Quaternions, Monge–Ampère structures and $k$-surfaces

5th October 2022

Graham Smith

Abstract: In $[15]$ Labourie develops a theory of immersed surfaces of
prescribed extrinsic curvature which has since found widespread applications
in hyperbolic geometry, general relativity, Teichmüller theory, and so on. In
this chapter, we present a quaternionic reformulation of these ideas. This
yields simpler proofs of the main results whilst pointing towards the higher-
dimensional generalisation studied by the author in $[25]$.

Classification AMS : 53A05, 12E15, 35J96

1 - Introduction. 1.1 - Introduction.  Immersed surfaces of prescribed
extrinsic curvature in $3$-dimensional manifolds have fascinated
mathematicians for almost two centuries. Following the pioneering work $[15]$
of Labourie, remarkable developments have been made in our understanding of
these objects, leading to striking applications across a broad range of
mathematical theoriesThe reader may consult, for example, $[2]$, $[3]$, $[4]$,
$[14]$, $[22]$, $[26]$, and so on, for a selection of applications of these
techniques.. In this chapter, we propose a quaternionic reformulation of
Labourie’s ideas. Not only will this yield simpler proofs of the main results,
but it will also point towards their higher-dimensional generalisations
studied by the author in $[25]$.

We first recall the main elements of Labourie’s work as it pertains to
immersed surfaces. Let $X:=(X,h)$ be a complete, oriented, $3$-dimensional
riemannian manifold, let $TX$ denote its tangent bundle, and let $SX\subseteq
TX$ denote its unit sphere bundle. We define an (oriented) immersed surface in
$X$ to be a pair $(S,e)$, where $S$ is an oriented surface, and
$e:S\rightarrow X$ is a smooth immersion. Given such a pair, we denote by
$\nu_{e}:S\rightarrow SX$ its unit normal vector field compatible with the
orientation, by ${\text{I}}_{e}$, ${\text{II}}_{e}$ and ${\text{III}}_{e}$ its
first, second and third fundamental forms respectively, by $A_{e}$ its shape
operator, and by $K_{e}:={\text{Det}}(A_{e})$ its extrinsic curvature
function. We say that the immersed surface is infinitesimally strictly convex
(ISC) whenever its second fundamental form is positive definite, we say that
it is quasicomplete whenever it is complete with respect to the riemannian
metric ${\text{I}}_{e}+{\text{III}}_{e}$, and, given a smooth function
$\kappa:SX\rightarrow\mathbb{R}$, we say that its extrinsic curvature is
prescribed by $\kappa$ whenever

$K_{e}:=\kappa\circ\nu_{e}.\eqnum{\text{{\rm(1.1)}}\hbox{}}$

We denote $\hat{e}:=\nu_{e}$, and we call the immersed surface $(S,\hat{e})$
the Gauss lift of $(S,e)$. Note that quasicompleteness of $(S,e)$ is
equivalent to completeness of its Gauss lift. Labourie’s key insight is that
the Gauss lift of any ISC immersed surface of prescribed extrinsic curvature
is a pseudo-holomorphic curve for some suitable almost complex structure. This
allows the powerful theory developed by Gromov in $[7]$ to be applied. The
first consequence is the following compactness result.

Theorem 1.1.1, Labourie’s compactness theorem Let
$\kappa:SX\rightarrow\mathbb{R}$ be a smooth, positive function, and let
$(S_{m},e_{m},p_{m})$ be a sequence of quasicomplete, pointed, ISC immersed
surfaces in $X$ of extrinsic curvature prescribed by $\kappa$. If the sequence
$(e_{m}(p_{m}))$ is precompact in $X$, then the sequence
$(S_{m},\hat{e}_{m},p_{m})$ of Gauss lifts is precompact in the smooth
Cheeger–Gromov topology.

Remark 1.1.1. This is proven in Theorem 3.3.3. The smooth Cheeger–Gromov
topology is described in Section 3.2. Remark 1.1.2. We have stated this result
in its simplest possible form. It may however be generalised in a number of
ways. For example, the ambient space $X$ as well as the function $\kappa$ can
be allowed to vary, the conditions of completeness of the ambient space and
quasicompleteness of the immersed surfaces can also be relaxed, and so on. We
refer the reader to Section 3.2, where potential generalisations are explained
in greater detail. The reader may likewise consult $[27]$ for recent
developments concerning the case with non-trivial boundary. Significantly,
Theorem 1.1.1 is only of limited use without an understanding of degenerate
limits, that is, those limits that are not Gauss lifts of immersed surfaces.
This is addressed by Labourie in his second key result. For any complete
geodesic $\Gamma$ in $X$, we denote by ${\text{N}}\Gamma$ its unit normal
bundle. We define a tube in $SX$ to be an immersed surface $(S,\hat{e})$ which
is a cover of ${\text{N}}\Gamma$ for some complete geodesic $\Gamma$.

Theorem 1.1.2, Labourie’s dichotomy With the notation of Theorem 1.1.1, every
accumulation point of the sequence $(S_{m},e_{m},p_{m})$ is either a tube or
the Gauss lift of some quasicomplete, pointed, ISC immersed surface in $X$ of
extrinsic curvature prescribed by $\kappa$.

Remark 1.1.3. This is proven in Theorem 3.3.3. These two remarkable results
and their variants have since become the basis of a rich theory of surfaces of
prescribed extrinsic curvature in riemannian and semi-riemannian manifolds. It
will be the object of this chapter to present a quaternionic framework within
which they can be proved, allowing us, on the one hand, to emphasize their
hyperkähler nature, and, on the other, to point towards their higher-
dimensional generalisations.

1.2 - Acknowledgements.  This paper was in part written whilst the author was
visiting the Institut des Hautes Études Scientifiques. The author is grateful
for the excellent working conditions enjoyed during that stay.

2 - Quaternions and Bernstein-type theorems. 2.1 - Quaternions.  We begin our
work with a detailed review of Hamilton’s theory of quaternions. Introduced in
1843, this theory has enjoyed a striking revival over the past forty years on
account of the remarkably simple approaches it provides to various deep
mathematical and physical phenomena (see, for example, $[6]$). Its application
to the study of prescribed curvature surfaces, which will be addressed in
detail in the sequel, presents yet another instance of the ubiquity of this
theory that makes it so intriguing.

Let $\mathbb{H}$ denote the algebra of quaternions. This is the associative,
unital algebra over $\mathbb{R}$ generated by the $3$ elements ${{i}}$,
${{j}}$ and ${{k}}$, with the relations

${{i}}^{2}={{j}}^{2}={{k}}^{2}={{i}}\cdot{{j}}\cdot{{k}}=-1.\eqnum{\text{{\rm(2.1)}}\hbox{}}$

Given an element $x\in\mathbb{H}$ of the form

$x:=a+b{{i}}+c{{j}}+d{{k}},\eqnum{\text{{\rm(2.2)}}\hbox{}}$

its conjugate is defined by

$\overline{x}:=a-b{{i}}-c{{j}}-d{{k}}.\eqnum{\text{{\rm(2.3)}}\hbox{}}$

Conjugation is an anti-involution of $\mathbb{H}$ in the sense that, for all
$x,y$,

$\overline{x\cdot
y}=\overline{y}\cdot\overline{x}.\eqnum{\text{{\rm(2.4)}}\hbox{}}$

Its $(+1)$-eigenspace, the space of real quaternions, is a $1$-dimensional
subspace which we denote by $\Cal{R}$. Its $(-1)$-eigenspace, the space of
imaginary quaternions, is a $3$-dimensional subspace which we denote by
$\Cal{I}$. We likewise denote by $\Cal{R}$ and $\Cal{I}$ the respective
projections onto these subspaces.

By (2.4), for all $x\in\mathbb{H}$,

$\overline{x\cdot\overline{x}}=\overline{\overline{x}}\cdot\overline{x}=x\cdot\overline{x},\eqnum{\text{{\rm(2.5)}}\hbox{}}$

so that $x\cdot\overline{x}$ is always real. We thus define an inner product
by

$\langle
x,y\rangle:=\Cal{R}(x\cdot\overline{y}).\eqnum{\text{{\rm(2.6)}}\hbox{}}$

With $x$ as in (2.2),

$\|x\|^{2}=a^{2}+b^{2}+c^{2}+d^{2},\eqnum{\text{{\rm(2.7)}}\hbox{}}$

so that (2.6) coincides with the standard inner product of $4$-dimensional
euclidian space. Furthermore, for all $x,y\in\mathbb{H}$,

$\|x\cdot y\|^{2}=x\cdot y\cdot\overline{x\cdot y}=x\cdot
y\cdot\overline{y}\cdot\overline{x}=\|x\|^{2}\|y\|^{2},\eqnum{\text{{\rm(2.8)}}\hbox{}}$

so that length is multiplicative.

For all $x,y,z\in\Cal{I}$,

$\Cal{R}(x\cdot y\cdot z)=\Cal{R}(\overline{x}\cdot\overline{y}\cdot
z)=\Cal{R}(\overline{y\cdot x}\cdot z)=\Cal{R}(y\cdot
x\cdot\overline{z})=-\Cal{R}(y\cdot x\cdot
z).\eqnum{\text{{\rm(2.9)}}\hbox{}}$

We thus define an alternating $3$-form over $\Cal{I}$ by

$\omega(x,y,z):=-\Cal{R}(x\cdot y\cdot z).\eqnum{\text{{\rm(2.10)}}\hbox{}}$

This is the volume form of $\langle\cdot,\cdot\rangle$ with orientation chosen
such that $({{i}},{{j}},{{k}})$ is a positive triple.

By (2.8), the sphere $\mathbb{S}^{3}$ of unit quaternions is a subgroup of
$\mathbb{H}$ with inverse given by conjugation.

Lemma & Definition 2.1.1 The homomorphism
$h:\mathbb{S}^{3}\times\mathbb{S}^{3}\rightarrow{\text{SO}}(\mathbb{H})$ given
by

$h(x,y)z:=x\cdot z\cdot\overline{y}.\eqnum{\text{{\rm(2.11)}}\hbox{}}$

is a double cover of ${\text{SO}}(\mathbb{H})$. In particular, it identifies
$\mathbb{S}^{3}\times\mathbb{S}^{3}$ with ${\text{Spin}}(\mathbb{H})$.

Proof: Since it is a continuous homomorphism between Lie groups of the same
dimension, it suffices to show that its kernel is

${\text{Ker}}(h)=\left\\{(1,1),(-1,-1)\right\\}.\eqnum{\text{{\rm(2.12)}}\hbox{}}$

However, suppose that $h(x,y)={\text{Id}}$. Substituting $z=1$ into (2.11)
yields $x\cdot\overline{y}=1$, so that $x=y$. Thus, for all $z$,

$z\cdot x=(h(x,x)\cdot z)\cdot x=x\cdot z\cdot\overline{x}\cdot x=x\cdot z,$

so that

$[x,z]=0.$

Since $z$ is arbitrary, $x$ is real, and since $x$ has unit norm, it is equal
$\pm 1$. This proves (2.12), and the result follows. $\square$

2.2 - Compatible complex structures.  Recall that a complex structure over
$\mathbb{H}$ is an $\mathbb{R}$-linear map $J:\mathbb{H}\rightarrow\mathbb{H}$
such that

$J^{2}=-{\text{Id}}.\eqnum{\text{{\rm(2.13)}}\hbox{}}$

We say, in addition, that $J$ is compatible whenever it preserves the metric,
that is, whenever, for all $x\in\mathbb{H}$,

$\|Jx\|=\|x\|.\eqnum{\text{{\rm(2.14)}}\hbox{}}$

We now proceed to identify all compatible complex structures over
$\mathbb{H}$.

Lemma 2.2.1 The set of square roots of $-1$ in $\mathbb{H}$ is the sphere
$\mathbb{S}^{3}\mathop{\cap}\Cal{I}$ of unit, imaginary quaternions.

Proof: Indeed, $x^{2}=-1$ if and only if $x^{-1}=-x$. This holds if and only
if $\|x\|=1$ and $x=-\overline{x}$, as desired. $\square$For every unit,
imaginary quaternion $x$, $h(x,1)$ and $h(1,x)$ are trivially compatible
complex structures over $\mathbb{H}$. We now verify that there are no others.

Lemma & Definition 2.2.2 If $J:\mathbb{H}\rightarrow\mathbb{H}$ is a
compatible complex structure, then $J$ is given by multiplication either on
the left or on the right by a unit, imaginary quaternion. We call the former
left complex structures and the latter right complex structures.

Remark 2.2.1. Every left complex structure trivially commutes with every right
complex structure. Note also that quaternionic conjugation sends left complex
structures into right complex structures and vice-versa. Proof: Since
$J\in{\text{SO}}(\mathbb{H})$, by Lemma 2.1.1, there exists
$(x,y)\in\mathbb{S}^{3}\times\mathbb{S}^{3}$ such that $J=h(x,y)$. Since

$-J^{2}=h(x^{2},-y^{2})={\text{Id}},$

it follows that

$x^{2}=-y^{2}=\pm 1.$

Without loss of generality, we may suppose that $(x^{2},y^{2})=(1,-1)$. It
then follows that $x=\pm 1$, $y\in\mathbb{S}^{3}\mathop{\cap}\Cal{I}$, and

$J=h(\pm 1,y)=h(1,\pm y),$

as desired. $\square$In particular, we obtain the following algebraic
characterisation of left and right complex structures.

Lemma 2.2.3 Let $J:\mathbb{H}\rightarrow\mathbb{H}$ be a right complex
structure. If $J^{\prime}:\mathbb{H}\rightarrow\mathbb{H}$ is another
compatible complex structure which commutes with $J$, then either $J^{\prime}$
is a left complex structure, or $J^{\prime}=\pm J$.

Remark 2.2.2. An analogous result trivially holds with the roles of left and
right complex structures inverted. Proof: It suffices to show that if
$J^{\prime}$ is a right complex structure which commutes with $J$, then
$J^{\prime}=\pm J$. However, let $x$ and $y$ be unit, imaginary quaternions
such that $J=h(1,x)$ and $J^{\prime}=h(1,y)$. Since $J$ and $J^{\prime}$
commute, so too do $x$ and $y$. Thus, since $x$ and $y$ are imaginary,

$\overline{x\cdot\overline{y}}=y\cdot\overline{x}=-y\cdot x=-x\cdot
y=x\cdot\overline{y},$

so that $x\cdot\overline{y}$ is real. Since $x$ and $y$ both have unit length,
this holds if and only if $x=\pm y$, and the result follows. $\square$

2.3 - Compatible quaternionic structures.  Let $E:=(E,g)$ be an inner-product
space. We define a compatible quaternionic structure over $E$ to be an algebra
homomorphism $\rho:\mathbb{H}\rightarrow{\text{End}}(E)$ such that, for all
$x\in\mathbb{H}$ and for all $u\in E$,

$\|\rho(x)u\|=\|x\|\cdot\|u\|.\eqnum{\text{{\rm(2.15)}}\hbox{}}$

We now proceed to identify all compatible quaternionic structures over
$\mathbb{H}$.

Lemma 2.3.1 If $\alpha:\mathbb{H}\rightarrow\mathbb{H}$ be an non-trivial
algebra homomorphism, then there exists a unit quaternion $z$ such that, for
all $x\in\mathbb{H}$,

$\alpha(x)=z\cdot x\cdot\overline{z}.\eqnum{\text{{\rm(2.16)}}\hbox{}}$

Proof: Since $\mathbb{H}$ is a skew field, $\alpha$ is injective and
$\alpha(\pm 1)=\pm 1$. In particular, $\alpha$ preserves $\Cal{R}$. It
likewise preserves the set $\mathbb{S}^{3}\mathop{\cap}\Cal{I}$ of square
roots of $-1$, and therefore also $\Cal{I}$. It follows that $\alpha$
preserves the decomposition $\mathbb{H}=\Cal{R}\oplus\Cal{I}$. It therefore
also preserves conjugation, and thus also the metric (2.6). Finally, by
(2.10), $\alpha$ preserves the orientation of $\Cal{I}$ and thus also of
$\mathbb{H}$, and is consequently an element of ${\text{SO}}(\mathbb{H})$. It
follows by Lemma 2.1.1 that there exist unit quaternions
$z,w\in\mathbb{S}^{3}$ such that $\alpha=h(w,z)$. Finally, since
$\alpha(1)=1$, $w=z$, and the result follows. $\square$

Lemma & Definition 2.3.2 If
$\alpha:\mathbb{H}\rightarrow{\text{End}}(\mathbb{H})$ is a compatible
quaternionic structure, then either  (1) there exists $z\in\mathbb{S}^{3}$
such that, for all $x,y\in\mathbb{H}$,

$\alpha(x)\cdot y=z\cdot x\cdot\overline{z}\cdot y,\
\text{or}\eqnum{\text{{\rm(2.17)}}\hbox{}}$

(2) there exists $z\in\mathbb{S}^{3}$ such that, for all $x,y\in\mathbb{H}$,

$\alpha(x)\cdot y=y\cdot
z\cdot\overline{x}\cdot\overline{z}.\eqnum{\text{{\rm(2.18)}}\hbox{}}$

We call the former left quaternionic structures and the latter right
quaternionic structures.

Remark 2.3.1. As before, every left quaternionic structure commutes with every
right quaternionic structure. Furthermore, quaternionic conjugation sends left
quaternionic structures to right quaternionic structures and vice-versa.
Proof: Since $\mathbb{H}$ is a skew field, $\alpha$ is injective and
$\alpha(\pm 1)=\pm{\text{Id}}$. It follows that $\alpha$ maps the set of unit,
imaginary quaternions to the set of compatible complex structures of
$\mathbb{H}$. By connectedness, we may suppose without loss of generality
that, for every unit, imaginary quaternion $x$, $\alpha(x)$ is a left complex
structure. It follows that, for all $x,y\in\mathbb{H}$,

$\alpha(x)\cdot y=\rho(x)\cdot y,$

for some non-trivial algebra homomorphism
$\rho:\mathbb{H}\rightarrow\mathbb{H}$. The result now follows by Lemma 2.3.1.
$\square$

2.4 - Calibrations.  The utility of quaternions to the theory of partial
differential equations arises from the theory of calibrations developed by
Harvey & Lawson in $[9]$. We now explain how this applies our setting. For
every unit, imaginary quaternion $x$, we define

$J_{x}:=h(x,1),\eqnum{\text{{\rm(2.19)}}\hbox{}}$

and we define the symplectic form $\omega_{x}$ over $\mathbb{H}$ by

$\omega_{x}:=\langle\cdot,J_{x}\cdot\rangle.\eqnum{\text{{\rm(2.20)}}\hbox{}}$

Lemma 2.4.1 Let $(x,y,z)$ be an orthonormal triplet of unit, imaginary
quaternions. Let $P\subseteq\mathbb{H}$ be a real plane. The area form dA of
$P$ satisfies, for all $\xi,\nu\in P$,

$dA(\xi,\nu)^{2}=\omega_{x}(\xi,\nu)^{2}+\omega_{y}(\xi,\nu)^{2}+\omega_{z}(\xi,\nu)^{2}.\eqnum{\text{{\rm(2.21)}}\hbox{}}$

Remark 2.4.1. In other words, the triplet $(\omega_{x},\omega_{y},\omega_{z})$
forms a calibration of $\mathbb{H}$ in the sense of $[9]$. Proof: We may
suppose that $\xi$ and $\nu$ both have unit length. Since
$(\nu,J_{x}\nu,J_{y}\nu,J_{z}\nu)$ is an orthonormal real basis of
$\mathbb{H}$,

$\langle\xi,\nu\rangle^{2}+\langle\xi,{\text{J}}_{x}\nu\rangle^{2}+\langle\xi,{\text{J}}_{y}\nu\rangle^{2}+\langle\xi,{\text{J}}_{z}\nu\rangle^{2}=\|\xi\|^{2}=1.$

Thus, if $\theta$ denotes the angle between $\xi$ and $\nu$, then

$dA(\xi,\nu)^{2}={\text{sin}}^{2}(\theta)=1-\langle\xi,\nu\rangle^{2}=\omega_{x}(\xi,\nu)^{2}+\omega_{y}(\xi,\nu)^{2}+\omega_{z}(\xi,\nu)^{2},$

as desired. $\square$In particular, we obtain the following result for real
planes in $\mathbb{H}$.

Lemma 2.4.2 Let $x$ be a unit, imaginary quaternion. Let $P$ be a real plane
in $\mathbb{H}$. $P$ is ${\text{J}}_{x}$-complex if and only if it is
$\omega_{y}$-lagrangian for every unit, imaginary quaternion $y$ orthogonal to
$x$.

Proof: Let $y$ and $z$ be unit, imaginary quaternions such that $(x,y,z)$ is
an orthonormal triplet, and let dA denote the area form of $P$. $P$ is
$J_{x}$-complex if and only if, for all $\xi,\nu\in P$,

${\text{dA}}(\xi,\nu)^{2}=\langle\xi,{\text{J}}_{x}\nu\rangle^{2}=\omega_{x}(\xi,\nu)^{2}.$

By (2.21), this holds if and only if $\omega_{y}$ and $\omega_{z}$ both vanish
over $P$, as desired. $\square$

2.5 - The Monge–Ampère equation.  We now apply our abstract framework to the
study of solutions of the real Monge–Ampère equation

${\text{Det}}({\text{Hess}}(u))=1.\eqnum{\text{{\rm(2.22)}}\hbox{}}$

To this end, we identify $\mathbb{C}$ with $\mathbb{R}^{2}$ in the natural
manner, and we introduce an explicit compatible quaternionic structure over
$\mathbb{C}\oplus\mathbb{C}=\mathbb{R}^{2}\oplus\mathbb{R}^{2}$ as follows.
First, let $J_{0}$ denote the operator of multiplication by $i$, so that

$J_{0}:=\begin{pmatrix}0\hfill&-1\hfill\cr
1\hfill&0\hfill\cr\end{pmatrix},\eqnum{\text{{\rm(2.23)}}\hbox{}}$

and let $g$ denote the standard metric over $\mathbb{C}\oplus\mathbb{C}$, that
is

$g((z,w)^{t},(z,w)^{t})=\left|z\right|^{2}+\left|w\right|^{2}.\eqnum{\text{{\rm(2.24)}}\hbox{}}$

We define

$\eqalign{I&:=\begin{pmatrix}J_{0}\hfill&0\hfill\cr
0\hfill&-J_{0}\hfill\cr\end{pmatrix},\cr
J&:=\begin{pmatrix}0\hfill&J_{0}\hfill\cr
J_{0}\hfill&0\hfill\cr\end{pmatrix},\ \text{and}\cr
K&:=\begin{pmatrix}0\hfill&-{\text{Id}}\hfill\cr{\text{Id}}\hfill&0\hfill\cr\end{pmatrix}.\cr}\eqnum{\text{{\rm(2.25)}}\hbox{}}$

Note that, in the spirit of Lemma 2.2.3, $I$, $J$ and $K$ are generators of
the set of compatible complex structures over $\mathbb{C}\oplus\mathbb{C}$
which commute with

$\hat{J}_{0}:=\begin{pmatrix}J_{0}\hfill&0\hfill\cr
0\hfill&J_{0}\hfill\cr\end{pmatrix}.\eqnum{\text{{\rm(2.26)}}\hbox{}}$

Let $(\omega_{i},\omega_{j},\omega_{k})$ denote the triplet of symplectic
forms corresponding to $(I,J,K)$, that is

$\eqalign{\omega_{i}&:=g(\cdot,I\cdot),\cr\omega_{j}&:=g(\cdot,J\cdot),\
\text{and}\cr\omega_{k}&:=g(\cdot,K\cdot).\cr}\eqnum{\text{{\rm(2.27)}}\hbox{}}$

We now examine the algebraic properties of different types of lagrangian
subspaces of $\mathbb{C}\oplus\mathbb{C}$. To this end, let
$P\subseteq\mathbb{C}\oplus\mathbb{C}$ be a real plane which is a graph over
the first component, that is

$P:=\left\\{(x,Ax)\ |\
x\in\mathbb{R}^{2}\right\\},\eqnum{\text{{\rm(2.28)}}\hbox{}}$

for some matrix $A\in{\text{End}}(2)$. The following two useful relations for
matrices $A\in{\text{End}}(2)$ may be verified by inspection.

$\eqalign{A^{t}J_{0}AJ_{0}&=-{\text{Det}}(A){\text{Id}},\ \text{and}\cr
A-A^{t}&=-{\text{Tr}}(AJ_{0})J_{0}.\cr}\eqnum{\text{{\rm(2.29)}}\hbox{}}$

Lemma 2.5.1 (1) $P$ is $\omega_{i}$-lagrangian if and only if
${\text{Det}}(A)=1$;  (2) $P$ is $\omega_{j}$-lagrangian if and only if
${\text{Tr}}(A)=0$; and  (3) $P$ is $\omega_{k}$-lagrangian if and only if
$A=A^{t}$.

Proof: Indeed, $P$ is $\omega_{i}$-lagrangian if and only if,

| $\displaystyle{{}\langle x,J_{0}y\rangle-\langle Ax,J_{0}Ay\rangle}$ | $\displaystyle{{}=0\ \forall x,y\in\mathbb{R}^{2}}$
---|---|---
$\displaystyle{\Leftrightarrow}\quad$ | $\displaystyle{{}\langle x,({\text{Id}}+A^{t}J_{0}AJ_{0})y\rangle}$ | $\displaystyle{{}=0\ \forall x,y\in\mathbb{R}^{2}.}$

This holds if and only if

$\displaystyle{\Leftrightarrow}\quad$ | $\displaystyle{{}{\text{Id}}+A^{t}J_{0}AJ_{0}}$ | $\displaystyle{{}=0}$
---|---|---
$\displaystyle{\Leftrightarrow}\quad$ | $\displaystyle{{}{\text{Det}}(A)}$ | $\displaystyle{{}=1,}$

where the last equivalence follows by (2.29). In a similar manner, using
(2.29) again, we show that $P$ is $\omega_{j}$-lagrangian if and only if
${\text{Tr}}(A)=0$. Finally, it is a standard result that $P$ is
$\omega_{k}$-lagrangian if and only if $A=A^{t}$, and this completes the
proof. $\square$We thus obtain the following key result which relates
solutions of the $2$-dimensional Monge–Ampère equation to pseudo-holomorphic
curves.

Lemma 2.5.2, Monge–Ampère = Pseudoholomorphic Let $\Omega$ be a simply-
connected open subset of $\mathbb{R}^{2}$. Let
$\alpha:\Omega\rightarrow\mathbb{R}^{2}$ be a smooth function and let
$S\subseteq\Omega\times\mathbb{R}^{2}$ denote its graph. Then $\alpha$ is the
derivative of a smooth solution $u:\Omega\rightarrow\mathbb{R}$ of the real
Monge–Ampère equation (2.22) if and only if $S$ is a $J$-holomorphic curve.

Proof: Indeed, by Lemma 2.4.2, $S$ is $J$-holomorphic if and only if it is
$\omega_{i}$\- and $\omega_{k}$-lagrangian. By Lemma 2.5.1, this holds if and
only if ${\text{Det}}(D\alpha)=1$ and $\alpha$ is closed. Since $\Omega$ is
simply-connected, $\alpha$ is closed if and only if it is the derivative of
some smooth function $u:\Omega\rightarrow\mathbb{R}$. The result now follows,
since ${\text{Det}}({\text{Hess}}(u))={\text{Det}}(D\alpha)=\pm 1$. $\square$

2.6 - Positivity I.  We now express convexity in the quaternionic framework.
To this end, let
$\pi_{1},\pi_{2}:\mathbb{C}\oplus\mathbb{C}\rightarrow\mathbb{C}$ denote
respectively the projections onto the first and second components, and denote

$V:={\text{Ker}}(\pi_{1})=\left\\{0\right\\}\oplus\mathbb{C}.\eqnum{\text{{\rm(2.30)}}\hbox{}}$

We call $V$ the vertical subspace of $\mathbb{C}\oplus\mathbb{C}$. Let $m$ be
the symmetric bilinear form defined over $\mathbb{C}\oplus\mathbb{C}$ by

$m((z_{1},w_{1})^{t},(z_{2},w_{2})^{t}):=\langle z_{1},w_{2}\rangle+\langle
z_{2},w_{1}\rangle.\eqnum{\text{{\rm(2.31)}}\hbox{}}$

Note that $m$ vanishes over $V$ and that

$m(\cdot,\cdot)=-m(I\cdot,I\cdot)=m(J\cdot,J\cdot)=-m(K\cdot,K\cdot).\eqnum{\text{{\rm(2.32)}}\hbox{}}$

It is worth noting that $m$ is, up to sign, uniquely determined by these
properties.

Lemma 2.6.1 Up to sign, $m$ is the unique symmetric bilinear form of unit norm
satisfying (2.32) which vanishes over $V$.

Proof: Indeed, let $m^{\prime}$ be another symmetric bilinear form which
vanishes over $V$ and which satisfies (2.32). Since $m^{\prime}$ vanishes over
$V$ and is $J$-invariant, it is uniquely determined by the restriction of
$\tilde{m}^{\prime}:=m(\cdot,J\cdot)$ to $V$. However, since $m^{\prime}$ is
$I$-anti-invariant, and since $I$ anticommutes with $J$, $\tilde{m}^{\prime}$
is also $I$-invariant, and its restriction to $V$ is thus unique up to a
scalar factor. Finally, since $m^{\prime}$ has unit norm, it follows that
$m^{\prime}=\pm m$, as desired. $\square$Define the subgroup
$\Sigma\subseteq{\text{SO}}(4)$ by

$\Sigma:=\left\\{\begin{pmatrix}A\hfill&0\hfill\cr
0\hfill&A\hfill\cr\end{pmatrix}\ \bigg{|}\
A\in{\text{SO}}(2)\right\\}.\eqnum{\text{{\rm(2.33)}}\hbox{}}$

Lemma 2.6.2 $\Sigma$ is the stabiliser subgroup in ${\text{SO}}(4)$ both of
$(I,J,K,m)$ and of $(J,V,\omega)$, where $\omega$ as any orientation form of
$V$.

Remark 2.6.1. In particular, the prescription of a triplet of the form
$(V,\omega,J)$ over any $4$-dimensional inner-product space $(E,g)$ is
equivalent to the prescription of a compatible quaternionic structure
$(I,J,K)$ together with a symmetric bilinear form $m$ satisfying (2.32).
Proof: Suppose that $M\in{\text{SO}}(4)$ preserves $(I,J,K,m)$. Define the
involution $\alpha$ of $\mathbb{C}\oplus\mathbb{C}$ by
$\alpha(z,w)=(z,\overline{w})$, and denote $\tilde{M}:=\alpha M\alpha$. Since
$(\alpha I\alpha)(z,w)=(iz,iw)$, $\tilde{M}\in{\text{U}}(2)$. Since, in
addition, $\tilde{M}$ preserves $(\alpha
K\alpha)(z,w)=(-\overline{w},\overline{z})$,

$M=\begin{pmatrix}a\hfill&b\hfill\cr-\overline{b}\hfill&\overline{a}\hfill\cr\end{pmatrix},$

for some $a,b\in\mathbb{C}$ such that
$\left|a\right|^{2}+\left|b\right|^{2}=1$. Finally, since $\tilde{M}$
preserves $m$, $b=0$, and it follows that $M$ is an element of $\Sigma$, as
desired.

Suppose now that $M\in{\text{SO}}(4)$ preserves $(J,V,\omega)$. In particular,
$M$ takes the form

$M=\begin{pmatrix}A\hfill&0\hfill\cr 0\hfill&B\hfill\cr\end{pmatrix},$

for some $A,B\in{\text{SO}}(2)$. Since $M$ preserves $J_{0}$, $A=B$, and it
follows that $M$ is an element of $\Sigma$, as desired. $\square$The
significance of the bilinear form $m$ for the study of convexity is given by
the following results.

Lemma & Definition 2.6.3 The restriction of $m$ to any $J$-complex line $L$ in
$\mathbb{C}\oplus\mathbb{C}$ is either positive-definite, negative-definite or
null. Furthermore, this restriction is null if and only if $L$ intersects $V$
non-trivially, that is, if and only if $\pi_{1}$ has non-trivial kernel over
$L$.

We say that a $J$-complex line $L$ in $\mathbb{C}\oplus\mathbb{C}$ is
positive, negative or null according to whether the restriction of $m$ to $L$
is positive-definite, negative-definite or null.

Proof: Let $L\subseteq\mathbb{R}^{2}\oplus\mathbb{R}^{2}$ be a $J$-complex
line. Since $m(J\cdot,J\cdot)=m$, $m$ has well-defined sign over $L$. Since
$L$ is $\omega_{k}$-lagrangian, for all $\xi,\nu\in L$,

$\langle\pi_{1}(\xi),\pi_{2}(\nu)\rangle=m(\xi,\nu)+\omega_{k}(\xi,\nu)=m(\xi,\nu).$

It follows that $m$ vanishes over $L$ if and only if one of
${\text{Ker}}(\pi_{1})\mathop{\cap}L$ or ${\text{Ker}}(\pi_{2})\mathop{\cap}L$
is non-trivial. However, since $J_{0}\pi_{1}=\pi_{2}J$, these spaces are
either both trivial or both non-trivial, and the result follows. $\square$

Lemma 2.6.4, Convexity = Positivity Let $\Omega$ be an open subset of
$\mathbb{R}^{2}$. Let $u:\Omega\rightarrow\mathbb{R}$ be a smooth solution of
the Monge–Ampère equation (2.22) and let
$S\subseteq\Omega\times\mathbb{R}^{2}$ denote the graph of its derivative. The
function $u$ is strictly convex if and only if every tangent plane of $S$ is
positive.

Proof: Indeed, the tangent planes of $S$ are positive if and only if, for all
$x\in\Omega$ and for all $\xi\in\mathbb{R}^{2}$,
$\langle\xi,{\text{Hess}}(u)(x)\xi\rangle>0$. $\square$

2.7 - Positivity II - a holomorphic approach.  Define
$\phi,\psi:\mathbb{C}\oplus\mathbb{C}\rightarrow\mathbb{C}$ by

$\eqalign{\phi(z,w)&:=z+w,\
\text{and}\cr\psi(z,w)&:=-\overline{z}+\overline{w},\cr}\eqnum{\text{{\rm(2.34)}}\hbox{}}$

and define

$\Phi:=(\phi,\psi).\eqnum{\text{{\rm(2.35)}}\hbox{}}$

Note that $\Phi$ is $\mathbb{C}$-linear with respect to the complex structure
$J$ of the domain and the complex structure $\hat{J}_{0}$ of the codomain. In
addition, for all $(z,w)\in\mathbb{C}^{2}$,

$(\Phi_{*}m)((z,w)^{t},(z,w)^{t})=\frac{1}{2}(\left|z\right|^{2}-\left|w\right|^{2}).\eqnum{\text{{\rm(2.36)}}\hbox{}}$

We thus consider $\phi$ and $\psi$ as the respective projections of
$\mathbb{C}\oplus\mathbb{C}$ onto the space and time components of $m$. This
has two useful consequences. The first is the following Lipschitz property.

Lemma 2.7.1 Let $P$ be a real plane in $\mathbb{C}\oplus\mathbb{C}$. If

$m|_{P}\geq 0,\eqnum{\text{{\rm(2.37)}}\hbox{}}$

then the restriction of $\phi$ to $P$ is $\sqrt{2}$-bilipschitz.

Proof: Indeed, for all $(z,w)\in P$,

$\left|\phi(z,w)\right|=\left|z+w\right|\leq\sqrt{2}\|z,w\|.$

Conversely,

$\left|\phi(z,w)\right|^{2}\geq\|z,w\|^{2}+m((z,w)^{t},(z,w)^{t})\geq\|z,w\|^{2},$

as desired. $\square$The second useful consequence is the following conformal
description of positivity. First, let $\hat{\mathbb{C}}$ denote the extended
complex plane and define the conformal diffeomorphism
$\alpha:\mathbb{C}\mathbb{P}^{1}\rightarrow\hat{\mathbb{C}}$ by

$\alpha([z:w]):=\frac{w}{z}.\eqnum{\text{{\rm(2.38)}}\hbox{}}$

Lemma 2.7.2 Let $L$ be a $J$-complex line in $\mathbb{C}\oplus\mathbb{C}$. $L$
is positive if and only if $\left|(\alpha\circ\Phi)(L)\right|<1$ and $L$ is
null if and only if $\left|(\alpha\circ\Phi)(L)\right|=1$.

Proof: Choose $(z,w)\in\Phi(L)$. By (2.36), $L$ is positive if and only if

$\left|w\right|^{2}<\left|z\right|^{2},$

which holds if and only if
$\left|\alpha(L)\right|=\left|\alpha([z:w])\right|<1$. Likewise, $L$ is null
if and only if

$\left|w\right|^{2}=\left|z\right|^{2},$

which holds if and only if $\left|\alpha(L)\right|=1$. $\square$

2.8 - Bernstein type theorems I.  The quaternionic formalism that we have
presented here now yields simple proofs of key Bernstein-type theorems. These
results are not only interesting in their own right, but also provide the
basis for the general compactness theorem that will be developed later.

Theorem 2.8.1 If $S$ is a complete, non-negative $J$-holomorphic curve in
$\mathbb{C}\oplus\mathbb{C}$, then $S$ is an affine plane.

Proof: Indeed, upon passing to the universal cover, we may assume that $S$ is
simply-connected. By Lemma 2.7.1, $\phi$ restricts to a bilipschitz,
holomorphic map from $S$ into $\mathbb{C}$. It is thus a conformal
diffeomorphism, and $S$ therefore has parabolic conformal type. Now define the
meromorphic function $\tau:S\rightarrow\hat{\mathbb{C}}$ by

$\tau(x):=(\alpha\circ\Phi)(T_{x}S).$

By Lemma 2.7.2, $\tau$ takes values in the closed unit disk. By Liouville’s
theorem, $\tau$ is constant, and $S$ is therefore an affine plane, as desired.
$\square$The Bernstein-type theorem $[11]$ of Jörgens is an immediate
corollary.

Theorem 2.8.2, Jörgens If $u:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a smooth
solution of the real Monge–Ampère equation (2.22), then $u$ is a quadratic
function.

Remark 2.8.1. Jörgens’ theorem was a key result in the study of the real
Monge–Ampère equations. It was generalised to Dimensions $3$ and $4$ by Calabi
in $[5]$, and then to arbitrary dimension by Pogorelov in $[20]$. The ideas
originated in these papers were later applied to the study of Monge-Ampère
equations in diverse settings. It is worth noting, however, that in the
higher-dimensional case, there is no useful analogue of the quaternionic
approach studied here. For such analogoues, we must instead look to the
special lagrangian potential equation, where a rich theory can indeed be
developed (see $[10]$ and $[25]$). Proof: Indeed, by Lemmas 2.5.2 and 2.6.3,
the graph $S$ of $du$ is a positive $J$-holomorphic curve. In addition, being
a graph over $\mathbb{R}^{2}$, it is complete, and the result now follows by
Theorem 2.8.1. $\square$Another nice application of Theorem 2.8.1 is the
following mild improvement of the classification result of complete flat
surfaces in $\mathbb{H}^{3}$ by Volkov–Vladimirova and Sasaki (see Chapter
$7.F$ of $[23]$). We say that an immersed surface $(S,e)$ in $\mathbb{H}^{3}$
is flat whenever it has constant extrinsic curvature equal to $1$.Flat
surfaces in $\mathbb{H}^{3}$ are studied in terms of their Gauss lifts in
$[12]$, $[13]$ and $[18]$. The terminology used in these papers differs from
our own: where they write “weakly complete”, we write “quasicomplete”; and the
objects which they refer to as “flat fronts” are for us the Gauss lifts of,
possibly singular, flat surfaces.

Theorem 2.8.3 The only quasicomplete, ISC, flat surfaces in $\mathbb{H}^{3}$
are the level sets of horospheres and the level sets of distance functions to
complete geodesics.

Remark 2.8.2. When the surface is complete, we recover the result of
Volkov–Vladimirova and Sasaki. Proof: By Gauss’ Theorem, every flat surface is
intrinsically flat and is thus locally isometric to $\mathbb{R}^{2}$. Let
$\Omega$ be an open subset of $\mathbb{R}^{2}$, let
$e:\Omega\rightarrow\mathbb{H}^{3}$ be an isometric immersion, and let
${\text{II}}_{e}$ denote its second fundamental form. By the Codazzi-Mainardi
equations, the derivative $({\text{D}}{\text{II}}_{e})_{ijk}$ of
${\text{II}}_{e}$ is symmetric, so that

${\text{II}}_{e}={\text{Hess}}(f),$

for some smooth function $f$, unique up to addition of a linear function. We
define the immersion
$\alpha:\Omega\rightarrow\mathbb{R}^{2}\oplus\mathbb{R}^{2}$ by

$\alpha(x):=(x,df(x)).$

Since ${\text{Det}}({\text{Hess}}(f))={\text{Det}}({\text{II}}_{e})=1$, it
follows by Lemma 2.5.2 that $\alpha$ is $J$-holomorphic. Furthermore,
$(\pi_{1}\circ\alpha)$ is a local isometry and $\alpha^{*}m=2{\text{II}}_{e}$,
so that $\alpha$ is also positive.

Now let $(S,e)$ be a quasicomplete, flat immersed surface in $\mathbb{H}^{3}$.
Upon taking the universal cover, we may suppose that $S$ is simply connected.
It then follows by the preceding discussion that there exists a
$J$-holomorphic immersion
$\alpha:S\rightarrow\mathbb{R}^{2}\oplus\mathbb{R}^{2}$ such that
$\beta:=(\pi_{1}\circ\alpha)$ is a local isometry and
$\alpha^{*}m=2{\text{II}}_{e}$. Furthermore, since $e$ is quasicomplete,
$\alpha$ is complete, and it follows by Theorem 2.8.1 that $(S,\alpha)$ is an
affine plane. In particular, $\beta$ defines a global isometry from $S$ into
$\mathbb{R}^{2}$. Upon composing with a rotation if necessary,
$e^{\prime}:=e\circ\beta^{-1}$ is an isometric immersion with second
fundamental form given by

${\text{II}}_{e^{\prime}}:=\begin{pmatrix}\lambda\hfill&0\hfill\cr
0\hfill&1/\lambda\hfill\cr\end{pmatrix},$

for some constant $\lambda\in]0,1]$. The result now follows by the fundamental
theorem of surface theory: when $\lambda=1$, $S$ is the level set of a
horofunction and, when $\lambda<1$, $S$ is the set of points lying at distance
${\text{arctanh}}(\lambda)$ from some complete geodesic in $\mathbb{H}^{3}$.
$\square$

2.9 - Bernstein type theorems II - non-trivial boundary.  The case of curves
with non-trivial boundary is similar, though more technical.

Theorem 2.9.1 Let $P\subseteq\mathbb{C}\oplus\mathbb{C}$ be a plane over which
$m$ has signature $(1,0)$. Let $(S,\partial
S)\subseteq\mathbb{C}\oplus\mathbb{C}$ be a complete, non-negative
$J$-holomorphic curve with boundary. If  (1) $\partial S\subseteq P$; and  (2)
there exists $\epsilon>0$ such that, for all $x\in\partial S$,

$m|_{T_{x}S}\geq\epsilon g|_{T_{x}S}.\eqnum{\text{{\rm(2.39)}}\hbox{}}$

then $S$ is an affine plane.

Proof: Upon taking the universal cover, we may suppose that $S$ is simply
connected. Let $\Cal{F}$ denote the foliation of $P$ by null lines and let
$L\subseteq P$ be any line transverse to this foliation. By Lemma 2.7.1,
$\phi$ restricts to a linear isomorphism from $P$ into $\mathbb{C}$. By
(2.39), there exists $\theta>0$ such that $\phi(\partial S)$ makes an angle of
at least $\theta$ with $\phi_{*}\Cal{F}$ at every point so that, by
completeness, $\phi(\partial S)$ is a union of complete Lipschitz graphs over
$L$. By Lemma 2.7.1 again, the restriction of $\phi$ to $S$ is a conformal
diffeomorphism onto its image. In particular $\phi(S)$ is a subset of
$\mathbb{C}$ bounded by at most $2$ complete Lipschitz graphs. $(S,\partial
S)$ is therefore conformally equivalent to one of
$\overline{\mathbb{D}}\setminus\left\\{-1\right\\}$ or
$\overline{\mathbb{D}}\setminus\left\\{\pm 1\right\\}$, and the Riemann
surface $\tilde{S}$, obtained by doubling $S$ along its boundary, is thus
conformally equivalent to one of $\mathbb{C}$ or
$\mathbb{C}\setminus\left\\{0\right\\}$. In both cases, $\tilde{S}$ is of
parabolic type.

Define the meromorphic function $\tau:S\rightarrow\hat{\mathbb{C}}$ by

$\tau(x):=(\alpha\circ\Phi)(T_{x}S).$

By Lemma 2.7.2, $\tau$ takes values in the closed unit disk. Furthermore, by
(2.39), there exists $\delta>0$ such that, for all $x\in\partial S$,

$\left|\tau(x)\right|\leq 1-\delta.$

Since the boundary $\partial S$ of $S$ has at most two isolated singularities,
and since $\tau$ is bounded, it follows by the Cauchy integral formula that,
for all $x\in S$,

$\left|\tau(x)\right|\leq 1-\delta.$

For all $r>0$, let $D_{r}$ and $C_{r}$ denote respectively the disk and circle
in $\mathbb{C}$ of radius $r$ about the origin. By definition of $P$, there
exists a circle $C$, which is a strict interior tangent to $C_{1}$ at some
point, such that, for all $x\in\partial S$,

$\tau(x)\in C.$

Let $R:\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}$ denote the conformal
reflection through $C$. By the Schwarz reflection principle, $\tau$ extends to
a holomorphic function $\tilde{\tau}:\tilde{S}\rightarrow\hat{\mathbb{C}}$
such that, for all $x\in S$,

$\tilde{\tau}(\overline{x}):=R\tau(x).$

In particular,

$\tilde{\tau}(\tilde{S})\subseteq\overline{D}_{1-\delta}\mathop{\cup}R\overline{D}_{1-\delta}=:\Omega.$

Since the complement of $\Omega$ in $\hat{\mathbb{C}}$ has non-trivial
interior, it follows by Liouville’s theorem that $\tilde{\tau}$, and therefore
also $\tau$, is constant. The surface $S$ is thus an affine half-plane, and
this completes the proof. $\square$In terms of the real Monge–Ampère equation,
this yields the following new Bernstein-type theorem for functions defined
over domains with boundary.

Theorem 2.9.2 Let $\phi:\mathbb{R}\times[0,\infty[\rightarrow\mathbb{R}$ be a
smooth solution of the real Monge–Ampère equation (2.22). If  (1) $\phi$
restricts to a non-trivial quadratic function over
$\mathbb{R}\times\left\\{0\right\\}$; and  (2) $\Delta\phi$ is bounded over
$\mathbb{R}\times\left\\{0\right\\}$, then $\phi$ is a quadratic function.

Proof: Indeed, by Lemmas 2.5.2 and 2.6.3, the graph $S$ of $du$ is a positive
$J$-holomorphic curve which satisfies the hypotheses of Theorem 2.9.1. The
result follows. $\square$

2.10 - Non-quadratic solutions.  We conclude this section with the following
interesting open problem. It is not clear that Theorems 2.9.1 and 2.9.2 are
optimal as stated. Although their first conditions are natural from the point
of view of Liouville’s theorem and the Schwarz reflection principle, their
second conditions are not. Nevertheless, the first conditions alone are not
sufficient, and the function

$\phi(x,y):=\frac{x^{2}}{y}+\frac{y^{3}}{12}\eqnum{\text{{\rm(2.40)}}\hbox{}}$

shows. Indeed, this function is a convex solution of the real Monge–Ampère
equation (2.22) over the open half-plane $\mathbb{R}\times]0,\infty[$ whose
restriction to every horizontal line is quadratic, but which is itself
trivially not quadratic.

The above function, together with many similar examples, is obtained by
directly solving the Monge–Ampère equation (2.22) with the ansatz

$f(x,y)=\alpha(y)x^{2}+\beta(y)x+\gamma(y).\eqnum{\text{{\rm(2.41)}}\hbox{}}$

However, it is also worth reviewing its properties from the quaternionic point
of view. Note first that

$\Phi(V)=\left\\{(z,\overline{z})\ |\
z\in\mathbb{C}\right\\}.\eqnum{\text{{\rm(2.42)}}\hbox{}}$

Let $\mathbb{H}^{+}$ denote the upper half-space in $\mathbb{C}$ and let
$\overline{\mathbb{H}}^{+}$ denote its closure. Consider a holomorphic
function
$F:=(f,g)^{t}:\overline{\mathbb{H}}^{+}\rightarrow\mathbb{C}\oplus\mathbb{C}$,
and suppose that

$F(\mathbb{R})\subseteq\Phi(V).\eqnum{\text{{\rm(2.43)}}\hbox{}}$

By (2.42), for all $x\in\mathbb{R}$,

$f(x)=\overline{g(x)}.\eqnum{\text{{\rm(2.44)}}\hbox{}}$

It follows by the Schwarz reflection principle that $f$ and $g$ extend to
entire functions such that, for all $z\in\mathbb{H}^{+}$,

$g(z)=\overline{f(\overline{z})}.\eqnum{\text{{\rm(2.45)}}\hbox{}}$

In particular, it is sufficient to prescribe only the first component $f$. In
order to obtain a complete immersed surface with boundary, it is also
necessary that $f$ define a conformal diffeomorphism of $\mathbb{H}^{+}$ onto
its image. This holds, for example, when $f$ is quadratic. We thus choose

$f(z):=-i-i(z+i)^{2},\eqnum{\text{{\rm(2.46)}}\hbox{}}$

so that

$F(z)=(-i-i(z+i)^{2},i+i(z-i)^{2})^{t}.\eqnum{\text{{\rm(2.47)}}\hbox{}}$

Composing with $\Phi^{-1}$ yields

$(\Phi^{-1}\circ
F)(z)=(2xy+2iy,2x+i(y^{2}-x^{2}))^{t},\eqnum{\text{{\rm(2.48)}}\hbox{}}$

where $x+iy:=z$, so that $(\Phi^{-1}\circ F)$ is the graph over
$\mathbb{R}\times]0,\infty[$ of the $1$-form

$\alpha(x,y):=\frac{2x}{y}dx+\bigg{(}\frac{y^{2}}{4}-\frac{x^{2}}{y^{2}}\bigg{)}dy,\eqnum{\text{{\rm(2.49)}}\hbox{}}$

which integrates to (2.40).

With a view towards the optimal formulations of Theorems 2.9.1 and 2.9.2, it
is tempting to conjecture that the only non-quadratic solutions of the
Monge–Ampère equation over the half-plane which restrict to quadratic
functions over the boundary are those described above. However, although we
have not been able to find other examples, we also have no proof of this
assertion.

3 - Monge–Ampère structures and $k$-surfaces. 3.1 - Monge–Ampère structures.
In $[15]$ Labourie uses a non-integrable generalisation of the structures
studied in Section 2 to derive compactness results for families of immersed
surfaces of prescribed extrinsic curvature in $3$-dimensional riemannian
manifolds. Labourie’s work now underlies a large part of the modern theory of
such surfaces, with applications in hyperbolic geometry, Teichmüller theory,
general relativity, and so on. We now provide an elementary presentation of
Labourie’s ideas.

Let $X$ be a manifold. Labourie defines a Monge–Ampère structure over $X$ to
be a quadruplet $(W,g,V,J)$ where  (1) $W$ is a smooth $4$-dimensional
subbundle of the tangent bundle of $X$ furnished with a riemannian metric $g$;
(2) $V$ is a smooth, oriented $2$-dimensional subbundle of $W$;  (3) $J$ is a
smooth section of ${\text{End}}(W)$ such that, for all $x$, $J_{x}$ restricts
to a compatible complex structure of the fibre $W_{x}$ with respect to which
the fibre $V_{x}$ is real. Remark 3.1.1. By Lemmas 2.6.1 and 2.6.2 and the
subsequent remark, at each point $x$ of $X$, the prescription of
$(V_{x},J_{x})$ is equivalent to the prescription of a compatible quaternionic
structure $(I_{x},J_{x},K_{x})$, together with a unit-length symmetric
bilinear form $m_{x}$ satisfying (2.32). The above geometric structure can
thus be expressed in purely quaternionic terms. This point of view is
particularly useful for its extension to the higher-dimensional case (see
$[25]$). We also highlight the parallels with the analogous theory of
Monge–Ampère structures developed in $[16]$, $[17]$ and $[21]$. The geometric
significance of $V$ is easier to understand when described in conformal terms.
Indeed, at every point $x$, the set of complex lines in $W_{x}$ which
intersect $V_{x}$ non-trivially defines a circle $C_{x}$ in the complex
projective space of $(W_{x},J_{x})$. Furthermore, the orientation of $V_{x}$
in turn yields an orientation of this circle which therefore has a well-
defined interior $D_{x}$. The prescription of $V$ is thus equivalent to the
prescription of a conformal disk bundle $D$ in the complex projective bundle
of $(W,J)$. In the spirit of Lemma 2.7.2 this yields a notion of positivity of
$J_{x}$-complex lines in $V_{x}$. Let $\overline{D}$ denote the bundle whose
fibre at $x$ is the topological closure of $D_{x}$.

A $J$-holomorphic curve in $X$ is a pair $(S,\phi)$ where $S$ is a Riemann
surface and $\phi:S\rightarrow X$ is a smooth function such that, for all $x$,
${\text{Im}}(D\phi(x))\subseteq W_{\phi(x)}$ and

$J_{\phi(x)}D\phi(x)=D\phi(x)j_{x},\eqnum{\text{{\rm(3.1)}}\hbox{}}$

where $j$ here denotes the complex structure of $S$. We define a Monge–Ampère
surface to be an immersed $J$-holomorphic curve $(S,\phi)$ in $X$ such that
(1) $(S,\phi)$ is tangent to $W$;  (2) $(S,\phi)$ is complete with respect to
$g$; and  (3) the tangent space to $(S,\phi)$ is at every point an element of
$\overline{D}$. We say that a Monge–Ampère surface is positive or null at some
point according to whether its tangent space at that point is an element of
$D$ or $\partial D=\overline{D}\setminus D$. Following Labourie, we define a
curtain surface to be a Monge–Ampère surface which is everywhere null.

Labourie’s framework consists of two parts, namely a compactness result, which
we will study in the next section, and a dichotomy result, given by the
following theorem.

Theorem 3.1.1 Let $(S,\phi)$ be a Monge–Ampère surface and let
$(S^{\prime},\phi^{\prime})$ be a curtain surface. If there exists $x\in S$
and $x^{\prime}\in S^{\prime}$ such that $\phi(x)=\phi^{\prime}(x^{\prime})$
and ${\text{Im}}(D\phi(x))={\text{Im}}(D\phi^{\prime}(x^{\prime}))$, then
there exists a neighbourhood $U$ of $x$ in $S$, a neighbourhood $V$ of
$x^{\prime}$ in $S^{\prime}$, and a conformal diffeomorphism
$\alpha:U\rightarrow V$ such that

$\phi=\phi^{\prime}\circ\alpha.\eqnum{\text{{\rm(3.2)}}\hbox{}}$

Theorem 3.1.1 motivates the following definition. We say that the Monge–Ampère
structure is integrable whenever every point of $\partial
D:=\overline{D}\setminus D$ is tangent to some curtain surface.

Theorem 3.1.2 If $(W,V,J)$ is an integrable Monge–Ampère structure then every
point of $\overline{D}\setminus D$ is tangent to a unique inextensible,
simply-connected curtain surface.

Proof: This follows immediately from Theorem 3.1.1 and unique continuation.
$\square$

Theorem 3.1.3, Labourie’s Dichotomy If $(W,V,J)$ is an integrable Monge–Ampère
structure, then every Monge–Ampère surface whose tangent bundle meets
$\partial D=\overline{D}\setminus D$ at some point is a curtain surface. That
is, every Monge–Ampère surface is either everywhere positive, or everywhere
null.

Proof: This likewise follows immediately from Theorem 3.1.1 and unique
continuation. $\square$Proof of Theorem 3.1.1: Suppose the contrary. For
$r>0$, let $\mathbb{D}_{r}$ and $\mathbb{R}_{r}^{m}$ denote the open disk and
the open ball, both of radius $r$, in $\mathbb{C}$ and $\mathbb{R}^{m}$
respectively. We parametrize a neighbourhood of
$y:=\phi(x)=\phi^{\prime}(x^{\prime})$ in $X$ by
$\mathbb{D}_{r}\times\mathbb{D}_{r}\times\mathbb{B}^{m}_{r}$ in such a manner
that $\mathbb{D}_{r}\times\left\\{(0,0)\right\\}$ identifies with a portion of
$\phi^{\prime}(S^{\prime})$ about $x^{\prime}$ and, for all
$z\in\mathbb{D}_{r}$,

$\eqalign{W_{(z,0,0)}&=\mathbb{C}\times\mathbb{C}\times\left\\{0\right\\},\
\text{and}\cr J_{(z,0,0)}&=\begin{pmatrix}i\hfill&0\hfill&0\hfill\cr
0\hfill&i\hfill&0\hfill\cr 0\hfill&0\hfill&0\hfill\cr\end{pmatrix}.\cr}$

Since $(S,\phi)$ is tangent to $(S^{\prime},\phi^{\prime})$ at $y$, a portion
of this surface about $x$ identifies with the graph of a smooth function
$F:\mathbb{D}_{r}\rightarrow\mathbb{D}_{r}\times\mathbb{B}_{r}^{m}$ which, by
hypothesis, is non-zero. Since both
$\mathbb{D}_{r}\times\left\\{(0,0)\right\\}$ and the graph of $F$ are
$J$-holomorphic curves, it follows by Aronszajn’s unique continuation theorem
(see $[1]$ and Theorem $2.3.4$ of $[19]$) that

$F(z)=(P(z),Q(z))+{\text{O}}(z^{k+2}),\eqnum{\text{{\rm(3.3)}}\hbox{}}$

for real homogeneous polynomials $P$ and $Q$, both of order $(k+1)$, where
$k\geq 1$. Furthermore, since the graph of $F$ is tangent to $W$,

$Q=0,\eqnum{\text{{\rm(3.4)}}\hbox{}}$

and, by $J$-holomorphicity again,

$P(z)=az^{k+1},\eqnum{\text{{\rm(3.5)}}\hbox{}}$

for some $a\in\mathbb{C}\setminus\left\\{0\right\\}$.

From this point, the result follows by an elementary topological argument. Let
$\Psi$ be a smooth function which, for all $(z,w,\xi)$, sends the complex
projective space of $(W_{(z,w,\xi)},J_{(z,w,\xi)})$ to that of
$(W_{(z,0,0)},J_{(z,0,0)})$. Since $W$ and $J$ are constant over
$\mathbb{D}_{t}\times\left\\{(0,0)\right\\}$, we think of $\Psi$ as sending
the complex projective bundle of $(W,J)$ to the trivial bundle with fibre
$\mathbb{C}\mathbb{P}^{1}$. For all $z$, let $\phi(z)$ denote the closest
point of the boundary of $\Psi(D_{(z,F(z))})$ to the origin. We claim that

$\phi(z)={\text{O}}(\left|z\right|^{k+1}).$

Indeed, since $\mathbb{D}_{r}\times\left\\{(0,0)\right\\}$ is a curtain
surface, for all $z\in\mathbb{D}_{r}$,

$0\in\Psi(D_{(z,0,0)}),$

and, since $F={\text{O}}(\left|z\right|^{k+1})$, the assertion follows by
smoothness of $\Psi$.

Consider now the curves $\gamma_{\epsilon}(\theta):=\epsilon e^{i\theta}$ and
$\delta_{\epsilon}(\theta):=\phi(\epsilon e^{i\theta})$. For all $\theta$, let
$\gamma_{\epsilon}^{\prime}(\theta)$ denote the image under $\Psi$ of the
tangent plane of the graph of $F$ at the point
$(\gamma_{\epsilon}(\theta),F(\gamma_{\epsilon}(\theta)))$. By (3.3), (3.4)
and (3.5),

$\gamma_{\epsilon}^{\prime}(\theta)=a(k+1)\epsilon^{k}e^{ik\theta}+{\text{O}}(\epsilon^{k+1}).$

It follows that, for sufficiently small $\epsilon$,
$(\gamma^{\prime}_{\epsilon}-\delta_{\epsilon})$ winds $k$ times around the
origin. However, this is absurd, since the tangent plane of $(S,\phi)$ is at
every point an element of $D$ whilst $\delta_{\epsilon}$ lies along its
boundary. The result follows. $\square$

3.2 - Compactness.  We now prove a compactness property for families of
Monge–Ampère surfaces. In order to correctly express the result, we require
the concept of smooth Cheeger–Gromov convergence. First, we define a pointed
riemannian manifold to be a triplet $(X,g,p)$, where $X$ is a smooth manifold,
$g$ is a riemannian metric and $p$ is a point of $X$. We say that a sequence
$(X_{m},g_{m},p_{m})$ of complete, pointed riemannian manifolds converges to
the complete, pointed riemannian manifold $(X_{\infty},g_{\infty},p_{\infty})$
in the smooth Cheeger–Gromov sense whenever there exists a sequence
$(\Phi_{m})$ of functions such that  (1) for all $m$,
$\Phi_{m}:X_{\infty}\rightarrow X_{m}$ and $\Phi_{\infty}(p_{\infty})=p_{m}$;
and for every relatively compact open subset $\Omega$ of $X_{\infty}$, there
exists $M$ such that  (2) for all $m\geq M$, the restriction of $\Phi_{m}$ to
$\Omega$ defines a smooth diffeomorphism onto its image; and  (3) the sequence
$((\Phi_{m}|_{\Omega})^{*}g_{m})_{m\geq M}$ converges to
$g_{\infty}|_{\Omega}$ in the $C^{\infty}$ sense. We call $(\Phi_{m})$ a
sequence of convergence maps of $(X_{m},g_{m},p_{m})$ with respect to
$(X_{\infty},g_{\infty},p_{\infty})$.

It is reassuring to observe that it defines a Hausdorff topology over the set
of isometry classes of complete, pointed riemannian manifolds.Strictly
speaking, the class of all complete, pointed riemannian manifolds is not a
set. However, the class of all embedded $n$-dimensional submanifolds of
$\mathbb{R}^{2n+2}$ furnished with complete riemannian metrics is. Since
Whitney’s theorem identifies these two classes up to isometry equivalence, the
family of isometry equivalence classes of complete, pointed riemannian
manifolds can be legitimately considered a set. Furthermore, although the
convergence maps are trivially non-unique, any two sequences $(\Phi_{m})$ and
$(\Phi_{m}^{\prime})$ of convergence maps are equivalent in the sense that
there exists an isometry $\Psi:X_{\infty}\rightarrow X_{\infty}$ preserving
$p_{\infty}$ such that, for any two relatively compact open subsets
$U\subseteq\overline{U}\subseteq V$ of $X_{\infty}$, there exists $M$ such
that  (1) for all $m\geq M$, the respective restrictions of $\Phi_{m}$ and
$\Phi_{m}^{\prime}\circ\Psi$ to $U$ and $V$ define smooth diffeomorphisms onto
their images;  (2) for all $m\geq M$,
$(\Phi_{m}^{\prime}\circ\Psi)(U)\subseteq\Phi_{m}(V)$; and  (3) the sequence
$((\Phi_{m}|_{V})^{-1}\circ\Phi_{m}^{\prime}\circ\Psi)$ converges in the
$C^{\infty}$ sense to the identity map over $U$. This condition trivially
defines an equivalence relation over the space of sequences of convergence
maps.

We now return to the case of immersed submanifolds. We say that a sequence
$(S_{m},e_{m},p_{m})$ of complete pointed immersed submanifolds in a complete
riemannian manifold $(X,g)$ converges to the complete pointed immersed
submanifold $(S_{\infty},e_{\infty},p_{\infty})$ in the smooth Cheeger–Gromov
sense whenever the sequence $(S_{m},e_{m}^{*}g,p_{m})$ converges to
$(S_{\infty},e_{\infty}^{*}g,p_{\infty})$ in the smooth Cheeger–Gromov sense
and, for one, and therefore for any, sequence $(\Phi_{m})$ of convergence
maps, the sequence $(e_{m}\circ\Phi_{m})$ converges to $e_{\infty}$ in the
$C^{\infty}_{\text{loc}}$ sense.

Smooth Cheeger–Gromov convergence can also be characterised in terms of
graphs. Indeed, let $NS_{\infty}$ denote the normal bundle of
$(S_{\infty},e_{\infty})$ in $\phi_{\infty}^{*}TX$. Recall that the
exponential map of $X$ defines a smooth function
${\text{Exp}}:NS_{\infty}\rightarrow X$. In particular, given a sufficently
small smooth section $f:\Omega\rightarrow NS_{\infty}$ defined over an open
subset $\Omega$ of $S_{\infty}$, the composition ${\text{Exp}}\circ f$ defines
a smooth immersion of $\Omega$ in $X$ which we call the graph of $f$. A
straightforward but technical argument shows that the sequence
$(S_{m},e_{m},p_{m})$ converges to $(S_{\infty},e_{\infty},p_{\infty})$ in the
smooth Cheeger–Gromov sense if and only if there exists a sequence
$(p_{m}^{\prime})$ of points in $S_{\infty}$ and sequences of functions
$(f_{m})$ and $(\alpha_{m})$ such that  (1) $(p_{m}^{\prime})$ converges to
$p_{\infty}$;  (2) for all $m$, $f_{m}$ maps $S_{\infty}$ into $NS_{\infty}$,
$\alpha$ maps $S_{\infty}$ into $S_{m}$ and $\alpha(p_{m}^{\prime})=p_{m}$;
and for every relatively compact open subset $\Omega$ of $S_{\infty}$, there
exists $M$ such that  (3) for all $m\geq M$, the restriction of $f_{m}$ to
$\Omega$ defines a smooth section of $NS_{\infty}$ over this set, the
restriction of $\alpha_{m}$ to $\Omega$ defines a smooth diffeomorphism onto
its image, and

${\text{Exp}}\circ f_{m}|_{\Omega}=e_{m}\circ\alpha|_{\Omega};\ \text{and}$

(4) the sequence $(f_{m})_{m\geq M}$ tends to zero in the $C^{\infty}$ sense.
These definitions are readily extended in a number of ways. For example, in
the case of a sequence $(X_{m},g_{m},p_{m})$ of pointed riemannian manifolds,
the hypothesis of completeness is unnecessary. Instead, it is sufficient to
assume that for all $R>0$, there exists $M$ such that, for all $m\geq M$, the
closed ball of radius $R$ about $p_{m}$ in $(X_{m},g_{m})$ is compact.
Likewise, in the case of immersed submanifolds, the target space can be
replaced with a sequence $(X_{m},g_{m},p_{m})$ of pointed riemannian manifolds
converging in the smooth Cheeger–Gromov sense to some complete pointed
riemannian manifold. Furthermore, it is not necessary to suppose that the
riemannian manifolds in this sequence are complete, and so on.

Elliptic regularity, together with the Arzela-Ascoli Theorem of $[24]$, yields
the following compactness result.

Lemma 3.2.1 Let $(X,g)$ be a complete riemannian manifold furnished with a
Monge–Ampère structure $(W,V,J)$. Let $(S_{m},e_{m},p_{m})$ be a sequence of
complete pointed $J$-holomorphic curves in $(X,g)$. If  (1) there exists a
compact subset $K$ of $X$ such that $e_{m}(p_{m})\in X$ for all $m$; and  (2)
for every compact subset $L\subseteq X$, there exists a constant $B>0$ such
that, for all $m$, the norm of the second fundamental form of $e_{m}$ is at
every point of $e_{m}^{-1}(L)$ bounded above by $B$, then there exists a
complete pointed Monge–Ampère surface $(S_{\infty},e_{\infty},p_{\infty})$ in
$(X,g)$ towards which this sequence subconverges in the smooth Cheeger–Gromov
sense.

Remark 3.2.1. This result readily extends in many ways. For example, it is not
necessary to assume that the $J$-holomorphic curves are complete. Likewise,
the target manifold with its Monge–Ampère structure may be replaced by
sequences converging to some limit, and so on. We now recall a simplified
version of the quasi-maximum lemma (see $[8]$).

Lemma 3.2.2 Let $X$ be a complete metric space and let
$f:X\rightarrow[1,\infty[$ be an upper semi-continuous function. For all $x\in
X$, there exists $y\in X$ such that  (1) $f(y)\geqslant f(x)$; and  (2) for
all $z\in B_{1/\sqrt{f(y)}}(y)$, $f(z)\leqslant 2f(y)$.

Proof: Indeed, otherwise, we recursively construct a sequence $(x_{m})$ of
points of $X$ such that, for all $m$, $f(x_{m+1})\geqslant 2f(x_{m})$ and
$d(x_{m+1},x_{m})\leqslant 1/\sqrt{f(x_{m})}$. We readily verify that
$(x_{m})$ is a Cauchy sequence which therefore converges to some limit
$x_{\infty}$, say. However, by upper semi-continuity,

$f(x_{\infty})\geqslant\mathop{{\text{LimSup}}}_{m\rightarrow\infty}f(x_{m})=\infty,$

which is absurd, and the result follows. $\square$

Lemma 3.2.3 Let $(X,g)$ be a complete riemannian manifold furnished with a
Monge–Ampère structure $(W,V,J)$. For every compact subset $K$ of $X$, there
exists $B>0$ such that every Monge–Ampère surface $(S,e)$ in $X$ has second
fundamental form bounded above by $B$ at every point of $e^{-1}(K)$.

Proof: We prove this result using a standard blow-up argument. Indeed, suppose
the contrary. There exists a sequence $(S_{m},e_{m},p_{m})$ of complete,
pointed Monge–Ampère surfaces and a sequence $(B_{m})$ of positive numbers
converging to $+\infty$ such that, for all $m$, $e_{m}(p_{m})\in K$ and

$\|{\text{II}}_{m}(p_{m})\|=B_{m},$

where, for all $m$, ${\text{II}}_{m}$ denotes the second fundamental form of
$e_{m}$. By Lemma 3.2.2, we may suppose that, for all $m$ and for all $y\in
B_{1/\sqrt{B_{m}}}(p_{m})$,

$\|{\text{II}}_{m}(y)\|\leq 2B_{m}.$

We extend $g$ to a riemannian metric defined over the whole of $X$ and, upon
rescaling, we consider the sequence $(X,B_{m}^{2}g,e_{m}(p_{m}))$ of pointed
riemannian manifolds. This sequence converges in the Cheeger–Gromov sense to
$(X_{\infty},g_{\infty},0)$, where
$X_{\infty}:=\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{m}$ and $g_{\infty}$
is the standard euclidean metric. In addition, we may suppose that the
Monge–Ampère structures of these manifolds converge to the constant
Monge–Ampère structure $(W_{\infty},V_{\infty},J_{\infty})$ given by

$\eqalign{W_{\infty}&:=\mathbb{C}\times\mathbb{C}\times\left\\{0\right\\},\vphantom{\bigg{)}}\cr
V_{\infty}&:=\mathbb{R}\times\mathbb{R}\times\left\\{0\right\\},\
\text{and}\vphantom{\bigg{)}}\cr
J_{\infty}&:=\begin{pmatrix}0\hfill&i\hfill\cr
i\hfill&0\hfill\cr\end{pmatrix}.\cr}$

For all $m$, let ${\text{II}}_{m}^{\prime}$ denote the shape operator of
$e_{m}$ with respect to the rescaled metric $B_{m}^{2}g$. For all $m$,

$\|{\text{II}}_{m}^{\prime}(p_{m})\|=1,$

and, for all $y\in B_{\sqrt{B_{m}}}(x_{m})$,
$\|{\text{II}}_{m}^{\prime}(y)\|\leq 2$. It follows by Theorem 3.2.1 and the
subsequent remark that there exists a Monge–Ampère surface
$(S_{\infty},e_{\infty},p_{\infty})$ in $X_{\infty}$ towards which
$(S_{m},e_{m},p_{m})$ subconverges in the smooth Cheeger–Gromov sense. In
particular, the norm of the second fundamental form of this surface at
$p_{\infty}$ is equal to $1$. However, since this surface is contained in
$\mathbb{C}\times\mathbb{C}\times\left\\{0\right\\}$, by Theorem 2.8.1, it is
an affine plane. This yields the desired contradiction and the result follows.
$\square$Combining Lemmas 3.2.1 and 3.2.3 yields Labourie’s compactness
theorem.

Theorem 3.2.4, Labourie’s compactness theorem Let $(X,g)$ be a complete
riemannian manifold furnished with a Monge–Ampère structure $(W,V,J)$. Let
$(S_{m},e_{m},p_{m})_{m\in\mathbb{N}}$ be a sequence of pointed Monge–Ampère
surfaces in $(X,g)$. If $(e_{m}(p_{m}))_{{m\in\mathbb{N}}}$ is precompact in
$X$, then $(S_{m},e_{m},p_{m})_{m\in\mathbb{N}}$ is precompact in the smooth
Cheeger–Gromov topology.

3.3 - Applications I - $k$-surfaces.  Let $(X,h)$ be a complete, oriented
$3$-dimensional riemannian manifold, let $SX$ denote its bundle of unit
tangent spheres, and furnish this manifold with the Sasaki metric. We
construct an integrable Monge–Ampère structure over $SX$ as follows. First,
let $VSX$ denote the vertical subbundle of $TSX$, and let $HSX$ denote the
horizontal bundle of the Levi–Civita connection. Recall that, at the point
$\xi:=\xi_{p}\in SX$, we have the natural identifications

$\eqalign{H_{\xi}SX&=T_{p}X,\ \text{and}\cr
V_{\xi}SX&=\langle\xi\rangle^{\perp},\cr}\eqnum{\text{{\rm(3.6)}}\hbox{}}$

where $\langle\xi\rangle^{\perp}$ here denotes the orthogonal complement of
$\xi_{p}$ in $T_{p}X$, so that $T_{\xi}SX$ decomposes as

$T_{\xi}SX=T_{p}X\oplus\langle\xi_{p}\rangle^{\perp}.\eqnum{\text{{\rm(3.7)}}\hbox{}}$

With respect to this decomposition, we define the subbundles $W$ and $V$ by

$\eqalign{W_{\xi}&:=\langle\xi_{p}\rangle^{\perp}\oplus\langle\xi_{p}\rangle^{\perp},\
\text{and}\cr
V_{\xi}&:=\left\\{0\right\\}\oplus\langle\xi_{p}\rangle^{\perp}.\cr}\eqnum{\text{{\rm(3.8)}}\hbox{}}$

Since $X$ is oriented, its tangent bundle carries a well-defined wedge
product, given by

$h(\nu_{p},\nu_{p}^{\prime}\wedge\nu_{p}^{\prime\prime}):={\text{dVol}}(\nu_{p},\nu_{p}^{\prime},\nu_{p}^{\prime\prime}).\eqnum{\text{{\rm(3.9)}}\hbox{}}$

Thus, for all $\xi:=\xi_{p}\in SX$, $\langle\xi\rangle^{\perp}$ carries a
well-defined complex structure $j_{\xi}$ given by

$j_{\xi}\cdot\nu_{p}=\xi_{p}\wedge\nu_{p}.\eqnum{\text{{\rm(3.10)}}\hbox{}}$

Given a smooth function $\phi:SX\rightarrow\mathbb{R}$, we define the complex
structure $J_{\phi}$ over $W$ by

$J_{\phi,\xi}\cdot(\nu_{p},\mu_{p}):=(e^{\phi}j_{\xi}\mu_{p},e^{-\phi}j_{\xi}\nu_{p}).\eqnum{\text{{\rm(3.11)}}\hbox{}}$

Since, in addition, $j_{\xi}$ defines an orientation over $V$,
$(W,V,J_{\phi})$ defines a Monge–Ampère structure over $X$. Note that the
symmetric bilinear form $m$ corresponding to this structure is, up to sign,
given by

$m_{\xi}((\nu_{p},\mu_{p}),(\nu_{p},\mu_{p}))=2\langle\nu_{p},\mu_{p}\rangle.\eqnum{\text{{\rm(3.12)}}\hbox{}}$

We now verify integrability of this Monge–Ampère structure. For every complete
geodesic $\Gamma$ in $X$, let ${\text{N}}\Gamma$ denote its unit normal bundle
in $SX$.

Lemma 3.3.1 The Monge–Ampère structure $(W,V,J_{\phi})$ is integrable and its
curtain surfaces are the covers of those surfaces of the form
${\text{N}}\Gamma$, for some complete geodesic $\Gamma$ in $X$.

Proof: Let $\Gamma$ be a complete geodesic in $X$. With respect to the
decomposition (3.7), given any $\xi:=\xi_{p}\in{\text{N}}_{p}\Gamma$, the
tangent space to ${\text{N}}\Gamma$ at this point is given by

${\text{T}}_{\xi}{\text{N}}\Gamma=\langle(\tau,0),(0,\tau\wedge\xi)\rangle,\eqnum{\text{{\rm(3.13)}}\hbox{}}$

where $\tau$ is here a unit tangent vector to $\Gamma$ at $p$. It immediately
follows that ${\text{N}}\Gamma$ is a curtain surface. Conversely, given a
point $\xi:=\xi_{p}$ in $SX$ and a vertical vector $(0,\nu_{p})\in W_{\xi}$,
let $\Gamma$ denote the unique geodesic tangent to $\xi_{p}\wedge\nu_{p}$ at
$p$. By (3.13), ${\text{N}}\Gamma$ is a curtain surface tangent to
$(0,\nu_{p})$ at $\xi_{p}$. Since $\xi$ is arbitrary, integrability follows,
and this completes the proof. $\square$It remains only to study the geometry
of positive Monge–Ampère surfaces. First, let $(S,e)$ be an ISC immersed
surface in $X$ and let $(S,\hat{e})$ denote its Gauss lift. Recall from the
introduction that, for any smooth function $\kappa:SX\rightarrow\mathbb{R}$,
the extrinsic curvature $K_{e}$ of $(S,e)$ is said to be prescribed by
$\kappa$ whenever

$K_{e}=\kappa\circ\hat{e}.\eqnum{\text{{\rm(3.14)}}\hbox{}}$

Lemma 3.3.2 An immersed surface in $SX$ is a positive Monge–Ampère surface if
and only if it is the Gauss lift of a quasicomplete, ISC, immersed surface in
$X$ of extrinsic curvature prescribed by $e^{2\phi}$.

Proof: We first show that every positive Monge–Ampère surface is the Gauss
lift of some immersed surface. Indeed, since $V$ is the kernel of the
canonical projection $\pi:SX\rightarrow X$, for every positive Monge–Ampère
surface $(S,\hat{e})$, the function $e:=\pi\circ\hat{e}$ is an immersion.
Observe, furthermore, that $\hat{e}$ is normal to $e$, so that the orientation
of $S$ can be chosen in such a manner that $(S,\hat{e})$ is the Gauss lift of
$(S,e)$, as desired.

Now let $(S,e)$ be an oriented, immersed surface in $X$ and let $(S,\hat{e})$
denote its Gauss lift. Trivially $(S,e)$ is quasicomplete if and only if
$(S,\hat{e})$ is complete. Next, with respect to the decomposition (3.7), the
derivative of $\hat{e}$ at a point $p\in S$ is given by

$D\hat{e}(p)\cdot\xi_{p}=(De(p)\cdot\xi_{p},De(p)\cdot A_{e}(p)\cdot\xi_{p}).$

Thus, by Lemma 2.5.1, $(S,\hat{e})$ is a $J$-holomorphic curve if and only if
$(S,e)$ has extrinsic curvature prescribed by $e^{2\phi}$. Likewise, by
(3.12), $(S,\hat{e})$ is positive if and only if $A_{e}$ is everywhere
positive-definite, that is, if and only if $(S,e)$ is ISC. The result now
follows. $\square$The theory developed in Sections 3.1 and 3.2 now yields the
following result.

Theorem 3.3.3, Labourie Let $\phi:SX\rightarrow\mathbb{R}$ be a smooth
function. Let $(S_{m},e_{m},p_{m})$ be a sequence of quasicomplete, pointed,
immersed surfaces of extrinsic curvature prescribed by $e^{2\phi}$. If
$(e_{m}(p_{m}))$ is precompact in $X$, then the sequence
$(S_{m},\hat{e}_{m},p_{m})$ of Gauss lifts is precompact in the smooth
Cheeger–Gromov sense. Furthermore, any accumulation point
$(S_{\infty},\hat{e}_{\infty},p_{\infty})$ of this sequence which is not a
curtain surface is the Gauss lift of some quasicomplete, pointed immersed
surface of constant extrinsic curvature prescribed by $e^{2\phi}$.

Remark 3.3.1. As in Section 3.2, we have stated this result in its simplest
form. More generally, the ambient space $X$ and that function $\phi$ may be
allowed to vary, the surfaces need not be quasicomplete, and so on.

3.4 - Applications II - Isometric immersions.  We conclude this paper by
reviewing a more sophisticated version of the construction of the preceding
section developed in $[14]$. By incorporating the domain explicitly into the
construction, we are able to study families of isometric immersions, as
opposed to reparametrisation equivalence classes, as is the case in the
preceding section.

We first review the corresponding linear framework. Let
${\text{Isom}}(\mathbb{R}^{2},\mathbb{R}^{3})$ denote the subset of
${\text{Lin}}(\mathbb{R}^{2},\mathbb{R}^{3})$ consisting of linear isometries
$\alpha:\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$. Note that this is an orbit
of ${\text{SO}}(3)$ in ${\text{Lin}}(\mathbb{R}^{2},\mathbb{R}^{3})$ upon
which this group acts with trivial stabiliser. It is therefore a smooth
$3$-dimensional submanifold, and its tangent bundle admits trivialisation

$\tau:{\text{Lin}}(\mathbb{R}^{2},\mathbb{R}^{3})\times{\mathfrak{so}}(3)\rightarrow{\text{T}}{\text{Isom}}(\mathbb{R}^{2},\mathbb{R}^{3});(\alpha,A)\mapsto(\alpha,A\circ\alpha).\eqnum{\text{{\rm(3.15)}}\hbox{}}$

In addition, viewing the wedge product linear isomorphism from
$\mathbb{R}^{3}$ to ${\mathfrak{so}}(3)$, a useful alternative form of this
trivialisation is

$\sigma:{\text{Lin}}(\mathbb{R}^{2},\mathbb{R}^{3})\times\mathbb{R}^{3}\rightarrow{\text{T}}{\text{Isom}}(\mathbb{R}^{2},\mathbb{R}^{3});(\alpha,x)\mapsto(\alpha,x\wedge\alpha).\eqnum{\text{{\rm(3.16)}}\hbox{}}$

We now return to the non-linear case. Let $(S,g)$ be an oriented riemannian
surface, and let $(X,h)$ be an oriented $3$-dimensional riemannian manifold.
Over the cartesian product $S\times X$, we consider the bundle
${\text{Lin}}(\pi_{1}^{*}TS,\pi_{2}^{*}TX)$, where $\pi_{1}$ and $\pi_{2}$
respectively denote the projections onto the first and second factors. Let
${\text{Isom}}(S\times X)$ denote its subbundle whose fibre at the point
$(p,q)$ is ${\text{Isom}}(T_{p}S,T_{q}X)$. We construct a Monge–Ampère
structure over the total space of this subbundle as follows. Let
${\text{V}}{\text{Isom}}(S\times X)$ denote the vertical subbundle of
${\text{T}}{\text{Isom}}(S\times X)$, and let ${\text{H}}{\text{Isom}}(S\times
X)$ denote the horizontal bundle of the Levi-Civita connection. Bearing in
mind the discussion of the preceding paragraph, at the point
$\alpha:=\alpha_{(p,q)}\in{\text{Isom}}_{(p,q)}(S\times X)$, we have the
natural identifications,

$\eqalign{H_{\alpha}{\text{Isom}}(S\times X)&:=T_{p}X\oplus T_{q}X,\
\text{and}\cr V_{\alpha}{\text{Isom}}(S\times
X)&:=T_{q}X,\cr}\eqnum{\text{{\rm(3.17)}}\hbox{}}$

so that $T_{\alpha}{\text{Isom}}(S\times X)$ decomposes as

${\text{T}}_{\alpha}{\text{Isom}}(S\times X)=T_{p}S\oplus T_{q}X\oplus
T_{q}X.\eqnum{\text{{\rm(3.18)}}\hbox{}}$

With respect to this decomposition, we define the subbundles $W$ and $V$ by

$\eqalign{W_{\alpha}&:=\left\\{(\xi_{p},\alpha\cdot\xi_{p},\alpha\cdot\nu_{p})\
|\ \xi_{p},\nu_{p}\in T_{p}S\right\\},\ \text{and}\cr
V_{\alpha}&:=\left\\{(0,0,\alpha\cdot\nu_{p})\ |\ \nu_{p}\in
T_{p}S\right\\}.\cr}\eqnum{\text{{\rm(3.19)}}\hbox{}}$

We furnish $W$ with the Sasaki metric $g$, and, given a smooth function
$\phi:{\text{Isom}}(S\times X)\rightarrow\mathbb{R}$, we define the complex
structure $J_{\phi}$ over $W$ by

$J_{\phi,\alpha}\cdot(\xi_{p},\alpha\cdot\xi_{p},\alpha\cdot\nu_{p}):=(e^{\phi}j_{p}\cdot\nu_{p},e^{\phi}\alpha\cdot
j_{p}\cdot\nu_{p},e^{-\phi}\alpha\cdot
j_{p}\cdot\xi_{p}),\eqnum{\text{{\rm(3.20)}}\hbox{}}$

where $j$ here denotes the complex structure of $(S,g)$. Since the orientation
of $S$ yields an orientation of $V$, the quadruplet $(W,g,V,J_{\phi})$ again
defines a Monge–Ampère structure over ${\text{Isom}}(S\times X)$. As before,
we verify that the associated symmetric, bilinear form $m$ is given by

$m_{\alpha}((\xi_{p},\alpha\cdot\xi_{p},\alpha\cdot\nu_{p}),(\xi_{p},\alpha\cdot\xi_{p},\alpha\cdot\nu_{p})):=2\langle\xi_{p},\nu_{p}\rangle.\eqnum{\text{{\rm(3.21)}}\hbox{}}$

We now show that this Monge–Ampère structure is integrable. To this end, let
$\gamma:\mathbb{R}\rightarrow S$ and $\delta:\mathbb{R}\rightarrow X$ be unit
speed parametrised geodesics and let $\alpha:T_{\gamma(0)}S\rightarrow
T_{\delta(0)}X$ be an isometry such that

$\alpha\cdot\dot{\gamma}(0)=\dot{\delta}(0).\eqnum{\text{{\rm(3.22)}}\hbox{}}$

We define
$e_{\gamma,\delta,\alpha}:\mathbb{R}^{2}\rightarrow{\text{Isom}}(S\times X)$
by

$e_{\gamma,\delta,\alpha}(s,t):=(\gamma(s),\delta(s),{\text{Exp}}(t\dot{\delta}(s)\wedge)\cdot\tau_{s}\alpha),\eqnum{\text{{\rm(3.23)}}\hbox{}}$

where, for all $s$, $\tau_{s}$ here denotes parallel transport along
$(\gamma,\delta)$ from $(\gamma(0),\delta(0))$ to $(\gamma(s),\delta(s))$.

Lemma 3.4.1 $(W,V,J_{\phi})$ is integrable, and its curtain surfaces are
quotients of immersed surfaces of the form
$(\mathbb{R}^{2},e_{\gamma,\delta,\alpha})$, with $\gamma$, $\delta$ and
$\alpha$ as above.

Proof: Indeed, with respect to the decomposition (3.18),

$\eqalign{De_{\gamma,\delta,\alpha}(s,t)\cdot\partial_{s}&=(\dot{\gamma}(s),\alpha\cdot\dot{\gamma}(s),0),\
\text{and}\cr
De_{\gamma,\delta,\alpha}(s,t)\cdot\partial_{t}&=(0,0,\alpha\cdot\dot{\gamma}(s)),\cr}\eqnum{\text{{\rm(3.24)}}\hbox{}}$

so that $(\mathbb{R}^{2},e_{\gamma,\delta,\alpha})$ is a curtain surface.
Conversely, given a point $\alpha:=\alpha_{(p,q)}\in{\text{Isom}}(S\times X)$
and a vertical vector $(0,0,\alpha\cdot\xi_{p})$ at this point, let
$\gamma:\mathbb{R}\rightarrow S$ and $\delta:\mathbb{R}\rightarrow X$ denote
the unique unit speed geodesics such that

$\eqalign{\dot{\gamma}(0)&:=\xi_{p},\
\text{and}\cr\dot{\delta}(0)&:=\alpha\cdot\xi_{p}.\cr}$

By (3.24), $(\mathbb{R}^{2},e_{\gamma,\delta,\alpha})$ is a curtain surface
tangent to $(0,0,\alpha\xi_{p})$ at $\alpha_{(p,q)}$. Integrability follows,
and this completes the proof. $\square$It remains only to describe the
geometric properties of positive Monge–Ampère surfaces. Let
$(S^{\prime},g^{\prime})$ be another riemannian surface and consider an
immersion $(e,f):S^{\prime}\rightarrow S\times X$. We say that
$(S^{\prime},(e,f))$ is admissable whenever both $e$ and $f$ are isometries.
When this holds, we define $E_{e,f}:S^{\prime}\rightarrow{\text{Isom}}(S\times
X)$ by

$E_{e,f}:=(e,f,Df\circ De^{-1}).\eqnum{\text{{\rm(3.25)}}\hbox{}}$

We call $(S^{\prime},E_{e,f})$ the Gauss lift of $(S^{\prime},(e,f))$. We say
that $(S^{\prime},(e,f))$ is infinitesimally strictly complex and
quasicomplete whenever $(S^{\prime},f)$ has these properties and, given a
smooth function $\kappa:{\text{Isom}}(S\times X)\rightarrow\mathbb{R}$, we say
that $(S^{\prime},(e,f))$ has extrinsic curvature prescribed by $\kappa$
whenever the extrinsic curvature $K_{f}$ of $f$ satisfies

$K_{f}=\kappa\circ E_{e,f}.\eqnum{\text{{\rm(3.26)}}\hbox{}}$

Lemma 3.4.2 An immersed surface in ${\text{Isom}}(S\times X)$ is a positive
Monge–Ampère surface if and only if it is the Gauss lift of a quasicomplete,
ISC, admissable immersed surface $(S^{\prime},(e,f))$ in $S\times X$ of
extrinsic curvature prescribed by $e^{2\phi}$.

Proof: As in the proof of Lemma 3.3.2, every positive Monge–Ampère surface is
the Gauss lift of some immersed surface. Now let $(S^{\prime},(e,f))$ be an
admissable immersed surface and let $(S^{\prime},E_{e,f})$ denote its Gauss
lift. Let $A_{f}$ denote the shape operator of $f$. Bearing in mind (3.25),
with respect to the decomposition (3.18), the derivative of $E_{e,f}$ at a
point $p\in S^{\prime}$ is given by

$\eqalign{DE_{e,f}(p)\cdot\xi_{p}&=(De(p)\cdot\xi_{p},Df(p)\cdot\xi_{p},Df(p)\cdot
A_{f}(p)\cdot\xi_{p})\cr&=(De(p)\cdot\xi_{p},\alpha\cdot
De(p)\cdot\xi_{p},\alpha\cdot De(p)\cdot A_{f}(p)\cdot\xi_{p}),\cr}$

where $\alpha:=E_{e,f}(p)$. It follows by Lemma 2.5.1 that
$(S^{\prime},E_{e,f})$ is a $J$-holomorphic curve if and only if
$(S^{\prime},(e,f))$ has extrinsic curvature prescribed by $e^{2\phi}$.
Likewise, by (3.21), $(S,\hat{e})$ is positive if and only if $A_{e}$ is
everywhere positive-definite, that is, if and only if $(S^{\prime},(e,f))$ is
ISC. The result now follows. $\square$This construction can be applied to the
study of isometric immersions in a variety of ways. Since the general result
is technical and unenlightening, we shall content ourselves here with an
illustrative example. Let $M:=(M,h)$ be a $3$-dimensional Cartan-Hadamard
manifold, and let $\mathbb{H}^{2}$ denote $2$-dimensional hyperbolic space
with its metric $g$. Let $(\omega_{m})$ be a sequence of bounded, smooth
functions over $\mathbb{H}^{2}$ converging in the $C^{\infty}_{\text{loc}}$
sense to the bounded, smooth function $\omega_{\infty}$. For all $m$, denote
$g_{m}:=e^{2\omega_{m}}g$, $S_{m}:=(\mathbb{H}^{2},g_{m})$,
$X_{m}:={\text{Isom}}(S_{m},M)$, and let $\phi_{m}:X_{m}\rightarrow\mathbb{R}$
be such that, for all $\alpha:=\alpha_{p,q}$,

$e^{2\phi_{m}(\alpha)}=\kappa_{m}(p)-\sigma({\text{Im}}(\alpha)),\eqnum{\text{{\rm(3.27)}}\hbox{}}$

where $\kappa_{m}$ here denotes the curvature of $g_{m}$, and $\sigma$ denotes
the sectional curvature of $h$. For all $m\in\mathbb{N}$, let
$e_{m}:S_{m}\rightarrow M$ be an isometric immersion and denote

$\hat{e}_{m}(x):=(x,e_{m}(x)).\eqnum{\text{{\rm(3.28)}}\hbox{}}$

Note that, for all $m$, $\hat{e}_{m}$ is admissable, quasicomplete, ISC, and
of extrinsic curvature prescribed by $e^{2\omega_{m}}$. Thus, for all $m$, the
Gauss lift $E_{m}$ of $\hat{E}_{m}$ is a $J_{\phi_{m}}$-holomorphic curve. The
theory developed in Sections 3.1 and 3.2 now yields the following result.

Theorem 3.4.3, Labourie Let $x_{0}\in\mathbb{H}^{2}$ be a fixed point. If
$(e_{m}(x_{0}))$ is precompact, then the sequence $(S_{m},E_{m},x_{0})$ of
Gauss lifts is precompact. Furthermore, any accumulation point
$(S_{\infty},E_{\infty},x_{0})$ is either a tube, or the Gauss lift of a
quasicomplete, ISC surface of the form $(\Omega,\hat{e}_{\infty},x_{\infty})$,
for some open subset $\Omega$ of $\mathbb{H}^{2}$.

Remark 3.4.1. Note, in particular, that the domain of the limit immersion need
not be the whole of $\mathbb{H}^{2}$.

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