# Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature
flow with circle symmetry

Jason D. Lotay  University of Oxford, U.K. http://people.maths.ox.ac.uk/lotay/
<EMAIL_ADDRESS>and  Goncalo Oliveira  Department of Mathematics
and CAMGSD, Instituto Superior Técnico, Portugal
https://sites.google.com/view/goncalo-oliveira-math-webpage/home
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article we consider the Lagrangian mean curvature flow of compact,
circle-invariant, almost calibrated Lagrangian surfaces in hyperkähler
4-manifolds with circle symmetry. We show that this Lagrangian mean curvature
flow can be continued for all time, through finite time singularities, and
converges to a chain of special Lagrangians, thus verifying various aspects of
Joyce’s conjectures [JoyceConjectures] in this setting. We show that the
singularities of the flow are neck pinches in the sense conjectured by Joyce
[JoyceConjectures]. We also give examples where such finite time singularities
are guaranteed to occur.

###### Contents

  1. 1 Introduction
  2. 2 The Gibbons–Hawking ansatz
  3. 3 Lagrangian mean curvature flow and modified curve shortening flow
  4. 4 Finite time singularities
  5. 5 Flow through singularities and long–time behaviour
  6. 6 Monotonicity of the Lagrangian angles

##  1\. Introduction

###  1.1. Context

A standing conjecture of Thomas [Thomas], motivated by mirror symmetry,
asserts that there is a stability condition for compact graded Lagrangians in
a Calabi–Yau manifold, which is expected to determine the existence (and
uniqueness) of a special Lagrangian in a given Hamiltonian isotopy class.
However, this stability condition is hard to work with and the conjecture has
so far remained unproven in its full generality. In [Lotay2020], the authors
proved the Thomas conjecture for circle-invariant Lagrangians in a large class
of hyperkähler 4-manifolds which includes all complete examples with finite
topology obtained from the Gibbons–Hawking ansatz. This contains all ALE and
ALF hyperkähler 4-manifolds admitting a tri-Hamiltonian circle action.

Soon after stating the above conjecture, Thomas–Yau [ThomasYau] proposed that
a similar stability condition on compact graded Lagrangians controls the long-
time existence and convergence of the Lagrangian mean curvature flow. Later
developments of Neves [NevesSingularities] showed explicitly that the initial
Lagrangian must at least be almost calibrated, which had implicitly been
assumed in [ThomasYau], otherwise finite time singularities of the flow are
inevitable. Under the almost calibrated assumption the authors proved, also in
[Lotay2020], the circle-invariant Thomas–Yau conjecture in the same class of
examples mentioned before.

Both the Thomas and Thomas–Yau conjectures pre-date Bridgeland’s definition of
stability conditions on triangulated categories [Bridgeland2007, Douglas2001],
which can be applied to the study of Lagrangians in Calabi–Yau manifolds by
using Fukaya categories where special Lagrangians become (semi-)stable
objects. Using this perspective, Joyce [JoyceConjectures] updated the
Thomas/Thomas–Yau conjectures by making use of the notion of Bridgeland
stability condition. As a way to tackle this conjecture, Joyce proposed the
use of Lagrangian mean curvature flow through finite time singularities
(possibly with surgeries).111See also [Solomon2014, Solomon2017, Solomon2020]
for an alternative proposal on how to approach the Thomas and Thomas–Yau
conjectures and also [Li2022] for some very recent developments. In
particular, Joyce’s program relies on a number of conjectures and principles,
most of which are still open, regarding what happens along the Lagrangian mean
curvature flow to an unstable Lagrangian. In this article, we shall prove
versions of Joyce’s conjectures for circle-invariant Lagrangians in the class
of hyperkähler 4-manifolds mentioned above, thus providing substantial
progress towards the full verification of Joyce’s program in this setting. Our
results give concrete examples of so-called “neck pinch” singularities and
establish exactly what happens for the flow of the Lagrangians mentioned
above: see Theorem 1.1. Our main result, Theorem 1.4, shows that the flow
through neck pinch singularities exists for all time and converges at infinity
to a chain of special Lagrangian submanifolds (with possibly different
phases).

###  1.2. Main results

In this article, we shall prove some of Joyce’s key conjectures on the
behaviour of the Lagrangian mean curvature flow starting at a (possibly
unstable) circle-invariant Lagrangian.

Our first main result explores the local behaviour at the finite time
singularities of Lagrangian mean curvature flow. The type of singularity we
find is known as a “neck pinch” as the resulting flow of Lagrangians locally
resembles a family of cylinders $S^{1}\times\mathbb{R}$ with
$S^{1}\times\\{0\\}$ contracted to a point when the singularity forms. This
proves versions of the conjectures stated as Principle 3.9(a), Problem
3.12(a), and Conjecture 3.16(ii)-(iii) in [JoyceConjectures].

To state the result, we say that a chart $\varphi:U\subseteq X\to
V\subseteq\mathbb{C}^{2}$ on a hyperkähler 4-manifold $X$, viewed as a
Calabi–Yau 2-fold, is a _pointed isomorphism at $p\in U$_ if $\varphi(p)=0\in
V$ and it identifies the Calabi–Yau structure in $T_{p}X$ with the standard
one in $\mathbb{C}^{2}$. We also recall that _Lawlor necks_ in
$\mathbb{C}^{2}$ are exact, embedded special Lagrangians which are
topologically $S^{1}\times\mathbb{R}$ and asymptotic to a pair of special
Lagrangian planes which intersect transversely at the origin.

######  Theorem 1.1 (“Neck pinch” finite time singularities).

Let $X$ be an ALE or ALF hyperkähler $4$-manifold admitting a tri-Hamiltonian
circle action. Let $L$ be an embedded, almost calibrated, circle-invariant
Lagrangian in $X$ which is either compact or asymptotic at infinity to a pair
of planes.

Let $\\{L_{t}\\}_{t\in[0,T)}$ be the unique smooth solution to Lagrangian mean
curvature flow in $X$ starting at $L$, so that $L_{t}$ has the same properties
as $L$ for each $t\in[0,T)$, for some $T>0$.

  1. (a)

Suppose that the flow $L_{t}$ develops a finite time singularity at $p\in X$
when $t\to T<\infty$. Then, there exist

     * •

open neighbourhoods $U$ of $p$ in $X$ and $V$ of $0$ in
$T_{p}X\cong\mathbb{C}^{2}$;

     * •

a pointed isomorphism $\varphi:U\to V$ at $p$;

     * •

a small $\delta>0$ and a smooth function
$\varepsilon:(T-\delta^{2},T)\to(0,\delta)$, with $\varepsilon(t)\searrow 0$
as $t\nearrow T$,

such $\varepsilon(t)^{-1}\varphi(L_{t}\cap U)$ converges on compact subsets of
$\mathbb{C}^{2}$ to a unique Lawlor neck.

  2. (b)

For any such $X\neq\mathbb{R}^{3}\times\mathbb{S}^{1}$ there is such a
Lagrangian $L$ so that the flow $L_{t}$ starting at $L$ develops a finite type
singularity as in _(a)_. Moreover, if $X$ contains a pair of special
Lagrangian spheres with different phases, even up to changing orientations on
the spheres, then $L$ can be chosen to be compact.

######  Remark 1.2.

As stated in [JoyceConjectures]*Remark 3.10, Theorem 1.1(a) contains more
information than stating that the (unique) Type II blow-up at the singularity
is a Lawlor neck. It can be interpreted as saying that there is a fixed Lawlor
neck $L_{\infty}$ in $\mathbb{C}^{2}$ such that
$\varphi^{-1}(\varepsilon(t)L_{\infty}\cap V)$ models the movement of $L_{t}$
in the fixed, definite size neighbourhood $U$ of $p\in X$.

Before stating our second main result we require the following definition.

######  Definition 1.3.

An ordered set $\\{L_{1},\ldots,L_{k}\\}$ of $k$ 2-spheres in $X$ is said to
be an _$A_{k}$ chain_ if for all $i\neq j$ their intersections satisfy
$L_{i}\cdot L_{j}=\delta_{i-1,j}+\delta_{i,j-1}$.

Our second main result shows that, in our setting, there are a finite number
of singular times along the flow and, moreover, a Lagrangian mean curvature
flow _through singularities_ exists for all time and converges to a finite
union of special Lagrangian spheres (with possibly different phases). This
proves, in our setting, several parts of the program in
[JoyceConjectures]*Section 3.2 & Conjecture 3.9.

######  Theorem 1.4 (Decomposition into special Lagrangians).

Let $X$ be an ALE or ALF hyperkähler $4$-manifold admitting a tri-Hamiltonian
circle action and $L$ a compact, connected, embedded, circle-invariant, almost
calibrated Lagrangian in $X$. There is a continuous family
$\\{L_{t}\\}_{t\in[0,+\infty)}$ of almost calibrated, circle-invariant,
Lagrangian integral currents with $L_{0}=L$ so that the following holds.

  * (a)

There is a finite number of singular times $0<T_{1}\leq\ldots\leq
T_{l}<\infty$ such that the family
$\\{L_{t}\\}_{t\in[0,\infty)\backslash\\{T_{1},\ldots,T_{l}\\}}$ satisfies
Lagrangian mean curvature flow.

  * (b)

At each singular time $T_{i}$ the flow undergoes a “neck pinch” singularity as
in Theorem 1.1.

  * (c)

There is $k\in\mathbb{N}$ and an $A_{k}$-chain of embedded, special
Lagrangians spheres $\\{L^{\infty}_{1},\ldots,L_{k}^{\infty}\\}$ such that
$L_{t}$ converges uniformly to $\cup_{j=1}^{k}L_{j}^{\infty}$ as $t\to+\infty$
and we have current convergence:

$\lim_{t\to+\infty}L_{t}=L_{1}^{\infty}+\ldots+L_{k}^{\infty}.$

  * (d)

If the grading on $L$ is a perfect Morse function, then the phases
$\theta_{j}$ of the special Lagrangians $L_{j}^{\infty}$ from _(c)_ can be
chosen to satisfy $\theta_{1}\geq\ldots\geq\theta_{k}$.

Furthermore, in _(c)_ we have that $k=1$ if $L$ is flow stable, and $k>1$ if
$L$ is flow unstable.

######  Remark 1.5.

The uniform convergence in (c) can always be improved to smooth convergence
unless in the $A_{k}$-chain of spheres
$\\{L_{1}^{\infty},\ldots,L_{k}^{\infty}\\}$ we have that the phase of two
adjacent spheres are equal. In this case there is the possibility of an
infinite time singularity of the flow: this is the so-called “semi-stable”
case. Therefore, if we assume that we are in the setting where no three
singularities of the potential $\phi$ lie on a line, then we will always have
smooth convergence. We can always make a small perturbation of the hyperkähler
structure on $X$ so that we are in this regime.

### Acknowledgements

The first author would like to thank Dominic Joyce for useful conversations.
The first author was partially supported by EPSRC grant EP/T012749/1 during
the course of this project. The second author wants to thank the members of
Hausel group at IST Austria who hosted him as part of this project was being
performed. The second author is currently funded by FCT 2021.02151.CEECIND and
previously by the NOMIS foundation. Both authors would like to thank the
Simons Laufer Mathematical Sciences Institute (formerly known as MSRI),
Berkeley for hospitality during the latter stages of this project.

##  2\. The Gibbons–Hawking ansatz

We provide a short description of the Gibbons–Hawking ansatz, and circle-
invariant Lagrangians and notions of stability in this context, as necessary
for our study. For further details, we refer the reader to [Lotay2020].

###  2.1. Hyperkähler 4-manifolds with tri-Hamiltonian circle action

Let $(X^{4},g)$ be hyperkähler and equipped with a circle action preserving
the three Kähler forms $\omega_{1},\omega_{2},\omega_{3}$, associated with the
orthogonal complex structures $I_{1},I_{2},I_{3}$ on $X$ satisfying the
quaternionic relations. Denote by $\xi$ the infinitesimal generator of the
circle action and let $\hat{X}=\\{x\in X\ |\ \xi_{x}\neq 0\\}$ be the open
dense set where the action is free. Then, $\hat{X}$ is a ${\rm U}(1)$-bundle

$\pi:\hat{X}\rightarrow Y^{3}$

over an open $3$-manifold $Y^{3}$. Equip this bundle with a connection
$\eta\in\Omega^{1}(\hat{X},\mathbb{R})$ whose horizontal spaces
$\ker(\eta)=\xi^{\perp}$ are $g$-orthogonal to $\xi$ and so that
$\eta(\xi)=1$. Note that $\iota_{\xi}d\eta=0$.

Consider the positive ${\rm U}(1)$-invariant function
$\phi:\hat{X}\to\mathbb{R}$ determined by $\phi^{-1}=|\xi|_{g}^{2}$ and define
the 1-forms $\alpha_{i}:=I_{i}(\phi^{-1}\eta)$, for $i=1,2,3$. The hyperkähler
metric $g$ may then be written on $\hat{X}$ as

(2.1) $g=\phi^{-1}\eta^{2}+\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}$

and the associated Kähler forms are given by:

(2.2) $\displaystyle\omega_{1}$
$\displaystyle=\phi^{-1/2}\eta\wedge\alpha_{1}+\alpha_{2}\wedge\alpha_{3},\quad\omega_{2}=\phi^{-1/2}\eta\wedge\alpha_{2}+\alpha_{3}\wedge\alpha_{1},\quad\omega_{3}=\phi^{-1/2}\eta\wedge\alpha_{3}+\alpha_{1}\wedge\alpha_{2}.$

With the orientation induced by the volume form
$\phi^{-1}\eta\wedge\alpha_{1}\wedge\alpha_{2}\wedge\alpha_{3}$, the forms
$\omega_{1},\omega_{2},\omega_{3}$ give a trivialization of
$\Lambda^{2}_{+}\hat{X}$, the bundle of self-dual 2-forms on $\hat{X}$.
Conversely, if we define 2-forms $\omega_{1},\omega_{2},\omega_{3}$ as in
(2.2) and fix the volume form
$\phi^{-1}\eta\wedge\alpha_{1}\wedge\alpha_{2}\wedge\alpha_{3}$, we can
recover the metric $g$ as in (2.1), and it follows from [Atiyah]*Lemma 4.1
that $g$ is hyperkähler if and only if $d\omega_{i}=0$ for $i=1,2,3$. Using
this characterization we have the following (cf. [Lotay2020]*Proposition 2.1).

######  Proposition 2.1.

Using the notation above, the metric $g$ in (2.1) equips $X^{4}$ with a
hyperkähler structure so that the ${\rm U}(1)$ action generated by $\xi$
preserves $g$ and $\omega_{1},\omega_{2},\omega_{3}$ in (2.2) if and only if
the following hold.

  1. (a)

The symmetric $2$-tensor

$g_{E}=\phi^{-2}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})$

is the pullback of a flat metric on $Y^{3}$.

  2. (b)

The pair $(\eta,\phi)$ is a Dirac monopole on $Y^{3}$, i.e.

(2.3) $\ast_{E}d\eta=-d\phi,$

where $\ast_{E}$ denotes the Hodge star operator associated with the metric
$g_{E}$ on $Y^{3}$ from _(a)_.

  3. (c)

There are local coordinates $(\mu_{1},\mu_{2},\mu_{3})$ on $Y^{3}$ such that
$\alpha_{i}=\phi^{\frac{1}{2}}d\mu_{i}$, and the hyperkähler metric can be
written as

(2.4)
$g=\frac{1}{\phi}\eta^{2}+\phi\left(d\mu_{1}^{2}+d\mu_{2}^{2}+d\mu_{3}^{2}\right).$

Moreover,

(2.5) $\omega_{1}=\eta\wedge d\mu_{1}+\phi d\mu_{2}\wedge
d\mu_{3},\quad\omega_{2}=\eta\wedge d\mu_{2}+\phi d\mu_{3}\wedge
d\mu_{1},\quad\omega_{3}=\eta\wedge d\mu_{3}+\phi d\mu_{1}\wedge d\mu_{2}.$

We shall consider the case when $Y^{3}$ is simply connected, in which case the
coordinates $(\mu_{1},\mu_{2},\mu_{3})$ can be taken to be global and form the
hyperkähler moment map

$\pi:X\rightarrow\mathbb{R}^{3}.$

In [Lotay2020]*Section 2.2 we give examples of hyperkähler manifolds arising
from this construction including the flat, Taub–NUT, Eguchi–Hanson,
Ooguri–Vafa and Anderson–Kronheimer–LeBrun examples. Here we shall simply
recall the multi-Taub–NUT and multi-Eguchi–Hanson examples.

######  Example 2.2.

Choose $k\geq 1$ points $p_{1},...,p_{k}$ in $\mathbb{R}^{3}$ and $m\geq 0$.
Set $Y=\mathbb{R}^{3}\backslash\\{p_{1},...,p_{k}\\}$ and

(2.6) $\phi=m+\sum_{i=1}^{k}\frac{1}{2|x-p_{i}|},$

where the norm is the Euclidean metric on $\mathbb{R}^{3}$. Then there is a
connection $\eta$ satisfying (2.3) and the metric $g$ given by (2.4) extends
smoothly across the points $p_{1},\ldots,p_{k}$ where the circle action
collapses, thus completing
$\hat{X}=\pi^{-1}(\mathbb{R}^{3}\backslash\\{p_{1},\ldots,p_{k}\\})$ to a
hyperkähler manifold $(X^{4},g)$ by adding in $k$ points.

Suppose that $m=0$. In this case, when $k=1$ we obtain the flat metric on
$\mathbb{R}^{4}$ and for $k=2$ we get the Eguchi–Hanson metric. For $k>2$ the
resulting metric is called the multi-Eguchi–Hanson metric. We always have
$\lim_{r\to\infty}\phi=0$ (where $r$ is the distance to a fixed point, say the
origin, in $\mathbb{R}^{3}$) and thus $g$ is asymptotic to the flat metric on
$\mathbb{R}^{4}/\mathbb{Z}_{k}$, so $(X^{4},g)$ is _asymptotically locally
Euclidean_ (ALE).

Suppose now that $m>0$. When $k=1$ we obtain the Taub–NUT metric on
$\mathbb{R}^{4}$ and so if $k>1$ we call the resulting metric multi-Taub–Nut.
(Note that if we allowed $k=0$ we would obtain the product metric on
$\mathbb{S}^{1}\times\mathbb{R}^{3}$.) In this situation
$\lim_{r\to\infty}\phi=m>0$ and so the circle generated by $\xi$ has finite
length at infinity. The metric $g$ is therefore asymptotic to one on a circle
bundle over $\mathbb{R}^{3}$ with the fibers having constant length
proportional to $m^{-1/2}$. Such metrics are called _asymptotically locally
flat_ (ALF).

######  Remark 2.3.

Example 2.2 describes all of the ALE and ALF gravitational instantons which
arise from the Gibbons–Hawking ansatz.

###  2.2. Notation

Throughout this article we will be working on a hyperkähler 4-manifold $X^{4}$
given by the Gibbons–Hawking ansatz which is an ALE or ALF gravitational
instanton. We will therefore use the notation in Proposition 2.1 and Example
2.2 for the remainder of the article.

###  2.3. Lagrangian submanifolds

On any hyperkähler 4-manifold, the twistor sphere defines a 2-sphere of Kähler
structures for which one can study Lagrangians. It is important to understand
this 2-sphere explicitly in our setting so that we may identify circle-
invariant Lagrangians in $X$ with certain curves in the base of the fibration
$\pi:X\to\mathbb{R}^{3}$. Throughout, we shall only consider connected
Lagrangians.

To this end, we see that the twistor sphere of $X$ can be identified with the
unit $2$-sphere $\mathbb{S}^{2}\subseteq\mathbb{R}^{3}$ as follows. Given
$v=(v_{1},v_{2},v_{3})\in\mathbb{S}^{2}\subseteq\mathbb{R}^{3}$, we have a
complex structure $I_{v}$ on $X$ such that the associated $2$-form using the
hyperkähler metric is (recalling (2.5))

(2.7) $\omega_{v}=\sum_{i=1}^{3}v_{i}\left(\eta\wedge d\mu_{i}+\phi\
d\mu_{j}\wedge d\mu_{k}\right),$

with $(i,j,k)$ denoting a cyclic permutation of $(1,2,3)$. For example, if
$v=(0,0,1)$ then $\omega_{v}=\omega_{3}$.

Any circle-invariant surface $L$ in $X$ corresponds, via
$\pi:X\to\mathbb{R}^{3}$, to a curve $\gamma\subseteq\mathbb{R}^{3}$. For
$v\in\mathbb{S}^{2}$, a short computation (cf. [Lotay2020]*Section 5.1)
yields, for $L=\pi^{-1}(\gamma)$,

$\omega_{v}|_{L}=\langle\gamma^{\prime},v\rangle\mathrm{vol}_{L},$

where $\gamma^{\prime}$ is the velocity of $\gamma$ with respect to Euclidean
arclength and $\langle.,.\rangle$ is the Euclidean inner product. We deduce
the following.

######  Lemma 2.4.

In the notation above, a circle-invariant surface $L^{2}=\pi^{-1}(\gamma)$ in
$X^{4}$ for a curve $\gamma\subseteq\mathbb{R}^{3}$ is Lagrangian with respect
to $\omega_{v}$ in (2.7) if and only if $\gamma$ lies in a plane orthogonal to
$v$.

Let $S_{\phi}=\\{p_{1},\ldots,p_{k}\\}\subseteq\mathbb{R}^{3}$ be the set of
singularities of $\phi$ as in Example 2.2. We then see that for
$L=\pi^{-1}(\gamma)$ to be compact and embedded $\gamma$ cannot have self
intersections and is either:

  * •

a simple closed curve not intersecting $S_{\phi}$, in which case
$L=\pi^{-1}(\gamma)\cong T^{2}$; or

  * •

a simple arc with end points $p_{i},p_{j}\in S_{\phi}$ with $i\neq j$ and
otherwise not meeting $S_{\phi}$, in which case $\pi^{-1}(\gamma)\cong S^{2}$.

######  Remark 2.5.

In our setting $S_{\phi}$ is finite, but more generally it can be infinite as
in the Ooguri–Vafa and Anderson–Kronheimer–LeBrun metrics (see
[Lotay2020]*Examples 2.7 & 2.8).

We will only be interested in graded Lagrangians, as we shall now define.

######  Definition 2.6.

Given a Calabi–Yau structure $(\omega,\Omega)$ on $X$, determined by a Kähler
form $\omega$ and holomorphic volume form $\Omega$, an oriented Lagrangian $L$
in $X$ is said to be _graded_ by $\theta:L\to\mathbb{R}$ if the restriction of
$\Omega$ to $L$ satisfies

(2.8) $e^{-i\theta}\Omega|_{L}=\mathrm{vol}_{L},$

where $L$ is the Riemannian volume form associated with the induced metric. We
will denote a Lagrangian $L$ graded by $\theta$ as $(L,\theta)$ where
necessary and we will refer to $\theta$ as the grading. The choice of grading
$\theta$ is also called the _Lagrangian angle_ of $L$.

######  Remark 2.7.

Notice, in particular, that for a compact graded Lagrangian $L$ the quantity

$\arg\int_{L}\Omega$

is well-defined up to integer multiples of $2\pi$. Moreover, given such an $L$
we can always multiply $\Omega$ by a unit complex number so that
$\arg\int_{L}\Omega=0$ modulo $2\pi$.

It will be useful to study a distinguished subclass of the graded Lagrangians
as follows.

######  Definition 2.8.

Let $(\omega,\Omega)$ be a Calabi–Yau structure on $X$. An oriented Lagrangian
$L$ in $(X,\omega,\Omega)$ is _almost calibrated_ if there is a choice of
grading $\theta$ on $L$ such that, for some $\delta>0$,

$\sup_{L}\theta-\inf_{L}\theta\leq\pi-\delta.$

If $L$ is compact, $L$ is almost calibrated if and only if there is a constant
$\theta_{0}$ so that $\mathop{\mathrm{Re}}(e^{-i\theta_{0}}\Omega)|_{L}>0$.

In our situation, given $v\in\mathbb{S}^{2}$ and $\omega_{v}$ as in (2.7),
there is a circle of holomorphic volume forms $\Omega_{v}$ so that
$(X,\omega_{v},\Omega_{v})$ is Calabi–Yau, namely

(2.9) $\Omega_{v}=\omega_{v_{1}}+i\omega_{v_{2}},$

where $\\{v,v_{1},v_{2}\\}$ is a positively oriented orthonormal basis for
$\mathbb{R}^{3}$. For example, if $v=(0,0,1)$ then
$\Omega_{v}=e^{i\alpha}(\omega_{1}+i\omega_{2})$ for some
$e^{i\alpha}\in\mathbb{S}^{1}$. We see that

$\Omega_{v}|_{L}=\left(\langle\gamma^{\prime},v_{1}\rangle+i\langle\gamma^{\prime},v_{2}\rangle\right)\mathrm{vol}_{L}$

and so the Lagrangian angle of $L$ satisfies
$\cos\theta=\langle\gamma^{\prime},v_{1}\rangle$ and
$\sin\theta=\langle\gamma^{\prime},v_{2}\rangle$, i.e. $\theta$ coincides (mod
$2\pi$) with the angle that $\gamma^{\prime}$ makes with $v_{1}$. See
[Lotay2020]*Section 5 for more details. As a consequence, any compact,
embedded, graded Lagrangian of the form $L=\pi^{-1}(\gamma)$ must have
$\gamma$ be a simple arc joining two singularities of $\phi$ and meeting no
other singularities, in which case $L$ is a $2$-sphere. Moreover, such an $L$
is almost calibrated if and only if the variation of the angle which
$\gamma^{\prime}$ makes with $v_{1}$ is strictly less than $\pi$: we shall
call such curves almost calibrated, by abuse of notation. Note that almost
calibrated curves are automatically embedded.

######  Remark 2.9.

Suppose that $L=\pi^{-1}(\gamma)$ is an embedded, compact, almost calibrated,
circle-invariant Lagrangian in $X$, for some choice of Calabi–Yau structure
$(\omega,\Omega)=(\omega_{v},\Omega_{v})$ for some $v\in\mathbb{S}^{2}$ as
above. With no loss of generality we can suppose that $\gamma$ is
perpendicular to the $\mu_{3}$-axis and $\Omega=\omega_{1}+i\omega_{2}$.
Furthermore, up to a translation we may set the initial point of $\gamma$ to
be $(0,0,0)$ and denote its final point by $(x,y,0)$. Then,

$\int_{L}\Omega=2\pi\int_{\gamma}(d\mu_{1}+id\mu_{2})=2\pi(x+iy),$

and so $\arg\int_{L}\Omega$ is the angle between the straight-line
$\overline{\gamma}$ connecting the endpoints of $\gamma$ and the
$\mu_{1}$-axis. Moreover,

$\left|\int_{L}\Omega\right|=2\pi\sqrt{x^{2}+y^{2}}=2\pi\mathrm{Length(\overline{\gamma})},$

and $2\pi\mathrm{Length(\overline{\gamma})}=\mathrm{Area}(\overline{L})$ where
$\overline{L}=\pi^{-1}(\overline{\gamma})$ is the area-minimizer in the
homology class of $L$. We also notice that $\arg\int_{L}\Omega=0$ if and only
if $y=0$ and $x>0$, i.e. the endpoint of $\gamma$ lies on the positive
$\mu_{1}$-axis.

###  2.4. Further notation

We shall from now on fix a circle-invariant Calabi–Yau structure
$(\omega,\Omega)=(\omega_{v},\Omega_{v})$ on $X$ for some $v\in\mathbb{S}^{2}$
as in (2.7) and (2.9). We see from our discussion above that for there to be
any compact, embedded, graded, circle-invariant Lagrangians in
$(X,\omega,\Omega)$ we require there to be at least 2 singularities of $\phi$,
i.e. we need $k>1$ in the notation of Example 2.2.

###  2.5. Stability and flow stability

We shall be using the notions of stability for Lagrangians introduced by
Thomas [Thomas] and Thomas–Yau [ThomasYau]. After the introduction of the
notion of Bridgeland stability conditions [Bridgeland2007] such notions may be
modified in order to use such a framework, but we shall not pursue this here.
A key reason for this is that it is not yet known whether Bridgeland stability
conditions exist in some version of the Fukaya category relevant for our
study. Moreover, it is part of Joyce’s programme [JoyceConjectures] that one
should use Lagrangian mean curvature flow to even define the Bridgeland
stability condition. For further discussion of the possible relation between
Bridgeland stability conditions and (modifications of) the Thomas–Yau
conjecture, we refer the reader to [JoyceConjectures, Li2022].

######  Definition 2.10 (Stability).

Let $(L,\theta)$ be a compact graded Lagrangian in $(X,\omega,\Omega)$. Then,
$L$ is _unstable_ if its Hamiltonian isotopy class can be decomposed as a
graded connect sum $L_{1}\\#L_{2}$, where $L_{1},L_{2}$ are compact graded
Lagrangians with variations of their gradings less than $2\pi$ and

$\arg\int_{L_{1}}\Omega\geq\arg\int_{L_{2}}\Omega.$

Moreover, if strict inequality can be achieved $(L,\theta)$ is said to be
_strictly unstable_. If only equality occurs, then $(L,\theta)$ is called
_semi-stable_.

Finally, the compact graded Lagrangian $(L,\theta)$ is called _stable_ if it
is not unstable.

######  Definition 2.11 (Flow stability).

Let $(L,\theta)$ be a compact, almost calibrated Lagrangian in
$(X,\omega,\Omega)$ satisfying $\arg\int_{L}\Omega=0$.222Recall by Remark 2.7
that there is essentially no loss of generality here. Then $(L,\theta)$ is
_flow stable_ if for any possible decomposition of the Hamiltonian isotopy
class of $(L,\theta)$ as a graded Lagrangian connect sum $L_{1}\\#L_{2}$ for
$L_{1}$ and $L_{2}$ compact and almost calibrated, we have

  * (a)

$\left[\arg\int_{L_{1}}\Omega,\arg\int_{L_{2}}\Omega\right]\nsubseteq(\inf_{L}\theta,\sup_{L}\theta)$,
or

  * (b)

$\mathrm{Area}(L)<|\int_{L_{1}}\Omega|+|\int_{L_{2}}\Omega|$ .

Note that the condition (b) implies that
$\mathrm{Area}(L)<\mathrm{Area}(L_{1})+\mathrm{Area}(L_{2})$. We say that
$(L,\theta)$ is _flow unstable_ if it is not flow stable (i.e. (a) and (b) are
both violated), and _strictly flow unstable_ if
$\arg\int_{L_{1}}\Omega<\arg\int_{L_{2}}\Omega$ in (a) or the reverse of the
inequality in (b) holds strictly. If $(L,\theta)$ is not strictly flow
unstable we say it is _flow semi-stable_.

Recall the notation from Remark 2.9, where $L=\pi^{-1}(\gamma)$ is a compact,
almost calibrated, embedded, circle-invariant Lagrangian in $X$ with
projection $\pi:X\to\mathbb{R}^{3}$. We shall now recast the notions of
stability/flow stability in Definitions 2.10 and 2.11 in this circle-invariant
setting, i.e. in terms of curves.

Consider any decomposition of the curve $\gamma$ as $\gamma_{1}\\#\gamma_{2}$,
for $\gamma_{1},\gamma_{2}$ almost calibrated, and denote by
$\overline{\gamma_{1}},\overline{\gamma_{2}}$ the straight-lines with the same
endpoints and orientations as $\gamma_{1}$ and $\gamma_{2}$ respectively. With
no loss of generality suppose that the endpoints of $\gamma$ are on the
$\mu_{1}$-axis, oriented so the $\mu_{1}$-coordinate increases from one
endpoint of $\gamma$ to the other. Let $\theta$, $\overline{\theta}_{1}$,
$\overline{\theta}_{2}$ denote the angles that $\gamma$,
$\overline{\gamma}_{1}$, $\overline{\gamma}_{2}$ make with the $\mu_{1}$-axis
respectively. Then we have the following observation.

######  Lemma 2.12.

The graded curve $\gamma$, equivalently $L=\mu^{-1}(\gamma)$, is stable if for
all decompositions $\gamma=\gamma_{1}\\#\gamma_{2}$ into graded curves we have
(in the notation above)

$\overline{\theta}_{1}<\overline{\theta}_{2}.$

Similarly, an almost calibrated curve $\gamma$ is flow stable if for all
decompositions $\gamma=\gamma_{1}\\#\gamma_{2}$ we have (in the notation
above)

  * (a)

$[\min\\{\overline{\theta}_{1},\overline{\theta}_{2}\\},\max\\{\overline{\theta}_{1},\overline{\theta}_{2}\\}]\nsubseteq(\inf_{\gamma}\theta,\sup_{\gamma}\theta)$,
or

  * (b)

$\mathrm{Length}(\gamma)<\mathrm{Length}(\overline{\gamma_{1}})+\mathrm{Length}(\overline{\gamma_{2}})$.

The notions of flow unstable, strictly flow unstable and flow semi-stable
clearly extend from Definition 2.11 to graded curves, following Lemma 2.12.

##  3\. Lagrangian mean curvature flow and modified curve shortening flow

The goal of this section is to give some preliminary general results regarding
the evolution of circle-invariant Lagrangians under mean curvature flow, or
equivalently curves under the modified curve shortening flow (3.1) below.

Recall, from [Lotay2020]*Proposition 4.5, that a circle-invariant Lagrangian
$L_{t}=\pi^{-1}(\gamma_{t})$ in $X$ evolves through the Lagrangian mean
curvature flow if and only if the planar curve $\gamma_{t}$ satisfies

(3.1) $\partial_{t}\gamma_{t}=\phi^{-1}\partial^{2}_{s}\gamma_{t},$

where $s$ the Euclidean arc-length parameter and $\phi$ is the potential on
$X$ as in Proposition 2.1. We will assume for ease of notation that
$\gamma_{t}\subset\\{0\\}\times\mathbb{R}^{2}\subseteq\mathbb{R}^{3}$ and we
identify $\\{0\\}\times\mathbb{R}^{2}\cong\mathbb{R}^{2}\cong\mathbb{C}$.

###  3.1. Evolution of the grading and curvature

Recall that for a graded Lagrangian $(L,\theta)$ the Lagrangian angle $\theta$
satisfies (2.8). Under Lagrangian mean curvature flow, the grading
$\theta_{t}$ of $L_{t}$ evolves via

(3.2) $\frac{\partial\theta_{t}}{\partial
t}=-\Delta\theta_{t}=-d^{*}d\theta_{t}.$

(Here, and throughout, we will use the “geometer’s Laplacian” $\Delta=d^{*}d$,
so that the evolution equation (3.2) for $\theta$ is the heat equation.) Using
(3.2) we quickly see that

$\displaystyle\left(\frac{\partial}{\partial
t}+\Delta\right)|d\theta_{t}|^{2}=-2|\nabla d\theta_{t}|^{2}.$

It turns out that, in our setting, the grading $\theta$ of a circle-invariant
Lagrangian $L=\pi^{-1}(\gamma)$ can be related to the curvature $\kappa$ of
$\gamma$, defined by $\partial_{s}^{2}\gamma=\kappa N$, where
$\\{\partial_{s}\gamma,N\\}$ is an oriented orthonormal basis of the plane
containing $\gamma$. Viewing $\theta$ as a function on $\gamma$, by
[Lotay2020]*Lemma 5.4 we have that

(3.3) $\kappa=\partial_{s}\theta.$

Hence, the evolution equation (3.2) for the grading $\theta_{t}$ of
$L_{t}=\pi^{-1}(\gamma_{t})$ should yield an equation for the evolution of the
curvature $\kappa_{t}$ of $\gamma_{t}$. This is stated in the next result.

######  Proposition 3.1.

Let $\gamma_{t}$ be a solution of (3.1) with curvature $\kappa_{t}$. Then,

(3.4)
$\partial_{t}\kappa_{t}=\partial_{s}^{2}(\phi^{-1}\kappa_{t})+\phi^{-1}\kappa_{t}^{3},$

and

(3.5)
$\partial_{t}(\phi^{-1}\kappa_{t})\geq\phi^{-1}\partial_{s}^{2}(\phi^{-1}\kappa_{t})+(\kappa_{t}-2\phi)(\phi^{-1}\kappa_{t})^{2}.$

###### Proof.

Recall that the evolution equation (3.1) can be written as

$\partial_{t}\gamma_{t}=\phi^{-1}\kappa_{t}N_{t}$

where, if $I$ denotes multiplication by $i$ in $\mathbb{C}$ and
${}^{\prime}=\partial_{s}$,

$N_{t}=I\gamma_{t}^{\prime}\quad\text{and}\quad\kappa_{r}=\langle\gamma_{t}^{\prime\prime},N_{t}\rangle.$

Now, in order to commute derivatives we use a fixed parameter $x(s)$ of
$\gamma_{t}$ independent of $t$. Furthermore, this may be done so that at a
fixed space-time point $(t_{0},x_{0})$ we have $x^{\prime}(s)=1$ and
$x^{\prime\prime}(s)=0$, or equivalently $|\partial_{x}\gamma_{t}|=1$ and
$\langle\partial_{x}^{2}\gamma_{t},\partial_{x}\gamma_{t}\rangle=0$. Using
such a parametrization we have

$\kappa_{t}=|\partial_{x}\gamma_{t}|^{-2}\langle\partial_{x}^{2}\gamma_{t},N_{t}\rangle$

and so

(3.6)
$\displaystyle\partial_{t}\kappa_{t}=|\partial_{x}\gamma_{t}|^{-2}\langle\partial_{t}\partial_{x}^{2}\gamma_{t},N_{t}\rangle+|\partial_{x}\gamma_{t}|^{-2}\langle\partial_{x}^{2}\gamma_{t},\partial_{t}N_{t}\rangle-2|\partial_{x}\gamma_{t}|^{-2}\langle\partial_{x}\gamma_{t},\partial_{t}\partial_{x}\gamma_{t}\rangle\kappa_{t}.$

Before continuing we make a few elementary observations which will prove
useful during the computation. As
$\langle\partial_{x}\gamma_{t},N_{t}\rangle=0$, we find that

$\langle\partial_{x}^{2}\gamma_{t},N_{t}\rangle+\langle\partial_{x}\gamma_{t},\partial_{x}N_{t}\rangle=0.$

Since $\langle N_{t},\partial_{x}N_{t}\rangle=0$, we deduce that

(3.7) $\displaystyle\partial_{x}N_{t}=-\kappa_{t}\partial_{x}\gamma_{t}.$

We now compute each term of (3.6) separately at the point $(t_{0},x_{0})$ in
question. For the first term we have, using (3.1) and (3.7),

$\displaystyle\langle\partial_{t}\partial_{x}^{2}\gamma_{t},N_{t}\rangle$
$\displaystyle=\langle\partial_{x}^{2}(\phi^{-1}\kappa_{t}N_{t}),N_{t}\rangle$
$\displaystyle=\partial_{x}\langle\partial_{x}(\phi^{-1}\kappa_{t}N_{t}),N_{t}\rangle-\langle\partial_{x}(\phi^{-1}\kappa_{t}N_{t}),\partial_{x}N_{t}\rangle$
$\displaystyle=\partial_{x}\left(\partial_{x}\langle\phi^{-1}\kappa_{t}N_{t},N_{t}\rangle-\langle\phi^{-1}\kappa_{t}N_{t},\partial_{x}N_{t}\rangle\right)-\phi^{-1}\kappa_{t}^{3}$
$\displaystyle=\partial_{x}^{2}(\phi^{-1}\kappa_{t})-\phi^{-1}\kappa_{t}^{3}$

at $(t_{0},x_{0})$. Since $\langle N_{t},\partial_{t}N_{t}\rangle=0$ and
$\partial^{2}_{x}\gamma_{t}$ is a multiple of $N_{t}$ at $(t_{0},x_{0})$, we
see that the second term in (3.6) vanishes there:

$\displaystyle\langle\partial_{x}^{2}\gamma_{t},\partial_{t}N_{t}\rangle=0.$

For the last term we again use (3.1) and (3.7) and find that at
$(t_{0},x_{0})$ we have

$\displaystyle\langle\partial_{x}\gamma_{t},\partial_{t}\partial_{x}\gamma_{t}\rangle$
$\displaystyle=\langle\partial_{x}\gamma_{t},\partial_{x}(\phi^{-1}\kappa_{t}N_{t})\rangle=\langle\partial_{x}\gamma_{t},\phi^{-1}\kappa_{t}\partial_{x}N_{t}\rangle=-\phi^{-1}\kappa_{t}^{2},$

so the last term in (3.6) is given by $2\phi^{-1}\kappa_{t}^{3}$. Inserting
all these formulae gives (3.4).

For the estimate (3.5) we observe first that, since

$\phi=m+\sum_{i=1}^{k}\frac{1}{2r_{i}}$

where $r_{i}$ is the Euclidean distance to $p_{i}$, we have

$|\nabla\phi|=|\sum_{i=1}^{k}\frac{dr_{i}}{2r_{i}^{2}}|\leq\sum_{i=1}^{k}\frac{1}{2r_{i}^{2}}\leq
2\phi^{2}$

because $|dr_{i}|=1$ for all $i$. Therefore, using this estimate together with
(3.1) and (3.4) we compute

$\displaystyle\partial_{t}(\phi^{-1}\kappa_{t})$
$\displaystyle=-\phi^{-2}(\partial_{t}\phi)\kappa_{t}+\phi^{-1}\partial_{t}\kappa_{t}$
$\displaystyle=-\phi^{-2}(\nabla_{\partial_{t}\gamma}\phi)\kappa_{t}+\phi^{-1}((\phi^{-1}\kappa_{t})^{\prime\prime}+\phi^{-1}\kappa_{t}^{3})$
$\displaystyle=-\phi^{-3}(\nabla_{N_{t}}\phi)\kappa_{t}^{2}+\phi^{-1}(\phi^{-1}\kappa_{t})^{\prime\prime}+\phi^{-2}\kappa_{t}^{3}$
$\displaystyle\geq-2\phi^{-1}\kappa_{t}^{2}+\phi^{-1}(\phi^{-1}\kappa_{t})^{\prime\prime}+\phi^{-2}\kappa_{t}^{3}.$

This gives (3.5). ∎

We want to show the relationship between Proposition 3.1 and convexity of
curves along the flow (3.1). To do this, we make a definition.

######  Definition 3.2.

Let $\gamma\subseteq\mathbb{R}^{3}$ be an embedded connected planar curve. We
say that $\gamma$ is _convex_ if it is a subset of a curve bounding a convex
region in the plane containing $\gamma$, which is equivalent to saying that
the curvature $\kappa$ of $\gamma$ is either everywhere non-negative or
everywhere non-positive. We say that $\gamma$ is _strictly convex_ if
$|\kappa|>0$ at every interior point of $\gamma$.

######  Remark 3.3.

Let $\gamma\subseteq\mathbb{R}^{3}$ be an embedded planar arc connecting two
singularities of $\phi$ and meeting no other singularities of $\phi$. By
(3.3), we see that $\gamma$ is strictly convex if and only if the Lagrangian
angle $\theta$ is a perfect Morse function on $L=\pi^{-1}(\gamma)$.

######  Proposition 3.4.

Let $\gamma_{0}\subseteq\mathbb{R}^{3}$ be an embedded planar arc connecting
two singularities $p_{1},p_{2}$ of $\phi$ and which meets no other
singularities of $\phi$. Suppose further that $\gamma_{0}$ is strictly convex.

Let $\\{\gamma_{t}\\}_{t\in[0,T]}$, for $T>0$, be a smooth solution of (3.1)
with fixed endpoints at $p_{1},p_{2}$ which meets no other singularities of
$\phi$. Then $\gamma_{t}$ is strictly convex for all $t\in[0,T]$.

###### Proof.

Suppose without loss of generality that $\kappa_{0}>0$ away from
$p_{1},p_{2}$. Since $\phi^{-1}>0$ away from $p_{1},p_{2}$, we see that in the
interior of the arc $\gamma_{t}$ we have $\kappa_{t}>0$ if and only if
$f_{t}=\phi^{-1}\kappa_{t}>0$.

Notice that we can rewrite (3.5) as (using ${}^{\prime}=\partial_{s}$)

$\partial_{t}f_{t}\geq\phi^{-1}f_{t}^{\prime\prime}+\kappa_{t}(\phi^{-1}\kappa_{t}-2)f_{t}.$

Since we are assuming that $\gamma_{t}$ is a smooth solution to (3.1), we know
that there exists some $c>0$ such that

$\kappa_{t}(\phi^{-1}\kappa_{t}-2)\geq-c\quad\text{for all $t\in[0,T]$}.$

Therefore, on the region of space-time where $f_{t}\geq 0$ (which includes
$t=0$ by assumption), we have

(3.8) $\partial_{t}f_{t}\geq\phi^{-1}f_{t}^{\prime\prime}-cf_{t}.$

Let $\varepsilon>0$ and define

$f^{\varepsilon}_{t}=f_{t}+\varepsilon t.$

We see from (3.8) that

$\displaystyle\partial_{t}f_{t}^{\varepsilon}$
$\displaystyle\geq\phi^{-1}(f_{t}^{\varepsilon})^{\prime\prime}-cf_{t}+\varepsilon$
$\displaystyle=\phi^{-1}(f_{t}^{\varepsilon})^{\prime\prime}-cf_{t}^{\varepsilon}+\varepsilon(ct+1)$
(3.9)
$\displaystyle>\phi^{-1}(f_{t}^{\varepsilon})^{\prime\prime}-cf_{t}^{\varepsilon}.$

Notice that $f_{0}^{\varepsilon}=f_{0}>0$ away from $p_{1},p_{2}$. Suppose
that $(t_{0},x_{0})$ is the first space-time point away from $p_{1},p_{2}$
where $f_{t}^{\varepsilon}=0$. Then $f_{t}^{\varepsilon}>0$ away from
$p_{1},p_{2}$ for all $t<t_{0}$ and $f_{t_{0}}^{\varepsilon}\geq 0$ with
$f_{t_{0}}(x_{0})=0$. Therefore, we must have that
$\partial_{t}f_{t}^{\varepsilon}\leq 0$ at $(t_{0},x_{0})$ whereas $x_{0}$ is
a local minimum of $f_{t_{0}}^{\varepsilon}$ and so
$(f_{t}^{\varepsilon})^{\prime\prime}\geq 0$ at $(t_{0},x_{0})$. However, we
see from (3.9) that, at $(t_{0},x_{0})$,

$0\geq\partial_{t}f_{t}^{\varepsilon}>\phi^{-1}(f_{t}^{\varepsilon})^{\prime\prime}-cf_{t}^{\varepsilon}\geq
0,$

which is a contradiction.

We deduce that

$f_{t}^{\varepsilon}=f_{t}+\varepsilon t>0$

away from $p_{1},p_{2}$ for all $t\in[0,T]$ for all $\varepsilon>0$. Letting
$\varepsilon$ tend to $0$ gives that $f_{t}$ and hence $\kappa_{t}$ is non-
negative on $[0,T]$.

Now we know that $f_{t}\geq 0$ everywhere on $[0,T]$ we have that the
inequality (3.9) holds for all $t\in[0,T]$. This is a parabolic inequality
away from $p_{1},p_{2}$ and so if $f_{0}>0$ away from $p_{1},p_{2}$ then by
the strong maximum principle we have that $f_{t}>0$ away from $p_{1},p_{2}$,
which gives the result. ∎

For possible future study we also record the following (easier) convexity
result which is immediate from Proposition 3.1 and the strong parabolic
maximum principle.

######  Proposition 3.5.

Let $\gamma_{0}\subseteq\mathbb{R}^{3}$ be an embedded planar curve meeting no
singularities of $\phi$ so that the curvature $\kappa_{0}$ of $\gamma_{0}$ is
non-negative. Let $\\{\gamma_{t}\\}_{t\in[0,T]}$, for $T>0$, be a smooth
solution of (3.1) which meets no other singularities of $\phi$. Then the
curvature $\kappa_{t}$ of $\gamma_{t}$ is positive for all $t\in[0,T]$.

###  3.2. Evolution of area of bounding holomorphic disks

We shall now consider a special situation which will be of interest in setting
up the analysis of singularities. Suppose we have a connected immersed minimal
Lagrangian $L_{\infty}$ (in particular, it could be the union of two special
Lagrangians with different phases intersecting at a point) and a solution
$L_{t}$ to Lagrangian mean curvature flow, with grading $\theta_{t}$, which
intersects $L_{\infty}$ at two points $p_{+}$ and $p_{-}$ for all $t$. Let $D$
be the unit disk in $\mathbb{C}$ and write

$\partial D\setminus\\{1,-1\\}=\partial D^{+}\sqcup\partial D^{-}$

as the disjoint union of two connected components with $D^{\pm}$ contained in
the upper/lower half-plane. Suppose further that $\sigma_{t}:D\to X$ is a
family of holomorphic disks with two marked points $\pm 1$ such that

$\gamma_{t}=\sigma_{t}(\partial D^{+})\subseteq
L_{t},\quad\gamma_{\infty}=\sigma_{t}(\partial D^{-})\subseteq
L_{\infty},\quad\sigma_{t}(\pm 1)=p_{\pm},$

where $\gamma_{\infty}$ is independent of $t$. (We could allow $p_{\pm}$ and
$\gamma_{\infty}$ to vary inside $L_{\infty}$ and we will obtain the same
answer, but this is not required for our purposes.) The situation of
particular interest to us is shown in Figure 3.1, where we call
$\sigma_{t}(D)$ a “holomorphic pacman disk” (for obvious reasons).

$p_{-}$$p_{+}$$\gamma_{\infty}\subseteq
L_{\infty}$$\sigma_{t}(D)$$\gamma_{t}\subseteq L_{t}$ Figure 3.1. Evolving
holomorphic pacman disk.

We may then compute the evolution of the area of the holomorphic disks
$\sigma_{t}(D)$ as follows.

######  Lemma 3.6.

Using the notation above, the area of the holomorphic disks $\sigma_{t}(D)$,
which is given by

(3.10) $A(t):=\int_{D}\sigma_{t}^{*}\omega,$

satisfies

(3.11) $\displaystyle\dot{A}(t)=\theta_{t}(p_{+})-\theta_{t}(p_{-}).$

###### Proof.

Differentiating (3.10) with respect to time yields (using the fact that
$d\omega=0$ and $\gamma_{\infty}$ is fixed)

$\displaystyle\dot{A}$
$\displaystyle=\int_{D}\sigma_{t}^{*}d(\iota_{\partial_{t}\sigma_{t}}\omega)=\int_{\partial
D}\sigma_{t}^{*}(\iota_{\partial_{t}\sigma_{t}}\omega)=\int_{\gamma_{t}}\iota_{\partial_{t}\gamma_{t}}\omega.$

Recall that $L_{t}$ is evolving via mean curvature flow

$\partial_{t}L_{t}=H_{t}=J\nabla\theta_{t}.$

Therefore, as $\gamma_{t}\subseteq L_{t}$, the component of
$\partial_{t}\gamma_{t}$ normal to $L$ must be $J\nabla\theta_{t}$. Moreover,
we do not see the component of $\partial_{t}\gamma_{t}$ tangential to $L_{t}$
in $\iota_{\partial_{t}\gamma_{t}}\omega$ since $L_{t}$ is Lagrangian for all
$t$. Therefore,

$\iota_{\partial_{t}\gamma_{t}}\omega=\omega(J\nabla\theta_{t},\cdot)=-g(\nabla\theta_{t},\cdot)=-d\theta_{t}.$

By the fundamental theorem of calculus, since we oriented $\gamma_{t}$ in the
anticlockwise direction so that it is compatible with the orientation induced
by $D$, we find that

$\displaystyle\dot{A}=-\left[\theta_{t}(\gamma_{t}(-1))-\theta_{t}(\gamma_{t}(1))\right]=\theta_{t}(p_{+})-\theta_{t}(p_{-}).$

as claimed. ∎

######  Remark 3.7.

In the simplest case, we shall be interested in applying this result when
$L_{t}$ is Hamiltonian isotopic through circle-invariant Lagrangians to
$L_{-}\\#L_{+}$ for two special Lagrangians $L_{-}$ and $L_{+}$ intersecting
transversely at a point $p$ and $\sigma_{t}(D)$ has boundary components on
$L_{t}$ and $L_{\infty}=L_{-}\cup L_{+}$, as suggested by Figure 3.1. Notice
that only the first of these boundary components needs to be smooth for the
computation in the proof of Lemma 3.6.

##  4\. Finite time singularities

This section establishes conditions under which finite time singularities of
the circle-invariant Lagrangian mean curvature flow develop and proves, in
this setting, a conjecture of Joyce on the local structure of such
singularities. This is stated as Theorem 4.6.

Along the way, we shall also prove, in Theorem 4.13, that the Lagrangian mean
curvature flow starting at a strictly unstable Lagrangian $L=\pi^{-1}(\gamma)$
with $\gamma$ convex develops finite time singularities. This will be improved
later in the article by dropping the hypothesis that $\gamma$ be convex.

###  4.1. Location of singularities

Before we begin we use some key results from [Lotay2020] to provide some
important tools for our singularity analysis. The first is the following
general result.

######  Proposition 4.1.

If the curve $\gamma_{t}\subseteq\mathbb{R}^{3}$ solves (3.1) then
$\phi^{-1}\kappa_{t}$ is the projection of the mean curvature of
$L_{t}=\pi^{-1}(\gamma_{t})$ to $\mathbb{R}^{3}$ and the flow exists as long
as the norm of the second fundamental form $L_{t}$ is bounded, which is
equivalent to requiring that

$\phi^{-1/2}|\kappa_{t}|\quad\text{and}\quad\phi^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$

are bounded on $\gamma_{t}$, where norms and $\nabla^{\perp}_{\mathbb{R}^{3}}$
are taken with respect to the Euclidean metric on $\mathbb{R}^{3}$.

###### Proof.

This follows from [Lotay2020]*Propositions 4.5 and 4.6, though it should be
noted that the factor of $\phi^{-1/2}$ was omitted from the second term there,
though this does not affect any of the arguments in that reference. ∎

Using Proposition 4.1 we have the following theorem based on
[Lotay2020]*Section 6.

######  Proposition 4.2.

Let $p_{1},\ldots,p_{k}\in\mathbb{R}^{3}$ for $k>1$ be the singularities of
$\phi$. Let $\gamma_{0}\subseteq\mathbb{R}^{3}$ be a compact, almost
calibrated, planar arc connecting $p_{1},p_{2}$ and meeting no other
singularities of $\phi$. Let $\gamma_{t}$ be the solution of (3.1) starting at
$\gamma_{0}$ with the fixed endpoints $p_{1},p_{2}$ and suppose that
$\gamma_{t}$ has a finite time singularity at a point $p$. Then $p=p_{i}$ for
some $i>2$ and $\phi^{-1}|\kappa_{t}|$ tends to zero at the singularity while
$\phi^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$ blows up.

###### Proof.

Suppose that $p\neq p_{i}$ for any $i>2$. By [Lotay2020]*Lemma 6.6, the almost
calibrated assumption ensures that $|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$
remains bounded as long as $\gamma_{t}$ never reaches any other singularities
of $\phi$. Hence, by Proposition 4.1, we must have that
$\phi^{-1/2}|\kappa_{t}|$ blows up at $p$.

However, [Lotay2020]*Lemmas 6.8, 6.9 and 6.10 can then be used to reach a
contradiction (these results do not require the stability assumption, only
that $\phi^{-1/2}|\kappa_{t}|$ blows up). Specifically, Lemma 6.8 states that
$p$ must be either $p_{1}$ or $p_{2}$, but then this possibility is ruled out
by Lemmas 6.9 and 6.10 which respectively show that in this situation we
cannot have $\phi^{-1}|\kappa_{t}|\to\infty$ at $p_{1},p_{2}$ or
$\phi^{-1}|\kappa_{t}|$ bounded at $p_{1},p_{2}$. This finishes the proof that
$p=p_{i}$ for some $i>2$.

We now see that, near $p=p_{i}$, we have that $\log\phi\sim\log|x-p|$ and so,
for points $\gamma_{t}(s)$ near $p$,

$\displaystyle\phi(\gamma_{t}(s))^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi(\gamma_{t}(s))|$
$\displaystyle\sim\frac{1}{|\gamma_{t}(s)-p|^{1/2}}|\langle\frac{\gamma_{t}(s)-p}{|\gamma_{t}(s)-p|},I\gamma_{t}^{\prime}(s)\rangle|$
(4.1)
$\displaystyle\leq\frac{1}{|\gamma_{t}(s)-p|^{1/2}}\sim\phi(\gamma_{t}(s))^{1/2}.$

with equality if and only if we are at a closest point to $p$ on $\gamma_{t}$.
Hence $\phi^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$ blows up at the
finite time singularity at $p=p_{i}$.

We now show that $\phi^{-1}|\kappa_{t}|$ tends to zero at the singularity. We
begin with the following.

######  Lemma 4.3.

The quantity $\phi^{-1}|\kappa|$ is bounded at the singularity.

###### Proof.

Suppose that we have a sequence of spacetime points $(s_{i},t_{i})$ with
$x_{i}:=\gamma_{t_{i}}(s_{i})\to p$ as $i\to\infty$ and

(4.2) $|\kappa_{t_{i}}(s_{i})||x_{i}-p|\to\infty\quad\text{as $i\to\infty$.}$

We deduce from (4.1) that $|\kappa_{t_{i}}(s_{i})|$ dominates
$\phi(\gamma_{t}(s))^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi(\gamma_{t_{i}}(s))|$
as $i\to\infty$ if the ratio

(4.3) $\frac{|x_{i}-p|}{|\gamma_{t_{i}}(s)-p|}$

is bounded as $i\to\infty$.

We can now adapt the argument of [Lotay2020]*Lemma 6.9 to reach a
contradiction. If we consider the balls $B_{i}=B(x_{i},|x_{i}-p|/2)$, then we
see that $p$ does not lie in $B_{i}$ (and nor does any other singularity of
$\phi$ for $i$ sufficiently large) and the ratio (4.3) is bounded for points
$\gamma_{t_{i}}(s)\in B_{i}$. Translating $x_{i}$ to the origin and rescaling
by $|\kappa_{t_{i}}(s_{i})|$ we obtain balls which exhaust $\mathbb{R}^{3}$ as
$i\to\infty$ by (4.2) and contain the image of the curve $\gamma_{t_{i}}$,
whose curvature is $1$ at the origin.

Recall that the norm of the second fundamental form of the Lagrangian
$\pi^{-1}(\gamma)$ is controlled by $\phi^{-1/2}|\kappa|$ (notice the extra
factor of $\phi^{-1/2}$ which tends to zero near $p$) and
$\phi^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$. Since the ratio (4.3)
is bounded on $B_{i}$ as $i\to\infty$, we deduce that the norm of the second
fundamental form of $\pi^{-1}(\gamma_{t_{i}})$ on $B_{i}$ is bounded above by
$|\kappa_{t_{i}}(s_{i})||\gamma_{t_{i}}(s_{i})-p|^{1/2}$. Dividing by
$|\kappa_{t_{i}}(s_{i})|$ then gives an upper bound which tends to zero as
$i\to\infty$.

We may then take a subsequential limit to obtain a non-straight curve
$\gamma_{\infty}$ (since the curvature is $1$ at the origin), which is the
projection of a totally geodesic circle-invariant Lagrangian $L_{\infty}$ in
$\mathbb{R}^{4}$, giving our required contradiction. ∎

Suppose now that $\phi^{-1}|\kappa|$ does not tend to zero at the singularity.
By Lemma 4.3, we know that, after possibly choosing a subsequence and
relabeling, there is a sequence $x_{i}=\gamma_{t_{i}}(s_{i})$ tending to $p$,
maximizing $\phi^{-1}|\kappa_{t_{i}}|$, and a constant $\delta>0$ such that

(4.4) $|\kappa_{t_{i}}(s_{i})||x_{i}-p|\to\delta>0\quad\text{as
$i\to\infty$.}$

Notice that this forces

(4.5) $|\kappa_{t_{i}}(s_{i})||x_{i}-p|^{1/2}\to\infty\quad\text{as
$i\to\infty$.}$

We then consider the annuli

(4.6) $A_{i}=B(x_{i},|x_{i}-p|^{1/2})\setminus\overline{B}(p,|x_{i}-p|/2).$

On these annuli, for large $i$, we have that
$\phi^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$ is bounded above by
$|x_{i}-p|^{-1/2}$, up to multiplication by a uniform constant. Hence, if we
translate $x_{i}$ to $0$ and scale by $\kappa_{t_{i}}(s_{i})$ we see that
$\phi^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$ is then bounded above
by $|\kappa_{t_{i}}(s_{i})|^{-1}|x_{i}-p|^{-1/2}$, which tends to zero by
(4.5). We also see that by our choice of spacetime points as maximizers of
$\phi^{-1}|\kappa_{t_{i}}|$ and our annuli in (4.6) we can now bound
$\phi^{-1/2}|\kappa|$ for points $\gamma_{t_{i}}(s)\in A_{i}$ for large $i$ as
follows:

$\displaystyle\phi^{-1/2}(\gamma_{t_{i}}(s))|\kappa_{t_{i}}(s)|$
$\displaystyle\sim|\gamma_{t_{i}}(s)-p|^{1/2}|\kappa_{t_{i}}(s)|$
$\displaystyle=\frac{|x_{i}-p|}{|\gamma_{t_{i}}(s)-p|^{1/2}}|\kappa_{t_{i}}(s_{i})|\frac{|\gamma_{t_{i}}(s)-p||\kappa_{t_{i}}(s)|}{|x_{i}-p||\kappa_{t_{i}}(s_{i})|}\leq\sqrt{2}|x_{i}-p|^{1/2}|\kappa_{t_{i}}(s_{i})|.$

Hence, after dividing by $|\kappa_{t_{i}}(s_{i})|$, we have an upper bound by
a constant multiple of $|x_{i}-p|^{1/2}$, which also tends to zero as
$i\to\infty$.

We observe that the outer radius of the rescaled and translated annuli now
tends to infinity after the rescaling by $|\kappa_{t_{i}}(s_{i})|$ by (4.5),
while after passing to a subsequence the inner ball converges to
$\overline{B}=\overline{B}(x,\delta/2)$ for some point $x\neq 0$ (the
subsequential limit of $|\kappa_{t_{i}}(s_{i})|(p-x_{i})$) by (4.4). Notice
that $0$ cannot lie in $\overline{B}$ by construction of $A_{i}$ in (4.6).

Therefore, after possibly passing to a further subsequence, we again obtain a
(possibly disconnected) non-straight limit curve $\gamma_{\infty}$ (now on
$\mathbb{R}^{3}\setminus\overline{B}$ with a point of curvature $1$ at the
origin) which is the projection of a totally geodesic circle-invariant
Lagrangian. This is another contradiction and so $\phi^{-1}|\kappa|$ does
indeed tend to zero as claimed. ∎

We can also deal with the case of some non-compact curves as follows.

######  Proposition 4.4.

Suppose that $p_{1},\dots,p_{k}\in\mathbb{R}^{3}$ for $k\geq 1$ are the
singularities of $\phi$ and let $\ell_{+},\ell_{-}$ be rays starting at
$p_{1}$ and meeting no other singularities of $\phi$. Let
$\gamma_{0}\subseteq\mathbb{R}^{3}$ be an almost calibrated planar arc which
lies in the same plane as $\ell_{-}\cup\ell_{+}$ and is asymptotic to
$\ell_{-}\cup\ell_{+}$ at infinity in the sense that, outside a compact set,
$\gamma_{0}$ is a smooth graph of a function $u$ over $\ell_{-}\cup\ell_{+}$
so that $|u|\to 0$ at infinity. Suppose further that $\gamma_{0}$ meets no
singularities of $\phi$.

There is a unique short-time solution $\gamma_{t}$ of (3.1) starting at
$\gamma_{0}$ which remains asymptotic to $\ell_{-}\cup\ell_{+}$ at infinity.
Moreover, if $\gamma_{t}$ has a finite time singularity at a point $p$ then
$p=p_{i}$ for some $i$, $\phi^{-1}|\kappa_{t}|$ tends to zero at the
singularity but $\phi^{-1/2}|\nabla^{\perp}_{\mathbb{R}^{3}}\log\phi|$ blows
up.

###### Proof.

Since there are only finitely many singularities of $\phi$, outside of some
compact set, the flow (3.1) is uniformly equivalent to the usual curve
shortening flow. Hence, we may apply the theory of the curve shortening flow,
including pseudolocality, to deduce that a unique solution $\gamma_{t}$ to
(3.1) exists and remains asymptotic to $\ell_{-}\cup\ell_{+}$ at infinity.

The proof now proceeds exactly as for Proposition 4.2 because all the analysis
of finite time singularities is local and there are no finite time
singularities outside of a compact set in $\mathbb{R}^{3}$. ∎

######  Remark 4.5.

The existence and uniqueness of the flow $\gamma_{t}$ in Proposition 4.4 can
also be deduced by considering the Lagrangian mean curvature flow
$L_{t}=\pi^{-1}(\gamma_{t})$ and using methods from [WeiBoSu].

###  4.2. Neck pinches

We now turn to the local structure at the finite time singularity of
Lagrangian mean curvature flow as considered thus far in this article. We
shall use the results in the previous subsection to show that by rescaling up
a finite size neighbourhood of the singular point, the Lagrangian mean
curvature flow approaches a fixed Lawlor neck. Consequently, the family of
rescaled Lawlor necks gives a first order approximation for the evolution of
the flow as it develops a finite time singularity. This result, given in the
two theorems below, proves Theorem 1.1(a). Recall the notation that $X$ is a
hyperkähler 4-manifold with a circle action as described in Subsection 2.2.

######  Theorem 4.6.

Let $L$ be an embedded, almost calibrated, circle-invariant Lagrangian in $X$
which is either compact or asymptotic at infinity to a pair of planes. Suppose
that $\\{L_{t}\\}_{t\in[0,T)}$ is an embedded, almost calibrated, circle-
invariant solution to Lagrangian mean curvature flow in $X$ starting at $L$,
that is also either compact or asymptotic to pair of planes respectively,
which develops a finite time singularity at $p\in X$ when $t\to T<\infty$.

Then, for any sequence of times $t_{i}\nearrow T$ as $i\to\infty$, after
passing to a subsequence which we also call $t_{i}$, there are:

  * •

open neighbourhoods $U$ of $p$ in $X$ and $V$ of $0$ in
$T_{p}X\cong\mathbb{C}^{2}$;

  * •

a pointed isomorphism $\varphi:U\to V$ at $p$;

  * •

a nullsequence $\varepsilon_{i}\searrow 0$

such that $\varepsilon_{i}^{-1}\varphi(L_{t_{i}}\cap U)$ converges on compact
subsets of $\mathbb{C}^{2}$ to a Lawlor neck
$\hat{L}=\pi_{H}^{-1}(\hat{\gamma})$, where
$\pi_{H}:\mathbb{C}^{2}\to\mathbb{R}^{3}$ is the radially extended Hopf
fibration and $\hat{\gamma}$ is a straight line at distance $1$ from the
origin.

###### Proof.

By assumption we know that $L_{t}=\pi^{-1}(\gamma_{t})$ where
$\\{\gamma_{t}\\}_{t\in[0,T)}$ is a family of planar curves in
$\mathbb{R}^{3}$ which is a solution to (3.1) of the type considered in
Proposition 4.2 or 4.4. Hence, if $\\{\gamma_{t}\\}_{t\in[0,T)}$ develops a
finite time singularity at time $T$, this must occur at a point
$p=p_{i}\in\mathbb{R}^{3}$ for $i>2$ and
$\phi^{-1/2}|\nabla_{\mathbb{R}^{3}}^{\perp}\log\phi|$ blows up there whilst
$\phi^{-1}|\kappa_{t}|$ stays bounded. It is also useful to recall the
estimate (4.1) for $\phi^{-1/2}|\nabla_{\mathbb{R}^{3}}^{\perp}\log\phi|$.

Parametrize the curves $\\{\gamma_{t}\\}_{t\in[0,T)}$ using their respective
arc-length parameters and take a sequence of space-time points $(s_{i},t_{i})$
such that $t_{i}\nearrow T$ as $i\to\infty$ and

(4.7)
$\lambda_{i}:=|\gamma_{t_{i}}(s_{i})-p|^{-1}=\max\\{|\gamma_{t}(s)-p|^{-1}\ :\
s\geq 0,\ t\leq t_{i}\\}.$

In particular, $\gamma_{t_{i}}(s_{i})$ is a closest point to $p$ on
$\gamma_{t_{i}}$. We have that $\lambda_{i}\nearrow+\infty$ and so we can find
$c>0$ such that the ball $B_{c}(p)\subseteq\mathbb{R}^{3}$ has the property
that $\gamma_{t_{i}}\cap B_{c}(p)$ is non-empty for all $i$ sufficiently large
and the metric on $\pi^{-1}(B_{c}(p))$ is close to the Euclidean metric on
$\mathbb{R}^{4}$. The latter property of $B_{c}(p)$ is equivalent to saying
that $2\phi(x)\sim|x-p|^{-1}$ on $B_{c}(p)$.

For $t\in[-\lambda_{i}^{2}t_{i},0)$ and $i$ sufficiently large we consider the
following curves on $B_{c\lambda_{i}}(0)\subseteq\mathbb{R}^{3}$:

(4.8) $\hat{\gamma}_{t}^{i}:=\lambda_{i}(\gamma_{t_{i}+\lambda_{i}^{-2}t}-p).$

We denote the corresponding Lagrangians in $\mathbb{C}^{2}$ by
$\hat{L}_{t}^{i}=\pi_{H}^{-1}(\hat{\gamma_{t}}^{i})$. Notice that we are
blowing up balls centered at $p$ rather than balls centered at
$\gamma_{t_{i}}(s_{i})$.

Observe that the curvature $\hat{\kappa}^{i}_{t}$ of $\hat{\gamma}_{t}^{i}$
satisfies

(4.9)
$|\hat{\kappa}^{i}_{t}|=\lambda_{i}^{-1}|\kappa_{t_{i}+\lambda_{i}^{-2}t}|\sim\frac{|\gamma_{t_{i}}(s_{i})-p|}{|\gamma_{t_{i}+\lambda_{i}^{-2}t}-p|}\phi(\gamma_{t_{i}+\lambda_{i}^{-2}t})^{-1}|\kappa_{t_{i}+\lambda_{i}^{-2}t}|$

by definition of $\lambda_{i}$ in (4.7) and our assumptions about the choice
of ball $B_{c}(p)$. By choice of $\gamma_{t_{i}}(s_{i})$ in (4.7) we see that
the quotient on the right-hand side of (4.9) is bounded above by $1$, since
$t<0$. By Propositions 4.2 and 4.4, we deduce from (4.9) that for all $i$
sufficiently large we have that $|\hat{\kappa}_{i}^{t}|$ is bounded above by a
uniform constant and tends to $0$ as $i\to\infty$.

As a result, after passing to a subsequence, we may extract a smooth limit
$\hat{\gamma}^{\infty}_{t}$ of the sequence $\hat{\gamma}_{t}^{i}$ of (4.8)
which must be a smooth embedded curve with curvature $0$ and so is a single
straight line $\hat{\gamma}$. Furthermore, under the translation and scaling
of $B_{c}(p)$, we see that $p$ gets mapped to $0$ and we let $y_{i}$ be the
image of $\gamma_{t_{i}}(s_{i})$. Then,

$|y_{i}|=\lambda_{i}^{-1}|\gamma_{t_{i}}(s_{i})-p|=1$

and by definition of $(s_{i},t_{i})$ in (4.7) we find that $y_{i}$ is the
closest point to the origin in $\hat{\gamma}^{i}_{t}$. Therefore,
$\hat{\gamma}$ is a straight line at distance $1$ from the origin. We deduce
that $\hat{L}=\pi_{H}^{-1}(\hat{\gamma})$ is a Lawlor neck, which is the limit
of the sequence $\hat{L}_{t}^{i}$. The result then follows. ∎

We can now use work from [FSS.neck] to improve Theorem 4.6 as follows.

######  Theorem 4.7.

Let $L\subseteq X$, $\\{L_{t}\\}_{t\in[0,T)}$, $p\in X$ and
$\pi_{H}:\mathbb{C}^{2}\to\mathbb{R}^{3}$ be as in Theorem 4.6. There are

  * •

open neighbourhoods $U$ of $p$ in $X$ and $V$ of $0$ in
$T_{p}X\cong\mathbb{C}^{2}$;

  * •

a pointed isomorphism $\varphi:U\to V$ at $p$;

  * •

a small $\delta>0$ and a smooth function
$\varepsilon:(T-\delta^{2},T)\to(0,\delta)$, with $\varepsilon(t)\searrow 0$
as $t\nearrow T$,

such $\varepsilon(t)^{-1}\varphi(L_{t}\cap U)$ converges on compact subsets of
$\mathbb{C}^{2}$ to a unique Lawlor neck $\hat{L}=\pi_{H}^{-1}(\hat{\gamma})$,
where $\hat{\gamma}$ is a unique straight line at distance $1$ from the
origin.

###### Proof.

Theorem 4.6 shows that at least one tangent flow at the singularity at $p$ is
a special Lagrangian union of two transverse planes in $\mathbb{C}^{2}$, which
are the asymptotics of the Lawlor neck $\hat{L}$ in the statement. Noting that
$L_{t}$ is almost calibrated, and thus zero Maslov, and also exact for all
$t$, this is precisely a finite time singularity of Lagrangian mean curvature
flow as studied in [FSS.neck].

As stated, the results of [FSS.neck] only apply to Lagrangian mean curvature
flow in $\mathbb{C}^{2}$ or in a compact Calabi–Yau 2-fold. Since our analysis
of the flow here takes place solely within a fixed small neighbourhood of $p$
in $X$, the analysis from [FSS.neck] carries over to this setting.

In particular, [FSS.neck]*Theorem 8.2 shows that the tangent flow at $p$ is in
fact unique. Then [FSS.neck]*Theorem 8.3 shows that for all times $t$ near $T$
we can find the scalings $\varepsilon(t)$ so that the rescaled flow
$\varepsilon(t)^{-1}\varphi(L_{t}\cap U)$ is a small $C^{1}$ graph over a
unique Lawlor neck of a given scale. This Lawlor neck must therefore be
$\hat{L}$ given in Theorem 4.6.

The only difference between the statements in [FSS.neck]*Theorem 8.3 and the
one claimed is that we work with the fixed ball centred at $p$, rather than
working with balls with different centres. This can be achieved since it can
be done for one sequence of times $t_{i}\nearrow T$ by Theorem 4.6. ∎

######  Remark 4.8.

Theorem 4.7 shows that a unique Lawlor neck $\hat{L}$ models
$\\{L_{t}\\}_{t\in[0,T)}$ in a fixed size neighbourhood of $p$ after rescaling
by $\varepsilon(t)^{-1}$. The neighbourhood in question is
$\pi^{-1}(B_{c}(p))$, where $c>0$ is fixed and $\pi:X\to\mathbb{R}^{3}$ is the
hyperkähler moment map, which is locally modelled on $\pi_{H}$ near $p$.
Alternatively, we can view $\varphi^{-1}(\varepsilon(t)\hat{L}\cap V)$ as
modelling $L_{t}$ on $U$, i.e. the scaled down Lawlor necks give an
approximation to the flow, once we identify a neighbourhood of $0$ in
$\mathbb{C}^{2}$ with a neighbourhood of $p$ in $X$.

###  4.3. Finite time singularities and the area of pacman disks

In this subsection we show that strictly unstable Lagrangians develop finite
time singularities under Lagrangian mean curvature flow. The main part of the
argument uses the maximum principle and a barrier construction. To formulate
the barriers, it is useful to introduce the following concepts.

######  Definition 4.9.

Let $p_{-},p_{+},p$ be singularities of $\phi$ lying in a plane and consider a
configuration of two oriented embedded planar curves $\gamma_{-}$,
$\gamma_{+}$ respectively starting and ending at $p_{-}$, $p$ and $p$,
$p_{+}$, such that $\gamma_{\pm}$ are unions of straight lines connecting
singularities of $\phi$. Let $\ell$ denote the oriented straight line from
$p_{-}$ to $p_{+}$. If we have an oriented embedded curve $\gamma$ starting at
$p_{-}$ and ending at $p_{+}$ so that the interior of the region bounded by
$\gamma\cup\gamma_{-}\cup\gamma_{+}$ is connected and $p$ lies in the interior
of the region bounded by $\gamma\cup\ell$, we shall call the triple
$(\gamma_{-},\gamma_{+},\gamma)$ a _triad with vertices $(p_{-},p_{+},p)$_:
see Figure 4.1.

Given such a triad, we let $\ell_{-},\ell_{+}$ denote the oriented straight
lines from $p_{-}$ to $p$ and $p$ to $p_{+}$ respectively and let
$\theta_{\pm}$ denote the angles made by $\ell_{\pm}$ with $\ell$. In Figure
4.1 we have that $\ell_{\pm}=\gamma_{\pm}$.

$p_{-}$$p_{+}$$p$$\gamma_{+}$$\gamma_{-}$$\gamma$$\ell$$\theta_{-}$$\theta_{+}$
Figure 4.1. The triad and its associated pacman disk.

######  Definition 4.10.

Let $(\gamma_{-},\gamma_{+},\gamma)$ be a triad with vertices
$(p_{-},p_{+},p)$ as in Definition 4.9. The interior of the planar region
enclosed by $\gamma$ and $\gamma_{\infty}=\gamma_{-}\cup\gamma_{+}$ will be
called a _pacman disk_ associated with the triad: see Figure 4.1.

We shall now show how the area of a pacman disk varies when the curve $\gamma$
in the triad evolves via the flow (3.1), utilizing Lemma 3.6.

######  Proposition 4.11.

Let $(\gamma_{-},\gamma_{+},\gamma_{0})$ be a triad with vertices
$(p_{-},p_{+},p)$ and let $\gamma_{t}$ evolve via the flow (3.1). Let
$\theta_{\pm}$ be as in Definition 4.9.

Then $(\gamma_{-},\gamma_{+},\gamma_{t})$ remains a triad with vertices
$(p_{-},p_{+},p)$ as long the flow does not meet another singularity of
$\phi$. If $A(t)$ is the area of the pacman disk associated with this triad as
in Definition 4.10, we have

(4.10) $\dot{A}\leq\theta_{+}-\theta_{-}.$

In particular, if $\theta_{+}<\theta_{-}$ then $A(t)\leq
A(0)-(\theta_{-}-\theta_{+})t$ and so $\gamma_{t}$ meets $p$ in finite time
$T\leq A(0)/(\theta_{-}-\theta_{+})$, if it does not reach some other
singularity of $\phi$ before $t=T$.

###### Proof.

We first observe that the conditions on $(\gamma_{-},\gamma_{+},\gamma_{0})$
being a triad mean that $\gamma_{0}$ does not intersect
$\gamma_{\infty}=\gamma_{-}\cup\gamma_{+}$ except at $p_{-},p_{+}$. Since
$\gamma_{\infty}$ is a union of straight lines, it remains stationary along
the flow (3.1). Hence, by the maximum principle, $\gamma_{\infty}$ acts as a
barrier along the flow, except at any points along $\gamma_{\infty}$ which are
singularities of $\phi$. Thus, under the assumption that the flow does not
meet a singularity of $\phi$, $(\gamma_{-},\gamma_{+},\gamma_{t})$ remains a
triad with vertices $(p_{-},p_{+},p)$. Note that a singularity cannot develop
at $p_{-},p_{+}$ by the work in [Lotay2020]*$\S$6, where only the graded
assumption (which is possible to make since $\gamma_{t}$ is embedded) is used
to rule out singularities at the endpoints.

Since the pacman disks associated with the triads
$(\gamma_{-},\gamma_{+},\gamma_{t})$ must lie in a fixed plane for all time,
we may assume without loss of generality they lie in the
$(\mu_{2},\mu_{3})$-plane. We may then lift these pacman disks to holomorphic
pacman disks in $X$ as in $\S$3.2, with boundary on curves contained in
$L_{t}=\pi^{-1}(\gamma_{t})$ and $L_{\infty}=\pi^{-1}(\gamma_{\infty})$ that
project to $\gamma_{t}$ and $\gamma_{\infty}$ under $\pi$. Since
$\gamma_{\infty}$ is a connected union of straight line segments, $L_{\infty}$
is a connected immersed minimal Lagrangian, and $L_{t}$ satisfies Lagrangian
mean curvature flow as $\gamma_{t}$ satisfies (3.1).

We may therefore apply Lemma 3.6 to give that

(4.11) $\dot{A}(t)=\theta_{t}(p_{+})-\theta_{t}(p_{-}).$

Since $\gamma_{\infty}$ is a barrier for the flow $\gamma_{t}$, as we argued
above, the lines $\ell_{\pm}$ are parts of this barrier. Therefore, there is a
constant $c\in\mathbb{R}$ (which is a multiple of $\pi$) such that
$\theta_{t}(p_{+})\leq\theta_{+}+c$ and $\theta_{t}(p_{-})\geq\theta_{-}+c$,
since this is true initially (i.e. for $t=0$) by the conditions on the triple
$(\gamma_{-},\gamma_{+},\gamma_{0})$ being a triad with vertices
$(p_{-},p_{+},p)$ as in Definition 4.9. The inequality (4.10) then follows
from (4.11). The final result is then an easy consequence of (4.10). ∎

A consequence of Proposition 4.11 is that we may now prove most of part (b) of
Theorem 1.1, which also constitutes a solution to a version of Problem 3.12(a)
in [JoyceConjectures].

######  Theorem 4.12.

Let $X\neq\mathbb{R}^{3}\times\mathbb{S}^{1}$ be an ALE or ALF hyperkähler
$4$-manifold constructed via the Gibbons–Hawking ansatz. Then, there is an
almost calibrated, embedded, circle-invariant Lagrangian $L_{0}\subset X$,
diffeomorphic to $\mathbb{S}^{1}\times\mathbb{R}$, such that the Lagrangian
mean curvature flow starting at $L_{0}$ develops a finite time singularity as
in Theorem 4.6.

###### Proof.

If $X\neq\mathbb{R}^{3}\times\mathbb{S}^{1}$ then $\phi$ has at least one
singularity which, with no loss of generality, we suppose to be located at the
origin in $\mathbb{R}^{3}$. Then, we define two infinite non-collinear planar
rays $\gamma_{-},\gamma_{+}$ emanating from $0$ which meet no other
singularities of $\phi$. We orient these rays so that $\gamma_{-}$ is directed
towards $0$ and $\gamma_{+}$ is oriented towards its noncompact end. Notice
that both $\pi^{-1}(\gamma_{\pm})$ are special Lagrangian $\mathbb{R}^{2}$’s
in $X$. Then, we set $\gamma_{0}\subset\mathbb{R}^{3}$ to be an infinite
planar curve asymptotic to $\gamma_{-}$ and $\gamma_{+}$ at its two ends such
that: it goes round 0 to connect its two ends through the side of $0$ that
makes an angle $\alpha>\pi$; and the area, $A$, of the infinite region
enclosed by $\gamma_{0}\cup\gamma_{-}\cup\gamma_{+}$ is finite. This may be
thought of as the triad depicted in Figure 4.1 with the points $p_{\pm}$ sent
to infinity. Then, the same argument as in Proposition 4.11 shows that

$\dot{A}\leq\theta_{+}-\theta_{-},$

where $\theta_{\pm}$ are the angles made by $\gamma_{\pm}$ with a fixed line
in the plane. We may always choose these rays so that $\theta_{-}>\theta_{+}$.
Then, the Lagrangian mean curvature flow starting at
$L_{0}=\pi^{-1}(\gamma_{0})$ must reach $0$ in a finite time less than
$A(0)/(\theta_{-}-\theta_{+})$ and so must develop a finite time singularity
at least by that time. Note that $L_{0}$ is topologically
$\mathbb{S}^{1}\times\mathbb{R}$ by construction. ∎

###  4.4. Unstable strictly convex curves

Proposition 4.11 has an important consequence: namely that circle-invariant
strictly unstable Lagrangians develop finite time singularities. Here we shall
not yet prove this result in its full generality but a simplified version for
a certain class of strictly unstable circle-invariant Lagrangians
$L=\pi^{-1}(\gamma)$ for which $\gamma$ is strictly convex. We shall see that,
as a consequence, we can complete the proof of Theorem 1.1(b). Recall the
notation that $X$ is a hyperkähler 4-manifold with a circle action as
described in Subsection 2.2.

######  Theorem 4.13.

Let $L=\pi^{-1}(\gamma)$ be a compact, embedded, almost calibrated, circle-
invariant Lagrangian in $X$. If $\gamma$ is strictly convex and strictly
unstable or flow unstable, then the Lagrangian mean curvature flow
$L_{t}=\pi^{-1}(\gamma_{t})$ starting at $L_{0}=L$ attains a finite time
singularity at $\pi^{-1}(p)$ for some singularity $p$ of $\phi$ in the region
bounded by $\gamma$ and the straight line connecting its endpoints.

###### Proof.

From the assumptions on $L$ and $\gamma$, we know that there are singularities
$p_{-},p_{+}$ of $\phi$ so that $\gamma$ is a planar embedded arc from $p_{-}$
to $p_{+}$ and that $\gamma$ meets no other singularities of $\phi$. If
$\gamma$ is strictly convex then, using the notation of Definition 4.9, the
region $\Delta$ bounded by $\gamma\cup\ell$ is convex.

As $\gamma$ is strictly unstable or flow unstable, there must be singularities
of $\phi$ in the interior of $\Delta$ which are not equal to $p_{-},p_{+}$;
let $\Delta_{\infty}\subseteq\Delta$ be the convex hull of the union of these
singularities with $p_{-},p_{+}$. Then $\Delta_{\infty}$ is a convex polygon
with boundary given by a union of straight lines connecting singularities of
$\phi$, with one side given by $\ell$.

Let $p$ be any singularity of $\phi$ on the boundary of $\Delta_{\infty}$ not
equal to $p_{-},p_{+}$ and let $\gamma_{-}$ be the oriented union of the sides
of $\Delta_{\infty}$ connecting $p_{-}$ to $p$, and let $\gamma_{+}$ be the
oriented union of the sides of $\Delta_{\infty}$ connecting $p$ to $p_{+}$. By
construction $(\gamma_{-},\gamma_{+},\gamma)$ is a triad with vertices
$(p_{-},p_{+},p)$ and since $\Delta_{\infty}$ is convex we know, in the
notation of Definition 4.9, that $\theta_{+}<\theta_{-}$.

Proposition 3.4 states that the solution $\gamma_{t}$ to (3.1) remains
strictly convex, as well as almost calibrated and embedded, as long as the
flow does not meet any other singularities of $\phi$. Therefore, the flow is
pointing into the region $\Delta$ for all $t$ for which the flow exists
smoothly. We deduce that $\gamma_{t}$ lies in $\Delta$ and
$(\gamma_{-},\gamma_{+},\gamma_{t})$ is a triad with vertices
$(p_{-},p_{+},p)$ for all $t$ before $\gamma_{t}$ reaches a singularity of
$\phi$ by Proposition 4.11.

Proposition 4.11 then implies that $\gamma_{t}$ reaches $p$ in finite time
unless it reaches some other singularity of $\phi$, which must be in $\Delta$,
before then. Since there are only a finite number of singularities of $\phi$
on the boundary of $\Delta_{\infty}$, there must be a first finite time $T$ so
that $\gamma_{t}$ reaches a singularity (which we now call $p$) of $\phi$ not
equal to $p_{\pm}$ lying on $\partial\Delta_{\infty}$.

Since $L_{T}$ would be given topologically by the union of at least two
spheres meeting at a point, whereas $L_{t}$ is a single sphere for $t<T$, the
flow must have a singularity at $p$ at time $t=T$. ∎

An immediately corollary of Theorem 4.13 is the following result which
guarantees the existence of a compact Lagrangian whose Lagrangian mean
curvature flow develops a neck-pinch singularity, and completes the proof of
Theorem 1.1(b).

######  Theorem 4.14.

Let $X$ be an ALE or ALF hyperkähler 4-manifold constructed via the
Gibbons–Hawking ansatz so that $\phi$ has at least three singular points that
lie in a plane but which are not collinear. Then, there is a compact, almost
calibrated, embedded, circle-invariant Lagrangian $L_{0}\subset X$,
diffeomorphic to $\mathbb{S}^{2}$, such that the Lagrangian mean curvature
flow starting at $L_{0}$ develops a finite time singularity as in Theorem 4.6.

###### Proof.

If we have the assumption that $\phi$ has three coplanar but not collinear
singularities, we may label them $p_{-},p_{+},p$ and arrange them in a
triangle as in Figure 4.1. Let $\ell$ be the straight line from $p_{-}$ to
$p_{+}$ We may then clearly choose a strictly convex curve $\gamma$ connecting
$p_{-}$ and $p_{+}$, which lies in the same plane as $p_{-},p_{+},p$, meets no
other singularities of $\phi$ and so that the region bounded by
$\gamma\cup\ell$ contains $p$. Since $p_{-},p_{+},p$ are not collinear, we
deduce that $\gamma$ is strictly unstable, and so we may apply Theorem 4.13.
Note that in this case, by the construction of $\gamma$, we have that
$\pi^{-1}(\gamma)$ is a 2-sphere. ∎

##  5\. Flow through singularities and long–time behaviour

In this section we shall prove that the Lagrangian mean curvature flow
starting at an embedded, almost calibrated, circle invariant Lagrangian exists
and can be continued through its finite time singularities, giving rise to a
flow that exists for all time. Furthermore, we prove that at infinite time,
such a Lagrangian mean curvature flow with surgeries converges to a union of
special Lagrangian submanifolds. The main result is stated as Theorem 5.9.

###  5.1. Flow of piecewise smooth curves

In this subsection we shall prove that for the modified curve shortening flow
(3.1) for suitable planar curves $\gamma_{t}\subset
P\cong\mathbb{R}^{2}\subset\mathbb{R}^{3}$, we can flow through each finite
time singularity which occurs. This flow of curves through singularities then
gives rise to a Lagrangian mean curvature flow $L_{t}:=\pi^{-1}(\gamma_{t})$
in the total space of $\pi:X\to\mathbb{R}^{3}$.

To state the properties of this flow with surgeries we make the following
definition.

######  Definition 5.1.

Let $I\subset\mathbb{R}$ be an interval. A continuous family of piecewise
smooth curves $\\{\gamma_{t}\\}_{t\in I}$ in $\mathbb{R}^{3}$ is said to be a
solution of the flow (3.1) if the following hold.

  * (i)

For all $t\in I$, any singular points of $\gamma_{t}$ are singularities of
$\phi$.

  * (ii)

Away from a finite set of times in $I$, each smooth component
$\gamma^{(i)}_{t}$ of $\gamma_{t}$ satisfies (3.1), i.e.

$\partial_{t}\gamma^{(i)}_{t}=\phi^{-1}(\gamma^{(i)}_{t})^{\prime\prime}.$

Note that a piecewise smooth, embedded, planar curve $\gamma$ has a grading
$\theta$, which is a lift of the angle that its tangent vector makes with a
fixed line, and that is only defined where $\gamma^{\prime}$ is well-defined.
We can then clearly extend the definition of almost calibrated to such curves.

The main result of this subsection is the following.

######  Proposition 5.2.

Let $\gamma_{0}$ be an almost calibrated planar curve in some $2$-plane
$P\subset\mathbb{R}^{3}$ and let $S$ be the singularities of $\phi$ in $P$.
Suppose that the $\gamma_{0}$ is either an embedded arc with endpoints in $S$
or asymptotic to a pair of distinct lines so that, in both cases, the interior
of $\gamma_{0}$ does not meet $S$. Then, there is a continuous family of
piecewise smooth almost calibrated curves $\\{\gamma_{t}\\}_{t\geq 0}\subset
P$, which is a solution of the flow (3.1). This family of curves is real
analytic in space-time except for a finite set of spatial points which lie in
$S$. Furthermore, there is a time $T$ such that for $t\geq T$ the number of
smooth components of $\gamma_{t}$ stays constant, so the flow has no further
singular times.

######  Remark 5.3.

An example of a flow through singularities produced by Proposition 5.2, with
three finite singular times, is shown in Figure 5.1.

$p_{1}$$p_{2}$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$\ell$$\ell_{1}$$\ell_{2}$$\ell_{3}$$\ell_{4}$$\ell_{5}$$\gamma_{0}$

$p_{1}$$p_{2}$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$\ell$$\ell_{1}$$\ell_{2}$$\ell_{3}$$\ell_{4}$$\ell_{5}$$\gamma_{T_{1}}$

$p_{1}$$p_{2}$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$\ell$$\ell_{1}$$\ell_{2}$$\ell_{3}$$\ell_{4}$$\ell_{5}$$\gamma_{T_{2}}$

$p_{1}$$p_{2}$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$\ell$$\ell_{1}$$\ell_{2}$$\ell_{3}$$\ell_{4}$$\ell_{5}$$\gamma_{T_{3}}=\gamma_{T}$

Figure 5.1. Flow with finite time singularities at $0<T_{1}<T_{2}<T_{3}=T$.

###### Proof.

Let $\gamma_{0}$ be an almost calibrated arc in $P$ and write
$S=\\{p_{1},\ldots,p_{k}\\}$, where $p_{1},p_{2}$ are the endpoints of
$\gamma_{0}$. Given a family of curves
$\\{\gamma_{t}\\}_{t\in[0,T_{1})}\subset P$ starting at $\gamma_{0}$ and
evolving through (3.1), Proposition 4.2 shows that the flow exists as long as
$\gamma_{t}$ does not meet $p_{i}$ for $i>2$.

Thus, since the flow (3.1) has short-time existence, the first finite time
singularity $T_{1}<\infty$ (if it exists) is characterized as the least
$T_{1}>0$ such that

$\limsup_{t\nearrow
T_{1}}\mathrm{dist}(\gamma_{t},S\backslash\\{p_{1},p_{2}\\})=0.$

Let $S^{T_{1}}$ be the subset of $S$ contained in the limit set of
$\gamma_{t}$ as $t\nearrow T_{1}$, i.e.

$S^{T_{1}}=S\cap\bigcap_{t\in[0,T_{1})}\overline{\bigcup_{t^{\prime}\in[t,T_{1})}\gamma_{t^{\prime}}},$

which we write as
$S^{T_{1}}=\\{p^{T_{1}}_{1},\ldots,p^{T_{1}}_{k_{1}+1}\\}\subset S$. Notice
that $k_{1}+1>2=|\\{p_{1},p_{2}\\}|$, since there is a singularity of the
original flow $\gamma_{t}$ at $p_{i}$ for $i>2$. Since $p_{1},p_{2}\in
S^{T_{1}}$, with no loss of generality we may set $p_{1}^{T_{1}}=p_{1}$ and
$p_{k_{1}+1}^{T_{1}}=p_{2}$ and note that $S\setminus
S^{T_{1}}=k-k_{1}-1<k-2$. Note also that, by Propositions 4.2, the speed of
the flow (3.1) for $\gamma_{t}$ tends to zero as $t\nearrow T_{1}$ at the
points of $S^{T_{1}}$.

We now describe the procedure for flowing past the singular time $T_{1}$.

  * (1)

We let $\gamma_{T_{1}}$ be the limit set of the curves $\gamma_{t}$ as
$t\nearrow T_{1}$, which is a piecewise smooth curve with smooth components
$\gamma_{T_{1}}^{(i)}$ with endpoints $p_{i}^{T_{1}},p_{i+1}^{T_{1}}\in
S^{T_{1}}$ for $i=1,\ldots,k_{1}$, i.e.

$\gamma_{T_{1}}:=\bigcap_{t\in[0,T_{1})}\overline{\bigcup_{t^{\prime}\in[t,T_{1})}\gamma_{t^{\prime}}}=\gamma^{(1)}_{T_{1}}\cup\gamma^{(2)}_{T_{1}}\cup\ldots\cup\gamma^{(k_{1})}_{T_{1}}.$

Note that $\gamma_{T_{1}}$ is $C^{1}$ near each point in $S^{T_{1}}$ by
Theorem 1.1 and [FSS.neck]*Theorem 1.2.

  * (2)

Note that each smooth arc $\gamma^{(i)}_{T_{1}}$ can be graded so that, away
from the points of $S^{T_{1}}\setminus\\{p_{1},p_{2}\\}$, a grading
$\theta(T_{1})$ of $\gamma_{T_{1}}$ is defined so that
$\theta(t)\to\theta(T_{1})$ as $t\nearrow T_{1}$. Given that $\gamma_{0}$ is
almost calibrated and the grading $\theta(t)$ of $\gamma_{t}$ evolves through
the heat equation $\gamma_{t}$ for $t<T_{1}$, there is a $\delta>0$ such that

$\sup_{\gamma_{t}}\theta(t)-\inf_{\gamma_{t}}\theta(t)\leq\sup_{\gamma_{0}}\theta(0)-\inf_{\gamma_{0}}\theta(0)\leq\pi-\delta.$

Therefore, by continuity, the grading $\theta(T_{1})$ of $\gamma_{T_{1}}$
satisfies the same inequality above. Hence, $\gamma_{T_{1}}$ is almost
calibrated, as are all of the smooth arcs $\gamma^{(i)}_{T_{1}}$ for
$i=1,\ldots,k_{1}$.

  * (3)

Let $i\in\\{1,\ldots,k_{1}\\}$. Since
$\phi^{-1}(\gamma^{(i)}_{T_{1}})^{\prime\prime}$ tends to zero at the
endpoints $p_{i}^{T_{1}},p_{i+1}^{T_{1}}$ of $\gamma^{(i)}_{T_{1}}$, we may
restart the flow (3.1) at time $T_{1}$ with initial condition the almost
calibrated arc $\gamma_{T_{1}}^{(i)}$, fixing the endpoints of the evolving
arcs, and it will remain almost calibrated. According to Definition 5.1, this
means that the piecewise smooth almost calibrated curves
$\gamma_{t}=\cup_{i=1}^{k}\gamma_{t}^{(i)}$, for $t>T_{1}$ solve (3.1) with
initial condition $\gamma_{T_{1}}$. Furthermore, as $\gamma_{t}$ solves a
parabolic equation away from the points of $S^{T_{1}}$, it is real analytic
except possibly at those points.

We now proceed by induction, applying the procedure above to each independent
flow $\gamma_{t}^{(i)}$ of smooth arcs with fixed endpoints lying in
$S^{T_{1}}$ until the next finite time singularity $T_{2}>T_{1}$ (if it
exists). Through this procedure, we obtain a continuous family
$\\{\gamma_{t}\\}_{t\geq 0}$ of piecewise smooth, almost calibrated curves
solving (3.1) (in the sense of Definition 5.1) with finite time singularities
at times $T_{1}<T_{2}<\ldots<T_{l}<\ldots$. Note that the grading defining the
almost calibrated condition flows through the singularities (cf.
[FSS.neck]*Theorem 1.2) and each element of $S$ can occur at most once as a
finite time singularity, otherwise $\gamma_{t}$ would contain a loop, which
would violate the almost calibrated condition. Therefore, using Proposition
4.2, we see that the subset $S^{T_{l}}$ of $S$ contained in the limit set of
$\gamma_{t}$ as $t\nearrow T_{1}$ satisfies $|S\setminus S^{T_{l}}|<k-l-1$, so
there can be at most $k-2$ finite time singularities. Hence, for sufficiently
large $t$, the number of smooth components of $\gamma_{t}$ stays constant,
which completes the proof in the case when $\gamma_{0}$ is an almost
calibrated planar embedded arc.

The case when $\gamma_{0}$ is instead an almost calibrated planar curve
asymptotic to a pair of distinct lines follows from the same argument with
minor modifications, using Proposition 4.4 in place of Proposition 4.2. ∎

######  Remark 5.4.

The proof of Proposition 5.2 shows that the maximal number of smooth
components of $\gamma_{t}$ for sufficiently large $t$ is $\\#S-1$ because
$\gamma_{t}$ cannot have loops.

###  5.2. Unstable curves

We are now in a position to improve our results on strictly convex, strictly
unstable, almost calibrated, compact curves in Theorem 4.13 in two different
directions. First, we shall investigate the long-time behaviour for the flow
of such unstable curves, and then use this to prove that all almost calibrated
unstable curves develop finite time singularities.

######  Corollary 5.5.

Let $P$ be a $2$-plane in $\mathbb{R}^{3}$ and let $S$ be the singularities of
$\phi$ in $P$. Let $\gamma_{0}$ be an almost calibrated, strictly convex arc
in $P$ with endpoints $p_{1},p_{2}\in S$, which otherwise does not meet $S$.
Let $\ell$ be the straight line connecting $p_{1},p_{2}$ and let $\mathcal{R}$
be the open region in $P$ bounded by $\gamma\cup\ell$. Then, there is a
ordered set of straight lines $\\{\ell_{1},\ldots,\ell_{k}\\}$ with
consecutive endpoints such that $\gamma_{t}$ converges uniformly to their
union,

$\lim_{t\to+\infty}\gamma_{t}=\ell_{1}\cup\ldots\cup\ell_{k},$

and $\ell\cup\ell_{1}\cup\ldots\cup\ell_{k}$ is the boundary of the convex
hull of $S\cap\mathcal{R}$ in $P$.

###### Proof.

After possibly applying a rotation and translation of $\mathbb{R}^{3}$ (which
each induce isometries of the metric on the ambient hyperkähler 4-manifold
$X$), we may assume that $\ell$ lies along the $x_{1}$-axis in
$\mathbb{R}^{3}$ and that $P$ is the plane with $x_{3}=0$. Without loss of
generality, we may assume that $p_{1}$ has $x_{1}$ coordinate less than
$p_{2}$ and that $\gamma$ lies in the region with $x_{2}\geq 0$.

Let $\\{\gamma_{t}\\}_{t\geq 0}$ be the piecewise smooth almost calibrated
curves solving (3.1) given by Proposition 5.2 and let $T\geq 0$ be so that the
number of smooth components of $\gamma_{t}$ stays constant for $t\geq T$ and
there are no further singular times. By Proposition 5.2 and its proof, we may
decompose $\gamma_{t}=\gamma_{t}^{(1)}\cup\ldots\cup\gamma_{t}^{(l)}$ for
$t\geq T$ into smooth components and let $q_{1},\ldots,q_{l-1}\in
S\setminus\\{p_{1},p_{2}\\}$ be ordered so that their $x_{1}$-coordinates are
increasing and so that the curves $\gamma_{t}^{(i)}$ connect $p_{1}$ to
$q_{1}$, $q_{1}$ to $q_{2}$ and so on until the last curve connects $q_{l-1}$
to $p_{2}$. Since strict convexity is preserved along the flow for finite time
by Proposition 3.4, we find that each of the curves $\gamma_{t}^{(i)}$ is
strictly convex. We then set $\ell_{i}$ to be the straight line in $P$ with
the same endpoints as $\gamma_{t}^{(i)}$ for $i=1,\ldots,l$.

Since the flows $\gamma_{t}^{(i)}$ are strictly convex and almost calibrated,
but have no finite time singularities for $t\geq T$, we deduce from Theorem
4.13 that $\gamma_{T}^{(i)}$ must be semi-stable and flow semi-stable. By
[Lotay2020]*Corollary 6.11, a flow stable curve will converge along the flow
(3.1) to the straight line connecting its endpoints. Therefore, if
$\gamma_{T}^{(i)}$ is flow stable, the flow $\gamma_{t}^{(i)}$ converges
smoothly to $\ell_{i}$. Hence, if all $\gamma_{T}^{(i)}$ are flow stable the
proof is complete with $k=l$.

Suppose that there is some $i\in\\{1,\ldots,l\\}$ such that $\gamma_{T}^{(i)}$
is flow semi-stable but not flow stable. The semi-stability of
$\gamma_{T}^{(i)}$ means there must be no singularities of $\phi$ in the
region bounded by $\gamma_{T}^{(i)}$ and $\ell_{i}$, but that there must be a
singularity of $\phi$ lying on $\ell_{i}$. We may then decompose $\ell_{i}$
into a union of straight lines
$\ell_{i}^{(1)}\cup\ldots\cup\ell_{i}^{(k_{i})}$ with consecutive endpoints
connecting singularities of $\phi$ so that there are no singularities of
$\phi$ in the interiors of each $\ell_{i}^{(j)}$. Notice that each
$\ell_{i}^{(j)}$ must have the same angle as $\ell_{i}$. Therefore, since
$\gamma_{t}^{(i)}$ exists smoothly for all $t\geq T$, the flow can only become
singular at singularities of $\phi$, and the Lagrangian angle evolves through
the heat equation (3.2), we see that the Lagrangian angle must converge
uniformly to a constant as $t\to\infty$ (cf. [ThomasYau]*p. 1109–1110), which
is the same angle as $\ell_{i}$. Hence, $\gamma_{t}^{(i)}$ converges uniformly
to the union of straight lines
$\ell_{i}=\ell_{i}^{(1)}\cup\ldots\cup\ell_{i}^{(k_{i})}$ as $t\to\infty$,
which completes the proof (with $k=\sum_{i=1}^{l}k_{i}$, setting $k_{i}=1$ if
$\gamma_{T}^{(i)}$ is flow stable). ∎

######  Remark 5.6.

Figure 5.1 gives an example of the result of Corollary 5.5, showing how an
initial curve converges to a union of five straight line segments
$\cup_{j=1}^{5}\ell_{j}$. Notice that $\ell_{2},\ell_{3}$ in Figure 5.1 have
the same angle, which gives an example of the flow semi-stable but not flow
stable situation considered at the end of the proof of Corollary 5.5.

######  Corollary 5.7.

Let $\gamma_{0}$ be an almost calibrated, strictly unstable, planar arc in
$\mathbb{R}^{3}$ so that its intersection with the singularities of $\phi$
consists of its endpoints. Then, the flow $\\{\gamma_{t}\\}_{t\in[0,+\infty)}$
obtained through Proposition 5.2 attains a finite time singularity.

###### Proof.

Let $S$ be singularities of $\phi$ in the 2-plane $P$ containing $\gamma_{0}$.
If $\gamma_{0}$ is strictly unstable, then there is a non-empty subset
$\\{q_{1},\ldots,q_{l}\\}\subseteq S$ contained in the interior of the region
bounded by $\gamma_{0}$ and the straight line $\ell$ connecting its endpoints
(notice that even if $\gamma_{0}$ and $\ell$ intersect, they enclose a
possibly disconnected region). Then, we choose an almost calibrated, strictly
convex, planar arc $\hat{\gamma}_{0}$ in $P$ with the same endpoints as $\ell$
(and $\gamma_{0}$), meeting no other elements of $S$ and bounding a segment of
$\gamma_{0}$ which, together with $\ell$, encloses some of the singularities
$\\{q_{1},\ldots,q_{l^{\prime}}\\}\subseteq\\{q_{1},\ldots,q_{l}\\}$ of
$\phi$. (Such a curve $\hat{\gamma}_{0}$ exists: see Figure 5.2 for an
example.)

$p_{1}$$p_{2}$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$\gamma_{0}$$\hat{\gamma}_{0}$$\ell$
Figure 5.2. A strictly convex barrier curve $\hat{\gamma}_{0}$ for a strictly
unstable curve $\gamma_{0}$.

Let $\\{\hat{\gamma}_{t}\\}_{t\geq 0}$ and $\\{\gamma_{t}\\}_{t\geq 0}$ be the
piecewise smooth solutions of the flow given by Proposition 5.2 starting at
$\hat{\gamma}_{0}$ and $\gamma_{0}$ respectively. The maximum principle
implies that $\hat{\gamma}_{t}$ and $\gamma_{t}$ do not intersect away from
$S$, hence $\hat{\gamma}_{t}$ can be taken as a barrier for $\gamma_{t}$.

Let $\Delta$ be the convex hull of the elements of $S$ contained in the
closure of the region bounded by $\hat{\gamma}\cup\ell$. Its boundary
$\partial\Delta$ is given by a union of straight lines
$\ell\cup\ell_{1}\cup\ldots\cup\ell_{k}$ and
$\lim_{t\to+\infty}\hat{\gamma}_{t}=\ell_{1}\cup\ldots\cup\ell_{k}$ by
Corollary 5.5. Suppose that $q_{i}\in\partial\Delta$ for some
$i\in\\{1,\ldots,l^{\prime}\\}$. Then, $\hat{\gamma}_{t}$ has a finite time
singularity at $q_{i}$ by Corollary 5.5 and its proof. The maximum principle
then also implies that $\gamma_{t}$ develops a finite time singularity at
$q_{i}$ as required.

Suppose instead that $\\{q_{1},\ldots,q_{l^{\prime}}\\}$ is contained in the
interior of $\Delta$. As
$\lim_{t\to+\infty}\hat{\gamma}_{t}=\ell_{1}\cup\ldots\cup\ell_{k}$ and
$\hat{\gamma}_{t}$ is a barrier for $\gamma_{t}$ we find that for sufficiently
large $T>0$ the curve $\gamma_{T}$ must pass through the interior of $\Delta$.
Hence, if $\gamma_{t}$ does not develop a finite time singularity before
$t=T$, we can find another strictly convex, almost calibrated curve
$\hat{\gamma}_{0}^{\prime}$ with the same endpoints as $\gamma_{t}$, totally
contained inside $\Delta$, and which bounds the segment of $\gamma_{T}$ itself
bounding $\\{q_{1},\ldots,q_{l^{\prime}}\\}$ together with $\ell$. We may then
replace $\hat{\gamma}_{0}$ with $\hat{\gamma}_{0}^{\prime}$ to produce a new
flow $\hat{\gamma}_{t}^{\prime}$ by Proposition 5.2 which serves as a new
barrier for $\gamma_{t}$. Since $S$ is finite, this procedure can be applied
iteratively until at least one element of $\\{q_{1},\ldots,q_{l^{\prime}}\\}$
is contained in the boundary of the convex hull of the singularities inside
such a novel barrier curve. We are then in the previous situation above and so
$\gamma_{t}$ does develop a finite time singularity as claimed. ∎

###  5.3. Long time behaviour for LMCF through singularities

In this final subsection we put together the results of the previous
subsections to prove Theorem 1.4. We begin with a natural definition following
Definition 5.1.

######  Definition 5.8.

Let $L_{0}=\pi^{-1}(\gamma_{0})$ be an embedded, circle-invariant, almost
calibrated Lagrangian in the hyperkähler 4-manifold $X$. Let
$\\{\gamma_{t}\\}_{t\geq 0}$ be a piecewise smooth solution of the flow
equation (3.1) in the sense of Definition 5.1. We say that the continuous
family $\\{L_{t}=\pi^{-1}(\gamma_{t})\\}_{t\geq 0}$ is a _Lagrangian mean
curvature flow through singularities_ starting at $L_{0}$.

We now prove one of our main results which, together with Theorem 1.1, will
account for items (a)–(d) in Theorem 1.4, except for the convergence of
currents in (c) of that result.

######  Theorem 5.9.

Let $L_{0}=\pi^{-1}(\gamma_{0})$ be a compact, connected, embedded, circle-
invariant, almost calibrated Lagrangian in the hyperkähler $4$-manifold $X$.

  * (a)

A compact, connected, circle-invariant, almost calibrated Lagrangian mean
curvature through singularities $\\{L_{t}\\}_{t\geq 0}$ exists for all time
and has a finite number of finite time singularities.

  * (b)

There is an $A_{k}$ chain $\\{L_{1}^{\infty},\ldots,L_{k}^{\infty}\\}$, in the
sense of Definition 1.3, of circle-invariant, embedded, special Lagrangian
spheres such that $\\{L_{t}\\}_{t\geq 0}$ uniformly converges to
$L_{1}^{\infty}\cup\ldots\cup L_{k}^{\infty}$. Moreover, if the grading on
$L_{0}$ is a perfect Morse function, then the phases of these special
Lagrangians can be arranged to be non-increasing.

  * (c)

The number $k$ of special Lagrangians in the $A_{k}$ chain is exactly one if
$\gamma_{0}$ is flow stable and strictly greater than one if $\gamma_{0}$ is
flow unstable.

###### Proof.

Let $\gamma_{0}$ be the planar curve in $\mathbb{R}^{3}$ such that
$L_{0}=\pi^{-1}(\gamma_{0})$ and let $\ell$ be the straight line connecting
its endpoints. By Proposition 5.2, the piecewise smooth flow
$\\{\gamma_{t}\\}_{t\geq 0}$ exists for all time. Hence, so does the
Lagrangian mean curvature flow through singularities $\\{L_{t}\\}_{t\geq 0}$,
which gives (a). Moreover, there is a finite time $T\geq 0$ so that the flow
$\gamma_{t}$ (and hence $L_{t}$) has no singularities and the number of smooth
components of $\gamma_{t}$ stays constant.

We may then decompose $\gamma_{t}$ for $t\geq T$ into smooth components as
$\gamma_{t}=\gamma_{t}^{(1)}\cup\ldots\cup\gamma_{t}^{(l)}$ with the
components ordered so that they have consecutive endpoints. For each
$i\in\\{1,\ldots,l\\}$, $\gamma_{t}^{(i)}$ is an almost calibrated planar arc
only meeting the singularities of $\phi$ at its endpoints and with no
singularity along the flow. Corollary 5.7 implies that $\gamma_{t}^{(i)}$ is
flow semi-stable for all $i$.

By the same arguments leading to the conclusion of Corollary 5.5 (which one
may notice do not use any convexity assumption on $\gamma_{t}$) we deduce that
there is an ordered set of straight lines $\\{\ell_{1},\ldots\ell_{k}\\}$ with
consecutive endpoints which are singularities of $\phi$ so that

$\gamma_{t}\to\ell_{1}\cup\ldots\cup\ell_{k}$

uniformly as $t\to\infty$. Setting $L_{i}^{\infty}=\pi^{-1}(\ell_{i})$ for
$i=1,\ldots,k$ gives the first part of (b).

If the Lagrangian angle on $L_{0}$ is a perfect Morse function, then
$\gamma_{0}$ is strictly convex by (3.3), so Corollary 5.5 applies and
$\ell\cup\ell_{1}\cup\ldots\cup\ell_{k}$ is the boundary of the convex hull of
the singularities of $\phi$ contained in the region enclosed by $\gamma_{0}$
and $\ell$. Hence the non-increasing property in (b) can indeed be arranged as
claimed.

The fact that $k=1$ if $\gamma_{0}$ is flow stable follows from the proof of
the circle-invariant Thomas–Yau conjecture in [Lotay2020]. The fact that $k>1$
if $\gamma_{0}$ is flow unstable is a consequence of Corollary 5.7 if
$\gamma_{0}$ is strictly flow unstable, and otherwise follows from the
argument at the end of the proof of Corollary 5.5 (again noticing that the
convexity is not used there) since we are then in the flow semi-stable but not
flow stable setting. ∎

######  Remark 5.10.

Figure 5.1 gives an example where the grading is a perfect Morse function and
the $A_{k}$ chain (where $k=5$ in the example, corresponding to the lines
$\\{\ell_{1},\ldots,\ell_{5}\\}$) is arranged so that the phases are non-
increasing. Notice that the phases corresponding to $\ell_{2},\ell_{3}$ are
equal, which shows that one cannot always ensure that the phases are
decreasing.

To complete the proof of Theorem 1.4, it remains to prove the statements about
the continuity and convergence as currents, which are consequences of the
results we have proven so far.

######  Proposition 5.11.

In the setting of Theorem 5.9, the family $\\{L_{t}\\}_{t\in[0,+\infty)}$
varies continuously as an integral Lagrangian current with the following
current convergence as $t\to\infty$:

$\lim_{t\to+\infty}L_{t}=L_{1}^{\infty}+\ldots+L_{k}^{\infty}.$

###### Proof.

By Theorem 5.9, there are $0<T_{1}<\ldots<T_{l}$ such that, for
$t\in[0,+\infty)\backslash\\{T_{1},\ldots,T_{l}\\}$, $L_{t}$ is a union of
smooth Lagrangians solving Lagrangian mean curvature flow, so the claimed
continuity of $t\mapsto L_{t}$ follows immediately for these times. We are
therefore left with proving continuity at the singular times $T_{i}$. For
this, for any compactly supported 2-form $\alpha$ on $X$, we show that

(5.1) $t\mapsto\int_{L_{t}}\alpha,$

is continuous at $t=T_{i}$ for $i=1,\ldots,l$. Recall by the proof of
Proposition 5.2 that, for $\varepsilon>0$ sufficiently small,

$L_{T_{i}}:=\bigcap_{s\in[T_{i}-\varepsilon,T_{i})}\overline{\bigcup_{s^{\prime}\in[s,T_{i})}L_{s^{\prime}}}.$

Hence, using the fact that $L_{t}=\pi^{-1}(\gamma_{t})$ and
$L_{T_{i}-\varepsilon}\cup-L_{T_{i}}$ is the boundary of
$\pi^{-1}(\overline{\cup_{s\in[T_{i}-\varepsilon,T_{i})}\gamma_{s}})$ and
Proposition 5.2, we find

$\big{|}\int_{L_{T_{i}-\varepsilon}}\alpha-\int_{L_{T_{i}}}\alpha\big{|}=\big{|}\int_{\partial\pi^{-1}(\overline{\cup_{s\in[T_{i}-\varepsilon,T_{i})}\gamma_{s}})}\alpha\big{|}=\big{|}\int_{\pi^{-1}(\overline{\cup_{s\in[T_{i}-\varepsilon,T_{i})}\gamma_{s}})}d\alpha\big{|}\lesssim\varepsilon\|d\alpha\|_{L^{\infty}},$

thus proving that (5.1) is continuous.

Finally, the statement that
$\lim_{t\to+\infty}L_{t}=L_{1}^{\infty}+\ldots+L_{k}^{\infty}$ as currents
follows from the uniform convergence of $L_{t}$ to
$L_{1}^{\infty}\cup\ldots\cup L_{k}^{\infty}$ from Theorem 5.9. ∎

##  6\. Monotonicity of the Lagrangian angles

Conjecture 3.34(e) in [JoyceConjectures] states that the decomposition into
special Lagrangians $L_{1}^{\infty},\ldots,L_{k}^{\infty}$, as in Theorem 1.4,
should be chosen to have their phases satisfying
$\theta_{1}\geq\ldots\geq\theta_{k}$. In Theorem 1.4(d), when the grading on
the initial Lagrangian is a perfect Morse function, we see that this ordering
by phase coincides with the ordering as an $A_{k}$-chain. In this section we
shall consider one example that shows how the decomposition of an initial
Lagrangian using our flow through singularities works using the monotonicity
of the Lagrangian angles, but which does not coincide with the ordering from
the $A_{k}$-chain.

We consider $X$ to be an ALE or ALF gravitational instanton obtained via the
Gibbons–Hawking ansatz with $\phi$ having at least 4 planar singularities
$\\{p_{1},p_{2},q_{1},q_{2}\\}$ arranged as in Figure 6.1 and let $\gamma_{0}$
be the planar almost calibrated arc indicated there.

$p_{1}$$p_{2}$$q_{1}$$q_{2}$$\gamma_{0}$$\ell_{1}$$\ell_{2}$$\ell_{3}$ Figure
6.1. A flow through singularities starting at $\gamma_{0}$ converges to
$\ell_{1}\cup\ell_{2}\cup\ell_{3}$.

By Theorem 5.9 there is a Lagrangian mean curvature flow through singularities
$L_{t}=\pi^{-1}(\gamma_{t})$ starting at $L_{0}=\pi^{-1}(\gamma_{0})$ which as
$t\to+\infty$ converges to an $A_{k}$ chain of special Lagrangian spheres. We
claim that this $A_{k}$ chain is $\cup_{j=1}^{3}L_{j}^{\infty}$ where
$L_{j}^{\infty}=\pi^{-1}(\ell_{j})$. To prove this, we define two triads:
$(\gamma^{1}_{-},\gamma^{1}_{+},\hat{\gamma}^{1})$ with vertices
$(p_{1},q_{2},q_{1})$ and $(\gamma^{2}_{-},\gamma^{2}_{+},\hat{\gamma}^{2})$
with vertices $(p_{2},q_{1},q_{2})$. These triads are chosen so that
$\hat{\gamma}^{j}$ is almost calibrated for $j=1,2$ and as shown in Figure
6.2, which also shows the pacman disks associated with the triads.

$p_{1}$$p_{2}$$q_{1}$$q_{2}$$\gamma_{0}$$\gamma^{1}_{-}$$\gamma^{1}_{+}$$\gamma^{2}_{-}$$\gamma_{+}^{2}$$\hat{\gamma}^{1}$$\hat{\gamma}^{2}$
Figure 6.2. Pacman disks for the flow through singularities starting at
$\gamma_{0}$.

The flows $\hat{\gamma}^{j}_{t}$ starting at $\hat{\gamma}^{j}$ have finite
time singularities at $q_{j}$ by Proposition 4.11. By construction, these
flows act as barriers for $\gamma_{t}$, and since the only places where
$\gamma_{t}$ can have finite time singularities are $q_{1},q_{2}$, the flow
$\gamma_{t}$ must have singularities at both points in finite time.

Hence, there are two singular times $T_{1}\leq T_{2}$ after which the family
$\gamma_{t}$ consists of a union of three smooth flows of embedded arcs which
exist for all time and converge to the straight lines connecting their
endpoints. We therefore have that $L_{t}$ converges to
$\cup_{j=1}^{3}L_{j}^{\infty}$ where $L_{j}^{\infty}=\pi^{-1}(\ell_{j})$ as
claimed, and moreover

$\lim_{t\to+\infty}L_{t}=L_{1}^{\infty}+L_{2}^{\infty}+L_{3}^{\infty}.$

Now, one needs to take care when exhibiting the limiting destabilizing
configuration for $L_{0}$ since, as in [JoyceConjectures]*Conjecture 3.34(e),
one requires a monotonicity condition on the Lagrangian angles of
$L_{j}^{\infty}$. Since the Lagrangian angles $\theta_{j}$ of $L_{j}^{\infty}$
are equal (up to the addition of a fixed constant, independent of $j$) to the
angle $\ell_{j}$ makes with the line from $p_{1}$ to $p_{2}$, we see that
$\theta_{1}>\theta_{3}>0>\theta_{2}$. Hence, there is no natural ordering of
the $L_{j}^{\infty}$ as a chain so that the desired monotonicity of the
Lagrangian angles is achieved.

Instead, notice that we can decompose the Hamiltonian isotopy class of $L_{0}$
into a graded Lagrangian connect sum of the special Lagrangian spheres as
$L_{1}^{\infty}\\#(L_{3}^{\infty}\\#L_{2}^{\infty})$. This shows that
$L_{1}^{\infty}$ destabilizes the class of $L_{0}$ with “quotient”
$L_{3}^{\infty}\\#L_{2}^{\infty}$, which is itself destabilized by
$L_{3}^{\infty}$ with “quotient” the stable special Lagrangian
$L_{2}^{\infty}$. This sequence of destabilizations occurs in the correct
order of monotonicity as $\theta_{1}>\theta_{3}>\theta_{2}$. The reader may
wonder what would happen if we had considered a similar configuration to
Figure 6.1 but with $\theta_{3}>\theta_{1}>\theta_{2}$ instead of
$\theta_{1}>\theta_{3}>\theta_{2}$. This is explained by the fact that we have
Hamiltonian isotopies

$L_{1}^{\infty}\\#(L_{3}^{\infty}\\#L_{2}^{\infty})\sim L_{0}\sim
L_{3}^{\infty}\\#(L_{1}^{\infty}\\#L_{2}^{\infty}),$

and so we also have destabilizations ordered
$L_{3}^{\infty},L_{1}^{\infty},L_{2}^{\infty}$ which is compatible with
$\theta_{3}>\theta_{1}>\theta_{2}$. Therefore, though we have two different
destabilizing configurations for $L_{0}$ using the special Lagrangians
$L_{j}^{\infty}$, only one is compatible with the required monotonicity of the
Lagrangian angles.

In conclusion, the Lagrangian mean curvature flow starting at
$L_{0}=\pi^{-1}(\gamma_{0})$ as in Figure 6.1 shows how the limit of the
conjectured Lagrangian mean curvature flow through singularities envisaged in
[JoyceConjectures] can be presented uniquely so that all of the conditions
required there are satisfied, particularly involving the monotonicity of the
Lagrangian angles.

## References