# On a Bers theorem for $\mathrm{SL}(3,\operatorname{\mathbb{C}})$

Christian El Emam  Christian El Emam: University of Luxembourg, Maison du
Nombre, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg.
<EMAIL_ADDRESS>and  Nathaniel Sagman  Nathaniel Sagman: University of
Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette,
Luxembourg<EMAIL_ADDRESS>
###### Abstract.

For $S$ a closed surface of genus at least $2$, let $\mathrm{Hit}_{3}(S)$ be
the Hitchin component of representations to $\mathrm{SL}(3,\mathbb{R}),$
equipped with the Labourie-Loftin complex structure. We construct a mapping
class group equivariant holomorphic map from a large open subset of
$\mathrm{Hit}_{3}(S)\times\overline{\mathrm{Hit}_{3}(S)}$ to the
$\mathrm{SL}(3,\operatorname{\mathbb{C}})$-character variety that restricts to
the identity on the diagonal and to Bers’ simultaneous uniformization on
$\mathcal{T}(S)\times\overline{\mathcal{T}(S)}$. The open subset contains
$\mathrm{Hit}_{3}(S)\times\overline{\mathcal{T}(S)}$ and
$\mathcal{T}(S)\times\overline{\mathrm{Hit}_{3}(S)}$, and the image includes
the holonomies of $\mathrm{SL}(3,\operatorname{\mathbb{C}})$-opers.

The map is realized by associating pairs of Hitchin representations to
immersions into $\operatorname{\mathbb{C}}^{3}$ that we call complex affine
spheres, which are equivalent to certain conformal harmonic maps into
$\mathrm{SL}(3,\operatorname{\mathbb{C}})/\mathrm{SO}(3,\operatorname{\mathbb{C}})$
and to new objects called bi-Higgs bundles. Complex affine spheres are
obtained by solving a second-order complex elliptic PDE that resembles both
the Beltrami and Tzitzéica equations. To study this equation we establish
analytic results that should be of independent interest.

###### Contents

  1. 1 Introduction
    1. 1.1 The main result
    2. 1.2 The space of complex affine spheres
    3. 1.3 Bi-Higgs bundles, opers, and future work
    4. 1.4 Analytic results
    5. 1.5 Outline of the paper
    6. 1.6 Related work
    7. 1.7 Acknowledgements
  2. 2 Preliminaries
    1. 2.1 Teichmüller space
    2. 2.2 Holomorphic Riemannian manifolds
    3. 2.3 Complex metrics
    4. 2.4 Bers’ theorem and Bers metrics
  3. 3 Complex affine spheres
    1. 3.1 Complex affine immersions
    2. 3.2 Blaschke affine immersions
    3. 3.3 The fundamental theorem for complex affine immersions
    4. 3.4 Complex affine spheres
    5. 3.5 Motivating examples
  4. 4 Complex Lie derivatives and transport
    1. 4.1 Complex Lie derivatives
    2. 4.2 Transport through complex vector fields
    3. 4.3 Transport of real analytic tensors
  5. 5 Complex elliptic operators
    1. 5.1 The Laplacian of a complex metric
    2. 5.2 Interior elliptic estimates
    3. 5.3 Proof of Theorem F
    4. 5.4 Proof of Theorem G
  6. 6 The holomorphic extension of the Labourie-Loftin parametrization
    1. 6.1 The Labourie-Loftin parametrization
    2. 6.2 Global analysis and the Gauss equation
    3. 6.3 The space $\mathbb{CAS}(S)$
    4. 6.4 The action by $\mathrm{MCG}(S)$ and the $\mathbb{C}^{*}$ twistor action
    5. 6.5 Proof of Theorem A
    6. 6.6 Proof of Theorem C
  7. 7 Bi-Higgs bundles and opers
    1. 7.1 Complex harmonic maps
    2. 7.2 Complex bi-gradings
    3. 7.3 bi-Higgs bundles
    4. 7.4 Bi-Hitchin bi-Higgs bundles
    5. 7.5 From bi-Higgs bundles to complex affine spheres
    6. 7.6 Opers
  8. A On the Beltrami operator
    1. A.1 Proof of Proposition 5.7
    2. A.2 Holomorphic dependence
    3. A.3 Proof of Proposition 2.9

##  1\. Introduction

Given a closed oriented surface $S$ of genus $g\geq 2$, let $\mathcal{T}(S)$
be the Teichmüller space of marked complex structures on $S$. By the
uniformization theorem, every point in $\mathcal{T}(S)$ determines a unique
conjugacy class of Fuchsian representations of $\pi_{1}(S)$ into
$\mathrm{PSL}(2,\mathbb{R})$.

Embedding $\mathrm{PSL}(2,\mathbb{R})$ inside its complexification
$\mathrm{PSL}(2,\operatorname{\mathbb{C}}),$ any Fuchsian representation
produces a representation to $\mathrm{PSL}(2,\operatorname{\mathbb{C}})$ that
preserves a round circle in $\mathbb{CP}^{1}.$ If we deform such a
representation inside the character variety
$\chi(\pi_{1}(S),\mathrm{PSL}(2,\operatorname{\mathbb{C}}))$ in a controlled
manner, the resulting representation is quasi-Fuchsian, which means it
preserves a Jordan curve in $\mathbb{CP}^{1}$. Quasi-Fuchsian representations
have a number of nice properties; for example, quasi-Fuchsian representations
are discrete and faithful, and hence the quotient of the action of a quasi-
Fuchsian representation on $\mathbb{H}^{3}$ is a quasi-Fuchsian $3$-manifold.
Given a quasi-Fuchsian representation $\rho$ preserving a Jordan curve
$\gamma$, $\rho$ also preserves and acts properly discontinuously on the two
connected domains $\Omega_{\rho}^{+},\Omega_{\rho}^{-}\subset\mathbb{CP}^{1}$
that are complementary to $\gamma$. The resulting quotients
$\faktor{\Omega_{\rho}^{+}}{\rho}$ and $\faktor{\Omega_{\rho}^{-}}{\rho}$
represent points in $\mathcal{T}(S)$ and $\mathcal{T}(\overline{S})$
respectively, where $\mathcal{T}(\overline{S})$ is the Teichmüller space of
the oppositely oriented surface $\overline{S}$. On the other hand, Bers proved
that one can associate a quasi-Fuchsian representation to any pair of
oppositely oriented complex structures, in a way that reverses the process
above. Moreover, if we denote by
$\mathcal{QF}(S)\subset{\chi}(\pi_{1}(S),\mathrm{PSL}(2,\operatorname{\mathbb{C}}))$
the open subset of quasi-Fuchsian representations, and give $\mathcal{T}(S)$
its natural complex structure, then Bers’ simultaneous uniformization theorem
(Theorem 2.7 below) says that this construction descends to a biholomorphism

$\mathcal{T}(S)\times\mathcal{T}(\overline{S})\to\mathcal{QF}(S).$

In another direction, for every $n\geq 2$, given a Fuchsian representation
into $\mathrm{PSL}(2,\mathbb{R})$, we can turn it into a representation to
$\mathrm{PSL}(n,\mathbb{R})$ by post-composing with Kostant’s principle
embedding from $\mathrm{PSL}(2,\mathbb{R})\to\mathrm{PSL}(n,\mathbb{R})$ (see
[Kos59] for details on this embedding). Any deformation of that representation
inside $\chi(\pi_{1}(S),\mathrm{PSL}(n,\mathbb{R}))$ is called a Hitchin
representation, and by definition such representations fill up a connected
component $\textrm{Hit}_{n}(S)$ of
$\chi(\pi_{1}(S),\mathrm{PSL}(n,\mathbb{R}))$ called a Hitchin component.
Introduced in Hitchin’s seminal paper [Hit92], Hitchin representations are
Anosov (hence discrete and faithful) [Lab06], and are central objects of study
in higher Teichmüller theory.

For $n$ odd, $\mathrm{SL}(n,\mathbb{R})=\mathrm{PSL}(n,\mathbb{R})$, so for
the rest of the paper we use $\mathrm{SL}(3,\cdot)$ over
$\mathrm{PSL}(3,\cdot)$. Let $\mathcal{M}_{3}(S)$ be the holomorphic vector
bundle over $\mathcal{T}(S)$ whose fiber over a class of a Riemann surface is
identified with its space of holomorphic cubic differentials, i.e., it
consists of pairs $[c,\operatorname{\mathrm{Q}}]$, where $c$ is a complex
structure on $S$ and $\operatorname{\mathrm{Q}}$ is a holomorphic cubic
differential on $(S,c)$. Labourie in [Lab07] and Loftin in [Lof01]
independently constructed a mapping class group equivariant and real analytic
diffeomorphism

$\mathcal{L}:\mathcal{M}_{3}(S)\xrightarrow{\sim}\mathrm{Hit}_{3}(S)\ .$

Both authors build this map using hyperbolic affine spheres. Labourie also
gave a way to see $\mathcal{L}$ using minimal surfaces in symmetric spaces
[Lab17], which generalizes to other higher Teichmüller spaces for Lie groups
of rank $2$ (see [CTT19]).

Similar to the case $n=2$, we can deform the Hitchin locus inside
${\chi}(\pi_{1}(S),\mathrm{SL}(3,\operatorname{\mathbb{C}})),$ and moreover we
can inquire to what extent a Bers’ phenomenon might hold for
$\mathrm{SL}(3,\operatorname{\mathbb{C}}),$ when we give $\mathrm{Hit}_{3}(S)$
the complex structure induced by $\mathcal{L}$. That is, one could–perhaps
optimistically–try to holomorphically identify points in
$\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})$ with representations
to $\mathrm{SL}(3,\operatorname{\mathbb{C}})$. In fact, embedding
$\mathcal{M}_{3}(S)$ as the “diagonal” inside
$\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})$, by analytic
continuing power series, one can find neighbourhoods $U$ of this diagonal and
$V$ of the Hitchin representations inside
${\chi}(\pi_{1}(S),\mathrm{SL}(3,\operatorname{\mathbb{C}}))$ on which
$\mathcal{L}$ extends uniquely to a biholomorphism

${\mathcal{L}}_{\operatorname{\mathbb{C}}}:U\to V.$

In this paper, we introduce maps from surfaces to
$\operatorname{\mathbb{C}}^{3}$, called complex affine spheres, that provide a
concrete and geometrically meaningful realization of
${\mathcal{L}}_{\operatorname{\mathbb{C}}}$. We use our affine spheres to show
that ${\mathcal{L}}_{\operatorname{\mathbb{C}}}$ extends surprisingly far into
$\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})$, well past the
diagonal as shown in Theorem A below, providing a more substantial and
interesting Bers’ phenomenon for $\mathrm{SL}(3,\operatorname{\mathbb{C}})$.
These results should have incarnations for more general higher Teichmüller
spaces.

###  1.1. The main result

The Labourie-Loftin parametrization via hyperbolic affine spheres comes from
the classical theory of affine differential geometry. See the book [NS94] for
references. It follows, from work of Cheng-Yau in [CY77] and [CY86] on affine
differential geometry, and Goldman in [Gol90] and Choi-Goldman in [CG93] on
geometric structures, that every Hitchin representation to
$\mathrm{SL}(3,\mathbb{R})$ is uniquely associated with an equivariant
hyperbolic affine sphere. Labourie and Loftin proved that every equivariant
hyperbolic affine sphere is determined by the data of a complex structure on
$S$ and an accompanying holomorphic cubic differential (see also Wang
[Wan91]), determining the map $\mathcal{L}$ above.

We define (positive hyperbolic) complex affine spheres to be immersions of
surfaces inside $\operatorname{\mathbb{C}}^{3}$ satisfying a geometric
condition and structural equation analogous to that of the real affine spheres
in $\mathbb{R}^{3}$ (see Definition 3.14 and the subsection below). We show in
Corollary 3.22 that a complex affine sphere is equivalent to a pair of complex
structures on $S$ with opposite orientations (not necessarily conjugate),
holomorphic cubic differentials for the two complex structures, and a solution
to an equation that depends on the complex structures and differentials.
Through this formalism, we explicitly construct and study
${\mathcal{L}}_{\operatorname{\mathbb{C}}}$. Denote the mapping class group by
$\mathrm{MCG}(S).$ The following is proved in Section 6.

######  Theorem A.

The Labourie-Loftin parametrization extends uniquely to a
$\mathrm{MCG}(S)$-equivariant holomorphic map

$\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})\supset\Omega\xrightarrow{\
{\mathcal{L}}_{\operatorname{\mathbb{C}}}\
}\chi(\pi_{1}(S),\mathrm{SL}(3,\operatorname{\mathbb{C}}))$

where $\Omega$ is a $\mathrm{MCG}(S)$-invariant open subset containing the
diagonal, $\mathcal{M}_{3}(S)\times\mathcal{T}(\overline{S})$, and
${\mathcal{T}}(S)\times\mathcal{M}_{3}(\overline{S})$. The image consists of
holonomies of complex affine spheres. Moreover,
${\mathcal{L}}_{\operatorname{\mathbb{C}}}$ agrees with Bers’ simultaneous
uniformization map on ${\mathcal{T}}(S)\times\mathcal{T}(\overline{S})$.

Above, we are using $\mathcal{T}(S)$ to denote the embedding of
$\mathcal{T}(S)$ into $\mathcal{M}_{3}(S)$ as the zero section. The fact that
$\Omega$ contains $\mathcal{M}_{3}(S)\times\mathcal{T}(\overline{S})$ and
$\mathcal{T}(\overline{S})\times\mathcal{M}_{3}(\overline{S})$ is very
interesting to us because, a priori,
${\mathcal{L}}_{\operatorname{\mathbb{C}}}$ is not expected to extend this far
from the diagonal. In fact, $\Omega$ can be constructed to be invariant under
what we call the $\operatorname{\mathbb{C}}^{*}$ twistor action, defined by
$\zeta\cdot([c_{1},\operatorname{\mathrm{Q}}_{1}],[\overline{c_{2}},\overline{\operatorname{\mathrm{Q}}_{2}}])=([c_{1},\zeta\operatorname{\mathrm{Q}}_{1}],[\overline{c_{2}},\zeta^{-1}\overline{\operatorname{\mathrm{Q}}_{2}}])$,
which also shows that $\Omega$ is large. Note we need to be precise when
talking about holomorphicity around the character variety (see Remarks 6.4 and
7.16).

We partially address what $\mathcal{L}_{\mathcal{C}}$ does on
$\mathcal{M}_{3}(S)\times\mathcal{T}(\overline{S})$ and its partner in Theorem
E below. See Section 1.3 for further comments.

We do not know whether the representations in the image of
${\mathcal{L}}_{\operatorname{\mathbb{C}}}$ are Anosov. Note that real affine
spheres descend to maps from $\widetilde{S}\to\mathbb{RP}^{2}$, which then
pull back convex projective structures to $S$. In light of this point, it
would be interesting to understand if one can use complex affine spheres to
see geometric structures.

###  1.2. The space of complex affine spheres

We prove Theorem A by constructing a finite dimensional moduli space of
complex affine spheres through which
${\mathcal{L}}_{\operatorname{\mathbb{C}}}$ factors.

The first ingredient in the construction is the notion of complex metrics on
$S$ (discussed in detail in Section 2). A complex metric on $S$ is a non-
degenerate bilinear symmetric form on the complexified tangent bundle
$\operatorname{\mathbb{C}}TS$, and it is called positive if it satisfies a
further non-degeneracy condition (see Definition 2.4). Given a positive
complex metric $g$, one can associate two complex structures $c_{1}$ and
$\overline{c_{2}}$ to $S$ of opposite orientations, and one can always locally
find coordinates $z$ for $c_{1}$ and $\overline{w}$ for $\overline{c_{2}}$,
and a function $\lambda$ to $\operatorname{\mathbb{C}}^{*}$ such that

$g=\lambda dzd\overline{w}.$

In [BE22], the authors formulate a covariant differential calculus for
positive complex metrics, analogous to that of Riemannian metrics, in which we
have Levi-Civita connections, curvature, Laplacians, and other usual objects.
Finally, for every pair of oppositely oriented complex structures
$(c_{1},\overline{c_{2}})$, one can find a special positive complex metric
${\bf\it h}_{(c_{1},\overline{c_{2}})}$ of constant curvature $-1$ in that
conformal class, which is called the Bers metric, and every positive complex
metric is conformally equivalent to a Bers metric.

Toward building ${\mathcal{L}}_{\operatorname{\mathbb{C}}}$, in Section 3 we
study (positive hyperbolic) complex affine spheres that can be deformed to
real affine spheres. Complex affine spheres carry a number of tensors that
generalize those from the real theory, such as (complex) Blaschke metrics and
Pick tensors, satisfying Gauss and Codazzi-type equations. If the Blaschke
metric has conformal class $(c_{1},\overline{c_{2}})$, then the Codazzi
equation is equivalent to the fact that the Pick tensor is of the form
$C=\operatorname{\mathrm{Q}}_{1}+\overline{\operatorname{\mathrm{Q}}_{2}}$,
with $\operatorname{\mathrm{Q}}_{1}$ and
$\overline{\operatorname{\mathrm{Q}}_{2}}$ holomorphic cubic differentials for
$c_{1}$ and $\overline{c_{2}}$ respectively (see Theorem 3.20). For a
distinctive subset of complex affine spheres, including those deformable to
the real locus, solving the Gauss equation corresponds to finding complex
solutions $u$ to the equation

$G(c_{1},\overline{c_{2}},\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}},u):=\Delta_{h}u-e^{2u}+\frac{1}{4}h(\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}})\cdot
e^{-4u}+1\ .$ (1)

where $h={\bf\it h}_{(c_{1},\overline{c_{2}})}$. Note that for a “diagonal”
point
$(c,\overline{c},\operatorname{\mathrm{Q}},\overline{\operatorname{\mathrm{Q}}})$,
Equation (1) reduces to the ordinary Tzitzéica equation for real hyperbolic
affine spheres, as studied by Labourie and Loftin.

Let $\mathcal{C}(S)$ be the space of complex structures on $S$, let
$\mathcal{CD}(S)$ be the bundle of cubic differentials over $\mathcal{C}(S)$,
and let $\mathcal{C}(\overline{S})$ and $\mathcal{CD}(\overline{S})$ be the
analogous spaces for $\overline{S}$. The product
$\mathcal{CD}(S)\times\mathcal{CD}(\overline{S})$ is a complex Banach
manifold, and when we take the quotient by
$\mathrm{Diff}_{0}(S)\times\mathrm{Diff}_{0}(S),$ where $\mathrm{Diff}_{0}(S)$
is the space of diffeomorphisms of $S$ isotopic to the identity, we get
$\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})$. Our distinctive
subset of affine spheres is parametrized by

$\mathrm{CAS}(S)=\\{(c_{1},\overline{c_{2}},\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}},u)\in\mathcal{CD}(S)\times\mathcal{CD}(\overline{S})\times
C^{\infty}(S,\operatorname{\mathbb{C}}):G(c_{1},\overline{c_{2}},\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}},u)=0\\}.$

We say that a complex affine sphere
$(c_{1},\overline{c_{2}},\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}},u)$
is infinitesimally rigid if linearization of $G$ in the direction of $u$ is an
isomorphism. We denote the space of infinitesimally rigid complex affine
spheres by $\mathrm{CAS}^{*}(S)$. The space
$\operatorname{\mathrm{CAS}}^{*}(S)$ is a complex Fréchet manifold for which
the natural projection
$\hat{\pi}\colon\operatorname{\mathrm{CAS}}^{*}(S)\to\operatorname{\mathcal{CD}}(S)\times\operatorname{\mathcal{CD}}(S)$
is a local biholomorphism and the holonomy map
$hol\colon\operatorname{\mathrm{CAS}}^{*}(S)\to\chi(\pi_{1}(S),\operatorname{\mathrm{SL}}(3,\operatorname{\mathbb{C}}))$
is holomorphic (see Section 6.2).

An important issue concerning the spaces $\operatorname{\mathrm{CAS}}(S)$ and
$\operatorname{\mathrm{CAS}}^{*}(S)$, which stems from a general issue around
complex-metric related objects, is that their quotients under the diagonal
action of $\mathrm{Diff}_{0}(S)$ are infinite-dimensional. To contend with
this matter, in Section 4 we introduce notions of complex Lie derivative and
transport through paths of complex vector fields, which allow to construct
local deformations through “double isotopies” (also see the introduction to
Section 4). Defining the transport amounts to solving a Cauchy problem (see
Equation (18)), which is analytically not so well-behaved. Precisely, the
Cauchy problem fails Hörmander’s criterion for local solvability (see Section
4.2).

Using complex Lie derivatives and transport, we manage to take a finite-
dimensional quotient $\mathbb{CAS}(S)$ of the space
$\operatorname{\mathrm{CAS}}^{*}(S)$ for which the fibers roughly correspond
to double isotopies of the cubic differentials. The space $\mathbb{CAS}(S)$
keeps track of the geometric information of the complex affine spheres, as
summarized in the following theorem (corresponding to the statements in
Section 6.3 of the text).

######  Theorem B.

The space $\mathbb{CAS}(S)$ admits the structure of a finite dimensional
complex manifold for which the quotient
$\operatorname{\mathrm{CAS}}^{*}(S)\to\mathbb{CAS}(S)$ is holomorphic.
Moreover, $hol$ descends holomorphically to a map
$hol\colon\mathbb{CAS}(S)\to\chi(\pi_{1}(S),\operatorname{\mathrm{SL}}(3,\operatorname{\mathbb{C}}))$
and $\hat{\pi}$ descends to a local biholomorphism
$\pi\colon\mathbb{CAS}(S)\to\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})$.

As we show in Section 6, the map ${\mathcal{L}}_{\operatorname{\mathbb{C}}}$
in Theorem Theorem A is the composition of $hol$ with a local inverse of
$\pi$.

A natural question is whether ${\mathcal{L}}_{\operatorname{\mathbb{C}}}$
admits a global extension. The following statement provides evidence towards a
negative answer.

######  Theorem C.

There is no holomorphic map

$\mathcal{CD}(S)\times\mathcal{CD}(\overline{S})\to\mathcal{CD}(S)\times\mathcal{CD}(\overline{S})\times
C^{\infty}(S,\operatorname{\mathbb{C}})$

with image in $\operatorname{\mathrm{CAS}}(S)$ that inverts $\hat{\pi}$ and
that coincides with the real affine spheres parametrization on the real locus.

In particular,
$\hat{\pi}\colon\operatorname{\mathrm{CAS}}^{*}(S)\to\operatorname{\mathcal{CD}}(S)\times\operatorname{\mathcal{CD}}(\overline{S})$
has no globally defined right inverse extending the real affine spheres
parametrization.

For input data
$(c,\overline{c},\operatorname{\mathrm{Q}},-\overline{\operatorname{\mathrm{Q}}})$,
Equation (1) is exactly the structural equation for minimal Lagrangians in
$\mathbb{CH}^{2}$. Theorem C is a consequence of the fact that for
$(c,\overline{c},\operatorname{\mathrm{Q}},-\overline{\operatorname{\mathrm{Q}}})$,
as $\operatorname{\mathrm{Q}}$ grows larger the linearization of Equation (1)
eventually develops a kernel, which was proved by Huang-Loftin-Lucia in their
paper about minimal Lagrangians [HLL13]. More discussion in Section 6.6.

###  1.3. Bi-Higgs bundles, opers, and future work

Labourie and Baraglia have shown that real hyperbolic affine spheres are
equivalent to certain minimal surfaces in the $\mathrm{SL}(3,\mathbb{R})$
Riemannian symmetric space (see [Lab07, Section 9] and [Bar10, Section 3.4]).
Moreover, if an affine sphere $\sigma$ is equivalent to a minimal surface $f$,
then the Higgs bundle of $f$ encodes the Pick tensor of $\sigma$. Naturally,
one might expect complex affine spheres to be equivalent to minimal surfaces
in some $\mathrm{SL}(3,\operatorname{\mathbb{C}})$-homogeneous space.

While working on this paper, by first studying complex affine spheres, we came
to realize a generalization of the theory of harmonic maps, minimal surfaces,
and Higgs bundles. Specifically, harmonic maps into holomorphic Riemannian
symmetric spaces such as
$\faktor{\mathrm{SL}(n,\operatorname{\mathbb{C}})}{\mathrm{SO}(n,\operatorname{\mathbb{C}})}=:\mathbb{Y}_{n}$
give rise to new objects that we call bi-Higgs bundles. In Section 7.3, we
define a class of harmonic maps into $\mathbb{Y}_{3}$, which we call bi-
Hitchin conformal harmonic maps, which are equivalent to certain bi-Higgs
bundles, and which come along with a pair of cubic differentials. We prove the
following in Sections 7.4 and 7.5. See Section 7 for more definitions and
precise details.

######  Theorem D.

There is a one-to-one correspondence, up to conjugation, between equivariant
positive hyperbolic complex affine spheres and bi-Hitchin conformal harmonic
maps to $\mathbb{Y}_{3}$, which preserves the cubic differentials. When the
affine sphere is deformable to a real affine sphere within the infinitesimally
rigid locus, the holonomies are the same.

Regarding the holonomies, one can certainly remove the additional assumption,
which we keep to simplify the exposition.

A complete development of the theory of bi-Higgs bundles would take us too far
afield from the aim of this paper, so we have chosen to defer a more extended
treatment to an upcoming paper. As well, writing this paper from the
perspective of complex affine spheres has its advantages: firstly, complex
affine spheres might shed more light on the geometry of representations to
$\mathrm{SL}(3,\operatorname{\mathbb{C}})$, and also, Equation (1), our main
tool in proving Theorem A, appears to arise more naturally using complex
affine spheres (see the end of Section 7.3).

Using bi-Higgs bundles, we also observe that a certain subset of
$\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})$ corresponds to
meaningful objects: opers. $\mathrm{SL}(n,\operatorname{\mathbb{C}})$-opers
are gauge theoretic objects that arise from globalizing differential equations
on Riemann surfaces (the full definition is in Section 7.6), and they are
equivalent to $\mathbb{CP}^{1}$-structures when $n=2$. The modern theory of
opers stems from Drinfield-Sokolov [DS84] and the work of Beilinson-Drinfield
on the Geometric Langlands Conjecture [BD96]. In the latter work, Beilinsion-
Drinfield showed that, over a Riemann surface $(S,c)$ with canonical bundle
$\mathcal{K}$, every connected component of the moduli space of
$\mathrm{SL}(n,\operatorname{\mathbb{C}})$-opers is parametrized by the
Hitchin base $\oplus_{i=2}^{n}H^{0}(S,\mathcal{K}^{i})$.

######  Theorem E.

Given $[c,S]\in\mathcal{M}_{3}(S)$,
${\mathcal{L}}_{\operatorname{\mathbb{C}}}([c,\operatorname{\mathrm{Q}}],[\overline{c},0])$
is represented by the holonomy of an
$\mathrm{SL}(3,\operatorname{\mathbb{C}})$-oper on $(S,c)$ with Hitchin
basepoint $(0,\operatorname{\mathrm{Q}})$, and
${\mathcal{L}}_{\operatorname{\mathbb{C}}}([c,0],[\overline{c},\overline{\operatorname{\mathrm{Q}}}])$
is represented by the holonomy of an oper on $(S,\overline{c})$ with Hitchin
basepoint $(0,\overline{\operatorname{\mathrm{Q}}}).$

Since Bers’ simultaneous uniformization procedure goes through constructing
$\mathbb{CP}^{1}$-structures with quasi-Fuchsian holonomies, it feels quite
natural that opers appear in the image of
${\mathcal{L}}_{\operatorname{\mathbb{C}}}.$ As well, from our understanding,
it’s been an open question whether opers have a geometric interpretation;
Theorems A and E seem to answer that question positively in a special case.

The relations between complex harmonic maps and opers seem worthy of further
study. As we mentioned above, we have not addressed the general case of what
${\mathcal{L}}_{\operatorname{\mathbb{C}}}$ does on
$\mathcal{M}_{3}(S)\times\mathcal{T}(\overline{S})$ and
$\mathcal{T}(S)\times\mathcal{M}_{3}(\overline{S})$. These loci suggest a
notion of “bi-opers,” which we plan to develop in future work. As well, it’s
worth pointing out that the $\operatorname{\mathbb{C}}^{*}$ twistor action is
really an artifact of a well-known $\operatorname{\mathbb{C}}^{*}$-action from
non-abelian Hodge theory. In [Gai14], Gaiotto conjectured, motivated by
physics, that the Hitchin component and the space of opers are holomorphically
identified by taking a “conformal limit” through this
$\operatorname{\mathbb{C}}^{*}$-action. This conjecture was proved for closed
surfaces in [Dum+21] (see also [CW19]). Thus, bi-Higgs bundles seem to provide
another perspective on Gaiotto’s conjecture and conformal limits, which we’re
eager to explore. We discuss this topic in more detail in Section 7.6.

###  1.4. Analytic results

As indicated above, proving Theorem A requires studying Equation (1). In
Section 5, we compute in local coordinates that if $g=\lambda
dzd\overline{w}$, and $\mu$ is the Beltrami form of $w$ as a function of $z$,
i.e., $\partial_{\overline{z}}w=\mu\partial_{z}w$, then the Laplace operator
of $g$ is

$\Delta_{g}=\frac{1}{\lambda\partial_{\overline{z}}\overline{w}}{\partial}_{\overline{z}}\left({\partial}_{z}-\overline{\mu}{\partial}_{\overline{z}}\right).$
(2)

Note that the factor ${\partial}_{z}-\overline{\mu}{\partial}_{\overline{z}}$
is the complex conjugation of the Beltrami operator. Since $|\mu|<1$,
$\Delta_{g}$ is a second order complex elliptic operator, i.e., it’s principle
symbol never vanishes. The general class of such PDE’s is not so well studied
(see [AIM09, §7.8] and [Beg94, §5] for references), and hence we have to
develop some general results ourselves. We record our two most substantial and
important results as theorems below.

Drawing on the Ahlfors-Bers analysis for the Beltrami operator, we prove
interior elliptic estimates for complex operators such as $\Delta_{g}$
(Proposition 5.5), from which we deduce Fredholm properties (Proposition 5.9).
Our first main analytic theorem shows that complex affine spheres with real
analytic input data are real analytic. Using this theorem, we’re able to use
our transport method, defined in Section 4, on complex affine spheres in the
proofs of Theorems A and B (we also use it in the proof of Theorem G below).

######  Theorem F.

Assume that $g$ is a real analytic positive complex metric on
$U\subset\operatorname{\mathbb{C}}$, let
$V\subset\operatorname{\mathbb{C}}\times\operatorname{\mathbb{C}}^{2}$ be an
open domain, let $F:U\times V\to\operatorname{\mathbb{C}}$ be a real analytic
function, and let $L$ be a semi-linear second order complex elliptic operator
of the form

$Lu(z)=\Delta_{g}u+F(z,u,Du).$

If a function $u\in C^{2}(U,\operatorname{\mathbb{C}})$ with $(u,Du)(U)\subset
V$ satisfies $Lu=0,$ then $u$ is real analytic.

We prove Theorem F by adapting a proof of the analogous result for real
ellitpic PDE’s, which is attributed to Kato [Kat96], Hashimoto [Has06], and
Blatt [Bla20]. There are some genuine differences in our complex setting (see
Remark 5.11), but still the idea pushes through.

Theorem G below is the key result in extending $\Omega$ to
$\mathcal{M}_{3}(S)\times\mathcal{T}(\overline{S})$ and
$\mathcal{M}_{3}(\overline{S})\times\mathcal{T}(S)$: it is used to show that
we have infinitesimally rigid complex affine spheres lying over this locus.
For the theorem below, see the more precise statement in Theorem 5.23.

######  Theorem G.

The operator $L_{{\bf\it
h}_{(c_{1},\overline{c_{2}})}}:C^{\infty}(S,\operatorname{\mathbb{C}})\to
C^{\infty}(S,\operatorname{\mathbb{C}})$ defined by

$L_{{\bf\it h}_{(c_{1},\overline{c_{2}})}}u=\Delta_{{\bf\it
h}_{(c_{1},\overline{c_{2}})}}u-2u\ $

is invertible for an open and dense subset of elements
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$,
including all the pairs $(c_{1},\overline{c_{2}})$ such that $c_{1}$ and
$\overline{c_{2}}$ induce the same real analytic structure on $S$.

Theorem G can be recast as saying that a dense class of Bers metrics are
infinitesimally rigid in their conformal class. In [BE22], it is proved that
Bers metrics form a connected component of the space of positive complex
metrics of constant curvature $-1$; this result is consistent with that one.

The proof of Theorem G falls outside the standard analytic toolkit: we use a
continuity argument, relying on the transport technique defined in Section 4
and on one of the main theorems proved in the first author’s work on
variations of positive complex metrics [Ema23, Theorem A], to connect pairs
$(c_{1},\overline{c_{2}})$ inducing the same real analytic structure to pairs
$(c,\overline{c})$, for which the result of Theorem G is elementary.

In fact, we believe that $L_{{\bf\it h}_{(c_{1},\overline{c_{2}})}}$ in
Theorem G is an isomorphism for all Bers metrics, but we didn’t need to prove
the result in this generality. While Theorem G shows that $2$ is not in the
spectrum, we do not know whether $u\mapsto\Delta_{h}u-ku$ is an isomorphism
for every $k>0$. Studying the spectrum of $\Delta_{h}$ further seems
interesting.

###  1.5. Outline of the paper

Toward understanding Theorem A, on a first reading one might want to focus on
Sections 2, 3, and 6, and take the more analytic results from Sections 4 and 5
(such as Theorems F and G) for granted. Theorems A, B, and C are proved in
Section 6, while in Section 7, a lot of which can be read independently of
Sections 3-6, we prove Theorem D, which we then use to prove Theorem E.

In Section 2 we give the preliminaries on Teichmüller theory and complex
metrics, among other things. Section 3 contains the geometry that underpins
the rest of the paper: we develop a general theory of complex affine
immersions, define complex affine spheres, and investigate their basic
properties. We also give a number of explicit examples of complex affine
immersions and complex affine spheres. Key results include Theorem 3.20, which
shows that every positive hyperbolic complex affine sphere determines a pair
of complex structures and cubic differentials, and Corollary 3.22 , which
shows that if one can solve Equation (1) built out of that data, then one can
recover a complex affine sphere.

In Section 4, a lot of which can be read independently of the rest of the
paper, we develop the notion of complex Lie derivatives and transport along
complex vector fields. The main takeaways toward our main theorems are Theorem
4.11 and Proposition 4.13, which are used in the proof of Theorem F, and then
eventually used to show in the proofs of Theorems A and C that local double
isotopy deformations preserve the holonomy classes of infinitesimally rigid
complex affine spheres.

In Section 5, which is independent of the theory of affine spheres, we carry
out our general analysis of complex elliptic operators. We first derive our
expression for $\Delta_{g}$, prove interior elliptic estimates, and Fredholm
properties. We then prove Theorem F and Theorem G.

With the essential tools developed, in Section 6 we return to complex affine
spheres. Here we formalize the strategy outlined in Section 1.2, construct the
moduli space of infinitesimally rigid complex affine spheres (thus proving
Theorem B), and then give the proof of Theorem A. We then prove Theorem C in
Section 6.6.

As indicated above, in Section 7, we define harmonic maps to holomorphic
Riemannian manifolds, show how bi-Higgs bundles arise out of them, and prove
Theorem D. After giving a brief introduction to opers in Section 7.6, we use
Theorem D to prove Theorem E.

To prove the interior elliptic interior estimates for $\Delta_{g}$ in Section
5, and to show that the map $\mathcal{L}_{\mathbb{C}}$ is holomorphic in
Section 6, we need a number of results on the Beltrami equation that should be
known to experts but which we couldn’t find in the literature. So we prove
these results in an appendix. Theorem A.5 should be independently valuable:
extending a classical result of Ahlfors and Bers [AB60, Theorem 11], Theorem
A.5 is the statement that the Riemann Mapping Theorem solutions to the
Beltrami equation vary holomorphically in higher Sobolev spaces.

###  1.6. Related work

There have been a number of works related to representations of surface groups
into complex Lie groups close to the Hitchin locus, although the perspectives
of all of the works so far are distinct from ours. See for example [DS20],
[DS21], [ADL21], [Ale+23], [Dav23], and in particular [DS21], where Dumas-
Sanders find a local continuation of Bers’ theorem in a setting that falls
more in line with the theory of Anosov representations.

We could frame the first half of Section 3 as the formulation of a basic
theory of affine real submanifolds of dimension $n$ inside $\mathbb{C}^{n+1}$,
which then leads to the definition of complex affine spheres. We’d like to
point out that there is an established theory of complex affine differential
geometry, but centered around complex hypersurfaces in
$\operatorname{\mathbb{C}}^{n+1}$, developed in [Abe91] and [DVV88] among
other sources.

There might be some interesting relations between bi-Higgs bundles and Hodge’s
thesis [Hod06], in which Hodge places some results about quasi-Fuchsian
representations in a Higgs bundle context, and also with the latter work
[LS20].

After writing this paper, we were notified about the paper [Kim07], in which
Kim studies a complex extension of the Riemannian Laplacian, which seems to
agree with our Bers Laplacians, and studies their determinants. Our analytic
results, such as elliptic estimates and Fredholmness for our Laplacians,
Theorem F, and especially Theorem G, can be seen as contributing to the study
initiated in [Kim07]. We also mention that Kim proves that the complex
Laplacians vary holomorphically in a $C^{2}$ sense, but for our aims, we still
need to prove (via Theorem A.5) that our operators vary holomorphically in our
specific Sobolev spaces.

Around the end of writing this paper, we learned of independent work of Rungi
and Tamburelli [RT24], who introduce complex minimal Lagrangian surfaces in
the bi-complex hyperbolic space, which generalize minimal Lagrangian surfaces
in $\mathbb{CH}^{2}$, hyperbolic affine spheres in $\mathbb{R}^{3}$, and Bers
embeddings in the holomorphic space form
$\mathbb{\operatorname{\mathbb{C}\mathbb{P}}}^{1}\times\mathbb{\operatorname{\mathbb{C}\mathbb{P}}}^{1}\backslash\Delta$.
In [RT24], the authors define what they call
$\mathrm{SL}(3,\operatorname{\mathbb{C}})$-bi-complex Higgs bundles, and they
give a parameterization of a subset of
$\mathrm{SL}(3,\operatorname{\mathbb{C}})$-Anosov representations by an open
set in $\mathcal{M}_{3}(S)\times\mathcal{M}_{3}(\overline{S})$, from which
they deduce that this subset of representations is endowed with a bi-complex
structure. The approach of their work seems to give a different point of view
and it would be interesting to analyze the interplay further. Nevertheless, we
underline that the two works are completely independent.

###  1.7. Acknowledgements

We thank Kari Astala, Simon Blatt, and Peter Smillie for helpful discussions
and correspondence. N.S. thanks the organizers and participants of the
conference _Advances in Higgs Bundles_ , which took place at the Brin
Mathematics Research Center at the University of Maryland in April 2024, for
conversations and sharing references from which this work benefited. We also
thank the Geometry and Topology group in Luxembourg for the support and for
creating such a nice working environment.

Both authors are funded by the FNR grant O20/14766753, Convex Surfaces in
Hyperbolic Geometry.

##  2\. Preliminaries

For the rest of the paper, let $S$ be a closed oriented surface of genus
$g\geq 2,$ and let $\overline{S}$ be that same surface with the opposite
orientation. Throughout the paper, we use $W^{k,p}(\cdot)$, $k\geq 1,$ $1\leq
p\leq\infty$, for Sobolev spaces. As we will see, the analysis of the
equations that we will eventually encounter require precise choices of Sobolev
spaces.

###  2.1. Teichmüller space

Fix a basepoint complex structure on $S$ that’s compatible with the
orientation, say $c_{0},$ and let $\mathcal{K}$ be the canonical bundle. A
Beltrami form $\mu$ on $S$ is an $L^{\infty}$-measurable section of the bundle
$\mathcal{K}^{*}\otimes\mathcal{K}^{-1}$ with $L^{\infty}$ norm strictly less
than $1$. Recall that marked complex structures are parametrized by Beltrami
forms. Explicitly, given a Beltrami form $\mu$ on $(S,c_{0})$, the Measurable
Riemann Mapping Theorem produces a unique pair consisting of a compatible
complex structure $c_{\mu}$ together with a quasiconformal map
$f_{\mu}:(S,c_{0})\to(S,c_{\mu})$ (see Section A.3 for further comments). We
say that a complex structure is in $W^{k,p}$ if the Beltrami form is
$W^{k,p}$. For a positive integer $l>1$, denote by $\mathcal{C}^{l}(S)$ the
space of $W^{l,\infty}$ complex structures on $S$ that are compatible with the
orientation. We write $\mathcal{C}(S)$ for the $C^{\infty}$ complex
structures. Let $\mathrm{Diff}_{+}(S)$ be the space of orientation preserving
diffeomorphisms of $S$ and let $\mathrm{Diff}_{0}(S)$ be the normal subgroup
of diffeomorphisms isotopic to the identity. The quotient
$\faktor{\mathrm{Diff}_{+}(S)}{\mathrm{Diff}_{0}(S)}$ is the mapping class
group $\mathrm{MCG}(S).$

######  Definition 2.1.

The Teichmüller space of marked complex structures on $S$ is the quotient
$\mathcal{T}(S)=\faktor{\mathcal{C}(S)}{\mathrm{Diff}_{0}(S)}$.

For $l<\infty,$ $\mathcal{C}^{l}(S)$ is a complex Banach manifold modelled on
the space of $W^{l,\infty}$ Beltrami forms, and for $l=\infty$,
$\mathcal{C}(S)$ is a complex Fréchet manifold, with semi-norms coming from
the $W^{l,\infty}$ norms (by Sobolev embedding theorems, this Fréchet
structure is biholomorphic to all of the more common ones). The following is
due to Earle-Eells. [EE69].

######  Theorem 2.2 (Earle-Eells).

Under the quotient topology, $\mathcal{T}(S)$ inherits a complex structure
with respect to which the map from $\mathcal{C}(S)\to\mathcal{T}(S)$ is
holomorphic and a principle $\mathrm{Diff}_{0}(S)$-fiber bundle.

Throughout the paper, we insist on viewing $\mathcal{C}(S)$ as a Fréchet
manifold, so that we can directly apply the result of Earle-Eells.

Note that the complex structure on the Teichmüller space
$\mathcal{T}(\overline{S})$ of the oppositely oriented surface $\overline{S}$,
identifies with the complex structure conjugate to the usual one on
$\mathcal{T}(S)$.

###  2.2. Holomorphic Riemannian manifolds

Let $\mathbb{X}$ be a complex manifold with almost complex structure
$\mathbb{J}$. A holomorphic Riemannian metric
$\operatorname{\langle\cdot,\cdot\rangle}$ on $\mathbb{X}$ is a holomorphic
assignment of a symmetric non-degenerate $\operatorname{\mathbb{C}}$-bilinear
form on each fiber of the holomorphic tangent bundle of $\mathbb{X}$, or more
concisely a holomorphic nowhere degenerate section of
$Sym_{2}(T^{1,0}(\mathbb{X}))$. These objects have been widely studied in
literature with different approaches, see for instance [DZ09].

A holomorphic affine connection $D$ on $T\mathbb{X}$ is an affine connection
such that $D\mathbb{J}=0$ and for local holomorphic vector fields
$Z_{1},Z_{2}$ one has
$D_{\mathbb{J}Z_{1}}Z_{2}=\mathbb{J}\left(D_{Z_{1}}Z_{2}\right)$. Holomorphic
Riemannian manifolds $(\mathbb{X},\operatorname{\langle\cdot,\cdot\rangle})$
have a natural notion of Levi-Civita connection $D$, which is a holomorphic
affine connection on $T\mathbb{X}$ that makes the metric parallel. Such a
connection induces a natural notion of $\operatorname{\mathbb{C}}$-multilinear
curvature tensor and, for all complex vector subspace $V<T_{p}\mathbb{X}$ such
that $dim_{\operatorname{\mathbb{C}}}V=2$ and that
$\operatorname{\langle\cdot,\cdot\rangle}|_{V}$ is non-degenerate, one can
define the sectional curvature.

Given a complex semisimple Lie group, its complex Killing form extends to the
whole tangent bundle by left or right translation and hence determines a
holomorphic Riemannian metric $\operatorname{\langle\cdot,\cdot\rangle}$. For
$G=\mathrm{SL}(n,\operatorname{\mathbb{C}})$, at
$A\in\mathrm{SL}(n,\operatorname{\mathbb{C}})$, the metric is given by, for
$V,W\in T_{A}\mathrm{SL}(n,\operatorname{\mathbb{C}})$, $\langle
V,W\rangle|_{A}=2n\mathrm{tr}(A^{-1}VA^{-1}W)$. We highlight two examples that
arise from taking quotients of complex Lie groups.

  1. (1)

$\mathbb{C}^{n}$ with the canonical symmetric
$\operatorname{\mathbb{C}}$-bilinear form given by
$\langle\underline{z},\underline{w}\rangle=\sum_{k=1}^{n}z_{k}w_{k}$, for
$\underline{z}=(z_{1},\dots,z_{n}),\underline{w}=(w_{1},\dots,w_{n})\in\operatorname{\mathbb{C}}^{n}$.
$(\mathbb{\operatorname{\mathbb{C}}}^{n},\operatorname{\langle\cdot,\cdot\rangle})$
is the unique simply connected complete holomorphic Riemannian metric with
constant sectional curvature zero [BE22, Theorem 2.6].

  2. (2)

The hyperboloid

$\mathbb{X}_{n}=\\{\underline{z}\in\operatorname{\mathbb{C}}^{n+1}\ |\
\langle\underline{z},\underline{z}\rangle=-1\\}$

is a complex submanifold of $\mathbb{C}^{n}$ and it inherits a holomorphic
Riemannian metric, under which it identifies with
$\faktor{\mathrm{SO}(n+1,\operatorname{\mathbb{C}})}{\mathrm{SO}(n,\operatorname{\mathbb{C}})}$.
For $n\geq 2$, $\mathbb{X}_{n}$ is the unique complete simply connected
holomorphic Riemannian manifold with constant sectional curvature -1 [BE22,
Theorem 2.6]. Moreover, for all $p,q$ with $p+q=n$,
$\mathbb{H}^{p,q}=\faktor{\mathrm{SO}(p,q)}{\mathrm{SO}(p)\times\mathrm{SO}(q)}$
embeds isometrically in $\mathbb{X}_{n}$ in a unique way up to composition
with ambient isometries.

Observe that, since all non-degenerate bilinear forms on
$\operatorname{\mathbb{C}}^{n}$ are isomorphic, one can define several
isometric models for $\operatorname{\mathbb{C}}^{n+1}$ and $\mathbb{X}_{n}$ by
changing the ambient bilinear form. For instance, in Example 3.24, we will
refer to the isometric model for
$(\operatorname{\mathbb{C}}^{n+1},\operatorname{\langle\cdot,\cdot\rangle})$
given by $\operatorname{\mathbb{C}}^{n,1}$, namely
$\operatorname{\mathbb{C}}^{n+1}$ endowed with the bilinear form
$\langle\underline{z},\underline{w}\rangle_{n,1}=-z_{n+1}w_{n+1}+\sum_{k=1}^{n}z_{k}w_{k}$.
Via the isometry between these two spaces one gets that $\mathbb{X}_{n}$ is
equivalent to
$(\hat{\mathbb{X}}_{n},\operatorname{\langle\cdot,\cdot\rangle}_{n,1})$ with

$\hat{\mathbb{X}}_{n}=\\{\underline{z}\in\operatorname{\mathbb{C}}^{n+1}\ |\
\langle\underline{z},\underline{z}\rangle_{n,1}=-1\\}\ .$ (3)

Finally, $\mathbb{X}_{2}$ models the space of oriented geodesics of hyperbolic
space $\mathbb{H}^{3}$, which will be relevant below. Indeed, identifying
$\mathrm{SO}(3,\operatorname{\mathbb{C}})$ with
$\operatorname{\mathrm{PSL}}(2,\operatorname{\mathbb{C}})$ and viewing
$\mathbb{X}_{2}$ as
$\faktor{\mathrm{PSL}(2,\operatorname{\mathbb{C}})}{\mathrm{SO}(2,\operatorname{\mathbb{C}})}$,
$\mathrm{SL}(2,\operatorname{\mathbb{C}})$ acts transitively on the space of
oriented geodesics, and the stabilizer of any oriented geodesic is conjugate
to $\mathrm{SO}(2,\operatorname{\mathbb{C}})$. By identifying a geodesic in
$\mathbb{H}^{3}$ with its endpoints in $\mathbb{CP}^{1}$, we obtain a
biholomorphism from

$\mathbb{X}_{2}\to\mathbb{G}:=\mathbb{CP}^{1}\times\mathbb{CP}^{1}\backslash\Delta.$

Under this biholomorphism, the pushed-forward holomorphic Riemannian metric
has the following description: given any complex affine chart $(U,z)$ on
$\operatorname{\mathbb{C}\mathbb{P}}^{1}$, the metric on $(U\times
U\setminus\Delta,(z_{1},z_{2}))$ has the form

$\operatorname{\langle\cdot,\cdot\rangle}_{\mathbb{G}}=-\frac{4}{(z_{1}-z_{2})^{2}}dz_{1}\cdot
dz_{2}\ .$

See [BE22, Section 2.4] for details. Several relations between immersions of
surfaces into $\mathbb{H}^{3}$ and $\mathbb{G}$ are treated in [ES22].

###  2.3. Complex metrics

Complex metrics first appeared in a paper by the first author on immersions of
(real) smooth manifolds into holomorphic Riemannian manifolds [BE22].
Throughout the paper, given a smooth manifold $M,$ we denote the complexified
tangent bundle $TM\otimes_{\mathbb{R}}\operatorname{\mathbb{C}}$ by
$\operatorname{\mathbb{C}}TM,$ and likewise for the complexified cotangent
bundle $\operatorname{\mathbb{C}}T^{*}M$.

######  Definition 2.3.

A complex metric on a smooth (real) manifold $M$ is a smooth section of
$\mathrm{Sym}^{2}(\mathbb{C}TM)$ that determines a non-degenerate symmetric
bilinear form in each fiber.

Of course, the bilinear extension of a Riemannian metric to the complexified
tangent bundle is a complex metric. Throughout the paper, we will view a
complex metric on a manifold $M$ as the same thing as an invariant complex
metric on the universal cover $\widetilde{M},$ and we won’t distinguish
notation.

A lot of the usual constructions in Riemannian geometry extend to the setting
of complex metrics and the proofs apply verbatim. Here are a few objects and
facts that we will use in this paper.

  1. (1)

Every complex metric $g$ has a natural Levi-Civita connection, namely an
affine connection
$\nabla^{g}\colon\Gamma(\operatorname{\mathbb{C}}TM)\to\Gamma(End(\operatorname{\mathbb{C}}TM))$
uniquely determined by the fact that it is torsion-free and compatible with
the metric, i.e. $\nabla^{g}g=0$.

  2. (2)

Through $g$ and its Levi-Civita connection $\nabla^{g}$, one can define the
$(0,4)$-type and the $(1,3)$-type curvature tensors, which we both denote by
$\mathrm{R}^{g}$ when there is no ambiguity, as

$\mathrm{R}^{g}(X,Y,Z,W)=g(\mathrm{R}^{g}(X,Y)Z,W):=g(\nabla^{g}_{X}\nabla^{g}_{Y}Z-\nabla^{g}_{Y}\nabla^{g}_{X}Z-\nabla^{g}_{[X,Y]}Z,W)$

for all $X,Y,Z,W\in\Gamma(\operatorname{\mathbb{C}}TM)$, which is
$\operatorname{\mathbb{C}}$-multilinear and has the usual symmetries as in the
Riemannian case. If $X$ and $Y$ span a $g$-non-degenerate $2$-plane in
$\operatorname{\mathbb{C}}TM$, we have the sectional curvature

$\mathrm{K}(X,Y)=\frac{\mathrm{R}^{g}(X,Y,Y,X)}{g(X,X)g(Y,Y)-(g(X,Y))^{2}}.$

When $M$ is a surface, this sectional curvature is called the Gauss curvature
and defines a function $\mathrm{K}_{g}:M\to\operatorname{\mathbb{C}}.$

  3. (3)

For any $C^{1}$ function $f\colon M\to\operatorname{\mathbb{C}}$, one can
define the gradient of $f$ as the unique vector field $\nabla^{g}f$ defined by
$df=g(\nabla^{g}f,\cdot)$. If $f$ is also $C^{2}$, we can define the Hessian
and the Laplacian of $f$ as

$\displaystyle\mathrm{Hess}_{g}(f)(X,Y)=g(\nabla^{g}_{X}\nabla^{g}f,Y),$
$\displaystyle\Delta_{g}(f)=tr_{g}(\mathrm{Hess}_{g}(f)).$

  4. (4)

Assume $M$ is a surface, and let $\varrho\colon
M\to\operatorname{\mathbb{C}}^{*}$. Then, the conformal metric
$\hat{g}=\varrho\cdot g$ satisfies
$\Delta_{\hat{g}}=\frac{1}{\varrho}\Delta_{g}$ and

$\mathrm{K}_{\hat{g}}=\frac{1}{\varrho}(\mathrm{K}_{g}-\frac{1}{2}\Delta_{g}\log(\varrho)).$
(4)

While the function $\log(\varrho)$ is only locally well-defined, up to
choices, $\Delta_{g}\log(\varrho)$ is globally well-defined.

In the following, we restrict to the case of surfaces $M=S$. The fact that a
complex metric $g$ on $S$ is non-degenerate implies that $g$ has two isotropic
directions at each point. In other words, for each point $p\in S$, the set
$\\{v\in\operatorname{\mathbb{C}}T_{p}S\ |\ g_{p}(v,v)=0\\}$ consists of two
complex lines of $\mathbb{C}T_{p}S$, corresponding to two points in
$\mathbb{P}\operatorname{\mathbb{C}}T_{p}S$. Observe that $\mathbb{P}T_{p}S$
embeds into the sphere $\mathbb{P}\operatorname{\mathbb{C}}T_{p}S$ as an
equatorial circle, which cuts $\mathbb{P}\operatorname{\mathbb{C}}T_{p}S$ into
two connected components.

######  Definition 2.4.

A complex metric $g$ is positive if for each point $p\in S$, no isotropic
direction of $g_{p}$ lies in $\mathbb{P}T_{p}S$, and the two isotropic
directions are contained in distinct components of
$\mathbb{P}\operatorname{\mathbb{C}}T_{p}S\setminus\mathbb{P}T_{p}S$.

Riemannian metrics complexify to positive complex metrics: the isotropic
directions are the eigen-lines of the almost complex structure and are hence
antipodal.

We denote the space of $C^{\infty}$ – resp. $W^{l,2}$ – positive complex
metrics on $S$ by $\mathcal{CM}(S)$ – resp. $\mathcal{CM}^{l}(S)$. Being open
subsets of the space of $C^{\infty}$ – resp. $W^{l,2}$ – sections of
$\mathrm{Sym}^{2}(\mathbb{C}TS)$, one can see that $\mathcal{CM}(S)$ is a
complex Fréchet manifold and $\mathcal{CM}^{l}(S)$ is a complex Banach
manifold.

We further define $\mathcal{CM}_{-1}(S)$ and $\mathcal{CM}_{-1}^{l}(S)$ to be
the subsets of positive complex metrics of constant curvature $-1$ in
$\mathcal{CM}(S)$ and $\mathcal{CM}^{l}(S)$ respectively.

The isotropic directions of a positive complex metric determine a pair of
oppositely oriented complex structures $c_{1}$ and $\overline{c_{2}}$ on $S$.
In fact, if $z$ and $\overline{w}$ are local holomorphic coordinates for
$c_{1}$ and $\overline{c_{2}}$ respectively, then
$g(\partial_{\overline{z}},\partial_{\overline{z}})=g(\partial_{w},\partial_{w})=0$,
and $g$ can be locally written as $g=\lambda dzd\overline{w}$, for some non-
vanishing complex valued function $\lambda$ (see [BE22, Section 6.1] for
details).

This defines a projection map

$({\bf\it c}_{+},{\bf\it
c}_{-})\colon\mathcal{CM}(S)\to\mathcal{C}(S)\times\mathcal{C}(\overline{S})$

whose fibers correspond to conformal classes of positive complex metrics.

The locally defined basis for $\operatorname{\mathbb{C}}TS$ given by
$\\{\partial_{w},\partial_{\overline{z}}\\}$ will be particularly useful for
computations, so we record a few elementary facts here. Let $\nabla^{g}$ be
the Levi-Civita connection for $g$, and let $\Pi_{1}$ and $\Pi_{2}$ be the
projections of $\operatorname{\mathbb{C}}TS$ onto the isotropic directions of
$g$ for $c_{1}$ and $\overline{c_{2}}$ respectively, which are locally spanned
by $\partial_{\overline{z}}$ and $\partial_{w}$ respectively.

######  Lemma 2.5.

For any locally defined complex vector field $X$,
$\nabla_{X}^{g}\partial_{\overline{z}}$ is parallel to
$\partial_{\overline{z}}$, and $\nabla_{X}^{g}\partial_{w}$ is parallel to
$\partial_{w}$.

###### Proof.

By metric compatibility,
$g(\nabla_{X}^{g}\partial_{\overline{z}},\partial_{\overline{z}})=\frac{1}{2}\nabla_{X}^{g}g(\partial_{\overline{z}},\partial_{\overline{z}})=0,$
and likewise for $\partial_{w}$. ∎

######  Lemma 2.6.

$\nabla_{\overline{z}}^{g}\partial_{w}=\Pi_{2}([\partial_{\overline{z}},\partial_{w}])$
and
$\nabla_{w}^{g}\partial_{\overline{z}}=\Pi_{1}([\partial_{w},\partial_{\overline{z}}]).$

###### Proof.

Since $\nabla^{g}$ is torsion free,
$\nabla_{\overline{z}}^{g}\partial_{w}-\nabla_{w}^{g}\partial_{\overline{z}}=[\partial_{\overline{z}},\partial_{w}]$.
We apply Lemma 2.5.

∎

###  2.4. Bers’ theorem and Bers metrics

In the introduction we defined quasi-Fuchsian representations. Bers’
Simultaneous Uniformization Theorem is more formally stated below. If $\rho$
is quasi-Fuchsian, preserving a Jordan curve $\gamma\subset\mathbb{CP}^{1}$,
the domains complementary to $\gamma$ are called the domains of discontinuity.
Recall that, identifying $\pi_{1}(S)$ with the group of covering
transformations of $\widetilde{S}\to S$, given an action $\rho$ of
$\pi_{1}(S)$ on a space $X,$ we say that $f:\widetilde{S}\to X$ is
$\rho$-equivariant if for all $\gamma\in\pi_{1}(S)$,
$f\circ\gamma=\rho(\gamma)\circ f$.

######  Theorem 2.7 (Bers’ Simultaneous Uniformization Theorem [Ber60]).

For all
$(c_{1},\overline{c_{2}})\in\mathcal{C}^{l}(S)\times\mathcal{C}^{l}(\overline{S})$,
there exists a quasi-Fuchsian representation
$\rho\colon\pi_{1}(S)\to\operatorname{\mathrm{PSL}}(2,\mathbb{C})$, unique up
to conjugation, with domains of discontinuity $\Omega_{\rho}^{+}$ and
$\Omega_{\rho}^{-}$, together with unique $\rho$-equivariant holomorphic
diffeomorphisms

${\bf\it
f_{+}}(c_{1},\overline{c_{2}})\colon(\widetilde{S},c_{1})\to\Omega_{\rho}^{+},\qquad\qquad{\bf\it\overline{f_{-}}}(c_{1},\overline{c_{2}})\colon(\widetilde{S},\overline{c_{2}})\to\Omega_{\rho}^{-}.$

This correspondence determines a biholomorphism

$\mathcal{T}(S)\times\mathcal{T}(\overline{S})\xrightarrow{\sim}\mathrm{QF}(S).$

Fix a pair
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$.
The two maps $f_{1}=\operatorname{{\bf\it f_{+}}}+(c_{1},\overline{c_{2}})$
and
$\overline{f_{2}}=\operatorname{{\bf\it\overline{f_{--}}}}(c_{1},\overline{c_{2}})$
define an immersion
$(f_{1},\overline{f_{2}})\colon\widetilde{S}\to\mathbb{G}\supset\Omega^{+}_{\rho}\times\Omega^{-}_{\rho}$,
and the pull-back of $\operatorname{\langle\cdot,\cdot\rangle}_{\mathbb{G}}$
is the positive complex metric

${\bf\it
h}_{(c_{1},\overline{c_{2}})}:=-\frac{4}{(f_{1}-\overline{f_{2}})^{2}}df_{1}d\overline{f_{2}}$
(5)

that, as a result of Theorem 6.7 in [BE22], has constant curvature $-1$. We
refer to the metrics obtained in this fashion as Bers metrics. In [BE22], the
first author with Bonsante proved that one can uniformize positive complex
metrics via Bers metrics.

######  Proposition 2.8 (Theorem 6.11 in [BE22]).

A complex metric $g$ is positive if and only if there exists a (unique) Bers
metric ${\bf\it h}_{(c_{1},\overline{c_{2}})}$ and a function $\varrho\colon
S\to\mathbb{C}^{*}$ such that

$g=\varrho\cdot{\bf\it h}_{(c_{1},\overline{c_{2}})}\ .$

As proved in [BE22], in the smooth category, Bers metrics form a connected
component of the subset of smooth complex metrics of constant curvature $-1$,
and $(c_{1},\overline{c_{2}})\mapsto{\bf\it h}_{(c_{1},\overline{c_{2}})}$
defines a bijective correspondence between pairs of complex structures and
Bers metrics. If the complex structures are $W^{l,\infty}$, then the Bers
metric, viewed as a metric on $S$, is a $W^{l,2}$ bilinear form on
$\operatorname{\mathbb{C}}TS.$ Indeed, briefly, by Proposition 5.7 stated
later on, the maps ${\bf\it{f_{+}}}$ and ${\bf\it\overline{f_{-}}}$ are
$W_{loc}^{l+1,p}$ for some small $p>2$, hence $W_{loc}^{l+1,2}$, and routinely
estimating the formula (5) using that $W^{l,2}(U)$ is a Banach algebra for any
$U\subset\operatorname{\mathbb{C}}$ (see Proposition 5.17), one sees that the
Bers metric on $\widetilde{S}$ is $W_{loc}^{l,2}$. Descending to $S$ gives the
result. A few times in the paper, especially in Section 6, we’ll need to know
the following.

######  Proposition 2.9.

The map
$\mathcal{C}^{l}(S)\times\mathcal{C}^{l}(\overline{S})\to\mathcal{CM}^{l}(S),$
defined by $(c_{1},\overline{c_{2}})\mapsto{\bf\it
h}_{(c_{1},\overline{c_{2}})}$, with image in $\mathcal{CM}_{-1}^{l}(S)$, is
holomorphic.

From the definition of the Fréchet structures on
$\mathcal{C}(S)\times\mathcal{C}(\overline{S})$ and $\mathcal{CM}(S),$ it
follows that the map $(c_{1},\overline{c_{2}})\mapsto{\bf\it
h}_{(c_{1},\overline{c_{2}})}$ is holomorphic in these spaces as well.
Proposition 2.9 follows from the analytic dependence of solutions to the
Beltrami equation in higher Sobolev spaces; since we couldn’t find a source
for the results we need, we prove these results, as well as Proposition 2.9,
in the Appendix A.

Finally, we record some results from [Ema23] on holomorphically perturbing
positive complex metrics, which we will use in the proof of Theorem G in
Section 5.3. For all $c\in\mathcal{C}(\mathrm{S})$ (resp.
$\overline{c}\in\mathcal{C}(\overline{\mathrm{S}})$), denote by
$\mathrm{QD}(c)$ (resp. $\mathrm{QD}(\overline{c})$) the space of holomorphic
quadratic differentials for $c$ (resp. $\overline{c}$). If $g$ is a positive
complex metric, then by openness of the positive property, for all
$q_{1}\in\mathrm{QD}(c)$ and $\overline{q_{2}}\in\mathrm{QD}(\overline{c})$
small enough, $g+q_{1}$ and $g+\overline{q_{2}}$ are positive metrics as well.

######  Proposition 2.10.

Let $g$ be a positive complex metric, with $(c_{1},\overline{c_{2}})=({\bf\it
c_{+}},{\bf\it c_{-}})(g)$, and $q_{1}\in\mathrm{QD}(c)$ and
$\overline{q_{2}}\in\mathrm{QD}(\overline{c})$ so that $g+q_{1}$ and
$g+\overline{q_{2}}$ are both positive. Then

$\mathrm{K}_{g}=\mathrm{K}_{g+q_{1}},\quad\mathrm{and}\quad{\bf\it
c_{+}}(g+q_{1})=c_{1},$

and

$\mathrm{K}_{g}=\mathrm{K}_{g+\overline{q_{2}}},\quad\mathrm{and}\quad{\bf\it
c_{-}}(g+\overline{q_{2}})=\overline{c_{2}}\ .$

This Proposition is stated as Proposition 3.8 in [Ema23] for Bers metrics, but
the proof works verbatim for any positive complex metric. Moreover, in [Ema23]
it is proved that deformations from $h$ to $h+q_{1}$ and $h+\overline{q_{2}}$
descend to give complex charts on Teichmüller space.

######  Theorem 2.11 (Theorems A and B in [Ema23]).

Let
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$,
$h={\bf\it h}_{(c_{1},\overline{c_{2}})}$, and let
$U_{(c_{1},\overline{c_{2}})}\subset\mathrm{QD}(c_{1})$ be an open
neighborhood of zero such that ${\bf\it h}_{(c_{1},\overline{c_{2}})}+q_{1}$
is a Bers metric. The map

$\displaystyle U$ $\displaystyle\to\mathcal{T}(\overline{S})$ $\displaystyle
q_{1}$ $\displaystyle\mapsto[{\bf\it{c_{-}}}(h+q_{1})]$

is a biholomorphism onto its image. The analogous statement holds for
deformations of the form ${\bf\it h}_{(c_{1},\overline{c_{2}})}\mapsto{\bf\it
h}_{(c_{1},\overline{c_{2}})}+\overline{q_{2}}$.

Although we won’t use this fact in the paper, it is worthwhile to note that
the charts from the theorem above are affine reparametrizations of the
Schwarzian parametrization.

##  3\. Complex affine spheres

In this section we develop the theory of complex affine immersions and complex
affine spheres as an extension of the theory of (real) affine immersions and
affine spheres. We don’t recall the real theory here, but we highlight it as a
special case of our construction (see Example 3.23, and also Section 6.1 later
on). Sources on the real theory include [NS94], [Lof01], and [Lab07].

In the section below, if regularity is unspecified, we will assume that all
the objects are regular enough for the operations to be well-defined.

###  3.1. Complex affine immersions

Let $\mathbb{X}$ be a complex manifold of complex dimension $n=m+1$ with
almost complex structure $\mathbb{J}$ and torsion free holomorphic affine
connection $D$. Starting in Section 3.4, $\mathbb{X}=\mathbb{C}^{3},$ equipped
with the Levi-Civita connection of its holomorphic Riemannian metric, so it
might be helpful to have $\mathbb{X}=\operatorname{\mathbb{C}}^{n}$ in mind as
the typical target space. We denote by $\text{Aff}(\mathbb{X},D)$ the space of
biholomorphisms of $\mathbb{X}$ that preserve $D$.

Now, let $M$ be a (real) smooth manifold of dimension $m$, let $\widetilde{M}$
be its universal cover, and identify $\pi_{1}(M)$ with the deck group of the
covering map. From now on until Section 3.3, fix a representation
$\rho\colon\pi_{1}(M)\to\text{Aff}(\mathbb{X},D)$ and let
$\sigma\colon\widetilde{M}\to\mathbb{X}$ be a equivariant map. We refer to
$\rho$ as the holonomy of $\sigma$. Note that one can see immersions
$M\to\mathbb{X}$ as equivariant immersions $\widetilde{M}\to\mathbb{X}$ with
the trivial holonomy $\rho\equiv id_{\mathbb{X}}$.

The tangent map $\sigma_{*}\colon T\widetilde{M}\to\sigma^{*}(T\mathbb{X})$
extends to a $\operatorname{\mathbb{C}}$-linear map
$\sigma_{*}\colon\operatorname{\mathbb{C}}T\widetilde{M}\to\sigma^{*}(T\mathbb{X})$
by defining $\sigma_{*}(v+iw):=\sigma_{*}(v)+\mathbb{J}(\sigma_{*}(w))$. In
the following, with $\sigma_{*}$ we will always denote the extension to
$\mathbb{C}T\widetilde{M}$.

######  Definition 3.1.

We say that an immersion $\sigma:\widetilde{M}\to\mathbb{X}$ is admissible if
the induced map
$\sigma_{*}:\operatorname{\mathbb{C}}T\widetilde{M}\to\sigma^{*}T\mathbb{X}$
is injective.

Admissibility means that $\sigma_{*}$ maps $\mathbb{R}$-linearly independent
vectors in $T_{p}\widetilde{M}$ to $\operatorname{\mathbb{C}}$-linearly
independent vectors in $\sigma^{*}(T\mathbb{X})$. The admissibility condition
allows the construction of several geometric tensors associated with the
immersion, imitating the theory of hypersurfaces in smooth manifolds.

Since $\sigma$ is $\rho$-equivariant, $\pi_{1}(M)$ acts on the bundle
$\sigma^{*}(T\mathbb{X})$ through $\rho$ preserving both $\sigma^{*}D$ and
$\sigma^{*}\mathbb{J}$: the quotient defines a bundle $E_{\rho}\to M$ that
inherits both a connection, which we still denote with $\sigma^{*}D$, and a
complex structure that we denote with $i$ to relax the notation. Assuming
$\sigma$ is admissible, $\mathbb{C}TM$ can be seen as a complex subbundle of
$E_{\rho}$ with complex co-rank 1. We abuse notation slightly and write this
subbundle as $\operatorname{\mathbb{C}}TM\subset E_{\rho}$ when the context is
clear.

######  Definition 3.2.

A $\rho$-equivariant complex affine immersion of $\widetilde{M}$ in
$\mathbb{X}$ is a pair $(\sigma,\xi)$ where $\sigma\colon M\to\mathbb{X}$ is a
$\rho$-equivariant admissible immersion, and $\xi$ is a smooth section of
$E_{\rho}$ called the transversal vector field, such that
$\mathbb{C}T\widetilde{M}\oplus\mathrm{Span}_{\mathbb{C}}(\xi)=E_{\rho}$.

Compare with [NS94, Definition II.1.1].

A complex affine connection $\nabla$ on $M$ is a $\mathbb{C}$-bilinear affine
connection on the bundle $\mathbb{C}TM$. Observe that all affine connections
on $TM$ can be extended by linearity to complex affine connections on
$\operatorname{\mathbb{C}}TM$. Notice that this notion, defined on smooth
manifolds, is very different from the notion of holomorphic connection on a
complex manifold introduced above.

According to the splitting
$\mathbb{C}TM\oplus\mathrm{Span}_{\mathbb{C}}(\xi)$, we get that for all
$\mathsf{X}\in\Gamma(TM)$, $Y\in\Gamma(\mathbb{C}TM)$

$\begin{cases}(\sigma^{*}D)_{\mathsf{X}}Y&=:\nabla_{\mathsf{X}}Y+g(\mathsf{X},Y)\cdot\xi\\\
(\sigma^{*}D)_{\mathsf{X}}\
\xi&=:-\mathsf{S}(\mathsf{X})+\tau(\mathsf{X})\cdot\xi.\end{cases}$ (6)

######  Remark 3.3.

One can see that:

  * •

for all $X,Y\in\Gamma(\mathbb{C}TM)$, with $X=\mathsf{X}_{1}+i\mathsf{X}_{2}$,
$\nabla_{X}Y:=\nabla_{\mathsf{X}_{1}}Y+i\nabla_{\mathsf{X}_{2}}Y$ is a
torsion-free complex affine connection on $\mathbb{C}TM$;

  * •

$g$ extends to a symmetric $\mathbb{C}$-bilinear form on $\mathbb{C}TM$,
which, by analogy with the theory of real affine spheres, we call the affine
fundamental form;

  * •

$\mathsf{S}$ is linear, and it extends to a section of
$\mathrm{End}(\mathbb{C}TM)$, called the shape operator;

  * •

$\tau$ is linear as well, and it extends to an element in
$\Omega^{1}(\mathbb{C}TM)$, called transversal connection form.

###  3.2. Blaschke affine immersions

In this subsection we construct preferred transversal vector fields called
affine normals. The theory here is roughly parallel to [NS94, §II.3]. However,
unlike the real case, we will see that there are topological obstructions to
the existence of such an affine normal. First of all, let us see how the
affine tensors defined in the previous subsection change according to a change
in the transversal vector field.

######  Lemma 3.4.

Let $(\sigma,\xi)$ be a complex affine immersion, with
$\nabla,g,\mathsf{S},\tau$ as in Equation (6). Let $\hat{\xi}=\alpha\xi+\eta$
be a different transversal vector field for $\sigma$, with $\alpha\in
C^{\infty}(M,\operatorname{\mathbb{C}}^{*})$ and
$\eta\in\Gamma(\operatorname{\mathbb{C}}TM)$. Then, the complex affine
immersion $(\sigma,\hat{\xi})$ induces
$\hat{\nabla},\hat{g},\hat{\mathsf{S}},\hat{\tau}$ according to:

  1. (1)

$\hat{g}=\frac{1}{\alpha}g$,

  2. (2)

$\hat{\nabla}_{X}Y=\nabla_{X}Y-\frac{1}{\alpha}g(X,Y)Z$,

  3. (3)

$\hat{\mathsf{S}}(X)=\alpha\mathsf{S}(X)-\nabla_{X}\eta+\tau(X)\eta+\frac{\partial_{X}\alpha+g(X,\eta)}{\alpha}\eta$,

  4. (4)

$\hat{\tau}(X)=\tau(X)+\frac{\partial_{X}\alpha+g(X,\eta)}{\alpha}$.

###### Proof.

For $X,Y,\eta\in\Gamma(TM)$, the proof follows by direct computation, as shown
for the real case in [NS94, Proposition II.2.5]. The equations for
$X,Y\in\Gamma(\mathbb{C}TM)$ follow by $\mathbb{C}$-linearity, while the
result for $\eta\in\Gamma(\mathbb{C}TM)$ follows by applying transformations
repeatedly, namely:

$\displaystyle\xi$
$\displaystyle\quad\leadsto\quad\xi+Re\left(\frac{\eta}{\alpha}\right)\quad\leadsto\quad-i\left(\xi+Re\left(\frac{\eta}{\alpha}\right)\right)+Im\left(\frac{\eta}{\alpha}\right)=-i\xi-i\frac{\eta}{\alpha}\quad\leadsto$
$\displaystyle\leadsto\quad
i\alpha\left(-i\xi-i\frac{\eta}{\alpha}\right)=\alpha\xi+\eta.$

∎

From now on, assume that $\mathbb{X}$ carries a holomorphic $D$-flat volume
form $\omega$, i.e. a nowhere vanishing holomorphic $n$-form $\omega$ such
that $D\omega=0$, and that our actions $\rho$ preserve this volume form. We
denote by Aff$(\mathbb{X},D,\omega)$ the space of biholomorphisms of
$\mathbb{X}$ that preserve both $D$ and $\omega$.

Given $(\sigma,\xi)$ a complex affine immersion of $\widetilde{M}$ into
$\mathbb{X}$, we define a nowhere vanishing section $\theta$ of
$\bigwedge^{k}\operatorname{\mathbb{C}}T^{*}\widetilde{M}$ on $\widetilde{M}$
by

$\theta(X_{1},\dots,X_{m})=(\sigma^{*}\omega)(X_{1},\dots,X_{m},\xi)\ .$ (7)

We call $\theta$ the induced complex volume form. By equivariance, the induced
complex volume form $\theta$ descends to a volume form on $M$.

######  Proposition 3.5.

For any $X\in\operatorname{\mathbb{C}}TM,$ $\nabla_{X}\theta=\tau(X)\theta.$

###### Proof.

The proof for $X\in\Gamma(TS)$ is identical to the proof of the analogous
result for real affine immersions [NS94, Proposition II.1.4], then the
statement follows by $\operatorname{\mathbb{C}}$-linearity of $\nabla\theta$
and $\tau$. ∎

Proposition 3.5 and Equation (6) tell us that $\theta$ is $\nabla$-parallel if
and only if $(\sigma^{*}D)\xi$ preserves the sub-bundle
$\mathbb{C}T\widetilde{M}$.

We say that $\sigma$ is non-degenerate if, given a transversal vector field
$\xi,$ the affine fundamental form $g$ is nowhere degenerate, i.e., it is a
complex metric. By Lemma 3.4, all choices of affine transversal sections
determine conformally equivalent fundamental forms, so this notion of non-
degeneracy is independent of our choice of the transversal section.

######  Remark 3.6.

Given a complex metric $g$ on a smooth manifold $\widetilde{M}$, on each
simply connected chart with local coordinates $(x_{i})_{i=1}^{m}$ one can
define a local metric volume form
$dV_{g}=\sqrt{det(g(\partial_{x_{i}},\partial_{x_{j}}))_{i,j}}\cdot
dx_{1}\wedge\dots\wedge dx_{n}$, which is well-defined up to a sign. Since
$\widetilde{M}$ is simply connected, there always exists a global metric
volume form on $\widetilde{M}$.

######  Definition 3.7.

Let $\rho\colon\pi_{1}(M)\to\text{Aff}(\mathbb{X},D,\omega)$ be a
homomorphism. A $\rho$-equivariant Blaschke affine immersion of
$\widetilde{M}$ in $\mathbb{X}$ is an (equivariant) affine immersion
$(\sigma,\xi)$ such that:

  1. (1)

the affine fundamental form $g$ is nowhere degenerate, and there exists a
global metric volume form $dV_{g}$ on $M$,

  2. (2)

$\nabla\theta=0$, hence $\tau=0$ by Proposition 3.5,

  3. (3)

$\theta=dV_{g}$.

The transversal vector field $\xi$ is called affine normal and $g$ is called
Blaschke metric.

######  Proposition 3.8.

Let $M=\widetilde{M}$ be simply connected. Given a non-degenerate affine
immersion $(\sigma,\xi)$ of $\widetilde{M}$ into $\mathbb{X}$, there exists a
transversal vector field $\hat{\xi}$, unique up to a sign, such that
$(\sigma,\hat{\xi})$ is a Blaschke affine immersion. $(\sigma,\hat{\xi})$ is
called the Blaschke normalization of $(\sigma,\xi)$.

###### Proof.

This proof is totally analogous to that of the corresponding result in the
real case (see [NS94, Theorem II.3.1]), but we include the full proof to show
where the topological assumption is used.

Since $\widetilde{M}$ is simply connected, the nondegenerate affine
fundamental form $g$ of $(\sigma,\xi)$ admits a global metric volume form
$dV_{g}$. Given $\alpha\in C^{\infty}(M,\operatorname{\mathbb{C}}^{*})$ and
$\eta\in\Gamma(\operatorname{\mathbb{C}}TM)$, if we transform $\xi$ to
$\hat{\xi}=\alpha\xi+\eta,$ the metric volume form $dV_{g}$ and the induced
complex volume form $\theta$ of $(\sigma,\xi)$ change according to
$dV_{\hat{g}}=\frac{1}{\alpha}dV_{g}$ and

$\hat{\theta}(X,Y)=\omega(X,Y,\alpha\xi+\eta)=\omega(X,Y,\alpha\xi)=\alpha\theta(X,Y).\
$

Since $M$ is simply connected, there exists a function $\alpha\in
C^{\infty}(\widetilde{M},\operatorname{\mathbb{C}}^{*}),$ unique up to
multiplication by $-1$, such that

$\alpha^{2}=\frac{dV_{g}}{\theta}.$

Choosing such $\alpha$ guarantees that the condition (3) holds, so it remains
to pick $\eta$ so that (2) holds. Since $g$ is non-degenerate, there exists a
unique complex vector field $\eta$ such that, as

$g(\eta,\cdot)=-(\alpha\tau+d\alpha).\ $

Using item (4) from Lemma 3.4, we see that $\hat{\xi}=\alpha\xi+\eta$
satisfies the requirements of the proposition. Observe that if we exchange
$\alpha$ with $-\alpha,$ then $\eta$ becomes $-\eta,$ and hence $\xi$ is
unique up to multiplication by $\pm 1$. ∎

######  Remark 3.9.

Looking into the proof of Proposition 3.8, one can see that, unlike the real
case, for general $M$ and for a general representation
$\rho\colon\pi_{1}(M)\to\mathrm{Aff}(\mathbb{X},D,\omega)$, a
$\rho$-equivariant non-degenerate affine immersion $(\sigma,\xi)$ might not
admit a $\rho$-equivariant Blaschke normalization. In fact, one needs to
assume that $(M,g)$ admits a global volume form (equivalently, a
$\pi_{1}(M)$-invariant metric volume form for $\widetilde{g}$ on
$\widetilde{M}$), and - even assuming this - the function
$\frac{dV_{g}}{\theta}$ might not admit a $\pi_{1}(M)$-invariant square root.

Nevertheless, the obstructions only depend on the topological properties of
$g$, $dV_{g}$, and $\theta$, so, if $\sigma$ is a deformation of a real affine
immersion, then it admits a Blaschke normalization.

######  Example 3.10 (Graph immersions).

Let $U$ a domain in $\mathbb{R}^{n}$ and
$F:\mathbb{R}^{n}\to\operatorname{\mathbb{C}}$ a $C^{1}$ function. A graph
immersion is an admissible immersion
$\sigma:U\to\operatorname{\mathbb{C}}^{n+1}$ of the form

$\sigma(x_{1},\dots,x_{n})=(x_{1},\dots,x_{n},F(x_{1},\dots,x_{n})).$

The choice of transversal vector field $\xi=(0,\dots,0,1)$ yields
$\mathsf{S}=0$ and $\tau=0$.

######  Example 3.11 (Complex paraboloids).

As a special example of a graph immersion, we have what we call complex
paraboloids. Consider, for instance,
$F(x,y)=(x,y,\frac{1}{2}(x^{2}+e^{i\theta}y^{2})).$ Taking $e^{i\theta}=1$
gives the well-known elliptic paraboloid, and taking $e^{i\theta}=-1$ gives a
hyperbolic paraboloid. It is not hard to compute that $\xi=(0,0,e^{-i\theta})$
produces a transversal vector field with $\tau=0$ and $\theta=dV_{g}.$

###  3.3. The fundamental theorem for complex affine immersions

From now on, we will assume that $\mathbb{X}=\mathbb{C}^{n+1}$ endowed with
the Levi-Civita connection of the holomorphic Riemannian metric, say $D$. The
standard equations from affine differential geometry admit complex extensions.

######  Proposition 3.12.

Consider an equivariant complex affine immersion $(\sigma,\xi)$ of
$\widetilde{M}$ into $(\mathbb{C}^{m+1},D)$ inducing the data
$\nabla,g,\mathsf{S},\tau$ on $M$ as in Equation (6).

The following equations hold for all $X,Y,Z\in\Gamma(\mathbb{C}TM)$.

Gauss equation: $\displaystyle\qquad
R^{\nabla}(X,Y)Z=g(Y,Z)\mathsf{S}(X)-g(X,Z)\mathsf{S}(Y),$ (8) Codazzi
equation 1:
$\displaystyle\qquad(\nabla_{X}g)(Y,Z)+\tau(X)g(Y,Z)=(\nabla_{Y}g)(X,Z)+\tau(Y)g(X,Z),$
(9) Codazzi equation 2:
$\displaystyle\qquad(\nabla_{X}\mathsf{S})(Y)-\tau(X)\mathsf{S}(Y)=(\nabla_{Y}\mathsf{S})(X)-\tau(Y)\mathsf{S}(X),$
(10) Ricci equation: $\displaystyle\qquad
g(X,\mathsf{S}(Y))-g(\mathsf{S}(X),Y)=d\tau(X,Y).$ (11)

###### Proof.

The corresponding results hold for (real) affine immersions between real
manifolds [NS94, Theorem II.2.1], as a result of the flatness of $D$. Indeed,
one can see explicitly from Equation (6) that for all
$\mathsf{X},{\mathsf{Y}},{\mathsf{Z}}\in\Gamma(TM)$

$\begin{split}\mathrm{R}^{D}(\mathsf{X},{\mathsf{Y}}){\mathsf{Z}}=&\mathrm{R}^{\nabla}(\mathsf{X},{\mathsf{Y}}){\mathsf{Z}}-g({\mathsf{Y}},{\mathsf{Z}})\mathsf{S}(\mathsf{X})+g(\mathsf{X},{\mathsf{Z}})\mathsf{S}({\mathsf{Y}})\\\
&+(\nabla_{\mathsf{X}}g)({\mathsf{Y}},{\mathsf{Z}})\xi-(\nabla_{\mathsf{Y}}g)({\mathsf{X}},{\mathsf{Z}})\xi+g({\mathsf{Y}},{\mathsf{Z}})\tau({\mathsf{X}})-g({\mathsf{X}},{\mathsf{Z}})\tau({\mathsf{Y}}),\\\
\mathrm{R}^{D}({\mathsf{X}},{\mathsf{Y}})\xi&=-(\nabla_{\mathsf{X}}\mathsf{S})({\mathsf{Y}})+(\nabla_{\mathsf{Y}}\mathsf{S})({\mathsf{X}})-\tau({\mathsf{X}})\mathsf{S}({\mathsf{Y}})+\tau({\mathsf{Y}})\mathsf{S}({\mathsf{X}})\\\
&-g({\mathsf{X}},\mathsf{S}({\mathsf{Y}}))\xi+g({\mathsf{Y}},\mathsf{S}({\mathsf{X}}))\xi+d\tau({\mathsf{X}},{\mathsf{Y}})\xi,\end{split}$
(12)

hence Equations (8)-(9)-(10)-(11) follow by
$\mathrm{R}^{D}(\mathsf{X},\mathsf{Y})Z=0=\mathrm{R}^{D}(\mathsf{X},\mathsf{Y})\xi$.
Finally, the statement follows for all
$X,Y,Z\in\Gamma(\operatorname{\mathbb{C}}TS)$ by
$\operatorname{\mathbb{C}}$-linearity of $\nabla$ and of the tensors. ∎

Conversely, the immersion data $\nabla,g,\mathsf{S},$ and $\tau$ determine
uniquely an affine immersion, as shown in the following statement.

######  Theorem 3.13.

Let $M$ be a smooth real $m$-manifold. Let
$\nabla\colon\Gamma(\operatorname{\mathbb{C}}TM)\to\Gamma(End_{\operatorname{\mathbb{C}}}(\operatorname{\mathbb{C}}TM))$
be a torsion-free affine connection, let $g$ be a complex metric, $\mathsf{S}$
an endomorphism of $\operatorname{\mathbb{C}}TM$, and
$\tau\in\Omega^{1}(\operatorname{\mathbb{C}}TM)$ such that Equations
(8)-(9)-(10)-(11) hold.

Then, there exists a $\pi_{1}(M)$-equivariant complex affine immersion
$(\sigma,\xi)\colon\widetilde{M}\to\mathbb{C}^{m+1}$ with affine connection
$\nabla$, affine fundamental form $g$, shape operator $\mathsf{S}$, and
transversal connection form $\tau$. Such affine immersions are unique up to
composition with elements in
$\mathrm{Aff}(\mathbb{C}^{m+1},D)=\mathrm{GL}(m+1,\operatorname{\mathbb{C}})\ltimes\operatorname{\mathbb{C}}^{m+1}$.

Moreover, if $\tau=0$ and $\nabla dV_{g}=0$, $(\sigma,\xi)$ can be chosen to
be an equivariant Blaschke affine immersion for
$\operatorname{\mathbb{C}}^{m+1}$ endowed with the canonical volume form
$\omega=dz_{1}\wedge\dots\wedge dz_{n+1}$, and its holonomy takes values in
$\operatorname{\mathrm{SL}}(m+1,\operatorname{\mathbb{C}})\ltimes\operatorname{\mathbb{C}}^{m+1}$.

Fo this statement, too, the computation follows as in the real case [NS94,
Theorem II.8.1], up to a few adjustments. We recall the essential ideas and
point out the main differences.

###### Proof.

_Step 1: Existence for $M=\widetilde{M}$._ Let
$N\colon\operatorname{\mathbb{C}}\times\widetilde{M}\to\widetilde{M}$ be the
trivial bundle, and fix $\hat{\xi}\in\Gamma(N)$ a nowhere zero section.
Consider the bundle $\mathrm{V}=\operatorname{\mathbb{C}}T\widetilde{M}\oplus
N$ and the connection $\hat{D}$ defined for all
$\mathsf{X}\in\Gamma(T\widetilde{M})$ and
$Y\in\Gamma(\operatorname{\mathbb{C}}T\widetilde{M})$ as

$\begin{cases}\hat{D}_{\mathsf{X}}Y&=\nabla_{\mathsf{X}}Y+g(\mathsf{X},Y)\hat{\xi}\\\
\hat{D}_{\mathsf{X}}\hat{\xi}&=-\mathsf{S}(\mathsf{X})+\tau(\mathsf{X})\hat{\xi}\\\
\hat{D}_{\mathsf{X}}(i\hat{\xi})&=-i\mathsf{S}(\mathsf{X})+i\tau(\mathsf{X})\hat{\xi}.\end{cases}$

Then, one gets the same equation as in (12) after replacing $D$ with
$\hat{D}$, and
$\mathrm{R}^{\hat{D}}(\mathsf{X},Y)(i\xi)=i\mathrm{R}^{\hat{D}}(\mathsf{X},Y)(\xi)$,
hence the hypotheses of the theorem imply that $R^{\hat{D}}=0$. Moreover,
$\hat{D}iY=i\hat{D}Y$, so there exists a bundle isomorphism

$\Phi\colon(\operatorname{\mathbb{C}}T\widetilde{M}\oplus
N,\hat{D})\to(\operatorname{\mathbb{C}}^{m+1}\times\widetilde{M},D)$

with $D$ denoting the standard connection on the trivial bundle, and $\Phi$
being a $\operatorname{\mathbb{C}}$-linear isomorphism fiberwise.

For all $X\in\Gamma(\operatorname{\mathbb{C}}T\widetilde{M})$, we can see
$\Phi(X_{p})=:(p,\alpha(X_{p}))$ with $\alpha$ a
$\operatorname{\mathbb{C}}^{m+1}$-valued 1-form on $\widetilde{M}$. With a
little abuse of notation, we will say that
$\alpha=\Phi_{|\operatorname{\mathbb{C}}T\widetilde{M}}$. Then, for all
$\mathsf{X},\mathsf{Y}\in\Gamma(T\widetilde{M})$,
$D_{\mathsf{X}}(\Phi(\mathsf{Y}))=\alpha(\mathsf{Y})$, and

$d\alpha(\mathsf{X},\mathsf{Y})=\partial_{\mathsf{X}}(\alpha(\mathsf{Y}))-\partial_{\mathsf{Y}}(\alpha(\mathsf{X}))-\alpha([\mathsf{X},\mathsf{Y}])=\Phi(\nabla_{\mathsf{X}}\mathsf{Y}-\nabla_{\mathsf{Y}}\mathsf{X}-[\mathsf{X},\mathsf{Y}])=0$

as a result of $\nabla$ being torsion-free. Hence, there exists
$\sigma\colon\widetilde{M}\to\operatorname{\mathbb{C}}^{m+1}$ such that
$d\sigma=\alpha$. As a consequence, $\sigma_{*}(X)=d\sigma(X)=\Phi(X)$ so
$\sigma_{*}(X)=0$ iff $X=0$, implying that $\sigma$ is an admissible
immersion.

Moreover, if we consider the pull-back bundle
$\sigma^{*}(\operatorname{\mathbb{C}}^{m+1})$ with the pull-back connection
$\sigma^{*}D$ and transversal vector field $\xi$ defined by
$\sigma_{*}(\xi)=\Phi(\hat{\xi})$, we get that, for all
$\mathsf{X}\in\Gamma(T\widetilde{M})$ and
$Y\in\Gamma(\operatorname{\mathbb{C}}T\widetilde{M})$, we have at each point
that

$\begin{cases}\sigma_{*}\left((\sigma^{*}D)_{\mathsf{X}}Y\right)=D_{\mathsf{X}}(\sigma_{*}(Y))=D_{\mathsf{X}}(\Phi(Y))=\Phi(\hat{D}_{\mathsf{X}}Y)=\Phi(\nabla_{\mathsf{X}}Y+g(\mathsf{X},Y)\hat{\xi})=\sigma_{*}(\nabla_{\mathsf{X}}Y)+g(\mathsf{X},Y)\sigma_{*}(\xi)\\\
\sigma_{*}\left((\sigma^{*}D)_{\mathsf{X}}\xi\right)=D_{X}(\sigma_{*}\xi)=D_{\mathsf{X}}(\Phi(\hat{\xi}))=\Phi(\hat{D}_{\mathsf{X}}\hat{\xi})=-\Phi(\mathsf{S}(\mathsf{X}))+\tau(\mathsf{X})\Phi(\hat{\xi})=\sigma_{*}(-\mathsf{S}(\mathsf{X})+\tau(\mathsf{X})\xi),\end{cases}$

which implies that $(\sigma,\xi)$ has immersion data given by
$\nabla,g,\mathsf{S},\tau$.

_Step 2: Uniqueness and the equivariant case._ Let $(\sigma,\xi)$ and
$(\sigma^{\prime},\xi^{\prime})$ be two affine immersions with the same
immersion data $\nabla,g,\mathsf{S},\tau$. We prove that there exists
$A\in\mathrm{GL}(m+1,\operatorname{\mathbb{C}})\ltimes\operatorname{\mathbb{C}}^{m+1}$
such that $\sigma^{\prime}=A\circ\sigma$ and
$\sigma^{\prime}_{*}\xi^{\prime}=A(\sigma_{*}\xi)$.

Define the function
$L\colon\widetilde{M}\to\mathrm{GL}(m+1,\operatorname{\mathbb{C}})$ as the
unique matrix such that, for all
$X\in\Gamma(\operatorname{\mathbb{C}}T\widetilde{M})$,
$\sigma^{\prime}_{*}(X)=L\circ\sigma_{*}(X)$ and
$\sigma^{\prime}_{*}(\xi^{\prime})=L(\sigma_{*}(\xi))$.

Then (recalling that
$\sigma_{*}((\sigma^{*}D)_{\mathsf{X}}Y)=\partial_{\mathsf{X}}(d\sigma(Y))=\partial_{\mathsf{X}}(\sigma_{*}(Y))$),
we have that for all $\mathsf{X}\in\Gamma(T\widetilde{M})$,
$Y\in\Gamma(\operatorname{\mathbb{C}}T\widetilde{M})$,

$\displaystyle(\partial_{\mathsf{X}}L)(\sigma_{*}Y)$
$\displaystyle=\partial_{\mathsf{X}}((L\circ\sigma_{*})(Y))-L(\partial_{\mathsf{X}}(\sigma_{*}Y))=\partial_{\mathsf{X}}(\sigma^{\prime}_{*}(Y))-L(\partial_{\mathsf{X}}(\sigma_{*}Y))$
$\displaystyle=\sigma^{\prime}_{*}(\nabla_{\mathsf{X}}Y)+g(\mathsf{X},Y)\xi^{\prime}-L(\sigma_{*}(\nabla_{\mathsf{X}}Y))-g(\mathsf{X},Y)L(\xi)=0,$
$\displaystyle(\partial_{\mathsf{X}}L)(\sigma_{*}\xi)$
$\displaystyle=\partial_{\mathsf{X}}(L(\sigma_{*}\xi))-L(\partial_{\mathsf{X}}(\sigma_{*}\xi))=\partial_{\mathsf{X}}(L(\sigma_{*}\xi))-L(\partial_{\mathsf{X}}(\sigma_{*}\xi))-L(\partial_{\mathsf{X}}(\sigma_{*}\xi))$
$\displaystyle=-\sigma^{\prime}_{*}(\mathsf{S}({\mathsf{X}}))+\tau({\mathsf{X}})\sigma^{\prime}_{*}(\xi)+L(\sigma_{*}(\mathsf{S}({\mathsf{X}})))+\tau({\mathsf{X}})L(\sigma_{*}(\xi))=0,$

hence $dL=0$, implying that $L$ is a constant. Moreover,
$d(\sigma^{\prime}-L\sigma)=0$, hence there exists
$b\in\operatorname{\mathbb{C}}^{m+1}$ such that
$\sigma^{\prime}=L\cdot\sigma+b$, proving uniqueness. The equivariant case
follows from uniqueness in the usual way.

_Step 3: Blaschke immersions._ Let $\omega=dz_{1}\wedge\dots\wedge dz_{m+1}$
be the standard $D$-flat volume form on $\operatorname{\mathbb{C}}^{m+1}$.
Assume $\tau=0$, and that $\nabla dV_{g}=0$ and construct a corresponding
equivariant affine immersion
$(\sigma,\xi)\colon\widetilde{M}\to\operatorname{\mathbb{C}}^{m+1}$, and let
$\theta=(\sigma^{*}\omega)(\cdot,\xi)$ be the pull-back form. Then, by
Proposition 3.5, $\nabla\theta=0$. Assuming $\nabla(dV_{g})=0$ as well, we get
that $dV_{g}=C\theta$ for some constant $C$ as a consequence of the fact that
$\bigwedge^{n}\operatorname{\mathbb{C}}T^{*}\widetilde{M}$ is rank-1. Up to
replacing now $(\sigma,\xi)$ with the affinely equivalent affine immersion
$(\frac{1}{C}\sigma,\frac{1}{C}\xi)$, we can assume that $C=1$, therefore
$(\sigma,\xi)$ is a Blaschke affine immersion in
$(\operatorname{\mathbb{C}}^{m+1},D,\omega)$. If $(\sigma,\xi)$ is
equivariant, its holonomy must have image in
$\operatorname{\mathrm{SL}}(m+1,\operatorname{\mathbb{C}})\ltimes\operatorname{\mathbb{C}}^{m+1}$.

∎

###  3.4. Complex affine spheres

We now restrict to the case where $M$ is a closed oriented surface, and
accordingly now write $M=S,$ and $\mathbb{X}=\mathbb{C}^{3}$ with the standard
affine connection $D$ and volume form $\omega=dz_{1}\wedge dz_{2}\wedge
dz_{3}$. We have that
$\mathrm{Aff}(\mathbb{C}^{3},D,\omega)=\operatorname{\mathrm{SL}}(3,\mathbb{C})\ltimes\mathbb{C}^{3}$.

######  Definition 3.14.

Let $\rho\colon\pi_{1}(M)\to\mathrm{Aff}(\mathbb{C}^{3},D,\omega)$, and let
$(\sigma,\xi)$ be a $\rho$-equivariant Blaschke affine immersion. We say that
$(\sigma,\xi)$ is a (proper) complex affine sphere if the shape operator has
the form $\mathsf{S}=\lambda\cdot id,$ where $\lambda\in\mathbb{C}^{*}$. A
complex affine sphere $(\sigma,\xi)$ is positive and hyperbolic if
$\lambda=-1$ and the Blaschke metric $g$ is a positive complex metric.

######  Remark 3.15.

As shown in [BE22, Theorem 6.5], every positive complex metric on an oriented
surface $S$ admits a metric volume form compatible with the orientation. We
will assume that the affine normal is compatible with the induced volume form.

######  Remark 3.16.

If an affine sphere $(\sigma,\xi)$ has shape operator in the form
$\mathsf{S}=\lambda\cdot id$ with $\lambda$ a function, by Codazzi Equation in
Proposition 3.12 one gets that $d\lambda=0$, hence $\lambda$ is a constant.

######  Remark 3.17.

If $(\sigma,\xi)$ is a proper complex affine sphere, then all of the affine
normals meet at a single point. Indeed, one can see that the map
$p\mapsto\sigma(p)+\frac{1}{\lambda}\xi(p)$ is a constant.

######  Convention.

By Remark 3.17, up to composing $\sigma$ with a translation, from now on we
will assume that the meeting point of the affine normals is
$\underline{0}\in\operatorname{\mathbb{C}}^{3}$, hence

$\sigma=-\frac{1}{\lambda}\xi\ $ (13)

and the data of $\sigma$ and $\lambda$ include the data of $\xi$. In this
setting, if $\sigma$ is an equivariant affine sphere, then its holonomy fixes
$\underline{0}$, and hence takes values in
$\operatorname{\mathrm{SL}}(3,\mathbb{C})$.

Pick tensors play a prominent role in the real theory, and we turn to them
now. Given $g,$ we have both the induced connection $\nabla$ and the Levi-
Civita connection of $g$, say $\nabla^{g}$. The difference tensor on
$\operatorname{\mathbb{C}}TS$ is

$K(X,Y)=K_{X}(Y)=\nabla_{X}Y-\nabla_{X}^{g}Y.$

Since both connections have no torsion, $K$ is symmetric. The Pick tensor is
the complex $3$-form defined by lowering indices,

$C(X,Y,Z)=-2g(K_{X}Y,Z).$

The proposition below can be proved as in the real case for real vector fields
(see [NS94, Section II.4]), and by $\operatorname{\mathbb{C}}$-linearity we
get the general case.

######  Proposition 3.18.

Let $(\sigma,\xi)$ be an equivariant proper complex affine sphere. Then, for
all $X,Y,Z,W\in\Gamma(\operatorname{\mathbb{C}}TS),$ we have that

  1. (1)

$C=\nabla g$,

  2. (2)

$\mathrm{tr}_{g}K_{X}=0$,

  3. (3)

$C,\nabla C,$ $\nabla K$, $\nabla^{g}C$, and $\nabla^{g}K$ are symmetric in
all of their entries,

  4. (4)

$g((\nabla^{g}_{X}K)(Y,Z),W)=g((\nabla^{g}_{X}K)(Y,W),Z)$,

  5. (5)

$\mathrm{R}^{\nabla}(X,Y)Z=\mathrm{R}^{g}(X,Y)Z+[K_{X},K_{Y}](Z).$

We use only items (2) and (3), but the other results above help motivate some
aspects of the proof of Theorem 3.20 below. Using that $\nabla^{g}C$ is
symmetric, we show that positive hyperbolic affine spheres come with
holomorphic cubic differentials, and thus we see the link to Theorem A.

######  Proposition 3.19.

Let $(\sigma,\xi)$ be an equivariant positive hyperbolic complex affine sphere
in $\mathbb{C}^{3}$. Let $g$ be the Blaschke metric, with conformal classes
given by
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$.
Then,

$C=\operatorname{\mathrm{Q}}_{1}+\overline{\operatorname{\mathrm{Q}}_{2}},$

where $\operatorname{\mathrm{Q}}_{1}$ is a $c_{1}$-holomorphic cubic
differential and $\overline{\operatorname{\mathrm{Q}}}_{2}$ is a
$\overline{c_{2}}$-holomorphic cubic differential.

###### Proof.

Let $z$ and $\overline{w}$ denote local coordinates for $c_{1}$ and
$\overline{c_{2}}$ respectively. By Proposition 3.18, $C$ is symmetric, so it
decomposes as

$C=pdz^{3}+qdz^{2}\cdot d\overline{w}+rdz\cdot
d\overline{w}^{2}+sd\overline{w}^{3}\ ,$

where $p,q,r,s$ are local $\mathbb{C}$-valued functions. By $(2)$ in
Proposition 3.18, $tr_{g}(K_{\partial_{\overline{z}}})=0$, hence

$0=tr_{g}(C(\partial_{\overline{z}},\cdot,\cdot))=2\frac{1}{g(\partial_{\overline{z}},\partial_{w})}C(\partial_{\overline{z}},\partial_{\overline{z}},\partial_{w})=\frac{\partial_{\overline{z}}\overline{w}\cdot\partial_{w}z}{h(\partial_{\overline{z}},\partial_{w})}r\
,$

hence $r=0$. Similarly, from $tr_{g}(K_{\partial_{w}})=0$ we deduce that
$q=0$.

We are left to prove that $p$ is $c_{1}$-holomorphic and $s$ is
$\overline{c_{2}}$-holomorphic. First, by Lemma 2.5, for all
$X\in\Gamma(\operatorname{\mathbb{C}}TS)$,
$\nabla^{g}_{X}\partial_{\overline{z}}\in\mathrm{Span}(\partial_{\overline{z}})$
and $\nabla^{g}_{X}\partial_{w}\in\Gamma(\mathrm{Span}(\partial_{w}))$. As a
result,

$(\nabla^{g}dz)(\partial_{\overline{z}})=d(dz(\partial_{\overline{z}}))-dz(\nabla^{g}\partial_{\overline{z}})=0-0=0.\
$

Moreover, by Cartan’s formula, for all
$X,Y\in\Gamma(\operatorname{\mathbb{C}}TS)$,

$\displaystyle(\nabla^{g}_{X}dz)(Y)-(\nabla^{g}_{Y}dz)(X)$
$\displaystyle=\partial_{X}(dz(Y))-dz(\nabla^{g}_{X}Y)-\partial_{Y}(dz(X))+dz(\nabla^{g}_{Y}X)=$
$\displaystyle=\partial_{X}(dz(Y))-\partial_{Y}(dz(X))-dz([X,Y])=(ddz)(X,Y)=0,$

hence $\nabla^{g}dz$ is commutative and
$\nabla^{g}dz\in\Gamma(\mathrm{Span}(dz^{2}))$. In the same fashion,
$\nabla^{g}d\overline{w}\in\Gamma(\mathrm{Span}(d\overline{w}^{2}))$.
Iterating the derivative, one can see that for all $m\in\mathbb{Z}_{+}$

$\nabla^{g}(dz^{m})\in\Gamma(\mathrm{Span}(dz^{m+1}))\qquad\qquad\qquad\nabla^{g}(d\overline{w}^{m})\in\Gamma(\mathrm{Span}(d\overline{w}^{m+1})).\
$

Now, by $(3)$ in Proposition 3.18, we have that $\nabla^{g}C$ is symmetric.
Hence, from

$\nabla^{g}C=dp\otimes dz^{3}+p\nabla^{g}(dz^{3})+ds\otimes
d\overline{w}^{3}+s\nabla^{g}(d\overline{w}^{3})$

we finally observe that

$\displaystyle\partial_{\overline{z}}p\cdot(\partial_{w}z)^{3}=(\nabla^{g}_{\partial_{\overline{z}}}C)(\partial_{w},\partial_{w},\partial_{w})$
$\displaystyle=(\nabla^{g}_{\partial_{w}}C)(\partial_{w},\partial_{w},\partial_{\overline{z}})=0,$
$\displaystyle\partial_{w}s\cdot(\partial_{\overline{z}}\overline{w})^{3}=(\nabla^{g}_{\partial_{w}}C)(\partial_{\overline{z}},\partial_{\overline{z}},\partial_{\overline{z}})$
$\displaystyle=(\nabla^{g}_{\partial_{\overline{z}}}C)(\partial_{\overline{z}},\partial_{\overline{z}},\partial_{w})=0,$

implying that $\partial_{\overline{z}}p=\partial_{w}s\equiv 0$. ∎

Conversely, the data of $g$ and $C$ completely characterize the complex affine
immersion, as described by the following theorem.

######  Theorem 3.20.

Let $g$ be a positive complex metric on a closed oriented surface $S$, with
conformal class
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$,
and let $\operatorname{\mathrm{Q}}_{1}$ and
$\overline{\operatorname{\mathrm{Q}}}_{2}$ be holomorphic cubic differentials
for $c_{1}$ and $\overline{c_{2}}$ respectively.

There exists an equivariant positive hyperbolic complex affine sphere
$\sigma\colon\widetilde{S}\to(\operatorname{\mathbb{C}}^{3},D,\omega)$ with
Blaschke metric $g$, Pick tensor
$C=\operatorname{\mathrm{Q}}_{1}+\overline{\operatorname{\mathrm{Q}}_{2}}$,
and with holonomy in $\operatorname{\mathrm{SL}}(3,\operatorname{\mathbb{C}})$
if and only if $g$ and $C$ satisfy

$\mathrm{K}_{g}=-1+\frac{1}{4}g(\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}}).$
(14)

Moreover, such a complex affine sphere is unique up to composition with
elements in $\operatorname{\mathrm{SL}}(3,\operatorname{\mathbb{C}})$.

In local holomorphic coordinates $z$ and $\overline{w}$ for $c_{1}$ and
$\overline{c_{2}}$ respectively, if $g=\lambda dzd\overline{w}$ and
$\operatorname{\mathrm{Q}}_{1}=\operatorname{\varphi}dz^{3},$
$\overline{\operatorname{\mathrm{Q}}_{2}}=\overline{\psi}d\overline{w}^{3},$
then

$\frac{1}{4}g(\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}})=2\frac{\operatorname{\varphi}\overline{\psi}}{\lambda^{3}}.$

As well, take note that, in terms of the Pick tensor,
$g(\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}})=\frac{1}{2}g(C,C),$
since $\operatorname{\mathrm{Q}}_{1}$ and
$\overline{\operatorname{\mathrm{Q}}_{2}}$ are $g$-isotropic.

###### Proof.

As above, let $z$ and $\overline{w}$ be coordinates for $c_{1}$ and
$\overline{c_{2}}$ respectively, and $g=\lambda dzd\overline{w}$,
$\operatorname{\mathrm{Q}}_{1}=\operatorname{\varphi}dz^{3},$ and
$\overline{\operatorname{\mathrm{Q}}_{2}}=\overline{\psi}d\overline{w}^{3}.$

Let $\nabla$ be the affine connection on $\operatorname{\mathbb{C}}TS$ defined
by $C=-2g(\nabla-\nabla^{g},\cdot)$. It suffices to show that $g$, $\nabla$,
$\mathsf{S}=id$, and $\tau=0$ satisfy the hypothesis of Theorem 3.13, as well
as $\nabla dV_{g}=0$, to produce an equivariant complex affine sphere. As in
Remark 3.17, by translating we can ensure that the holonomy lies in
$\operatorname{\mathrm{SL}}(3,\operatorname{\mathbb{C}})$.

Observe that Equations (10) and (11) hold trivially. We prove that Equation
(9) holds. Since $\nabla^{g}g=0$, for all
$X,Y,Z\in\Gamma(\operatorname{\mathbb{C}}TS)$,

$\displaystyle(\nabla_{X}g)(Y,Z)-(\nabla_{Y}g)(X,Z)$
$\displaystyle=(\nabla_{X}g)(Y,Z)-(\nabla_{Y}g)(X,Z)-(\nabla^{g}_{X}g)(Y,Z)-(\nabla^{g}_{Y}g)(X,Z)$
$\displaystyle=-g(\nabla_{X}Y,Z)-g(Y,\nabla_{X}Z)+g(\nabla_{Y}X,Z)+g(X,\nabla_{Y}Z)$
$\displaystyle=\frac{1}{2}(C(X,Y,Z)+C(X,Z,Y)-C(Y,X,Z)-C(Y,Z,X))=0,$

where in the last line we used the symmetry of $C$.

We next prove that $\nabla dV_{g}=0$. Suggestively defining the symmetric
tensor $K$ by $K_{X}Y=\nabla_{X}Y-\nabla^{g}_{X}Y$, we observe that for all
$X,$

$0=C(\cdot,\partial_{\overline{z}},\partial_{w})=-2g(K_{\cdot}\partial_{\overline{z}},\partial_{w}),$

and hence
$K_{X}\partial_{\overline{z}}=K_{\partial_{\overline{z}}}X\in\Gamma(\mathrm{Span}(\partial_{w}))$.
Similarly,
$K_{X}\partial_{w}=K_{\partial_{w}}X\in\Gamma(\mathrm{Span}(\partial_{\overline{z}}))$.
One can compute the tensor $K$ explicitly:

$\begin{split}&K_{\partial_{w}}\partial_{w}=-\frac{1}{2}\frac{(\partial_{w}z)^{3}}{g(\partial_{\overline{z}},\partial_{w})}\operatorname{\varphi}\partial_{\overline{z}},\\\
&K_{\partial_{\overline{z}}}\partial_{\overline{z}}=-\frac{1}{2}\frac{(\partial_{\overline{z}}\overline{w})^{3}}{g(\partial_{\overline{z}},\partial_{w})}\overline{\psi}\partial_{w},\\\
&K_{\partial_{\overline{z}}}\partial_{w}=K_{\partial_{w}}\partial_{\overline{z}}=0.\end{split}$
(15)

Observe that, for all 1-forms $\theta$, one has
$K_{X}(\theta):=\nabla_{X}\theta-\nabla_{X}^{g}\theta=-\theta(K_{X})$. Hence,
$K_{\partial_{w}}dz=-dz(K_{\partial_{w}})=0$, implying that
$Kdz\in\mathrm{Span}(d\overline{w}^{2})$ and
$Kd\overline{w}\in\mathrm{Span}(dz^{2})$.

As a result, writing $dV_{g}$ in the form $dV_{g}=\delta dz\wedge
d\overline{w}$, we have that for all $X\in\Gamma(\operatorname{\mathbb{C}}TS)$

$\nabla_{X}dV_{g}=\nabla_{X}dV_{g}-\nabla^{g}_{X}dV_{g}=\delta K_{X}(dz)\wedge
d\overline{w}+\delta dz\wedge K_{X}(d\overline{w})=0,\ $

as desired.

Now we prove that Equation (8) holds. First of all, observe that $\nabla^{g}C$
is symmetric in all of its entries. In fact,

$\nabla^{g}C=\partial_{z}\operatorname{\varphi}dz^{4}+\operatorname{\varphi}\nabla^{g}(dz^{3})+\partial_{\overline{w}}\overline{\psi}d\overline{w}^{4}+\overline{\psi}\nabla^{g}(d\overline{w}^{3})$

and, as seen in the proof of Proposition 3.19,
$\nabla^{g}(dz^{3})\in\Gamma(\mathrm{Span}(dz^{4}))$ and
$\nabla^{g}(d\overline{w}^{3})\in\Gamma(\mathrm{Span}(d\overline{w}^{4}))$. As
a consequence, $\nabla^{g}K$ is symmetric in all of its entries because one
can easily check that

$(\nabla^{g}_{X}C)(Y,Z,W)-(\nabla^{g}_{Y}C)(X,Z,W)=-2g((\nabla^{g}_{X}K)(Y,Z),W)+2g((\nabla^{g}_{Y}K)(X,Z),W)\
$

for all $X,Y,Z\in\Gamma(\operatorname{\mathbb{C}}TS).$ Now, from
$\nabla=\nabla^{g}+K$, we have that

$\displaystyle\mathrm{R}^{\nabla}(X,Y)Z$
$\displaystyle=\mathrm{R}^{g}(X,Y)Z+(\nabla^{g}_{X}K)(Y,Z)-(\nabla^{g}_{Y}K)(X,Z)+[K_{X},K_{Y}]Z$
$\displaystyle=\mathrm{R}^{g}(X,Y)Z+[K_{X},K_{Y}]Z,\ $

so the Gauss Equation (8) for $\mathsf{S}=-id$ is equivalent to

$\mathrm{R}^{g}(X,Y)Z=-[K_{X},K_{Y}]Z-g(Y,Z)X+g(X,Z)Y,$

hence to

$g(\mathrm{R}^{g}(X,Y)Z,W)=-g([K_{X},K_{Y}]Z,W)-g(Y,Z)g(X,W)+g(X,Z)g(Y,Z)$
(16)

for all $X,Y,Z,W\in\Gamma(\operatorname{\mathbb{C}}TS)$. It is now easy to
check, using that $K$ is $g$-self-adjoint, that in Equation (16) the right-
hand side has the same symmetries as the left-hand side, so it is sufficient
to check that the equality holds for $X=W=\partial_{\overline{z}}$ and
$Y=Z=\partial_{w}$. Finally writing out the Gauss curvature as

$\mathrm{K}_{g}=\frac{g(\mathrm{R}^{g}(\partial_{\overline{z}},\partial_{w})\partial_{w},\partial_{\overline{z}})}{-(g(\partial_{\overline{z}},\partial_{w})^{2})}=\frac{g(K_{\partial_{\overline{z}}}K_{\partial_{w}}\partial_{w},\partial_{\overline{z}})-g(K_{\partial_{w}}K_{\partial_{\overline{z}}}\partial_{w},\partial_{\overline{z}})-(g(\partial_{\overline{z}},\partial_{w}))^{2}}{(g(\partial_{\overline{z}},\partial_{w}))^{2}},$

Equation (15) returns that the Gauss equation (8) is equivalent to

$\mathrm{K}_{g}=\frac{g(K_{\partial_{w}}\partial_{w},K_{\partial_{\overline{z}}}\partial_{\overline{z}})}{(g(\partial_{\overline{z}},\partial_{w}))^{2}}-1=\frac{1}{4}\operatorname{\varphi}\overline{\psi}\cdot\frac{(\partial_{w}z)^{3}(\partial_{\overline{z}}\overline{w})^{3}}{(g(\partial_{\overline{z}},\partial_{w}))^{3}}-1=2\frac{\operatorname{\varphi}\overline{\psi}}{\lambda^{3}}-1.$

∎

From now on, we will be interested only in Blaschke metrics $g$ of the form
$g=e^{2u}h$, with $h$ being a Bers metric and
$u:S\to\operatorname{\mathbb{C}}$ a function. Any Blaschke metric that’s
connected to a Riemannian metric can be written in this way.

######  Example 3.21.

We are not providing a full construction, but there are in fact positive
complex metrics $g$ that have constant curvature $-1$ and that are conformal
multiples of Bers metrics by a conformal multiple that does not admit a
logarithm; by 3.20, $g$ and $C=0$ define a positive hyperbolic complex affine
sphere. To construct such a metric, one can imagine starting from a Fuchsian
type immersion $(f,\overline{f})\colon\widetilde{S}\to\mathbb{G}$ and applying
to $f$ a $2\pi$-grafting around a closed non-separating curve $\gamma$:
precomposing with a suitable isotopy before grafting, we get a new developing
map $\hat{f}\colon\widetilde{S}\to\operatorname{\mathbb{C}\mathbb{P}}^{1}$
with the same holonomy and such that $\hat{f}(p)\neq\overline{f}(p)$ for all
$p\in\widetilde{S}$, hence
$(\hat{f},\overline{f})\colon\widetilde{S}\to\mathbb{G}$ is an equivariant
immersion. The induced complex metric $g$ now has curvature $-1$ by Theorem
6.7 in [BE22], and it is such that, assuming $\dot{\gamma}\neq 0$,
$g(\dot{\gamma},\dot{\gamma})$ defines a homotopically nontrivial loop on
$\mathbb{C}^{*}$, defining an example as above.

As a consequence of Theorem 3.20 and of the formula for the curvature of
conformal metrics in Section 2.3, we have the following corollary.

######  Corollary 3.22.

Let $h={\bf\it h}_{(c_{1},\overline{c_{2}})}$ be a Bers metric, with
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$.
Let $\operatorname{\mathrm{Q}}_{1}$ and
$\overline{\operatorname{\mathrm{Q}}_{2}}$ be holomorphic cubic differentials
for $c_{1}$ and $\overline{c_{2}}$ respectively. Then $g=e^{2u}\cdot h$, with
$u\colon S\to\operatorname{\mathbb{C}}$, and
$C=\operatorname{\mathrm{Q}}_{1}+\overline{\operatorname{\mathrm{Q}}_{2}}$ are
the Blaschke metric and the Pick tensor of an equivariant positive hyperbolic
complex affine sphere in $\operatorname{\mathbb{C}}^{3}$ if and only if

$G(c_{1},\overline{c_{2}},\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}},u):=\Delta_{h}u-e^{2u}+\frac{1}{4}{h(\operatorname{\mathrm{Q}}_{1},\overline{\operatorname{\mathrm{Q}}_{2}})}\cdot
e^{-4u}+1=0.$ (17)

###  3.5. Motivating examples

Our first two examples show that complex affine spheres generalize both real
affine spheres and the quasiconformal maps from the Bers theorem.

######  Example 3.23 (Real affine spheres).

Affine differential geometry is a special case of the theory outlined here.

By embedding $\mathbb{R}^{3}$ inside $\operatorname{\mathbb{C}}^{3}$, any
immersion $\sigma:\widetilde{S}\to\mathbb{R}^{3}$ can be viewed as an
admissible immersion $\sigma:\widetilde{S}\to\operatorname{\mathbb{C}}^{3}$.
Moreover, if we can pick a section $\xi$ of $\sigma^{*}T\mathbb{R}^{3}$ that
is transverse to $TS$, in which case $\sigma$ is an affine immersion in the
traditional sense [NS94, Definition II.1.1], then
$\sigma:\widetilde{S}\to\operatorname{\mathbb{C}}^{3}$ is a complex affine
immersion. If we demand that the image of $\sigma$ is locally strictly convex,
then one can find a global real transverse normal. Choosing the transverse
normal to be the complexification of a real transverse normal, all of the
tensors that come into play are complexifications of real tensors. Moreover,
the Blaschke metric is Riemannian, the Pick tensor is real, and the immersion
data from Theorem 3.20 is always of the form
$(c,\overline{c},\operatorname{\mathrm{Q}},\overline{\operatorname{\mathrm{Q}}})$.
Equation (17) is a real semilinear elliptic PDE,

$\Delta_{h}u=e^{2u}-\frac{1}{4}h(\operatorname{\mathrm{Q}},\overline{\operatorname{\mathrm{Q}}})e^{-4u}-1,$

which is the ordinary Tzitzéica equation and is known to admit a unique real
solution [Lof01, Proposition 4.0.2], [Lab07, Lemma 4.1.2]. If one takes that
real solution and integrates it out to an affine sphere using Theorem 3.20,
then by uniqueness that affine sphere lives in an isometrically and totally
geodesically embedded copy of $\mathbb{R}^{3}$.

######  Example 3.24 (Bers maps).

Consider $\operatorname{\mathbb{C}}^{2,1}$, namely
$\operatorname{\mathbb{C}}^{3}$ with the complex bilinear form
$\operatorname{\langle\cdot,\cdot\rangle}$ defined by
$\langle\underline{z},\underline{w}\rangle=z_{1}w_{1}+z_{2}w_{2}-z_{3}w_{3}$,
and define $\mathbb{\hat{X}}_{2}=\\{\underline{z}\in\mathbb{C}^{3}\ |\
\langle\underline{z},\underline{z}\rangle=-1\\}\ $ with the inherited
holomorphic Riemannian metric. With reference to Section 2.2, one has
$\hat{\mathbb{X}}_{2}\cong\mathbb{X}_{2}\cong\mathbb{G}$.

We show that positive hyperbolic complex affine spheres with Pick tensor $C=0$
correspond to equivariant immersions into ${\mathbb{\hat{X}}}_{2}$ with pull-
back metric being the Blaschke metric (which, by Theorem 3.20, has curvature
-1). Moreover, in the identification of
$\operatorname{\mathrm{PSL}}(2,\operatorname{\mathbb{C}})$ with
$Isom_{0}(\mathbb{\hat{X}}_{2})=Isom_{0}(\operatorname{\mathbb{C}}^{2,1})=\mathrm{SO}(2,1,\operatorname{\mathbb{C}})$,
if the Blaschke metric is a Bers metric, then the holonomy is quasi-Fuchsian,
which shows that the construction is compatible with Bers’ theorem.

As a consequence of Theorem 6.7 in [BE22], for all positive complex metric $g$
of curvature $-1$ on $S$, there exists a
$(\pi_{1}(S),\mathrm{SO}(3,\mathbb{C}))$-equivariant admissible immersion
$\sigma\colon\widetilde{S}\to\mathbb{\hat{X}}_{2}$ with pull-back metric $g$.
Moreover, such an immersion is totally geodesic, namely
$(\sigma^{*}D^{\mathbb{\hat{X}}_{2}})_{X}Y=\nabla^{g}_{X}Y$ where
$D^{\mathbb{\hat{X}}_{2}}$ denotes the Levi-Civita connection of
$\mathbb{\hat{X}}_{2}$. On the other hand, $N_{\underline{z}}=i\underline{z}$
is a normal vector field for $\mathbb{\hat{X}}_{2}$ inside $\mathbb{C}^{2,1}$
(namely, at each point it is the unique vector - up to a sign - with norm 1
orthogonal to $T\mathbb{\hat{X}}_{2}$). As a result, one can easily see that
$(\sigma^{*}D)_{\mathsf{X}}N=i\mathsf{X}$ and the second fundamental form of
the immersion inside $\mathbb{C}^{3}$ is $-ig\otimes
N=g\otimes\underline{\sigma(p)}$, where $\underline{\sigma(p)}$ denotes
$\sigma(p)$ seen as a vector of $\mathbb{C}^{3}$. Hence, denoting
$\xi_{p}=\sigma^{*}(\underline{\sigma(p)})$, we have that

$\begin{cases}(\sigma^{*}D)_{\mathsf{X}}Y&=\nabla^{g}_{\mathsf{X}}Y+g(\mathsf{X},Y)\xi\\\
(\sigma^{*}D)_{\mathsf{X}}\xi&=\mathsf{X}.\end{cases}\ $

Fix the canonical volume form $\omega$ on $\operatorname{\mathbb{C}}^{2,1}$
given by the determinant. One can easily see that
$\omega(\cdot,\cdot,\underline{z})$ on $\mathbb{\hat{X}}_{2}$ is a volume,
i.e. it is compatible with the holomorphic Riemannian metric. Up to composing
$\sigma$ with the map $-id$ on $\mathbb{\hat{X}}_{2}$ (which does not lie in
$\mathrm{SO}(2,1,\operatorname{\mathbb{C}})$), we have that
$dV_{g}=\sigma^{*}(\omega(\cdot,\cdot,z))$. We therefore conclude that
$\sigma$ is a positive hyperbolic complex affine sphere. Since the affine
connection coincides with $\nabla^{g}$, the Pick tensor is zero, and by
uniqueness, we conclude that this is the unique positive hyperbolic complex
affine sphere with Blaschke metric $g$ and Pick tensor $C=0$. Finally, in the
identification of
$\operatorname{\mathrm{PSL}}(2,\operatorname{\mathbb{C}})\cong
Isom_{0}(\mathbb{G})\cong
Isom_{0}(\mathbb{\hat{X}}_{2})\cong\mathrm{SO}(2,1,\operatorname{\mathbb{C}})$,
one has trivially that the holonomy
$\pi_{1}(S)\to\mathrm{SO}(2,1,\operatorname{\mathbb{C}})$ of the positive
hyperbolic complex affine sphere with Blaschke metric being the Bers metric
${\bf\it h}_{(c_{1},\overline{c_{2}})}$ and Pick tensor $C=0$ is the quasi-
Fuchsian representation $\rho(c_{1},\overline{c_{2}})$.

######  Example 3.25 (Complex Tzitzéica surfaces).

The Tzitzéica surface plays an important role in studying high energy harmonic
maps to symmetric spaces [DW15]. We can explicitly construct a complex
analogue. We won’t refer to this example again, but we hope it conveys that
one can make a lot of examples of positive hyperbolic complex affine spheres
by hand.

Let $z=z(x,y),w=w(x,y):\mathbb{R}^{2}\to\operatorname{\mathbb{C}}$ be
diffeomorphisms, and $\zeta=e^{2\pi\frac{i}{3}}$ the third root of unity. For
each $c\in\operatorname{\mathbb{C}}^{*},$ we consider the surface given by,

$f(x,y)=c(e^{z+\overline{w}},e^{\zeta^{2}z+\zeta\overline{w}},e^{\zeta
z+\zeta^{2}\overline{w}}).$

Note that the image lies inside the complex surface

$\\{z=(z_{1},z_{2},z_{3})\in\operatorname{\mathbb{C}}^{3}:z_{1}z_{2}z_{3}=c\\}.$

When $z=w,$ the parametrization becomes

$f(x,y)=c(e^{\mathrm{Re}z},e^{\mathrm{Re}z\zeta^{-1}},e^{\mathrm{Re}z\zeta^{-2}}),$

and if we assume as well that $c\in\mathbb{R},$ then this surface lives in
$\mathbb{R}^{3}$ and is called the Tzitzéica surface. In this real case, the
map $f$ describes a connected component of the surface in $\mathbb{R}^{3}$
defined implicitly by $xyz=c.$ For this reason, when $z\neq w$, we call our
surfaces complex Tzitzéica surfaces. We leave it to the reader to compute that
$f$ defines a positive hyperbolic complex affine sphere, and that the Blaschke
metric is $2dzd\overline{w}$ and the Pick tensor is
$C=dz^{3}+d\overline{w}^{3}$.

##  4\. Complex Lie derivatives and transport

The space of complex structures on a surface is an infinite-dimensional Banach
manifold. Still, its quotient up to isotopy, namely Teichmüller space, is
finite-dimensional and the holonomy map factors through it. Generally
speaking, taking the quotient up to isotopy is a powerful tool for
constructing finite-dimensional configuration spaces while preserving their
geometric significance. This is also the case for the moduli space of real
hyperbolic affine spheres (discussed in Section 6.1).

Since $\mathcal{C}(S)\times\mathcal{C}(\overline{S})$ is the complexification
of $\mathcal{C}(S),$ the natural group acting on
$\mathcal{C}(S)\times\mathcal{C}(\overline{S})$ is the product
$\mathrm{Diff}_{0}(S)\times\mathrm{Diff}_{0}(S)$, which returns a finite
dimensional quotient, namely $\mathcal{T}(S)\times\mathcal{T}(\overline{S})$
(note that if we naively act by $\mathrm{Diff}_{0}(S)$ on the left or right or
diagonally, we get an infinite dimensional quotient). The map
$(c_{1},\overline{c_{2}})\mapsto{\bf\it h}_{(c_{1},\overline{c_{2}})}$ pushes
forward the action of $\mathrm{Diff}_{0}(S)\times\mathrm{Diff}_{0}(S)$ onto
the space of Bers metrics, but the meaning of the induced quotient is not
clear. For instance, if $\phi\in\mathrm{Diff}_{0}(S),$ it is not
straightforward to see in what sense or why
$-\frac{4}{(z-\overline{w})^{2}}dzd\overline{w}$ and
$-\frac{4}{(z-\phi^{*}(\overline{w}))^{2}}dz\phi^{*}(d\overline{w})$ should be
identified.

In this section, we define the notions of complex Lie derivative and of
transport of tensors through paths of complex vector fields. As we will see,
these tools allow us to study the
$\mathrm{Diff}_{0}(S)\times\mathrm{Diff}_{0}(S)$ action and will motivate the
construction of the finite-dimensional space $\mathbb{CAS}(S)$ in Section 6.3.
Moreover, this approach will be a key tool in proving Theorem G.

###  4.1. Complex Lie derivatives

Let $M$ be a smooth manifold. Denote by

$Tens^{p,q}(M)=\Gamma\left((TM)^{p_{\otimes}}\otimes(T^{*}M)^{q_{\otimes}}\right)$

the space of $(p,q)$-type tensors on $M$. The Lie derivative along a vector
field can be seen as a bilinear map

$\begin{split}\Gamma(TM)\times Tens^{p,q}(M)&\to Tens^{p,q}(M),\\\
(X,\alpha)&\mapsto\mathscr{L}_{X}\alpha.\end{split}\ $

We extend it $\operatorname{\mathbb{C}}$-bilinearly to a map

$\Gamma(\mathbb{C}TM)\times(Tens^{p,q}(M)\otimes\operatorname{\mathbb{C}})\to
Tens^{p,q}(M)\otimes\operatorname{\mathbb{C}}.\ $

That is, we extend by the rule

$\mathscr{L}_{Z}\alpha=\mathscr{L}_{ReZ}(Re(\alpha))-\mathscr{L}_{ImZ}(Im(\alpha))+i\mathscr{L}_{ReZ}(Im(\alpha))+i\mathscr{L}_{ImZ}(Re(\alpha)).$

We call $\mathscr{L}_{Z}\alpha$ the complex Lie derivative of $\alpha$ along
the complex vector field $Z$. $\mathscr{L}_{Z}$ is characterized by the
following algebraic properties:

  1. (1)

for all $C^{1}$ function $f\colon S\to\mathbb{C}$,
$\mathscr{L}_{Z}(f)=\partial_{Z}f,$

  2. (2)

for all complex vector fields $W\in\Gamma(\mathbb{C}TM)$,
$\mathscr{L}_{Z}W=[Z,W],$

  3. (3)

for all complex differential forms
$\omega\in\Omega^{r}(M)\otimes\operatorname{\mathbb{C}}$,
$\mathscr{L}_{Z}\omega=d\omega(Z,\cdot)+d(\omega(Z,\cdot)),$

  4. (4)

for all $\alpha,\beta$ smooth complex tensors,
$\mathscr{L}_{Z}(\alpha\otimes\beta)=(\mathscr{L}_{Z}\alpha)\otimes\beta+\alpha\otimes(\mathscr{L}_{Z}\beta),$

  5. (5)

one has the recursive formula for every $\alpha\in
Tens^{p,q}(M)\otimes\mathbb{C}$ given by

$\displaystyle(\mathscr{L}_{Z}\alpha)(\omega_{1},\dots,\omega_{p},W_{1},\dots,W_{q})$
$\displaystyle=\partial_{Z}(\alpha(\omega_{1},\dots,\omega_{p},W_{1},\dots,W_{q}))-\alpha(\mathscr{L}_{Z}\omega_{1},\dots,\omega_{p},W_{1},\dots,W_{q}))-\dots$
$\displaystyle-\alpha(\omega_{1},\dots,\mathscr{L}_{Z}\omega_{p},W_{1},\dots,W_{q}))-\alpha(\omega_{1},\dots,\mathcal{\omega}_{p},\mathscr{L}_{Z}W_{1},\dots,W_{q}))-\dots$
$\displaystyle-\alpha(\omega_{1},\dots,\mathcal{\omega}_{p},W_{1},\dots,\mathscr{L}_{Z}W_{q})),$

where $\omega_{1},\dots,\omega_{p}\in\Omega^{1}(\mathbb{C}TM)$, and
$W_{1},\dots,W_{q}\in\Gamma(\mathbb{C}TM)$.

######  Remark 4.1.

We show a way to define the complex Lie derivative of a complex structure
$c_{1}\in\mathcal{C}(S)$ for which $\mathscr{L}_{Z}{c_{1}}\in
T_{c_{1}}\mathcal{C}(S)$ lies in the kernel of the projection
$\mathcal{C}(S)\to\mathcal{T}(S)$.

We define it as follows. Let $Z\in\Gamma(\operatorname{\mathbb{C}}TS)$ and let
$z$ be any local holomorphic coordinate for $c_{1}$, and decompose $Z$ with
respect to the basis $\\{\partial_{z},\partial_{\overline{z}}\\}$, hence
$Z=pr_{\partial_{z}}(Z)+pr_{\partial_{\overline{z}}}(Z)$. We define

$\mathscr{L}_{Z}c_{1}:=\mathscr{L}_{pr_{\partial_{z}}(Z)+\overline{pr_{\partial_{z}}(Z)}}c_{1}$

where the RHS is a standard real Lie derivative, corresponding to the
infinitesimal deformation through the isotopy $\phi_{t}$ with
$\dot{\phi}_{0}=pr_{\partial_{z}}(Z)+\overline{pr_{\partial_{z}}(Z)}$.

A motivation for this choice is that

$\mathscr{L}_{Z}z=\mathscr{L}_{pr_{\partial_{z}}(Z)}z=\mathscr{L}_{pr_{\partial_{z}}(Z)+\overline{pr_{\partial_{z}}(Z)}}z=\frac{d}{dt}\big{|}_{t=0}(z\circ\phi_{t}),$

and also $\mathscr{L}_{Z}dz=\mathscr{L}_{\dot{\phi}_{0}}dz$. Moreover, by the
properties of the complex Lie derivative, one has that for all
$c_{1}$-holomorphic $n$-th differential $\alpha$,
$\mathscr{L}_{Z}\alpha=\mathscr{L}_{\dot{\phi}_{0}}\alpha$.

Using the same approach on $\overline{S}$, one defines the complex Lie
derivative of any element in $\mathcal{C}(\overline{S})$.

######  Example 4.2.

Here is a key example to motivate the complex Lie derivative. Let $S$ be a
closed surface, let $\phi_{t}:[0,1]\times S\to S$ be an isotopy, and let
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$
with Bers metric ${\bf\it h}_{(c_{1},\overline{c_{2}})}$. Let $z$ and
$\overline{w}$ be local coordinates for $c_{1}$ and $\overline{c_{2}}$
respectively, and consider the path of Bers metrics given locally by

$h_{t}={\bf\it
h}_{(c_{1},\phi_{t}^{*}(\overline{c_{2}}))}=-\frac{4}{(z-\phi_{t}^{*}(\overline{w}))^{2}}dz\cdot(\phi_{t}^{*}(d\overline{w})).\
$

The variation $\dot{h}_{t}=\frac{d}{dt}\big{|}_{0}h_{t}$ defines a bilinear
form on $\operatorname{\mathbb{C}}TS$. We claim that this variation can be
seen as a complex Lie derivative of $h_{t}$. Indeed, let
$\overline{w_{t}}:=\phi_{t}^{*}(\overline{w})$ be a local coordinate for
$\phi_{t}^{*}(\overline{c_{2}})$, denote
$\dot{\phi}_{t}=:\gamma_{t}\partial_{z}+\overline{\gamma_{t}}\partial_{\overline{z}}$,
and consider the complex vector field

$Z_{t}:=\left(\gamma_{t}\frac{\partial_{z}\overline{w}_{t}}{\partial_{\overline{z}}\overline{w}_{t}}+\overline{\gamma_{t}}\right)\partial_{\overline{z}}.\
$

One can easily check that this description of $Z_{t}$ does not depend on the
choice of the local holomorphic coordinates $z$ and $\overline{w}_{t}$, so
$Z_{t}\in\Gamma(\mathbb{C}TS)$.

By construction, one has that:

$\displaystyle\mathscr{L}_{Z_{t}}c_{1}=0,$
$\displaystyle\mathscr{L}_{Z_{t}}\phi_{t}^{*}(\overline{c_{2}})=\gamma_{t}\partial_{z}\overline{w_{t}}+\overline{\gamma_{t}}\partial_{\overline{z}}\overline{c_{2}}=\mathscr{L}_{\dot{\phi_{t}}}\phi_{t}^{*}(\overline{c_{2}}).$

In particular, $\mathscr{L}_{Z_{t}}z=0$, $\mathscr{L}_{Z_{t}}dz=0$,
$\mathscr{L}_{Z_{t}}\overline{w}_{t}=\mathscr{L}_{\dot{\phi}_{t}}w_{t}$ and
$\mathscr{L}_{Z_{t}}d\overline{w_{t}}=\mathscr{L}_{\dot{\phi}_{t}}d\overline{w_{t}}$.
In conclusion, by applying the properties of the complex Lie derivative, one
gets that

$\dot{h}_{t}=\mathscr{L}_{Z_{t}}h_{t}.\ $

Recall the quotient map $\mathcal{C}(S)\to\mathcal{T}(\overline{S})$, which we
label as $\Psi.$ As Example 4.2 suggests, Lie derivatives through complex
vector fields allow to deform simultaneously pairs of complex structures. This
is made clearer by the following Proposition which can be seen as a complex
extension of the natural description for $T\mathcal{C}(S)$ through the bundle
$\Psi$.

######  Proposition 4.3.

Let $(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$
and consider the map

$d_{c_{1}}\Psi\oplus
d_{\overline{c_{2}}}\overline{\Psi}:T_{c_{1}}\mathcal{C}(S)\times
T_{\overline{c_{2}}}\mathcal{C}(\overline{S})\to
T_{[c_{1}]}\mathcal{T}(S)\times
T_{[\overline{c_{2}}]}\mathcal{T}(\overline{S}).$

The map

$\displaystyle\Gamma(\operatorname{\mathbb{C}}TS)$ $\displaystyle\to
T_{c_{1}}\mathcal{C}(S)\times T_{\overline{c_{2}}}\mathcal{C}(\overline{S})$
$\displaystyle Z$
$\displaystyle\mapsto(\mathscr{L}_{Z}c_{1},\mathscr{L}_{Z}\overline{c_{2}})$

defines a linear isomorphism onto the kernel of $d\Psi_{c_{1}}\oplus
d\overline{\Psi}_{\overline{c_{2}}}$ .

As a result, one can see

$T_{(c_{1},\overline{c_{2}})}\mathcal{C}(S)\times\mathcal{C}(\overline{S})\cong
T_{[c_{1}]}\mathcal{T}(S)\times
T_{[\overline{c_{2}}]}\mathcal{C}(\overline{S})\times\Gamma(\operatorname{\mathbb{C}}TS).$

###### Proof.

Use the notation of Remark 4.1 and Example 4.2. The map is injective: since
$\mathrm{Span}(\partial_{\overline{z}})\cap\mathrm{Span}(\partial_{w})=\\{0\\}$,
$pr_{\partial_{z}}Z=0=pr_{\partial_{\overline{w}}}Z$ implies $Z=0$.

By Remark 4.1, we see that the image of the map is in fact contained in the
kernel.

Let $\dot{\phi}_{1},\dot{\phi}_{2}\in T_{id}\mathrm{Diff}(S)$, hence
$\mathscr{L}_{\dot{\phi}_{1}}c_{1}\in T_{c_{1}}\mathcal{C}(S)$ and
$\mathscr{L}_{\dot{\phi}_{2}}\overline{c_{2}}\in
T_{\overline{c_{2}}}\mathcal{C}(\overline{S})$. As in Example 4.2, one can
find $Z_{1},Z_{2}\in\Gamma(\operatorname{\mathbb{C}}TS)$ such that

$\displaystyle\mathscr{L}_{Z_{1}}c_{1}=\mathscr{L}_{\dot{\phi}_{1}}c_{1}\qquad$
$\displaystyle\text{and}\qquad\mathscr{L}_{Z_{1}}\overline{c_{2}}=0,\ $
$\displaystyle\mathscr{L}_{Z_{1}}c_{1}=0\qquad$
$\displaystyle\text{and}\qquad\mathscr{L}_{Z_{1}}\overline{c_{2}}=\mathscr{L}_{\dot{\phi}_{2}}\overline{c_{2}},$

hence

$(\mathscr{L}_{Z_{1}+Z_{2}}c_{1},\mathscr{L}_{Z_{1}+Z_{2}}\overline{c_{2}})=(\mathscr{L}_{\dot{\phi}_{1}}c_{1},\mathscr{L}_{\dot{\phi}_{2}}\overline{c_{2}}),$

proving surjectivity. ∎

######  Corollary 4.4.

Let $h_{t}={\bf\it h}_{(c_{1}^{t},\overline{c_{2}}^{t})}$ be a $C^{1}$ path of
Bers metrics with constant holonomy, then
$\dot{h}_{t}=\mathscr{L}_{Z_{t}}h_{t}$, where $Z_{t}$ is such that
$(\dot{c}_{1}^{t},\dot{\overline{c_{2}}}^{t})=(\mathscr{L}_{Z_{t}}c_{1}^{t},\mathscr{L}_{Z_{t}}\overline{c_{2}}^{t})$.

###### Proof.

Applying the same method as in example 4.2, denoting by $z_{t}$ and
$\overline{w_{t}}$ complex coordinates of $c_{1}^{t}$ and
$\overline{c_{2}}^{t}$, we get

$\frac{d}{dt}h_{t}=\frac{8(\partial_{Z_{t}}z_{t}-\partial_{Z_{t}}\overline{w_{t}})}{(z_{t}-\overline{w}_{t})^{3}}dz_{t}d\overline{w_{t}}-\frac{4}{(z_{t}-\overline{w}_{t})^{2}}((\mathscr{L}_{Z_{t}}dz)\cdot
d\overline{w}_{t}+dz\cdot(\mathscr{L}_{Z_{t}}d\overline{w}_{t}))=\mathscr{L}_{Z_{t}}h_{t}.\
$

∎

###  4.2. Transport through complex vector fields

Unlike real Lie derivatives, realizing complex Lie derivatives as
infinitesimal deformations is difficult. More precisely, given a path of
complex vector fields $Z_{t}\in\Gamma(\operatorname{\mathbb{C}}TM)$,
$t\in[0,T]$ and a tensor $\alpha$, we are interested in solutions $\alpha_{t}$
to the system

$\begin{cases}\mathscr{L}_{Z_{t}}\alpha_{t}=\dot{\alpha}_{t}\\\
\alpha_{0}=\alpha,\end{cases}$ (18)

and understanding the existence and uniqueness for this Cauchy problem (18) is
not in general straightforward. A simple case is as follows: on the unit disk,
for $g(z,t)\partial_{\overline{z}}$ a time-dependent vector field and $f$ a
given function, find $f_{t}$ such that

$\begin{cases}g(z,t)\partial_{\overline{z}}f_{t}(z)=\frac{d}{dt}f_{t}(z)\\\
f_{0}(z)=f(z).\end{cases}$ (19)

As we will see below, by the Cauchy-Kovalevskaya and Holmgren theorems, when
$g$ and $f$ are real analytic, we can find a unique short time solution. But
in fact, as can be verified through Hörmander’s classical criterion, the
problem (19) is not locally solvable, which means one can find
$C_{0}^{\infty}$ input data for which no short time solution even exists. For
more on local solvability, see [Fol95] and [Tre70].

######  Definition 4.5.

We will call a solution to $\eqref{eq: transport problem}$ a transport of
$\alpha$ through $Z_{\bullet}$ and we denote it by
$\alpha_{t}=:\mathscr{T}_{Z_{t}}\alpha$.

We will also use the notation $\mathscr{T}_{Z_{t}}c$ to denote paths of
complex structures such that
$\frac{d}{dt}\mathscr{T}_{Z_{t}}c=\mathscr{L}_{Z_{t}}(\mathscr{T}_{Z_{t}}c)$,
in the sense of Remark 4.1.

For the rest of this subsection, we assume that one can solve the problem
(18), and in the next subsection we discuss existence in the real analytic
case.

######  Example 4.6.

In fact, the theory of Bers metrics suggests that we shouldn’t hope for long
time existence of the transport either. Using the unit disk as a toy model,
consider a map
$(f_{1},\overline{f_{2}}):\mathbb{D}\to\mathbb{CP}^{1}\times\mathbb{CP}^{1}\backslash\Delta$
for which there exists points $p,q\in\mathbb{D}$ such that
$f_{1}(p)=\overline{f_{2}}(q).$ Then let $\phi_{t}$ be any isotopy of the disk
taking $p$ to $q$ at a time $t_{0}.$ For $t<t_{0}$, we have a family of
positive complex metrics given by

$-\frac{4}{(f_{1}-\phi_{t}^{*}\overline{f_{2}})^{2}}df_{1}\cdot(\phi_{t}^{*}d(\overline{f_{2}})),$

which is tangent to a complex Lie derivative at time zero (see Example 4.2),
but which also blows up at $p$ as $t$ increases to $t_{0}.$

######  Remark 4.7.

By the definition of complex Lie derivative, we can deduce some natural
properties of the transport.

  1. (1)

If $\omega\in\Omega^{p}(S)\otimes\operatorname{\mathbb{C}}$ is transportable
through $Z_{\bullet}$, then

${\textstyle\mathstrut}\mathscr{T}_{Z_{t}}(d\omega)=d({\textstyle\mathstrut}\mathscr{T}_{Z_{t}}\omega).$

  2. (2)

For all complex tensors $\alpha,\beta$ which can be transported through
$Z_{\bullet}$ one has

$\mathscr{T}_{Z_{t}}(\alpha\otimes\beta)=(\mathscr{T}_{Z_{t}}\alpha)\otimes(\mathscr{T}_{Z_{t}}\beta).$

  3. (3)

If $\alpha$ in $Tens^{p,q}(S)\otimes\mathbb{C}$, $W_{1},\dots
W_{q}\in\Gamma(\mathbb{C}TS)$, and
$\omega_{1},\dots\omega_{p}\in\Omega^{1}(\mathbb{C}TS)$ can all be transported
through $Z_{\bullet}$, then

${\textstyle\mathstrut}\mathscr{T}_{Z_{t}}\left(\alpha(\omega_{1},\dots,\omega_{p},W_{1},\dots,W_{q})\right)=({\textstyle\mathstrut}\mathscr{T}_{Z_{t}}\alpha)({\textstyle\mathstrut}\mathscr{T}_{Z_{t}}(\omega_{1}),\dots,{\textstyle\mathstrut}\mathscr{T}_{Z_{t}}(\omega_{p}),{\textstyle\mathstrut}\mathscr{T}_{Z_{t}}(W_{1}),\dots,{\textstyle\mathstrut}\mathscr{T}_{Z_{t}}(W_{q})).$

######  Remark 4.8.

Let
$(c_{1},\overline{c_{2}})\in\mathcal{C}(S)\times\mathcal{C}(\overline{S})$,
let $\phi^{1}_{t},\phi^{2}_{t}$ be isotopies. Let
$Z_{t}\in\Gamma(\operatorname{\mathbb{C}}TS)$ be such that
$\frac{d}{dt}((\phi^{1}_{t})^{*}c_{1},(\phi^{2}_{t})^{*}\overline{c_{2}})=(\mathscr{L}_{Z_{t}}c_{1}^{t},\mathscr{L}_{Z_{t}}\overline{c_{2}}^{t})$
as in Proposition 4.3. Then, for all $c_{1}$-holomorphic (resp.
$\overline{c_{2}}$-holomorphic) tensor $\alpha$, one has
$(\phi_{1}^{t})^{*}\alpha={\textstyle\mathstrut}\mathscr{T}_{Z_{t}}\alpha$
(resp.
$(\phi_{2}^{t})^{*}\alpha={\textstyle\mathstrut}\mathscr{T}_{Z_{t}}\alpha$).
Moreover, by Corollary 4.4, ${\bf\it
h}_{((\phi_{1}^{t})^{*}{c_{1}},(\phi_{2}^{t})^{*}\overline{c_{2}})}={\textstyle\mathstrut}\mathscr{T}_{Z_{t}}{\bf\it
h}_{(c_{1},\overline{c_{2}})}$.

Here are a few more remarks about the transport through complex vector fields.
Observe that the hypothesis of the following proposition holds for
$Z_{t}\in\Gamma(\operatorname{\mathbb{C}}TS)$ corresponding to
$\frac{d}{dt}((\phi_{1}^{t})^{*}c_{1},(\phi_{2}^{t})^{*}{\overline{c_{2}}})$
and for $\mathscr{T}_{t}({\bf\it h}_{(c_{1},\overline{c_{2}})})={\bf\it
h}_{((\phi_{1}^{t})^{*}c_{1},(\phi_{2}^{t})^{*}{\overline{c_{2}}})}$.

######  Proposition 4.9.

Denote by $\mathscr{T}_{t}={\textstyle\mathstrut}\mathscr{T}_{Z_{t}}$ the
transport of tensors through the path of vector fields $t\mapsto
Z_{t}\in\Gamma(\operatorname{\mathbb{C}}TS)$, $t\in[0,T]$. Assume that $g$ is
a positive complex metric on $S$ for which $\mathscr{T}_{t}g$ exists and is a
positive complex metric, and that $({\bf\it c}_{+},{\bf\it
c}_{-})(\mathscr{T}_{t}g)=(\mathscr{T}_{t}({\bf\it
c}_{+}(g)),\mathscr{T}_{t}({\bf\it c}_{-}(g)))$.

  1. (1)

Up to reducing $T$, there exists a local orthonormal frame $(e_{1},e_{2})$ for
$g$ which admits a transport through $Z_{\bullet}$ and such that
$(\mathscr{T}_{t}e_{1},\mathscr{T}_{t}e_{2})$ is orthonormal for
$\mathscr{T}_{t}g$.

  2. (2)

If $\alpha\in Tens^{p,q}(S)\otimes\mathbb{C}$ can be transported through
$Z_{\bullet}$,

$\mathscr{T}_{t}(g(\alpha,\alpha))=(\mathscr{T}_{t}g)(\mathscr{T}_{t}\alpha,\mathscr{T}_{t}\alpha).$

  3. (3)

If $X,Y\in\Gamma(\mathbb{C}TS)$ can be transported through $Z_{\bullet}$,

$\mathscr{T}_{t}(\nabla^{g}_{X}Y)=\nabla^{\mathscr{T}_{t}g}_{\mathscr{T}_{t}X}(\mathscr{T}_{t}Y),$

where $\nabla^{\mathscr{T}_{t}(g)}$ denotes the Levi-Civita connection of
$\mathscr{T}_{t}(g)$.

  4. (4)

The curvature changes according to
$\mathscr{T}_{t}(\mathrm{R}^{g})=\mathrm{R}^{\mathscr{T}_{t}(g)}$ and
$\mathscr{T}_{t}(\mathrm{K}_{g})=\mathrm{K}_{\mathscr{T}_{t}g}$.

  5. (5)

If $\alpha$ can be transported through $Z_{\bullet}$,

$\mathscr{T}_{t}(\nabla^{g}\alpha)=\nabla^{\mathscr{T}_{t}g}(\mathscr{T}_{t}\alpha).\
$

  6. (6)

If $f\in C^{r}(S,\mathbb{C})$ can be transported through $Z_{\bullet}$,
higher-order derivatives change according to

$\mathscr{T}_{t}\left((\nabla^{g})^{r}f\right)=\left(\nabla^{\mathscr{T}_{t}g}\right)^{r}(\mathscr{T}_{t}f).\
$

In particular, the Hessian and the Laplacian change according to

$\mathscr{T}_{t}(\mathrm{Hess}_{g}(f))=\mathrm{Hess}_{\mathscr{T}_{t}(g)}(\mathscr{T}_{t}f),$
$\mathscr{T}_{t}(\Delta_{g}(f))=\Delta_{\mathscr{T}_{t}(g)}(\mathscr{T}_{t}f).\
$

###### Proof.

In the proof, we denote $g_{t}=\mathscr{T}_{t}g$ and
$(c_{1},\overline{c_{2}})=({\bf\it c}_{+},{\bf\it c}_{-})(g)$, with $z$ and
$\overline{w}$ local holomorphic coordinates for $c_{1}$ and
$\overline{c_{2}}$ respectively. Observe that since $\mathscr{T}_{t}c_{1}$ and
$\mathscr{T}_{t}\overline{c_{2}}$ correspond, by Proposition 4.3, to
deformation by isotopies, up to shrinking the charts and reducing $T$, we can
take $z_{t}=\mathscr{T}_{t}z$ and
$\overline{w_{t}}=\mathscr{T}_{t}\overline{w}$ local holomorphic coordinates
for $\mathscr{T}_{t}c_{1}$ and $\mathscr{T}_{t}\overline{c_{2}}$.

  1. (1)

We first determine a transportable orthonormal coframe. Use $g_{t}$ to denote
also the dual bilinear forms on $\operatorname{\mathbb{C}}T^{*}S$. By
hypothesis,
$g_{t}(dz_{t},dz_{t})=g_{t}(d\overline{w}_{t},d\overline{w}_{t})=0$. It is
therefore easy to check that

$\theta^{1}_{t}=\frac{dz_{t}+d\overline{w}_{t}}{\sqrt{2g_{t}(dz_{t},d\overline{w}_{t})}}\qquad\quad\theta^{2}_{t}=i\frac{dz_{t}-d\overline{w}_{t}}{\sqrt{2g_{t}(dz_{t},d\overline{w}_{t})}}$

determine an orthonormal coframe for $g_{t}$, and satisfy
$\frac{d}{d}\theta^{j}_{t}=\mathscr{L}_{Z_{t}}\theta^{j}_{t}$, hence
$\theta^{j}_{t}=\mathscr{T}_{t}\theta^{j}_{0}$. (Here the square roots exist
up to shrinking to a contractible neighborhood.)

Denote with $(e_{1}^{t},e_{2}^{t})$ the dual frame to
$(\theta^{1}_{t},\theta^{2}_{t})$, which is therefore $g_{t}$-orthonormal.
Then, applying $\frac{d}{dt}-\mathscr{L}_{Z_{t}}$ to the equations
$\theta^{j}_{t}(e_{k}^{t})=\delta_{k}^{j}$, we get that
$\frac{d}{dt}e_{k}^{t}-\mathscr{L}_{Z_{t}}e_{k}^{t}$ lies in the kernel of
both $\theta^{1}_{t}$ and $\theta^{2}_{t}$, therefore it is zero and
$e_{k}^{t}=\mathscr{T}_{t}e_{k}^{0}$.

  2. (2)

Let $(e_{1},e_{2})$ be a $g$-orthonormal frame as in item 1. Denote the dual
frame $(\theta^{1},\theta^{2})$, which is therefore such that there exist
transports for which
$(\mathscr{T}_{t}\theta^{1}_{t},\mathscr{T}_{t}\theta^{2}_{t})$ is an
orthonormal frame for $g_{t}$. Then,

$\displaystyle\mathscr{T}_{t}\left(g\left(\theta^{m_{1}}\otimes\theta^{m_{p}}\otimes
e_{n_{1}}\otimes
e_{n_{q}},\theta^{m^{\prime}_{1}}\otimes\theta^{m^{\prime}_{p}}\otimes
e_{n^{\prime}_{1}}\otimes e_{n^{\prime}_{q}}\right)\right)$
$\displaystyle=\delta_{m_{1},m^{\prime}_{1}}\cdot\dots\cdot\delta_{m_{p},m^{\prime}_{p}}\cdot\delta_{n_{1},n^{\prime}_{1}}\cdot\dots\cdot\delta_{n_{q},n^{\prime}_{q}}$
$\displaystyle=(\mathscr{T}_{t}g)\left(\mathscr{T}_{t}\big{(}\theta^{m_{1}}\otimes\dots\otimes\theta^{m_{p}}\otimes
e_{n_{1}}\otimes\dots\otimes
e_{n_{q}}\big{)},\mathscr{T}_{t}\left(\theta^{m^{\prime}_{1}}\otimes\dots\otimes\theta^{m^{\prime}_{p}}\otimes
e_{n^{\prime}_{1}}\otimes\dots\otimes e_{n^{\prime}_{q}}\right)\right).$

To conclude the proof, observe that applying
$\frac{d}{dt}-\mathscr{L}_{Z_{t}}$ to the basis decomposition
$\mathscr{T}_{t}\alpha=\sum_{I,J}a^{t}_{I,J}\mathscr{T}_{t}(\theta^{I})\otimes\mathscr{T}_{t}(e_{J})$,
with $I,J$ multi-indices, one gets that
$a^{t}_{I,J}=\mathscr{T}_{t}a_{I,J}^{0}$, hence the statement follows by
linearity.

  3. (3)

Use the notation as in the previous steps. As shown in [BE22], for complex
metrics one has a natural notion of Levi-Civita connection forms that extend
the Riemannian notion. So the metric $g_{t}$ and the orthonormal frame
$(\mathscr{T}_{t}e_{1},\mathscr{T}_{t}e_{2})$ coframe
$(\mathscr{T}_{t}\theta^{1},\mathscr{T}_{t}\theta^{2})$ induce the forms
$(\omega_{i}^{j})_{t}$ defined by

$\nabla^{g_{t}}\mathscr{T}_{t}(e_{i})=\sum_{j}(\omega_{i}^{j})_{t}\otimes\mathscr{T}_{t}(e_{j}),$

and which are characterized as the unique 1-forms such that

$\begin{cases}\mathscr{T}_{t}(d\theta^{i})=-\sum_{j}(\omega_{j}^{i})_{t}\wedge\mathscr{T}_{t}(\theta^{j})\\\
(\omega_{j}^{i})_{t}=-(\omega_{i}^{j})_{t}.\end{cases}$

Applying $\frac{d}{dt}-\mathscr{L}_{Z_{t}}$ to the equation on top, one gets
by Remark 4.7 that

$\left(\frac{d}{dt}-\mathscr{L}_{Z_{t}}\right)(\omega^{i}_{j})_{t}\wedge(\mathscr{T}_{t}\theta^{j})=0=-\left(\frac{d}{dt}-\mathscr{L}_{Z_{t}}\right)(\omega_{j}^{i})_{t}\wedge(\mathscr{T}_{t}\theta^{i}),$

hence $(\frac{d}{dt}-\mathscr{L}_{Z_{t}})(\omega_{j}^{i})_{t}=0$, thus
$\mathscr{T}_{t}(\nabla^{g}{e_{j}})=\nabla^{\mathscr{T}_{t}g}(\mathscr{T}_{t}e_{j})$