# Neck pinches along the Lagrangian mean curvature flow of surfaces

Jason D. Lotay  Mathematical Institute, University of Oxford, Oxford OX2 6GG,
United Kingdom<EMAIL_ADDRESS>,  Felix Schulze  Mathematics
Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
<EMAIL_ADDRESS>and  Gábor Székelyhidi  Department of Mathematics,
Northwestern University, Evanston IL 60208, USA<EMAIL_ADDRESS>
###### Abstract.

Let $L_{t}$ be a zero Maslov, rational Lagrangian mean curvature flow in a
compact Calabi–Yau surface, and suppose that at the first singular time a
tangent flow is given by the static union of two transverse planes. We show
that in this case the tangent flow is unique, and that the flow can be
continued past the singularity as an immersed, smooth, zero Maslov, rational
Lagrangian mean curvature flow. Furthermore, if $L_{0}$ is a sphere that is
stable in the sense of Thomas–Yau, then such a singularity cannot form.

##  1\. Introduction

The question of the existence of special Lagrangian submanifolds is an
important problem in complex and symplectic geometry. Special Lagrangians play
a central role in the Strominger–Yau–Zaslow conjecture [25] on mirror
symmetry, and are of interest in the variational problem of finding area-
minimizing Lagrangians, studied extensively by Schoen–Wolfson [19]. Smoczyk
[24] showed that the mean curvature flow preserves the class of Lagrangian
submanifolds in Calabi–Yau manifolds, and so a natural expectation is that a
suitable Lagrangian can be deformed into a special Lagrangian using the flow.
The Thomas–Yau conjecture [28], motivated by mirror symmetry [27], predicts
that this is indeed the case, assuming that the initial Lagrangian satisfies a
certain stability condition. More recently Joyce [12] formulated a detailed
conjectural picture relating singularity formation along the Lagrangian mean
curvature flow to Bridgeland stability conditions on Fukaya categories.

It was shown by Neves [17] that singularities are, in a sense, unavoidable
along the Lagrangian mean curvature flow, even if the initial Lagrangian is a
small Hamiltonian perturbation of a special Lagrangian. At the same time,
Neves [15] shows that for the flow of zero Maslov Lagrangians any tangent flow
at a singular point is a union of special Lagrangian cones. This means that
Type I singularities – which are typically easier to analyse – do not exist.
In this paper we study the simplest kind of singularities, called neck pinches
in [12, Conjecture 3.16], in the two-dimensional case. Our main result is the
following, which we state in the setting of a compact ambient Calabi–Yau
surface, though it also works in $\mathbb{C}^{2}$. Note that here, and
throughout, we allow our Lagrangians to be immersed, which is important in the
context of Lagrangian mean curvature flow.

######  Theorem 1.1.

Let $X$ be a compact Calabi–Yau surface, and $L\subset X$ a zero Maslov,
rational Lagrangian. Let $L_{t}$ be the mean curvature flow starting from $L$
for $t\in[0,T)$, where $T$ is the first finite singular time. Let
$(\mathbf{x}_{T},T)$ be a singular point, and suppose that a tangent flow at
$(\mathbf{x}_{T},T)$ is given by the transverse union of two multiplicity one
planes. The tangent flow at $(\mathbf{x}_{T},T)$ is then unique.

In Theorem 1.1 the assumption is that one tangent flow is given by a union of
multiplicity one transverse planes $P_{1}\cup P_{2}$, with corresponding
Lagrangian angles $\theta_{1},\theta_{2}$. We note here that by Neves [16,
Corollary 4.3] the flow cannot form a singularity unless
$\theta_{1}=\theta_{2}$. Therefore throughout the article we will only be
concerned with the case when $P_{1}$ and $P_{2}$ have the same Lagrangian
angle.

The uniqueness of tangent flows is a fundamental problem for analysing the
singularities of mean curvature flow, and there have been several important
results in this direction recently [20, 5, 3]. A major new difficulty in
Theorem 1.1 is that it is the first example of uniqueness for a tangent flow
that is singular. The proof crucially exploits several aspects of the
Lagrangian setting, and does not apply in the general setting of mean
curvature flow.

Theorem 1.1 allows us to analyse the behavior of the flow at the singularity.
First, we have the following, showing that the flow can be continued past the
singular time if all singularities at time $T$ are modelled on two transverse,
multiplicity one planes. Recall that the grading of a zero Maslov Lagrangian
corresponds to a global choice of function representing the Lagrangian angle,
see Definition 2.1.

######  Theorem 1.2.

Suppose that $X,L$ are as in Theorem 1.1 and assume that at each singular
point $(\mathbf{x},T)$ a tangent flow is a static union of two multiplicity
one, transverse planes. Then $L_{t}$ converges to an immersed Lagrangian
$C^{1}$-submanifold $L_{T}$ in the sense of currents as $t\to T$, and the flow
can be restarted as a smooth, zero Maslov, rational Lagrangian mean curvature
flow with initial condition $L_{T}$. Furthermore, the extended flow is smooth
(together with its grading) through the singular time, away from the singular
points.

A slight extension of the ideas involved in proving uniqueness of the tangent
flow also allows us to show that if the tangent flow is given by the union of
two transverse planes, then close to the singularity the flow looks like the
two transverse planes, desingularized by a Lawlor neck which “pinches off”:
see Theorem 8.3. This has the following consequence.

######  Theorem 1.3.

Suppose that $X,L$ are as in Theorem 1.1. For $t<T$ sufficiently close to the
singular time we can write $L_{t}$ as a graded self-connected sum of an
immersed Lagrangian $M$ at a self-intersection point.

If $M$ is not connected, then we can write it is a graded connected sum
$M=M_{1}\\#M_{2}$ and the following holds:

(1.1)
$\mathrm{vol}(L)>\left|\int_{M_{1}}\Omega\right|+\left|\int_{M_{2}}\Omega\right|,$

where $\Omega$ is the holomorphic volume form on $X$. If in addition $L$ is
almost calibrated, then we also have

(1.2) $\phi(M_{1}),\phi(M_{2})\subset(\inf_{L}\theta,\sup_{L}\theta),$

where $\phi(M_{i})$ is the “cohomological” Lagrangian angle of $M_{i}$ defined
in (9.3). (See Section 9 for detailed definitions.)

This result provides some evidence for Thomas–Yau’s Conjecture 7.3 in [28].
Indeed, their conjecture states that if the flow has a finite time
singularity, then $L$ can be decomposed into a graded connected sum
$M_{1}\\#M_{2}$ satisfying the conditions in (1.1). Our result shows that this
is one of the possible scenarios when the tangent flow at the first singular
time is given by two transverse planes. In particular, the decomposition as a
graded connect sum is guaranteed if $L$ is a sphere. Note that Joyce’s
conjectural picture [12] predicts that other singularities could still form,
notably those with tangent flows given by two static planes meeting along a
line. It is an important problem to understand what we can say about the flow
in the presence of such singularities and some progress towards this was made
in the authors’ previous work [14]. An optimistic expectation is that for a
generic initial surface, the only tangent flows that appear at singularities
are of these two types, i.e. two multiplicity one planes meeting either at a
point or along a line.

###  1.1. Outline

To conclude this introduction we give a brief outline of the contents of the
paper, along with some of the main ingredients of the proofs. In most of the
paper we will consider the flow in $\mathbb{C}^{2}$ and we will only discuss
the necessary changes in the case of a compact ambient space in Section 9.

The technical heart of our results is an analysis of rescaled Lagrangian mean
curvature flows $M_{\tau}\subset\mathbb{C}^{2}$ which are close to the union
$V=P_{1}\cup P_{2}$ of two transverse planes on a time interval $t\in[0,1]$,
say. We introduce a distance function $D_{V}(M_{\tau})$ from $V$ to
$M_{\tau}$, which incorporates both an $L^{2}$-type distance, as well as the
difference in Lagrangian angles. (See Definition 3.4 for the precise
definition.) Although the non-compactness introduces some technical
difficulties, the main difficulty when compared to earlier studies is the
presence of the singularity at the origin. The first step for dealing with
this is to observe that the function $|zw|^{2}$ satisfies a useful
differential inequality along the flow: here $z,w$ are complex coordinates,
for a suitable hyperkähler-rotated complex structure, such that
$V=\\{zw=0\\}$. The differential inequality is exploited in Lemma 3.5,
allowing us to convert bounds on $D_{V}(M_{\tau})$ to pointwise distance
bounds at a later time.

In general the knowledge that $M_{\tau}$ is close to $V$ in the Hausdorff
sense does not imply graphicality of $M_{\tau}$ over $V$, even on regions away
from the origin, because of possible multiplicity. In Proposition 4.5 we show
that good graphicality of $M_{\tau}$ over $V$ on a fixed annulus
$B_{2}\setminus B_{1}$ can be propagated out to larger annular regions
$B_{R}\setminus B_{r}$ at later times, in the presence of our pointwise
distance bounds. This graphicality estimate is then used in Proposition 4.6 to
derive a crucial estimate

$|\mathcal{A}(M_{\tau})|\leq D_{V}(M_{\tau-1})^{1+\alpha_{1}}$

for the excess $\mathcal{A}$ (defined in (2.4)) in terms of the distance,
where $\alpha_{1}>0$.

Our next task, in Sections 5 and 6, is to derive a three-annulus type lemma
for the distance function $D_{V}$. The main result is Proposition 6.2, and the
proof relies on an analysis of solutions of the drift heat equation on a plane
with some mild singularities at the origin, together with the non-
concentration estimates in Lemma 3.5.

The technical heart of the paper is the proof of the decay estimate,
Proposition 7.3, for the distance function. Given a three-annulus lemma as in
Proposition 6.2, the usual strategy for controlling the flow $M_{\tau}$ is to
show that at each scale the flow must decay towards the “best fit” cone of the
form $V^{\prime}=P_{1}^{\prime}\cup P_{2}^{\prime}$. The difficulty is that
our estimates only apply when $V^{\prime}$ is special Lagrangian, i.e. the
planes $P_{1}^{\prime}$ and $P_{2}^{\prime}$ have the same Lagrangian angle.
Since in general these angles might be different, the situation is somewhat
reminiscent of the case of non-integrable tangent flows, which is typically
dealt with using the Lojasiewicz–Simon inequality [22]. Our approach is quite
different from this, and can be thought of as a quantitative version of
Neves’s result [16, Corollary 4.3] stating that at a singularity the tangent
flow cannot be the union of two planes with different Lagrangian angles. These
considerations lead to the alternative (ii) in Proposition 7.3.

We give the proofs of the main applications in Section 8. Given the decay
estimate in Proposition 7.3, the uniqueness result, Theorem 1.1, follows
standard arguments. The proof of the existence of a $C^{1}$ limiting surface
$L_{T}$ in Theorem 1.2 is similar to [3, Corollary 1.2]. We can then restart
the flow using the approach of Wang [29].

The main ingredient for proving Theorem 1.3 is to show that one can find small
Lawlor necks, i.e. surfaces of the form $\\{zw=\pm\epsilon\\}$, in the
presence of a tangent flow $V$ given by two transverse planes. This relies on
showing that if between different scales the flow stays close to possibly
moving pairs of planes, then these planes have to stay very close to each
other, similarly to how the uniqueness of the tangent flow is proved. A
related result was shown by Edelen [8, Theorem 13.1] in the context of minimal
hypersurfaces.

Finally in Section 9 we will discuss the changes needed when working in a
compact ambient space $X$ instead of in $\mathbb{C}^{2}$. We follow the
approach found for instance in White [30, Section 4], isometrically embedding
$X\subset\mathbb{R}^{N}$, and writing the (rescaled) mean curvature flow in
$X$ as a (rescaled) mean curvature flow in $\mathbb{R}^{N}$ with an additional
forcing term. The quantities, such as $|zw|$, that we used in $\mathbb{C}^{2}$
can be defined by projecting to the tangent space $T_{p}X$ at the point $p$
where the singularity forms. Along the rescaled flow this projection as well
as the forcing term introduces additional errors when compared to the
calculations in $\mathbb{C}^{2}$, however these errors decay exponentially
fast and so the geometric conclusions still hold.

###  1.2. Acknowledgements

We thank Dominic Joyce and Yang Li for their interest in this work and helpful
comments. JDL and FS were partially supported by a Leverhulme Trust Research
Project Grant RPG-2016-174. GSz was supported in part by NSF grant
DMS-2203218.

##  2\. Preliminaries

In this section we introduce various key definitions and notation that we
shall require throughout the article.

###  2.1. Lagrangians in $\mathbb{C}^{2}$

We first recall some basic definitions concerning Lagrangian submanifolds in
$\mathbb{C}^{2}$.

######  Definition 2.1.

An oriented Lagrangian submanifold $L$ in $\mathbb{C}^{2}$ is _zero Maslov_ if
there exists a function $\theta$ on $L$ (called the _Lagrangian angle_) so
that

$\mathbf{H}=J\nabla\theta,$

where $\mathbf{H}$ is the mean curvature vector of $L$ and $J$ is the complex
structure on $\mathbb{C}^{2}$. The choice of function $\theta$ is called a
_grading_ of $L$. We further say that $L$ is _almost calibrated_ if $\theta$
can be chosen so that, for some $\epsilon>0$,

$\sup\theta-\inf\theta\leq\pi-\epsilon.$

######  Definition 2.2.

An oriented Lagrangian $L$ in $\mathbb{C}^{2}$ is _exact_ if there exists a
function $\beta$ on $L$ so that

$J\mathbf{x}^{\perp}=\nabla\beta,$

where $\mathbf{x}^{\perp}$ is the normal projection of the position vector
$\mathbf{x}\in\mathbb{C}^{2}$. Equivalently,

(2.1) $\displaystyle\mathrm{d}\beta=\lambda|_{L},$

where $\lambda$ is the Liouville form on $\mathbb{C}^{2}$, which is a 1-form
on $\mathbb{C}^{2}$ so that $\frac{1}{2}\lambda$ is a primitive for the Kähler
form $\omega$ on $\mathbb{C}^{2}$. The Lagrangian $L$ is _rational_ if the set
$\lambda(H_{1}(L,\mathbb{Z}))$ is discrete in $\mathbb{R}$. Note that exact
Lagrangians are rational.

###  2.2. Lagrangian mean curvature flow

In most of this article we will consider a smooth, zero Maslov solution to
Lagrangian mean curvature flow (LMCF)

$[0,T)\ni t\mapsto L_{t}\subset\mathbb{C}^{2}$

which evolves with normal speed given by $\mathbf{H}$. See Section 9 for the
setting of a compact ambient space.

Throughout we will assume that the Lagrangian angle $\theta$ of $L_{t}$ along
the flow is uniformly bounded: $|\theta|<C_{0}$. In addition we assume that
$L_{t}$ has uniformly bounded area ratios, i.e. there exists $C_{1}>0$ such
that

$\sup_{\mathbf{x},t}\mathcal{H}^{2}(L_{t}\cap B(\mathbf{x},r))\leq
C_{1}r^{2}\text{ for all }r>0,$

where $B(\mathbf{x},r)$ is the Euclidean ball of radius $r$ about
$\mathbf{x}\in\mathbb{C}^{2}$. We call

$\mathcal{M}:=\\{L_{t}\times\\{t\\}\,|\,t\in[0,T)\\}\subset\mathbb{C}^{2}\times\mathbb{R}$

the spacetime track of the flow, and write $\mathcal{M}(t)=L_{t}\,.$

It will be useful to perform parabolic rescalings of our flows, so we shall
introduce the following notation.

######  Definition 2.3.

For $\lambda>0$ we shall denote the _parabolic rescaling_

$\mathcal{D}_{\lambda}:\mathbb{C}^{2}\times\mathbb{R}\to\mathbb{C}^{2}\times\mathbb{R},(\mathbf{x},t)\mapsto(\lambda\mathbf{x},\lambda^{2}t)\,.$

Note that for a (Lagrangian) mean curvature flow $\mathcal{M}$, it holds that
$\mathcal{D}_{\lambda}\mathcal{M}$ is again a (Lagrangian) mean curvature
flow.

We recall Huisken’s monotonicity formula [10]:

(2.2)
$\begin{split}\frac{d}{dt}\int_{L_{t}}f\rho_{\mathbf{x}_{0},t_{0}}\,d\mathcal{H}^{2}&=\int_{L_{t}}(\partial_{t}f-\Delta
f)\rho_{\mathbf{x}_{0},t_{0}}\,d\mathcal{H}^{2}\\\
&\qquad-\int_{L_{t}}f\,\bigg{|}\,\mathbf{H}-\frac{(\mathbf{x}-\mathbf{x}_{0})^{\perp}}{2(t_{0}-t)}\bigg{|}^{2}\rho_{\mathbf{x}_{0},t_{0}}\,d\mathcal{H}^{2}\,,\end{split}$

for $t<t_{0}$, where $f$ is a function on $L_{t}$ with polynomial growth
(locally uniform in $t$), and

$\rho_{\mathbf{x}_{0},t_{0}}(\mathbf{x},t)=(4\pi(t_{0}-t))^{-1}\exp\left(-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4(t_{0}-t)}\right)$

is the backwards heat kernel. The _entropy_ $\lambda(L)$ defined by Colding-
Minicozzi [4] is given by

$\lambda(L)=\sup_{\mathbf{x}_{0}\in\mathbb{C}^{2},\,r>0}\frac{1}{4\pi
r}\int_{L}e^{-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4r}}\mathrm{d}\mathcal{H}^{2}.$

By virtue of Huisken’s monotonicity formula, $t\mapsto\lambda(L_{t})$ is non-
increasing along any $2$-dimensional mean curvature flow in $\mathbb{C}^{2}$.

We will be studying the behaviour of the flow close to a singularity at
$(\mathbf{x}_{0},T)$. It is convenient to shift the flow in space-time such
that $(\mathbf{x}_{0},T)$ is the origin $(\mathbf{0},0)$, i.e. we consider
instead the flow

$\tilde{\mathcal{M}}:=\mathcal{M}-(\mathbf{x}_{0},T)\,,$

defined for $t\in[-T,0)$. For ease of notation we will drop the tilde in the
following.

A tangent flow at $(\mathbf{0},0)$ is defined to be any weak limit of a
sequence of rescalings $\mathcal{D}_{\lambda_{k}}\mathcal{M}$, with
$\lambda_{k}\to\infty$. According to the structure theorem due to Neves [15,
Theorem A], in our setting of a zero Maslov flow with bounded Lagrangian angle
in two dimensions, the tangent flows are all given by unions of Lagrangian
planes with multiplicities. In addition the Lagrangian angle $\theta$ along
the sequence of rescalings converges in a suitable sense to the angles of the
planes. We recall that the assumptions of Theorem 1.1 mean that one tangent
flow at $(\mathbf{0},0)$ is given by a union of multiplicity one transverse
planes $P_{1}\cup P_{2}$ with the same Lagrangian angle.

It turns out to be helpful to consider a further rescaling, which turns self-
similarly shrinking solutions into static points of the flow.

######  Definition 2.4.

The _rescaled flow_ is

$[-\log(T),+\infty)\ni\tau\mapsto
M_{\tau}:=e^{\frac{\tau}{2}}\mathcal{M}(-e^{-\tau})=e^{\frac{\tau}{2}}L_{-e^{-\tau}},$

which evolves with normal speed

(2.3) $\mathbf{H}+\frac{\mathbf{x}^{\perp}}{2}\,.$

In terms of the rescaled flow the tangent flows of $\mathcal{M}$ can be
studied by taking limits of sequences $M_{\tau_{k}}$ as $\tau_{k}\to\infty$.

###  2.3. Set-up

For the majority of this article we will consider a rescaled Lagrangian mean
curvature flow $M_{\tau}$ in $\mathbb{C}^{2}$, for $\tau\in[T_{0},T_{1}]$,
close in a suitable sense to the static flow of the transverse union of two
planes. The flow is assumed to have uniformly bounded Lagrangian angle and
area ratios as above. Two additional conditions will play an important role,
the first of which involves the following key quantity.

######  Definition 2.5.

Let $M$ be a graded Lagrangian in $\mathbb{C}^{2}$. We define the _excess_ of
$M$ to be

(2.4)
$\mathcal{A}(M)=\int_{M}e^{-|\mathbf{x}|^{2}/4}-2\int_{\mathbb{R}^{2}}e^{-|\mathbf{x}|^{2}/4}+\inf_{\theta_{0}}\int_{M}|\theta-\theta_{0}|^{2}e^{-|\mathbf{x}|^{2}/4}.$

Note that we allow $\mathcal{A}$ to be negative. We also observe that
$\mathcal{A}$ is monotonically decreasing along the rescaled flow by Huisken’s
montonicity formula, since it is an infimum of a family of decreasing
functions.

In the conditions below and throughout we let $B_{R}(\mathbf{x})$ denote the
ball of radius $R$ about $\mathbf{x}\in\mathbb{C}^{2}$ and let
$B_{R}=B_{R}(\mathbf{0})$.

#### Condition (‡)

We assume that there is a small $c_{0}>0$ (to be chosen later) so that

(‡) $\mathcal{A}(M_{T_{0}})-\mathcal{A}(M_{T_{1}})<c_{0}.$

#### Condition (*)

We assume that $M$ is exact in $B_{1}$, i.e. the Liouville form $\lambda$
satisfies

($\ast$) $\int_{\gamma}\lambda=0$

for every closed loop $\gamma\subset M\cap B_{1}$. In addition we assume $M$
is connected in $B_{1}$, i.e. $M\cap B_{1}$ cannot be written as the union of
two nonempty submanifolds.

We shall see later that Conditions (‡ ‣ 2.3) and ($\ast$ ‣ 2.3) hold along the
rescaled flow when our original Lagrangian mean curvature flow develops a
finite time singularity with tangent flow given by a special Lagrangian union
of two transverse planes: see Lemma 8.1.

##  3\. Distance to planes

We will now suppose that the flow $M_{\tau}$, defined for
$\tau\in[T_{0},T_{1}]$, is close to a special Lagrangian transverse union
$V_{0}=P_{0,1}\cup P_{0,2}$ of planes through $\mathbf{0}$ in
$\mathbb{C}^{2}$, in a way that will be specified later.

###  3.1. Nearby pairs of planes and complex lines

It will be useful to restrict the set of pairs of planes we are considering to
be those sufficiently close to $V_{0}$ in the following sense.

######  Definition 3.1.

Fix a small number $c_{1}>0$ and let $\mathcal{V}$ be the space of all special
Lagrangian unions $V=P_{1}\cup P_{2}$ of two planes through the origin, such
that the angles between $P_{i}$ and $P_{0,i}$ are smaller than $c_{1}$, i.e.
the cones in $\mathcal{V}$ are small deformations of $V_{0}$. We further
assume that $c_{1}$ is chosen sufficiently small so that elements $P_{1}\cup
P_{2}$ in $\mathcal{V}$ are bounded away from the multiplicity two planes,
i.e. there exists $c_{2}>0$ so the angle between $P_{1}$ and $P_{2}$ is at
least $c_{2}$. Below we may further shrink $c_{1}$, however it will always be
a constant that depends on the choice of $V_{0}$ only.

For any $V\in\mathcal{V}$ we denote the Lagrangian angle of $V$ by
$\theta_{V}$. We also let $\mathcal{V}^{\prime}\subset\mathcal{V}$ be defined
in the same way as $\mathcal{V}$ but using the constant $c_{1}/2$ instead of
$c_{1}$.

We can choose a hyperkähler rotation of the complex structure of
$\mathbb{C}^{2}$ depending on $\theta_{V}$, and complex coordinates $z,w$,
such that $V$ is defined by $zw=0$. We have the following basic observation,
where the notion of Lagrangian and Lagrangian angle refers to the original
Calabi–Yau structure on $\mathbb{C}^{2}$.

######  Lemma 3.2.

There is a constant $C>0$, independent of $V\in\mathcal{V}$, such that on any
oriented Lagrangian $2$-plane $P$ in $\mathbb{C}^{2}$ with Lagrangian angle
$\theta$ we have

$|\nabla z\cdot\nabla w|\leq C|\theta-\theta_{V}|.$

###### Proof.

Let $\mathcal{L}$ denote the space of oriented Lagrangian 2-planes in
$\mathbb{C}^{2}$, and let $\theta:\mathcal{L}\to S^{1}$ denote the Lagrangian
angle function. Then $\mathrm{d}\theta$ is nowhere vanishing.

Let $P\in\mathcal{L}$ with $\theta=\theta_{V}$. After the hyperkähler rotation
which makes $V$ a pair of complex lines, $P$ also becomes a complex line on
which $z,w$ and $zw$ are all holomorphic, and in particular harmonic. We
therefore have

$2\nabla z\cdot\nabla w=\Delta(zw)-z\Delta(w)-w\Delta(z)=0.$

Thus, $\nabla z\cdot\nabla w$ vanishes on the zero set of $\theta$.

Since $\mathrm{d}\theta$ is nowhere vanishing, i.e. $\theta-\theta_{V}$
vanishes to order 1 along its zero set, the result follows. ∎

We also record the following evolution equations.

######  Lemma 3.3.

We have

(3.1) $(\partial_{t}-\Delta)|zw|^{2}\leq 4|zw||\nabla z\cdot\nabla w|\,.$

Furthermore, for $\delta>0$ consider
$u_{\delta}:=\sqrt{\delta^{2}+|\theta-\theta_{0}|^{2}}$ and
$v_{\delta}:=\sqrt{\delta^{2}+|zw|^{2}}$. Then

(3.2) $(\partial_{t}-\Delta)u_{\delta}\leq
0\quad\text{and}\quad(\partial_{t}-\Delta)v_{\delta}\leq 2|\nabla z\cdot\nabla
w|\,.$

###### Proof.

We have that $z,w$ satisfy the heat equation, so

$(\partial_{t}-\Delta)zw=-2\nabla z\cdot\nabla w\,.$

Thus we can compute

(3.3)
$\begin{split}(\partial_{t}-\Delta)|zw|^{2}&=(\partial_{t}-\Delta)(\overline{zw}\,zw)\\\
&=-2(\overline{\nabla z\cdot\nabla w}\,zw+\overline{zw}\,\nabla z\cdot\nabla
w)-2\overline{\nabla(zw)}\cdot\nabla(zw)\\\ &\leq 4|zw||\nabla z\cdot\nabla
w|\,.\end{split}$

For $v_{\delta}$ we compute using (3.3) and Kato’s inequality

$\begin{split}(\partial_{t}-\Delta)v_{\delta}&=\frac{1}{2v_{\delta}}(\partial_{t}-\Delta)|zw|^{2}+\frac{1}{4v_{\delta}^{3}}|\nabla|zw|^{2}|^{2}\\\
&=-\frac{1}{v_{\delta}}\left((\overline{\nabla z\cdot\nabla
w}\,zw+\overline{zw}\,\nabla z\cdot\nabla
w)+\overline{\nabla(zw)}\cdot\nabla(zw)\right)\\\
&\quad+\frac{|zw|^{2}}{v^{3}_{\delta}}|\nabla|zw||^{2}\\\
&\leq-\frac{1}{v_{\delta}}\left((\overline{\nabla z\cdot\nabla
w}\,zw+\overline{zw}\,\nabla z\cdot\nabla w)\right)\leq 2|\nabla z\cdot\nabla
w|\,.\end{split}$

The computation for $u_{\delta}$ is analogous. ∎

###  3.2. Distance to planes and monotonicity

We now introduce two notions of distance to a pair of planes
$V\in\mathcal{V}$.

######  Definition 3.4.

Given any $V\in\mathcal{V}$, let $d_{V}$ denote the distance function in
$\mathbb{C}^{2}$ to $V$ and define the $L^{2}$-distance $I_{V}(M_{\tau})$ of
$M_{\tau}$ to $V$ by

$I_{V}(M_{\tau})^{2}=\int_{M_{\tau}}\Big{(}|\mathbf{x}|^{2}d_{V}^{2}+|\theta-\theta_{V}|^{2}\Big{)}\,e^{-|\mathbf{x}|^{2}/4}.$

It is convenient to introduce a variant $D_{V}$ of this, which also encodes
graphicality. Suppose that on the annulus $B_{2}\setminus B_{1}$ the surface
$M_{\tau}$ is the graph of a (vector-valued) function $u$ over $V$ with
$|u|,|\nabla u|\leq c_{1}$, for a small $c_{1}$ as in Definition 3.1. Then we
set $D_{V}(M_{\tau})=I_{V}(M_{\tau})$. If $M_{\tau}$ is not such a small graph
over $V$ on the annulus, then we let $D_{V}(M_{\tau})=\infty$.

Lemma 3.3 leads to the following properties of $I_{V}$.

######  Lemma 3.5.

The $L^{2}$-distance to $V\in\mathcal{V}$ has the following properties.

  1. (1)

There is $C>0$, depending only on $\mathcal{V}$, so that

$I_{V}(M_{\tau+s})\leq CI_{V}(M_{\tau})\quad\text{for any $s\in[0,1]$.}$

  2. (2)

For any $0<\delta\leq s\leq 1$ there are $p>1$ and $C>0$ (both depending on
$\delta$), such that on $M_{\tau+s}$ we have

$|\mathbf{x}|^{2}d_{V}^{2}+|\theta-\theta_{V}|^{2}\leq
Ce^{\frac{|\mathbf{x}|^{2}}{4p}}I_{V}(M_{\tau})^{2}.$

  3. (3)

There is a $C>0$ satisfying the following. For any $\gamma>0$ there is a
compact subset $K_{\gamma}\subset\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$ such
that if

$\int_{K_{\gamma}\cap
M_{\tau+1}}(|\mathbf{x}|^{2}d_{V}^{2}+|\theta-\theta_{V}|^{2})e^{-|\mathbf{x}|^{2}/4}\leq\epsilon^{2},$

then

$I_{V}(M_{\tau+1})\leq C\big{(}\epsilon+\gamma I_{V}(M_{\tau})\big{)}.$

###### Proof.

As in Lemma 3.2 we choose complex coordinates $z,w$ for a hyperkähler-rotated
complex structure on $\mathbb{C}^{2}$ such that $V$ is given by $zw=0$. It
follows from Lemma 3.2 and (3.3) that

$\begin{split}(\partial_{t}-\Delta)|zw|^{2}&\leq 4|zw||\nabla z\cdot\nabla
w|\\\ &\leq C|zw||\theta-\theta_{V}|\\\ &\leq
C(|zw|^{2}+|\theta-\theta_{V}|^{2}),\end{split}$

and so

(3.4) $(\partial_{t}-\Delta)(|zw|^{2}+|\theta-\theta_{V}|^{2})\leq
C(|zw|^{2}+|\theta-\theta_{V}|^{2}).$

The monotonicity formula along the unrescaled flow then implies that for
$-T\leq t_{1}<t_{2}<0$

$\int_{L_{t_{2}}}(|zw|^{2}+|\theta-\theta_{V}|^{2})\rho_{\mathbf{0},0}\leq
e^{C(t_{2}-t_{1})}\int_{L_{t_{1}}}(|zw|^{2}+|\theta-\theta_{V}|^{2})\rho_{\mathbf{0},0}\,.$

Rescaling the flow parabolically such that $\tau$ corresponds to $t_{1}=-1$
(i.e. translating $\tau$ to $0$ for the rescaled flow) this implies that for
the rescaled flow

$\int_{M_{\tau+s}}(e^{-2s}|zw|^{2}+|\theta-\theta_{V}|^{2})e^{-|\mathbf{x}|^{2}/4}\leq
e^{C}\int_{M_{\tau}}(|zw|^{2}+|\theta-\theta_{V}|^{2})e^{-|\mathbf{x}|^{2}/4}\,.$

To deduce property (1) we claim that $|{\bf x}|d_{V}$ is uniformly equivalent
to $|zw|$, with a constant depending on $c_{2}$ in the definition of
$\mathcal{V}$. Indeed, the distance $d_{V}$ is uniformly equivalent to
$\min\\{|z|,|w|\\}$, while $|{\bf x}|$ is uniformly equivalent to
$\max\\{|z|,|w|\\}$, and $\min\\{|z|,|w|\\}\max\\{|z|,|w|\\}=|zw|$.

Let $s>0$. A sharper estimate as in property (1) is obtained by using Ecker’s
log-Sobolev inequality [7, Theorem 3.4] along the rescaled flow $M_{\tau}$. We
first note that, similar to above, we can use (3.2) and Lemma 3.2 to estimate

$(\partial_{t}-\Delta)(u_{\delta}+v_{\delta})\leq C(u_{\delta}+v_{\delta})\,.$

The results in [7] then imply that for some $\hat{p}>2$ depending on $s$, we
have

$\left(\int_{M_{\tau+s/2}}(u_{\delta}+v_{\delta})^{\hat{p}}e^{-|\mathbf{x}|^{2}/4}\right)^{1/\hat{p}}\leq
C\left(\int_{M_{\tau}}(u_{\delta}+v_{\delta})^{2}e^{-|\mathbf{x}|^{2}/4}\right)^{\frac{1}{2}}$

for $C$ depending on $\mathcal{V}$ and $\delta_{0}$. Letting $\delta\searrow
0$ then yields for $f=|zw|^{2}+|\theta-\theta_{V}|^{2}$ and $p=\hat{p}/2>1$
that

(3.5) $\left(\int_{M_{\tau+s/2}}f^{p}e^{-|\mathbf{x}|^{2}/4}\right)^{1/p}\leq
C\int_{M_{\tau}}fe^{-|\mathbf{x}|^{2}/4}\leq CI_{V}(M_{\tau})^{2}\,.$

Let us now consider the mean curvature flow $L_{t}$, with initial condition
$L_{-1}=M_{\tau+s/2}$. Consider any $(\mathbf{x}_{0},t_{0})$ with
$t_{0}\in(-1,0)$ and let $\rho_{\mathbf{x}_{0},t_{0}}$ be the backwards heat
kernel centered at $(\mathbf{x}_{0},t_{0})$. Using the monotonicity formula
again for $f^{p}$ with $f=|zw|^{2}+|\theta-\theta_{V}|^{2}$, together with
(3.4), we have the pointwise estimate

(3.6) $f(\mathbf{x}_{0},t_{0})^{p}\leq
C\int_{L_{-1}}f^{p}\rho_{\mathbf{x}_{0},t_{0}}.$

At the same time from (3.5) we have that

(3.7) $\int_{L_{-1}}f^{p}\rho_{\mathbf{0},0}\leq CI_{V}(M_{\tau})^{2p}.$

To estimate $f(\mathbf{x}_{0},t_{0})^{p}$ we therefore need to bound
$\rho_{\mathbf{x}_{0},t_{0}}/\rho_{\mathbf{0},0}$ at $t=-1$. We have

(3.8)
$\frac{\rho_{\mathbf{x}_{0},t_{0}}}{\rho_{\mathbf{0},0}}(\mathbf{x},-1)=C_{t_{0}}\exp\Big{(}-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4(t_{0}+1)}+\frac{|\mathbf{x}|^{2}}{4}\Big{)},$

for a $t_{0}$-dependent constant $C_{t_{0}}$ (which is uniformly bounded as
long as $t_{0}$ is bounded away from $-1$). Since

(3.9)
$\displaystyle(t_{0}+1)|\mathbf{x}|^{2}-|\mathbf{x}-\mathbf{x}_{0}|^{2}=t_{0}|\mathbf{x}+t_{0}^{-1}\mathbf{x}_{0}|^{2}-t_{0}^{-1}(t_{0}+1)|\mathbf{x}_{0}|^{2}$

and $t_{0}\in(-1,0)$, we have

(3.10)
$-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4(t_{0}+1)}+\frac{|\mathbf{x}|^{2}}{4}\leq-\frac{|\mathbf{x}_{0}|^{2}}{4t_{0}}.$

It follows from (3.6), (3.7), (3.8) and (3.10) that

$f(\mathbf{x}_{0},t_{0})^{p}\leq
C_{t_{0}}\exp\left(-\frac{|\mathbf{x}_{0}|^{2}}{4t_{0}}\right)\int_{L_{-1}}f^{p}\rho_{\mathbf{0},0}\leq
C_{t_{0}}\exp\left(-\frac{|\mathbf{x}_{0}|^{2}}{4t_{0}}\right)I_{V}(M_{\tau})^{p}.$

Scaling this estimate back to the rescaled flow $M_{\tau}$ (i.e. scaling
$\mathbf{\bar{x}}=(-t_{0})^{-1/2}\mathbf{x}$ at $t=t_{0}$ and $e^{-s}=-t_{0}$)
we then have the following estimate on $M_{\tau+s}$:

$(e^{2s}|zw|^{2}+|\theta-\theta_{V}|^{2})^{p}\leq
Ce^{|\mathbf{x}|^{2}/4}I_{V}(M_{\tau})^{2p},$

and so

$|zw|^{2}+|\theta-\theta_{V}|^{2}\leq
Ce^{|\mathbf{x}|^{2}/4p}I_{V}(M_{\tau})^{2}$

as claimed in property (2).

To see property (3) we use (3.5) again. With $s=2$ it implies that for any
compact set $K$, with suitable $p,p^{\prime}>1$ we have

(3.11) $\displaystyle\int_{M_{\tau+1}\setminus K}fe^{-|\mathbf{x}|^{2}/4}$
$\displaystyle\leq\left(\int_{M_{\tau+1}\setminus
K}e^{-|\mathbf{x}|^{2}/4}\right)^{1/{p^{\prime}}}\left(\int_{M_{\tau+1}\setminus
K}f^{p}e^{-|\mathbf{x}|^{2}/4}\right)^{1/p}$
$\displaystyle\leq\mathrm{Vol}(M_{\tau+1}\setminus
K,e^{-|\mathbf{x}|^{2}/4})^{1/{p^{\prime}}}CI_{V}(M_{\tau})^{2}.$

Together with the integral bound on $K$ at $\tau+1$ assumed in the statement
of property (3), if we choose $K$ sufficiently large, depending on $\gamma$,
we get

$I_{V}(M_{\tau+1})^{2}\leq\epsilon^{2}+C\gamma^{2}I_{V}(M_{\tau})^{2},$

as required. ∎

Next we control the growth of the distance $D_{V}$.

######  Proposition 3.6.

There is a constant $C>0$ such that if the constant $c_{0}$ in Condition (‡ ‣
2.3) and $D_{V}(M_{\tau})$ are sufficiently small, then

(3.12) $D_{V}(M_{\tau+s})\leq CD_{V}(M_{\tau})\quad\text{for $s\in[0,1]$,}$

as long as $\tau\in[T_{0}+1,T_{1}-2]$.

###### Proof.

Given the growth bound for $I_{V}$ in Lemma 3.5, we only need to ensure that
if $M_{\tau}$ is a $C^{1}$-small graph over $V$ on the annulus $B_{2}\setminus
B_{1}$, then so is $M_{\tau+s}$ for $s\in[0,1]$. We will show that this
follows from the estimate for the excess, defined in Definition 2.5.

Recall that $\theta$ satisfies the heat equation along the Lagrangian mean
curvature flow and $|\nabla\theta|=|\mathbf{H}|$. It follows from Huisken’s
monotonicity formula that we have

(3.13)
$\displaystyle\mathcal{A}(M_{T_{0}})-\mathcal{A}(M_{T_{1}})\geq\int_{T_{0}}^{T_{1}}\int_{M_{\tau}}\left(2|\mathbf{H}|^{2}+\left|\mathbf{H}+\frac{\mathbf{x}^{\perp}}{2}\right|^{2}\right)e^{-|\mathbf{x}|^{2}/4}.$

We can then argue by contradiction, and suppose that we have a sequence of
rescaled flows $M^{i}_{\tau}$, satisfying Condition (‡ ‣ 2.3) with
corresponding constants $c_{0}\to 0$, however the conclusion (3.12) does not
hold. We can assume that we are working at $\tau=0$ and that also
$D_{V}(M^{i}_{0})\to 0$.

As in Neves [15, Theorem A], up to choosing a subsequence, the flows $M^{i}$
converge to a static limit flow $M^{\infty}$ given by a union of planes
$V^{\prime}$ for $\tau\in(-1,2)$. The assumption that $D_{V}(M^{i}_{0})\to 0$
implies that $M^{i}_{0}$ is the graph of a (vector-valued) function $u_{i}$
over $V$ on the annulus $B_{2}\setminus B_{1}$, with $|u_{i}|,|\nabla
u_{i}|<c_{0}$ and $|u_{i}|\to 0$. Note that this implies that $V^{\prime}=V$.
By White’s regularity theorem [30] the convergence is smooth on
$(B_{3}\setminus B_{1/2})\times(-1/2,3/2)$. This implies that, for $i$
sufficiently large, $M^{i}_{\tau}$ is the graph of a function $|u_{i}|,|\nabla
u_{i}|<c_{0}$ over $V$ on $B_{2}\setminus B_{1}$ for all $t\in[0,1]$.

Thus Lemma 3.5 implies $D_{V}(M^{i}_{s})\leq CD_{V}(M^{i}_{0})$ for all
$s\in[0,1]$ as required. ∎

######  Remark 3.7.

From now on we will assume that $c_{0}>0$ in Condition (‡ ‣ 2.3) is small
enough for Proposition 3.6 to hold.

For later use, we will need to compare the distances $D_{V}$ as we vary the
cone $V\in\mathcal{V}$. For $V,V^{\prime}\in\mathcal{V}$ let $d(V,V^{\prime})$
denote the Hausdorff distance between $V\cap B_{1}$ and $V^{\prime}\cap
B_{1}$.

######  Lemma 3.8.

There is a constant $C$ such that if $V,V^{\prime}\in\mathcal{V}$ and
$D_{V}(M_{\tau})$ is sufficiently small, then

$D_{V^{\prime}}(M_{\tau})\leq C(D_{V}(M_{\tau})+d(V,V^{\prime})).$

###### Proof.

For any $\mathbf{x}\in B_{1}$ we have $d_{V^{\prime}}(\mathbf{x})\leq
d_{V}(\mathbf{x})+d(V,V^{\prime})$, so by scaling we have
$d_{V^{\prime}}(\mathbf{x})\leq d_{V}(\mathbf{x})+|\mathbf{x}|d(V,V^{\prime})$
for $\mathbf{x}\in\mathbb{C}^{2}$. We also have
$|\theta_{V}-\theta_{V^{\prime}}|\leq Cd(V,V^{\prime})$ for a constant $C$.
Combining these observations we get

$I_{V^{\prime}}(M_{\tau})\leq C(I_{V}(M_{\tau})+d(V,V^{\prime})^{2}).$

To get the same estimate for $D_{V},D_{V^{\prime}}$ we just need to ensure
that $M_{\tau}$ is graphical over $V^{\prime}$ on $B_{2}\setminus B_{1}$ for
$u$ with $|u|,|\nabla u|$ sufficiently small as in Definition 3.4. This
follows using Condition (‡ ‣ 2.3) as in the proof of Proposition 3.6. ∎

##  4\. Graphicality

###  4.1. Graphicality and distance to planes

We want to see that an estimate for $D_{V}(M_{\tau-1})$ can be used to show
graphicality of $M_{\tau}$ on a much larger region than just the fixed annulus
$B_{2}\setminus B_{1}$. For this we first need the following.

######  Lemma 4.1.

Let $L_{t}$ be a mean curvature flow of surfaces in the ball
$B_{2}\subset\mathbb{C}^{2}$ for $t\in[-4,0]$, with uniform area bound
$\mathcal{H}^{2}(L_{t})\leq C$. Suppose that $S$ is an embedded smooth surface
passing through the origin, and with second fundamental form satisfying
$|A|\leq\delta$ for some $\delta>0$. Assume that in addition we have the
following.

  1. (1)

$L_{t}$ is contained in the $\delta$-neighbourhood of $S$.

  2. (2)

On the parabolic ball $[-1/4,0]\times B_{1/2}$ the flow $L_{t}$ is the graph
of a (vector-valued) function $u$ over $S$ with $|u|<\delta$ and $|\nabla
u|<1$.

  3. (3)

We have the estimate

(4.1) $\int_{-4}^{0}\int_{L_{t}\cap B_{2}}|\mathbf{H}|^{2}<\delta.$

Let $\varepsilon>0$ be given. Then if $\delta$ is chosen sufficiently small,
the flow $L_{t}$ is the graph of $u$ over $S$ on the parabolic ball
$[-1,0]\times B_{1}$, with $|\nabla u|\leq\varepsilon$.

###### Proof.

We argue by contradiction. Suppose that we have a sequence of flows
$L_{t}^{i}$ satisfying the assumptions with corresponding constants
$\delta_{i}\to 0$, and surfaces $S_{i}$, but the claimed graphicality fails
for all $i$. Up to choosing a subsequence we can assume that the $S_{i}$
converge to a plane $S_{\infty}$. We can furthermore assume that the flows
$(L^{i}_{t})_{-4\leq t\leq 0}$ converge to a unit regular Brakke flow
$(\mu_{t})_{-4\leq t\leq 0}$ on $B_{2}$, supported on $S_{\infty}$. The
constancy theorem implies that $\mu_{t}$ agrees with $S_{\infty}$ up to an
integer multiplicity, which is monotonically decreasing in time.

Note that the graphicality assumption (2) together with interior parabolic
estimates implies that $L^{i}_{t}\to S_{\infty}$ smoothly on $[-1/4,0]\times
B_{1/2}$ , and thus $\mu_{t}$ agrees with $S_{\infty}$ with multiplicity 1 on
$[-1/4,0]\times B_{2}$.

At the same time, for a cutoff function $\chi$ supported in $B_{2}$ we have

$\bigg{|}\partial_{t}\int_{L^{i}_{t}}\chi\bigg{|}=\bigg{|}\int_{L^{i}_{t}}-\chi|\mathbf{H}|^{2}+\langle
D\chi,\mathbf{H}\rangle\bigg{|}\leq C\int_{L^{i}_{t}\cap
B_{2}}|\mathbf{H}|^{2}+|\mathbf{H}|,$

so for any $t_{0}<t_{1}$ we have

$\bigg{|}\int_{L^{i}_{t_{0}}}\chi-\int_{L^{i}_{t_{1}}}\chi\bigg{|}\leq
C\int_{-4}^{0}\int_{L_{t}^{i}\cap B_{2}}|\mathbf{H}|^{2}+|\mathbf{H}|\to 0.$

It follows that $(\mu_{t})$ is static on $[-4,0]$ and thus agrees with
$S_{\infty}$ with multiplicity one. From White’s regularity theorem [30] we
deduce that for sufficiently large $i$ the flows $L^{i}_{t}$ converge smoothly
on $[-1,0]\times B_{1}$, which implies the required graphicality. ∎

Using Lemma 4.1 repeatedly we can extend the graphicality of our flow to
larger and larger regions, as long as it stays in a small neighbourhood of a
smooth surface and we can control the integral of $|\mathbf{H}|^{2}$. For the
latter we have the following result.

######  Lemma 4.2.

There are constants $C,r_{0},\alpha>0$ and $p>1$ satisfying the following.
Suppose that $L_{t}$ is a Lagrangian mean curvature flow for $t\in[-1,0)$,
where $L_{-1}=M_{\tau}$ for some $\tau$. Then whenever $r\leq r_{0}$ and
$t_{0}\in(-3/4,-1/4)$ with $[t_{0}-r^{2},t_{0}+r^{2}]\subset[-3/4,-1/4]$ and

(4.2) $\displaystyle
D_{V}(M_{\tau-1})^{2}\exp\Big{(}-\frac{|\mathbf{x}_{0}|^{2}}{4pt_{0}}\Big{)}\leq
1,$

we have

(4.3) $\displaystyle r^{-2}\int_{t_{0}-r^{2}}^{t_{0}}\int_{L_{t}\cap
B_{r}(\mathbf{x}_{0})}|\mathbf{H}|^{2}\leq CD_{V}(M_{\tau-1})^{\alpha}.$

###### Proof.

We consider the monotonicity formula applied with the backwards heat kernel
$\rho_{\mathbf{x}_{0},t_{0}+r^{2}}$ centered at
$(\mathbf{x}_{0},t_{0}+r^{2})$. Using $|\nabla\theta|^{2}=|\mathbf{H}|^{2}$ we
have that

(4.4)
$\displaystyle\int_{t_{0}-r^{2}}^{t_{0}}\int_{L_{t}}2|\mathbf{H}|^{2}\rho_{\mathbf{x}_{0},t_{0}+r^{2}}\leq\int_{L_{-1}}|\theta-\theta_{V}|^{2}\rho_{\mathbf{x}_{0},t_{0}+r^{2}}.$

By the pointwise estimate in Lemma 3.5 there is some $p_{1}>1$ close to $1$
and $C>0$ such that on $L_{-1}$ we have

(4.5) $\displaystyle|\theta-\theta_{V}|^{2}\leq
Ce^{|\mathbf{x}|^{2}/4p_{1}}D_{V}(M_{\tau-1})^{2}.$

Therefore we need to estimate the integral

$\int_{L_{-1}}\exp\Big{(}\frac{|\mathbf{x}|^{2}}{4p_{1}}-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4(t_{0}+r^{2}+1)}\Big{)}.$

Let $p_{2}\in(1,p_{1})$. Arguing as in (3.10) we find that

(4.6)
$\displaystyle\frac{|\mathbf{x}|^{2}}{4p_{2}}-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4(t_{0}+r^{2}+1)}\leq\frac{|\mathbf{x}_{0}|^{2}}{4(p_{2}-1-t_{0}-r^{2})}.$

The integral of
$\exp\Big{(}\frac{|\mathbf{x}|^{2}}{4p_{1}}-\frac{|\mathbf{x}|^{2}}{4p_{2}}\Big{)}$
on $L_{-1}$ is uniformly bounded, so combining (4.4), (4.5) and (4.6) gives

$\int_{t_{0}-r^{2}}^{t_{0}}\int_{L_{t}}|\mathbf{H}|^{2}\rho_{\mathbf{x}_{0},t_{0}+r^{2}}\leq
CD_{V}(M_{\tau-1})^{2}\exp\Big{(}\frac{|\mathbf{x}_{0}|^{2}}{4(p_{2}-1-t_{0}-r^{2})}\Big{)}.$

It remains to choose $p$ close enough to 1 in the bound (4.2) and
$\alpha,r_{0}>0$ small enough so that for $r\leq r_{0}$ we have

$D_{V}(M_{\tau-1})^{2-\alpha}\exp\Big{(}\frac{|\mathbf{x}_{0}|^{2}}{4(p_{2}-1-t_{0}-r^{2})}\Big{)}\leq
1.$

Using (4.2), this follows if

(4.7)
$\displaystyle\frac{2-\alpha}{2}\frac{1}{4pt_{0}}+\frac{1}{4(p_{2}-1-t_{0}-r^{2})}<0.$

Note that in the limiting case, when $\alpha$ and $r_{0}$ are both $0$ and
$p=1$, (4.7) reduces to the inequality $p_{2}>1$, which is satisfied. We can
therefore arrange that (4.7) also holds for suitable $\alpha,r_{0},p$.
Combining this with the fact that on $[t_{0}-r^{2},t_{0}]\times
B_{r}(\mathbf{x}_{0})$ the function $\rho_{\mathbf{x}_{0},t_{0}+r^{2}}$ is
bounded from below by a positive multiple of $r^{-2}$ yields (4.3). ∎

###  4.2. Graphicality scale

Note that under the correspondence between the mean curvature flow and its
rescaled version, if $L_{-1}=M_{\tau}$, then $M_{\tau+1}=e^{1/2}L_{-e^{-1}}$.
In particular, setting $t_{0}=-e^{-1}$ and, for $\mathbf{x}_{0}\in
L_{-e^{-1}}$, letting $\tilde{\mathbf{x}}_{0}=e^{1/2}\mathbf{x}_{0}$, we see
that $|\mathbf{x}_{0}|^{2}/4pt_{0}=|\tilde{\mathbf{x}}_{0}|^{2}/4p$. This
motivates the following.

######  Definition 4.3.

We choose $p_{0}>1$ smaller than the $p>1$ in both Lemma 3.5 for $s=1$ and
Lemma 4.2. For any (small) $d>0$, we define $R_{d}>0$ so that

(4.8) $d^{2}e^{R_{d}^{2}/4p_{0}}=1.$

The $R_{d}$ just defined will be the radius up to which we can obtain good
graphicality of our flow. This motivates the following definition.

######  Definition 4.4.

We say that $M_{s}$ has _good graphicality_ on an annulus
$A:=B_{r_{2}}\setminus B_{r_{1}}$ for $0<r_{1}<r_{2}<\infty$ over $V$ if
$M_{s}\cap A$ is the graph of a (vector-valued) function $u$ over $V$ with
$|u|,|\nabla u|\leq c_{1}$, where $c_{1}$ is as in Definition 3.1.

We have the following.

######  Proposition 4.5.

Use the notation of Definition 4.3. There are constants $\epsilon,A>0$ and
$p>p_{0}$ such that if $D_{V}(M_{\tau-1})=d<\epsilon$, then $M_{\tau+1}$ is
the graph of a (vector-valued) function $u$ over $V$ on the annulus
$B_{R_{d}}\setminus B_{Ad^{1/2}}$, satisfying the following estimates:

  * •

for $1<|\mathbf{x}|<R_{d}$ we have $|u|,|\nabla u|\leq
Ae^{|\mathbf{x}|^{2}/8p}d$;

  * •

for $Ad^{1/2}<|\mathbf{x}|<2$ we have $|\mathbf{x}|^{-1}|u|,|\nabla u|\leq
Ad|\mathbf{x}|^{-2}$.

###### Proof.

Note that by Proposition 3.6, $M_{s}$ has good graphicality over $V$ (in the
sense of Definition 4.4) on $B_{2}\setminus B_{1}$ for $s\in[\tau,\tau+1]$. In
addition, property (2) in Lemma 3.5 shows that there is $C>0$ and $p>p_{0}$
such that

(4.9) $\displaystyle d_{V}^{2}\leq
C|\mathbf{x}|^{-2}e^{|\mathbf{x}|^{2}/4p}d^{2}$

on $M_{s}$ for $s\in[\tau,\tau+1]$. Therefore, by the definition of $R_{d}$ in
(4.8), if $1<|\mathbf{x}|<R_{d}$ we have

(4.10) $d_{V}^{2}\leq Ce^{R_{d}^{2}(p_{0}-p)/4pp_{0}}.$

In particular, recalling that $p>p_{0}$, if we let $d$ be small, so that
$R_{d}$ is large by (4.8), we can ensure that $d_{V}$ is as small as we like
on the annulus $B_{R_{d}}\setminus B_{1}$ along the rescaled flow $M_{s}$ for
$s\in[\tau,\tau+1]$.

Applying Lemma 4.2 with $t_{0}=-e^{-1}$ we see that for all $\mathbf{x}_{0}$
with $\exp(-\tfrac{|\mathbf{x}_{0}|^{2}}{4pt_{0}})\leq d^{-2}$ we have

(4.11) $\displaystyle r_{0}^{-2}\int_{t_{0}-r_{0}^{2}}^{t_{0}}\int_{L_{t}\cap
B_{r_{0}}(\mathbf{x}_{0})}|\mathbf{H}|^{2}\leq Cd^{\alpha}.$

Rescaling the flow parabolically such that $L_{-1}=M_{\tau+1}$ (i.e. scaling
parabolically by $e^{1/2}$) this implies that for any $\mathbf{x}_{0}\in
B_{R_{d}}$ we have a backwards parabolic ball $[-1-r_{0}^{2},-1]\times
B_{r_{0}}(\mathbf{x}_{0})$ on which the spacetime integral of
$|\mathbf{H}|^{2}$ is bounded by $Cr_{0}^{2}d^{\alpha}$. Note that the
constant changes by a controlled factor due to the rescaling.

Recall that the second fundamental form of $V$ vanishes and that $M_{s}$ has
good graphicality over $V$ on $B_{2}\setminus B_{1}$ for $s\in[\tau,\tau+1]$.
Since $L_{-1}=M_{\tau+1}$ this implies that $L_{t}$ is the graph of a (vector-
valued) function $\hat{u}$ over $V$ on $\sqrt{-t}(B_{2}\setminus B_{1})$ with
$|\hat{u}|\leq\sqrt{-t}\cdot c_{1},|\nabla\hat{u}|\leq c_{1}$ for
$t\in[-e,-1]$.

From (4.11), scaling the flow by $4r_{0}^{-1}$ and shifting the origin
accordingly, by taking $d$ sufficiently small, we see for every
$\mathbf{x}_{0}\in B_{4r_{0}^{-1}R_{d}}\setminus B_{4r_{0}^{-1}}$ that (4.11)
implies that the hypotheses (1)–(3) of Lemma 4.1 (with $\varepsilon=c_{1}$)
are satisfied. Thus, undoing the scaling and shifting of the origin, the good
graphicality of $L_{t}$ for $t\in[-1-r_{0}^{2},-1]$ over $V$ can be propagated
out from the ball $\sqrt{-t}(B_{2}\setminus B_{1})$ if $d$ is sufficiently
small, to the annulus $\sqrt{-t}(B_{R_{d}}\setminus B_{1})$ on which $L_{t}$
is still in a sufficiently small neighbourhood of $V$. Note that once we have
good graphicality of the flow on a parabolic ball, then on a smaller parabolic
ball $|\nabla\hat{u}|$ can be bounded in terms of $|\hat{u}|$ by standard
parabolic theory. This introduces the constant $A$ in the claimed estimates.

The argument for extending graphicality to the annulus $B_{2}\setminus
B_{Ad^{1/2}}$ is similar. Here the distance bound (4.9) implies that
$d_{V}\leq C|\mathbf{x}|^{-1}d$ on $L_{t}$ for $t\in[-1,-e]$ (recalling that
$L_{-1}=M_{\tau+1}$). Suppose that $\mathbf{x}_{0}\in V$ with
$Ad^{1/2}\leq|\mathbf{x}_{0}|<2$, and let $r=A^{-1}|\mathbf{x}_{0}|$. If $A$
is sufficiently large, then $V\cap B_{r}(\mathbf{x}_{0})$ is a plane, so it
has zero second fundamental form. In addition for $t\in[-1,e]$ the distance
from $L_{t}$ to $V$ on $B_{r}(\mathbf{x}_{0})$ is bounded by
$C|\mathbf{x}_{0}|^{-1}d$, where $C$ can be chosen independent of sufficiently
large $A$. After scaling up by $r^{-1}$ the distance from $r^{-1}L_{t}$ to $V$
on $r^{-1}B_{r}(\mathbf{x}_{0})$ is bounded above by
$CA|\mathbf{x}_{0}|^{-2}d\leq CA^{-1}$ since $|\mathbf{x}_{0}|\geq Ad^{1/2}$.
By choosing $A$ large enough, we can again ensure that Lemma 4.1 can be
applied repeatedly to extend the region of good graphicality. The required
estimate for $|u|$ follows from the pointwise bound for $d_{V}$, while the
estimate for $|\nabla u|$ in the annular region $B_{2}\setminus B_{Ad^{1/2}}$
follows by standard parabolic theory and scaling parabolic balls of the form
$[t-r^{2},t]\times B_{r}(\mathbf{x})$ to unit size, where $r=|\mathbf{x}|/2$.
∎

###  4.3. Excess and distance

The graphicality bound from Proposition 4.5 implies the following estimate for
the excess $\mathcal{A}$ in (2.4) in terms of $D_{V}(M)$.

######  Proposition 4.6.

There is a small $\alpha_{1}>0$ such that if $D_{V}(M_{\tau-1})$ is
sufficiently small, then

$|\mathcal{A}(M_{\tau})|\leq D_{V}(M_{\tau-1})^{1+\alpha_{1}}.$

###### Proof.

By Definition 3.4 and Proposition 3.6 we have

$\int_{M_{\tau}}|\theta-\theta_{V}|^{2}e^{-|\mathbf{x}|^{2}/4}\leq
D_{V}(M_{\tau})^{2}\leq CD_{V}(M_{\tau-1})^{2}.$

Therefore, by formula (2.4) we only need to estimate the difference between
the Gaussian areas of $M_{\tau}$ and $V$ (recalling that $V$ is a pair of
planes).

Let $d=D_{V}(M_{\tau-1})$ and recall $p_{0}>1$ and $R_{d}>0$ given in
Definition 4.3. We also recall the constants $\epsilon,A>0$ from Proposition
4.5 and we assume that $d<\epsilon$. We further assume that $d$ is
sufficiently small so that $d^{1/10}>Ad^{1/2}$.

We study four regions separately.

####  (a) $|\mathbf{x}|>R_{d}$.

By the area growth bounds for $M_{\tau}$ we have constants $C,k>0$ such that

$\int_{M_{\tau}\setminus B_{R_{d}}}e^{-|\mathbf{x}|^{2}/4}\leq
CR_{d}^{k}e^{-R_{d}^{2}/4}.$

Once $R_{d}$ is sufficiently large,

$CR_{d}^{k}e^{-R_{d}^{2}/4}\leq e^{-R_{d}^{2}/4p_{0}}=d^{2}.$

The same integral bound also holds on $V\setminus B_{R_{d}}$ so the required
estimate (4.6) holds on this region with $\alpha_{1}=1$.

####  (b) $1<|\mathbf{x}|<R_{d}$.

Here Proposition 4.5 states that $M_{\tau}$ is the graph of $u$ over $V$, with
$|u|,|\nabla u|\leq Ae^{|x|^{2}/8p}d$, for some $p>p_{0}>1$. By the definition
of $R_{d}$ in (4.8), we have $Ae^{R_{d}^{2}/8p}d\to 0$ as $d\to 0$. It follows
that the area form $dA_{M_{\tau}}$ of $M_{\tau}$, pulled back to $V$, can be
compared to the area form $dA_{V}$ of $V$ as follows:

$\left|\frac{dA_{M_{\tau}}}{dA_{V}}-1\right|\leq
Ce^{|\mathbf{x}|^{2}/4p}d^{2},$

for some constant $C$. Note that the difference in the area forms is at least
quadratic in $u$ since $V$ has zero mean curvature. Integrating, we have

$\left|\int_{M_{\tau}\cap(B_{R_{d}}\setminus
B_{1})}e^{-|\mathbf{x}|^{2}/4}-\int_{V\cap(B_{R_{d}}\setminus
B_{1})}e^{-|\mathbf{x}|^{2}/4}\right|\leq Cd^{2}\int_{V\cap(B_{R_{d}}\setminus
B_{1})}e^{|\mathbf{x}|^{2}/4p}e^{-|\mathbf{x}|^{2}/4}.$

Since $p>1$ the last integral is bounded independently of $d$, and so the
required estimate (4.6) holds on this region too, with $\alpha_{1}=1$.

####  (c) $d^{1/10}<|\mathbf{x}|<1$.

Since we have assumed that $d^{1/10}>Ad^{1/2}$, Proposition 4.5 implies that
on this region $M_{\tau}$ is the graph of $u$ over $V$, with
$|\mathbf{x}|^{-1}|u|,|\nabla u|\leq Ad|\mathbf{x}|^{-2}$. Similarly to (b) we
can again compare the area forms:

$\left|\frac{dA_{M_{\tau}}}{dA_{V}}-1\right|\leq Cd^{2}|\mathbf{x}|^{-4}.$

Integrating, we have

$\displaystyle\Bigg{|}\int_{M_{\tau}\cap(B_{1}\setminus
B_{d^{1/10}})}e^{-|\mathbf{x}|^{2}/4}$
$\displaystyle-\int_{V\cap(B_{1}\setminus
B_{d^{1/10}})}e^{-|\mathbf{x}|^{2}/4}\Bigg{|}$ $\displaystyle\leq
Cd^{2}\int_{V\cap(B_{1}\setminus
B_{d^{1/10}})}|\mathbf{x}|^{-4}e^{-|\mathbf{x}|^{2}/4}$ $\displaystyle\leq
Cd^{2}d^{-4/10}=Cd^{8/5}.$

The required estimate (4.6) therefore holds with $\alpha_{1}=3/5$.

####  (d) $|\mathbf{x}|<d^{1/10}$

Let us write $r_{0}=d^{1/10}$ for the radius of this ball for simplicity. As
in (c), Proposition 4.5 implies that the cross section $M_{\tau}\cap\partial
B_{r_{0}}$ is an exponential normal graph in the sphere $\partial B_{r_{0}}$
of a function $\tilde{u}$ over $V\cap\partial B_{r_{0}}$, where $\tilde{u}$
satisfies $r_{0}^{-1}|\tilde{u}|,|\nabla\tilde{u}|\leq Adr_{0}^{-2}=Ad^{4/5}$.
The cross section $V\cap\partial B_{r_{0}}$ of $V$ is minimal in the sphere (a
union of geodesics), so the cross section of $M_{\tau}$ has length

(4.12) $\displaystyle\Big{|}\mathcal{H}^{1}(M_{\tau}\cap\partial
B_{r_{0}})-\mathcal{H}^{1}(V\cap\partial B_{r_{0}})\Big{|}\leq
Cd^{8/5}r_{0}=Cd^{17/10}.$

Let $V_{\tau}$ be the cone over $M_{\tau}\cap\partial B_{r_{0}}$. By (4.12),
the area of $V_{\tau}$ then satisfies

(4.13) $\displaystyle\Big{|}\mathcal{H}^{2}(V_{\tau}\cap
B_{r_{0}})-\mathcal{H}^{2}(V\cap B_{r_{0}})\Big{|}\leq
Cr_{0}d^{17/10}=Cd^{9/5}.$

On $M_{\tau}$ we have $|\theta-\theta_{V}|\leq Cd$ by property (2) in Lemma
3.5, noting that the exponential factor in property (2) can be bounded
independently of $d$ for $d$ small. Hence, up to rotating the holomorphic
volume form $\Omega$ so that we can assume $\theta_{V}=0$, we have that

$\mathrm{Re}\,\Omega|_{M_{\tau}}\leq
dA_{M_{\tau}}\leq(1+Cd^{2})\mathrm{Re}\,\Omega|_{M_{\tau}}.$

Therefore

(4.14) $\displaystyle\Bigg{|}\int_{M_{\tau}\cap
B_{r_{0}}}e^{-|\mathbf{x}|^{2}/4}dA_{M_{\tau}}$
$\displaystyle-\int_{M_{\tau}\cap
B_{r_{0}}}e^{-|\mathbf{x}|^{2}/4}\mathrm{Re}\,\Omega|_{M_{\tau}}\Bigg{|}$
$\displaystyle\leq Cd^{2}\left|\int_{M_{\tau}\cap
B_{r_{0}}}e^{-|\mathbf{x}|^{2}/4}\mathrm{Re}\,\Omega|_{M_{\tau}}\right|.$

At the same time, there is a hypersurface $U_{\tau}$ in $B_{r_{0}}$ bounded by
$M_{\tau}$ and $V_{\tau}$ so that

(4.15) $\int_{M_{\tau}\cap
B_{r_{0}}}e^{-|\mathbf{x}|^{2}/4}\,\mathrm{Re}\,\Omega=\int_{V_{\tau}\cap
B_{r_{0}}}e^{-|\mathbf{x}|^{2}/4}\,\mathrm{Re}\,\Omega+\int_{U_{\tau}}\mathrm{d}(e^{-|\mathbf{x}|^{2}/4})\wedge\mathrm{Re}\,\Omega,$

since $\mathrm{d}\Omega=0$. We have
$\mathrm{d}(e^{-|\mathbf{x}|^{2}/4})=-\frac{|\mathbf{x}|}{2}e^{-|\mathbf{x}|^{2}/4}\mathrm{d}|\mathbf{x}|$,
so it follows that

(4.16)
$\left|\int_{U_{\tau}}\mathrm{d}(e^{-|\mathbf{x}|^{2}/4})\wedge\mathrm{Re}\,\Omega\right|\leq
Cr_{0}\mathcal{H}^{3}(U_{\tau})=Cd^{1/10}\mathcal{H}^{3}(U_{\tau}).$

Let us write $U_{\tau}=U_{\tau,1}\cup U_{\tau,2}$, where

$U_{\tau,1}=U_{\tau}\cap B_{Ad^{1/2}}\quad\text{and}\quad
U_{\tau,2}=U_{\tau}\setminus B_{Ad^{1/2}}.$

Note that outside of $B_{Ad^{1/2}}$ the surface $M_{\tau}$ is still the graph
of some $u$ over $V$ satisfying $|\mathbf{x}|^{-1}|u|,|\nabla u|\leq
Ad|\mathbf{x}|^{-2}$ by Proposition 4.5. The cone $V_{\tau}$ is also the graph
of a function $v$ over $V$ with $|\mathbf{x}|^{-1}|v|,|\nabla v|\leq Cd^{4/5}$
by construction. It follows that, once $d$ is small, $M_{\tau}$ is the graph
of some $\tilde{u}$ over $V_{\tau}$ with
$|\mathbf{x}|^{-1}|\tilde{u}|,|\nabla\tilde{u}|\leq Ad|\mathbf{x}|^{-2}$
(recalling that $|\mathbf{x}|^{2}<d^{1/5}$ in the region under consideration).
We can then choose $U_{\tau}$ so that $U_{\tau,2}$ is the hypersurface swept
out by the graphs of $s\tilde{u}$ over $V_{\tau}$ for $s\in[0,1]$. We estimate
the volume of $U_{\tau,2}$ by the integral

$\mathcal{H}^{3}(U_{\tau,2})\leq
C\int_{Ad^{1/2}}^{d^{1/10}}\frac{Ad}{r}\,r\,dr\leq Cd^{11/10}.$

Finally we choose $U_{\tau,1}$ to minimize area, such that its boundary is
given by the union of $U_{\tau,2}\cap\partial B_{Ad^{1/2}}$, $M_{\tau}\cap
B_{Ad^{1/2}}$ and $V_{\tau}\cap B_{Ad^{1/2}}$. The isoperimetric inequality
then implies that

$\mathcal{H}^{3}(U_{\tau,1})\leq C\mathcal{H}^{2}(\partial U_{\tau,1})^{3/2}.$

To estimate $\mathcal{H}^{2}(\partial U_{\tau,1})$, consider the three pieces
of the boundary. In the sphere $\partial B_{Ad^{1/2}}$ the cross sections of
$M_{\tau}$ and $V_{\tau}$ are graphs of functions bounded by $d^{1/2}$, so
$\mathcal{H}^{2}(U_{\tau,2}\cap\partial B_{Ad^{1/2}})\leq Cd$. From (4.13) we
know that the boundary piece $V_{\tau}\cap B_{Ad^{1/2}}$ also has area bounded
by $Cd$. To control $\mathcal{H}^{2}(M_{\tau}\cap B_{Ad^{1/2}})$ it is enough
to use that $M_{\tau}\cap B_{Ad^{1/2}}$ is almost calibrated, which follows
from the fact that $|\theta-\theta_{V}|\leq Cd$ and $d$ is small. As in [15,
Lemma 7.1] we have

$\mathcal{H}^{2}(M_{\tau}\cap B_{Ad^{1/2}})\leq
C\mathcal{H}^{1}(M_{\tau}\cap\partial B_{Ad^{1/2}})^{2}\leq Cd.$

In sum, it follows that with this choice of $U_{\tau}$ we have

$\mathcal{H}^{3}(U)\leq C(d^{3/2}+d^{11/10})\leq Cd^{11/10}.$

Therefore, using (4.14), (4.15) and (4.16), we have

$\int_{M_{\tau}\cap
B_{r_{0}}}e^{-|\mathbf{x}|^{2}/4}\,dA_{M_{\tau}}\leq(1+Cd^{2})\left(\int_{V\cap
B_{r_{0}}}e^{-|\mathbf{x}|^{2}/4}+Cd^{18/10}+Cd^{12/10}\right).$

Here the term involving $d^{18/10}$ is obtained from comparing the area forms
of $V_{\tau}$ and $V$.

Combining our estimates on the different regions (a)-(d) we have

$\left|\int_{M_{\tau}}e^{-|\mathbf{x}|^{2}/4}-\int_{V}e^{-|\mathbf{x}|^{2}/4}\right|\leq
Cd^{12/10},$

and so the required estimate (4.6) for $\mathcal{A}(M_{\tau})$ holds with
$\alpha_{1}=1/10$, once $d$ is sufficiently small. ∎

##  5\. Limiting solutions of drift heat equation

In this section we show that from a sequence of rescaled flows whose initial
conditions are getting closer and closer to the pair of planes $V$, we can
extract in the limit a solution to the drift heat equation which, after
removing a leading order singular term, is defined on each plane $P_{1},P_{2}$
in $V$. We also show that this solution will satisfy good estimates.

###  5.1. Sequences of rescaled flows

We will need to pass to limits along sequences of rescaled flows. It is here
that the local exactness imposed in Condition ($\ast$ ‣ 2.3) will begin to
play a role. First we have the following, showing that Condition ($\ast$ ‣
2.3) is preserved along the flow as long as $D_{V}(M_{\tau})$ stays
sufficiently small.

######  Lemma 5.1.

There is an $\epsilon>0$ satisfying the following. Suppose the flow $M_{\tau}$
satisfies Condition (‡ ‣ 2.3) for $\tau\in[-1,T]$, with $T>0$, and
$D_{V}(M_{\tau})<\epsilon$ for $\tau\in[-1,T]$. If $M_{0}$ satisfies Condition
($\ast$ ‣ 2.3) then $M_{\tau}$ satisfies Condition ($\ast$ ‣ 2.3) for
$\tau\in[0,T]$.

###### Proof.

If $\epsilon$ is sufficiently small, then $M_{\tau}$ is a smooth graph over
$V$ on the annulus $B_{2}\setminus B_{1/2}$ for $\tau\in[0,T]$. If follows
that no additional component of the flow can appear in $B_{1}$ at any time
$\tau>0$ if $M_{0}\cap B_{1}$ is connected. As for the exactness, if
$\gamma_{t}$ denotes the evolution of a closed loop $\gamma$ along the
(unrescaled) mean curvature flow, then
$\partial_{t}\int_{\gamma_{t}}\lambda=0$. Moreover, by the graphicality
statement above, any closed loop $\gamma$ in $M_{\tau}\cap B_{2}$ is homotopic
to a closed loop in $M_{\tau}\cap B_{1}$, for $\tau\in[0,T]$. It follows from
this that if $\int_{\gamma}\lambda=0$ for any loop $\gamma\in B_{1}\cap
M_{0}$, then the same holds for any loop $\gamma\in B_{1}\cap M_{\tau}$ for
$\tau\in[0,T]$. In particular $M_{\tau}$ satisfies Condition ($\ast$ ‣ 2.3)
for $\tau\in[0,T]$. ∎

######  Proposition 5.2.

There is a constant $C>0$, depending only on $\mathcal{V}$, satisfying the
following. Let $T>0$ and let $M^{i}_{\tau}$ be a sequence of flows defined for
$\tau\in[-1,T+2]$ which satisfy Condition (‡ ‣ 2.3), and such that $M^{i}_{0}$
satisfy Condition ($\ast$ ‣ 2.3). Suppose that $D_{V}(M^{i}_{0})=:d_{i}\to 0$.

  1. (1)

There exist compact sets $K_{i}\subset\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$
exhausting $\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$, satisfying the
following. For $s\in[1,T]$ the surface $M^{i}_{s}$ is the graph of $u_{i}(s)$
over $V$ on $K_{i}$ such that, up to choosing a subsequence, the
$d_{i}^{-1}u_{i}$ converge locally smoothly on $[1,T]\times
V\setminus\\{\mathbf{0}\\}$ to a solution $u(s)$ of the drift heat equation

(5.1) $\partial_{s}u=\Delta u+\frac{1}{2}(u-\mathbf{x}\cdot\nabla u).$

  2. (2)

The limit $u$ can be identified with an exact $1$-form on
$V\setminus\\{\mathbf{0}\\}$. We write $u=(u_{1},u_{2})$, where $u_{j}$ is the
restriction of $u$ to the plane $P_{j}\setminus\\{\mathbf{0}\\}$ in terms of
$V=P_{1}\cup P_{2}$.

We can further decompose

(5.2) $\displaystyle
u=e^{s}u_{0}+\tilde{u}=e^{s}(a_{1}\mathrm{d}\ln|\mathbf{x}|,a_{2}\mathrm{d}\ln|\mathbf{x}|)+(\tilde{u}_{1},\tilde{u}_{2}),$

where $a_{1},a_{2}$ are constants such that $|a_{1}|,|a_{2}|\leq C$, and the
$\tilde{u}_{j}$ extend smoothly across the origin.

  3. (3)

We have the following estimates at $s=1$:

(5.3)
$\displaystyle\int_{V\setminus\\{\mathbf{0}\\}}|\mathbf{x}|^{2}|u|^{2}e^{-|\mathbf{x}|^{2}/4}$
$\displaystyle\leq C,$ $\displaystyle\sup_{B_{1}\cap
V\setminus\\{\mathbf{0}\\}}|\mathbf{x}||u|+|\mathrm{d}^{*}u|$
$\displaystyle\leq C.$

###### Proof.

It follows from Proposition 4.5 that $M^{i}_{s}$ is the graph of $u_{i}$ over
$V$ for $s\in[1,T]$ on the annuli $B_{R_{d_{i}}}\setminus B_{Ad_{i}^{1/2}}$,
where $R_{d_{i}}\to\infty$ as $d_{i}\to 0$ by Definition 4.3, and hence on
larger and larger compact annuli $K_{i}$. Moreover, on any fixed compact set
$K\subset V\setminus\\{\mathbf{0}\\}$ we have uniform bounds for
$d_{i}^{-1}u_{i}$ and $d_{i}^{-1}\nabla u_{i}$ as $i\to\infty$. Standard
parabolic estimates imply that, up to choosing a subsequence, the $u_{i}$
converge locally smoothly to a solution of (5.1) on
$V\setminus\\{\mathbf{0}\\}$.

Using that the $M^{i}_{\tau}$ are Lagrangian it follows that $u$ can be
identified with a closed 1-form. Condition ($\ast$ ‣ 2.3) implies that the
integral of $u$ along the two circles $V\cap\partial B_{1}$ vanishes, and so
$u$ is actually exact. Writing $u=(u_{1},u_{2})$ as in (b), we then have
$u_{j}=\mathrm{d}f_{j}$ for functions $f_{j}$ on
$P_{j}\setminus\\{\mathbf{0}\\}$.

The estimates (5.3) follow directly from the definition of $I_{V}(M^{i}_{1})$
together with the bounds given in Lemma 3.5. For the bound on
$\mathrm{d}^{*}u$ note that when we locally view $M^{i}_{1}$ as the graph of
the 1-form $u_{i}$ over a plane, then the difference $\theta-\theta_{V}$ in
the Lagrangian angle is given by $\mathrm{d}^{*}u_{i}$ up to lower order terms
in $u_{i}$.

It remains to show the claimed decomposition (5.2). For this we focus on one
of the planes $P=P_{j}$, and the corresponding solution $u=\mathrm{d}f$ of the
drift heat equation. By rescaling

$U(x,t)=\sqrt{-t}u(x/\sqrt{-t},-\ln(-t))$

we obtain a solution $U$ of the heat equation on a time interval
$[T_{0}^{\prime},T_{1}^{\prime}]$ on $P\setminus\\{\mathbf{0}\\}$. We have
$U=\mathrm{d}F$ for $F:P\setminus\\{\mathbf{0}\\}\to\mathbb{R}$ and we can
arrange that $F$ also satisfies the heat equation. The bound
$|\mathrm{d}^{*}u|\leq C$ on $B_{1}\setminus\\{\mathbf{0}\\}$ implies that we
also have a uniform bound $|\Delta F|\leq C$ on
$[T_{0}^{\prime},T_{1}^{\prime}]\times
B_{r}(\mathbf{0})\setminus\\{\mathbf{0}\\}$ for some $r>0$. Since $\Delta F$
also satisfies the heat equation, it follows that $\Delta F$ extends smoothly
across the origin in $P$. Using that $\Delta F=\partial_{t}F$ on
$P\setminus\\{\mathbf{0}\\}$, for any $t\in[T_{0}^{\prime},T_{1}^{\prime}]$ we
have

(5.4) $F(t)-F(T_{0}^{\prime})=v(t)\text{ on }P\setminus\\{\mathbf{0}\\},$

where $v(t)=\int_{T_{0}^{\prime}}^{t}\partial_{s}F\,ds$ is smooth across the
origin. Since $\Delta F(T_{0}^{\prime})$ is smooth across the origin, there is
a smooth function $g$ such that $F(T_{0}^{\prime})-g$ is harmonic on
$P\setminus\\{\mathbf{0}\\}$. Note that the bound $|\mathrm{d}F|\leq
C|\mathbf{x}|^{-1}$ near the origin implies that

$|F(T_{0}^{\prime})-g|\leq C|\ln|\mathbf{x}||$

near the origin. This implies that $F(T_{0}^{\prime})-g=a\ln|x|$ for a
constant $a$ satisfying $|a|\leq C$, up to modifying $g$ by a smooth function.
Using this in (5.4) we have that

$F(t)=a\ln|\mathbf{x}|+\tilde{F}(t),$

where $\tilde{F}(t)$ extends smoothly across the origin, and therefore solves
the heat equation on all of $P$. Scaling $F$ and hence $U$ back to give
$u=\mathrm{d}f$, this shows the required decomposition (5.2). ∎

###  5.2. Estimates for solutions of the drift heat equation

We will use the following estimates for the smooth part of the limiting
solution of the drift heat equation obtained in Proposition 5.2.

######  Proposition 5.3.

Suppose that $u$ is an exact $1$-form valued solution of the drift heat
equation (5.1) on $\mathbb{R}^{2}$ on the time interval $[0,\infty)$.

  1. (1)

Suppose that at $s=0$ we have the bounds

$\displaystyle\int_{\mathbb{R}^{2}}|\mathbf{x}|^{2}|u|^{2}e^{-|\mathbf{x}|^{2}/4}$
$\displaystyle\leq 1,$ $\displaystyle|\mathrm{d}^{*}u|$ $\displaystyle\leq
1,\text{ on }B_{1}.$

Then there is a uniform constant $C>0$ so that at $s=0$ we also have

$\int_{\mathbb{R}^{2}}|u|^{2}e^{-|\mathbf{x}|^{2}/4}\leq C.$

  2. (2)

If at $s=0$ we have $\int_{\mathbb{R}^{2}}|u|^{2}e^{-|\mathbf{x}|^{2}/4}\leq
1$, then at time $s=1$ we have

$|u|^{2},|\nabla u|^{2},|\nabla^{2}u|^{2}\leq Ce^{|\mathbf{x}|^{2}/4p}$

for some constants $C>0$ and $p>1$.

###### Proof.

To prove (1), it is enough to show that under the assumptions we have a
uniform bound $|u|\leq C$ on $B_{1}$ at $s=1$. We can write $u=\mathrm{d}f$,
with $|\Delta f|\leq 1$ and $f(p)=0$ for a basepoint $p\in\partial B_{1}$.
Elliptic estimates for the system $\mathrm{d}u=0,|\mathrm{d}^{*}u|\leq 1$
together with the integral bound for $u$ imply that we have a uniform bound
$|u|\leq C$ on the annulus $B_{2}\setminus B_{1/2}$, and thus $f$ satisfies a
gradient bound there. It follows that we have $|f|\leq C$ on $\partial B_{1}$.
Since $|\Delta f|\leq 1$ on $B_{1}$, the maximum principle then implies a
uniform bound $|f|\leq C$ on $B_{1}$ and so we also have a uniform gradient
bound $|\mathrm{d}f|\leq C$ on $B_{3/4}$. The required estimate for $|u|$
follows.

To prove (2), we first argue as in the proof of Lemma 3.5 to obtain the
pointwise estimate $|u|^{2}\leq Ce^{|\mathbf{x}|^{2}/4p}$ for $s\in[1/4,1]$
for some $C>0$ and $p>1$. In order to estimate $\nabla u$, we can consider the
evolution of $f=|u|^{2}+s|\nabla u|^{2}$. In terms of the drift Laplacian

(5.5) $\displaystyle\mathcal{L}_{0}=\Delta-\frac{1}{2}\mathbf{x}\cdot\nabla$

we have (recalling that $s\geq 0$)

$(\partial_{s}-\mathcal{L}_{0})(|u|^{2}+s|\nabla u|^{2})=|u|^{2}-2|\nabla
u|^{2}-2s|\nabla^{2}u|^{2}+|\nabla u|^{2}\leq|u|^{2}.$

It follows, using an estimate analogous to (3.5) (see also [2, Theorem
1.6.2]), that at time $s=1/2$ we have a bound

$\int_{\mathbb{R}^{2}}|\nabla u|^{2p}e^{-|\mathbf{x}|^{2}/4}\leq C,$

for some $p>1$. Arguing again as in the proof of Lemma 3.5 we obtain the
required pointwise bounds for $|\nabla u|$ at $s=1$. The bound for
$|\nabla^{2}u|$ is similar. ∎

##  6\. Three-annulus lemmas

In this section we prove two versions of the 3-annulus lemma. The first is for
solutions of the drift heat equation. The second is for our distance $D_{V}$
to the planes $V$.

###  6.1. Drift heat equation

We show the following 3-annulus lemma for solutions of the drift heat equation
given by Proposition 5.2. Note that this is slightly stronger than log-
convexity of the norm proved by Colding–Minicozzi [6]. The proof is similar to
Simon [23, Lemma 3.3].

######  Lemma 6.1.

There are small $0<\lambda_{1}<\lambda_{2}<1$ satisfying the following. Let
$u=e^{s}a_{0}\mathrm{d}\ln|\mathbf{x}|+\tilde{u}$ be a solution of the drift
heat equation (5.1) on $\mathbb{R}^{2}\setminus\\{\mathbf{0}\\}$, where
$\tilde{u}$ extends smoothly across the origin. We define the norm

(6.1)
$\displaystyle\|u(\tau)\|^{2}=|a_{0}|^{2}e^{2\tau}+\int_{\mathbb{R}^{2}}|\tilde{u}(\tau)|^{2}e^{-|\mathbf{x}|^{2}/4}$

and observe that we have a decomposition

(6.2)
$u(s)=a_{0}e^{s}\mathrm{d}\ln|\mathbf{x}|+\sum_{i>0}a_{i}e^{\mu_{i}s}\phi_{i},$

where the $\phi_{i}$ are orthonormal eigenfunctions of the drift Laplacian
$\mathcal{L}_{0}$ in (5.5).

  1. (1)

If $\|u(1)\|\geq e^{\lambda_{1}}\|u(0)\|$ then $\|u(2)\|\geq
e^{\lambda_{2}}\|u(1)\|$.

  2. (2)

If $u\neq 0$ has no homogeneous degree zero component, i.e. no term
corresponding to $\mu_{i}=0$ in (6.2), then we must have

either $\|u(2)\|\geq e^{\lambda_{1}}\|u(1)\|$ or $\|u(1)\|\leq
e^{-\lambda_{1}}\|u(0)\|$.

###### Proof.

By (6.2) and the definition of the norm, if we set $\mu_{0}=1$ then we have

$\|u(s)\|^{2}=\sum_{i=0}^{\infty}a_{i}^{2}e^{2\mu_{i}t}.$

Fix a small $a>0$ so that if $\mu_{i}\in[-10a,10a]$ for some $i$, then
$\mu_{i}=0$. We have

$\displaystyle\|u(0)\|^{2}$ $\displaystyle=\sum a_{i}^{2},$ $\displaystyle
e^{-2a}\|u(1)\|^{2}$ $\displaystyle=\sum a_{i}^{2}e^{2\mu_{i}-2a},$
$\displaystyle e^{-4a}\|u(2)\|^{2}$ $\displaystyle=\sum
a_{i}^{2}e^{4\mu_{i}-4a}.$

It follows that

$\displaystyle\frac{1}{2}(\|u(0)\|^{2}+e^{-4a}\|u(2)\|^{2})$
$\displaystyle=\sum
a_{i}^{2}e^{2\mu_{i}-2a}\frac{1}{2}(e^{2a-2\mu_{i}}+e^{2\mu_{i}-2a})$
$\displaystyle=\sum a_{i}^{2}e^{2\mu_{i}-2a}\cosh(2\mu_{i}-2a).$

By our choice of $a$ we have $|2\mu_{i}-2a|\geq a$ for all $i$, so we get

(6.3)
$\frac{1}{2}(\|u(0)\|^{2}+e^{-4a}\|u(2)\|^{2})\geq(1+c)e^{-2a}\|u(1)\|^{2},$

for some $c>0$. We choose $\lambda_{1}<a<\lambda_{2}$ with $\lambda_{j}$ very
close to $a$. If $\|u(1)\|^{2}\geq e^{2\lambda_{1}}\|u(0)\|^{2}$ then from
(6.3) we have

$(1+c)e^{-2a}\|u(1)\|^{2}\leq\frac{1}{2}e^{-4a}\|u(2)\|^{2}+\frac{1}{2}e^{-2\lambda_{1}}\|u(1)\|^{2}.$

Rearranging this we have

$\|u(2)\|^{2}\geq
e^{2a}\Big{(}2(1+c)-e^{2a-2\lambda_{1}}\Big{)}\|u(1)\|^{2}\geq
e^{2\lambda_{2}}\|u(1)\|^{2},$

if the $\lambda_{j}$ are sufficiently close to $a$. This shows (1). The proof
of (2), choosing $\lambda_{1},\lambda_{2}$ closer to zero if necessary, is
similar. ∎

###  6.2. Distance

Using Lemma 6.1 we can show a 3-annulus lemma for the distance $D_{V}$, by a
contradiction argument.

######  Proposition 6.2.

Let $\lambda_{1},\lambda_{2}$ be as in Lemma 6.1. Let
$0<\lambda_{1}^{\prime}<\lambda_{2}^{\prime}$ be such that
$\lambda_{j}^{\prime}\in(\lambda_{1},\lambda_{2})$. There is a large $N_{0}>0$
satisfying the following. Given an integer $N>N_{0}$, suppose that the flow
satisfies Condition (‡ ‣ 2.3) for $\tau\in[-1,2N+10]$, and $M_{0}$ satisfies
Condition ($\ast$ ‣ 2.3). There is an $\epsilon>0$ depending on $N$ such that
if $D_{V}(M_{0})<\epsilon$, then

$D_{V}(M_{N})\geq e^{\lambda_{1}^{\prime}N}D_{V}(M_{0})\text{ implies
}D_{V}(M_{2N})\geq e^{\lambda_{2}^{\prime}N}D_{V}(M_{N}).$

###### Proof.

The proof is by contradiction, similar to the proof of [22, Lemma 2, p. 549],
using property (3) in Lemma 3.5 to deal with the singularity of $V$ at the
origin and its noncompactness. See [26, Proposition 5.12] for a related
argument.

Suppose that the result fails for a given large integer $N$, so that we have a
sequence of flows $M^{i}_{\tau}$ with $D_{V}(M^{i}_{0})\to 0$ such that the
conclusion fails. By Proposition 3.6 we have $d_{i}=D_{V}(M^{i}_{N})\to 0$ and
our hypothesis can be written:

(6.4) $\displaystyle D_{V}(M^{i}_{0})$ $\displaystyle\leq
e^{-\lambda_{1}^{\prime}N}d_{i},$ $\displaystyle D_{V}(M^{i}_{2N})$
$\displaystyle<e^{\lambda_{2}^{\prime}N}d_{i}.$

In particular $d_{i}>0$. Using Proposition 5.2 we can write $M^{i}_{s}$ as the
graphs of $u_{i}(s)$ for $s\in[1,2N+8]$ over $V$ on larger and larger compact
sets $K_{i}\subset V\setminus\\{\mathbf{0}\\}$. The inequalities (6.4) and
Proposition 3.6 imply that $D_{V}(M^{i}_{s})\leq C_{N}d_{i}$ for
$s\in[1,2N+8]$.

As in Proposition 5.2, up to choosing a subsequence, we can assume that the
$d_{i}^{-1}u_{i}$ converge locally smoothly to a limit solution $u$ of (5.1)
on $V\setminus\\{\mathbf{0}\\}$. We can write
$u=e^{s}a_{0}\mathrm{d}\ln|\mathbf{x}|+\tilde{u}$ by Proposition 5.2, where
$\tilde{u}$ is smooth across the origin and $a_{0}$ is constant on each plane
in $V$. Using (5.3) and (6.4) the limit satisfies the estimates

(6.5)
$\displaystyle\int_{V}|\mathbf{x}|^{2}|\tilde{u}(1)|^{2}e^{-|\mathbf{x}|^{2}/4}\leq
Ce^{-2\lambda_{1}^{\prime}N},\quad\int_{V}|\mathbf{x}|^{2}|\tilde{u}(2N+1)|^{2}e^{-|\mathbf{x}|^{2}/4}\leq
Ce^{2\lambda_{2}^{\prime}N},$

for $C>0$ (independent of $N$). In addition the $\mathrm{d}\ln|\mathbf{x}|$
component of $u$ satisfies $|a_{0}|^{2}e^{2(2N+1)}\leq
Ce^{2\lambda_{2}^{\prime}N}$, and so

$|a_{0}|\leq Ce^{(\lambda_{2}^{\prime}-2)N}.$

Recall the norm in (6.1). Proposition 5.2, Proposition 5.3 and (6.5) imply
that

(6.6) $\|\tilde{u}(2)\|\leq
Ce^{-\lambda_{1}^{\prime}N},\quad\|\tilde{u}(2N+2)\|\leq
Ce^{\lambda_{2}^{\prime}N}.$

Let $\kappa>0$. We now have the following using (6.6) together with Lemma 6.1.

######  Claim 6.3.

For $N$ sufficiently large (depending on $\kappa$) we have
$\|\tilde{u}(N-1)\|\leq\kappa$.

###### Proof.

To see this note that we have two possibilities.

  * •

If $\|\tilde{u}(k+1)\|\leq e^{\lambda_{1}}\|\tilde{u}(k)\|$ for
$k=2,\ldots,N-2$, then we have

$\|\tilde{u}(N-1)\|\leq e^{(N-3)\lambda_{1}}\|\tilde{u}(2)\|\leq
Ce^{(N-3)\lambda_{1}-\lambda_{1}^{\prime}N}.$

Since $\lambda_{1}^{\prime}>\lambda_{1}>0$ this implies that
$\|\tilde{u}(N-1)\|\leq\kappa$ if $N$ is chosen sufficiently large.

  * •

If $\|\tilde{u}(k+1)\|\geq e^{\lambda_{1}}\|\tilde{u}(k)\|$ for some $k\leq
N-2$, then by Lemma 6.1 we have $\|\tilde{u}(k+1)\|\geq
e^{\lambda_{2}}\|\tilde{u}(k)\|$ for $k=N-1,\ldots,2N+1$. This implies

$\|\tilde{u}(N-1)\|\leq e^{-(N+3)\lambda_{2}}\|\tilde{u}(2N+2)\|\leq
Ce^{-(N+3)\lambda_{2}+\lambda_{2}^{\prime}N}.$

Since $\lambda_{2}>\lambda_{2}^{\prime}>0$, we have
$\|\tilde{u}(N-1)\|\leq\kappa$ if $N$ is sufficiently large.∎

Given Claim 6.3, let us assume therefore that $N$ is large enough that
$\|\tilde{u}(N-1)\|\leq\kappa$. The estimates in Proposition 5.3 then imply
that we have pointwise bounds

(6.7) $\displaystyle|\tilde{u}(N)|^{2},|\nabla\tilde{u}(N)|^{2}\leq
C\kappa^{2}e^{|\mathbf{x}|^{2}/4p},$

for some $C>0$ and $p>1$. The logarithmic component of $u$ also satisfies

$\Big{|}e^{N}a_{0}\mathrm{d}\ln|\mathbf{x}|\Big{|}\leq
Ce^{(\lambda_{2}^{\prime}-1)N}|\mathbf{x}|^{-1}\leq\kappa|\mathbf{x}|^{-1},$

for large $N$ (we can assume that $\lambda_{2}^{\prime}<1$).

We now use the local smooth convergence of the $d_{i}^{-1}u_{i}$ to $u$. For
any fixed compact set $K\subset\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$ this
implies that, as $i\to\infty$, the functions $d_{i}^{-1}d_{V}$ and
$d_{i}^{-1}(\theta-\theta_{V})$ on $M^{i}_{N}$ converge to $|u|$ and
$\mathrm{d}^{*}\tilde{u}$ on $V\cap K$. Using the estimates (6.7) and the fact
that $p>1$ it follows that for a given $K$, if we choose $i$ sufficiently
large (depending on $K,\kappa$), we have (note that we can assume that
$d_{i}^{-2}d_{V}^{2}$ differs from $|u|^{2}$ by at most $\kappa$ on $K$ for
large $i$)

(6.8) $\displaystyle\int_{M^{i}_{N}\cap
K}(|\mathbf{x}|^{2}d_{V}^{2}+|\theta-\theta_{V}|^{2})e^{-|\mathbf{x}|^{2}/4}$
$\displaystyle\leq d_{i}^{2}\kappa^{2}$
$\displaystyle\quad+d_{i}^{2}\int_{V\cap
K}(|\mathbf{x}|^{2}|u|^{2}+|\mathrm{d}^{*}\tilde{u}|^{2})e^{-|\mathbf{x}|^{2}/4}$
$\displaystyle\leq
d_{i}^{2}\kappa^{2}+Cd_{i}^{2}\kappa^{2}\int_{V}(|\mathbf{x}|^{2}+1)e^{\frac{|\mathbf{x}|^{2}}{4p}-\frac{|\mathbf{x}|^{2}}{4}}$
$\displaystyle\leq Cd_{i}^{2}\kappa^{2}.$

We will now apply part (3) of Lemma 3.5 to estimate $I_{V}(M^{i}_{N})$ in
terms of $I_{V}(M^{i}_{N-1})$ together with the integral bound (6.8) for
suitable $K$. Note first that by (6.4) and part (1) of Lemma 3.5 we have

$I_{V}(M^{i}_{N-1})=D_{V}(M^{i}_{N-1})\leq C_{N}d_{i}\,,$

for an $N$-dependent constant, while
$I_{V}(M^{i}_{N})=D_{V}(M^{i}_{N})=d_{i}$. Let $\gamma>0$. Assuming $d_{i}>0$
is sufficiently small (depending on $\gamma$), from Lemma 3.5 we have a
compact set $K_{\gamma}\subset\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$ such
that the integral estimate (6.8) on $K=K_{\gamma}$ implies

(6.9) $d_{i}=I_{V}(M^{i}_{N})\leq C(d_{i}\kappa+\gamma C_{N}d_{i}).$

We first choose $\kappa$ such that $C\kappa<1/4$. The choice of $\kappa$
determines an $N_{0}$, such that for $N>N_{0}$ we have the estimate
$\|\tilde{u}(N-1)\|\leq\kappa$ from Claim 6.3. Choosing $N>N_{0}$ then
determines the constant $C_{N}$, and we choose $\gamma$ such that
$CC_{N}\gamma<1/4$. This choice determines the set $K_{\gamma}$, and then for
sufficiently large $i$ we have the estimate (6.8) on $K=K_{\gamma}$. For such
large $i$ the inequality (6.9) holds, and it implies $d_{i}\leq d_{i}/2$,
which is a contradiction as $d_{i}>0$. ∎

##  7\. Decay estimates

We first define a variant of the excess from Definition 2.5 and show that it
satisfies a monotonicity formula.

######  Definition 7.1.

Recall the excess $\mathcal{A}(M)$ from Definition 2.5. For any $\alpha>0$ we
let

$\mathcal{A}_{\alpha}(M)=|\mathcal{A}(M)|^{\alpha-1}\mathcal{A}(M),$

i.e. $|\mathcal{A}_{\alpha}|=|\mathcal{A}|^{\alpha}$, but
$\mathcal{A}_{\alpha}$ has the same sign as $\mathcal{A}$.

######  Lemma 7.2.

For any $\tau_{1}<\tau_{2}$ and for $\alpha\in(0,1)$ we have

$\mathcal{A}_{\alpha}(M_{\tau_{1}})-\mathcal{A}_{\alpha}(M_{\tau_{2}})\geq\alpha\int_{\tau_{1}}^{\tau_{2}}|\mathcal{A}(M_{s})|^{\alpha-1}\int_{M_{s}}\left(2|\mathbf{H}|^{2}+\left|\mathbf{H}+\frac{\mathbf{x}^{\perp}}{2}\right|^{2}\right)e^{-|\mathbf{x}|^{2}/4}\,ds.$

###### Proof.

From Huisken’s monotonicity formula we know that $\mathcal{A}_{\alpha}(M_{s})$
is monotonically decreasing with $s$, being the infimum of a family of
decreasing functions as we vary $\theta_{0}$ in the definition of
$\mathcal{A}$ in (2.4). In particular $\mathcal{A}_{\alpha}(M_{s})$ is
differentiable almost everywhere, and at these points the derivative satisfies

$\frac{d}{ds}\mathcal{A}_{\alpha}(M_{s})\leq-\alpha|\mathcal{A}(M_{s})|^{\alpha-1}\int_{M_{s}}\left(2|\mathbf{H}|^{2}+\left|\mathbf{H}+\frac{\mathbf{x}^{\perp}}{2}\right|^{2}\right)e^{-|\mathbf{x}|^{2}/4}.$

The required inequality follows by integrating with respect to
$s\in[\tau_{1},\tau_{2}]$. ∎

The main technical result of this section is the following. Recall that we
defined $\mathcal{V}^{\prime}\subset\mathcal{V}$ in the same way as
$\mathcal{V}$ in Definition 3.1, just with the constant $c_{1}/2$ instead of
$c_{1}$ measuring closeness to the fixed pair of planes $V_{0}$.

######  Proposition 7.3.

There are $\epsilon_{0},C,N_{1}>0$ and $\alpha\in(0,1)$ such that if $N>N_{1}$
is an integer, $D_{V}(M_{0})<\epsilon_{0}$, $M_{1}$ satisfies Condition
($\ast$ ‣ 2.3), and the flow exists for $\tau\in[-1,3N^{2}+2]$ satisfying
Condition (‡ ‣ 2.3), then we have the following. If
$V\in\mathcal{V}^{\prime}$, then there is a $V^{\prime}\in\mathcal{V}$
satisfying $d(V,V^{\prime})\leq CD_{V}(M_{0})$ together with one of the
following conditions:

  * (i)

$D_{V^{\prime}}(M_{N})\leq\frac{1}{2}D_{V}(M_{0})$,

  * (ii)

$D_{V^{\prime}}(M_{N})\leq\mathcal{A}_{\alpha}(M_{N-3})-\mathcal{A}_{\alpha}(M_{N})$,

  * (iii)

$D_{V}(M_{N^{2}})\geq e^{\lambda_{1}^{\prime}N^{2}}D_{V}(M_{0})$,

where $\lambda_{1}^{\prime}$ is given by Proposition 6.2.

###### Proof.

We prove the result by contradiction. Suppose that for some large integer $N$
we have a sequence of flows $M^{i}_{\tau}$ for $\tau\in[-1,3N^{2}+2]$, and
$D_{V_{i}}(M^{i}_{0})=d_{i}\to 0$ for some $V_{i}\in\mathcal{V}^{\prime}$. We
will show that if none of the conditions (i)–(iii) hold, then we reach a
contradiction if $N$ is sufficiently large. First note that up to choosing a
subsequence we can replace the sequence $V_{i}$ by a single $V$.

Using Proposition 3.6 and Lemma 5.1 we know that for sufficiently large $i$
the surfaces $M^{i}_{\tau}$ satisfy Condition ($\ast$ ‣ 2.3) for
$\tau\in[1,3N^{2}+2]$. Using Proposition 5.2 we can find compact sets $K_{i}$
exhausting $\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$ so that $M^{i}_{s}$ is
the graph of $u_{i}(s)$ over $V\cap K_{i}$ for $s\in[1,3N^{2}]$. In addition,
up to choosing a subsequence, the rescaled functions $d_{i}^{-1}u_{i}$
converge locally smoothly to a solution $u$ of the drift heat equation (5.1)
on $[1,3N^{2}]\times V\setminus\\{\mathbf{0}\\}$.

Let $u_{0}$ be the static component of $\tilde{u}$ in the decomposition (6.2),
corresponding to the kernel of the drift Laplacian

$\mathcal{L}_{0}+\frac{1}{2}=\Delta+\frac{1}{2}(1-\mathbf{x}\cdot\nabla).$

We can thus write $u_{0}=(\mathrm{d}f_{1},\mathrm{d}f_{2})$ for homogeneous
degree 2 functions $f_{1},f_{2}$ on $\mathbb{R}^{2}$ with respect to the
splitting $V=P_{1}\cup P_{2}$ into a pair of planes. We then let
$a_{j}=\mathrm{d}^{*}\mathrm{d}f_{j}$ for $j=1,2$ (which are constants), set
$a=\frac{1}{8}(a_{1}-a_{2})$ and define

$\displaystyle u_{00}$
$\displaystyle=\Big{(}\mathrm{d}(f_{1}+a|\mathbf{x}|^{2}),\mathrm{d}(f_{2}-a|\mathbf{x}|^{2})\Big{)},$
$\displaystyle u_{01}$
$\displaystyle=\Big{(}\mathrm{d}(-a|\mathbf{x}|^{2}),\mathrm{d}(a|\mathbf{x}|^{2})\Big{)},$

again using the splitting $V=P_{1}\cup P_{2}$. Note that

$\displaystyle\mathrm{d}^{*}u_{00}$
$\displaystyle=(a_{1}-4a,a_{2}+4a)=(\textstyle\frac{1}{2}(a_{1}+a_{2}),\textstyle\frac{1}{2}(a_{1}+a_{2})),$
$\displaystyle\mathrm{d}^{*}u_{01}$
$\displaystyle=(4a,-4a)=(\textstyle\frac{1}{2}(a_{1}-a_{2}),\textstyle\frac{1}{2}(a_{2}-a_{1})).$

We have therefore decomposed $u$ as

(7.1) $u=u_{0}+u^{\perp}=u_{00}+u_{01}+u^{\perp}.$

The purpose of (7.1) is that we can deform the cone $V$ in the direction of
$u_{00}$ while keeping it special Lagrangian, since
$\mathrm{d}^{*}u_{00}=\frac{1}{2}(a_{1}+a_{2})$ on both planes. The directions
$u_{01}$ however correspond to deformations of $V$ into non-special Lagrangian
directions, whenever $a_{1}\neq a_{2}$. More precisely, let $V^{\prime}_{i}$
denote the graph of $d_{i}u_{00}$ over $V$. Using Lemma 3.8 we have
$D_{V^{\prime}_{i}}(M^{i}_{0})\leq Cd_{i}$, and we can argue as above to write
$M^{i}_{s}$ as the graph of $u^{\prime}_{i}(s)$ over larger and larger subsets
of $V_{i}^{\prime}$. The rescaled functions $d_{i}^{-1}u_{i}^{\prime}$ then
converge to $u^{\prime}=u_{01}+u^{\perp}$ in the decomposition (7.1).

Fix a $\kappa>0$. We first use (2) in Lemma 6.1 to show the following.

######  Claim 7.4.

If $N$ is sufficiently large (depending on $\kappa$), then either
$\|u^{\perp}(N-4)\|\leq\kappa$ or (iii) holds.

###### Proof.

Recall that we consider $u^{\perp}$ for $\tau\in[1,3N^{2}]$ and that
$u^{\perp}$ has no homogeneous degree zero component. Suppose that
$\|u^{\perp}(N-4)\|>\kappa$. If $N$ is sufficiently large, there must be some
$k<N-4$ for which $\|u^{\perp}(k+1)\|>e^{-\lambda_{1}}\|u^{\perp}(k)\|$, since
otherwise we would have $\|u^{\perp}(N-4)\|\leq Ce^{-(N-5)\lambda_{1}}$, which
for large $N$ is less than $\kappa$. Lemma 6.1 (2) now implies that
$\|u^{\perp}(k+2)\|\geq e^{\lambda_{1}}\|u^{\perp}(k+1)\|$. However, Lemma 6.1
(1) then implies that

$\|u^{\perp}(l+1)\|\geq e^{\lambda_{2}}\|u^{\perp}(l)\|\quad\text{for }l\geq
k+2\,.$

It follows that $\|u^{\perp}(N-3)\|\geq e^{\lambda_{1}}\kappa$ and
$\|u^{\perp}(l+1)\|\geq e^{\lambda_{2}}\|u^{\perp}(l)\|$ for all $k\geq N-3$.
Iterating this, we find that

$\|u^{\perp}(N^{2})\|\geq e^{\lambda_{1}+(N^{2}+3-N)\lambda_{2}}\kappa.$

We can split $u^{\perp}$ further, writing

(7.2) $\displaystyle
u^{\perp}=e^{s}a_{0}\mathrm{d}\ln|\mathbf{x}|+\tilde{u}^{\perp}$

such that $\tilde{u}^{\perp}$ is a smooth solution to the drift heat equation
on $V$. Ecker’s log-Sobolev inequality [7] then implies that there is $p>2$
such that

$\left(\int_{V}(\tilde{u}^{\perp}(N^{2}))^{p}e^{-|\mathbf{x}|^{2}/4}\right)^{1/p}\leq
C\|\tilde{u}^{\perp}(N^{2}-1)\|\leq C\|\tilde{u}^{\perp}(N^{2})\|\,.$

Combined with the estimates from Proposition 5.3 (2) for $\tilde{u}^{\perp}$
and (7.2), we deduce that there is $r_{0}>0$ such that

$\int_{V\cap(B_{1/r_{0}}\setminus
B_{r_{0}})}(u^{\perp}(N^{2}))^{2}e^{-|\mathbf{x}|^{2}/4}\geq\frac{1}{2}\|u^{\perp}(N^{2})\|^{2}\,.$

The definition of $D_{V}$ then implies that for sufficiently large $i$ we have

$D_{V}(M^{i}_{N^{2}})\geq
C^{-1}d_{i}e^{\lambda_{1}+(N^{2}+3-N)\lambda_{2}}\kappa\geq
e^{\lambda_{1}^{\prime}N^{2}}d_{i},$

for sufficiently large $N$, since $\lambda_{2}>\lambda_{1}^{\prime}$. Hence
(iii) holds. ∎

Let us suppose from now on that (iii) does not hold. Suppose that
$\|u_{01}\|=\kappa_{1}$ for some small $\kappa_{1}\geq 0$ (note that $u_{01}$
is $s$-independent). We have $\kappa_{1}<C$ for a uniform constant $C$. We
also assume that for a given small $\kappa>0$, $N$ is chosen large enough so
that $\|u^{\perp}(N-4)\|\leq\kappa$ by Claim 7.4. We now show the following,
from which we will deduce that (i) holds if $\kappa_{1}$ is sufficiently
small.

######  Claim 7.5.

If $i$ is sufficiently large,

(7.3) $D_{V^{\prime}_{i}}(M^{i}_{N-3})\leq C(\kappa_{1}+\kappa)d_{i}.$

###### Proof.

By Proposition 3.6 together with Lemma 3.8 we have
$D_{V^{\prime}_{i}}(M^{i}_{N-4})\leq C_{N}d_{i}$ for $C_{N}$ depending on $N$.
As in the proof of Proposition 6.2 we can write
$u=e^{s}a_{0}\mathrm{d}\ln|\mathbf{x}|+\tilde{u}$ such that we have pointwise
bounds of the form

(7.4) $|\tilde{u}(s)|^{2},|\nabla^{k}\tilde{u}(s)|^{2}\leq
C(\kappa_{1}+\kappa)^{2}e^{|\mathbf{x}|^{2}/4p},\text{ for }k=1,2,3\text{ and
}s\in[N-3,N],$

for some $C>0$ and $p>1$. We can then use the local smooth convergence of
$d_{i}^{-1}u_{i}\to u$ together with property (3) of Lemma 3.5 to ensure that
(7.3) holds. ∎

It follows from Claim 7.5, using Proposition 3.6, that for a larger $C$ we
have $D_{V^{\prime}_{i}}(M^{i}_{s})\leq C(\kappa_{1}+\kappa)d_{i}$ for
$s\in[N-3,N]$. If now $C\kappa_{1}<1/4$, then by choosing $\kappa=C^{-1}/4$ we
will have $D_{V^{\prime}_{i}}(M^{i}_{N})\leq\frac{1}{2}d_{i}$, i.e. (i) holds
for large enough $i$.

We therefore assume further that (i) does not hold, $\kappa_{1}=\|u_{01}\|\geq
C^{-1}/4$, and we choose $\kappa<\kappa_{1}$. In particular, the value of
$\mathrm{d}^{*}u_{01}$ on the two planes differs by at least $C^{-1}$ for some
$C>0$.

In the rest of the proof our goal is to show that if $\kappa$ is sufficiently
small (i.e. $N$ is large), and (ii) also does not hold, then we get a
contradiction. The basic idea is that in this case the flow $M^{i}_{s}$ for
$s\in[N-3,N]$ would have distance of order $\kappa d_{i}$ from a pair of
planes whose Lagrangian angles differ by $\kappa_{1}d_{i}$. If $M^{i}_{N}$ is
connected and $\kappa\ll\kappa_{1}$, then as in [15, Theorem B], one might
expect that this leads to a contradiction. The difficulty is that we need a
quantitative version of this idea, which works uniformly as $d_{i}\to 0$.

We are assuming that (ii) fails, therefore

$\mathcal{A}_{\alpha}(M^{i}_{N-3})-\mathcal{A}_{\alpha}(M^{i}_{N})<D_{V^{\prime}_{i}}(M^{i}_{N})\leq
C_{N}d_{i}.$

Using Lemma 7.2 this implies

(7.5)
$\int_{N-3}^{N}|\mathcal{A}(M^{i}_{s})|^{\alpha-1}\int_{M^{i}_{s}}(|\mathbf{H}|^{2}+|\mathbf{x}^{\perp}|^{2})e^{-|\mathbf{x}|^{2}/4}\,ds\leq\alpha^{-1}Cd_{i}.$

From Proposition 4.6 there is a small $\alpha_{1}>0$ so that

$|\mathcal{A}(M^{i}_{s})|\leq
D_{V^{\prime}_{i}}(M^{i}_{s})^{1+\alpha_{1}}\text{ for }s\in[N-3,N].$

Using this in (7.5) for $\alpha\in(0,1)$ we get

$\int_{N-3}^{N}\int_{M^{i}_{s}}(|\mathbf{H}|^{2}+|\mathbf{x}^{\perp}|^{2})e^{-|\mathbf{x}|^{2}/4}\,ds\leq
C_{\alpha}d_{i}^{1+(1-\alpha)(1+\alpha_{1})},$

for an $\alpha$-dependent constant $C_{\alpha}>0$. For $\alpha$ sufficiently
small and $0<2\alpha_{2}<\alpha_{1}/2$, we see that

$1+(1-\alpha)(1+\alpha_{1})>2+\frac{\alpha_{1}}{2}>2+2\alpha_{2}.$

Therefore, for sufficiently large $i$ we have

$\int_{N-3}^{N}\int_{M^{i}_{s}\cap
B_{2}}|\mathbf{H}|^{2}+|\mathbf{x}^{\perp}|^{2}\,ds\leq
d_{i}^{2+2\alpha_{2}},$

where we have removed the Gaussian weight by restricting to $M^{i}_{s}\cap
B_{2}$.

Let $\sigma>0$ be small, to be chosen later, independent of $i$. We can find
$s^{i}_{1},s^{i}_{2}\in[N-2,N-1]$ such that

(7.6) $\frac{\sigma}{2}<s^{i}_{2}-s^{i}_{1}<\sigma,$

and, in addition, at $s^{i}_{j}$ for $j=1,2$ we have

(7.7) $\int_{M_{s^{i}_{j}}\cap
B_{2}}|\mathbf{H}|^{2}+|\mathbf{x}^{\perp}|^{2}\leq d_{i}^{2+\alpha_{2}}$

for $i$ sufficiently large (depending on $\sigma$ as well).

Note that because of the bound $D_{V_{i}^{\prime}}(M^{i}_{N-3})\leq
C_{N}d_{i}$ and Proposition 4.5 we have that $M^{i}_{s}$ has good graphicality
over $V_{i}^{\prime}$ on the annulus $B_{R_{C_{N}d_{i}}}\setminus
B_{C_{0}d_{i}^{1/2}}$ for $s\in[N-2,0]$. This graphicality and Condition
($\ast$ ‣ 2.3) implies that the integral of the Liouville form $\lambda$
vanishes on any loop in $M^{i}_{s}\cap B_{R_{C_{N}d_{i}}}$, so we can define
primitives $\beta$ satisfying $\mathrm{d}\beta=\lambda$ on $M^{i}_{s}\cap
B_{R_{C_{N}d_{i}}}$. Restating [15, Proposition 6.1] for the rescaled flow, we
can choose the primitives $\beta$ along the flows $M^{i}_{s}$ such that
$e^{-s}(\beta+2\theta)$ satisfies the drift heat equation (5.1).

Our next goal is to estimate $\beta$ at the times $s^{i}_{1},s^{i}_{2}$. First
we consider what happens on the ball $B_{C_{0}d_{i}^{1/2}}$. For simplicity we
write $M$ for $M^{i}_{s^{i}_{1}}$ or $M^{i}_{s^{i}_{2}}$.

######  Claim 7.6.

There is a constant $\beta_{0}$ (depending on $s,i$) such that
$|\beta-\beta_{0}|<\kappa d_{i}$ on $M\cap B_{C_{0}d_{i}^{1/2}}$.

###### Proof.

Let $\tilde{M}=d_{i}^{-1/2}M$ and let $\tilde{\beta}$ on $\tilde{M}$ be given
by $\tilde{\beta}(p)=d_{i}^{-1}\beta(d_{i}^{1/2}p)$. Then $\tilde{M}$ is a
connected, almost calibrated Lagrangian, with uniform area bounds, satisfying

$\int_{\tilde{M}\cap B_{C_{0}}}|\nabla\tilde{\beta}|^{2}=d_{i}^{-2}\int_{M\cap
B_{C_{0}d_{i}^{1/2}}}|\nabla\beta|^{2}\leq d_{i}^{\alpha_{2}},$

using (7.7) and $|\nabla\beta|=|\mathbf{x}^{\perp}|$. (Recall that
$\alpha_{2}>0$.) We also have a uniform bound

$|\nabla\tilde{\beta}(p)|_{\tilde{M}}=d_{i}^{-1/2}|\nabla\beta(d_{i}^{1/2}p)|_{M}\leq
C$

for $p\in B_{C_{0}}$, so [16, Lemma 3.7], together with the connectivity
assumption in Condition ($\ast$ ‣ 2.3), implies that
$\mathrm{osc}\,\tilde{\beta}\to 0$ as $i\to\infty$. It follows that on $M\cap
B_{C_{0}d_{i}^{1/2}}$ we have $\mathrm{osc}\,(\beta)<\kappa d_{i}$ for
sufficiently large $i$. Setting $\beta_{0}=\beta(q)$ for some $q\in
B_{C_{0}d_{i}^{1/2}}$ yields the claim. ∎

Next we extend this pointwise bound on $B_{C_{0}d_{i}^{1/2}}$ to an integral
bound on the (larger) ball $B_{1}$, by using the integral estimate for
$|\nabla\beta|^{2}$ from (7.7) again.

######  Claim 7.7.

For sufficiently large $i$,

(7.8) $\int_{M\cap B_{1}}|\beta-\beta_{0}|^{2}\leq 9\kappa^{2}d_{i}^{2}\,.$

###### Proof.

Since on the annulus $B_{1}\setminus B_{C_{0}d_{i}^{1/2}}$ the surface $M$ has
good graphicality over $V^{\prime}_{i}$, we can view $\beta$ as a function $b$
on two copies of the annulus $B_{1}\setminus
B_{C_{0}d_{i}^{1/2}}\subset\mathbb{R}^{2}$, where we use polar coordinates
$r,\phi$. Using (7.7) we then have

$\int_{B_{1}\setminus B_{C_{0}d_{i}^{1/2}}}|\nabla b|^{2}\leq
2d_{i}^{2+\alpha_{2}}$

for sufficiently large $i$. For each $r\in[C_{0}d_{i}^{1/2},1]$ and
$\phi\in[0,2\pi]$ we have

$|b(r,\phi)-\beta_{0}|\leq\kappa d_{i}+\int_{C_{0}d_{i}^{1/2}}^{r}|\nabla
b|(s,\phi)\,ds,$

where $\beta_{0}$ is the constant given by Claim 7.6. Therefore,

(7.9) $\displaystyle\int_{0}^{2\pi}\int_{C_{0}d_{i}^{1/2}}^{1}$
$\displaystyle|b(r,\phi)-\beta_{0}|^{2}\,r\,dr\,d\phi$ $\displaystyle\leq
2\pi\kappa^{2}d_{i}^{2}+2\int_{0}^{2\pi}\int_{C_{0}d_{i}^{1/2}}^{1}\left(\int_{C_{0}d_{i}^{1/2}}^{r}|\nabla
b|(s,\phi)\,ds\right)^{2}\,r\,dr\,d\phi.$

Note that using Hölder’s inequality we have

$\displaystyle\int_{0}^{2\pi}\int_{C_{0}d_{i}^{1/2}}^{1}\Bigg{(}\int_{C_{0}d_{i}^{1/2}}^{r}|\nabla
b|$ $\displaystyle(s,\phi)\,ds\Bigg{)}^{2}\,r\,dr\,d\phi$ $\displaystyle\leq
C\int_{0}^{2\pi}\int_{C_{0}d_{i}^{1/2}}^{1}\int_{C_{0}d_{i}^{1/2}}^{r}|\nabla
b|^{2}(s,\phi)\,s\,ds\,r^{2}dr\,d\phi$ $\displaystyle\leq
C\int_{0}^{2\pi}\int_{C_{0}d_{i}^{1/2}}^{1}|\nabla b|^{2}(s,\phi)\,ds\,d\phi$
$\displaystyle\leq Cd_{i}^{2+\alpha_{2}}\leq\kappa^{2}d_{i}^{2},$

once $i$ is large enough (so that $d_{i}$ is small). The result follows for
$i$ sufficiently large, combined with Claim 7.6 and the uniform area ratio
bounds. ∎

We now show that we can get a similar pointwise estimates to Claim 7.6 on
$K_{i}\cap M$, where we recall the compact sets $K_{i}$ given at the start of
the proof.

######  Claim 7.8.

Up to replacing the compact sets $K_{i}$ by smaller sets (still exhausting
$\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$ in the limit as $i\to\infty$), there
is a constant $C>0$ and $p>1$ such that

(7.10) $|\beta-\beta_{0}|\leq C\kappa d_{i}e^{|\mathbf{x}|^{2}/8p}\text{ on
}K_{i}\cap M.$

###### Proof.

Recall the decomposition of $u$ in (7.1). We now show that $\nabla\beta$ is of
order $\kappa d_{i}$ on compact sets away from $0$, because the term $u_{01}$
in (7.1) does not contribute to $\mathbf{x}^{\perp}$, being homogeneous of
degree 1. More precisely, recall that $M^{i}_{s}$ is the graph of $u_{i}(s)$
over $K_{i}\cap V$ and note that $d_{i}^{-1}\mathbf{x}^{\perp}$ on $M^{i}_{s}$
converges locally smoothly as $i\to\infty$ to $u-\mathbf{x}\cdot\nabla u$ on
$V\setminus\\{\mathbf{0}\\}$. Since $u_{00},u_{01}$ have degree 1, it follows
that they have no contribution to $u-\mathbf{x}\cdot\nabla u$. Therefore, as
$i\to\infty$ we have
$d_{i}^{-1}|\nabla\beta|\to|u^{\perp}-\mathbf{x}\cdot\nabla u^{\perp}|$
locally smoothly. At the same time, by Claim 7.4 we have
$\|u^{\perp}(N-4)\|\leq C\kappa$ and so, by Proposition 5.3, for $s\in[N-3,N]$
we also have pointwise bounds

(7.11) $|u^{\perp}|,|\nabla u^{\perp}|\leq C\kappa e^{|\mathbf{x}|^{2}/8p}$

for some $C>0$ and $p>1$. Returning to the setting where $M=M^{i}_{s^{i}_{j}}$
for $j=1,2$ and using (7.8) we can integrate the estimate we have for
$|\nabla\beta|$ to find that up to replacing $K_{i}$ by smaller sets and
decreasing $p$ we get (7.10). ∎

We also need a more global estimate for $\beta$ and $\theta$, up to the good
graphicality radius $R_{Cd_{i}}$ (from Definition 4.3) on $M^{i}_{s}$ for
$s\in[N-3,N]$.

######  Claim 7.9.

Recall Definition 4.3. There is $C>0$ and $p>p_{0}>1$ such that for
$s\in[N-3,N]$ we have

(7.12) $\displaystyle|\beta-\beta_{0}|$ $\displaystyle\leq
Cd_{i}e^{|\mathbf{x}|^{2}/8p}\text{ on }M^{i}_{s}\cap
B_{R_{C_{N}d_{i}}}\setminus B_{1/2},$
$\displaystyle|\theta-\theta_{V_{i}^{\prime}}|$ $\displaystyle\leq
Cd_{i}e^{|\mathbf{x}|^{2}/8p}\text{ on }M^{i}_{s}\cap
B_{R_{C_{N}d_{i}}}\setminus B_{1/2},$

once $i$ is sufficiently large.

###### Proof.

By the smooth convergence of $d_{i}^{-1}u_{i}\to u$ on the annulus
$B_{2}\setminus B_{1/2}$, together with the bounds (7.4), we have estimates

$|\nabla\beta|,|\Delta\beta|,|\theta-\theta_{V^{\prime}}|,|\nabla\theta|,|\Delta\theta|\leq
Cd_{i}$

on $M^{i}_{s}\cap B_{2}\setminus B_{1/2}$ for $s\in[N-3,N]$. Using the
evolution equation for $\beta$ and (7.10) it then follows that

(7.13) $\displaystyle|\beta-\beta_{0}|\leq Cd_{i}\quad\text{ on }M^{i}_{s}\cap
B_{2}\setminus B_{1/2}$

for $s\in[N-3,N]$. To extend this estimate out to distance $R_{C_{N}d_{i}}$ we
first observe that, by Proposition 4.5, on $B_{R_{C_{N}d_{i}}}\setminus B_{1}$
the surface $M^{i}_{s}$ is the graph of $u_{i}$ over $V_{i}^{\prime}$
satisfying $|u_{i}|,|\nabla u_{i}|\leq Cd_{i}e^{|\mathbf{x}|^{2}/8p}$ for some
$p>p_{0}>1$. Moreover, by the definition of $R_{C_{N}d_{i}}$ in (4.8) and the
fact that $p>p_{0}$ we know that $d_{i}e^{R_{C_{N}d_{i}}^{2}/8p}\to 0$ as
$i\to\infty$. Since the leading order term in $\mathbf{x}^{\perp}$ is
$u_{i}-\mathbf{x}\cdot\nabla u_{i}$ for $i$ large, we see that on
$M^{i}_{s}\cap B_{R_{C_{N}d_{i}}}\setminus B_{1}$ for $s\in[N-3,N]$ we have
(decreasing $p>p_{0}$ if necessary)

$|\nabla\beta|=|\mathbf{x}^{\perp}|\leq Cd_{i}e^{|\mathbf{x}|^{2}/8p}$

for sufficiently large $i$. Integrating this and using our bound (7.13) on
$B_{2}\setminus B_{1/2}$ implies that for an even smaller $p>p_{0}>1$ we have
the estimate (7.12) for $i$ sufficiently large and $s\in[N-3,N]$. The bound
for $\theta$ in (7.12) follows similarly, since to leading order
$\theta-\theta_{V_{i}^{\prime}}$ is given by $\mathrm{d}^{*}u_{i}$ for $i$
large. ∎

Recall the times $s^{i}_{2}>s^{i}_{1}$ which satisfy (7.6) depending on some
small $\sigma>0$, which we are free to choose. We will now use that
$e^{-s}(\beta+2\theta)$ and $\theta$ satisfy the drift heat equation (5.1) to
derive pointwise estimates in $B_{2}$ for $\theta$ at time $s^{i}_{2}$ in
terms of an integral estimate for $\beta$ at time $s^{i}_{1}$.

######  Claim 7.10.

Let

$h=e^{s^{i}_{1}-s}(\beta-\beta_{0}+2(\theta-\theta_{V_{i}^{\prime}}))-2(\theta-\theta_{V_{i}^{\prime}})$

on $M^{i}_{s}$ so that $h(s^{i}_{1})=\beta-\beta_{0}$. There is some $C>0$
independent of $i$ such that

(7.14) $\displaystyle\sup_{M^{i}_{s}\cap B_{2}}h^{2}\leq
C\kappa^{2}d_{i}^{2}\quad\text{ at }s=s^{i}_{2}$

and hence

(7.15) $\displaystyle\mathrm{osc}\,\theta\leq C\kappa d_{i}\quad\text{ on
}M^{i}_{s^{i}_{2}}\cap B_{2}\setminus B_{1/2}.$

###### Proof.

Since we have only defined $\beta$ on the ball $B_{R_{Cd_{i}}}$ we need to
incorporate a cutoff function. In fact even if $\beta$ were defined globally
we would need such a cutoff if we do not assume that $\beta$ has uniform
polynomial growth bounds.

To that end, let $\chi_{0}:[0,\infty)\to\mathbb{R}$ denote a cutoff function
with $\chi_{0}(t)=1$ for $t<(R_{C_{N}d_{i}}-1)^{2}$ and $\chi_{0}(t)=0$ for
$t>R_{C_{N}d_{i}}^{2}$. We can arrange that
$\chi,\chi^{\prime},\chi^{\prime\prime}$ are uniformly bounded independently
of $i$. Let $\chi(x)=\chi_{0}(|x|^{2})$. Note that $\chi^{2}h^{2}$ is then
defined globally.

Recall $\mathcal{L}_{0}$ given in (5.5). Along the rescaled flow (for
surfaces) we have

(7.16) $(\partial_{s}-\mathcal{L}_{0})|\mathbf{x}|^{2}=|\mathbf{x}^{\rm
T}|^{2}-4$

and so

(7.17)
$(\partial_{s}-\mathcal{L}_{0})e^{s^{i}_{1}-s}|\mathbf{x}|^{2}\leq-2\quad\text{
for }s\in[s^{i}_{1},s^{i}_{2}].$

We also have

$(\partial_{s}-\mathcal{L}_{0})\chi=\chi_{0}^{\prime}(|\mathbf{x}|^{2})(\partial_{s}-\mathcal{L}_{0})|\mathbf{x}|^{2}-\chi_{0}^{\prime\prime}(|\mathbf{x}|^{2})|\nabla|\mathbf{x}|^{2}|^{2}.$

Hence,

(7.18) $|(\partial_{s}-\mathcal{L}_{0})\chi|\leq C|\mathbf{x}|^{2}$

and $(\partial_{s}-\mathcal{L}_{0})\chi$ is supported on the set where
$R_{C_{N}d_{i}}-1<|\mathbf{x}|<R_{C_{N}d_{i}}$.

Since $h$ satisfies the drift heat equation (5.1), we may compute

$(\partial_{s}-\mathcal{L}_{0})(\chi^{2}h^{2})=2\chi
h^{2}(\partial_{s}-\mathcal{L}_{0})\chi-2|\nabla\chi|^{2}h^{2}-2\chi^{2}|\nabla
h|^{2}-8\chi h\nabla\chi\cdot\nabla h.$

We use the inequality $|8\chi h\nabla\chi\cdot\nabla h|\leq 2\chi^{2}|\nabla
h|^{2}+8h^{2}|\nabla\chi|^{2}$ and the estimate (7.18) to get

(7.19)
$(\partial_{s}-\mathcal{L}_{0})(\chi^{2}h^{2})\leq\begin{cases}C|\mathbf{x}|^{2}h^{2}&\text{
for }R_{C_{N}d_{i}}-1<|\mathbf{x}|<R_{C_{N}d_{i}},\\\ 0&\text{
otherwise.}\end{cases}$

By Claim 7.9, there is $p>p_{0}>1$ so that we also have the bound

(7.20) $\displaystyle h^{2}\leq Cd_{i}^{2}e^{R_{Cd_{i}}^{2}/4p}\quad\text{ for
}1<|\mathbf{x}|<R_{Cd_{i}}.$

We now define the function

$\Theta=\Big{(}e^{s_{1}^{i}-s}|\mathbf{x}|^{2}-e^{-\sigma}(R_{Cd_{i}}-1)^{2}\Big{)}_{+},$

where $(\ldots)_{+}$ means the positive part of the function and $\sigma$ is
the constant in (7.6). Using (7.17) and $s\in[s_{1}^{i},s_{2}^{i}]$ we have
that

(7.21)
$\displaystyle(\partial_{s}-\mathcal{L}_{0})\Theta\leq\begin{cases}-2&\text{
when }e^{s_{1}^{i}-s}|\mathbf{x}|^{2}>e^{-\sigma}(R_{Cd_{i}}-1)^{2},\\\
0&\text{ otherwise, }\end{cases}$

in the distributional sense. Note that by (7.6) we have
$e^{s-s^{i}_{1}-\sigma}<1$ for $s\in[s_{1}^{i},s_{2}^{i}]$. We deduce from
(7.19), (7.20) and (7.21) that there is some $C_{1}>0$ so that

(7.22)
$(\partial_{s}-\mathcal{L}_{0})\Big{(}\chi^{2}h^{2}+C_{1}d_{i}^{2}R_{Cd_{i}}^{2}e^{R_{Cd_{i}}^{2}/4p}\Theta\Big{)}\leq
0\quad\text{ for }s\in[s^{i}_{1},s^{i}_{2}].$

At $s=s^{i}_{1}$ we have $\chi^{2}h^{2}=\chi^{2}(\beta-\beta_{0})^{2}$, and so
using (7.8), (7.12), together with the uniform area ratio bounds for
$M^{i}_{s}$, we can ensure that

(7.23) $\int_{M^{i}_{s^{i}_{1}}}\chi^{2}h^{2}e^{-|\mathbf{x}|^{2}/4}\leq
C\kappa^{2}d_{i}^{2},$

once $i$ is large enough. To estimate the integral of $\Theta$ at
$s=s^{i}_{1}$, note that $\Theta(\mathbf{x})=0$ if $s=s^{i}_{1}$ and
$|\mathbf{x}|<e^{-\sigma/2}(R_{Cd_{i}}-1)$, which holds if
$|\mathbf{x}|<e^{-\sigma}R_{Cd_{i}}$ once $i$ is large. For all $\mathbf{x}$
we have $\Theta(\mathbf{x})\leq|\mathbf{x}|^{2}$ and so by our previous
observation we have

$\int_{M^{i}_{s^{i}_{1}}}\Theta
e^{-|\mathbf{x}|^{2}/4}\leq\int_{M^{i}_{s^{i}_{1}}\setminus
B_{e^{-\sigma}R_{Cd_{i}}}}|\mathbf{x}|^{2}e^{-|\mathbf{x}|^{2}/4}\leq
Ce^{-2\sigma}R_{Cd_{i}}^{2}e^{-e^{-2\sigma}R_{Cd_{i}}^{2}/4}.$

If $\sigma$ is chosen sufficiently small, so that $e^{2\sigma}<p$ for the
$p>1$ in Claim 7.9, see (7.20), then we will have

(7.24)
$\displaystyle\int_{M^{i}_{s^{i}_{1}}}d_{i}^{2}R_{Cd_{i}}^{2}e^{R_{Cd_{i}}^{2}/4p}\Theta\,e^{-|\mathbf{x}|^{2}/4}\leq
Cd_{i}^{2}R_{Cd_{i}}^{4}\exp\left(\frac{R_{Cd_{i}}^{2}}{4p}-\frac{R_{Cd_{i}}^{2}}{4e^{2\sigma}}\right)\leq\kappa^{2}d_{i}^{2}$

for sufficiently large $i$. Combining this with (7.23) and using (7.22) we can
apply the monotonicity formula to obtain pointwise estimates for
$\chi^{2}h^{2}$ at $s=s^{i}_{2}$. Since $s^{i}_{2}>\sigma/2$, and $\sigma$
only depends on $p$ in (7.24), we obtain the estimate (7.14) for $h^{2}$.

At $s=s^{i}_{2}$ we have

$\begin{split}h&=e^{s^{i}_{1}-s_{2}^{i}}(\beta-\beta_{0}+2(\theta-\theta_{V_{i}^{\prime}}))-2(\theta-\theta_{V_{i}^{\prime}})\\\
&=e^{s^{i}_{1}-s_{2}^{i}}\beta+2(e^{s^{i}_{1}-s_{2}^{i}}-1)\theta-e^{s^{i}_{1}-s_{2}^{i}}(\beta_{0}+2\theta_{V_{i}^{\prime}})+2\theta_{V_{i}^{\prime}}\,.\end{split}$

and at this time the oscillation of $\beta$ on $B_{2}\setminus B_{1/2}$ is
bounded by $C\kappa d_{i}$ from (7.10) for $i$ large. Noting (7.6), it follows
that the oscillation of $\theta$ is also bounded by $C\kappa d_{i}$ on this
annulus for $i$ sufficiently large as claimed. ∎

To complete the proof, recall the decomposition (7.1) and that on
$B_{2}\setminus B_{1}$ we know that $M^{i}_{s^{i}_{2}}$ is the graph of
$u_{i}$ over $V^{\prime}_{i}$, where $d_{i}^{-1}u_{i}\to u_{01}+u^{\perp}$,
and so
$d_{i}^{-1}(\theta-\theta_{V^{\prime}_{i}})\to\mathrm{d}^{*}(u_{01}+u^{\perp})$.
From (7.11) we have $|\mathrm{d}^{*}u^{\perp}|\leq C\kappa$ on $B_{2}\setminus
B_{1}$, while the value of $\mathrm{d}^{*}u_{01}$ on the two planes differs by
at least $C^{-1}$. Therefore the oscillation of
$\mathrm{d}^{*}(u_{01}+u^{\perp})$ on $B_{2}\setminus B_{1}$ is at least
$C^{-1}-C\kappa>C^{-1}/2$ if we choose $\kappa$ sufficiently small. The
oscillation of $\theta$ on $B_{2}\setminus B_{1/2}$ is therefore at least
$C^{-1}d_{i}/4$ for large $i$. This contradicts the bound (7.15) if $\kappa$
is sufficiently small. ∎

###  7.1. Closeness to planes

We next show that if condition (iii) in Proposition 7.3 holds, then we can
still arrange that the flow remains close to the original pair of planes
$V_{0}$ as long as the change in the excess $\mathcal{A}$ is controlled. From
now on we fix $N_{1}>0$ and $\alpha\in(0,1)$ such that Proposition 7.3
applies, and we assume that $N>N_{1}$ is a fixed integer large enough so that
Proposition 6.2 applies to $N^{2}$ and such that

(7.25) $\displaystyle e^{-\lambda_{1}^{\prime}N^{2}}<1/2\,.$

Again recall that $\lambda_{1}^{\prime}>0$ is given by Proposition 6.2.

######  Proposition 7.11.

Let $\delta_{1}>0$. There is an $\epsilon_{1}>0$ depending on $\delta_{1}$
such that, if we have a flow $M_{\tau}$ satisfying Condition (‡ ‣ 2.3) for
$\tau\in[-1,T+10]$, $M_{1}$ satisfies Condition ($\ast$ ‣ 2.3), and

  1. (1)

$\mathcal{A}_{\alpha}(M_{0})-\mathcal{A}_{\alpha}(M_{T})<\epsilon_{1}$,

  2. (2)

$D_{V}(M_{0})<\epsilon_{1}$, for some $V\in\mathcal{V}^{\prime}$,

  3. (3)

$D_{V}(M_{N^{2}})\geq e^{\lambda_{1}^{\prime}N^{2}}D_{V}(M_{0})$,

then $D_{V}(M_{T})<\delta_{1}$.

###### Proof.

Using assumptions (2), (3), if $\epsilon_{1}$ is sufficiently small then we
can apply Proposition 6.2 to deduce $D_{V}(M_{2N^{2}})\geq
e^{\lambda_{1}^{\prime}N^{2}}D_{V}(M_{N^{2}})$. We can keep iterating this
estimate, to get

$D_{V}(M_{iN^{2}})\geq e^{\lambda_{1}^{\prime}N^{2}}D_{V}(M_{(i-1)N^{2}})$

for $i=3,4,\ldots,k$, where $k$ is the largest integer which still satisfies

$D_{V}(M_{(k-3)N^{2}})<\epsilon_{0},\text{ and }kN^{2}<T,$

for a constant $\epsilon_{0}$ that is smaller than the $\epsilon$ in
Proposition 6.2.

If $(k+1)N^{2}\geq T$, then Proposition 3.6 implies that $D_{V}(M_{T})\leq
C\epsilon_{0}$. If $\epsilon_{0}$ is chosen sufficiently small (and
$\epsilon_{1}<\epsilon_{0}$), then this implies $D_{V}(M_{T})<\delta_{1}$ as
required.

We therefore assume that $(k+1)N^{2}<T$ and
$D_{V}(M_{(k-2)N^{2}})\geq\epsilon_{0}$. We have $D_{V}(M_{(k-3)N^{2}})\leq
e^{-\lambda_{1}^{\prime}N^{2}}D_{V}(M_{(k-2)N^{2}})$ so using (7.25) we have

(7.26)
$D_{V}(M_{(k-2)N^{2}})-D_{V}(M_{(k-3)N^{2}})\geq(1-e^{-\lambda_{1}^{\prime}N^{2}})D_{V}(M_{(k-2)N^{2}})\geq\frac{\epsilon_{0}}{2}.$

We claim that (7.26) together with condition (1) and
$D_{V}(M_{(k-3)N^{2}})<\epsilon_{0}$ leads to a contradiction if
$\epsilon_{0}$ is chosen small, and $\epsilon_{1}$ is sufficiently small
depending on $\epsilon_{0}$.

Shifting $\tau=(k-3)N^{2}$ to $\tau=0$, we can thus suppose that we have a
sequence of flows $M_{\tau}^{i}$ satisfying Condition (‡ ‣ 2.3) for
$\tau\in[-N^{2},N^{2}]$, satisfying

$\displaystyle\mathcal{A}_{\alpha}(M_{-N^{2}}^{i})-\mathcal{A}_{\alpha}(M_{N^{2}}^{i})<\epsilon_{1,i},$
$\displaystyle
D_{V}(M^{i}_{N^{2}})-D_{V}(M^{i}_{0})\geq\frac{\epsilon_{0}}{2},$
$\displaystyle D_{V}(M^{i}_{-N^{2}})<\epsilon_{0},$

for a sequence $\epsilon_{1,i}\to 0$. From this, together with the same
argument as in [15, Theorem A], we know that as $i\to\infty$, along a
subsequence, these flows converge to a static flow given by a union of planes.
By the bound on $D_{V}(M^{i}_{-N^{2}})$ these planes must be given by some
$V^{\prime}\in\mathcal{V}$ if $\epsilon_{0}$ is small enough. In particular
for any $\tau\in[0,N^{2}]$ the $M^{i}_{\tau}$ converge locally smoothly to
$V^{\prime}$ on compact sets in $\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$.

Since $M^{i}_{N^{2}}$ and $M^{i}_{0}$ both converge to $V^{\prime}$, for any
compact set $K\subset\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$ we have as
$i\to\infty$:

$\int_{M^{i}_{N^{2}}\cap
K}(|\mathbf{x}|^{2}d_{V}^{2}+|\theta-\theta_{V}|^{2})e^{-|\mathbf{x}|^{2}/4}-\int_{M^{i}_{0}\cap
K}(|\mathbf{x}|^{2}d_{V}^{2}+|\theta-\theta_{V}|^{2})e^{-|x|^{2}/4}\to 0.$

We can use the uniform bound $D_{V}(M^{i}_{-N^{2}})<\epsilon_{0}$ and argue as
in the proof of Proposition 6.2, in particular in (6.9), to show
$D_{V}(M^{i}_{N^{2}})-D_{V}(M^{i}_{0})\to 0$ which gives our required
contradiction. ∎

We can now prove our main result controlling the distance of flows close to
the union of two transverse planes by combining Propositions 7.3 and 7.11.

######  Proposition 7.12.

Let $\delta_{2}>0$. There is an $\epsilon_{2}>0$ depending on $\delta_{2}$,
such that if we have a flow $M_{\tau}$ satisfying Condition (‡ ‣ 2.3) for
$\tau\in[-1,T+10]$, $M_{1}$ satisfies Condition ($\ast$ ‣ 2.3), and

  1. (1)

$\mathcal{A}_{\alpha}(M_{0})-\mathcal{A}_{\alpha}(M_{T})<\epsilon_{2}$,

  2. (2)

$D_{V_{0}}(M_{0})<\epsilon_{2}$,

then $D_{V_{0}}(M_{T})<\delta_{2}$.

Note that $T$ is independent of the constants $\delta_{2},\epsilon_{2}$ and,
in particular, can be large.

###### Proof.

We iterate the cases (i), (ii) in Proposition 7.3 as long as possible,
starting with $V_{0}$. We obtain a sequence $V_{1},\ldots,V_{k}\in\mathcal{V}$
together with numbers $e_{i}=D_{V_{i}}(M_{iN})$ such that
$d(V_{i},V_{i+1})\leq Ce_{i}$. We can continue this iteration and define
$V_{k+1}$ unless one of the following occurs:

  * (a)

$kN+3N^{2}+2>T$,

  * (b)

$V_{k}\not\in\mathcal{V}^{\prime}$ or $D_{V_{k}}(M_{kN})\geq\epsilon_{0}$,

  * (c)

$D_{V_{k}}(M_{kN+N^{2}})\geq e^{\lambda_{1}^{\prime}N^{2}}D_{V_{k}}(M_{kN})$.

We show that if $\epsilon_{2}$ is sufficiently small, then (b) cannot occur
before (a) or (c) does. To see this we can argue as in [26, Theorem 6.7] to
control the sum of the $e_{i}$, to find

(7.27) $\displaystyle\sum_{i=0}^{k}e_{i}\leq
2e_{0}+2C(\mathcal{A}_{\alpha}(M_{0})-\mathcal{A}_{\alpha}(M_{T}))\leq
C\epsilon_{2}.$

In particular this implies that both $e_{k}$ and $d(V_{0},V_{k})$ are bounded
above by $C\epsilon_{2}$, so we can ensure that (b) does not occur for
$\epsilon_{2}$ sufficiently small. In addition, using Lemma 5.1, Condition
($\ast$ ‣ 2.3) is preserved.

If (a) occurs first, then we have

(7.28) $\displaystyle D_{V_{k}}(M_{T})\leq CD_{V_{k}}(M_{kN})\leq
C\epsilon_{2}$

by Proposition 3.6, and by Lemma 3.8 we get $D_{V_{0}}(M_{T})\leq
C\epsilon_{2}$. If (c) occurs first, then from Proposition 7.11 we conclude
that $D_{V_{k}}(M_{T})<\delta_{1}$ if we choose $\epsilon_{2}<\epsilon_{1}$,
and Lemma 3.8 implies $D_{V_{0}}(M_{T})\leq C(\delta_{1}+\epsilon_{2})$.

If we choose $\delta_{1}>0$ sufficiently small (determining a value for
$\epsilon_{1}>0$), and then choose $\epsilon_{2}>0$ small so that also
$\epsilon_{2}<\epsilon_{1}$, then in either case we will have
$D_{V_{0}}(M_{T})<\delta_{2}$ as required. ∎

##  8\. Neck pinches

In this section we give the main geometric applications of the estimates we
have obtained. We suppose that $\mathcal{M}(t)$ is a rational, zero Maslov
Lagrangian mean curvature flow in $\mathbb{C}^{2}$, with uniformly bounded
area ratios and uniformly bounded Lagrangian angle.

###  8.1. Uniqueness of tangent flows

Our first result is the uniqueness of tangent flows given by a union of
transverse planes. Note that by [16, Corollary 4.3] the two planes must have
the same Lagrangian angle if a singularity forms. We first have the following,
ensuring that Conditions (‡ ‣ 2.3) and ($\ast$ ‣ 2.3) hold along the
corresponding rescaled flow.

######  Lemma 8.1.

Let $\mathcal{M}(t)$ be a mean curvature flow in $\mathbb{C}^{2}$, with
initial condition given by a rational, zero Maslov Lagrangian with uniformly
bounded area ratios and uniformly bounded Lagrangian angle. Suppose that the
flow develops a singularity at $(\mathbf{0},0)$, with a tangent flow given by
the static flow $V_{0}$, where $V_{0}$ is a special Lagrangian union of two
transverse planes. Let $M_{\tau}=e^{\tau/2}\mathcal{M}(-e^{-\tau})$ denote the
corresponding rescaled flow. Then there is a sequence $\tau_{i}\to\infty$
satisfying

  1. (1)

$D_{V_{0}}(M_{\tau_{i}})\to 0$,

  2. (2)

$M_{\tau}$ satisfies Condition (‡ ‣ 2.3) for $\tau\in[\tau_{0},\infty)$,

  3. (3)

$M_{\tau_{i}+1}$ satisfies Condition ($\ast$ ‣ 2.3).

###### Proof.

Note that the uniform bounds on area ratios and the bound for the Lagrangian
angle is preserved along the flow. Condition (‡ ‣ 2.3) holds on
$[\tau_{0},\infty)$ for sufficiently large $\tau_{0}$ by the monotonicity of
$\mathcal{A}(M_{\tau})$. The fact that $D_{V_{0}}(M_{\tau_{i}})\to 0$ follows
from the assumption that one tangent flow at $(\mathbf{0},0)$ is given by
$V_{0}$.

It remains to show Condition ($\ast$ ‣ 2.3) for $M_{\tau_{i}+1}$ for large
enough $i$. Let us first consider the connectedness of $B_{1}\cap
M_{\tau_{i}+1}$. Note that by the assumption $D_{V_{0}}(M_{\tau_{i}})\to 0$ we
have that $M_{\tau_{i}+1}$ has good graphicality over $V_{0}$ on
$B_{2}\setminus B_{1/2}$ for large $i$. For large $i$ the pointwise bounds in
Lemma 3.5 also imply that $B_{2}\cap M_{\tau_{i}+1}$ is almost calibrated.
Since there are no compact almost calibrated Lagrangians in $\mathbb{C}^{2}$,
this implies that $B_{1}\cap M_{\tau_{i}+1}$ has either 1 or 2 connected
components. If it has 2 components, then for sufficiently large $i$ we can
argue as in [16, Corollary 4.3] to show that in fact the original flow
$\mathcal{M}(t)$ does not have a singularity at $(\mathbf{0},0)$. Therefore
for sufficiently large $i$, $B_{1}\cap M_{\tau_{i}+1}$ is connected.

Finally consider the exactness part of Condition ($\ast$ ‣ 2.3). The
rationality assumption is preserved along the flow, see [15, Section 6]. It
follows that there is a constant $a>0$ such that, after rescaling, for any
loop $\gamma\subset M_{\tau_{i}+1}$ we have

(8.1) $\int_{\gamma}\lambda\in 2\pi ae^{\tau_{i}+1}\mathbb{Z}.$

Although we might only be able to define a multivalued function $\beta$
satisfying $\mathrm{d}\beta=\lambda$ on $M_{\tau}$, (8.1) implies that
$f=\sin(e^{-\tau_{i}-1}a^{-1}\beta)$ is single valued. Without loss of
generality we can assume that $f(\mathbf{x}_{0})=0$ for a basepoint
$\mathbf{x}_{0}\in B_{1}\cap M_{\tau_{i}+1}$. We have

$\nabla f=e^{-\tau_{i}-1}a^{-1}\cos(e^{-\tau_{i}-1}a^{-1}\beta)\nabla\beta,$

so using $|\nabla\beta|=|\mathbf{x}^{\perp}|$ we have $|\nabla f|\leq
e^{-\tau_{i}-1}a^{-1}$ on $B_{1}$. For large $i$, $B_{2}\cap M_{\tau_{i}+1}$
is almost calibrated, and so as in the proof of [15, Lemma 7.2], we have a
uniform lower bound $\mathcal{H}^{2}(\hat{B}_{i}(\mathbf{x}_{1},1))>K$ for the
intrinsic unit balls in $M_{\tau_{i}+1}$ centred at any $\mathbf{x}_{1}\in
B_{1}\cap M_{\tau_{i}+1}$. We also have an upper bound for the area of
$B_{1}\cap M_{\tau_{i}+1}$, using the bound for the area ratios. Together with
the connectedness this implies that there is a uniform constant $C_{1}$ such
that for any $\mathbf{x}_{1}\in B_{1}\cap M_{\tau_{i}+1}$, the points
$\mathbf{x}_{0},\mathbf{x}_{1}$ can be connected by a curve in $B_{1}\cap
M_{\tau_{i}+1}$ of length at most $C_{1}$, for large $i$. The gradient bound
for $f$, together with $f(\mathbf{x}_{0})=0$, then implies that $|f|\leq
C_{1}e^{-\tau_{i}-1}a^{-1}$ on $B_{1}\cap M_{\tau_{i}+1}$. For sufficiently
large $i$ we can then define a single-valued function $\beta$ on $B_{1}\cap
M_{\tau_{i}+1}$ satisfying $\mathrm{d}\beta=\lambda$, so
$\int_{\gamma}\lambda=0$ follows for any loop $\gamma\subset B_{1}\cap
M_{\tau_{i}+1}$. ∎

######  Theorem 8.2.

Suppose that $\mathcal{M}(t)$ satisfies the same assumptions as in Lemma 8.1.
Then all tangent flows at $(\mathbf{0},0)$ are given by $V_{0}$.

###### Proof.

Let $M_{\tau}$ be the rescaled flow around $(\mathbf{0},0)$. Using Lemma 8.1
we have that Condition (‡ ‣ 2.3) holds on a time interval of the form
$[\tau_{0},\infty)$, and in addition we have a sequence $\tau_{i}\to\infty$
such that $D_{V_{0}}(M_{\tau_{i}})\to 0$ and $M_{\tau_{i}+1}$ satisfies
Condition ($\ast$ ‣ 2.3).

The uniqueness of the tangent flow then follows directly from Proposition
7.12. Indeed, from the monotonicity of $\mathcal{A}_{\alpha}$ from Lemma 7.2
we also have

$\mathcal{A}_{\alpha}(M_{\tau_{i}})-\lim_{\tau\to\infty}\mathcal{A}_{\alpha}(M_{\tau})\to
0\quad\text{ as }i\to\infty.$

It follows from Proposition 7.12 that

$\lim_{i\to\infty}\sup_{\tau>\tau_{i}}D_{V_{0}}(M_{\tau})=0,$

which implies that $M_{\tau}\to V_{0}$ locally smoothly on compact sets in
$\mathbb{C}^{2}\setminus\\{\mathbf{0}\\}$ as $\tau\to\infty$. ∎

###  8.2. Lawlor necks

We now show that tangent flows given by a special Lagrangian union of two
transverse planes can only form if for sufficiently small times before the
singularity the flow looks locally like the two transverse planes,
desingularised by a shrinking Lawlor neck. Recall that, given a special
Lagrangian pair of transverse planes $V_{0}$ in $\mathbb{C}^{2}$, up to scale
there are two (exact) Lawlor necks $N_{\pm}$ asymptotic to these planes
(corresponding to $zw=\pm 1$ in suitable complex coordinates $(z,w)$ under
hyperkähler rotation). Using ideas of Seidel (cf. [21]), one can see that
$N_{\pm}$ are not Hamiltonian isotopic (using compactly supported isotopies),
but we shall not use this fact in our result below.

######  Theorem 8.3.

Under the hypotheses of Theorem 8.2, for every $\varepsilon>0$ there is
$r_{0}>0$ and a smooth function $r:[-r_{0}^{2},0)\to(0,r_{0})$ with $r(t)\to
0$ as $t\to 0$ and points $\mathbf{x}_{0}(t)\to 0$ such that
$\mathcal{M}(t)\cap\big{(}B_{r_{0}}(\mathbf{0})\setminus
B_{r(t)}(\mathbf{x}_{0}(t))\big{)}$ is a $C^{1}$-graph over $V_{0}$ with
$C^{1}$-norm bounded by $\varepsilon$. Furthermore,

(8.2) $\displaystyle r(t)^{-1}(\mathcal{M}(t)-\mathbf{x}_{0}(t))$

converge locally smoothly on $\mathbb{C}^{2}$ to a unique choice of Lawlor
neck (either $N_{+}$ or $N_{-}$) asymptotic to $V_{0}$ at infinity of maximal
neck size such that, outside of $B_{1}(\mathbf{0})$, $N$ can be written as a
$C^{1}$-graph over $V_{0}$ with $C^{1}$-norm bounded by $\varepsilon$.

To prove this result, we suppose $\mathcal{M}$ satisfies the hypotheses of
Theorem 8.2 and consider basepoints
$X=(\mathbf{x},t)\in\mathbb{C}^{2}\times[-1,0]$ close to $(\mathbf{0},0)$ at
which to discuss closeness of $\mathcal{M}$ to some $V\in\mathcal{V}$.

######  Definition 8.4.

Let $\varepsilon_{0}\in(0,1)$, $\varepsilon\in(0,\varepsilon_{0})$ and
$V\in\mathcal{V}$. We say that $\mathcal{M}$ is _$\varepsilon$ -close_ to $V$
at $X=(\mathbf{x},t)$ if the flow $\mathcal{M}^{\prime}=\mathcal{M}-X$ is a
Lagrangian $C^{1}$-graph with $C^{1}$-norm bounded by $\varepsilon$ over $V$
on
$(B_{\varepsilon^{-1}}(\mathbf{0})\setminus\bar{B}_{\varepsilon}(\mathbf{0}))\times[-\varepsilon^{-2},-\varepsilon^{2}]$.
We assume $\varepsilon_{0}$ is chosen sufficiently small (depending on
$\mathcal{V}$) such that
$\mathcal{M}^{\prime}\cap((B_{\varepsilon^{-1}}(\mathbf{0})\setminus\bar{B}_{\varepsilon}(\mathbf{0}))\times[-\varepsilon^{-2},-\varepsilon^{2}])$
is the union of two disjoint embedded annuli.

######  Remark 8.5.

We will assume further that $\varepsilon_{0}>0$ is sufficiently small such
that pseudolocality [11] implies that for every $\delta>0$ there is
$C_{\delta}\gg 1$ and $0<\varepsilon\leq\varepsilon_{0}$ such that if
$\mathcal{M}(t_{0})\cap B_{C_{\delta}}(\mathbf{x})$ is a $C^{1}$-graph over a
2-plane $P$ with $C^{1}$-norm bounded by $\varepsilon$, then
$\mathcal{M}(t)\cap B_{1}(\mathbf{x})$ is a $C^{1}$-graph over $P$ with
$C^{1}$-norm bounded by $\delta$ for $t\in[t_{0},t_{0}+1]\cap[-1,0)$.

Next we identify the range of scales at which the flow is close to some
$V\in\mathcal{V}$. We fix a small $\varepsilon\in(0,\epsilon_{0})$.

######  Definition 8.6.

Suppose that $\mathcal{M}$ is $\varepsilon$-close at $X$ to some
$V\in\mathcal{V}$. We define $\lambda_{\text{min}}(X),\lambda_{\text{max}}(X)$
to be the endpoints of the maximal interval

$1\in(\lambda_{\text{min}}(X),\lambda_{\text{max}}(X))\subseteq(0,\infty)$

such that for all
$\lambda\in(\lambda_{\text{min}}(X),\lambda_{\text{max}}(X))$ we have that
$\mathcal{D}_{\lambda^{-1}}(\mathcal{M}-X)$ is $\varepsilon$-close at
$(\mathbf{0},0)$ to $V_{\lambda}$ for some $V_{\lambda}\in\mathcal{V}$.

Note that $\lambda_{\text{min}}(X),\lambda_{\text{max}}(X)$ are continuous in
the base-point $X$.

######  Remark 8.7.

Since we can assume that all tangent flows of $\mathcal{M}$ at
$(\mathbf{0},0)$ are $\mathcal{M}_{V_{0}}$, for any sequence
$\lambda_{i}\searrow 0$ we have

(8.3)
$\mathcal{M}_{i}:=\mathcal{D}_{\lambda_{i}^{-1}}\mathcal{M}\rightharpoonup\mathcal{M}_{V_{0}}\,.$

Note that along the sequence $\mathcal{M}_{i}$ this implies that for all
points $X=(\mathbf{x},t)$ sufficiently close to
$\\{\mathbf{0}\\}\times(-\infty,0)$ we have that $\lambda_{\text{min}}(X)\to
0$ and $\lambda_{\text{max}}(X)\to\infty$.

We now wish to rule out the possibility that $\lambda_{\text{min}}(X)=0$.

######  Lemma 8.8.

Assume that for some $X=(\mathbf{x},t)$ with $t<0$ we have along
$\mathcal{M}_{i}$ in (8.3) that $\lambda_{\text{min}}(X)=0$. Then, locally
around $X$, the flow $\mathcal{M}$ is the smooth flow of two immersed planes.

###### Proof.

We first note that Remark 8.5 implies that we have smooth control on the flow
forward in time in
$B_{\lambda(\varepsilon^{-1}-C_{\delta})}(\mathbf{x})\setminus
B_{\lambda(C_{\delta}+\varepsilon)}(\mathbf{x})$ up to time $t$ as one goes
down with the scale $\lambda$ from $1$ down to $\lambda_{\text{min}}$, i.e. in
$B_{(\varepsilon^{-1}-C_{\delta})}(\mathbf{x})\setminus
B_{\lambda_{\text{min}}(C_{\delta}+\varepsilon)}(\mathbf{x})$. So if
$\lambda_{\text{min}}(X)=0$ we have that $\mathcal{M}_{i}(t)$ is locally
around $\mathbf{x}$ the union of two smooth embedded flows, where each is a
small $C^{1}$-graph over the Lagrangian planes $P_{1}$ and $P_{2}$
respectively (for $V_{0}=P_{1}\cup P_{2}$). Since the flow $\mathcal{M}_{i}$
is smooth, this has to be true forwards in time until $t=0$, and thus there is
no singularity at $(\mathbf{0},0)$ for $\mathcal{M}_{i}$ and thus for
$\mathcal{M}$. ∎

Given Lemma 8.8, we can thus always assume that $\lambda_{\text{min}}(X)>0$.
We now argue that there is more or less a ‘unique’ point where
$\lambda_{\text{min}}$ is minimised.

######  Lemma 8.9.

For $\mathcal{M}_{i}$ as in (8.3) consider points
$X_{i}(t)=(\mathbf{x}_{i}(t),t)$ which minimise $\lambda_{\text{min}}$
relative to other points $X=(\mathbf{x},t)$. Then

$\lambda_{\text{min}}(\mathbf{x},t)>\lambda_{\text{min}}(X_{i}(t))>0$

for
$(\mathbf{x},t)\in\big{(}B_{\lambda_{\text{max}}(X_{i})(\varepsilon^{-1}-C)}(\mathbf{x}_{i}(t))\setminus
B_{\lambda_{\text{min}}(X_{i})(C+\varepsilon)}(\mathbf{x}_{i}(t))\big{)}\times\\{t\\}$,
where $C=C_{\delta}$ for a suitable $\delta>0$ in Remark 8.5.

###### Proof.

This follows by a similar argument to the proof of Lemma 8.8 since Remark 8.5
implies that we have smooth control on the flow forward in time in
$B_{\lambda(\varepsilon^{-1}-C)}(\mathbf{x})\setminus
B_{\lambda(C+\varepsilon)}(\mathbf{x})$ up to time $t$ as one goes down with
the scale $\lambda$ from $\lambda_{\text{max}}$ down to
$\lambda_{\text{min}}$. ∎

######  Lemma 8.10.

There is $0<\varepsilon_{1}\leq\varepsilon_{0}$ such that for
$0<\varepsilon\leq\varepsilon_{1}$ the following holds. If there is
$\lambda_{0}\in(\lambda_{\text{min}}(X),\lambda_{\text{max}}(X))$ such that we
can choose $V_{\lambda_{0}}=V_{0}$, then $V_{\lambda}\in\mathcal{V}^{\prime}$
for all $\lambda\in(\lambda_{\text{min}}(X),\lambda_{0}]$.

###### Proof.

We consider the rescaled flow $(\hat{M}_{\tau})_{\tau\geq 0}$ for
$\hat{\mathcal{M}}:=\mathcal{D}_{\lambda_{0}^{-1}}(\mathcal{M}-X)$ and choose
$\delta_{2}>0$ and $\varepsilon_{0}>0$ small such that
$\hat{M}_{-2\log(\lambda)+2\log(\lambda_{0}(X))}$ for
$\lambda\in(\lambda_{\text{min}}(X),\lambda_{0}]$ being $\varepsilon$-close to
$V_{\lambda}$ (for $0<\varepsilon\leq\varepsilon_{0}$) and
$D_{V_{0}}(\hat{M}_{-2\log(\lambda)+2\log(\lambda_{0}(X))})\leq\delta_{2}$
implies that $V_{\lambda}\in\mathcal{V}^{\prime}$. This fixes $\epsilon_{2}>0$
in Proposition 7.12. We can then choose $\varepsilon_{1}>0$ sufficiently small
such that

$\hat{M}_{0},\hat{M}_{-2\log(\lambda_{\text{min}}(X))-2\log(\lambda_{0})}$

being $\varepsilon$-close to $V_{0}$ and $V_{\lambda_{\text{min}}(X)}$
respectively implies that condition (1) in Proposition 7.12 is met. Applying
Proposition 7.12 yields the statement. ∎

###  8.3. Finding Lawlor necks

We consider $0<\varepsilon\leq\varepsilon_{1}$, where $\varepsilon_{1}$ as in
Lemma 8.10. Consider a sequence $\lambda_{i}\searrow 0$ and let
$\mathcal{M}_{i}$ be as in (8.3). We fix $t<0$ and let $X_{i}(t)$ be as in
Lemma 8.9. We consider the flows

(8.4)
$\mathcal{M}^{\prime}_{i,t}:=\mathcal{D}_{\lambda^{-1}_{\text{min}}(X_{i}(t))}(\mathcal{M}_{i}-X_{i}(t))\
.$

We can assume that
$\mathcal{M}^{\prime}_{i,t}\rightharpoonup\widehat{\mathcal{M}}$, where
$\widehat{\mathcal{M}}$ is an ancient unit-regular Brakke flow such that
$\mathcal{D}_{\lambda^{-}1}\widehat{\mathcal{M}}$ is $\varepsilon$-close to
some $V_{\lambda}\in\mathcal{V}$ for $\lambda\in[1,\infty)$, but not for a
sequence $\lambda_{i}\nearrow 1$. Furthermore, $\widehat{\mathcal{M}}$ is
(locally) the limit of smooth, exact, almost calibrated Lagrangian mean
curvature flows with uniformly bounded Lagrangian angle and uniformly bounded
area ratios.

######  Lemma 8.11.

The flow $\widehat{\mathcal{M}}$ is a static Lawlor neck $N$ asymptotic to
$V^{\prime}\in\mathcal{V}^{\prime}$, where $V^{\prime}$ is $\varepsilon$-close
to $V_{0}$, where the centre and the scale of $N$ are such that there is no
point $X=(\mathbf{x},t)$ with $\lambda_{\text{min}}(X)<1$.

###### Proof.

We first note that $\widehat{\mathcal{M}}$ has entropy bounded by two.
Furthermore, from the argument in the proof of Lemma 8.8 we see that outside
of
$(B_{C_{\delta}+1}(\mathbf{0})\times(-\infty,0))\cup\bigcup_{t<0}B_{\sqrt{-t}(C_{\delta}+1)}(\mathbf{0})\times\\{t\\}$
the flow is a smooth Lagrangian which is a controlled $C^{1}$-graph over
$V_{0}$. Let $\widehat{\mathcal{M}}^{\prime}$ be a tangent flow of
$\widehat{\mathcal{M}}$ at $-\infty$. The discussion before implies that
$\mathcal{D}_{\lambda}\widehat{\mathcal{M}}^{\prime}$ is $\varepsilon$-close
to some $V_{\lambda}\in\mathcal{V}^{\prime}$ for all $\lambda>0$ as well as
that $\widehat{\mathcal{M}}^{\prime}$ is a smooth Lagrangian and controlled
$C^{1}$-graph over $V_{0}$ outside of
$\bigcup_{t<0}B_{\varepsilon\sqrt{-t}}(\mathbf{0})\times\\{t\\}$. Furthermore,
the proof of [18, Theorem 3.1], see also [13, Theorem 3.1], directly extends
to Brakke flows which are limits of smooth Lagrangian mean curvature flows
with uniformly bounded Lagrangian angle and uniformly bounded area ratios.
Thus we can assume that $\widehat{\mathcal{M}}^{\prime}$ is a static pair of
planes $V^{\prime}\in\mathcal{V}^{\prime}$. Note that if there is a point
where $\widehat{\mathcal{M}}$ has Gaussian density two, then the flow is
static (since it is unit regular) and up to a translation equal to
$\mathcal{M}_{V^{\prime}}$. However, this would imply that there are points
$\tilde{X}_{i}=(\tilde{\mathbf{x}}_{i},t)$ close to $X_{i}(t)$ with
$\lambda_{\text{min}}(\tilde{X}_{i})<\lambda_{\text{min}}(X_{i}(t))$ where
$X_{i}(t)$ is as in Lemma 8.9. This yields a contradiction.

We may therefore assume that all Gaussian densities are less than two, which
implies the convergence
$\mathcal{M}^{\prime}_{i,t}\rightharpoonup\widehat{\mathcal{M}}$ is smooth and
thus $\widehat{\mathcal{M}}$ is a smooth, ancient, exact, almost calibrated
Lagrangian mean curvature flow with uniformly bounded Lagrangian angle and
uniformly bounded area ratios. Since $\widehat{\mathcal{M}}$ cannot be a union
of Lagrangian planes, by [13, Theorem 1.1] it is up to rigid motions a static
Lawlor neck. Arguing similarly as before we see that there cannot be a point
$X=(\mathbf{x},t)$ with $\lambda_{\text{min}}(X)<1$. ∎

###### Proof of Theorem 8.3.

Consider $0<\varepsilon\leq\varepsilon_{1}$, where $\varepsilon_{1}$ as in
Lemma 8.10. Replacing $\mathcal{M}$ by $D_{\lambda^{-1}}\mathcal{M}$ for some
sufficiently small $\lambda>0$ we can assume that for all $t\in[-1,0)$ there
are points $X(t)$ as in Lemma 8.9 such that
$\lambda_{\text{min}}(X(t))<1\leq\lambda_{\text{max}}(X(t))$, minimising
$\lambda_{\text{min}}$ in their time slice, and with $V_{\lambda}=V_{0}$ for
$\lambda=1$. Lemma 8.10 then implies that for each $t\in(0,1]$ and
$\lambda\in[\lambda_{\text{min}}(X(t)),1]$ we have that
$\mathcal{D}_{\lambda^{-1}}(\mathcal{M}-X(t))$ is $\varepsilon$-close to
$V_{\lambda,t}\in\mathcal{V}^{\prime}$. Applying Remark 8.5 we see that
$\mathcal{M}(t)-\mathbf{x}(t)$ is a small $C^{1}$-graph over $V_{0}$ on
$B_{1}(\mathbf{0})\setminus B_{C\lambda_{\text{min}}(X(t))}(\mathbf{0})$ for a
suitable fixed $C>0$ (depending only on $\varepsilon>0$).

Restricting to $-\delta\leq t<0$ we can apply Lemma 8.11 to find that the
rescaled flow
$\mathcal{D}_{(\lambda_{\text{min}}(X(t))^{-1}}(\mathcal{M}-X(t))$ has to be a
small $C^{1}$-graph over a Lawlor neck $N$ over a large compact set. Note that
by continuity of the flow (and assuming sufficient $C^{1}$-closeness to $N$)
this has to be either $N_{+}$ or $N_{-}$ for all $-\delta\leq t<0$. The full
convergence to $N$ as $t\nearrow 0$ then follows by considering a sequence
$\varepsilon_{i}\to 0$. ∎

###  8.4. Continuing the flow past the singularity

We can now argue that the uniqueness of the tangent flow implies that at the
singular time $t=0$, in a neighbourhood of $\mathbf{0}\in\mathbb{C}^{2}$, the
flow limits to the union of two Lagrangian graphs such that we can restart the
flow as a Lagrangian mean curvature flow.

######  Lemma 8.12.

There is $r_{0}>0$ such that on
$B_{r_{0}}(\mathbf{0})\setminus\\{\mathbf{0}\\}$ the flow $\mathcal{M}(t)$
converges as $t\to 0$ locally to two Lagrangian graphs $L_{1},L_{2}$ over
$P_{1}$ and $P_{2}$ respectively (where $V_{0}=P_{1}\cup P_{2}$). Moreover,
using Lagrangian neighbourhoods for $P_{i}$, we have that
$L_{i}=\text{graph}_{P_{i}}(\mathrm{d}f_{i})$ for $i=1,2$, where $f_{i}\in
C^{2}\big{(}P_{i}\cap B_{r_{0}}(\mathbf{0})\big{)}$ is smooth away from
$\mathbf{0}$ with
$f_{i}(\mathbf{0})=\mathrm{d}f_{i}(\mathbf{0})=\nabla_{P_{i}}(\mathrm{d}f_{i})(\mathbf{0})=0$.

###### Proof.

Note that Theorem 8.2 implies that there is $0<\lambda_{0}\leq 1$ and a
continuous increasing function $\varepsilon:(0,\lambda_{0}]\to(0,1)$ with
$\varepsilon(\lambda)\to 0$ as $\lambda\to 0$ such that
$\mathcal{D}_{\lambda^{-1}}\mathcal{M}$ is $\varepsilon(\lambda)$-close to
$V_{0}$. Combining this with Remark 8.5 as in the proof of Theorem 8.3 then
yields that there is a smooth Lagrangian limiting surface $L$ on
$B_{r_{0}}(\mathbf{0})\setminus\\{\mathbf{0}\\}$ such that $\lambda^{-1}L$ is
$\delta(\lambda)$-close to $V_{0}$ for some continuous decreasing function
$\delta:(0,\lambda_{0}]\to(0,1)$ with $\delta(\lambda)\to 0$ as $\lambda\to
0$. This implies the claimed convergence and that we can decompose
$L=L_{1}\cup L_{2}$ such that each $L_{i}$ is a small $C^{1}$-graph over
$P_{i}\cap B_{r_{0}}(\mathbf{0})$ of a vector-valued function $u_{i}$, which
is smooth away from $\mathbf{0}$ and $C^{1}$ across $\mathbf{0}$ with
$u_{i}(\mathbf{0})=\nabla_{P_{i}}u_{u}(\mathbf{0})=0$. Applying the Lagrangian
neighbourhood theorem to each $P_{i}$, we see that $u_{i}$ may be identified
with a closed 1-form on $P_{i}\cap B_{r_{0}}(\mathbf{0})$, which is then
necessarily exact. The Poincaré lemma then gives a $C^{2}$-function $f_{i}$
with $\mathrm{d}f_{i}=u_{i}$ as claimed. ∎

###### Proof of Theorem 1.2.

By Lemma 8.12 there can be only finitely many singularities at time $T$ where