisebox{-0.5pt}{\small{2}}⃝}\
\underbrace{-\operatorname{Tr}_{{\boldsymbol{g}}}\langle{\boldsymbol{S}}_{\xi}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}),{\boldsymbol{S}}_{\xi}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\rangle}_{\raisebox{-0.5pt}{\small{3}}⃝}\
\underbrace{-\operatorname{Tr}_{{\boldsymbol{g}}}\langle
R_{M}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},\xi){\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},\xi\rangle}_{\raisebox{-0.5pt}{\small{4}}⃝}\leavevmode\nobreak\
,$

where every “$\pm$” is taken to be $+$ or $-$ when $\xi$ is a section of
${\mathcal{N}}^{+}$ or ${\mathcal{N}}^{-}$, respectively. We shall prove (5.4)
by showing that the terms \raisebox{-0.5pt}{\small{1}}⃝,
\raisebox{-0.5pt}{\small{3}}⃝ and \raisebox{-0.5pt}{\small{4}}⃝ have the
correct sign for trivial reason (similarly as in the proof of Proposition
3.2), whereas the term \raisebox{-0.5pt}{\small{2}}⃝ is controlled by
\raisebox{-0.5pt}{\small{4}}⃝ by virtue of Hypothesis ($\star$ ‣ 5.1).

For the term \raisebox{-0.5pt}{\small{1}}⃝, since ${\mathcal{N}}^{+}$ and
${\mathcal{N}}^{-}$ are positive and negative definite respectively, we
immediately have $\raisebox{-0.5pt}{\small{1}}⃝\geq 0$ and
$\raisebox{-0.5pt}{\small{1}}⃝\leq 0$ for $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{+})$ and $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{-})$, respectively.

For the term \raisebox{-0.5pt}{\small{3}}⃝, recall from Remark 4.2 that the
shape operator ${\boldsymbol{S}}_{\xi}\in
C^{\infty}(\Sigma,\operatorname{End}(\mathsf{T}\Sigma))$ assigned to a normal
vector field $\xi$ is identically zero when $\xi$ is orthogonal to
${\mathcal{L}}_{2}$. In particular, we have ${\boldsymbol{S}}_{\xi}=0$ if
$\xi\in C^{\infty}(\Sigma,{\mathcal{N}}^{+})$. Therefore, given an orthonormal
local frame $(e_{1},e_{2})$ of $\mathsf{T}\Sigma$, we have

$\raisebox{-0.5pt}{\small{3}}⃝=-\sum_{i=1}^{2}\langle{\boldsymbol{S}}_{\xi}(e_{i}),{\boldsymbol{S}}_{\xi}(e_{i})\rangle\
\left\\{\begin{array}[]{l}=0\ \text{ if $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{+})$},\\\ \leq 0\ \text{ if $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{-})$}.\end{array}\right.$

For the term \raisebox{-0.5pt}{\small{4}}⃝, note that at any $z\in\Sigma$ with
$\xi(z)\neq 0$, the sectional curvature of $M$ along the tangent $2$-plane
spanned by $e_{i}$ and $\xi$ is $K(e_{i}\wedge\xi)=\frac{\langle
R_{M}(e_{i},\xi)e_{i},\xi\rangle}{\langle\xi,\xi\rangle}=-1$. Therefore, at
the point $z$, we have

$\raisebox{-0.5pt}{\small{4}}⃝=-\sum_{i=1}^{2}\langle
R_{M}(e_{i},\xi)e_{i},\xi\rangle=2\langle\xi,\xi\rangle.$

This clearly holds at the points where $\xi=0$ as well, hence holds throughout
$\Sigma$. In particular, we have $\raisebox{-0.5pt}{\small{4}}⃝\geq 0$ and
$\raisebox{-0.5pt}{\small{4}}⃝\leq 0$ for $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{+})$ and $\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{-})$, respectively.

In the rest of the proof, we show that the absolute value of the term
\raisebox{-0.5pt}{\small{2}}⃝ is controlled by that of
\raisebox{-0.5pt}{\small{4}}⃝:

(5.6)
$|\operatorname{Tr}_{{\boldsymbol{g}}}\langle\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle|\leq
2|\langle\xi,\xi\rangle|\ \text{ for any }\xi\in
C^{\infty}(\Sigma,{\mathcal{N}}^{\pm}),$

and moreover the inequality is strict on an open dense subset of $\\{\xi\neq
0\\}$. This implies the required inequality (5.4) and prove the theorem.

In order to show (5.6), we first deduce from the decomposition (4.2) of the
connection $\nabla$ on $\mathsf{T}M|_{\Sigma}$ the following more explicit
expression of $\Gamma_{0}^{+}$ (which is just the $\Gamma$ in the proof of
Theorem 4.13 with the first row and first column removed):

$\Gamma_{0}^{+}=\scalebox{0.9}{$\begin{pmatrix}{\boldsymbol{\alpha}}_{3}^{*}&&&\\\
{\boldsymbol{\alpha}}_{4}&{\boldsymbol{\alpha}}_{5}^{*}&&\\\
&\ddots&\ddots&\\\ &&\ddots&{\boldsymbol{\alpha}}_{n-1}^{*}\\\
&&&{\boldsymbol{\alpha}}_{n}\end{pmatrix}$}\text{ if $n$ is
even,}\quad\Gamma_{0}^{+}=\scalebox{0.9}{$\begin{pmatrix}{\boldsymbol{\alpha}}_{3}^{*}&&&&\\\
{\boldsymbol{\alpha}}_{4}&{\boldsymbol{\alpha}}_{5}^{*}&&&\\\
&\ddots&\ddots&&\\\ &&\ddots&{\boldsymbol{\alpha}}_{n-2}^{*}&\\\
&&&{\boldsymbol{\alpha}}_{n-1}&{\boldsymbol{\alpha}}_{n}^{*}\end{pmatrix}$}\text{
if $n$ is odd.}$

We proceed to compute
$\operatorname{Tr}_{\boldsymbol{g}}\langle\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle$.
Pick an orthonormal local frame $(u_{i},v_{i})$ of each ${\mathcal{L}}_{i}$ as
in the proof of Theorem 4.10, so that ${\boldsymbol{\alpha}}_{j}$ can be
viewed as a $2\times 2$-matrix of $1$-forms with the explicit expression given
in that proof, while ${\boldsymbol{\alpha}}_{j}^{*}$ and $\Gamma_{0}^{-}$ are
the transposes of ${\boldsymbol{\alpha}}_{j}$ and $\Gamma_{0}^{+}$
respectively. We also use the frame to view a section $\xi$ of
${\mathcal{N}}^{+}$ or ${\mathcal{N}}^{-}$ as a column vector of functions.
Then we may write
$\langle\xi,\xi\rangle=\pm\leftidx{{}^{\mathsf{t}}}\\!{\xi}\,\xi$ and
$\langle\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\,\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle=\mp\leftidx{{}^{\mathsf{t}}}\\!{\xi}\
\leftidx{{}^{\mathsf{t}}}\\!{\Gamma_{0}}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\
\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi$ for
any $\xi\in C^{\infty}(\Sigma,{\mathcal{N}}^{\pm})$. Thus, when $n$ is even,
we get by matrix computations that

$\langle\Gamma_{0}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle=-\leftidx{{}^{\mathsf{t}}}\\!{\xi}\scalebox{0.9}{$\begin{pmatrix}\scalebox{0.85}{\parbox{34.14322pt}{${\boldsymbol{\alpha}}_{3}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}{\boldsymbol{\alpha}}_{4}$}}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{5}&&\\\\[8.5359pt]
{\boldsymbol{\alpha}}_{5}{\boldsymbol{\alpha}}_{4}&\scalebox{0.85}{\parbox{34.14322pt}{${\boldsymbol{\alpha}}_{5}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{5}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{6}{\boldsymbol{\alpha}}_{6}$}}&\ddots&\\\\[8.5359pt]
&\ddots&\ddots&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\\\\[8.5359pt]
&&{\boldsymbol{\alpha}}_{n}{\boldsymbol{\alpha}}_{n-1}&\scalebox{0.85}{\parbox{45.52458pt}{${\boldsymbol{\alpha}}_{n-1}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}{\boldsymbol{\alpha}}_{n}$}}\end{pmatrix}$}\xi\quad\text{
for }\xi\in C^{\infty}(\Sigma,{\mathcal{N}}^{+}),$
$\langle\Gamma_{0}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle=\leftidx{{}^{\mathsf{t}}}\\!{\xi}\scalebox{0.9}{$\begin{pmatrix}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}{\boldsymbol{\alpha}}_{3}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}&&&\\\\[8.5359pt]
{\boldsymbol{\alpha}}_{4}{\boldsymbol{\alpha}}_{3}&\scalebox{0.85}{\parbox{34.14322pt}{${\boldsymbol{\alpha}}_{4}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{5}{\boldsymbol{\alpha}}_{5}$}}&\ddots&&\\\\[8.5359pt]
&\ddots&\ddots&\ddots&\\\\[8.5359pt]
&&\ddots&\scalebox{0.85}{\parbox{54.06006pt}{${\boldsymbol{\alpha}}_{n-2}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-2}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}{\boldsymbol{\alpha}}_{n-1}$}}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\\\\[8.5359pt]
&&&{\boldsymbol{\alpha}}_{n}{\boldsymbol{\alpha}}_{n-1}&{\boldsymbol{\alpha}}_{n}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\end{pmatrix}$}\xi\quad\text{
for }\xi\in C^{\infty}(\Sigma,{\mathcal{N}}^{-}),$

where
${\boldsymbol{\alpha}}_{3}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}$
is understood as the $2\times 2$-matrix-valued $2$-tensor
${\boldsymbol{\alpha}}_{3}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})$,
and similarly for the other terms in the matrices. When $n$ is odd, the
results become

$\langle\Gamma_{0}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle=-\leftidx{{}^{\mathsf{t}}}\\!{\xi}\scalebox{0.9}{$\begin{pmatrix}\scalebox{0.85}{\parbox{34.14322pt}{${\boldsymbol{\alpha}}_{3}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}{\boldsymbol{\alpha}}_{4}$}}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{5}&&\\\\[8.5359pt]
{\boldsymbol{\alpha}}_{5}{\boldsymbol{\alpha}}_{4}&\ddots&\ddots&\\\\[8.5359pt]
&\ddots&\scalebox{0.85}{\parbox{54.06006pt}{${\boldsymbol{\alpha}}_{n-2}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-2}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}{\boldsymbol{\alpha}}_{n-1}$}}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\\\\[8.5359pt]
&&{\boldsymbol{\alpha}}_{n}{\boldsymbol{\alpha}}_{n-1}&{\boldsymbol{\alpha}}_{n}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\end{pmatrix}$}\xi\quad\text{
for }\xi\in C^{\infty}(\Sigma,{\mathcal{N}}^{+}),$
$\langle\Gamma_{0}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle=\leftidx{{}^{\mathsf{t}}}\\!{\xi}\scalebox{0.9}{$\begin{pmatrix}{\boldsymbol{\alpha}}_{3}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}&&\\\\[8.5359pt]
{\boldsymbol{\alpha}}_{4}{\boldsymbol{\alpha}}_{3}&\scalebox{0.85}{\parbox{34.14322pt}{${\boldsymbol{\alpha}}_{4}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{4}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{5}{\boldsymbol{\alpha}}_{5}$}}&\ddots&\\\\[8.5359pt]
&\ddots&\ddots&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\\\\[8.5359pt]
&&{\boldsymbol{\alpha}}_{n}{\boldsymbol{\alpha}}_{n-1}&\scalebox{0.85}{\parbox{54.06006pt}{${\boldsymbol{\alpha}}_{n-1}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n-1}$\\\
$+\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}{\boldsymbol{\alpha}}_{n}$}}\end{pmatrix}$}\xi\quad\text{
for }\xi\in C^{\infty}(\Sigma,{\mathcal{N}}^{-}).$

The trace of each $2$-tensor in these matrices with respect to
${\boldsymbol{g}}$ (not to be confused with the matrix trace considered in the
proof of Theorem 4.13) is computed as follows:

######  Lemma 5.7.

Let $I$ denote the $2\times 2$ identity matrix. For $2\leq i,j\leq n-1$, we
have

$\displaystyle\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}{\boldsymbol{\alpha}}_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{j}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=0\quad\text{(the
zero $2\times 2$ matrix)},$
$\displaystyle\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}{\boldsymbol{\alpha}}_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=2\|{\boldsymbol{\alpha}}_{i}\|^{2}I,$
$\displaystyle\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}{\boldsymbol{\alpha}}_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{i}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=0,$
$\displaystyle\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}){\boldsymbol{\alpha}}_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=2(\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}+\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2})I,$
$\displaystyle\operatorname{Tr}_{{\boldsymbol{g}}}\big{[}{\boldsymbol{\alpha}}_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=2\left[(\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}+\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2})I+\tfrac{2}{h_{1}h_{n-1}}\scalebox{0.9}{
$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{n}^{+}\overline{\alpha_{n}^{-}})&\operatorname{\mathsf{Im}}(\alpha_{n}^{+}\overline{\alpha_{n}^{-}})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{n}^{+}\overline{\alpha_{n}^{-}})&-\operatorname{\mathsf{Re}}(\alpha_{n}^{+}\overline{\alpha_{n}^{-}})\end{pmatrix}$}\right]$

(see Theorem 4.10 and Remark 4.11 for the notations). Moreover, the
eigenvalues of the right-hand side of the last equation are
$2(\|{\boldsymbol{\alpha}}_{n}^{+}\|\pm\|{\boldsymbol{\alpha}}_{n}^{-}\|)^{2}$.

###### Proof.

The expressions of the ${\boldsymbol{\alpha}}_{j}$’s given in the proof of
Theorem 4.10 can be rewritten as

$\displaystyle{\boldsymbol{\alpha}}_{j}$
$\displaystyle=\tfrac{\sqrt{h_{j}}}{\sqrt{h_{j-1}}}\left[\scalebox{0.9}{$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{j})&-\operatorname{\mathsf{Im}}(\alpha_{j})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{j})&\operatorname{\mathsf{Re}}(\alpha_{j})\end{pmatrix}$}{\mathsf{d}x}+\scalebox{0.9}{$\begin{pmatrix}-\operatorname{\mathsf{Im}}(\alpha_{j})&-\operatorname{\mathsf{Re}}(\alpha_{j})\\\\[2.84544pt]
\operatorname{\mathsf{Re}}(\alpha_{j})&-\operatorname{\mathsf{Im}}(\alpha_{j})\end{pmatrix}$}{\mathsf{d}y}\right]\quad(2\leq
j\leq n-1),$ $\displaystyle{\boldsymbol{\alpha}}_{n-1}$
$\displaystyle={\boldsymbol{\alpha}}_{n}^{+}+{\boldsymbol{\alpha}}_{n}^{-},\text{
where}$ $\displaystyle{\boldsymbol{\alpha}}_{n}^{+}$
$\displaystyle=\tfrac{\sqrt{h_{n}}}{\sqrt{h_{n-1}}}\left[\scalebox{0.9}{$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{n}^{+})&-\operatorname{\mathsf{Im}}(\alpha_{n}^{+})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{n}^{+})&\operatorname{\mathsf{Re}}(\alpha_{n}^{+})\end{pmatrix}$}{\mathsf{d}x}+\scalebox{0.9}{$\begin{pmatrix}-\operatorname{\mathsf{Im}}(\alpha_{n}^{+})&-\operatorname{\mathsf{Re}}(\alpha_{n}^{+})\\\\[2.84544pt]
\operatorname{\mathsf{Re}}(\alpha_{n}^{+})&-\operatorname{\mathsf{Im}}(\alpha_{n}^{+})\end{pmatrix}$}{\mathsf{d}y}\right],$
$\displaystyle{\boldsymbol{\alpha}}_{n}^{-}$
$\displaystyle=\tfrac{1}{\sqrt{h_{n-1}h_{n}}}\left[\scalebox{0.9}{$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{n}^{-})&-\operatorname{\mathsf{Im}}(\alpha_{n}^{-})\\\\[2.84544pt]
-\operatorname{\mathsf{Im}}(\alpha_{n}^{-})&-\operatorname{\mathsf{Re}}(\alpha_{n}^{-})\end{pmatrix}$}{\mathsf{d}x}+\scalebox{0.9}{$\begin{pmatrix}-\operatorname{\mathsf{Im}}(\alpha_{n}^{-})&-\operatorname{\mathsf{Re}}(\alpha_{n}^{-})\\\\[2.84544pt]
-\operatorname{\mathsf{Re}}(\alpha_{n}^{-})&\operatorname{\mathsf{Im}}(\alpha_{n}^{-})\end{pmatrix}$}{\mathsf{d}y}\right].$

Taking account of the fact that

$\operatorname{Tr}_{\boldsymbol{g}}\big{(}{\mathsf{d}x}\otimes{\mathsf{d}x}\big{)}=\operatorname{Tr}_{\boldsymbol{g}}\big{(}{\mathsf{d}y}\otimes{\mathsf{d}y}\big{)}=\tfrac{1}{h_{1}}\leavevmode\nobreak\
,\quad\operatorname{Tr}_{\boldsymbol{g}}\big{(}{\mathsf{d}x}\otimes{\mathsf{d}y}\big{)}=\operatorname{Tr}_{\boldsymbol{g}}\big{(}{\mathsf{d}y}\otimes{\mathsf{d}x}\big{)}=0$

(because
${\boldsymbol{g}}=\operatorname{\mathsf{Re}}{\boldsymbol{h}}_{1}=h_{1}({\mathsf{d}x}^{2}+{\mathsf{d}y}^{2})$),
we obtain the required equalities and the eigenvalues by direct computations.
∎

Using the expressions of
$\langle\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle$
and Lemma 5.7, we get expressions for
$\operatorname{Tr}_{\boldsymbol{g}}\langle\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle$,
from which it is clear that at any point of $\Sigma$, both of the quadratic
form
$\xi\mapsto-\operatorname{Tr}_{\boldsymbol{g}}\langle\Gamma_{0}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{+}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle$
and
$\xi\mapsto\operatorname{Tr}_{\boldsymbol{g}}\langle\Gamma_{0}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{-}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle$
have eigenvalues belonging to the list

$\displaystyle 2\|{\boldsymbol{\alpha}}_{3}\|^{2},\quad
2\|{\boldsymbol{\alpha}}_{3}\|^{2}+2\|{\boldsymbol{\alpha}}_{4}\|^{2},\quad
2\|{\boldsymbol{\alpha}}_{4}\|^{2}+2\|{\boldsymbol{\alpha}}_{5}\|^{2},\
\cdots$ $\displaystyle\quad\cdots,\
2\|{\boldsymbol{\alpha}}_{n-2}\|^{2}+2\|{\boldsymbol{\alpha}}_{n-1}\|^{2},\quad
2\|{\boldsymbol{\alpha}}_{n-1}\|^{2}+2\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}+2\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2},\quad
2(\|{\boldsymbol{\alpha}}_{n}^{+}\|\pm\|{\boldsymbol{\alpha}}_{n}^{-}\|)^{2}.$

But Hypothesis ($\star$ ‣ 5.1) implies that the following inequalities hold on
an open dense set $U\subset\Sigma$:

(5.7)
$\tfrac{1}{2}(\|{\boldsymbol{\alpha}}_{n}^{+}\|\pm\|{\boldsymbol{\alpha}}_{n}^{-}\|)^{2}<\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}+\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2}<\|{\boldsymbol{\alpha}}_{n-1}\|^{2}<\
\cdots\
<\|{\boldsymbol{\alpha}}_{3}\|^{2}<\|{\boldsymbol{\alpha}}_{2}\|^{2}<\tfrac{1}{2}.$

Therefore, all the eigenvalues are strictly less than $2$ on $U$. This implies
the required inequality (5.6) and completes the proof. ∎

This theorem can be roughly understood as saying that A-surfaces satisfying
the assumptions are saddle type critical points for the area functional. Such
critical points are nondegenerate. In fact, it is a simple linear algebraic
fact that if a symmetric matrix $L$ has the form

$L=\begin{pmatrix}L_{+}&*\\\ *&L_{-}\end{pmatrix}$

such that the blocks $L_{+}$ and $L_{-}$ are positive definite and negative
definite respectively, then $L$ is nondegenerate. Generalizing this to
infinite dimensions and considering the Jacobi operator $L_{\Sigma}$ in place
of $L$, we obtain:

######  Corollary 5.8.

Under the hypotheses of Theorem 5.5, $\Sigma$ does not admit any nonzero
compactly supported Jacobi field.

###### Proof.

Let $\xi\in C^{\infty}(\Sigma,{\mathcal{N}})$ be a compactly supported Jacobi
field and write $\xi=\xi_{+}+\xi_{-}$, where $\xi_{\pm}$ is a compactly
supported section of ${\mathcal{N}}^{\pm}$. Since $L_{\Sigma}\xi=0$, we have

$\langle L_{\Sigma}\xi_{+},\,\xi_{+}\rangle+\langle
L_{\Sigma}\xi_{-},\,\xi_{+}\rangle=\langle
L_{\Sigma}\xi,\,\xi_{+}\rangle=0,\quad\langle
L_{\Sigma}\xi_{-},\,\xi_{-}\rangle+\langle
L_{\Sigma}\xi_{+},\,\xi_{-}\rangle=\langle L_{\Sigma}\xi,\,\xi_{-}\rangle=0.$

So the following equalities (between functions on $\Sigma$) hold:

$\langle L_{\Sigma}\xi_{-},\,\xi_{+}\rangle=-\langle
L_{\Sigma}\xi_{+},\,\xi_{+}\rangle,\quad\langle
L_{\Sigma}\xi_{+},\,\xi_{-}\rangle=-\langle
L_{\Sigma}\xi_{-},\,\xi_{-}\rangle.$

The second variation formula (see Theorem 3.1) says that for any compactly
supported $\eta\in C^{\infty}(\Sigma,{\mathcal{N}})$, the integral of
$-\langle L_{\Sigma}\eta,\eta\rangle$ on $\Sigma$ is exactly the second
variation of volume associated to $\eta$. Therefore, by Theorem 5.5, the
integral of the first (resp. second) function above is $\geq 0$ (resp. $\leq
0$), with strict inequality unless $\xi_{+}\equiv 0$ (resp. $\xi_{-}\equiv
0$). But the two integrals coincide because $L_{\Sigma}$ is self-adjoint (see
§3.2). So we conclude that $\xi_{\pm}$ are both identically zero. ∎

In particular, if $\Sigma$ is closed, then it does not admit any nonzero
Jacobi field at all. Note that we still assume Hypothesis ($\star$ ‣ 5.1)
here. Applying the sufficient condition for ($\star$ ‣ 5.1) in terms of
divisors from Theorem 5.2 (1) (with
${\boldsymbol{\alpha}}_{1}=\frac{1}{\sqrt{2}}$), we obtain:

######  Corollary 5.9.

Let $\Sigma$ be a closed A-surface of genus $\geq 2$ in an orientable pseudo-
Riemannian manifold $M$ locally modeled on $\mathbb{H}^{p,q}$ with
$n=\frac{p+q}{2}\geq 3$. Suppose $\Sigma$ is not locally contained in any
pseudo-hyperbolic subspace of codimension $4$ and the holomorphic forms
${\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{\pm}$
in its structural data satisfy
$({\boldsymbol{\alpha}}_{2})\prec\cdots\prec({\boldsymbol{\alpha}}_{n-1})\prec\min\big{\\{}({\boldsymbol{\alpha}}_{n}^{+}),({\boldsymbol{\alpha}}_{n}^{-})\big{\\}}$.
Then $\Sigma$ does not admit any nonzero Jacobi field.

###  5.3. Uniqueness of A-surface with given Gauss map

An A-surface $\Sigma\subset\mathbb{H}^{p,q}$ is not completely determined by
its Gauss map $f:\Sigma\to X_{\mathrm{SO}_{0}(p,q+1)}$, as the antipodal
surface $-\Sigma$ has the same Gauss map (more precisely, the latter is the
composition of $f$ with the antipodal map $-\Sigma\to\Sigma$; this can be
easily seen by using the setup in the proof of Theorem 4.13). Again under
Hypothesis ($\star$ ‣ 5.1), we may show that this is the only ambiguity:

######  Theorem 5.10.

Let $\Sigma\subset\mathbb{H}^{p,q}$ be an A-surface, where
$n=\frac{p+q}{2}\geq 3$. Suppose $\Sigma$ is not contained in any pseudo-
hyperbolic subspace of codimension $4$ and satisfies ($\star$ ‣ 5.1). Let
$\iota^{\prime}:\Sigma\to\mathbb{H}^{p,q}$ be an A-surface immersion with the
same Gauss map as $\Sigma$. Then $\iota^{\prime}$ is the restriction of either
the identity map or the antipodal map of $\mathbb{H}^{p,q}$ to $\Sigma$.

###### Proof.

Let $\iota,u_{1},v_{1},\cdots,u_{n},v_{n}$ be locally defined maps from
$\Sigma$ to $\mathbb{R}^{p,q+1}$ as in the proofs of Theorem 4.6 and Theorem
4.13 (where $\iota$ is the inclusion), so that the matrix-valued function

$\widetilde{f}:=(u_{1},\,v_{1},\,u_{3},\,v_{3},\,\cdots,\
\iota,\,u_{2},\,v_{2},\,u_{4},\,v_{4})$

is a locally defined lift of the Gauss map $f$ of $\Sigma$ and we have

(5.8)
$\widetilde{f}^{-1}\mathsf{d}\widetilde{f}=\begin{pmatrix}*&\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}\\\
\Gamma&*\end{pmatrix},$

where $\Gamma$ has the expression (4.29) in terms of the structural $1$-forms
${\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n}$ of $\Sigma$.

For any another A-surface immersion $\iota^{\prime}$, we again have locally
defined maps
$u^{\prime}_{1},v^{\prime}_{1},\cdots,u^{\prime}_{n},v^{\prime}_{n}$ from
$\Sigma$ to $\mathbb{R}^{p,q+1}$ such that

$\widetilde{f}^{\prime}:=(u^{\prime}_{1},\,v^{\prime}_{1},\,u^{\prime}_{3},\,v^{\prime}_{3},\,\cdots,\
\iota^{\prime},\,u^{\prime}_{2},\,v^{\prime}_{2},\,u^{\prime}_{4},\,v^{\prime}_{4})$

lifts the Gauss map $f^{\prime}$ of $\iota^{\prime}$ and that

(5.9)
$\widetilde{f}^{\prime-1}\mathsf{d}\widetilde{f}^{\prime}=\begin{pmatrix}*&\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}^{\prime}\\\
\Gamma^{\prime}&*\end{pmatrix},$

with $\Gamma^{\prime}$ given by the same expression (4.29) as $\Gamma$ except
that each ${\boldsymbol{\alpha}}_{j}$ is replaced by the $1$-forms
${\boldsymbol{\alpha}}_{j}^{\prime}$ coming from $\iota^{\prime}$, which might
be different from ${\boldsymbol{\alpha}}_{j}$ a priori.

Now assume $f^{\prime}=f$. Then there is a locally defined map
$g=(g_{1},g_{2})$ from $\Sigma$ to $\mathrm{SO}(p)\times\mathrm{SO}(q+1)$ such
that $\widetilde{f}^{\prime}=\widetilde{f}g$, and it follows that
$\widetilde{f}^{\prime-1}\mathsf{d}\widetilde{f}^{\prime}=g^{-1}(\widetilde{f}^{-1}\mathsf{d}\widetilde{f})g+g^{-1}\mathsf{d}g$.
Comparing with (5.8) and (5.9), we infer that
$\Gamma^{\prime}=g_{2}^{-1}\Gamma g_{1}$, and as a consequence,

(5.10)
$\operatorname{Tr}_{\boldsymbol{g}}\big{[}\Gamma^{\prime}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\,\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}^{\prime}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=g_{2}^{-1}\operatorname{Tr}_{\boldsymbol{g}}\big{[}\Gamma({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\,\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}g_{2},$

where we take the tensor traces of matrix-valued $2$-tensors with respect to
the first fundamental form ${\boldsymbol{g}}$ (as in the proof of Theorem 5.5)
to produce matrix-valued functions. These traces can be calculated similarly
as the calculation of
$\operatorname{Tr}_{\boldsymbol{g}}\langle\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi,\Gamma_{0}^{\pm}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\xi\rangle$
in the proof of Theorem 5.5 (the $\Gamma_{0}^{+}$ therein is just $\Gamma$
with the first row and column removed). The result is

(5.11)
$\operatorname{Tr}_{\boldsymbol{g}}\big{[}\Gamma^{\prime}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\,\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}^{\prime}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=\begin{pmatrix}2&\\\
&\Lambda^{\prime}\end{pmatrix},\quad\operatorname{Tr}_{\boldsymbol{g}}\big{[}\Gamma({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\,\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}=\begin{pmatrix}2&\\\
&\Lambda\end{pmatrix},$

where the matrix $\Lambda$ is diagonal except for a possible $2\times 2$ block
and its eigenvalues belong to the list

$\displaystyle
2\|{\boldsymbol{\alpha}}_{2}\|^{2}+2\|{\boldsymbol{\alpha}}_{3}\|^{2},\quad
2\|{\boldsymbol{\alpha}}_{3}\|^{2}+2\|{\boldsymbol{\alpha}}_{4}\|^{2},\
\cdots$ $\displaystyle\quad\cdots,\
2\|{\boldsymbol{\alpha}}_{n-2}\|^{2}+2\|{\boldsymbol{\alpha}}_{n-1}\|^{2},\quad
2\|{\boldsymbol{\alpha}}_{n-1}\|^{2}+2\|{\boldsymbol{\alpha}}_{n}^{+}\|^{2}+2\|{\boldsymbol{\alpha}}_{n}^{-}\|^{2},\quad
2(\|{\boldsymbol{\alpha}}_{n}^{+}\|\pm\|{\boldsymbol{\alpha}}_{n}^{-}\|)^{2}.$

The matrix $\Lambda^{\prime}$ is the same except that
${\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{\pm}$
are replaced by
${\boldsymbol{\alpha}}_{2}^{\prime},\cdots,{\boldsymbol{\alpha}}_{n-1}^{\prime},{\boldsymbol{\alpha}}_{n}^{\prime}{\\!\\!}^{\pm}$.

We may now apply condition ($\star$ ‣ 5.1), which implies inequalities (5.7)
on an open dense set $U\subset\Sigma$. It follows that at every point in $U$,
the sole largest eigenvalue of
$\operatorname{Tr}_{\boldsymbol{g}}\big{[}\Gamma({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\,\leftidx{{}^{\mathsf{t}}}\\!{\Gamma}({\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,})\big{]}$
is $2$. In view of (5.10) and (5.11), this forces the
$\mathrm{SO}(q+1)$-valued function $g_{2}$ to have the form

$g_{2}=\begin{pmatrix}\pm 1&\\\ &*\end{pmatrix}$

on $U$. By continuity, it has the same form throughout $\Sigma$. Since
$\widetilde{f}^{\prime}=\widetilde{f}g$, we conclude that
$\iota^{\prime}=\pm\iota$. ∎

##  6\. Application to cyclic $\mathrm{SO}_{0}(n,n+1)$-Higgs bundles

In this section, we review some backgrounds on cyclic
$\mathrm{SO}_{0}(n,n+1)$-Higgs bundle before showing in Theorem 6.3 ($\approx$
Theorem B) that such Higgs bundles yield A-surfaces, then we use the
infinitesimal rigidity result to prove Theorem D (restated as Theorem 6.6) and
finally show in Proposition 6.8 that in the $G_{2}^{\prime}$ case these
A-surfaces belong to the type from Proposition 4.15.

###  6.1. Cyclic $\mathrm{SO}_{0}(n,n+1)$-Higgs bundles and A-surfaces

We refer to [3, 19, 21, 30] for the definition of $G$-Higgs bundles for a
general real reductive group $G$ and the special case of
$\mathrm{SO}_{0}(p,q)$-Higgs bundles. We only consider here the following
particular type of $\mathrm{SO}_{0}(n,n+1)$-Higgs bundles:

######  Definition 6.1.

Let $\Sigma$ be a closed Riemann surface. Given holomorphic line bundles
${\mathcal{L}}_{1},\cdots,{\mathcal{L}}_{n}$ ($n\geq 2$) on $\Sigma$ and
holomorphic sections ${\boldsymbol{\alpha}}_{1}$, ${\boldsymbol{\alpha}}_{j}$
($2\leq j\leq n-1$) and ${\boldsymbol{\alpha}}_{n}^{\pm}$ of the line bundles
${\mathcal{K}}\\!{\mathcal{L}}_{1}$,
${\mathcal{K}}\\!{\mathcal{L}}_{j-1}^{-1}{\mathcal{L}}_{j}$ and
${\mathcal{K}}\\!{\mathcal{L}}_{n-1}^{-1}{\mathcal{L}}_{n}^{\pm 1}$,
respectively, with
${\boldsymbol{\alpha}}_{1},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{+}\not\equiv
0$, we definite the _cyclic $\mathrm{SO}_{0}(n,n+1)$-Higgs bundle_ associated
to these data as the $\mathrm{SO}_{0}(n,n+1)$-Higgs bundle
$({\mathcal{V}},Q_{\mathcal{V}},{\mathcal{W}},Q_{\mathcal{W}},\eta)$ given by

$\displaystyle{\mathcal{V}}$
$\displaystyle={\mathcal{L}}_{-n+1}\oplus{\mathcal{L}}_{-n+3}\oplus\ \cdots\
\oplus{\mathcal{L}}_{n-3}\oplus{\mathcal{L}}_{n-1},\quad
Q_{\mathcal{V}}=\scalebox{0.8}{$\begin{pmatrix}&&1\\\\[-2.84544pt]
&\iddots&\\\\[-2.84544pt] 1&&\end{pmatrix}$},$ $\displaystyle{\mathcal{W}}$
$\displaystyle={\mathcal{L}}_{-n}\oplus{\mathcal{L}}_{-n+2}\oplus\ \cdots\
\oplus{\mathcal{L}}_{n-2}\oplus{\mathcal{L}}_{n},\quad
Q_{\mathcal{W}}=\scalebox{0.8}{$\begin{pmatrix}&&1\\\\[-2.84544pt]
&\iddots&\\\\[-2.84544pt] 1&&\end{pmatrix}$},$ $\displaystyle\eta$
$\displaystyle=\scalebox{0.9}{$\begin{pmatrix}*&&&*\\\\[-2.84544pt]
&\ddots&&\\\\[-2.84544pt] &&*&\end{pmatrix}$}\in
H^{0}(\Sigma,{\mathcal{K}}\otimes\mathrm{Hom}({\mathcal{W}},{\mathcal{V}})),$

where we set ${\mathcal{L}}_{0}:={\mathcal{O}}$,
${\mathcal{L}}_{m}:={\mathcal{L}}_{-m}^{-1}$ for $m<0$, and let each “$*$” in
the matrix expression of $\eta$ be one of the above holomorphic sections,
namely the unique one which fits in that slot (for example, the first row is
$({\boldsymbol{\alpha}}_{n}^{+},0,\cdots,0,{\boldsymbol{\alpha}}_{n}^{-})$,
the second is $(0,{\boldsymbol{\alpha}}_{n-2},0,\cdots,0)$, etc.).

Alternatively, we may define the above
$({\mathcal{V}},Q_{\mathcal{V}},{\mathcal{W}},Q_{\mathcal{W}},\eta)$ as the
$\mathrm{SO}_{0}(n,n+1)$-Higgs bundle whose underlying
$\mathrm{GL}(2n+1,\mathbb{C})$-Higgs bundle is

$\displaystyle{\mathcal{E}}={\mathcal{V}}\oplus{\mathcal{W}}={\mathcal{L}}_{n}^{-1}\oplus\cdots\oplus{\mathcal{L}}_{1}^{-1}\oplus{\mathcal{O}}\oplus{\mathcal{L}}_{1}\oplus\cdots\oplus{\mathcal{L}}_{n},$
$\displaystyle\hskip
7.96674pt\Phi=\scalebox{0.85}{$\left(\begin{array}[]{ccccc|c|ccccc}0&&&&&&&&&{\boldsymbol{\alpha}}_{n}^{-}&\\\
{\boldsymbol{\alpha}}_{n}^{+}&0&&&&&&&&&{\boldsymbol{\alpha}}_{n}^{-}\\\
&{\boldsymbol{\alpha}}_{n-1}&\ddots&&&&&&&&\\\ &&\ddots&\ddots&&&&&&&\\\
&&&{\boldsymbol{\alpha}}_{2}&0&&&&&&\\\
\hline\cr&&&&{\boldsymbol{\alpha}}_{1}&0&&&&&\\\
\hline\cr&&&&&{\boldsymbol{\alpha}}_{1}&0&&&&\\\
&&&&&&{\boldsymbol{\alpha}}_{2}&\ddots&&&\\\ &&&&&&&\ddots&\ddots&&\\\
&&&&&&&&{\boldsymbol{\alpha}}_{n-1}&0&\\\
&&&&&&&&&{\boldsymbol{\alpha}}_{n}^{+}&0\end{array}\right)$}.$

We emphasize that each of the sub-diagonal holomorphic forms
${\boldsymbol{\alpha}}_{1},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{+}$
is assumed not to be identically zero, although
${\boldsymbol{\alpha}}_{n}^{-}$ can be zero. Also note that the trivial line
bundle ${\mathcal{O}}$ is a summand in ${\mathcal{V}}$ (resp. ${\mathcal{W}}$)
when $n$ is odd (resp. even) and we can choose a global holomorphic section
$\underline{1}$ of ${\mathcal{O}}$ with
$Q_{\mathcal{V}}(\underline{1}\,,\,\underline{1})\equiv 1$ (resp.
$Q_{\mathcal{W}}(\underline{1}\,,\,\underline{1})\equiv 1$). This section is
unique up to sign. We let $\underline{\mathbb{R}}:=\mathbb{R}\underline{1}$ be
the real line subbundle of ${\mathcal{O}}$ generated by it.

In general, a polystable $G$-Higgs bundle admits a canonical $K$-reduction
($K\subset G$ denotes the maximal compact subgroup), namely the one solving
the Hitchin equation, as well as a canonical flat $G$-connection $D$, namely
the Hitchin connection. Applying the parallel transportation of $D$ to the
$K$-reduction, we get a harmonic map $f$ from $\widetilde{\Sigma}$ to the
symmetric space $X_{G}=G/K$ which is equivariant with respect to the holonomy
representation of $D$. For the above particular $\mathrm{SO}_{0}(n,n+1)$-Higgs
bundles, $f$ is a conformal minimal immersion and these objects have a more
concrete description:

######  Proposition/Definition 6.2.

The above $({\mathcal{E}},\Phi)$ is stable if and only if either
${\boldsymbol{\alpha}}_{n}^{-}\not\equiv 0$ or $\deg({\mathcal{L}}_{n})<0$. In
this case, there is a unique hermitian metric ${\boldsymbol{h}}_{i}$ on each
${\mathcal{L}}_{i}$ such that the hermitian metric

$H:={\boldsymbol{h}}_{n}^{-1}\oplus\cdots\oplus{\boldsymbol{h}}_{1}^{-1}\oplus
1\oplus{\boldsymbol{h}}_{1}\oplus\cdots\oplus{\boldsymbol{h}}_{n}$

on ${\mathcal{E}}$ (where the middle $1$ denotes the hermitian metric on
${\mathcal{O}}$ under which the section $\underline{1}$ has unit norm)
satisfies the _Hitchin equation_

(6.1) $F_{\nabla^{H}}+[\Phi,\Phi^{*}]=0.$

Here, $\nabla^{H}$ is the Chern connection of $({\mathcal{E}},H)$ and
$\Phi^{*}$ is the $\operatorname{End}({\mathcal{E}})$-valued $(0,1)$-form
$H$-adjoint to $\Phi$.

Eq.(6.1) coincides with equations (5.2) or (5.3) in Theorem 5.2 and is
equivalent to the condition that the real connection

$D^{H}:=\nabla^{H}+\Phi+\Phi^{*}$

on ${\mathcal{E}}$ is flat. The metric $H$ and the connection $D^{H}$ are
called the _harmonic metric_ and _Hitchin connection_ of
$({\mathcal{E}},\Phi)$, respectively. We also refer to ${\boldsymbol{h}}_{i}$
as the harmonic metric on ${\mathcal{L}}_{i}$. Further let

  * •

$Q_{\mathcal{E}}$ be the holomorphic quadratic form
$Q_{\mathcal{V}}\oplus(-Q_{\mathcal{W}})$ on ${\mathcal{E}}$;

  * •

$\lambda_{i}$ be the anti-linear involution of
${\mathcal{L}}_{i}^{-1}\oplus{\mathcal{L}}_{i}$ given by

$\lambda_{i}(\ell_{i}^{-1})=\tfrac{1}{h_{i}}\ell_{i},\quad\lambda_{i}(\ell_{i})=h_{i}\ell_{i}^{-1}$

(where $\ell_{i}$ is a holomorphic local frame of ${\mathcal{L}}_{i}$,
$\ell_{i}^{-1}$ is the corresponding frame of ${\mathcal{L}}_{i}^{-1}$, and
$h_{i}:={\boldsymbol{h}}_{i}(\ell_{i},\ell_{i})$; the definition of
$\lambda_{i}$ is independence of the choice of $\ell_{i}$);

  * •

$\lambda$ be the anti-linear involution of ${\mathcal{E}}$ which is the direct
sum of $\lambda_{1},\cdots,\lambda_{n}$ and the anti-linear involution of
${\mathcal{O}}$ fixing $\underline{\mathbb{R}}\,$;

  * •

${\mathcal{E}}_{\mathbb{R}}$, ${\mathcal{V}}_{\mathbb{R}}$,
${\mathcal{W}}_{\mathbb{R}}$ be the real subbundles formed by the fixed points
of $\lambda$ in ${\mathcal{E}}$, ${\mathcal{V}}$, ${\mathcal{W}}$,
respectively.

Then the connection $D^{H}$ preserves both $Q_{\mathcal{E}}$ and
${\mathcal{E}}_{\mathbb{R}}$, whereas $Q_{\mathcal{E}}$ restricts to a real
quadratic form of signature $(n,n+1)$ on ${\mathcal{E}}_{\mathbb{R}}$, denoted
by
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle$,
under which ${\mathcal{V}}_{\mathbb{R}}$ (resp. ${\mathcal{W}}_{\mathbb{R}}$)
is a positive (resp. negative) definite subbundle of rank $n$ (resp. $n+1$).
Moreover, if we choose a base point $z_{0}\in\Sigma$, let

$\rho:\pi_{1}(\Sigma,z_{0})\to\mathrm{SO}_{0}({\mathcal{E}}_{\mathbb{R}}|_{z_{0}},\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle)\cong\mathrm{SO}_{0}(n,n+1)$

be the holonomy representation of $D^{H}$ and

$\displaystyle f:\widetilde{\Sigma}\to X_{\mathrm{SO}_{0}(n,n+1)}$
$\displaystyle\cong\big{\\{}\text{$n$-dimensional positive definite subspaces
of ${\mathcal{E}}_{\mathbb{R}}|_{z_{0}}$}\big{\\}}$
$\displaystyle\cong\big{\\{}\text{$(n+1)$-dimensional negative definite
subspaces of ${\mathcal{E}}_{\mathbb{R}}|_{z_{0}}$}\big{\\}}$

be the $\rho$-equivariant map given by translating the fiber of
${\mathcal{V}}_{\mathbb{R}}$ or ${\mathcal{W}}_{\mathbb{R}}$ at each point
into ${\mathcal{E}}_{\mathbb{R}}|_{z_{0}}$ via the parallel transportation of
$D^{H}$, then $f$ is a conformal minimal immersion.

###### Outline of proof.

If we assume $({\mathcal{E}},\Phi)$ to be stable, then every assertion can be
either verified directly or deduced from the general theory of Higgs bundles
[22, 25, 40, 56] and cyclic Higgs bundles [6, 20]. For the stability statement
at the beginning, the “if” part can be shown by the same argument as Lemma 4.5
and Lemma 5.11 in [19]; the “only if” part is because polystablity implies
solvability of the Hitchin equation, but by the form (5.2) or (5.3) of the
Hitchin equation, it is clear that if ${\boldsymbol{\alpha}}_{n}^{-}\equiv 0$
then the Chern form $\frac{\boldsymbol{i}}{2\pi}{\bar{\partial}}\partial\log
h_{n}=-\tfrac{1}{\pi}\partial_{z\bar{z}}^{2}\log
h_{n}{\mathsf{d}x}\wedge{\mathsf{d}y}$ of ${\mathcal{L}}_{n}$ is negative away
from the zeros of ${\boldsymbol{\alpha}}_{n}^{+}$, hence
$\deg({\mathcal{L}}_{n})=\int_{\Sigma}\frac{\boldsymbol{i}}{2\pi}{\bar{\partial}}\partial\log
h_{n}<0$ (this also shows that if $({\mathcal{E}},\Phi)$ is polystable then it
is stable). ∎

When ${\mathcal{L}}_{1}={\mathcal{K}}^{-1}$ and
${\boldsymbol{\alpha}}_{1}=\tfrac{1}{\sqrt{2}}\in
H^{0}(\Sigma,\mathbb{C})=H^{0}(\Sigma,{\mathcal{K}}\\!{\mathcal{L}}_{1})$, the
Hitchin equation (5.2) (5.3) are exactly the equations satisfied by the
structural data of A-surfaces (see Theorem 4.10). In this case, we show that
$({\mathcal{E}},\Phi)$ yields an A-surfaces in $\mathbb{H}^{p,q}$ (with
$(p,q)$ given by $n$ as before; see (1.3)) in a similar way as how maximal
$\mathrm{SO}_{0}(2,q+1)$-Higgs bundles yield maximal surfaces in
$\mathbb{H}^{2,q}$ (see [21]):

######  Theorem 6.3.

In the setting of Proposition/Definition 6.2, further assume

  * •

${\mathcal{L}}_{1}={\mathcal{K}}^{-1}$ and
${\boldsymbol{\alpha}}_{1}=\tfrac{1}{\sqrt{2}}$;

  * •

$({\boldsymbol{\alpha}}_{2})\prec\cdots\prec({\boldsymbol{\alpha}}_{n-1})\prec\min\big{\\{}({\boldsymbol{\alpha}}_{n}^{+}),({\boldsymbol{\alpha}}_{n}^{-})\big{\\}}$
(see §5.1).

Then there exists a unique antipodal pair $\pm F$ of $\rho$-equivariant
A-surface immersions of $\widetilde{\Sigma}$ into $\mathbb{H}^{p,q}$ whose
Gauss map is $f$. Moreover, after possibly swapping $\pm F$, the structural
data of $F$ is exactly the holomorphic data
$({\mathcal{L}}_{1},\cdots,{\mathcal{L}}_{n},{\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{\pm})$
used to construct $({\mathcal{E}},\Phi)$ along with the harmonic metric
${\boldsymbol{h}}_{i}$ on each ${\mathcal{L}}_{i}$.

###### Proof.

In this proof, when $n$ is odd, we multiply $Q_{\mathcal{E}}$ by $-1$ and swap
the notations of ${\mathcal{V}}$ and ${\mathcal{W}}$. Under this new notation,
${\mathcal{W}}$ is the direct sum of all ${\mathcal{L}}_{k}$ with $k\in[-n,n]$
even, including ${\mathcal{L}}_{0}={\mathcal{O}}$. The signature of
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle$
is exactly $(p,q+1)$, while ${\mathcal{V}}_{\mathbb{R}}$ (resp.
${\mathcal{W}}_{\mathbb{R}}$) is a positive (resp. negative) definite
subbundle of rank $p$ (resp. $q+1$) in ${\mathcal{E}}_{\mathbb{R}}$.

Identifying $\mathbb{H}^{p,q}$ as the hypersurface in the fiber
${\mathcal{E}}_{\mathbb{R}}|_{z_{0}}$ formed by all
$x\in{\mathcal{E}}_{\mathbb{R}}|_{z_{0}}$ with $\langle x,x\rangle=-1$, we let
$F$ be the $\rho$-equivariant map from $\widetilde{\Sigma}$ to
$\mathbb{H}^{p,q+1}$ given by translating the section $\underline{1}$ of
$\underline{\mathbb{R}}$ at each point into
${\mathcal{E}}_{\mathbb{R}}|_{z_{0}}$ (similarly as the construction of $f$).
We shall show that $F$ is an A-surface immersion with Gauss map $f$ and has
the required structural data. Combining with the uniqueness result in Theorem
5.10, this implies the theorem.

Let ${\mathcal{L}}^{\prime}_{i}$ denote the real subbundle of
${\mathcal{L}}^{-1}_{i}\oplus{\mathcal{L}}_{i}$ fixed by $\lambda_{i}$. Then
${\mathcal{L}}^{\prime}_{i}$ is positive (resp. negative) definite with
respect to
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle$
exactly when $i$ is odd (resp. even), and we have orthogonal splittings

${\mathcal{E}}_{\mathbb{R}}=\underline{\mathbb{R}}\oplus{\mathcal{L}}_{1}^{\prime}\oplus{\mathcal{L}}_{2}^{\prime}\oplus\cdots\oplus{\mathcal{L}}_{n}^{\prime},\quad{\mathcal{W}}_{\mathbb{R}}=\underline{\mathbb{R}}\oplus{\mathcal{L}}_{2}^{\prime}\oplus{\mathcal{L}}_{4}^{\prime}\oplus\cdots.$

Choose a holomorphic local frame $\ell_{i}$ of ${\mathcal{L}}_{i}$ and
consider the following orthonormal local frame $(u_{i},v_{i})$ of
${\mathcal{L}}_{i}^{\prime}$ with respect to the positive definite metric
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle_{i}:=(-1)^{i+1}\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle|_{{\mathcal{L}}_{i}^{\prime}}$:

$u_{i}:=\tfrac{1}{\sqrt{2}}\big{(}h_{i}^{\frac{1}{2}}\ell_{i}^{-1}+h_{i}^{-\frac{1}{2}}\ell_{i}\big{)},\quad
v_{i}:=\tfrac{1}{\sqrt{2}}\big{(}-\boldsymbol{i}h_{i}^{\frac{1}{2}}\ell_{i}^{-1}+\boldsymbol{i}h_{i}^{-\frac{1}{2}}\ell_{i}\big{)}.$

We may identify ${\mathcal{L}}_{i}^{\prime}$ with ${\mathcal{L}}_{i}$ through
the $\sqrt{2}$-scaled projection from
${\mathcal{L}}_{i}^{-1}\oplus{\mathcal{L}}_{i}$ to ${\mathcal{L}}_{i}$, and
view ${\mathcal{L}}_{i}^{\prime}$ as a hermitian holomorphic line bundle in
this way. Note that this identifies
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle_{i}$
with the real part of the hermitian metric ${\boldsymbol{h}}_{i}$ and
identifies $(u_{i},v_{i})$ with the frame
$h_{i}^{-\frac{1}{2}}(\ell_{i},\boldsymbol{i}\ell_{i})$ of
${\mathcal{L}}_{i}$. In particular, ${\mathcal{L}}_{1}^{\prime}$ gets
identified with ${\mathcal{L}}_{1}={\mathcal{K}}^{-1}$, which can be viewed as
the tangent bundle $\mathsf{T}\Sigma$ (see the paragraph preceding Theorem
4.6), while the metric
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle_{1}\cong\operatorname{\mathsf{Re}}{\boldsymbol{h}}_{1}$
corresponds to the conformal Riemannian metric
${\boldsymbol{g}}=h_{1}|{\mathsf{d}z}|^{2}$ on $\Sigma$.

To prove the required statement about $F$, it suffices to show that the
Hitchin connection $D^{H}$ (restricted to ${\mathcal{E}}_{\mathbb{R}}$) is
expressed under the local frame
$(\underline{1},u_{1},v_{1},\cdots,u_{n},v_{n})$ of
${\mathcal{E}}_{\mathbb{R}}$ as

(6.2) $D^{H}=\mathsf{d}+\Omega,\quad\text{ where }\Omega=\scalebox{0.9}{
$\begin{pmatrix}&\leftidx{{}^{\mathsf{t}}}\\!{\theta}&&&\\\\[2.84544pt]
\theta&\Omega_{1}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{2}&&\\\\[2.84544pt]
&{\boldsymbol{\alpha}}_{2}&\Omega_{2}&\ddots&\\\
&&\ddots&\ddots&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{n}\\\\[2.84544pt]
&&&{\boldsymbol{\alpha}}_{n}&\Omega_{n}\end{pmatrix}$}.$

Here, $\theta=\leftidx{{}^{\mathsf{t}}}\\!{(}\theta_{1},\theta_{2})$ is an
orthonormal local frame of $\mathsf{T}^{*}\Sigma$ with respect to
${\boldsymbol{g}}$, $\Omega_{i}$ represents the Chern connection of
${\mathcal{L}}_{i}^{\prime}\cong{\mathcal{L}}_{i}$, and we abuse the notation
to let ${\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1}$ and
${\boldsymbol{\alpha}}_{n}={\boldsymbol{\alpha}}_{n}^{+}+{\boldsymbol{\alpha}}_{n}^{-}$
be the matrices of $1$-forms given in the proof of Theorem 4.10, which
represent the holomorphic forms
${\boldsymbol{\alpha}}_{2},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{\pm}$
used to construct $({\mathcal{E}},\Phi$). In fact, once (6.2) is established,
by the parallel transportation process in the construction of $f$ and $F$, we
may infer that there are locally defined maps $U_{1},V_{1},\cdots,U_{n},V_{n}$
from $\widetilde{\Sigma}$ to $\mathbb{R}^{p,q+1}$ (given by parallel
transporting $u_{1},v_{1},\cdots,u_{n},v_{n}$) such that the matrix valued
function $P:=(F,U_{1},V_{1},\cdots,U_{n},V_{n})$ satisfies
$P^{-1}\mathsf{d}P=\Omega$, whereas the function
$(U_{1},V_{1},U_{3},V_{3},\cdots,F,U_{2},V_{2},U_{4},V_{4})$, which takes
values in $\mathrm{SO}_{0}(p,q+1)$, is a lift of $f$. But this implies that
$F$ has the required structural data and $f$ is the Gauss map of $F$ (c.f. the
proof of Theorems 4.6 and 4.13).

To show (6.2), we start from the expression of $D^{H}$ under the frame
$(\ell_{n}^{-1},\cdots,\ell_{1}^{-1},\underline{1},\ell_{1},\cdots,\ell_{n})$
of ${\mathcal{E}}$:

$\displaystyle
D^{H}=\nabla^{H}+\Phi+\Phi^{*}=\mathsf{d}+\Omega^{\prime},\text{ where}$
(6.14) $\displaystyle\Omega^{\prime}=\left(\scalebox{0.85}{
$\begin{array}[]{ccccc|c|ccccc}-\tfrac{\partial
h_{n}}{h_{n}}&\tfrac{h_{n}\overline{\alpha_{n}^{+}}}{h_{n-1}}{\mathsf{d}\bar{z}}&&&&&&&&\alpha_{n}^{-}{\mathsf{d}z}&\\\\[8.5359pt]
\alpha_{n}^{+}{\mathsf{d}z}&-\tfrac{\partial
h_{n-1}}{h_{n-1}}&\tfrac{h_{n-1}\overline{\alpha_{n-1}}}{h_{n-2}}{\mathsf{d}\bar{z}}&&&&&&&&\alpha_{n}^{-}{\mathsf{d}z}\\\\[8.5359pt]
&\alpha_{n-1}{\mathsf{d}z}&\ddots&\ddots&&&&&&&\\\\[8.5359pt]
&&\ddots&\ddots&\tfrac{h_{2}\overline{\alpha_{2}}}{h_{1}}{\mathsf{d}\bar{z}}&&&&&&\\\\[8.5359pt]
&&&\alpha_{2}{\mathsf{d}z}&-\tfrac{\partial
h_{1}}{h_{1}}&\tfrac{h_{1}{\mathsf{d}\bar{z}}}{\sqrt{2}}&&&&&\\\
\hline\cr&&&&\tfrac{{\mathsf{d}z}}{\sqrt{2}}&0&\tfrac{h_{1}{\mathsf{d}\bar{z}}}{\sqrt{2}}&&&&\\\
\hline\cr&&&&&\tfrac{{\mathsf{d}z}}{\sqrt{2}}&\tfrac{\partial
h_{1}}{h_{1}}&\tfrac{h_{2}\overline{\alpha_{2}}}{h_{1}}{\mathsf{d}\bar{z}}&&&\\\\[8.5359pt]
&&&&&&\alpha_{2}{\mathsf{d}z}&\ddots&\ddots&&\\\\[8.5359pt]
&&&&&&&\ddots&\ddots&\tfrac{h_{n-1}\overline{\alpha_{n-1}}}{h_{n-2}}{\mathsf{d}\bar{z}}&\\\\[8.5359pt]
\tfrac{\overline{\alpha_{n}^{-}}\,{\mathsf{d}\bar{z}}}{h_{n-1}h_{n}}&&&&&&&&\alpha_{n-1}{\mathsf{d}z}&\tfrac{\partial
h_{n-1}}{h_{n-1}}&\tfrac{h_{n}\overline{\alpha_{n}^{+}}}{h_{n-1}}{\mathsf{d}\bar{z}}\\\\[8.5359pt]
&\tfrac{\overline{\alpha_{n}^{-}}\,{\mathsf{d}\bar{z}}}{h_{n-1}h_{n}}&&&&&&&&\alpha_{n}^{+}{\mathsf{d}z}&\tfrac{\partial
h_{n}}{h_{n}}\end{array}$ }\right).$

The frame for (6.2) is
$(\underline{1},u_{1},v_{1},\cdots,u_{n},v_{n})=(\ell_{n}^{-1},\cdots,\ell_{1}^{-1},\underline{1},\ell_{1},\cdots,\ell_{n})R$
with change-of-basis matrix

$R=\tfrac{1}{\sqrt{2}}\left(\scalebox{0.87}{
$\begin{array}[]{c|cc|cc|cc|cc}&&&&&&&h_{n}^{\frac{1}{2}}&-\boldsymbol{i}h_{n}^{\frac{1}{2}}\\\
&&&&&&\iddots&&\\\\[-2.84544pt] &&&&&\iddots&&&\\\
&&&h_{2}^{\frac{1}{2}}&-\boldsymbol{i}h_{2}^{\frac{1}{2}}&&&&\\\
&h_{1}^{\frac{1}{2}}&-\boldsymbol{i}h_{1}^{\frac{1}{2}}&&&&&&\\\
\hline\cr\sqrt{2}&&&&&&&&\\\
\hline\cr&h_{1}^{-\frac{1}{2}}&\boldsymbol{i}h_{1}^{-\frac{1}{2}}&&&&&&\\\
&&&h_{2}^{-\frac{1}{2}}&\boldsymbol{i}h_{2}^{-\frac{1}{2}}&&&&\\\
&&&&&\ddots&&&\\\\[-2.84544pt] &&&&&&\ddots&&\\\
&&&&&&&h_{n}^{-\frac{1}{2}}&\boldsymbol{i}h_{n}^{-\frac{1}{2}}\end{array}$
}\right).$

So the matrix of $D^{H}$ under this frame is
$R^{-1}\Omega^{\prime}R+R^{-1}\mathsf{d}R$, and we only need to show that it
equals $\Omega$ in (6.2). This is a straightforward but lengthy matrix
computation. We give below the details of some key components of the
computation:

  * •

The ${\mathcal{L}}_{i}^{\prime}$-diagonal block of
$R^{-1}\Omega^{\prime}R+R^{-1}\mathsf{d}R$ is

$\scalebox{0.85}{
$\begin{pmatrix}h_{i}^{\frac{1}{2}}&-\boldsymbol{i}h_{i}^{\frac{1}{2}}\\\\[5.69046pt]
h_{i}^{-\frac{1}{2}}&\boldsymbol{i}h_{i}^{-\frac{1}{2}}\end{pmatrix}^{-1}$
}\begin{pmatrix}-\tfrac{\partial h_{i}}{h_{i}}&\\\ &\tfrac{\partial
h_{i}}{h_{i}}\end{pmatrix}\scalebox{0.85}{
$\begin{pmatrix}h_{i}^{\frac{1}{2}}&-\boldsymbol{i}h_{i}^{\frac{1}{2}}\\\\[5.69046pt]
h_{i}^{-\frac{1}{2}}&\boldsymbol{i}h_{i}^{-\frac{1}{2}}\end{pmatrix}$
}+\scalebox{0.85}{
$\begin{pmatrix}h_{i}^{\frac{1}{2}}&-\boldsymbol{i}h_{i}^{\frac{1}{2}}\\\\[5.69046pt]
h_{i}^{-\frac{1}{2}}&\boldsymbol{i}h_{i}^{-\frac{1}{2}}\end{pmatrix}^{-1}$
}\scalebox{0.85}{
$\mathsf{d}\begin{pmatrix}h_{i}^{\frac{1}{2}}&-\boldsymbol{i}h_{i}^{\frac{1}{2}}\\\\[5.69046pt]
h_{i}^{-\frac{1}{2}}&\boldsymbol{i}h_{i}^{-\frac{1}{2}}\end{pmatrix}$
}=\begin{pmatrix}&\tfrac{\mathsf{d}^{\mathrm{c}}h_{i}}{2h_{i}}\\\
-\tfrac{\mathsf{d}^{\mathrm{c}}h_{i}}{2h_{i}}&\end{pmatrix},$

where
$\mathsf{d}^{\mathrm{c}}h:=\boldsymbol{i}(\partial-{\bar{\partial}})h=\partial_{y}h{\mathsf{d}x}-\partial_{x}h{\mathsf{d}y}$.
The result is exactly the matrix expression of the Chern connection of
$({\mathcal{L}}_{i},{\boldsymbol{h}}_{i})$ under the frame
$h_{i}^{-\frac{1}{2}}(\ell_{i},\boldsymbol{i}\ell_{i})$, as required.

  * •

The $\underline{\mathbb{R}}$-to-${\mathcal{L}}_{1}^{\prime}$ block of
$R^{-1}\Omega^{\prime}R+R^{-1}\mathsf{d}R$ is

$\scalebox{0.85}{
$\begin{pmatrix}h_{1}^{\frac{1}{2}}&-\boldsymbol{i}h_{1}^{\frac{1}{2}}\\\\[5.69046pt]
h_{1}^{-\frac{1}{2}}&\boldsymbol{i}h_{1}^{-\frac{1}{2}}\end{pmatrix}^{-1}$
}\begin{pmatrix}\tfrac{h_{1}{\mathsf{d}\bar{z}}}{\sqrt{2}}\\\\[5.69046pt]
\frac{{\mathsf{d}z}}{\sqrt{2}}\end{pmatrix}\cdot\sqrt{2}=h_{1}^{\frac{1}{2}}\begin{pmatrix}{\mathsf{d}x}\\\
{\mathsf{d}y}\end{pmatrix}$

(the term $R^{-1}\mathsf{d}R$ is block-diagonal and does not contribute here;
similarly below). This is indeed an orthonormal local frame of
$\mathsf{T}^{*}\Sigma$ under
${\boldsymbol{g}}=h_{1}({\mathsf{d}x}^{2}+{\mathsf{d}y}^{2})$, as required.

  * •

The ${\mathcal{L}}_{j-1}^{\prime}$-to-${\mathcal{L}}_{j}^{\prime}$ block
($2\leq j\leq n-1$) and the
${\mathcal{L}}_{n-1}^{\prime}$-to-${\mathcal{L}}_{n}^{\prime}$ block are
respectively

$\displaystyle\scalebox{0.85}{
$\begin{pmatrix}h_{j}^{\frac{1}{2}}&-\boldsymbol{i}h_{j}^{\frac{1}{2}}\\\\[5.69046pt]
h_{j}^{-\frac{1}{2}}&\boldsymbol{i}h_{j}^{-\frac{1}{2}}\end{pmatrix}^{-1}$
}\begin{pmatrix}\tfrac{h_{j}\overline{\alpha_{j}}}{h_{j-1}}{\mathsf{d}\bar{z}}&\\\
&\alpha_{j}{\mathsf{d}z}\end{pmatrix}\scalebox{0.85}{
$\begin{pmatrix}h_{j-1}^{\frac{1}{2}}&-\boldsymbol{i}h_{j-1}^{\frac{1}{2}}\\\\[5.69046pt]
h_{j-1}^{-\frac{1}{2}}&\boldsymbol{i}h_{j-1}^{-\frac{1}{2}}\end{pmatrix}$
}=\tfrac{\sqrt{h_{j}}}{\sqrt{h_{j-1}}}\scalebox{0.9}{$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})&-\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{j}{\mathsf{d}z})&\operatorname{\mathsf{Re}}(\alpha_{j}{\mathsf{d}z})\end{pmatrix}$},$
$\displaystyle\hskip
14.22636pt=\tfrac{\sqrt{h_{n}}}{\sqrt{h_{n-1}}}\scalebox{0.9}{$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{n}^{+}{\mathsf{d}z})&-\operatorname{\mathsf{Im}}(\alpha_{n}^{+}{\mathsf{d}z})\\\\[2.84544pt]
\operatorname{\mathsf{Im}}(\alpha_{n}^{+}{\mathsf{d}z})&\operatorname{\mathsf{Re}}(\alpha_{n}^{+}{\mathsf{d}z})\end{pmatrix}$}+\tfrac{1}{\sqrt{h_{n-1}h_{n}}}\scalebox{0.9}{$\begin{pmatrix}\operatorname{\mathsf{Re}}(\alpha_{n}^{-}{\mathsf{d}z})&-\operatorname{\mathsf{Im}}(\alpha_{n}^{-}{\mathsf{d}z})\\\\[2.84544pt]
-\operatorname{\mathsf{Im}}(\alpha_{n}^{-}{\mathsf{d}z})&-\operatorname{\mathsf{Re}}(\alpha_{n}^{-}{\mathsf{d}z})\end{pmatrix}$}.$

These are exactly the matrices representing ${\boldsymbol{\alpha}}_{j}$ and
${\boldsymbol{\alpha}}_{n}={\boldsymbol{\alpha}}_{n}^{+}+{\boldsymbol{\alpha}}_{n}^{-}$
under the frame $h_{i}^{-\frac{1}{2}}(\ell_{i},\boldsymbol{i}\ell_{i})$ (see
the proof of Theorem 4.10), as required.

In summary, we obtain the expression (6.2) of $D^{H}$ by computations and this
completes the proof. ∎

###  6.2. Collier’s components

We briefly review here the construction of Collier’s components [19], which
leads to the statement of Theorem D, and then give the proof.

We henceforth fix integers $g\geq 2$ and $n\geq 3$. Given $k\geq 1$, the Hodge
bundle ${\mathcal{H}}_{k}$ is the holomorphic vector bundle over the
Teichmüller space ${{\mathcal{T}}}_{g}$ whose fiber
${\mathcal{H}}_{k}|_{[\Sigma]}$ at any $[\Sigma]\in{{\mathcal{T}}}_{g}$ is the
vector space $H^{0}(\Sigma,{\mathcal{K}}^{k})$ of holomorphic
$k$-differentials on the Riemann surface $\Sigma$ (marked by $S_{g}$). Given
$0<d\leq(2g-2)n$, we also define a fiber bundle ${\mathcal{F}}_{d}$ over
${{\mathcal{T}}}_{g}$ in such a way that the fiber
${\mathcal{F}}_{d}|_{[\Sigma]}$ is the moduli space
$\widetilde{{\mathcal{F}}}_{d}|_{[\Sigma]}\big{/}\sim$, where

  * •

$\widetilde{{\mathcal{F}}}_{d}|_{[\Sigma]}$ is the space of triples
$({\mathcal{M}},{\boldsymbol{\mu}},{\boldsymbol{\nu}})$ consisting of a
holomorphic line bundle ${\mathcal{M}}$ of degree $d$ on $\Sigma$ and
holomorphic sections ${\boldsymbol{\mu}}\in
H^{0}(\Sigma,{\mathcal{K}}^{n}\\!\\!\\!{\mathcal{M}}^{-1})\setminus\\{0\\}$,
${\boldsymbol{\nu}}\in H^{0}(\Sigma,{\mathcal{K}}^{n}\\!\\!\\!{\mathcal{M}})$;

  * •

the equivalence relation
$({\mathcal{M}},{\boldsymbol{\mu}},{\boldsymbol{\nu}})\sim({\mathcal{M}}^{\prime},{\boldsymbol{\mu}}^{\prime},{\boldsymbol{\nu}}^{\prime})$
means that there are line bundle isomorphisms
$\phi_{0}:{\mathcal{K}}^{-n+1}\to{\mathcal{K}}^{-n+1}$ and
$\phi_{1}:{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ which, along with the
isomorphism $\phi_{-1}:{\mathcal{M}}^{-1}\to{\mathcal{M}}^{\prime}{}^{-1}$
induced by $\phi_{1}$, fit into a commutative diagram

Here, an horizontal arrow such as
${\mathcal{L}}_{1}\overset{{\boldsymbol{\alpha}}}{\to}{\mathcal{L}}_{2}$ means
that ${\boldsymbol{\alpha}}$ is a
$\mathrm{Hom}({\mathcal{L}}_{1},{\mathcal{L}}_{2})$-valued holomorphic
$1$-form, or equivalently a holomorphic section of
${\mathcal{K}}\\!{\mathcal{L}}_{1}^{-1}\\!{\mathcal{L}}_{2}$.

######  Remark 6.4.

The $\widetilde{{\mathcal{F}}}_{d}|_{[\Sigma]}$ here is slightly different
from the one in [19]444Note that everything in [19] is based on a fix Riemann
surface $\Sigma$, so the notation
“$\,{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\,|_{[\Sigma]}$” is not
included therein. in that our ${\mathcal{M}}$ is a concrete line bundle rather
than a point in $\mathrm{Pic}^{d}(\Sigma)$, but the resulting quotient
${\mathcal{F}}_{d}|_{[\Sigma]}$ is the same. As noticed in [19],
${\mathcal{F}}_{d}|_{[\Sigma]}$ identifies with the total space of a rank
$d+(2n-1)(g-1)$ holomorphic vector bundle over the symmetric product
$\mathrm{Sym}^{(2g-2)n-d}\Sigma$.

The direct sum fiber bundle
${\mathcal{H}}_{2}\oplus{\mathcal{H}}_{4}\oplus\cdots\oplus{\mathcal{H}}_{2n-2}\oplus{\mathcal{F}}_{d}$
over ${{\mathcal{T}}}_{g}$ can be viewed as a data space whose each point
determines an equivalence class of stable $\mathrm{SO}_{0}(n,n+1)$-Higgs
bundles. More specifically, from
$\big{(}{\boldsymbol{\phi}}_{2},\cdots,{\boldsymbol{\phi}}_{2n-2},({\mathcal{M}},{\boldsymbol{\mu}},{\boldsymbol{\nu}})\big{)}\in\big{(}{\mathcal{H}}_{2}\oplus{\mathcal{H}}_{4}\oplus\cdots\oplus{\mathcal{H}}_{2n-2}\oplus\widetilde{{\mathcal{F}}}_{d}\big{)}|_{[\Sigma]}$,
we get the following $\mathrm{SO}_{0}(n,n+1)$-Higgs bundle
$({\mathcal{V}},Q_{\mathcal{V}},{\mathcal{W}},Q_{\mathcal{W}},\eta)$ on
$\Sigma$:

$\displaystyle{\mathcal{V}}={\mathcal{K}}^{n-1}\oplus{\mathcal{K}}^{n-3}\oplus\cdots\oplus{\mathcal{K}}^{-n+3}\oplus{\mathcal{K}}^{-n+1},\quad
Q_{\mathcal{V}}=\scalebox{0.9}{$\begin{pmatrix}&&1\\\\[-2.84544pt]
&\iddots&\\\\[-2.84544pt] 1&&\end{pmatrix}$},$
$\displaystyle{\mathcal{W}}={\mathcal{M}}\oplus{\mathcal{K}}^{n-2}\oplus{\mathcal{K}}^{n-4}\oplus\cdots\oplus{\mathcal{K}}^{-n+4}\oplus{\mathcal{K}}^{-n+2}\oplus{\mathcal{M}}^{-1},\quad
Q_{\mathcal{W}}=\scalebox{0.8}{$\begin{pmatrix}&&1\\\\[-2.84544pt]
&\iddots&\\\\[-2.84544pt] 1&&\end{pmatrix}$},$
$\displaystyle\eta=\scalebox{0.85}{$\begin{pmatrix}{\boldsymbol{\mu}}&{\boldsymbol{\phi}}_{2}&{\boldsymbol{\phi}}_{4}&\cdots&{\boldsymbol{\phi}}_{2n-2}&{\boldsymbol{\nu}}\\\
&1&\ddots&\ddots&\vdots&\\\\[-2.84544pt]
&&\ddots&\ddots&{\boldsymbol{\phi}}_{4}&\\\ &&&1&{\boldsymbol{\phi}}_{2}&\\\
&&&&\tfrac{1}{\sqrt{2}}&\end{pmatrix}$}\in
H^{0}(\Sigma,{\mathcal{K}}\otimes\mathrm{Hom}({\mathcal{W}},{\mathcal{V}}));$

while replacing $({\mathcal{M}},{\boldsymbol{\mu}},{\boldsymbol{\nu}})$ by an
equivalent triple
$({\mathcal{M}}^{\prime},{\boldsymbol{\mu}}^{\prime},{\boldsymbol{\nu}}^{\prime})$
yields an equivalent Higgs bundle (the $\frac{1}{\sqrt{2}}$ here can be
replaced by $1$ as mentioned in the introduction).

Let
${\mathcal{X}}:=\mathrm{Hom}(\pi_{1}(S_{g}),\mathrm{SO}_{0}(n,n+1))/\mathrm{SO}_{0}(n,n+1)$
be the $\mathrm{SO}_{0}(n,n+1)$-character variety. Combining the above
construction with the non-abelian Hodge correspondence, which assigns to every
equivalence class of stable $\mathrm{SO}_{0}(n,n+1)$-Higgs bundles a point of
${\mathcal{X}}$, we obtain a map

$\Psi:{\mathcal{H}}_{2}\oplus{\mathcal{H}}_{4}\oplus\cdots\oplus{\mathcal{H}}_{2n-2}\oplus{\mathcal{F}}_{d}\to{\mathcal{X}}.$

For any fixed Riemann surface $\Sigma$, Collier [19] studied the restriction
of $\Psi$ to the fiber of the domain bundle at $[\Sigma]$ and found that it
parametrizes certain remarkable components of ${\mathcal{X}}$:

######  Theorem 6.5 ([19]).

Given $d\in(0,(2g-2)n]$ and a Riemann surface $\Sigma$, the map $\Psi$
restricts to a diffeomorphism from
$\big{(}{\mathcal{H}}_{2}\oplus{\mathcal{H}}_{4}\oplus\cdots\oplus{\mathcal{H}}_{2n-2}\oplus{\mathcal{F}}_{d}\big{)}|_{[\Sigma]}$
to a connected component ${\mathcal{X}}_{d}$ of ${\mathcal{X}}$.

On the other hand, the motivation behind Labourie’s conjecture is to give
certain components of character varieties _mapping-class-group-equivariant_
parametrizations. For the component ${\mathcal{X}}_{d}$ from Theorem 6.5, the
major candidate for such a parametrization is the restriction of $\Psi$ to the
domain fiber bundle with the summand ${\mathcal{H}}_{2}$ removed. However,
although the existing techniques [47, 9, 36] imply that this restriction is
surjective, the injectivity is a difficult problem, especially in view of the
new twist [52, 53] in the study of Labourie’s conjecture for hermitian-type
Lie groups. Our goal here is to provide the following partial result which
extends Labourie’s work [48] for Hitchin components. The proof is closely
modeled on the argument in [48, §7.3.4], only with the infinitesimal rigidity
of cyclic surfaces replaced by that of A-surfaces.

######  Theorem 6.6.

Given $d\in(0,(2g-2)n]$, $\Psi$ restricts to an immersion from the fiber
bundle ${\mathcal{F}}_{d}$ to ${\mathcal{X}}_{d}$.

###### Proof.

Fix
$q=([\Sigma],{\mathcal{M}},{\boldsymbol{\mu}},{\boldsymbol{\nu}})\in\widetilde{{\mathcal{F}}}_{d}$
and consider a smooth variation
$q_{t}=([\Sigma_{t}],{\mathcal{M}}_{t},{\boldsymbol{\mu}}_{t},{\boldsymbol{\nu}}_{t})$
with $q_{0}=q$, $t\in(-\epsilon,\epsilon)$. Let $({\mathcal{E}}_{t},\Phi_{t})$
be the Higgs bundle constructed from $q_{t}$ as above, with
${\boldsymbol{\phi}}_{2},\cdots,{\boldsymbol{\phi}}_{2n-2}=0$. Then
$({\mathcal{E}}_{t},\Phi_{t})$ is the special case of the Higgs bundles
studied in §6.1 with ${\mathcal{L}}_{i}={\mathcal{K}}^{-i}$ for $i\leq n-1$,
${\mathcal{L}}_{n}={\mathcal{M}}^{-1}$,
${\boldsymbol{\alpha}}_{1}=\frac{1}{\sqrt{2}}$,
${\boldsymbol{\alpha}}_{2}=\cdots={\boldsymbol{\alpha}}_{n-1}=1$,
${\boldsymbol{\alpha}}_{n}^{+}={\boldsymbol{\mu}}_{t}$ and
${\boldsymbol{\alpha}}_{n}^{-}={\boldsymbol{\nu}}_{t}$. By Prop./Def. 6.2 and
Theorem 6.3, we get from $({\mathcal{E}}_{t},\Phi_{t})$ a holonomy
representation
$\rho_{t}:\pi_{1}(S_{g})=\pi_{1}(\Sigma_{t})\to\mathrm{SO}_{0}(n,n+1)$ and a
$\rho_{t}$-equivariant A-surface immersion
$F_{t}:\widetilde{\Sigma}_{t}\to\mathbb{H}^{p,q}$, while the map $\Psi$ is
given by $\Psi([q_{t}])=[\rho_{t}]\in{\mathcal{X}}_{d}$. To prove the theorem,
we assume that the path $([\rho_{t}])_{t\in(-\epsilon,\epsilon)}$ in
${\mathcal{X}}_{d}$ is stationary at $t=0$, and only need to show that the
path $([q_{t}])$ in ${\mathcal{F}}_{d}$ is stationary at $t=0$ as well.

After conjugating each $\rho_{t}$ by some element of $\mathrm{SO}_{0}(n,n+1)$,
we may assume that $\rho_{t}$ itself (rather than the conjugacy class
$[\rho_{t}]$) is stationary, namely
$\accentset{\,\mbox{\large.}}{\rho}_{0}=0$. As a consequence, taking the
derivative in $t$ of the $\rho_{t}$-equivariance relation
$F_{t}(\gamma.z)=\rho_{t}(\gamma).F_{t}(z)$ ($\gamma\in\pi_{1}(\Sigma_{t})$,
$z\in\widetilde{\Sigma}$), we obtain, at $t=0$,

$\accentset{\,\mbox{\large.}}{F}_{0}(\gamma.z)=\rho_{0}(\gamma).\accentset{\,\mbox{\large.}}{F}_{0}(z)\
\ \text{ for all }\gamma\in\pi_{1}(\Sigma),\ z\in\widetilde{\Sigma}.$

Namely, $\accentset{\,\mbox{\large.}}{F}_{0}$ is $\rho_{0}(\Sigma)$-invariant.
In particular, the normal component
$\accentset{\,\mbox{\large.}}{F}_{0}^{\mathcal{N}}$ of
$\accentset{\,\mbox{\large.}}{F}_{0}$, which can be viewed as a Jacobi field
on the A-surface $F_{0}(\widetilde{\Sigma})\subset\mathbb{H}^{p,q}$ (see
§3.3), descents to a Jacobi field on the closed A-surface
$F_{0}(\widetilde{\Sigma})/\rho_{0}(\pi_{1}(\Sigma))$ in the quotient
$M:=\mathbb{H}^{p,q}/\rho_{0}(\pi_{1}(\Sigma))$. Therefore, we may apply
Corollary 5.9 to conclude that
$\accentset{\,\mbox{\large.}}{F}_{0}^{\mathcal{N}}\equiv 0$, or in other
words, $\accentset{\,\mbox{\large.}}{F}_{0}$ is tangent to $F$.

Since the first fundamental form of the immersion $F_{t}$ is conformal to the
Riemann surface structure $\Sigma_{t}$, by the first variation formula of
first fundamental form (see e.g. [57]), the variation of this Riemann surface
structure at $t=0$ is given by its Lie derivative with respect to the tangent
vector field of $\Sigma$ corresponding to
$\accentset{\,\mbox{\large.}}{F}_{0}$. It follows that the path
$([\Sigma_{t}])$ in ${{\mathcal{T}}}_{g}$ is stationary at $t=0$. Therefore,
we may assume from the beginning that $\Sigma_{t}$ is the fixed Riemann
surface $\Sigma$ and only the triple
$({\mathcal{M}}_{t},{\boldsymbol{\mu}}_{t},{\boldsymbol{\nu}}_{t})$ on
$\Sigma$ vary. By Lemma 3.3, the connection $\nabla^{{\mathcal{N}}_{t}}$ on
the normal bundle ${\mathcal{N}}_{t}\subset
F_{t}^{*}\mathsf{T}\mathbb{H}^{p,q}$ is stationary at $t=0$. But on the other
hand, by definition of A-surfaces,
$({\mathcal{N}}_{t},\nabla^{{\mathcal{N}}_{t}})$ is isomorphic to
${\mathcal{K}}^{-2}\oplus\cdots\oplus{\mathcal{K}}^{-n+1}\oplus{\mathcal{M}}^{-1}_{t}$
(as a real vector bundle) endowed with the connection

$\nabla=\scalebox{0.9}{$\begin{pmatrix}\nabla_{2}&1^{*}&&&\\\\[2.84544pt]
1&\nabla_{3}&\ddots&&\\\ &\ddots&\ddots&1^{*}&\\\\[5.69046pt]
&&1&\nabla_{n-1}&{\boldsymbol{\mu}}_{t}^{*}+{\boldsymbol{\nu}}_{t}^{*}\\\\[2.84544pt]
&&&{\boldsymbol{\mu}}_{t}+{\boldsymbol{\nu}}_{t}&\nabla_{n}\end{pmatrix}$}$

(where each $1\in\mathbb{C}\cong
H^{0}(\Sigma,{\mathcal{K}}\otimes\mathrm{Hom}_{\mathbb{C}}({\mathcal{K}}^{-j+1},{\mathcal{K}}^{-j}))$
is viewed as a
$\mathrm{Hom}_{\mathbb{R}}({\mathcal{K}}^{-j+1},{\mathcal{K}}^{-j})$-valued
$1$-form). In particular, the equivalence class of the holomorphic chain

is determined by the connection $\nabla^{{\mathcal{N}}_{t}}$. Therefore, we
conclude that this equivalence class is stationary at $t=0$. By definition of
the equivalence relation in $\widetilde{{\mathcal{F}}}_{d}|_{[\Sigma]}$, this
means that the point in
${\mathcal{F}}_{d}|_{[\Sigma]}=\widetilde{{\mathcal{F}}}_{d}|_{[\Sigma]}/\sim$
represented by
$({\mathcal{M}}_{t},{\boldsymbol{\mu}}_{t},{\boldsymbol{\nu}}_{t})$ is
stationary at $t=0$, as required. ∎

###  6.3. $G_{2}^{\prime}$-Hitchin component

We now turn to the special case of Def. 6.1 and Prop./Def. 6.2 with

(6.15) $n=3,\quad{\mathcal{L}}_{i}={\mathcal{K}}^{-i}\
(i=1,2,3),\quad{\boldsymbol{\alpha}}_{1}=\tfrac{1}{\sqrt{2}},\quad{\boldsymbol{\alpha}}_{2}={\boldsymbol{\alpha}}_{3}^{+}=1.$

The $({\mathcal{E}},\Phi)$’s in this case are parametrized by
${\boldsymbol{\alpha}}_{n}^{-}\in H^{0}(\Sigma,{\mathcal{K}}^{6})$ and are
exactly the Higgs bundles in the $G_{2}^{\prime}$-Hitchin component
constructed from the principal $3$-dimensional subalgebra
$\mathfrak{s}\subset\mathfrak{g}_{2}^{\mathbb{C}}$ in §2.7. By using the
description in Remark 2.11 of the $\mathrm{SO}(4)$-module
$\mathfrak{p}\subset\mathfrak{g}_{2}^{\prime}$, one may check that
$({\mathcal{E}},\Phi)$ is indeed a $G_{2}^{\prime}$-Higgs bundles, so the
Hitchin connection $D^{H}$ on
${\mathcal{E}}={\mathcal{K}}^{3}\oplus\cdots\oplus{\mathcal{K}}^{-3}$ is a
$G_{2}^{\prime}$-connection. We omit the details here, as it involves the
lengthy definition of $G$-Higgs bundles for a general $G$. Alternatively, we
now use Proposition 2.16 to identify $D^{H}$ as a $G_{2}^{\prime}$-connection
in a direct way.

Recall from Proposition 2.15 that $\mathfrak{g}_{2}^{\mathbb{C}}$ can be
identified as the Lie algebra of matrices infinitesimally preserving the
$3$-form
$\varpi:=\boldsymbol{i}\big{(}\varepsilon^{147}+\varepsilon^{246}-\varepsilon^{345}-\scalebox{0.8}{$2\sqrt{2}$}\,\varepsilon^{156}-\tfrac{1}{\sqrt{2}}\varepsilon^{237}\big{)}\in\bigwedge^{3}\mathbb{C}^{7*}$.
But each term $\varepsilon^{ijk}$ in $\varpi$ (note that $i+j+k=12$) can be
viewed as a holomorphic section of $\bigwedge^{3}{\mathcal{E}}^{*}$ through

$\varepsilon^{ijk}(\xi_{1},\xi_{2},\xi_{3}):=\sum_{\sigma\in\mathfrak{S}_{3}}\mathrm{sign}(\sigma)\
\xi_{\sigma(1)}^{(i)}\otimes\xi_{\sigma(2)}^{(j)}\otimes\xi_{\sigma(3)}^{(k)}$

for any holomorphic local sections $\xi_{1},\xi_{2},\xi_{3}$ of
${\mathcal{E}}$, where $\mathfrak{S}_{3}$ is the permutation group of the
indices $\\{1,2,3\\}$ and the superscript “$(i)$” means taking the $i$th
component in
${\mathcal{E}}={\mathcal{K}}^{3}\oplus\cdots\oplus{\mathcal{K}}^{-3}$, namely
the ${\mathcal{K}}^{4-i}$-component; the tensor product yields a holomorphic
section of
${\mathcal{K}}^{4-i}\otimes{\mathcal{K}}^{4-j}\otimes{\mathcal{K}}^{4-k}={\mathcal{O}}$,
which is just a holomorphic function. Therefore, we may consider $\varpi$ as a
holomorphic section of $\bigwedge^{3}{\mathcal{E}}^{*}$ as well. It allows us
to deduce from Proposition 2.16 and Prop./Def. 6.2:

######  Corollary 6.7.

In the case (6.15), the following statements about the objects in Prop./Def.
6.2 hold:

  * •

the harmonic metric
$H={\boldsymbol{h}}_{3}^{-1}\oplus{\boldsymbol{h}}_{2}^{-1}\oplus{\boldsymbol{h}}_{1}^{-1}\oplus
1\oplus{\boldsymbol{h}}_{1}\oplus{\boldsymbol{h}}_{2}\oplus{\boldsymbol{h}}_{3}$
satisfies ${\boldsymbol{h}}_{3}={\boldsymbol{h}}_{1}{\boldsymbol{h}}_{2}/4$;

  * •

the Hitchin connection $D^{H}$ preserves $\varpi\in
H^{0}(\Sigma,\bigwedge^{3}{\mathcal{E}}^{*})$;

  * •

$\varpi$ restricts to a real $3$-form on each fiber of
${\mathcal{E}}_{\mathbb{R}}$ which, together with the metric
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle$,
makes the fiber isomorphic to $\mathbb{R}^{3,4}$ endowed with the $3$-form
$\varphi$ associated to the cross product $\times$ (see §2.3).

As a consequence, the representation $\rho$ takes values in the subgroup
$G_{2}^{\prime}$ of
$\mathrm{SO}_{0}({\mathcal{E}}_{\mathbb{R}}|_{z_{0}},\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle)\cong\mathrm{SO}_{0}(3,4)$
formed by linear transformations of ${\mathcal{E}}_{\mathbb{R}}|_{z_{0}}$
preserving
$\langle{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,},{\,\raisebox{0.8pt}{\scalebox{0.65}{$\bullet$}}\,}\rangle$
and the $3$-form, whereas the minimal immersion $f$ takes values in the
corresponding submanifold $X_{G_{2}^{\prime}}$ of $X_{\mathrm{SO}_{0}(3,4)}$.

###### Proof.

It is a general fact for Higgs bundles in the Hitchin component (see [41, 45])
that the Cartan involution $\rho$ (induced by the harmonic metric) on each
fiber of the underlying Lie algebra bundle commutes with the holomorphic
involution $\sigma$ (induced by the principal $3$-dimensional subalgebra), and
in the cyclic case $\rho$ furthermore preserves $\mathfrak{h}$ . Therefore,
the statement on $H$ follows from Proposition 2.16. For the statement on
$D^{H}$, we note that in the current setting, the expression (6.14) of $D^{H}$
under the local frame
$(\ell_{n}^{-1},\cdots,\ell_{1}^{-1},\underline{1},\ell_{1},\cdots,\ell_{n})$
of ${\mathcal{E}}$ (here $n=3$ and we may take $\ell_{i}={\mathsf{d}z}^{-i}$)
becomes

(6.16) $D^{H}=\mathsf{d}+\left(\scalebox{0.9}{
$\begin{array}[]{ccc|c|ccc}-\tfrac{\partial
h_{3}}{h_{3}}&\tfrac{h_{3}}{h_{2}}{\mathsf{d}\bar{z}}&&&&\alpha_{3}^{-}{\mathsf{d}z}&\\\\[8.5359pt]
{\mathsf{d}z}&-\tfrac{\partial
h_{2}}{h_{2}}&\tfrac{h_{2}}{h_{1}}{\mathsf{d}\bar{z}}&&&&\alpha_{3}^{-}{\mathsf{d}z}\\\\[8.5359pt]
&{\mathsf{d}z}&-\tfrac{\partial
h_{1}}{h_{1}}&\tfrac{h_{1}{\mathsf{d}\bar{z}}}{\sqrt{2}}&&&\\\\[8.5359pt]
\hline\cr&&\tfrac{{\mathsf{d}z}}{\sqrt{2}}&0&\tfrac{h_{1}{\mathsf{d}\bar{z}}}{\sqrt{2}}&&\\\
\hline\cr&&&\tfrac{{\mathsf{d}z}}{\sqrt{2}}&\tfrac{\partial
h_{1}}{h_{1}}&\tfrac{h_{2}}{h_{1}}{\mathsf{d}\bar{z}}&\\\\[8.5359pt]
\tfrac{\overline{\alpha_{3}^{-}}{\mathsf{d}\bar{z}}}{h_{2}h_{3}}&&&&{\mathsf{d}z}&\tfrac{\partial
h_{2}}{h_{2}}&\tfrac{h_{3}}{h_{2}}{\mathsf{d}\bar{z}}\\\\[8.5359pt]
&\tfrac{\overline{\alpha_{3}^{-}}{\mathsf{d}\bar{z}}}{h_{2}h_{3}}&&&&{\mathsf{d}z}&\tfrac{\partial
h_{3}}{h_{3}}\end{array}$ }\right),\text{ where }h_{3}=\tfrac{h_{1}h_{2}}{4}.$

This matrix has the form (2.12) in Proposition 2.15, hence $D^{H}$ preserves
$\varpi$. The statement on the restriction of $\varpi$ to
${\mathcal{E}}_{\mathbb{R}}$ is again contained in Proposition 2.16. These
statements imply that $D^{H}$ is a $G_{2}^{\prime}$-connection on
${\mathcal{E}}_{\mathbb{R}}$, whence the statement on $\rho$ and $f$ follows.
∎

The A-surfaces underlying $({\mathcal{E}},\Phi)$ belong to the subclass
discussed in Proposition 4.15:

######  Proposition 6.8.

In the above setting, the A-surface immersion
$F:\widetilde{\Sigma}\to\mathbb{H}^{4,2}$ given in Theorem 6.3 is
${\boldsymbol{J}}$-holomorphic. As a consequence, $-F$ is
${\boldsymbol{J}}$-anti-holomorphic and the images of $\pm F$ are
${\boldsymbol{J}}$-holomorphic curve with nowhere vanishing second fundamental
form and timelike osculation lines.

###### Proof.

Consider the holomorphic local frame
$(\ell_{3}^{-1},\ell_{2}^{-1},\ell_{1}^{-1},\underline{1},\ell_{1},\ell_{2},\ell_{3})=({\mathsf{d}z}^{3},{\mathsf{d}z}^{2},\cdots,{\mathsf{d}z}^{-3})$
of ${\mathcal{E}}$. In the proof of Theorem 6.3, we have shown that under the
frame
$(\underline{1},u_{1},v_{1},u_{2},v_{2},u_{3},v_{3})=(\ell_{3}^{-1},\cdots,\ell_{3})R$
of ${\mathcal{E}}_{\mathbb{R}}$ given by

$R=\tfrac{1}{\sqrt{2}}\left(\scalebox{0.9}{$\begin{array}[]{c|cc|cc|cc}&&&&&h_{3}^{\frac{1}{2}}&-\boldsymbol{i}h_{3}^{\frac{1}{2}}\\\\[5.69046pt]
&&&h_{2}^{\frac{1}{2}}&-\boldsymbol{i}h_{2}^{\frac{1}{2}}&&\\\\[5.69046pt]
&h_{1}^{\frac{1}{2}}&-\boldsymbol{i}h_{1}^{\frac{1}{2}}&&&&\\\
\hline\cr\scalebox{0.9}{$\sqrt{2}$}&&&&&&\\\
\hline\cr&h_{1}^{-\frac{1}{2}}&\boldsymbol{i}h_{1}^{-\frac{1}{2}}&&&&\\\\[5.69046pt]
&&&h_{2}^{-\frac{1}{2}}&\boldsymbol{i}h_{2}^{-\frac{1}{2}}&&\\\\[5.69046pt]
&&&&&h_{3}^{-\frac{1}{2}}&\boldsymbol{i}h_{3}^{-\frac{1}{2}}\end{array}$}\right)$

(note that $h_{3}=\frac{h_{1}h_{2}}{4}$ by Corollary 6.7), the Hitchin
connection is represented by the matrix

$\Omega=\scalebox{0.9}{
$\begin{pmatrix}&\leftidx{{}^{\mathsf{t}}}\\!{\theta}&&\\\\[2.84544pt]
\theta&\Omega_{1}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{2}&\\\\[2.84544pt]
&{\boldsymbol{\alpha}}_{2}&\Omega_{2}&\leftidx{{}^{\mathsf{t}}}\\!{{\boldsymbol{}}}\alpha_{3}\\\\[2.84544pt]
&&{\boldsymbol{\alpha}}_{3}&\Omega_{n}\end{pmatrix}$};$

and as a consequence, there are maps $U_{i},V_{i}$ ($i=1,2,3$) from
$\widetilde{\Sigma}$ to $\mathbb{R}^{3,4}$ such that the matrix-valued map
$P:=(F,U_{1},V_{1},U_{2},V_{2},U_{3},V_{3})$ satisfies
$P^{-1}\mathsf{d}P=\Omega$. In particular,
$\mathsf{d}F_{z}:\mathsf{T}_{z}\widetilde{\Sigma}\to\mathsf{T}_{F(z)}\mathbb{H}^{4,2}\cong
F(z)^{\perp}$ sends the unit tangent vectors
$h_{1}^{-\frac{1}{2}}\partial_{x},h_{1}^{-\frac{1}{2}}\partial_{y}\in\mathsf{T}_{z}\widetilde{\Sigma}$
to $U_{1}$ and $V_{1}$, respectively.

On the other hand, by the proof of Proposition 2.16, the modified frame
$(\underline{1},u_{2},v_{2},u_{1},v_{1},-v_{3},u_{3})$ (which equals
$(\ell_{3}^{-1}\,\cdots,\ell_{3})B$ with $B$ from that proof) is a
$G_{2}^{\prime}$-frame of ${\mathcal{E}}_{\mathbb{R}}$ in the sense of
Definition 2.3, so the matrix $(F,U_{2},V_{2},U_{1},V_{1},-V_{3},U_{3})$ is in
$G_{2}^{\prime}$. Thus, if we re-denote the column vectors of this matrix by
$e_{1},\cdots,e_{7}$ successively, then their cross products are given by
Table 2. In particular, we have $F\times U_{1}=e_{1}\times e_{4}=e_{5}=V_{1}$.
By definition of the almost complex structure ${\boldsymbol{J}}$ (see §2.6),
this means that ${\boldsymbol{J}}$ sends $U_{1}(z)\in
F(z)^{\perp}=\mathsf{T}_{F(z)}\mathbb{H}^{4,2}$ to $V_{1}(z)$ for all
$z\in\widetilde{\Sigma}$. Combining with the above description of
$\mathsf{d}F_{z}$, we conclude that
$\mathsf{d}F_{z}\circ{\boldsymbol{j}}={\boldsymbol{J}}\circ\mathsf{d}F_{z}$
(where ${\boldsymbol{j}}$ denotes the complex structure on
$\widetilde{\Sigma}$), namely $F$ is ${\boldsymbol{J}}$-holomorphic. ∎

##  Appendix A Proof of Theorem 5.2

The proof here is adapted from [23, §5]. In order to make it easier for the
reader to compare with the original work, we restate the theorem as follows,
where the assumption from Remark 5.4 has been added and the notation has been
reset (see the remark below) so as to be consistent with [23].

######  Theorem.

Let $\Sigma$ be a closed Riemann surface of genus $\geq 2$,
${\mathcal{L}}_{1},\cdots,{\mathcal{L}}_{n}$ ($n\geq 2$) be holomorphic line
bundles on $\Sigma$ and ${\boldsymbol{h}}_{i}$ be a hermitian metric on
${\mathcal{L}}_{i}$ ($i=1,\cdots,n$). Let ${\boldsymbol{\gamma}}_{j}$
($j=1,\cdots,n-1$), ${\boldsymbol{\gamma}}_{n}$ and ${\boldsymbol{\beta}}$ be
holomorphic sections of the line bundles
${\mathcal{K}}\\!{\mathcal{L}}_{j}^{-1}{\mathcal{L}}_{j+1}$,
${\mathcal{K}}\\!{\mathcal{L}}_{n}^{-1}$ and
${\mathcal{K}}\\!{\mathcal{L}}_{1}{\mathcal{L}}_{2}$, respectively, such that
none of ${\boldsymbol{\gamma}}_{1},\cdots,{\boldsymbol{\gamma}}_{n}$ is
identically zero and the following equations hold on $\Sigma$:

(A.1) $\displaystyle\text{if $n=2$: }\begin{cases}\partial_{z\bar{z}}^{2}\log
h_{1}=-\tfrac{h_{2}}{h_{1}}|\gamma_{1}|^{2}+h_{1}h_{2}|\beta|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{2}=\tfrac{h_{2}}{h_{1}}|\gamma_{1}|^{2}-\tfrac{1}{h_{2}}|\gamma_{2}|^{2}+h_{1}h_{2}|\beta|^{2};\end{cases}$
(A.2) $\displaystyle\text{if $n\geq 3$:
}\begin{cases}\partial_{z\bar{z}}^{2}\log
h_{1}=-\tfrac{h_{2}}{h_{1}}|\gamma_{1}|^{2}+h_{1}h_{2}|\beta|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{2}=\tfrac{h_{2}}{h_{1}}|\gamma_{1}|^{2}-\tfrac{h_{3}}{h_{2}}|\gamma_{2}|^{2}+h_{1}h_{2}|\beta|^{2},\\\\[5.69046pt]
\partial_{z\bar{z}}^{2}\log
h_{k}=\tfrac{h_{k}}{h_{k-1}}|\gamma_{k-1}|^{2}-\tfrac{h_{k+1}}{h_{k}}|\gamma_{k}|^{2}\quad(3\leq
k\leq n-1),\\\\[5.69046pt] \partial_{z\bar{z}}^{2}\log
h_{n}=\tfrac{h_{n}}{h_{n-1}}|\gamma_{n-1}|^{2}-\tfrac{1}{h_{n}}|\gamma_{n}|^{2}.\end{cases}$

Then the following statements hold.

  1. (1)

If $({\boldsymbol{\gamma}}_{1})\prec({\boldsymbol{\beta}})\neq\emptyset$, then
$\partial_{z\bar{z}}^{2}\log h_{1}\leq 0$ on $\Sigma$, with strict inequality
away from the zeros of ${\boldsymbol{\gamma}}_{1}$.

  2. (2)

If
$({\boldsymbol{\gamma}}_{n})\prec\cdots\prec({\boldsymbol{\gamma}}_{2})\prec\min\big{\\{}({\boldsymbol{\gamma}}_{1}),({\boldsymbol{\beta}})\big{\\}}$,
then $\partial_{z\bar{z}}^{2}\log h_{i}\leq 0$ on $\Sigma$ for $i=2,\cdots,n$,
with strict inequality away from the zeros of ${\boldsymbol{\gamma}}_{i}$.

######  Remark.

The $({\mathcal{L}}_{i},{\boldsymbol{h}}_{i})$ here corresponds to
$({\mathcal{L}}_{n+1-i}^{-1},{\boldsymbol{h}}_{n+1-i}^{-1})$ in Theorem 5.2.
In particular, $\log h_{1}$, $\cdots$, $\log h_{n}$ correspond to $-\log
h_{n},\cdots,-\log h_{1}$ in Theorem 5.2, respectively. Meanwhile,
${\boldsymbol{\gamma}}_{n},\cdots,{\boldsymbol{\gamma}}_{2},{\boldsymbol{\gamma}}_{1},{\boldsymbol{\beta}}$
correspond to
${\boldsymbol{\alpha}}_{1},\cdots,{\boldsymbol{\alpha}}_{n-1},{\boldsymbol{\alpha}}_{n}^{+},{\boldsymbol{\alpha}}_{n}^{-}$,
respectively. In fact, while the equations in Theorem 5.2 are the Hitchin
equation for the Higgs bundle $({\mathcal{E}},\Phi)$ exhibited in §6.1, the
above (A.1) and (A.2) are the Hitchin equation for the Higgs bundle

$\displaystyle\hskip
2.84544pt{\mathcal{E}}={\mathcal{L}}_{1}\oplus{\mathcal{L}}_{2}\cdots\oplus{\mathcal{L}}_{n}\oplus{\mathcal{O}}\oplus{\mathcal{L}}_{n}^{-1}\oplus\cdots{\mathcal{L}}_{2}^{-1}\oplus{\mathcal{L}}_{1}^{-1},$
$\displaystyle\Phi=\scalebox{0.83}{$\left(\begin{array}[]{cccc|c|cccc}0&&&&&&&{\boldsymbol{\beta}}&\\\
{\boldsymbol{\gamma}}_{1}&0&&&&&&&{\boldsymbol{\beta}}\\\
&\ddots&\ddots&&&&&&\\\ &&{\boldsymbol{\gamma}}_{n-1}&0&&&&&\\\
\hline\cr&&&{\boldsymbol{\gamma}}_{n}&0&&&&\\\
\hline\cr&&&&{\boldsymbol{\gamma}}_{n}&0&&&\\\
&&&&&{\boldsymbol{\gamma}}_{n-1}&\ddots&&\\\ &&&&&&\ddots&0&\\\
&&&&&&&{\boldsymbol{\gamma}}_{1}&0\end{array}\right)$}.$

###### Proof.

When ${\boldsymbol{\beta}}\equiv 0$, (1) is trivial and (2) can be proved by a
simple adaptation of the argument below for ${\boldsymbol{\beta}}\not\equiv
0$. So let us assume ${\boldsymbol{\beta}}\not\equiv 0$ from now on and only
outline the adaption for ${\boldsymbol{\beta}}\equiv 0$ at the end.

Pick a background conformal metric ${\boldsymbol{g}}=h_{0}|{\mathsf{d}z}|^{2}$
on $\Sigma$. After being divided by $h_{0}$, each term in (A.2) is a globally
defined function on $\Sigma$. The terms on the right-hand side are

$\|{\boldsymbol{\gamma}}_{j}\|^{2}=\tfrac{h_{j+1}}{h_{0}h_{j}}|\gamma_{j}|^{2}\quad(j=1,\cdots,n-1),\quad\|{\boldsymbol{\gamma}}_{n}\|^{2}=\tfrac{1}{h_{0}h_{n}}|\gamma_{n}|^{2},\quad\|{\boldsymbol{\beta}}\|^{2}=\tfrac{h_{1}h_{2}|\beta|^{2}}{h_{0}}.$

In what follows, we extend the definition of the real logarithm and
exponential functions by putting $\log 0:=-\infty$ and $e^{-\infty}:=0$.
Consider the functions
$u_{0},\cdots,u_{n}:\Sigma\to\\{-\infty\\}\cup\mathbb{R}$ given by

$u_{0}:=\log\|{\boldsymbol{\beta}}\|^{2},\quad
u_{i}:=\log\|{\boldsymbol{\gamma}}_{i}\|^{2}\quad(i=1,\cdots,n).$

Note that $u_{0}$ (resp. $u_{i}$) is smooth away from the discrete set
$\\{u_{0}=-\infty\\}$ (resp. $\\{u_{i}=-\infty\\}$), which is just the zero
locus of ${\boldsymbol{\beta}}$ (resp. ${\boldsymbol{\gamma}}_{i}$).

Let $\Delta=\frac{4}{h_{0}}\partial_{z\bar{z}}^{2}$ be the Laplacian
associated to the metric ${\boldsymbol{g}}$. Since
$\partial_{z\bar{z}}^{2}\log|\phi|^{2}=0$ for any holomorphic function
$\phi(z)$, by (A.1) and the first two equations in (A.2), away from
$\\{u_{0}=-\infty\\}$, we have

$\displaystyle\tfrac{1}{4}\Delta u_{0}$
$\displaystyle=\tfrac{1}{h_{0}}\partial_{z\bar{z}}^{2}\big{(}\log|\beta|^{2}+\log
h_{1}+\log h_{2}-\log h_{0}\big{)}=\tfrac{1}{h_{0}}\partial_{z\bar{z}}^{2}\log
h_{1}+\tfrac{1}{h_{0}}\partial_{z\bar{z}}^{2}\log h_{2}+\tfrac{\kappa}{2}$
$\displaystyle=\big{(}-\|{\boldsymbol{\gamma}}_{1}\|^{2}+\|{\boldsymbol{\beta}}\|^{2}\big{)}+\big{(}\|{\boldsymbol{\gamma}}_{1}\|^{2}-\|{\boldsymbol{\gamma}}_{2}\|^{2}+\|\beta\|^{2}\big{)}+\tfrac{\kappa}{2}$
$\displaystyle=2\|{\boldsymbol{\beta}}\|^{2}-\|{\boldsymbol{\gamma}}_{2}\|^{2}+\tfrac{\kappa}{2}=2e^{u_{0}}-e^{u_{2}}+\tfrac{\kappa}{2}$

where $\kappa=-\frac{2}{h_{0}}\partial_{z\bar{z}}^{2}\log h_{0}$ is the
curvature of ${\boldsymbol{g}}$. By similar computations, we obtain

$\text{if $n=3$: }\begin{cases}\tfrac{1}{4}\Delta
u_{0}=2e^{u_{0}}-e^{u_{2}}+\tfrac{\kappa}{2}\leavevmode\nobreak\ ,\\\
\tfrac{1}{4}\Delta u_{1}=2e^{u_{1}}-e^{u_{2}}+\tfrac{\kappa}{2};\\\
\tfrac{1}{4}\Delta u_{2}=-e^{u_{0}}-e^{u_{1}}+e^{u_{2}}+\tfrac{\kappa}{2};\\\
\end{cases}\ \text{if $n\geq 3$: }\begin{cases}\tfrac{1}{4}\Delta
u_{0}=2e^{u_{0}}-e^{u_{2}}+\tfrac{\kappa}{2}\leavevmode\nobreak\ ,\\\
\tfrac{1}{4}\Delta
u_{1}=2e^{u_{1}}-e^{u_{2}}+\tfrac{\kappa}{2}\leavevmode\nobreak\ ,\\\
\tfrac{1}{4}\Delta
u_{2}=-e^{u_{0}}-e^{u_{1}}+2e^{u_{2}}-e^{u_{3}}+\tfrac{\kappa}{2}\leavevmode\nobreak\
,\\\ \tfrac{1}{4}\Delta
u_{k}=-e^{u_{k-1}}+2e^{u_{k}}-e^{u_{k+1}}+\tfrac{\kappa}{2}\quad(3\leq k\leq
n-1),\\\ \tfrac{1}{4}\Delta
u_{n}=-e^{u_{n-1}}+e^{u_{n}}+\tfrac{\kappa}{2}.\end{cases}$

Taking differences, we get equations for $\tfrac{1}{4}\Delta(u_{i}-u_{i+1})$
away from $\\{u_{i}=-\infty\\}\cup\\{u_{i+1}=-\infty\\}$:

(A.3) if $n=2$:
$\displaystyle\begin{cases}\tfrac{1}{4}\Delta(u_{0}-u_{1})=2e^{u_{0}}-2e^{u_{1}},\\\
\tfrac{1}{4}\Delta(u_{1}-u_{2})=e^{u_{0}}+3e^{u_{1}}-2e^{u_{2}};\end{cases}\text{if
$n=3$: }\begin{cases}\tfrac{1}{4}\Delta(u_{0}-u_{1})=2e^{u_{0}}-2e^{u_{1}},\\\