# A note on modified $J$-Flow with the Calabi Ansatz

P. Sivaram  Department of Mathematics, Indian Institute of Science, Bangalore,
India - 560012

###### Abstract.

We study the modified $J$-flow introduced in [15], particularly the
singularities of the flow using the Calabi symmetry. In [20], on toric
manifolds the convergence of modified $J$-flow to the smooth solution was
proven under the assumption of positivity of certain intersection numbers. In
the case of the Calabi ansatz we show that if some of those intersection
numbers are not positive, then the modified $J$-flow blows up along some
variety and away from the variety we prove the convergence to the solution.

As in [10], we also prove that the convergence behavior of the modified
$J$-flow with Calabi symmetry depends on the topological constants $c$ and the
minimum of the Hamiltonian function.

##  1\. Introduction

There has been considerable progress in recent years on solvability of inverse
Hessian equations going beyond the more classically studied complex Monge-
Ampere equations. An equation that has been of particular interest, in part
due to its relevance to constructing constant scalar curvature Kähler (cscK)
metrics, is the $J$-equation. The $J$-equation on Kähler manifolds was
introduced by Donaldson [9] in the moment map setting and by X. Chen [3] as
the Euler-Lagrange equation for a functional that appears in his formula for
the Mabuchi functional. Let $(X,\omega)$ be a Kähler manifolds, and let $\chi$
be another Kähler metric. The $J$-equation seeks a Kähler metric
$\chi_{\varphi}:=\chi+\sqrt{-1}\partial\overline{\partial}\varphi$ in the
class of $[\chi]$ satisfying

$\Lambda_{\chi_{\varphi}}\omega:=n\frac{\omega\wedge\chi_{\varphi}^{n-1}}{\chi_{\varphi}^{n}}=c,$

where $c$ is necessarily given by

$c=n\frac{[\chi]^{n-1}\cdot[\omega]}{[\chi]^{n}}.$

A consequence of Chen’s formula for the Mabuchi energy (cf. [17]) is that
solvability of the $J$-equation in the class $[\chi]=c_{1}(K_{M})$ implies
existence of constant scalar curvature metrics.

In [4], X. Chen introduced a natural flow, the so-called $J$-flow, to study
existence of solutions to the $J$-equation and proved it’s long-time existence
and convergence under a bi-sectional curvature lower bound. The $J$-flow is
defined as follows:

$\begin{cases}\frac{\partial\varphi}{\partial
t}=c-\Lambda_{\chi_{\varphi}}\omega\\\
\varphi_{t}|_{t=0}=\varphi_{0}\in\mathcal{H},\end{cases}$ (1.1)

where
$\mathcal{H}=\\{\varphi:\chi_{\varphi}=\chi+\sqrt{-1}\partial\overline{\partial}\varphi>0\\}.$
Note that on Kähler surfaces, the $J$-equation reduces to a complex Monge-
Ampere equation, and hence by Yau’s celebrated resolution of the Calabi
conjecture (cf. [23]), a solution exists if and only if $[c\chi-\omega]$ is a
Kähler class. More generally, in [17], Song and Weinkove proved that a
solution to the $J$-equation in the class $[\chi]$ exists if and only if there
exists a $\varphi\in\mathcal{H}$ such that

$c\chi_{\varphi}^{n-1}-(n-1)\omega\wedge\chi_{\varphi}^{n-2}>0.$

The interested readers can refer to [21, 22] for earlier results, and [18] for
an extension of this theorem to more general inverse Hessian equations. While
this is an optimal PDE result, the condition above (called the cone condition)
is a pointwise condition, and is generally difficult to verify. Inspired by
the work of Demailly and Paun [8] and the close analogy between complex Monge-
Ampere equations and the $J$-equation, Lejmi and Szekelyhidi [13] conjectured
that a solution to the $J$-equation exists if and only if the following Nakai
type criteria holds:

$\int_{V}(c\chi^{p}-p\omega\wedge\chi^{p-1})>0,$

for any $p$-dimensional sub-variety $V$ of the manifold. Note that the above
condition is a numerical or topological condition, and at least in principle,
much easier to verify. A uniform version of this conjecture was proved by G.
Chen in [2]. Based on the work of Chen, the full Lejmi-Szekelyhidi conjecture
was established by Datar-Pingali [7] on projective manifolds and Song [16] for
general Kähler manifolds.

In analogy with the role of the $J$-equation in studying the Mabuchi
functional, the modified $J$-equation was introduced by Li-Shi [15] to study
the modified Mabuchi functional. The setting is as follows: Suppose now, that
$\omega$ and $\chi$ are invariant under the action of a real torus $T\subset
Aut_{0}(X)$. Let

$\mathcal{H}^{T}=\\{\varphi\in
C^{\infty}(X,\mathbb{R})^{T}:\chi_{\varphi}=\chi+\sqrt{-1}\partial\overline{\partial}\varphi>0\\}$

be the $T$-invariant Kähler potentials of $\chi.$ Let $\xi$ be a holomorphic
vector field with $Im(\xi)\in\mathfrak{t}.$ We define the Hamiltonian function
of $\xi$ with respect to the metric $\chi$ as the real valued function
uniquely determined by the following properties:

$i_{\xi}\chi=\frac{\sqrt{-1}}{2\pi}\overline{\partial}\theta_{\xi}(\chi)\text{
and }\int_{X}\theta_{\xi}(\chi)\chi^{n}=0.$

The modified $J$-equation is defined as

$n\frac{\omega\wedge\chi_{\varphi}^{n-1}}{\chi_{\varphi}^{n}}=c+\theta_{\xi}(\chi_{\varphi}),$
(1.2)

where $c$ is as before. Note that the above equation reduces to the
$J$-equation if $T$ is trivial. In [15] Li-Shi proved that if there exist a
$\hat{\chi}\in[\chi]$ such that

$(c+\theta_{\xi}(\hat{\chi}))\hat{\chi}^{n-1}-(n-1)\omega\wedge\hat{\chi}^{n-2}>0,$

then there exist a unique $\varphi\in\mathcal{H}^{T}$ which satisfies the
modified $J$-equation (1.2). They proved the result using the parabolic flow
method by defining the modified $J$-flow by adapting the arguments in [17].The
modified $J$ flow is defined as

$\begin{cases}\frac{\partial\varphi}{\partial
t}=c+\theta_{\xi}(\chi_{\varphi})-n\frac{\omega\wedge\chi_{\varphi}^{n-1}}{\chi_{\varphi}^{n}}\\\
\varphi_{t}|_{t=0}=\varphi_{0}\in\mathcal{H}^{T}.\end{cases}$ (1.3)

In [20] Takahshi conjectured that the modified $J$-equation has solution in
the class $[\chi]$ if and only if $c+\theta_{\xi}(\chi)>0$ and the following
Nakai criteria holds: For all $p$-dimensional toric sub-varieties $V\subset
M$,

$\int_{V}((c+\theta_{\xi}(\chi))\chi^{p}-p\omega\wedge\chi^{p-1})>0,$

and verified this conjecture for toric manifolds (cf. [5] for an analogous
result for the $J$-equation).

The main goal of this note is to study the behaviour of the modified $J$-flow
on the blow-ups $X_{n}=\mathrm{Bl}_{x_{0}}\mathbb{P}^{n}$ of $\mathbb{P}^{n}$.
By Takahashi’s result, the existence problem is completely settled, and so we
focus instead on the case when the Nakai criteria fails. These are also
examples of ruled surfaces. Indeed
$X_{n}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1)\oplus\mathcal{O})$, and
one can study the modified $J$-flow using the Calabi ansatz. For the $J$-flow,
Fang and Lai obtained a complete description of the behaviour of the $J$-flow
on these manifolds using the Calabi ansatz in [10]. In particular, that even
when the Nakai criteria fails, the flow still converges on the complement of
the exceptional divisor $E$ to a solution of the $J$-equation on
$X_{n}\setminus E$ with a different slope. Our main goal is to extend these
results to the modified $J$-flow on these manifolds.

To state our main result, we introduce some notation. Given real numbers
$a,b>1$ and $k\geq 0$, we let

$c_{k}=n\frac{ab^{n-1}-1}{b^{n}-1}-\frac{nk}{n+1}\frac{b^{n+1}-1}{b^{n}-1}.$

When $k\geq 0$, we characterise the behaviour of the modified $J$ flow using
the constant $c_{k}+k$.

######  Theorem 1.1.

Let $X_{n}=\mathbb{P}^{n}\\#\overline{\mathbb{P}^{n}}$ be the blow up of
$\mathbb{P}^{n}$ at one point with two Kähler metrics $\omega\in
a[E_{\infty}]-[E_{0}]$ and $\chi\in b[E_{\infty}]-[E_{0}]$, where $E_{0}$ and
$E_{\infty}$ be the exceptional divisor and the pull-back of the divisor
associated to $\mathcal{O}_{\mathbb{P}^{n}}(1)$ respectively and
$\xi=kw\frac{\partial}{\partial w}$,where $w$ is the fiberwise coordinate of
$\mathcal{O}_{\mathbb{P}^{n-1}}(-1)$ and $k\geq 0$. And let $\chi_{t}$ be the
solution of the modified $J$-flow. Then the following three cases characterize
the convergence behavior of the modified $J$-flow:

  1. (1)

If $c_{k}+k>n-1$, then $\chi_{t}\rightarrow\chi_{\infty}$ as $t\to\infty$ on
$X_{n}$ smoothly and $\chi_{\infty}$ is the solution of the modified $J$
equation.

  2. (2)

If $c_{k}+k=n-1,$ then $\chi_{t}\rightarrow\chi_{\infty}$ as $t\to\infty$ on
$X_{n}\smallsetminus E_{0}$ smoothly, where
$\chi_{\infty}=\chi+\sqrt{-1}\partial\overline{\partial}\varphi,\text{ for
some }\varphi\in L^{\infty}(X_{n}),$ a singular Kähler metric has a conical
singularity along $E_{0}$ and smooth everywhere else. And the flow converges
to the solution of the equation

$n\frac{\omega\wedge\chi_{\infty}^{n-1}}{\chi_{\infty}^{n}}=c+\theta_{\xi}(\chi_{\infty}),\text{
on }X_{n}\smallsetminus E_{0}.$

  3. (3)

If $0<c_{k}+k<n-1$, then $\chi_{t}\to\hat{\chi}_{\infty}+(\lambda-1)[E_{0}]$,
a Kähler current as $t\to\infty$ on $X_{n}\smallsetminus E_{0}$ smoothly,
where $\lambda\in(1,b)$ is unique such that

$n\frac{ab^{n-1}-\lambda^{n-1}}{b^{n}-\lambda^{n}}-\frac{nk}{n+1}\frac{b^{n+1}-\lambda^{n+1}}{b^{n}-\lambda^{n}}+k\lambda=\frac{n-1}{\lambda}.$

And the flow converges to the solution of the equation

$n\frac{\omega\wedge\hat{\chi}_{\infty}^{n-1}}{\hat{\chi}_{\infty}^{n}}=n\frac{ab^{n-1}-\lambda^{n-1}}{b^{n}-\lambda^{n}}+\theta_{\xi}(\hat{\chi}_{\infty}),\text{
on }X_{n}\smallsetminus E_{0}.$

To conclude this section, we make a few remarks on some ongoing work, and
possible future directions. Firstly we note that in a work in progress, we
introduce a notion of Futaki invariant for the modified $J$-equation, and
prove a Atiyah-Bott type lower bound for an $L^{2}$-energy in terms of the
Futaki invariant, obtaining a modified $J$-equation analogue of a similar
result due to Lejmi and Szekelyhidi. As in [13], using the above convergence
result, we can improve the lower bound to an equality in the case of the
manifolds $X_{n}$. In another direction, one can also obtain a similar
convergence result for more general projective bundles where the zero section
may have a higher co-dimension. In [6], the authors develop a program for
obtaining weak solutions to the $J$-equation even when the Nakai criteria
fails, and obtain weak solutions on Kähler surfaces. It would be interesting
to develop an analogous program for the modified $J$-equation. Unlike in the
case of the $J$-equation, finding smooth solutions to the modified equation on
Kähler surfaces is already non-trivial (cf. [15]). In light of this, one would
expect additional difficulties in adapting the arguments in [6] to the
modified setting.

##  2\. Modified $J$-flow with the Calabi Ansatz

###  2.1. Modified $J$-equation with the Calabi Ansatz:

We first review the ansatz of Calabi to construct Kähler metrics on certain
ruled surfaces. The interested reader can refer to [1, 12, 19] for more
details. Let $X_{n}$ denote blow-up of $\mathbb{P}^{n}$ at one point.
Equivalently,
$X_{n}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1)\oplus\mathcal{O})$. Let
$h$ be a metric on $\mathcal{O}_{\mathbb{P}^{n}}(-1)$ with curvature
$-\sqrt{-1}2\pi\omega_{FS}$, and write $s=\log|.|_{h}$ for the log of the
fiberwise norm. Denote by $[E_{0}]$ and $[E_{\infty}]$ the zero section
(exceptional divisor) and the infinity section respectively. The Kähler cone
is then given by

$\mathcal{K}_{X_{n}}=\\{\beta[E_{\infty}]-\alpha[E_{0}]:\beta>\alpha>0\\}.$

For an appropriate choice of strictly increasing convex function
$f\in\mathcal{C}^{\infty}(\mathbb{R})$ we can write down a Kähler metric

$\omega=\sqrt{-1}\partial\overline{\partial}f(s)$

on $X_{n}\smallsetminus(E_{0}\cup
E_{\infty})\cong\mathbb{C}^{n}\smallsetminus\\{0\\}.$ At a point choose local
coordinates $z=(z_{1},z_{2},\dots,z_{n-1})$ on $\mathbb{P}^{n-1}$ and a
fiberwise coordinate $w$ such that $d\log h(z)=0.$ At this point we then have
that

$\omega=\sqrt{-1}f^{\prime}(s)\omega_{FS}+f^{\prime\prime}(s)\frac{\sqrt{-1}dw\wedge
d\bar{w}}{|w|^{2}}.$

In order to extend $\omega$ to a smooth Kähler metric on $X_{n}$, the
following asymptotic properties of $f$ are required:

  1. (i)

$F_{0}(t):=f(\log t)-\alpha\log t$ extends to a smooth function at $t=0$, and
$F_{0}^{\prime}(0)>0.$

  2. (ii)

$F_{\infty}(t):=f(-\log t)+\beta\log t$ extends to a smooth function at $t=0$,
and $F_{\infty}^{\prime}(0)>0.$

By the asymptotic behavior of $f,$ we have

$\lim_{s\to-\infty}f^{\prime}(s)=\alpha,\lim_{s\to\infty}f^{\prime}(s)=\beta,$

and the Kähler class of $\omega$ is then given by
$\beta[E_{\infty}]-\alpha[E_{0}].$

We can repeat this construction for a different convex function
$g\in\textbf{C}^{\infty}(\mathbb{R})$ and consider

$\chi=\sqrt{-1}\partial\overline{\partial}g(s).$

We normalize the Kähler classes so that

$\omega\in a[E_{\infty}]-[E_{0}],\chi\in b[E_{\infty}]-[E_{0}];\quad a,b>1.$

We now write down the modified $J$-equation using Calabi ansatz. We introduce
a moment map coordinate $\tau=f^{\prime}(s),$ and define the strictly
increasing function $\psi\colon[1,b]\to[1,a]$ by letting

$\psi(f^{\prime}(s))=g^{\prime}(s),\forall~{}s\in\mathbb{R}.$

We consider the $\mathbb{S}^{1}$ action on $X_{n}$ of the form

$e^{\sqrt{-1}\theta}\cdot(z,w)=(z,e^{\sqrt{-1}k\theta}w).$

The vector field generating the action is given by
$\xi=kw\frac{\partial}{\partial w}.$ Since

$i_{\xi}\chi=i_{\xi}\left(f^{\prime\prime}(s)\frac{\sqrt{-1}dw\wedge
d\bar{w}}{|w|^{2}}\right)=\sqrt{-1}~{}\overline{\partial}(kf^{\prime}(s)),$

the normalized Hamiltonian function in the momentum coordinate is

$\theta_{\xi}(\chi)=k\left(\tau-\frac{n}{n+1}\frac{b^{n+1}-1}{b^{n}-1}\right)$

then the modified $J$ equation can be written as

$\psi^{\prime}(\tau)+(n-1)\frac{\psi(\tau)}{\tau}=c_{k}+k\tau,$ (2.1)

where

$c_{k}=c-\frac{kn}{n+1}\frac{b^{n+1}-1}{b^{n}-1}\text{ and
}c=n\frac{ab^{n-1}-1}{b^{n}-1}.$

A solution must be strictly increasing and also satisfy the boundary
conditions $\psi(1)=1$ and $\psi(b)=a.$ If we take the derivative of the
equation (2.1) with respect to $\tau$ then we get a second order ODE
associated to the modified $J$ equation

$\psi^{\prime\prime}(\tau)+(n-1)\frac{\psi^{\prime}(\tau)}{\tau}-(n-1)\frac{\psi(\tau)}{\tau^{2}}-k=0.$
(2.2)

With the boundary conditions $\psi(1)=1$ and $\psi(b)=a,$ we obtain a unique
solution

$\tilde{\psi}(\tau)=\frac{k\tau^{2}}{n+1}+\frac{c_{k}}{n}\tau+\left(\frac{n}{n+1}-\frac{c_{k}}{n}\right)\frac{1}{\tau^{n-1}}.$
(2.3)

We will use the equation (2.2) to define the modified $J$-flow and we also see
that the presence of the quadratic term in the solution (2.3) makes the
analysis of the equation different from the $J$-flow ODE. Note that if we take
$k=0$, then we recover the $J$-equation.

From now onwards we assume that $k\geq 0$.

######  Remark 2.1.

  1. (a)

For the $J$-equation with the Calabi ansatz, solvability of the equation
depends on the topological constant
$c=n\frac{[\omega]\cdot[\chi]^{n-1}}{[\chi]^{n}}$. Analogous to that for the
modified $J$-equation with Calabi ansatz, solvability depends on the minimum
of the normalized hamiltonian and the constant $c$, that is we get a necessary
and sufficient condition for the existence of a smooth solution to the
modified $J$-equation from

$\int_{E_{0}}\left((c+\theta_{\xi}(\chi))\chi^{n-1}-(n-1)\omega\wedge\chi^{n-2}\right)=c_{k}+k-(n-1)>0$

and

$c_{k}+k=\min_{X_{n}}\big{(}c+\theta_{\xi}(\chi)\big{)}.$

  2. (b)

By Lemma 2.1 of [20], the constant $\min_{X_{n}}(c+\theta_{\xi}(\chi))$ is
independent of the metric $\chi$ we pick from the class $[\chi]$ in general.

######  Theorem 2.2.

Let $X_{n}$ be a blow up of $\mathbb{P}^{n}$ at one point and $\omega\in
a[E_{\infty}]-[E_{0}]$ and $\chi\in b[E_{\infty}]-[E_{0}]$ to be two Kähler
classes. If $c_{k}+k>n-1,$ then there exist a solution to the modified
$J$-equation (2.2).

###### Proof.

The solution of the ODE (2.2) is given by

$\psi(\tau)=\frac{k\tau^{2}}{n+1}+\frac{c_{k}}{n}\tau+\left(\frac{n+1-k}{n+1}-\frac{c_{k}}{n}\right)\frac{1}{\tau^{n-1}}.$
(2.4)

Given that $c_{k}>n-(k+1).$ Then

$\displaystyle\psi^{\prime}(\tau)$
$\displaystyle=\frac{2k\tau}{n+1}+\frac{c_{k}}{n}-(n-1)\left(\frac{n+1-k}{n+1}-\frac{c_{k}}{n}\right)\frac{1}{\tau^{n}}$
$\displaystyle>\frac{2k}{n+1}+\frac{n-(k+1)}{n}-(n-1)\left(\frac{n+k+1}{n(n+1)}\right)$
$\displaystyle=0.$

This implies that $\psi$ has an inverse. So,
$f^{\prime}(s)=\psi^{-1}(g^{\prime}(s)),\text{ for all }s\in\mathbb{R}.$ Since
$f$ satisfies the properties of the Calabi ansatz we can get a metric which
solves the modified $J$-equation (1.2). ∎

As a consequence we have the following rotationally symmetric version of
Conjecture 1.4 of [20].

######  Corollary 2.3.

Let $X_{n}$ be the blow up of $\mathbb{P}^{n}$ at one point and $\omega\in
a[E_{\infty}]-[E_{0}]$ and $\chi\in b[E_{\infty}]-[E_{0}]$ to be two Kähler
classes. Then the following are equivalent

  1. (1)

The modified $J$-equation (1.2) has a solution.

  2. (2)

$c+\theta_{\xi}(\chi)>0$ and for each $p\in\\{1,2,\dots,n-1\\}$ and
$p$-dimensional irreducible sub-variety $V^{p}\subset X$,

$\int_{V^{p}}\left((c+\theta_{\xi}(\chi))\chi^{p}-p\omega\wedge\chi^{p-1}\right)>0.$

  3. (3)

$c_{k}+k>n-1.$

###### Proof.

For $(1)$ implies $(2)$, by assumption there exist $\chi\in
b[E_{\infty}]-[E_{0}]$ which solves the modified $J$-equation (1.2)

$n\frac{\omega\wedge\chi^{n-1}}{\chi^{n}}=c+\theta_{\xi}(\chi).$

Let us choose a coordinate at a point $y\in X_{n}$ such that
$\omega_{ij}=\delta_{ij}$ and $\chi_{ij}=\mu_{i}\delta_{ij},\mu_{i}>0\text{
for }1\leq i,j\leq n.$ At this point the modified $J$-equation will be of the
form

$\sum_{i=1}^{n}\frac{1}{\mu_{i}}=c+\theta_{\xi}(\chi).$

This implies that $c+\theta_{\xi}(\chi)>0$, since $y\in X_{n}$ was arbitrary
and for any $\\{i_{1},i_{2},\dots,i_{p}\\}\subset\\{1,2,\dots,n\\}$ we have

$\sum_{r=1}^{p}\frac{1}{\mu_{i_{r}}}<c+\theta_{\xi}(\chi)$

So $(c+\theta_{\xi}(\chi))\chi^{p}-(p-1)\omega\wedge\chi^{p-1}>0.$ And if we
integrate the form
$(c+\theta_{\xi}(\chi))\chi^{p}-(p-1)\omega\wedge\chi^{p-1}$ over any
$p$-dimensional irreducible sub-variety $V^{p}$ we get

$\int_{V^{p}}(c+\theta_{\xi}(\chi))\chi^{p}-(p-1)\omega\wedge\chi^{p-1}>0.$

For $(2)$ implies $(3)$, we have $[E_{\infty}]|_{E_{0}}\equiv 0$ and
$[E_{0}]|_{E_{0}}\equiv-\omega_{FS}$ and we can take $V^{n-1}=E_{0}.$ Then

$\displaystyle\int_{E_{0}}\left((c+\theta_{\xi}(\chi))\chi^{n-1}-(n-1)\omega\wedge\chi^{n-1}\right)$
$\displaystyle=\int_{E_{0}}\left((c_{k}+k)\chi^{n-1}-(n-1)\omega\wedge\chi^{n-2}\right)$
$\displaystyle=c_{k}+k-(n-1).$

Thus $c_{k}+k>n-1$ follows from the given hypothesis. And $(3)$ implies $(1)$
follows from Theorem 2.2. ∎

######  Remark 2.4.

  1. (a)

An important point to note here is that for the $J$-equation if $\omega$ and
$\chi$ belong to the same Kähler class, then we can always solve the
$J$-equation. But this is not the case for the modified $J$-equation. For
example on $X_{2}$ with $k=1$, there is no $\omega,\chi\in
3[E_{\infty}]-[E_{0}]$ such that

$2\frac{\omega\wedge\chi}{\chi^{2}}=2+\theta_{\xi}(\chi).$

Since $c_{1}=\frac{-1}{6}<0$ in this case. Also consider $\omega\in
7[E_{\infty}]-[E_{0}]$ and $\chi\in 6[E_{\infty}]-[E_{0}]$ in the blow up of
$\mathbb{P}^{2}$ at a point. Then $c_{1}<0.$ Thus the modified $J$-equation
can not have a solution in $[\chi]$. Hence $a\geq b$ does not guarantee the
solution for the modified $J$ equation, unlike the $J$-equation.

  2. (b)

If $(\omega,\chi)$ is $J$-unstable, that is $c<n-1$, then we can not have
solution for the modified $J$-equation on that particular pair for any $k\geq
0$, since

$k\left(1-\frac{n}{n+1}\frac{b^{n+1}-1}{b^{n}-1}\right)<0.$

  3. (c)

When $k\geq 0,$ we have $c_{k}+k>n-1$ as a necessary and sufficient condition
for the existence of the solution of the modified $J$-equation. This provides
the upper bound on the value of $k$, that is

$0\leq
k<\left(1+n\frac{ab^{n-1}-1}{b^{n}-1}-n\right)\left(\frac{n}{n+1}\frac{b^{n+1}-1}{b^{n}-1}-1\right)^{-1}.$

  4. (d)

If we choose the $k$ to be negative, then to get a sufficient condition for
the existence of the increasing function we have to integrate over the sub-
variety $E_{\infty}.$

Now we turn our attention to the unstable case, that is when $c_{k}+k<n-1.$ In
this case the general solution(2.4) of equation (2.2) is not increasing in
$[1,b]$. Instead we consider the following auxiliary family of equations

$\begin{cases}\psi^{\prime\prime}(\tau)+(n-1)\frac{\psi^{\prime}(\tau)}{\tau}-(n-1)\frac{\psi(\tau)}{\tau^{2}}-k=0,&\tau\in(s,b)\\\
\psi(s)=1\text{ and }\psi(b)=a,&\text{ where }s\in(1,b).\end{cases}$ (2.5)

Solution of the above equation is

$\psi_{s}(\tau)=\frac{\tau^{2}}{n+1}+\frac{C_{k}(s)}{n}\tau+\frac{A_{k}(s)}{\tau^{n-1}}.$

Using the boundary conditions to solve for $C_{k}(s)\text{ and }A_{k}(s)$, we
get

$C_{k}(s)=n\frac{ab^{n-1}-s^{n-1}}{b^{n}-s^{n}}-\frac{kn}{n+1}\frac{b^{n+1}-s^{n+1}}{b^{n}-s^{n}}\text{
and }A_{k}(s)=s^{n-1}\frac{n+1-ks^{2}}{n+1}-s^{n}\frac{C_{k}(s)}{n}.$ (2.6)

Let

$\mathcal{S}=\\{s\in(1,b):\text{ there exist }\psi_{s}\text{ solves
}\eqref{sjeqn}\text{ with }\psi_{s}^{\prime}(\tau)>0,\text{ for all
}\tau\in[s,b]\\},$

and

$\lambda=\inf\mathcal{S}.$

Since the solution (2.4) $\psi$ of the ODE (2.2) is convex and
$\psi^{\prime}(1)<0$ in the case $c_{k}+k<n-1$ there exist a unique $\bar{s}$
such that $\psi^{\prime}(\bar{s})=0$ and there exist a unique
$s\in(\bar{s},b)$ such that $\psi(s)=1.$ Hence by definition $s$ belong to the
set $\mathcal{S}$ and $1<\lambda\leq s$.

Then $\lambda$ satisfies the following equation

$C_{k}(\lambda)+k\lambda=\frac{n-1}{\lambda}$

which follows from the equation $\psi_{\lambda}^{\prime}(\lambda)=0$, by the
definition of $\lambda.$

The left hand side of the above equation (in analogy to the $J$-equation) is
related to the topological constant of the modified $J$-equation as follows:

Let $\chi_{\lambda}\in b[E_{\infty}]-\lambda[E_{0}]$ and $\omega\in
a[E_{\infty}]-[E_{0}]$. Then

$C_{k}(\lambda)+k\lambda=\min_{X_{n}}\left(n\frac{[\omega]\cdot[\chi_{\lambda}]^{n-1}}{[\chi_{\lambda}]^{n}}+\theta_{\xi}(\chi_{\lambda})\right).$

Suppose for $[\chi]=b[E_{\infty}]-[E_{0}],[\omega]=a[E_{\infty}]-[E_{0}]$ and
vector field $\xi=kw\frac{\partial}{\partial w}(k\geq 0)$, we get that
$0<c_{k}+k<n-1$, we claim that we can perturb the Kähler class of $\chi$ so
that we can solve the modified $J$-equation in the perturbed class. And that
perturbation is measured by the constant $\lambda$ that we introduced above.

For any $s\in(1,b)$, let $\chi_{s}\in b[E_{\infty}]-s[E_{0}]$ be any Kähler
metric and we know that

$C_{k}([\chi_{s}],[\omega])+ks=\min_{X_{n}}\left(n\frac{[\omega]\cdot[\chi_{s}]^{n-1}}{[\chi_{s}]^{n}}+\theta_{\xi}(\chi_{s})\right).$

###  2.2. Geometric description of the constant $\lambda$ in the case
$c_{k}+k<n-1$:

In this subsection we give a geometric description of the constant $\lambda$.

Let $\chi_{s}\in b[E_{\infty}]-s[E_{0}]$ and

$\mathcal{S}=\left\\{s\in(1,b):C_{k}([\chi_{s}],[\omega])+ks>\frac{n-1}{s}\right\\}.$

Let us look at the constant $C_{k}([\chi_{s}],[\omega])+ks$,

$C_{k}([\chi_{s}],[\omega])+ks=n\frac{ab^{n-1}-s^{n-1}}{b^{n}-s^{n}}-\frac{nk}{n+1}\frac{b^{n}+b^{n-1}s+\cdots+s^{n-1}}{b^{n-1}+b^{n-2}s+\cdots+s^{n-1}}+ks.$

As $s\to b$ the first term in the above sum goes to infinity and the second
and third term are bounded for all $s\in(1,b)$, the set $\mathcal{S}$ is
nonempty.

We claim that the condition $C_{k}([\chi_{s}],[\omega])+ks>\frac{n-1}{s}$ is
the condition for the existence of the solution of the modified $J$-equation
in the Kähler class $[\chi_{s}]$. Let $\alpha_{s}\in
b_{1}[E_{\infty}]-[E_{0}],$ where $b_{1}=bs^{-1}$ for $s\in\mathcal{S}$ be any
Kähler metric, $k_{1}=s^{2}k$ and $\xi^{\prime}=k_{1}w\frac{\partial}{\partial
w}=s^{2}\xi$. Then

$C_{k}([\chi_{s}],[\omega])+ks>\frac{n-1}{s}\text{ if and only if
}C_{k_{1}}([\alpha_{s}],[\omega])+k_{1}>n-1.$

By Theorem 2.2, there exist $\hat{\alpha}_{s}\in[\alpha_{s}]$ such that

$n\frac{\omega\wedge\hat{\alpha}_{s}^{n-1}}{\hat{\alpha}_{s}^{n}}=n\frac{[\omega]\cdot[\hat{\alpha}_{s}]^{n-1}}{[\hat{\alpha}_{s}]^{n}}+\theta_{\xi^{\prime}}(\hat{\alpha}_{s}).$
(2.7)

Letting $\hat{\chi}:=s\hat{\alpha}_{s}\in b[E_{\infty}]-s[E_{0}]$, if we
multiply the equation (2.7) by $s^{-1}$ we get

$n\frac{\omega\wedge\hat{\chi}^{n-1}}{\hat{\chi}^{n}}=n\frac{[\omega]\cdot[\hat{\chi}]^{n-1}}{[\hat{\chi}]^{n}}+\theta_{\xi}(\hat{\chi}),$

which proves our claim. In particular by the definition of $\lambda$ we have

$C_{k}(\lambda)+k\lambda=\frac{n-1}{\lambda}\text{ and
}\lambda=\inf\mathcal{S}.$

Note that since $c_{k}+k<n-1$ by Corolloary 2.3 we also have that $\lambda>1.$

Moreover, if we take any $s\in(1,\lambda)$, then by definition of $\lambda$ we
get that

$C_{k}([\chi_{s}],[\omega])+ks<\frac{n-1}{s}$

and this is equivalent to

$C_{k_{1}}([\alpha_{s}],[\omega])+k_{1}<n-1.$

Thus we can not have a smooth solution to the modified $J$-equation in the
class $[\chi_{s}].$

###  2.3. Modified $J$-flow with the Calabi ansatz:

Recall that the modified $J$-flow is defined as

$\begin{cases}\frac{\partial\varphi}{\partial
t}=c-n\frac{\omega\wedge\chi_{\varphi}^{n-1}}{\chi_{\varphi}^{n}}+\theta_{\xi}(\chi_{\varphi})\\\
\varphi_{t}|_{t=0}=\varphi_{0}\in\mathcal{H}^{T},\end{cases}$ (2.8)

where $\theta_{\xi}(\chi_{\varphi})$ is the Hamiltonian function of the action
of $\xi$ with respect to the metric $\chi_{\varphi}$ which is normalized so
that $\int_{X}\theta_{\xi}(\chi_{\varphi})\chi_{\varphi}^{n}=0$ and
$c=n\frac{[\omega].[\chi]^{n-1}}{[\chi]^{n}}.$ Differentiating equation (2.8)
with respect to $t$, we get

$\frac{\partial}{\partial t}\left(\frac{\partial\varphi}{\partial
t}\right)=\tilde{\Delta}\left(\frac{\partial\varphi}{\partial
t}\right)+\xi\left(\frac{\partial\varphi}{\partial t}\right),$

where $\tilde{\Delta}f=h^{i\bar{j}}\partial_{i}\partial_{\bar{j}}f$ and
$h^{i\bar{j}}=\chi^{i\bar{k}}g_{l\bar{k}}\chi^{l\bar{j}}.$ Then the maximum
principle implies that

$\min_{X_{n}}\left.\frac{\partial\varphi}{\partial
t}\right|_{t=0}\leq\frac{\partial\varphi}{\partial
t}\leq\max_{X_{n}}\left.\frac{\partial\varphi}{\partial t}\right|_{t=0}.$

In particular,

$\Lambda_{\chi_{\varphi}}\omega\leq\max_{X_{n}}\Lambda_{\chi_{0}}\omega+\max_{X_{n}}\theta_{\xi}(\chi_{\phi})-\min_{X_{n}}\theta_{\xi}(\chi_{0}).$
(2.9)

By Corollary 5.3 of [24], $\theta_{\xi}(\chi_{\varphi})$ is uniformly bounded.
Then the right hand side of the above equation is bounded above by strictly
positive constant $A.$ Thus

$\chi_{\varphi}\geq A\omega,$ (2.10)

as long as the flow exists.

Let $\omega=\sqrt{-1}\partial\overline{\partial}g(s)$ and
$\chi_{t}=\sqrt{-1}\partial\overline{\partial}f(s,t)$, for $t\geq 0$ satisfy
the Calabi Ansatz from the previous section. Then the modified $J$ flow (2.8)
becomes

$\frac{\partial f(s,t)}{\partial
t}=c-(n-1)\frac{g^{\prime}(s)}{f^{\prime}(s,t)}-\frac{g^{\prime\prime}(s)}{f^{\prime\prime}(s,t)}+kf^{\prime}(s,t)-\frac{kn}{n+1}\frac{b^{n+1}-1}{b^{n}-1}.$
(2.11)

As before, we define strictly increasing functions in the variable
$\tau\in[1,b]$ for all $t\geq 0$, that is $\psi:[1,b]\times[0,\infty)\to[1,a]$
by

$\psi(f^{\prime}(s,t),t)=g^{\prime}(s).$

######  Proposition 2.5.

The function $\psi=\psi(\tau,t)$ defined above is a classical solution of the
initial-boundary value problem

$\begin{cases}\frac{\partial\psi}{\partial t}=Q(\psi)P[\psi],\\\
\psi(1,t)=1,\quad\psi(b,t)=a,\forall t\in(0,\infty)\end{cases}$ (2.12)

where

$P[\psi]=\psi^{\prime\prime}(\tau,t)+(n-1)\frac{\psi^{\prime}(\tau,t)}{\tau}-(n-1)\frac{\psi(\tau,t)}{\tau^{2}}-k,$
(2.13)

and $Q\in\textbf{C}^{\infty}([1,a],[0,\infty))$ is uniquely determined by $g$
such that $Q(x)=0$ if and only if $x=1$ and $x=b$. Moreover, the initial data
$\psi(\tau,0)=\psi_{0}(\tau)$ is any smooth monotone function satisfying the
above boundary conditions.

###### Proof.

We have the equation $\psi(f^{\prime}(s,t))=g^{\prime}(s).$ Take derivative
with respect to the variable $t.$

$\frac{\partial\psi}{\partial\tau}\frac{\partial f^{\prime}}{\partial
t}+\frac{\partial\psi}{\partial t}=0$ (2.14)

And if we take derivative of the equation (2.11) with respect to the variable
$s$, we get

$\frac{\partial f^{\prime}}{\partial
t}=-(n-1)\left(\frac{g^{\prime}}{f^{\prime}}\right)^{\prime}-\left(\frac{g^{\prime\prime}}{f^{\prime\prime}}\right)^{\prime}+kf^{\prime\prime}(s,t)$

Using this in equation (2.14), we get

$\displaystyle\frac{\partial\psi}{\partial t}$
$\displaystyle=\psi^{\prime}(f^{\prime}(s,t))\left[(n-1)\left(f^{\prime\prime}\frac{\psi^{\prime}(\tau)}{f^{\prime}}-\frac{\psi(\tau)}{f^{\prime
2}}f^{\prime\prime}\right)+\psi^{\prime\prime}(\tau)f^{\prime\prime}-kf^{\prime\prime}\right]$
$\displaystyle=\psi^{\prime}(f^{\prime}(s,t))f^{\prime\prime}(s,t)\left[(n-1)\left(\frac{\psi^{\prime}(\tau)}{\tau}-\frac{\psi(\tau)}{\tau^{2}}\right)+\psi^{\prime\prime}(\tau)-k\right]$
$\displaystyle=g^{\prime\prime}(s)\left(\psi^{\prime\prime}(\tau)+(n-1)\frac{\psi^{\prime}(\tau)}{\tau}-(n-1)\frac{\psi(\tau)}{\tau^{2}}-k\right)$

Since $g$ is convex and $g^{\prime}>0$, we can get that
$g^{\prime\prime}(s)=g^{\prime\prime}((g^{\prime})^{-1}(\psi))$. Let
$Q(\psi)=g^{\prime\prime}((g^{\prime})^{-1}(\psi)).$ Then by the asymptotic
properties of $g$, we can get that $Q(1)=Q(b)=0$ and $Q(\tau)>0$, for all
$\tau\in(1,b).$

∎

##  3\. Convergence of the flow with the special initial data

In this section we prove a weaker version of our main theorem for a special
choice of initial condition. For the most part, our proof runs parallel to the
arguments in [10] and [6]. There is however one key point of difference that
we now wish to explain. Recall that the proof of [10] and [6] also proceeds by
first proving convergence for a special initial Kähler form, namely the Kähler
form for which the function $\psi$ is linear. With this choice of initial data
one can then prove a comparison principle along the flow and a monotonicity
property along the flow. Unfortunately, as we explain below (cf. Remark 3.3),
for the modified $J$-flow, the choice of a linear initial function does not
quite work. We instead have to choose another concave, strictly increasing
function. The next lemma summarises the key properties that we need from the
initial function.

######  Lemma 3.1.

For $c_{k}+k\leq n-1$, there exists a smooth, strictly monotonic and concave
function $\psi_{0}:[1,b]\rightarrow[1,a]$ with the following properties:

  1. (1)

$\psi_{0}\geq\sigma$, where we define

$\sigma(\tau)=\begin{cases}1,&1\leq\tau\leq\lambda;\\\
\psi_{\lambda}(\tau),&\lambda\leq\tau\leq b,\end{cases}\text{ and
$\lambda\in(1,b)$ satisfies }C_{k}(\lambda)+k\lambda=\frac{n-1}{\lambda}.$
(3.1)

  2. (2)

$P[\psi_{0}]\leq 0$, where $P$ is the $2^{nd}$ order linear differential
operator given by (2.13).

###### Proof.

We define $\psi_{0}$ an increasing, smooth and concave function by

$\psi_{0}(\tau)=\frac{a-1}{b^{-n}-1}(\tau^{-n}-1)+1.$

We know that the $\lambda$ satisfies the equation
$C_{k}(\lambda)+k\lambda=\frac{n-1}{\lambda}$ this implies that

$\frac{n+1-k\lambda^{2}}{n+1}-\lambda\frac{C_{k}(\lambda)}{n}>0,$

therefore $\sigma$ is convex on $[1,b].$ Since $\psi_{0}$ is concave we have

$1\leq\sigma(\tau)<\psi_{0}(\tau)<a,\text{ for all }\tau\in(1,b).$

Next, we compute

$\displaystyle P[\psi_{0}]$
$\displaystyle=\frac{a-1}{b^{-n}-1}n(n+1)\frac{1}{\tau^{n+2}}-(n-1)\frac{a-1}{b^{-n}-1}\frac{n}{\tau^{n+2}}-(n-1)\frac{\psi_{0}(\tau)}{\tau^{2}}-k$
$\displaystyle=-\frac{2n(a-1)}{1-b^{-n}}\frac{1}{\tau^{n+2}}-(n-1)\frac{\psi_{0}(\tau)}{\tau^{2}}-k$
$\displaystyle<0$

∎

######  Proposition 3.2.

Let $0<c_{k}+k\leq n-1$ and suppose $\psi(\tau,t)$ be the solution of the
equation (2.13) with the initial value $\psi_{0}(\tau)=\psi(\tau,0)$,
constructed in the previous lemma, then

$\psi(\tau,t)\xrightarrow[C_{loc}^{1}((\lambda,b))]{C^{0}}\sigma(\tau)=\begin{cases}1,&1\leq\tau\leq\lambda\\\
\psi_{\lambda}(\tau),&\lambda\leq\tau\leq b\end{cases},$ (3.2)

where $\psi_{\lambda}$ is a solution of the ODE (2.2) with
$\psi_{\lambda}(\lambda)=1$ and $\psi_{\lambda}(b)=a$ and $\lambda$ can be
described as

$\lambda=\inf\\{s:\text{ there exist }\psi_{s}\text{ satisfies
}\eqref{ode},\psi_{s}(s)=1,\psi_{s}(b)=a\text{ and
}\psi_{s}^{\prime}(\tau)>0,\text{ for all }\tau\in(s,b)\\}.$

###### Proof.

The proof consists of several steps.

  * •

First we claim that $\psi(\tau,t)\geq\sigma(\tau).$ Define
$h(\tau,t)=\psi(\tau,t)-\sigma(\tau)$ and it satisfies the following equation
on $[\lambda,b]$

$\frac{\partial h}{\partial
t}=Q(\psi)\left(h^{\prime\prime}(\tau,t)+(n-1)\frac{h^{\prime}(\tau,t)}{\tau}-(n-1)\frac{h(\tau,t)}{\tau^{2}}\right).$
(3.3)

Since $\psi(\tau,t)$ is an increasing function for each $t$, $\psi(\tau,t)>1$,
for all $(\tau,t)\in(1,b]\times[0,\infty),$ hence the claim is equivalent to
proving that

$\inf_{[\lambda,b]\times[0,T]}h(\tau,t)\geq 0,\text{ for all }T>0.$

Suppose the claim is not true, that is, there exist a $T>0$ such that
$\inf_{[\lambda,b]\times[0,T]}h(\tau,t)<0.$ Let
$h(x,t_{0})=\inf_{[\lambda,b]\times[0,T]}h(\tau,t),$ for some
$(x,t_{0})\in[\lambda,b]\times[0,T].$

Since $h(\tau,0)=\psi_{0}(\tau)-\sigma(\tau)\geq 0$ (by Lemma 3.1), $h(b,t)=0$
and $h(\lambda,t)>0$ we can see that $t_{0}>0$ and $x\in(\lambda,b).$ And we
have

$Q(\psi(x,t_{0}))>0,\frac{\partial h}{\partial t}(x,t_{0})\leq
0,h^{\prime}(x,t_{0})=0\text{ and }h^{\prime\prime}(x,t_{0})\geq 0.$

Hence, at $(x,t_{0})$ we have

$\displaystyle\Bigg{(}\frac{\partial h}{\partial
t}-Q(\psi)\left(h^{\prime\prime}+(n-1)\frac{h^{\prime}}{\tau}-(n-1)\frac{h}{\tau^{2}}\right)\Bigg{)}(x,t_{0})$
$\displaystyle\leq(n-1)Q(\psi(x,t_{0}))\frac{h(x,t_{0})}{x^{2}}$
$\displaystyle<0$

which is a contradiction to the equation (3.3) satisfied by $h$ on
$[\lambda,b].$ Hence we have

$\psi(\tau,t)\geq\sigma(\tau),\text{ for all }\tau\in[1,b].$

  * •

Next we claim that $\psi(\tau,t)$ is decreasing in $t.$ Let
$H(\tau,t)=\frac{\partial\psi}{\partial t}(\tau,t)$. Firstly note that by our
choice of $\psi_{0}$ we have that by Lemma 3.1

$H(\tau,0)=Q(\psi_{0})P[\psi_{0}]\leq 0.$

The claim is equivalent to proving that

$\sup_{[1,b]\times[0,T]}H(\tau,t)\leq 0,\text{ for all }T>0.$

Suppose this is not true, then there exist a $T>0$ such that
$\sup_{[1,b]\times[0,T]}H(\tau,t)>0.$ Let
$H(x,t_{0})=\sup_{[1,b]\times[0,T]}H(\tau,t)$ and consider

$F(\tau,t)=e^{-Ct}H(\tau,t),\text{ where
}C=\sup_{[1,b]\times[0,T]}Q^{\prime}(\psi(\tau,t))\frac{H(\tau,t)}{Q(\psi(\tau,t))}.$

Since

$\frac{H(\tau,t)}{Q(\psi(\tau,t))}=\psi^{\prime\prime}(\tau,t)+(n-1)\frac{\psi^{\prime}(\tau,t)}{\tau}-(n-1)\frac{\psi(\tau,t)}{\tau^{2}}-k,$

the constant $C$ is finite.

Let $F(y,t_{1})=\sup_{[1,b]\times[0,T]}F(\tau,t).$ Since
$F(\tau,0)=H(\tau,0)\leq 0$ and $F(1,t)=F(b,t)=0$ it should be the case that
$t_{1}>0$ and $y\in(1,b)$. Then at $(y,t_{1})$ we have

$\displaystyle F(y,t_{1})>0,Q(\psi(y,t_{1}))>0,H(y,t_{1})>0,\frac{\partial
F}{\partial t}(y,t_{1})\geq 0,$ $\displaystyle
F^{\prime}(y,t_{1})=H^{\prime}(y,t_{1})=0,F^{\prime\prime}(y,t_{1})\leq
0\text{ and }H^{\prime\prime}(y,t_{1})\leq 0.$

This implies that

$0\leq\Big{(}\frac{\partial F}{\partial
t}-Q(\psi)F^{\prime\prime}\Big{)}(y,t_{1})=e^{-Ct_{1}}\left(-CH(y,t_{1})+\frac{\partial
H}{\partial t}(y,t_{1})-Q(\psi(y,t_{1}))H^{\prime\prime}(y,t_{1})\right)$

so

$0\leq-e^{-Ct_{1}}\Bigg{(}H(y,t_{1})^{2}\frac{Q^{\prime}(\psi)}{Q(\psi)}-\frac{\partial
H}{\partial t}(y,t_{1})\\\
+Q(\psi)\Big{(}H^{\prime\prime}(y,t_{1})+(n-1)\frac{H^{\prime}(y,t_{1})}{y}-(n-1)\frac{H(y,t_{1})}{y^{2}}\Big{)}+(n-1)Q(\psi)\frac{H(y,t_{1})}{y^{2}}\Bigg{)}$

Thus

$H(y,t_{1})^{2}\frac{Q^{\prime}(\psi)}{Q(\psi)}-\frac{\partial H}{\partial
t}(y,t_{1})+Q(\psi)\Big{(}H^{\prime\prime}(y,t_{1})\\\
+(n-1)\frac{H^{\prime}(y,t_{1})}{y}-(n-1)\frac{H(y,t_{1})}{y^{2}}\Big{)}\leq-(n-1)Q(\psi)\frac{H(y,t_{1})}{y^{2}}<0$

which is a contradiction to the fact that $H$ actually satisfies the following
equation

$\frac{\partial H}{\partial
t}(\tau,t)=H(\tau,t)^{2}\frac{Q^{\prime}(\psi)}{Q(\psi)}+Q(\psi)\Big{(}H^{\prime\prime}(\tau,t)+(n-1)\frac{H^{\prime}(\tau,t)}{\tau}-(n-1)\frac{H(\tau,t)}{\tau^{2}}\Big{)}.$

Hence

$\frac{\partial\psi}{\partial t}\leq 0,$

and $\psi(\tau,t)$ is a decreasing function in $t$.

  * •

Now we prove the $C^{0}$ convergence. The proof is very similar to that of
[6], so we give only sketch of the proof and for the details we refer the
reader to [6]. Since
$\Lambda_{\chi_{\varphi(t)}}\omega=\psi^{\prime}(\tau,t)+(n-1)\frac{\psi(\tau,t)}{\tau}$
and by the inequality (2.9) we get that $0<\psi^{\prime}(\tau,t)<C$ for some
positive constant C, together this with the previous claim implies that
$\psi(\tau,t)$ converges uniformly on $[1,b]$ to its pointwise limit
$\psi_{\infty}(\tau)$, an increasing continuous function.

  * •

Next, we prove the $C^{1}_{loc}$ convergence of $\psi(\tau,t)$ and
differentiability of $\psi_{\infty}$ on $(s,b)$, where

$s=\sup\\{x:\psi_{\infty}(x)=1\\}.$

Since $\psi(\tau,t)\geq\sigma(\tau),$ we get
$\psi_{\infty}(\tau)\geq\sigma(\tau)$. Next, as in [6], we consider the
evolving point-wise slope

$\eta(\tau,t):=\psi^{\prime}(\tau,t)+(n-1)\frac{\psi(\tau,t)}{\tau}-k\tau.$

Then

$\frac{\partial\psi}{\partial t}(\tau,t)=Q(\psi)\eta^{\prime}(\tau,t).$

So for each $t$, $\eta(\tau,t)$ is decreasing on $[1,b].$ Let us fix
$0<\delta<<1$ and $T>0$ . Since $\psi(\tau,t)$ converges uniformly to
$\psi_{\infty}$ we get $Q(\psi(\tau,t))$ is uniformly bounded below, for all
$(\tau,t)\in[s+\delta,b-\delta]\times[T,\infty)$, and this implies

$\int_{T}^{\infty}\int_{s+\delta}^{b-\delta}\left|\frac{\partial\psi}{\partial
t}\right|d\tau
dt=\int_{s+\delta}^{b-\delta}\Big{(}\psi(\tau,T)-\psi(\tau,\infty)\Big{)}d\tau\leq
D$ (3.4)

for some constant $D>0$ independent of $\delta.$ This implies that

$\int_{T}^{\infty}\Big{(}\eta(1+\delta,t)-\eta(b-\delta,t)\Big{)}dt=-\int_{T}^{\infty}\int_{s+\delta}^{b-\delta}\eta^{\prime}(\tau,t)d\tau
dt\leq C(\delta),$ (3.5)

for some $C(\delta)>0$. Therefore, there exist a sequence $\\{t_{j}\\}$ such
that $\eta(\tau,t_{j})$ converges to $\eta_{\infty}$ and by the monotinicity
of $\eta$ the limit $\eta_{\infty}$ will be a constant. And the uniform
convergence of $\psi(\tau,t)$ gives the uniform convergence of
$\psi^{\prime}(\tau,t_{j})$ on $(s,b).$

Now, let $x,\tau\in(s,b)$, then

$\eta_{\infty}(\tau-x)=\lim_{j\to\infty}\int_{x}^{\tau}\eta(y,t_{j})dy=\psi_{\infty}(\tau)-\psi_{\infty}(x)+\int_{x}^{\tau}\Big{(}(n-1)\frac{\psi_{\infty}(y)}{y}-ky\Big{)}dy.$

Hence $\psi_{\infty}$ is differentiable on $(s,b)$ and it satisfies the
following equation

$\psi_{\infty}^{\prime}(\tau)+(n-1)\frac{\psi_{\infty}(\tau)}{\tau}-k\tau=\eta_{\infty}.$

  * •

Next we claim that
$\lim_{t\to\infty}\psi(\tau,t)=\psi_{\infty}(\tau)=\sigma(\tau).$ Since
$\psi_{\infty}(\tau)\geq\sigma(\tau)$ and
$1\leq\sigma(\tau)<\psi_{0}(\tau)<a,\text{ for all }\tau\in(1,b)$ the function
$\psi_{\infty}$ should be of the following form

$\psi_{\infty}(\tau)=\begin{cases}1,&1\leq\tau\leq s;\\\
\rho(\tau),&s\leq\tau\leq b,\end{cases}$

where $s\in[1,b)$ and $\rho$ is a solution of the ODE (2.2) in the interval
$[s,b]$ with the boundary condition $\rho(s)=1\text{ and }\rho(b)=a$ and
$\rho^{\prime}(\tau)>0,\text{ for all }\tau\in(s,b)$ which follows from the
last claim and the fact that $\psi_{\infty}$ is increasing. This implies that
$s\leq\lambda.$ From the definition of $\lambda$ we can get that $s=\lambda.$
And the uniqueness of the solution of the ODE (2.2) on the interval
$[\lambda,b]$ implies that

$\psi_{\infty}(\tau)=\sigma(\tau).$

Therefore $\psi(\tau,t)$ converges in $C^{1}_{loc}((\lambda,b))$ to its
limiting function $\sigma(\tau)=\lim_{t\to\infty}\psi(\tau,t)$, follows from
the fact that $\psi_{\infty}^{\prime}(\tau)>0$ on any compact
$K\subset(\lambda,b).$

Finally, since $\psi_{\infty}$ satisfies the ODE (2.2) on $(\lambda,b)$, with
the boundary conditions $\psi_{\infty}(\lambda)=1$ and $\psi_{\infty}(b)=a$,
we get $\eta_{\infty}=C_{k}(\lambda).$ ∎

######  Remark 3.3.

We now explain why the straight line path does not work for the general
modified $J$ flow. The main issue that the sign of $P[\psi_{0}]$ could be
indeterminate, which is a key starting point in establishing the monotonicity
of $\psi(\tau,t)$ in the time variable. For instance, consider the following
example on $X_{2}$, let $a=12,b=6$ and $k=1$, then in this case $0<c_{k}+k<1$
and for the straight line path $\psi_{0}$ we have $P[\psi_{0}](1)>0$ and
$P[\psi_{0}](6)<0.$

######  Remark 3.4.

For the case $c_{k}+k>n-1,$ the solution of the ODE (2.2), $\tilde{\psi}$ can
be neither convex nor concave. For example, on $X_{3}$, if we take $a=4,b=2$
and $k=1$, then the solution $\tilde{\psi}$ is neither convex nor concave. And
we can not choose the straight line path as a intial value of the flow, as
$\tilde{\psi}$ neither lies above nor lies below the straight line path
completely. So, we choose
$\psi_{0}(\tau)=\tilde{\psi}+\frac{(\tau-1)(b-\tau)}{N\tau^{n}},$ for some
sufficiently large $N,$ as the initial data of the flow, then $\psi_{0}$
satisfies all the properties mentioned in Lemma 3.1 and then we can proceed as
in the last propostion to prove the convergence in the case of $c_{k}+k>n-1.$

##  4\. Uniqueness and higher order estimates

Uniqueness and higher order estimates for the $J$-flow with the Calabi ansatz
in the unstable case is done in [6]. In this section we derive the uniform
higher order estimates and prove the uniqueness of the limit of the modified
$J$-flow for the cases $c_{k}+k\leq n-1,$ for this we need some results which
are very similar to the results proved in [6] with necessary modifications.
So, just for the readers convenience we state those results and give sketch of
the proof for some of the results which needed the necessary modifications.

In the following proposition we prove that on a given compact subset
$K\subset\mathbb{R}$ the normalized ansatz Kähler potentials actually
converge. Let

$\Phi:[1,a]\times[0,\infty)\to[1,b],\text{ defined by
}\Phi(\tau,t):=\psi^{-1}(\tau,t).$

That is $f^{\prime}(s,t)=\Phi(g^{\prime}(s),t)$ and we define
$\Phi_{\infty}:=(\psi_{\infty}|_{[\lambda,b]})^{-1}:[1,a]\to[\lambda,b].$

######  Proposition 4.1.

For any compact $K\subset\mathbb{R}$,

$f^{\prime}(s,t)\to(\Phi_{\infty}\circ g^{\prime})(s),$

increasingly in $L^{\infty}(K)$ as $t\to\infty.$

###### Proof.

For a given compact subset $K\subset\mathbb{R}$ we can find a
$B\subset\subset(\lambda,b)$ such that $K=(\Phi_{\infty}\circ
g^{\prime})^{-1}(B)$, and we have $\psi_{\infty}^{\prime}(\tau)>0,$ for all
$\tau\in B.$ Then by Proposition 3.2, there exists $C,T>0$ such that for any
$t>T$ and $\tau\in B$, we have

$C^{-1}<\psi^{\prime}(\tau,t)<C.$

Since $\psi$ decreases in $t$ and by definition of $\Phi$ we get that
$\Phi(.,t)$ converges uniformly to $\Phi_{\infty}$ on $\Phi_{\infty}^{-1}(B)$
increasingly in $t$. Thus, for any compact $K\subset\mathbb{R}$,
$f^{\prime}(s,t)$ converges increasingly to $(\Phi_{\infty}\circ
g^{\prime})(s)$, for $s\in K.$ ∎

We define the normalized ansatz potentials by

$\hat{f}(s,t):=f(s,t)-f(0,t)=\int_{0}^{s}f^{\prime}(r,t)dr$

and

$f_{\infty}(s):=\int_{0}^{s}(\Phi_{\infty}\circ g^{\prime})(r)dr.$

By the above proposition we get that $\hat{f}(s,t)$ converges uniformly to
$f_{\infty}(s)$ on any compact set $K\subset\mathbb{R}$ as $t\to\infty.$

######  Lemma 4.2.

For all $t\geq 0,$ we have

$\hat{f}(s,t)\leq f_{\infty}(s)\text{ when }s\geq 0,\text{ and
}\hat{f}(s,t)\geq f_{\infty}(s)\text{ when }s\leq 0.$

###### Proof.

Since
$\hat{f}(s,t)-f_{\infty}(s)=\int_{0}^{s}(f^{\prime}(r,t)-(\Phi_{\infty}\circ
g^{\prime})(r))dr$, the lemma follows from the Proposition 4.1. ∎

Let

$\phi(s,t):=f(s,t)-f_{0}(s),\phi_{\infty}(s):=f_{\infty}(s)-f_{0}(s),$

where $f_{0}$ is the global potential for
$\chi_{0}=\sqrt{-1}\partial\overline{\partial}f_{0}(s)\in
b[E_{\infty}]-[E_{0}]$ so $f^{\prime}_{0}(-\infty)=1$ and
$f_{0}^{\prime}(\infty)=b$ also since, $f_{\infty}$ satisfies the modified $J$
equation on $[\lambda,b]$ with $f^{\prime}_{\infty}(-\infty)=\lambda$ and
$f^{\prime}_{\infty}(\infty)=b$,
$\sqrt{-1}\partial\overline{\partial}f_{\infty}\in
b[E_{\infty}]-\lambda[E_{0}].$

Therefore $\phi(\cdot,t)$ is a Kähler potential with respect to the metric
$\chi_{0}$, for all $t\geq 0$ and it satisfies the following equation

$\frac{\partial\phi}{\partial
t}(s,t)=c_{k}-(n-1)\frac{g^{\prime}(s)}{f^{\prime}(s,t)}-\frac{g^{\prime\prime}(s)}{f^{\prime\prime}(s,t)}+kf^{\prime}(s,t)$

Let $\nu:\mathbb{R}\to(-\infty,0]$ be a smooth increasing function satisfying

$\displaystyle\nu(s)=(\lambda-1)s\text{ when }s\in(-\infty,-1)\text{ and }$
$\displaystyle\nu(s)=0\text{ when }s\in(0,\infty).$

######  Lemma 4.3.

Let

$\hat{\phi}_{\infty}:=\phi_{\infty}-\nu,$

then $\hat{\phi}_{\infty}$ is bounded on $\mathbb{R}.$

###### Proof.

By definition $\hat{\phi}_{\infty}=f_{\infty}-(f_{0}+\nu)$ and
$\sqrt{-1}\partial\overline{\partial}f_{\infty}\in
b[E_{\infty}]-\lambda[E_{0}]$ is a Kähler current and
$\sqrt{-1}\partial\overline{\partial}(f_{0}+\nu)$ is a Kähler metric, thus
$\hat{\phi}_{\infty}$ is bounded in $\mathbb{R}.$ ∎

######  Lemma 4.4.

There exist a $C>0$ such that for all $(s,t)\in\mathbb{R}\times[0,\infty)$, we
have

$\nu(s)-C\leq\phi(s,t)-\phi(0,t)\leq C$

and

$\lim_{t\to\infty}\frac{\partial\phi}{\partial t}(0,t)=c_{k}-C_{k}(\lambda).$

###### Proof.

Without loss of generality we can assume $f_{0}(0)=0$ and by the definition of
$\phi(s,t)$, we have

$\phi(s,t)-\phi(0,t)=f(s,t)-f_{0}(s)-[f(0,t)-f_{0}(0)]=\hat{f}(s,t)-f_{0}(s).$

and the first equation of the lemma follows from Lemma 4.2 and Lemma 4.3. And
by Proposition 4.1 and Proposition 3.2 for any $K\subset\subset\mathbb{R}$, we
have

$\lim_{t\to\infty}\left|\frac{g^{\prime\prime}}{f^{\prime\prime}}+(n-1)\frac{g^{\prime}}{f^{\prime}}-kf^{\prime}-C_{k}(\lambda)\right|_{L^{\infty}(K)}=0.$

Hence

$\lim_{t\to\infty}\frac{\partial\phi}{\partial t}(0,t)=c_{k}-C_{k}(\lambda).$

∎

We chose $\phi(s,t)$ as a solution of the flow with the special initial
metric, now let $\varphi$ be the solution of the modified $J$ flow with any
initial data $\varphi(.,0)$ with Calabi symmetry. And let

$\vartheta(s,t)=\varphi(s,t)-f_{0}(s)\text{ and
}\chi_{0}=\sqrt{-1}\partial\overline{\partial}f_{0}.$

Then $\vartheta(.,t)$ are Kähler potentials of $[\chi_{0}]$ and the modified
$J$ flow is

$\frac{\partial\vartheta}{\partial
t}=c+\theta_{\xi}(\chi_{\vartheta})-\Lambda_{\chi_{\vartheta}}\omega.$

######  Lemma 4.5.

There exists a $C>0$ such that on $X_{n}\times[0,\infty)$,

$\phi_{\infty}(s)-C\leq\vartheta(s,t)-\phi(0,t)\leq C.$

###### Proof.

We know that $\theta_{\xi}(\chi_{\varphi})=\theta_{\xi}(\chi)+\xi(\varphi).$
If we take difference between the flows with initial values $\phi(\cdot,0)$
and $\vartheta(\cdot,0)$, then $\phi(x,t)-\vartheta(x,t)$ satisfies the
following equation

$\frac{\partial}{\partial
t}\left(\phi(x,t)-\vartheta(x,t)\right)=F^{i\bar{j}}[(1-s_{t})\chi_{\phi(t)}+s_{t}\chi_{\vartheta(t)}](\phi(t)-\vartheta(t))_{i\bar{j}}+\xi(\phi(t)-\vartheta(t)),$

where $0<s_{t}<1.$ By the maximum principle we have

$|\phi(x,t)-\vartheta(x,t)|_{C^{0}}\leq|\phi(x,0)-\vartheta(x,0)|_{C^{0}}\leq
C,\text{ for all }t\in[0,\infty).$

This together with the last lemma we can see that

$\nu(s)-C_{1}\leq\vartheta(s,t)-\phi(0,t)\leq C_{1}.$

And by Lemma 4.3 we can get that $\nu(s)-C_{2}\leq\phi(s,t)\leq\nu(s)+C_{2}$
this implies that

$\phi_{\infty}(s)-C\leq\vartheta(s,t)-\phi(0,t)\leq C.$

∎

The Kähler potential $\phi_{\infty}$ of $\chi_{0}$ is not a strict subsolution
of the modified $J$ equation for the new Kähler classes
$[\chi_{\infty}]=b[E_{\infty}]-\lambda[E_{0}]$ and $[\omega]$ and the constant
$C_{k}(\lambda),$ because $\chi_{\infty}$ blows up in the normal direction to
$E_{0}.$

Let $f_{\infty}$ be the limiting solution corresponding to the new Kähler
classes. Then $f_{\infty}^{\prime}(-\infty)=\lambda>1$ and

$\frac{g^{\prime\prime}(s)}{f_{\infty}^{\prime\prime}(s)}+(n-1)\frac{g^{\prime}(s)}{f_{\infty}^{\prime}(s)}-kf_{\infty}^{\prime}(s)=C_{k}(\lambda)$

this implies that

$(n-2)\frac{g^{\prime}(s)}{f_{\infty}^{\prime}(s)}+\frac{g^{\prime\prime}(s)}{f_{\infty}^{\prime\prime}(s)}-kf_{\infty}^{\prime}(s)\leq
C_{k}(\lambda)-\frac{1}{b},~{}(n-1)\frac{g^{\prime}(s)}{f_{\infty}^{\prime}(s)}-kf_{\infty}^{\prime}(s)\leq
C_{k}(\lambda)$

and the equality holds at $-\infty$ in the second inequality as

$\lim_{s\to-\infty}e^{-s}f_{\infty}^{\prime\prime}(s)=\infty.$

Our aim is to get a strict subsolution so we modify $f_{\infty}$ using the
barrier function $\nu$ which is increasing and smooth and defined in the
following way

$\nu(s)=s\text{ when }s\leq-1\text{ and }\nu(s)=0\text{ when }s>0.$

Let

$V_{\infty}(s)=f_{\infty}(s)+\varepsilon\nu(s)$

for sufficiently small $\varepsilon>0.$ Then $V_{\infty}$ defines a Kähler
metric $\Omega=\sqrt{-1}\partial\overline{\partial}V_{\infty}\in
b[E_{\infty}]-(\lambda+\varepsilon)[E_{0}].$ Further, we can choose a constant
$d>0$ so that

$\begin{split}(n-2)\frac{g^{\prime}(s)}{V^{\prime}_{\infty}(s)}+\frac{g^{\prime\prime}(s)}{V^{\prime\prime}_{\infty}(s)}-kV_{\infty}^{\prime}(s)<&C_{k}(\lambda)-2d\varepsilon\\\
\text{ and
}\quad(n-1)\frac{g^{\prime}(s)}{V^{\prime}_{\infty}(s)}-kV_{\infty}^{\prime}(s)<&C_{k}(\lambda)-2d\varepsilon.\end{split}$
(4.1)

We will use this subsolution to prove the second order estimate, as Lemma 4.7.

Now, let

$\hat{\vartheta}(s,t):=\vartheta(s,t)-\phi(0,t).$

Then

$\displaystyle\frac{\partial\hat{\vartheta}}{\partial t}=$ $\displaystyle
c+\theta_{\xi}(\chi_{\hat{\vartheta}(t)})-\Lambda_{\chi_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\vartheta}}(\omega)-\frac{\partial\phi}{\partial
t}(0,t)$ $\displaystyle=$ $\displaystyle
C_{k}(\lambda)+\delta(t)+k\vartheta^{\prime}(s,t)+kf_{0}^{\prime}(s)-\Lambda_{\chi_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\vartheta}}(\omega),$

where $\delta(t):=c_{k}-C_{k}(\lambda)-\frac{\partial\phi}{\partial t}(0,t)\to
0$ as $t\to\infty$ by Lemma 4.4.

Let

$u:=\hat{\vartheta}-(V_{\infty}-f_{0}).$

Then
$u(s,t)=\hat{\vartheta}(s,t)-(f_{\infty}(s)+\varepsilon\nu(s)-f_{0}(s))=\hat{\vartheta}(s,t)-(\phi_{\infty}(s)-\varepsilon\nu(s)),$
thus $u$ is smooth on $X_{n}\smallsetminus E_{0}$ and tends to $\infty$ along
$E_{0}.$

######  Lemma 4.6.

There exist a constant $C>0$ such that

$\inf_{X_{n}\times[0,\infty)}u>-C.$

###### Proof.

By Lemma 4.5 we have

$u(s,t)=\hat{\vartheta}(s,t)-(\phi_{\infty}(s)-\varepsilon\nu(s))\geq\phi_{\infty}(s)-C-(\phi_{\infty}(s)-\varepsilon\nu(s))\geq-C.$
∎

The following lemma is analogous to Lemma 3.1 of Song-Weinkove in [17] which
guarantee us the second order estimates away from the exceptional divisor.

######  Lemma 4.7.

Let $\chi=\chi_{0}+\sqrt{-1}\partial\overline{\partial}\vartheta$ be the
Kähler metrics corresponding to the solution of the $J$ flow. Then there exist
$A,C>0$ such that on $X_{n}\smallsetminus E_{0},$ we have

$\Lambda_{\omega}\chi\leq Ce^{Au}.$

###### Proof.

As in [6], we can apply the maximum principle argument to
$H=\log\Lambda_{\omega}\chi-Au$ with the perturbed heat operator and then use
the subsolution construction to get the required estimate.

The evolution for $\log\Lambda_{\omega}\chi$ is given by the following general
estimate

$\left(\frac{\partial}{\partial
t}-\Delta_{t}\right)\log\Lambda_{\omega}\chi\leq-\frac{n}{\Lambda_{\omega}\chi}\left(h^{k\bar{l}}R_{k\bar{l}}^{i\bar{j}}\chi_{i\bar{j}}-\chi^{k\bar{l}}R_{k\bar{l}}\right),$

where
$h^{i\bar{j}}=\chi^{i\bar{l}}\omega_{k\bar{l}}\chi^{k\bar{j}},\Delta_{t}:=n^{-1}h^{i\bar{j}}\partial_{j}\overline{\partial}_{j},$
and $R$ is the curvature tensor for $\omega.$

Also

$\displaystyle\left(\frac{\partial}{\partial t}-\xi-\Delta_{t}\right)u=$
$\displaystyle\frac{\partial\hat{\vartheta}}{\partial
t}-\xi(\hat{\vartheta})+\xi(V_{\infty}-f_{0})-\Delta_{t}u$ $\displaystyle=$
$\displaystyle\eta_{\infty}+\delta(t)-\Lambda_{\chi_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\vartheta}}(\omega)+k\vartheta^{\prime}+kf_{0}^{\prime}-k\vartheta^{\prime}$
$\displaystyle+kV_{\infty}^{\prime}-kf_{0}^{\prime}-h^{i\bar{j}}\vartheta_{i\bar{j}}+h^{i\bar{j}}\Omega_{i\bar{j}}-h^{i\bar{j}}f_{0_{i\bar{j}}}$
$\displaystyle=$
$\displaystyle\eta_{\infty}+\delta(t)-2\chi^{i\bar{j}}\omega_{i\bar{j}}+\chi^{i\bar{l}}\omega_{k\bar{l}}\chi^{k\bar{j}}\Omega_{i\bar{j}}+kV_{\infty}^{\prime}$

Suppose the maximum of $H$ on $X_{n}\times[0,t_{0}]$ is achieved at
$(p_{0},t_{0}),$ with $t_{0}>0$, otherwise the estimate follows trivially.
Then applying maximum principle, we have at $(p_{0},t_{0})$

$0\leq\left(\frac{\partial}{\partial t}-\xi-\Delta_{t}\right)H\\\
\leq-(n\Lambda_{\omega}\chi)^{-1}(h^{k\bar{l}}R_{k\bar{l}}^{i\bar{j}}\chi_{i\bar{j}}-\chi^{k\bar{l}}R_{k\bar{l}}+n^{-1}\xi(\Lambda_{\omega}\chi))\\\
-A(\eta_{\infty}+\delta(t)-2\chi^{i\bar{j}}\omega_{i\bar{j}}+\chi^{i\bar{l}}\omega_{k\bar{l}}\chi^{k\bar{j}}\Omega_{i\bar{j}}+kV_{\infty}^{\prime})$
(4.2)

Since $\Lambda_{\omega}\chi$ is uniformly bounded below(by inequality (2.10)),
for a given $2d\varepsilon>0$ we can choose sufficiently large $A$ such that
at $(p_{0},t_{0}),$ we have

$\eta_{\infty}+\delta(t)+h^{i\bar{j}}\Omega_{i\bar{j}}-2\chi^{i\bar{j}}\omega_{i\bar{j}}+kV_{\infty}^{\prime}\leq-(nA\Lambda_{\omega}\chi)^{-1}(h^{k\bar{l}}R_{k\bar{l}}^{i\bar{j}}\chi_{i\bar{j}}-\chi^{k\bar{l}}R_{k\bar{l}}-n\xi(\Lambda_{\omega}\chi))<2d\varepsilon$

As $\delta(t)\to 0$ when $t\to\infty$, we can find a $T$ such that
$|\delta(t)|<d\varepsilon,$ for all $t>T.$ And, we can take $t_{0}>T$,
otherwise, the estimate follows trivially. This implies that

$\eta_{\infty}+h^{i\bar{j}}\Omega_{i\bar{j}}-2\chi^{i\bar{j}}\omega_{i\bar{j}}+kV_{\infty}^{\prime}<d\varepsilon$
(4.3)

From (4.1) we already have that

$2d\varepsilon<\eta_{\infty}-(n-2)\frac{g^{\prime}}{V_{\infty}^{\prime}}-\frac{g^{\prime\prime}}{V_{\infty}^{\prime\prime}}+kV_{\infty}^{\prime}\text{
and
}2d\varepsilon<\eta_{\infty}-(n-1)\frac{g^{\prime}}{V_{\infty}^{\prime}}+kV_{\infty}^{\prime}.$

Note that
$[h^{i\bar{j}}]=diag\left[\frac{g^{\prime}}{(f^{\prime})^{2}},\dots,\frac{g^{\prime}}{(f^{\prime})^{2}},\frac{g^{\prime\prime}}{(f^{\prime\prime})^{2}}\right]$,
if we apply this in (4.2) and using (4.3) and the above inequality we get

$\displaystyle d\varepsilon\geq$
$\displaystyle\eta_{\infty}+(n-1)\frac{g^{\prime}V_{\infty}^{\prime}}{(f^{\prime})^{2}}+\frac{g^{\prime\prime}V_{\infty}^{\prime\prime}}{(f^{\prime\prime})^{2}}-2(n-1)\frac{g^{\prime}}{f^{\prime}}-2\frac{g^{\prime\prime}}{f^{\prime\prime}}+kV_{\infty}^{\prime}$
$\displaystyle=$
$\displaystyle(n-1)g^{\prime}V_{\infty}^{\prime}\left(\frac{1}{f^{\prime}}-\frac{1}{V_{\infty}^{\prime}}\right)^{2}+\frac{g^{\prime\prime}V_{\infty}^{\prime\prime}}{(f^{\prime\prime})^{2}}-2\frac{g^{\prime\prime}}{f^{\prime\prime}}+\left(\eta_{\infty}-(n-1)\frac{g^{\prime}}{V_{\infty}^{\prime}}+kV_{\infty}^{\prime}\right)$
$\displaystyle\geq$ $\displaystyle
2d\varepsilon-2\frac{g^{\prime\prime}}{f^{\prime\prime}}$

and

$\displaystyle d\varepsilon\geq$
$\displaystyle\eta_{\infty}+(n-1)\frac{g^{\prime}V_{\infty}^{\prime}}{(f^{\prime})^{2}}+\frac{g^{\prime\prime}V_{\infty}^{\prime\prime}}{(f^{\prime\prime})^{2}}-2(n-1)\frac{g^{\prime}}{f^{\prime}}-2\frac{g^{\prime\prime}}{f^{\prime\prime}}+kV_{\infty}^{\prime}$
$\displaystyle=$
$\displaystyle(n-2)g^{\prime}V_{\infty}^{\prime}\left(\frac{1}{f^{\prime}}-\frac{1}{V_{\infty}^{\prime}}\right)^{2}+g^{\prime\prime}V_{\infty}^{\prime\prime}\left(\frac{1}{f^{\prime\prime}}-\frac{1}{V_{\infty}^{\prime\prime}}\right)^{2}+\frac{g^{\prime\prime}V_{\infty}^{\prime\prime}}{(f^{\prime\prime})^{2}}-2\frac{g^{\prime}}{f^{\prime}}$
$\displaystyle\qquad+\left(\eta_{\infty}-(n-2)\frac{g^{\prime}}{V_{\infty}^{\prime}}-\frac{g^{\prime\prime}}{V_{\infty}^{\prime\prime}}+kV_{\infty}^{\prime}\right)$
$\displaystyle\geq$ $\displaystyle
2d\varepsilon-2\frac{g^{\prime}}{f^{\prime}}$

Combinig the above estimates, we have at $(p_{0},t_{0})$,

$\Lambda_{\omega}\chi=(n-1)\frac{f^{\prime}}{g^{\prime}}+\frac{f^{\prime\prime}}{g^{\prime\prime}}\leq\frac{2n}{d\varepsilon}$

and so for any $p\in X_{n}$ and $t\in[0,t_{0}]$ we have

$\displaystyle\log\Lambda_{\omega}\chi(p,t)=$ $\displaystyle H(p,t)+Au(p,t)$
$\displaystyle\leq$
$\displaystyle\log\Lambda_{\omega}\chi(p_{0},t_{0})-Au(p_{0},t_{0})+Au(p,t)$
$\displaystyle\leq$ $\displaystyle-\log d\varepsilon+\log
2n+Au(p,t)-A\inf_{X_{n}}u(p,t_{0}).$

Taking exponential on both sides and using the Lemma 4.6 we get the required
inequality. ∎

######  Lemma 4.8.

For any $K\subset\subset X_{n}\smallsetminus E_{0}$ and $l>0,$ there exists
$C:=C(K,l)$ such that

$\|\hat{\vartheta}\|_{C^{l}(K)}\leq C.$

###### Proof.

In section 3 of [17], Song and Weinkove derived the $C^{0}$ estimate by
sequence of lemmas using just the $C^{2}$ estimate and not any equation. As we
have $C^{2}$ estimate in Lemma 4.7, using the arguments from [17] we can
derive the $C^{0}$ estimate. By Schauder estimates and Evans-Krylov estimate
we can obtain the uniform $C^{2,\alpha}$ estimate and by bootstrapping we get
the higher order estimates. ∎

We got all the necessary estimates to pass the limit, now we prove in the next
theorem that irrespective of the initial value the flow always converges to a
unique limiting solution.

######  Theorem 4.9.

Let $\vartheta(t)$ be the solution of the modified $J$-flow, then
$\Lambda_{\chi_{\vartheta(t)}}\omega-\theta_{\xi}(\chi_{\vartheta(t)})$
converges to $C_{\lambda}$ smoothly on $X_{n}\smallsetminus E_{0}$ as
$t\to\infty,$ where
$C_{\lambda}=n\frac{ab^{n-1}-\lambda^{n-1}}{b^{n}-\lambda^{n}}.$

###### Proof.

Let us define a modified $J$-energy functional as

$E(t)=\int_{X_{n}}\left(\Lambda_{\chi_{\vartheta(t)}}\omega-\theta_{\xi}(\chi_{\vartheta(t)})\right)^{2}\chi_{\vartheta(t)}^{n}.$

In [15] Li-Shi proved that the energy decreases along the modified $J$-flow by
showing that the energy functional along the flow satisfies the following
equation:

$\frac{d}{dt}E(t)=-2\int_{X_{n}}\left|\nabla^{\chi_{\vartheta(t)}}\left(\Lambda_{\chi_{\vartheta(t)}}\omega-\theta_{\xi}(\chi_{\vartheta(t)})\right)\right|_{\omega}^{2}\chi_{\vartheta(t)}^{n}$

The proof follows from the similar arguments of the proof of the result in
[6]. Hence, the modified $J$-flow always converges to a unique limit and it
satisfies the modified $J$-equation in the new Kähler class. ∎

##  5\. Proof of the main theorem

###### Proof of Theorem 1.1.

######  Case 1.

It follows from the Case 1. of Theorem 3.2. that the solution of the ODE (2.2)
$\tilde{\psi}$ satisfies $\tilde{\psi}^{\prime}(\tau)>0,\text{ for all
}\tau\in[1,b].$ By Corollary 2.3 that the condition $c_{k}+k>n-1$ is
equivalent to

$(c+\theta_{\xi}(\chi))\chi^{n-1}-(n-1)\omega\wedge\chi^{n-2}>0.$

Then by Theorem 3.3 of Li-Shi [15], the modified $J$-flow converges smoothly
to the unique solution of the modified $J$-equation irrespective of the
initial value we choose. Thus $\chi_{t}\to\chi_{\infty}$ as $t\to\infty$
smoothly on $X_{n}.$ Therefore we get

$n\frac{\omega\wedge\chi_{\infty}^{n-1}}{\chi_{\infty}^{n}}=c+\theta_{\xi}(\chi_{\infty}).$

######  Case 2.

And by the arguments similar to the proof of Lemma 4.8 will give us the
uniform higher order estimates on compact set $K\subset X_{n}\smallsetminus
E_{0}$. So the flow converges to the critical equation away from $E_{0}.$ This
implies that on $X_{n}\smallsetminus E_{0}$ we have

$n\frac{\omega\wedge\chi_{\infty}^{n-1}}{\chi_{\infty}^{n}}=c+\theta_{\xi}(\chi_{\infty}).$

Since the convergence of $f(\cdot,t)$ to $f_{\infty}$ as $t\to\infty$ and it
derivatives are uniform on compact subsets $K\subset X_{n}\smallsetminus
E_{0}$, the function $f_{\infty}$ is smooth away from the exceptional divisor
$E_{0}$ and it is continuous on $X_{n}$ and
$f^{\prime}_{\infty}(s)=\psi^{-1}(g^{\prime}(s))$. This implies that

$\chi_{\infty}=\sqrt{-1}\partial\overline{\partial}f_{\infty}=\chi_{0}+\sqrt{-1}\partial\overline{\partial}\phi_{\infty},\text{
where }\phi_{\infty}=f_{\infty}-f_{0}\in L^{\infty}(X_{n}).$

And $\phi_{\infty}$ is smooth away from the exceptional divisor. In this case
we can extend the metric $\chi_{\infty}$ to $E_{0}.$ And Theorem 4.9 gives us
that the limiting solution is unique.

######  Case 3.

In this case we have

$\lim_{s\to-\infty}f^{\prime}_{\infty}(s)=\lambda>1.$

This implies that
$\hat{\chi}_{\infty}=\sqrt{-1}\partial\overline{\partial}f_{\infty}\in
b[E_{\infty}]-\lambda[E_{0}].$

So $\chi_{t}\to\hat{\chi}_{\infty}+(\lambda-1)[E_{0}]$ as $t\to\infty.$ And
$f^{\prime}_{\infty}(s)=\psi^{-1}(g^{\prime}(s))$ where $\psi$ is a solution
of the ODE (2.2) in the interval $[\lambda,b]$ with
$\psi^{\prime}(\lambda)=0.$ By the argument similar to Case 2. we get the
higher order estimates away form $E_{0}.$ So the flow converges to the
critical equation away from $E_{0}.$ This implies that on $X_{n}\smallsetminus
E_{0}$ we have

$n\frac{\omega\wedge\hat{\chi}_{\infty}^{n-1}}{\hat{\chi}_{\infty}^{n}}=n\frac{ab^{n-1}-\lambda^{n-1}}{b^{n}-\lambda^{n}}+\theta_{\xi}(\hat{\chi}_{\infty}).$

And by Proposition 3.2 the $\lambda\in(1,b)$ is unique and satisfies the
equation

$n\frac{ab^{n-1}-\lambda^{n-1}}{b^{n}-\lambda^{n}}-\frac{nk}{n+1}\frac{b^{n+1}-\lambda^{n+1}}{b^{n}-\lambda^{n}}+k\lambda=\frac{n-1}{\lambda}.$

And, the uniqueness of limiting solution follows from Theorem 4.9.

∎

######  Remark 5.1.

Suppose we are given with two Kähler classes $\chi\in
b[E_{\infty}]-b_{0}[E_{0}]$, $\omega\in a[E_{\infty}]-a_{0}[E_{0}]$ and the
vector field $\xi_{k}=kw\frac{\partial}{\partial w}(k\geq 0)$, then we
consider the new Kähler classes $\frac{b}{b_{0}}[E_{\infty}]-[E_{0}]$ and
$\frac{a}{a_{0}}[E_{\infty}]-[E_{0}]$ and let $f_{1}(s,t)$ be the solution of
the modified $J$-flow for the scaled classes with
$\sqrt{-1}\partial\overline{\partial}g_{1}(s)\in\frac{a}{a_{0}}[E_{\infty}]-[E_{0}].$

As the modified $J$-equation has the normalized Hamiltonian on its right side
we have to scale the vector field by $\frac{b_{0}^{2}}{a_{0}}$ that is
$\xi^{\prime}=\frac{b_{0}^{2}}{a_{0}}\xi.$ Then the usual parabolic scaling
gives us the flow for the original Kähler classes. That is if we take

$f(s,t)=b_{0}f_{1}(s,b_{0}^{-2}a_{0}t)\text{ and }g(s)=a_{0}g_{1}(s),$

then this satisfies the modified $J$-flow on the Kähler classes $\chi$ and
$\omega.$

### Acknowledgments:

The author would like to thank his PhD. advisor Ved Datar for suggesting this
problem, for the helpful discussions, and the suggestions for improving the
first draft of the current work. He would also like to thank Professor Vamsi
P. Pingali for useful comments on the first draft. This research was supported
by an NBHM fellowship and the graduate program of the Indian Institute of
Science.

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