# Singularity formation along the line bundle mean curvature flow

Yu Hin Chan<EMAIL_ADDRESS>and  Adam Jacob<EMAIL_ADDRESS>Department of Mathematics, University of California Davis, 1 Shields Ave.,
Davis, CA, 95616

###### Abstract.

The line bundle mean curvature flow is a complex analogue of the mean
curvature flow for Lagrangian graphs, with fixed points solving the deformed
Hermitian-Yang-Mills equation. In this paper we construct two distinct
examples of singularities along the flow. First, we find a finite time
singularity, ruling out long time existence of the flow in general. Next we
show long time existence of the flow with a Calabi symmetry assumption on the
blowup of $\mathbb{P}^{n}$, $n\geq 3$, if one assumes supercritical phase.
Using this, we find an example where a singularity occurs at infinite time
along the destabilizing subvariety in the semi-stable case.

∗Supported in part by a Simons Collaboration Grant.

##  1\. Introduction

Let $(X,\omega)$ be a compact Kähler manifold, and $[\alpha]\in
H^{1,1}(X,\mathbb{R})$ a real cohomology class. The deformed Hermitian-Yang-
Mills (dHYM) equation seeks a representative $\alpha\in[\alpha]$ satisfying

(1.1) ${\rm Im}(e^{-i\hat{\theta}}(\omega+i\alpha)^{n})=0$

for a fixed constant $e^{i\hat{\theta}}\in S^{1}$. Recently this equation has
garnered significant attention, and extensive work has centered around the
relationship between existence of a solution and notions of geometric
stability [2, 3, 4, 5, 8, 14, 20]. Although much of this work has been done
with elliptic methods, substantial progress has been made following a
parabolic approach as well [10, 12, 15, 29, 30].

In this paper we focus on one such parabolic method, known as the line bundle
mean curvature flow. Fix a background form $\alpha_{0}\in[\alpha]$, and define
$\alpha_{t}:=\alpha_{0}+i\partial\bar{\partial}\phi_{t}$. At any point $p\in
X$ one can choose coordinates so $\omega^{-1}\alpha_{t}$ is diagonal with
eigenvalues $\\{\lambda_{1},...,\lambda_{n}\\}$. The line bundle mean
curvature flow can be expressed as

(1.2) $\dot{\phi}_{t}=\sum_{k}{\rm arctan}(\lambda_{k})-\hat{\theta},$

where $\hat{\theta}$ is some choice of a lift of $e^{i\hat{\theta}}$ to
$\mathbb{R}$. This parabolic flow is the complex analogue of the Lagrangian
mean curvature flow in the graphical setting, with the distinction being that
the mean curvature flow is given by eigenvalues of the real Hessian of a
function, as opposed to the complex Hessian (we direct the reader to [21, 22,
23, 31, 32] for further background on the Lagrangian case). By the complex
formulation of arctan, one sees $\sum_{k}{\rm arctan}(\lambda_{k})$ is the
argument of the top dimensional form $(\omega+i\alpha)^{n}$, and so solutions
to (1.1) are fixed points of (1.2). We denote this argument by
$\Theta(\alpha_{t})=\sum_{k}{\rm arctan}(\lambda_{k})$.

Developed by the second author and S.-T. Yau, the flow (1.2) was used to prove
existence of a solution to (1.1) under the assumption of hypercritical phase,
defined by $\Theta(\alpha_{0})>(n-1)\frac{\pi}{2}$, in addition to the
assumption that $(X,\omega)$ has non-negative orthogonal bisectional curvature
[15]. The phase assumption is useful for two reasons. First, it ensures
convexity of the operator $\Theta(\cdot)$. Second, it allows a natural choice
of a lift of $\hat{\theta}$, which is a priori defined up to a multiple of
$2\pi$. In fact, being able to choose such a lift is a major difficulty in the
study of (1.1), and one would not expect the flow to converge without making
the appropriate choice of a lift at the start.

Given the cohomological obstructions to the existence of solutions to (1.1)
from [3], it is evident that the flow (1.2) can not converge if one choses an
initial, unstable class. However, it was previously not known if the flow
exists for all time, or if a finite time singularity could occur. The first
goal of our paper is to construct an explicit example of a finite time
singularity, ruling out long time existence.

######  Theorem 1.1.

Let $X$ be the blowup of $\mathbb{P}^{n}$ at a point. There exists a Kähler
form $\omega$, and cohomology class $[\alpha]\in H^{1,1}(X,\mathbb{R})$
admitting a representative $\alpha_{0}$, for which the flow 1.2 achieves a
finite-time singularity. Specifically, if $\lambda_{Max}(p,t)$ denotes the
largest eigenvalue of $\omega^{-1}\alpha_{t}$ at a point $p\in X$, then there
exists a sequence of points $\\{x_{k}\\}\subset X$ and times $t_{k}\rightarrow
T<\infty$ such that

$\lim_{k\rightarrow\infty}\lambda_{Max}(x_{k},t_{k})=\infty.$

This example is constructed using a particular type of symmetry on $X$, called
Calabi Symmetry, which is described in Section 2. The symmetry allows the dHYM
equation to be written as an ODE, and the flow (1.2) is reduced to a parabolic
PDE with one spacial variable. Due to similarities with the curve shortening
flow, we construct subsolutions which, along with a particular choice of an
initial condition, force a singularity to happen. Our example can be
constructed on classes that admit a solution to the dHYM equation,
demonstrating that finite time singularities can not be ruled out by class
conditions alone. In fact, we believe similar examples of finite time
singularities can be constructed on any pair of classes $[\omega]$ and
$[\alpha]$ on the blowup of $\mathbb{P}^{n}$. Thus finite time singularities
will remain an integral part of the study of 1.2, and will need to be ruled
out by choosing appropriate initial conditions.

Our next goal is to demonstrate that the flow can also become singular at
infinite time, and to find an example where we can predict exactly where this
singularity will occur from the initial classes $[\alpha]$ and $[\omega]$.
Using the same Calabi Symmetry setup as above, we first show that if the
initial form satisfies supercritical phase then the flow exists for all time:

######  Theorem 1.2.

Let $(X,\omega)$ be the blowup of $\mathbb{P}^{n}$ at a point, $n\geq 3$, and
consider a class $[\alpha]\in H^{1,1}(X,\mathbb{R})$. Assume $\omega$,
$\alpha_{0}\in[\alpha]$ have Calabi-symmetry, and furthermore assume
$\alpha_{0}$ has supercritical phase, that is
$\Theta(\alpha_{0})>(n-2)\frac{\pi}{2}$. Then the flow (1.2) beginning at
$\alpha_{0}$ exists for all time.

Note that Takahashi proved long time existence for the line bundle mean
curvature flow in the hypercritical phase case [29], which implies that
$\alpha_{t}$ stays a Kähler form and the operator $\Theta(\cdot)$ is convex.
For our result we use the weaker supercritical phase assumption, which does
not imply convexity of the operator is $\Theta(\cdot)$. However, it does imply
the level sets are convex [3], which is enough to apply Evans-Krylov.

To see where the long time singularity occurs, we turn to the conjectured
relationship between solutions to the dHYM equation and stability. Following
the work of Lejmi-Székelyhidi on the $J$-equation [17], the second author,
along with T.C. Collins and S.-T. Yau, integrated a certain positivity
condition along subvarieties to develop a necessary class condition for
existence, and conjectured it was a sufficient condition as well [3].
Specifically, for any irreducible analytic subvariety $V\subseteq X$, define
the complex number

$Z_{[\alpha][\omega]}(V):=-\int_{V}e^{-i\omega+\alpha},$

where by convention we only integrate the term in the expansion of order ${\rm
dim}(V)$. Under the supercritical phase assumption $Z_{[\alpha][\omega]}(X)$
lies in the upper half plane $\mathbb{H}$. The conjecture of Collins-J.-Yau
posits that a solution to the dHYM equation exists if and only if

(1.3) $\pi>{\rm arg}Z_{[\alpha][\omega]}(V)>{\rm arg}Z_{[\alpha][\omega]}(X).$

Later, when $n=3$, Collins-Xie-Yau demonstrated a necessary Chern number
inequality [6] (which has since been extended to $n=4$ [11]), which is also
useful for defining the lifted angle $\hat{\theta}$ algebraically. Collins-Yau
further conjectured that such a Chern number inequality in higher dimension
was needed [7]. Indeed, recently when $n=3$ an example was found where the
stability inequality (1.3) holds, but the Chern number inequality does not,
and no solution to the dHYM equation exists [33]. We note that slightly weaker
versions of the Collins-J.-Yau conjecture have been solved by Chen [2]
(assuming uniform stability), and Chu-Lee-Takahashi [8] (in the projective
case). These results all rest on the supercritical phase assumption.

Some of the few results without the supercritical phase assumption are due to
the second author and Sheu [14] (and later [13]), who take advantage of the
same Calabi-Symmetry used in this paper. They demonstrate that the
inequalities (1.3) can be reinterpreted as stating whether two points in
$\mathbb{C}$ lie on the same level set of a harmonic polynomial, from which it
follows that solutions to the dHYM exists. Since the stability conditions are
necessary, and supercritical phase impiles long time existence of the flow in
this setting, the unstable case ends up being the perfect setup to construct a
singularity for $\alpha_{t}$ as $t$ approaches infinity. In particular we
demonstrate:

######  Theorem 1.3.

Let $(X,\omega)$ be the blowup of $\mathbb{P}^{n}$ at a point, $n\geq 3$.
There exists classes $[\alpha]$ and $[\omega]$, which are semi-stable in the
sense of (1.3), where the flow (1.2) starting at an initial representative
$\alpha_{0}$ exists for all time and becomes singular at time $t=\infty$ along
the destabilizing subvariety.

Similar to the proof of Theorem 1.1, we utilize explicit subsolutions of a
modified curve shortening flow to force a singularity at infinite time.

Here we briefly discuss two other parabolic flows in the literature for which
solutions to (1.1) are fixed points. The first is the tangent Lagrangian phase
flow, introduced by Takahashi in [30]. Defined by $\dot{\phi}_{t}={\rm
tan}(\Theta(\alpha_{t})-\hat{\theta}),$ this flow is the the gradient flow of
the Kempf-Ness functional arising from the infinite dimensional GIT picture
for stability and the dHYM equation, as developed by Collins-Yau [7]. As a
result, this flow is well behaved with respect to many important functionals,
and it could be useful when exploring if some type of limiting Harder-
Narasimhan filtration exists in the unstable case. One downside of this flow
is that it is only defined for “almost calibrated” potentials, when the angle
$\Theta(\alpha_{t})$ varies from the target angle by less than
$\frac{\pi}{2}$. The second flow was introduced by Fu-Yau-Zhang and is defined
by the equation $\dot{\phi}_{t}={\rm
cot}(n\frac{\pi}{2}-\Theta(\alpha_{t}))-{\rm
cot}(n\frac{\pi}{2}-\hat{\theta})$ [10]. This flow has the advantage that
${\rm cot}(n\frac{\pi}{2}-\Theta(\alpha_{t}))$ is concave under the
supercritical phase assumption. Additionally the form of the flow allows for
some useful estimates for subsolutions. However, this flow is only defined for
supercritical phase, since otherwise one may end up taking the cotangent of
zero. Note that in comparison to the above flows, the line bundle mean
curvature flow is always defined for short time.

The singularity examples we construct point towards many new interesting
problems to explore. One question is whether similar singularities can be
constructed on more general Kähler manifolds, perhaps with some sort of gluing
technique. We also wonder if there are any examples which can be related to
singularities of the graphical Lagrangian mean curvature flow, given the
formal similarities between the two flows. In the above examples, the highest
eigenvalue of $\omega^{-1}\alpha_{t}$ approaches infinity while the
derivatives of the eigenvalues stay bounded. Thus the analogue of the
graphical “Lagrangian” (given by one derivative of the potential) is tilting
up to achieve vertical slope. It would be interesting if one could find
examples with higher order singularities, which would allow for a richer
blowup analysis. Finally, in our long time singularity example, the
singularity occurs along the destabilizing subvariety, and one would expect
this relationship between stability and singularity formation to hold in more
general settings. We hope to explore these problems in future work.

The paper is organized as follows. In Section 2 we introduce the Calabi
symmetry assumption, which is used in constructing our examples. In Section 3
we construct our finite time singularity. Section 4 contains our proof of long
time existence in the supercritical phase case. We conclude with our example
of a long time singularity in Section 5.

Acknowledgements. This problem arose at the American Institute of Mathematics
workshop “Stability in mirror symmetry,” and the second author would like to
thank the organizers and the institute for providing a productive research
environment. In particular special thanks to Tristan C. Collins, Jason Lotay,
and Felix Schulze for some helpful discussion. This work was funded in part by
a Simons collaboration grant.

##  2\. Calabi Symmetry

Throughout this paper we work on the blowup of $\mathbb{P}^{n}$ at one point
$p$, which we denote by $X$. This manifold admits $(1,1)$ forms that satisfy a
symmetry ansatz, originally defined by Calabi in [1]. We include a short
introduction to this ansatz here, and direct the reader to [1, 9, 14, 24, 25,
26, 27, 28] for further details.

Let $E$ denote the exceptional divisor, and $H$ the pullback of the hyperplane
divisor from $\mathbb{P}^{n}$. These two divisors span
$H^{1,1}(X,\mathbb{R})$, and any Kähler class will lie in $a_{1}[H]-a_{2}[E]$
with $a_{1}>a_{2}>0$. Normalizing, assume $X$ admits a Kähler form $\omega$ in
the class

$[\omega]=a[H]-[E],$

with $a>1$. Furthermore, assume our class $[\alpha]$ satisfies

$[\alpha]=p[H]-q[E],$

for a choice of $p,q\in\mathbb{R}$.

On $X\backslash(H\cup E)\cong\mathbb{C}^{n}\backslash\\{0\\}$, set $\rho={\rm
log}(|z|^{2})$. If $u(\rho)\in C^{\infty}(\mathbb{R})$ satisfies
$u^{\prime}(\rho)>0,$ $u^{\prime\prime}(\rho)>0$, then
$\omega=i\partial\bar{\partial}u$ defines a Kähler form on
$\mathbb{C}^{n}\backslash\\{0\\}$. For $\omega$ to extend to a Kähler form on
$X$ in the class $a[H]-[E]$, $u$ must satisfy boundary asymptotics.
Specifically, define $U_{0},U_{\infty}:[0,\infty)\rightarrow\mathbb{R}$ via

$U_{0}(r):=u({\rm log}r)-{\rm log}r\qquad{\rm and}\qquad
U_{\infty}(r):=u(-{\rm log}r)+a{\rm log}r.$

Assume $U_{0}$ and $U_{\infty}$ extend by continuity to smooth functions at
$r=0$, with both $U_{0}^{\prime}(0)>0$ and $U_{\infty}^{\prime}(0)>0$. This
fixes the asymptotic behavior of $u$

$\lim_{\rho\rightarrow-\infty}u^{\prime}(\rho)=1,\qquad\lim_{\rho\rightarrow\infty}u^{\prime}(\rho)=a,$

and ensures $\omega=i\partial\bar{\partial}u$ extends to a Kähler form on $X$
in the correct class.

Next, given a function $v(\rho)\in C^{\infty}(\mathbb{R})$, the Hessian
$i\partial\bar{\partial}v(\rho)$ defines a $(1,1)$ form $\alpha$ on
$\mathbb{C}^{n}\backslash\\{0\\}$. In order for $\alpha$ to extend to $X$ in
the class $[\alpha]=p[H]-q[E]$, we require similar asymptotics without the
positivity assumptions, as $\alpha$ need not be a Kähler form. Consider the
functions $V_{0},V_{\infty}:[0,\infty)\rightarrow\mathbb{R}$ defined via

$V_{0}(r):=v({\rm log}r)-q{\rm log}r\qquad{\rm and}\qquad
V_{\infty}(r):=v(-{\rm log}r)+p{\rm log}r.$

Assume that $V_{0}$ and $V_{\infty}$ extend by continuity to smooth functions
at $r=0$, which implies $v(\rho)$ satisfies:

$\lim_{\rho\rightarrow-\infty}v^{\prime}(\rho)=q,\qquad\lim_{\rho\rightarrow\infty}v^{\prime}(\rho)=p.$

Then $i\partial\bar{\partial}v$ extends to a smooth (1,1) form on $X$ in the
class $[\alpha]$.

We refer to forms $\omega$ and $\alpha$ constructed in the above manner as
having Calabi Symmetry. Restricting to $\mathbb{C}^{n}\backslash\\{0\\}$, one
can check that in this case the eigenvalues of $\omega^{-1}\alpha$ are
$\frac{v^{\prime}}{u^{\prime}}$ with multiplicity $(n-1)$, and
$\frac{v^{\prime\prime}}{u^{\prime\prime}}$ with multiplicity one (for a proof
of this see [9]). Furthermore, because $u^{\prime\prime}>0$, the first
derivative $u^{\prime}$ is monotone increasing, allowing us to use Legendre
transform coordinates and view $u^{\prime}$ as a real variable, denoted by
$x$, which ranges from $1$ to $a$. One can then write $v^{\prime}$ as a graph
$f$ over $x\in(1,a)$, so we have $f(x)=f(u^{\prime}(\rho))=v^{\prime}(\rho)$.
Taking the derivative of both sides with respect to $\rho$ gives

$f^{\prime}(x)u^{\prime\prime}(\rho)=v^{\prime\prime}(\rho).$

We allow the slight abuse of notation where $f^{\prime}$ denotes the
derivative of $f$ with respect to the variable $x$, and $u^{\prime\prime}$ and
$v^{\prime\prime}$ denote the second derivatives with respect to the variable
$\rho$. By the above, the eigenvalues of $\omega^{-1}\alpha$ are

$\frac{v^{\prime}}{u^{\prime}}=\frac{f}{x}\,(\text{with
multiplicity}\,n-1)\qquad{\rm
and}\qquad\frac{v^{\prime\prime}}{u^{\prime\prime}}=f^{\prime}.$

As $x\rightarrow 1$, we have $\rho\rightarrow-\infty$, while $x\rightarrow a$
implies $\rho\rightarrow\infty$. Thus the asymptotics of $v(\rho)$ imply

$\lim_{x\rightarrow 1^{+}}f(x)=q,\qquad\lim_{x\rightarrow a^{-}}f(a)=p,$

and we extend $f(x)$ to the boundary $[1,a]$ by continuity.

In this form, the dHYM equation can be written as an ODE

(2.1)
$\displaystyle\textnormal{Im}\left(e^{-i\hat{\theta}}\left(1+i\frac{f}{x}\right)^{n-1}\left(1+if^{\prime}\right)\right)=0$

subject to the boundary constraints $f(1)=q$, $f(a)=p$. Furthermore the
Lagrangian angle given by the eigenvalues of $\omega^{-1}\alpha$ can be
expressed as

$\Theta(x):=(n-1)\arctan\left(\frac{f}{x}\right)+\arctan\left(f^{\prime}\right).$

Because $\alpha=i\partial\bar{\partial}v$, in our setting we can write line
bundle mean curvature flow as

$\dot{v}=\Theta(x)-\hat{\theta}.$

In order to arrive at an equation for $f$ we take the derivative of both sides
with respect to $\rho$ and see

$\frac{d\dot{v}}{d\rho}=\frac{d\Theta}{dx}\frac{dx}{d\rho}.$

This now becomes

(2.2)
$\displaystyle\dot{f}=L(f):=u^{\prime\prime}\left(\frac{f^{\prime\prime}}{1+f^{\prime
2}}+(n-1)\frac{xf^{\prime}-f}{x^{2}+f^{2}}\right)=u^{\prime\prime}\Theta^{\prime}.$

We have now defined second order parabolic equation for $f$, to which a
solution can be integrated in $\rho$ to arrive at a solution of the line
bundle mean curvature flow (1.2). Note that
$u^{\prime\prime}(1)=u^{\prime\prime}(a)=0$, so the flow fixes the boundary
values of $f$. One interesting consequence of taking an extra derivative to
define the flow is that it is no longer necessary to take a lift
$\hat{\theta}$ of the average angle. In this way the flow is more analogous to
a how graph evolves by the mean curvature vector rather than how a potential
evolves by the Lagrangian angle.

The flow (2.2) is defined on graphs over $[1,a]$ with fixed boundary. However,
we can generalize it to curves, which is useful in order to construct
barriers. Consider the region $D:=\\{(x,y)\in\mathbb{R}^{2}\,|\,1\leq x\leq
a\\}$. Let $\gamma_{t}(s):I\subseteq\mathbb{R}\to D$ be a family of smooth
curves, and let $s$ be the arc-length parameter. Let

$\kappa=\frac{d}{ds}\arctan{\gamma^{\prime}}$

denote the usual plane curvature, and let

$\xi=\frac{d}{ds}\arctan{\gamma}$

be an extrinsic quantity. Consider the flow

(2.3)
$\displaystyle\dot{\gamma}=u^{\prime\prime}(\gamma)\left(\kappa+(n-1)\xi\right)\textbf{N},$

where the normal vector N is defined by $e^{i\frac{\pi}{2}}\gamma^{\prime}$,
and $u^{\prime\prime}(\gamma)$ is defined to be the function
$u^{\prime\prime}$ applied to the $x$-coordinate of $\gamma$. Notice the
relationship between this flow and the curve shortening flow
$\dot{\gamma}=\kappa\textbf{N}.$

In the case where $\gamma(x)=(x,f(x))$ is a graph of a function, we have
$ds=\sqrt{1+f^{\prime 2}}dx$. Simple computations show

$\langle\dot{\gamma},\textbf{N}\rangle=\frac{\dot{f}}{\sqrt{1+f^{\prime
2}}},\qquad\kappa=\frac{f^{\prime\prime}}{(1+f^{\prime
2})^{3/2}},\qquad\xi=\frac{1}{\sqrt{1+f^{\prime
2}}}\frac{xf^{\prime}-f}{x^{2}+f^{2}}.$

Hence, (2.3) reduces to (2.2) in this case, and thus is the correct
generalization to curves.

##  3\. A finite time singularity

Consider a real number $R>1$ (to be determined later), and set $a=6R$. As
above, consider a function $u:\mathbb{R}\rightarrow\mathbb{R}$ so that
$\omega:=i\partial\bar{\partial}u$ extends from
$\mathbb{C}^{n}\backslash\\{0\\}$ to a Kähler form on $X$ in the class
$a[H]-[E]$. Furthermore, assume that $u^{\prime\prime}<R$, and that there
exists a small constant $k$ so that $u^{\prime\prime}(x)\geq k(x-1)(a-x)$ on
$[1,a]$ (which is possible since by the Calabi symmetry assumptions
$u^{\prime\prime}(x)$ approaches the boundary linearly in $x$). We remark that
one should be able to construct a similar singularity example for any Kähler
form satisfying Calabi symmetry, however we include our extra assumptions on
$u$ and $R$ for the ease of presentation.

The idea is as follows. Consider a class $[\alpha]=p[H]-q[E]$ and assume
$p\geq a$. Define a representative $\alpha_{0}$ via the function $f_{0}(x)$,
which has a graph such as in Figure 1. We construct a family of shrinking
circles, and a traveling family of hyperbolas, which are subsolutions to
(2.3). If $f_{t}$ is the evolution of $f_{0}$ via the line bundle MCF (2.2),
and $f_{0}$ avoids the initial circle and hyperbola at time $t=0$, then by the
maximum principle $f_{t}$ must avoid these families for all time. The
hyperbolas push out past the center of the circles before they shrink to a
point, forcing $f_{t}$ to achieve vertical slope at some finite time.

Figure 1. The graph of a function $f_{0}$ which forms a singularity.

We first construct our family of hyperbolas. Observe that both $\kappa$ and
$\xi$ are invariant under orthogonal transformation. Hence, by interchanging
the $x$ and $y$ coordinates, we have the following lemma.

######  Lemma 3.1.

Suppose $y=f_{t}(x)$ satisfies the flow (2.2). If the inverse
$x=f_{t}^{-1}(y)=:h_{t}(y)$ exists, then $h_{t}(y)$ satisfies

(3.1)
$\displaystyle\dot{h}=u^{\prime\prime}(h(y))\left(\frac{h^{\prime\prime}}{1+h^{\prime
2}}+(n-1)\frac{yh^{\prime}-h}{y^{2}+h^{2}}\right).$

######  Lemma 3.2.

Suppose $b(t):[0,T)\to\mathbb{R}$ satisfies the initial value problem:

(3.2)
$\displaystyle\dot{b}=-\frac{kb_{\infty}(b_{\infty}-1)(a^{2}-b_{0}^{2})(b-b_{\infty})b^{3}}{a(a^{2}-b_{\infty}^{2})(2a^{2}-b_{\infty}^{2})^{2}}$

where $1<b_{\infty}<b_{0}<a$ are constants. Then

$g_{t}(y):=\sqrt{\frac{a^{2}-b^{2}}{a^{2}-b_{\infty}^{2}}y^{2}+b^{2}}$

is a sub-solution to the equation 3.1 for
$y\in\left[-\sqrt{a^{2}-b_{\infty}^{2}},\sqrt{a^{2}-b_{\infty}^{2}}\right]$.

###### Proof.

For simplicity, write

$m=\frac{a^{2}-b^{2}}{a^{2}-b_{\infty}^{2}},\qquad
1-m=\frac{b^{2}-b_{\infty}^{2}}{a^{2}-b_{\infty}^{2}}.$

We also write $g=g_{t}$ for notational simplicity. Notice that $b_{0}\geq
b>b_{\infty}$ from the initial value problem, so $m<1$. We compute

$g^{\prime}=\frac{my}{\sqrt{my^{2}+b^{2}}}=\frac{my}{g},$

which in turn gives

$g^{\prime\prime}=\frac{m}{g}-\frac{myg^{\prime}}{g^{2}}=\frac{m}{g}-\frac{m^{2}y^{2}}{g^{3}}=\frac{mg^{2}-m^{2}y^{2}}{g^{3}}=\frac{mb^{2}}{g^{3}}.$

Furthermore the two expressions from (3.1) can be written as

$\frac{g^{\prime\prime}}{1+g^{\prime 2}}=\frac{mb^{2}}{g(g^{2}+m^{2}y^{2})},$

and

$\frac{yg^{\prime}-g}{y^{2}+g^{2}}=\frac{my^{2}-g^{2}}{g(y^{2}+g^{2})}=\frac{-b^{2}}{{g(g^{2}+y^{2})}}.$

Thus

$\displaystyle L(g)$
$\displaystyle:=u^{\prime\prime}(g(y))\left(\frac{g^{\prime\prime}}{1+g^{\prime
2}}+(n-1)\frac{yg^{\prime}-g}{y^{2}+g^{2}}\right)$
$\displaystyle=u^{\prime\prime}(g(y))\left(\frac{mb^{2}}{g(g^{2}+m^{2}y^{2})}-(n-1)\frac{b^{2}}{{g(g^{2}+y^{2})}}\right)$
$\displaystyle\leq
u^{\prime\prime}(g(y))\left(\frac{mb^{2}}{g(g^{2}+m^{2}y^{2})}-\frac{b^{2}}{{g(g^{2}+y^{2})}}\right)$
$\displaystyle=u^{\prime\prime}(g(y))\frac{-(1-m)b^{4}}{g(g^{2}+m^{2}y^{2})(g^{2}+y^{2})}$
$\displaystyle\leq
k(g-1)(a-g)\frac{-(1-m)b^{4}}{g(g^{2}+m^{2}y^{2})(g^{2}+y^{2})}\leq 0,$

where the last inequality follows from our assumption $u^{\prime\prime}(x)\geq
k(x-a)(a-x)$.

Now, observe that

$(a-g)(a+g)=a^{2}-my^{2}-b^{2}=m\left(\frac{a^{2}-b^{2}}{m}-y^{2}\right)=m(a^{2}-b_{\infty}^{2}-y^{2}).$

The right hand side is non-negative when
$y\in\left[-\sqrt{a^{2}-b_{\infty}^{2}},\sqrt{a^{2}-b_{\infty}^{2}}\right]$.
As a result

$\displaystyle L(g)$
$\displaystyle\leq\frac{-k(g-1)m(a^{2}-b_{\infty}^{2}-y^{2})(1-m)b^{4}}{g(a+g)(g^{2}+m^{2}y^{2})(g^{2}+y^{2})}$
$\displaystyle=\frac{-k(a^{2}-b_{\infty}^{2}-y^{2})(g-1)(a^{2}-b^{2})(b+b_{\infty})(b-b_{\infty})b^{4}}{g(a+g)(a^{2}-b_{\infty}^{2})^{2}(g^{2}+m^{2}y^{2})(g^{2}+y^{2})},$

where we plugged in the definition of $m$ and $(1-m)$. Because the above
expression is negative, the inequalities $m<1$, $b_{\infty}\leq g\leq a$, and
$b_{\infty}<b\leq b_{0}$, allow us to conclude

$L(g)\leq\frac{-k(a^{2}-b_{\infty}^{2}-y^{2})(b_{\infty}-1)(a^{2}-b_{0}^{2})2b_{\infty}(b-b_{\infty})b^{4}}{g(2a)(a^{2}-b_{\infty}^{2})^{2}(2a^{2}-b_{\infty}^{2})^{2}}.$

Next, we turn to the evolution of $g$:

$\displaystyle\dot{g}=\frac{\dot{m}y^{2}+2b\dot{b}}{2g}=\frac{\frac{-2b\dot{b}}{a^{2}-b_{\infty}^{2}}y^{2}+2b\dot{b}}{2g}$
$\displaystyle=\frac{b\dot{b}}{g}\left(1-\frac{y^{2}}{a^{2}-b_{\infty}^{2}}\right)$
$\displaystyle=\frac{b\dot{b}(a^{2}-b_{\infty}^{2}-y^{2})}{g(a^{2}-b_{\infty}^{2})}\leq
0.$

Putting everything together we arrive at

$\displaystyle\dot{g}-L(g)\geq\frac{b(a^{2}-b_{\infty}^{2}-y^{2})}{g(a^{2}-b_{\infty}^{2})}\left(\dot{b}+\frac{k(b_{\infty}-1)(a^{2}-b_{0}^{2})b_{\infty}(b-b_{\infty})b^{3}}{a(a^{2}-b_{\infty}^{2})^{2}(2a^{2}-b_{\infty}^{2})}\right).$

The right hand side is zero by the initial value problem. Hence we have
demonstrated $\dot{g}-L(g)\geq 0$.

∎

We now solve the initial value problem (3.2), and compute the time for which
the hyperbola pushes out a specified distance. Set $b_{\infty}=R$, and
$b_{0}=5R$. Recall $a=6R$, so $1<b_{\infty}<b_{0}<a$. Note there exists a
constant $M>0$ such that

$C_{1}:=k\frac{b_{\infty}(b_{\infty}-1)(a^{2}-b_{0}^{2})}{a(a^{2}-b_{\infty}^{2})(2a^{2}-b_{\infty}^{2})^{2}}\geq\frac{1}{MR^{3}}.$

The differential equation 3.2 is separable, yielding

$-C_{1}dt=\frac{db}{(b-b_{\infty})b^{3}},$

which has the solution

$-C_{1}t+C_{0}=\frac{b_{\infty}^{2}+2b^{2}\log(b-b_{\infty})+2b_{\infty}b-2b^{2}\log(b)}{2b_{\infty}^{3}b^{2}}$

where $C_{0}$ is given by the initial value $b(0)=b_{0}=5R$. Plugging in $t=0$
we see directly that

$C_{0}=\frac{11/50+{\rm log}(4/5)}{R^{3}}.$

Let $T$ be the time such that $b(T)=2R$. Then, we have

(3.3) $\displaystyle T=\frac{1}{C_{1}}\left(\frac{11/50+{\rm
log}(4/5)}{R^{3}}-\frac{5/8-{\rm log}(2)}{R^{3}}\right)\leq A$

for some constant $A$.

######  Proposition 3.3.

Let $\gamma(t)$ satisfy (2.3). If $\gamma(0)$ does not intersect the hyperbola
$g_{0}(y)$, then $\gamma(t)$ does not intersect $g_{t}(y)$ for as long as the
flow is defined.

###### Proof.

Suppose $p=(x_{0},y_{0})$ is the first point of intersection of the two
curves, occurring at time $t_{0}$. Since the hyperbola $g_{t}$ never achieves
horizontal slope, we can assume near $p$ that $\gamma(t)$ is a graph of a
function $h_{t}(y)$ over the ball $B_{\delta}(y_{0})$ in $y$-axis solving
(3.1). Without loss of generality, for $0\leq t<t_{0}$ assume that
$g_{t}(y)>h_{t}(y)$ over $B_{\delta}(y_{0})$. Then over the region
$B_{\delta}(y_{0})\times[0,t_{0}),$ we see $(\frac{d}{dt}-L)(g-h)\geq 0$, yet
$g-h>0$ on the parabolic boundary. The result follows from the maximum
principle. ∎

Next we turn to the family of shrinking circles which act as a barrier. Since
$\xi$ is relatively small for a curve far away from the origin, (2.3) behaves
similarly to the curve shortening flow in this case. The idea is to consider a
family of circles far away from the origin which evolve slightly faster than
curve shortening flow, in order to absorb the small $\xi$ term.

######  Proposition 3.4.

For $R=a/6>1$ as above, assume the graph of $f_{0}(x)$ does not intersect the
ball $B_{R}(3R,y_{0})$. Then, for $y_{0}$ sufficiently negative, the family of
shrinking balls $B_{\sqrt{R^{2}-4Rt}}(3R,y_{0})$ does not intersect the family
of graphs of $f_{t}(x)$ evolving via (2.2), as long as the flow is defined.

###### Proof.

Locally, we can write $\phi_{t}(x)=-\sqrt{r(t)^{2}-(x-3R)^{2}}+y_{0}$ as
equation representing the lower boundary of the shrinking balls, where
$r(t)=\sqrt{R^{2}-4Rt}$. Direct computation gives

$\displaystyle u^{\prime\prime}\frac{\phi^{\prime\prime}}{1+\phi^{\prime
2}}-\dot{\phi}$
$\displaystyle=\frac{u^{\prime\prime}-2R}{\sqrt{r^{2}-(x-3R)^{2}}}<\frac{-R}{\sqrt{r^{2}-(x-3R)^{2}}},$

since by assumption $u^{\prime\prime}<R$. Suppose $t=t_{0}$ is the first time
the graph of $\phi_{t}$ intersects $f_{t}$ from above at a point $x_{0}$. At
this point of intersection we have
$f^{\prime}_{t_{0}}(x_{0})=\phi^{\prime}_{t_{0}}(x_{0})$,
$\dot{f}_{t_{0}}(x_{0})\geq\dot{\phi}_{t_{0}}(x_{0})$, and we can assume
$f_{t}(x)<\phi_{t}(x)$ for all $t<t_{0}$, and so
$f^{\prime\prime}_{t_{0}}(x_{0})\leq\phi^{\prime\prime}_{t_{0}}(x_{0})$. Then
at $t=t_{0}$, $x=x_{0}$, we have

$\displaystyle\dot{f}-\dot{\phi}$
$\displaystyle=-\dot{\phi}+u^{\prime\prime}\left(\frac{f^{\prime\prime}}{1+f^{\prime
2}}+(n-1)\frac{x_{0}f^{\prime}-f}{x_{0}^{2}+f^{2}}\right)$
$\displaystyle\leq-\dot{\phi}+u^{\prime\prime}\left(\frac{\phi^{\prime\prime}}{1+\phi^{\prime
2}}+(n-1)\frac{x_{0}\phi^{\prime}-\phi}{x_{0}^{2}+\phi^{2}}\right)$
$\displaystyle<-\frac{R}{\sqrt{r^{2}-(x_{0}-3R)^{2}}}+u^{\prime\prime}(n-1)\frac{x_{0}\phi^{\prime}-\phi}{x_{0}^{2}+\phi^{2}}.$

To achieve a contradiction we need to show that for $y_{0}$ sufficiently
negative the right hand side above is negative. To control the $\phi^{\prime}$
term we can compute directly

$-\frac{R}{\sqrt{r^{2}-(x_{0}-3R)^{2}}}+\frac{u^{\prime\prime}(n-1)x_{0}\phi^{\prime}}{x_{0}^{2}+\phi^{2}}=\frac{-R(x_{0}^{2}+\phi^{2})+(n-1)u^{\prime\prime}(x_{0}-3R)}{(x_{0}^{2}+\phi^{2})\sqrt{r^{2}-(x_{0}-3R)^{2}}}.$

Recall that by assumption $u^{\prime\prime}<R$. Choose $y_{0}$ sufficiently
negative to ensure
$-(x_{0}^{2}+\phi^{2})+(n-1)(x_{0}-3R)\leq-\frac{1}{2}(x_{0}^{2}+\phi^{2})$.
Then

$-\frac{R}{\sqrt{r^{2}-(x_{0}-3R)^{2}}}+\frac{u^{\prime\prime}(n-1)x_{0}\phi^{\prime}}{x_{0}^{2}+\phi^{2}}\leq\frac{-R}{2\sqrt{r^{2}-(x_{0}-3R)^{2}}}<-\frac{1}{2}$

since $r<R$. We have now demonstrated that

$\dot{f}-\dot{\phi}<-\frac{1}{2}-\frac{u^{\prime\prime}(n-1)\phi}{x_{0}^{2}+\phi^{2}}.$

The function $\phi$ is negative, so the second term on the right hand side
above is positive. However, we can choose $y_{0}$ sufficiently negative so
that this term is less than $\frac{1}{2}$, and the result follows. ∎

We now demonstrate the existence of a singularity using our two subsolutions
constructed above.

Figure 2. The maximum principle forces $f_{t}$ to achieve vertical slope.

For $R>1$, set $b_{\infty}=R$, $b_{0}=5R$, and $a=6R.$ Consider the circle of
radius $R$ centered around $(3R,y_{0})$, with $y_{0}$ sufficiently negative so
that the hypothesis of Proposition 3.4 is satisfied. The right side of the
circle lies on the line $x=4R$. Note that the vertex of the hyperbola
$g_{0}(y)$ lies on the line $x=b_{0}=5R$. Furthermore, the hyperbola
intersects $x=a$ at $y=\pm\sqrt{35R^{2}}$. Since $p>a=6R,$ we see $(a,p)$ lies
above the top of the hyperbola $g_{0}(y)$. Thus, it is possible to choose a
function $f_{0}:[1,a]\rightarrow\mathbb{R}$ with $f_{0}(1)=q$, $f_{0}(a)=p$,
such that $f_{0}$ goes below $B_{R}(3R,y_{0})$, then increases above the
hyperbola $g_{0}(y)$ before arriving at $(a,p)$.

Let $f_{t}(x)$ be the solution of (2.2) starting at $f_{0}(t)$. By Proposition
3.3 and Proposition 3.4, $f_{t}(x)$ can not intersect $g_{t}(y)$ nor
$B_{\sqrt{R^{2}-4Rt}}(3R,y_{0})$ as long as the flow is defined. Note it takes
time $t=R/4$ for $B_{\sqrt{R^{2}-4Rt}}(3R,y_{0})$ to shrink a point. Also, if
$T$ is the time the hyperbola $g_{T}(y)$ has pushed out to the line $x=2R$, as
we have seen by (3.3) there exists a constant $A$ such that $T\leq A$. Hence,
choose $R$ large enough to ensure $A<R/4$ and thus $T<R/4$, which implies the
hyperbola will push past the center of the shrinking circles before they
completely disappear. This forces $f_{t}$ to first have a vertical tangency,
as illustrated in Figure 2, demonstrating the existence of a finite time
singularity and proving Theorem 1.1.

##  4\. Long time existence

The example above shows that a finite-time singularity for the flow (2.2) can
occur in the interior of the interval $(1,a)$. In particular, one can not
always expect $\sup_{(1,a)}|f_{t}^{\prime}(x)|$ to stay bounded for finite
time. The main goal of this section is to rule out a finite time singularity
if one chooses a sufficiently nice initial function $f_{0}(x)$, specifically
where the corresponding $(1,1)$ form $\alpha_{0}$ has supercritical phase.
This is an important step towards the construction of a singularity at
infinite time.

As a first step we show that along the flow, the first derivative
$|f_{t}^{\prime}(x)|$ stays bounded at the boundary points $x=1$ and $x=a$. In
fact, our boundary estimate does not need the supercritical phase assumption.

######  Proposition 4.1.

Suppose $f_{t}(x)$ is defined on $(t,x)\in[0,T)\times[1,a]$. Then, there
exists uniform constants $A,B$ so that

$|f_{t}^{\prime}(1)|+|f_{t}^{\prime}(a)|\leq Ae^{Bt}.$

###### Proof.

We will show $|f^{\prime}(1)|<C(T)$, as the other boundry point is treated
similarly. Consider $g_{t}(x)=q+Ae^{B(n-1)t}(x-1)$. Choose $A\gg 0$
sufficiently large to ensure both $Ae^{B(n-1)t}\geq 2\max\\{|q|,|q|^{-1}\\}$
and $f_{0}<g_{0}$ for all $x\in(1,a]$. Choose $B\gg 0$ so that
$u^{\prime\prime}<B(x-1)$. We claim that $f_{t}<g_{t}$ for all time
$t\in[0,T)$.

Suppose not, and assume the curves touch for the first time at $x=x_{0}>1$ and
$t=t_{0}$. Then, $f_{t_{0}}(x_{0})=g_{t_{0}}(x_{0})$,
$f^{\prime}_{t_{0}}(x_{0})=g^{\prime}_{t_{0}}(x_{0})$,
$f^{\prime\prime}_{t_{0}}(x_{0})\leq g^{\prime\prime}_{t_{0}}(x_{0})$, and
$\dot{f}_{t_{0}}(x_{0})\geq\dot{g}_{t_{0}}(x_{0})$. Thus, when $x=x_{0}$,
$t=t_{0}$ we have

$\displaystyle\dot{f}$
$\displaystyle=u^{\prime\prime}\left(\frac{f^{\prime\prime}}{1+f^{\prime
2}}+(n-1)\frac{x_{0}f^{\prime}-f}{x_{0}^{2}+f^{2}}\right)$ $\displaystyle\leq
B(x_{0}-1)\left(\frac{g^{\prime\prime}}{1+g^{\prime
2}}+(n-1)\frac{x_{0}g^{\prime}-g}{x_{0}^{2}+g^{2}}\right)$
$\displaystyle=B(x_{0}-1)(n-1)\frac{Ax_{0}e^{B(n-1)t_{0}}-q-Ae^{B(n-1)t_{0}}(x_{0}-1)}{x_{0}^{2}+(q+Ae^{B(n-1)t_{0}}(x_{0}-1))^{2}}$
$\displaystyle<ABe^{B(n-1)t}(x_{0}-1)(n-1)\frac{1-A^{-1}e^{-B(n-1)t_{0}}q}{1+q^{2}},$

since $x_{0}^{2}+(q+Ae^{B(n-1)t_{0}}(x_{0}-1))^{2}>1+q^{2}$. Furthermore by
assumption on $A$ we have $-A^{-1}e^{-B(n-1)t_{0}}q\leq\frac{1}{2}q^{2}$, and
so

$\frac{1-A^{-1}e^{-B(n-1)t_{0}}q}{1+q^{2}}\leq 1.$

Hence,

$\dot{f}<ABe^{B(n-1)t_{0}}(x_{0}-1)(n-1)=\dot{g},$

a contradiction. Thus $g_{t}$ serves as a barrier giving an upper bound for
the derivative $f^{\prime}(1)\leq Ae^{B(n-1)t}$. The lower bound is treated
similarly. ∎

We now turn to the case where we do have long time existence, namely when
$n\geq 3$ and $\alpha_{0}$ has supercritical phase, i.e.
$\Theta(\alpha_{0})>(n-2)\frac{\pi}{2}$.

######  Lemma 4.2.

The supercritical phase condition is preserved along the flow.

###### Proof.

On a general Kähler manifold $(X,\omega)$, set
$\omega=ig_{\bar{k}j}dz^{j}\wedge d\bar{z}^{k}$ and
$\alpha=i\alpha_{\bar{k}j}dz^{j}\wedge d\bar{z}^{k}$. Consider the metric
$\eta_{\bar{k}j}=g_{\bar{k}j}+\alpha_{\bar{k}p}g^{\bar{q}p}\alpha_{\bar{q}j}$.
By equation (5.4) in [15], the angle $\Theta(\alpha_{t})$ evolves via the heat
equation

(4.1) $\dot{\Theta}(\alpha_{t})=\Delta_{\eta}\Theta(\alpha_{t}),$

and thus the result follows from the maximum principle. ∎

######  Lemma 4.3.

If $f_{0}(x)$ satisfies the supercritical phase assumption, $f_{t}(x)>0$ for
all $t\geq 0$.

###### Proof.

Suppose there exists a time $t_{0}$ and a point $x_{0}$ where
$f_{t_{0}}(x_{0})\leq 0$. This implies arctan$\left(\frac{f}{x_{0}}\right)\leq
0$. Yet
$\Theta(x_{0}):=(n-1)\arctan\left(\frac{f}{x_{0}}\right)+\arctan\left(f^{\prime}\right),$
and so the super critical phase assumption implies

$\arctan\left(f^{\prime}\right)>(n-2)\frac{\pi}{2},$

which is impossible for $n\geq 3$. ∎

######  Lemma 4.4.

Under the supercritical phase assumption there exists a uniform constant $C$
so that $f_{t}^{\prime}(x)>-C$ for all $t\geq 0$.

###### Proof.

By the supercritical phase condition

$\arctan\left(f_{t}^{\prime}\right)>(n-2)\frac{\pi}{2}-(n-1)\arctan\left(\frac{f_{t}}{x}\right).$

Since $x\geq 1$ and $f_{t}\leq C$ by the maximum principle, there exists an
$\epsilon>0$ so that
$\arctan\left(\frac{f_{t}}{x}\right)<\frac{\pi}{2}-\epsilon$. Thus

$\arctan\left(f_{t}^{\prime}\right)>-\frac{\pi}{2}+(n-1)\epsilon.$

This gives a lower bound for $f^{\prime}_{t}$. ∎

######  Proposition 4.5.

Under the supercritical phase assumption, a solution $f_{t}(x)$ to (2.2) has
bounded first derivative for all times $T<\infty$. In particular, there exists
uniform constants $A,B$ so that

$\sup_{x\in[1,a]}|f^{\prime}_{t}(x)|\leq A(1+t)e^{Bt}.$

###### Proof.

By the previous lemma we only need an upper bound for $f_{t}^{\prime}$. By
Proposition 4.1 we have

$A^{-1}e^{-Bt}\left(|f_{t}^{\prime}(1)|+|f_{t}^{\prime}(a)|\right)\leq 1.$

As a result if $\sup_{x\in[1,a]}A^{-1}e^{-Bt}|f^{\prime}_{t}(x)|$ is large,
this supremum must be achieved at an interior point. Let $x_{0}$ be the
interior max. At this point we have $f^{\prime}_{t}(x_{0})>0$,
$f^{\prime\prime}_{t}(x_{0})=0$, and $f^{\prime\prime\prime}_{t}(x_{0})\leq
0$. By direct computation at $x_{0}$ it holds

$\displaystyle\dot{f}^{\prime}$
$\displaystyle=\frac{d}{dx}\left(u^{\prime\prime}\left(\frac{f^{\prime\prime}}{1+f^{\prime
2}}+(n-1)\frac{xf^{\prime}-f}{x+f^{2}}\right)\right)$
$\displaystyle\leq\frac{du^{\prime\prime}}{dx}(n-1)\frac{x_{0}f^{\prime}-f}{x_{0}^{2}+f^{2}}+u^{\prime\prime}\frac{d}{dx}\left(\frac{f^{\prime\prime}}{1+f^{\prime
2}}+(n-1)\frac{xf^{\prime}-f}{x^{2}+f^{2}}\right)$ $\displaystyle\leq
Cf^{\prime}+u^{\prime\prime}\left(\frac{f^{\prime\prime\prime}}{1+f^{\prime
2}}-(n-1)\frac{2(x_{0}f^{\prime}-f)(x_{0}+ff^{\prime})}{(x_{0}^{2}+f^{2})^{2}}\right),$

where we repeatedly plugged in that $f^{\prime\prime}(x_{0})=0$. Since $f$ is
positive the term $-2x_{0}f(f^{\prime})^{2}$ is negative, and thus

$\dot{f}^{\prime}\leq
Cf^{\prime}+u^{\prime\prime}2(n-1)\frac{fx_{0}+f^{2}f^{\prime}-x_{0}^{2}f^{\prime}}{(x_{0}^{2}+f^{2})^{2}}\leq
Cf^{\prime}+C$

for some constant $C.$

Now, consider the function $A^{-1}e^{-Bt}f^{\prime}_{t}(x)-Ct$. By making $B$
larger, if necessary, we can assume $B\geq C$. At an interior maximum we see

$\frac{d}{dt}\left(A^{-1}e^{-Bt}f^{\prime}-Ct\right)\leq 0,$

from which the result follows. ∎

We remark that the above proof fails when the function $f$ is not positive,
since then the term $-2x_{0}f(f^{\prime})^{2}$ is positive. Thus the best
inequality one can derive in this case is $\dot{f}^{\prime}\leq Cf^{\prime
2}$, which is certainly not enough to prevent a finite time singularity, as we
have demonstrated. We are now ready to prove our second main result.

###### Proof of Theorem 1.2.

Let $\alpha_{t}:=\alpha_{0}+i\partial\bar{\partial}\phi_{t}$, be the solution
to (1.2) starting at $\alpha_{0}$, and assume the flow is defined for
$t\in[0,T)$ for some time $T<\infty$. By proposition 4.5, all the eigenvalues
of $\omega^{-1}\alpha_{t}$ are bounded uniformly by a constant $C_{T}$. From
here the result follows from the argument outlined in Proposition 5.2 in [15].

The idea is that once the eigenvalues are bounded, the operator
$\Delta_{\eta}$ is uniformly elliptic. Given $\Theta(\alpha_{t})$ solves the
heat equation (4.1), the parabolic estimates of Krylov-Safonov ([16] Theorem
11, Section 4.2) imply $\Theta(\alpha_{t})$ is in $C^{\alpha}$ in time which
gives $\phi_{t}$ is uniformly bounded in $C^{1,\alpha}$ in time. Now, the
uniform eigenvalue bounds also imply $\phi_{t}$ has bounded $C^{2}$ norm. The
supercritical phase assumption implies the operator $\Theta(\cdot)$ has convex
level sets, which allows us to apply Evans-Krylov theory (see Section 6 of
[3]). This gives uniform $C^{2,\alpha}$ bounds for $\phi_{t}$ which can be
bootstrapped to higher order estimates. Thus we get smooth convergence
$\phi_{t}\rightarrow\phi_{T}$ to some limit, which allows us to continue the
flow past the time $T$. ∎

##  5\. Singular behavior at $t=\infty$

We now construct an example where the line bundle mean curvature flow develops
a singularity at infinite time along a destabilizing subvariety. Recall from
Section 2 that if one assumes Calabi-symmetry at an initial time, then (1.2)
can be reformulated as a flow of curves (2.3). As a first step, we construct a
family of subsolutions to (2.3) in polar coordinates that converges to a
stationary solution $\gamma_{\infty}$. By [14], we know such a solution must
lie on a level set of the harmonic polynomial ${\rm
Im}(e^{-i\hat{\theta}}z^{n})$. Write
$\gamma_{\infty}(\theta)=(x_{\infty}(\theta),y_{\infty}(\theta))=(r_{\infty}(\theta)\cos\theta,r_{\infty}(\theta)\sin\theta)$,
with $\theta\in[\theta_{\min},\theta_{\max}]$. We also assume

(5.1) $\displaystyle 1\leq x_{\infty}(\theta)\leq a\quad\textnormal{and}\quad
x_{\infty}(\theta_{\min})=x_{\infty}(\theta_{\max})=a.$

This leads to the following result.

######  Proposition 5.1.

Under the assumptions

(5.2) $r_{\infty}^{\prime}\geq
0\quad\textit{and}\quad\frac{r_{\infty}^{\prime}}{r_{\infty}}\leq
2\tan\theta,$

there exists a subsolution
$\gamma_{t}(\theta)=(r_{t}(\theta)\cos\theta,r_{t}(\theta)\sin\theta)$ to
(2.3) such that $\gamma_{t}\to\gamma_{\infty}$ uniformly as $t\to\infty$.

###### Proof.

We first write down (2.3) in polar coordinates. Note that
$\dot{\gamma}=(\dot{r}\cos\theta,\dot{r}\sin\theta)$, with the normal vector
to $\gamma$ given by

$\textbf{N}=\frac{1}{(r^{\prime
2}+r^{2})^{1/2}}(-r^{\prime}\sin\theta-r\cos\theta,r^{\prime}\cos\theta-r\sin\theta).$

Thus $\langle\dot{\gamma},{\textbf{N}}\rangle=-\frac{\dot{r}r}{(r^{\prime
2}+r^{2})^{1/2}}$. In this case the extrinsic quantity $\xi$ is simply
$\xi=\frac{d}{ds}\theta=\frac{1}{(r^{\prime 2}+r^{2})^{1/2}}$. The curvature
of a plane curve in polar coordinates is given by $\kappa=\frac{2r^{\prime
2}-rr^{\prime\prime}+r^{2}}{(r^{\prime 2}+r^{2})^{\frac{3}{2}}}$. Hence taking
the dot product of (2.3) with N we arrive at

$\dot{r}r=-u^{\prime\prime}\left(\frac{2r^{\prime
2}-rr^{\prime\prime}+r^{2}}{r^{\prime 2}+r^{2}}+(n-1)\right).$

Because $\gamma_{\infty}$ is stationary, we see (2.3) is equivalent to

(5.3) $\displaystyle\frac{2r_{\infty}^{\prime
2}-r_{\infty}r_{\infty}^{\prime\prime}+r_{\infty}^{2}}{r_{\infty}^{\prime
2}+r_{\infty}^{2}}+(n-1)=0.$

Now, let $b=b(t):[0,\infty)\to\mathbb{R}$ be an increasing function to be
determined later. We use $b(t)$ to define $r_{t}(\theta)$ by

(5.4)
$\frac{1}{r_{t}^{2}(\theta)}=\frac{1}{1+b}\left(\frac{b}{r_{\infty}^{2}(\theta)}+\frac{\cos^{2}\theta}{a^{2}}\right).$

For an appropriate choice of $b(t)$, we will show that the family of curves
$\gamma_{t}(\theta)=(r_{t}(\theta)\cos\theta,r_{t}(\theta)\sin\theta)$, which
form an interpolation between $\gamma_{0}$ and $\gamma_{\infty}$, gives a
subsolution to (2.3).

Figure 3. $\gamma_{t}$ being the interpolation between $\gamma_{0}$ and
$\gamma_{\infty}$.

Differentiating (5.4) with respect to $\theta$, and suppressing dependence on
$t$ and $\theta$ from our notation for simplicity, we have

$\frac{r^{\prime}}{r^{3}}=\frac{1}{1+b}\left(\frac{br_{\infty}^{\prime}}{r_{\infty}^{3}}+\frac{\sin(2\theta)}{2a^{2}}\right)$

as well as

$\frac{r^{\prime\prime}}{r^{3}}-\frac{3r^{\prime
2}}{r^{4}}=\frac{1}{1+b}\left(\frac{br_{\infty}^{\prime\prime}}{r_{\infty}^{3}}-\frac{3br_{\infty}^{\prime
2}}{r_{\infty}^{4}}+\frac{\cos(2\theta)}{a^{2}}\right).$

So,

$\displaystyle\frac{2r^{\prime 2}-rr^{\prime\prime}+r^{2}}{r^{4}}$
$\displaystyle=-\left(\frac{r^{\prime\prime}}{r^{3}}-\frac{3r^{\prime
2}}{r^{4}}\right)+\frac{1}{r^{2}}-\left(\frac{r^{\prime}}{r^{3}}\right)^{2}r^{2}$
$\displaystyle=\frac{1}{1+b}\left(-\frac{br_{\infty}^{\prime\prime}}{r_{\infty}^{3}}+\frac{3br_{\infty}^{\prime
2}}{r_{\infty}^{4}}-\frac{\cos(2\theta)}{a^{2}}+\frac{b}{r_{\infty}^{2}}+\frac{\cos^{2}\theta}{a^{2}}\right.$
$\displaystyle\qquad\qquad\left.-\left(\frac{br_{\infty}^{\prime}}{r_{\infty}^{3}}+\frac{\sin(2\theta)}{2a^{2}}\right)^{2}\left(\frac{b}{r_{\infty}^{2}}+\frac{\cos^{2}\theta}{a^{2}}\right)^{-1}\right).$

By (5.3),

$-\frac{br_{\infty}^{\prime\prime}}{r_{\infty}^{3}}+\frac{3br_{\infty}^{\prime
2}}{r_{\infty}^{4}}+\frac{b}{r_{\infty}^{2}}=\frac{-b}{r_{\infty}^{4}}\left((n-1)(r_{\infty}^{\prime
2}+r_{\infty}^{2})-r_{\infty}^{\prime 2}\right).$

Now, for notational simplicity, set

$A=\frac{br_{\infty}^{\prime}}{r_{\infty}^{3}}+\frac{\sin(2\theta)}{2a^{2}},\;\;\;B=\frac{b}{r_{\infty}^{2}}+\frac{\cos^{2}\theta}{a^{2}}.$

Then returning to the above we see

$\displaystyle\frac{2r^{\prime 2}-rr^{\prime\prime}+r^{2}}{r^{\prime
2}+r^{2}}$ $\displaystyle=\frac{1}{r^{2}}\frac{2r^{\prime
2}-rr^{\prime\prime}+r^{2}}{r^{4}}\left(\left(\frac{r^{\prime}}{r^{3}}\right)^{2}+\left(\frac{1}{r^{2}}\right)^{2}\right)^{-1}$
$\displaystyle=\frac{B}{A^{2}+B^{2}}\left(\frac{-b}{r_{\infty}^{4}}\left((n-1)(r_{\infty}^{\prime
2}+r_{\infty}^{2})-r_{\infty}^{\prime
2}\right)+\frac{\sin^{2}\theta}{a^{2}}-\frac{A^{2}}{B}\right).$

Hence

$\displaystyle\frac{2r^{\prime 2}-rr^{\prime\prime}+r^{2}}{r^{\prime
2}+r^{2}}+(n-1)$
$\displaystyle=\frac{1}{A^{2}+B^{2}}\left(-(n-1)\frac{Bb}{r_{\infty}^{2}}-(n-2)\frac{Bbr_{\infty}^{\prime
2}}{r_{\infty}^{4}}\right.$
$\displaystyle\quad\quad\left.+\frac{B\sin^{2}\theta}{a^{2}}+(n-2)A^{2}+(n-1)B^{2}\right).$

We now compute

$-(n-1)\frac{Bb}{r_{\infty}^{2}}+(n-1)B^{2}=(n-1)B\left(B-\frac{b}{r_{\infty}^{2}}\right)=(n-1)B\frac{\cos^{2}\theta}{a^{2}},$

and

$A^{2}-\frac{Bbr_{\infty}^{\prime
2}}{r_{\infty}^{4}}=\frac{br_{\infty}^{\prime}\sin(2\theta)}{a^{2}r_{\infty}^{3}}+\frac{\sin^{2}(2\theta)}{4a^{4}}-\frac{br_{\infty}^{\prime
2}\cos^{2}\theta}{a^{2}r_{\infty}^{4}}.$

Combining these, we have

$\displaystyle\frac{2r^{\prime 2}-rr^{\prime\prime}+r^{2}}{r^{\prime
2}+r^{2}}+(n-1)$
$\displaystyle=\frac{(n-1)B\cos^{2}\theta+B\sin^{2}\theta}{a^{2}(A^{2}+B^{2})}+\frac{(n-2)\sin^{2}(2\theta)}{4a^{4}(A^{2}+B^{2})}$
$\displaystyle+\frac{n-2}{A^{2}+B^{2}}\left(\frac{br_{\infty}^{\prime}\sin(2\theta)}{a^{2}r_{\infty}^{3}}-\frac{br_{\infty}^{\prime
2}\cos^{2}\theta}{a^{2}r_{\infty}^{4}}\right).$

By assumption,

$r_{\infty}^{\prime}\geq
0\quad\textnormal{and}\quad\frac{r_{\infty}^{\prime}}{r_{\infty}}\leq
2\tan\theta,$

which implies

$\frac{br_{\infty}^{\prime}\sin(2\theta)}{a^{2}r_{\infty}^{3}}-\frac{br_{\infty}^{\prime
2}\cos^{2}\theta}{a^{2}r_{\infty}^{4}}\geq 0.$

Additionally, $r_{\infty},$ $\sin\theta$, and $\cos\theta$, are all bounded
above and below away from zero. This implies there exists a constant $C_{1}$
so that, for large $b$,

$\frac{2r^{\prime 2}-rr^{\prime\prime}+r^{2}}{r^{\prime
2}+r^{2}}+(n-1)\geq\frac{C_{1}}{b}.$

Returning to (5.4), we take the derivative of both sides in $t$

$-\frac{2\dot{r}}{r^{3}}=-\frac{\dot{b}}{(1+b)^{2}}\left(\frac{b}{r_{\infty}^{2}}+\frac{\cos^{2}\theta}{a^{2}}-\frac{1+b}{r_{\infty}^{2}}\right)=-\frac{\dot{b}}{(1+b)^{2}}\left(\frac{\cos^{2}\theta}{a^{2}}-\frac{1}{r_{\infty}^{2}}\right).$

Multiplying by $-r^{4}$ and plugging in the square of (5.4) for $r^{4}$ gives

$\displaystyle 2r\dot{r}$
$\displaystyle=\left(\frac{\cos^{2}\theta}{a^{2}}-\frac{1}{r_{\infty}^{2}}\right)\left(\frac{b}{r_{\infty}^{2}}+\frac{\cos^{2}\theta}{a^{2}}\right)^{-2}\dot{b}$
$\displaystyle=(r_{\infty}x-ra)\left(\frac{r_{\infty}x+ra}{a^{2}r^{2}r_{\infty}^{2}}\right)\left(\frac{b}{r_{\infty}^{2}}+\frac{\cos^{2}\theta}{a^{2}}\right)^{-2}\dot{b}$
$\displaystyle\geq(r_{\infty}x-ra)\frac{C_{2}}{b^{2}}\dot{b}$

for some $C_{2}>0$ whenever $b$ is large. Note that the polar curves
$r(\theta)$ intersect the line $x=a$ to the zeroth order, which implies there
exists a constant $C_{3}>0$ for which

(5.5) $\displaystyle
0\geq\inf_{x\in[a-\epsilon,a]}\left(u^{\prime\prime-1}(r_{\infty}x-ra)\right)+\inf_{x\in[1,a]}(r_{\infty}x-ra)\geq-
C_{3}.$

Next, we use the same assumption on the background Kähler form as Section 3,
namely, for $x\in[1,a-\epsilon]$ we assume $u^{\prime\prime}(x)\geq k(x-1).$
This implies

$\displaystyle u^{\prime\prime}$ $\displaystyle\geq k(r\cos\theta-1)$
$\displaystyle=k\left(\sqrt{(1+b)\left(\frac{b}{r_{\infty}^{2}}+\frac{\cos^{2}\theta}{a}\right)^{-1}}\cos\theta-1\right)$
$\displaystyle=k\left(\sqrt{(1+b)\left(\frac{b}{(r_{\infty}\cos\theta)^{2}}+\frac{1}{a}\right)^{-1}}-1\right)$
$\displaystyle\geq
k\left(\sqrt{(1+b)\left(b+\frac{1}{a}\right)^{-1}}-1\right).$

For simplicity, write the right hand side above as $C(b)$, which is a smooth
positive function approaching $0$ as $b\to\infty$. Combining with (5.5) we
arrive at,

$\displaystyle\frac{2}{u^{\prime\prime}}r\dot{r}\geq-\frac{C_{2}C_{3}}{b^{2}}\left(1+\frac{1}{C(b)}\right)\dot{b}.$

If $b$ solves the initial value problem

$\dot{b}=2\left(1+\frac{1}{C(b)}\right)^{-1}\frac{C_{1}}{C_{2}C_{3}}b;\qquad
b_{0}\gg 0,$

then $r_{t}(\theta)$ defines a subsolution:

$\displaystyle\frac{1}{u^{\prime\prime}}r\dot{r}+\left(\frac{2r^{\prime
2}-rr^{\prime\prime}+r^{2}}{r^{\prime 2}+r^{2}}+(n-1)\right)\geq 0.$

Notice that we require $b_{0}\gg 0$. Thus the subsolution does not start at
$\gamma_{0}$ (given by the vertical line in Figure 3), but rather a curve
starting closer to $\gamma_{\infty}$ in the interpolation. It then sweeps out
to $\gamma_{\infty}$ as $t\rightarrow\infty$. ∎

We now show that the assumptions on $r_{\infty}$ in Proposition 5.1 can be
satisfied with an explicit example. As we have stated above, in [14] it was
demonstrated that under the Calabi-Symmetry assumption, solutions to the dHYM
equation correspond to functions $f:[1,a]\rightarrow\mathbb{R}$, satisfying
the boundary conditions $f(1)=q$, $f(a)=p$, so that the graph $(x,f(x))$ lies
on a level curve of ${\rm Im}(e^{-i\hat{\theta}}z^{n})$. Furthermore, the
proof of Theorem 1 from [14] uses that if the level curve through $(1,q)$ has
vertical slope, then the class $[\alpha]$ is semi-stable with respect to the
stability condition (1.3), with the exceptional divisor $E$ being the
destabilizing subvariety. Thus in this case any graph $f_{\infty}(x)$ lying on
the level curve is singular with unbounded derivative at $(1,q)$, and by
construction the corresponding representative of $[\alpha]$ will be singular
precisely along $E$. It is this singular graph that will be the limiting curve
to the line bundle mean curvature flow.

######  Lemma 5.2.

There exists a Kähler class $[\omega]$ and a semi-stable class $[\alpha]$ with
a stationary solution $\gamma_{\infty}$ which satisfies
$\gamma_{\infty}(\theta_{0})=(1,q)$, $\gamma_{\infty}(\theta_{\max})=(a,p)$,
and where the corresponding polar function $r_{\infty}$ satisfies the
assumptions of Proposition 5.1.

###### Proof.

Choose $\gamma_{\infty}$ lying on a level curve of ${\rm
Im}(e^{-i\hat{\theta}}z^{n})$ so that $\gamma_{\infty}(\theta_{0})=(1,q)$ and
$\gamma_{\infty}^{\prime}(\theta_{0})$ is vertical, for some $\theta_{0}$.
This guarantees we are working with a semi-stable class. The corresponding
polar function $r_{\infty}(\theta)$ will now satisfy (5.3). Define $\beta$ by

$\displaystyle\tan\beta:=\frac{y^{\prime}(\theta)}{x^{\prime}(\theta)}$
$\displaystyle=\frac{r_{\infty}^{\prime}\sin\theta+r_{\infty}\cos\theta}{r_{\infty}^{\prime}\cos\theta-
r_{\infty}\sin\theta}=\frac{r_{\infty}^{\prime}r_{\infty}^{-1}\tan\theta+1}{r_{\infty}^{\prime}r_{\infty}^{-1}-\tan\theta}.$

As a result

$\frac{r_{\infty}^{\prime}}{r_{\infty}}=\cot(\beta-\theta)=\tan(\pi/2-\beta+\theta).$

Now, choose $q\gg 0$. Because $\gamma_{\infty}^{\prime}(\theta_{0})$ is
vertical, we know $\beta(\theta_{0})=\pi/2$. In particular, at this point

$r_{\infty}^{\prime}(\theta_{0})>0\quad\textnormal{and}\quad\frac{r_{\infty}^{\prime}(\theta_{0})}{r_{\infty}(\theta_{0})}=\tan(\theta_{0})<2\tan(\theta_{0}).$

Thus, there exists a neighborhood of $\theta_{0}$ where (5.2) holds.

We now check (5.1). At $\theta=\theta_{0}$,

$\displaystyle x_{\infty}^{\prime}$
$\displaystyle=r_{\infty}\cos\theta\left(\frac{r_{\infty}^{\prime}}{r_{\infty}}-\tan\theta\right)=0$
$\displaystyle x_{\infty}^{\prime\prime}$
$\displaystyle=\frac{\cos\theta}{r_{\infty}}\left(-2r_{\infty}r_{\infty}^{\prime}\tan\theta+r_{\infty}r_{\infty}^{\prime\prime}-r_{\infty}^{2}\right)$
$\displaystyle=\frac{\cos\theta}{r_{\infty}}\left(-2r_{\infty}^{\prime
2}+r_{\infty}r_{\infty}^{\prime\prime}-r_{\infty}^{2}\right)>0,$

where last inequality follows from (5.3). Hence, $x_{\infty}$ achieves local
minimum at $\theta=\theta_{0}$. We choose $a$ slightly greater than $1$ such
that $x_{\infty}(\theta_{\min})=x_{\infty}(\theta_{\max})=a$. This
demonstrates the assumptions of Proposition 5.1. ∎

We are now ready to complete the proof of Theorem 1.3. Consider the classes
$[\omega]$ and $[\alpha]$ discussed in the above lemma. Let $f_{\infty}(x)$
denote the graphical portion of $\gamma_{\infty}$ that connects $(1,q)$ to
$(a,p)$. Since the assumptions of Proposition 5.1 are satisfied, there exists
a subsolution $\gamma_{t}$ pushing out towards $\gamma_{\infty}$. In the proof
of Proposition 5.1 we saw the subsolution condition is not satisfied unless
$b$ is sufficiently large, and so the subsolution starts at some time $t_{0}$,
with $\gamma_{t_{0}}$ already pushed out towards $\gamma_{\infty}$.

Consider a function $f_{t_{0}}$ satisfying $f_{t_{0}}(1)=q$, and
$f_{t_{0}}(a)=p$, which lies above the curve $\gamma_{t_{0}}$, but below
$\gamma_{\infty}$, as in Figure 4. This function defines an initial
representative $\alpha_{0}\in[\alpha]$, and its angle is given by

$\Theta(\alpha_{0})=(n-1){\rm arctan}\left(\frac{f_{t_{0}}}{x}\right)+{\rm
arctan}(f_{t_{0}}^{\prime}).$

The supercritical phase assumption in Theorem 1.2 is satisfied if we choose
$q$ large enough so that ${\rm arctan}\left(\frac{f_{t_{0}}}{x}\right)$ is
sufficiently close to $\pi/2$. Thus if we consider a solution $\alpha_{t}$ to
the line bundle mean curvature flow starting at $\alpha_{0}$, the flow exists
for all time. Let $f_{t}$ be graph corresponding to $\alpha_{t}$. By the
maximum principle, $f_{t}$ must stay below $f_{\infty}$ and above $\gamma_{t}$
for all time.

Figure 4. A singularity at $(1,q)$ at $t=\infty$.

Because the subsolution $\gamma_{t}$ sweeps out to $\gamma_{\infty}$ as
$t\rightarrow\infty$, the solution to the flow $f_{t}$ must converge to
$f_{\infty}$ in $C^{0}$. In particular, it can not develop an infinite time
singularity at any point other than $(1,q)$, where it will achieve vertical
tangency. By construction, this point corresponds to the exceptional divisor
$E$, which is precisely the destabilizing subvariety. Thus, the corresponding
forms $\alpha_{t}$ along the line bundle mean curvature flow will blow up
along $E$ at infinite time.

## References

  * [1] E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Vol. 102 of Ann. Math. Studies, Princeton Univ. Press, Princeton, N.J. (1982), 259-290.
  * [2] G. Chen, The J-equation and the supercritical deformed Hermitian-Yang-Mills equation. Invent. Math. 225 (2021), no. 2, 529-602.
  * [3] T.C. Collins, A. Jacob, and S.-T. Yau, (1,1) forms with specified Lagrangian phase. Camb. J. Math. 8 (2020), no. 2, 407-452.
  * [4] T.C. Collins, J. Lo, Y. Shi, and S.-T. Yau, Bridgeland stable line bundles of fiber degree 1 on Weierstrass elliptic K3 surface. arXiv:2306.05620 .
  * [5] T.C. Collins and Y. Shi, Stability and the deformed Hermitian-Yang-Mills equation. Surveys in Differential Geometry. 24 (2019), 1-38.
  * [6] T.C. Collins, D. Xie, and S.-T. Yau, The deformed Hermitian-Yang-Mills equation in geometry and physics, Geometry and Physics, Volume I: A Festschrift in honour of Nigel Hitchin, Oxford University Press, December, 2018.
  * [7] T.C. Collins and S.-T. Yau, Moment maps, nonlinear PDE, and stability in mirror symmetry, Ann. PDE 7 (2021), no. 1, Paper No. 11, 73 pp.
  * [8] J. Chu, M.-C. Lee, and R. Takahashi, A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation arXiv:2105.10725.
  * [9] H. Fang and M. Lai, Convergence of general inverse $\sigma_{k}$-flow on Kähler manifolds with Calabi ansatz, Trans. Amer. Math. Soc. 365 (2013), no. 12, 6543-6567.
  * [10] J. Fu, S.-T. Yau, and D. Zhang, A deformed Hermitian-Yang-Mills flow. arXiv:2105.13576
  * [11] X. Han and X Jin, Chern number inequalities of deformed Hermitian-Yang-Mills metrics on four dimensional Kähler manifolds. arXiv:2008.06862.
  * [12] X. Han and X Jin, Stability of line bundle mean curvature flow, arXiv: 2001.07406.
  * [13] A. Jacob, The Deformed Hermitian-Yang-Mills Equation and Level Sets of Harmonic Polynomials. arXiv:2204.01875.
  * [14] A. Jacob and N. Sheu, The deformed Hermitian-Yang-Mills equation on the blowup of $\mathbb{P}^{n}$.
  * [15] A. Jacob and S.-T. Yau, A special Lagrangian type equation for holomorphic line bundles, Math. Ann. 369 (2017). no 1-2, 869-898.
  * [16] N.V. Krylov, Nonlinear elliptic and parabolic equations of the second order. Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987. xiv+462 pp. ISBN: 90-277-2289-7.
  * [17] M. Lejmi, and G. Székelyhidi, The $J$-flow and stability, Advances in Math. 274 (2015), 404-431.
  * [18] C. Leung, S.-T. Yau, and E. Zaslow, From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 209-225, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI, 2001.
  * [19] M. Marino, R. Minasian, G. Moore, and A. Strominger, Nonlinear Instantons from Supersymmetric p-Branes, hep-th/9911206.
  * [20] P. Pingali, The deformed Hermitian Yang-Mills equation on three-folds. Anal. PDE 15 (2022), no. 4, 921-935.
  * [21] K. Smoczyk, Angle theorems for the Lagrangian mean curvature flow, Math. Z. 240 (2002), 849-883.
  * [22] K. Smoczyk, Longtime existence of the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 20 (2004), 25-46.
  * [23] K. Smoczyk and M.-T. Wang, Mean curvature flows of Lagrangians submanifolds with convex potentials, J. Differential Geom. 62 (2002), 243-257.
  * [24] J. Song, Ricci flow and birational surgery, arXiv:1304.2607.
  * [25] J. Song and B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow, Duke Math. J. 162 (2013), no. 2, 367-415.
  * [26] J. Song and B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow II, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1529-1561.
  * [27] J. Song and B. Weinkove, The Kähler-Ricci flow on Hirzebruch surfaces, J. Reine Angew. Math. 659 (2011), 141-168.
  * [28] J. Song and Y. Yuan, Metric flips with Calabi ansatz, Geom. Func. Anal. 22 (2012), no. 1, 240-265.
  * [29] R. Takahashi, Collapsing of the line bundle mean curvature flow on Kähler surfaces. Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 27, 18 pp.
  * [30] R. Takahashi, Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow. Internat. J. Math. 31 (2020), no. 14, 2050116, 26 pp.
  * [31] C.-J. Tsai, M.-P. Tsui, and M.-T. Wang, Mean curvature flows of two-convex Lagrangians. arXiv:2302.02512.
  * [32] M.-T. Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension, Invent. math. 148 (2002) 3, 525-543.
  * [33] J. Zhang, A note on the supercritical deformed Hermitian-Yang-Mills equation. arXiv:2302.06592.