# Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and
$\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations

Nicholas Rungi  NR: Université Grenoble Alpes (Institut Fourier), Grenoble,
France<EMAIL_ADDRESS>and  Andrea Tamburelli  AT:
Department of Mathematics, University of Pisa, Italy.
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this paper we introduce complex minimal Lagrangian surfaces in the bi-
complex hyperbolic space and study their relation with representations in
$\mathrm{SL}(3,\mathbb{C})$. Our theory generalizes at the same time minimal
Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres
in $\mathbb{R}^{3}$, and Bers embeddings in the holomorphic space form
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}\setminus\Delta$. If
these surfaces are equivariant under representations in
$\mathrm{SL}(3,\mathbb{C})$, our approach generalizes the study of almost
$\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$, Hitchin
representations in $\mathrm{SL}(3,\mathbb{R})$, and quasi-Fuchsian
representations in $\mathrm{SL}(2,\mathbb{C})$. Moreover, we give a
parameterization of $\mathrm{SL}(3,\mathbb{C})$-_quasi-Fuchsian_
representations by an open set in the product of two copies of the bundle of
holomorphic cubic differentials over the Teichmüller space of $S$, from which
we deduce that this space of representations is endowed with a bi-complex
structure. In the process, we introduce bi-complex Higgs bundles as a new tool
for studying representations into semisimple complex Lie groups.

###### Contents

  1. 1 Background material
  2. 2 Bi-complex hyperbolic plane
  3. 3 Complex Lagrangian minimal surfaces in $\mathbb{CH}_{\tau}^{2}$
  4. 4 A digression on minimal Lagrangians in $\mathbb{H}^{2}_{\tau}$
  5. 5 The geometry of quasi-Fuchsian representations into $\mathrm{SL}(3,\mathbb{C})$

## Introduction

The study of immersed surfaces in space forms has always been a topic of great
interest both geometrically and analytically. In fact, much work has been
produced concerning the investigation of minimal surfaces in Riemannian
symmetric spaces and their analogue in pseudo-Riemannian homogeneous spaces.
The present paper has two main objectives: the first is to study a special
class of immersed surfaces, called _minimal complex Lagrangian_ , into the bi-
complex hyperbolic plane which has not been defined and considered before. We
will explain how this theory generalizes at the same time minimal Lagrangians
in complex hyperbolic plane ([LM13, HLL13, LM19]), hyperbolic affine spheres
in $\mathbb{R}^{3}$ ([Lof01],[Lab07]) and another special class of immersed
surfaces into a holomorphic space form studied by Bonsante and El-Emam
([BEE22]). The second objective is an application to the so-called _higher
rank Teichmüller theory_ ([Wie18]), which concerns the study of
representations of surface groups into a non-compact semisimple Lie group $G$
of arbitrary rank, aiming to generalize Teichmüller theory that considers the
case $G=\mathbb{P}\mathrm{SL}(2,\mathbb{R})$. Although classically arising
with the case of $G$ being real, recently a great amount of interest has
developed around the case of complex Lie groups
([Wie16],[DS20],[ADL21],[DS21],[AMTW23],[FPV24]). In this regard, we show how
the space of equivariant complex Lagrangian minimal surfaces parameterizes a
neighborhood of the Fuchsian locus into the space of quasi-Hitchin
representations in $\mathrm{SL}(3,\mathbb{C})$, consisting of
$B^{\mathbb{C}}$-Anosov representations, with $B^{\mathbb{C}}$ being a Borel
subgroup. Moreover, the dynamical property of Anosov representations will
allow us to describe the boundary at infinity of the surface in terms of
boundary maps. In the next paragraphs we describe the prerequisites necessary
to state and explain in more detail the main results of the article.
_Bi-complex hyperbolic plane and complex Lagrangian minimal surfaces_
Let $\mathbb{C}_{\tau}$ be the algebra of bi-complex numbers, namely the
$4$-dimensional real algebra generated by $\\{1,\tau,i,j\\}$ where
$\tau^{2}=1,i^{2}=j^{2}=-1$ and $j=\tau i=i\tau$. By definition, any number
$z\in\mathbb{C}_{\tau}$ can be written as $z=z_{1}+\tau z_{2}$, where
$z_{1},z_{2}$ are complex numbers with respect to $i$. Even though
$\mathbb{C}_{\tau}$ is a real algebra endowed with zero divisors, it is
possible to define, as in the complex case, a hyperbolic plane modeled on it
by taking the quadric in $\mathbb{C}^{3}_{\tau}$ of negative norm vectors with
respect to a suitable $\mathbb{C}$-bilinear form $\mathbf{q}$ (Section 2.2).
It will be denoted by $\mathbb{CH}^{2}_{\tau}$ and named the bi-complex
hyperbolic plane. This space is a four-dimensional complex manifold and, from
a differential geometric perspective, it is endowed with a special metric
structure. As in the usual setting, one can identify the tangent space
$T_{z}\mathbb{CH}^{2}_{\tau}\subset\mathbb{C}^{3}_{\tau}$ with the
$\mathbf{q}$-orthogonal to the $\mathbb{C}_{\tau}$-linear span of the vector
$z\in\mathbb{C}^{3}_{\tau}$, resulting in an isomorphism
$T_{z}\mathbb{CH}^{2}_{\tau}\cong\mathbb{C}^{2}_{\tau}$. Therefore,
multiplication by $i$ on $\mathbb{C}^{2}_{\tau}$ induces a complex structure
$\mathbf{I}$ on $\mathbb{CH}^{2}_{\tau}$ and multiplication by $\tau$ induces
a para-complex structure $\mathbf{P}$. In particular, the product
$\mathbf{J}:=\mathbf{I}\mathbf{P}$ defines another complex structure. In
addition, the $\mathbb{C}$-bilinear form $\mathbf{q}$ allows one to define two
other objects: a metric $\hat{g}$ such that
$\hat{g}(\mathbf{P}\cdot,\mathbf{P}\cdot)=-\hat{g}$ and a symplectic form
$\hat{\omega}$, related by $\hat{\omega}=\hat{g}(\cdot,\mathbf{P}\cdot)$,
which are both holomorphic with respect to the complex structure $\mathbf{I}$.
In the end, the space $\mathbb{CH}^{2}_{\tau}$ is endowed with a so-called
_holomorphic para-Kähler structure_
$(\mathbf{I},\mathbf{P},\hat{g},\hat{\omega})$ (Section 2.5) whose complex
para-holomorphic sectional curvature is negative constant (Section 2.7).

Throughout the paper, we are interested in immersions
$\sigma:U\to\mathbb{CH}^{2}_{\tau}$, where $U$ is a simply-connected domain of
real dimension $2$. Using recent techniques developed by Bonsante and El-Emam
([BEE22]), we can treat them as having complex codimension 2, in order to
introduce a notion of complex Lagrangian which we now briefly describe. By
extending in a $\mathbb{C}$-linear way the differential $\sigma_{\ast}$ on
$\mathbb{C}TU$ using the complex structure $\mathbf{I}$, one can consider the
classical objects of the real case: first fundamental form $h$, Levi-Civita
connection $\nabla$ and second fundamental form $\mathrm{I}\mathrm{I}$,
extended by $\mathbb{C}$-linearity. The immersions satisfying
$\mathrm{tr}_{\hat{g}}\mathrm{I}\mathrm{I}=0$ and $\sigma^{*}\hat{\omega}=0$
will be called _minimal complex Lagrangian_ , for which we are able to
describe their structure equations:

######  Theorem A.

Let $\sigma:U\to\mathbb{CH}^{2}_{\tau}$ be a complex Lagrangian minimal
surface, then the complex first fundamental form $h$ and the complex second
fundamental form $\mathrm{I}\mathrm{I}$ satisfy the following structure
equations:

$\displaystyle d^{\nabla}\mathrm{I}\mathrm{I}=0\ \ \ \ \ \ \ \ \ \ \ $
(Codazzi equation) $\displaystyle K_{h}+\|\mathrm{I}\mathrm{I}\|^{2}=-1\ \ \ \
\ $ (Gauss equation),

where $K_{h}$ denotes the complex Gaussian curvature of the metric $h$ and
$\|\mathrm{I}\mathrm{I}\|^{2}$ the norm of $\mathrm{I}\mathrm{I}$.

Now let $S$ be a closed surface of genus $g\geq 2$ so that, in the above
theorem, $U$ is the universal cover of $S$ and the immersion $\sigma$ is
equivariant. We will prove that a pair $(h,\mathrm{I}\mathrm{I})$ as in the
Theorem A is equivalent to a $\mathbb{C}$-complex structure $\mathcal{J}$,
i.e. an automorphism of $\mathbb{C}TS$ such that $\mathcal{J}^{2}=-1$, and a
Codazzi tensor $C$ compatible with $\mathcal{J}$. If we denote by
$\mathcal{C}^{\mathbb{C}}(S)$ the space of $\mathbb{C}$-complex structures and
by $\mathcal{Q}^{\mathbb{C}}(S)$ the bundle over $\mathcal{C}^{\mathbb{C}}(S)$
whose fiber in a point $\mathcal{J}$ is given by the space of Codazzi tensors
compatible with $\mathcal{J}$, then its total space can be identified with
$\mathcal{Q}_{3}(S)\times\mathcal{Q}_{3}(S)$ the product of two copies of the
bundle of holomorphic cubic differentials over the space of complex structures
on $S$. In other words, the data $(\mathcal{J},C)$ is equivalent to two
complex structures $(J_{1},J_{2})$ inducing the same orientation and two
$(J_{1},J_{2})$-holomorphic cubic differentials $q_{1},q_{2}$ such that
$C=q_{1}+\bar{q}_{2}$. We will call the _Hitchin locus_ the subset of
$\mathcal{Q}^{\mathbb{C}}(S)$ given by the preimage of the diagonal in
$\mathcal{Q}_{3}(S)\times\mathcal{Q}_{3}(S)$ via the aforementioned
identification.

######  Theorem B.

There is a neighborhood $\mathcal{U}$ of the Hitchin locus in
$\mathcal{Q}_{3}^{\mathbb{C}}(S)$ such that all pairs
$(\mathcal{J},C)\in\mathcal{U}$ can be realized as embedding data of an
equivariant complex Lagrangian minimal surface in $\mathbb{CH}^{2}_{\tau}$.

In contexts similar to ours, the theory of Higgs bundles and the existence and
uniqueness of the solution of Hitchin equations have often been used to study
the integrability of the Gauss-Codazzi equations. In our setting of complex
metrics, we introduce the new notion of $\mathrm{SL}(3,\mathbb{C})$-bi-complex
Higgs bundles and explain how this is related to a complex Lagrangian minimal
immersion $\sigma:\widetilde{S}\to\mathbb{CH}^{2}_{\tau}$ in such a way that
the analogues of Hitchin equations for bi-complex Higgs bundles are equivalent
to the structure equations of $\sigma$. However, since we no longer have
existence and uniqueness of solutions to Hitchin equations in this context, an
analytical study of the involved differential operators was necessary. On the
other hand, we will use $\mathrm{SL}(3,\mathbb{C})$-bi-complex Higgs bundles
in the proof of another result that will be explained in the next paragraphs.
This new notion of bi-complex Higgs bundles can be easily generalized to other
complex semisimple Lie groups $G_{\mathbb{C}}$ and we believe could be a
useful tool for the study of representations in $G_{\mathbb{C}}$. Finally, the
open set for which we have existence and uniqueness of complex Lagrangian
minimal surfaces in $\mathbb{CH}^{2}_{\tau}$ is actually larger than the one
given by Theorem B, because a similar approach can be applied to the preimage
of a different locus inside $\mathcal{Q}_{3}^{\mathbb{C}}(S)$ obtained by
picking $J_{1}=J_{2}$ and $q_{2}=-\bar{q}_{1}$.
_A unified point of view_
One of the reasons for starting the study of complex Lagrangian minimal
immersions in $\mathbb{CH}^{2}_{\tau}$ was to find a unified viewpoint with
other seemingly unrelated situations which, as we will explain, are simply a
particular case of a more general phenomenon. From the point of view of Lie
groups, the automorphism group that preserves the holomorphic para-Kähler
structure on $\mathbb{CH}^{2}_{\tau}$ is isomorphic to
$\mathrm{SL}(3,\mathbb{C})$, which admits two non-compact real forms:
$\mathrm{SL}(3,\mathbb{R})$ and $\mathrm{SU}(2,1)$. It is known in the
literature that $\mathrm{SU}(2,1)$ can be seen as the group of biholomorphisms
of the complex hyperbolic space $\mathbb{CH}^{2}$. Actually, there is another
hyperbolic space modeled on the algebra of para-complex numbers and denoted by
$\mathbb{H}^{2}_{\tau}$, which admits a para-Kähler structure with the group
of para-biholomorphisms given by $\mathrm{SL}(3,\mathbb{R})$. Using the
complex structure $\mathbf{J}$ and the para-complex structure $\mathbf{P}$ on
$\mathbb{CH}^{2}_{\tau}$, it can be seen that $\mathbb{CH}^{2}$ is realized as
the submanifold $\mathbb{CH}^{2}_{\tau}\cap(\mathbb{R}^{3}\oplus
j\mathbb{R}^{3})$ and $\mathbb{H}^{2}_{\tau}$ is realized as
$\mathbb{CH}^{2}_{\tau}\cap(\mathbb{R}^{3}\oplus\tau\mathbb{R}^{3})$.
This allows us to observe that if we have a complex Lagrangian minimal
immersion $\sigma:\widetilde{S}\to\mathbb{CH}^{2}_{\tau}$, equivariant with
respect to a representation $\rho:\pi_{1}(S)\to\mathrm{SL}(3,\mathbb{C})$,
whose image is contained in $\mathbb{CH}^{2}$, then the surface is actually
equivariant with respect to a representation $\mathrm{SU}(2,1)$. Moreover, if
$\rho$ is sufficiently close to the Fuchsian locus (_almost
$\mathbb{R}$-Fuchsian_ in $\mathrm{SU}(2,1)$), then it preserves the unique
minimal Lagrangian immersion studied by Loftin and McIntosh ([LM13]). On the
other hand, if the image is contained in $\mathbb{H}^{2}_{\tau}$, then the
representation takes values in $\mathrm{SL}(3,\mathbb{R})$ and preserves a
minimal Lagrangian immersion in the para-complex hyperbolic space. In this
case, it is equivalent to having a hyperbolic affine sphere in
$\mathbb{R}^{3}$ equivariant with respect to a Hitchin representation. In
particular, we obtain a new variational description of hyperbolic affine
spheres as stable critical points of an area functional. A similar result was
obtained by Hildebrand ([Hil11]), by studying minimal Lagrangian immersions in
a space called the cross-ratio manifold, which turns out to be isometric to
$\mathbb{H}^{2}_{\tau}$. As a final observation, the submanifold given by
$\mathbb{CH}^{2}_{\tau}\cap(\mathbb{R}^{3}\oplus i\mathbb{R}^{3})$ can be
identified with the holomorphic space form
$\mathbb{X}:=\\{\underline{z}\in\mathbb{C}^{3}\ |\
z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=-1\\}$, which has been shown to be isometric to
the complex manifold
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}\setminus\Delta$, being
$\Delta$ the diagonal ([BEE22],[EES22]). If the image of the complex
Lagrangian minimal immersion is contained in $\mathbb{X}$, then we recover the
_Bers embeddings_ studied by Bonsante and El-Emam ([BEE22]) whose holonomy
representation takes value into $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$. The
picture to keep in mind can be summarized in the following two diagrams:

${{\mathrm{SU}(2,1)}}$${\mathbb{CH}^{2}}$${{\mathbb{P}\mathrm{SL}(2,\mathbb{C})}}$${{\mathrm{S}\mathrm{L}(3,\mathbb{C})}}$${\mathbb{X}}$${\mathbb{CH}^{2}_{\tau}}$${{\mathrm{SL}(3,\mathbb{R})}}$${\mathbb{H}^{2}_{\tau}}$$\scriptstyle{\text{almost}\mathbb{\
}\mathbb{R}\text{-Fuchsian}}$minimal LagrangianirreducibleBers
metricHitchin$\scriptstyle{\begin{subarray}{c}\text{hyperbolic affine}\\\
\text{sphere}\end{subarray}}$

In brief, our equivariant complex Lagrangian minimal surfaces in
$\mathbb{CH}^{2}_{\tau}$ generalize equivariant minimal Lagrangian surfaces in
$\mathbb{CH}^{2}$, hyperbolic affine spheres in $\mathbb{R}^{3}$ and Bers
embeddings in $\mathbb{X}$. This allows for a unified treatment of almost
$\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$, Hitchin
representations in $\mathrm{SL}(3,\mathbb{R})$ and quasi-Fuchsian
representations in $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$, as well as their
small deformations.

_Quasi-Hitchin representations and parameterization close to the Fuchsian
locus_
As previously mentioned, the second motivation for studying complex Lagrangian
minimal immersions into $\mathbb{CH}^{2}_{\tau}$ comes from higher Teichmüller
theory, more specifically from the study of quasi-Hitchin representations in
complex Lie groups, which we now recall. Let
$\eta:\pi_{1}(S)\to\mathbb{P}\mathrm{SL}(2,\mathbb{R})$ be an injective
representation with discrete image. Using the irreducible embedding
$\iota:\mathbb{P}\mathrm{SL}(2,\mathbb{R})\hookrightarrow\mathrm{SL}(3,\mathbb{R})$
we can define a representation $\iota\circ\eta$ into
$\mathrm{SL}(3,\mathbb{R})$. The set of all possible deformations of
$\iota\circ\eta$ form a connected component in the corresponding character
variety, which is called _Hitchin component_. In fact, it was initially
discovered by Hitchin using Higgs bundles techniques ([Hit92]), proving that
it is an open ball of dimension $16g-16$ and contains a copy of Teichmüller
space coming from the representations $\iota\circ\eta$ as above. A remarkable
property of Hitchin representations is that of being $B$-Anosov, where
$B<\mathrm{SL}(3,\mathbb{R})$ is a Borel subgroup ([Lab06],[FG06]). Roughly
speaking, a representation $\rho:\pi_{1}(S)\to\mathrm{SL}(3,\mathbb{R})$ is
$B$-Anosov if there exists a unique $\rho$-equivariant continuous map $\xi$
from $\partial_{\infty}\pi_{1}(S)$ to the space of full flags in
$\mathbb{R}^{3}$, which is endowed with a transversality property: for any
$x\neq y$ in $\partial_{\infty}\pi_{1}(S)$ one has
$\xi_{x}\oplus\xi_{y}=\mathbb{R}^{3}$; and a _dynamic preserving_ property
satisfied whenever the representation is irreducible. A similar definition can
be given for representations into the complex Lie group
$\mathrm{SL}(3,\mathbb{C})$, which we will call $B^{\mathbb{C}}$-Anosov, being
$B^{\mathbb{C}}$ the complexified Borel subgroup. Deformations, inside
$\mathrm{SL}(3,\mathbb{C})$, of Hitchin representations into
$\mathrm{SL}(3,\mathbb{R})$ that remain $B^{\mathbb{C}}$-Anosov are called
_quasi-Hitchin_ and form a space denoted with
$\mathrm{QHit}_{3}^{\mathbb{C}}(S)$. Even though it is probably already known
in the literature, in Section 5.2 we include a proof that irreducible
representations in $\mathrm{QHit}_{3}^{\mathbb{C}}(S)$ are smooth points, thus
obtaining, that Hitchin representations in $\mathrm{SL}(3,\mathbb{R})$, almost
$\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$ and quasi-Fuchsian
representations in $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ post-composed with
the irreducible embedding
$\iota^{\mathbb{C}}:\mathbb{P}\mathrm{SL}(2,\mathbb{C})\hookrightarrow\mathbb{P}\mathrm{SL}(3,\mathbb{C})$,
which are all examples of quasi-Hitchin representations, are all contained in
the smooth locus of the $\mathrm{SL}(3,\mathbb{C})$-representation variety.
The third main result of the paper, and perhaps the most interesting, concerns
a parameterization of the space of $\mathrm{SL}(3,\mathbb{C})$-_quasi-
Fuchsian_ representations, obtained by small deformations of representations
of the form
$\pi_{1}(S)\xrightarrow{\eta}\mathbb{P}\mathrm{SL}(2,\mathbb{R})\xhookrightarrow{\iota}\mathrm{SL}(3,\mathbb{R})\hookrightarrow\mathrm{SL}(3,\mathbb{C})$,
where $\eta$ is discrete and faithful, which has been object of recent study
([DS20]). We define a _holonomy map_ $\widetilde{\mathrm{hol}}$, which
associates to each point
$(\mathcal{J},C)\in\mathcal{U}\subset\mathcal{Q}_{3}^{\mathbb{C}}(S)$ the
holonomy of the unique equivariant complex Lagrangian minimal immersion
determined by the data $(h,\mathrm{I}\mathrm{I})$ which is equivalent to
$(\mathcal{J},C)$. In order to promote this map to a local diffeomorphism, we
need to restrict the domain. We consider the subset
$\mathcal{R}\subset\mathcal{Q}^{\mathbb{C}}_{3}(S)$ of pairs $(\mathcal{J},C)$
such that, if $\mathcal{J}=(J_{1},J_{2})$ and $g_{1}$ and $g_{2}$ denote the
unique hyperbolic metric in the conformal class of $J_{1}$ and $J_{2}$, then
the identity $(S,g_{1})\rightarrow(S,g_{2})$ is harmonic. We call such pairs
$(\mathcal{J},C)$ _renormalized pairs_. Note that the space of renormalized
pairs is still infinite dimensional and the group $\mathrm{Diff}_{0}(S)$ of
diffeomorphisms of $S$ isotopic to the identity acts diagonally on
$\mathcal{R}$ by pull-back with quotient
$Q^{\mathbb{C}}_{3}(S):=Q_{3}(S)\times Q_{3}(S)$ isomorphic to two copies of
the bundle of holomorphic cubic differentials over the Teichmüller space of
$S$. We call _Fuchsian locus_ the pre-image in $Q^{\mathbb{C}}_{3}(S)$ of the
diagonal inside $\mathrm{Teich}(S)\times\mathrm{Teich}(S)\subset
Q_{3}(S)\times Q_{3}(S)$ under the isomorphism described previously. The map
$\widetilde{\mathrm{hol}}$ clearly descends to the quotient and defines a map

$\mathrm{hol}:U:=(\mathcal{U}\cap\mathcal{R})/\mathrm{Diff}_{0}(S)\rightarrow\mathrm{Hom}(\pi_{1}(S),\mathrm{SL}(3,\mathbb{C}))/\mathrm{SL}(3,\mathbb{C})\
.$

######  Theorem C.

$\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations that are
sufficiently close to Fuchsian are parameterized by a neighborhood of the
Fuchsian locus inside the product $Q_{3}^{\mathbb{C}}(S)=Q_{3}(S)\times
Q_{3}(S)$.

The proof of the theorem uses Goldman symplectic form $\omega$ ([Gol84]). More
precisely, we show that the pull-back form $\text{hol}^{*}\omega$, under the
map $\mathrm{hol}$ that associates to the embedding data of an equivariant
complex Lagrangian minimal surface its holonomy representation, is non-
degenerate at all points $(\mathcal{J},C)$ such that $\mathcal{J}=(J,J)$ and
$C=0$, by using the explicit description of the bi-complex Higgs bundle
associated with a complex Lagrangian minimal immersion, and then conclude by
the inverse function theorem.

_Boundary maps of Anosov representations_
The last result we obtain concerns the Anosov property of
$\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations and their
correlation with the complex Lagrangian minimal surface given by Theorem C.
More specifically, one can define an isometric model of the bi-complex
hyperbolic plane as the subspace of
$\mathbb{C}\mathbb{P}^{2}\times(\mathbb{C}\mathbb{P}^{2})^{*}$ formed by all
pairs $(v,\varphi)$ such that $\varphi(v)\neq 0$, where $\varphi$ is
considered as a linear functional. Thanks to this description, an analogous
notion of boundary at infinity for $\mathbb{CH}^{2}_{\tau}$ can be defined,
and it is identified with the space of full flags in $\mathbb{C}^{3}$, that
is, the homogeneous space $\mathrm{SL}(3,\mathbb{C})/B^{\mathbb{C}}$.
Therefore, if $\rho$ is an $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian
representation and
$\xi:\partial_{\infty}\pi_{1}(S)\to\mathrm{SL}(3,\mathbb{C})/B^{\mathbb{C}}$
is its boundary map, we obtain

######  Theorem D.

The $\rho$-equivariant complex Lagrangian minimal surface
$\sigma:\widetilde{S}\to\mathbb{CH}^{2}_{\tau}$ bounds the image of the
boundary map $\xi$.

We believe that this relation about the boundary map $\xi$ and the boundary of
the equivariant minimal Lagrangian surface is evidence that this is the
correct setting to study quasi-Hitchin representations in
$\mathrm{SL}(3,\mathbb{C})$.

_Comparison with other work on representations into complex Lie groups_
In recent years, the interest in surface group representations into a complex
semisimple Lie group $G^{\mathbb{C}}$ of higher rank has steadily increased.
Dumas and Sanders ([DS20], [DS21]) carefully studied the cohomological and
analytical properties of the domain of discontinuity $\Omega_{\rho}$ of a
$G^{\mathbb{C}}$-quasi-Fuchsian representation $\rho$ acting on flag varieties
and conjectured that the quotient $\Omega_{\rho}/\rho(\pi_{1}(S))$ is a fiber
bundle over $S$, which they verified for
$G^{\mathbb{C}}=\mathrm{SL}(3,\mathbb{C})$. It is conceivable that the use of
our complex Lagrangian minimal surfaces in $\mathbb{CH}^{2}_{\tau}$ combined
with the techniques developed by Collier-Tholozan-Toulisse ([CTT19]) could
simplify the proof of their result considerably. More recently, Alessandrini-
Maloni-Tholozan-Wienhard ([AMTW23]) proved Dumas and Sanders’ conjecture in
full generality and described, in particular, the topology of
$\Omega_{\rho}/\rho(\pi_{1}(S))$ for quasi-Hitchin representations in
$\mathrm{Sp}(4,\mathbb{C})$ acting on the space of complex Lagrangians in
$\mathbb{C}^{4}$. However, in all the aforementioned works, a description of
the geometry and topology of the space of $G^{\mathbb{C}}$-quasi-Fuchsian
representations is not available. Our work fills this gap, discovering, in
addition, a bi-complex structure in the space of
$\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations close to the
Fuchsian locus, which, in our opinion, is very surprising, since neither from
the algebraic nor from the Higgs bundle perspective the existence of a para-
complex structure is expected. Finally, in the process of writing this paper,
we have been informed that El-Emam and Sagman have also been studying the
geometry associated to representations of surface groups into
$\mathrm{SL}(3,\mathbb{C})$ ([EES24]). They define a notion of complex affine
spheres in $\mathbb{C}^{3}$, which naturally correspond to some harmonic maps
in $\mathrm{SL}(3,\mathbb{C})/\mathrm{SO}(3,\mathbb{C})$ and to new objects,
called bi-Higgs bundles, which extend the usual Higgs bundles. These objects
allow to define a holomorphic analog of Bers theorem for
$\mathrm{SL}(3,\mathbb{C})$, namely a holomorphic map from a large open subset
of $\mathrm{Hit}_{3}(S)\times\overline{\mathrm{Hit}_{3}(S)}$ – containing
$\mathrm{Hit}_{3}(S)\times\mathrm{Teich}(\bar{S})$ and
$\mathrm{Teich}(S)\times\mathrm{Hit}_{3}(S)$ – to the character variety of
$\mathrm{SL}(3,\mathbb{C})$ extending the inclusion of the diagonal. It would
be interesting to study the connection between ours and their strategy in the
future. At the time of writing, the two works are completely independent.

### Outline of the paper

For the sake of clarity, we will carefully explain where to find the various
notions and results that the reader might be interested in.
Section 1 is dedicated to the review of the theory of complex and holomorphic
metrics on manifolds, and then to the introduction of differential geometric
structures with special curvature properties. Section 2 is entirely dedicated
to the introduction and study of the bi-complex hyperbolic plane. In this
regard, in Sections 2.2 and 2.3, we provide two different models of the space;
in Section 2.4, we define its boundary at infinity and give its homeomorphism
type; in Section 2.5, we show how its holomorphic para-Kähler structure can be
obtained through an adaptation of the Marsden-Weinstein symplectic reduction;
in Section 2.6, we explain how to obtain the submanifolds of
$\mathbb{CH}^{2}_{\tau}$ mentioned in the introduction, and how the
restriction of the holomorphic para-Kähler structure behaves on these
submanifolds; in Section 2.7, we study and show that the complex para-
holomorphic curvature of $\mathbb{CH}^{2}_{\tau}$ is negative constant.
Section 3 includes the study of complex Lagrangian minimal immersions in
$\mathbb{CH}^{2}_{\tau}$, and Theorem A is proved in Section 3.1. The most
analytical part is Section 3.2, where the Gauss equation and the Laplacian of
complex metrics are computed in local coordinates. As for Section 3.3, we
introduce the so-called bi-complex Higgs bundles and explain how to construct
such object from a complex Lagrangian minimal surface. In Section 4 we focus
on immersions whose image is contained in the para-complex hyperbolic space
$\mathbb{H}^{2}_{\tau}$ by showing the equivalence with hyperbolic affine
spheres in $\mathbb{R}^{3}$, and obtaining as a new result that they are
stable critical points of a certain area functional. Finally, making use of
all the theory developed in the previous sections, Theorem B is proved in
Section 5.3, Theorem C in Section 5.4 and Theorem D in Section 5.5.

### Acknowledgement

The authors are grateful to Andrea Seppi for useful conversations, for
pointing out the references [Hil11, Tre19] and for comments on an earlier
draft, and to Jérémy Toulisse for a helpful remark on boundary maps. A.T.
acknowledges support from the National Science Foundation with grant
DMS-2005551. N.R. is funded by the European Union (ERC, GENERATE, 101124349).
Views and opinions expressed are however those of the author(s) only and do
not necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them.

##  1\. Background material

In this section we recall some general notions about complex metrics on
manifolds and para-complex structures. Most of the material is classical in
the setting of (pseudo)-Riemannian manifolds (see for instance ([KN96]) and we
adapt it here to the context of complex metrics. Part of it can already be
found in the literature ([DGMY09],[BW11],[LS17]).

###  1.1. Complex metrics on complex manifolds

Let $(M,\mathbf{I})$ be a complex manifold of complex dimension $n$. We think
of $\mathbf{I}$ as an integrable almost complex structure on $M$, i.e. an
automorphism of $TM$ such that $\mathbf{I}^{2}=-\mathrm{Id}$ for which the
Nijenhuis tensor

$N_{\mathbf{I}}(X,Y):=[X,Y]+\mathbf{I}([\mathbf{I}X,Y]+[X,\mathbf{I}Y])-[\mathbf{I}X,\mathbf{I}Y]$

vanishes for all $X,Y\in\Gamma(TM)$. It is well-known that this condition is
equivalent to the existence of local coordinates
$(x_{1},y_{1},\dots,x_{n},y_{n})$ in a neighborhood of every point such that
for all $i=1,\dots,n$

$\mathbf{I}\left(\frac{\partial}{\partial
x_{i}}\right)=\frac{\partial}{\partial y_{i}}\ \ \ \text{and}\ \ \
\mathbf{I}\left(\frac{\partial}{\partial
y_{i}}\right)=-\frac{\partial}{\partial x_{i}}\ .$

A _complex metric_ on $(M,\mathbf{I})$ is the datum of a non-degenerate
complex-valued symmetric bi-linear form $g_{p}:T_{p}M\times
T_{p}M\rightarrow\mathbb{C}$ that varies smoothly in $p$ such that
$g_{p}(X,\mathbf{I}Y)=g_{p}(\mathbf{I}X,Y)=ig_{p}(X,Y)$ for all $X,Y\in
T_{p}M$. We say that a complex metric is _holomorphic_ if for all holomorphic
vector fields $Z,Z^{\prime}$ on $M$ the function $g(Z,Z^{\prime})$ is
holomorphic.

A complex metric induces a canonical connection $D$ on $TM$, which has the
same properties as the standard Levi-Civita connection in a (pseudo)-Kähler
manifold:

######  Theorem 1.1 ([LB83]).

Given a complex metric $g$ on a complex manifold $(M,\mathbf{I})$, there is a
unique connection $D$ on $TM$, called Levi-Civita connection, such that for
all $X,Y,Z\in\Gamma(TM)$ the following conditions hold:

  * •

$X\cdot g(Y,Z)=g(D_{X}Y,Z)+g(Y,D_{X}Z)$ (metric compatibility)

  * •

$[X,Y]=D_{X}Y-D_{Y}X$ (torsion free).

Moreover, $D\mathbf{I}=0$.

Moreover, if $g$ is holomorphic the real and imaginary part of $g$ induce two
pseudo-Riemannian metrics on $M$ with neutral signature $(n,n)$ and the
connection $D$ is the common Levi-Civita connection of both $\mathrm{Re}(g)$
and $\mathrm{Im}(g)$.

The notion of Levi-Civita connection for the complex metric $g$ leads to the
standard definition of the Riemann curvature tensor

$R^{D}(X,Y)Z:=D_{X}D_{Y}Z-D_{Y}D_{X}Z-D_{[X,Y]}Z\ \ \ \ \ X,Y,Z\in\Gamma(TM)\
.$

We will often use the $(4,0)$ Riemann tensor

$R^{D}(X,Y,Z,W):=g(R^{D}(X,Y)Z,W)\ .$

We use the same letter to denote both tensors, as it will be clear from
context which one we are using. The Riemann tensor satisfies the usual
symmetries:

$\displaystyle R^{D}(X,Y,Z,W)=-R^{D}(Y,X,Z,W)$ $\displaystyle
R^{D}(X,Y,Z,W)=R^{D}(Z,W,X,Y)$ $\displaystyle
R^{D}(X,Y,Z,W)+R^{D}(Y,Z,X,W)+R^{D}(Z,X,Y,W)=0$

for all $X,Y,Z,W\in\Gamma(TM)$. In particular, since $R^{D}$ is clearly
$\mathbb{C}$-linear in the last two entries, i.e.
$R^{D}(X,Y,Z,\mathbf{I}W)=R^{D}(X,Y,\mathbf{I}Z,W)=iR^{D}(X,Y,Z,W)$, the
Riemann tensor is actually $\mathbb{C}$-linear in all entries. As a
consequence, the sectional curvature of a complex metric $g$, defined as usual
as

$\mathrm{Sec}_{g}(X,Y):=-\frac{R^{D}(X,Y,X,Y)}{g(X,X)g(Y,Y)-g(X,Y)^{2}}\ ,$

only depends on the complex plane spanned by $X$ and $Y$. Thus, we will refer
to $\mathrm{Sec}_{g}$ as _complex sectional curvature_ of $g$. Clearly, the
complex sectional curvature of a complex plane spanned by $X$ and $Y$ is well
defined only when the restriction of $g$ to this plane is non-degenerate.

######  Proposition 1.2.

Let $g$ be a complex metric on $(M,\mathbf{I})$ with constant complex
sectional curvature $k$. Let $D$ be the Levi-Civita connection of $g$. Then
its Riemann tensor is

$R^{D}(X,Y,Z,W)=k(g(X,Z)g(Y,W)-g(Y,Z)g(X,W))$

for all $X,Y,Z,W\in\Gamma(TM)$.

###### Proof.

The standard proof in the context of (pseudo)-Riemannian manifolds can be
adapted word-by-word: the tensor $T(X,Y,Z,W)$ defined as the right-hand side
of the formula in the statement satisfies the same symmetries as the Riemann
tensor and $T(X,Y,X,Y)=k(g(X,X)g(Y,Y)-g(X,Y)^{2})=R^{D}(X,Y,X,Y)$ for all
$X,Y\in\Gamma(TM)$, since $g$ has constant complex sectional curvature $k$.
Then one shows, as in the classical case, that two $(4,0)$ tensors that have
the same symmetries as the Riemann tensor are uniquely determined by the
values they take on quadruples of the form $(X,Y,X,Y)$. ∎

###  1.2. Positive complex metrics on surfaces

Complex metrics can be defined on a real manifold $N$ as well: consider the
complexified tangent bundle $\mathbb{C}TN$ with the standard complex structure
given by the multiplication by $i$ and then define a non-degenerate symmetric
$\mathbb{C}$-bilinear form on each $\mathbb{C}T_{p}N$ that changes smoothly
with $p$. Theorem 1.1 still holds and provides now a Levi-Civita
$\mathbb{C}$-bilinear connection $D$. The notions of curvature are then
defined exactly in the same way. In particular, when $N$ is a surface, the
curvature tensor is uniquely determined by one complex sectional curvature
(since there is only one complex plane in $\mathbb{C}TN$), which we call
_complex Gaussian curvature_.

In the case of surfaces, the theory of complex metrics has been developed
further in [BEE22] and a notion of _positive_ complex metrics has been
introduced that has similarities with the classical conformal geometry of
surfaces.

Let $S$ be an oriented surface. The natural inclusion
$TS\hookrightarrow\mathbb{C}TS$ factors to a bundle inclusion
$\mathbb{P}_{\mathbb{R}}(TS)\hookrightarrow\mathbb{P}_{\mathbb{C}}(\mathbb{C}TS)$
of bundles over $S$. At every $p\in S$, each fiber
$\mathbb{P}_{\mathbb{R}}(T_{p}S)$ is mapped homeomorphically into a circle in
$\mathbb{P}_{\mathbb{C}}(\mathbb{C}T_{p}S)\cong S^{2}$ whose complement is the
disjoint union of two open discs.

######  Definition 1.3.

A _$\mathbb{C}$ -complex structure_ on $\mathbb{C}TS$ is a $\mathbb{C}$-linear
bundle endomorphism $\mathcal{J}:\mathbb{C}TS\rightarrow\mathbb{C}TS$ such
that

  * •

$\mathcal{J}^{2}=-\mathrm{Id}$;

  * •

for all $p\in S$, the eigenspaces $V_{i}(\mathcal{J})$ and
$V_{-i}(\mathcal{J})$ of $\mathcal{J}$ relative to the eigenvalues $\pm i$
have trivial intersection with $T_{p}S$ and the points
$\mathbb{P}_{\mathbb{C}}(V_{i}(\mathcal{J}))$ and
$\mathbb{P}_{\mathbb{C}}(V_{-i}(\mathcal{J}))$ lie in different connected
components of
$\mathbb{P}_{\mathbb{C}}(\mathbb{C}T_{p}S)\setminus\mathbb{P}_{\mathbb{R}}(T_{p}S)$.

Clearly a complex structure on $S$ extends to a $\mathbb{C}$-complex
structure. However, not all $\mathbb{C}$-complex structures can be obtained in
this way since they may not restrict to an endomorphism of $TS$, in other
words $\mathcal{J}(\overline{X})\neq\overline{\mathcal{J}(X)}$ in general.
Nevertheless, a $\mathbb{C}$-complex structure induces two complex structures
$J_{1}$ and $J_{2}$ on $S$ defined by the conditions

$V_{i}(\mathcal{J})=V_{i}(J_{2})=\overline{V_{-i}(J_{2})}\ \ \ \text{and}\ \ \
V_{-i}(\mathcal{J})=V_{-i}(J_{1})=\overline{V_{i}(J_{1})}$

which uniquely characterize $J_{1}$ and $J_{2}$. It can be checked ([BEE22])
that $J_{1}$ and $J_{2}$ induce the same orientation on $S$. Therefore, a
$\mathbb{C}$-complex structure determines an orientation on $S$ and the set
$\mathcal{C}^{\mathbb{C}}(S)$ of $\mathbb{C}$-complex structures on $S$
compatible with the given orientation is in bijection with
$\mathcal{C}(S)\times\mathcal{C}(S)$, i.e. with pairs of complex structures on
$S$, via the map that associates to $\mathcal{J}$ the pair $(J_{1},J_{2})$.
$\mathbb{C}$-complex structures that are $\mathbb{C}$-linear extensions of
complex structures on $S$ are mapped to the diagonal of
$\mathcal{C}(S)\times\mathcal{C}(S)$ under the above correspondence.

Like Riemannian metrics on $S$ determine a complex structure, a positive
complex metric $g$ pins down a $\mathbb{C}$-complex structure.

######  Definition 1.4.

Let $g$ be a complex metric on $S$ and denote by $\mathrm{Iso}(g_{p})$ the set
of isotropic vectors of $g$ on $\mathbb{C}T_{p}S$. We say that $g$ is
_positive_ if the two points $\mathbb{P}_{\mathbb{C}}(\mathrm{Iso}(g_{p}))$
lie in different connected components of
$\mathbb{P}_{\mathbb{C}}(\mathbb{C}T_{p}S)\setminus\mathbb{P}_{\mathbb{R}}(T_{p}S)$
for all $p\in S$.

The connection between positive complex metrics and $\mathbb{C}$-complex
structures on $S$ is explained in the following result:

######  Proposition 1.5 ([BEE22]).

Let $g$ be a positive complex metric on an oriented surface $S$. Then there
exists a unique $\mathbb{C}$-complex structure $\mathcal{J}$ such that

  * •

it induces the fixed orientation on $S$;

  * •

it is compatible with $g$ in the sense that $g(\mathcal{J},\mathcal{J})=g$.

It follows that if $\mathcal{J}$ is the $\mathbb{C}$-complex structure
provided by Proposition 1.5 and corresponds to the pair of complex structures
$(J_{1},J_{2})$, then the positive complex metric $g$ can be written locally
as

$g=g\left(\frac{\partial}{\partial
w_{1}},\frac{\partial}{\partial\overline{w}_{2}}\right)dw_{1}d\overline{w}_{2}\
,$

where $w_{1}$ and $w_{2}$ are local holomorphic coordinates for $J_{1}$ and
$J_{2}$.

A positive complex metric $g$ has also a canonical (up to a sign) area form
$d\mathrm{A}_{g}=g(\mathcal{J}\cdot,\cdot)$ characterized by the property that
it evaluates to $\pm 1$ on all $g$-orthonormal frames $\\{X,Y\\}$ of
$\mathbb{C}TS$. In local coordinates, if $g=\rho dw_{1}d\overline{w}_{2}$,
then

$d\mathrm{A}_{g}=\frac{i}{2}\rho dw_{1}\wedge d\overline{w}_{2}\ .$ (1)

###  1.3. Complex para-Kähler and complex para-Hermitian structures

A _para-complex structure_ on a complex $n$-manifold $(M,\mathbf{I})$ is a
bundle endomorphism $\mathbf{P}:TM\rightarrow TM$ such that

  * •

$\mathbf{P}^{2}=\mathrm{Id}$;

  * •

the eigendistributions $\mathcal{D}_{\pm}$ relative to the $\pm 1$ eigenvalues
of $\mathbf{P}$ have the same dimension and are integrable.

This last condition is equivalent to the vanishing of its Nijenhuis tensor

$N_{\mathbf{P}}(X,Y):=-[X,Y]+\mathbf{P}([\mathbf{P}X,Y]+[X,\mathbf{P}Y])-[\mathbf{P}X,\mathbf{P}Y]$

or, equivalently, to the existence of a local product structure, in the sense
that around any point $p\in M$ we can find local coordinates
$(x_{1},\dots,x_{n},y_{1},\dots,y_{n})$ such that for all $i=1,\dots,n$

$\mathbf{P}\left(\frac{\partial}{\partial
x_{i}}\right)=\frac{\partial}{\partial x_{i}}\ \ \ \text{and}\ \ \
\mathbf{P}\left(\frac{\partial}{\partial
y_{i}}\right)=-\frac{\partial}{\partial y_{i}}\ .$

Let us now endow $(M,\mathbf{I})$ with a complex metric $g$ with Levi-Civita
connection $D$. We say that the para-complex structure $\mathbf{P}$ is
compatible with $g$ if $g(\mathbf{P}\cdot,\mathbf{P}\cdot)=-g$. In this case,
the form $\omega_{\mathbf{P}}:=g(\cdot,\mathbf{P}\cdot)$ is skew-symmetric and
we can extend $g$ to a complex para-Hermitian metric by defining the tensor
$q:=g+\tau\omega_{\mathbf{P}}$, where $\tau$ is a non-real element such that
$\tau^{2}=1$.

In general, a _complex para-Hermitian metric_ on $(M,\mathbf{I},\mathbf{P})$
is the datum of a bilinear form $q_{p}$ on $T_{p}M$ with values in the algebra
of bi-complex numbers $\mathbb{C}_{\tau}:=\mathbb{C}[\tau]$ (see Section 2.1
for more information about this algebra) that varies smoothly with $p$ and
satisfies the following properties:

  * •

$q(\mathbf{I}X,Y)=q(X,\mathbf{I}Y)=iq(X,Y)$

  * •

$q(\mathbf{P}X,Y)=-q(X,\mathbf{P}Y)=\tau q(X,Y)$

  * •

$q(X,Y)=\overline{q(Y,X)}^{\tau}$

for all $X,Y\in\Gamma(TM)$. It is straightforward to check that, if $q$ is a
complex para-Hermitian metric on $(M,\mathbf{I},\mathbf{P})$, then
$g=\frac{1}{2}(q+\bar{q}^{\tau})$ is a complex metric compatible with
$\mathbf{P}$ and $\frac{1}{2}(q-\bar{q}^{\tau})$ is a skew-symmetric
$\mathbb{C}$-bilinear form that coincides with $\omega_{\mathbf{P}}$.
Therefore, all para-Hermitian metrics can actually be obtained from the above
construction and we will refer to the quadruple
$(\mathbf{I},\mathbf{P},g,\omega_{\mathbf{P}})$ as a _complex para-Hermitian
structure_ on $M$. Moreover, we say that the complex para-Hermitian structure
is _holomorphic_ if the complex metric $g$ and the symplectic form
$\omega_{\mathbf{P}}$ are holomorphic.

We say that a para-complex structure $\mathbf{P}$ on $(M,\mathbf{I},g)$ is
_complex para-Kähler_ if $\omega_{\mathbf{P}}$ is a non-degenerate symplectic
form; if, moreover, $g$ and $\omega_{\mathbf{P}}$ are holomorphic, we call it
_holomorphic para-Kähler_. As in the classical setting, the Kähler condition
is equivalent to $D\mathbf{P}=0$.

A _bi-complex_ structure on $M$ is a complex para-Hermitian structure
$(\mathbf{I},\mathbf{P},g,\omega_{\mathbf{P}})$ such that
$\mathbf{J}:=\mathbf{I}\mathbf{P}=\mathbf{P}\mathbf{I}$ is an integrable
complex structure. We say that it is _bi-Kähler_ if $\mathbf{P}$ (and hence
$\mathbf{J}$) is parallel for the Levi-Civita connection of $g$. Moreover, the
quadruple $(\mathbf{I},\mathbf{P},g,\omega_{\mathbf{P}})$ gives rise to a
_holomorphic bi-Kähler_ structure if, in addition, the metric $g$ is
holomorphic.

######  Remark 1.6.

Our definition of holomorphic bi-Kähler structure coincides with the notion of
_holomorphic bi-complex Kähler_ structure given by Loustau and Sanders ([LS17,
Definition 2.14]). They also showed that it is equivalent to a structure
called _complex bi-Lagrangian_ which consists of the data of a complex
symplectic form $\omega$ on $(M,\mathbf{I})$ and an ordered pair
$(\mathcal{L}_{1},\mathcal{L}_{2})$ of transverse complex Lagrangian
foliations ([LS17, Proposition 2.15]). In particular, bi-complex structures
appear naturally in the complexification of complex manifolds ([LS17, Theorem
3.13])

On a complex para-Hermitian manifold
$(M,\mathbf{I},\mathbf{P},g,\omega_{\mathbf{P}})$ with Levi-Civita connection
$D$, we can consider the complex sectional curvature of complex planes spanned
by $X$ and $\mathbf{P}X$:

$\mathrm{Sec}_{g}(X,\mathbf{P}X)=\frac{R^{D}(X,\mathbf{P}X,X,\mathbf{P}X)}{g(X,X)^{2}}\
.$

We call it _para-holomorphic complex sectional curvature_ and we say that $g$
has constant para-holomorphic complex sectional curvature if
$\mathrm{Sec}_{g}(X,\mathbf{P}X)$ is a constant function independent of $X$,
whenever $X$ is non isotropic. A result similar to Proposition 1.2 holds in
this setting:

######  Proposition 1.7.

Let $(M,\mathbf{I},\mathbf{P},g,\omega_{\mathbf{P}})$ be a complex para-
Hermitian manifold with constant para-holomorphic complex sectional curvature
$k$. Then the Riemann tensor of $g$ can be written as

$\displaystyle R^{D}(X,Y,Z,W)=-\frac{k}{4}\big{(}$ $\displaystyle
g(X,Z)g(Y,W)-g(Y,Z)g(X,W)+g(X,\mathbf{P}Z)g(\mathbf{P}Y,W)$
$\displaystyle-g(Y,\mathbf{P}Z)g(\mathbf{P}X,W)+2g(X,\mathbf{P}Y)g(\mathbf{P}Z,W)\big{)}$

for all $X,Y,Z,W\in\Gamma(TM)$.

###### Proof.

The same argument that is used in the pseudo-Riemannian setting works
([BCGR+98]). We give only a brief sketch. Let $T(X,Y,Z,W)$ denote the tensor
defined by the right hand side of the formula in the statement. It is
straightforward to verify that

$\displaystyle T(X,Y,Z,W)=-T(Y,X,Z,W)=-T(X,Y,W,Z)$ $\displaystyle
T(X,Y,Z,W)=T(Z,W,X,Y)$ (2) $\displaystyle T(X,Y,Z,W)+T(Y,Z,X,W)+T(Z,X,Y,W)=0$
$\displaystyle
T(X,Y,Z,W)=-T(\mathbf{P}X,\mathbf{P}Y,Z,W)=-T(X,Y,\mathbf{P}Z,\mathbf{P}W)$

for all $X,Y,Z,W\in\Gamma(TM)$. Moreover, we note that
$T(X,\mathbf{P}X,X,\mathbf{P}X)=kg(X,X)^{2}$
$=R^{D}(X,\mathbf{P}X,X,\mathbf{P}X)$ for all non-isotropic $X\in\Gamma(TM)$,
because $g$ has constant para-holomorphic complex sectional curvature $k$.
Then one proves that a $(4,0)$-tensor enjoying the symmetries in Equation
(1.3) is uniquely determined by the values it takes on quadruples of vectors
of the form $(X,\mathbf{P}X,X,\mathbf{P}X)$. ∎

In the case of bi-complex structures, Proposition 1.7 implies that $g$ has
also constant _$\mathbf{J}$ -holomorphic complex sectional curvature_ in the
sense that all non-degenerate complex planes spanned $X$ and $\mathbf{J}X$
have the same complex sectional curvature. Indeed,

$\displaystyle R^{D}(X,\mathbf{J}X,X,\mathbf{J}X)$
$\displaystyle=-\frac{k}{4}\big{(}g(X,X)g(\mathbf{J}X,\mathbf{J}X)-g(\mathbf{J}X,X)^{2}+g(X,\mathbf{P}X)g(\mathbf{P}\mathbf{J}X,\mathbf{J}X)$
$\displaystyle\ \ \ \
-g(\mathbf{J}X,\mathbf{P}X)^{2}+2g(X,\mathbf{P}\mathbf{J}X)g(\mathbf{P}X,\mathbf{J}X)\big{)}$
$\displaystyle=-kg(X,X)^{2}$

hence

$\mathrm{Sec}_{g}(X,\mathbf{J}X)=-\frac{{R}^{D}(X,\mathbf{J}X,X,\mathbf{J}X)}{g(X,X)^{2}}=k\
$

for all $X\in\Gamma(TM)$ not isotropic for $g$.

##  2\. Bi-complex hyperbolic plane

In this section we introduce bi-complex numbers and the bi-complex hyperbolic
space $\mathbb{CH}_{\tau}^{2}$, which generalizes both the complex hyperbolic
space $\mathbb{CH}^{2}$ and the para-complex hyperbolic space
$\mathbb{H}_{\tau}^{2}$, and we will describe its basic properties.

###  2.1. Bi-complex numbers

We review here some basic properties of bi-complex numbers and fix some
notation that we will use throughout the paper.

The algebra of bi-complex numbers $\mathbb{C}_{\tau}$ is the commutative
algebra over $\mathbb{C}$ generated by $1$ and $\tau$, where $\tau$ is a non-
complex number such that $\tau^{2}=1$. It is a $4$-dimensional vector space
over the reals with basis $\\{1,\tau,i,j:=\tau i\\}$. Here we denoted by $i$
the standard imaginary unit in $\mathbb{C}$. Note that $j$ is also a square
root of $-1$ since $i$ and $\tau$ commute.

Given a bi-complex number $w=z_{1}+\tau z_{2}$, we will use the standard
notation $\bar{w}$ to denote complex conjugation, so that
$\bar{w}=\bar{z}_{1}+\tau\bar{z}_{2}$. We then define

$\mathrm{Re}(w):=\frac{w+\bar{w}}{2}\ \ \ \ \ \text{and}\ \ \ \ \
\mathrm{Im}(w):=\frac{w-\bar{w}}{2i}\ .$

These are para-complex numbers, i.e. elements of the real subalgebra
$\mathbb{R}_{\tau}\subset\mathbb{C}_{\tau}$, with the property that
$w=\mathrm{Re}(w)+i\mathrm{Im}(w)$. In a similar way, we denote by
$\bar{w}^{\tau}$ the conjugate of $w$ with respect to the para-complex unit
$\tau$, in other words $\bar{w}^{\tau}:=z_{1}-\tau z_{2}$. We then define

$\mathrm{Re}_{\tau}(w):=\frac{w+\bar{w}^{\tau}}{2}\ \ \ \ \ \text{and}\ \ \ \
\ \mathrm{Im}_{\tau}(w):=\frac{w-\bar{w}^{\tau}}{2\tau}\ $

as the $\tau$-real and $\tau$-imaginary part of $w$. Note that, despite the
name, $\mathrm{Re}_{\tau}(w)$ and $\mathrm{Im}_{\tau}(w)$ are complex numbers.

The $\tau$-norm of a bi-complex number $w=z_{1}+\tau z_{2}$ is the complex
number

$|w|_{\tau}:=w\bar{w}^{\tau}=z_{1}^{2}-z_{2}^{2}\ .$

The non-vanishing of the $\tau$-norm characterizes invertible bi-complex
numbers.

For what follows, it will be useful to write a bi-complex number $w=z_{1}+\tau
z_{2}$ with respect to the basis of idempotents $\\{e_{+},e_{-}\\}$, where

$e_{+}:=\frac{1+\tau}{2}\ \ \ \ \ \text{and}\ \ \ \ \ e_{-}:=\frac{1-\tau}{2}\
.$

By a straightforward computation, we can rewrite $w=z^{+}e_{+}+z^{-}e_{-}$
with

$z^{+}=z_{1}+z_{2}\ \ \ \ \ \text{and}\ \ \ \ \ z^{-}=z_{1}-z_{2}\ .$

######  Remark 2.1.

We want to emphasize that our definition of the algebra of bi-complex numbers
might a-priori seem different from what can be found in the literature (see
[LESSV12]), but in fact they are equivalent. It will become clear in the next
sections why the definition we have given is useful for our purposes.

###  2.2. Bi-complex hyperbolic plane: the hyperboloid model

Consider the vector space $\mathbb{C}^{3}_{\tau}$ endowed with the
$\mathbb{C}$-bilinear para-Hermitian form

$\mathbf{q}(z,w):=z^{t}Q\bar{w}^{\tau}\ ,$

where $Q=\mathrm{diag}(1,1,-1)$.

######  Definition 2.2.

The bi-complex hyperbolic plane is

$\mathbb{CH}^{2}_{\tau}=\\{z\in\mathbb{C}^{3}_{\tau}\ |\
\mathbf{q}(z,z)=-1\\}/\sim$

where $z\sim z^{\prime}$ if there exists $w\in\mathbb{C}_{\tau}$ of unit
$\tau$-norm such that $z=wz^{\prime}$.

It follows immediately from the definition that $\mathbb{CH}^{2}_{\tau}$ is a
complex manifold of dimension $4$. Moreover, given
$z\in\mathbb{CH}^{2}_{\tau}$, we can identify the tangent space
$T_{z}\mathbb{CH}^{2}_{\tau}$ of $\mathbb{CH}^{2}_{\tau}$ with the
$\mathbf{q}$-orthogonal complement of $z$ obtaining thus a decomposition

$T_{z}\mathbb{C}^{3}_{\tau}=\mathbb{C}_{\tau}z\oplus
T_{z}\mathbb{CH}^{2}_{\tau}\ .$ (3)

This shows that $T_{z}\mathbb{CH}^{2}_{\tau}$ is naturally isomorphic to
$\mathbb{C}_{\tau}^{2}$, hence it is equipped with a complex structure
$\mathbf{I}$, inherited from the complex coordinates on
$\mathbb{C}^{3}_{\tau}$ and corresponding to the multiplication by $i$ on
$\mathbb{C}_{\tau}^{2}$, a para-complex structure $\mathbf{P}$ induced by the
multiplication by $\tau$, and another complex structure by composition
$\mathbf{J}:=\mathbf{I}\mathbf{P}=\mathbf{P}\mathbf{I}$. This explains the
reason why we called this space bi-complex hyperbolic space. In addition, the
restriction of $\mathrm{Re}_{\tau}(\mathbf{q})$ to
$T_{z}\mathbb{CH}^{2}_{\tau}$ defines a holomorphic metric $\hat{g}$ such that
$\hat{g}(\mathbf{P}\cdot,\mathbf{P}\cdot)=-\hat{g}$. Therefore, the pairing
$\hat{\omega}=\hat{g}(\cdot,\mathbf{P}\cdot)$ defines a holomorphic symplectic
form and the $\mathbb{C}_{\tau}$-valued tensor
$\hat{q}:=\hat{g}+\tau\hat{\omega}$ defines a holomorphic para-Hermitian
metric on $\mathbb{CH}^{2}_{\tau}$.

The isometry group of $\mathbb{CH}^{2}_{\tau}$ consists of all
$\mathbb{C}$-linear transformations of $\mathbb{C}_{\tau}^{3}$ that preserve
$\mathbf{q}$. We denote such group with $\mathrm{U}(2,1,\mathbb{C}_{\tau})$.

######  Proposition 2.3.

The map

$\displaystyle\Phi:\mathrm{GL}(3,\mathbb{C})$
$\displaystyle\rightarrow\mathrm{U}(2,1,\mathbb{C}_{\tau})$ $\displaystyle A$
$\displaystyle\mapsto Ae_{+}+Q(A^{-1})^{t}Qe_{-}$

is an isomorphism of groups. In particular, the subgroup
$\mathrm{SU}(2,1,\mathbb{C}_{\tau})$ of matrices with unit determinant
preserving $\mathbf{q}$ is isomorphic to $\mathrm{SL}(3,\mathbb{C})$.

###### Proof.

Let $X\in\mathrm{U}(2,1,\mathbb{C}_{\tau})$. Rewriting each entry of $X$ in
the basis of idempotents, we can decompose $X=X_{+}e_{+}+X_{-}e_{-}$ with
$X_{\pm}\in\mathfrak{gl}(3,\mathbb{C})$. Now, the matrix $X$ preserves the
$\mathbb{C}$-bilinear $\tau$-Hermitian form $\mathbf{q}$ if and only if
$X^{t}Q\overline{X}^{\tau}=Q$. Unraveling this equation we get

$\displaystyle Q$ $\displaystyle=X^{t}Q\overline{X}^{\tau}$
$\displaystyle=(X_{+}e_{+}+X_{-}e_{-})^{t}Q(\overline{X_{+}e_{+}+X_{-}e_{-}}^{\tau})$
$\displaystyle=(X_{+}^{t}e_{+}+X_{-}^{t}e_{-})Q(X_{+}e_{-}+X_{-}e_{+})$
($\overline{e_{\pm}}^{\tau}=e_{\mp}$)
$\displaystyle=X_{+}^{t}QX_{-}e_{+}+X_{-}^{t}QX_{+}e_{-}$ ($e_{+}e_{-}=0$)
$\displaystyle=\frac{1}{2}(X_{+}^{t}QX_{-}+X_{-}^{t}QX_{+})+\frac{\tau}{2}(X_{+}^{t}QX_{-}-X_{-}^{t}QX_{+})\
,$

which is satisfied if and only if

$\begin{cases}X_{+}^{t}QX_{-}-X_{-}^{t}QX_{+}=0\\\
X_{+}^{t}QX_{-}+X_{-}^{t}QX_{+}=2Q\ .\end{cases}$

We deduce that $X_{+}^{t}QX_{-}=X_{-}^{t}QX_{+}$ and
$X_{-}=Q(X_{+}^{-1})^{t}Q$. This shows that $\Phi$ is surjective. Since
$\mathrm{Ker}(\Phi)=\\{\mathrm{Id}\\}$, the map $\Phi$ is an isomorphism of
groups.
Finally, observing that

$\displaystyle\det(X)$
$\displaystyle=\det(X_{+}e_{+}+X_{-}e_{-})=\det(X_{+})e_{+}+\det(X_{-})e_{-}$
$\displaystyle=\det(X_{+})e_{+}+\det(X_{+})^{-1}e_{-}\ ,$

a matrix $X\in\mathrm{U}(2,1,\mathbb{C}_{\tau})$ has unit determinant if and
only if $\det(X_{+})=1$. Therefore,
$\Phi^{-1}(\mathrm{SU}(2,1,\mathbb{C}_{\tau}))=\mathrm{SL}(3,\mathbb{C})$. ∎

######  Remark 2.4.

The equation $X^{t}Q\overline{X}^{\tau}=Q$ satisfied by an
$X\in\mathrm{U}(2,1,\mathbb{C}_{\tau})$ implies that $\det(X)$ is a unitary
bi-complex number. As a result, we can identify
$\mathrm{SU}(2,1,\mathbb{C}_{\tau})$ with the quotient of
$\mathrm{U}(2,1,\mathbb{C}_{\tau})$ by the action by scalar multiplication of
elements of $\mathbb{C}_{\tau}$ with $\tau$-norm equal to $1$.

It is straightforward to verify that $\mathrm{SL}(3,\mathbb{C})$ acts
transitively on $\mathbb{CH}_{\tau}^{2}$ with stabilizer isomorphic to
$\mathrm{GL}(2,\mathbb{C})$. Thus, we can identify $\mathbb{CH}_{\tau}^{2}$
with the homogeneous space
$\mathrm{SL}(3,\mathbb{C})/\mathrm{GL}(2,\mathbb{C})$. This suggests a
relation between the bi-complex hyperbolic plane and the complex projective
plane, which we explain in the next subsection.

###  2.3. Bi-complex hyperbolic plane: relation with incidence geometry

We give now another model of $\mathbb{CH}^{2}_{\tau}$ that will play an
important role in bringing a connection between minimal Lagrangian surfaces in
$\mathbb{CH}^{2}_{\tau}$ and hyperbolic affine spheres (see [Tre19, §8.5] for
a related discussion on a similar case).

Using the basis of idempotents $\\{e_{+},e_{-}\\}$, we can see
$\mathbb{C}^{3}_{\tau}$ as the product
$\mathbb{C}^{3}e_{+}\oplus\mathbb{C}^{3}e_{-}$. In the sequel, we will
identify the second $\mathbb{C}^{3}$ factor with the dual of $\mathbb{C}^{3}$
with the isomorphism being induced by the bilinear form $Q$ via Riesz
representation theorem; in other words we will identify

$\displaystyle(\mathbb{C}^{3})^{*}$
$\displaystyle\longleftrightarrow\mathbb{C}^{3}$ $\displaystyle\varphi$
$\displaystyle\longleftrightarrow v_{\varphi}\ ,$

where $v_{\varphi}$ is the unique vector in $\mathbb{C}^{3}$ such that
$\varphi(w)=v_{\varphi}^{t}Qw$ for all $w\in\mathbb{C}^{3}$.

######  Proposition 2.5.

There is a diffeomorphism between $\mathbb{CH}^{2}_{\tau}$ and the set

$\mathbb{P}\mathcal{L}=\\{(v,\varphi)\in\mathbb{C}\mathbb{P}^{2}\times(\mathbb{C}\mathbb{P}^{2})^{*}\
|\ \varphi(v)\neq 0\\})\ ,$

consisting of pairs of projective classes of transverse lines and planes in
$\mathbb{C}^{3}$.

###### Proof.

We first show that the change of coordinates

$\displaystyle\alpha:\mathbb{C}^{3}_{\tau}\rightarrow\mathbb{C}^{3}e_{+}\oplus\mathbb{C}^{3}e_{-}$

composed with the above identification between the second
$\mathbb{C}^{3}$-factor and its dual induces a diffeomorphism between the
quadric

$V_{-1}=\\{z\in\mathbb{C}^{3}_{\tau}\ |\ \mathbf{q}(z,z)=-1\\}$

and

$\mathcal{L}_{-1}=\\{(v,\varphi)\in\mathbb{C}^{3}\times(\mathbb{C}^{3})^{*}\
|\ \varphi(v)=-1\\}.$

Indeed, if $x,y\in\mathbb{C}^{3}$, then $\alpha(x+\tau
y)=(x+y)e_{+}+(x-y)e_{-}$. The functional corresponding to $(x-y)$ is
represented by the row vector $(x-y)^{t}Q$. Therefore, if $z=x+\tau y\in
V_{-1}$, then

$\displaystyle-1=\mathbf{q}(z,z)$ $\displaystyle=z^{t}Q\bar{z}^{\tau}$
$\displaystyle=x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}-x_{3}^{2}+y_{3}^{2}$
$\displaystyle=(x-y)^{t}Q(x+y)\ ,$

and the last expression equals exactly the evaluation of the functional
$(x-y)^{t}Q$ on the vector $(x+y)$, thus the pair $(x+y,(x-y)^{t}Q)$ belongs
to $\mathcal{L}_{-1}$. This shows that $\mathcal{L}_{-1}$ is diffeomorphic to
$V_{-1}$, as it is clear that the map constructed above is smooth and with
smooth inverse.
Now, if $z\in V_{-1}$ is multiplied by $w\in\mathbb{C}_{\tau}$ with
$|w|_{\tau}=1$, then $w=u_{+}e_{+}+u_{-}e_{-}$ with $u_{+}u_{-}=1$ and the
corresponding pair $(x+y,(x-y)^{t}Q,)\in\mathcal{L}_{-1}$ changes to
$(u_{+}(x+y),u_{-}(x-y)^{t}Q)$. Therefore, the diffeomorphism defined before
induces one between $\mathbb{CH}^{2}_{\tau}$ and the quotient of
$\mathcal{L}_{-1}$ by the action of $\mathbb{C}^{*}$ given by
$\lambda\cdot(v,\varphi):=(\lambda v,\lambda^{-1}\varphi)$, which is exactly
the set $\mathbb{P}\mathcal{L}$ of the statement. ∎

###  2.4. Topology and the boundary at infinity of $\mathbb{CH}^{2}_{\tau}$

Here we explain the topological type of the bi-complex hyperbolic space and we
introduce its boundary at infinity.

######  Proposition 2.6.

The space $\mathbb{CH}^{2}_{\tau}$ is homeomorphic to the tangent bundle
$T\mathbb{C}\mathbb{P}^{2}$. In particular, it is simply connected and the
(real) para-complex hyperbolic space $\mathbb{H}^{2}_{\tau}$ is homeomorphic
to $T\mathbb{R}\mathbb{P}^{2}$.

###### Proof.

Let us denote with $||\cdot||$ and with $\langle\cdot,\cdot\rangle$
respectively the Euclidean norm and scalar product on $\mathbb{R}^{6}$. The
space

$\mathcal{V}:=\\{(u,v)\in\mathbb{R}^{6}\times\mathbb{R}^{6}\ |\ ||u||=||v||,\
\langle u,v\rangle=1\\}$

is homeomorphic to the tangent bundle of $S^{5}$ via the map

$\displaystyle\Phi:$ $\displaystyle\ \mathcal{V}\longrightarrow TS^{5}$
$\displaystyle(u,v)\longmapsto\bigg{(}\frac{u+v}{||u+v||},u-v\bigg{)}\ ,$

being the vectors $u+v$ and $u-v$ orthogonal (see also [GMA89]). After
identifying $\mathbb{R}^{6}$ with $\mathbb{C}^{3}$ we can think of
$\mathcal{V}$ as a subset of $\mathbb{C}^{3}\times\mathbb{C}^{3}$. It carries
an $S^{1}$-action given by $\lambda\cdot(u,v):=(\lambda u,\lambda^{-1}v)$,
which is nothing but the restriction on $\mathcal{V}$ of the standard
$(\mathbb{C}^{*}\times\mathbb{C}^{*})$-action on
$\mathbb{C}^{3}\times\mathbb{C}^{3}$. The quotient space is identified with
the set of projective classes of transverse lines and planes in
$\mathbb{C}^{3}$, which is an equivalent model for $\mathbb{CH}^{2}_{\tau}$
(Proposition 2.5). The above $S^{1}$-action is induced on $TS^{5}$ via the
homeomorphism $\Phi$ and the corresponding quotient space is identified with
$T\mathbb{C}\mathbb{P}^{2}$, where $\mathbb{C}\mathbb{P}^{2}\cong
S^{5}/S^{1}$. The claim regarding $\mathbb{H}^{2}_{\tau}$ comes from the
observation
$\mathbb{H}^{2}_{\tau}=\mathbb{CH}^{2}_{\tau}\cap\big{(}\mathbb{R}^{3}\oplus\tau\mathbb{R}^{3}\big{)}$
(see also Section 2.6). ∎

Using the model of $\mathbb{CH}_{\tau}^{2}$ in terms of incidence geometry
(see Section 2.3), it is natural to define the boundary at infinity of
$\mathbb{CH}^{2}_{\tau}$ as the topological boundary of
$\mathbb{CH}^{2}_{\tau}$ inside
$\mathbb{C}\mathbb{P}^{2}\times(\mathbb{C}\mathbb{P}^{2})^{*}$. Therefore, we
set

$\partial_{\infty}\mathbb{CH}^{2}_{\tau}:=\\{(v,\varphi)\in\mathbb{C}\mathbb{P}^{2}\times(\mathbb{C}\mathbb{P}^{2})^{*}\
|\ \varphi(v)=0\\}\ .$

We will see in Section 5 that this notion of boundary is well-suited for the
study of quasi-Hitchin representations in $\mathrm{SL}(3,\mathbb{C})$, as we
will be able to make a connection between the boundary map provided by the
Anosov property of quasi-Hitchin representations and the boundary at infinity
of an equivariant complex Lagrangian minimal surface in
$\mathbb{CH}^{2}_{\tau}$, which we introduce in Section 3.

###  2.5. The bi-complex hyperbolic plane as a holomorphic para-Kähler
reduction

Here we want to realize the space $\mathbb{CH}^{2}_{\tau}$ as a holomorphic
para-Kähler quotient using the classical Marsden-Weinstein theory. In contrast
with the classical symplectic reduction theory ([MW74],[Wei80]), the ambient
space $\mathbb{C}^{3}_{\tau}$ is endowed with a holomorphic para-Kähler
structure $\mathbf{q}=\mathbf{g}+\tau\bm{\omega}$, where $\mathbf{g}$ and
$\bm{\omega}$ are a holomorphic metric and a holomorphic symplectic form,
respectively. Therefore, the procedure will have two main differences: the
first one regards the (real) para-Kähler reduction, namely the fact that we
want to induce a pseudo-Riemannian metric to the quotient, and this requires
further verification ([Run23, Theorem B.6]); the second one is that the
structure we want to induce must be holomorphic, so we are looking for a
holomorphic Hamiltonian action of a complex Lie group on
$\mathbb{C}^{3}_{\tau}$ whose moment map is also holomorphic (see [Kob87,
§7.5] for more details).

Let us consider the function $f:\mathbb{C}^{3}_{\tau}\rightarrow\mathbb{C}$
given by $f(z)=\mathbf{q}(z,z)$. We denote by $\mathcal{U}$ the subgroup of
$\mathbb{C}_{\tau}$ of bi-complex numbers with unitary $\tau$-norm. We notice
that the action of $\mathcal{U}$ by multiplication is holomorphic and
preserves the level sets of the function $f$: indeed, if $u\in\mathcal{U}$,
then

$f(uz)=\mathbf{q}(uz,uz)=|u|_{\tau}\mathbf{q}(z,z)=f(z)\ .$

Since $u=u_{1}+\tau u_{2}\in\mathcal{U}$ satisfies $u_{1}^{2}-u_{2}^{2}=1$, we
can parameterize $\mathcal{U}$ using complex hyperbolic trigonometric
functions

$\mathcal{U}=\\{\cosh(v)+\tau\sinh(v)\ |\ v\in\mathbb{C},\
\mathrm{Im}(v)\in[0,2\pi)\\}\ .$

We observe that the complex metric $\mathbf{g}=\mathrm{Re}_{\tau}(\mathbf{q})$
restricted to the orbits of $\mathcal{U}$ on the non-zero level sets of $f$ is
non-degenerate: indeed, for a $z\in f^{-1}(k)$, we have

$\frac{d}{dv}_{|_{v=0}}(\cosh(v)+\tau\sinh(v))z=\tau z$

and $\mathbf{g}(\tau z,\tau z)=\mathrm{Re}_{\tau}(\mathbf{q}(\tau z,\tau
z))=-k\neq 0$. A straightforward computation shows that the action of
$\mathcal{U}$ is Hamiltonian with respect to the holomorphic symplectic form
$\hat{\omega}$ with moment map $\mu(z)=\frac{1}{2}f(z)$. Indeed, we already
computed that the infinitesimal generator of the action of $\mathcal{U}$ is
the vector field $U(z)=\tau z$ and for all $v\in T_{z}\mathbb{C}_{\tau}^{3}$
we have

$\displaystyle\bm{\omega}(U(z),v)$
$\displaystyle=\mathrm{Im}_{\tau}(\mathbf{q}(\tau z,v))$
$\displaystyle=\mathrm{Re}_{\tau}(\mathbf{q}(z,v))$
$\displaystyle=\frac{1}{2}\frac{d}{dt}_{|_{t=0}}\mathbf{q}(z+tv,z+tv)=d\mu_{z}(v)\
.$

Therefore, holomorphic symplectic reduction theory ([Kob87, §7.5]) gives, for
all $k\in\mathbb{C}^{*}$, a non-degenerate holomorphic symplectic form
$\hat{\omega}_{k}$ on the quotient $f^{-1}(k)/\mathcal{U}$. In particular, the
bi-complex hyperbolic space $\mathbb{CH}^{2}_{\tau}=f^{-1}(-1)/\mathcal{U}$
has one that we denote by $\hat{\omega}:=\hat{\omega}_{-1}$. Moreover, from
the decomposition

$T_{z}\mathbb{C}^{3}_{\tau}=T_{z}\mathbb{CH}^{2}_{\tau}\oplus
T(\mathcal{U}\cdot z)\oplus\tau T(\mathcal{U}\cdot z)\ ,$

we see that the para-complex structure given by the multiplication by $\tau$
descends to a para-complex structure $\mathbf{P}$ on $\mathbb{CH}^{2}_{\tau}$,
such that $\hat{\omega}(\mathbf{P}\cdot,\mathbf{P}\cdot)=-\hat{\omega}$.
Hence, the holomorphic metric $\mathbf{g}=\bm{\omega}(\cdot,\mathbf{P}\cdot)$
also descends to $\mathbb{CH}^{2}_{\tau}$ to a holomorphic metric $\hat{g}$,
so that the triple $(\hat{\omega},\mathbf{P},\hat{g})$ defines a holomorphic
para-Kähler structure on $\mathbb{CH}^{2}_{\tau}$. We note, moreover, that
vectors tangent to the orbit of the action of $\mathcal{U}$ are
$\bm{\omega}$-orthogonal to vectors in the kernel of $d\mu$, in other words to
vectors tangent to a level set of $f$.

###  2.6. Remarkable submanifolds in $\mathbb{CH}_{\tau}^{2}$

Our interest in the bi-complex hyperbolic plane arose as a way of giving a
unified point of view for the study of surface group representations into
$\mathrm{SL}(3,\mathbb{C})$. It is well-known that, if $S$ denotes a closed,
connected, oriented surface of genus at least $2$, to any Hitchin
representation $\rho:\pi_{1}(S)\rightarrow\mathrm{SL}(3,\mathbb{R})$, there is
a unique $\rho$-equivariant affine sphere in $\mathbb{R}^{3}$. As we will see
later in Section 4.1 and partially by work of Hildebrand ([Hil11]), hyperbolic
affine spheres in $\mathbb{R}^{3}$ are equivalent to minimal Lagrangian
surfaces in the (real) para-complex hyperbolic space
$\mathbb{H}^{2}_{\tau}:=\mathbb{CH}^{2}_{\tau}\cap\mathbb{R}^{3}_{\tau}$,
which is a para-Kähler, totally real, and totally geodesic submanifold in
$\mathbb{CH}^{2}_{\tau}$. In particular, if $\hat{q}=\hat{g}+\tau\hat{\omega}$
denotes the holomorphic para-Hermitian structure on $\mathbb{CH}^{2}_{\tau}$,
then its (real) para-Hermitian restriction to $\mathbb{H}^{2}_{\tau}$ is given
by
$\mathrm{Re}(\hat{g})|_{\mathbb{H}^{2}_{\tau}}+\tau\mathrm{Re}(\hat{\omega})|_{\mathbb{H}^{2}_{\tau}}$
where $\mathrm{Re}(\hat{g})|_{\mathbb{H}^{2}_{\tau}}$ is a pseudo-Riemannian
metric of neutral signature $(2,2)$.

On the other hand, if $\varrho:\pi_{1}(S)\rightarrow\mathrm{SU}(2,1)$ is
sufficiently close to a Fuchsian representation (i.e. to one that is faithful
and discrete and factors through $\mathrm{SO}_{0}(2,1)$), there is a unique
$\varrho$-equivariant minimal Lagrangian surface in the complex hyperbolic
space $\mathbb{CH}^{2}$. This can also be found as a totally geodesic
submanifold of $\mathbb{CH}^{2}_{\tau}$ by considering the intersection
$\mathbb{CH}^{2}_{\tau}\cap(\mathbb{R}\oplus i\tau\mathbb{R})^{3}$, where
$\mathbb{R}\oplus i\tau\mathbb{R}$ denotes the subspace of bi-complex numbers
in which the $\tau$-real part is a real number and the $\tau$-imaginary part
is a purely imaginary complex number. Note that the complex structure on this
copy of $\mathbb{CH}^{2}$ is inherited from $\mathbf{J}$. Let us now explain
how the $\mathbb{C}$-linear para-Hermitian structure $\mathbf{q}$ restricts to
a (real) $\mathbf{J}$-Hermitian structure on $\mathbb{CH}^{2}$, which induces
the usual Kähler metric. First we decompose $\mathbf{q}$ with respect to the
real basis $\\{1,i,j,\tau\\}$ of the algebra of bi-complex numbers

$\displaystyle\mathbf{q}$
$\displaystyle=\mathrm{Re}(\mathbf{g})+i\mathrm{Im}(\mathbf{g})+\tau\mathrm{Re}(\bm{\omega})+\tau
i\mathrm{Im}(\bm{\omega})$ ($j=\tau i$)
$\displaystyle=\big{(}\mathrm{Re}(\mathbf{g})+j\mathrm{Im}(\bm{\omega})\big{)}+\tau(\mathrm{Re}(\bm{\omega})+j\mathrm{Im}(\mathbf{g}))$
$\displaystyle=:\tilde{\mathbf{g}}+\tau\tilde{\bm{\omega}}\ .$

It is then clear from the definition of $\mathbb{CH}^{2}$ as the intersection
of $\mathbb{CH}^{2}_{\tau}$ with $(\mathbb{R}\oplus\tau i\mathbb{R})^{3}$,
that $\tilde{\bm{\omega}}$ vanishes identically when restricted to
$\mathbb{CH}^{2}$. Then we see that $\tilde{\mathbf{g}}$ defines a
$\mathbf{J}$-Hermitian form because

$\displaystyle\tilde{\mathbf{g}}(\mathbf{J}\cdot,\mathbf{J}\cdot)$
$\displaystyle=\mathrm{Re}(\mathbf{g})(\mathbf{J}\cdot,\mathbf{J}\cdot)+j\mathrm{Im}(\bm{\omega})(\mathbf{J}\cdot,\mathbf{J}\cdot)$
$\displaystyle=\mathrm{Re}(\mathbf{g})(\mathbf{I}\mathbf{P}\cdot,\mathbf{I}\mathbf{P}\cdot)+j\mathrm{Im}(\bm{\omega})(\mathbf{I}\mathbf{P}\cdot,\mathbf{I}\mathbf{P}\cdot)$
$\displaystyle=-\mathrm{Re}(\mathbf{g})(\mathbf{P}\cdot,\mathbf{P}\cdot)-j\mathrm{Im}(\bm{\omega})(\mathbf{P}\cdot,\mathbf{P}\cdot)$
($\mathbf{q}(\mathbf{I}\cdot,\mathbf{I}\cdot)=-\mathbf{q}$)
$\displaystyle=\mathrm{Re}(\mathbf{g})+j\mathrm{Im}(\bm{\omega})=\tilde{\mathbf{g}}$
($\mathbf{q}(\mathbf{P}\cdot,\mathbf{P}\cdot)=-\mathbf{q}$)

From this discussion, we conclude that the $\mathbf{J}$-Hermitian structure on
$\mathbb{CH}^{2}$ is indeed induced from the restriction of the bi-complex
structure of the ambient space.

There is another interesting submanifold to consider in
$\mathbb{CH}^{2}_{\tau}$, which is found by looking at the intersection
$\mathbb{CH}^{2}_{\tau}\cap\mathbb{C}^{3}$. This space, denoted by
$\mathbb{X}$, is endowed with the complex structure $\mathbf{I}$ and a
holomorphic Riemannian metric $\mathbf{g}$ given by the restriction of the
$\mathbb{C}$-linear para-Hermitian form $\mathbf{q}$. Explicitly it is
realized as the hypersurface $\\{\underline{z}\in\mathbb{C}^{3}\ |\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2}=-1\\}$ quotient out by all the complex numbers
that square to one, namely $\\{\pm 1\\}$. Since all non-degenerate bilinear
form over $\mathbb{C}$ are conjugate to each other, this space is isometric to
the space of (maximal, oriented, unparametrized) geodesics of the three-
dimensional real hyperbolic space $\mathbb{H}^{3}$ first studied by Bonsante
and El-Emam ([BEE22]) and then by El-Emam and Seppi ([EES22]), which is
further identified with
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}\setminus\Delta$, being
$\Delta$ the diagonal. The subgroup of $\mathrm{SL}(3,\mathbb{C})$ that
preserves $\mathbb{X}$ is isomorphic to $\mathrm{SO}(3,\mathbb{C})$ and acts
by holomorphic isometries for its holomorphic Riemannian metric. A special
class of immersed surfaces in $\mathbb{X}$, called _Bers embeddings_ , has
been studied in [BEE22], whose holonomy representation takes value in
$\mathrm{SL}(2,\mathbb{C})\cong\mathrm{SO}(3,\mathbb{C})$: the theory we will
describe in Section 3 will generalize part of their work as well.

We will see in Section 3 that hyperbolic affine spheres in $\mathbb{R}^{3}$,
minimal Lagrangian surfaces in $\mathbb{CH}^{2}$ and Bers embeddings in
$\mathbb{X}$ are instances of more general minimal Lagrangian surfaces in
$\mathbb{CH}^{2}_{\tau}$, thus giving a unified point of view to these
seemingly different notions.

###  2.7. Differential geometry of $\mathbb{CH}^{2}_{\tau}$

We describe here the main geometric features of $\mathbb{CH}^{2}_{\tau}$ that
will be useful in the rest of the paper.

Recall that $\mathbb{CH}^{2}_{\tau}$ is a complex manifold of dimension $4$.
We denoted by $\mathbf{I}$ its standard complex structure, seen as a bundle
endomorphism of $T\mathbb{CH}^{2}_{\tau}$ such that
$\mathbf{I}^{2}=-\mathrm{Id}$. The bi-complex hyperbolic space is also endowed
with a holomorphic metric $\hat{g}$. This induces a natural connection
$\hat{D}$ of $T\mathbb{CH}^{2}_{\tau}$ by Theorem 1.1. Considering the
$\mathbf{q}$-orthogonal decomposition

$T_{z}\mathbb{C}^{3}_{\tau}=\mathbb{C}_{\tau}\cdot z\oplus z^{\perp}\ ,$

for $z\in\mathbb{C}^{3}_{\tau}$ such that $\mathbf{q}(z,z)=-1$, after
identifying $T_{z}\mathbb{CH}^{2}_{\tau}$ with $z^{\perp}$, it is
straightforward to check, using the properties that characterize uniquely the
Levi-Civita connection of $\hat{g}$, that $\hat{D}$ is exaclty the projection
onto $z^{\perp}$ of the standard flat connection $\mathbf{D}$ defined on
$T\mathbb{C}_{\tau}^{3}$.

######  Lemma 2.7.

The bi-complex hyperbolic space has constant para-holomorphic complex
sectional curvature $-2$.

###### Proof.

Let $z\in\mathbb{C}^{3}_{\tau}$ such that $\mathbf{q}(z,z)=-1$. By the
previous discussion, the $\mathbf{q}$-orthogonal splitting
$T_{z}\mathbb{C}^{3}_{\tau}=T_{z}\mathbb{CH}^{2}_{\tau}\oplus\mathbb{C}_{\tau}\cdot
z$ allows to decompose the flat connection $\mathbf{D}$ as

$\displaystyle\mathbf{D}_{X}Y$
$\displaystyle=\hat{D}_{X}Y+\hat{\mathrm{I}\mathrm{I}}(X,Y)$
$\displaystyle\mathbf{D}_{X}\xi$
$\displaystyle=-\hat{B}_{\xi}X+\hat{\nabla}^{N}_{X}\xi$

where $X,Y\in\Gamma(T\mathbb{CH}_{\tau})$ and
$\xi\in\Gamma(\mathbb{C}_{\tau}\cdot z)$. Therefore, if $\mathbf{R}$ and
$\hat{R}$ denote the Riemann curvature tensor of $\mathbf{D}$ and $\hat{D}$
respectively, we have, for all non-isotropic
$X\in\Gamma(T_{z}\mathbb{CH}^{2}_{\tau})$,

$\displaystyle 0$
$\displaystyle=\mathbf{g}(\mathbf{R}(X,\mathbf{P}X)X,\mathbf{P}X)$
$\displaystyle=\mathbf{g}(\mathbf{D}_{\mathbf{P}X}\mathbf{D}_{X}X-\mathbf{D}_{X}\mathbf{D}_{\mathbf{P}X}-\mathbf{D}_{[\mathbf{P}X,X]}X,\mathbf{P}X)$
$\displaystyle=\hat{g}(\hat{D}_{\mathbf{P}X}\hat{D}_{X}X-\hat{D}_{X}\hat{D}_{\mathbf{P}X}X-\hat{D}_{[\mathbf{P}X,X]},\mathbf{P}X)$
$\displaystyle\ \ \
+\mathbf{g}(-\hat{B}_{\hat{\mathrm{I}\mathrm{I}}(X,X)}\mathbf{P}X,\mathbf{P}X)+\hat{g}(\hat{B}_{\hat{\mathrm{I}\mathrm{I}}(X,\mathbf{P}X)},\mathbf{P}X)$
$\displaystyle=\hat{g}(\hat{R}(X,\mathbf{P}X)X,\mathbf{P}X)$ $\displaystyle\ \
\
-\mathbf{g}(\hat{\mathrm{I}\mathrm{I}}(X,X),\hat{\mathrm{I}\mathrm{I}}(\mathbf{P}X,\mathbf{P}X))+\mathbf{g}(\hat{\mathrm{I}\mathrm{I}}(X,\mathbf{P}X),\hat{\mathrm{I}\mathrm{I}}(X,\mathbf{P}X))$
$\displaystyle=\mathrm{Sec}_{\hat{g}}(X,\mathbf{P}X)\|X\|^{4}-2\|X\|^{4}$

and the statement follows. In the last step we use the following identities

$\hat{\mathrm{I}\mathrm{I}}(X,X)=\|X\|^{2}z\ \ \
\hat{\mathrm{I}\mathrm{I}}(X,\mathbf{P}X)=\|X\|^{2}\mathbf{P}z\ \ \
\hat{\mathrm{I}\mathrm{I}}(\mathbf{P}X,\mathbf{P}X)=-\|X\|^{2}z\ ,$

which follow immediately from the definition. ∎

######  Remark 2.8.

For reasons that will be apparent later, we re-scale the metric $\hat{g}$ of
$\mathbb{CH}^{2}_{\tau}$ so that is has para-holomorphic complex sectional
curvature equal to $-4$. Note that, by the discussion after Proposition 1.7,
this implies that the totally geodesic copy of $\mathbb{CH}^{2}$ inside
$\mathbb{CH}^{2}_{\tau}$ defined in Section 2.6 also has holomorphic sectional
curvature $-4$, which is the most common convention.

From the construction of $\mathbb{CH}^{2}_{\tau}$ as holomorphic para-Kähler
reduction (see Section 2.5), the bi-complex hyperbolic space comes equipped
with an integrable para-holomorphic structure $\mathbf{P}$ and a symplectic
form $\hat{\omega}$, which are related to the holomorphic metric $\hat{g}$ by
the equation $\hat{\omega}=\hat{g}(\cdot,\mathbf{P}\cdot)$. Note that, since
the symplectic form $\hat{\omega}$ is closed and the para-complex structure
$\mathbf{P}$ is integrable, we can conclude, as in the classical setting of
Kähler manifolds, that $\hat{D}\mathbf{P}=0$. In particular, the
eigendistributions $\hat{\mathcal{D}}_{\pm}$ relative to the $\pm 1$
eigenvalues of $\mathbf{P}$ not only are integrable but are also
$\hat{D}$-parallel.

##  3\. Complex Lagrangian minimal surfaces in $\mathbb{CH}_{\tau}^{2}$

In this section we study smooth immersions
$\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ of a simply connected
$2$-dimensional domain $U$. As an immersion of smooth manifolds, $\sigma$ has
codimension $6$. However, we use recent techniques introduced by Bonsante and
El-Emam ([BEE22]) to treat them as surfaces of (complex) codimension $2$.

###  3.1. Embedding data and structural equations

Let $\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ be as above. We extend the
differential of $\sigma$ by $\mathbb{C}$-linearity to obtain a map

$\displaystyle\sigma_{*}:\mathbb{C}TU$ $\displaystyle\rightarrow
T\mathbb{CH}^{2}_{\tau}$ $\displaystyle V+iW$ $\displaystyle\mapsto
d\sigma(V)+\mathbf{I}d\sigma(W)\ .$

The pull-back $\sigma^{*}\hat{g}$ is a $\mathbb{C}$-bilinear form on
$\mathbb{C}TU$. We denote it by $h$ and we call it complex-valued first
fundamental form. From now on, we will consider only immersions for which $h$
is non-degenerate.

Consider now the pull-back vector bundle
$\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$. This is a complex vector bundle over
$U$ with complex structure given by

$i\sigma^{*}(V):=\sigma^{*}(\mathbf{I}V)\ \ \
V\in\Gamma(T\mathbb{CH}^{2}_{\tau})\ .$

We endow $\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$ with the pull-back complex
bilinear form $g:=\sigma^{*}\hat{g}$. Under the non-degenerate assumption, we
can consider $\mathbb{C}TU$ as a complex subbundle of
$\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$. Note that the restriction of $g$ to
$\mathbb{C}TU$ is exactly the complex-valued first fundamental form $h$. We
have a $g$-orthogonal decomposition

$\sigma^{*}(T\mathbb{CH}_{\tau}):=\mathbb{C}TU\oplus\mathbb{C}NU\ ,$

where $\mathbb{C}NU$ is, by definition, the $g$-orthogonal complement of
$\mathbb{C}TU$ in $\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$. It is a complex
bundle of complex rank $2$, which plays the same role as the normal bundle to
the immersion. The Levi-Civita connection $\hat{D}$ on
$T\mathbb{CH}^{2}_{\tau}$ pulls back to an $\mathbb{R}$-bilinear connection
$D$ on $\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$. We can extend it by
$\mathbb{C}$-linearity by setting

$D_{\sigma^{*}(V)+i\sigma^{*}(V^{\prime})}\sigma^{*}(W):=D_{\sigma^{*}(V)}\sigma^{*}(W)+iD_{\sigma^{*}(V^{\prime})}\sigma^{*}(W)$

for all $V,V^{\prime},W\in\Gamma(T\mathbb{CH}^{2}_{\tau})$. Since
$\hat{D}\mathbf{I}=0$, the connection $D$ is actually $\mathbb{C}$-bilinear
because

$D_{\sigma^{*}(V)}(i\sigma^{*}(W))=D_{\sigma^{*}(V)}(\sigma^{*}(\mathbf{I}W))=\hat{D}_{V}(\mathbf{I}W)=\mathbf{I}\hat{D}_{V}W=iD_{\sigma^{*}(V)}\sigma^{*}(W)\
,$

for all $V,W\in\Gamma(T\mathbb{CH}^{2}_{\tau})$. It thus makes sense to write
$D_{X}Y$ for any $X,Y\in\Gamma(\mathbb{C}TU)$ and we can decompose it into a
tangential and normal component

$D_{X}Y=\nabla_{X}Y+\mathrm{I}\mathrm{I}(X,Y)\ .$

It is straightforward to check that $\nabla$ is the Levi-Civita connection of
$h$ and $\mathrm{I}\mathrm{I}$ is a $\mathbb{C}$-bilinear symmetric tensor,
called complex second fundamental form.

######  Definition 3.1.

We say that an immersion $\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ is
complex Lagrangian if the pull-back $\sigma^{*}\hat{\omega}$ of the
holomorphic symplectic form on $\mathbb{C}TU$ vanishes identically.

In particular, if $\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ is a complex
Lagrangian immersion, then its complex-valued first fundamental form $h$ can
be obtained by restriction of the pull-back $q:=\sigma^{*}\hat{q}$, since
$\hat{q}=\hat{g}+\tau\hat{\omega}$. In this case the non-degeneracy of $h$ can
be verified by looking at the relative position between $\mathbb{C}TU$ and the
distributions $\mathcal{D}^{\pm}:=\sigma^{*}(\hat{\mathcal{D}}_{\pm})$:

######  Lemma 3.2.

Let $\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ be a smooth complex Lagrangian
immersion. Then $h=\sigma^{*}\hat{g}$ is non-degenerate if and only if the
distributions $\mathcal{D}_{\pm}$ are transverse to $\mathbb{C}TU$.

###### Proof.

Since $\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$ can be written as direct sum of
$\mathcal{D}_{\pm}$, the subspace $\mathbb{C}TU$ is transverse to both
$\mathcal{D}_{\pm}$ if and only if $\mathbb{C}TU$ is a graph of a
$\mathbb{C}$-linear invertible map
$\mathcal{L}:\mathcal{D}_{+}\rightarrow\mathcal{D}_{-}$. Since
$\mathcal{D}_{\pm}$ are the eigenspaces of the para-complex structure
$\mathbf{P}$, they are both totally isotropic for the metric $g$. However,
because $g$ is non-degenerate, we can find a basis
$\\{e_{1}^{+},e_{2}^{+},e_{1}^{-},e_{2}^{-}\\}$ of
$\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$ as complex vector space such that
$e_{i}^{\pm}\in\mathcal{D}_{\pm}$ and $g(e_{i}^{+},e_{j}^{-})=\delta_{ij}$. By
construction, we then have
$\mathbb{C}TU=\mathrm{Span}(e_{1}^{+}+\mathcal{L}e_{1}^{+},e_{2}^{+}+\mathcal{L}e_{2}^{+})$.
Let $L$ be the matrix associated to $\mathcal{L}$ with respect to the basis
$\\{e_{1}^{+},e_{2}^{+}\\}$ in the domain and $\\{e_{1}^{-},e_{2}^{-}\\}$ in
the target. It is straightforward to verify that the restriction of $g$ and
$\omega$ to $\mathbb{C}TU$ are represented by the matrices $L+L^{t}$ and
$L-L^{t}$, respectively. Since $\sigma$ is a complex Lagrangian immersion, we
deduce that $L$ is symmetric and thus $L$ is invertible if and only if the
restriction $h$ of $g$ to $\mathbb{C}TU$ is non-degenerate. ∎

######  Lemma 3.3.

Let $\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ be a complex Lagrangian
immersion. Then the restriction of the para-complex structure $\mathbf{P}$
induces a $\mathbb{C}$-linear isomorphism
$\mathbf{P}:\mathbb{C}TU\rightarrow\mathbb{C}NU$. The same holds for the
complex structure $\mathbf{J}$.

###### Proof.

Since $\mathbf{P}$ is an involution, it is sufficient to check that
$\mathbf{P}X\in\mathbb{C}NU$ for all $X\in\mathbb{C}TU$. Now, by the previous
remark, for all $Y\in\mathbb{C}TU$, we have

$g(\mathbf{P}X,Y)=\mathrm{Re}_{\tau}(q(\mathbf{P}X,Y))=\mathrm{Re}_{\tau}(\tau
q(X,Y))=\mathrm{Re}_{\tau}(\tau g(X,Y))=0\ .$

Since $\mathbf{J}=\mathbf{P}\mathbf{I}$ and $\mathbf{I}$ is an automorphism of
$\mathbb{C}TU$, the same conclusion holds for $\mathbf{J}$. ∎

The Levi-Civita connection $D$ on $\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$
induces a connection $\nabla^{N}$ on the normal bundle $\mathbb{C}NU$ by
decomposing

$D_{X}\mathbf{P}Y=-B_{\mathbf{P}Y}X+\nabla^{N}_{X}\mathbf{P}Y\ ,$

where $-B_{\mathbf{P}Y}X$ is by definition the tangential component of
$D_{X}\mathbf{P}Y$ for all $X,Y\in\Gamma(\mathbb{C}TU)$. The complex shape
operator $B:\mathbb{C}TU\times\mathbb{C}NU\rightarrow\mathbb{C}TU$ is complex
bi-linear and, as a consequence of the compatibility of the connection $D$
with the metric $g$, it satisfies

$g(\mathrm{I}\mathrm{I}(X,Y),\mathbf{P}Z)=g(B_{\mathbf{P}Z}X,Y)$ (4)

for all $X,Y,Z\in\Gamma(\mathbb{C}TU)$.

######  Definition 3.4.

Let $\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ be a smooth immersion. We say
that $\sigma$ is minimal if and only if its complex second fundamental form
$\mathrm{I}\mathrm{I}$ is traceless, namely given an $h$-orthonormal basis
$\\{X_{1},X_{2}\\}$ of $\mathbb{C}TU$ we have
$\mathrm{I}\mathrm{I}(X_{1},X_{1})=-\mathrm{I}\mathrm{I}(X_{2},X_{2})$.

######  Proposition 3.5.

Let $\mathrm{I}\mathrm{I}$ and $B$ be the complex second fundamental form and
complex shape operator of a smooth, complex Lagrangian minimal immersion into
$\mathbb{CH}^{2}_{\tau}$. Then the following identities hold:

  1. i)

$\mathbf{P}(\mathrm{I}\mathrm{I}(X,Y))=-B_{\mathbf{P}Y}X$

  2. ii)

$\mathbf{P}(\nabla_{X}Y)=\nabla^{N}_{X}\mathbf{P}Y$ ,

for all $X,Y,Z\in\Gamma(\mathbb{C}TU)$. Moreover, the tensors defined by

$C^{\pm}(X,Y,Z):=\pm g(\mathrm{I}\mathrm{I}(X,Y),\mathbf{P}Z)$

are totally symmetric.

###### Proof.

Recall that, by definition of the Levi-Civita connection $\nabla$ and the
complex second fundamental form, we have
$D_{X}Y=\nabla_{X}Y+\mathrm{I}\mathrm{I}(X,Y)$ for all
$X,Y\in\Gamma(\mathbb{C}TU)$. Therefore,

$\displaystyle\mathbf{P}(\nabla_{X}Y+\mathrm{I}\mathrm{I}(X,Y))$
$\displaystyle=\mathbf{P}(D_{X}Y)$ $\displaystyle=D_{X}(\mathbf{P}Y)$
($D\mathbf{P}=0$) $\displaystyle=-B_{\mathbf{P}Y}X+\nabla^{N}_{X}\mathbf{P}Y.$

By Lemma 3.3, the operator $\mathbf{P}$ sends tangent vectors to normal
vectors and viceversa, thus, by transversality of $\mathbb{C}TU$ and
$\mathbb{C}NU$ we can conclude that
$\mathbf{P}(\mathrm{I}\mathrm{I}(X,Y))=-B_{\mathbf{P}Y}X$ and
$\mathbf{P}(\nabla_{X}Y)=\nabla^{N}_{X}\mathbf{P}Y$ .
As for the tensor $C^{+}$, it is clear that $C^{+}$ is symmetric in the first
two entries, because the complex second fundamental form is a symmetric
operator. It remains to show that $C^{+}$ is symmetric in the second and third
entry. This follows from an explicit computation:

$\displaystyle C^{+}(X,Z,Y)$
$\displaystyle=g(\mathrm{I}\mathrm{I}(X,Z),\mathbf{P}Y)$
$\displaystyle=-g(\mathbf{P}(\mathrm{I}\mathrm{I}(X,Z)),Y)$
($g(\mathbf{P},\mathbf{P})=-g$) $\displaystyle=g(B_{\mathbf{P}Z}X,Y)$
(relation $i)$)
$\displaystyle=g(\mathrm{I}\mathrm{I}(X,Y),\mathbf{P}Z)=C^{+}(X,Y,Z)\ .\ $
(Equation (4))

∎

Because we are only considering immersions for which the complex-valued first
fundamental form is non-degenerate, Lemma 3.2 furnishes two decompositions of
$\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$:

$\sigma^{*}(T\mathbb{CH}^{2}_{\tau})=\mathbb{C}TU\oplus\mathcal{D}_{+}=\mathbb{C}TU\oplus\mathcal{D}_{-}\
.$

These induce two $\mathbb{C}$-bilinear connections $\nabla^{\pm}$ on
$\mathbb{C}TU$ by splitting the ambient connection $D$ into a tangential
component and a part $\alpha^{\pm}$ belonging to $\mathcal{D}_{\pm}$: in other
words,

$D_{X}Y=\nabla^{+}_{X}Y+\alpha^{+}(X,Y)=\nabla^{-}_{X}Y+\alpha^{-}(X,Y)$ (5)

for all $X,Y\in\mathbb{C}TU$. Note that $\alpha^{\pm}$ are symmetric and
$\mathbb{C}$-bilinear. Another important property of the distributions
$\mathcal{D}_{\pm}$ is closely related to the holomorphic para-Hermitian
structure $\hat{q}$ of $\mathbb{CH}^{2}_{\tau}$. In this regard, we introduce
the following new tensor $\hat{Q}:=\hat{g}+\hat{\omega}$ whose symmetric part
coincides with $\hat{g}$ and the skew-symmetric part with $\hat{\omega}$. In
particular, it is parallel with respect to the Levi-Civita connection of
$\hat{g}$, i.e. $\hat{D}\hat{Q}=0$. Let $Q=\sigma^{*}\hat{Q}$ be the
corresponding tensor on $\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$.

######  Lemma 3.6.

For any $X\in\mathcal{D}_{-},Y\in\mathcal{D}_{+}$ and
$Z\in\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$ we have

$Q(X,Z)=Q(Z,Y)=0\ .$ (6)

Moreover, if $X,Y\in\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$ satisfy
$Q(X,Z)=Q(Z,Y)=0$ for any $Z\in\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$, then
$X\in\mathcal{D}_{-}$ and $Y\in\mathcal{D}_{+}$.

###### Proof.

Let $X\in\mathcal{D}_{-}$ and $Z\in\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$, then

$\displaystyle Q(X,Z)$ $\displaystyle=g(X,Z)+\omega(X,Z)$
$\displaystyle=g(X,Z)+g(\mathbf{P}X,Z)$
($\omega(\cdot,\cdot)=g(\mathbf{P}\cdot,\cdot)$)
$\displaystyle=g(X,Z)-g(X,Z)=0\ .$ ($\mathbf{P}X=-X$)

Observing that $Y\in\mathcal{D}_{+}$ if and only if $\mathbf{P}Y=Y,$ with a
similar computation and using that
$g(\mathbf{P}\cdot,\cdot)=-g(\cdot,\mathbf{P}\cdot)$, we obtain $Q(Z,Y)=0$.
Finally, suppose that given $X,Y\in\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$ we
have $Q(X,Z)=Q(Z,Y)=0$ for any $Z\in\sigma^{*}(T\mathbb{CH}^{2}_{\tau})$. This
is equivalent to

$g(X+\mathbf{P}X,Z)=g(Z,Y-\mathbf{P}Y)=0\ ,\ \ \ \forall
Z\in\sigma^{*}(T\mathbb{CH}^{2}_{\tau})\ ,$

and since the complex metric $g$ is non-degenerate, we conclude that
$\mathbf{P}X=-X$ and $\mathbf{P}Y=Y$ as required. ∎

The above result has direct application to our case of interest.

######  Lemma 3.7.

Let $\sigma:U\rightarrow\mathbb{CH}^{2}_{\tau}$ be a complex Lagrangian
immersion with non-degenerate complex-valued first fundamental form $h$. Then

  1. i)

the tensors

$T^{\pm}(X,Y,Z):=(\nabla^{\pm}_{X}h)(Y,Z)$

are totally symmetric;

  2. ii)

the Levi-Civita connection $\nabla$ of $h$ satisfies

$\nabla=\frac{1}{2}(\nabla^{+}+\nabla^{-})$

###### Proof.

Let us start with point $i)$: the symmetry of $T^{+}(X,Y,Z)$ in the second and
third entries follows from its definition, in fact for any vector field $X$,
the tensor $\nabla^{+}_{X}h(\cdot,\cdot)$ is still symmetric. On the other
hand, since $\sigma$ is a Lagrangian immersion we have $Q=h$, therefore

$\displaystyle T^{+}(X,Y,Z)$ $\displaystyle=(\nabla^{+}_{X}Q)(Y,Z)$
$\displaystyle=X\cdot Q(Y,Z)-Q(\nabla^{+}_{X}Y,Z)-Q(Y,\nabla^{+}_{X}Z)$
$\displaystyle=X\cdot
Q(Y,Z)-Q(D_{X}Y-\alpha^{+}(X,Y),Z)-Q(Y,D_{X}Z-\alpha^{+}(X,Z))$
$\displaystyle=Q(\alpha^{+}(X,Y),Z)+Q(Y,\alpha^{+}(X,Z))$ ($DQ=0$)
$\displaystyle=Q(\alpha^{+}(X,Y),Z)\ ,$ ($\alpha^{+}(X,Z)\in\mathcal{D}_{+}$
and Equation (6))

in particular $T^{+}$ is also symmetric in the first and second entries. The
proof for the symmetry of $T^{-}$ is analogous.
Regarding point $ii)$ we already know that the connection
$\frac{1}{2}(\nabla^{+}+\nabla^{-})$ is torsion-free, thus we only need to
prove that it preserves the induced metric. First notice that

$\displaystyle X\cdot h(Y,Z)$ $\displaystyle=X\cdot Q(Y,Z)$
$\displaystyle=Q(D_{X}Y,Z)+Q(Y,D_{X}Z)$ ($DQ=0$)
$\displaystyle=Q(\nabla_{X}^{-}Y+\alpha^{-}(X,Y),Z)+Q(Y,\nabla_{X}^{+}Z+\alpha^{+}(X,Z))$
$\displaystyle=Q(\nabla_{X}^{-}Y,Z)+Q(Y,\nabla_{X}^{+}Z)$ (Equation (6))
$\displaystyle=h(\nabla_{X}^{-}Y,Z)+h(Y,\nabla_{X}^{+}Z)\ .$

Then, finally

$\displaystyle\frac{1}{2}(\nabla_{X}^{+}+\nabla_{X}^{-})(h)(Y,Z)$
$\displaystyle=\frac{1}{2}(X+X)\cdot
h(Y,Z)-\frac{1}{2}h(\nabla_{X}^{+}Y+\nabla_{X}^{-}Y,Z)+$
$\displaystyle-\frac{1}{2}h(Y,\nabla_{X}^{+}Z+\nabla_{X}^{-}Z)$
$\displaystyle=X\cdot h(Y,Z)-X\cdot h(Y,Z)=0\ ,$

where in the last line we used the above computation.

∎

As a consequence, the tensors $C^{\pm}$ defined in Proposition 3.5 can be
written in terms of the connections $\nabla^{\pm}$:

######  Corollary 3.8.

Under the same assumptions as Lemma 3.7 we have

$2C^{\pm}(X,Y,Z)=T^{\pm}(X,Y,Z)$

###### Proof.

We prove the relation for $T^{+}$; the one for $T^{-}$ easily follows using
Lemma 3.7 part $ii)$ since $\nabla h=0$.
Averaging the decompositions

$D_{X}Y=\nabla^{+}_{X}Y+\alpha^{+}(X,Y)=\nabla^{-}_{X}Y+\alpha_{-}(X,Y)\ ,$

we deduce that for all $X,Y\in\Gamma(\mathbb{C}TU)$ we have

$\displaystyle D_{X}Y$
$\displaystyle=\frac{1}{2}(\nabla^{+}_{X}Y+\nabla^{-}_{X}Y)+\frac{1}{2}(\alpha^{+}(X,Y)+\alpha_{-}(X,Y))$
$\displaystyle=\nabla_{X}Y+\frac{1}{2}(\alpha^{+}(X,Y)+\alpha_{-}(X,Y))\ .$
(Lemma 3.7 part $ii)$)

Therefore, $\frac{1}{2}(\alpha^{+}(X,Y)+\alpha_{-}(X,Y))$ is the normal
component of $D_{X}Y$ and thus coincides with $\mathrm{I}\mathrm{I}(X,Y)$.
Let us now consider the $\mathrm{End}(\mathbb{C}TU)$-valued 1-form

$A^{+}:=\nabla^{+}-\nabla\ .$

Then, for all $X,Y,Z\in\Gamma(\mathbb{C}TU)$,

$\displaystyle C^{+}(X,Y,Z)$
$\displaystyle=-h(\mathbf{P}(\mathrm{I}\mathrm{I}(X,Y)),Z)$
$\displaystyle=-\frac{1}{2}h(\mathbf{P}(\alpha^{+}(X,Y))+\mathbf{P}(\alpha^{-}(X,Y)))$
$\displaystyle=-\frac{1}{2}h(\alpha^{+}(X,Y)-\alpha^{-}(X,Y))$
$\displaystyle=\frac{1}{2}h(D_{X}Y-\nabla^{+}_{X}Y-D_{X}Y+\nabla^{-}_{X}Y,Z)$
$\displaystyle=-\frac{1}{2}h(-\nabla^{+}_{X}Y+2\nabla_{X}Y-\nabla^{+}_{X}Y,Z)$
($\nabla^{-}=2\nabla-\nabla^{+}$) $\displaystyle=g(A^{+}(X)Y,Z)\ .$

Since $C^{+}$ is totally symmetric, it follows that $A^{+}(X)Y=A^{+}(Y)X$ for
all $X,Y\in\mathbb{C}TU$ and that $A^{+}(X)$ is $h$-self-adjoint for all
$X\in\mathbb{C}TU$. Therefore,

$\displaystyle T^{+}(X,Y,Z)$ $\displaystyle=(\nabla^{+}_{X}h)(Y,Z)$
$\displaystyle=(\nabla^{+}_{X}-\nabla_{X})(h)(Y,Z)$ ($\nabla h=0$)
$\displaystyle=h(A^{+}(X)Y,Z)+h(Y,A^{+}(X)Z)=2C^{+}(X,Y,Z)\ .$

∎

######  Remark 3.9.

From the relation $C^{+}(X,Y,Z)=h(A^{+}(X)Y,Z)$ proved in Corollary 3.8, we
deduce that $A^{+}=-\mathbf{P}\mathrm{I}\mathrm{I}$.

######  Theorem 3.10.

The complex-valued first fundamental form $h$ and the complex second
fundamental form $\mathrm{I}\mathrm{I}$ of a smooth complex Lagrangian minimal
immersion satisfy

$\displaystyle d^{\nabla}\mathrm{I}\mathrm{I}=0\ \ \ \ \ \ \ \ \ \ \ $
(Codazzi equation) $\displaystyle K_{h}+\|\mathrm{I}\mathrm{I}\|^{2}=-1\ \ \ \
\ $ (Gauss equation),

where $K_{h}$ denotes the complex Gaussian curvature of the metric $h$ and the
norm of $\mathrm{I}\mathrm{I}$ is defined as

$\|\mathrm{I}\mathrm{I}\|^{2}=g(\mathrm{I}\mathrm{I}(e_{1},e_{1}),\mathrm{I}\mathrm{I}(e_{1},e_{1}))+g(\mathrm{I}\mathrm{I}(e_{2},e_{1}),\mathrm{I}\mathrm{I}(e_{2},e_{1}))\
,$

with $\\{e_{1},e_{2}\\}$ being an orthonormal basis of $\mathbb{C}TU$.

###### Proof.

Let $X,Y\in\Gamma(\mathbb{C}TU)$. The decompositions
$D_{X}Y=\nabla_{X}Y+\mathrm{I}\mathrm{I}(X,Y)$ and
$D_{X}\xi=-B_{\xi}X+\nabla^{N}_{X}\xi$ for all $X,Y\in\Gamma(\mathbb{C}TU)$
and $\xi\in\Gamma(\mathbb{C}NU)$ imply that

$\displaystyle\begin{split}R^{D}(X,Y)Z&=D_{X}D_{Y}Z-D_{Y}D_{X}Z-D_{[X,Y]}Z\\\
&=D_{X}(\nabla_{Y}Z+\mathrm{I}\mathrm{I}(Y,Z))-D_{Y}(\nabla_{X}Z+\mathrm{I}\mathrm{I}(X,Z))\\\
&\ \ \ -\nabla_{[X,Y]}Z-\mathrm{I}\mathrm{I}([X,Y],Z)\\\
&=\nabla_{X}\nabla_{Y}Z+D_{X}(\mathrm{I}\mathrm{I}(Y,Z))+\mathrm{I}\mathrm{I}(X,\nabla_{Y}Z)\\\
&\ \ \
-\nabla_{Y}\nabla_{X}Z-D_{Y}(\mathrm{I}\mathrm{I}(X,Z))-\mathrm{I}\mathrm{I}(Y,\nabla_{X}Z)\\\
&\ \ \ -\nabla_{[X,Y]}Z-\mathrm{I}\mathrm{I}([X,Y],Z)\\\
&=R^{\nabla}(X,Y)Z-B_{\mathrm{I}\mathrm{I}(Y,Z)}X+B_{\mathrm{I}\mathrm{I}(X,Z)}Y\\\
&\ \ \
+\nabla^{N}_{X}(\mathrm{I}\mathrm{I}(Y,Z))-\nabla^{N}_{Y}(\mathrm{I}\mathrm{I}(X,Z))+\mathrm{I}\mathrm{I}(X,\nabla_{Y}Z)-\mathrm{I}\mathrm{I}(Y,\nabla_{X}Z)\\\
&\ \ \
-\mathrm{I}\mathrm{I}(\nabla_{X}Y,Z)+\mathrm{I}\mathrm{I}(\nabla_{Y}X,Z)\\\
&=\underbrace{R^{\nabla}(X,Y)Z-B_{\mathrm{I}\mathrm{I}(Y,Z)}X+B_{\mathrm{I}\mathrm{I}(X,Z)}Y}_{\in\mathbb{C}TU}+\underbrace{d^{\nabla}\mathrm{I}\mathrm{I}(X,Y,Z)}_{\in\mathbb{C}NU}\
.\\\ \end{split}$

On the other hand, by Proposition 1.7, Lemma 2.7 and Remark 2.8, we can write

$\displaystyle R^{D}(X,Y)Z$
$\displaystyle=\underbrace{g(X,Z)Y-g(Y,Z)X}_{\in\mathbb{C}TU}+\underbrace{g(X,\mathbf{P}Z)\mathbf{P}Y-g(Y,\mathbf{P}Z)\mathbf{P}X+2g(X,\mathbf{P}Y)\mathbf{P}Z}_{=0}$

because $\mathbf{P}(\mathbb{C}TU)=\mathbb{C}NU$. Equating the two expressions
above, we conclude that $d^{\nabla}\mathrm{I}\mathrm{I}=0$. Moreover, if
$\\{X,Y\\}$ is an $h$-orthonormal basis of $\mathbb{C}TU$, then

$\displaystyle K_{h}$ $\displaystyle=-R^{\nabla}(X,Y,X,Y)$
$\displaystyle=-R^{D}(X,Y,X,Y)-g(B_{\mathrm{I}\mathrm{I}(Y,X)}X,Y)+g(B_{\mathrm{I}\mathrm{I}(X,X)}Y,Y)$
$\displaystyle=-1-g(\mathrm{I}\mathrm{I}(Y,X),\mathrm{I}\mathrm{I}(Y,X))+g(\mathrm{I}\mathrm{I}(X,X),\mathrm{I}\mathrm{I}(Y,Y))$
$\displaystyle=-1-g(\mathrm{I}\mathrm{I}(Y,X),\mathrm{I}\mathrm{I}(Y,X))-g(\mathrm{I}\mathrm{I}(X,X),\mathrm{I}\mathrm{I}(X,X))$
($\mathrm{tr}(\mathrm{I}\mathrm{I})=0$)
$\displaystyle=-1-\|\mathrm{I}\mathrm{I}\|^{2}$

and the proof is complete. ∎

######  Remark 3.11.

We observe that if the complex Lagrangian minimal surface $\sigma(U)$ is
totally geodesic, namely $\mathrm{I}\mathrm{I}=0$, and the first fundamental
form is a positive complex metric $h$ (Definition 1.4), then it satisfies
$K_{h}=-1$. Therefore, it can be realized as the special class of immersed
surfaces in $\mathbb{X}$ studied by Bonsante and El Emam ([BEE22, Theorem
6.9]).

###  3.2. Computations in local coordinates

We will now restrict to the case where the complex-valued first fundamental
form $h$ is a positive complex metric. We saw in Section 1.2 that, under this
assumption, we can find a unique compatible $\mathbb{C}$-complex structure
$\mathcal{J}$ corresponding to a pair $(J_{1},J_{2})$ of complex structures on
$S$ and local holomorphic coordinates $z$ and w on $S$ such that
$h=2e^{2\psi}dzd\bar{w}$. This implies that the vector fields
$\partial_{\bar{z}}$ and $\partial_{w}$ form an $h$-isotropic basis of
$\mathbb{C}TU$. On the other hand, since $\mathbb{C}TU$ can also be generated
by $\partial_{z}$ and $\partial_{\bar{z}}$, the vector field $\partial_{w}$
can be written as a $\mathbb{C}$-linear combination of $\partial_{z}$ and
$\partial_{\bar{z}}$. Indeed,

$\partial_{w}=(\partial_{w}z)(\partial_{z}+\bar{\mu}\partial_{\bar{z}})\ ,$

where $\mu$ is the _Beltrami differential_ of $w$ with respect to the local
coordinate $z$, in other words

$\mu=-\frac{\partial_{\bar{z}}w}{\partial_{z}w}\ .$

In particular, we observe that $\partial_{\bar{z}}$ and $\partial_{w}$ do not
commute. In fact, the following holds:

######  Lemma 3.12.

Let $\partial_{\bar{z}}$ and $\partial_{w}$ be the vector fields defined
above. Then

$[\partial_{\bar{z}},\partial_{w}]=\partial_{\bar{z}}(\log(\partial_{w}z))\partial_{w}-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\partial_{\bar{z}}\
.$

###### Proof.

Let $f:S\rightarrow\mathbb{C}$ be a smooth function. In the computation that
follows we will make repeated use of the following identities, which are easy
consequences of the definition of the Beltrami differential $\mu$:

$\bar{\mu}=-\frac{\partial_{z}\bar{w}}{\partial_{\bar{z}}\bar{w}}\ \ \ \ \ \ \
\
\bar{\mu}_{\bar{z}}=-\partial_{z}(\log(\partial_{\bar{z}}\bar{w}))-\bar{\mu}\partial_{\bar{z}}(\log(\partial_{\bar{z}}\bar{w}))=-\frac{\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))}{\partial_{w}z}$

By definition of Lie bracket of vector fields, we have

$\displaystyle[\partial_{\bar{z}},\partial_{w}](f)$
$\displaystyle=[\partial_{\bar{z}},(\partial_{w}z)(\partial_{z}+\bar{\mu}\partial_{\bar{z}})](f)$
$\displaystyle=\partial_{\bar{z}}[(\partial_{w}z)(\partial_{z}+\bar{\mu}\partial_{\bar{z}})(f)]-(\partial_{w}z)(\partial_{z}+\bar{\mu}\partial_{\bar{z}})f_{\bar{z}}$
$\displaystyle=\partial_{\bar{z}}[(\partial_{w}z)(f_{z}+\bar{\mu}f_{\bar{z}})]-(\partial_{w}z)(f_{z\bar{z}}+\bar{\mu}f_{\bar{z}\bar{z}})$
$\displaystyle=(\partial_{w}z)(f_{\bar{z}z}+\bar{\mu}_{\bar{z}}f_{\bar{z}}+\bar{\mu}f_{\bar{z}\bar{z}})+(\partial_{\bar{z}}\partial_{w}z)(f_{z}+\bar{\mu}f_{\bar{z}})-(\partial_{w}z)(f_{z\bar{z}}+\bar{\mu}f_{\bar{z}\bar{z}})$
$\displaystyle=(\partial_{w}z)\bar{\mu}_{\bar{z}}f_{\bar{z}}+(\partial_{\bar{z}}\partial_{w}z)f_{z}+(\partial_{\bar{z}}\partial_{w}z)\bar{\mu}f_{\bar{z}}$
$\displaystyle=-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))f_{\bar{z}}+(\partial_{\bar{z}}\partial_{w}z)f_{z}+(\partial_{\bar{z}}\partial_{w}z)\bar{\mu}f_{\bar{z}}$
$\displaystyle=-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))f_{\bar{z}}+(\partial_{\bar{z}}\partial_{w}z)\left(\frac{f_{w}}{\partial_{w}z}-\bar{\mu}f_{\bar{z}}\right)+(\partial_{\bar{z}}\partial_{w}z)\bar{\mu}f_{\bar{z}}$
($\partial_{z}=\frac{\partial_{w}}{\partial_{w}z}-\bar{\mu}\partial_{\bar{z}}$)
$\displaystyle=-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))f_{\bar{z}}+\partial_{\bar{z}}(\log(\partial_{w}z))f_{w}$

and the statement follows. ∎

######  Definition 3.13.

Let $h$ be a positive complex metric with Levi-Civita connection $\nabla$.
Given a smooth function $\phi:S\rightarrow\mathbb{C}$, we define the
$h$-Laplacian of $\phi$ as

$\Delta_{h}\phi:=\mathrm{tr}_{h}(\nabla^{2}\phi)=\mathrm{tr}_{h}((X,Y)\mapsto\nabla_{X}\nabla_{Y}\phi-\nabla_{\nabla_{X}Y}\phi)\
.$

######  Proposition 3.14.

In local coordinates, if $h=2e^{2\psi}dzd\bar{w}$, we have

$\Delta_{h}\phi=\frac{e^{-2\psi}}{(\partial_{\bar{z}}\bar{w})(\partial_{w}z)}\bigg{[}\phi_{w\bar{z}}+\phi_{\bar{z}w}-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\phi_{\bar{z}}-\partial_{\bar{z}}(\log(\partial_{w}z))\phi_{w}\bigg{]}\
.$

###### Proof.

We start by computing the Christoffel symbols of the Levi-Civita connection of
$h$. We set

$\displaystyle\nabla_{\partial_{\bar{z}}}\partial_{\bar{z}}$
$\displaystyle=\Gamma_{\bar{z}\bar{z}}^{\bar{z}}\partial_{\bar{z}}+\Gamma_{\bar{z}\bar{z}}^{w}\partial_{w}\
\ \ \ \ \
\nabla_{\partial_{w}}\partial_{\bar{z}}=\Gamma_{w\bar{z}}^{\bar{z}}\partial_{\bar{z}}+\Gamma_{w\bar{z}}^{w}\partial_{w}$
$\displaystyle\nabla_{\partial_{\bar{z}}}\partial_{w}$
$\displaystyle=\Gamma_{\bar{z}w}^{\bar{z}}\partial_{\bar{z}}+\Gamma_{\bar{z}w}^{w}\partial_{w}\
\ \ \ \
\nabla_{\partial_{w}}\partial_{w}=\Gamma_{ww}^{\bar{z}}\partial_{\bar{z}}+\Gamma_{ww}^{w}\partial_{w}\
.$

Since $\nabla$ is compatible with the metric, we have

$0=\partial_{\bar{z}}(h(\partial_{\bar{z}},\partial_{\bar{z}}))=\frac{1}{2}h(\nabla_{\partial_{\bar{z}}}\partial_{\bar{z}},\partial_{\bar{z}})=\frac{1}{2}h(\Gamma_{\bar{z}\bar{z}}^{\bar{z}}\partial_{\bar{z}}+\Gamma_{\bar{z}\bar{z}}^{w}\partial_{w},\partial_{\bar{z}})=\frac{1}{2}\Gamma_{\bar{z}\bar{z}}^{w}e^{2\psi}(\partial_{w}z)(\partial_{\bar{z}}\bar{w})$

and we conclude that $\Gamma_{\bar{z}\bar{z}}^{w}=0$. With a similar
computation, we get that
$\Gamma_{\bar{z}w}^{\bar{z}}=\Gamma_{w\bar{z}}^{w}=\Gamma_{ww}^{\bar{z}}=0$.
Moreover,

$\displaystyle\Gamma_{\bar{z}\bar{z}}^{\bar{z}}h(\partial_{\bar{z}},\partial_{w})+\Gamma_{\bar{z}w}^{w}h(\partial_{\bar{z}},\partial_{w})$
$\displaystyle=h(\nabla_{\partial_{\bar{z}}}\partial_{\bar{z}},\partial_{w})+h(\partial_{\bar{z}},\nabla_{\partial_{\bar{z}}}\partial_{w})$
$\displaystyle=\partial_{\bar{z}}(h(\partial_{\bar{z}},\partial_{w}))=\partial_{\bar{z}}(e^{2\psi}(\partial_{\bar{z}}\bar{w})(\partial_{w}z))$
$\displaystyle=e^{2\psi}[2\psi_{\bar{z}}(\partial_{\bar{z}}\bar{w})(\partial_{w}z)+(\partial_{\bar{z}}\partial_{\bar{z}}\bar{w})(\partial_{w}z)+(\partial_{\bar{z}}\bar{w})(\partial_{\bar{z}}\partial_{w}z)]$

and dividing both sides by
$h(\partial_{\bar{z}},\partial_{w})=e^{2\psi}(\partial_{\bar{z}}\bar{w})(\partial_{w}z)$
we obtain

$\Gamma_{\bar{z}\bar{z}}^{\bar{z}}+\Gamma_{\bar{z}w}^{w}=2\psi_{\bar{z}}+\partial_{\bar{z}}(\log(\partial_{\bar{z}}\bar{w}))+\partial_{\bar{z}}(\log(\partial_{w}z))\
.$ (7)

Similarly,

$\displaystyle\Gamma_{w\bar{z}}^{\bar{z}}h(\partial_{\bar{z}},\partial_{w})+\Gamma_{ww}^{w}h(\partial_{\bar{z}},\partial_{w})$
$\displaystyle=h(\nabla_{w}\partial_{\bar{z}},\partial_{w})+h(\partial_{\bar{z}},\nabla_{\partial_{w}}\partial_{w})$
$\displaystyle=\partial_{w}(h(\partial_{\bar{z}},\partial_{w}))=\partial_{w}(e^{2\psi}(\partial_{\bar{z}}\bar{w})(\partial_{w}z))$
$\displaystyle=e^{2\psi}[2\psi_{w}(\partial_{\bar{z}}\bar{w})(\partial_{w}z)+(\partial_{w}\partial_{\bar{z}}\bar{w})(\partial_{w}z)+(\partial_{\bar{z}}\bar{w})(\partial_{w}\partial_{w}z)]$

so that

$\Gamma_{w\bar{z}}^{\bar{z}}+\Gamma_{ww}^{w}=2\psi_{w}+\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))+\partial_{w}(\log(\partial_{w}z))\
.$ (8)

In addition, since $\nabla$ is torsion-free, we have

$\Gamma_{\bar{z}w}^{w}\partial_{w}-\Gamma_{w\bar{z}}^{\bar{z}}\partial_{\bar{z}}=\nabla_{\partial_{\bar{z}}}\partial_{w}-\nabla_{\partial_{w}}\partial_{\bar{z}}=[\partial_{\bar{z}},\partial_{w}]\
.$

Comparing this last equation with Lemma 3.12, we deduce that

$\Gamma_{\bar{z}w}^{w}=\partial_{\bar{z}}(\log(\partial_{w}z))\ \ \
\text{and}\ \ \
\Gamma_{w\bar{z}}^{\bar{z}}=\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\ $
(9)

and, replacing these Christoffel symbols into Equation (7) and (8), we finally
get

$\Gamma_{\bar{z}\bar{z}}^{\bar{z}}=2\psi_{\bar{z}}+\partial_{\bar{z}}(\log(\partial_{\bar{z}}\bar{w}))\
\ \ \text{and}\ \ \
\Gamma_{ww}^{w}=2\psi_{w}+\partial_{w}(\log(\partial_{w}z))\ .$ (10)

It is now straightforward to verify that the matrix representing
$\nabla^{2}\phi$ in the basis $\\{\partial_{\bar{z}},\partial_{w}\\}$ is

$\nabla^{2}\phi=\begin{pmatrix}\phi_{\bar{z}\bar{z}}-[2\psi_{\bar{z}}+\partial_{\bar{z}}(\log(\partial_{\bar{z}}\bar{w}))]\phi_{\bar{z}}&\phi_{\bar{z}w}-\partial_{\bar{z}}(\log(\partial_{w}z))\phi_{w}\\\
\phi_{w\bar{z}}-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\phi_{\bar{z}}&\phi_{ww}-[2\psi_{w}+\partial_{w}(\log(\partial_{w}z))]\phi_{w}\end{pmatrix}$

and, since $h$ in the same basis can be expressed by

$h=e^{2\psi}(\partial_{\bar{z}}\bar{w})(\partial_{w}z)\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\ ,$

we conclude that

$\displaystyle\Delta_{h}\phi$
$\displaystyle=\mathrm{tr}_{h}(\nabla^{2}\phi)=\mathrm{tr}(h^{-1}\nabla^{2}\phi)$
$\displaystyle=\frac{e^{-2\psi}}{(\partial_{\bar{z}}\bar{w})(\partial_{w}z)}\bigg{[}\phi_{w\bar{z}}+\phi_{\bar{z}w}-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\phi_{\bar{z}}-\partial_{\bar{z}}(\log(\partial_{w}z))\phi_{w}\bigg{]}\
.$

∎

######  Remark 3.15.

This is the same operator defined in [Kim07], which extends holomorphically
the Laplacian of a Riemannian metric on $S$.

Similar to the case of Riemannian metrics on $S$, the Laplacian operator is
related to its compatible $\mathbb{C}$-complex structure $\mathcal{J}$:

######  Proposition 3.16.

Let $h$ be a positive complex metric on $S$ with compatible
$\mathbb{C}$-complex structure $\mathcal{J}$ and area form $dA_{h}$. Then for
any smooth function $\phi:S\rightarrow\mathbb{C}$, we have

$d(d\phi\circ\mathcal{J})=(\Delta_{h}\phi)dA_{h}\ .$

###### Proof.

Since

$\displaystyle(d\phi\circ\mathcal{J})(\partial_{\bar{z}})$
$\displaystyle=d\phi(\mathcal{J}\partial_{\bar{z}})=i\phi_{\bar{z}}$
$\displaystyle(d\phi\circ\mathcal{J})(\partial_{w})$
$\displaystyle=d\phi(\mathcal{J}\partial_{\bar{z}})=-i\phi_{w}\ ,$

the $1$-form $d\phi\circ\mathcal{J}$ can be written in local coordinates as

$d\phi\circ\mathcal{J}=-i\frac{\phi_{w}}{\partial_{w}z}dz+i\frac{\phi_{\bar{z}}}{\partial_{\bar{z}}\bar{w}}d\bar{w}\
.$

Now, if $\alpha=Adz+B\bar{w}$ is any $1$-form, then

$\displaystyle d\alpha(\partial_{\bar{z}},\partial_{w})$
$\displaystyle=\partial_{\bar{z}}(\alpha(\partial_{w}))-\partial_{w}(\alpha(\partial_{\bar{z}}))-\alpha([\partial_{\bar{z}},\partial_{w}])$
$\displaystyle=\partial_{\bar{z}}(A(\partial_{w}z))-\partial_{w}(B(\partial_{\bar{z}}\bar{w}))-\alpha([\partial_{\bar{z}},\partial_{w}])$
$\displaystyle=A_{\bar{z}}(\partial_{w}z)+A(\partial_{\bar{z}}\partial_{w}z)-B_{w}(\partial_{\bar{z}}\bar{w})-B(\partial_{w}\partial_{\bar{z}}\bar{w})$
$\displaystyle\ \ \
-\alpha(\partial_{\bar{z}}(\log(\partial_{w}z))\partial_{w}-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\partial_{\bar{z}})$
(Lemma 3.12)
$\displaystyle=A_{\bar{z}}(\partial_{w}z)-B_{w}(\partial_{\bar{z}}\bar{w})\ .$

Therefore, the local expression of $d\alpha$ is

$d\alpha=\left[\frac{B_{w}}{\partial_{w}z}-\frac{A_{\bar{z}}}{\partial_{\bar{z}}\bar{w}}\right]dz\wedge
d\bar{w}\ .$

Applying this formula to $\alpha=d\phi\circ\mathcal{J}$, where
$A=-i\frac{\phi_{w}}{\partial_{w}z}$ and
$B=i\frac{\phi_{\bar{z}}}{\partial_{\bar{z}}\bar{w}}$, we get

$\displaystyle d(d\phi\circ\mathcal{J})$
$\displaystyle=\frac{i}{(\partial_{w}z)(\partial_{\bar{z}}\bar{w})}\bigg{[}\phi_{\bar{z}w}+\phi_{w\bar{z}}-\partial_{\bar{z}}(\log(\partial_{\bar{z}}\bar{w}))\phi_{\bar{z}}-\partial_{\bar{z}}(\log(\partial_{w}z))\phi_{w}\bigg{]}dz\wedge
d\bar{w}$ $\displaystyle=(\Delta_{h}\phi)dA_{h}$ (Proposition 3.14 and
Equation (1))

as claimed. ∎

Finally, the formula for the curvature of a conformal Riemannian metric on a
surface can be extended to positive complex metrics:

######  Theorem 3.17.

Let $h=2e^{2\psi}dzd\bar{w}$ be a positive complex metric on $S$. Then its
complex Gaussian curvature can be computed as

$K_{h}=-\Delta_{h}\psi=-\frac{1}{2}\Delta_{h}\log(h)\ .$

###### Proof.

We first compute the Riemann tensor in local coordinates:

$\displaystyle
R^{\nabla}(\partial_{\bar{z}},\partial_{w},\partial_{\bar{z}},\partial_{w})$
$\displaystyle=h(\nabla_{\partial_{\bar{z}}}\nabla_{\partial_{w}}\partial_{\bar{z}},\partial_{w})-h(\nabla_{\partial_{w}}\nabla_{\partial_{\bar{z}}}\partial_{\bar{z}},\partial_{w})-h(\nabla_{[\partial_{\bar{z}},\partial_{w}]}\partial_{\bar{z}},\partial_{w})$
$\displaystyle=h(\nabla_{\partial_{\bar{z}}}(\Gamma_{w\bar{z}}^{\bar{z}}\partial_{\bar{z}}),\partial_{w})-h(\nabla_{\partial_{w}}(\Gamma_{\bar{z}\bar{z}}^{\bar{z}}\partial_{\bar{z}},\partial_{w})-h(\nabla_{\Gamma_{\bar{z}w}^{w}\partial_{w}-\Gamma_{w\bar{z}}^{\bar{z}}\partial_{\bar{z}}}\partial_{\bar{z}},\partial_{w})$
$\displaystyle=h(\partial_{\bar{z}},\partial_{w})\left[\partial_{\bar{z}}\Gamma_{w\bar{z}}^{\bar{z}}-\partial_{w}\Gamma_{\bar{z}\bar{z}}^{\bar{z}}+\Gamma_{w\bar{z}}^{\bar{z}}\Gamma_{\bar{z}\bar{z}}^{\bar{z}}-\Gamma_{\bar{z}w}^{w}\Gamma_{w\bar{z}}^{\bar{z}}\right]\
.$

Then, by Equation (9) and (10), we find that

$\displaystyle K_{h}$
$\displaystyle=\frac{R^{\nabla}(\partial_{\bar{z}},\partial_{w},\partial_{\bar{z}},\partial_{w})}{h(\partial_{\bar{z}},\partial_{w})^{2}}=\frac{1}{h(\partial_{\bar{z}},\partial_{w})}\left[\partial_{\bar{z}}\Gamma_{w\bar{z}}^{\bar{z}}-\partial_{w}\Gamma_{\bar{z}\bar{z}}^{\bar{z}}+\Gamma_{w\bar{z}}^{\bar{z}}\Gamma_{\bar{z}\bar{z}}^{\bar{z}}-\Gamma_{\bar{z}w}^{w}\Gamma_{w\bar{z}}^{\bar{z}}\right]$
$\displaystyle=\frac{1}{h(\partial_{\bar{z}},\partial_{w})}\bigg{[}\partial_{\bar{z}}\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))-2\psi_{w\bar{z}}-\partial_{w}\partial_{\bar{z}}(\log(\partial_{\bar{z}}\bar{w}))$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\Big{(}2\psi_{\bar{z}}+\partial_{\bar{z}}\log(\partial_{\bar{z}}\bar{w})-\partial_{\bar{z}}(\log(\partial_{w}z))\Big{)}\bigg{]}$
$\displaystyle=\frac{-1}{h(\partial_{\bar{z}},\partial_{w})}[2\psi_{w\bar{z}}-2\psi_{\bar{z}}\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))]$
$\displaystyle=\frac{-1}{h(\partial_{\bar{z}},\partial_{w})}[\psi_{w\bar{z}}+[\partial_{w},\partial_{\bar{z}}](\psi)+\psi_{\bar{z}w}-2\psi_{\bar{z}}\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))]$
$\displaystyle=\frac{-1}{h(\partial_{\bar{z}},\partial_{w})}[\psi_{w\bar{z}}+\psi_{\bar{z}w}-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\psi_{\bar{z}}-\partial_{\bar{z}}(\log(\partial_{w}z))\psi_{w}]$
(Lemma 3.12)
$\displaystyle=\frac{-1}{e^{2\psi}(\partial_{\bar{z}}\bar{w})(\partial_{w}z)}[\psi_{w\bar{z}}+\psi_{\bar{z}w}-\partial_{w}(\log(\partial_{\bar{z}}\bar{w}))\psi_{\bar{z}}-\partial_{\bar{z}}(\log(\partial_{w}z))\psi_{w}]$
$\displaystyle=-\Delta_{h}\psi$ (Proposition 3.14)

∎

######  Corollary 3.18.

Let $h=e^{2\phi}g$ be a positive complex metric conformal to $g$. Then

$K_{h}=e^{-2\phi}(K_{g}-\Delta_{g}\phi)$

###### Proof.

In local coordinates, $g=2e^{2\psi}dzd\bar{w}$ so that

$K_{h}=-\Delta_{h}(\psi+\phi)=-e^{-2\phi}\Delta_{g}(\psi+\phi)=e^{-2\phi}(K_{g}-\Delta_{g}\phi)\
.$

∎

We now turn our attention to finding a local expression for the tensor
$C^{+}(\cdot,\cdot,\cdot)=-h(\mathbf{P}\mathrm{I}\mathrm{I}(\cdot,\cdot),\cdot)$
introduced in Proposition 3.5 (see also Remark 3.9). We start by showing a
compatibility relation between $C^{+}$ and $\mathcal{J}$:

######  Lemma 3.19.

The tensor $C^{+}$ satisfies

$C^{+}(\mathcal{J}\cdot,\mathcal{J}\cdot,\mathcal{J}\cdot)=-C^{+}(\mathcal{J}\cdot,\cdot,\cdot)\
.$

###### Proof.

For any $X,Y,Z\in\Gamma(\mathbb{C}TU)$, we have

$\displaystyle C^{+}(\mathcal{J}X,\mathcal{J}Y,\mathcal{J}Z)$
$\displaystyle=-h(\mathbf{P}\mathrm{I}\mathrm{I}(\mathcal{J}X,\mathcal{J}Y),\mathcal{J}Z)$
$\displaystyle=h(A^{+}(\mathcal{J}X)\mathcal{J}Y,\mathcal{J}Z)$
($A^{+}=-\mathbf{P}\mathrm{I}\mathrm{I}$)
$\displaystyle=-h(\mathcal{J}A^{+}(\mathcal{J}X)\mathcal{J}Y,Z)\ .$
($h(\mathcal{J},\mathcal{J})=h$)

Comparing this last expression with

$-C^{+}(\mathcal{J}X,Y,Z)=-h(A^{+}(\mathcal{J}X)Y,Z)\ ,$

we see that the statement is proved if we show that $A^{+}(\cdot)$ anti-
commutes with $\mathcal{J}$. Now, since $\mathrm{I}\mathrm{I}$ is traceless,
also $A^{+}(\cdot)$ is traceless, and we already know that $A^{+}(\cdot)$ is
$h$-self-adjoint from the proof of Corollary 3.8. This implies that, in the
$h$-isotropic basis $\\{\partial_{\bar{z}},\partial_{w}\\}$, the endomorphism
$A^{+}(\cdot)$ is represented by a matrix of the form

$A^{+}(\cdot)=\begin{pmatrix}0&*\\\ **&0\end{pmatrix}\ ,$

where $*$ and $**$ denote a arbitrary $\mathbb{C}$-valued smooth functions.
Since, in the same basis, $\mathcal{J}=\mathrm{diag}(i,-i)$, anti-
commutativity of $A^{+}(\cdot)$ and $\mathcal{J}$ follows by an easy direct
computation. ∎

######  Theorem 3.20.

There is a $J_{1}$-holomorphic cubic differential $q_{1}$ and a
$J_{2}$-holomorphic cubic differential $q_{2}$ such that

$C^{+}=q_{1}+\overline{q}_{2}\ .$

As a consequence, in local coordinates

$C^{+}=\alpha(z)dz^{3}+\overline{\beta(w)}d\bar{w}^{3},$

for some $\alpha(z)$ and $\beta(w)$ holomorphic functions in their respective
variable.

###### Proof.

We do this computation in $h$-isotropic normal coordinates
$\\{\partial_{\bar{u}},\partial_{v}\\}$, which exist by applying a change of
basis to the usual normal coordinates of the Levi-Civita connection ([NS94]).
This means that locally $h=\rho dud\bar{v}$ and all Christoffel symbols vanish
at a given point $p\in S$. Note that the $\mathbb{C}$-complex structure
$\mathcal{J}$ must satisfy
$\mathcal{J}\partial_{\bar{u}}=-i\partial_{\bar{u}}$ and
$\mathcal{J}\partial_{v}=i\partial_{v}$ because it is compatible with $h$.
Since $C^{+}$ is a $(0,3)$-tensor, it must be written as

$C^{+}=a(u,\bar{v})du^{3}+b(u,\bar{v})dud\bar{v}^{2}+c(u,\bar{v})du^{2}d\bar{v}+d(u,\bar{v})d\bar{v}^{3}$

for some smooth functions $a,b,c$ and $d$. On the other hand, Lemma 3.19
implies that $b=c=0$. Indeed,

$\displaystyle ic(z,\bar{w})(\partial_{w}z)^{2}(\partial_{\bar{z}}\bar{w})$
$\displaystyle=iC^{+}(\partial_{\bar{z}},\partial_{w},\partial_{w})=-C^{+}(\mathcal{J}\partial_{\bar{z}},\partial_{w},\partial_{w})$
$\displaystyle=C^{+}(\mathcal{J}\partial_{\bar{z}},\mathcal{J}\partial_{w},\mathcal{J}\partial_{w})=-iC^{+}(\partial_{\bar{z}},\partial_{w},\partial_{w})$
$\displaystyle=-ic(z,\bar{w})(\partial_{w}z)^{2}(\partial_{\bar{z}}\bar{w})\ $

and the computation for $b$ is analogous.
Now, since $h^{-1}C^{+}=-\mathbf{P}\mathrm{I}\mathrm{I}$, the Codazzi equation
of $\mathrm{I}\mathrm{I}$ and $\nabla\mathbf{P}=\nabla h=0$ imply that
$d^{\nabla}C^{+}=0$, in other words

$(\nabla_{\partial_{\bar{u}}}C^{+})(\partial_{v},\cdot,\cdot)=(\nabla_{\partial_{v}}C^{+})(\partial_{\bar{u}},\cdot,\cdot)\
.$

Because we are working in normal coordinates, the equation above evaluated at
$p$ is equivalent to

$\partial_{\bar{u}}(C^{+}(\partial_{v},\cdot,\cdot))(p)=\partial_{v}(C^{+}(\partial_{\bar{u}},\cdot,\cdot))(p)\
.$

As a consequence,

$\begin{cases}\partial_{\bar{u}}(C^{+}(\partial_{v},\partial_{v},\partial_{v}))(p)=0\\\
\partial_{v}(C^{+}(\partial_{\bar{u}},\partial_{\bar{u}},\partial_{\bar{u}})(p)=0\end{cases}\Rightarrow\begin{cases}\partial_{\bar{u}}(a(u,\bar{v})(\partial_{v}u)^{3})(p)=0\\\
\partial_{v}(d(u,\bar{v})(\partial_{\bar{u}}\bar{v})^{3})(p)=0\end{cases}$

Since the vector fields $\partial_{\bar{u}}$ and $\partial_{v}$ commute in $p$
the last two equations simplify into

$\begin{cases}0=\partial_{\bar{u}}(a(u,\bar{v}))(p)=\partial_{\bar{v}}a(u,\bar{v})(p)\partial_{\bar{u}}\bar{v}(p)\\\
0=\partial_{v}(d(u,\bar{v}))(p)=\partial_{u}d(u,\bar{v})(p)\partial_{v}u(p)\end{cases}$

which imply that $\partial_{\bar{u}}(a(u,\bar{v}))(p)=0$ and
$\partial_{v}(d(u,\bar{v}))(p)=0$. This means that $a$ is a
$J_{1}$-holomorphic function and $d$ is $J_{2}$-anti-holomorphic. Therefore,
$C^{+}$ is the sum of a $J_{1}$-holomorphic and a $J_{2}$-anti-holomorphic
cubic differential. Since this last statement is independent of the chosen
coordinate system, the theorem is proved. ∎

We conclude this section by relating the norm of the second fundamental form
$\mathrm{I}\mathrm{I}$ to the holomorphic cubic differentials $q_{1}$ and
$q_{2}$.

######  Proposition 3.21.

Let $C^{+}=-h^{-1}\mathbf{P}\mathrm{I}\mathrm{I}=q_{1}+\bar{q}_{2}$. Then

$\|\mathrm{I}\mathrm{I}\|^{2}=-8\|C^{+}\|_{h}^{2}\ ,$

where the norm of $C^{+}$ is defined as

$\|C^{+}\|_{h}^{2}:=\frac{q_{1}\bar{q}_{2}}{h^{3}}\ .$

###### Proof.

In local $h$-isotropic coordinates $\\{\partial_{\bar{z}},\partial_{w}\\}$, we
write

$h=2e^{2\psi}dzd\bar{w}\ \ \ \ \text{and}\ \ \ \
C^{+}=q_{1}+\overline{q}_{2}=\alpha dz^{3}+\overline{\beta}d\bar{w}^{3}\ .$

Let
$s^{2}=h(\partial_{\bar{z}},\partial_{w})=e^{2\psi}(\partial_{w}z)(\partial_{\bar{z}}\bar{w})$
and consider the $h$-orthonormal frame

$\bigg{\\{}e_{1}:=\frac{1}{s\sqrt{2}}(\partial_{\bar{z}}+\partial_{w}),e_{2}:=\frac{i}{s\sqrt{2}}(\partial_{\bar{z}}-\partial_{w})\bigg{\\}}\
.$

Now,

$\mathrm{I}\mathrm{I}(e_{1},e_{1})=h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{1}),e_{1})\mathbf{P}e_{1}+g(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{1}),e_{2})\mathbf{P}e_{2}$

and, since $\mathbf{P}e_{i}$ have norm $-1$, we get

$\|\mathrm{I}\mathrm{I}(e_{1},e_{1})\|^{2}=-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{1}),e_{1})^{2}-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{1}),e_{2})^{2}\
.$

Similarly,

$\|\mathrm{I}\mathrm{I}(e_{1},e_{2})\|^{2}=-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{2}),e_{1})^{2}-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{2}),e_{2})^{2}\
.$

Each of these terms can be expressed using $\alpha$ and $\beta$. For instance,

$\displaystyle-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{1}),e_{1})$
$\displaystyle=C^{+}(e_{1},e_{1},e_{1})$
$\displaystyle=\frac{1}{2\sqrt{2}s^{3}}C^{+}(\partial_{\bar{z}}+\partial_{w},\partial_{\bar{z}}+\partial_{w},\partial_{\bar{z}}+\partial_{w})$
$\displaystyle=\frac{1}{2\sqrt{2}s^{3}}[(\partial_{\bar{z}}\bar{w})^{3}\overline{\beta}+(\partial_{w}z)^{3}\alpha]$

and, with similar computations,

$\displaystyle-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{1}),e_{2})$
$\displaystyle=\frac{i}{2\sqrt{2}s^{3}}[(\partial_{\bar{z}}\bar{w})^{3}\overline{\beta}-(\partial_{w}z)^{3}\alpha]$
$\displaystyle-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{1},e_{2}),e_{1})$
$\displaystyle=\frac{i}{2\sqrt{2}s^{3}}[(\partial_{\bar{z}}\bar{w})^{3}\overline{\beta}-(\partial_{w}z)^{3}\alpha]$
$\displaystyle-h(\mathbf{P}\mathrm{I}\mathrm{I}(e_{2},e_{2}),e_{2})$
$\displaystyle=-\frac{1}{2\sqrt{2}s^{3}}[(\partial_{\bar{z}}\bar{w})^{3}\overline{\beta}+(\partial_{w}z)^{3}\alpha]\
.$

Therefore,

$\displaystyle\|\mathrm{I}\mathrm{I}\|^{2}$
$\displaystyle=\|\mathrm{I}\mathrm{I}(e_{1},e_{1})\|^{2}+\|\mathrm{I}\mathrm{I}(e_{1},e_{2})\|^{2}=-\frac{(\partial_{w}z)^{3}(\partial_{\bar{z}}\bar{w})^{3}\alpha\overline{\beta}}{s^{6}}$
$\displaystyle=-\frac{\alpha\overline{\beta}}{e^{6\psi}}=-8\frac{q_{1}\overline{q}_{2}}{h^{3}}=-8\|C^{+}\|^{2}_{h}\
.$

∎

######  Corollary 3.22.

The Gauss equation of a smooth complex Lagrangian minimal immersion with
positive complex-valued first fundamental form $h=2e^{2\psi}dzd\bar{w}$ and
with holomorphic cubic differentials $q_{1}$ and ${q}_{2}$ is given in local
isotropic coordinates by

$\Delta_{h}\psi+q_{1}\overline{q}_{2}e^{-6\psi}-1=0$

###### Proof.

It follows immediately from Theorem 3.10, Theorem 3.17 and Proposition 3.21. ∎

###  3.3. Bi-complex Higgs bundles

In this section we introduce $\mathrm{SL}(n,\mathbb{C})$-bi-complex Higgs
bundles and use them to show that solving the Gauss-Codazzi equations found in
Theorem 3.10 is sufficient to find a complex Lagrangian minimal surface in
$\mathbb{CH}^{2}_{\tau}$. This notion can be easily extended to any semisimple
complex Lie group, but we restrict to $\mathrm{SL}(n,\mathbb{C})$ in order to
keep the length of this paper somewhat limited.

Let $S$ be a closed connected oriented surface endowed with a
$\mathbb{C}$-complex structure $\mathcal{J}$. It is useful to consider the bi-
complexified tangent space
$\mathbb{C}_{\tau}TS:=\mathbb{C}TS\otimes_{\mathbb{C}}\mathbb{C}_{\tau}$ and
cotangent space
$\mathbb{C}_{\tau}T^{*}S:=\mathbb{C}T^{*}S\otimes_{\mathbb{C}}\mathbb{C}_{\tau}$.
We will call sections of $\mathbb{C}_{\tau}TS$ and of
$\mathbb{C}_{\tau}T^{*}S$ bi-complex vector fields and bi-complex $1$-forms,
respectively. The $\mathbb{C}$-complex structure $\mathcal{J}$ extends to
$\mathbb{C}_{\tau}TS$ by $\mathbb{C}_{\tau}$-linearity so that we can
decompose $\mathbb{C}_{\tau}TS$ into $\mathcal{J}$-eigenspaces

$\mathbb{C}_{\tau}TS=V_{i}(\mathcal{J})\oplus
V_{-i}(\mathcal{J})=\mathbb{C}_{\tau}TS^{(1,0)}\oplus\mathbb{C}_{\tau}TS^{(0,1)}\
.$

A similar decomposition can be obtained for $\mathbb{C}_{\tau}T^{*}S$ by
considering the action of the adjoint of $\mathcal{J}$.

######  Definition 3.23.

A bi-complex vector bundle of rank $k$ over $S$ is a complex vector bundle $E$
over $S$ with fiber $\mathbb{C}^{2k}$ endowed fiberwise with a para-complex
structure $\mathbf{T}$ such that locally, over an open set $U$ of $S$, we have
a diffeomorphism

$\displaystyle\varphi_{U}:E_{|_{U}}\rightarrow U\times\mathbb{C}^{k}_{\tau}$

such that for all $p\in U$, its restriction $\varphi_{p}$ at each fiber is a
para-complex $\mathbb{C}$-linear isomorphism, i.e.
$\varphi_{p}(\mathbf{T}v)=\tau\varphi_{p}(v)$ for all $v\in E_{p}$.

######  Remark 3.24.

We will often think of a bi-complex vector bundle of rank $k$ simply as a
vector bundle with fiber $\mathbb{C}^{k}_{\tau}$. Moreover, we remark that the
fibers of a bi-complex vector bundle are naturally equipped with a
$\mathbb{C}$-complex structure induced by the multiplication by
$j\in\mathbb{C}_{\tau}$.

It follows immediately from the definition that transition functions of a bi-
complex vector bundle of rank $k$ belong to a subgroup of
$\mathrm{GL}(2k,\mathbb{C})$ isomorphic to
$\mathrm{GL}(k,\mathbb{C})\times\mathrm{GL}(k,\mathbb{C})\cong\mathrm{GL}(k,\mathbb{C}_{\tau})$.

######  Definition 3.25.

We say that a bi-complex vector bundle $E$ over $S$ is
$\mathcal{J}$-holomorphic if its transition functions
$g_{\alpha\beta}:U_{\alpha}\cap
U_{\beta}\rightarrow\mathrm{GL}(k,\mathbb{C}_{\tau})$ are
$\mathcal{J}$-holomorphic, in the sense that they satisfy

$dg_{\alpha\beta}(\mathcal{J}v)=jdg_{\alpha\beta}(v)\ \ \ \ {\text{for all}\
v\in\mathbb{C}TS}\ .$

On a $\mathcal{J}$-holomorphic bi-complex vector bundle we can define an
analog of a $\bar{\partial}$-operator as follows. First, given a smooth
function $f:U\subset S\rightarrow\mathbb{C}^{k}_{\tau}$, we denote by
$\bar{\partial}^{\mathcal{J}}f$ the $\mathcal{J}$-antiliner part of $df$, in
other words

$\bar{\partial}^{\mathcal{J}}f:=\frac{1}{2}\bigg{(}df-
jdf(\mathcal{J})\bigg{)}\ .$

Now, if $s=\sum_{i=1}^{k}s_{i}e_{i}$ is a local section of $E$, where
$s_{i}:U\rightarrow\mathbb{C}^{k}_{\tau}$ are smooth functions and
$\\{e_{1},\dots,e_{k}\\}$ is a basis of $\mathbb{C}^{k}_{\tau}$ as
$\mathbb{C}_{\tau}$-module, since the transition functions of $E$ are
$\mathcal{J}$-holomorphic, the operator

$\bar{\partial}^{E}(s):=\sum_{i=1}^{k}(\bar{\partial}^{\mathcal{J}}s_{i})\otimes
e_{i}\ $

does not depend on the local trivialization and thus it is defined globally as
a map
$\bar{\partial}^{E}:\Gamma(E)\rightarrow\Omega^{0,1}(S,\mathbb{C}_{\tau})\otimes\Gamma(E)$.
We say that a section $s$ of $E$ is $\mathcal{J}$-holomorphic if
$\bar{\partial}^{E}s=0$.

######  Definition 3.26.

A $\mathbb{C}_{\tau}$-orthogonal structure $Q$ on a bi-complex vector bundle
$E$ is a smooth choice of non-degenerate $\mathbb{C}_{\tau}$-bilinear inner
products on each fiber.

######  Definition 3.27.

An $\mathrm{SL}(n,\mathbb{C})$-bi-complex Higgs bundle on $(S,\mathcal{J})$ is
a triple $(E,Q,\phi)$ such that

  * •

$E$ is a $\mathcal{J}$-holomorphic bi-complex vector bundle of rank $n$ with
$\bigwedge^{2n}E\cong\mathcal{O}$;

  * •

$Q$ is a $\mathbb{C}_{\tau}$-orthogonal structure on $E$;

  * •

$\phi$ is a $\mathcal{J}$-holomorphic section of
$\mathrm{End}_{0}(E,\mathbf{T})\otimes\mathbb{C}_{\tau}T^{*}S$ such that
$\phi^{t}Q+Q\phi=0$, which we call _bi-complex Higgs field_.

Here $\mathrm{End}_{0}(E,\mathbf{T})$ denotes the bundle of traceless
$\mathbb{C}$-linear automorphismsm of $E$ that preserve $\mathbf{T}$, or,
equivalently, after indentifying each fiber of $E$ with
$\mathbb{C}^{n}_{\tau}$, the bundle of traceless $\mathbb{C}_{\tau}$-linear
automorphisms of $E$.

From an $\mathrm{SL}(n,\mathbb{C})$-bi-complex Higgs bundle, it is possible to
build a flat $\mathrm{SL}(n,\mathbb{C})$-connection following ideas similar to
the classical theory of Higgs bundles.

######  Definition 3.28.

A complex $\mathcal{J}$-Hermitian metric $H$ on a bi-complex vector bundle $E$
is a smooth choice of non-degenerate $\mathbb{C}$-bilinear
$\mathcal{J}$-Hermitian inner products on each fiber. We say that a
$\mathbb{C}_{\tau}$-linear connection $D^{H}$ on $E$ is compatible with $H$ if

$V(H(s_{1},s_{2}))=H(D^{H}_{V}s_{1},s_{2})+H(s_{1},D^{H}_{\overline{V}^{\tau}}s_{2})$

for all sections $s_{1},s_{2}$ of $E$ and for all bi-complex vector fields
$V$.

######  Theorem 3.29.

Let $E$ be a $\mathcal{J}$-holomorphic vector bundle endowed with a complex
$\mathcal{J}$-Hermitian metric. Then there exists a unique
$\mathbb{C}_{\tau}$-linear connection $D^{H}$ on $E$, called _Chern
connection_ , that is compatible with $H$ and whose $(0,1)$-part coincides
with $\bar{\partial}^{E}$.

###### Proof.

Let $V$ be a $\mathcal{J}$-holomorphic bi-complex vector field. Then
$D^{H}_{\overline{V}^{\tau}}=\bar{\partial}^{E}_{\overline{V}^{\tau}}$, so the
connection $D^{H}$ is completely determined once we find $D^{H}_{V}$. Since
$D^{H}$ must be compatible with the metric, we must have

$H(D^{H}_{V}s_{1},s_{2})=V(H(s_{1},s_{2}))-H(s_{1},\bar{\partial}^{E}_{\overline{V}^{\tau}}s_{2})$

for all sections $s_{1}$ and $s_{2}$ of $E$. Since $H$ is non-degenerate, this
equation characterizes $D^{H}_{V}$ uniquely. ∎

Given an $\mathrm{SL}(n,\mathbb{C})$-bi-complex Higgs bundle $(E,Q,\phi)$ we
look for a complex $\mathcal{J}$-Hermitian metric $H$ on $E$ that is
compatible with $Q$ (i.e. $H$ commutes with $Q$ if seen as isomorphisms
between $E$ and $E^{*}$) such that the connection

$\nabla=D^{H}+\phi+\phi^{*H}$

is flat, in other words $H$ satisfies Hitchin’s equation

$F^{\nabla}=F^{H}+\phi\wedge\phi^{*H}=0\ .$ (11)

Clearly, in this setting Hitchin ([Hit87]) and Simpson’s ([Sim92]) work cannot
be applied and the existence of such metric $H$ is not guaranteed. However, if
a solution to Equation (11) can be found, then, using the same argument as in
the classical Higgs bundle theory, the holonomy of $\nabla$ lies in a subgroup
of $\mathrm{GL}(n,\mathbb{C}_{\tau})$ isomorphic to
$\mathrm{SL}(n,\mathbb{C})$.

We will now see how we can associate to an equivariant complex Lagrangian
minimal surface in $\mathbb{CH}^{2}_{\tau}$ an $\mathrm{SL}(3,\mathbb{C})$-bi-
complex Higgs bundle in such a way that Gauss-Codazzi equations correspond to
Hitchin’s equation and to the $\mathcal{J}$-holomorphicity of the bi-complex
Higgs field.

Let $\sigma:\tilde{S}\rightarrow\mathbb{CH}^{2}_{\tau}$ be an equivariant
complex Lagrangian minimal surface. We assume that its complex first
fundamental form $h$ is a positive complex metric and we fix local holomorphic
coordinates $z$ and $w$ such that locally $h=2e^{2\psi}dzd\bar{w}$ for some
complex-valued smooth function $\psi$. We will still denote with $\mathcal{J}$
the $\mathbb{C}$-complex structure compatible with $h$. By an argument similar
to that in Lemma 4.6, we can find a lift
$\tilde{\sigma}:\tilde{S}\rightarrow\mathbb{C}_{\tau}^{3}$ such that
$\mathbf{q}(\tilde{\sigma},\tilde{\sigma})=1$ and
$\mathbf{q}(\tilde{\sigma}_{\bar{z}},\tilde{\sigma})=\mathbf{q}(\tilde{\sigma}_{w},\tilde{\sigma})=0$.
Moreover, it will be convenient to set

$s^{2}=\mathbf{q}(\tilde{\sigma}_{\bar{z}},\tilde{\sigma}_{w})=e^{2\psi}(\partial_{\bar{z}}\bar{w})(\partial_{w}z)\
.$

We also write the tensor $C^{+}$ in local coordinates as $C^{+}=\alpha
dz^{3}+\bar{\beta}d\bar{w}^{3}$. Now, the set
$\\{e_{1}:=\tilde{\sigma}_{\bar{z}}/s,e_{2}:=\tilde{\sigma}_{w}/s,\tilde{\sigma}\\}$
is a reference frame for $\mathbb{C}_{\tau}^{3}$ as a
$\mathbb{C}_{\tau}$-module. The pull-back of the standard flat connection on
$\mathbb{C}_{\tau}^{3}$ can be represented in this frame by the endomorphisms-
valued $1$-form $\Omega:=F^{-1}dF$, where $F$ is the matrix whose columns are
given by the vectors $e_{1},e_{2}$, and $\tilde{\sigma}$ in this order.
Exploiting the geometry of complex Lagrangian minimal surfaces, one can
compute that
$\Omega=\frac{\hat{A}}{\partial_{w}z}dz+\frac{\hat{B}}{\partial_{\bar{z}}\bar{w}}d\bar{w}$,
where

$\hat{A}=\begin{pmatrix}-\psi_{w}+\frac{1}{2}\partial_{w}\log(\partial_{\bar{z}}\bar{w})-\frac{1}{2}\partial_{w}\log(\partial_{w}z)&-\frac{1}{s^{2}}\tau\alpha(\partial_{w}z)^{3}&0\\\
0&\psi_{w}-\frac{1}{2}\partial_{w}\log(\partial_{\bar{z}}\bar{w})+\frac{1}{2}\partial_{w}\log(\partial_{w}z)&s\\\
s&0&0\end{pmatrix}\ \ \ $

and

$\hat{B}=\begin{pmatrix}\psi_{\bar{z}}+\frac{1}{2}\partial_{\bar{z}}\log(\partial_{\bar{z}}\bar{w})-\frac{1}{2}\partial_{\bar{z}}\log(\partial_{w}z)&0&s\\\
-\frac{1}{s^{2}}\tau\bar{\beta}(\partial_{\bar{z}}\bar{w})^{3}&-\psi_{\bar{z}}-\frac{1}{2}\partial_{\bar{z}}\log(\partial_{\bar{z}}\bar{w})+\frac{1}{2}\partial_{\bar{z}}\log(\partial_{w}z)&0\\\
0&s&0\end{pmatrix}\ .$

We explain how to find the first column of $\hat{A}$; the other entries of the
two matrices can be found with similar computations and are left to the
interested reader. First, we observe that
$\hat{A}=\Omega(\partial_{w})=F^{-1}F_{w}$, thus the entries of its first
column are the coordinates of $\partial_{w}e_{1}$ with respect to our
reference frame. Simple linear algebra shows (recall that $e_{1}$ and $e_{2}$
are isotropic and their pairing is equal to $1$, whereas
$\mathbf{q}(\tilde{\sigma},\tilde{\sigma})=-1$) that

$\partial_{w}e_{1}=\mathbf{q}(\partial_{w}e_{1},e_{2})e_{1}+\mathbf{q}(\partial_{w}e_{1},e_{1})e_{2}-\mathbf{q}(\partial_{w}e_{1},\tilde{\sigma})\tilde{\sigma},$

thus it is enough to compute the $\mathbf{q}$-pairing between
$\partial_{w}e_{1}$ and the vectors of the frame:

  1. i)

$\mathbf{q}(\partial_{w}e_{1},\tilde{\sigma})=\partial_{w}\mathbf{q}(e_{1},\tilde{\sigma})-\mathbf{q}(e_{1},se_{2})=-s$

  2. ii)

$\mathbf{q}(\partial_{w}e_{1},e_{1})=-\frac{1}{s}s_{w}\mathbf{q}(e_{1},e_{2})+\frac{1}{s^{2}}\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\tilde{\sigma}_{\bar{z}})=0$
because

     * •

$2\mathrm{Re}_{\tau}(\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\tilde{\sigma}_{\bar{z}}))=\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\tilde{\sigma}_{\bar{z}})+\mathbf{q}(\tilde{\sigma}_{\bar{z}},\tilde{\sigma}_{w\bar{z}})=\partial_{w}\mathbf{q}(\tilde{\sigma}_{\bar{z}},\tilde{\sigma}_{\bar{z}})=0$
;

     * •

$\\!\begin{aligned}
-\mathrm{Im}_{\tau}(\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\tilde{\sigma}_{\bar{z}}))&=\mathrm{Re}_{\tau}(\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\tau\tilde{\sigma}_{\bar{z}}))=\mathrm{Re}_{\tau}(\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\mathbf{P}\tilde{\sigma}_{\bar{z}}))\\\
&=g(\mathrm{I}\mathrm{I}(\partial_{w},\partial_{\bar{z}}),\mathbf{P}\tilde{\sigma}_{\bar{z}})=C^{+}(\partial_{w},\partial_{\bar{z}},\partial_{\bar{z}})=0\end{aligned}$

  3. iii)

$\mathbf{q}(\partial_{w}e_{1},e_{2})=-\frac{1}{s}s_{w}\mathbf{q}(e_{1},e_{2})+\frac{1}{s^{2}}\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\tilde{\sigma}_{w})=-\frac{1}{2}\partial_{w}\log(s^{2})+\partial_{w}\log(\partial_{\bar{z}}\bar{w})$,
because, by Lemma 3.12,
$\mathbf{q}(\tilde{\sigma}_{w\bar{z}},\tilde{\sigma}_{w})=\mathbf{q}(\tilde{\sigma}_{\bar{z}w},\tilde{\sigma}_{w})+s^{2}\partial_{w}\log(\partial_{\bar{z}}\bar{w})$
and $\mathbf{q}(\tilde{\sigma}_{\bar{z}w},\tilde{\sigma}_{w})$ can be computed
as follows:

     * •

$\mathrm{Re}_{\tau}(\mathbf{q}(\tilde{\sigma}_{\bar{z}w},\tilde{\sigma}_{w}))=\frac{1}{2}\big{(}\mathbf{q}(\tilde{\sigma}_{\bar{z}w},\tilde{\sigma}_{w})+\mathbf{q}(\tilde{\sigma}_{w},\tilde{\sigma}_{\bar{z}w})\big{)}=\partial_{\bar{z}}\mathbf{q}(\tilde{\sigma}_{w},\tilde{\sigma}_{w})=0$
;

     * •

$\\!\begin{aligned}
-\mathrm{Im}_{\tau}(\mathbf{q}(\tilde{\sigma}_{\bar{z}w},\tilde{\sigma}_{w}))&=\mathrm{Re}_{\tau}(\mathbf{q}(\tilde{\sigma}_{\bar{z}w},\tau\tilde{\sigma}_{w}))=g(\mathrm{I}\mathrm{I}(\partial_{\bar{z}},\partial_{w}),\mathbf{P}\tilde{\sigma}_{w})\\\
&=C^{+}(\partial_{\bar{z}},\partial_{w},\partial_{w})=0\end{aligned}$

Flatness of the standard flat connection translates into the Maurer-Cartan
equation for the form $\Omega$

$d\Omega+\Omega\wedge\Omega=0\ .$