# Octonionic Calabi-Yau theorem.

Semyon Alesker111Partially supported by ISF grants 865/16 and 743/22 and the
US - Israel BSF grant 2018115.
Department of Mathematics
Tel Aviv University, Ramat Aviv
69978 Tel Aviv, Israel
e-mail<EMAIL_ADDRESS>Peter V. Gordon222Supported in a part by the US
- Israel BSF grant 2020005 and Simons Foundation Collaboration Grant for
Mathematicians 317882.
Department of Mathematical Sciences
Kent State University
Kent, Ohio 44242, USA
e-mail<EMAIL_ADDRESS>
###### Abstract

On a certain class of 16-dimensional manifolds a new class of Riemannian
metrics, called octonionic Kähler, is introduced and studied. It is an
octonionic analogue of Kähler metrics on complex manifolds and of HKT-metrics
of hypercomplex manifolds. Then for this class of metrics an octonionic
version of the Monge-Ampère equation is introduced and solved under
appropriate assumptions. The latter result is an octonionic version of the
Calabi-Yau theorem from Kähler geometry.

###### Contents

  1. 1 Introduction.
  2. 2 Algebra of octonions.
  3. 3 Octonionic projective line $\mathbb{O}\mathbb{P}^{1}$.
  4. 4 Octonionic Hessian.
  5. 5 Octonionic linear algebra.
  6. 6 The groups $SL_{2}(\mathbb{O})$ and $GL_{2}(\mathbb{O})$.
  7. 7 Further properties of the octonionic Hessian.
  8. 8 $G$-affine manifolds.
  9. 9 Octonionic analogue of Kähler metrics.
  10. 10 Integration by parts in $\mathbb{O}^{2}$.
  11. 11 The Monge-Ampère equation and the main result.
  12. 12 Uniqueness of solution.
  13. 13 Zero order estimate.
  14. 14 Elementary inequality.
  15. 15 A few elementary identities.
  16. 16 Diagonalization.
  17. 17 Second order estimates.
  18. 18 A weak Harnack inequality.
  19. 19 An auxiliary estimate for oPSH functions.
  20. 20 A proposition from linear algebra.
  21. 21 $C^{2,\alpha}$-estimate.
  22. 22 Proof of main Theorem 11.3.
  23. 23 Appendix

##  1 Introduction.

  1. 1

Real and complex Monge-Ampère (MA) equations play a central role in several
areas of analysis and geometry. In particular, the celebrated Calabi-Yau
theorem [58] establishes existence of Ricci flat Kähler metrics on compact
Kähler manifolds reducing the problem to solving a complex MA equation. This
result is a powerful analytic tool to study geometry and topology of compact
Kähler manifolds.

  2. 2

There were several attempts to generalize the notion of MA equation and of
plurisubharmonic function to a few other contexts. Namely, the Calabi-Yau
theorem was successfully extended to the more general class of compact
Hermitian manifolds by Tosatti-Weinkove [54] in full generality and previously
by Cherrier [20] in special cases.

Furthermore, the first author [3] introduced and studied the quaternionic MA
operator and the class of quaternionic plurisubharmonic (qpsh) functions on
the flat quaternionic space $\mathbb{H}^{n}$. Later he introduced and solved,
under appropriate assumptions, the Dirichlet problem for quaternionic MA
equation in [4]. The latter problem was solved under weaker assumptions by Zhu
[59] and Kolodziej-Sroka [52].

M. Verbitsky and the first author [12] extended the definition of the
quaternionic MA operator and the class of qpsh functions to the broader class
of hypercomplex manifold; then the first author [8] extended them to even
broader class of quaternionic manifolds. M. Verbitsky and the first author
[13] introduced a quaternionic version of the Calabi problem. So far this
problem is open in full generality, but it was proven in a few special cases,
see [9], [26], [21]. Some relevant a priori estimates were obtained in [13],
[10], [11], [51].

A number of results in quaternionic pluripotential theory were obtained in
[57], [55], [56].

  3. 3

The MA operator and a class of plurisubharmonic functions in two octonionic
variables were introduced by the first author in [7], where also a few
properties of them were established and an application to valuations on convex
sets was given.

  4. 4

At the same time there is a parallel development by Harvey and Lawson, e.g.
[31], [32], [33], [34]. The notion of plurisubharmonic function is developed
in great generality, as well as the theory of homogeneous MA equation. Being
specialized to the quaternionic situation, this theory has an overlap with
some of the results mentioned above.

  5. 5

Let us describe the main results of this paper.

As mentioned above, the MA operator in two octonionic variables was introduced
by the first author in [7]. Based on this operator, the present paper
introduces and studies the octonionic MA equation analogous to the Calabi
problem in Kähler geometry.

The MA equation is stated for a special class of 16-dimensional manifolds with
a kind of octonionic structure. We call such manifolds
$GL_{2}(\mathbb{O})$-manifolds, they are introduced in Section 8. Basic
examples of such manifolds are torii $\mathbb{O}^{2}/\Lambda$, where
$\mathbb{O}^{2}\simeq\mathbb{R}^{16}$ is the octonionic plane, and
$\Lambda\subset\mathbb{O}^{2}$ is a lattice.

Furthermore in Section 9 we introduce on any $GL_{2}(\mathbb{O})$-manifold a
class of Riemannian metrics which are octonionic analogues of Kähler metrics.
We call them octonionic Kähler metrics. In local coordinates, i.e. on
$\mathbb{O}^{2}$, such metrics are characterized by two conditions. First, the
restriction of each such metric on any (right) octonionic line is proportional
to the standard Euclidean metric on $\mathbb{O}^{2}\simeq\mathbb{R}^{16}$.
Second, such metrics satisfy a system of linear first order PDEs (see (9.1))
analogous to the closeness of Kähler form on a complex manifold. We also prove
an analogue of the local $dd^{c}$-lemma from complex analysis (see Theorem
9.2): locally any such metric can be given by a smooth potential $u$, i.e. the
metric can be identified with the octonionic Hessian of $u$. The converse is
also true provided the Hessian of $u$ is pointwise positive definite.

Next we introduce an octonionic MA equation on a $GL_{2}(\mathbb{O})$-manifold
$M$. Let $G_{0}$ be an octonionic Kähler ($C^{\infty}$-smooth) metric; in
local coordinates it can be identified with a positive definite octonionic
Hermitian matrix. Let $f\colon M\longrightarrow\mathbb{R}$ be a
$C^{\infty}$-smooth function. Consider the equation which in local coordinates
reads

$\displaystyle\det(G_{0}+Hess_{\mathbb{O}}(\phi))=e^{f}\det(G_{0}),$ (1.1)

where $Hess_{\mathbb{O}}$ is the octonionic Hessian (see Section 4 for the
precise definition), $\det$ is the determinant of octonionic Hermitian
$2\times 2$ matrices (see (5.10)). The main result of this paper, Theorem
11.3, says that on the subclass of compact connected so called
$Spin(9)$-manifolds (example: tori $\mathbb{O}^{2}/\Lambda$ where
$\Lambda\subset\mathbb{O}^{2}$ is a lattice) there exists a
$C^{\infty}$-smooth solution $\phi$ of (1.1) if and only if the function $f$
satisfies the normalization condition

$\int_{M}(e^{f}-1)\det(G_{0})\cdot\Theta=0,$

where $\Theta$ is a parallel non-vanishing 3/4-density on $M$ (it always
exists on $Spin(9)$-manifolds and is unique up to multiplication by a
constant).

  6. 6

The class of metrics on $GL_{2}(\mathbb{O})$-manifolds we mentioned is
analogous not only to Kähler metrics on complex manifolds, but also to Hessian
metrics on affine manifolds [19], and to HKT-metrics on hypercomplex manifolds
[36], [28]. Our main result on the solvability of the MA equation is analogous
to and is motivated by the Calabi-Yau theorem [58] for Kähler manifolds. Also
it is analogous to the Cheng-Yau theorem [19] for Hessian manifolds, and to
the similar conjecture on quaternionic MA equation on HKT-manifolds due to M.
Verbitsky and the author [13] which has been proven so far in a few special
cases mentioned above.

  7. 7

In literature there are quite a few investigations of geometric structures
related to octonions. Thus Gibbons, Papadopoulos, and Stelle [24] introduced a
class of Octonionic Kähler with Torsion metrics on $8k$-dimensional manifolds
in connection with black hole moduli spaces.

Friedrich [23] introduced the notion of nearly parallel $Spin(9)$-structure on
16-manifolds whose structure group is reduced to $Spin(9)$. The role of
$Spin(9)$ in octonionic geometry is reviewed by Parton and Piccinni [47] (see
also [38]).

Very recently Kotrbatý and Wannerer [39] described the multiplicative
structure on $Spin(9)$-invariant valuations on convex sets on $\mathbb{O}^{2}$
and gave applications to integral geometry.

  8. 8

Acknowledgements. We thank F. Nazarov and M. Sodin for useful discussions and
G. Grantcharov, G. Papadopoulos, I. Soprunov, and A. Swann for supplying us
with some references.

##  2 Algebra of octonions.

  1. 1

The octonions $\mathbb{O}$ form an 8-dimensional algebra over the reals
$\mathbb{R}$ which is neither associative nor commutative. The product on
$\mathbb{O}$ can be described as follows. $\mathbb{O}$ has a basis
$e_{0},e_{1},\dots,e_{7}$ over $\mathbb{R}$ where $e_{0}=1$ is the unit, and
the product of basis elements is given by the following multiplication table:

$\begin{array}[]{c|c|c|c|c|c|c|c|}&e_{1}&e_{2}&e_{3}&e_{4}&e_{5}&e_{6}&e_{7}\\\
\hline\cr e_{1}&-1&e_{4}&e_{7}&-e_{2}&e_{6}&-e_{5}&-e_{3}\\\ \hline\cr
e_{2}&-e_{4}&-1&e_{5}&e_{1}&-e_{3}&e_{7}&-e_{6}\\\ \hline\cr
e_{3}&-e_{7}&-e_{5}&-1&e_{6}&e_{2}&-e_{4}&e_{1}\\\ \hline\cr
e_{4}&e_{2}&-e_{1}&-e_{6}&-1&e_{7}&e_{3}&-e_{5}\\\ \hline\cr
e_{5}&-e_{6}&e_{3}&-e_{2}&-e_{7}&-1&e_{1}&e_{4}\\\ \hline\cr
e_{6}&e_{5}&-e_{7}&e_{4}&-e_{3}&-e_{1}&-1&e_{2}\\\ \hline\cr
e_{7}&e_{3}&e_{6}&-e_{1}&e_{5}&-e_{4}&-e_{2}&-1\\\ \hline\cr\end{array}$

  2. 2

There is also an easier alternative way to reproduce the product rule using
the so called Fano plane (see Figure 1 below). In Figure 1 each pair of
distinct points lies in a unique line (the circle is also considered to be a
line). Each line contains exactly three points, and these points are
cyclically oriented. If $e_{i},e_{j},e_{k}$ are cyclically oriented in this
way then

$e_{i}e_{j}=-e_{j}e_{i}=e_{k}.$

We have to add two more rules:

$\bullet$ $e_{0}=1$ is the identity element;

$\bullet$ $e_{i}^{2}=-1$ for $i>0$.

All these rules define uniquely the algebra structure of $\mathbb{O}$. The
center of $\mathbb{O}$ is equal to $\mathbb{R}$.

Figure 1:  Fano plane, the figure is taken from [16]

  3. 3

Every octonion $q\in\mathbb{O}$ can be written uniquely in the form

$q=\sum_{i=0}^{7}x_{i}e_{i}$

where $x_{i}\in\mathbb{R}$. The summand $x_{0}e_{0}=x_{0}$ is called the real
part of $q$ and is denoted by $Re(q)$.

One defines the octonionic conjugate of $q$ by

$\bar{q}:=x_{0}-\sum_{i=1}^{7}x_{i}e_{i}.$

It is well known that the conjugation is an anti-involution of $\mathbb{O}$:

$\overline{\overline{q}}=q,\,\overline{a+b}=\bar{a}+\bar{b},\,\overline{ab}=\bar{b}\bar{a}.$

Let us define a norm on $\mathbb{O}$ by $|q|:=\sqrt{q\bar{q}}$. Then $|\cdot|$
is a multiplicative norm on $\mathbb{O}$: $|ab|=|a||b|$. The square of the
norm $|\cdot|^{2}$ is a positive definite quadratic form. Its polarization is
a positive definite scalar product $<\cdot,\cdot>$ on $\mathbb{O}$ which is
given explicitly by

$<x,y>=Re(x\bar{y}).$

Furthermore $\mathbb{O}$ is a division algebra: any $q\neq 0$ has a unique
inverse $q^{-1}$ such that $qq^{-1}=q^{-1}q=1$. In fact

$q^{-1}=|q|^{-2}\bar{q}.$

  4. 4

We denote by $\mathbb{H}$ the usual quaternions. It is associative division
algebra. We will fix once and for all an imbedding of algebras
$\mathbb{H}\subset\mathbb{O}$. Let us denote by $i,j\in\mathbb{H}$ the usual
quaternionic units, and $k=ij$. Then $i,j,k$ are pairwise orthogonal with
respect to the scalar product $<\cdot,\cdot>$. Let us fix once and for all an
octonionic unit $l\in\mathbb{O},l^{2}=-1$ which is orthogonal to $i,j,k$. Then
$l$ anti-commutes with $i,j,k$. Every element $q\in\mathbb{O}$ can be written
uniquely in the form

$q=x+yl\mbox{ with }x,y\in\mathbb{H}.$

The multiplication of two such octonions is given by the formula

$\displaystyle(x+yl)(w+zl)=(xw-\bar{z}y)+(zx+y\bar{w})l$ (2.1)

where $x,y,w,z\in\mathbb{H}$.

  5. 5

We have the following weak forms of the associativity in octonions. A proof
can be found in [7], Lemma 1.1.1.

######  2.1 Lemma.

Let $a,b,c\in\mathbb{O}$. Then

(i) $Re((ab)c)=Re(a(bc))$ (this real number will be denoted by $Re(abc)$).

(ii) $a(bc)+\bar{b}(\bar{a}c)=(ab+\bar{b}\bar{a})c$.

(iii) $(ca)b+(c\bar{b})\bar{a}=c(ab+\bar{b}\bar{a})$.

(iv) Any subalgebra of $\mathbb{O}$ generated by any two elements and their
conjugates is associative. It is always isomorphic either to $\mathbb{R}$,
$\mathbb{C}$, or $\mathbb{H}$.

(v) $Re((\bar{a}b)(ca))=|a|^{2}Re(bc).$

The following identities are called Moufang identities (see e.g. [48]).

######  2.2 Lemma.

Let $x,y,z\in\mathbb{O}$. Then

$\displaystyle(zx)(yz)=z((xy)z)=(z(xy))z.$

Polarizing the Moufang identities we immediately get a corollary.

######  2.3 Corollary.

Let $x,y,z_{1},z_{2}\in\mathbb{O}$. Then

$\displaystyle(z_{1}x)(yz_{2})+(z_{2}x)(yz_{1})=z_{1}((xy)z_{2})+z_{2}((xy)z_{1})=(z_{1}(xy))z_{2}+(z_{2}(xy))z_{1}.$

##  3 Octonionic projective line $\mathbb{O}\mathbb{P}^{1}$.

  1. 1

The results of this section are well known and elementary, we provide them for
completeness of presentation.

######  3.1 Definition.

An octonionic (right) line in $\mathbb{O}^{2}$ is a real 8-dimensional
subspace $L\subset\mathbb{O}^{2}$ either of the form
$\\{(q,aq)|q\in\mathbb{O}\\}$ with arbitrary $a\in\mathbb{O}$, or of the form
$\\{(0,q)|q\in\mathbb{O}\\}$.

The latter option is considered as the line at infinity. The set of all
octonionic lines is denoted by $\mathbb{O}\mathbb{P}^{1}$. It is easy to see
that it is a compact subset of the Grassmannian of 8-dimensional linear
subspaces of $\mathbb{R}^{16}$. Moreover it is a smooth submanifold
diffeomorphic to the 8-sphere $\SS^{8}$: the line at infinity corresponds to
the pole of this sphere, and the rest is parameterized by octonions
$a\in\mathbb{O}$ is diffeomorphic to $\mathbb{R}^{8}$.

  2. 2

Let us summarize a few other basic and well known properties of
$\mathbb{O}\mathbb{P}^{1}$.

######  3.2 Lemma.

(i) For any $a\in\mathbb{O}$ the set $\\{(aq,q)|q\in\mathbb{O}\\}$ is also an
octonionic line.
(ii) Any two octonionic lines either intersect trivially or coincide.
(iii) For any non-zero vector $\xi\in\mathbb{O}^{2}$ there is a unique
octonionic line containing $\xi$. We call it the line spanned by $\xi$.
(iv) Let $\xi,\eta\in\mathbb{O}$ be two vectors of norm one. The lines spanned
by these vectors are equal if and only if one has for the $2\times 2$-matrices

$\xi\cdot\xi^{*}=\eta\cdot\eta^{*}.$

In this case we write $\xi\sim\eta$.

The proof of this lemma is elementary and straightforward.

##  4 Octonionic Hessian.

In this section we remind the octonionic Hessian introduced in [7]. For
$i=1,2$ write

$q_{i}=\sum_{a=0}^{7}x_{i}^{a}e_{a}\mbox{ with }x_{i}^{a}\in\mathbb{R}.$

Let $F\colon\mathbb{O}^{2}\longrightarrow\mathbb{O}$ be a sufficiently smooth
function. Define for $i,j=1,2$

$\displaystyle\frac{\partial
F}{\partial\bar{q}_{i}}:=\sum_{a=0}^{7}e_{a}\frac{\partial F}{\partial
x_{i}^{a}},$ $\displaystyle\frac{\partial F}{\partial
q_{j}}:=\sum_{a=0}^{7}\frac{\partial F}{\partial x_{j}^{a}}\bar{e}_{a}.$

In general the operators $\frac{\partial}{\partial\bar{q}_{i}}$ and
$\frac{\partial}{\partial q_{j}}$ do not commute. However, they do commute
when applied to real valued functions.

Let $f\colon\mathbb{O}^{2}\longrightarrow\mathbb{R}$ be a $C^{2}$-smooth real
valued function. Define its octonionic Hessian by

$\displaystyle
Hess_{\mathbb{O}}(f):=\left(\frac{\partial^{2}f}{\partial\bar{q}_{i}\partial
q_{j}}\right)_{i,j=1,2},$ (4.1)

where the operators $\frac{\partial}{\partial\bar{q}_{i}}$ and
$\frac{\partial}{\partial q_{j}}$ are applied in either order. Then
$Hess_{\mathbb{O}}(f)$ is an octonionic Hermitian $2\times 2$ matrix.

##  5 Octonionic linear algebra.

  1. 1

Let us denote

$\mathbb{O}^{m}:=\\{(q_{1},\dots,q_{m})|\,q_{i}\in\mathbb{O}\\}.$

For $\xi=(q_{1},\dots,q_{m})\in\mathbb{O}^{m}$, $x\in\mathbb{O}$ we will
denote by $\xi\cdot x$ the $m$-tuple $(q_{1}x,\dots,q_{m}x)\in\mathbb{O}^{m}$.
Notice that usually we will write elements of $\mathbb{O}^{m}$ as $m$-columns
rather than rows.

Let us denote by ${\cal H}_{n}(\mathbb{R})$ the space of real symmetric
$(n\times n)$-matrices. The space ${\cal H}_{n}(\mathbb{R})$ is naturally
identified with the space of real valued quadratic forms on $\mathbb{R}^{n}$.

Let us denote by ${\cal H}_{m}(\mathbb{O})$ the space of octonionic Hermitian
$(m\times m)$-matrices. By definition, an $(m\times m)$-matrix $A=(a_{ij})$
with octonionic entries is called Hermitian if $a_{ij}=\overline{a_{ji}}$ for
any $i,j$. For a matrix $A=(a_{ij})$ denote also $A^{*}:=(\overline{a_{ji}})$.
In what follows we will be only interested in $2\times 2$ octonionic Hermitian
matrices. In this case we have the following explicit description of such
matrices. Namely,

$\displaystyle{\cal H}_{2}(\mathbb{O})=\left\\{\left[\begin{array}[]{cc}a&q\\\
\bar{q}&b\end{array}\right]\big{|}a,b\in\mathbb{R},q\in\mathbb{O}\right\\}.$
(5.3)

  2. 2

We have the natural $\mathbb{R}$-linear map

$\displaystyle j\colon{\cal H}_{2}(\mathbb{O})\longrightarrow{\cal
H}_{16}(\mathbb{R}),$ (5.4)

which is defined as follows: for any $A\in{\cal H}_{2}(\mathbb{O})$ the value
of the quadratic form $j(A)$ on any octonionic 2-column
$\xi\in\mathbb{O}^{2}=\mathbb{R}^{16}$ is given by $j(A)(\xi)=Re(\xi^{*}A\xi)$
(note that the bracketing inside the right hand side of the formula is not
important due to Lemma 2.1(i)). It is easy to see that the map $j$ is
injective. Via this map $j$ we will identify ${\cal H}_{2}(\mathbb{O})$ with a
subspace of ${\cal H}_{16}(\mathbb{R})$.

  3. 3

Let us construct now a linear map

$\theta\colon{\cal H}_{16}(\mathbb{R})\longrightarrow{\cal H}_{2}(\mathbb{O})$

in the opposite direction and such that $\theta\circ j=Id$; it will be useful
later. For any $B\in{\cal H}_{16}(\mathbb{R})$ let us denote by $b$ the
corresponding quadratic form on $\mathbb{R}^{16}=\mathbb{O}^{2}$, namely

$b(x)=\sum_{i,j=1}^{16}x_{i}B_{ij}x_{j}.$

Define

$\displaystyle\theta(B):=\frac{1}{16}Hess_{\mathbb{O}}(b)=\frac{1}{16}\left(\frac{\partial^{2}b}{\partial\bar{q}_{i}\partial
q_{j}}\right)_{i,j=1}^{2}.$ (5.5)

Note that the matrix in the right hand side of the last formula is independent
of a point in $\mathbb{O}^{2}$.

The next lemma is proved in [7], Corollary 1.2.2, but we present a proof for
the sake of completeness.

######  5.1 Lemma.

The following identity holds.

$\theta\circ j=Id.$

Proof. Let $B=\left[\begin{array}[]{cc}a&q\\\ \bar{q}&b\end{array}\right]$
where $a,b\in\mathbb{R},\,q\in\mathbb{O}$. Below we denote
$\Delta_{i}=\sum_{a=0}^{7}\frac{\partial^{2}}{\partial(x_{i}^{a})^{2}}$. We
have

$\displaystyle(\theta\circ
j)(B)=\frac{1}{16}\left(Hess_{\mathbb{O}}(Re(\xi^{*}B\xi)\right)=$
$\displaystyle\frac{1}{16}Hess_{\mathbb{O}}\left(a|\xi_{1}|^{2}+b|\xi_{2}|^{2}+2Re(\bar{\xi}_{1}q\xi_{2})\right)=$
$\displaystyle\frac{1}{16}\left[\begin{array}[]{cc}a\Delta_{1}(|\xi_{1}|^{2})&2\frac{\partial^{2}}{\partial\bar{\xi}_{1}\partial\xi_{2}}Re(\bar{\xi}_{1}q\xi_{2})\\\
2\frac{\partial^{2}}{\partial\bar{\xi}_{2}\partial\xi_{1}}Re(\bar{\xi}_{1}q\xi_{2})&b\Delta_{2}(|\xi_{2}|^{2})\end{array}\right]=$
$\displaystyle\left[\begin{array}[]{cc}a&\frac{1}{8}\frac{\partial^{2}}{\partial\bar{\xi}_{1}\partial\xi_{2}}Re(\bar{\xi}_{1}q\xi_{2})\\\
\frac{1}{8}\frac{\partial^{2}}{\partial\bar{\xi}_{2}\partial\xi_{1}}Re(\bar{\xi}_{1}q\xi_{2})&b\end{array}\right].$

We have to show that the term above the diagonal equals $q$ (the term under
the diagonal is its conjugate). For

$\displaystyle\frac{1}{8}\frac{\partial^{2}}{\partial\bar{\xi}_{1}\partial\xi_{2}}Re(\bar{\xi}_{1}q\xi_{2})=\sum_{a,b=0}^{7}e_{a}\left(\frac{\partial^{2}}{\partial
x_{1}^{a}\partial x_{2}^{b}}Re(\bar{\xi}_{1}\cdot
q\cdot\xi_{2})\right)\bar{e}_{b}=$
$\displaystyle\frac{1}{8}\sum_{a,b,c,d=0}^{7}e_{a}\left(\frac{\partial^{2}}{\partial
x_{1}^{a}\partial x_{2}^{b}}Re(x_{1}^{c}\bar{e}_{c}\cdot q\cdot
x_{2}^{d}e_{d})\right)\bar{e}_{b}=$
$\displaystyle\frac{1}{8}\sum_{a,b,c,d=0}^{7}e_{a}\left(Re(\delta_{ac}\bar{e}_{c}\cdot
q\cdot\delta_{bd}e_{d})\right)\bar{e}_{b}=$
$\displaystyle\frac{1}{8}\sum_{a,b=0}^{7}e_{a}(Re(\bar{e}_{a}qe_{b}))\bar{e}_{b}=\frac{1}{8}\sum_{b}\left(\sum_{a}e_{a}(Re(\bar{e}_{a}(qe_{b})))\right)\bar{e}_{b}=$
$\displaystyle\frac{1}{8}\sum_{b}(qe_{b})\bar{e}_{b}=\frac{1}{8}\sum_{b}q(e_{b}\bar{e}_{b})=q.$

Lemma follows. Q.E.D.

  4. 4

There is a useful notion of positive definite octonionic Hermitian matrix.

######  5.2 Definition.

Let $A\in{\cal H}_{2}(\mathbb{O})$. $A$ is called positive definite (resp.
non-negative definite) if for any 2-column
$\xi\in\mathbb{O}^{2}\backslash\\{0\\}$

$Re(\xi^{*}A\xi)>0\,\,(\mbox{resp. }Re(\xi^{*}A\xi)\geq 0).$

For a positive definite (resp. non-negative definite) matrix $A$ one writes as
usual $A>0$ (resp. $A\geq 0$).

  5. 5

On the class of octonionic Hermitian $2\times 2$-matrices there is a nice
notion of determinant which is defined by

$\displaystyle\det\left(\left[\begin{array}[]{cc}a&q\\\
\bar{q}&b\end{array}\right]\right):=ab-|q|^{2}.$ (5.10)

######  5.3 Remark.

It turns out that a nice notion of determinant does exist also on octonionic
Hermitian matrices of size 3, see e.g. Section 3.4 of [16]. Note also that a
nice notion of determinant does exist for quaternionic Hermitian matrices of
any size: see the survey [14], the article [25], and for applications to
quaternionic plurisubharmonic functions see [3], [4], [5], [6].

  6. 6

The following result is a version of the Sylvester criterion for octonionic
matrices of size two, see [7], Proposition 1.2.5.

######  5.4 Proposition.

Let $A=\left[\begin{array}[]{cc}a&q\\\ \bar{q}&b\end{array}\right]\in{\cal
H}_{2}(\mathbb{O})$. Then $A>0$ if and only if $a>0$ and $\det A>0$.

  7. 7

Now we remind the notion of mixed determinant of two octonionic Hermitian
matrices in analogy to the classical real case (see e.g. [49]). First observe
that the determinant $\det{\cal H}_{2}(\mathbb{O})\longrightarrow\mathbb{R}$
is a homogeneous polynomial of degree 2 on the real vector space ${\cal
H}_{2}(\mathbb{O})$. Hence it admits a unique polarization: a bilinear
symmetric map

$D\colon{\cal H}_{2}(\mathbb{O})\times{\cal
H}_{2}(\mathbb{O})\longrightarrow\mathbb{R}$

such that $D(A,A)=\det A$ for any $A\in{\cal H}_{2}(\mathbb{O})$. This map $D$
is called the mixed determinant. By the abuse of notation it will be denoted
again by $\det$. Explicitly if $A=(a_{ij})_{i,j=1}^{2}$,
$B=(b_{ij})_{i,j=1}^{2}$ are octonionic Hermitian matrices then

$\displaystyle\det(A,B)=\frac{1}{2}\left(a_{11}b_{22}+a_{22}b_{11}-2Re(a_{12}b_{21})\right).$
(5.11)

######  5.5 Lemma.

If $A,B\in{\cal H}_{2}(\mathbb{O})$ are positive (resp. non-negative) definite
then

$\det(A,B)>0\,\,(\mbox{resp. }\det(A,B)\geq 0).$

Proof. Let us assume that $A,B$ are positive definite. Then by Proposition 5.4
we have

$\displaystyle a_{11}>0,a_{22}>0,|a_{12}|<\sqrt{a_{11}a_{22}},$ $\displaystyle
b_{11}>0,b_{22}>0,|b_{12}|<\sqrt{b_{11}b_{22}}.$

These inequalities imply

$\displaystyle
Re(a_{12}b_{21})\leq|a_{12}|\cdot|b_{21}|<\sqrt{a_{11}a_{22}b_{11}b_{22}}.$
(5.12)

Substituting (5.12) into (5.11) we get

$\displaystyle\det(A,B)>\frac{1}{2}\left(a_{11}b_{22}+a_{22}b_{11}-2\sqrt{a_{11}a_{22}b_{11}b_{22}}\right)\geq
0$

where the last estimate is the arithmetic-geometric mean inequality. Thus
$\det(A,B)>0$ for positive definite matrices $A,B$. For non-negative definite
matrices the result follows by approximtion. Q.E.D.

######  5.6 Remark.

One can also prove the following version of the Aleksandrov inequality for
mixed determinants [2]: The mixed determinant $\det(\cdot,\cdot)$ is a non-
degenerate quadratic form on the real vector space ${\cal H}_{2}(\mathbb{O})$;
its signature type has one plus and the rest are minuses. Consequently, if
$A>0$ then for any $B\in{\cal H}_{2}(\mathbb{O})$ one has

$\det(A,B)^{2}\geq\det A\cdot\det B$

and the equality holds if and only if $A$ is proportional to $B$ with a real
coefficient.

We do not present a detailed proof of this result since we are not going to
use it. Notice only that this result can be deduced formally from the
corresponding result for quaternionic matrices proved in [3]. Indeed all the
entries of $A$ and $B$ together contain at most two non-real octonions, hence
the field generated by them is associative by Lemma 2.1(iv).

##  6 The groups $SL_{2}(\mathbb{O})$ and $GL_{2}(\mathbb{O})$.

  1. 1

We recall in this section the definitions and basic properties of the groups
$SL_{2}(\mathbb{O})$ and $GL_{2}(\mathbb{O})$. We refer to [53] for the proofs
and further details.

  2. 2

An octonionic $2\times 2$-matrix is called traceless if the sum of its
diagonal elements is equal to zero. Every $2\times 2$-matrix $A$ with
octonionic entries defines an $\mathbb{R}$-linear operator on $\mathbb{O}^{2}$
by $\xi\mapsto A\cdot\xi$. However, the space of such operators is not closed
under the commutator due to lack of associativity. One denotes by
$\mathfrak{sl}_{2}(\mathbb{O})$ the Lie subalgebra of
$\mathfrak{gl}_{16}(\mathbb{R})$ generated by $\mathbb{R}$-linear operators on
$\mathbb{O}^{2}\simeq\mathbb{R}^{16}$ determined by all traceless octonionic
matrices. This Lie algebra $\mathfrak{sl}_{2}(\mathbb{O})$ turns our to be
semi-simple [53] (see Theorem 6.1 below for details). Note that any semi-
simple Lie subalgebra of an algebraic group is a Lie algebra of a closed
algebraic subgroup (see e.g. [44], Ch. 3, §3.3). In our case this subgroup is
denoted by $SL_{2}(\mathbb{O})\subset GL(16,\mathbb{R})$.

######  6.1 Theorem ([53]).

(i) The Lie algebra $\mathfrak{sl}_{2}(\mathbb{O})$ is isomorphic to the Lie
algebra $\mathfrak{so}(9,1)$.

(ii) The Lie group $SL_{2}(\mathbb{O})$ is isomorphic to the group $Spin(9,1)$
(which is the universal covering of the identity component of the pseudo-
orthogonal group $O(9,1)$).

######  6.2 Remark.

A maximal compact subgroup of $SL_{2}(\mathbb{O})\simeq Spin(9,1)$ is
isomorphic to the group $Spin(9)$ which is the universal covering of the
special orthogonal group $SO(9)$.

######  6.3 Remark.

In fact $SL_{2}(\mathbb{O})\subset SL_{16}(\mathbb{R})$. Indeed since the Lie
algebra $\mathfrak{sl}_{2}(\mathbb{O})$ is semi-simple, it is equal to its own
commutant. Hence
$\mathfrak{sl}_{2}(\mathbb{O})\subset\mathfrak{sl}_{16}(\mathbb{R})$. Since
$SL_{2}(\mathbb{O})$ is connected it follows that $SL_{2}(\mathbb{O})\subset
SL(16,\mathbb{R})$.

  3. 3

Let us denote by
$\mathfrak{gl}_{2}(\mathbb{O}):=\mathbb{R}\oplus\mathfrak{sl}_{2}(\mathbb{O})\subset\mathfrak{gl}_{16}(\mathbb{R})$.
It is the Lie algebra of the Lie group
$GL_{2}(\mathbb{O}):=\mathbb{R}^{\ast}\cdot SL_{2}(\mathbb{O})$ which is the
group generated by the group of homotheties $\mathbb{R}^{\ast}$ and
$SL_{2}(\mathbb{R})$. $GL_{2}(\mathbb{O})$ is connected since $-I_{2}\in
SL_{2}(\mathbb{O})$ (the latter follows from Lemma 14.77 in [30]).

  4. 4

$\mathfrak{sl}_{2}(\mathbb{O}),\mathfrak{gl}_{2}(\mathbb{O})$ and
$SL_{2}(\mathbb{O}),GL_{2}(\mathbb{O})$ come with their fundamental
representations on $\mathbb{O}^{2}=\mathbb{R}^{16}$.

Furthermore the Lie algebra $\mathfrak{sl}_{2}(\mathbb{O})$ acts on the space
${\cal H}_{2}(\mathbb{O})$. This action is uniquely characterized by the
following property (see [53] for the details): if $A$ is a traceless matrix it
acts by $A\colon X\mapsto-A^{*}X-XA$. Since the group $SL_{2}(\mathbb{O})$ is
connected and simply connected, this representation of
$\mathfrak{sl}_{2}(\mathbb{O})$ integrates to a representation of the group
$SL_{2}(\mathbb{O})$ on ${\cal H}_{2}(\mathbb{O})$.

This action naturally extends to $\mathfrak{gl}_{2}(\mathbb{O})$: for
$\lambda\in\mathbb{R}\in\mathfrak{gl}_{2}(\mathbb{O})$ one has

$\lambda(X)=-2\lambda X.$

The latter action integrates to the representation of $GL_{2}(\mathbb{O})$ on
${\cal H}_{2}(\mathbb{O})$ such that for any $\lambda\in\mathbb{R}^{\ast}$

$\lambda(X)=\lambda^{-2}X.$

This representation of $GL_{2}(\mathbb{O})$ in ${\cal H}_{2}(\mathbb{O})$ will
be denoted by $\rho$.

  5. 5

The following result is well known, see e.g., in [7], Proposition 1.4.3.

######  6.4 Proposition.

The group $GL_{2}(\mathbb{O})$ preserves the cone of positive definite
octonionic Hermitian matrices.

  6. 6

Let us identify the (real) dual space to $\mathbb{O}^{2}=\mathbb{R}^{16}$ with
itself using the standard inner product
$\mathbb{O}^{2}\times\mathbb{O}^{2}\longrightarrow\mathbb{R}$ which can be
described as follows

$\displaystyle\langle\xi,\eta\rangle=Re(\xi^{*}\cdot\eta).$ (6.1)

Let $Mat_{2\times 2}(\mathbb{O})$ denote the space of square octonionic
matrices of size 2.

######  6.5 Lemma.

(i) The (Euclidean) dual operator to the operator $\xi\mapsto A\xi$, where
$\xi\in\mathbb{O}^{2}$ and $A\in Mat_{2\times 2}(\mathbb{O})$ is given by

$\eta\mapsto A^{*}\eta.$

(ii) The involutive automorphism of the group $GL_{16}(\mathbb{R})$ given by
$X\mapsto(X^{t})^{-1}$ preserves the subgroup $GL_{2}(\mathbb{O})$. Its
restriction to this subgroup will be denoted by

$\mathcal{S}\colon
GL_{2}(\mathbb{O})\tilde{\longrightarrow}GL_{2}(\mathbb{O}).$

Proof. (i) We have

$\displaystyle\langle\xi,A\eta\rangle:=Re(\xi^{*}A\eta)=Re((A^{*}\xi)^{*}\eta)=\langle
A^{*}\xi,\eta\rangle.$

(ii) Recall that $GL_{2}(\mathbb{O})=\mathbb{R}^{*}\cdot SL_{2}(\mathbb{O})$.
Clearly for $\lambda\in\mathbb{R}^{*}$ one has ${\cal
S}(\lambda)=\lambda^{-1}\in GL_{2}(\mathbb{O})$. Now it suffices to show that
${\cal S}$ preserves the Lie algebra $\mathfrak{sl}_{2}(\mathbb{O})$ of
$SL_{2}(\mathbb{O})$.

The action of ${\cal S}$ on the Lie algebra $\mathfrak{gl}_{16}(\mathbb{R})$
is given by ${\cal S}(X)=-X^{t}$. The Lie algebra
$\mathfrak{sl}_{2}(\mathbb{O})$ is generated by transformations of the form
$\hat{A}(\xi)=A\xi$, where $A\in Mat_{2\times 2}(\mathbb{O})$ is traceless,
i.e. the sum of its diagonal entries vanishes. Then, by part (i) one has

${\cal S}(\hat{A})(\xi)=-A^{*}\xi.$

Thus ${\cal S}(\hat{A})\in\mathfrak{sl}_{2}(\mathbb{O})$, and the lemma
follows. Q.E.D.

  7. 7

The following lemma is essentially contained in [41], and we present a proof
for convenience of the reader.

######  6.6 Lemma ([41]).

The linear map ${\cal T}\colon
Sym_{\mathbb{R}}^{2}(\mathbb{O}^{2})\longrightarrow{\cal H}_{2}(\mathbb{O})$
given by

${\cal
T}(\xi\otimes\eta)\mapsto\frac{1}{2}(\xi\cdot\eta^{*}+\eta\cdot\xi^{*})$

is $GL_{2}(\mathbb{O})$-equivariant, where the action of $GL_{2}(\mathbb{O})$
on the source space is the second symmetric power of the representation ${\cal
S}$, and its action on the target is the representation $\rho$.

Proof. The equivariance of the map under $\mathbb{R}^{*}\subset
GL_{2}(\mathbb{O})$ is evident. It suffices to prove the equivariance under
the action of the Lie algebra $\mathfrak{sl}_{2}(\mathbb{O})$. Moreover it
suffices to check that for a set of generators of the Lie algebra
$\mathfrak{sl}_{2}(\mathbb{O})$. For let $A\in Mat_{2\times 2}(\mathbb{O})$ be
a traceless matrix. It suffices to show that for any traceless $A\in
Mat_{2\times 2}(\mathbb{O})$ and any 2-column $\xi\in\mathbb{O}^{2}$ one has
the equality of $2\times 2$ matrices

$\displaystyle(A^{*}\xi)\xi^{*}+\xi(A^{*}\xi)^{*}=A^{*}(\xi\xi^{*})+(\xi\xi^{*})A.$
(6.2)

Let us write explicitly

$\displaystyle\xi=\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right],\,\,\,A=\left[\begin{array}[]{cc}a&b\\\
c&-a\end{array}\right].$

The left hand side of (6.2) is then equal to

$\displaystyle lhs=\left(\left[\begin{array}[]{cc}\bar{a}&\bar{c}\\\
\bar{b}&-\bar{a}\end{array}\right]\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right]\right)[\bar{\xi}_{1},\bar{\xi}_{2}]+\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right]\left([\bar{\xi}_{1},\bar{\xi}_{2}]\left[\begin{array}[]{cc}a&b\\\
c&-a\end{array}\right]\right)=$
$\displaystyle\left[\begin{array}[]{cc}(\bar{a}\xi_{1}+\bar{c}\xi_{2})\bar{\xi}_{1}&(\bar{a}\xi_{1}+\bar{c}\xi_{2})\bar{\xi}_{2}\\\
(\bar{b}\xi_{1}-\bar{a}\xi_{2})\bar{\xi}_{1}&(\bar{b}\xi_{1}-\bar{a}\xi_{2})\bar{\xi}_{2}\end{array}\right]+\left[\begin{array}[]{cc}\xi_{1}(\bar{\xi}_{1}a+\bar{\xi}_{2}c)&\xi_{1}(\bar{\xi}_{1}b-\bar{\xi}_{2}a)\\\
\xi_{2}(\bar{\xi}_{1}a+\bar{\xi}_{2}c)&\xi_{2}(\bar{\xi}_{1}b-\bar{\xi}_{2}a)\end{array}\right]=$
$\displaystyle\left[\begin{array}[]{cc}2|\xi_{1}|^{2}Re(a)+2Re(\xi_{1}\bar{\xi}_{2}c)&\bar{c}|\xi_{2}|^{2}+b|\xi_{1}|^{2}+(\bar{a}\xi_{1})\bar{\xi}_{2}-\xi_{1}(\bar{\xi}_{2}a)\\\
c|\xi_{2}|^{2}+\bar{b}|\xi_{1}|^{2}+\xi_{2}(\bar{\xi}_{1}a)-(\bar{a}\xi_{2})\bar{\xi}_{1}&-2|\xi_{2}|^{2}Re(a)+2Re(\xi_{2}\bar{\xi}_{1}b)\end{array}\right].$

Let us compute the right hand side of (6.2):

$\displaystyle rhs=\left[\begin{array}[]{cc}\bar{a}&\bar{c}\\\
\bar{b}&-\bar{a}\end{array}\right]\left[\begin{array}[]{cc}|\xi_{1}|^{2}&\xi_{1}\bar{\xi}_{2}\\\
\xi_{2}\bar{\xi}_{1}&|\xi_{2}|^{2}\end{array}\right]+\left[\begin{array}[]{cc}|\xi_{1}|^{2}&\xi_{1}\bar{\xi}_{2}\\\
\xi_{2}\bar{\xi}_{1}&|\xi_{2}|^{2}\end{array}\right]\left[\begin{array}[]{cc}a&b\\\
c&-a\end{array}\right]=$
$\displaystyle\left[\begin{array}[]{cc}\bar{a}|\xi_{1}|^{2}+\bar{c}(\xi_{2}\bar{\xi}_{1})&\bar{a}(\xi_{1}\bar{\xi}_{2})+\bar{c}|\xi_{2}|^{2}\\\
\bar{b}|\xi_{1}|^{2}-\bar{a}(\xi_{2}\bar{\xi}_{1})&\bar{b}(\xi_{1}\bar{\xi}_{2})-\bar{a}|\xi_{2}|^{2}\end{array}\right]+\left[\begin{array}[]{cc}a|\xi_{1}|^{2}+(\xi_{1}\bar{\xi}_{2})c&b|\xi_{1}|^{2}-(\xi_{1}\bar{\xi}_{2})a\\\
(\xi_{2}\bar{\xi}_{1})a+c|\xi_{2}|^{2}&(\xi_{2}\bar{\xi}_{1})b-a|\xi_{2}|^{2}\end{array}\right]=$
$\displaystyle\left[\begin{array}[]{cc}2|\xi_{1}|^{2}Re(a)+2Re(\xi_{1}\bar{\xi}_{2}c)&\bar{c}|\xi_{2}|^{2}+b|\xi_{1}|^{2}+\bar{a}(\xi_{1}\bar{\xi}_{2})-(\xi_{1}\bar{\xi}_{2})a\\\
c|\xi_{2}|^{2}+\bar{b}|\xi_{1}|^{2}+(\xi_{2}\bar{\xi}_{1})a-\bar{a}(\xi_{2}\bar{\xi}_{1})&-2|\xi_{2}|^{2}Re(a)+2Re(\xi_{2}\bar{\xi}_{1}b)\end{array}\right].$

We have to show that $lhs=rhs$. For the diagonal terms of these matrices the
identity is obvious. Since both matrices are Hermitian it suffices to show the
equality of the terms with indices $(1,2)$, that is

$(\bar{a}\xi_{1})\bar{\xi}_{2}-\xi_{1}(\bar{\xi}_{2}a)=\bar{a}(\xi_{1}\bar{\xi}_{2})-(\xi_{1}\bar{\xi}_{2})a.$

To simplify the notation we have to show that

$\displaystyle(\bar{a}b)c-b(ca)=\bar{a}(bc)-(bc)a\mbox{ for all
}a,b,c\in\mathbb{O}.$ (6.10)

For $a\in\mathbb{R}$ both sides of (6.10) vanish. Thus by linearity we may and
will assume that $a$ is purely imaginary, i.e. $\bar{a}=-a$. We have to prove
than that

$(ab)c+b(ca)=a(bc)+(bc)a.$

Denote $[a,b,c]:=(ab)c-a(bc)$. The the last equality is equivalent to

$\displaystyle[a,b,c]=[b,c,a].$ (6.11)

By Theorem 15.11(iii) of [1] it is alternating function of 3 variables. Hence
(6.11) holds, and the lemma follows. Q.E.D.

  8. 8

The following lemma is essentially contained in [41], and we present a proof
for convenience of the reader as in [7].

######  6.7 Lemma ([41]).

(i) The group $SL_{2}(\mathbb{O})$ acts naturally on
$\mathbb{O}\mathbb{P}^{1}$, namely for any $\phi\in SL_{2}(\mathbb{O})$ and
any $L\in\mathbb{O}\mathbb{P}^{1}$ the subspace $\phi(L)$ is an octonionic
projective line. This action is transitive.

(ii) For any $L\in\mathbb{O}\mathbb{P}^{1}$ and any $\phi\in
SL_{2}(\mathbb{O})$ the restriction

$\phi|_{L}\colon L\longrightarrow\phi(L)$

is a conformal linear map.

Proof. Let $\phi\in SL_{2}(\mathbb{O})$. We first show that if for two unit
vectors $\xi,\eta\in\mathbb{O}^{2}$ one has $\xi\sim\eta$ (as defined in Lemma
3.2(iv)) then $|\phi(\xi)|=|\phi(\eta)|$. To do so observe that for any
$v\in\mathbb{O}^{2}$ one has

$|v|^{2}=v^{*}v=Tr(vv^{*}),$

where $Tr$ stands for the trace and denotes sum of the diagonal elements.
Therefore

$\displaystyle|\phi(\xi)|^{2}=Tr(\phi(\xi)\cdot\phi(\xi)^{*})\overset{\mbox{Lemma
}\ref{m-sch1}}{=}Tr(\phi(\xi\cdot\xi^{*}))=$ $\displaystyle
Tr(\phi(\eta\cdot\eta^{*}))\overset{\mbox{Lemma
}\ref{m-sch1}}{=}Tr(\phi(\eta)\cdot\phi(\eta)^{*})=|\phi(\eta)|^{2}.$

Thus $|\phi(\xi)|=|\phi(\eta)|$. To prove both parts of the lemma (except
transitivity) it remains to show that for $\xi,\eta$ as above one has
$\phi(\xi)\sim\phi(\eta)$, namely
$\phi(\xi)\cdot\phi(\xi)^{*}=\phi(\eta)\cdot\phi(\eta)^{*}$. Applying Lemma
6.6 twice we get

$\phi(\xi)\cdot\phi(\xi)^{*}=\phi(\xi\xi^{*})=\phi(\eta\eta^{*})=\phi(\eta)\cdot\phi(\eta)^{*}.$

Finally let us prove the transitivity of the action of $SL_{2}(\mathbb{O})$ on
$\mathbb{O}\mathbb{P}^{1}$. It suffices to show that for any non-zero vector
$(\alpha,b)\in\mathbb{O}^{2}$ with $\alpha\in\mathbb{R}$ there exists $\phi\in
SL_{2}(\mathbb{O})$ such that

$\displaystyle\phi\left(\left[\begin{array}[]{c}1\\\
0\end{array}\right]\right)=\left[\begin{array}[]{c}\alpha\\\
b\end{array}\right].$ (6.16)

The octonion $b$ can be written $b=\beta+s\gamma$, where
$\beta,\gamma\in\mathbb{R}$, and $s$ is a purely imaginary octonion of norm 1.
Thus $s^{2}=-1$. The subalgebra generated by $s$ and $1$ is isomorphic to
$\mathbb{C}$. It suffices to construct a matrix $\phi$ with entries in this
subalgebra. Thus for simplicity of notation we may assume that
$b\in\mathbb{C}$. It is well known that $SL_{2}(\mathbb{C})$ acts transitively
on $\mathbb{C}^{2}\backslash\\{0\\}$. Hence there is $\phi\in
SL_{2}(\mathbb{C})$ satisfying (6.16). Furthermore it is well known that
$\exp\colon\mathfrak{sl}_{2}(\mathbb{C})\longrightarrow SL_{2}(\mathbb{C})$ is
onto (this can be seen using the Jordan form of a complex matrix). Thus
$\phi=\exp A$ where
$A\in\mathfrak{sl}_{2}(\mathbb{C})\subset\mathfrak{sl}_{2}(\mathbb{O})$.
Q.E.D.

  9. 9

Recall that the maps $j$ and $\theta$ were defined in (5.4) and (5.5)
respectively. Let us denote by ${\cal H}_{16}^{0}(\mathbb{R})$ the subspace of
${\cal H}_{16}(\mathbb{R})$ consisting of real quadratic forms whose
restriction to any projective line is proportional to the square of the
standard Euclidean norm on $\mathbb{O}^{2}=\mathbb{R}^{16}$. Lemma 6.7 easily
implies:

######  6.8 Lemma.

${\cal H}^{0}_{16}(\mathbb{R})$ is a $GL_{2}(\mathbb{O})$-invariant subspace
of ${\cal H}_{16}(\mathbb{R})$.

  10. 10

######  6.9 Lemma.

The following inclusion holds

$j({\cal H}_{2}(\mathbb{O}))\subset{\cal H}_{16}^{0}(\mathbb{R}).$

Proof. Observe that any octonionic line (except the one at infinity) is
spanned by a vector $\xi=\left[\begin{array}[]{cc}1\\\ a\end{array}\right]$.
For any such $\xi$ and any $A=\left[\begin{array}[]{cc}\alpha&x\\\
\bar{x}&\beta\end{array}\right]\in{\cal H}_{2}(\mathbb{O})$ we have

$\displaystyle j(A)=\left(\left[\begin{array}[]{cc}q\\\
aq\end{array}\right]\right)=Re\left([\bar{q},\bar{q}\bar{a}]\left[\begin{array}[]{cc}\alpha&x\\\
\bar{x}&\beta\end{array}\right]\left[\begin{array}[]{cc}q\\\
aq\end{array}\right]\right)=$ $\displaystyle
Re(\alpha|q|^{2}+\bar{q}x(aq)+(\bar{q}\bar{a})\bar{x}q+\beta|aq|^{2})=$
$\displaystyle|q|^{2}(\alpha+\beta|a|^{2})+2Re(\bar{q}x(aq))\overset{\mbox{Lemma
}\ref{L:oalg1}(v)}{=}$ $\displaystyle|q|^{2}(\alpha+\beta|a|^{2}+2Re(ax)),$

where the last equality follows from the fact that any subalgebra generated by
any two octonions is associative. The lemma follows. Q.E.D.

  11. 11

######  6.10 Lemma.

Let $B\in{\cal H}_{16}(\mathbb{R})$. Then, for any
$\xi\in\mathbb{O}^{2},\,\xi\neq 0$ one has

$\displaystyle((j\circ\theta)(B))(\xi)=Re(\xi^{*}\theta(B)\xi)=\left(\int_{\SS_{\xi}}b(\eta)d\eta\right)\cdot|\xi|^{2},$
(6.18)

where $\SS_{\xi}$ is the unit 7-sphere (with respect to the standard Euclidean
metric on $\mathbb{O}^{2}=\mathbb{R}^{16}$) in the octonionic line spanned by
$\xi$, and $d\eta$ is the probability Lebesgue measure on this sphere.

Proof. The first equality follows from the definition, see Section 5,
paragraph 2. In [7], Lemma 1.2.1, it was shown that if
$\xi=\left[\begin{array}[]{c}1\\\ a\end{array}\right]$ one has

$\displaystyle Re(\xi^{*}\theta(B)\xi)=\int_{x\in\SS^{7}}b(\xi\cdot x)dx,$

where the integration is with respect to the rotation invariant probability
measure on $\SS^{7}\subset\mathbb{O}$. The last integral is clearly equal to
$\left(\int_{\SS_{\xi}}b(\eta)d\eta\right)\cdot|\xi|^{2}$. Thus, the lemma
holds for vectors $\xi$ with the first coordinate equal to 1.

By Lemma 6.9 one has $j({\cal H}_{2}(\mathbb{O}))\subset{\cal
H}_{16}^{0}(\mathbb{R})$. The integral in (6.18) defines a function on
$\mathbb{O}^{2}$ which is proportional to the square of the standard Euclidean
norm on each octonionic line which coincides with $(j\circ\theta)(B)$ on each
vector $\xi=\left[\begin{array}[]{c}1\\\ a\end{array}\right]$. Since any
octonionic line contains a vector of such a form, these two functions (the
integral in (6.18) and $(j\circ\theta)(B)$) coincide on $\mathbb{O}^{2}$.
Q.E.D.

######  6.11 Corollary.

The linear maps

$\displaystyle j\colon{\cal H}_{2}(\mathbb{O})\longrightarrow{\cal
H}_{16}^{0}(\mathbb{R}),$ $\displaystyle\theta\big{|}_{{\cal
H}_{16}^{0}(\mathbb{R})}\colon{\cal
H}_{16}^{0}(\mathbb{R})\longrightarrow{\cal H}_{2}(\mathbb{O})$

are isomorphisms and inverse to each other.

Proof. First $\theta\circ j=id$ by Lemma 5.1. Hence, it remains to show that

$j\circ\theta\big{|}_{{\cal H}_{16}^{0}(\mathbb{R})}\colon{\cal
H}_{16}^{0}(\mathbb{R})\longrightarrow{\cal H}_{16}^{0}(\mathbb{R}),$

is the identity.

For a real quadratic form $b$ on $\mathbb{R}^{16}$ we denote by $B\in{\cal
H}_{16}(\mathbb{R})$ its matrix in the standard basis. By Lemma 6.10 for any
$\xi\in\mathbb{O}^{2}\backslash\\{0\\}$ one has

$((j\circ\theta)(B))(\xi)=\left(\int_{\SS_{\xi}}b(\eta)d\eta\right)\cdot|\xi|^{2}.$

Since $B\in{\cal H}_{16}^{0}(\mathbb{R})$, the function under the integral is
constant on the unit sphere $\SS_{\xi}$. Hence the right hand side is equal to
$b(\xi)$. The result follows. Q.E.D.

  12. 12

######  6.12 Lemma.

The map

$j\circ\theta\colon{\cal H}_{16}(\mathbb{R})\longrightarrow{\cal
H}_{16}^{0}(\mathbb{R})$

is $GL_{2}(\mathbb{O})$-equivariant.

Proof. For any $\xi\in\mathbb{O}^{2}\backslash\\{0\\}$ and any $g\in
GL_{2}(\mathbb{O})$ one has

$\displaystyle g((j\circ\theta)(B))(\xi)\overset{\mbox{Lemma
}\ref{L-average}}{=}\left(\int_{\SS_{g^{-1}\xi}}b(\eta)d\eta\right)\cdot|g^{-1}\xi|^{2}=$
$\displaystyle\left(\int_{\SS_{\xi}}b\left(\frac{g^{-1}\zeta}{|g^{-1}\zeta|}\right)d\zeta\right)|g^{-1}\xi|^{2}=\left(\int_{\SS_{\xi}}b\left(\frac{g^{-1}\zeta}{g^{-1}(\xi/|\xi)|}\right)d\zeta\right)|g^{-1}\xi|^{2}=$
$\displaystyle\left(\int_{\SS_{\xi}}b(g^{-1}\zeta)d\zeta\right)|\xi|^{2}=\left(\int_{\SS_{\xi}}(gb)(\zeta)d\zeta\right)|\xi|^{2}\overset{\mbox{Lemma
}\ref{L-average}}{=}$ $\displaystyle((j\circ\theta)(gB))(\xi).$

Q.E.D.

  13. 13

######  6.13 Proposition.

The maps $\theta\colon{\cal H}_{16}(\mathbb{R})\longrightarrow{\cal
H}_{2}(\mathbb{O})$ and $j\colon{\cal H}_{2}(\mathbb{O})\longrightarrow{\cal
H}_{16}(\mathbb{R})$ are $GL_{2}(\mathbb{O})$-equivariant with the action of
$GL_{2}(\mathbb{O})$ on ${\cal H}_{2}(\mathbb{O})$ as described in paragraph 4
of this subsection, and its action on ${\cal H}_{16}(\mathbb{R})$ is the
standard action on quadratic forms: $(gb)(\xi)=b(g^{-1}\xi)$.

Proof. The equivariance of $j$ was announced without proof in [7], Remark
1.5.3. We present a proof below. Recall that
$GL_{2}(\mathbb{O})=\mathbb{R}^{*}\cdot SL_{2}(\mathbb{O})$. Clearly $j$ is
$\mathbb{R}^{*}$-equivariant, hence it suffices to prove
$SL_{2}(\mathbb{O})$-equivariance. Namely, we have to show that for any $g\in
SL_{2}(\mathbb{O})$ and any $X\in{\cal H}_{2}(\mathbb{O})$ one has

$j(gX)=g(jX).$

This is equivalent that for any $\xi\in\mathbb{R}^{16}=\mathbb{O}^{2}$ the
following identity holds

$Re(\xi^{*}(gX)\xi)=Re\left((g^{-1}\xi)^{*}X(g^{-1}\xi)\right).$

It suffices to prove the infinitesimal version of this equality for generators
of the Lie algebra $\mathfrak{sl}_{2}(\mathbb{O})$. Namely, one has to show
that for any $A\in\mathfrak{sl}_{2}(\mathbb{O})$

$Re(\xi^{*}(A^{*}X+XA)\xi)=Re((A\xi)^{*}X\xi+\xi^{*}X(A\xi)).$

Since in each side the second summand is the Hermitian conjugate of the first
one this is equivalent to

$\displaystyle Re(\xi^{*}(XA)\xi)=Re(\xi^{*}X(A\xi)).$ (6.19)

Let us compute the right hand side of (6.19) for $A$ being a traceless matrix,
i.e. $A=\left[\begin{array}[]{cc}z&x\\\ y&-z\end{array}\right]$, and let
$X=\left[\begin{array}[]{cc}\alpha&q\\\
\bar{q}&\beta\end{array}\right]\in{\cal H}_{2}(\mathbb{O})$:

$\displaystyle\mbox{rhs of
}(\ref{E:jjj})=Re\left\\{[\bar{\xi}_{1},\bar{\xi}_{2}]\left[\begin{array}[]{cc}\alpha&q\\\
\bar{q}&\beta\end{array}\right]\left(\left[\begin{array}[]{cc}z&x\\\
y&-z\end{array}\right]\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right]\right)\right\\}=$ $\displaystyle
Re(\bar{\xi}_{1}(\alpha z)\xi_{1}+\bar{\xi}_{1}q(y\xi_{1})-\bar{\xi}_{2}(\beta
z)\xi_{2}+\bar{\xi}_{2}\bar{q}(x\xi_{2})+$ $\displaystyle\bar{\xi}_{1}\alpha
x\xi_{2}+\bar{\xi}_{2}\beta(y\xi_{1})-\bar{\xi}_{1}q(z\xi_{2})+\bar{\xi}_{2}\bar{q}(z\xi_{1}))=$
$\displaystyle Re((\alpha
z)\xi_{1}\bar{\xi}_{1}+q(y\xi_{1})\bar{\xi}_{1}-(\beta
z)\xi_{2}\bar{\xi}_{2}+\bar{q}(x\xi_{2})\bar{\xi}_{2})+$ $\displaystyle
Re[\bar{\xi}_{1}\alpha
x\xi_{2}+(\bar{\xi}_{1}\bar{y})\beta\xi_{2}-(\bar{\xi}_{1}q)(z\xi_{2})+(\bar{\xi}_{1}\bar{z})(q\xi_{2})]=$
$\displaystyle Re(\alpha z+qy)|\xi_{1}|^{2}+Re(\bar{q}x-\beta
z)|\xi_{2}|^{2}+Re[\bar{\xi}_{1}(\alpha
x+\beta\bar{y})\xi_{2}-(\bar{\xi}_{1}q)(z\xi_{2})+(\bar{\xi}_{1}\bar{z})(q\xi_{2})].$

For the left hand side of (6.19) we have

$\displaystyle\mbox{lhs of
}(\ref{E:jjj})=Re\left([\bar{\xi}_{1},\bar{\xi}_{2}]\left(\left[\begin{array}[]{cc}\alpha&q\\\
\bar{q}&\beta\end{array}\right]\left[\begin{array}[]{cc}z&x\\\
y&-z\end{array}\right]\right)\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right]\right)=$ $\displaystyle
Re\left([\bar{\xi}_{1},\bar{\xi}_{2}]\left[\begin{array}[]{cc}\alpha
z+qy&\alpha x-qz\\\ \bar{q}z+\beta y&\bar{q}x-\beta
z\end{array}\right]\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right]\right)=$ $\displaystyle Re\left(\bar{\xi}_{1}(\alpha
z+qy)\xi_{1}+\bar{\xi}_{2}(\bar{q}x-\beta z)\xi_{2}+\bar{\xi}_{1}(\alpha
x-qz)\xi_{2}+\bar{\xi}_{2}(\bar{q}z+\beta y)\xi_{1}\right)=$ $\displaystyle
Re(\alpha z+qy)|\xi_{1}|^{2}+Re(\bar{q}x-\beta
z)|\xi_{2}|^{2}+Re(\bar{\xi}_{1}(\alpha x+\beta\bar{y}-qz+\bar{z}q)\xi_{2}).$

Comparing this with the right hand side of (6.19) it remains to show that

$\displaystyle
Re[-(\bar{\xi}_{1}q)(z\xi_{2})+(\bar{\xi}_{1}\bar{z})(q\xi_{2})]=Re[\bar{\xi}_{1}(-qz+\bar{z}q)\xi_{2}].$
(6.23)

Note that if $z\in\mathbb{R}$ then obviously both sides vanish. Thus let us
assume that $z$ is purely imaginary, i.e. $\bar{z}=-z$. In this case (6.23)
reads

$Re[(\bar{\xi}_{1}q)(z\xi_{2})+(\bar{\xi}_{1}z)(q\xi_{2})]=Re[\bar{\xi}_{1}(qz+zq)\xi_{2}].$

This identity holds by Corollary 2.3.

The equivariance of $\theta$ was proven in [7], Lemma 1.5.2. However the proof
there contains a gap, so we present a complete proof below.

By Lemma 6.12 the map $j\circ\theta\colon{\cal
H}_{16}(\mathbb{R})\longrightarrow{\cal H}_{16}^{0}(\mathbb{R})$ is
$GL_{2}(\mathbb{O})$-equivariant, as well as $j$ as was shown above. Hence,
$\theta=j^{-1}\circ(j\circ\theta)$ is $GL_{2}(\mathbb{O})$-equivariant. Q.E.D.

##  7 Further properties of the octonionic Hessian.

######  7.1 Proposition ([7], Prop. 1.5.1).

Let $f\colon\mathbb{O}^{2}\longrightarrow\mathbb{R}$ be a $C^{2}$-smooth
function. Let $A\in SL_{2}(\mathbb{O})$. Then,

$\left(\frac{\partial^{2}}{\partial\bar{q}_{i}\partial
q_{j}}(f(A^{-1}q))\right)=A\left(\frac{\partial^{2}f}{\partial\bar{q}_{i}\partial
q_{j}}(A^{-1}q)\right),$

where $A$ in the right hand side denotes the induced action of $A$ on ${\cal
H}_{2}(\mathbb{O})$.

Proof. By translation it is enough to check the above equality at $q=0$.
Moreover we may and will assume that $f$ is a quadratic form. By the
definition of $\theta$, the proposition reduces to the
$GL_{2}(\mathbb{O})$-equivariance of $\theta$; that holds thanks to
Proposition 6.13. Q.E.D.

##  8 $G$-affine manifolds.

  1. 1

The results of this section are probably folklore. Let $G\subset
GL(n,\mathbb{R})$ be a subgroup. We introduce a rather special class of
manifolds which we call $G$-affine.

######  8.1 Definition.

(1) We say that an atlas of charts $\\{(U_{\alpha},\phi_{\alpha})\\}$ on a
smooth manifold $M^{n}$, where $\phi_{\alpha}\colon
U_{\alpha}\longrightarrow\phi_{\alpha}(U_{\alpha})\subset\mathbb{R}^{n}$ is a
diffeomorphism onto an open subset of $\mathbb{R}^{n}$, defines a $G$-affine
structure on $M$ if for any $\alpha,\beta$ the diffeomorphism

$\phi_{\beta}\circ\phi_{\alpha}^{-1}\colon\phi_{\alpha}(U_{\alpha}\cap
U_{\beta})\longrightarrow\phi_{\beta}(U_{\alpha}\cap U_{\beta})$

is a composition of a transformation from $G$ and a translation.

(2) Two atlases of charts on $M$ are called equivalent if their union defines
a $G$-affine structure (in other words, the transition maps between charts
from different atlases are compositions of maps from $G$ and translations.

(3) A $G$-affine structure on $M$ is an equivalence class of atlases of
charts.

The following example is relevant for the current study.

######  8.2 Example.

Let $G=\\{Id\\}$ be the trivial group. Then a torus $\mathbb{R}^{n}/\Lambda$
is a $G$-affine manifold, where $\Lambda\subset\mathbb{R}^{n}$ is a lattice.

######  8.3 Remark.

In this work the groups of interests are
$G=GL_{2}(\mathbb{O}),SL_{2}(\mathbb{O})\simeq Spin(1,9),Spin(9)\subset
GL_{16}(\mathbb{R})$.

  2. 2

The following proposition is obvious.

######  8.4 Proposition.

On a $G$-affine manifold there exists a unique torsion free flat connection on
the tangent bundle such that on each chart of the given atlas
$\\{(U_{\alpha},\phi_{\alpha})\\}$ this connection is the trivial one. The
holonomy of this connection is contained in $G$.

  3. 3

On a $G$-affine manifold $M$ the real Hessian of a $C^{2}$-smooth function is
well defined and takes values in the vector bundle $Sym^{2}(T^{*}M)$ whose
fiber over $x\in M$ can be identified with the space of quadratic forms on the
tangent space $T_{x}M$.

  4. 4

Recall that densities are sections of the lines bundles of top degree
differential forms and of the orientation bundle. Compactly supported
densities can be integrated over the manifold. The following result is well
known, also it can be easily proved by the reader.

######  8.5 Proposition.

Assume that the group $G$ is contained in the group of (linear) volume
preserving transformations. Then on any $G$-affine manifold there exists
unique nowhere vanishing density which is parallel with respect to the
connection from Proposition 8.4.

  5. 5

Let us remind the notion of $\lambda$-density on a smooth manifold for a real
number $\lambda$. Let $X^{n}$ be a smooth manifold (not necessarily
$G$-affine). Let $(V_{\alpha},\psi_{\alpha})$ be an atlas of charts. We have
diffeomorphisms

$\displaystyle\psi_{\beta}\circ\psi_{\alpha}^{-1}\colon\psi_{\alpha}(V_{\alpha}\cap
V_{\beta})\longrightarrow\psi_{\beta}(V_{\alpha}\cap V_{\beta}).$ (8.1)

Let $j_{\alpha\beta}$ denote the absolute value of the determinant of the
inverse of the differential of this map. Clearly

$\displaystyle j_{\alpha\beta}\cdot j_{\beta\gamma}=j_{\alpha\gamma}.$ (8.2)

Let us define the real line bundle of $\lambda$-densities over $X$ as follows.
Over each chart $V_{\alpha}$ we choose the trivial line bundle, and the
transition map from $V_{\alpha}$ to $V_{\beta}$ is equal to
$(j_{\alpha\beta})^{\lambda}$. Due to (8.2) this defines the line bundle. Its
sections are called $\lambda$-densities. The $1$-densities coincide with
densities from the previous paragraph.

The line bundle of $\lambda$-densities is canonically oriented: in any chart
its orientation coincides with the natural one of $\mathbb{R}$. A section of
this line bundle is called positive (resp. non-negative) if its value at any
point is positive (resp. non-negative) with respect to the given orientation.

Any nowhere vanishing $\lambda$-density can be raised to a real power $\mu$,
and the result is a $\lambda\mu$-density.

  6. 6

Let $M^{16}$ be an $GL_{2}(\mathbb{O})$-manifold in the sense of Definition
8.1. The tangent bundle $TM$ can be described as follows.

Let us fix the trivial bundle with fiber $\mathbb{O}^{2}$ over each map
$V_{\alpha}$. Let us glue them over each pairwise intersection $V_{\alpha}\cap
V_{\beta}$ according to the transition map (8.1) as follows. The differential
$d(\psi_{\beta}\circ\psi_{\alpha}^{-1})$ is a locally constant matrix with
values in $GL_{2}(\mathbb{O})$. We identify the restriction to $V_{\alpha}\cap
V_{\beta}$ of the trivial bundle $V_{\alpha}\times\mathbb{O}^{2}$ with the
restriction to the same intersection of the trivial bundle
$V_{\beta}\times\mathbb{O}^{2}$ via the map

$(x,X)\mapsto\left(x,d_{x}(\psi_{\beta}\circ\psi_{\alpha}^{-1})\right)X.$

  7. 7

The cotangent bundle $T^{*}M$ over the $GL_{2}(\mathbb{O})$-manifold $M$ is
described similarly identifying the restriction to $V_{\alpha}\cap V_{\beta}$
of the trivial bundle $V_{\alpha}\times\mathbb{O}^{2}$ with the restriction to
the same intersection of the trivial bundle $V_{\beta}\times\mathbb{O}^{2}$
via the map

$(x,X)\mapsto\left(x,{\cal
S}(d_{x}(\psi_{\beta}\circ\psi_{\alpha}^{-1}))X\right),$

where ${\cal S}(g)=(g^{t})^{-1}$ is the map from Lemma 6.5(ii).

  8. 8

For a $GL_{2}(\mathbb{O})$-manifold $M$ let us define a vector bundle
$\underline{{\cal H}_{2}}(M)$ over $M$ with the fiber ${\cal
H}_{2}(\mathbb{O})$ which is the space of octonionic Hermitian $2\times 2$
matrices. We will see that $\underline{{\cal H}_{2}}(M)$ can be considered as
a sub-bundle of the bundle $Sym^{2}_{\mathbb{R}}(T^{*}M)$ of real quadratic
forms on the tangent bundle $TM$ and the octonionic Hessian of a real valued
function is a section of $\underline{{\cal H}_{2}}(M)$.

Let us fix the trivial bundle with fiber ${\cal H}_{2}(\mathbb{O})$ over each
map $U_{\alpha}$. Let us glue them over each pairwise intersection
$U_{\alpha}\cap U_{\beta}$ according to the transition map (8.1) as follows.
Let $\rho$ be the representation of $GL_{2}(\mathbb{O})$ in ${\cal
H}_{2}(\mathbb{O})$ introduced in Subsection 6, paragraph 4. Let us identify
the restriction to $U_{\alpha}\cap U_{\beta}$ of the trivial bundle
$U_{\alpha}\times{\cal H}_{2}(\mathbb{O})$ with the restriction to the same
intersection of the trivial bundle $U_{\beta}\times{\cal H}_{2}(\mathbb{O})$
via the map

$(x,X)\mapsto\left(x,\rho\left(d_{x}(\psi_{\beta}\circ\psi_{\alpha}^{-1})\right)X\right).$

These identifications define the vector bundle over $M$ which we denote by
$\underline{{\cal H}_{2}}(M)$.

  9. 9

The $GL_{2}(\mathbb{O})$-equivariant linear map ${\cal
T}\colon\mathbb{O}\otimes_{\mathbb{R}}\mathbb{O}\longrightarrow{\cal
H}_{2}(\mathbb{O})$ from Lemma 6.6 induces the morphism of vector bundles

$\displaystyle{\cal T}\colon
Sym_{\mathbb{R}}^{2}(T^{*}M)\longrightarrow\underline{{\cal H}_{2}}(M),$ (8.3)

where $T^{*}M$ is the cotangent bundle of $M$.

  10. 10

By Proposition 6.4 the group $GL_{2}(\mathbb{O})$ preserves the cone of
positive definite octonionic Hermitian matrices. Hence in each fiber of the
bundle $\underline{{\cal H}_{2}}(M)$ there is a convex cone which coincides
with the latter cone in each coordinate chart. Hence one can talk of positive
definite and non-negative definite (say, continuous) sections of
$\underline{{\cal H}_{2}}(M)$.

  11. 11

It is easy to see that the octonionic Hessian of a real smooth function $f$,
which is locally written as $Hess_{\mathbb{O}}f$, is a section of
$\underline{{\cal H}_{2}}(M)$.

Furthermore its determinant $\det Hess_{\mathbb{O}}f$ is a section of the line
bundle of $1/4$-densities.

  12. 12

If the manifold $M$ is $SL_{2}(\mathbb{O})$-manifold, rather than
$GL_{2}(\mathbb{O})$, then there is a family of $\lambda$-densities
$\\{\Theta^{\lambda}\\}_{\lambda\in\mathbb{R}}$ of $\lambda$-densities which
is unique up to a proportionality and
(a) $\Theta^{\lambda}$ never vanishes;
(b) $\Theta^{\lambda}$ is parallel with respect to the connection from
Proposition 8.4;
(c) $(\Theta^{\lambda})^{\mu}=\Theta^{\lambda\mu}$ for any
$\lambda,\mu\in\mathbb{R}$.

##  9 Octonionic analogue of Kähler metrics.

  1. 1

Let us introduce on $GL_{2}(\mathbb{O})$-manifolds a class of Riemannian
metrics which can be considered as an octonionic analogue of Kähler metrics on
complex manifolds, or of Hessian metrics on affine manifolds, or and
HyperKähler with Torsion (HKT) metrics on hypercomplex manifolds. The
considerations are purely local, so we may and will replace our
$GL_{2}(\mathbb{O})$-manifold with a domain $\Omega\subset\mathbb{O}^{2}$.

Let us introduce the first order differential operators on $\mathbb{O}$-valued
functions acting on a open subset $\Omega\subset\mathbb{O}^{2}$ for $i=1,2$:

$\displaystyle\overset{\longrightarrow}{\partial_{\bar{i}}}F=\sum_{p=0}^{7}e_{p}\frac{\partial
F}{\partial x_{i}^{p}},$ $\displaystyle
F\overset{\leftarrow}{\partial_{\bar{i}}}=\sum_{p=0}^{7}\frac{\partial
F}{\partial x_{i}^{p}}e_{p}.$

Note that the first operator coincides with
$\frac{\partial}{\partial\bar{q}_{i}}$.

######  9.1 Definition.

Let $T\colon\Omega\longrightarrow{\cal H}_{2}(\mathbb{O})$ be a
$C^{\infty}$-smooth function. We call $T$ a closed current it is satisfies the
following system of first order linear differential equations:

$\displaystyle
T_{\bar{1}2}\overset{\leftarrow}{\partial}_{\bar{2}}=\overset{\longrightarrow}{\partial_{\bar{1}}}T_{\bar{2}2},\,\,\,T_{\bar{2}1}\overset{\leftarrow}{\partial}_{\bar{1}}=\overset{\longrightarrow}{\partial_{\bar{2}}}T_{\bar{1}1}$
(9.1)

The equations (9.1) being specialised to the case of 2 complex variables are
equivalent to the definition of a closed (smooth) $(1,1)$-current.

  2. 2

The main result of this section is

######  9.2 Theorem.

Let $\Omega\subset\mathbb{O}^{2}=\mathbb{R}^{16}$ be an open convex subset.
Let $T\colon\Omega\longrightarrow{\cal H}_{2}(\mathbb{O})$ be a
$C^{\infty}$-smooth function. Then $T$ satisfies the system (9.1) if and only
if there exists a $C^{\infty}$-smooth function
$u\colon\Omega\longrightarrow\mathbb{R}$ such that

$T=Hess_{\mathbb{O}}(u).$

Note that in this theorem $T$ defines a metric if and only if $T>0$, or
equivalently $u$ is strictly octonionic plurisubharmonic in the sense of [7].

  3. 3

A complex version of Theorem 9.2 is classical and is called local
$dd^{c}$-lemma, see e.g. [43], Proposition 8.8. A quaternionic version of this
result was first obtained by Michelson-Strominger [42] for the flat
quaternionic space $\mathbb{H}^{n}$, and was proven by Banos-Swann [17] for
arbitrary hypercomplex manifolds.

  4. 4

As we will see, the proof of the theorem is a straightforward combination of
two main ingredients: an Ehrenpreis’ theorem which is a general result from
linear PDE with constant coefficients [35], Theorem 7.6.13, and a computation
from commutative algebra which we performed on a computer using Macauley2.

Let us describe the first mentioned ingredient. This is Theorem 7.6.13 in
Hörmander’s book [35]. According to this book, this result was first announced
by Ehrenpreis [22] with further contributions by Malgrange [40] and Palamodov
[45], [46].

Let $P_{1},\dots,P_{J}$ be linear differential operators on $\mathbb{R}^{n}$
with constant complex coefficients; thus they are complex polynomials in
$\partial_{1},\dots,\partial_{n}$.

######  9.3 Theorem.

Let $\Omega\subset\mathbb{R}^{n}$ be an open convex subset. Let
$f_{1},\dots,f_{j}\colon\Omega\longrightarrow\mathbb{C}$ be
$C^{\infty}$-smooth functions. Then there exists a $C^{\infty}$-smooth
function $u\colon\Omega\longrightarrow\mathbb{C}$ satisfying the system of
equations

$\displaystyle P_{j}u=f_{j},\mbox{ for all }j=1,\dots,J,$ (9.2)

if and only if for any $J$-tuple of linear differential operatorsa with
constant complex coefficient $Q_{1},\dots,Q_{J}$ such that

$\displaystyle\sum_{j=1}^{J}Q_{j}P_{j}=0$ (9.3)

one has

$\displaystyle\sum_{j=1}^{J}Q_{j}f_{j}=0.$ (9.4)

######  9.4 Remark.

In our application to Theorem 9.2 it will be important to have a version of
Theorem 9.3 where all the differential operators $P_{j}$ have real
coefficients, and all the functions $f_{j}$ and $u$ are real valued. Then
exactly the same theorem holds where the polynomials $Q_{j}$ also have real
coefficients. This form is easily deduced from the above one, in particular a
complex solution $u$ is replaced by its real part which is also a solution
(9.2).

  5. 5

We are going to apply Theorem 9.3 (taking into account Remark 9.4) to the
system

$\displaystyle Hess_{\mathbb{O}}(u)=T,$ (9.5)

where $T\colon\Omega\longrightarrow{\cal H}_{2}(\mathbb{O})$. There are $n=16$
variables and $J=10$ (real) equations.

  6. 6

First let us observe that the ring of linear differential operators with
constant real coefficients is identified with the ring of real polynomials via
the substitution $\partial_{i}\mapsto x_{i}$. In order to apply Theorem 9.3 in
a concrete situation one has to describe $J$-tuple of polynomials
$(Q_{1},\dots,Q_{J})$ satisfying (9.3). Let us also observe that this set is a
submodule of the free module of rank $J$ over the polynomial ring
$A:=\mathbb{R}[x_{1},\dots,x_{n}]$ (now we work with real coefficients). It is
well known that every submodule of a finitely generated $A$-module is finitely
generated (see e.g. Proposition 6.5 and Corollary 7.6 in [15]). Clearly it
suffices then to verify (9.4) only on generators of that submodule.

The computer software Macaulay2 allows to compute easily generators of such a
module. There is a special command ’kernel’ which is applied to a matrix whose
entries are polynomials with rational coefficients. This matrix is considered
a morphism between free modules over the polynomial ring. The output of the
’kernel’ command is generators of the kernel of this morphism.

This computer computation leads to system (9.1). We discuss implementation of
this procedure and provide a printout of Macaulay2 program in the Appendix.
Let us notice that Macaulay2 works with polynomials with rational (rather than
real or complex) coefficients and computes generators of a submodule over the
ring $\mathbb{Q}[x_{1},\dots,x_{n}]$. In our situation the polynomials
$P_{1},\dots,P_{J}$ have indeed rational coefficients. One has the following
easy lemma which implies that it suffices to find generators of a submodule
over polynomials with rational coefficients.

######  9.5 Lemma.

Let $P_{1},\dots,P_{J}\in\mathbb{Q}[x_{1},\dots,x_{n}]$. Denote

$\displaystyle
N_{\mathbb{R}}=\\{(Q_{1},\dots,Q_{J})|\,\,Q_{j}\in\mathbb{R}[x_{1},\dots,x_{J}]\,\&\,\sum_{j=1}^{J}Q_{j}P_{j}=0\\},$
$\displaystyle
N_{\mathbb{Q}}=\\{(Q_{1},\dots,Q_{J})|\,\,Q_{j}\in\mathbb{Q}[x_{1},\dots,x_{J}]\,\&\,\sum_{j=1}^{J}Q_{j}P_{j}=0\\}.$

Then

$N_{\mathbb{R}}=\mathbb{R}\cdot
N_{\mathbb{Q}}:=\\{\sum_{\alpha}\lambda_{\alpha}(Q_{1}^{\alpha},\dots,Q_{J}^{\alpha})|\,\lambda_{\alpha}\in\mathbb{R}\,\&\,(Q_{1}^{\alpha},\dots,Q_{J}^{\alpha})\in
N_{\mathbb{Q}}\\}.$

The lemma follows immediately from existence of a basis of $\mathbb{R}$ over
$\mathbb{Q}$.

  7. 7

Let us introduce a class of Riemannian metrics on
$GL_{2}(\mathbb{O})$-manifolds, called octonionic Kähler metrics, analogous to
Kähler metrics on complex manifold, Hessian metrics on affine manifolds, and
HKT-metrics on hypercomplex manifolds.

######  9.6 Definition.

A Riemannian metric on a $GL_{2}(\mathbb{O})$-manifold $M$ is called
octonionic Hermitian if for any point $x\in M$ and some (equivalently, any)
chart containing $x$ the restriction of the metric to any octonionic line
contained in the tangent space $T_{x}M\simeq\mathbb{O}^{2}$ (the latter
identification uses the chart) is proportional to the standard Euclidean
metric on $\mathbb{O}^{2}$.

Note that independence of this definition of a chart follows from Lemma 6.7.
Octonionic Hermitian metrics can be described explicitly in a chart as
follows. Let $\Omega\subset\mathbb{O}^{2}$ be an open subset. Let
$T=(T_{\bar{i}j})_{i,j=1}^{2}$ be a $C^{\infty}$-smooth function on $\Omega$
with values in the space ${\cal H}_{2}(\mathbb{O})$ of $2\times 2$ octonionic
Hermitian matrices. Assume that $T>0$ pointwise in $\Omega$. Thus $T$ defines
a Riemannian metric $g$ in $\Omega$ as follows

$|\xi|^{2}_{g}=\sum_{i,j=1}^{2}Re(\bar{\xi}_{i}T_{\bar{i}j}\xi_{j})$

for any $\xi\in\mathbb{O}^{2}$. By Corollary 6.11 this metric $g$ is
octonionic Hermitian, and any octonionic Hermitian metric is given by this
formula for a unique $T$ as above.

######  9.7 Definition.

An octonionic Hermitian metric $g$ on $M$ is called octonionic Kähler if in
some (equivalently, any) chart the ${\cal H}_{2}(\mathbb{O})$-valued function
$T$ corresponding to $g$ satisfies the system of equations (9.1).

######  9.8 Remark.

The independence of the last definition of a chart can be proved by showing
that the system (9.1) is invariant under $GL_{2}(\mathbb{O})$ transformations.
Alternatively, by Theorem 9.2 $g$ is octonionic Kähler if and only if locally
$T=Hess_{\mathbb{O}}(u)$. But the latter equation is invariant under
$GL_{2}(\mathbb{O})$ transformations by Proposition 7.1.

##  10 Integration by parts in $\mathbb{O}^{2}$.

  1. 1

In this section we prove a few identities for integrals over
$SL_{2}(\mathbb{O})$-manifolds to be used later. We start with necessary
results from linear algebra.

Let $A=\left[\begin{array}[]{cc}a_{\bar{1}1}&a_{\bar{1}2}\\\
a_{\bar{2}1}&a_{\bar{2}2}\end{array}\right]$ be an octonionic Hermitian
matrix. Like in the commutative case we denote by

$adj(A):=\left[\begin{array}[]{cc}a_{\bar{2}2}&-a_{\bar{1}2}\\\
-a_{\bar{2}1}&a_{\bar{1}1}\end{array}\right]$

its adjoint matrix. It is also octonionic Hermitian. One can verify by a
direct computation that like in the commutative case if $A$ is invertible then

$adj(A)=(\det A)\cdot A^{-1}.$

It is easy to see that if $A\geq 0$ then $adj(A)\geq 0$. The following
elementary identity is proved by a straightforward computation:

$\displaystyle\det(A,B)=\frac{1}{2}Re(Tr(adj(A)\cdot B))$ (10.1)

for any $A,B\in{\cal H}_{2}(\mathbb{O})$, where $Tr$ denotes the sum of the
diagonal elements.

  2. 2

For any $2\times 2$ octonionic matrices $X,Y$ one clearly has

$\displaystyle Re(Tr(X))=Re(Tr(X^{*})),$ (10.2) $\displaystyle
Re(Tr(XY))=Re(Tr(YX)).$ (10.3)

Let $C\in{\cal H}_{2}(\mathbb{O})$ and let $B$ be any $2\times 2$ octonionic
matrix. Then

$\displaystyle
Re(Tr(CB))\overset{(\ref{o2-01})}{=}Re(Tr(B^{*}C))\overset{(\ref{o2-02})}{=}Re(Tr(CB^{*})).$

Hence $Re(Tr(CB))=Re(Tr(C\frac{(B+B^{*})}{2}))$. This and (10.1) imply

$\displaystyle\det(A,\frac{B+B^{*}}{2})=\frac{1}{2}Re(Tr(adj(A)\cdot B)).$
(10.4)

  3. 3

The following proposition is a version of integration by parts. Here and below
we denote

$\Delta_{i}:=\sum_{a=0}^{7}\partial^{2}_{x_{i}^{a}},\,i=1,2.$

######  10.1 Proposition.

Let $u,v,f\colon\mathbb{O}^{2}\longrightarrow\mathbb{R}$ be
$C^{\infty}$-smooth functions, and assume that one of them is compactly
supported. Then

$\int_{\mathbb{O}^{2}}\det(Hess_{\mathbb{O}}(u),Hess_{\mathbb{O}}(v))fdx=-\frac{1}{2}Re\left(\int_{\mathbb{O}^{2}}\left([f_{1},f_{2}]adj(Hess_{\mathbb{O}}(v))\left[\begin{array}[]{c}u_{\bar{1}}\\\
u_{\bar{2}}\end{array}\right]\right)dx\right).$

Proof. Let us rewrite the left hand side in the proposition multiplied by 2 as
follows:

$\displaystyle\int
Re\left(u_{\bar{1}1}v_{\bar{2}2}+u_{\bar{2}2}v_{\bar{1}1}-(u_{\bar{1}2}v_{\bar{2}1}+u_{\bar{2}1}v_{\bar{1}2})\right)fdx=$
$\displaystyle-\int\left(Re\\{u_{\bar{1}}\Delta_{2}v_{1}\\}+Re\\{u_{\bar{2}}\Delta_{1}v_{2}\\}-Re\\{u_{\bar{1}}\underset{=\Delta_{2}v_{1}}{\underbrace{\overset{\longrightarrow}{\partial_{2}}(v_{\bar{2}1})}}\\}-Re\\{u_{\bar{2}}\underset{=\Delta_{1}v_{2}}{\underbrace{\overset{\longrightarrow}{\partial_{1}}(v_{\bar{1}2})}}\\}\right)fdx-$
$\displaystyle-\int\left(Re\\{u_{\bar{1}}f_{1}v_{\bar{2}2}\\}+Re\\{u_{\bar{2}}f_{2}v_{\bar{1}1}\\}-Re\\{u_{\bar{1}}f_{2}v_{\bar{2}1}\\}-Re\\{u_{\bar{2}}f_{1}v_{\bar{1}2}\\}\right)dx=$
$\displaystyle-\int\left(Re\\{f_{1}v_{\bar{2}2}u_{\bar{1}}\\}+Re\\{f_{2}v_{\bar{1}1}u_{\bar{2}}\\}-Re\\{f_{2}v_{\bar{2}1}u_{\bar{1}}\\}-Re\\{f_{1}v_{\bar{1}2}u_{\bar{2}}\\}\right)dx=$
$\displaystyle-\int
Re\left([f_{1},f_{2}]\left[\begin{array}[]{cc}v_{\bar{2}2}&-v_{\bar{1}2}\\\
-v_{\bar{2}1}&v_{\bar{1}1}\end{array}\right]\left[\begin{array}[]{c}u_{\bar{1}}\\\
u_{\bar{2}}\end{array}\right]\right)dx.$

The result follows. Q.E.D.

  4. 4

We are going now to rewrite the statement of Proposition 10.1 in a different
language so that it will make sense on any $SL_{2}(\mathbb{O})$-manifold.

First we have

$\displaystyle
Re\left([f_{1},f_{2}]adj(Hess_{\mathbb{O}}(v))\left[\begin{array}[]{c}u_{\bar{1}}\\\
u_{\bar{2}}\end{array}\right]\right)=$ (10.8) $\displaystyle
ReTr\left(adj(Hess_{\mathbb{O}}(v))\left[\begin{array}[]{c}u_{\bar{1}}\\\
u_{\bar{2}}\end{array}\right][f_{1},f_{2}]\right)\overset{(\ref{mixed-
adj-2})}{=}$ (10.11) $\displaystyle 2\det\left(Hess_{\mathbb{O}}(v),{\cal
T}\left(\left[\begin{array}[]{c}u_{\bar{1}}\\\
u_{\bar{2}}\end{array}\right]\otimes\left[\begin{array}[]{c}f_{\bar{1}}\\\
f_{\bar{2}}\end{array}\right]\right)\right),$ (10.16)

where ${\cal T}\colon Sym_{\mathbb{R}}^{2}(\mathbb{O}^{2})\longrightarrow{\cal
H}_{2}(\mathbb{O})$ is the map from Lemma 6.6.

Note that the differential $du\in(\mathbb{R}^{16})^{*}$ is identified with
$\left[\begin{array}[]{c}u_{\bar{1}}\\\ u_{\bar{2}}\end{array}\right]$ when
$\mathbb{R}^{16}$ is identified with $\mathbb{O}^{2}$ in the usual sense, i.e.

$(x_{0},\dots,x_{7},y_{0},\dots,y_{7})\mapsto(\sum_{p=0}^{7}x_{p}e_{p},\sum_{p=0}^{7}y_{p}e_{p}),$

and the dual space $(\mathbb{R}^{16})^{*}$ is identified with
$\mathbb{R}^{16}$ in the standard coordinate way.

Hence, under these identifications, (10.16) is equal to
$2\det\left(Hess_{\mathbb{O}}(v),{\cal T}(du\otimes df)\right)$. Thus we
obtain

$\displaystyle
Re\left([f_{1},f_{2}]adj(Hess_{\mathbb{O}}(v))\left[\begin{array}[]{c}u_{\bar{1}}\\\
u_{\bar{2}}\end{array}\right]\right)=2\det\left(Hess_{\mathbb{O}}(v),{\cal
T}(du\otimes df)\right).$ (10.19)

Note that the left hand side of (10.19) is defined in local coordinates, while
the right hand side is defined globally on $M$.

This identification allows us to generalize Proposition 10.1 to arbitrary
$SL_{2}(\mathbb{O})$-manifolds.

######  10.2 Proposition.

Let $M^{16}$ be an $SL_{2}(\mathbb{O})$-manifold with a parallel non-vanishing
3/4-density $\Theta$. Let $u,v,f\colon M\longrightarrow\mathbb{R}$ be
$C^{\infty}$-smooth functions, and assume that one of them is compactly
supported. Then

$\displaystyle\mbox{(a)
}\int_{M}f\det(Hess_{\mathbb{O}}(u),Hess_{\mathbb{O}}(v))\cdot\Theta=-\int_{M}\det\left(Hess_{\mathbb{O}}(v),{\cal
T}(du\otimes df)\right)\cdot\Theta,$

where now ${\cal T}\colon
Sym_{\mathbb{R}}^{2}(T^{*}M)\longrightarrow\underline{{\cal H}}_{2}(M)$ is the
map defined in Section 8, paragraph 9;
(b) $\int_{M}f\det(u,v)\cdot\Theta=\int_{M}u\det(f,v)\cdot\Theta.$

Part (a) follows from 10.1 using partition of unity. Part (b) immediately
follows from part (a).

######  10.3 Corollary.

Let $M^{16}$ be an $SL_{2}(\mathbb{O})$-manifold with a parallel non-vanishing
3/4-density $\Theta$. Let $v,f\colon\mathbb{O}^{2}\longrightarrow\mathbb{R}$
be $C^{\infty}$-smooth functions, and assume that one of them is compactly
supported. Let $G_{0}$ be a $C^{\infty}$-smooth section of $\underline{{\cal
H}}_{2}(M)$ satisfying Theorem 9.2. Then,

$\mbox{(a)
}\int_{M}f\det(Hess_{\mathbb{O}}(u),G_{0})\cdot\Theta=-\int_{M}\det\left(G_{0},{\cal
T}(du\otimes df)\right)\cdot\Theta,$ $\mbox{(b)
}\int_{M}f\det(Hess_{\mathbb{O}}(v),G_{0})\cdot\Theta=\int_{M}v\det(Hess_{\mathbb{O}}(f),G_{0})\cdot\Theta.$

Proof. Clearly (b) follows from (a) since the right hand side of (a) is
symmetric in $u$ and $f$. To prove (b) observe that there exists a locally
finite open covering $\\{U_{\alpha}\\}$ by charts with a subordinate to it
partition of unity $\\{\chi_{\alpha}\\}$ such that

$G_{0}=Hess_{\mathbb{O}}(v_{\alpha})\mbox{ on }U_{\alpha},$

where $v_{\alpha}\colon U_{\alpha}\longrightarrow\mathbb{R}$ is a
$C^{\infty}$-smooth function. We have

$\displaystyle\int_{M}f\det(Hess_{\mathbb{O}}(v),G_{0})\cdot\Theta=$
$\displaystyle\sum_{\alpha}\int_{U_{\alpha}}f\chi_{\alpha}\det(Hess_{\mathbb{O}}(u),Hess_{\mathbb{O}}(v_{\alpha}))\cdot\Theta\overset{\mbox{Prop.
}\ref{P:integration-parts2}(a)}{=}$
$\displaystyle-\sum_{\alpha}\int_{U_{\alpha}}\det(Hess_{\mathbb{O}}(v_{\alpha}),{\cal
T}(du\otimes d(f\chi_{\alpha})))\cdot\Theta=$
$\displaystyle-\sum_{\alpha}\int_{U_{\alpha}}\det(G_{0},{\cal T}(du\otimes
d(f\chi_{\alpha})))\cdot\Theta=$ $\displaystyle-\int_{M}\det(G_{0},{\cal
T}(du\otimes df))\cdot\Theta.$

Q.E.D.

##  11 The Monge-Ampère equation and the main result.

  1. 1

Let $M$ be a $GL_{2}(\mathbb{O})$-manifold. Using the language described in
Section 8 we can interpret the Monge-Ampère equation globally. Let $G_{0}$ be
a $C^{\infty}$-smooth section of the bundle $\underline{{\cal H}_{2}}(M)$. Let
$u,f$ be $C^{\infty}$-smooth functions. The Monge-Ampère equation which is
written in local coordinates as

$\det(G_{0}+Hess_{\mathbb{O}}(\phi))=e^{f}\det(G_{0}),$

is well defined globally when both sides are interpreted as $1/4$-densities.

To prove solvability of the last equation we will assume that $G_{0}$
satisfies Theorem 9.2: for every point $p\in M$ there exists a neighborhood
$U$ and a $C^{\infty}$-function $u$ such that

$\displaystyle G_{0}(x)=Hess_{\mathbb{O}}u(x)\mbox{ for any }x\in U.$ (11.1)

  2. 2

Let us state a general conjecture a special case of which we will prove in
this paper. This conjecture is obviously motivated by the Calabi-Yau theorem
for Kähler manifolds [58].

######  11.1 Conjecture.

Let $M^{16}$ be a compact $GL_{2}(\mathbb{O})$-manifold. Let $G_{0}$ be a
$C^{\infty}$-smooth positive section of the bundle $\underline{{\cal
H}_{2}}(M)$ satisfying condition (11.1). Let $f\colon
M\longrightarrow\mathbb{R}$ be a $C^{\infty}$-smooth function. Then there
exist a $C^{\infty}$-smooth function $u\colon M\longrightarrow\mathbb{R}$ and
a constant $A>0$ such that

$\displaystyle\det(G_{0}+Hess_{\mathbb{O}}(\phi))=Ae^{f}\det(G_{0})\mbox{ on
}M.$ (11.2)

  3. 3

Let us assume that $M$ is an $SL_{2}(\mathbb{O})$-manifold rather than
$GL_{2}(\mathbb{O})$. Then by the results of Section 8, paragraph 12, there
exists a parallel and non-vanishing $3/4$-density $\Theta^{3/4}$, which will
be denoted just $\Theta$ in the rest of the paper.

######  11.2 Lemma.

Let $M^{16}$ be a compact $SL_{2}(\mathbb{O})$-manifold with a non-vanishing
parallel $3/4$-density $\Theta$. Let a $C^{4}$-smooth function $\phi$ and a
constant $A$ satisfy (11.2). Then,

$A=\frac{\int_{M}\det G_{0}\cdot\Theta}{\int_{M}e^{f}\det G_{0}\cdot\Theta}.$

Proof. Let us multiply both sides of (11.2) by $\Theta$ and integrate over
$M$. Using the assumption (11.1) and Corollary 10.3 we get in the left hand
side

$\displaystyle\int_{M}(\det
G_{0}+2\det(G_{0},Hess_{\mathbb{O}}\phi)+\det(Hess_{\mathbb{O}}\phi,Hess_{\mathbb{O}}\phi)\cdot\Theta=$
$\displaystyle\int_{M}(\det
G_{0}+\phi\det(G_{0},Hess_{\mathbb{O}}1)+\phi\det(Hess_{\mathbb{O}}\phi,Hess_{\mathbb{O}}1)\cdot\Theta=$
$\displaystyle\int_{M}\det G_{0}\cdot\Theta.$

In the right hand side we have $A\int_{M}e^{f}\det G_{0}\cdot\Theta.$ Q.E.D.

  4. 4

From now on dealing with $SL_{2}(\mathbb{O})$-manifolds we will replace $f$
with $f+\ln A$ thus getting the equation

$\displaystyle\det(G_{0}+Hess_{\mathbb{O}}(\phi))=e^{f}\det(G_{0}),$ (11.3)

where

$\displaystyle\int_{M}(e^{f}-1)\det G_{0}\cdot\Theta=0.$ (11.4)

  5. 5

Let us formulate the main result of this paper. Note first that a
$Spin(9)$-manifold is $SL_{2}(\mathbb{O})$-manifold, and hence it carries a
non-vanishing parallel $3/4$-density $\Theta$.

######  11.3 Theorem.

Let $M^{16}$ be a compact $Spin(9)$-manifold. Let $f\colon
M\longrightarrow\mathbb{R}$ be a $C^{\infty}$-smooth function. Let $G_{0}$ be
a $C^{\infty}$-smooth positive section of $\underline{{\cal H}_{2}}(M)$
satisfying (11.1). Assume that $f,G_{0}$ satisfy (11.4). Then there exists a
unique (up to an additive constant) $C^{\infty}$-smooth function $\phi\colon
M\longrightarrow\mathbb{R}$ satisfying the Monge-Ampère equation (11.3).

##  12 Uniqueness of solution.

######  12.1 Theorem.

Let $M^{16}$ be a compact connected $GL_{2}(\mathbb{O})$-manifold. Let $G_{0}$
be a continuous positive section of $\underline{{\cal H}}_{2}(M)$. Let
$f\colon M\longrightarrow\mathbb{R}$ be a continuous function. Let
$\phi_{1},\phi_{2}\colon M\longrightarrow\mathbb{R}$ be $C^{2}$-smooth
functions satisfying

$\det(G_{0}+Hess_{\mathbb{O}}(\phi_{1,2}))=e^{f}\det(G_{0}).$

Then $\phi_{1}-\phi_{2}$ is a constant.

We start with the following lemma.

######  12.2 Lemma.

Let $M$ be a compact connected $GL_{2}(\mathbb{O})$-manifold. Let $G$ be a
continuous section of $\underline{{\cal H}_{2}}(M)$ such that $\det G>0$
everywhere on $M$, and $G>0$ at least at one point. Then $G>0$ everywhere.

Proof. Set ${\cal U}:=\\{z\in M|G(u)>0\\}$. By assumption ${\cal U}$ is non-
empty. Clearly ${\cal U}$ is open. It suffices to show that ${\cal U}$ is
closed. Let $z\in\bar{\cal U}$. Then $G(z)\geq 0$. If $G(z)$ is not positive
definite then $\det G(z)=0$ contradicting the assumption. To show the latter
statement note that we may assume the $G(z)$ is a complex Hermitian matric
since its entries are contained in a subfield of $\mathbb{O}$ isomorphic to
$\mathbb{C}$. Then $G(z)\geq 0$ but not positive definite also as a complex
Hermitian matrix by the Sylvester criterion (Proposition 5.4)). For complex
matrices that implies that $\det G(z)=0$. Q.E.D.

Proof of Theorem 12.1. Subtracting the equations we get

$\displaystyle\det(Hess_{\mathbb{O}}(\phi_{1}-\phi_{2}),(G_{0}+Hess_{\mathbb{O}}(\phi_{1}))+(G_{0}+Hess_{\mathbb{O}}(\phi_{2})))=0.$

Since $G_{0}+Hess_{\mathbb{O}}(\phi_{1,2})>0$ by Lemma 12.2, the function
$\phi_{1}-\phi_{2}$ satisfies a second order linear elliptic equation with no
free term and with continuous coefficients. By the strong maximum principle
(see [27], Theorem 3.5) this function has to be constant. Q.E.D.

##  13 Zero order estimate.

The main result of this section is Theorem 13.5 which is a $C^{0}$-estimate of
a solution of the Monge-Ampère equation (11.3). The method of the proof is
close to the original one due to Yau [58] with further modifications explained
in [37].

Throughout this section we assume the following: $M^{16}$ is a compact
connected $SL_{2}(\mathbb{O})$-manifold with a positive parallel $3/4$-density
$\Theta$; $f\in C^{\infty}(M,\mathbb{R})$; $G_{0}$ is a $C^{\infty}$-smooth
section of the bundle $\underline{{\cal H}_{2}}(M)$ satisfying condition
(11.1). In addition assume that the condition (11.4) is satisfied and $\phi$
is a $C^{2}$-smooth solution of (11.3).

######  13.1 Lemma.

The solution $\phi$ satisfies the following estimates
(a) $||\nabla|\phi|^{p/2}||_{L^{2}}^{2}\leq
c_{1}\cdot\frac{p^{2}}{(p-1)}\int_{M}(1-e^{f})\phi|\phi|^{p-2}\det
G_{0}\cdot\Theta\mbox{ for }p>2,$,
(b)$||\nabla\phi||_{L^{2}}^{2}\leq c_{1}\int_{M}(1-e^{f})\phi\det
G_{0}\cdot\Theta,$
where the constant $c_{1}$ depends on $M,G_{0}$, and $\Theta$ only.

Proof. Let us fix a finite triangulation $\\{{\cal S}_{\alpha}\\}_{\alpha}$ of
$M$ into simplices such that each simplex is contained in a coordinate chart.
Note that

$(\phi|\phi|^{p-2})_{x_{i}^{a}}=(p-1)\phi_{x_{i}^{a}}|\phi|^{p-2}.$

We have

$\displaystyle\int_{M}(1-e^{f})\phi|\phi|^{p-2}\det G_{0}\cdot\Theta=$
$\displaystyle\int_{M}\phi|\phi|^{p-2}(\det
G_{0}-\det(G_{0}+Hess_{\mathbb{O}}\phi))\cdot\Theta$
$\displaystyle-\int_{M}\phi|\phi|^{p-2}\det(2G_{0}+Hess_{\mathbb{O}}\phi,Hess_{\mathbb{O}}\phi)\cdot\Theta\overset{\mbox{Corol.
\ref{CoR:cor-by-parts}(a)}}{=}$
$\displaystyle(p-1)\int_{M}|\phi|^{p-2}\det\left(2G_{0}+Hess_{\mathbb{O}}(\phi),{\cal
T}(d\phi\otimes d\phi)\right)\cdot\Theta\overset{(\ref{E:integr-by-parts-
octon})}{=}$ $\displaystyle\frac{(p-1)}{2}\sum_{\alpha}\int_{{\cal
S}_{\alpha}}|\phi|^{p-2}\left(\left[\phi_{1},\phi_{2}\right](adj(2G_{0}+Hess_{\mathbb{O}}(\phi)))\left[\begin{array}[]{c}\phi_{\bar{1}}\\\
\phi_{\bar{2}}\end{array}\right]\right)(dq)_{\alpha},$

where $(dq)_{\alpha}$ denotes a Lebesgue measure on a coordinate chart
containing ${\cal S}_{\alpha}$; its normalization depends on the restriction
of $\Theta$ to ${\cal S}_{\alpha}$. Since for $2\times 2$ matrices
$adj(A+B)=adj(A)+adj(B)$, the last expression can be rewritten as

$\displaystyle\frac{(p-1)}{2}\sum_{\alpha}\int_{{\cal
S}_{\alpha}}|\phi|^{p-2}\left(\left[\phi_{1},\phi_{2}\right](adj(G_{0})+\underset{\geq
0}{\underbrace{adj(G_{0}+Hess_{\mathbb{O}}(\phi))}})\left[\begin{array}[]{c}\phi_{\bar{1}}\\\
\phi_{\bar{2}}\end{array}\right]\right)(dq)_{\alpha}.$

Than last expression is greater than or equal to

$\displaystyle\frac{(p-1)}{2}\sum_{\alpha}\int_{{\cal
S}_{\alpha}}|\phi|^{p-2}\left(\left[\phi_{1},\phi_{2}\right]adj(G_{0})\left[\begin{array}[]{c}\phi_{\bar{1}}\\\
\phi_{\bar{2}}\end{array}\right]\right)(dq)_{\alpha}.$ (13.5)

Taking $p=2$ in (13.5) we have

$\displaystyle\frac{1}{2}\sum_{\alpha}\int_{{\cal
S}_{\alpha}}\left(\left[\phi_{1},\phi_{2}\right]adj(G_{0})\left[\begin{array}[]{c}\phi_{\bar{1}}\\\
\phi_{\bar{2}}\end{array}\right]\right)(dq)_{\alpha}\geq
const||\nabla\phi||^{2}_{L^{2}},$

where $const$ depends only on $M,G_{0},\Theta$, and the triangulation only.

Now let us take $p>2$ in (13.5). Since

$(|\phi|^{p/2})_{x_{i}^{a}}=\pm\frac{p}{2}\phi_{x_{i}^{a}}|\phi|^{p/2-1}$

pointwise, we get

$\displaystyle(\ref{E:ddd5})=\frac{2(p-1)}{p^{2}}\sum_{\alpha}\int_{{\cal
S}_{\alpha}}\left(\left[(|\phi|^{p/2})_{1},(|\phi|^{p/2})_{2}\right]adj(G_{0})\left[\begin{array}[]{c}(|\phi|^{p/2})_{\bar{1}}\\\
(|\phi|^{p/2})_{\bar{2}}\end{array}\right]\right)(dq)_{\alpha}\geq$
$\displaystyle const\frac{(p-1)}{p^{2}}||\nabla(|\phi|^{p/2})||_{L^{2}}^{2},$

where $const$ again depends only on $M,G_{0},\Theta$, and the triangulation
only. Q.E.D.

######  13.2 Lemma.

There exists a constant $c_{2}$ depending on $M,G_{0}$, and $\Theta$ only,
such that for any function $\psi$ from the Sobolev space $W^{1,2}(M)$,

$||\psi||^{2}_{L^{\frac{16}{7}}}\leq
c_{2}(||\nabla\psi||_{L^{2}}^{2}+||\psi||^{2}_{L^{2}}).$

Moreover, if $\psi$ satisfies $\int_{M}\psi\det G_{0}\cdot\Theta=0$, one has

$||\psi||^{2}_{L^{\frac{16}{7}}}\leq c_{2}||\nabla\psi||_{L^{2}}^{2}.$

Proof: By the Sobolev imbedding theorem there exists a constant $C^{\prime}$
such that, for any function $\psi\in W^{1,2}(M)$, one has

$||\psi||^{2}_{L^{\frac{16}{7}}}\leq
C^{\prime}(||\nabla\psi||^{2}_{L^{2}}+||\psi||^{2}_{L^{2}}).$

That gives the first inequality in the lemma.

If the function $\psi$ satisfies $\int_{M}\psi\det G_{0}\cdot\Theta=0$ then
one has $||\psi||_{L^{2}}^{2}\leq\tilde{C}||\nabla\psi||_{L^{2}}^{2}$ since
the second eigenvalue of the Laplacian on $M$ is strictly positive. This gives
Lemma 13.2 proven. Q.E.D.

######  13.3 Lemma.

In the notation of the beginning of this section, there exists a constant
$c_{3}$ depending on $M,G_{0},\Theta$ and $||f||_{C^{0}}$ only such that for
any $p\in[2,\frac{16}{7}]$ one has $||\phi||_{L^{p}}\leq c_{3}$.

Proof: By Lemma 13.1(b) we get

$\displaystyle||\nabla\phi||^{2}_{L^{2}}\leq$ $\displaystyle
const(1+\exp({||f||_{C^{0}}}))||\phi||_{L^{1}}$ $\displaystyle\leq$
$\displaystyle const^{\prime}\cdot(1+\exp({||f||_{C^{0}}}))||\phi||_{L^{2}}$

where the second inequality follows from the Hölder inequality. Since

$\int_{M}\phi\det G_{0}\cdot\Theta=0,$

we have

$||\phi||_{L^{2}}\leq c_{2}||\nabla\phi||_{L^{2}}.$

Hence $||\nabla\phi||_{L^{2}}\leq
c_{2}const^{\prime}\cdot(1+\exp({||f||_{C^{0}}}))$. Therefore by Lemma 13.2
there exists a constant $c_{3}^{\prime}$ depending on $M,G_{0},\Theta$, and
$||f||_{C^{0}}$ only such that

$||\phi||_{L^{\frac{16}{7}}}\leq c_{3}^{\prime}.$

Hence by the Hölder inequality $||\phi||_{L^{p}}\leq c_{3}$ for
$p\in[2,\frac{16}{7}]$. Q.E.D.

######  13.4 Lemma.

There exist positive constants $Q,c_{4}$ depending on $M,$ $G_{0},\Theta$, and
$||f||_{C^{0}}$ only such that for any $p\geq 2$

$||\phi||_{L^{p}}\leq Q(c_{4}p)^{-\frac{8}{p}}.$

Proof. Obviously there exists a constant $c_{5}>0$ depending only on
$M,G_{0},\Theta$ such that

$\displaystyle(\int_{M}\det G_{0}\cdot\Theta)^{\frac{1}{p}}<c_{5}\mbox{ for
all }p\geq 2.$ (13.8)

Define

$c_{4}:=(8/7)^{7}c_{2}(4c_{1}c_{5}e^{||f||_{C^{0}}}+1)$

where $c_{1},c_{2}$ are from Lemmas 13.1, 13.2. Choose $Q$ so that
$Q>c_{3}(c_{4}p)^{\frac{8}{p}}$ for $2\leq p\leq\frac{16}{7}$ and
$Q>(c_{4}p)^{\frac{8}{p}}$ for $2\leq p<\infty$.

We will prove the result by induction on $p$. By Lemma 13.3, if $2\leq
p\leq\frac{16}{7}$ then $||\phi||_{L^{p}}\leq c_{3}\leq
Q(c_{4}p)^{-\frac{8}{p}}$. For the inductive step, suppose that

$||\phi||_{L^{p}}\leq Q(c_{4}p)^{-\frac{8}{p}}\mbox{ for }2\leq p\leq
k,\,\mbox{ where }k\geq\frac{16}{7}\mbox{ is a real number}.$

We will show that

$||\phi||_{L^{q}}\leq Q(c_{4}q)^{-\frac{8}{q}}\mbox{ for
}2<q\leq\frac{16}{7}k,$

and, therefore, by induction Lemma 13.4 will be proven. Let $p\in(2,k]$. By
Lemma 13.1(a) we get

$\displaystyle||\nabla|\phi|^{p/2}||_{L^{2}}^{2}\leq
c_{1}\frac{p^{2}}{(p-1)}(1+e^{||f||_{C^{0}}})||\phi||^{p-1}_{L^{p-1}}.$ (13.9)

Applying Lemma 13.2 to $\psi=|\phi|^{p/2}$ we get

$\displaystyle||\phi||^{p}_{L^{\frac{8}{7}p}}\leq
c_{2}(||\nabla|\phi|^{p/2}||^{2}_{L^{2}}+||\phi||^{p}_{L^{p}}).$ (13.10)

Combining (13.10) and (13.9) we obtain

$||\phi||^{p}_{L^{\frac{8}{7}p}}\leq
c_{2}(4pc_{1}e^{||f||_{C^{0}}}||\phi||^{p-1}_{L^{p-1}}+||\phi||^{p}_{L^{p}}).$

Let $q:=\frac{8}{7}p$. Since $2<p\leq k$ we have by the induction assumption

$\displaystyle||\phi||_{L^{p}}\leq Q(c_{4}p)^{-\frac{8}{p}}.$ (13.11)

By the Hölder inequality
$||\phi||_{L^{p-1}}^{p-1}\leq||\phi||_{L^{p}}^{p-1}(\int_{M}\det
G_{0}\cdot\Theta)^{\frac{1}{p}}\overset{(\ref{E:c-5-def})}{\leq}c_{5}||\phi||_{L^{p}}^{p-1}$.

Hence we get

$\displaystyle||\phi||^{p}_{L^{q}}\leq
c_{2}\left(4pc_{1}c_{5}e^{||f||_{C^{0}}}||\phi||^{p-1}_{L^{p}}+||\phi||_{L^{p}}^{p}\right)\leq$
$\displaystyle
c_{2}\left(4pc_{1}c_{5}e^{||f||_{C^{0}}}(Q(c_{4}p)^{-\frac{8}{p}})^{p-1}+\left(Q(c_{4}p)^{-\frac{8}{p}}\right)^{p}\right).$

We have by the choice of $Q$

$Q(c_{4}p)^{-\frac{8}{p}}\geq 1.$

Hence,

$||\phi||^{p}_{L^{q}}\leq
c_{2}Q^{p}(c_{4}p)^{-8}(4pc_{1}c_{5}e^{||f||_{C^{0}}}+1).$

It remains to show that the last expression is at most
$Q^{p}(c_{4}q)^{-\frac{8}{q}p}.$ Equivalently it suffices to check that

$c_{2}(c_{4}p)^{-8}(4pc_{1}c_{5}e^{||f||_{C^{0}}}+1)\leq(c_{4}q)^{-7}=(\frac{8}{7}c_{4}p)^{-7}.$

In view that the left hand side is at most
$c_{2}(c_{4}p)^{-8}p(4c_{1}c_{5}e^{||f||_{C^{0}}}+1)$, it is enough to check
that

$c_{2}c_{4}^{-8}(4c_{1}c_{5}e^{||f||_{C^{0}}}+1)\leq(\frac{8}{7}c_{4})^{-7}.$

Namely,

$c_{4}\geq(8/7)^{7}c_{2}(4c_{1}c_{5}e^{||f||_{C^{0}}}+1)$

which holds by the definition of $c_{4}$. Q.E.D.

When $p\longrightarrow+\infty$ we get the main result of this section.

######  13.5 Theorem.

Let $M^{16}$ be a compact $SL_{2}(\mathbb{O})$-manifold with a positive
parallel $3/4$-density $\Theta$. Let $G_{0}$ be a $C^{\infty}$-smooth positive
definite section of the bundle $\underline{{\cal H}_{2}}(M)$ satisfying
condition (11.1). Let $\phi$ be a $C^{2}$-smooth solution of the Monge-Ampère
equation (11.3) satisfying $\int_{M}\phi\det G_{0}\cdot\Theta=0$. Then there
exists a constant $C$ depending on $M,G_{0},\Theta$, and $||f||_{C^{0}}$ such
that

$||\phi||_{C^{0}}\leq C.$

##  14 Elementary inequality.

The goal of this section is to prove Proposition 14.1 which will be used later
on in the proof of the second order estimates.

First recall that a $C^{2}$-smooth function
$u\colon\Omega\longrightarrow\mathbb{R}$ in a domain
$\Omega\subset\mathbb{O}^{2}$ is called strictly octonionic plurisubharmonic
(strictly oPSH) provided $Hess_{\mathbb{O}}(u)>0$ in $\Omega$ pointwise.

######  14.1 Proposition.

Let $u\in C^{4}$ be a strictly oPSH function such that at a given point $z$
its octonionic Hessian $(u_{\bar{i}j})$ is diagonal. Then at this point $z$
one has

$\displaystyle\sum_{p=0}^{7}\sum_{i,k=1}^{2}\frac{|u_{\bar{k}kx^{p}_{i}}|^{2}}{u_{\bar{i}i}u_{\bar{k}k}}\leq
4\sum_{p=0}^{7}\sum_{i,k,l=1}^{2}\frac{|u_{\bar{k}ix^{p}_{l}}|^{2}}{u_{\bar{i}i}u_{\bar{k}k}}.$
(14.1)

Proof. Let us fix now the indices $i,k$ and compare the summands containing
this pair of indices in both sides of (14.1).

First consider the case $i=k$. In the left hand side we have

$\displaystyle\sum_{p=0}^{7}\frac{|u_{\bar{k}kx^{p}_{k}}|^{2}}{u_{\bar{k}k}u_{\bar{k}k}}.$
(14.2)

In the right hand side of (14.1) we have

$\displaystyle
4\sum_{p=0}^{7}\sum_{l=1}^{2}\frac{|u_{\bar{k}kx^{p}_{l}}|^{2}}{u_{\bar{k}k}u_{\bar{k}k}}$
(14.3)

It is clear that $(\ref{C50})\leq(\ref{C51})$.

Let us consider the case now $i\neq k$. The left hand side of (14.1) contains
two summands with the pair $i,k$:

$\displaystyle\frac{1}{u_{\bar{i}i}u_{\bar{k}k}}\sum_{p}(|u_{\bar{k}kx^{p}_{i}}|^{2}+|u_{\bar{i}ix^{p}_{k}}|^{2}).$
(14.4)

The right hand side of (14.1) contains two summands with the pair $i,k$:

$\displaystyle\frac{4}{u_{\bar{i}i}u_{\bar{k}k}}\sum_{p,l}(|u_{\bar{k}ix^{p}_{l}}|^{2}+|u_{\bar{i}kx^{p}_{l}}|^{2})=\frac{8}{u_{\bar{i}i}u_{\bar{k}k}}\sum_{p,l}|u_{\bar{k}ix^{p}_{l}}|^{2}.$
(14.5)

To finish the proof of proposition, it suffices now to show that
$(\ref{C52})\leq(\ref{C53})$. Explicitly, after cancelling out the term
$u_{\bar{i}i}u_{\bar{k}k}$ on both sides, it reduces to

$\displaystyle\sum_{p}(|u_{\bar{k}kx^{p}_{i}}|^{2}+|u_{\bar{i}ix^{p}_{k}}|^{2})\leq
8\sum_{p,l}|u_{\bar{k}ix^{p}_{l}}|^{2}.$ (14.6)

In order to show such a general inequality it suffices to sum up in the right
hand side over $l=i,k$ only. Thus (14.6) follows from

$\displaystyle\sum_{p}(|u_{\bar{k}kx^{p}_{i}}|^{2}+|u_{\bar{i}ix^{p}_{k}}|^{2})\leq
8\sum_{p}(|u_{\bar{k}ix^{p}_{i}}|^{2}+|u_{\bar{k}ix^{p}_{k}}|^{2}).$ (14.7)

In the last inequality we may separate summands containing derivatives $kki$
and $kii$. These two inequalities are completely symmetric and obtained one
from the other by exchange $i$ by $k$. Thus it suffices to show that

$\displaystyle\sum_{p}|u_{\bar{k}kx^{p}_{i}}|^{2}\leq
8\sum_{p}|u_{\bar{k}ix^{p}_{k}}|^{2}=8\sum_{p}|u_{\bar{i}kx_{k}^{p}}|^{2}.$
(14.8)

Let us define two operators
$\overset{\leftarrow}{\partial_{k}},\overset{\leftarrow}{\partial_{\bar{k}}}$
acting on the space of octonionic valued functions:

$\displaystyle\Phi\overset{\leftarrow}{\partial_{k}}:=\sum_{p=0}^{7}\frac{\partial\Phi}{\partial
x^{p}_{k}}\bar{e}_{p},$
$\displaystyle\Phi\overset{\leftarrow}{\partial_{\bar{k}}}:=\sum_{p=0}^{7}\frac{\partial\Phi}{\partial
x^{p}_{k}}e_{p},$

where $\bar{e}_{p}$ denotes the octonionic conjugate of the octonionic unit
$e_{p}$ (here $p=0,...,7$).

Recall that $\Delta_{k}=\sum_{p=0}^{7}\frac{\partial^{2}}{(\partial
x^{k}_{p})^{2}}$ is the Laplacian with respect to the $k$-th octonionic
variable. Note that for real valued function $u$ one has
$\Delta_{k}u=u_{\bar{k}k}$.

Let us show that for an $\mathbb{O}$-valued function $\Psi$ one has

$\displaystyle\Delta_{k}\Psi=(\Psi\overset{\leftarrow}{\partial_{\bar{k}}})\overset{\leftarrow}{\partial_{k}}=(\Psi\overset{\leftarrow}{\partial_{k}})\overset{\leftarrow}{\partial_{\bar{k}}}.$
(14.9)

By Lemma 2.1(iii) for any octonions $a,b,c$ one has

$(ca)b+(c\bar{b})\bar{a}=c(ab+\bar{b}\bar{a}).$

Using this identity we have

$\displaystyle(\Psi\overset{\leftarrow}{\partial_{k}})\overset{\leftarrow}{\partial_{\bar{k}}}=\sum_{p,q=0}^{7}(\Psi_{x_{k}^{p}x_{k}^{q}}\bar{e}_{p})e_{q}=\frac{1}{2}\sum_{p,q=0}^{7}((\Psi_{x_{k}^{p}x_{k}^{q}}\bar{e}_{p})e_{q}+(\Psi_{x_{k}^{p}x_{k}^{q}}\bar{e}_{q})e_{p})=$
$\displaystyle\frac{1}{2}\sum_{p,q=0}^{7}\Psi_{x_{k}^{p}x_{k}^{q}}(\bar{e}_{p}e_{q}+\bar{e}_{q}e_{p})=$
$\displaystyle\sum_{p}\Psi_{x_{k}^{p}x_{k}^{p}}+\frac{1}{2}\sum_{p\neq
q}\Psi_{x_{k}^{p}x_{k}^{q}}(\bar{e}_{p}e_{q}+\bar{e}_{q}e_{p})=\Delta_{k}\Psi+\sum_{p\neq
q}\Psi_{x_{k}^{p}x_{k}^{q}}Re(\bar{e}_{p}e_{q})=$
$\displaystyle\Delta_{k}\Psi.$

Thus (14.9) is proven.

Let us take $\Phi=u_{\bar{i}}(=\sum_{q}e_{q}u_{x^{q}_{i}})$. Then (14.8) can
be rewritten

$\displaystyle|\Delta_{k}\Phi|^{2}\leq
8\sum_{p}|\Phi_{x^{p}_{k}}\overset{\leftarrow}{\partial_{k}}|^{2}.$ (14.10)

We have

$\Delta_{k}\Phi=(\Phi\overset{\leftarrow}{\partial_{k}})\overset{\leftarrow}{\partial_{\bar{k}}}=\sum_{q}(\Phi\overset{\leftarrow}{\partial_{k}})_{x^{q}_{k}}e_{q}.$

Denote $\Psi:=\Phi\overset{\leftarrow}{\partial_{k}}$. Then (14.10) takes the
form:

$\displaystyle|\sum_{q}\Psi_{x^{q}_{k}}e_{q}|\leq\sqrt{8\sum_{q}|\Psi_{x^{q}_{k}}|^{2}}.$
(14.11)

Indeed by the Cauchy-Schwartz inequality we have

$|\sum_{q=0}^{7}\Psi_{x^{q}_{k}}e_{q}|\leq\sum_{q=0}^{7}|\Psi_{x^{q}_{k}}|\leq\sqrt{8}\cdot\sqrt{\sum_{q=0}^{7}|\Psi_{x^{q}_{k}}|^{2}}.$

That completes the proof. Q.E.D.

##  15 A few elementary identities.

######  15.1 Claim.

Let $X\in{\cal H}_{2}(\mathbb{O})$. Then there exists a unique $Y\in{\cal
H}_{2}(\mathbb{O})$ such that $XY=YX=I_{2}$ if and only if $\det X\neq 0$.
(Such $Y$ is denoted by $X^{-1}$ as usual.)

Proof. The subalgebra generated by elements of $X$ is commutative, thus is
isomorphic either to $\mathbb{R}$ or $\mathbb{C}$. Over these field $X$ is
invertible, and the inverse matrix is necessarily Hermitian.

Conversely, let $XY=YX=I_{2}$. The subalgebra generated by elements of $X$ and
$Y$ is associative, thus one may assume that it is isomorphic to $\mathbb{H}$.
Over $\mathbb{H}$ the uniqueness of $X^{-1}$ is well known and proved as
usual. Thus elements of $Y$ belong to the commutative field generated by
elements of $X$, in fact $Y=\frac{1}{\det X}adj(X)$. This proves the
uniqueness of $Y$. Q.E.D.

We denote by $f_{a}$ the derivative of a function $f$ with respect to a real
direction $a$. Furthermore we denote

$tr(X):=Re(Tr(X)),$

where $Tr$ is the sum of diagonal elements as previously.

######  15.2 Claim.

Let $U$ be a smooth function with values in ${\cal H}_{2}(\mathbb{O})$. Then,

(1) $(\log(\det U))_{a}=tr(U^{-1}U_{a})$;

(2) $(U^{-1})_{a}=-U^{-1}U_{a}U^{-1}$.333The brackets in the right hand side
are not important since the algebra generated by elements of two Hermitian
matrices is associative.

Proof. (1) We have

$\displaystyle(\log(\det U))_{a}=(\det U)^{-1}\cdot
2\det(U,U_{a})\overset{(\ref{mixed-adj})}{=}$ $\displaystyle(\det U)^{-1}\cdot
tr(adj(U)U_{a})=tr(U^{-1}U_{a}).$

(2) is elementary. Q.E.D.

The proof of the following claim is also standard.

######  15.3 Claim.

Assume $\det U=F$. Then
(1) $tr(U^{-1}U_{a})=(\log F)_{a}$.
(2) $tr(U^{-1}U_{ab})-tr((U^{-1}U_{b}U^{-1})U_{a})=(\log F)_{ab}.$

######  15.4 Remark.

Given 3 matrices $X,Y,Z\in{\cal H}_{2}(\mathbb{O})$, $tr(XYXZ)$ does not
depend on bracketing inside. This follows from two facts: (1) $Re(abc)$ does
not depend on bracketing for $a,b,c\in\mathbb{O}$; (2) $XYX$ does not depend
on bracketing for $X,Y\in{\cal H}_{2}(\mathbb{O})$.

##  16 Diagonalization.

The following proposition seems to be folklore.

######  16.1 Proposition.

(1) For any $A\in{\cal H}_{2}(\mathbb{O})$ there exists a transformation from
$Spin(9)$ which maps $A$ to a real diagonal matrix.
(2)Let $A,B\in{\cal H}_{2}(\mathbb{O})$, $A>0$. Then there exists a
transformation from $Spin(1,9)$ which maps $A$ to a matrix proportional to
$I_{2}$, and $B$ to a diagonal matrix.

Proof. (1) The entries of $A$ are contained in a subfield isomorphic to
$\mathbb{C}$. Let us assume, without loss of generality and for simplicity of
notation, that the latter subfield is $\mathbb{C}$.

Recall that any $L\in\mathfrak{sl}_{2}(\mathbb{O})$ acts on $A$ as
$A\mapsto-L^{*}A-AL=:\hat{L}(A)$. If
$L\in\mathfrak{sl}_{2}(\mathbb{C})\subset\mathfrak{sl}_{2}(\mathbb{O})$ it
follows that

$e^{\hat{L}}(A)=(e^{-L})^{*}Ae^{-L}.$

Hence the subgroup $SL_{2}(\mathbb{C})\subset SL_{2}(\mathbb{O})$ acts on
${\cal H}_{2}(\mathbb{O})$ as

$X(A)=(X^{-1})^{*}AX^{-1}.$

Note that any complex Hermitian matrix can be diagonalized using such
transformation with above $X\in SU(2)$ being a complex matrix. The latter
subgroup is contained in $Spin(9)$.

(2) By part (1) we may apply a transformation from $Spin(9)$ which
diagonalizes $A$, i.e. makes it
$A^{\prime}=\left[\begin{array}[]{cc}\lambda_{1}&0\\\
0&\lambda_{2}\end{array}\right]$ where $\lambda_{1,2}>0$. Then replace
$A^{\prime}$ with $\left[\begin{array}[]{cc}r&0\\\
0&r^{-1}\end{array}\right]A^{\prime}\left[\begin{array}[]{cc}r&0\\\
0&r^{-1}\end{array}\right]$ with $r\in\mathbb{R}\backslash\\{0\\}$. For an
appropriate $r$ we can get $cI_{2}$. Since $Spin(9)$ preserves the identity
matrix $I_{2}$, we may apply further an appropriate transformation from
$Spin(9)$ and get the first matrix equal to $cI_{2}$ and the other one
diagonal. Q.E.D.

##  17 Second order estimates.

  1. 1

Let $M$ be a compact connected $SL_{2}(\mathbb{O})$-manifold with a parallel
positive $3/4$-density $\Theta$. Let $G_{0}$ be a $C^{\infty}$-smooth positive
definite section of $\underline{{\cal H}_{2}}(M)$. Also assume that $f\in
C^{\infty}(M,\mathbb{R})$ and $\phi\in C^{4}(M,\mathbb{R})$ satisfy

$\displaystyle\det(G_{0}+Hess_{\mathbb{O}}(\phi))=e^{f}\det(G_{0}).$ (17.1)

In this section we make the following strong assumption on $M$: $M$ is a
$Spin(9)$-manifold. It follows that there exists a positive
$C^{\infty}$-smooth section $G_{00}$ of $\underline{{\cal H}}_{2}(M)$ which is
flat with respect to the natural (corresponding to the
$GL_{2}(\mathbb{O})$-structure) connection on $M$.

  2. 2

Let us define the second order linear differential operator on real valued
functions

$\Delta f=tr(G_{00}^{-1}Hess_{\mathbb{O}}(f)),$

(recall that $tr=Re\,Tr$). $\Delta$ is globally defined. Indeed, this operator
is equal to the derivative of $U\mapsto\log\frac{\det U}{\Theta^{1/3}}$ at
$U=G_{00}$ in the direction $Hess_{\mathbb{O}}(f)$.

By Proposition 16.1 locally there exist coordinates (consistent with the
$GL_{2}(M)$\- rather than the $Spin(9)$-structure on $M$) in which
$G_{00}=Id$. In such coordinates the operator $\Delta$ is the usual flat
Laplacian

$\Delta f=f_{\bar{1}1}+f_{\bar{2}2}.$

Let us denote $U:=G_{0}+Hess_{\mathbb{O}}\phi$. Lemma 12.2 implies that $U>0$.
Define another linear second order differential operator

$\Delta^{\prime}f=Tr(U^{-1}Hess_{\mathbb{O}}(f)).$

Both $\Delta$ and $\Delta^{\prime}$ are elliptic with vanishing free term.

  3. 3

Below we will make some local computations in coordinates system where
$G_{00}=Id$ and $G_{0}+Hess_{\mathbb{O}}\phi=U=Hess_{\mathbb{O}}u$. Thus $u$
is strictly oPSH. Clearly, $\Delta u=tr(U)$.

In such coordinate system we have

$\Delta^{\prime}(f)=\sum_{i,j=1}^{2}Re((U^{-1})_{\bar{i}j}f_{\bar{j}i}).$

######  17.1 Proposition.

Let $u\in C^{4}$ be a strictly oPSH function. Let $\gamma$ be a smooth
function of one real variable. Choose (locally) coordinates such that
$G_{00}\equiv Id$. Then, in these coordinates,

$\displaystyle\,\,[\gamma(\Delta u)]_{ab}=\gamma^{\prime\prime}(\Delta u)\cdot
tr(U_{a})\cdot tr(U_{b})+\gamma^{\prime}(\Delta u)tr(U_{ab}).$

If moreover at a point $z$ the matrix $U(z)=(u_{\bar{i}j}(z))$ is diagonal
then at this point $z$ one has

$\displaystyle\Delta^{\prime}(\gamma(\Delta u))=\gamma^{\prime\prime}(\Delta
u)\sum_{i,p}\frac{1}{u_{\bar{i}i}}(trU_{x^{p}_{i}})^{2}+\gamma^{\prime}(\Delta
u)\sum_{i,k}\frac{u_{\bar{i}i\bar{k}k}}{u_{\bar{i}i}}.$

Proof. The first part is straightforward. For the second part, using the first
part, we have

$\displaystyle\Delta^{\prime}(\gamma(\Delta u))=\sum_{i}\frac{(\gamma(\Delta
u))_{\bar{i}i}}{u_{\bar{i}i}}=\sum_{i,p}\frac{(\gamma(\Delta
u))_{x_{i}^{p}x_{i}^{p}}}{u_{\bar{i}i}}=$
$\displaystyle\gamma^{\prime\prime}(\Delta
u)\sum_{i,p}\frac{(Tr(U_{x_{i}^{p}}))^{2}}{u_{\bar{i}i}}+\gamma^{\prime}(\Delta
u)\sum_{i,p}\sum_{k}\frac{(u_{\bar{k}k})_{x_{i}^{p}x_{i}^{p}}}{u_{\bar{i}i}}=$
$\displaystyle\gamma^{\prime\prime}(\Delta
u)\sum_{i,p}\frac{(Tr(U_{x_{i}^{p}}))^{2}}{u_{\bar{i}i}}+\gamma^{\prime}(\Delta
u)\sum_{i,k}\frac{u_{\bar{i}i\bar{k}k}}{u_{\bar{i}i}}$

Q.E.D.

  4. 4

######  17.2 Lemma.

(1) Let $\det U=F$. Then one has444The bracketing in the expression below is
not important.

$Tr(U^{-1}\Delta
U)-\sum_{i,p}Tr(U^{-1}U_{x_{i}^{p}}U^{-1}U_{x_{i}^{p}})=\Delta(\log F).$

(2) Assume that at a point $z$ the matrix $U(z)=(u_{\bar{i}j}(z))$ is
diagonal. Then, the equality above is equivalent to

$\sum_{i,k}\frac{u_{\bar{i}i\bar{k}k}}{u_{\bar{i}i}}-\sum_{i,l,n,p}\frac{|u_{\bar{l}nx_{i}^{p}}|^{2}}{u_{\bar{l}l}u_{\bar{n}n}}=\Delta(\log
F).$

Proof. Part (2) follows directly from part (1). Part (1) follows from Claim
15.3(2) by taking there $a=b$ corresponding to $x_{i}^{p}$ and summing over
$i,p$. Q.E.D.

######  17.3 Proposition.

Let $u\in C^{4}$ be strictly oPSH. At any point $z\in M$ one has

$\Delta^{\prime}((\Delta u)^{3/4})\geq\frac{3}{4}(\Delta u)^{-1/4}\Delta(\log
F),$

where $F$ is as in Lemma 17.2.

Proof. Fix $z\in M$. By Proposition 16.1 we may and will choose coordinates
near $z$ such that $G_{00}(z)=Id$ and $U(z)$ is diagonal.

Combining results of Lemma 17.2 into Proposition 17.1 we get

$\displaystyle\Delta^{\prime}(\gamma(\Delta u))=\gamma^{\prime\prime}(\Delta
u)\sum_{i,p}\frac{1}{u_{\bar{i}i}}(TrU_{x_{p}^{i}})^{2}+\gamma^{\prime}(\Delta
u)(\sum_{i,l,n,p}\frac{|u_{\bar{l}nx_{i}^{p}}|^{2}}{u_{\bar{l}l}u_{\bar{n}n}}+\Delta(\log
F)).$

Equivalently

$\displaystyle\Delta^{\prime}(\gamma(\Delta u))=\gamma^{\prime\prime}(\Delta
u)\sum_{i,p}\frac{1}{u_{\bar{i}i}}(\sum_{k}u_{\bar{k}kx_{i}^{p}})^{2}+\gamma^{\prime}(\Delta
u)(\sum_{i,l,n,p}\frac{|u_{\bar{l}nx_{i}^{p}}|^{2}}{u_{\bar{l}l}u_{\bar{n}n}}+\Delta(\log
F)).$ (17.2)

Let us assume that $\gamma$ is concave, i.e. $\gamma^{\prime\prime}\leq 0$.
Then,

$\displaystyle\Delta^{\prime}(\gamma(\Delta u))=\gamma^{\prime\prime}(\Delta
u)\sum_{i,p}\frac{1}{u_{\bar{i}i}}\left(\sum_{k}\sqrt{u_{\bar{k}k}}\frac{u_{\bar{k}kx_{i}^{p}}}{\sqrt{u_{\bar{k}k}}}\right)^{2}+\gamma^{\prime}(\Delta
u)(\sum_{i,l,n,p}\frac{|u_{\bar{l}nx_{i}^{p}}|^{2}}{u_{\bar{l}l}u_{\bar{n}n}}+\Delta(\log
F))\geq$ $\displaystyle\gamma^{\prime\prime}(\Delta
u)\sum_{i,p}\frac{1}{u_{\bar{i}i}}\cdot\Delta
u\cdot\left(\sum_{k}\frac{|u_{\bar{k}kx_{i}^{p}}|^{2}}{u_{\bar{k}k}}\right)+\gamma^{\prime}(\Delta
u)(\sum_{i,l,n,p}\frac{|u_{\bar{l}nx_{i}^{p}}|^{2}}{u_{\bar{l}l}u_{\bar{n}n}}+\Delta(\log
F))=$ $\displaystyle\Delta u\cdot\gamma^{\prime\prime}(\Delta
u)\sum_{i,k,p}\frac{|u_{\bar{k}kx_{i}^{p}}|^{2}}{u_{\bar{i}i}u_{\bar{k}k}}+\gamma^{\prime}(\Delta
u)\left(\sum_{i,l,n,p}\frac{|u_{\bar{l}nx_{i}^{p}}|^{2}}{u_{\bar{l}l}u_{\bar{n}n}}+\Delta(\log
F)\right).$

Let us finally take $\gamma(t)=t^{3/4}$. It is concave for $t>0$. Then we get

$\displaystyle\Delta^{\prime}((\Delta u)^{3/4})\geq\frac{3}{4}(\Delta
u)^{-1/4}\left(\sum_{i,l,n,p}\frac{|u_{\bar{l}nx_{i}^{p}}|^{2}}{u_{\bar{l}l}u_{\bar{n}n}}-\frac{1}{4}\sum_{i,k,p}\frac{|u_{\bar{k}kx_{i}^{p}}|^{2}}{u_{\bar{i}i}u_{\bar{k}k}}\right)+\frac{3}{4}(\Delta
u)^{-1/4}\Delta(\log F).$

By Proposition 14.1 the first summand is non-negative. The result follows.
Q.E.D.

  5. 5

Let us state the main result of this section.

######  17.4 Theorem.

Let $M^{16}$ be a compact connected $Spin(9)$-manifold. Let $G_{00}$ be a
$C^{\infty}$-smooth positive definite section $\underline{{\cal H}_{2}}(M)$
parallel with respect to the flat connection corresponding to the affine
structure of $M$.555Such a section $G_{00}$ does exist on any
$Spin(9)$-manifold. It necessarily satisfies the condition (11.1). Let $G_{0}$
be another $C^{\infty}$-smooth positive definite section of $\underline{{\cal
H}_{2}}(M)$ satisfying the condition (11.1). Assume also that $f\in
C^{2}(M,\mathbb{R})$ and $\phi\colon M\longrightarrow\mathbb{R}$ is a
$C^{4}$-smooth solution of the quaternionic Monge-Ampère equation

$\displaystyle\det(G_{0}+Hess_{\mathbb{H}}\phi)=e^{f}\det G_{0}.$ (17.3)

Then there exists a constant $C$ depending on $M,G_{00},G_{0}$, and
$||f||_{C^{2}(M)}$ such that

$||\Delta\phi||_{C^{0}}\leq C.$

Proof. At the point $z_{0}$ of minimum of $\phi$ one has
$Hess_{\mathbb{O}}(\phi(z_{0}))\geq 0$. Hence
$G_{0}(z_{0})+Hess_{\mathbb{O}}\phi(z_{0})\geq 0$. But
$\det(G_{0}+Hess_{\mathbb{O}}(\phi))>0$ on $M$. Hence by Lemma 12.2

$\displaystyle Hess_{\mathbb{O}}(\phi)>-G_{0}.$

Hence it remains to obtain an upper bound on $\Delta\phi$.

Let us consider the function

$T:=\frac{4}{3}[tr(G^{-1}_{00}\cdot(G_{0}+Hess_{\mathbb{O}}(\phi)))]^{3/4}-\phi.$

This function is globally defined (this can be proven in the same way as in
paragraph 2 of this section for $\Delta$. In order to prove the theorem, it
suffices to show that it is bounded from above. Let $z\in M$ be a point of
maximum of the function $T$. We clearly have

$\Delta^{\prime}(T)(z)\leq 0.$

Near $z$ let us represent $G_{0}=Hess_{\mathbb{O}}(u_{0})$. Set
$u:=u_{0}+\phi$ near the point $z$; this function is strictly oPSH. Denote
$U:=Hess_{\mathbb{O}}(u)=G_{0}+Hess_{\mathbb{O}}(\phi)$.

Let us choose coordinates near $z$ such that $G_{00}$ is equal to $I_{2}$, and
$U(z)$ is diagonal; this is possible thanks to Proposition 16.1.

Then, near $z$ one has $T=\frac{4}{3}(\Delta u)^{3/4}-\phi.$ Consequently,

$\displaystyle 0\geq\Delta^{\prime}(\frac{4}{3}(\Delta
u(z))^{3/4})-\Delta^{\prime}\phi=\Delta^{\prime}(\frac{4}{3}(\Delta
u(z))^{3/4})-tr(U^{-1}(U-G_{0}))|_{z}=$
$\displaystyle\Delta^{\prime}(\frac{4}{3}(\Delta
u(z))^{3/4})-2+tr(U^{-1}G_{0})\big{|}_{z}=\Delta^{\prime}(\frac{4}{3}(\Delta
u(z))^{3/4})-2+\sum_{i=1}^{2}\frac{(G_{0})_{\bar{i}i}}{u_{\bar{i}i}}\big{|}_{z}\geq$
$\displaystyle\Delta^{\prime}(\frac{4}{3}(\Delta
u(z))^{3/4})-2+C_{1}\sum_{i=1}^{2}\frac{1}{u_{\bar{i}i}(z)},$

where $C_{1}>0$. Applying now Proposition 17.3 we get for $C_{2}>0$

$\displaystyle 0\geq(\Delta u(z))^{-1/4}(\Delta\log
F)-2+C_{1}\sum_{i}\frac{1}{u_{\bar{i}i}(z)}\geq-C_{2}(\Delta
u(z))^{-1/4}-2+C_{1}\sum_{i}\frac{1}{u_{\bar{i}i}(z)},$

where $F=e^{f}\det G_{0}$ as above. This is equivalent to

$\displaystyle(u_{\bar{1}1}(z)+u_{\bar{2}2}(z))^{-1/4}\geq
C_{3}(\frac{1}{u_{\bar{1}1}(z)}+\frac{1}{u_{\bar{2}2}(z)})-C_{4},$ (17.4)

where $C_{3},C_{4}>0$. Recall also that

$\displaystyle u_{\bar{1}1}(z)u_{\bar{2}2}(z)=F(z).$ (17.5)

Without loss of generality we may assume that

$\displaystyle u_{\bar{1}1}(z)\geq u_{\bar{2}2}(z).$ (17.6)

Then (17.4) implies that

$\displaystyle[u_{\bar{1}1}(z)]^{-1/4}\geq\frac{C_{3}}{F}u_{\bar{1}1}(z)-C_{4}.$
(17.7)

If $u_{\bar{1}1}(z)\leq\frac{C_{4}+1}{C_{3}}F$ then the theorem is proven. Let
us assume otherwise. Then (17.7) implies

$[u_{\bar{1}1}(z)]^{-1/4}\geq 1.$

Hence $u_{\bar{1}1}(z)\leq 1$ and the theorem is proven again. Q.E.D.

##  18 A weak Harnack inequality.

Let us recall now the weak Harnack inequality (see e.g. [27], Theorem 8.18, or
[29], Theorem 4.15). Below we normalize everywhere the Lebesgue measure on
$\mathbb{R}^{N}$ by $vol(B_{1})=1$ where $B_{1}$ is Euclidean unit ball.

######  18.1 Theorem (weak Harnack inequality).

Let $B_{R}\subset\mathbb{R}^{N}$ be an open Euclidean ball of radius $R$. Let
$(a_{ij})_{i,j=1}^{N}\in L^{\infty}(B_{R})\cap C^{1}(B_{R})$ be a symmetric
matrix satisfying uniform elliptic estimate

$\lambda||\xi||^{2}\leq\sum_{i,j}a_{ij}(x)\xi_{i}\xi_{j}\leq\Lambda||\xi||^{2},\mbox{
for all }\xi\in\mathbb{R}^{N},$

with $\lambda,\Lambda>0$. Let $v\in C^{2}(B_{R})$ be a function satisfying

$\displaystyle v\geq 0,$ $\displaystyle\sum_{i,j}D_{j}(a_{ij}D_{i}v)\leq\psi,$

where $\psi\in L^{\infty}(B_{R})$. Then, for any $0<\theta<\tau<1$ we have

$\inf_{B_{\theta R}}v+R||\psi||_{L^{\infty}(B_{R})}\geq
C\left(R^{-N}\int_{B_{\tau R}}v\right),$

where the constant $C$ depends only on $\lambda,\Lambda,\theta,\tau,N$.

######  18.2 Remark.

Theorem 18.1 also holds in the following weaker form. We will take
$R=4r,\theta=1/4,\tau=1/2$. Then,

$\displaystyle r^{-N}\int_{B_{r}}v\leq C^{\prime}(\inf_{B_{r}}v+r),$ (18.1)

where the constant $C^{\prime}$ depends on
$\lambda,\Lambda,||\psi||_{L^{\infty}(B_{4r})},N$ only.

In the next Lemma 18.3 and Proposition 18.4 we show that the operator ${\cal
D}h:=\det U\cdot tr(U^{-1}\cdot
Hess_{\mathbb{O}}(h))=\det(U,Hess_{\mathbb{O}}(h))$, where
$U=Hess_{\mathbb{O}}(u)$, can be written in the divergence form, and hence
later we will be able to apply to it Theorem 18.1.

######  18.3 Lemma.

Let $u,h\colon\mathbb{O}^{2}\longrightarrow\mathbb{R}$ be smooth functions.
Denote $U=Hess_{\mathbb{O}}(u),H=Hess_{\mathbb{O}}(h)\in{\cal
H}_{2}(\mathbb{O})$. Then,

$\det U\cdot
tr(U^{-1}H)=u_{\bar{2}2}h_{\bar{1}1}+u_{\bar{1}1}h_{\bar{2}2}-2\sum_{r,s}Re\left(\bar{e}_{r}u_{\bar{1}2}e_{s}\right)h_{x_{1}^{r}x_{2}^{s}}.$

Proof. By Section 10, paragraph 1, we have

$(\det U)\cdot
U^{-1}=adj(U)=\left[\begin{array}[]{cc}u_{\bar{2}2}&-u_{\bar{1}2}\\\
-u_{\bar{2}1}&u_{\bar{1}1}\end{array}\right].$

Hence the left hand side of the above equality is equal to

$\displaystyle tr\left(\left[\begin{array}[]{cc}u_{\bar{2}2}&-u_{\bar{1}2}\\\
-u_{\bar{2}1}&u_{\bar{1}1}\end{array}\right]\cdot\left[\begin{array}[]{cc}h_{\bar{1}1}&h_{\bar{1}2}\\\
h_{\bar{2}1}&h_{\bar{2}2}\end{array}\right]\right)=u_{\bar{2}2}h_{\bar{1}1}+u_{\bar{1}1}h_{\bar{2}2}-2Re(u_{\bar{1}2}h_{\bar{2}1})=$
$\displaystyle
u_{\bar{2}2}h_{\bar{1}1}+u_{\bar{1}1}h_{\bar{2}2}-2Re(u_{\bar{1}2}(\sum_{rs}e_{s}h_{x_{1}^{r}x_{2}^{s}}\bar{e}_{r})).$

The lemma follows. Q.E.D.

Let ${\cal D}(h)=\sum_{m,n}a_{mn}\partial_{m}\partial_{n}h$ be a second order
differential operator in a real space $\mathbb{R}^{N}$. We say that ${\cal D}$
can be written in the divergence form if

${\cal D}(h)=\sum_{m,n}\partial_{m}(a_{mn}\partial_{n}h),$

or equivalently

$\displaystyle\sum_{m}\partial_{m}a_{mn}=0\mbox{ for any }n.$ (18.3)

######  18.4 Proposition.

Let $U=Hess_{\mathbb{O}}(u)$ where $u$ is strictly oPSH. Let

${\cal D}(h)=\det U\cdot tr(U^{-1}\cdot Hess_{\mathbb{O}}(h)).$

Then ${\cal D}$ can be written in the divergence form.

Proof. First assume in (18.3) that $x_{n}=x_{1}^{r}$. Then the left hand side
in the expression above is equal to

$\displaystyle\partial_{x_{1}^{r}}u_{\bar{2}2}-\sum_{s}\partial_{x_{2}^{s}}\left(Re(\bar{e}_{r}u_{\bar{1}2}e_{s})\right)=$
$\displaystyle\Delta_{2}u_{x_{1}^{r}}-Re(\bar{e}_{r}(\sum_{s}u_{\bar{1}2,x_{2}^{s}}e_{s}))=$
$\displaystyle\Delta_{2}u_{x_{1}^{r}}-Re\left[\bar{e}_{r}\left((u_{\bar{1}}\overset{\leftarrow}{\partial_{2}})\overset{\leftarrow}{\partial_{\bar{2}}}\right)\right]\overset{(\ref{E:derivatives})}{=}$
$\displaystyle\Delta_{2}u_{x_{1}^{r}}-Re(\bar{e}_{r}\Delta_{2}u_{\bar{1}})=$
$\displaystyle\Delta_{2}u_{x_{1}^{r}}-\Delta_{2}(Re(\bar{e}_{r}u_{\bar{1}}))=0.$

The case $x_{n}=x_{2}^{s}$ is considered in the same manner. Q.E.D.

##  19 An auxiliary estimate for oPSH functions.

Let $L\subset\mathbb{O}^{2}$ be an affine octonionic line. Let
$v_{0},v_{1},\dots,v_{7}$ be an orthonormal basis of the linear subspace
parallel to $L$. Consider the linear differential operator

$\Delta_{L}:=\sum_{p=0}^{7}\frac{\partial^{2}}{\partial v_{p}^{2}}.$

Clearly $\Delta_{L}$ is independent of the choice of a basis.

######  19.1 Lemma.

Let $u\in C^{4}$ be strictly oPSH function on an open subset ${\cal
O}\subset\mathbb{O}^{2}$ such that

$MA_{\mathbb{O}}(u)=F>0,$

where $F\in C^{\infty}(M,\mathbb{R})$. Then for any affine octonionic line $L$
we have

$\displaystyle tr(U^{-1}\cdot\Delta_{L}U)\geq\Delta_{L}(\log F).$ (19.1)

Proof. Let $v_{0},\dots,v_{7}$ be an orthonormal basis of the linear subspaces
parallel to $L$. Then $\Delta_{L}=\sum_{p=0}^{7}\frac{\partial^{2}}{(\partial
v_{p})^{2}}$. It is enough to show that

$\displaystyle tr\left(U^{-1}\cdot\frac{\partial^{2}U}{\partial
v_{p}^{2}}\right)\geq\frac{\partial^{2}(\log F)}{\partial v_{p}^{2}}$ (19.2)

for any $p=0,\dots,7$. By Claim 15.3(2) we have

$\displaystyle tr(U^{-1}\cdot\frac{\partial^{2}U}{\partial
v_{p}^{2}})=\frac{\partial^{2}(\log F)}{\partial
v_{p}^{2}}+tr(U^{-1}\frac{\partial U}{\partial v_{p}}U^{-1}\frac{\partial
U}{\partial v_{p}}).$

In order to prove the lemma, it suffices to show that $tr(U^{-1}\frac{\partial
U}{\partial v_{p}}U^{-1}\frac{\partial U}{\partial v_{p}})\geq 0$. More
generally let us show that if $A,B\in{\cal H}_{2}(\mathbb{O})$ and $A>0$ then

$tr(A^{-1}BA^{-1}B)\geq 0.$

All elements of $A,B$ belong to the associative subfield generated by
$a_{\bar{1}2},b_{\bar{1}2}$. Any such subfield is contain in a subfield
isomorphic to quaternions $\mathbb{H}$. Note that the matrices $A$ and $B$ can
be diagonalized simultaneously over $\mathbb{H}$. More precisely there exists
an invertible quaternionic matrix $X$ and a real diagonal matrix $D$ such that
$A=X^{*}X$ and $B=X^{*}DX$.

Then

$\displaystyle tr(A^{-1}BA^{-1}B)=tr(X^{-1}D^{2}X)=tr(D^{2})\geq 0.$

Lemma is proved. Q.E.D.

##  20 A proposition from linear algebra.

  1. 1

Let $A\in{\cal H}_{2}(\mathbb{O})$. Its characteristic polynomial
$p_{A}(t)=\det(A-tI_{2})$ has two real roots which are called spectrum of $A$.
Indeed, the subfield generated by $a_{\bar{1}2}$ is associative and
commutative, and the result follows from the case of complex Hermitian
matrices.

The action of $Spin(9)$ on ${\cal H}_{2}(\mathbb{O})$ preserves $\det$ and
hence the characteristic polynomial. Hence spectrum of a matrix does not
change under the $Spin(9)$-action.

  2. 2

Let $\zeta\in\mathbb{O}^{2}$ be a vector. Define the octonionic Hermitian
matrix

$\zeta\otimes\zeta^{*}:=(\zeta_{i}\bar{\zeta}_{j}).$

######  20.1 Claim.

The matrix $\zeta\otimes\zeta^{*}$ is non-negative definite.

Proof. It suffices to show that for any $\varepsilon>0$ the matrix

$\zeta\otimes\zeta^{*}+\varepsilon
I_{2}=\left[\begin{array}[]{cc}|\zeta_{1}|^{2}+\varepsilon&\zeta_{1}\bar{\zeta}_{2}\\\
\zeta_{2}\bar{\zeta}_{1}&|\zeta_{2}|^{2}+\varepsilon\end{array}\right]$

is positive definite. This follows from Proposiiton 5.4 (the Sylvester
criterion). Indeed the diagonal entries are positive, and the determinant is
equal to

$(|\zeta_{1}|^{2}+\varepsilon)(|\zeta_{2}|^{2}+\varepsilon)-|\zeta_{1}\bar{\zeta}_{2}|^{2}=\varepsilon^{2}+\varepsilon(|\zeta_{1}|^{2}+|\zeta_{2}|^{2})>0.$

Q.E.D.

  3. 3

We will need the following linear algebraic proposition which is completely
analogous to the real, complex, and quaternionic cases (see [27], Lemma 17.13,
for the real case; for the complex case [50], p. 103, or [18], Lemma 5.17; and
for the quaternionic case see [9], Lemma 4.9.

######  20.2 Proposition.

Fix $0<\lambda<\Lambda<\infty$. Then there exist a natural number $N$, unit
vectors $\zeta_{1},\dots,\zeta_{N}\in\mathbb{O}^{2}$ such that at least one of
the two coordinates of each $\zeta_{i}$ is real, and
$0<\lambda_{*}<\Lambda_{*}<\infty$ depending on $\lambda,\Lambda$ only such
that every $A\in{\cal H}_{2}(\mathbb{O})$ whose spectrum lies in
$[\lambda,\Lambda]$ can be written

$\sum_{k=1}^{N}\beta_{k}\zeta_{k}\otimes\zeta_{k}^{*}\mbox{ with
}\beta_{k}\in[\lambda_{*},\Lambda_{*}].$

Moreover, the vectors $\zeta_{1},\dots,\zeta_{N}$ can be chosen to contain the
vectors $(1,0),(0,1)\in\mathbb{O}^{2}$.

Proof. The proof is standard in most part. For convenience of the reader we
outline the argument. Let us denote by $K$ the set of octonionic Hermitian
matrices whose spectrum lies in $[\lambda,\Lambda]$. This is a compact subset
of the interior of the cone of positive definite octonionic Hermitian
matrices; we denote the latter open cone by ${\cal C}$. Then there exists a
convex compact polytope $P\subset{\cal C}$ which contains a neighborhood of
$K$. Using a diagonalization (over a subfield of $\mathbb{O}$ isomorphic to
$\mathbb{C}$), every matrix $X\in{\cal C}$, in particular any vertex of $P$,
can be written in the form

$\displaystyle
X=\sum_{i=1}^{2}\alpha_{i}(X)\left(\xi_{i}(X)\otimes\xi_{i}^{*}(X)\right),$
(20.1)

where $\alpha_{i}(X)>0$, and $\xi_{i}(X)\in\mathbb{O}^{2}$ are unit vectors
with one of the two coordinates being real. Let us fix such a decomposition
for any vertex of $P$. Let us define a new finite subset $S_{1}\subset{\cal
C}$ as follows

$\displaystyle
S_{1}:=\left\\{\left(\sum_{j}\alpha_{j}(X)\right)\cdot\left(\xi_{i}(X)\otimes\xi_{i}^{*}(X)\right)|\,X\in
Vert(P),\alpha_{j}(X),\xi_{i}(X)\mbox{ satisfy }(\ref{E:2-1})\right\\}.$

Let us add to $S_{1}$ matrices of the form $e_{i}\otimes e_{i}^{*}$, $i=1,2$,
where $e_{1}=(1,0),e_{2}=(0,1)$. This set we denote by $S$. It is clear that
$P\subset conv(S)$. Hence $conv(S)$ contains a neighborhood of $K$. Now the
lemma follows from the following general fact which is left to the reader (and
where one takes $Q=conv(S)$): Let $K$ be a compact subset of a finite
dimensional real vector space which is contained in the interior of a compact
convex polytope $Q$. Then there exists $\varepsilon>0$ such that any point
$x\in K$ can be written as a convex combination of vertices of $Q$:

$x=\sum_{v}\beta(v)v\mbox{ with }\beta(v)>\varepsilon,$

where the sum runs over all vertices of $Q$, $\sum_{v}\beta(v)=1$. Q.E.D.

Let us state one more estimate on matrices to be used later.

######  20.3 Lemma.

Let $A,B\in{\cal H}_{2}(\mathbb{O})$ and $A,B>0$. Then

$tr(A^{-1}(A-B))\leq\frac{2}{\sqrt{\det A}}(\sqrt{\det A}-\sqrt{\det B}).$

Proof. The elements of $A,B$ generate an associative subfield contained in a
subfield isomorphic to $\mathbb{H}$. Over $\mathbb{H}$ the statement was
proved in Lemma 4.8 in [9] (using simultaneous diagonalization of $A$ and $B$
and directly applying the arithmetic-geometric mean inequality). Q.E.D.

##  21 $C^{2,\alpha}$-estimate.

  1. 1

In this section $M$ is a compact connected $Spin(9)$-manifold with a flat
$C^{\infty}$-smooth positive definite section $G_{00}$ of $\underline{{\cal
H}_{2}}(M)$. We denote as in Section 17, paragraph 2, the operator

$\Delta f=tr(G_{00}^{-1}Hess_{\mathbb{O}}(f)).$

$\Delta$ is elliptic, and one can choose local coordinates consistent with the
$Spin(9)$-structure such that $\Delta$ is proportional to the usual flat
Laplacian on $\mathbb{O}^{2}=\mathbb{R}^{16}$.

Furthermore we assume as previously that $f\in C^{\infty}(M,\mathbb{R})$, and
$G_{0}$ is a $C^{\infty}$-smooth positive definite section of
$\underline{{\cal H}_{2}}(M)$. Let $\phi\in C^{4}$ satisfy the Monge-Ampère
equation

$\det(G_{0}+Hess_{\mathbb{O}}(\phi))=e^{f}\det G_{0}.$

  2. 2

The main result of this section is

######  21.1 Theorem.

There exist $\alpha\in(0,1)$ and $C>0$, both depending on $M$,$G_{0}$,
$||f||_{C^{2}(M)}$,$||\phi||_{C^{0}(M)}$,$||\Delta\phi||_{C^{0}(M)}$ only,
such that

$||\phi||_{C^{2,\alpha}(M)}\leq C.$

This theorem immediately follows from the below local estimate.

######  21.2 Theorem.

Let $u\in C^{4}$ be a strictly oPSH function in an open subset ${\cal
O}\subset\mathbb{O}^{2}$ satisfying

$\displaystyle\det(Hess_{\mathbb{O}}(u))=F>0$ (21.1)

with $F\in C^{2}({\cal O})$. Let ${\cal O}^{\prime}\subset{\cal O}$ be a
relatively compact open subset. Then there exist $\alpha\in(0,1)$ depending on
$||u||_{C^{0}({\cal O})},||\Delta u||_{C^{0}({\cal O})},$ $||\log
F||_{C^{2}({\cal O})}$ only (here $\Delta$ is the flat Laplacian on
$\mathbb{O}^{2}=\mathbb{R}^{16}$) and a constant $C>0$ depending in addition
on $dist({\cal O}^{\prime},\partial{\cal O})$ such that

$||u||_{C^{2,\alpha}({\cal O}^{\prime})}\leq C.$

  3. 3

First let us introduce a notation. Let $\zeta\in\mathbb{O}^{2}$ be a unit
vector such that at least one of its coordinates is real. Let $\Delta_{\zeta}$
be the Laplacian on the octonionic line (or its translates) $\\{\zeta\cdot
q|\,q\in\mathbb{O}\\}$ spanned by $\zeta$ (see Section 19). The following
lemma was proved in [7], see equality (28) there.

######  21.3 Lemma.

For any smooth real values function $f$ one has

$\Delta_{\zeta}f(x)=Re(\zeta^{*}Hess_{\mathbb{O}}(f)\zeta).$

  4. 4

Proof of Theorem 21.2.
Step 1. Lemma 20.3 implies that for any point $x,y\in{\cal O}$

$\displaystyle F(y)tr\left(U^{-1}(y)(U(y)-U(x))\right)\leq$ (21.2)
$\displaystyle 2F^{1/2}(y)(F^{1/2}(y)-F^{1/2}(x))\leq C_{1}||x-y||,$ (21.3)

where $C_{1}$ depends on $||\log F||_{C^{1}({\cal O})}$ only.

Step 2. Let ${\cal D}$ be the operator from Proposition 18.4, i.e.

${\cal D}(h)=\det U\cdot tr(U^{-1}\cdot Hess_{\mathbb{O}}(h))=F\cdot
tr(U^{-1}\cdot Hess_{\mathbb{O}}(h)).$

We claim that ${\cal D}$ is uniformly elliptic in ${\cal O}$ with constants
$\lambda,\Lambda$ depending on $||\log F||_{C^{0}({\cal O})},||\Delta
u||_{C^{0}({\cal O})}$ only.

Indeed, fix a point $z\in{\cal O}$. By Propositions 7.1 and 16.1 we can make
change of variables $\xi^{\prime}=T\xi$ where $T\in Spin(9)$ and assume that
$U(z)$ is diagonal. Then

${\cal D}h(z)=u_{\bar{1}1}(z)h_{\bar{2}2}(z)+u_{\bar{2}2}(z)h_{\bar{1}1}(z).$

The uniform ellipticity follows from this, the estimate
$0<u_{\bar{i}i}(z)\leq||\Delta u||_{C^{0}({\cal O})}$, and the equality
$u_{\bar{1}1}(z)u_{\bar{2}2}(z)=F(z)$.

Step 3. Fix a point $z_{0}\in{\cal O}^{\prime}$ and a ball
$B_{4r}=B(z_{0},4r)\subset{\cal O}$. We are going to apply the weak Harnack
inequality to the operator ${\cal D}$ acting on the function

$v:=\sup_{B_{4r}}\Delta_{\zeta}u-\Delta_{\zeta}u.$

The function $v$ satisfies by Lemma 19.1

${\cal D}v\leq-\Delta_{\zeta}(\log F)$

where we have used that
$\Delta_{\zeta}(u_{\bar{i}j})=(\Delta_{\zeta}u)_{\bar{i}j}$.

Applying the weak Harnack inequality in the form given in Remark 18.2 we get:

$\displaystyle
r^{-16}\int_{B_{r}}(\sup_{B_{4r}}\Delta_{\zeta}u-\Delta_{\zeta}u)\leq
C(\sup_{B_{4r}}\Delta_{\zeta}u-\sup_{B_{r}}\Delta_{\zeta}u+r)$ (21.4)

where $C$ depends on $||\log F||_{C^{2}({\cal O})}$ and $||\Delta
u||_{C^{0}({\cal O})}$ only. We apply this inequality shortly.

Step 4. The eigenvalues of $U=Hess_{\mathbb{O}}(u)$ belong to
$[\lambda_{1},\Lambda_{1}]$ with $0<\lambda_{1}<\Lambda_{1}<\infty$ depending
on $||\Delta u||_{C^{0}({\cal O})},||\log F||_{C^{0}({\cal O})}$ only. Hence
the eigenvalues of $FU^{-1}$ are in $[\lambda_{2},\Lambda_{2}]$ with
$0<\lambda_{2}<\Lambda_{2}<\infty$ are under control.666In fact one can take
$\lambda_{2}=\lambda_{1}$ and $\Lambda_{2}=\Lambda_{1}$. By Proposition 20.2
there exist a positive integer $N$, unit vectors
$\zeta_{1},\dots,\zeta_{N}\in\mathbb{O}^{2}$ such that each $\zeta_{k}$ has at
least one real coordinate, and $0<\lambda_{*}<\Lambda_{*}<\infty$ (under
control) such that for any $y\in{\cal O}$

$F(y)U^{-1}(y)=\sum_{i=1}^{N}\beta_{k}(y)\zeta_{k}\otimes\zeta_{k}^{*}\mbox{
with }\beta_{k}(y)\in[\lambda_{*},\Lambda_{*}].$

Observe also that for a unit vector $\zeta\in\mathbb{O}^{2}$ such that at
least one of its coordinates is real one has

$tr((\zeta\otimes\zeta^{*})(Hess_{\mathbb{O}}(u)))=tr(\zeta^{*}(Hess_{\mathbb{O}}(u))\zeta)\overset{\mbox{Lemma
}\ref{L:laplace-line}}{=}\Delta_{\zeta}u.$

This and (21.2)-(21.3) imply

$\displaystyle\sum_{k=1}^{N}\beta_{k}(y)(\Delta_{\zeta_{k}}u(y)-\Delta_{\zeta_{k}}u(x))\leq
C_{1}||x-y||\mbox{ for }x,y\in{\cal O}.$ (21.5)

Step 5. Let ${\cal O}^{\prime\prime}$ be a compact neighborhood of ${\cal
O}^{\prime}$ in ${\cal O}$. Let us fix now $z_{0}\in{\cal O}^{\prime\prime}$
and a ball $B_{4r}=B(z_{0},4r)\subset{\cal O}$ centered at $z_{0}$. Let us
denote

$M_{k,r}:=\sup_{B_{r}}\Delta_{\zeta_{k}}u,\,\,\,m_{k,r}:=\inf_{B_{r}}\Delta_{\zeta_{k}}u,$
$\eta(r):=\sum_{k=1}^{N}(M_{k,r}-m_{k,r}).$

We will show that for some $\alpha\in(0,1),r_{0}>0,C>0$ under control

$\displaystyle\eta(r)\leq Cr^{\alpha}\mbox{ for }0<r<r_{0}.$ (21.6)

Since $\zeta_{1},\dots,\zeta_{N}$ can be chosen to contain the vectors
$(1,0),(0,1)\in\mathbb{O}^{2}$, this will imply an estimate on $||\Delta
u||_{C^{\alpha}({\cal O}^{\prime\prime})}$. Then the Schauder estimates ([27],
Theorem 4.6) will imply an estimate on $||u||_{C^{2,\alpha}({\cal
O}^{\prime})}$.

The condition $\eta(r)\leq Cr^{\alpha}$ is equivalent to

$\eta(r)\leq\delta\eta(4r)+r,\,0<r<r_{1}$

where $\delta\in(0,1),r_{1}$ are under control ([27], Lemma 8.23). Summing up
(21.4) over $\zeta=\zeta_{l}$ with $l\neq k$ for any fixed $k$ we get

$\displaystyle r^{-16}\int_{B_{r}}\sum_{l\neq
k}(M_{l,4r}-\Delta_{\zeta_{l}}u)\leq C_{3}(\eta(4r)-\eta(r)+r).$ (21.7)

By (21.5) we have for any $x\in B_{4r},y\in B_{r}$

$\displaystyle\beta_{k}(y)(\Delta_{\zeta_{k}}u(y)-\Delta_{\zeta_{k}}u(x))\leq
C_{1}||x-y||+\sum_{l\neq
k}\beta_{l}(y)(\Delta_{\zeta_{l}}u(x)-\Delta_{\zeta_{l}}u(y))\leq$
$\displaystyle C_{4}r+\Lambda_{*}\sum_{l\neq
k}(M_{l,4r}-\Delta_{\zeta_{l}}u(y)).$

Optimizing in $x$ we get

$\Delta_{\zeta_{k}}u(y)-m_{k,4r}\leq\frac{1}{\lambda_{*}}\left(C_{4}r+\Lambda_{*}\sum_{l\neq
k}(M_{l,4r}-\Delta_{\zeta_{l}}u(y))\right).$

Integrating the last inequality over $y\in B_{r}$ and using (21.7) we obtain

$\displaystyle r^{-16}\int_{B_{r}}(\Delta_{\zeta_{k}}u(y)-m_{k,4r})\leq
C_{5}(\eta(4r)-\eta(r)+r).$ (21.8)

Let us estimate the left hand side of (21.8) from below. Since we have
normalized the Lebesgue measure on $\mathbb{O}^{2}$ so that $vol(B_{1})=1$, we
have

$\displaystyle
r^{-16}\int_{B_{r}}(\Delta_{\zeta_{k}}u(y)-m_{k,4r})\overset{(\ref{harnack-
main})}{\geq}-m_{k,4r}+M_{k,4r}-C(M_{k,4r}-M_{k,r}+r)=$ $\displaystyle
C(M_{k,r}-m_{k,r})-(C-1)(M_{k,4r}-m_{k,4r})+C(m_{k,r}-m_{k,4r})-Cr\geq$
$\displaystyle C(M_{k,r}-m_{k,r})-(C-1)(M_{k,4r}-m_{k,4r})-Cr.$

Substituting this back into (21.8) we obtain

$\displaystyle C(M_{k,r}-m_{k,r})-(C-1)(M_{k,4r}-m_{k,4r})-Cr\leq
C_{5}(\eta(4r)-\eta(r)+r).$

Summing this up over $k$ we get

$\displaystyle C\eta(r)-(C-1)\eta(4r)\leq C_{6}(\eta(4r)-\eta(r)+r).$

Hence,

$\eta(r)\leq\frac{C+C_{6}-1}{C+C_{6}}\eta(4r)+r.$

Theorem 21.2 is proved. Q.E.D.

##  22 Proof of main Theorem 11.3.

  1. 1

For any integer $k\geq 1$ and $\beta\in(0,1)$ let us define

$\displaystyle{\cal U}^{k,\beta}:=\\{\phi\in C^{k,\beta}(M)|\,\det
G_{0}+Hess_{\mathbb{O}}\phi>0\mbox{ and }\int_{M}\phi\cdot\det
G_{0}\cdot\Theta=0\\},$ $\displaystyle{\cal V}^{k,\beta}:=\\{\chi\in
C^{k,\beta}(M)|\,\chi>0\mbox{ and }\int_{M}(\chi-1)\cdot\det
G_{0}\cdot\Theta=0\\},$

and set

${\cal M}(\phi):=\frac{\det(G_{0}+Hess_{\mathbb{O}}(\phi))}{\det G_{0}}.$

The continuity of

${\cal M}\colon{\cal U}^{k+2,\beta}\longrightarrow{\cal V}^{k,\beta}$

is obvious. The only thing to check is that for any $\phi\in C^{k+2,\beta}(M)$
one has

$\displaystyle\int_{M}({\cal M}(\phi)-1)\det G_{0}\cdot\Theta=0.$

This easily follows from Lemma 11.2.

  2. 2

Note that ${\cal U}^{k+2,\beta}$ and ${\cal V}^{k,\beta}$ are open subsets in
Banach spaces with the induced Hölder norms. The following proposition holds.

######  22.1 Proposition.

Let $M^{16}$ be a compact connected $SL_{2}(\mathbb{O})$-manifold with a
parallel $3/4$-density $\Theta$. Let $k\geq 2$ be an integer, and let
$\beta\in(0,1)$. Then the map ${\cal M}\colon{\cal
U}^{k+2,\beta}\longrightarrow{\cal V}^{k,\beta}$ is locally a diffeomorphism
of Banach spaces, and in particular its image is an open subset.

Proof. By the inverse function theorem for Banach spaces, it suffices to show
that the differential of ${\cal M}$ at any $\phi\in{\cal U}^{k+2,\beta}$ is an
isomorphism of tangent spaces. The tangent space to ${\cal U}^{k+2,\beta}$ at
any point is

$\tilde{C}^{k+2,\beta}(M):=\\{\phi\in C^{k+2,\beta}(M)|\,\int_{M}\phi\cdot\det
G_{0}\cdot\Theta=0\\}.$

The tangent space to ${\cal V}^{k,\beta}$ at ${\cal M}(\phi)$ is

$\tilde{C}^{k,\beta}(M):=\\{\chi\in C^{k,\beta}(M)|\,\int_{M}\chi\cdot\det
G_{0}\cdot\Theta=0\\}.$

The differential of ${\cal M}$ at $\phi$ is equal to

$\displaystyle D{\cal
M}_{\phi}(\psi)=2\frac{\det(G_{0}+Hess_{\mathbb{O}}\phi,Hess_{\mathbb{O}}\psi)}{\det
G_{0}}.$

Being defined by this formula we consider $D{\cal M}_{\phi}$ as a map
$C^{k+2,\beta}(M)\longrightarrow C^{k,\beta}(M)$ (without restricting to
$\tilde{C}$). Then obviously $D{\cal M}_{\phi}$ is a linear differential
second order elliptic operator. It has no free term (i.e. $D{\cal
M}_{\phi}(1)=0$), and its coefficients belong to $C^{k,\beta}$.

By the strong maximum principle (see e.g. [27], Theorem 8.19) and since $M$ is
connected, the kernel of $D{\cal M}_{\phi}$ consists only of constant
functions; in particular it is one dimensional. The image of $D{\cal