# The Preservation of Convexity by Geodesics in the Space of Kähler Potentials
on Complex Affine Manifolds

Jingchen Hu

###### Abstract

On a compact complex affine manifold with a constant coefficient Kähler metric
$\omega_{0}$, we introduce a concept: $(S,\omega_{0})$-convexity and show that
$(S,\omega_{0})$-convexity is preserved by geodesics in the space of Kähler
potentials. This implies that if two potentials are both strictly
$(S,\omega_{0})$-convex, then the metrics along the geodesic connecting them
are non-degenerate.

##  1 Introduction

Results of this paper provide partial answers to the following questions:
First, in the space of Kähler potentials, with the metric introduced by
Semmes-Mabuchi-Donaldson, any two points can be joined by a weak geodesic, but
metrics along the geodesic may be degenerate. The question is can we pose some
conditions on two points, so that metrics along the geodesic connecting them
do not degenerate? Second, the maximum rank problem has been extensively
studied for a general class of fully nonlinear elliptic equations. But the
situation for degenerate elliptic equations has remained unexplored, for
example, degenerate complex Monge-Ampère equation. One question is whether,
under some conditions, maximum rank property holds for solutions of degenerate
complex Monge-Ampère equations?

In this paper, on a complex affine manifold with a constant coefficient metric
$\omega_{0}$, we introduce a concept $(S,\omega_{0})$-convexity, and show that
if two potentials are both strictly $(S,\omega_{0})$-convex, then they can be
connected by a geodesic with non-degenerate metric. Under similar condition,
we can show the Hessian of the solution to the homogenous complex Monge-Ampère
equation on an $n+1$ dimensional product space has rank $n$.

In section 1.1 we introduce some basic concepts and present some former
results; in section 1.2 we introduce the concept of
$(S,\omega_{0})$-convexity, an elliptic perturbation of the homogenous complex
Monge-Ampère equation and also present our main results; in section 1.3 we
provide a bird’s-eye view of results of each section and show the structure of
the paper; in section 1.4, more notation and convention are introduced.

###  1.1 Background

Given an $n$ dimensional Kähler manifold $(V,\omega_{0})$, we define the space
of Kähler potentials:

$\displaystyle\mathcal{H}=\\{\phi\in
C^{\infty}(V)|\omega_{0}+\sqrt{-1}\partial\overline{\partial}\phi>0\\}.$ (1.1)

A Riemannian metric can be introduced to this space. For $\psi_{1},\psi_{2}\in
T_{\phi}\mathcal{H}$, let

$\displaystyle<\psi_{1},\psi_{2}>_{\phi}=\int_{V}\psi_{1}\psi_{2}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\phi)^{n}.$
(1.2)

With the Riemannian metric above, for a curve
$\\{\varphi(t)|t\in[0,1]\\}\subset\mathcal{H}$, its energy is

$\displaystyle
E(\varphi)=\int_{0}^{1}\int_{\mathcal{V}}\varphi_{t}^{2}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi)^{n}dt.$
(1.3)

In this paper we denote
$\omega_{0}=\sqrt{-1}b_{\alpha{\overline{\beta}}}dz^{\alpha}\wedge\overline{dz^{\beta}}$,
$\alpha,\beta\in\\{1,...,n\\}$, and
$g_{\alpha{\overline{\beta}}}=b_{\alpha{\overline{\beta}}}+\varphi_{\alpha{\overline{\beta}}},\
g^{\theta{\overline{\beta}}}g_{\alpha{\overline{\beta}}}=\delta_{\theta\alpha}$.
Then the Euler-Lagrange equation for the energy above is

$\displaystyle\varphi_{tt}=\varphi_{t\alpha}g^{\alpha{\overline{\beta}}}\varphi_{t{\overline{\beta}}}.$
(1.4)

When $\omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi>0$, equation
(1.4) is equivalent to

$\displaystyle\det\left(\begin{array}[]{cc}\varphi_{tt}&\varphi_{t{\overline{\beta}}}\\\
\varphi_{\alpha
t}&\varphi_{\alpha{\overline{\beta}}}+b_{\alpha{\overline{\beta}}}\end{array}\right)=0.$
(1.7)

Here the curve $\varphi$ is considered as a function defined on $[0,1]\times
V$. Let

$\displaystyle\mathcal{S}=\\{\tau=t+\sqrt{-1}\theta|0\leq t\leq
1\\}\subset\mathbb{C}.$ (1.8)

We can consider $\varphi$ as a function on $\mathcal{S}\times V$ by letting
$\varphi(\tau)=\varphi(t)$. Then equation (1.7) becomes a homogenous complex
Monge-Ampère equation:

$\displaystyle\det\left(\begin{array}[]{cc}\varphi_{\tau\overline{\tau}}&\varphi_{\tau{\overline{\beta}}}\\\
\varphi_{\alpha{\overline{\tau}}}&\varphi_{\alpha{\overline{\beta}}}+b_{\alpha{\overline{\beta}}}\end{array}\right)=0.$
(1.11)

Denote the projection from $\mathcal{S}\times V$ to $V$ by $\pi_{V}$ and
denote $\pi_{V}^{\ast}(\omega_{0})$ by $\Omega_{0}$. Then equation (1.11)
becomes

$\displaystyle(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi)^{n+1}=0.$
(1.12)

This leads to the study of the following Dirichlet problem on
$\mathcal{S}\times V$.

######  Problem 1.1 (Geodesic Problem).

Given $\varphi_{0},\varphi_{1}\in\mathcal{H}$, find $\Phi\in
C^{1,1}(\mathcal{S}\times V)$, satisfying

$\displaystyle(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi)^{n+1}=0,$
$\displaystyle\ \ \ \ \text{ in }\mathcal{S}\times V;$ (1.13) $\displaystyle\
\ \ \Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi\geq 0,$
$\displaystyle\ \ \ \ \text{ in }\mathcal{S}\times V;$ (1.14) $\displaystyle\
\ \ \ \ \ \ \Phi_{\theta}=0,$ $\displaystyle\ \ \ \ \text{ in
}\mathcal{S}\times V;$ (1.15) $\displaystyle\ \ \ \ \ \ \ \Phi=\varphi_{0},$
$\displaystyle\ \ \ \ \text{ on }\\{t=0\\}\times V;$ (1.16) $\displaystyle\ \
\ \ \ \ \ \Phi=\varphi_{1},$ $\displaystyle\ \ \ \ \text{ on }\\{t=1\\}\times
V.$ (1.17)

For a solution $\Phi$ to Problem 1.1, $\Phi(t,\ast)$ may not be in
$\mathcal{H}$, so we consider it as a weak or generalized geodesic connection
$\varphi_{0}$ and $\varphi_{1}$.

More generally, we can replace $\mathcal{S}$ by a Riemann surface
$\mathcal{R}$ and consider the following Dirichlet problem. In this paper we
only consider the case where $\mathcal{R}$ is a bounded domain in $\mathbb{C}$
with smooth boundary.

######  Problem 1.2 (A Homogenous Monge-Ampère Equation on General Product
Spaces).

Given $\mathcal{R}$, a bounded domain in $\mathbb{C}$ with smooth boundary,
and $F\in C^{\infty}(\partial\mathcal{R}\times V)$ satisfying

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}(F(\tau,\ast))>0,\
\ \ \ \text{ for any }\tau\in\partial\mathcal{R},$ (1.18)

find $\Phi\in C^{1,1}(\mathcal{R}\times V)$ which satisfies

$\displaystyle(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi)^{n+1}=0,$
$\displaystyle\ \ \ \ \text{ in }\mathcal{R}\times V;$ (1.19) $\displaystyle\
\ \ \Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi\geq 0,$
$\displaystyle\ \ \ \ \text{ in }\mathcal{R}\times V;$ (1.20) $\displaystyle\
\ \ \ \ \ \ \Phi=F,$ $\displaystyle\ \ \ \ \text{ on
}\partial\mathcal{R}\times V.$ (1.21)

######  Remark 1.1.

Problem 1.1 can be reduced to Problem 1.2. Let $f$ be a holomorphic covering
map from $\mathcal{S}$ to an annulus $\\{\tau|\ 1<|\tau|<2\\}.$ If $\Phi$ is a
solution to Problem 1.2 with $\mathcal{R}$ being the annulus $\\{\tau|\
1<|\tau|<2\\}$ and $F|_{\\{|\tau|=1\\}}=\varphi_{0}$,
$F|_{\\{|\tau|=2\\}}=\varphi_{1}$, then $\Phi(f(\tau),z)$ is a solution to
Problem 1.1.

Problem 1.2 and 1.1 were introduced by [M87] [S92] [D99]. The existence of
$C^{1,1}$ solution was established by [C00] [B12] [CTW19] [CTW17] and it was
also shown by [LV13] and [DL12] that the optimal regularity of general
solutions is $C^{1,1}$.

Besides regularity, we may ask if

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(\tau,\ast)>0,\
\ \ \ \ \ \text{for all }\tau\in\mathcal{R},$ (1.22)

for a solution $\Phi$ to Problem 1.2. This is similar to the maximum rank
problem, which asks if

$\displaystyle\text{rank}(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(\tau,\ast))=n.$
(1.23)

It’s easy to see that (1.22) implies (1.23). But the inverse implication may
not be true. It turns out that, for a general solution, (1.22) or (1.23) may
not be valid. An example was constructed in [RN], when $\mathcal{R}$ is a disc
and $V$ is $\mathbb{C}P^{1}$. In this example, a solution $\Phi$ was
constructed, which satisfies

$\displaystyle\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi=0,$ (1.24)

in an open set in $\mathcal{R}\times V$. However, we may ask is it possible to
find some conditions for the boundary value $F$, so that if they are satisfied
then (1.22) is valid.

Theorem 1 of [D02] says that, when $\mathcal{R}$ is a disc, the set of smooth
functions $F$ for which a smooth solution to Problem 1.2 exists is open in
$C^{\infty}(\partial\mathcal{R}\times V)$. Actually the proof implies that if
the boundary value of $\Phi$ is in this set, then (1.22) is satisfied. The
proof also suggests that this set is open in $C^{2}$ topology. The proof made
use of the foliation structure associated to a solution to homogenous complex
Monge-Ampère equations. This technique was also used in [L81], to construct
pluri-complex Green’s function. In [CFH20], by partially generalizing this
technique to the case where $\mathcal{R}$ is an annulus , we proved that if
$|\varphi_{0}|_{5}+|\varphi_{1}|_{5}$ is small enough, then the geodesic
connecting $\varphi_{0}$ and $\varphi_{1}$ are $C^{4}$ and

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(t,\ast)>0,\
\ \ \ \ \ \text{for all }t\in[0,1].$ (1.25)

In a recent paper [H22], the author improved the result above, reducing the
$C^{5}$ smallness condition to a $C^{2}$ smallness condition.

However, [GP12M] [LV13] [H21] and the Appendix A of [H22] all suggest that a
proper condition on $F$, which implies (1.22), may be a convexity condition.
In [GP12M], it was shown that, when $V$ is a 1-dimensional complex flat torus,
if $\varphi_{1}$ and $\varphi_{0}$ both satisfy a convexity condition, then
the geodesic connecting them has non-degenerate metric. In the Appendix A of
[H22], a similar result was proved, for solutions to Problem 1.2, with a very
different method. Furthermore, computations of [LV13] and [H21] suggest that
the convexity condition is also necessary.

In this paper, when $V$ is a compact complex affine manifold with a constant
coefficient metric, we introduce a concept: $(S,\omega_{0})$-convexity and
show that boundary values satisfying $(S,\omega_{0})$-convexity implies
(1.22).

###  1.2 Notation, Constructions and Main Results

In this paper, we will discuss the situation, where $V$ is a compact complex
affine manifold with a constant coefficient Kähler metric $\omega_{0}$. By the
definition of complex affine manifold, $V$ is equipped with an atlas, so that
all transition maps are affine and holomorphic. Furthermore, we require that
$\omega_{0}$ is a constant metric, which we mean, in any coordinate
neighborhood, with coordinates $\\{z^{\alpha}\\}_{\alpha=1}^{n}$, if

$\displaystyle\omega_{0}=\sqrt{-1}b_{\alpha{\overline{\beta}}}dz^{\alpha}\wedge\overline{dz^{\beta}},$
(1.26)

then $b_{\alpha{\overline{\beta}}}$ is constant for any
$\alpha,\beta\in\\{1,...,n\\}.$

With these preparations, we can introduce the concept of
$\omega_{0}$-convexity.

######  Definition 1.1 ($\omega_{0}$-Convexity and Strict
$\omega_{0}$-Convexity).

A function $\varphi\in C^{0}(V)$ is (strictly) $\omega_{0}$-convex if, in any
coordinate chart, with coordinates $\\{z^{\alpha}\\}$,

$\displaystyle\varphi+b_{\alpha{\overline{\beta}}}z^{\alpha}\overline{z^{\beta}}$
(1.27)

is a (strictly) convex function.

The convexity above has been widely used in many works related to Hessian
manifolds, for example, it was called local convexity in [CV01] and called
g-convexity in [GT21]. However, to estimate the convexity of solutions to
Problem 1.2, it’s necessary to extend this concept and introduce the following
concept of $(S,\omega_{0})$-convexity.

######  Definition 1.2 ($(S,\omega_{0})$-Convexity and Strict
$(S,\omega_{0})$-Convexity for $C^{0}$ Function).

Suppose $S$ is a constant section of $T_{2,0}^{\ast}(V)$. Then a function
$\varphi\in C^{0}(V)$ is (strictly) $(S,\omega_{0})$-convex if, in any
coordinate chart, with coordinates $\\{z^{\alpha}\\}$,

$\displaystyle\varphi+b_{\alpha{\overline{\beta}}}z^{\alpha}\overline{z^{\beta}}+\text{Re}(S_{{\alpha\beta}}z^{\alpha}z^{\beta})$
(1.28)

is a (strictly) convex function.

For a constant section, we mean, in any coordinate chart, the tensor
components of $S$ are constant. Obviously, when $S=0$,
$(S,\omega_{0})$-convexity is exactly $\omega_{0}$-convexity. Furthermore, to
gauge the convexity, we introduce the concept of modulus of convexity.

######  Definition 1.3 (Modulus of $(S,\omega_{0})$-Convexity).

Suppose $S$ is a constant section of $T_{2,0}^{\ast}(V)$. Then a function
$\varphi\in C^{0}(V)$ is $(S,\omega_{0})$-convex of modulus $\geq\mu$ if, in
any coordinate chart, with coordinates $\\{z^{\alpha}\\}$,

$\displaystyle\varphi+(1-\mu)b_{\alpha{\overline{\beta}}}z^{\alpha}\overline{z^{\beta}}+\text{Re}(S_{{\alpha\beta}}z^{\alpha}z^{\beta})$
(1.29)

is a convex function. It’s $(S,\omega_{0})$-convex of modulus $>\mu$ if (1.29)
is strictly convex.

######  Remark 1.2.

It’s easy to see that if $\varphi$ is $(S,\omega_{0})$-convex of modulus
$>\mu$, for $\mu\geq 0$, then it’s strictly $(S,\omega_{0})$-convex. Another
fact is that $\varphi$ is $(S,\omega_{0})$-convex of modulus $>\mu$ implies
that it’s $(S,\omega_{0})$-convex of modulus $\geq\mu$. In addition, since $V$
is compact, $\varphi$ is $(S,\omega_{0})$-convex of modulus $\geq\mu$ implies
that, for any $\mu^{\prime}<\mu$, $\varphi$ is $(S,\omega_{0})$-convex of
modulus $>\mu^{\prime}$.

When the function $\varphi$ is $C^{2}$ continuous, the strict
$(S,\omega_{0})$-convexity can be defined using complex second order
derivatives. Using Lemma A.1, we will see Definition 1.2 is equivalent to the
following.

######  Definition 1.4 (Strict $(S,\omega_{0})$-Convexity for $C^{2}$
Functions).

Suppose $S$ is a constant section of $T_{2,0}^{\ast}(V)$. Then a function
$\varphi\in C^{2}(V)$ is strictly $(S,\omega_{0})$-convex if

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi>0$ (1.30)

and the maximal eigenvalue of the tensor

$\displaystyle
K=(\varphi_{\theta\nu}-S_{\theta\nu})g^{\nu{\overline{\zeta}}}\overline{(\varphi_{\eta\zeta}-S_{\eta\zeta})}g^{\gamma{\overline{\eta}}}dz^{\theta}\otimes\frac{\partial}{\partial
z^{\gamma}}$ (1.31)

is smaller than 1.

######  Remark 1.3.

Here $\varphi_{\alpha\beta}dz^{\alpha}\otimes dz^{\beta}$ is a well defined
section of $T_{2,0}^{\ast}(V)$, because the coordinate transition functions
between charts are affine. In addition, using basic linear algebra, we know
eigenvalues of $K$ are real and non-negative.

We introduce another measurement of the convexity:

######  Definition 1.5 (Degree of Convexity).

Suppose $S$ is a constant section of $T^{\ast}_{2,0}(V)$ and $\delta$ is a
positive number. Then a function $\varphi\in C^{2}(V)$ is
$(S,\omega_{0})$-convex of degree $>\delta$ if, for any constant section
$\Theta$ of $T_{2,0}^{\ast}(V)$ with

$\displaystyle\text{maximum eigenvalue of
}\left(\Theta_{\alpha\eta}b^{\eta{\overline{\gamma}}}\
\overline{\Theta_{\rho\gamma}}b^{\beta{\overline{\rho}}}dz^{\alpha}\otimes\frac{\partial}{\partial
z^{\beta}}\right)\leq\delta^{2},$ (1.32)

$\varphi$ is a strictly $(S+\Theta,\omega_{0})$-convex function.

It turns out that the degree of convexity and the modulus of convexity
coincide. In Lemma A.2, we show that, for $\varphi\in C^{2}(V)$ and
$\delta\geq 0$, $\varphi$ is $(S,\omega_{0})$-convex of degree $>\delta$ if
and only if it is $(S,\omega_{0})$-convex of modulus $>\delta$.

The main results of the paper are the following.

######  Theorem 1.1 (Estimates for Geodesics).

Given $\varphi_{0},\varphi_{1}\in\mathcal{H}$, suppose that there is an $S$
which is a constant section of $T^{\ast}_{2,0}(V)$, so that $\varphi_{0}$ and
$\varphi_{1}$ are both $(S,\omega_{0})$-convex of modulus $>\mu$, for $\mu>0$.
Let $\\{\varphi_{t}|\ t\in[0,1]\\}$ be the geodesic connecting $\varphi_{0}$
and $\varphi_{1}$. Then, for any $t\in(0,1)$, $\varphi_{t}$ is
$(S,\omega_{0})$-convex of modulus $\geq\mu$ and, by definition, this implies

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}\geq\mu\omega_{0},$
(1.33)

in the weak sense.

The theorem above is a particular case of the following theorem, according to
Remark 1.1.

######  Theorem 1.2 (Estimates on Product Space).

Suppose $F$ is a $C^{\infty}$ function on $\partial\mathcal{R}\times V$ and,
for a constant $\mu>0$ and a constant section $S$ of $T^{\ast}_{2,0}(V)$,

$F(\tau,\ast)$ is $(S,\omega_{0})$-convex of modulus $>\mu$, for any
$\tau\in\partial\mathcal{R}$. (1.34)

Let $\Phi$ be the solution to Problem 1.2 with boundary value $F$. Then
$\Phi(\tau,\ast)$ is $(S,\omega_{0})$-convex of modulus $\geq\mu$, for any
$\tau\in\mathcal{R}$, and, as a consequence

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}[\Phi(\tau,\ast)]\geq\mu\omega_{0},$
(1.35)

in the weak sense.

As we discussed, solutions to Problem 1.2 may only be $C^{1,1}$, but to
implement our method we need to use up to 4-th order derivatives. Therefore,
we consider an elliptic perturbation of Problem 1.2, for which there are
smooth solutions. In many previous works, the following problem was studied:

######  Problem 1.3 (Non-Degenerate Monge-Ampère Equation on Product Space).

Given $\mathcal{R}$, a bounded domain in $\mathbb{C}$ with smooth boundary,
and $F\in C^{\infty}(\partial\mathcal{R}\times V)$ satisfying

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}(F(\tau,\ast))>0,$
$\displaystyle\ \ \ \ \text{ for any }\tau\in\partial\mathcal{R},$ (1.36)

find $\Phi\in C^{1,1}(\mathcal{R}\times V)$ which satisfies

$\displaystyle(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi)^{n+1}$
$\displaystyle=\varepsilon\sqrt{-1}d\tau\wedge\overline{d\tau}\wedge\Omega_{0}^{n},$
$\displaystyle\ \ \ \ \text{ in }\mathcal{R}\times V;$ (1.37) $\displaystyle\
\ \ \Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi$ $\displaystyle>0,$
$\displaystyle\ \ \ \ \text{ in }\mathcal{R}\times V;$ (1.38) $\displaystyle\
\ \ \ \ \ \ \Phi$ $\displaystyle=F,$ $\displaystyle\ \ \ \ \text{ on
}\partial\mathcal{R}\times V.$ (1.39)

In above, $\varepsilon$ is a positive constant.

However, in this paper we introduce a different perturbation.

######  Problem 1.4 (An Elliptic Perturbation of Homogenous Complex Monge-
Ampère Equation).

Suppose $F\in C^{\infty}(\partial\mathcal{R}\times V)$ satisfies that

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}(F(\tau,\ast))>0,\
\ \ \ \text{ for any }\tau\in\partial\mathcal{R}.$ (1.40)

Find $\Phi\in C^{\infty}(\mathcal{R}\times V)$ satisfying:

$\displaystyle(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi)^{n+1}=\epsilon\sqrt{-1}d\tau\wedge\overline{d\tau}\wedge\Omega_{0}\wedge(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi)^{n-1},$
$\displaystyle\text{ in\ \ \ }\mathcal{R}\times V;$ (1.41) $\displaystyle\ \ \
\ \ \ \ \ \ \ \Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi>0,$
$\displaystyle\text{ in\ \ \ }\mathcal{R}\times V;$ (1.42) $\displaystyle\ \ \
\ \ \ \ \ \ \ \ \ \ \ \Phi=F,$ $\displaystyle\text{ on \
}\partial\mathcal{R}\times V.$ (1.43)

Here $\epsilon$ is a positive constant.

######  Remark 1.4.

Equation (1.41) is equivalent to

$\displaystyle\Phi_{\tau{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}=\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\beta}}},$ (1.44)

providing

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}(\Phi(\tau,\ast))>0,\
\ \ \ \text{ for any }\tau\in\mathcal{R}.$ (1.45)

We also notice that, when $n=1$, Problem 1.4 is exactly Problem 1.3, with
$\varepsilon=\epsilon$. In a previous paper [H22] we proved a particular
1-dimensional case of Theorem 1.2, by working with solutions to Problem 1.3.

In Section 4, we prove that under the condition of Theorem 1.2 a smooth
solution to Problem 1.4 exists. However, we don’t know if a smooth solution
always exists for general boundary value $F$. The convergence of solutions as
$\epsilon$ goes to zero is discussed in Section 5.

###  1.3 Structure of the Paper

In section 2, by differentiating equation (1.44), we find second order
derivatives of $\Phi$,

$\displaystyle
A_{{\alpha{\overline{\beta}}}}=\Phi_{\alpha{\overline{\beta}}}+b_{\alpha{\overline{\beta}}}\
\ \ \ \ \ \text{and}\ \ \ \ \ \ B_{{\alpha\beta}}=\Phi_{\alpha\beta},$ (1.46)

satisfy two non-linear equations (2.2) and (2.3).

In section 3.1, using $A$ and $B$, we construct $M_{S}$, which measures the
convexity, and $Q^{[p]}_{S}$, which is a smooth approximation of $M_{S}$.
Then, using equation (2.2) and (2.3), we show, for an elliptic operator
$L^{i\overline{j}}\partial_{i\overline{j}}$ and
$Q^{<p>}_{S}=\left(Q^{[p]}_{S}\right)^{p}$,

$\displaystyle
L^{i\overline{j}}\partial_{i\overline{j}}\left(Q^{<p>}_{S}\right)\geq 0,$
(1.47)

providing $Q^{[p]}_{S}\leq 1-\frac{1}{2p}$.

In section 3.2, using a continuity argument, we show if $M_{S}<1$ on
${\partial\mathcal{R}\times V}$, then $M_{S}<1$ in ${\mathcal{R}\times V}$.
This means the strict $(S,\omega_{0})$-convexity on $\partial\mathcal{R}\times
V$ implies the strict $(S,\omega_{0})$-convexity in $\mathcal{R}\times V$.

In section 3.3, by altering $S$, we show the $(S,\omega_{0})$-convexity of
modulus $>\mu$ on $\partial\mathcal{R}\times V$ implies the
$(S,\omega_{0})$-convexity of modulus $>\mu$ in $\mathcal{R}\times V$.

We point out that estimates in section 3.1, 3.2 and 3.3 all depends on some
apriori assumptions. These assumptions can be removed after we prove the
existence of smooth solutions to Problem 1.4.

In section 4, we establish $C^{0}$, $C^{1}$, $C^{2}$ and $C^{2,\alpha}$
estimates. The methods are standard in PDE. In the proof, the result of
section 3.3 plays an import role. Actually, it basically says equation (1.44)
is a uniform elliptic equation, providing $\epsilon>0$. With these estimates,
we prove existence of $C^{\infty}$ solution by the method of continuity in
section 4.6.

Finally, we prove estimates for solutions to Problem 1.1 and 1.2, by letting
$\epsilon\rightarrow 0.$

###  1.4 A Convention for Tensor Contraction

In this paper, we need to contract a sequence of rank $2$ tensors to form new
tensors. The computation can be simplified by converting rank $2$ tensors to
matrices. In this sections, we explain how to do this.

Suppose $A$ is a section of $T_{1,1}^{\ast}(V)$ and $B$ is a section of
$T_{2,0}^{\ast}(V)$. In a coordinate chart, $A$ and $B$ can be considered as
matrix valued functions, which we still denote by $A$ and $B$. Let

$\displaystyle A=(A_{{\alpha{\overline{\beta}}}}),\ \ \ \ \ \
B=(B_{\alpha\beta}),$ (1.48)

where $\alpha$ is the row index and $\beta$ is the column index. As a matrix,
when $A$ is invertible, we denote

$\displaystyle(A^{{\alpha{\overline{\beta}}}})=A^{-1},$ (1.49)

where $\alpha$ is the column index and $\beta$ is the row index. Then we have

$\displaystyle
A^{\alpha{\overline{\beta}}}A_{\alpha{\overline{\theta}}}=\delta_{\theta\beta}$
(1.50)

and we know

$\displaystyle A^{\alpha{\overline{\beta}}}\frac{\partial}{\partial
z^{\alpha}}\otimes\overline{\frac{\partial}{\partial z^{\beta}}}$ (1.51)

is a section of $T_{1,1}(V)$.

Let

$\displaystyle
K_{\alpha}^{\beta}=B_{{\alpha\theta}}\overline{A^{\mu{\overline{\theta}}}}\
\overline{B_{\mu{\overline{\rho}}}}A^{\beta{\overline{\rho}}},$ (1.52)

then $K_{\alpha}^{\beta}dz^{\alpha}\otimes\frac{\partial}{\partial z^{\beta}}$
is a section of $T_{1,0}^{\ast}(V)\otimes T_{1,0}(V)$. Locally, we can
consider $K$ as a matrix, with

$\displaystyle K=(K_{\alpha}^{\beta}),$ (1.53)

where $\alpha$ is the row index and $\beta$ is the column index. Then

$\displaystyle K=B\overline{A^{-1}}\ {\overline{B}}{A^{-1}}.$ (1.54)

It’s easy to see that eigenvalues of $K$ do not change under coordinate
transformations. Furthermore, for $p\in\mathbb{Z}^{+}$, we need to use
$K^{p}$, which is also a section of $T_{1,0}^{\ast}(V)\otimes T_{1,0}(V)$.

In addition, for a section $\Theta$ of $T_{1,0}^{\ast}(V)\otimes T_{1,0}(V)$,
we denote

$\displaystyle\Theta<C,$ (1.55)

for a constant $C$, if, at any point $p\in V$, the maximum eigenvalue of
$\Theta(p)$ is smaller than $C$.

##  2 Equations for Second Order Derivatives

In this paper we will mainly work on the product space $\mathcal{R}\times V$,
where $\mathcal{R}$ is a compact domain in $\mathbb{C}$ with smooth boundary
and $V$ is the complex affine manifold. We denote the coordinate on
$\mathcal{R}$ by $\tau$ and denote the coordinates on $V$ by
$\\{z^{\alpha}\\}_{\alpha=1}^{n}$. Coordinates on $V$ are indexed by Greek
letters, except $\tau$. The coordinate $\tau$ on $\mathcal{R}$ will be
considered as the $0$-th coordinate and, in some situation, we denote it by
$z^{0}$. Thus, the coordinates on $\mathcal{R}\times V$ will be indexed by
Roman letters, running from $0$ to $n$.

Suppose $\Phi$ is a solution to Problem 1.4. Let

$\displaystyle
A_{\alpha{\overline{\beta}}}=\Phi_{\alpha{\overline{\beta}}}+b_{\alpha{\overline{\beta}}},\
\ \ \ \ \ \ \ B_{\alpha\beta}=\Phi_{\alpha\beta}.$ (2.1)

As described in section 1.4, $A,B$ can be considered as matrices locally. In
this section, by differentiating (1.44), we show, as matrices, $A,B$ satisfy
the following equations:

$\displaystyle
L^{i{\overline{j}}}\partial_{i{\overline{j}}}A=L^{i{\overline{j}}}(\partial_{i}A)A^{-1}(\partial_{\overline{j}}A)+L^{i\overline{j}}(\partial_{\overline{j}}B)\overline{A^{-1}}(\partial_{i}\overline{B}),$
(2.2) $\displaystyle
L^{i{\overline{j}}}\partial_{i{\overline{j}}}B=L^{i{\overline{j}}}(\partial_{i}A)A^{-1}(\partial_{\overline{j}}B)+L^{i\overline{j}}(\partial_{\overline{j}}B)\overline{A^{-1}}(\partial_{i}\overline{A}).$
(2.3)

Here $L=L^{i{\overline{j}}}\partial_{i\overline{j}}$ is an elliptic operator
on $\mathcal{R}\times V$, with

$\displaystyle\left(\begin{array}[]{cc}L^{0\overline{0}}&L^{0{\overline{\beta}}}\\\
L^{\alpha\overline{0}}&L^{\alpha{\overline{\beta}}}\end{array}\right)=\left(\begin{array}[]{cc}1&-\Phi_{\mu{\overline{\tau}}}g^{\mu{\overline{\beta}}}\\\
-\Phi_{\tau{\overline{\mu}}}g^{\alpha{\overline{\mu}}}&\Phi_{\tau{\overline{\mu}}}g^{\alpha{\overline{\mu}}}\Phi_{\mu{\overline{\tau}}}g^{\mu{\overline{\beta}}}+\epsilon
b_{\eta{\overline{\zeta}}}g^{\alpha{\overline{\zeta}}}g^{\eta{\overline{\beta}}}\end{array}\right).$
(2.8)

The computations in this section are similar to the computations of Section
2.1 and 2.2 of [H22]. However, the computations here are much simpler because
we are working on a flat affine manifold so $\Phi_{\alpha\beta}$ are
coordinate derivatives while in [H22] we need to compute covariant
derivatives.

Apply $\partial_{\theta}$ to (1.44), we get

$\displaystyle\Phi_{\theta\tau{\overline{\tau}}}-\Phi_{\theta{\overline{\beta}}\tau}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}\theta}=-\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}.$
(2.9)

Then apply $\partial_{\gamma}$ to (2.9), we get

$\displaystyle\Phi_{\theta\gamma\tau\overline{\tau}}-\Phi_{\theta\gamma{\overline{\beta}}\tau}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}+\Phi_{\theta{\overline{\beta}}\tau}g^{\alpha{\overline{\mu}}}\Phi_{\gamma{\overline{\mu}}\rho}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}-\Phi_{\theta{\overline{\beta}}\tau}g^{\alpha{\overline{\beta}}}\Phi_{\alpha\gamma{\overline{\tau}}}+\Phi_{\tau\gamma{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}$
(2.10)
$\displaystyle-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\eta}}}\Phi_{\zeta{\overline{\eta}}\gamma}g^{\zeta{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\theta\gamma\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\eta}}}\Phi_{\zeta{\overline{\eta}}\gamma}g^{\zeta{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}$
(2.11)
$\displaystyle+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}\Phi_{\alpha\gamma{\overline{\tau}}}-\Phi_{\tau\gamma{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{\alpha\theta{\overline{\tau}}}+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\zeta{\overline{\mu}}\gamma}g^{\zeta{\overline{\beta}}}\Phi_{\theta\alpha{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{\alpha\theta\gamma{\overline{\tau}}}$
(2.12) $\displaystyle\ \ =\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\zeta}}}\Phi_{\eta{\overline{\zeta}}\gamma}g^{\eta{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}-\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\theta\gamma\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}+\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\zeta}}}\Phi_{\eta{\overline{\zeta}}\gamma}g^{\eta{\overline{\beta}}}.$
(2.13)

We will use the following convention. $(\ast.\ast)_{k}$ stands for the $k$-th
term in the line $(\ast.\ast)$, including the sign. For example,

$\displaystyle(\ref{eq:20221030-4})_{2}=-\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\theta\gamma\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}},\
\ \ \ (\ref{eq:20221030-1})_{1}=\Phi_{\theta\gamma\tau{\overline{\tau}}},\ \ \
\
(\ref{eq:20221030-3})_{4}=-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{\alpha\theta\gamma{\overline{\tau}}}.$
(2.14)

It’s straightforward to verify the following equality:

$\displaystyle(\ref{eq:20221030-1})_{1}+(\ref{eq:20221030-1})_{2}+(\ref{eq:20221030-2})_{2}+(\ref{eq:20221030-3})_{4}-(\ref{eq:20221030-4})_{2}$
$\displaystyle=\Phi_{\theta\gamma i{\overline{j}}}L^{i{\overline{j}}},$ (2.15)
$\displaystyle(\ref{eq:20221030-1})_{3}+(\ref{eq:20221030-1})_{4}+(\ref{eq:20221030-2})_{1}+(\ref{eq:20221030-3})_{1}-(\ref{eq:20221030-4})_{1}$
$\displaystyle=-\Phi_{\theta{\overline{\beta}}i}g^{\alpha{\overline{\beta}}}\Phi_{\alpha\gamma{\overline{j}}}L^{i{\overline{j}}},$
(2.16)
$\displaystyle(\ref{eq:20221030-1})_{5}+(\ref{eq:20221030-2})_{3}+(\ref{eq:20221030-3})_{2}+(\ref{eq:20221030-3})_{3}-(\ref{eq:20221030-4})_{3}$
$\displaystyle=-\Phi_{\theta\rho{\overline{j}}}g^{\rho{\overline{\zeta}}}\Phi_{\gamma{\overline{\zeta}}i}L^{i{\overline{j}}}.$
(2.17)

This gives us (2.3).

Similarly, we apply $\partial_{\overline{\gamma}}$ to (2.9) and get

$\displaystyle\Phi_{\theta{\overline{\gamma}}\tau\overline{\tau}}-\Phi_{\theta{\overline{\gamma}}{\overline{\beta}}\tau}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}+\Phi_{\theta{\overline{\beta}}\tau}g^{\alpha{\overline{\mu}}}\Phi_{{\overline{\gamma}}{\overline{\mu}}\rho}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}-\Phi_{\theta{\overline{\beta}}\tau}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\gamma}}{\overline{\tau}}}+\Phi_{\tau{\overline{\gamma}}{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}$
(2.18)
$\displaystyle-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\eta}}}\Phi_{\zeta{\overline{\eta}}{\overline{\gamma}}}g^{\zeta{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\theta{\overline{\gamma}}\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\eta}}}\Phi_{\zeta{\overline{\eta}}{\overline{\gamma}}}g^{\zeta{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}$
(2.19)
$\displaystyle+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\gamma}}{\overline{\tau}}}-\Phi_{\tau{\overline{\gamma}}{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{\alpha\theta{\overline{\tau}}}+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\zeta{\overline{\mu}}{\overline{\gamma}}}g^{\zeta{\overline{\beta}}}\Phi_{\theta\alpha{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{\alpha\theta{\overline{\gamma}}{\overline{\tau}}}$
(2.20) $\displaystyle\ \ =\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\zeta}}}\Phi_{\eta{\overline{\zeta}}{\overline{\gamma}}}g^{\eta{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\beta}}}-\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\theta{\overline{\gamma}}\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}+\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\mu}}\theta}g^{\rho{\overline{\zeta}}}\Phi_{\eta{\overline{\zeta}}{\overline{\gamma}}}g^{\eta{\overline{\beta}}}.$
(2.21)

It’s straightforward to verify the following equality

$\displaystyle(\ref{eq:20221032-1})_{1}+(\ref{eq:20221032-1})_{2}+(\ref{eq:20221032-2})_{2}+(\ref{eq:20221032-3})_{4}-(\ref{eq:20221032-4})_{2}$
$\displaystyle=\Phi_{\theta{\overline{\gamma}}i{\overline{j}}}L^{i{\overline{j}}},$
(2.22)
$\displaystyle(\ref{eq:20221032-1})_{3}+(\ref{eq:20221032-1})_{4}+(\ref{eq:20221032-2})_{1}+(\ref{eq:20221032-3})_{1}-(\ref{eq:20221032-4})_{1}$
$\displaystyle=-\Phi_{\theta{\overline{\beta}}i}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\gamma}}{\overline{j}}}L^{i{\overline{j}}},$
(2.23)
$\displaystyle(\ref{eq:20221032-1})_{5}+(\ref{eq:20221032-2})_{3}+(\ref{eq:20221032-3})_{2}+(\ref{eq:20221032-3})_{3}-(\ref{eq:20221032-4})_{3}$
$\displaystyle=-\Phi_{\theta\rho{\overline{j}}}g^{\rho{\overline{\zeta}}}\Phi_{{\overline{\gamma}}{\overline{\zeta}}i}L^{i{\overline{j}}}.$
(2.24)

This gives us (2.2).

##  3 Apriori Estimates

In this section, we will prove some estimates for solutions to Problem 1.4,
with some apriori assumptions. These assumptions can be removed after we prove
the existence of $C^{\infty}$ solutions to Problem 1.4 in section 4. One
estimate in this section, Prop 3.3, will play an indispensable role in our
proof of the existence result.

Suppose $S$ is a constant section of $T_{2,0}^{\ast}(V)$ and $F\in
C^{\infty}(\partial\mathcal{R}\times V)$ satisfies

$\displaystyle F(\tau,\ast)\text{ is strictly $(S,\omega_{0})$-convex, for any
$\tau\in\partial\mathcal{R}$. }$ (3.1)

For a solution $\Phi$ to Problem 1.4 with boundary value $F$, let

$\displaystyle B$ $\displaystyle=(\Phi_{\alpha\beta}),$ (3.2) $\displaystyle
A$
$\displaystyle=(\Phi_{\alpha{\overline{\beta}}}+b_{\alpha{\overline{\beta}}}),$
(3.3) $\displaystyle K_{S}$ $\displaystyle=(B-S)\overline{A^{-1}}\
\overline{(B-S)}{A^{-1}},$ (3.4)

and, for any $(\tau,z)\in\mathcal{R}\times V$,

$\displaystyle M_{S}(\tau,z)=\text{ Maximum Eigenvalue of }K_{S}(\tau,z).$
(3.5)

Condition (3.1) implies that

$\displaystyle M_{S}<1,\text{ on }\partial\mathcal{R}\times V.$ (3.6)

We want to show that

$\displaystyle M_{S}<1,\text{ in }\mathcal{R}\times V.$ (3.7)

This implies that $\Phi(\tau,\ast)$ is strictly $(S,\omega_{0})$-convex for
any $\tau\in\partial\mathcal{R}$. However, it’s difficult to directly work
with $M_{S}$, since it may not be differentiable. We introduce the following
approximation of $M_{S}$. Let

$\displaystyle Q^{<p>}_{S}=\text{tr}(K_{S}^{p}),$ (3.8) $\displaystyle
Q^{[p]}_{S}=\left(Q^{<p>}_{S}\right)^{\frac{1}{p}}.$ (3.9)

According to basic calculus, for $\lambda_{1},\ ...\ ,\lambda_{n}\geq 0$,

$\displaystyle\lim_{p\rightarrow+\infty}\left(\lambda_{1}^{p}+\ ...\
+\lambda_{n}^{p}\right)^{1/p}=\max\\{\lambda_{1},\ ...\ ,\lambda_{n}\\},$
(3.10)

so

$\displaystyle\lim_{p\rightarrow+\infty}Q^{[p]}_{S}=M_{S}.$ (3.11)

If we can show for $p$ big enough,

$\displaystyle Q^{[p]}_{S}\leq\max_{{\partial\mathcal{R}\times
V}}Q^{[p]}_{S},\ \ \ \ \text{ in }\mathcal{R}\times V,$ (3.12)

then we can let $p$ go to $\infty$ and prove (3.7).

In section 3.1, we prove that for the elliptic operator
$L=L^{i\overline{j}}\partial_{i\overline{j}}$ introduced in section 2,

$\displaystyle L^{i\overline{j}}\left(Q^{<p>}_{S}\right)_{i\overline{j}}\geq
0,\ \ \ \ \ \text{ in }\mathcal{R}\times V,$ (3.13)

providing $K_{S}\leq 1-\frac{1}{2p}$. In section 3.2, using a continuity
argument we prove

$\displaystyle Q^{[p]}_{S}\leq\max_{{\partial\mathcal{R}\times
V}}Q^{[p]}_{S},\ \ \ \ \text{ in }\mathcal{R}\times V.$ (3.14)

In section 3.3, by altering $S$, we prove a convexity estimate, which implies
a metric lower bound estimate

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(\tau,\ast)\geq\mu\omega_{0},$
(3.15)

for a constant $\mu>0$.

In section 3.2 and 3.3, we need an apriori assumption that for any
$\lambda\in[0,1]$, Problem 1.4 has a solution $\Phi^{\lambda}$ with boundary
value $\lambda F$ and $\\{\Phi^{\lambda}|\lambda\in[0,1]\\}$ is a continuous
curve in $C^{4}(\overline{\mathcal{R}}\times V)$ in $C^{2}$ topology.

###  3.1 Computation

Suppose $\Phi$ is a $C^{4}$ solution to Problem 1.4. In a local coordinate
chart, let

$\displaystyle K$ $\displaystyle=B\overline{A^{-1}}\ {\overline{B}}{A^{-1}}$
(3.16)

and, for a positive integer $p$,

$\displaystyle Q^{<p>}=\text{tr}(K^{p}).$ (3.17)

As matrices, $B,A$ and $K$ depends on the choice of coordinate, while
$Q^{<p>}$ is a well defined function on $\mathcal{R}\times V$. In this
section, we show

$\displaystyle L^{i\overline{j}}\left(Q^{<p>}\right)_{i\overline{j}}\geq 0,\ \
\ \text{providing }K\leq 1-\frac{1}{2p}.$ (3.18)

After proving this, we know for

$\displaystyle K_{S}=(B-S)\overline{A^{-1}}\ \overline{(B-S)}{A^{-1}}$ (3.19)

and

$\displaystyle Q^{<p>}_{S}=\text{tr}(K_{S}^{p}),$ (3.20) $\displaystyle
L^{i\overline{j}}\left(Q^{<p>}_{S}\right)_{i\overline{j}}\geq 0,\ \ \ \
\text{providing }K_{S}\leq 1-\frac{1}{2p}.$ (3.21)

This is because $B-S,A$ satisfy the same set of equations as $B,A$ do. Similar
to (2.2) (2.3), we have:

$\displaystyle L^{i{\overline{j}}}\partial_{i{\overline{j}}}A$
$\displaystyle=L^{i{\overline{j}}}(\partial_{i}A)A^{-1}\partial_{\overline{j}}A+L^{i\overline{j}}\partial_{\overline{j}}(B-S)\overline{A^{-1}}\partial_{i}\overline{(B-S)};$
(3.22) $\displaystyle L^{i{\overline{j}}}\partial_{i{\overline{j}}}(B-S)$
$\displaystyle=L^{i{\overline{j}}}(\partial_{i}A)A^{-1}\partial_{\overline{j}}(B-S)+L^{i\overline{j}}\partial_{\overline{j}}(B-S)\overline{A^{-1}}(\partial_{i}\overline{A}).$
(3.23)

Equations above are equivalent to (2.2) (2.3) because $S$ is a constant
section and all derivatives of $S$ are zero.

Before the computation of
$L^{i\overline{j}}\left(Q^{<p>}\right)_{i\overline{j}}$, we do some
preparation. First, we note that the conjugate of equation (2.3) is equivalent
to

$\displaystyle L^{i\overline{j}}\left(\overline{A^{-1}}\
{\overline{B}}_{i}{A^{-1}}\right)_{{\overline{j}}}=0.$ (3.24)

Then we introduce a quantity

$\displaystyle{\mathcal{B}_{i}}=B_{i}-A_{i}{A^{-1}}B-B\overline{A^{-1}}\
\overline{A}_{i}.$ (3.25)

The reason to introduce ${\mathcal{B}_{i}}$ is to combine some terms with
$B_{i}$ to simplify the computation. Even ${\mathcal{B}_{i}}$ can be
considered as a tensor, we just need to consider it as a symmetric matrix
valued function defined in a local coordinate chart. (3.25) is equivalent to

$\displaystyle
A^{-1}{\mathcal{B}_{i}}\overline{A^{-1}}=\partial_{i}\left({A^{-1}}B\overline{A^{-1}}\right).$
(3.26)

When ${\mathcal{B}_{i}}$ is differentiated by
$L^{i\overline{j}}\partial_{{\overline{j}}}$, using (2.2), (2.3) and the
conjugate of (2.2), we find

$\displaystyle
L^{i\overline{j}}\partial_{{\overline{j}}}{\mathcal{B}_{i}}=\left(-B_{{\overline{j}}}\overline{A^{-1}}\
{\overline{B}}_{i}{A^{-1}}B-B\overline{A^{-1}}\
{\overline{B}}_{i}{A^{-1}}B_{{\overline{j}}}\right).$ (3.27)

Now we start to compute
$L^{i\overline{j}}\left(Q^{<p>}\right)_{i\overline{j}}$. In the expression of
$Q^{<p>}$, we combine some terms together to simplify the computation:

$\displaystyle Q^{<p>}$ $\displaystyle=\text{tr}\left(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}}\right)^{p}$ (3.28)
$\displaystyle=\text{tr}\left[({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\cdot...\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\right].$
(3.29)

In (3.29), $({A^{-1}}B\overline{A^{-1}})$ and $\overline{B}$ both appear $p$
times. When $\partial_{i}$ act on any of $({A^{-1}}B\overline{A^{-1}})$ (or
$\overline{B}$), we get the same result. So

$\displaystyle\partial_{i}Q^{<p>}=$ $\displaystyle
p\cdot\text{tr}\left[\partial_{i}({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\cdot...\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\right]$
(3.30) $\displaystyle+$ $\displaystyle
p\cdot\text{tr}\left[({A^{-1}}B\overline{A^{-1}})\cdot\partial_{i}{\overline{B}}\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\cdot...\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\right].$
(3.31)

We plug (3.26) into (3.30) and get

$\displaystyle\partial_{i}Q^{<p>}=$ $\displaystyle
p\cdot\text{tr}\left[{A^{-1}}{\mathcal{B}_{i}}\overline{A^{-1}}\cdot{\overline{B}}\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\cdot...\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\right]$
(3.32) $\displaystyle+$ $\displaystyle
p\cdot\text{tr}\left[({A^{-1}}B\overline{A^{-1}})\cdot\partial_{i}{\overline{B}}\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\cdot...\cdot({A^{-1}}B\overline{A^{-1}})\cdot{\overline{B}}\right].$
(3.33)

We reorganize terms in the product of (3.32) (3.33):

$\displaystyle\partial_{i}Q^{<p>}=$ $\displaystyle
p\cdot\text{tr}\left[{\mathcal{B}_{i}}\cdot(\overline{A^{-1}}\
{\overline{B}}{A^{-1}})\cdot B\cdot(\overline{A^{-1}}\
\overline{B}{A^{-1}})\cdot...\cdot B\cdot(\overline{A^{-1}}\
\overline{B}{A^{-1}})\right]$ (3.34) $\displaystyle+$ $\displaystyle
p\cdot\text{tr}\left[(\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}})\cdot
B\cdot(\overline{A^{-1}}\ \overline{B}{A^{-1}})\cdot
B\cdot...\cdot(\overline{A^{-1}}\ \overline{B}{A^{-1}})\cdot B\right].$ (3.35)

When apply $L^{i\overline{j}}\partial_{{\overline{j}}}$ to $Q^{<p>}_{i}$,
$L^{i\overline{j}}\partial_{\overline{j}}$ acts on 6 kinds of terms:

  * (i)

$L^{i\overline{j}}\partial_{\overline{j}}$ acts on ${\mathcal{B}_{i}}$ in
(3.34);

  * (ii)

$L^{i\overline{j}}\partial_{\overline{j}}$ acts on $(\overline{A^{-1}}\
{\overline{B}}{A^{-1}})$ in (3.34);

  * (iii)

$L^{i\overline{j}}\partial_{\overline{j}}$ acts on $B$ in (3.34);

  * (iv)

$L^{i\overline{j}}\partial_{\overline{j}}$ acts on
$(\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}})$ in (3.35);

  * (v)

$L^{i\overline{j}}\partial_{\overline{j}}$ acts on $B$ in (3.35);

  * (vi)

$L^{i\overline{j}}\partial_{\overline{j}}$ acts on $(\overline{A^{-1}}\
\overline{B}{A^{-1}})$ in (3.35).

In the following we do the computation separately.

(i) When $L^{i\overline{j}}\partial_{\overline{j}}$ acts on
${\mathcal{B}_{i}}$ in (3.34), the result is

$\displaystyle
p\cdot\text{tr}\left[L^{i\overline{j}}\partial_{\overline{j}}{\mathcal{B}_{i}}(\overline{A^{-1}}\
\overline{B}{A^{-1}})\cdot\left(B\cdot(\overline{A^{-1}}\
\overline{B}{A^{-1}})\right)^{p-1}\right]L^{i\overline{j}}$ (3.36)
$\displaystyle=$ $\displaystyle
p\cdot\text{tr}\left[-B_{\overline{j}}\overline{A^{-1}}\
{\overline{B}}_{i}{A^{-1}}\cdot\left(B\cdot(\overline{A^{-1}}\
\overline{B}{A^{-1}})\right)^{p}\right]L^{i\overline{j}}$ (3.37)
$\displaystyle+$ $\displaystyle p\cdot\text{tr}\left[-\overline{A^{-1}}\
\overline{B}_{i}{A^{-1}}B_{\overline{j}}\cdot\left((\overline{A^{-1}}\
\overline{B}{A^{-1}})\cdot B\right)^{p}\right]L^{i\overline{j}}.$ (3.38)

To get this, we need to use equation (3.27).

(ii) When $L^{i\overline{j}}\partial_{\overline{j}}$ acts on
$(\overline{A^{-1}}\ {\overline{B}}{A^{-1}})$ in (3.34), we need to use the
conjugate of (3.26):

$\displaystyle\overline{A^{-1}}\
\overline{{\mathcal{B}_{j}}}{A^{-1}}=\partial_{{\overline{j}}}(\overline{A^{-1}}\
\overline{B}{A^{-1}}).$ (3.39)

The result is the sum of $p$ terms:

$\displaystyle p\cdot\text{tr}\left[{\mathcal{B}_{i}}(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{0}\overline{A^{-1}}\
\overline{{\mathcal{B}_{j}}}{A^{-1}}(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{p-1}\right]L^{i\overline{j}}$ (3.40) $\displaystyle+$
$\displaystyle p\cdot\text{tr}\left[{\mathcal{B}_{i}}(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{1}\overline{A^{-1}}\
\overline{{\mathcal{B}_{j}}}{A^{-1}}(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{p-2}\right]L^{i\overline{j}}$ (3.41) $\displaystyle\
\ \ ...$ $\displaystyle+$ $\displaystyle
p\cdot\text{tr}\left[{\mathcal{B}_{i}}(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{p-1}\overline{A^{-1}}\
\overline{{\mathcal{B}_{j}}}{A^{-1}}(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{0}\right]L^{i\overline{j}}.$ (3.42)

In above, fictitious terms $(B\overline{A^{-1}}\ {\overline{B}}{A^{-1}})^{0}$
and $(\overline{A^{-1}}\ \overline{B}{A^{-1}}B)^{0}$ are added in to make the
pattern clearer. We will do this in other parts of the computation either.

(iii) When $L^{i\overline{j}}\partial_{\overline{j}}$ acts on $B$ in (3.34),
we don’t need to use other equations. The result is the sum of $p-1$ terms:

$\displaystyle p\cdot\text{tr}\left[{\mathcal{B}_{i}}(\overline{A^{-1}}\
\overline{B}{A^{-1}})(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{0}B_{{\overline{j}}}(\overline{A^{-1}}\
\overline{B}{A^{-1}})(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{p-2}\right]L^{i\overline{j}}$ (3.43) $\displaystyle+$
$\displaystyle p\cdot\text{tr}\left[{\mathcal{B}_{i}}(\overline{A^{-1}}\
\overline{B}{A^{-1}})(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{1}B_{{\overline{j}}}(\overline{A^{-1}}\
\overline{B}{A^{-1}})(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{p-3}\right]L^{i\overline{j}}$ (3.44) $\displaystyle\
\ \ \ ...$ $\displaystyle+$ $\displaystyle
p\cdot\text{tr}\left[{\mathcal{B}_{i}}(\overline{A^{-1}}\
\overline{B}{A^{-1}})(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{p-2}B_{{\overline{j}}}(\overline{A^{-1}}\
\overline{B}{A^{-1}})(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{0}\right]L^{i\overline{j}}.$ (3.45)

(iv) When $L^{i\overline{j}}\partial_{\overline{j}}$ acts on
$(\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}})$ in (3.35), the result
is zero. This is because of the equation (3.24).

(v) When $L^{i\overline{j}}\partial_{\overline{j}}$ acts on $B$ in (3.35), we
don’t need to use other equations. Simply differentiating $B$, we get the
following result which is the sum of $p$ terms:

$\displaystyle
p\cdot\text{tr}\left[\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}}(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{0}B_{{\overline{j}}}(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{p-1}\right]L^{i\overline{j}}$ (3.46) $\displaystyle+$
$\displaystyle
p\cdot\text{tr}\left[\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}}(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{1}B_{{\overline{j}}}(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{p-2}\right]L^{i\overline{j}}$ (3.47) $\displaystyle\ \
\ \ ...$ $\displaystyle+$ $\displaystyle
p\cdot\text{tr}\left[\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}}(B\overline{A^{-1}}\
{\overline{B}}{A^{-1}})^{p-1}B_{{\overline{j}}}(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{0}\right]L^{i\overline{j}}.$ (3.48)

(vi) When $L^{i\overline{j}}\partial_{\overline{j}}$ acts on
$(\overline{A^{-1}}\ \overline{B}{A^{-1}})$ in (3.35), we need to use (3.39).
The result is the sum of $p-1$ terms

$\displaystyle
p\cdot\text{tr}\left[(\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}})B(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{0}(\overline{A^{-1}}\
\overline{{\mathcal{B}_{j}}}{A^{-1}})B(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{p-2}\right]L^{i\overline{j}}$ (3.49) $\displaystyle+$
$\displaystyle
p\cdot\text{tr}\left[(\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}})B(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{1}(\overline{A^{-1}}\
\overline{{\mathcal{B}_{j}}}{A^{-1}})B(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{p-3}\right]L^{i\overline{j}}$ (3.50) $\displaystyle\ \
\ \ ...$ $\displaystyle+$ $\displaystyle
p\cdot\text{tr}\left[(\overline{A^{-1}}\partial_{i}{\overline{B}}{A^{-1}})B(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{p-2}(\overline{A^{-1}}\
\overline{{\mathcal{B}_{j}}}{A^{-1}})B(\overline{A^{-1}}\
\overline{B}{A^{-1}}B)^{0}\right]L^{i\overline{j}}.$ (3.51)

At a point $(\tau_{0},z_{0})\in\mathcal{R}\times V$, we change coordinate on
$V$ to diagonalize $A,B$. As discussed in section 1.4, we need to find $P$, so
that

$\displaystyle PAP^{\ast}=I,\ \ \ PBP^{T}=\Lambda=\text{diag}(\Lambda_{1},\
...\ ,\Lambda_{n})\geq 0.$ (3.52)

This can be done by Lemma A.4. Denote that

$\displaystyle{\mathcal{B}_{i}}=(\mathcal{B}_{i;\alpha\beta}),\ \ \ \
\partial_{i}{\overline{B}}=(\overline{B}_{i;\alpha\beta}).$ (3.53)

With these notation, results of (i)-(vi) can be simplified: Result of (i) is:

$\displaystyle
p\left(-\overline{B}_{i;\alpha\beta}\overline{\overline{B}_{j;\alpha\beta}}(\Lambda_{\alpha}^{2p}+\Lambda_{\beta}^{2p})\right)L^{i\overline{j}};$
(3.54)

result of (ii) is

$\displaystyle
p\left(\mathcal{B}_{i;\alpha\beta}\overline{\mathcal{B}_{j;\alpha\beta}}(\Lambda_{\alpha}^{2p-2}+\Lambda_{\alpha}^{2p-4}\Lambda_{\beta}^{2}+\
...\ +\Lambda_{\beta}^{2p-2})\right)L^{i\overline{j}};$ (3.55)

result of (iii) is

$\displaystyle
p\left(\mathcal{B}_{i;\alpha\beta}\overline{\overline{B}_{j;\alpha\beta}}(\Lambda_{\alpha}^{2p-3}\Lambda_{\beta}+\Lambda_{\alpha}^{2p-5}\Lambda_{\beta}^{3}\
...\ +\Lambda_{\alpha}\Lambda_{\beta}^{2p-3})\right)L^{i\overline{j}};$ (3.56)

result of (iv) is still $0$; result of (v) is

$\displaystyle
p\left(\overline{B}_{i;\alpha\beta}\overline{\overline{B}_{j;\alpha\beta}}(\Lambda_{\alpha}^{2p-2}+\Lambda_{\alpha}^{2p-4}\Lambda_{\beta}^{2}+\
...\ +\Lambda_{\beta}^{2p-2})\right)L^{i\overline{j}};$ (3.57)

result of (vi) is

$\displaystyle
p\left(\overline{B}_{i;\alpha\beta}\overline{\mathcal{B}_{j;\alpha\beta}}(\Lambda_{\alpha}^{2p-3}\Lambda_{\beta}+\Lambda_{\alpha}^{2p-5}\Lambda_{\beta}^{3}\
...\ +\Lambda_{\alpha}\Lambda_{\beta}^{2p-3})\right)L^{i\overline{j}}.$ (3.58)

Summing up results of (i)-(vi), we get

$\displaystyle
L^{i\overline{j}}\left(Q^{<p>}\right)_{i\overline{j}}=p\sum_{i,j}\sum_{\alpha,\beta}L^{i\overline{j}}(\mathcal{B}_{i;\alpha\beta},\overline{B}_{i;\alpha\beta})\mathcal{W}_{\alpha\beta}\overline{\left(\begin{array}[]{cc}\mathcal{B}_{j;\alpha\beta}\\\
\overline{B}_{j;\alpha\beta}\end{array}\right)},$ (3.61)

where

$\displaystyle\mathcal{W}_{\alpha\beta}=\left(\begin{array}[]{cc}\sum_{k=0}^{p-1}\Lambda_{\alpha}^{2k}\Lambda_{\beta}^{2p-2-2k}&\sum_{k=0}^{p-2}\Lambda_{\alpha}^{2k+1}\Lambda_{\beta}^{2p-3-2k}\\\
\sum_{k=0}^{p-2}\Lambda_{\alpha}^{2p-3-2k}\Lambda_{\beta}^{2k+1}&\sum_{k=0}^{p-1}\Lambda_{\alpha}^{2k}\Lambda_{\beta}^{2p-2-2k}-\Lambda_{\alpha}^{2p}-\Lambda_{\beta}^{2p}\end{array}\right).$
(3.64)

It’s obvious that when $\mathcal{W}_{\alpha\beta}\geq 0$, (3.61)$\geq 0$. In
the following, we show when
$\Lambda_{\alpha},\Lambda_{\beta}\leq\sqrt{1-\frac{1}{2p}}$,
$\mathcal{W}_{\alpha\beta}\geq 0.$

According to linear algebra, $\mathcal{W}_{\alpha\beta}\geq 0$ if and only if
$\text{tr}(\mathcal{W}_{\alpha\beta})\geq 0$ and
$\det(\mathcal{W}_{\alpha\beta})\geq 0$.
$\text{tr}(\mathcal{W}_{\alpha\beta})$ is easy to compute:

$\displaystyle\text{tr}(\mathcal{W}_{\alpha\beta})\geq\Lambda_{\alpha}^{2p-2}+\Lambda_{\beta}^{2p-2}-\Lambda_{\alpha}^{2p}-\Lambda_{\beta}^{2p}.$
(3.65)

It’s greater than $0$ providing

$\displaystyle\Lambda_{\alpha},\Lambda_{\beta}<1.$ (3.66)

To investigate the sign of $\det(\mathcal{W}_{\alpha\beta})$, we need to
simplify the right-hand side of (3.64). When
$\Lambda_{\alpha}=\Lambda_{\beta}$, denoting
$\Lambda_{\alpha}=\Lambda_{\beta}=\lambda$,

$\displaystyle\mathcal{W}_{\alpha\beta}=\left(\begin{array}[]{cc}p\lambda^{2p-2}&(p-1)\lambda^{2p-2}\\\
(p-1)\lambda^{2p-2}&p\lambda^{2p-2}-2\lambda^{2p}\end{array}\right).$ (3.69)

The determinant follows by direct computation:

$\displaystyle\det(\mathcal{W}_{\alpha\beta})=(2p-1)\lambda^{4p-4}\left[1-\frac{2p}{2p-1}\lambda^{2}\right].$
(3.70)

It’s non-negative proving

$\displaystyle\lambda^{2}\leq 1-\frac{1}{2p}.$ (3.71)

When $\Lambda_{\alpha}\neq\Lambda_{\beta}$, we use the summation formula for
geometric series to compute the summation in (3.64). We assume
$\Lambda_{\alpha}^{2}>\Lambda_{\beta}^{2}$. Note that we have
$\Lambda_{\alpha},\Lambda_{\beta}\geq 0$, so when
$\Lambda_{\alpha}\neq\Lambda_{\beta}$,
$\Lambda^{2}_{\alpha}\neq\Lambda^{2}_{\beta}$. The result is

$\displaystyle\mathcal{W}_{\alpha\beta}=\left(\begin{array}[]{cc}\frac{\Lambda_{\beta}^{2p}-\Lambda_{\alpha}^{2p}}{\Lambda^{2}_{\beta}-\Lambda^{2}_{\alpha}}&-\frac{\Lambda_{\beta}\Lambda_{\alpha}^{2p-1}-\Lambda_{\beta}^{2p-1}\Lambda_{\alpha}}{\Lambda^{2}_{\beta}-\Lambda^{2}_{\alpha}}\\\
-\frac{\Lambda_{\beta}\Lambda_{\alpha}^{2p-1}-\Lambda_{\beta}^{2p-1}\Lambda_{\alpha}}{\Lambda^{2}_{\beta}-\Lambda^{2}_{\alpha}}&\frac{\Lambda_{\beta}^{2p}-\Lambda_{\alpha}^{2p}}{\Lambda^{2}_{\beta}-\Lambda^{2}_{\alpha}}-\Lambda_{\alpha}^{2p}-\Lambda_{\beta}^{2p}\end{array}\right).$
(3.74)

The determinant follows by straightforward computation:

$\displaystyle\det(\mathcal{W}_{\alpha\beta})=\frac{(-\Lambda^{4p}_{\beta}+\Lambda^{4p-2}_{\beta})-(-\Lambda^{4p}_{\alpha}+\Lambda^{4p-2}_{\alpha})}{\Lambda^{2}_{\beta}-\Lambda^{2}_{\alpha}}.$
(3.75)

Let $f(x)=-x^{2p}+x^{2p-1}.$ Then (3.75) can be simplified as

$\displaystyle\det(\mathcal{W}_{\alpha\beta})=\frac{f(\Lambda_{\beta}^{2})-f(\Lambda_{\alpha}^{2})}{\Lambda^{2}_{\beta}-\Lambda^{2}_{\alpha}}.$
(3.76)

It’s non-negative, providing $\Lambda_{\alpha}^{2}$ and $\Lambda_{\beta}^{2}$
stay on an interval where $f$ is non-decreasing. By computing the derivative
of $f$ we know $f$ is non-decreasing on $[0,1-\frac{1}{2p}]$. Combining with
(3.66) (3.71), we know $\det(\mathcal{W}_{\alpha\beta})\geq 0$, providing
$\Lambda_{\alpha}^{2},\Lambda_{\beta}^{2}\leq 1-\frac{1}{2p}.$ Therefore,

$\displaystyle L^{i\overline{j}}\left(Q^{<p>}\right)_{i\overline{j}}\geq 0,\ \
\ \ \text{ providing }K\leq 1-\frac{1}{2p}.$ (3.77)

In the computation above, we can replace $B$ by $B-S$ and get

$\displaystyle L^{i\overline{j}}\left(Q^{<p>}_{S}\right)_{i\overline{j}}\geq
0,\ \ \ \ \text{ providing }K_{S}\leq 1-\frac{1}{2p}.$ (3.78)

As a result, we have the following proposition:

######  Proposition 3.1.

Suppose $\Phi$ is a $C^{4}$ solution to Problem 1.4 and $S$ is a constant
section of $T_{2,0}^{\ast}(V)$. For $L,\ K_{S},\ Q^{<p>}_{S}$ and $\
Q^{[p]}_{S}$, defined by (2.8) (3.4) (3.8) (3.9) respectively, we have

$\displaystyle
L^{i\overline{j}}\partial_{i\overline{j}}\left(Q^{<p>}_{S}\right)\geq 0,$
(3.79)

providing $K_{S}\leq 1-\frac{1}{2p},$ and, as a consequence,

$\displaystyle Q^{[p]}_{S}\leq\max_{{\partial\mathcal{R}\times
V}}Q^{[p]}_{S},\ \ \ \ \ \ \text{ in }\mathcal{R}\times V,$ (3.80)

providing $K_{S}\leq 1-\frac{1}{2p},$ in $\mathcal{R}\times V$.

######  Remark 3.1.

Let

$\displaystyle Q^{[\rho,p]}_{S}=\left(\text{tr}(K_{S}^{\rho
p})\right)^{\frac{1}{p}}.$ (3.81)

With more complicated computations we can show

$\displaystyle
L^{i\overline{j}}\left(Q^{[\rho,p]}_{S}\right)_{i\overline{j}}\geq 0,$ (3.82)

providing $K_{S}\leq 1-\frac{1}{2\rho}$. So, by letting $p$ go to $\infty$, we
know

$\displaystyle L^{i\overline{j}}\left(M_{S}^{\rho}\right)_{i\overline{j}}\geq
0,$ (3.83)

providing $K_{S}\leq 1-\frac{1}{2\rho}$, in the sense of viscosity solution
(see section 6 of [CIL92]). However, the current result (3.77) is enough for
our use. We can also consider

$\displaystyle\text{tr}\left(e^{pK_{S}}\right)$ (3.84)

and achieve a similar result. In [H22], (3.84) is used in $n=1$ case.

###  3.2 Preservation of $(S,\omega_{0})$-Convexity by the Method of
Continuity

In addition to (3.1), in this section, we make the following assumption.

######  Assumption 3.1.

For any $\sigma\in[0,1]$, Problem 1.4 with boundary value $\sigma F$ has a
solution $\Phi^{\sigma}$ and $\\{\Phi^{\sigma}|\sigma\in[0,1]\\}$ is a
continuous curve in $C^{4}(\overline{\mathcal{R}}\times V)$ in $C^{2}$
topology.

######  Remark 3.2.

According to the ellipticity of equation (1.44), which is proved in section
4.1, the solution to Problem 1.4 is unique. As a consequence, $\Phi^{0}$ must
equal to $0$.

Let

$\displaystyle A_{\sigma}$
$\displaystyle=(\Phi^{\sigma}_{\alpha{\overline{\beta}}}+b_{{\alpha{\overline{\beta}}}}),$
(3.85) $\displaystyle B_{S,\sigma}$
$\displaystyle=(\Phi^{\sigma}_{\alpha\beta}+\sigma S_{{\alpha\beta}}),$ (3.86)
$\displaystyle K_{S,\sigma}$
$\displaystyle=B_{S,\sigma}\overline{A_{\sigma}^{-1}}\
\overline{B_{S,\sigma}}A_{\sigma}^{-1},$ (3.87) $\displaystyle
Q^{<p>}_{S,\sigma}$ $\displaystyle=\text{tr}(K_{S,\sigma}^{p}),$ (3.88)
$\displaystyle Q^{[p]}_{S,\sigma}$
$\displaystyle=\left(Q^{<p>}_{S,\sigma}\right)^{\frac{1}{p}},$ (3.89)

and for any $(\tau,z)\in\mathcal{R}\times V$,

$\displaystyle M_{S,\sigma}(\tau,z)=\text{Maximum Eigenvalue of
}K_{S,\sigma}(\tau,z).$ (3.90)

According to the assumption (3.1),

$\displaystyle\max_{\partial\mathcal{R}\times V}M_{S,1}<1.$ (3.91)

So we can choose $p$ large enough, such that

$\displaystyle\max_{\partial\mathcal{R}\times V}Q^{[p]}_{S,1}<1-\frac{1}{2p}.$
(3.92)

This can be done because when $p\rightarrow+\infty$,

$\displaystyle 1-\frac{1}{2p}\rightarrow 1$ (3.93)

and

$\displaystyle\max_{\partial\mathcal{R}\times
V}Q^{[p]}_{S,1}\rightarrow\max_{\partial\mathcal{R}\times V}M_{S,1}<1.$ (3.94)

According to Lemma A.7, for any $(\tau,z)\in\partial\mathcal{R}\times V$,
$Q^{[p]}_{S,\sigma}(\tau,z)$ is a monotone non-decreasing function of
$\sigma$, so $\max_{\partial\mathcal{R}\times V}Q^{[p]}_{S,\sigma}$ is a
monotone non-decreasing function of $\sigma$. Thus

$\displaystyle\max_{\partial\mathcal{R}\times
V}Q^{[p]}_{S,\sigma}<1-\frac{1}{2p},$ (3.95)

for any $\sigma\in[0,1]$. Therefore, for no $\sigma\in[0,1]$,

$\displaystyle\max_{{\overline{\mathcal{R}}}\times
V}Q^{[p]}_{S,\sigma}\in\left(\max_{\partial\mathcal{R}\times
V}Q^{[p]}_{S,\sigma},1-\frac{1}{2p}\right).$ (3.96)

This is because if (3.96) is valid, for $\sigma=\sigma_{0}$, then

$\displaystyle\max_{{\overline{\mathcal{R}}}\times
V}Q^{[p]}_{S,\sigma_{0}}\leq 1-\frac{1}{2p},$ (3.97)

and, as a consequence,

$\displaystyle\max_{{\overline{\mathcal{R}}}\times V}M_{S,\sigma_{0}}\leq
1-\frac{1}{2p}.$ (3.98)

So, using Proposition 3.1, we know

$\displaystyle\max_{{\overline{\mathcal{R}}}\times
V}Q^{[p]}_{S,\sigma_{0}}\leq\max_{\partial\mathcal{R}\times
V}Q^{[p]}_{S,\sigma_{0}},$ (3.99)

which contradicts with (3.96).

The function $\max_{{\overline{\mathcal{R}}}\times V}Q^{[p]}_{S,\sigma}$ is a
continuous function of $\sigma$, because of Assumption 3.1. And
$\max_{{\overline{\mathcal{R}}}\times V}Q^{[p]}_{S,0}=0$, according to the
uniqueness of $C^{2}$ solution. So

$\displaystyle\max_{{\overline{\mathcal{R}}}\times
V}Q^{[p]}_{S,\sigma}\leq\max_{\partial\mathcal{R}\times V}Q^{[p]}_{S,\sigma},\
\ \ \ \text{for any }\sigma\in[0,1].$ (3.100)

Figure 1: Method of Continuity

As illustrated by Figure 1, the dashed curve

$\displaystyle\left\\{(\sigma,\max_{\partial\mathcal{R}\times
V}Q^{[p]}_{S,\sigma})\big{|}\sigma\in[0,1]\right\\}$ (3.101)

raises up from left to right and stays below the line
$\\{m=1-\frac{1}{2p}\\}$. The continuous curve

$\displaystyle\left\\{(\sigma,\max_{{\overline{\mathcal{R}}}\times
V}Q^{[p]}_{S,\sigma})\big{|}\sigma\in[0,1]\right\\},$ (3.102)

whose left endpoint is $(0,0)$, cannot intersect with the shadowed area, so it
has to stay below the shadowed area. Actually, it has to coincide with the
dashed curve.

Therefore, we have, for any $\sigma\in[0,1]$,

$\displaystyle M_{S,\sigma}\leq\max_{\overline{\mathcal{R}}\times
V}Q^{[p]}_{S,\sigma}=\max_{\partial\mathcal{R}\times
V}Q^{[p]}_{S,\sigma}\leq\max_{\partial\mathcal{R}\times V}Q^{[p]}_{S,1},\ \ \
\ \ \ \text{ in }\mathcal{R}\times V.$ (3.103)

Let $p\rightarrow 0$, we get

$\displaystyle M_{S,\sigma}\leq\max_{\partial\mathcal{R}\times V}M_{S,1},\ \ \
\ \ \ \text{ in }\mathcal{R}\times V.$ (3.104)

In sum, we have the following Proposition.

######  Proposition 3.2.

Suppose $S$ is a constant section of $T_{2,0}^{\ast}(V)$ and $F\in
C^{2}({\partial\mathcal{R}\times V})$ satisfies that

$\displaystyle F(\tau,\ast)\text{ is strictly $(S,\omega_{0})$-convex, for any
$\tau\in\partial\mathcal{R}$.}$ (3.105)

In addition, we assume that the Assumption 3.1 is satisfied. Then the solution
$\Phi$ to Problem 1.4, with boundary value $F$, satisfies

$\displaystyle\Phi(\tau,\ast)\text{ is strictly $(S,\omega_{0})$-convex, for
any $\tau\in\mathcal{R}$.}$ (3.106)

###  3.3 Convexity and Metric Lower Bound Estimates by Altering $S$

In this section, we assume that the conditions of Proposition 3.2 are
satisfied. In addition, we assume that, for a constant $\delta>0$,

$\displaystyle F(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of modulus
$>\delta$, for any $\tau\in\partial\mathcal{R}$},$ (3.107)

or equivalently, according to Lemma A.2,

$\displaystyle F(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of degree
$>\delta$, for any $\tau\in\partial\mathcal{R}$}.$ (3.108)

Condition (3.108) says, for any constant section $\Theta$ of
$T_{2,0}^{\ast}(V)$, with

$\displaystyle\Theta\overline{W^{-1}}\ \overline{\Theta}W^{-1}\leq\delta^{2},$
(3.109)

we have

$\displaystyle F(\tau,\ast)\text{ is strictly $(S+\Theta,\omega_{0})$-convex,
for any $\tau\in\partial\mathcal{R}$}.$ (3.110)

Here $W=(b_{\alpha{\overline{\beta}}})$. So, by Proposition 3.2, for any
constant section $\Theta$ of $T_{2,0}^{\ast}(V)$, satisfying (3.109), we have

$\displaystyle\Phi(\tau,\ast)\text{ is strictly
$(S+\Theta,\omega_{0})$-convex, for any $\tau\in\mathcal{R}$}.$ (3.111)

Therefore,

$\displaystyle\Phi(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of degree
$>\delta$, for any $\tau\in\mathcal{R}$}.$ (3.112)

By Lemma A.2, we know (3.112) is equivalent to

$\displaystyle\Phi(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of modulus
$>\delta$, for any $\tau\in\mathcal{R}$}.$ (3.113)

Then, by the definition of modulus of convexity, Definition 1.3, we know

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(\tau,\ast)>\delta\omega_{0},\
\ \ \ \ \ \text{for any }\tau\in\mathcal{R}.$ (3.114)

As a result, we have the following convexity and metric lower bound estimate:

######  Proposition 3.3 (Apriori Convexity and Metric Lower Bound Estimate).

Suppose that, for a constant $\delta>0$ and a constant section $S$ of
$T_{2,0}^{\ast}(V)$, $F\in C^{\infty}({\partial\mathcal{R}\times V})$
satisfies

$\displaystyle F(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of degree
$>\delta$, for any $\tau\in\partial\mathcal{R}$}.$ (3.115)

In addition, Assumption 3.1 is satisfied. Then a solution $\Phi$ to Problem
1.4, with boundary value $F$ satisfies

$\displaystyle\Phi(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of degree
$>\delta$, for any $\tau\in\mathcal{R}$}$ (3.116)

and, as a consequence,

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(\tau,\ast)>\delta\omega_{0}\text{,
\ \ \ \ \ \ for any $\tau\in\mathcal{R}$}.$ (3.117)

##  4 Existence of Solutions to the Perturbed Equation

In this section, we prove the existence of smooth solutions to Problem 1.4.

In section 4.1, we discuss some basic properties of equation (1.44), including
ellipticity and concavity. Then in section 4.2, we derive a directional
partial $C^{2}$ estimate, in the direction of the affine manifold. The
estimate allows us to derive $C^{0}$ and $C^{1}$ estimates, in section 4.3.
Then with the metric lower bound estimate, Proposition 3.3, we prove $C^{2}$
and $C^{2,\alpha}$ estimates in section 4.4 and 4.5. Finally, in section 4.6,
we prove the existence of smooth solutions.

###  4.1 Basic Properties of the Elliptic Perturbation Equation

First, equation (1.44) is elliptic. This has been indicated by (2.9). More
precisely, let $\Phi^{\lambda}$ be a family of solutions of equation (1.44)
with $\Phi^{0}=\Phi$ and $\frac{d}{d\lambda}\Phi^{\lambda}=\Psi$, at
$\lambda=0$. Then differentiating

$\displaystyle\Phi^{\lambda}_{\tau\overline{\tau}}-\Phi^{\lambda}_{\tau{\overline{\beta}}}g^{{\alpha{\overline{\beta}}}}_{\lambda}\Phi^{\lambda}_{{\overline{\tau}}\alpha}=\epsilon
b_{{\alpha{\overline{\beta}}}}g^{{\alpha{\overline{\beta}}}}_{\lambda}$ (4.1)

with respect to $\lambda$, at $\lambda=0$, gives

$\displaystyle L^{i\overline{j}}\Psi_{i\overline{j}}=0,$ (4.2)

where $L^{i\overline{j}}$ was introduced in section 2 by (2.8). In (4.1),
$g_{\lambda}^{{\alpha{\overline{\beta}}}}$ is the inverse of
$b_{{\alpha{\overline{\beta}}}}+\Phi^{\lambda}_{{\alpha{\overline{\beta}}}}$.

Then we consider the concavity. We will show

$\displaystyle
F(\Phi_{i\overline{j}})=\log(\Phi_{\tau{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha\overline{\beta}}\Phi_{\alpha{\overline{\tau}}})-\log(b_{\alpha\overline{\beta}}g^{\alpha\overline{\beta}})$
(4.3)

is a concave function of $\Phi_{i\overline{j}}$, providing

$\displaystyle\left(\begin{array}[]{cc}\Phi_{\tau{\overline{\tau}}}&\Phi_{\tau{\overline{\beta}}}\\\
\Phi_{\alpha{\overline{\tau}}}&\Phi_{{\alpha{\overline{\beta}}}}+b_{\alpha{\overline{\beta}}}\end{array}\right)>0.$
(4.6)

Actually, if we denote

$\displaystyle F_{1}(\Phi_{i\overline{j}})$
$\displaystyle=\log(\Phi_{\tau{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha\overline{\beta}}\Phi_{\alpha{\overline{\tau}}}),$
(4.7) $\displaystyle F_{2}(\Phi_{i\overline{j}})$
$\displaystyle=-\log(b_{\alpha\overline{\beta}}g^{\alpha\overline{\beta}}),$
(4.8)

then we can show $F_{1}$ and $F_{2}$ are both concave. These computations are
in Appendix A.3.

Suppose $\Phi$ is a $C^{4}$ solution to Problem 1.4 and $X$ is a constant
vector field in $\mathcal{R}\times V$. We apply $\partial_{X}$ to equation
(1.44) and get

$\displaystyle\Phi_{X\tau{\overline{\tau}}}-\Phi_{X{\overline{\beta}}\tau}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}+\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{X\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\tau}}}-\Phi_{\tau{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{X\alpha{\overline{\tau}}}=-\epsilon
b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{X\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}.$
(4.9)

With the linearized operator $L^{i\overline{j}}\partial_{i{\overline{j}}}$,
(4.9) is simplified to

$\displaystyle L^{i\overline{j}}\partial_{i{\overline{j}}}(\Phi_{X})=0.$
(4.10)

Then apply $\partial_{X}$ to equation (4.9), we get

$\displaystyle L^{i\overline{j}}\partial_{i{\overline{j}}}(\Phi_{XX})\geq 0.$
(4.11)

This is because of the concavity of (4.3). We can also get (4.11) directly by
replacing $\partial_{\theta}$ and $\partial_{\overline{\gamma}}$ in
(2.10)-(2.13) by $\partial_{X}$.

###  4.2 Affine-Manifold-Directional $C^{2}$ Estimates

Suppose $X$ is a constant real vector field in $\mathcal{R}\times V$, parallel
to $V$. This is to say, if we denote the projection from $\mathcal{R}\times V$
to $\mathcal{R}$ by $\pi_{\mathcal{R}}$, then
$(\pi_{\mathcal{R}})_{\ast}(X)=0$. By equation (4.11), we know

$\displaystyle\Phi_{XX}\leq\max_{\partial\mathcal{R}\times V}\Phi_{XX},\ \ \ \
\ \text{ in }\mathcal{R}\times V.$ (4.12)

Because $\omega_{0}$ has a lower bound and
$\sqrt{-1}\partial\overline{\partial}F(\tau,\ast)$ has a uniform upper bound,
we can find a constant $C>0$, so that

$\displaystyle\max_{\partial\mathcal{R}\times
V}\Phi_{XX}=\max_{\partial\mathcal{R}\times V}F_{XX}\leq C\omega_{0}(X,JX).$
(4.13)

Therefore,

$\displaystyle\sqrt{-1}\partial\overline{\partial}\Phi(X,JX)=\frac{\Phi_{XX}+\Phi_{JXJX}}{2}\leq
C\omega_{0}(X,JX).$ (4.14)

This implies

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(\tau,\ast)\leq(1+C)\omega_{0},\
\ \ \ \ \ \text{for any }\tau\in\partial\mathcal{R}$ (4.15)

and equivalently

$\displaystyle(g_{\alpha\overline{\beta}})\leq(1+C)b_{\alpha\overline{\beta}}.$
(4.16)

As a consequence, we have

$\displaystyle
b_{\alpha\overline{\beta}}g^{\alpha\overline{\beta}}\cdot\det(g_{\alpha\overline{\beta}})\leq
n(1+C)^{n-1}\cdot\det(b_{\alpha\overline{\beta}}).$ (4.17)

###  4.3 $C^{0}$ and $C^{1}$ Estimates

To do the $C^{0}$ and boundary $C^{1}$ estimates, we construct $\Psi$ and
$\Phi^{0}$, so that

$\displaystyle\Psi\leq\Phi\leq\Phi^{0},\ \ \ \ \ \text{ in }\mathcal{R}\times
V,$ (4.18)

and

$\displaystyle\Psi=\Phi=\Phi^{0},\ \ \ \ \ \text{ on
}\partial\mathcal{R}\times V.$ (4.19)

We let $\Phi^{0}$ be the solution to Problem 1.2, with the boundary condition

$\displaystyle\Phi^{0}=\Phi,\ \ \ \ \ \ \text{ on }{\partial\mathcal{R}\times
V}.$ (4.20)

We easily know that

$\displaystyle\Phi\leq\Phi^{0},\ \ \ \ \ \text{ in }\mathcal{R}\times V,$
(4.21)

because $\Phi_{0}$ is a maximal $\Omega_{0}$-PSH function.

For the construction of $\Psi$, we need to use the estimate (4.17). Locally,
we have

$\displaystyle\det(h_{i\overline{j}})=\epsilon
b_{\alpha\overline{\beta}}g^{\alpha\overline{\beta}}\cdot\det(g_{\alpha\overline{\beta}})\leq\epsilon
n(1+C)^{n-1}\cdot\det(b_{\alpha\overline{\beta}}).$ (4.22)

So, for a solution $\Psi$ to Problem 1.3, with $\varepsilon=\epsilon
n(1+C)^{n-1}$, we have

$\displaystyle(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi)^{n+1}\leq(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Psi)^{n+1}.$
(4.23)

Thus $\Psi\leq\Phi$, given the boundary condition $\Psi=\Phi$ on
${\partial\mathcal{R}\times V}$. The global $C^{0}$ and $C^{1}$ estimates for
$\Psi$ are known by [CTW17] and [B12]. Therefore, we have the global $C^{0}$
estimate and boundary $C^{1}$ estimate for $\Phi$.

The $C^{1}$ interior estimate can be derived from boundary estimates with
equation (4.10).

###  4.4 $C^{2}$ Estimates

We first prove the boundary $C^{2}$ estimate, then we use equation (4.11) to
derive the interior estimate.

To do the boundary estimate, we need to flatten the boundary. Around
$\tau_{0}\in\partial\mathcal{R}$, find a holomorphic map

$\displaystyle
f:B_{\delta^{\prime}}(\tau_{0})\cap\overline{\mathcal{R}}\rightarrow\mathbb{C}=\\{\zeta=\xi+\sqrt{-1}\eta\\},$
(4.24)

for a small $\delta^{\prime}$. We want that $f^{\prime}\neq 0$,
$f(\tau_{0})=0,$

$\displaystyle
f(\partial\mathcal{R})\subset\left\\{\zeta|\text{Im}(\zeta)=0\right\\}$ (4.25)

and

$\displaystyle
f(B_{\delta^{\prime}}(\tau_{0})\cap\overline{\mathcal{R}})\supset
B_{\delta}^{+}(0)=\\{|\text{Re}(\zeta)|<\delta,0\leq\text{Im}(\zeta)<\delta\\},$
(4.26)

for a small $\delta$. For a point $p_{0}\in V$, let $\\{z^{\alpha}\\}$ be a
set of coordinates in $B_{r}(p_{0})\subset V$, for a small $r$. Without loss
of generality, we can assume that $p_{0}=0$ in this coordinate chart. We also
require that the coordinate $z^{\alpha}$ is properly chosen so that the
natural metric on $B_{r}(0)$, as a subset of $\mathbb{C}^{n}$, is the metric
$\omega_{0}$. In the following, we will work in the coordinate chart
$B^{+}_{\delta}(0)\times B_{r}(0)$ and estimate second order derivatives at
$(0,0)$. For convenience, we denote $B^{+}_{\delta}(0)\times B_{r}(0)$ by
$\mathcal{D}$ and denote $\\{\text{Im}(\zeta)=0\\}\times B_{r}(0)$ by
$\Gamma$.

In the coordinate chart $\mathcal{D}$, $\Phi(\zeta,\vec{z})$ satisfies

$\displaystyle\Phi_{\zeta{\overline{\zeta}}}-\Phi_{\zeta{\overline{\beta}}}g^{\alpha\overline{\beta}}\Phi_{\alpha{\overline{\zeta}}}=\epsilon\cdot
k(\zeta)\cdot b_{\alpha\overline{\beta}}g^{\alpha\overline{\beta}}.$ (4.27)

Here $k=\frac{1}{|f^{\prime}|^{2}}$, so it is a positive and smooth function
$\zeta$.

$\displaystyle\frac{1}{K}<k<K,$ (4.28)

for a constant $K$.

The second order derivatives of $\Phi$ at $(0,0)$ in $\Gamma$ directions are
known, they depend on the boundary value $F$. Let $X$ be a constant vector
field parallel to $\Gamma$ and with $|X|=1$. We need to estimate
$\Phi_{X\eta}$, then using equation we can control $\Phi_{\eta\eta}$.

The method of estimating $\Phi_{X\eta}$ is similar to the method used in
[G98]. According to the estimate of section 4.3, there is a constant $C_{1}$,
so that

$\displaystyle|\Phi_{X}|\leq C_{1}.$ (4.29)

In the following, we will show $\Phi_{X\eta}(0,0)\leq C_{2}$, for a constant
$C_{2}$.

First, we need to derive an equation satisfied by $\Phi_{X}$. Applying
$\partial_{X}$ to equation (4.27) gives

$\displaystyle\Phi_{X\zeta{\overline{\zeta}}}-\Phi_{X{\overline{\beta}}\zeta}g^{\alpha{\overline{\beta}}}\Phi_{\alpha{\overline{\zeta}}}+\Phi_{\zeta{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{X\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}\Phi_{\alpha{\overline{\zeta}}}-$
$\displaystyle\Phi_{\zeta{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\Phi_{X\alpha{\overline{\zeta}}}$
(4.30) $\displaystyle=-\epsilon
kb_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\mu}}}\Phi_{X\rho{\overline{\mu}}}g^{\rho{\overline{\beta}}}+\epsilon(\partial_{X}k)b_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\beta}}}.$
(4.31)

We introduce the following operator $\mathcal{L}$, which is a scalar function
multiple of $L$, after coordinate transformation. Equivalently, $\mathcal{L}$
can also be considered as the linearization operator of (4.27):

$\displaystyle\mathcal{L}=\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}},$
(4.32)

with

$\displaystyle\left(\mathcal{L}^{i\overline{j}}\right)=\left(\begin{array}[]{cc}\mathcal{L}^{0\overline{0}}&\mathcal{L}^{0{\overline{\beta}}}\\\
\mathcal{L}^{{\overline{\alpha}}0}&\mathcal{L}^{\alpha{\overline{\beta}}}\end{array}\right)=\left(\begin{array}[]{cc}1&-\Phi_{\rho{\overline{\zeta}}}g^{\rho{\overline{\beta}}}\\\
-\Phi_{\zeta{\overline{\mu}}}g^{\alpha{\overline{\mu}}}&\epsilon
kb_{\mu{\overline{\rho}}}g^{\mu{\overline{\beta}}}g^{\alpha{\overline{\rho}}}+\Phi_{\zeta{\overline{\mu}}}g^{\alpha{\overline{\mu}}}\Phi_{\rho{\overline{\zeta}}}g^{\rho{\overline{\beta}}}\end{array}\right).$
(4.37)

Here $i,j$ run from $0$ to $n$, and the $0$-th coordinate is $\zeta$. With
this operator, equation (4.31) becomes

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Phi_{X})=\epsilon(\partial_{X}k)b_{{\alpha{\overline{\beta}}}}g^{{\alpha{\overline{\beta}}}}.$
(4.38)

Since we have the metric lower bound estimate, Proposition 3.3, we know the
right-hand side of (4.38) is bounded. We can assume, for a constant
$\tilde{C}$,

$\displaystyle-\tilde{C}\leq\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Phi_{X}).$
(4.39)

Then we construct a barrier function $u$, so that $u\geq\Phi_{X}$ and
$u(0,0)=\Phi_{X}(0,0)$. Thus we can get an upper bound for $\Phi_{X\eta}$. The
barrier function is

$\displaystyle u=l+C_{3}\left(|z^{\alpha}|^{2}+\xi^{2}\right)+C_{4}\eta-
C_{5}\eta^{2}+C_{6}(\Phi-\Psi).$ (4.40)

In above, $l$ is the $\Gamma$-directional linearization of $\Phi_{X}$ at
$(0,0)$. That’s to say,

$\displaystyle l(0,0)$ $\displaystyle=\Phi_{X}(0,0),$ (4.41)
$\displaystyle\partial_{\eta}l$ $\displaystyle=0$ (4.42)

and

$\displaystyle\nabla_{\Gamma}l(0,0)=\nabla_{\Gamma}\Phi_{X}(0,0).$ (4.43)

$\Psi$ is a solution to Problem 1.3 with $\varepsilon=\epsilon n(1+C)^{n-1}+1$
and

$\displaystyle\Psi=\Phi,\ \ \ \ \ \ \text{ on }{\partial\mathcal{R}\times V}.$
(4.44)

The $\Psi$ constructed here is even smaller than the $\Psi$ constructed in
section (4.3), so $\Psi\leq\Phi$. And according to the $C^{2}$ estimate for
$\Psi$ [B12], we have

$\displaystyle\left(\begin{array}[]{cc}\Psi_{\zeta{\overline{\zeta}}}&\Psi_{\zeta{\overline{\beta}}}\\\
\Psi_{\alpha{\overline{\zeta}}}&\Psi_{{\alpha{\overline{\beta}}}}+b_{\alpha{\overline{\beta}}}\end{array}\right)>\frac{1}{C_{7}}\left(\begin{array}[]{cc}1&\\\
&b_{\alpha{\overline{\beta}}}\end{array}\right).$ (4.49)

In the following, we show that we can properly choose parameters
$C_{3},C_{4},C_{5},C_{6}$, so that

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}u\leq-\tilde{C},\
\ \ \ \ \text{ in }\Omega,$ (4.50)

and

$\displaystyle u\geq\Phi_{X},\ \ \ \ \ \ \text{ on }\partial\Omega.$ (4.51)

Then the comparison principle implies that $u\geq\Phi_{X}$ in $\Omega$.

We compute and estimate $\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}u$
term by term:

(i) $\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(l+C_{4}\eta)=0$,
because $l+C_{4}\eta$ is a linear function.

(ii)Using the metric lower bound estimate, Proposition 3.3, we have

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(|z^{\alpha}|^{2}+\xi^{2})=\epsilon
kb_{\theta{\overline{\gamma}}}g^{\theta{\overline{\alpha}}}g^{\alpha{\overline{\gamma}}}+g^{\theta{\overline{\alpha}}}\Phi_{\theta{\overline{\zeta}}}g^{\alpha{\overline{\gamma}}}\Phi_{\zeta{\overline{\gamma}}}+\frac{1}{2}\leq
C_{8}\epsilon
K+g^{\theta{\overline{\alpha}}}\Phi_{\theta{\overline{\zeta}}}g^{\alpha{\overline{\gamma}}}\Phi_{\zeta{\overline{\gamma}}}.$
(4.52)

Here $C_{8}$ depends on the metric lower bound.

(iii)
$\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(-\eta^{2})=-\frac{1}{2}.$

(iv) For $\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Phi-\Psi)$, we
split it into two terms:
$\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Phi)+\mathcal{L}^{{\alpha{\overline{\beta}}}}b_{{\alpha{\overline{\beta}}}}$
and
$-(\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Psi)+\mathcal{L}^{{\alpha{\overline{\beta}}}}b_{{\alpha{\overline{\beta}}}})$.
Using
$b_{\alpha{\overline{\beta}}}+\Phi_{\alpha{\overline{\beta}}}=g_{\alpha{\overline{\beta}}}$,
we get

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Phi)+\mathcal{L}^{{\alpha{\overline{\beta}}}}b_{{\alpha{\overline{\beta}}}}=2\epsilon
kb_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\beta}}}.$ (4.53)

For the right-hand side of (4.53), we use the metric lower bound estimate and
get

$\displaystyle 2\epsilon
kb_{\alpha{\overline{\beta}}}g^{\alpha{\overline{\beta}}}\leq 2\epsilon
KC_{9}.$ (4.54)

For
$\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Psi)+\mathcal{L}^{{\alpha{\overline{\beta}}}}b_{{\alpha{\overline{\beta}}}}$,
we use (4.49) and get

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Psi)+\mathcal{L}^{{\alpha{\overline{\beta}}}}b_{{\alpha{\overline{\beta}}}}\geq\frac{1}{C_{7}}\left(g^{\alpha{\overline{\eta}}}\Phi_{\zeta{\overline{\eta}}}g^{\mu{\overline{\beta}}}\Phi_{\mu{\overline{\zeta}}}b_{\alpha{\overline{\beta}}}\right).$
(4.55)

Note that we have already made the assumption, when choosing the coordinate
chart, that, in $\mathcal{D}$,
$b_{{\alpha{\overline{\beta}}}}=\delta_{\alpha{\overline{\beta}}}$. So

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Psi)+\mathcal{L}^{{\alpha{\overline{\beta}}}}b_{{\alpha{\overline{\beta}}}}\geq\frac{1}{C_{7}}\left(g^{\alpha{\overline{\eta}}}\Phi_{\zeta{\overline{\eta}}}g^{\mu{\overline{\alpha}}}\Phi_{\mu{\overline{\zeta}}}\right).$
(4.56)

The right-hand side of (4.56) can be used to control the right-hand side of
(4.52).

In sum, we have

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}u\leq(C_{3}\cdot\epsilon
KC_{8}+C_{6}\cdot 2\epsilon
KC_{9}-\frac{C_{5}}{2})+g^{\alpha{\overline{\eta}}}\Phi_{\zeta{\overline{\eta}}}g^{\mu{\overline{\alpha}}}\Phi_{\mu{\overline{\zeta}}}(C_{3}-\frac{C_{6}}{C_{7}}).$
(4.57)

We need to choose $C_{3},C_{5},C_{6}$ so that

$\displaystyle C_{6}\geq C_{3}\cdot C_{7}$ (4.58)

and

$\displaystyle C_{5}\geq 2(C_{3}\cdot 2\epsilon KC_{8}+C_{6}\cdot 2\epsilon
KC_{9}+\tilde{C}).$ (4.59)

We also want $u\geq\Phi_{X}$ on $\partial\Omega$. We need to choose $C_{3}$
big enough, so that

$\displaystyle
C_{3}(\sum_{\alpha}|z^{\alpha}|^{2}+\xi^{2})+l\geq\Phi_{X}+(\sum_{\alpha}|z^{\alpha}|^{2}+\xi^{2}),\
\ \ \ \text{ on }\Gamma.$ (4.60)

This requires

$\displaystyle C_{3}\geq\max_{\Gamma}|D^{2}(\Phi_{X}|_{\Gamma})|+1.$ (4.61)

We note that

$\displaystyle C_{4}\eta-C_{5}\eta^{2}+C_{6}(\Phi-\Psi)=0,\ \ \ \ \ \ \text{
on }\Gamma,$ (4.62)

so $u\geq\Phi_{X}$ on $\Gamma$, given (4.60) is valid. To make $u\geq\Phi_{X}$
on $\partial\Omega-\Gamma$, we choose $C_{4}$ big enough. Given (4.60), we
have

$\displaystyle u\geq\Phi_{X}+\delta_{1},\ \ \ \ \ \ \text{ on
}\partial\Gamma,$ (4.63)

for a small positive constant $\delta_{1}$. Then for a small
$\delta_{2}\in(0,r)$,

$\displaystyle
l+C_{3}\left(|z^{\alpha}|^{2}+\xi^{2}\right)-C_{5}\eta^{2}+C_{6}(\Phi-\Psi)\geq\Phi_{X},\
\ \ \ \ \ \text{ on }\\{\eta\leq\delta_{2}\\}\cap\partial\Omega.$ (4.64)

$\delta_{2}$ depends on $\delta_{1},C_{5},C_{6}$, second order derivatives of
$F$ and the norms of gradients of $\Psi,\Phi$. We also have

$\displaystyle
l+C_{3}\left(|z^{\alpha}|^{2}+\xi^{2}\right)-C_{5}\eta^{2}+C_{6}(\Phi-\Psi)>\Phi_{X}-C_{10},\
\ \ \ \ \ \text{ in }\Omega,$ (4.65)

for a constant $C_{10}$, depending on $C_{5},C_{6}$, second order derivatives
of $F$, the $C^{1}$ norm of $\Phi$ and the $C^{0}$ norm of $\Psi$. We can
choose

$\displaystyle C_{4}>\frac{C_{10}}{\delta_{2}}.$ (4.66)

Then $u\geq\Phi_{X}$ on $\partial\mathcal{D}$.

In sum, we choose $C_{3}$ large enough with condition (4.61), then choose
$C_{6}$ with condition (4.58), then choose $C_{5}$ with condition (4.59) and
finally choose $C_{4}$ according to condition (4.66).

Thus, we have an upper bound for $\Phi_{X\eta}$. To get the lower bound, we
simply replace $\partial_{X}\Phi$ by $\partial_{-X}\Phi$.

Then we can use equation (4.27) to get the estimate of $\Phi_{\eta\eta}$. Just
note that $4\Phi_{\zeta{\overline{\zeta}}}=\Phi_{\xi\xi}+\Phi_{\eta\eta}$. So

$\displaystyle\Phi_{\eta\eta}=-\Phi_{\xi\xi}+4\Phi_{\zeta{\overline{\beta}}}g^{\alpha\overline{\beta}}\Phi_{\alpha{\overline{\zeta}}}+4\epsilon\cdot
k(\zeta)\cdot b_{\alpha\overline{\beta}}g^{\alpha\overline{\beta}}.$ (4.67)

The estimate of the right-hand side of (4.67) depends on the boundary value
$F$, the metric lower bound estimate and the estimate of $\Phi_{X\eta}$.

Given the boundary estimate, we can go back to the original $(\tau,\vec{z})$
coordinates and use equation (4.11). Similar to section 4.2, for any constant
vector $X$ in $\mathcal{R}\times V$, we have

$\displaystyle\Phi_{XX}\leq\max_{\partial\mathcal{R}\times V}\Phi_{XX},\ \ \ \
\ \ \text{ in }\mathcal{R}\times V.$ (4.68)

For the lower bound estimate of $\Phi_{XX}$, we have

$\displaystyle-\Phi_{XX}$
$\displaystyle=\Phi_{JXJX}-2\sqrt{-1}\partial\overline{\partial}\Phi(X,JX)$
(4.69)
$\displaystyle=\Phi_{JXJX}-2(\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi)(X,JX)+2\Omega_{0}(X,JX).$
(4.70)

Then, using $\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi\geq 0$, we
get the lower bound of $\Phi_{XX}$.

In sum, we get

$\displaystyle|\Phi|_{C^{2}(\overline{\mathcal{R}}\times V)}\leq C_{11},$
(4.71)

for a constant $C_{11}$, which depends on $\epsilon$,
$|F|_{C^{3}(\overline{\mathcal{R}}\times V)}$, $\omega_{0}$, the metric lower
bound estimate and the boundary of $\mathcal{R}$. In particular, when
$\epsilon\rightarrow 0$, the constant $C_{11}$ does not go to $\infty$.
However, we don’t need to use this fact.

###  4.5 $C^{2,\alpha}$ Estimates

With the $C^{2}$ estimate in the previous section, we know the operator
$\mathcal{L}$ and $L$ are uniform elliptic. We only need to prove the boundary
$C^{2,\alpha}$ estimate. Then, with the uniform ellipticity, concavity,
$C^{2}$ estimate and the boundary $C^{2,\alpha}$ estimate, we can derive the
interior $C^{2,\alpha}$ estimate with standard method [H16].

For the boundary $C^{2,\alpha}$ estimate we need to flatten the boundary
again. Adopting notation of section 4.4, we know that $\Phi$ satisfies the
equation

$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Phi_{X})=\epsilon(\partial_{X}k)b_{{\alpha{\overline{\beta}}}}g^{{\alpha{\overline{\beta}}}}\
\ \ \ \ \ \text{ in $\mathcal{D}$}.$ (4.72)

We construct a function

$\displaystyle\mathcal{F}(\zeta,\vec{z})=\partial_{X}F(\text{Re}(\zeta),\vec{z}).$
(4.73)

Then

$\displaystyle\Phi_{X}-F$ $\displaystyle=0,$ $\displaystyle\text{ on }\Gamma;$
(4.74)
$\displaystyle\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}(\Phi_{X}-F)$
$\displaystyle=\epsilon(\partial_{X}k)b_{{\alpha{\overline{\beta}}}}g^{{\alpha{\overline{\beta}}}}-\mathcal{L}^{i\overline{j}}\partial_{i\overline{j}}\mathcal{F},$
$\displaystyle\text{ in }\mathcal{D}.$ (4.75)

The right-hand side of (4.75) is bounded, according to the $C^{2}$ estimate in
the previous section, so we can use Theorem 1.2.16 of [H16] and get the
$C^{\alpha}$ estimate for $\partial_{\eta}(\Phi_{X}-\mathcal{F})=\Phi_{X\eta}$
in a small neighborhood of $0$ in $\Gamma$, for an $\alpha\in(0,1)$. Using
equation (4.67), we get the $C^{\alpha}$ estimate for $\Phi_{\eta\eta}$.

###  4.6 Existence of Smooth Solutions by the Method of Continuity

Suppose $F\in C^{\infty}({\partial\mathcal{R}\times V})$ satisfies condition
(3.1), for a constant section $F$ of $T_{2,0}^{\ast}(V)$. Then according to
Lemma A.7,

$\displaystyle\sigma F(\tau,\ast)\text{ is strictly $(S,\omega_{0})$-convex
for any $\tau\in\mathcal{R}$ and any $\sigma\in[0,1]$ }.$ (4.76)

Consider the set

$\displaystyle\mathscr{S}=\\{\sigma\in[0,1]\big{|}$ Problem 1.4 with boundary
value $s\cdot F$ has a solution $\Phi^{s}$, for any $s\leq\sigma$, (4.77)
$\displaystyle\ \ \ \text{and $\Phi^{s}$ is a continuous curve in
$C^{4}(\overline{\mathcal{R}}\times V)$ with $C^{2}$ topology}\\}.$ (4.78)

Obviously, $\mathscr{S}$ is non-empty, since it contains $0$. So, if
$\mathscr{S}$ is both open and closed, then $\mathscr{S}=[0,1]$

Before we prove the openness and closeness, we point out that if a solution
$\Phi$ is in $C^{2,\alpha}(\overline{\mathcal{R}}\times V)$ then we can use
the standard bootstrap technique (Theorem 5.1.9 and 5.1.10 of [H16])) to show
that $\Phi$ is actually in $C^{\infty}(\overline{\mathcal{R}}\times V)$. This
is because of the condition (1.42) and the ellipticity of the equation (1.41).

The openness can be proved with standard implicit function theorem. This is
because of the condition (1.42) and the ellipticity. Without such condition,
the openness can be quite difficult to prove. For example, in [CFH20], we used
Nash-Moser inverse function theorem to prove an openness result for geodesic
equations.

For the closeness, given
$\\{\sigma_{i}\\}_{i\in\mathbb{Z}^{+}}\subset\mathscr{S}$ with
$\lim_{i\rightarrow\infty}\sigma_{i}=\sigma_{\infty}$, we need to show that
$\sigma_{\infty}\in\mathscr{S}$. According to the $C^{2,\alpha}$ estimate, the
sequence of solutions $\Phi^{\sigma_{i}}$ satisfy

$\displaystyle|\Phi^{\sigma_{i}}|_{C^{2,\alpha}}\leq C$ (4.79)

and

$\displaystyle\Omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi^{\sigma_{i}}\geq\frac{1}{C}(\Omega_{0}+\sqrt{-1}d\tau\wedge\overline{d\tau}),$
(4.80)

for a constant $C$, which depends on $\epsilon$. It’s easy to know that
$\Phi^{\sigma_{i}}$ is a Cauchy sequence in
$C^{0}(\overline{\mathcal{R}}\times V)$. So, using interpolation, we can find
$\Phi^{\sigma_{\infty}}\in C^{2,\frac{\alpha}{2}}(\overline{\mathcal{R}}\times
V)$ and

$\displaystyle\Phi^{\sigma_{i}}\rightarrow\Phi^{\sigma_{\infty}},\ \ \ \ \ \ \
\ \text{ in $C^{2,\frac{\alpha}{2}}$ norm.}$ (4.81)

The $C^{2,\frac{\alpha}{2}}$ convergence implies $\Phi^{\sigma_{\infty}}$
satisfies equation (1.41) and condition (1.42). Therefore,
$\Phi^{\sigma_{\infty}}$ is a solution to Problem 1.4, with boundary value
${\sigma_{\infty}}F$.

It remains to show that, for any $\lambda\in[0,\sigma_{\infty})$, there is a
solution $\Phi^{\lambda}$ and
$\\{\Phi^{\lambda}|\lambda\in[0,\sigma_{\infty}]\\}$ is a continuous curve
with $C^{2}$ topology.

In the following, when we say solution $\Phi^{\theta}$, we always mean a
solution to Problem 1.4 with boundary value $\theta F$. For any
$\nu<\sigma_{\infty}$, let $\delta_{\nu}=\frac{\sigma_{\infty}-\nu}{2}$. There
is a $\sigma_{k}>\nu+\delta$ because $\sigma_{k}\rightarrow\sigma_{\infty}$.
$\sigma_{k}$ being in $\mathscr{S}$ implies solution $\Phi^{\nu}$ with
boundary value $\nu F$ exists and $\\{\Phi^{s}|s\in[0,\nu+\delta_{\nu}]\\}$ is
a continuous curve with $C^{2}$ topology. So

$\displaystyle\\{\Phi^{\lambda}|\lambda\in[0,\sigma_{\infty})\\}=\bigcup_{\nu<\sigma_{\infty}}\\{\Phi^{\lambda}|\lambda\in[0,\nu+\delta_{\nu})\\}$
(4.82)

is $C^{2}$ continuous everywhere. Therefore, for any sequence
$\nu_{k}\rightarrow\sigma_{\infty}$, solution $\Phi^{\nu_{k}}$ has uniform
$C^{2,\alpha}$ estimate. It’s easy to know $\Phi^{\nu_{k}}$ converges to
$\Phi^{\sigma_{\infty}}$ in $C^{0}$ norm, so, by interpolation, we know
$\Phi^{\nu_{k}}$ converges to $\Phi^{\sigma_{\infty}}$ in $C^{2}$ norm and the
curve $\\{\Phi^{\lambda}|\lambda\in[0,\sigma_{\infty}]\\}$ is $C^{2}$
continuous everywhere.

In sum, we have the following.

######  Theorem 4.1 (Existence of Smooth Solutions to Problem 1.4 and
Convexity Estimates).

Suppose that, for a constant $\delta>0$ and a constant section $S$ of
$T_{2,0}^{\ast}(V)$, $F\in C^{\infty}({\partial\mathcal{R}\times V})$
satisfies

$\displaystyle F(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of modulus
$>\delta$, for any $\tau\in\partial\mathcal{R}$}.$ (4.83)

Then Problem 1.4 with boundary value $F$ has a unique and smooth solution
$\Phi$. In addition,

$\displaystyle\Phi(\tau,\ast)\text{ is $(S,\omega_{0})$-convex of modulus
$>\delta$, for any $\tau\in\mathcal{R}$},$ (4.84)

and, consequently,

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi(\tau,\ast)>\delta\omega_{0}\text{,
\ \ \ \ \ \ for any $\tau\in\mathcal{R}$}.$ (4.85)

##  5 Estimates For Homogenous Monge-Ampère Equations

In this section, we prove estimates for solutions to Problem 1.2 and Problem
1.1. Solutions we talk about in this section all have the same boundary value
$F$ which satisfies condition (1.34).

Suppose $\Phi^{\epsilon}$ is the solution to Problem 1.4 and $\Phi^{0}$ is the
solution to Problem 1.2. We will show that $\Phi^{\epsilon}$ converges to
$\Phi^{0}$ in $C^{0}$ norm and $\Phi^{0}$ satisfies estimates, which are
satisfied by $\Phi^{\epsilon}$.

Let $\Psi^{\epsilon}$ be the solution to Problem 1.3 with
$\varepsilon=\epsilon n(1+C)^{n-1}$, where the constant $C$ is from (4.22).
Then we know

$\displaystyle\Psi^{\epsilon}\leq\Phi^{\epsilon}\leq\Phi^{0},\ \ \ \ \ \
\text{ in }{\mathcal{R}\times V}.$ (5.1)

According to estimates, in [C00] or [B12], for solutions to Problem 1.3,
$\Psi^{\epsilon}\rightarrow\Phi^{0}$ in $C^{0}$ norm, so
$\Phi^{\epsilon}\rightarrow\Phi^{0}$ in $C^{0}$ norm. In particular, for any
$\tau\in\mathcal{R}$,

$\displaystyle\Phi^{\epsilon}(\tau,\ast)\rightarrow\Phi^{0}(\tau,\ast),\ \ \ \
\ \ \text{ in }C^{0}\text{ norm. }$ (5.2)

According to Theorem 4.1, in every local coordinate chart,

$\displaystyle\Phi^{\epsilon}(\tau,\ast)+(1-\mu)b_{\alpha{\overline{\beta}}}z^{\alpha}\overline{z^{\beta}}+\text{Re}\left(S_{\alpha\overline{\beta}}z^{\alpha}\overline{z^{\beta}}\right)$
(5.3)

is a convex function, for any $\tau\in\mathcal{R}$. Then, because of the
$C^{0}$ convergence of $\Phi^{\epsilon}\rightarrow\Phi^{0}$, we can replace
$\Phi^{\epsilon}$ by $\Phi^{0}$ in (5.3) and get

$\displaystyle\Phi^{0}(\tau,\ast)+(1-\mu)b_{\alpha{\overline{\beta}}}z^{\alpha}\overline{z^{\beta}}+\text{Re}\left(S_{\alpha\overline{\beta}}z^{\alpha}\overline{z^{\beta}}\right)$
(5.4)

is a convex function, for any $\tau\in\mathcal{R}$. Thus $\Phi^{0}(\tau,\ast)$
is $(S,\omega_{0})$-convex of modulus $\geq\mu$, for any $\tau\in\mathcal{R}$.

For the metric lower bound estimate, the proof is standard, we only need to do
an integration by parts. Theorem 4.1 implies that, for any positive function
$\eta$,

$\displaystyle\int_{V}(\omega_{0}(1-\mu)+\sqrt{-1}\partial\overline{\partial}\Phi^{\epsilon}(\tau,\ast))\wedge\omega_{0}^{n-1}\eta\geq
0,$ (5.5)

for any $\tau\in\mathcal{R}$. Then for any $\eta$ with sufficiently small
support, we can find $\rho_{0}$, so that

$\displaystyle\omega_{0}=\sqrt{-1}\partial\overline{\partial}\rho_{0},\ \ \ \
\ \ \text{ in the support of $\eta$}.$ (5.6)

Thus

$\displaystyle\int_{V}(\rho_{0}(1-\mu)+\Phi^{\epsilon}(\tau,\ast))\wedge\omega_{0}^{n-1}\wedge\sqrt{-1}\partial\overline{\partial}\eta\geq
0,$ (5.7)

for any $\tau\in\mathcal{R}$. Let $\epsilon\rightarrow 0$, we get

$\displaystyle\int_{V}(\rho_{0}(1-\mu)+\Phi^{0}(\tau,\ast))\wedge\omega_{0}^{n-1}\wedge\sqrt{-1}\partial\overline{\partial}\eta\geq
0,$ (5.8)

for any $\tau\in\mathcal{R}$. So

$\displaystyle\omega_{0}+\sqrt{-1}\partial\overline{\partial}\Phi^{0}(\tau,\ast)\geq\mu\omega_{0},$
(5.9)

in the weak sense, for any $\tau\in\mathcal{R}$.

Thus, Theorem 1.2 is proved.

######  Remark 5.1.

Here, we cannot use Lemma A.2 to directly derive metric lower bound estimate
from convexity estimate, because $\Phi^{0}(\tau)$ may not be $C^{2}$
continuous and the degree of $(S,\omega_{0})$-convexity may not be well
defined.

##  Appendix A Algebra Lemmas

###  A.1 Lemmas for Convexity Estimate

In this appendix, we show that if $\varphi$ is $C^{2}$ then Definition 1.2 and
Definition 1.4 are equivalent. Furthermore, the modulus of
$(S,\omega_{0})$-convexity and the degree of $(S,\omega_{0})$-convexity also
coincide. The main results are Lemma A.1 and Lemma A.2.

######  Lemma A.1 (Equivalent Definitions of Strict
$(S,\omega_{0})$-Convexity).

Suppose that $\varphi$ is a $C^{2}$ continuous function on $V$. Then it
satisfies the condition of Definition 1.2 if and only if it satisfies the
condition of Definition 1.4

######  Lemma A.2 (Equivalence between Modulus of Convexity and Degree of
Convexity).

Suppose that $\varphi$ is a $C^{2}$ continuous function on $V$ and $S$ is a
constant section of $T_{2,0}^{\ast}(V)$. Then $\varphi$ is
$(S,\omega_{0})$-convex of degree $>\delta$ if and only if it is
$(S,\omega_{0})$-convex of modulus $>\delta$.

In the proof of these Lemmas and also in other parts of the paper, we need to
use the following Autonne-Takagi factorization and its corollary Lemma A.4.
The following Autonne-Takagi factorization is the Corollary 4.4.4(c) of
[HJ13].

######  Lemma A.3 (Autonne-Takagi Factorization).

Given a complex valued symmetric matrix $S$, there is a unitary matrix $U$
such that

$\displaystyle S=U^{T}\Sigma U$ (A.1)

in which $\Sigma$ is a non-negative diagonal matrix. And obviously the
Hermitian matrix $S\overline{S}$ has a decomposition

$\displaystyle S\overline{S}=U^{T}\Sigma^{2}\overline{U}.$ (A.2)

With Autonne-Takagi Factorization, we can prove the following lemma.

######  Lemma A.4.

Suppose $A$ is an $n\times n$ positive definite Hermitian matrix and $B$ is a
complex $n\times n$ symmetric matrix. Then we can find an $n\times n$
invertible matrix $P$, so that

$\displaystyle PAP^{\ast}=I,\ \ \ \ \ \ \ \ \ PBP^{T}=\Lambda,$ (A.3)

in which $\Lambda$ is a non-negative diagonal matrix.

###### Proof of Lemma A.4.

Since $A$ is a positive definite Hermitian matrix, we can find an invertible
matrix $Q$ so that

$\displaystyle QAQ^{\ast}=I.$ (A.4)

Then we apply Autonne-Takagi facorization to $QBQ^{T}$ and find $R\in U(n)$ so
that

$\displaystyle R(QBQ^{T})R^{T}=\Lambda.$ (A.5)

$P=RQ$ satisfies condition (A.3). ∎

The proof of Lemma A.1 essentially depends on the following lemma.

######  Lemma A.5.

Suppose $U,V,W$ are real valued $n\times n$ matrices and $U,V$ are symmetric.
Let

$\displaystyle A=\frac{1}{4}(U+W)+\frac{\sqrt{-1}}{4}(V-V^{T}),$ (A.6)
$\displaystyle B=\frac{1}{4}(U-W)-\frac{\sqrt{-1}}{4}(V+V^{T}).$ (A.7)

Then

$\displaystyle\left(\begin{array}[]{cc}U&V\\\ V^{T}&W\end{array}\right)>0$
(A.10)

if and only if

$\displaystyle A>0\text{\ \ \ and \ \ \ }B\overline{A^{-1}}\
{\overline{B}}{A^{-1}}<1.$ (A.11)

We first prove Lemma A.5, then Lemma A.1 follows easily.

###### Proof of Lemma A.5.

($\Rightarrow$) Given $U,V,W$ satisfying A.10, we can construct a strictly
convex quadratic polynomial on $\mathbb{C}^{n}$, with coordinate
$z^{\alpha}=x^{\alpha}+\sqrt{-1}y^{\alpha}$,

$\displaystyle H({\bf
z})=U_{\alpha\beta}x^{\alpha}x^{\beta}+2V_{\alpha\beta}x^{\alpha}y^{\beta}+W_{\alpha\beta}y^{\alpha}y^{\beta}.$
(A.12)

$H$ is a strictly PSH function, since it’s strictly convex. And it’s
straightforward to check that

$\displaystyle\partial_{\alpha{\overline{\beta}}}H=A_{\alpha{\overline{\beta}}}\text{
\ \ \ and \ \ \ \ }\partial_{\alpha\beta}H=B_{\alpha\beta}.$ (A.13)

Therefore, we know $A>0$. Then using Lemma A.4 we find matrix $P$, so that

$\displaystyle PAP^{\ast}=I,\ \ \ \ \ \ \ \ \ PBP^{T}=\Lambda,$ (A.14)

in which $\Lambda=\text{diag}(\lambda_{1},\lambda_{2},\ ...\ ,\lambda_{n})$ is
non-negative. We consider a new set of coordinates $\\{\zeta^{\alpha}\\}$,
with

$\displaystyle z^{\alpha}=P^{\alpha}_{\beta}\zeta^{\beta}.$ (A.15)

With this new coordinate

$\displaystyle(\partial_{\zeta^{\alpha}\zeta^{\beta}}H)=(P_{\alpha}^{\mu}P_{\beta}^{\rho}\partial_{z^{\mu}z^{\rho}}H)=PBP^{T}=\Lambda;$
(A.16)
$\displaystyle(\partial_{\zeta^{\alpha}\zeta^{\overline{\beta}}}H)=(P_{\alpha}^{\mu}\overline{P_{\beta}^{\rho}}\partial_{z^{\mu}z^{\overline{\rho}}}H)=PAP^{\ast}=I.$
(A.17)

In above, $P=(P_{\alpha}^{\beta})$, where $\alpha$ is the row index and
$\beta$ is the column index. It’s obvious that the linear change of
coordinates (A.15) does not affect the fact that $H$ is a convex function. So
the restriction of $H$ to a complex line

$\displaystyle L_{\alpha}=\\{\zeta^{\alpha}=\tau,\zeta^{\mu}=0,\text{ for
}\mu\neq\alpha|\tau\in\mathbb{C}\\}$ (A.18)

is still a convex function. Therefore,

$\displaystyle
H|_{L_{\alpha}}=\tau{\overline{\tau}}+\lambda_{\alpha}\frac{\tau^{2}+{\overline{\tau}}^{2}}{2}$
(A.19)

is a convex function of $\tau$. This implies $|\lambda_{\alpha}|<1$. So
$\Lambda^{2}<1$ and

$\displaystyle B\overline{A^{-1}}\
{\overline{B}}{A^{-1}}=P^{-1}\Lambda^{2}P<1.$ (A.20)

($\Leftarrow$) For another direction, we use Lemma A.4 to diagonalize $A,B$
simultaneously. Since $A>0$, we can find $R$ so that

$\displaystyle B=R\Lambda R^{T},\ \ \ \ \text{and }\ \ \ \ \ A=RR^{\ast}.$
(A.21)

Let $R=R_{1}+\sqrt{-1}R_{2}.$ Then

$\displaystyle B=R_{1}\Lambda R_{1}^{T}-R_{2}\Lambda
R_{2}^{T}+\sqrt{-1}(R_{2}\Lambda R_{1}^{T}+R_{1}\Lambda R_{2}^{T});$ (A.22)
$\displaystyle
A=R_{1}R_{1}^{T}+R_{2}R_{2}^{T}+\sqrt{-1}(R_{2}R_{1}^{T}-R_{1}R_{2}^{T}).$
(A.23)

We can get $U,V,W$ by adding (A.6) and (A.7) or subtracting one from another:

$\displaystyle U=2\text{Re}(A+B),\ \ W=2\text{Re}(A-B),\ \ V=2\text{Im}(A-B).$
(A.24)

Then plug (A.22) and (A.23) into (A.24), we get

$\displaystyle\left(\begin{array}[]{cc}U&V\\\
V^{T}&W\end{array}\right)=2\left(\begin{array}[]{cc}R_{1}&R_{2}\\\
-R_{2}&R_{1}\end{array}\right)\left(\begin{array}[]{cc}I+\Lambda&\\\
&I-\Lambda\end{array}\right)\left(\begin{array}[]{cc}R_{1}^{T}&-R_{2}^{T}\\\
R_{2}^{T}&R_{1}^{T}\end{array}\right).$ (A.33)

Therefore, $\Lambda^{2}<1$ implies (A.10). Similar to (A.20), we have

$\displaystyle B\overline{A^{-1}}\ {\overline{B}}{A^{-1}}=R\Lambda^{2}R^{-1}.$
(A.34)

So $B\overline{A^{-1}}\ {\overline{B}}{A^{-1}}<1$ implies $\Lambda^{2}<1$. ∎

Lemma A.1 follows immediately, by letting
$A=(\varphi_{\alpha{\overline{\beta}}})$ and $B=(\varphi_{{\alpha\beta}})+S.$

The following linear algebra lemma is essential for the equivalence between
modulus of convexity and degree of convexity.

######  Lemma A.6.

Suppose $A,G$ are positive definite $n\times n$ Hermitian matrices, $B$ is a
complex valued symmetric matrix and $\mu$ is a positive constant. Then

$\displaystyle A>\mu G\text{,\ \ and\ \ }B\overline{(A-\mu G)^{-1}}\
\overline{B}(A-\mu G)^{-1}<1$ (A.35)

if and only if

$\displaystyle A>0,\text{\ and\ }(B-\Theta)\overline{A^{-1}}\
\overline{B-\Theta}A^{-1}<1,\text{\ for any symmetric $\Theta$, with
}\Theta\overline{G^{-1}}\ \overline{\Theta}G^{-1}\leq\mu^{2}.$ (A.36)

###### Proof of Lemma A.6.

We first provide a proof in the $n=1$ case. The lemma in this case is very
intuitive and it best demonstrates the idea. In this case, $B,\Theta$ are
complex numbers, $A,G$ are positive numbers, and, without loss of generality,
we can assume $G=1$. As illustrated by Figure 2, the set

$\displaystyle\\{(B,A)\big{|}A>0,|B-\Theta|<A\\}$ (A.37)

is a cone with the corner at $(\Theta,0)$.

Figure 2: Metric Lower Bound Estimate

The condition that, for all $\Theta$ with $|\Theta|\leq\mu$,

$\displaystyle(B,A)\subset\\{(B,A)\big{|}A>0,|B-\Theta|<A\\}$ (A.38)

is equivalent to

$\displaystyle(B,A)\subset\bigcap_{|\Theta|\leq\mu}\\{(B,A)\big{|}A>0,|B-\Theta|<A\\}.$
(A.39)

The elementary geometry tells us that

$\displaystyle\bigcap_{|\Theta|\leq\mu}\\{(B,A)\big{|}A>0,|B-\Theta|<A\\}=\\{A>\mu+|B|\\}.$
(A.40)

Therefore, (A.39) is equivalent to

$\displaystyle A>\mu\ \ \ \ \ \ \text{and}\ \ \ \ \ \
\left|\frac{B}{A-\mu}\right|<1,$ (A.41)

which is condition (A.35).

In general dimension, we need to construct quadratic polynomials from $A,B,G$,
similar to the proof of Lemma A.1. Given an Hermitian matrix $H$ and a
symmetric matrix $S$, let

$\displaystyle K_{S}^{H}({\bf
z})=H_{{\alpha{\overline{\beta}}}}z^{\alpha}z^{{\overline{\beta}}}+\text{Re}(S_{\alpha\beta}z^{\alpha}z^{\beta}).$
(A.42)

According to Lemma A.1, $A,B,G$ and $\mu$ satisfying (A.35) is equivalent to
that

$\displaystyle K_{B}^{A-\mu G}\text{\ is a strictly convex function on
$\mathbb{C}^{n}$};$ (A.43)

$A,B,G$ and $\mu$ satisfying (A.36) is equivalent to that

$\displaystyle K_{B-\Theta}^{A}\text{\ is a strictly convex function on
$\mathbb{C}^{n}$, for any symmetric $\Theta$, with $\Theta\overline{G^{-1}}\
\overline{\Theta}G^{-1}\leq\mu^{2}$}.$ (A.44)

Because $K_{B-\Theta}^{A}$ and $K_{B}^{A-\mu G}$ are both quadratic
polynomials, they are strictly convex if and only if they are positive on
$\mathbb{C}^{n}-\\{0\\}$. So we need to show

$\displaystyle K_{B}^{A-\mu G}>0,\text{\ on \
}\mathbb{C}^{n}\backslash\\{0\\}$ (A.45)

if and only if

$\displaystyle K_{B-\Theta}^{A}>0,\text{\ on \
}\mathbb{C}^{n}\backslash\\{0\\}\text{, for any symmetric $\Theta$, with
$\Theta\overline{G^{-1}}\ \overline{\Theta}G^{-1}\leq\mu^{2}$}.$ (A.46)

Equivalently, we need to show

$\displaystyle K_{B}^{A}({\bf z})>\mu
G_{\alpha{\overline{\beta}}}z^{\alpha}z^{\overline{\beta}},\text{\ \ on \ \
}\mathbb{C}^{n}\backslash\\{0\\}$ (A.47)

if and only if

$\displaystyle K_{B}^{A}({\bf
z})>\text{Re}(\Theta_{\alpha\beta}z^{\alpha}z^{\beta}),\text{\ on \
}\mathbb{C}^{n}\backslash\\{0\\}\text{, for any symmetric $\Theta$, with
$\Theta\overline{G^{-1}}\ \overline{\Theta}G^{-1}\leq\mu^{2}$}.$ (A.48)

This is valid because

$\displaystyle\sup_{\Theta\text{ symmetric, }\Theta\overline{G^{-1}}\
\overline{\Theta}G^{-1}\leq\mu^{2}}\text{Re}(\Theta_{\alpha\beta}z^{\alpha}z^{\beta})=\mu
G_{\alpha{\overline{\beta}}}z^{\alpha}z^{\overline{\beta}}.$ (A.49)

To prove (A.49), we find $P$, so that $PGP^{\ast}=I$, and we let

$\displaystyle\frac{P\Theta P^{T}}{\mu}=S.$ (A.50)

Then we find (A.49) is equivalent to

$\displaystyle\sup_{S\text{ symmetric, }S\overline{S}\ \leq
1}\text{Re}(S_{\alpha\beta}z^{\alpha}z^{\beta})=\delta_{\alpha{\overline{\beta}}}z^{\alpha}z^{\overline{\beta}}.$
(A.51)

(A.51) can be easily proved with basic linear algebra.

∎

Lemma A.2 follows immediately, by letting
$A=(\varphi_{\alpha{\overline{\beta}}})$ and $B=(\varphi_{{\alpha\beta}})+S.$

###  A.2 Monotonicity

In this section, we prove the following algebra lemma.

######  Lemma A.7 (A Monotonicity Lemma).

Suppose $A_{0}$ and $A$ are Hermitian matrices satisfying

$\displaystyle A_{0}>0\ \ \ \text{and}\ \ \ A_{0}+A>0,$ (A.52)

and $B$ is a symmetric matrix. Let

$\displaystyle K_{t}=B\overline{(A_{0}+tA)^{-1}}\
{\overline{B}}(A_{0}+tA)^{-1}.$ (A.53)

Then $t^{2p}\text{tr}(K_{t}^{p})$ and the maximum eigenvalue of $t^{2}K_{t}$
are both non-decreasing functions of $t$. Here $p$ is any positive integer.

###### Proof.

First of all, we note that condition (A.52) implies

$\displaystyle A_{0}+tA>0,\ \ \ \ \text{ for any }t\in(0,1).$ (A.54)

So, in (A.53), ${(A_{0}+tA)^{-1}}$ exists. In the following we compute

$\displaystyle\frac{d}{dt}\left[t^{2p}\text{tr}(K_{t}^{p})\right]$ (A.55)

and show it’s non-negative. We need to simultaneously diagonalize $A$ and
$A_{0}+tA$ to simplify the computation.

For $t_{0}\in[0,1]$, find $P$ so that

$\displaystyle PAP^{\ast}=\Lambda,\ \ \ \ P(A_{0}+t_{0}A)P^{\ast}=I.$ (A.56)

Here $\Lambda=\text{diag}(\lambda_{1},\ ...\ ,\lambda_{n})$, with
$\lambda_{\alpha}\in\mathbb{R}$. Let

$\displaystyle H=PBP^{T},$ (A.57)

then $H$ is a symmetric matrix. Plug (A.56) and (A.57) into (A.53) to simplify
the expression of $K_{t_{0}}$. We get, at $t=t_{0}$,

$\displaystyle K_{t_{0}}=P^{-1}HH^{\ast}P,$ (A.58)

and so

$\displaystyle PK_{t_{0}}^{p-1}P^{-1}=(HH^{\ast})^{p-1}.$ (A.59)

Then we compute the derivative,

$\displaystyle\frac{d}{dt}\left[\text{tr}(K_{t}^{p})t^{2p}\right]$ (A.60)
$\displaystyle=$
$\displaystyle\frac{d}{dt}\left[t^{2p}\text{tr}\left(B\overline{(A_{0}+tA)^{-1}}\
{\overline{B}}(A_{0}+tA)^{-1}\right)^{p}\right]$ (A.61) $\displaystyle=$
$\displaystyle 2p\cdot t^{2p-1}\text{tr}(K_{t}^{p})-p\cdot
t^{2p}\text{tr}\left[B\overline{(A_{0}+tA)^{-1}}\ {\overline{A}}\
\overline{(A_{0}+tA)^{-1}}\ {\overline{B}}(A_{0}+tA)^{-1}K_{t}^{p-1}\right]$
(A.62) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -p\cdot
t^{2p}\text{tr}\left[B\overline{(A_{0}+tA)^{-1}}\ {\overline{B}}\
{(A_{0}+tA)^{-1}}\ A(A_{0}+tA)^{-1}K_{t}^{p-1}\right].$ (A.63)

Plug (A.56) (A.57) and (A.59) into the expression above, we get, at $t=t_{0}$,

$\displaystyle\frac{d}{dt}\left[t^{2p}\text{tr}(K_{t}^{p})\right]=p\cdot
t_{0}^{2p-1}\left(2\cdot\text{tr}\left[(HH^{\ast})^{p}\right]-t_{0}\cdot\text{tr}\left[(H^{\ast}H)^{p}\Lambda\right]-t_{0}\cdot\text{tr}\left[(HH^{\ast})^{p}\Lambda\right]\right).$
(A.64)

Using the fact that $H$ is symmetric, we know

$\displaystyle\text{tr}\left[(H^{\ast}H)^{p}\Lambda\right]=\text{tr}\left[\Lambda(HH^{\ast})^{p}\right]=\text{tr}\left[(HH^{\ast})^{p}\Lambda\right].$
(A.65)

So

$\displaystyle\frac{d}{dt}\left[t^{2p}\text{tr}(K_{t}^{p})\right]$
$\displaystyle=2p\cdot
t_{0}^{2p-1}\left(\text{tr}\left[(HH^{\ast})^{p}\right]-t_{0}\cdot\text{tr}\left[(HH^{\ast})^{p}\Lambda\right]\right)$
(A.66) $\displaystyle=2p\cdot
t_{0}^{2p-1}\left(\text{tr}\left[(HH^{\ast})^{p}(I-t_{0}\Lambda)\right]\right).$
(A.67)

It’s obvious that $HH^{\ast}$ is semi-positive definite, and, according to
(A.56),

$\displaystyle I-t_{0}\Lambda=PA_{0}P^{\ast}>0.$ (A.68)

Therefore, we get

$\displaystyle\frac{d}{dt}\left[t^{2p}\text{tr}(K_{t}^{p})\right]\geq 0.$
(A.69)

∎

###  A.3 Concavity

In this appendix, we show the operator (4.3) is concave. Suppose $A$ is an
$(n+1)\times(n+1)$ positive definite Hermitian matrix and $\mathcal{G}$ is an
$n\times n$ positive definite Hermitian matrix. Denote the lower right
$n\times n$ block of $A$ by $\mathcal{A}$. Let

$\displaystyle
F_{1}(A)=\log\left(A_{0\overline{0}}-A_{0{\overline{\beta}}}\mathcal{A}^{\alpha{\overline{\beta}}}A_{\alpha\overline{0}}\right)$
(A.70)

and

$\displaystyle
F_{2}(A)=-\log\left(\mathcal{G}_{\alpha{\overline{\beta}}}\mathcal{A}^{\alpha{\overline{\beta}}}\right).$
(A.71)

We will prove

######  Lemma A.8.

$F_{1}$ is a concave function of $A$ in the space of positive definite
$(n+1)\times(n+1)$ Hermitian matrices; $F_{2}$ is a concave function of
$\mathcal{A}$ in the space of positive definite $n\times n$ Hermitian
matrices.

###### Proof.

For the concavity of $F_{1}$, actually, we can show

$\displaystyle
f_{1}(A)=A_{0\overline{0}}-A_{0{\overline{\beta}}}\mathcal{A}^{\alpha{\overline{\beta}}}A_{\alpha\overline{0}}$
(A.72)

is a concave function of $A$. Let $H$ be an $(n+1)\times(n+1)$ Hermitian
matrix. Similar to $A$, denote the lower-right block of $H$ by $\mathcal{H}$.
Let

$\displaystyle q(t)=f_{1}(A+tH),$ (A.73)

for $t$ close to $0$. We will show that $q^{\prime\prime}(0)\leq 0.$ To
simplify the computation, we diagonalize $\mathcal{A}$ and $\mathcal{H}$
simultaneously. Find an $n\times n$ matrix $\mathcal{P}$, so that

$\displaystyle\mathcal{P}\mathcal{A}\mathcal{P}^{\ast}=I;\ \ \ \ \ \
\mathcal{P}\mathcal{H}\mathcal{P}^{\ast}=\Lambda=\text{diag}(\lambda_{1},\
...\ ,\lambda_{n}).$ (A.74)

Let

$\displaystyle P=\left(\begin{array}[]{cc}1&0\\\
0&\mathcal{P}\end{array}\right).$ (A.77)

Note that another expression for $f_{1}$ is

$\displaystyle f_{1}(A)=\frac{\det A}{\det{\mathcal{A}}}.$ (A.78)

So

$\displaystyle
q(t)=\frac{\det(A+tH)}{\det(\mathcal{A}+t\mathcal{H})}=\frac{\det[P({{A}}+tH)P^{\ast}]}{\det[\mathcal{P}({\mathcal{A}}+t\mathcal{H})\mathcal{P}^{\ast}]}.$
(A.79)

Denote that

$\displaystyle
PAP^{\ast}=\left(\begin{array}[]{cccc}a_{0\overline{0}}&\cdots&u_{0{\overline{\beta}}}&\cdots\\\
\vdots&&&\\\ u_{\alpha\overline{0}}&&I\\\ \vdots&\end{array}\right),\ \ \ \ \
PHP^{\ast}=\left(\begin{array}[]{cccc}h_{0\overline{0}}&\cdots&v_{0{\overline{\beta}}}&\cdots\\\
\vdots&&&\\\ v_{\alpha\overline{0}}&&\Lambda\\\ \vdots&\end{array}\right).$
(A.88)

With these simplifications,

$\displaystyle
q(t)=a_{0\overline{0}}+th_{0\overline{0}}-\sum_{\alpha}\frac{(u_{0{\overline{\alpha}}}+tv_{0{\overline{\alpha}}})(u_{\alpha\overline{0}}+tv_{\alpha\overline{0}})}{1+t\lambda_{\alpha}}.$
(A.89)

Straightforward computation gives

$\displaystyle
q^{\prime\prime}(0)=-\sum_{\alpha}(u_{0{\overline{\alpha}}}\lambda_{\alpha}-v_{0{\overline{\alpha}}})(u_{\alpha\overline{0}}\lambda_{\alpha}-v_{\alpha\overline{0}})\leq
0.$ (A.90)

Therefore, $f_{1}$ is concave, and, consequently, $F_{1}=\log(f_{1})$ is
concave.

For the concavity of $F_{2}$, actually, we can show

$\displaystyle f_{2}=\frac{1}{\text{tr}(\mathcal{G}\mathcal{A}^{-1})}$ (A.91)

is a concave function of $\mathcal{A}$. This is a simple consequence of the
well-known fact that

$\displaystyle\frac{1}{\text{tr}(\mathcal{A}^{-1})}=\frac{\det(\mathcal{A})}{\sigma_{n-1}(\mathcal{A})}$
(A.92)

is a concave function of $\mathcal{A}$. Therefore, $F_{2}=\log(f_{2})$ is
concave.

∎

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
(No. 12288201) and the Project of Stable Support for Youth Team in Basic
Research Field, CAS, (No. YSBR-001). The author would like to thank Li Chen,
Xiuxiong Chen, Jianchun Chu, Jiyuan Han, Laszlo Lempert, Long Li, Yu Li,
Guohuan Qiu, Zaijiu Shang, Li Sheng, Bing Wang, Youde Wang, Bin Xu for very
helpful discussions.

## References

  * [B12] Z. Blocki, On Geodesics in the Space of Kähler Metrics. Adv. Lect. Math. 21, Int. Press, Somerville, MA, 3-19(2012).
  * [CV01] L. Caffarelli, J. Viaclovsky, On the Regularity of Solutions to Monge-Ampère Equations on Hessian Manifolds, Communications in Partial Differential Equations, 26:11-12, 2339-2351(2001).
  * [C00] X. Chen, The Space of Kähler Metrics, J. Differential Geom. , 56 (2000) 189-234.
  * [CFH20] X. Chen, M. Feldman, J. Hu, Geodesic Convexity of Small Neighborhood in the Space of Kähler Potentials.  J. Functional Analysis 2020, 279(7):108603.
  * [CTW19] J. Chu, V. Tosatti, B. Weinkove, The Monge-Ampère Equation for non-Integrable Almost Complex Structures. J. Eur. Math. Soc. Volume 21, Issue 7, 2019, pp. 1949–1984.
  * [CTW17] J. Chu, V. Tosatti, B. Weinkove, On the $C^{1,1}$ Regularity of Geodesics in the Space of Kähler Metrics. Ann. PDE 3, 15 (2017).
  * [CIL92] M. Crandal, H. Ishi, P. Lions, User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bulletin of the American Mathematical Society. V. 26, No. 1, July 1992, Page 1-67.
  * [DL12] T. Darvas, L. Lempert, Weak Geodesics in the Space of Kähler Metrics. Math. Res. Lett. 19 (2012), no. 5, 1127-1135.
  * [D99] S. K. Donaldson, Symmetric Spaces, Kähler Geometry and Hamiltonian Dynamics, Amer. Math. Soc. Transl. (2) Vol 196. 13-33, 1999.
  * [D02] S. K. Donaldson, Holomorphic Discs and Complex Monge-Ampère Equation, J. Symplectic Geom. Volume 1, Number 2, 171-196, 2002.
  * [GT98] D. Gilbarg, N. Trudinger Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 ed. Springer, 2001.
  * [G98] B. Guan, The Dirichlet Problem for Complex Monge-Ampère Equations and Regularity of the Pluri-Complex Green Function. Communications in Analysis and Geometry, Volume 6, Number 4, 687-703, 1998.
  * [GP12M] P. Guan, D. H. Phong, A Maximum Rank Problem for Degenerate Elliptic Fully Nonlinear Equations,  Math. Ann. 354 (2012), no. 1, 147-169.
  * [GP12P] P. Guan, D. H. Phong, Partial Legendre Transforms of Non-Linear Equations,  Proceedings of the American Mathematical Society, Vol. 140, No. 11, Nov. 2012, Pages 3831-3842.
  * [GT21] V. Guedj, T. D.Tô, Monge-Ampère Equations on Compact Hessian Manifold, arXiv:2106.14740v1.
  * [H16] Q. Han, Nonlinear Elliptic Equations of the Second Order, GSM 171, American Mathematical Society, 2016.
  * [HJ13] R. Horn, C. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, Second Edition 2013.
  * [H21] J. Hu, An Obstacle for Higher Regularity of Geodesics in the Space of Kähler Potentials, International Mathematics Research Notices, Volume 2021, Issue 15, August 2021, Pages 11493–11513.
  * [H22] J. Hu, A Metric Lower Bound Estimate for Geodesics in the Space of Kähler Potentials, arXiv:2208.13651.
  * [K91] M. Klimek, Pluripotential Theory, Oxford Sciences Publications, Clarendon Press, 1991.
  * [L81] L. Lempert, La métrique de Kobayashi et la représentation des domain sur la boule, Bull. Soc. Math. Frence, tome 109 (1981), 427-474.
  * [LV13] L. Lempert, L. Vivas, Geodesics in the Space of Kähler Metrics, Duke Math. J. , Vol. 162, No. 7, 2013.
  * [M87] T. Mabuchi, Some symplectic geometry on compact Kähler manifolds, I. Osaka J. Math. 24 (1987), no. 2, 227-252.
  * [M79] R. Moriyón, Regularity of the Dirichlet Problem for the Complex Monge-Ampère Equation, Proc. Natl. Acad. Sci. USA, Vol. 76, No. 3, pp. 1022-1023, March 1979, Mathematics.
  * [M82] R. Moriyón, Regularity of the Dirichlet Problem for the Degenerate Complex Monge-Ampère Equation, Comm. Pure Appl. Math., Vol. XXXV, 1-27(1982).
  * [N82] K. Nomizu, The Lorentz-Poincaré metric on the upper half-space and its extension,  Hokkaido Math. J. , Vol. 11 (1982) p. 253-261.
  * [RN] J. Ross, D. W. Nyström, On the Maximal Rank Problem for the Complex Homogenous Monge-Ampère Equation, Analysis and PDE, Vol. 12 Issue 2 p. 493-504.
  * [S92] S. Semmes, Complex Monge-Ampère and Symplectic Manifolds, Amer. J. Math. , Vol. 114, No. 3. (Jun., 1992), pp. 495-550.
  * [S95] S. Semmes, The Homogeneous Complex Monge-Ampère Equation and the Infinite Dimensional Versions of Classic Symmetric Spaces, The Gelfand Mathematical Seminars, 1993–1995. Birkhäuser Boston.
  * [CS43] C. Siegel, Symplectic Geometry, American Journal of Mathematics Vol. 65, No. 1 (Jan., 1943), pp. 1-86.

Jingchen Hu

Loo-Keng Hua Center for Mathematical Sciences
Email<EMAIL_ADDRESS>