# Lagrangian mean curvature flow in the
complex projective plane

Christopher G. Evans

###### Abstract.

We prove a Thomas–Yau-type conjecture for monotone Lagrangian tori satisfying
a symmetry condition in the complex projective plane $\mathbb{CP}^{2}$. We
show that such tori exist for all time under Lagrangian mean curvature flow
with surgery, undergoing at most a finite number of surgeries before flowing
to a minimal Clifford torus in infinite time. Furthermore, we show that we can
construct a torus with any finite number of surgeries before convergence.
Along the way, we prove many interesting subsidiary results and develop
methods which should be useful in studying Lagrangian mean curvature flow in
non-Calabi–Yau manifolds, even in non-symmetric cases.

##  1\. Introduction

Starting from the Clifford torus, Vianna ([vianna_2014],[Vianna2016])
constructs by an iterative sequence of mutations an infinite family
$\mathcal{F}$ of monotone Lagrangian tori in the complex projective plane
$\mathbb{CP}^{2}$, with no two members of $\mathcal{F}$ Hamiltonian isotopic.
In this paper, we make and explore a simple observation: Vianna’s mutation is
exactly the surgery procedure one expects at the prototypical Lawlor neck-type
singularity in Lagrangian mean curvature flow. By restricting to equivariant
examples of $\mathcal{F}$, those satisfying an
$(S^{1}\times\mathbb{Z}_{2})$-symmetry to be specified later, we define a
surgery called neck-to-neck surgery at all possible singularities and are able
to prove a Thomas–Yau-type conjecture ([Thomas2002],[joyce_2015]):

######  Theorem 1.1 (Main theorem).

Let $F:L\to\mathbb{CP}^{2}$ be an embedded equivariant monotone Lagrangian
torus in the complex projective plane with the standard Fubini–Study Kähler
metric. Then under Lagrangian mean curvature flow with surgery, $L$ exists for
all time and, after undergoing at most a finite number of surgeries, converges
to a minimal Clifford torus in infinite time.

This is a natural extension of the classical result that an embedded circle in
$\mathbb{CP}^{1}=S^{2}$ that divides $S^{2}$ into equal area pieces exists for
all time and converges to an equator. Unlike in the curve-shortening case,
singularities are an inevitable feature of Lagrangian mean curvature flow in
higher dimensions, and therefore developing an understanding of surgeries at
these singularities is of crucial importance. We highlight that to the
author’s knowledge, this is the first example of Lagrangian mean curvature
flow with surgery in the literature.

In the course of proving the main theorem, we prove some interesting side
results, a selection of which we now highlight.

Wang [Wang2001] proved that almost-calibrated Lagrangians in Calabi–Yau
manifolds do not attain type I singularities. This result provides a clear
distinction between Lagrangian mean curvature flow and hypersurface mean
curvature flow where type I singularities are commonplace and type II
singularities are rare. We prove an analogue of this result for monotone
Lagrangians in Fano manifolds, which we take to mean Kähler–Einstein manifolds
with positive Einstein constant $\kappa$:

######  Theorem 1.2.

Let $F_{t}:L^{n}\to M^{2n}$ be a monotone Lagrangian mean curvature flow in a
Kähler–Einstein manifold $M$ with Einstein constant $\kappa>0$. Then $F_{t}$
does not attain any type I singularities.

By studying the equation for the equivariant mean curvature, which we derive
quickly and explicitly by a novel method, we are able to show that:

######  Theorem 1.3.

There exists a countably infinite family of complete immersed minimal
Lagrangian tori in $\mathbb{CP}^{2}$.

The immersed minimal tori constructed are similar to the self-shrinking
solutions to curve-shortening flow discovered by Abresch–Langer
[abresch_langer_1986], see Figure 1.

Figure 1. The profile curves of a selection of complete immersed minimal
equivariant Lagrangian tori in $\mathbb{CP}^{2}$ of varying period.

The equivariant restriction in the main theorem leaves us with two classes
Lagrangian tori: up to Hamiltonian isotopy, they are the Clifford torus and
the Chekanov torus. The main theorem above might seem to imply that a Clifford
torus does not have singularities under Lagrangian mean curvature flow.
However, this is not the case:

######  Theorem 1.4.

There exists a Clifford torus $L$ in $\mathbb{CP}^{2}$ such that under mean
curvature flow $L$ has a finite-time singularity and surgery at the
singularity makes $L$ a Chekanov torus.

By iterating the explicit construction we provide in a relatively
straightforward manner, it is also clear that for any $n\geq 0$, one can find
a Lagrangian torus $L$ in $\mathbb{CP}^{2}$ with exactly $n$ singularities
before converging to a minimal Clifford torus.

Thus the behaviour we observe is of a cyclical nature: a Clifford torus can
collapse to become a Chekanov torus, which then exists for some time after
surgery before collapsing to a Clifford torus again. This process can repeat
any finite number of times before eventually becoming a stable Clifford torus.
We note that this type of “flip-flopping” behaviour is unusual in mean
curvature flow, and to the best of the author’s knowledge has not been
observed before. We conjecture that it is in fact not unusual behaviour for
Lagrangian mean curvature flow, and that we should expect to see similar
behaviour in Calabi–Yau manifolds as well.

Despite the restrictive nature of the symmetry used, we believe that the ideas
and principles used to guide the proofs here have the potential to see further
application in the non-symmetric case. The primary reason for the symmetry is
not for symplectic or topological reasons, but to reduce the type of
singularities that can occur to a case which is well-understood by the work of
Wood ([Wood2019],[wood_2020]). Our main contribution on this front is to
define a surgery using the Scale Lemma (Lemma 5.4), showing that singularity
formation happens on an arbitrarily small scale. This allows us to categorise
the behaviour in the proof of Theorem 1.1 by observing that the number of
intersections with a specific pair of real projective planes decreases under
the flow with surgery. Fascinatingly, no such result exists in the Calabi–Yau
case and the method of proof employed here cannot possibly be generalised.

Let us discuss the proof of the main theorem, in the process giving a guide to
the paper.

In Section 4.2, we begin by calculating the governing equation for equivariant
mean curvature flow in $\mathbb{CP}^{2}$. Here, we use a novel method to avoid
raw computation: we relate the mean curvature of a Lagrangian to the relative
Lagrangian angle it forms with the standard toric fibration of
$\mathbb{CP}^{2}$ by Lagrangian Clifford tori. This vastly simplifies the
calculations.

In Section 4.3, we introduce one of the key ideas which enables the majority
of the rest of the paper. Using a mild generalisation of the
Cieliebak–Goldstein theorem [Cieliebak2004] to include Lagrangians with
corners, we consider evolution equations of areas of $J$-holomorphic triangles
bounded by segments of our flowing Lagrangian tori and Lagrangian cones given
by unions of totally geodesic real projective planes in $\mathbb{RP}^{2}$. The
key insight is that by considering such triangles between Hamiltonian non-
isotopic objects, we are considering objects which are in a Floer-theoretic
sense non-trivial. Thus the behaviour of these triangles is geometric rather
than topological, and hence can be measured with respect to the mean curvature
flow.

The main mathematical debt of this paper is owed to the work of Neves
([Neves2007],[Neves2010]) and the work of Wood ([Wood2019],[wood_2020]) on
singularity formation in Lagrangian mean curvature flow. In Section 5.2, we
replicate their results in the positive curvature setting. In doing so, we
restrict the class of singularities that can form to simply one type: Lawlor
neck singularities at the origin with type I blow-up given by a specific cone
of opening angle $\pi/2$, namely $C^{0}_{\pi/2}$. A full description of the
singular behaviour is given in Section 5.2, although since the proof of this
fact is long and rather tedious, we refer the reader to the author’s doctoral
thesis for a full proof. In addition, we prove the Scale Lemma using
properties of the minimal equivariant Lagrangians constructed in Section 4.4.

Next, we study equivariant Clifford and Chekanov tori. We show in Section 5.3
that Clifford tori satisfying a graphical condition have long-time existence
and convergence to the minimal equivariant Clifford torus. We then show in
Section 5.4 that any Chekanov torus has a finite-time singularity under mean
curvature flow. In the process, we show that there is no minimal equivariant
Chekanov torus. The method of proof, using a triangle calculation as described
above, is tantalisingly close to being generalisable to non-equivariant tori.

In Section 5.5, we define a neck-to-neck surgery using the Scale Lemma that
strictly decreases the number of intersections with the cone $C^{0}_{\pi/2}$.
This allows us to handle the remaining cases, and prove the main theorem.

######  Acknowledgments.

This paper is a reorganised and abridged version of the author’s doctoral
thesis [Evans_2022].

I would like to thank my supervisors Jason Lotay and Felix Schulze for their
constant support and encouragement. Many thanks are owed to Jonny Evans for
introducing me to the work of Vianna, and encouraging me to study it in the
context of mean curvature flow. I would also like to thank my thesis examiners
for their many useful comments and corrections.

Thank you to Emily Maw for patiently explaining almost-toric fibrations to me.
Thank you to Ben Lambert for many helpful talks over the years, and for
struggling through some particularly heinous calculations to verify one of my
results. Finally, thank you to Albert Wood, who not only explained in careful
detail a great number of his results to me, but also read preliminary versions
of my thesis and provided important feedback.

This work was supported by the Leverhulme Trust Research Project Grant
RPG-2016-174.

##  2\. Preliminaries

###  2.1. Lagrangian tori in $\mathbb{CP}^{2}$

Let $(M,g,J,\omega)$ be a Kähler–Einstein manifold with Einstein constant
$\kappa$. We call $M$ a Fano manifold if $\kappa>0$. This definition is non-
standard in the literature, but is most appropriate from the point of view of
Lagrangian mean curvature flow. A half-dimensional immersed submanifold
$F:L^{n}\to M^{2n}$ is called Lagrangian if the symplectic form $\omega$
vanishes on $L$, i.e. $F^{*}\omega=0$. We will frequently refer to the
immersed Lagrangian $F(L)$ by $L$ where it is unlikely to cause confusion.

Lagrangians contain a great deal of information about the symplectic topology
of the manifold they live in, but since symplectic geometry is always trivial
locally, it is often necessary to restrict to a subclass of Lagrangians in
order to obtain interesting results. For Fano manifolds, the most important
subclass is that of monotone Lagrangians. An embedded Lagrangian $L$ is called
monotone if for any disc $D\in\pi_{2}(M,L)$, the Maslov class of $D$ is
related to its holomorphic area by

$\kappa\int_{D}\omega=\pi\mu(D).$

The prototypical and arguably most important Fano manifolds are the complex
projective spaces $\mathbb{CP}^{n}$, realised as quotients of Euclidean space
$\mathbb{C}^{n+1}$ by the Hopf fibration

$(z_{0},\dots,z_{n})\mapsto[z_{0}:\cdots:z_{n}],$

where $[z_{0}:\cdots:z_{n}]=[w_{0}:\cdots:w_{n}]$ if and only if
$(w_{0},\dots,w_{n})=\lambda(z_{0},\dots,z_{n})$, for some
$\lambda\in\mathbb{C}$. Of course, the Hopf fibration can also be taken from
the unit $S^{2n+1}$ sphere in $\mathbb{C}^{n+1}$.

The first complex projective space, the complex projective line
$\mathbb{CP}^{1}$ is the round 2-sphere $S^{2}$ with Einstein constant
$\kappa=4$. Since we only have 2 real dimensions, it is easy to find all the
Lagrangians: any curve in $\mathbb{CP}^{1}$ is Lagrangian since $F^{*}\omega$
is always zero. It is also easy to spot the monotone Lagrangians: an embedded
circle $F:S^{1}\to\mathbb{CP}^{1}$ is monotone if and only if $F(S^{1})$
divides the sphere into two pieces of equal area. All monotone Lagrangians are
Hamiltonian isotopic.

The second complex projective space, the complex projective plane
$\mathbb{CP}^{2}$, has Einstein constant $\kappa=6$, which immediately
distinguishes it from the $4$-sphere $S^{4}$. From a symplectic point of view,
$\mathbb{CP}^{2}$ is vastly more complicated than $\mathbb{CP}^{1}$, as
evidenced by the abundance of interesting monotone Lagrangians one can
construct. The first encountered monotone Lagrangian is the Clifford torus
$L_{\text{Cl}}$ given by the projection of the “equator” of the $5$-sphere
$S^{5}\subset\mathbb{C}^{3}$:

$(e^{i\theta_{0}},e^{i\theta_{1}},e^{i\theta_{2}})\mapsto[e^{i\theta_{0}}:e^{i\theta_{1}}:e^{i\theta_{2}}].$

Unlike $\mathbb{CP}^{1}$, there are a great wealth of embedded monotone tori
not Hamiltonian isotopic to $L_{\text{Cl}}$. The first is the Chekanov torus,
discovered in the 90s [schlenk_chekanov_2010]. One way to obtain the Chekanov
torus is from the Clifford torus by a process known as a mutation, which we
now discuss.

####  2.1.1. Vianna’s exotic tori in $\mathbb{CP}^{2}$

Vianna constructs in [Vianna2016] an infinite family $\mathcal{F}$ of monotone
Lagrangian tori in $\mathbb{CP}^{2}$, with no two tori Hamiltonian isotopic.
We present some details of this construction here as it forms the main
motivating example for the rest of the paper.

The first member of the family $\mathcal{F}$ is the Clifford torus

$L_{\text{Cl}}=\\{[x:y:z]:|x|=|y|=|z|=1\\},$

which is realised as the barycentric fibre in the standard toric fibration of
$\mathbb{CP}^{2}$. From here, Vianna constructs the next member of
$\mathcal{F}$ by a topological procedure known as a mutation:

  1. (1)

Introduce a nodal fibre at one of the corners by a nodal trade. The corner of
the base diagram is now a circle and the fibre above the cross is a Lagrangian
torus pinched to create a nodal singularity. The barycentric fibre is still a
Clifford torus, and the metric is still the Fubini–Study metric.

  2. (2)

Rescale a neighbourhood of the line at $\infty$ (i.e. the $\mathbb{CP}^{1}$
given by $\\{[x:y:0]:x,y\in\mathbb{C}\\}$) until the barycentre has passed
over the nodal fibre. The barycentre is now a Chekanov torus and the metric is
no longer the Fubini–Study metric.

  3. (3)

Isotope the metric back to the Fubini–Study metric using Moser’s trick. The
barycentre remains a Chekanov torus.

Items 2 and 3 together are called a nodal slide, and the full mutation is
illustrated in Figure 2.

Figure 2. The mutation procedure. The fibre above the barycentres (red dots)
are Clifford tori on the left-hand side, and Chekanov tori on the right-hand
side.

Vianna then iterates this procedure, introducing new nodal fibres at different
corners of the moment polytope. Since the two corners $(1,0)$ and $(0,1)$ are
the same after the first mutation, the Chekanov torus $L_{\text{Ch}}$ becomes
a unique new torus $L_{(1,4,25)}$. Iterating further from this point generates
two new tori every time. Vianna indexes this family by integer triples
$(a^{2},b^{2},c^{2})$ with $a^{2}+b^{2}+c^{2}=3abc$, which are known as Markov
triples, and shows that a torus $L_{(a^{2},b^{2},c^{2})}$ is realised as the
barycentric fibre of a degeneration of the weighted projective space
$\mathbb{CP}^{2}(a^{2},b^{2},c^{2})$, though we shall not use this perspective
in this paper.

${\cdots}$${L_{(1,25,169)}}$${\cdots}$${L_{\text{Cl}}}$${L_{\text{Ch}}}$${L_{(1,4,25)}}$${\cdots}$${L_{(4,25,841)}}$${\cdots}$
Figure 3. Vianna’s exotic tori, indexed by Markov triples
$(a^{2},b^{2},c^{2})$.

We focus for the rest of the paper only on the first level of this procedure,
which has been known since the work of Chekanov–Schlenk
[schlenk_chekanov_2010].

We can distinguish tori of Clifford-type $L_{\text{Cl}}$ from tori of
Chekanov-type $L_{\text{Ch}}$ by counting $J$-holomorphic disc classes in
$H_{2}(\mathbb{CP}^{2},L)$. Following the results of Auroux [Auroux_2007], we
have that there are 3 classes of Maslov 2 discs with boundary on
$L_{\text{Cl}}$. Denote by $\alpha_{1}$ the disc class $w\mapsto[w:1:1]$, and
by $\alpha_{2}$ the disc class $w\mapsto[1:w:1]$. Then
$H_{2}(\mathbb{CP}^{2},L_{\text{Cl}})$ is generated by
$\\{\alpha_{1},\alpha_{2},Q-\alpha_{1}-\alpha_{2}\\}$ where
$Q=[\mathbb{CP}^{1}]$ is the hyperplane class.

On the other hand, consider the Chekanov torus in
$\mathbb{C}^{2}=\mathbb{CP}^{2}-\\{z=0\\}$ given by

$L_{\text{Ch}}=\left\\{\left(\gamma(s)e^{i\alpha},\gamma(s)e^{-i\alpha}\right):s,\alpha\in\mathbb{R}\right\\},$

where $\gamma(s)\in\mathbb{C}$ is an embedded closed curve not enclosing the
origin. By an abuse of notation, let $\alpha$ denote the disc class of a disc
with boundary given by the $\alpha$ coordinate, and by $\beta$ the disc class
given by the $s$ coordinate. Then $H_{2}(\mathbb{CP}^{2},L_{\text{Ch}})$ is
generated by $\\{\alpha,\beta,Q\\}$. However, the class $\alpha$ does not
contain any holomorphic representatives - this is precisely the same reason
the corresponding class can collapse for the Clifford torus $L_{\text{Cl}}$
under mean curvature flow in $\mathbb{C}^{2}$ as demonstrated in [Evans2018].
In fact, the Maslov 2 classes on $L_{\text{Ch}}$ are precisely

$\left\\{\beta,Q-2\beta+\alpha,Q-2\beta,Q-2\beta-\alpha\right\\},$

each occurring with moduli space of holomorphic discs of dimension 1 except
for the $Q-2\beta$ class which has dimension 2.

###  2.2. Lagrangian mean curvature flow

It is a fact first proven by Smoczyk [Smoczyk1996] that the gradient-descent
flow of area, i.e. mean curvature flow, preserves the Lagrangian condition in
Kähler–Einstein manifolds. This gives rise to Lagrangian mean curvature flow.
The preservation of the Lagrangian condition is no coincidence; for a
Lagrangian in a Kähler–Einstein manifold, the mean curvature $\vec{H}$ can be
written as a closed $1$-form $H$ on $L$, making use of the isomorphism between
the tangent, normal and cotangent bundles of a Lagrangian given by the Kähler
condition.

Within the Lagrangian class, there are further preserved conditions. If the
ambient manifold is Calabi–Yau, the most important preserved types are zero-
Maslov and almost-calibrated, which have attracted a great deal of interest.
For Fano manifolds, the monotone condition is the most important preserved
quantity (we provide an elementary proof that the monotone condition is
preserved in Section 2.5.)

Singularities of the flow occur at times $T$ when $|A|^{2}\to\infty$ as $t\to
T$, where $A$ is the second fundamental form. As in hypersurface mean
curvature flow, singularities for Lagrangian mean curvature flow can be
divided into two types. A type I singularity occurs when the rate at which
$|A|^{2}$ blows up is at most parabolic in time. All other singularities are
of type II. As a general principle, the important preserved classes mentioned
above do not have type I singularities; this statement is due to Wang
[Wang2001] in the Calabi–Yau case. We prove that monotone Lagrangians in Fano
manifolds do not attain type I singularities in Section 3.

As an example, consider the case of Lagrangians in $\mathbb{CP}^{1}=S^{2}$. It
is well-known that an embedded circle has a finite-time singularity if and
only if it is not monotone. In this case, the singularity is of type I. On the
other hand, monotone Lagrangians in $\mathbb{CP}^{1}$, i.e. circles dividing
the sphere into equal area pieces, exist for all time under mean curvature
flow and converge to a geodesic equator.

Results of the latter type are known as Thomas–Yau-type results. The general
idea is that the important preserved classes should exist for all time and
converge to minimal Lagrangians. However, this is known to be false since for
higher-dimensional examples, singularities are inevitable. Finite-time
singularities are unavoidable, hence attention has turned to methods to
resolve finite-time singularities via surgery in order to continue the flow.
Geometric flows with surgery are well-studied, for instance the ground-
breaking work of Perelman [Perelman2002] on Ricci flow with surgery, or the
work of Huisken–Sinistrarri [Huisken2009] on mean convex mean curvature flow
with surgery. However, without proper understanding of singularity formation,
it is impossible to perform surgery. This has presented the most difficult
obstacle to defining a Lagrangian mean curvature flow with surgery.

We note one final property of singularities of mean curvature flow. In
general, geometric flows may have infinite-time singularities; indeed, this
occurs in Ricci flow and in Yang–Mills flow, amongst others. In mean curvature
flow however, one can rule out infinite-time singularities in certain cases.
For simplicity, we state the following result of Chen–He [chen_he_2010] as we
need it in this paper, though it applies in a far wider generality.

######  Proposition 2.1.

Let $L$ be a Lagrangian mean curvature flow of a compact Lagrangian in a
compact Kähler–Einstein manifold $M$ with $\kappa>0$. Then $L$ either attains
a finite-time singularity or has uniformly bounded $|A|^{2}$ for all time and
converges subsequentially to a minimal Lagrangian submanifold in $M$ in
infinite time.

###  2.3. Holomorphic volume forms and the Lagrangian angle

Let $\Omega$ be a holomorphic volume form defining a Lagrangian angle by
$\Omega_{L}=e^{i\theta}\operatorname{vol}_{L}$. If $\Omega$ is parallel, as is
the case in Calabi–Yau manifolds, we have the following important result:

######  Proposition 2.2.

Let $L$ be an oriented Lagrangian in a Calabi–Yau manifold $M$ with mean
curvature 1-form $H$, where $H(X)=\omega(X,\vec{H})$. Then $H=d\theta$, where
$\theta$ is the Lagrangian angle.

###### Proof.

Taking the ambient gradient $\overline{\nabla}_{X}$ of $\Omega$ with respect
to any tangent vector $X$ yields the equality:

$\overline{\nabla}_{X}\Omega=-id\theta(X)\cdot\Omega+iH(X).$

We refer the reader to Thomas–Yau [Thomas2002] for the full calculation, which
is first attributed to Oh [Oh1994]. Since $\Omega$ is parallel, the result
follows. ∎

Note that the above proof implies that for any holomorphic volume form
$\Omega$ defining a Lagrangian angle by
$\Omega_{L}=e^{i\theta}\operatorname{vol}_{L}$, we have

$H(X)\cdot\Omega=d\theta(X)\cdot\Omega-i\nabla_{X}\Omega$

for any $X\in T_{p}L$, even without the parallel condition.

Let $\\{L_{\alpha}\\}_{\alpha\in I}$ be a Lagrangian torus fibration of a
subset $U\in M$ of a Kähler–Einstein manifold. For each $\alpha$, define a
holomorphic volume form $\Omega_{L_{\alpha}}$ along $L_{\alpha}$, i.e. a unit
section of the canonical bundle $K_{M|_{L_{\alpha}}}$, by

$\displaystyle\Omega_{L}(X_{1},\dots,X_{n})$
$\displaystyle=\operatorname{vol}_{L}(X_{1},\dots,X_{n}),$
$\displaystyle\Omega_{L}(JX_{1},X_{2},\dots,X_{n})$
$\displaystyle=i\operatorname{vol}_{L}(X_{1},\dots,X_{n}),\quad\text{etc.}$

for tangent vectors $X_{i}\in T_{p}L$. Now let $x\in U$. There is a unique
$\alpha(x)\in I$ such that $x\in L_{\alpha(x)}$, so we define a section of
$K_{M|_{U}}$ by

$\Omega(x)(X_{1},\dots,X_{n})=\Omega_{L_{\alpha(x)}}(X_{1},\dots,X_{n}),$

for $X_{i}\in T_{x}M$. We call $\Omega$ a relative holomorphic volume form (to
the fibration $L_{\alpha}$).

In contrast to the Calabi–Yau case where $\Omega$ was always parallel, the
volume form defined here is in general not parallel. In [Lotay2018], the form
$\Omega_{L}$ is differentiated in tangent and normal directions. For tangent
vector fields $X\in\Gamma(TL)$ we have

$\overline{\nabla}_{X}\Omega_{L}=iH_{L}(X)\,\Omega_{L}$ (2.1)

where $H_{L}$ is the mean curvature 1-form on $L$. On the other hand, if
$JY=\frac{\partial A_{s}}{\partial s}|_{s=0}$ is the normal vector field
corresponding to a 1-parameter family $A_{s}:L\to M$ of Lagrangians immersions
then the normal derivative is

$\overline{\nabla}_{JY}\Omega_{L}=-i\operatorname{div}_{L}(Y)\,\Omega_{L}.$
(2.2)

Now suppose the fibration $\\{L_{\alpha}\\}$ are the level sets of a moment
map for an isometric Hamiltonian $T^{n}$-action on $U$. Since the action is an
isometry, any vector field $\tilde{X}$ generated by the subgroup $\exp(tX)$ of
$T^{n}$ has
$\mathcal{L}_{\tilde{X}}\operatorname{vol}_{L_{\alpha}}=\operatorname{div}_{L_{\alpha}}(\tilde{X})\operatorname{vol}_{L_{\alpha}}=0$.
Furthermore, since the action is Hamiltonian, $J\tilde{X}$ is a normal vector
field corresponding to a 1-parameter family of Lagrangian immersions. So we
have shown the following:

######  Theorem 2.3.

Let $\Omega$ be a holomorphic volume form on an open subset $U\subset M$ of a
Kähler–Einstein manifold. If $L$ is a Lagrangian in $U$, then for any $X\in
T_{p}L$ we have

$H(X)\cdot\Omega=d\theta(X)\cdot\Omega-i\nabla_{X}\Omega,$

where $H$ is the mean curvature 1-form and $\theta$ is the Lagrangian angle of
$L$ with respect to $\Omega$.

Suppose now that $M$ is an isometric toric manifold, that is to say there is
an isometric Hamiltonian action of $T^{n}$ on $M^{2n}$. Away from the singular
points of the action, the level sets $\\{L_{\alpha}\\}$ are a Lagrangian
fibration, and we can define $\Omega$ such that $\theta(L_{\alpha})=0$ for all
$\alpha$. Then for any $X=Y+JZ$ with $Y,Z\in T_{p}L$ we have

$H(X)=d\theta(X)+H_{L_{\alpha}(p)}(Y),$

where $H_{L_{\alpha}(p)}$ is the mean curvature 1-form of the unique
Lagrangian $L_{\alpha}$ passing through $p$.

This allows us to relate the curvature of a Lagrangian to the curvature of a
fibration via the Lagrangian angle. Choosing a fibration with easily
calculable curvature, this will vastly simplify the calculation of mean
curvature. We use this technique in Section 4.2.

###  2.4. Evolution equations

The following calculation appears in Thomas–Yau [Thomas2002], but the results
were known by Oh [Oh1994] and Smoczyk [Smoczyk1999]. For any holomorphic
volume form $\Omega$ defining a Lagrangian angle by
$\Omega_{L}=e^{i\theta}\operatorname{vol}_{L}$, we have

$\displaystyle i\frac{\partial}{\partial t}\theta
e^{i\theta}\operatorname{vol}_{L}+e^{i\theta}\frac{\partial}{\partial
t}\operatorname{vol}_{L}=$ $\displaystyle\frac{\partial}{\partial
t}\left(e^{i\theta}\operatorname{vol}_{L}\right)=\mathcal{L}_{V}\Omega_{L}=d\left(V\lrcorner\Omega_{L}\right)=-i\,d\left(e^{i\theta}(JV)\lrcorner\operatorname{vol}_{L}\right)$
$\displaystyle=$ $\displaystyle
e^{i\theta}d\theta\wedge\left(JV\lrcorner\operatorname{vol}_{L}\right)-ie^{i\theta}d^{\dagger}(JV)\operatorname{vol}_{L}$

In Calabi–Yau manifolds, we have that $\vec{H}=J\nabla\theta$, and hence under
mean curvature flow where $V=\vec{H}$ we have the evolution equations

$\begin{split}\frac{\partial}{\partial
t}\theta&=d^{\dagger}d\theta=\Delta\theta\\\ \frac{\partial}{\partial
t}\operatorname{vol}_{L}&=-|\vec{H}|^{2}\operatorname{vol}_{L}\,.\end{split}$
(2.3)

The mean curvature 1-form $H$ satisfies the evolution equation

$\frac{\partial}{\partial t}H=dd^{\dagger}H+\kappa H,$ (2.4)

where $\kappa$ is the Einstein constant, i.e. $\rho=\kappa\omega$. It is clear
then that the cohomology class $[He^{-\kappa t}]$ is preserved under the flow.
In particular, $H$ exact is preserved.

###  2.5. The Cieliebak–Goldstein theorem

Recall that the space of Lagrangian subspaces $\mathcal{L}(n)$ in
$\mathbb{R}^{2n}$ is isomorphic to $U(n)/O(n)$, and hence $\det^{2}$ induces
an isomorphism from $\mu:\pi_{1}(\mathcal{L}(n))\to\mathbb{Z}$, called the
Maslov index. The Maslov class of a disc is defined to be the Maslov index of
the boundary under any local trivialisation. Then we have the following
theorem of Cieliebak–Goldstein [Cieliebak2004], which is fundamental to the
rest of this paper:

######  Theorem 2.4 (Cieliebak–Goldstein).

In a Kähler–Einstein manifold $M$ with Einstein constant $\kappa$, the mean
curvature 1-form $H$ of a Lagrangian $F:L\to M$ is related to the Maslov class
$\mu$ of a disc $u:(D,\partial D)\to(M,L)$ by 111Here and throughout the rest
of this paper, we abuse notation by conflating forms with their pullbacks and
curves in the image of a Lagrangian with their pre-image. For instance, in
(2.5),  $\int_{D}\omega=\int_{D}u^{*}\omega,\quad\int_{\partial
D}H=\int_{F^{-1}(u(\partial D))}H.$

$\kappa\int_{D}\omega-\pi\mu(D)=-\int_{\partial D}H.$ (2.5)

We call a Lagrangian submanifold monotone if for any disc $u:(D,\partial
D)\to(M,L)$,

$\int_{D}\omega=c\mu(D),$ (2.6)

for a constant $c$ dependent on $M$ and $L$ but not $u$. We call a disc $u$
Maslov $m$ if the $\mu(u)=m$. In the case of a monotone Lagrangian in an exact
Calabi–Yau manifold (i.e. $\omega=d\lambda$), and in view of (2.5), we see
that (2.6) is equivalent to

$\int_{\partial D}\lambda=\int_{D}\omega=c\mu(D)=\frac{c}{\pi}\int_{\partial
D}H=\frac{c}{\pi}\int_{\partial D}d\theta,$

hence in the literature for Lagrangian mean curvature flow where the
Calabi–Yau case (specifically $\mathbb{C}^{n}$) is frequently the primary
focus, the definition of monotone is often taken as

$[\lambda]=C[d\theta].$

######  Remark 2.5.

The Cieliebak–Goldstein formula is a generalisation of the Gauss-Bonnet
formula. When $M$ is a surface, $\omega$ is the Riemannian volume form on $M$,
and so $M$ Einstein implies that

$\int_{D}K\operatorname{vol}(D)=\kappa\int_{D}\omega,$

where $K$ is the Gauss curvature. Moreover, all curves are Lagrangian so

$\int_{\partial D}k_{g}\operatorname{vol(\partial D)}=\int_{\partial D}H$

where $k$ is the geodesic curvature. The Euler characteristic of a disc is 1,
and the Maslov class of a holomorphic disc in a symplectic surface is 1 by
definition.

The above remark helps to motivate a mild generalisation of the
Cieliebak–Goldstein formula to include $J$-holomorphic polygons with boundary
on multiple intersecting Lagrangians, comparable to generalising Gauss–Bonnet
with a smooth boundary to a piecewise-smooth boundary with corners and turning
angles.

######  Theorem 2.6.

Let $L_{1},\dots,L_{m}$ be Lagrangian in $M$ and let

$u:(D,(\partial D_{1},\dots,\partial D_{m}))\to(M,(L_{1},\dots,L_{m}))$

denote a map from the unit disc with $m$ marked points $p_{i}$ on the boundary
to $M$, mapping $p_{i}$ to $L_{i}\cap L_{i-1}$ and mapping the arc $\partial
D_{i}$ from $p_{i}$ to $p_{i+1}$ to $L_{i}$. Then

$\kappa\int_{D}\omega-\pi\tilde{\mu}(D)=-\sum_{i}\int_{\partial
D_{i}}H_{L_{i}},$ (2.7)

where $\tilde{\mu}$ is the Maslov class of $u:(D,(\partial
D_{1},\dots,\partial D_{m}))\to(M,(L_{1},\dots,L_{m}))$, defined in the proof
below.

###### Proof.

The proof is equivalent to the generalisation of the Gauss–Bonnet formula from
surfaces without corners to surfaces with corners. We refer the reader to
[Evans_2022] for a complete description. ∎

Let us now consider the implications of the Cieliebak–Goldstein formula for
Lagrangian mean curvature flow. From (2.5) and the evolution equation (2.4) we
obtain

$\begin{split}\frac{\partial}{\partial
t}\int_{D}\omega=-\frac{1}{\kappa}\int_{\partial D}dd^{\dagger}H+\kappa
H&=-\int_{\partial D}H\\\ &=\kappa\int_{\partial
D}\omega-\pi\mu(D).\end{split}$ (2.8)

$L$ is monotone when $\mu(D)$ is proportional to $\int_{D}\omega$, so we note
two immediate corollaries for $\kappa\neq 0$.

######  Corollary 2.7.

Let $L$ be a Lagrangian in a Kähler–Einstein manifold with $\kappa\neq 0$. $H$
is exact if and only if $L$ is monotone with monotone constant $\pi/\kappa$.

######  Corollary 2.8.

Monotone Lagrangians are preserved under mean curvature flow. When $\kappa\neq
0$, the monotone constant $\pi/\kappa$ is invariant under the flow.

###### Proof.

The result has been shown already for $\kappa=0$. For $\kappa\neq 0$, the
result follows from Corollary 2.7 and the fact that exactness of $H$ is
preserved by equation (2.4). ∎

To illustrate the theory so far, we consider the best understood example of
Lagrangian mean curvature flow in non-Ricci-flat manifolds.

Consider the two sphere $S^{2}=\mathbb{CP}^{1}$ with the standard Kähler
metric and let $\gamma$ be an embedded closed curve in $S^{2}$. Then there
are, up to reparametrisation, exactly two $J$-holomorphic discs
$u_{1},u_{2}:D\to S^{2}$ with $u_{i}(\partial D)=\gamma$. We have that
$\gamma$ is monotone when

$\int_{D}u_{1}^{*}\omega=\int_{D}u_{2}^{*}\omega,$

where $\omega$ is the standard Fubini–Study form on $\mathbb{CP}^{1}=S^{2}$,
i.e. when $\gamma$ divides $S^{2}$ into two pieces of equal area. Then we have
two behaviours:

######  Proposition 2.9.

  1. (1)

If $\gamma$ is not monotone, $\gamma$ attains a type I singularity in finite
time with blow-up a self-shrinking circle.

  2. (2)

If $\gamma$ is monotone, mean curvature flow exists for all time and converges
in infinite time to a great circle.

###### Proof.

Recall Grayson’s theorem [grayson_1989]: curve-shortening flow in surfaces
either attains finite-time singularities with type I blow-up a shrinking
circle, or exists for all times and converges to a geodesic. This the result
follows from (2.8) in both cases. ∎

##  3\. Type I singularities in Fano manifolds

We saw that monotone curves did not attain type I singularities:
heuristically, any type I singularity would require the collapsing of one of
the disc classes, which is prohibited by the monotone condition. We now
generalise this to higher dimensions. First, we can classify all zero-Maslov
self-shrinkers that may arise as a type I blow-up by a result of
Groh–Schwarz–Smoczyk–Zehmisch [Groh2007]:

######  Theorem 3.1.

If $F:L^{n}\to\mathbb{C}^{n}$ is a zero-Maslov Lagrangian self-shrinker
arising as a result of a type I blow-up, then $L$ is a minimal Lagrangian
cone.

This follows directly from [Groh2007, Theorem 1.9], noting that type I blow-
ups have bounded area ratios.

Since type I blow-ups are smooth, embedded self-shrinkers for type I
singularities, this implies there are no zero-Maslov type I blow-ups for type
I singularities. Since any type I model is locally symplectomorphic to the
standard unit ball, this excludes the possibility of type I singularities for
monotone Lagrangians:

######  Theorem 3.2.

Let $F_{t}:L^{n}\to M^{2n}$ be a monotone Lagrangian mean curvature flow,
$\kappa\neq 0$. Then $F_{t}$ does not attain any type I singularities.

###### Proof.

Suppose for a contradiction that $F_{t}$ attains a type I singularity at time
$T$. Any sequence $\eta_{i}\to\infty$ subsequentially defines a type I blow-up

$\tilde{F}_{s}:=\lim_{i\to\infty}\tilde{F}^{\eta_{i}}_{s}=\lim_{i\to\infty}\eta_{i}F_{T+\eta_{i}^{-2}s}.$

Since the singularity is type I, $\tilde{F}(L):=\tilde{F}_{-1}(L)$ is a non-
planar embedded Lagrangian self-shrinker, and hence by Theorem 3.1 has non-
zero Maslov class.

Let $\tilde{D}\in H_{2}(\mathbb{C}^{n},\tilde{F}(L))$ have $\mu(\tilde{D})>0$.
The convergence of $\tilde{F}_{s}$ to a type I blow-up is smooth and the
Maslov class is topological, so for all sufficiently large $i$, there exists
$\tilde{D}_{i}\in H_{2}(\mathbb{C}^{n},\tilde{F}_{-1}^{\eta_{i}})$ with
$\mu(\tilde{D}_{i})=\mu(\tilde{D})>0$ and $\tilde{D}_{i}\to\tilde{D}$ as
$i\to\infty$. Furthermore, $\tilde{D}^{i}$ are the images under the parabolic
rescaling of discs
$D_{i}=\eta_{i}^{-1}\tilde{D}_{i}\in\pi_{2}(W,F_{T-\eta_{i}^{-2}}(L))$. Since
$L$ is monotone and the Maslov class is invariant under rescaling,

$\int_{D_{i}}\omega=\frac{\pi}{\kappa}\mu(\tilde{D}_{i})=\frac{\pi}{\kappa}\mu(\tilde{D})>0$

for all $i$, but

$\lim_{i\to\infty}\int_{D_{i}}\omega=\lim_{i\to\infty}\int_{\eta_{i}^{-1}\tilde{D}_{i}}\omega=0,$

a contradiction. ∎

This theorem is the positive curvature equivalent of the result of Wang
[Wang2001] showing that almost-calibrated Lagrangians do not attain type I
singularities in Calabi–Yau manifolds. This strengthens the perspective that
monotone submanifolds are the correct class of submanifolds to study to find
positive curvature analogues of the Thomas–Yau conjecture. The rest of the
paper will be devoted to exploring what a Thomas–Yau conjecture looks like in
the prototypical Fano surface $\mathbb{CP}^{2}$.

##  4\. Equivariant Lagrangians in $\mathbb{CP}^{2}$

###  4.1. Clifford and Chekanov tori in Lefschetz fibrations

Our goal is to study the behaviour of Clifford and Chekanov tori in
$\mathbb{CP}^{2}$ under mean curvature flow, but this presents a number of
difficulties. The main problem is the class of potential singularities is too
great. Heuristically, singular behaviour is local and since $\mathbb{CP}^{2}$
looks flat on sufficiently small scales, we expect that a priori any singular
behaviour observed for zero-Maslov Lagrangians in $\mathbb{C}^{2}$ should also
occur for monotone Lagrangians in $\mathbb{CP}^{2}$. In particular, zero-
object singularities222As a note on the terminology: The obvious surgery at
such a singularity bubbles off an immersed Lagrangian sphere with a single
transverse self-intersection. Since such a sphere represents a zero object in
the Fukaya category, it seems sensible to call these singularities which are
collapsing zero-homotopic curves zero-object singularities. See Joyce
[joyce_2015, Section 3.7] for a more detailed description. like those studied
in Neves [Neves2007, Figure 3] (Figure 4) can occur and currently we have
little understanding about the nature of these singularities. A second issue
is that there is no control over where the singularity happens and what
Lagrangian cone the type I blow-up produces, even under the assumption that we
obtain Lawlor neck singularities. Since these are general problems in
Lagrangian mean curvature flow, we choose a symmetric subclass of Lagrangians
in $\mathbb{CP}^{2}$ which cannot have the zero-object singularities and where
we have strong control over the location and type of the singularities.

Figure 4. An example of a zero-object singularity - A Lagrangian plane
attaining a type II singularity.

We consider two rational maps $\mathbb{CP}^{2}\to\mathbb{CP}^{1}$. The first
is the Lefschetz fibration

$f([x:y:z])=[xy:z^{2}]$

in the complement of the anti-canonical divisor $D=\\{(xy-z^{2})z=0\\}$. The
second is the projection

$\pi([x:y:z])=[y:z].$

This extends to a foliation of $\mathbb{CP}^{2}$ by holomorphic spheres each
intersecting at a single point $[0:0:1]$ with intersection number 1.

We call a subset $U\subset\mathbb{C}$ point-symmetric if $x\in U$ if and only
if $-x\in U$. For a point-symmetric curve $\gamma(s)\in\mathbb{C}$, define

$L_{\gamma}^{0}=\left\\{\left[\gamma(s)e^{i\alpha}:\gamma(s)e^{-i\alpha}:1\right]\,\big{|}\;\alpha\in\mathbb{R},s\in\mathbb{R}\right\\}$

and notice that since $\gamma$ is point-symmetric,
$f(L_{\gamma}^{0})=\\{[\gamma(s)^{2}:1]:s\in\mathbb{R}\\}$ is an embedded
curve in $\mathbb{CP}^{1}$ if $\gamma(s)$ is embedded in $\mathbb{C}$. We will
also allow unions of two smooth non-intersecting curves such that the union is
point-symmetric. By an abuse of notation, we refer to such a curve as
$\gamma(s)$ where the parameter $s$ is now allowed to vary over two intervals
or circles.

First, we identify various Lagrangians in this format. Let
$\gamma(s)=s\in\mathbb{R}\subset\mathbb{C}$. Then

$\\{[se^{i\alpha}:se^{-i\alpha}:1]\\}$

lies above $\gamma$ and is compactified by the circle
$[e^{i\alpha}:e^{-i\alpha}:0]$ at infinity. The resulting manifold is

$L_{\mathbb{R}}^{0}:=\\{[se^{i\alpha}:se^{-i\alpha}:1]\\}\cup\\{[e^{i\alpha}:e^{-i\alpha}:0]\\}=\\{[1:e^{-2i\alpha}:re^{-i\alpha}]\\}\cup\\{[0:0:1]\\}$

or equivalently, using the substitution $s=\cot\phi$,

$L_{\mathbb{R}}^{0}=\\{[\cos\phi e^{i\alpha}:\cos\phi
e^{-i\alpha}:\sin\phi]:\phi\in[0,\pi/2],\alpha\in\mathbb{R}\\}.$

Note that $L_{\mathbb{R}}^{0}$ is fixed under the anti-symplectic involution
$X:[x:y:z]\mapsto[\bar{y}:\bar{x}:\bar{z}]$, hence is isomorphic to
$\mathbb{RP}^{2}$. The same applies for any other line through the origin in
$\mathbb{C}$.

The curve $\gamma_{r}(s)=re^{is}$ lifts to a Lagrangian torus of Clifford-
type, which is monotone and minimal if and only if $r=1$. Furthermore, any
point-symmetric closed curve enclosing the origin lifts to a torus of
Clifford-type, monotone if and only if the symplectic area contained is equal
to $4\pi/6=2\pi/3.$ This follows from the Cieliebak–Goldstein formula (2.5),
$\kappa=6$ and the fact that the disc is Maslov 4. Any closed circle $\gamma$
not enclosing the origin and its point-symmetric image $-\gamma$ together lift
to a torus of Chekanov-type (provided $\gamma$ does not intersect $-\gamma$),
monotone if and only if the area contained is $2\pi/6=\pi/3$. The fact that
these Lagrangians are Clifford and Chekanov respectively can be checked by
observing their images under the Lefschetz fibration $f$ and comparing with
the standard definitions in Auroux, for instance, [Auroux_2007].

We will distinguish between Clifford tori and Chekanov tori by their
intersections with real projective planes $\mathbb{RP}^{2}$. Immediately we
observe that any closed curve $\gamma$ enclosing the origin intersects any
line $l$ through the origin in at least two points, hence any equivariant
Clifford torus $L_{\gamma}\cong L_{\text{Cl}}$ intersects
$L_{l}\cong\mathbb{RP}^{2}$ in at least one circle. Indeed, this result is
generalisable: $L_{\text{Cl}}$ is non-displaceable from $\mathbb{RP}^{2}$, as
can be shown in multiple different ways (see for instance [biran_cornea_2009]
or [entov_polterovich_2009]). Indeed, Amorim and Alston [alston_amorim_2011]
give a lower bound of 2 for the number of intersections between a Clifford
torus and $\mathbb{RP}^{2}$. On the other hand, one can easily observe that
there exists a pair of point-symmetric circles $\gamma(s)\in\mathbb{C}$ each
containing a disc of area 2 and not intersecting the imaginary axis
$i\mathbb{R}\in\mathbb{C}$. Hence Chekanov tori are displaceable from
$\mathbb{RP}^{2}$.

In the sequel, it will be useful to consider cones of real projective planes
intersecting our flowing Lagrangian tori, so we make the following definition:

######  Definition 4.1.

Denote by $l_{b}$ the line $\\{se^{ib}:s\in\mathbb{R}\\}\subset\mathbb{C}$.
For $a\in(0,\pi)$, let $C^{b}_{a}$ be a cone of opening angle $a$ about
$l_{b}$, i.e. the union of $l_{b-a/2}$ and $l_{b+a/2}$. We say that a point-
symmetric pair of closed curves $\gamma$ is contained in $C^{b}_{a}$ if
$\arg(\gamma(s))\in(b-a/2,b+a/2)\cup(-b-a/2,-b+a/2)$ for all $s$.

Finally, we define the symmetry condition we will be using.

######  Definition 4.2.

A Lagrangian $L_{\gamma}$ is called equivariant if $\gamma$ is point-symmetric
and $\mathbb{Z}_{2}$-symmetric with respect to the real axis.

The point-symmetry is an $S^{1}$-symmetry on the level of $L_{\gamma}$, so the
equivariance considered is an $(S^{1}\times\mathbb{Z}_{2})$-symmetry. The main
reason for this symmetry condition is to greatly restrict the variety of
singularities that can occur. Specifically, we want to have only Lawlor neck
singularities occurring at the origin, with type I blow-up given by
$C^{0}_{\pi/2}$. We shall see how the equivariance gives this in Section 5.2.

###  4.2. Mean curvature of equivariant Lagrangians

Before proceeding to the proofs of the main theorems, we calculate the
evolution equation satisfied by the profile curve $\gamma$ under mean
curvature flow. Despite being the governing equation for the rest of the
results in the paper, we do not need the precise formulation frequently: it is
only necessary for the explicit construction of various barriers. However, the
derivation of the evolution equation for $\gamma$ is interesting in its own
right since we calculate the mean curvature of $L_{\gamma}$ by a novel method.

Recall the fibration $\\{L_{\alpha}\\}$ by Clifford-type tori given by the
fibres of the moment map

$\mu([x:y:z])=\frac{1}{|x|^{2}+|y|^{2}+|z|^{2}}\left(|x|^{2},|y|^{2}\right).$

The equivariant fibres are

$L_{r}=\\{L_{re^{i\phi}}:r>0\\}$

and for the rest of this paper, we denote by $\Omega$ the holomorphic volume
form relative to $\\{L_{\alpha}\\}$. We first calculate the mean curvature of
$L_{r}$, then we calculate the mean curvature of any other equivariant torus
$L_{\gamma}$ by calculating the relative Lagrangian angle between $L_{\gamma}$
and $L_{r}$ using Theorem 2.3. Recall that Theorem 2.3 implies that the
relative Lagrangian angle $\theta=\theta_{\operatorname{rel}}$ defined by
$\Omega$ satisfies

$H_{L_{\gamma}}(X)=d\theta_{\operatorname{rel}}(X)+H_{L_{r}}(\pi X)$

where $\pi$ is the projection onto the tangent bundle of $L_{r}$.

We calculate the mean curvature 1-form of Clifford tori $L_{r}$ indirectly.
The curve $\gamma(s)=re^{is}$ bounds a $J$-holomorphic disc which lifts to
$\mathbb{CP}^{2}$ giving a disc

$u:z\mapsto[rz:rz:1]$

with boundary on $L_{r}$ of Maslov index 4. Cieliebak–Goldstein gives

$-\int_{\partial D}H_{L_{r}}=6\int_{D}\omega-4\pi$

since $\kappa=6$ for $\mathbb{CP}^{2}$ with the Fubini–Study metric. We
calculate $\int_{D}\omega$ directly. We have that in radial coordinates
$x=r_{1}e^{i\theta_{1}}$, $y=r_{2}e^{i\theta_{2}}$, the Kähler form is

$\displaystyle\omega=\frac{1}{\left(1+r_{1}^{2}+r_{2}^{2}\right)^{2}}\Big{(}$
$\displaystyle r_{1}(1+r_{2}^{2})dr_{1}\wedge
d\theta_{1}-r_{1}r_{2}^{2}dr_{1}\wedge d\theta_{2}$ $\displaystyle-
r_{1}^{2}r_{2}dr_{2}\wedge d\theta_{1}+r_{2}(1+r_{1}^{2})dr_{2}\wedge
d\theta_{2}\Big{)},$

so

$\int_{D}\omega=2\pi\int_{0}^{r}\frac{2\tilde{r}}{(1+2\tilde{r}^{2})^{2}}\,d\tilde{r}=\pi\frac{2r^{2}}{1+2r^{2}}.$
(4.1)

Hence using Cieliebak–Goldstein, we have

$-\int_{\partial
D}H_{L_{r}}=6\pi\frac{2r^{2}}{1+2r^{2}}-4\pi=4\pi\left(\frac{r^{2}-1}{1+2r^{2}}\right).$

Note that $r=1$ is the monotone flat Clifford torus. Then by the symmetry of
the tori $L_{r}$, we have that

$H_{L_{r}}=-2\left(\frac{r^{2}-1}{1+2r^{2}}\right)ds$ (4.2)

as a 1-form on $L_{r}$.

Next we calculate the relative Lagrangian angle. If
$\gamma(s)=r(s)e^{i\phi(s)}$, then $L_{\gamma}$ is given by the embedding

$F_{\gamma}:(s,\alpha)\to\left[r(s)e^{i\phi(s)}e^{i\alpha}:r(s)e^{i\phi(s)}e^{-i\alpha}:1\right].$

Identifying the tangent space of $\mathbb{CP}^{2}$ in the coordinate patch
where $z=1$ with $\mathbb{C}^{2}$ in the obvious way, we find that

$\omega\left(\frac{\partial F_{\gamma}}{\partial s},\frac{\partial
F_{\gamma}}{\partial\alpha}\right)=\omega\left(\partial_{r_{1}}+\partial_{r_{2}},\partial_{\theta_{1}}-\partial_{\theta_{2}}\right)=0,$

which verifies that $L_{\gamma}$ is Lagrangian, and furthermore, we have

$\Omega_{L_{r}}\left(\frac{\partial F_{\gamma}}{\partial s},\frac{\partial
F_{\gamma}}{\partial\alpha}\right)=\Omega_{L_{r}}\left(-r^{-1}r^{\prime}J\partial_{\phi}+\phi^{\prime}\partial_{\phi},\frac{\partial
F_{\gamma}}{\partial\alpha}\right),$

where $\partial_{\phi}=\partial_{\theta_{1}}+\partial_{\theta_{2}}$ and we
have used $J\partial_{\theta_{i}}=-r_{i}\partial_{r_{i}}$. Since
$\frac{\partial F_{\gamma}}{\partial\alpha}$ is tangent to $L_{r}$, the
Lagrangian angle $\theta$ relative to $\Omega_{L_{r}}$ is given by

$\theta=\arg\left(\phi^{\prime}-ir^{\prime}r^{-1}\right)=-\tan^{-1}\left(\frac{r^{\prime}}{r\phi^{\prime}}\right)$

and hence

$d\theta=\frac{-r^{\prime\prime}r\phi^{\prime}+r^{\prime
2}\phi^{\prime}+r^{\prime}r\phi^{\prime\prime}}{r^{\prime 2}+r^{2}\phi^{\prime
2}}ds.$ (4.3)

But the Euclidean planar curvature $k$ of $\gamma$ is

$\begin{split}k&=\frac{-r^{\prime\prime}r\phi^{\prime}+2r^{\prime
2}\phi^{\prime}+r^{\prime}r\phi^{\prime\prime}+r^{2}\phi^{\prime
3}}{\left(r^{\prime 2}+r^{2}\phi^{\prime 2}\right)^{3/2}}\\\
&=\left(\frac{-r^{\prime\prime}r\phi^{\prime}+r^{\prime
2}\phi^{\prime}+r^{\prime}r\phi^{\prime\prime}}{\left(r^{\prime
2}+r^{2}\phi^{\prime 2}\right)}+\phi^{\prime}\right)\frac{1}{\sqrt{r^{\prime
2}+r^{2}\phi^{\prime 2}}}\end{split}$ (4.4)

We have that the projection of $\frac{\partial F_{\gamma}}{\partial s}$ onto
$L_{r}$ is

$\pi\left(\frac{\partial F_{\gamma}}{\partial
s}\right)=\frac{\omega\left(\frac{\partial F_{\gamma}}{\partial
s},J\frac{\partial F_{r}}{\partial s}\right)}{\omega\left(\frac{\partial
F_{r}}{\partial s},J\frac{\partial F_{r}}{\partial s}\right)}\frac{\partial
F_{r}}{\partial s}=\phi^{\prime}\frac{\partial F_{r}}{\partial s}$

so we are led to conclude that

$H_{L_{r}}\left(\pi\left(\frac{\partial F_{\gamma}}{\partial
s}\right)\right)=-2\left(\frac{r^{2}-1}{1+2r^{2}}\right)\phi^{\prime}.$ (4.5)

Combining the above equations, we obtain

$\displaystyle H_{L_{\gamma}}=d\theta+H_{L_{r}}(\pi(\cdot))$
$\displaystyle=\left(k\sqrt{r^{\prime 2}+r^{2}\phi^{\prime
2}}-\phi^{\prime}-2\left(\frac{r^{2}-1}{1+2r^{2}}\right)\phi^{\prime}\right)\,ds$
$\displaystyle=\left(k\sqrt{r^{\prime 2}+r^{2}\phi^{\prime
2}}+\left(\frac{1-4r^{2}}{1+2r^{2}}\right)\phi^{\prime}\right)\,ds.$

Hence we have that

$\omega\left(\frac{\partial F_{\gamma}}{\partial
s},\vec{H}_{L_{\gamma}}\right)=k\sqrt{r^{\prime 2}+r^{2}\phi^{\prime
2}}+\left(\frac{1-4r^{2}}{1+2r^{2}}\right)\phi^{\prime},$

but

$\displaystyle\omega\left(\frac{\partial F_{\gamma}}{\partial
s},J\frac{\partial F_{\gamma}}{\partial s}\right)$
$\displaystyle=\omega\left(r^{\prime}(\partial_{r_{1}}+\partial_{r_{2}})+\phi^{\prime}(\partial_{\theta_{1}}+\partial_{\theta_{2}}),r^{\prime}r^{-1}(\partial_{\theta_{1}}+\partial_{\theta_{2}})-r\phi^{\prime}(\partial_{r_{1}}+\partial_{r_{2}})\right)$
$\displaystyle=\left(r^{\prime 2}r^{-1}+r\phi^{\prime
2}\right)\omega\left(\partial_{r_{1}}+\partial_{r_{2}},\partial_{\theta_{1}}+\partial_{\theta_{2}}\right)$
$\displaystyle=2\frac{r^{\prime 2}+r^{2}\phi^{\prime 2}}{(1+2r^{2})^{2}}.$

So we conclude that

$\vec{H}_{L_{\gamma}}=\frac{1}{2}\left(1+2r^{2}\right)^{2}\left(k+\left(\frac{1-4r^{2}}{1+2r^{2}}\right)\frac{\phi^{\prime}}{\sqrt{r^{\prime
2}+r^{2}\phi^{\prime 2}}}\right)DF_{\gamma}(\nu)$

where $\nu$ is the Euclidean normal to $\gamma$ in $\mathbb{C}$. Since
$\langle\gamma,\nu\rangle=-r^{2}\phi^{\prime}/|\gamma^{\prime}|$, we have that
the mean curvature flow of $F_{\gamma}$ in $\mathbb{CP}^{2}$ induces an
equivariant flow on $\gamma$ given by

$\frac{\partial\gamma}{\partial
t}=\frac{1}{2}\left(1+2r^{2}\right)^{2}\left(k-\left(\frac{1-4r^{2}}{1+2r^{2}}\right)\frac{\langle\gamma,\nu\rangle}{r^{2}}\right)\nu$
(4.6)

###  4.3. Triangle calculations using Cieliebak–Goldstein

In order to prove the main results of this paper, we apply the generalised
Cieliebak–Goldstein theorem (Theorem 2.6) to certain $J$-holomorphic polygons
with boundary on flowing Lagrangians. The most important are triangles with
one vertex at the origin. Since these triangle calculations are ubiquitous and
essential in the sequel, we review the methods involved here.

######  Example 4.3.

Let $L_{\gamma}$ be an equivariant Lagrangian in $\mathbb{CP}^{2}$
intersecting the cone $C^{0}_{\psi}$ at points $p^{+},p^{-}$, see Figure 5,
with Euclidean turning angle $\xi$ at $p^{+},p^{-}$. Consider the
$J$-holomorphic triangle $P$ with boundary on $L_{\gamma}$ given by the
horizontal lift of the Euclidean triangle (also denoted $P$) with boundary on
$\gamma$, $C^{0}_{\psi}$ and vertices at $0,p^{+}$ and $p^{-}$.

Figure 5. The triangles $P$ and $Q$ considered in Example 4.3

We first calculate $\tilde{\mu}(P)$. The Maslov number can be broken down into
two components: a component from the topology of the triangle, and a component
from the angles at the vertices. By smoothing the corners of the triangle so
that the resulting triangle does not intersect the origin, we see that the
topological component is 2. The contribution from the turning angles at the
either of the corners $p^{\pm}$ is given by
$(-\theta_{1}(p^{\pm})+\theta_{2}(p^{\pm}))/\pi$, which is equivalent to the
difference in Euclidean Lagrangian angle between the Lagrangian planes
$T_{p_{1}}L_{1}$ and $T_{p_{1}}L_{2}$. So we have

$\tilde{\mu}(P)=2-\frac{2}{\pi}\xi-A(\psi),$

where $A(\psi)$ is some function of the opening angle at the origin to be
determined.

We could calculate this directly by calculating the difference in Lagrangian
angle between $l_{\psi/2}$ and $l_{-\psi/2}$. For the purposes of intuition
however, we calculate indirectly using the example where $\gamma(s)=e^{is}$ is
the minimal Clifford torus. We have that $\xi=\pi/2$, so

$\tilde{\mu}(P)=1-A(\psi).$

Furthermore, the area of $P$ is given by

$\int_{P}\omega=\frac{\psi}{2\pi}\frac{4\pi}{6}=\frac{\psi}{3}$

since the area is $4\pi/6$ when $\psi=2\pi$. Since $H_{L_{\gamma}}=0$, (2.7)
implies that

$A(\psi)=1-\frac{\kappa}{\pi}\int_{P}\omega=\frac{1}{\pi}\left(\pi-2\psi\right).$

Since the contribution of $\psi$ at the origin is independent of the choice of
$\gamma$, we have that

$\tilde{\mu}(P)=2-\frac{2}{\pi}\xi-\frac{1}{\pi}\left(\pi-2\psi\right).$ (4.7)

In the important special case where $\xi=\pi$, i.e. $\gamma$ is tangent to the
cone $C^{0}_{\psi}$ at the points $p^{+}$ and $p^{-}$, the sign of
$\tilde{\mu}(P)$ is controlled by the opening angle $\psi$. We have that

$\tilde{\mu}(P)=-\frac{1}{\pi}\left(\pi-2\psi\right)$

and hence $\tilde{\mu}(P)$ is negative for $\psi<\pi/2$ and positive for
$\psi>\pi/2$.

####  4.3.1. Evolution equations for polygons

Since they are important in the sequel, we recall the key formulae concerning
$H$ and $\theta$. By Theorem 2.3, we have that

$H=d\theta+\alpha,$

where $\alpha$ is the 1-form $H_{L_{r}}(\pi(\cdot))$, where $\pi$ is
projection to the tangent bundle of $L_{r}$. Furthermore, $\theta$ defined in
this way satisfies the evolution equation

$\frac{\partial}{\partial t}\theta=\Delta\theta+d^{\dagger}\alpha,$

by the same calculation that yielded (2.3), and the mean curvature 1-form $H$
satisfies

$\frac{\partial}{\partial t}H=dd^{\dagger}H+\kappa H.$

Recall that for a polygon $P$ with no corners, the Maslov number is the Maslov
class and is invariant under mean curvature flow, and so we have

$\frac{\partial}{\partial
t}\int_{P}\omega=-\frac{1}{\kappa}\frac{\partial}{\partial t}\int_{\partial
P}H=-\frac{1}{\kappa}\int_{\partial P}dd^{\dagger}H+\kappa
H=\kappa\int_{P}\omega-\pi\mu(P).$

It initially seems reasonable to conjecture then that for a polygon $P$ with
corners,

$\frac{\partial}{\partial
t}\int_{P}\omega=\kappa\int_{P}\omega-\pi\tilde{\mu}(P).$

However, this does not hold for two reasons. Firstly, we obtain boundary terms
from integrating $dd^{\dagger}H$. Secondly, when differentiating, we must
account for potential tangential motion of the vertices of the polygon under
mean curvature flow.

For these reasons, we only consider the evolution equations in the context of
Example 4.3. We note that in this case we have that the sides of the triangle
on the cone are constant angle and minimal.

To that end, let $L_{\gamma}$ be a flowing equivariant Lagrangian,
intersecting the cone $C^{0}_{\psi}$ at points $p^{\pm}$, forming a triangle
$P$ as in Example 4.3. Initially, we assume the intersections are transverse.
Writing $\theta$ for the relative Lagrangian angle of $L_{\gamma}$ and
$H=H_{L_{\gamma}}$ for the mean curvature 1-form, by differentiating (2.7) we
obtain

$\displaystyle\frac{\partial}{\partial t}\int_{P}\omega$
$\displaystyle=\frac{\partial}{\partial
t}\left(\frac{\pi}{\kappa}\tilde{\mu}(P)-\frac{1}{\kappa}\int_{\gamma}H\right)$
$\displaystyle=\frac{1}{\kappa}\frac{\partial}{\partial
t}\left(\theta(p^{-})-\theta(p^{+})\right)-\frac{1}{\kappa}\frac{\partial}{\partial
t}\int_{\gamma}H$

From each term, we obtain a normal and tangential term to account for the
tangential movement of the intersection points $p^{\pm}$ along $C^{0}_{\psi}$
under the flow. Writing the mean curvature flow as

$\frac{\partial}{\partial t}X=\vec{H}+V$

for a tangential diffeomorphism $V$ to be determined, we have that

$\frac{\partial}{\partial
t}\left(\theta(p^{\pm})\right)=\Delta\theta(p^{\pm})+d^{\dagger}\alpha(p^{\pm})+\langle\nabla\theta,V\rangle(p^{\pm}),$

and

$\displaystyle\frac{\partial}{\partial t}\int_{\gamma}H=$
$\displaystyle\int_{\gamma}\left(dd^{\dagger}H+\kappa
H\right)+\left\langle\nabla\int_{\gamma}H,V\right\rangle$ $\displaystyle=$
$\displaystyle\kappa\int_{\gamma}H-\Delta\theta(p^{-})+\Delta\theta(p^{+})-d^{\dagger}\alpha(p^{-})+d^{\dagger}\alpha(p_{2})$
$\displaystyle-\langle\nabla\theta,V\rangle(p^{-})+\langle\nabla\theta,V\rangle(p^{+})-\alpha(V)(p^{-})+\alpha(V)(p^{+})$

where we have used that $H=d\theta+\alpha$ and hence
$d^{\dagger}H=\Delta\theta+d^{\dagger}\alpha$, where $\alpha$ is the closed
1-form on $L$ defined by $\alpha(X)=H_{L_{r}}(\pi X)$. Combining the above
equations and applying (2.7), we obtain

$\begin{split}\frac{\partial}{\partial
t}\int_{P}\omega=&\kappa\int_{P}\omega-\pi\tilde{\mu}(P)\\\
&+\frac{1}{\kappa}\left(-\alpha(V)(p^{-})+\alpha(V)(p^{+})\right).\end{split}$
(4.8)

Since the intersection is transversal, we can write the tangential vector
field $V$ as $\vec{H}+V=W$, for some vector field $W$ on $L_{\gamma}$ tangent
to $C^{0}_{\psi}$. The vector field $W$ then gives the motion of $p^{\pm}$
along the cone, and we have that

$\alpha(V)=\alpha(W-\vec{H})=\alpha(-\vec{H}).$

Note that while $V$ is not well-defined when the intersection is not
transversal, $\alpha(-\vec{H})$ is well-defined everywhere on $L_{\gamma}$.
Thus it is tempting to claim that

$\begin{split}\frac{\partial}{\partial
t}\int_{P}\omega=&\kappa\int_{P}\omega-\pi\tilde{\mu}(P)\\\
&+\frac{1}{\kappa}\left(-\alpha(-\vec{H})(p^{-})+\alpha(-\vec{H})(p^{+})\right).\end{split}$
(4.9)

even when the intersection is non-transversal. The most important case of this
is characterised in the following lemma, where $\psi$ is a local maximum
opening angle, allowed to vary in time.

######  Lemma 4.4.

Let $L_{\gamma}$ be an equivariant Lagrangian mean curvature flow in
$\mathbb{CP}^{2}$ on a time interval $[t_{1},t_{2}]$, with $\gamma$ not
passing through the origin. Suppose that for $t\in[t_{1},t_{2}]$, $L_{\gamma}$
has a local maximum opening angle $\psi(t)$ on $[t_{1},t_{2}]$, where
$\psi(t)$ is a smooth function of $t$. Then the triangle $P$ defined by the
cone $C^{0}_{\psi}$ and $\gamma$, with vertices at $p^{\pm}$ and the origin,
satisfies

$\frac{d}{dt}\int_{P}\omega\leq\kappa\int_{P}\omega+(\pi-2\psi)$ (4.10)

###### Proof.

Let $\gamma(s)$ be parametrised by some variable $s$. Then there exists a
smooth function $S(t)$ such that $\gamma(S(t))$ attains the maximum opening
angle $\psi(t)$.

Figure 6. Three triangles $P_{s}$, ordered with increasing $s$. In the right
diagram, the area is counted with sign, i.e. the green area is counted
positively and the purple area negatively.

Let $A(s,t)=\int_{P_{s}}\omega$, where $P_{s}$ is the triangle intersecting
$\gamma$ at $\gamma(s)$. Here, the integral is the signed integral of
$\omega$, see Figure 6. Then we have to calculate the time-derivative of $A$
at $S(t)$ for $t\in(t_{1},t_{2})$. By choosing a sufficiently small time
neighbourhood $(t_{-},t_{+})\subset(t_{1},t_{2})$ of $t$, we can find a time-
independent space neighbourhood $(s_{-},s_{+})$ of $S(t)$ for all $t$ such
that $\gamma(s)$ intersects the cone transversally for all $s\neq S(t)$.

For any fixed opening angle $\chi$ with transversal intersections with
$\gamma$ at $p^{\pm}_{\chi}$, we have that

$\frac{\partial}{\partial
t}\int_{P_{\chi}}\omega=\kappa\int_{P_{\chi}}\omega-\pi\tilde{\mu}(P_{\chi})+\frac{1}{\kappa}\left(-\alpha(-\vec{H})(p_{\chi}^{+})+\alpha(-\vec{H})(p_{\chi}^{-})\right),$

where $P_{\chi}$ is the triangle of opening angle $\chi$, again calculated
with sign. Now allowing that the opening angle $\chi=\chi(s,t)$ may evolve
with $s$,

$\frac{d}{dt}A(s,t)=\frac{\partial}{\partial
t}\int_{P_{\chi}}\omega+\frac{d\chi}{dt}(s,t)\frac{\partial
A}{\partial\chi}(s,t)$

and taking limits as $s\to S(t)$ gives

$\displaystyle\frac{d}{dt}A(S(t),t)=$
$\displaystyle\kappa\int_{P_{\chi}}\omega-\pi\tilde{\mu}(P_{\chi})+\frac{1}{\kappa}\left(-\alpha(-\vec{H})(p_{\chi}^{+})+\alpha(-\vec{H})(p_{\chi}^{-})\right)$
$\displaystyle+\frac{d\chi}{dt}(S(t),t)\frac{\partial
A}{\partial\chi}(S(t),t)+\frac{dS}{dt}(t)\frac{\partial A}{\partial
s}(S(t),t).$

But since $S(t)$ is a local maximum of the area by assumption, we have that

$\frac{\partial A}{\partial s}(S(t),t)=0.$

Furthermore, the maximum opening angle is decreasing in time, so

$\frac{d\chi}{dt}(S(t),t)\leq 0,$

and $A$ is always increasing in $\chi$ for $\chi<\psi$, so

$\frac{d\chi}{dt}(S(t),t)\geq 0.$

Finally,

$-\alpha(-\vec{H})(p^{+})+\alpha(-\vec{H})(p^{-})<0$

since the direction of the mean curvature is fixed by the assumption that
$p^{\pm}$ are at the maximum opening angle. Since the Maslov number satisfies
$\pi\tilde{\mu}(P)=-(\pi-2\psi)$, we conclude that

$\frac{d}{dt}A(S(t),t)\leq\kappa\int_{P_{\chi}}\omega+(\pi-2\psi),$

as desired. ∎

###  4.4. Minimal equivariant Lagrangians

The main result of this section is Theorem 1.3, which we slightly expand on
now that we have the relevant terminology from Section 4.2.

######  Theorem 4.5.

There exists a countably infinite family of complete immersed minimal
equivariant equivariant Lagrangians. In particular, given any radius $R$ with
$0<R<1$, there exists a complete immersed minimal equivariant torus
$L_{\gamma}$ with $0<\min_{\gamma}r(\gamma)\leq R$, where $r$ is the Euclidean
radius function on $\gamma$.

Since the method of proof is rather long and calculational, we provide an
abridged version, referring the reader to the author’s doctoral thesis
[Evans_2022][Section 4.6] for full details.

From equation (4.6), any minimal equivariant Lagrangian must satisfy

$k-\left(\frac{1-4r^{2}}{1+2r^{2}}\right)\frac{\langle\gamma,\nu\rangle}{r^{2}}=0.$
(4.11)

Away from the origin, equation (4.11) is a non-linear 2nd order ODE. Given any
point $x\in\mathbb{C}$ and an initial velocity $v\in T_{x}\mathbb{C}$, there
is a unique local solution to (4.11) passing through $x$ with velocity $v$.
The proof is identical to the equivalent statement for existence and
uniqueness of geodesics.

Two classes of solutions to (4.11) are immediately apparent. First, either
from the derivation of (4.11) or by direct calculation, one can see that the
Clifford torus $L_{1}:=L_{e^{is}}$ given by the unit circle is a minimal
submanifold. Second, any straight line through the origin
$l_{b}=\\{se^{ib}:s\in\mathbb{R}\\}$ has $k=0$ and $\langle
l_{b},\nu\rangle=0$, and hence gives a minimal submanifold of
$\mathbb{CP}^{2}$, topologically a real projective plane. Furthermore, the
existence and uniqueness implies that if a solution $\gamma$ at any point has
$\langle\gamma,\nu\rangle=0$, then it is a line $l_{b}$ everywhere.

We now restrict attention to point-symmetric solutions that are graphs over
sections of the unit circle, i.e. $\gamma(s)=r(s)e^{is}$ with
$r(s)\in(0,\infty)$.

From (4.11), we have that $r$ satisfies

$-r^{\prime\prime}r+2r^{\prime
2}+r^{2}+\left(\frac{1-4r^{2}}{1+2r^{2}}\right)(r^{\prime 2}+r^{2})=0.$

Rearranging, we obtain

$-r^{\prime\prime}r+\left(\frac{3}{1+2r^{2}}\right)r^{\prime
2}+2\left(\frac{1-r^{2}}{1+2r^{2}}\right)r^{2}=0.$ (4.12)

From here, we can derive a first integral of the equation by use of the
substitution $f(s)=\log(r(s))$. Skipping the derivation, one can simply
observe that

$f^{\prime 2}=C\frac{e^{4f}}{\left(1+2e^{2f}\right)^{3}}-1=:B(f,C).$ (4.13)

is a first integral of equation (4.12).

The theory of roots of cubics provided by Descartes’ rule of signs gives that
for any $C>27$, $B(f,C)$ has exactly 1 positive and 1 negative real zeroes.
After taking the exponential of $f$, this gives two zeroes $r_{1}$ and $r_{2}$
with $0<r_{1}<1<r_{2}<\infty$. Theses are the minimum and maximum values of
$r$ for out solution $\gamma$. As $C\to 27$, $r_{i}\to 1$. Thus the Clifford
torus is the solution with $C=27$. As $C\to\infty$, $r_{1}\to 0$ and
$r_{2}\to\infty$ monotonically.

Next we approach the question of periodicity. Solutions are bounded between
$r_{1}$ and $r_{2}$, hence they oscillate between the two with some period
$\psi_{C}$, dependent on the constant $C$. The first integral implies that

$\psi_{C}=2\int_{\log r_{1}}^{\log
r_{2}}\sqrt{\frac{\left(1+2e^{2f}\right)^{3}}{Ce^{4f}-\left(1+2e^{2f}\right)^{3}}}df=\int_{r_{1}}^{r_{2}}\sqrt{\frac{\left(1+2r^{2}\right)^{3}}{Cr^{4}-\left(1+2r^{2}\right)^{3}}}\frac{1}{r}dr.$

If we can find an integer pair $(m,k)$ and a corresponding value $C(m,k)>27$
such that

$m\psi_{C(m,k)}=2\pi k,$

then we have found a complete solution to (4.12). Unfortunately the above
integral cannot be evaluated explicitly using standard methods.

We aim instead to analyse the limiting behaviour as $C\to\infty$, illustrated
in Figure 7. We show the following:

######  Lemma 4.6.

The period $\psi_{C}$ converges to $3\pi/2$ as $C\to\infty$.

Figure 7. The inner period converges to $\pi/2$ and the outer period converges
to $\pi$ as $C\to\infty$.

###### Proof.

We omit many details which can be found in [Evans_2022], providing only a
sketch.

  1. (1)

We separate the period $\psi_{C}$ into two parts. The inner period
$\psi_{C}^{-}$, i.e. the period where $r(s)<1$, and the outer period, i.e. the
period where $r(s)>1$. We estimate each separately using similar methods.

Beginning with the inner period, we use a geometric inequality illustrate in
Figure reffig-uppervslower.

Figure 8. The area of $P$ (the blue region) is bounded above by the area of
$A$ (the green region) and bounded below by the area of $B$ (the orange
region).

We estimate the $J$-holomorphic biangle $P$ bounded between the unit circle
and our solution $\gamma$ from above by the $J$-holomorphic quadrangle $A$
with sides on the $L_{1}$, $L_{r_{1}}$ and the cone $C^{0}_{\psi_{C}^{-}}$. We
also estimate $P$ from below by the $J$-holomorphic triangle with sides on
$L_{1}$ and the two “radial straight lines” $\eta^{\pm}$ joining the minimum
to the intersection of $\gamma$ with $L_{1}$, where

$\eta^{\pm}(s)=\left(2\frac{1-r_{1}}{\psi_{C}^{-}}s+r_{1}\right)e^{\pm is}.$

Then we have the geometric inequalities

$\kappa\int_{A}\omega>\kappa\int_{P}\omega>\kappa\int_{B}\omega,$ (4.14)

where the second inequality holds since $r$ is convex in $s$ for $r\leq 1$.

By using Cieliebak–Goldstein, we can calculate that using the upper bound that

$\psi_{C}^{-}>\frac{1+2r_{1}^{2}}{1-r_{1}^{2}}\left(\pi-\tan^{-1}\left(\sqrt{\frac{1}{27}\frac{(1+2r_{1}^{2})^{3}}{r_{1}^{4}}-1}\right)\right),$

so $\psi_{C}^{-}>\pi/2$ for all $C>27$ and

$\lim_{C\to\infty}\psi_{C}^{-}=\lim_{r_{1}\to 0}\psi_{C}^{-}\geq\pi/2.$ (4.15)

For the lower bound, we instead calculate by performing the integral directly,
and after a lengthy calculation we derive that

$\lim_{C\to\infty}\psi_{C}^{-}=\lim_{r_{1}\to 0}\psi_{C}^{-}\leq\pi/2.$ (4.16)

Combining (4.15) and (4.16) gives

$\lim_{C\to\infty}\psi_{C}^{-}=\lim_{r_{1}\to 0}\psi_{C}^{-}=\pi/2.$

  2. (2)

Now we calculate the upper period, which we claim satisfies

$\lim_{C\to\infty}\psi_{C}^{+}=\lim_{r_{2}\to\infty}\psi_{C}^{+}=\pi.$

If the geometric inequalities held, all the calculations would proceed as
above and the result would follow. However, though the upper bound does hold
as before, the lower bound does not a priori. This is since the $r$ is not
concave for all $r>1$. However, the radius $\tilde{r}$ of the inflection point
of $r$ is much smaller than $r_{2}$, i.e.

$1<\tilde{r}<<r_{2}$

for all $C$, and a fairly lengthy but not challenging application of
Cieliebak–Goldstein yields the geometric inequality as desired. This completes
the proof.

∎

We can now prove the main theorem of this section.

######  Theorem 4.7.

There exists a countably infinite family of complete immersed minimal
equivariant Lagrangians. In particular, for any $R>0$, there exists a complete
immersed minimal equivariant Lagrangian $L_{\gamma}$ with
$\min_{L_{\gamma}}r\leq R$.

###### Proof.

For $C=54$, we show that $\psi_{54}>3\pi/2$ by an explicit calculation. We
estimate the integral by separating the integrand into two parts. The first
part contains no poles in the interval $[r_{1},r_{2}]$ and hence can be
estimated directly. The second part is a well-known elliptic integral which we
can evaluate explicitly. The result follows.

Note that the period $\psi_{C}$ of a solution $\gamma_{C}$ to (4.12) depends
continuously upon the initial condition. Since $\psi_{C}\to 3\pi/2$ by Lemma
4.6 and by the above $\psi_{C}>3\pi/2$ for some $C>27$, we have that there
exists $\delta>0$ such that for every $\psi\in(3\pi/2,3\pi/2+\delta)$ there
exists a $C>27$ such that $\psi_{C}=\psi$. In particular, we can find
infinitely many integer pairs $(m,k)$ and values $C(m,k)>27$ such that

$m\psi_{C(m,k)}=2\pi k.$

Then the minimal equivariant Lagrangians $\gamma_{C(m,k)}$ described by
$C(m,k)$ are complete immersed minimal equivariant Lagrangian. This is an
infinitely large family of unique solutions since for every sufficiently large
prime $k$, we can obtain at least one solution.

Since this argument also applies to any $\delta^{\prime}<\delta$, we can
construct $L_{\gamma}$ satisfying the second part of the theorem. ∎

Figure 1 in the introduction illustrates the spirograph-like shape of the
complete immersed minimal equivariant Lagrangians.

While we are on the subject, we prove one final property of the minimal
surfaces which will be useful in proving Lemma 5.4. The idea of the proof is
similar to Lemma 4.6, but a slightly different geometric estimate is required.
Instead of estimating using a disc $B$ with boundary on a radial straight
line, we use a disc $B$ with boundary on a Euclidean straight line, see Figure
9

Figure 9. Radial straight lines (orange) used in Lemma 4.6 compared to the
Euclidean straight line (red) used in Proposition 4.8. Both give lower bounds
for the region contained between $\gamma$ (blue) and the circle of radius
$R_{C}$ (purple).

######  Proposition 4.8.

Let $\gamma_{C}(s)=r(s)e^{is}$, $C>27$, be a solution to equation (4.12) with
initial condition $r^{\prime}(0)=0$ with $r(0)=r_{1}<1$. Let
$R_{C}:=r(\pi/4)$, i.e. the radius of intersection with the cone
$C^{0}_{\pi/2}$. Then $R_{C}\to 0$ as $C\to\infty$.

We need a lemma to prove Proposition 4.8, which guarantees the Euclidean
straight lines give lower bounds for sufficiently large $C>27$.

######  Lemma 4.9.

Let $\gamma_{C}$ be as in the statement of Proposition 4.8. Then $R_{C}<1/2$
for $C$ sufficiently large.

###### Proof.

The idea of the proof is to find a subsolution (a hyperbola) with the desired
behaviour, and then use the comparison principle to obtain the result.

Consider the hyperbolas $\gamma_{a,c}$ given by

$a^{2}x^{2}-y^{2}=c^{2}$

for constants $a,c>0$. For sufficiently small $c>0$ and $a>0$ given by

$a^{2}=\left(2c^{2}+\frac{1}{2}\right)+\sqrt{\left(2c^{2}+\frac{1}{2}\right)^{2}+2c^{2}},$

we have that $\gamma_{a,c}$ intersects $C^{0}_{\pi/2}$ at $R<1/2$. We leave
the verification of this fact to the interested reader, or alternatively refer
them to [Evans_2022].

Let $C$ be any constant such that $\gamma_{C}$ has $r_{1}\leq c$. We claim
this implies that $\gamma_{C}$ intersects the cone $C^{0}_{\pi/2}$ at a radius
less than $R<1/2$. Suppose not. Then $\gamma_{C}$ intersects $\gamma_{a,c}$ at
two points inside the cone $C^{0}_{\pi/2}$. Consider the quasilinear elliptic
operator $Q(f)$ given by

$Q(f)=-f^{\prime\prime}+2\frac{1-r^{2}}{1+2r^{2}}f^{\prime
2}+2\frac{1-r^{2}}{1+2r^{2}}$

where $f=\log r$, see equation (4.12. Let $f_{C}$ and $f_{a,c}$ be the
logarithms of the radius functions of $\gamma_{C}$ and $\gamma_{a,c}$
respectively. Then we have that $Q(f_{C})=0$ and $Q(f_{a,c})<0$. Furthermore,
we have that for $s\in[-S,S]\subset(-\pi/4,\pi/4)$, $f_{C}(s)\leq f_{a,c}(s)$
with $f_{C}(S)=f_{a,c}(S)$. Since

$\partial_{r}\left(\frac{1-r^{2}}{1+2r^{2}}\right)<0$

for all $r$, we can apply the comparison principle for quasilinear elliptic
operators to deduce that

$f_{C}(s)\geq f_{a,c}(s)$

for all $s\in[-S,S]$, a contradiction. ∎

###### Proof of Proposition 4.8.

By Lemma 4.9, we have that $r_{1}<R_{C}<1/2$ for sufficiently large $C>27$,
and hence $r(s)<1/2$ for all $s\in[-\pi/4,\pi/4]$. Consider now $\gamma$ as a
graph over the $y$-axis, i.e. $\gamma(y)=x(y)+iy$ for $y$ in an interval $I$
containing $0$. Elementary calculation gives that $x(y)$ is convex as a
function of $y$ for all $r<1/2$. Hence the Euclidean straight line
$\eta^{\pm}$ connecting the minimum value $r_{1}$ with $R_{C}e^{\pm i\pi/4}$
does not intersect $\gamma_{C}$ for $s\in(0,\pi/4)$. Similar to the proof of
Lemma 4.6, denote by $P$ the $J$-holomorphic biangle bounded by $\gamma$ and
the circle $L_{R_{C}}$, and by $B$ the $J$-holomorphic triangle with boundary
on $\eta^{\pm}$ and $L_{R_{C}}$.

Then we have the geometric inequality

$\kappa\int_{P}\omega>\kappa\int_{B}\omega.$

Proceeding in a similar manner to Lemma 4.6, we find after some calculation
(see [Evans_2022] for details) that the geometric inequality gives

$\pi
R_{C}^{2}\left(\frac{3}{1+2R_{C}^{2}}-\frac{3}{2}\right)>\pi-2\tan^{-1}\left(\sqrt{\frac{(1+2r_{1}^{2})^{3}}{r_{1}^{4}}\frac{R_{C}^{4}}{(1+2R_{C}^{2})^{3}}-1}\right)-3\sqrt{2}r_{1}R_{C}.$

If $R_{C}$ is bounded below by $\varepsilon>0$, the right-hand side converges
to 0 as $C\to\infty$ while the left-hand side is strictly less than 0, a
contradiction. This completes the proof. ∎

##  5\. Lagrangian mean curvature flow of equivariant Lagrangian tori in
$\mathbb{CP}^{2}$

###  5.1. A Thomas–Yau-type theorem

The goal for this section is to prove Theorem 1.1. We give now an overview of
the method of proof, the details of which will be contained in the following
sections. First, we state a more detailed version of Theorem 1.1.

######  Theorem 5.1.

Let $L$ be an equivariant Lagrangian torus in $\mathbb{CP}^{2}$ which
intersects the cone $C^{0}_{\pi/2}$ a total of $4n$ times, for $n\geq 1$. With
$L$ evolving by mean curvature flow, the following hold:

  1. (1)

If $n=1$, then $L$ becomes graphical over the minimal Clifford torus $L_{1}$
in finite time, after which $L$ exists for all time and converges to $L_{1}$
in infinite time.

  2. (2)

If $L$ is a Chekanov torus with $n=n_{0}>1$, $L$ has a finite-time singularity
at the origin. By performing a neck-to-neck surgery before the singular time,
$L$ becomes a Clifford torus with $n\leq n_{0}-1$.

  3. (3)

If $L$ is a Clifford torus with $n=n_{0}>1$, then after a sufficiently large
time, $L$ has either has $n=1$ or $L$ has attained a finite-time singularity
at the origin. In the latter case, by performing a neck-to-neck surgery before
the singular time, $L$ becomes a Chekanov torus with $n\leq n_{0}-1$.

###### Proof.

The proof is the sum total of the results in the following sections.

  1. (1)

The main results of Section 5.2 imply number of intersections with
$C^{0}_{\pi/2}$ is strictly decreasing under the flow and that any singularity
occurs at the origin with blow-up given by $C^{0}_{\pi/2}$.

  2. (2)

In Section 5.3, we prove that graphical Clifford tori exist for all time and
converge to $L_{0}$.

  3. (3)

In Section 5.4, we prove that Chekanov tori always have finite-time
singularities.

  4. (4)

In Section 5.5, we define a neck-to-neck surgery procedure which by definition
strictly decreases $n$, and complete the proof by dealing with the case of
Clifford tori with $n>1$.

∎

###  5.2. Singularities for equivariant tori

In this section we replicate many of the results which are known for
equivariant flows in $\mathbb{C}^{2}$ by the work of Neves ([Neves2007],
[Neves2010]) and Wood ([Wood2019], [wood_2020])333For the readers convenience,
we summarise some background information on type II singularities and blow-ups
in Lagrangian mean curvature flow in Appendix A.. The corresponding section
[Evans_2022][Section 4.7] comprises one of the longest and most technical
sections of the paper, but the overall heuristic is simple. Since singular
behaviour of a flow is a local phenomenon, and any blow-up procedure “blows
away” the ambient curvature, we should expect any results that hold in
$\mathbb{C}^{2}$ to also hold in $\mathbb{CP}^{2}$.

As a first and important example of this, we show there is a direct analogue
of the monotone version of Neves’ Theorem B [Neves2010] that holds in
$\mathbb{CP}^{2}$.

######  Lemma 5.2.

Let $L$ be a monotone Lagrangian mean curvature flow in $\mathbb{CP}^{2}$ with
a finite-time singularity at $T<\infty$. For any sequence $(L_{j}^{s})$ of
rescaled flows, the following property holds for all $R>0$ and almost all
$s<0$:

For any sequence of connected components $\Sigma_{j}$ of $B_{4R}(0)\cap
L_{j}^{s}$ that intersect $B_{R}(0)$, there exists a special Lagrangian cone
$\Sigma$ in $B_{2R}(0)$ with Lagrangian angle $\bar{\theta}$ such that, after
passing to a subsequence,

$\lim_{j\to\infty}\int_{\Sigma_{j}}f(\exp(i\theta_{j}^{s}))\phi
d\mathcal{H}^{2}=mf(\exp(i\bar{\theta}))\mu(\phi)$

for every $f\in C(S^{1})$ and every smooth $\phi$ compactly supported on
$B_{2R}(0)$, where $\mu$ and $m$ denote the Radon measure of the support of
$\Sigma$ and its multiplicity respectively.

###### Proof.

By the work of Castro–Lerma ([castro_lerma_miquel_2018]) (see section
[Evans_2022][Section 3.3] for a concise explanation), a monotone torus in
$\mathbb{CP}^{2}$ lifts to a monotone spherical Lagrangian $3$-torus in
$S^{5}\subset\mathbb{C}^{3}$. The flow also lifts, becoming a flow with a
singularity at a time $\tilde{T}<1/2$ (recall that $t=1/2$ is the singular
time of the unit sphere $S^{5}$). The singularity in the lift occurs along an
$S^{1}$, the Hopf fibre above the singular point of the original flow.

At this point we can already apply [Neves2010, Theorem A] to show that we have
convergence to a finite set of special Lagrangian cones with angles
$\theta_{k}$. We want to show that we can instead apply [Neves2010, Theorem
B], which a priori only applies in $\mathbb{C}^{2}$. Indeed, the only part of
the proof of Theorem B which does not hold in higher dimensions is [Neves2010,
Lemma 5.2]. So we have to show that for all $j$ sufficiently large, there
exists some $C>0$ such that

$\left(\mathcal{H}^{3}(A)\right)^{2/3}\leq C\mathcal{H}^{2}(\partial A),$
(5.1)

for any open subset $A$ of $L^{j}_{s}\cap B_{6R}(0)$ with rectifiable
boundary. The proof of [Neves2010, Lemma 5.2] doesn’t hold immediately since
the dimension is too high to apply the Michael–Simon Sobolev inequality
directly in $\mathbb{C}^{3}$. Instead, we apply the Michael–Simon Sobolev
inequality for Riemannian manifolds with positive curvature to the flow in
$\mathbb{CP}^{2}$ and, by arguing that the Hopf fibre is non-collapsing at the
final time, we are able to lift the resulting inequality to $\mathbb{C}^{3}$
up to a constant. Then we can apply the Michael–Simon Sobolev inequality in
$\mathbb{C}^{3}$ and deduce the result. See [Evans_2022] for details. ∎

We now state the main results

######  Proposition 5.3.

Suppose $L_{\gamma}$ is an equivariant monotone Clifford or Chekanov torus in
$\mathbb{CP}^{2}$. Let $T\in(0,\infty]$ be the maximal existence time for
$L_{\gamma}$.

  1. (1)

If $\gamma$ is initially embedded, it is embedded for all $t\in[0,T)$. If
$L_{\gamma_{1}},L_{\gamma_{2}}$ are two initial conditions with finite number
of intersections, then the number of intersections of $\gamma_{1}$ and
$\gamma_{2}$ is a decreasing function in $t$. Similarly, the number of
intersections of $\gamma$ with any cone $C^{a}_{b}$ is also a decreasing
function in $t$.

  2. (2)

If $L_{\gamma}$ has a finite-time singularity, then it must occur at the
origin.

  3. (3)

The type I blow-up of any singularity is the cone $C^{0}_{\pi/2}$.

  4. (4)

The type II blow-up is a Lawlor neck asymptotic to the type I blow-up. The
blow-up is independent of rescaling sequence.

  5. (5)

Any sequence of connected components as in the statement of Theorem 5.2
converges to a multiplicity 1 copy of the cone $C^{0}_{\pi/2}$.

###### Proof.

See [Evans_2022] for more detail on the following.

  1. (1)

All 3 statements may be proven by variations on the same argument, which dates
back to Angenent [angenent_1991] applying a classical result of Sturm
[sturm_1836] on the zeroes of a uniformly parabolic PDE (see [angenent_1991,
Proposition 1.2]).

  2. (2)

Suppose $L_{\gamma}$ has a finite-time singularity away from the origin.
Taking a type II blow-up at that point, we see that the origin is blown away
to infinity, hence the $S^{1}$ symmetry becomes a translational symmetry.
Therefore the type II blow-up is an eternal flow that splits as
$\tau\times\mathbb{R}$ for some curve $\tau$.

Since the singularity must be type II, the second fundamental form $A$ of the
type II blow-up has $|A|=1$ at the space-time point $(0,0)$ by definition. So
the geodesic curvature of $\tau$ is non-zero, and therefore the mean curvature
of the type II blow-up is non-zero. However, application of Theorem B implies
that we obtain a blow-up with constant Lagrangian angle, and hence the blow-up
is minimal, a contradiction.

  3. (3)

By Theorem B, any connected component converges to a special Lagrangian cone.
The $\mathbb{Z}_{2}$ symmetry restricts the possible blow-ups to
$C^{0}_{\pi/2}$ and $C^{\pi/4}_{\pi/2}$, with angles 0 or $\pi$, and
$\pm\pi/2$ respectively.

We then have to eliminate the possibility of $C^{\pi/4}_{\pi/2}$ occurring. We
present the basic idea here, referring the reader to [Evans_2022] for the
complicated details.

If a connected component $\sigma^{i}$ converging to $C^{\pi/4}_{\pi/2}$
intersects the real (or equivalently imaginary axis), then $\sigma^{i}$ is
$\mathbb{Z}_{2}$ reflected across the real axis. But then the angle on one
side of the axis is strictly determined by the angle on the other side, and it
is relatively straightforward to show that such a connected component cannot
converge in angle, violating Theorem B. So any connected component cannot
intersect either the real or imaginary axis.

Now that $\sigma^{i}$ is trapped in, for example, the positive quadrant, we
aim to show that $\sigma^{i}$ converges to a Lawlor neck-type singularity. To
do so, we first have to show no higher multiplicity can arise from a single
connected component $\sigma^{i}$.

A long but somewhat standard density argument using Huisken’s monotonicity
formula yields a bound on the density on any annulus: Let $\sigma^{i}_{k}$ be
any connected component of $\sigma^{i}\cap A(R,2R)$. Then we can show that

$\lim_{i\to\infty}\frac{\mathcal{H}^{1}(\sigma^{i}_{k}\cap A(R,2R))}{R}=2.$
(5.2)

Now the topology of the situation implies that a higher multiplicity cannot
arise from a single connected component $\sigma^{i}$.

Hence the component $\sigma^{i}$ converges to a single density copy of
$C^{\pi/4}_{\pi/2}$. But its mirror $\bar{\sigma}^{i}$ has the same
convergence, so we have a double density copy of $C^{\pi/4}_{\pi/2}$. In this
case however, we can guarantee that each component intersects the real axis
and the imaginary axis within some small ball about the origin. This is the
content of the Scale Lemma, which we prove immediately after this proof since
it is important in the rest of the paper.

  4. (4)

The proof is similar to that found in Wood [wood_2020].

  5. (5)

Let $\Sigma_{j}$ be a sequence of connected components converging to a
multiplicity 2 copy of the cone $C^{0}_{\pi/2}$. Since $\Sigma_{j}$ are
connected within a ball $B_{R}$ of small radius $R$ for sufficiently large
$j$, and since they converge to a multiplicity 2 copy of the cone
$C^{0}_{\pi/2}$, then within $B_{R}$, either $\Sigma_{j}$ intersects the
positive real axis greater than once, or $\Sigma_{j}$ intersects both the real
axis and the imaginary axis. Since they are connected within $B_{R}$ and are
equivariant, we must have that they bound a disc $D\subset B_{R}$. But
$D\subset B_{R}$ implies
$\kappa\int_{D}\omega<\kappa\int_{B_{R}}\omega<<2\pi$, so $L$ is not monotone
before the singular time, a contradiction.

∎

We conclude this section with an essentially important lemma, which enables us
to define surgery at singularities. The same proof but with the cone
$C^{\pi/4}_{\pi/2}$ completes the proof of the third part of Proposition 5.3.

######  Lemma 5.4 (Scale lemma).

Let $L_{\gamma}$ be a monotone equivariant Lagrangian mean curvature flow in
$\mathbb{CP}^{2}$ with a finite-time singularity at the origin at time
$T<\infty$. Suppose the type I blow-up is the cone $C^{0}_{\pi/2}$ and the
connected component converging to $C^{0}_{\pi/2}$ intersects the real axis.

Then for any $R>0$, $\varepsilon_{0}>0$, $\delta>0$, there exists an
$\varepsilon$ with $0<\varepsilon<\varepsilon_{0}$ and a time $t^{\prime}$
with $T-\delta<t^{\prime}<T$ such that $L_{\gamma}\cap B_{R}$ intersects
$C^{0}_{\pi/2+2\varepsilon}$ at the time $t^{\prime}$, where $B_{R}:=B_{R}(0)$
is a ball of (Euclidean) radius $R$ at the origin.

In essence, the scale lemma guarantees that singularity formation happens on
an arbitrarily small scale. When we do surgery, this allows us to reduce the
number of intersections with the cone $C^{0}_{\pi/2}$, thus controlling the
total number of surgeries any flow can undergo. Currently, no such result
exists for Lagrangian mean curvature flow in Euclidean space. Indeed, the
proof we give relies heavily upon barriers that cannot exist in Euclidean
space.

###### Proof.

Figure 10. In order to form a singularity, $\gamma$ must eventually intersect
$\gamma_{C}$ within $B_{R/2}$. But to do so, it must first intersect
$\gamma_{C}$ within the annulus $A(R/2,R)$, and hence intersects a cone
$C^{0}_{\pi/2+2\varepsilon}$ for $\varepsilon>0$.

Suppose not. Then there exists $R>0$, $\varepsilon_{0}>0$, $\delta>0$ such
that for any $\varepsilon$ with $0<\varepsilon<\varepsilon_{0}$ and
$t^{\prime}$ with $T-\delta<t^{\prime}<T$, $L_{\gamma}\cap B_{R}$ does not
intersect $C^{0}_{\pi/2+2\varepsilon}$. Since $T$ is the first singular time,
we can, by taking a smaller $R>0$ if necessary, have that $L_{\gamma}\cap
B_{R}$ is empty at time $T-\delta$. We may also freely assume $R<1$.

Note that since $\gamma$ has a finite-time singularity at the origin, we have
that there exists some $Q>0$ such that $\max_{p\in\gamma}r(p)<Q$ for all time
$t\in[0,T)$, where $r(p)$ is the Euclidean distance to the origin.

Consider the complete immersed minimal equivariant surfaces $L_{\gamma_{C}}$
constructed in Theorem 4.7, where $\gamma_{C}(s)=r_{C}(s)e^{is}$ has initial
values $r(0)=r_{1},r^{\prime}(0)=0$. Since $L_{\gamma_{C}}$ is complete,
$\gamma_{C}$ intersects $\gamma$ a finite number of times, non-increasing
under the flow. By choosing $C>27$ sufficiently large, we can find a curve
$\gamma_{C}$ such that

  1. (1)

$\gamma_{C}$ has maximum $r_{2}>Q$.

  2. (2)

$r_{C}(s)<R/2$ for all $s\in[-\pi/4,\pi/4]$, see Proposition 4.8.

  3. (3)

The inner period $\psi^{-}_{C}$ (see Lemma 4.6) satisfies
$\psi^{-}_{C}<\pi/2+2\varepsilon_{0}$.

Consider the set

$\mathcal{S}=\left\\{p\in\gamma\,:\,p=\gamma_{C}(s)\text{ for
}s\in[0,\psi_{C}/2]\right\\},$

where $\psi_{C}$ is the period of $\gamma_{C}$. Since $\gamma$ is monotone and
both $\gamma$ and $\gamma_{C}$ are complete, this set is non-empty and finite
for all time. Furthermore, if $p\in\mathcal{S}$, then $r(p)\in[R,Q]$ for all
time by assumption, since otherwise we would have found a point $p\in
L_{\gamma}\cap B_{R}$ intersecting $C^{0}_{\pi/2+2\varepsilon}$ for some
$\varepsilon$ with $0<\varepsilon<\varepsilon_{0}$. Since the number of
intersections between $\gamma$ and $\gamma_{C}$ is non-increasing,
$L_{\gamma}\cap L_{\gamma_{C}}\cap B_{R}$ is empty for all time.

However, since the connected component giving $C^{0}_{\pi/2}$ in the blow-up
contains the real axis, we have that the intersection with the real axis
converges to 0 since otherwise the blow-up would have to contain a line
$C^{0}_{a}$ with $a<\pi/4$. In particular, there must exist some time
$\tilde{t}$ with $T-\delta<\tilde{t}<T$ when $L_{\gamma}\cap
L_{\gamma_{C}}\cap B_{R}$ is non-empty, a contradiction. ∎

######  Remark 5.5.

The final step of the above proof can be simplified using the assumption that
the Lawlor neck is the type II blow-up.

###  5.3. Clifford tori

A natural condition to impose on solutions of the equivariant flow (4.6) is
that $\gamma$ is graphical over the minimal equivariant Clifford torus
$L_{1}$. We have already studied minimal solutions of (4.6): the only embedded
minimal solution is $L_{1}$. Thus, by Proposition 2.1, if we can prove that
$\gamma$ has long-time existence, then we obtain convergence to $L_{1}$ in
infinite time.

######  Remark 5.6.

Recall the fibration $\\{L_{\alpha}\\}$ by Clifford-type tori given by the
moment map

$\mu([x:y:z])=\frac{1}{|x|^{2}+|y|^{2}+|z|^{2}}\left(|x|^{2},|y|^{2}\right).$

The equivariant fibres are

$L_{r}=\\{L_{re^{i\phi}}:r>0\\}$

and from here on we denote by $\Omega$ the holomorphic volume form relative to
$L_{r}$. Then for a Lagrangian $L$, we defined the Lagrangian angle $\theta$
relative to $\Omega$ by

$\Omega_{L}=e^{i\theta}\operatorname{vol}_{L}.$

Recall that in the Calabi–Yau case, if $\theta$ can be chosen to be a real-
valued function, we call $L$ zero-Maslov with respect to $\Omega$.
Furthermore, we call $L$ almost-calibrated with respect to $\Omega$ if there
exists some $\delta>0$ such that

$\cos\theta>\delta>0.$

Note that $\gamma$ is graphical over $L_{1}$ if and only if $\gamma$ is
almost-calibrated with respect to $\Omega$.

However, unlike in the Calabi–Yau case, $\theta$ defined in this way satisfies
the evolution equation

$\frac{\partial}{\partial t}\theta=\Delta\theta+d^{\dagger}\alpha,$

and hence the parabolic maximum principle does not imply that $\cos\theta$ is
increasing in time. Indeed, consider the following setup: let $\gamma$ be a
small ellipse with eccentricity $0<e<1$ centred on the origin. Then $\gamma$
has a finite-time singularity at the origin. Furthermore, if $e$ is
sufficiently close to 1, the singularity is type II and has type II blow-up a
Lawlor neck. In particular, there is no constant $\delta>0$ such that
$\cos\theta>\delta$ for all time. So almost-calibrated is not preserved in
general.

Furthermore, even in the case where $L_{\gamma}$ is monotone, we should not
expect almost-calibrated to be preserved locally. Indeed, we construct an
example later in the paper where a non-graphical Clifford torus forms a
finite-time singularity. However the construction seems to indicate that the
almost-calibrated condition breaks locally in this case.

These examples illustrates two ideas. Firstly, we should consider $\gamma$ as
graphical rather than almost-calibrated. Secondly, we should only expect
graphical to be preserved in the case that $\gamma$ gives a monotone torus
$L_{\gamma}$. This concludes the remark.

######  Proposition 5.7.

Let $L_{\gamma}$ be a monotone equivariant Clifford torus, graphical over the
minimal equivariant Clifford torus $L_{1}$. Then under mean curvature flow,
$L_{\gamma}$ exists for all time and converges to $L_{1}$ in infinite time.

###### Proof.

As mentioned above, it suffices to show that we have long-time existence.
First, note that the graphical condition is preserved up to any potential
singular time since the number of intersections of $\gamma$ with any cone
$C^{a}_{b}$ is decreasing in time by Proposition 5.3.

Suppose for a contradiction that we have a finite time singularity at time
$T$. Any finite-time singularity must occur at the origin by Proposition 5.3,
and must have blow-up given by $C^{0}_{\pi/2}$.

For any graphical $\gamma$ and $\varepsilon>0$, the cone
$C^{0}_{\pi/2-2\varepsilon}$ intersects $\gamma$ 4 times, dividing the Maslov
4 disc into 4 triangles. Denote the triangles intersecting the positive and
negative real axes by $P^{+}_{\varepsilon}$ and $P^{-}_{\varepsilon}$, and the
triangles intersecting the positive and negative imaginary axes by
$Q^{+}_{\varepsilon}$ and $Q^{-}_{\varepsilon}$. Note that since $L_{\gamma}$
is monotone,

$\int_{P^{+}_{\varepsilon}+P^{-}_{\varepsilon}+Q^{+}_{\varepsilon}+Q^{-}_{\varepsilon}}\omega=4\pi/6=2\pi/3.$

Suppose the type II blow-up of a connected component is a Lawlor neck
intersecting the real axis (this assumption is reasonable and simplifies the
proof, but can be removed, see Remark 5.5). Then for any $\varepsilon>0$, we
have that

$\int_{P^{+}_{\varepsilon}}\omega\to 0$

as $t\to T$, where $P_{\varepsilon}$ is the $J$-holomorphic triangle bounded
by $\gamma$ and $C^{0}_{\pi/2-2\varepsilon}$, intersecting the positive real
axis and with a vertex at 0. But the total area contained outside the cone is
bounded above by $\pi/2+2\varepsilon$, so

$\int_{Q^{+}_{\varepsilon}+Q^{-}_{\varepsilon}}\omega<\pi/2+2\varepsilon.$

Let $\varepsilon=\pi/48$. We can find a time $t$ close to $T$ such that

$\int_{P^{+}_{\varepsilon}+P^{-}_{\varepsilon}}\omega<\pi/24.$

Then at $t$,

$\int_{P^{+}_{\varepsilon}+P^{-}_{\varepsilon}+Q^{+}_{\varepsilon}+Q^{-}_{\varepsilon}}\omega<\pi/2+\pi/12=7\pi/12<2\pi/3,$

which contradicts $L_{\gamma}$ being monotone. ∎

######  Remark 5.8.

As in Remark 5.5, the assumption that the type II blow-up is a Lawlor neck
simplifies the proof, but is not necessary.

###  5.4. Chekanov tori

In the following, we will analyse the behaviour of equivariant Chekanov tori
under mean curvature flow. Note first of all that any equivariant Chekanov
torus does not intersect either the imaginary or real axis. Without loss of
generality assume the former. Then there exists a cone $C^{0}_{\psi}$ of
maximal opening angle $\psi$ such that $C^{0}_{\psi}\cap L_{\gamma}$ is non-
empty. Since $L_{\gamma}$ is monotone and the area inside the cone
$C^{0}_{\psi}$ is $\psi/2$, we must have that $\psi>2\pi/3$.

We begin by proving there are no minimal equivariant Chekanov tori. This is
interesting in its own right, and the method of proof suggests it may
generalise to the non-equivariant case.

######  Proposition 5.9.

There is no minimal equivariant Chekanov torus.

###### Proof.

The result is immediate by the classification of equivariant tori in Section
4.4. However, we present a different proof since we believe the idea of the
proof may be more widely applicable.

Let $L_{\gamma}$ be an equivariant Chekanov torus. Without loss of generality,
we are free to restrict to the subclass of Chekanov tori $L_{\gamma}$ for
which $\gamma$ does not intersect the imaginary axis in $\mathbb{C}$.

Since $\gamma$ does not intersect the imaginary axis and $L_{\gamma}$ is
equivariant, there is some maximal angle $\psi$ such that $C^{0}_{\psi}$
intersects $\gamma$. Since $\gamma$ does not pass through the origin and does
not intersect the imaginary axis, $\psi=\pi-\delta$ for some $\delta>0$.
Denote the first points of intersection of $L_{\gamma}$ and $C^{0}_{\psi}$ by
$p^{+}$ and $p^{-}$, i.e. if $p\in C^{0}_{\psi}\cap L_{\gamma}$, then
$r(p^{\pm})\leq r(p)$.

Figure 11. The triangle $P$ collapses in finite time for a Chekanov torus.

Consider the $J$-holomorphic triangle $P$ with boundary on
$l_{\psi/2},l_{-\psi/2}$ and $\gamma$ with one vertex at the origin and the
other two vertices at $p^{+}$ and $p^{-}$. As in Example 4.3,

$\tilde{\mu}(P)=-\frac{1}{\pi}\left(\pi-2\psi\right),$

and hence by (2.7) we have that

$-\int_{\gamma_{0}}H=\kappa\int_{P}\omega+\pi-2\psi,$

where $\gamma_{0}$ is the arc of $\gamma$ running from $p^{+}$ to $p^{-}$. But

$\int_{P}\omega<\frac{\psi}{2}-\frac{\pi}{3}$

since the area of $P$ is bounded by the difference between the total area
contained in the cone and the area of the Maslov 2 disc bounded by $\gamma$.
So

$-\int_{\gamma_{0}}H<3\psi-2\pi+\pi-2\psi=\psi-\pi=-\delta<0.$

Hence $L_{\gamma}$ is not minimal. ∎

######  Corollary 5.10.

Let $L_{\gamma}$ be an equivariant Chekanov torus in $\mathbb{CP}^{2}$. Then
under mean curvature flow, $L_{\gamma}$ has a finite-time singularity at the
origin $[0:0:1]$.

###### Proof.

By Proposition 5.9, there is no minimal equivariant representative in the
Hamiltonian isotopy class of $L_{\gamma}$. Proposition 2.1 therefore implies
that we have a finite-time singularity, which must occur at the origin by
Proposition 5.3. ∎

We also proffer an alternative proof using the evolution equation derived in
Lemma 4.4.

###### Proof.

Suppose there is no finite-time singularity. We are in the situation of Lemma
4.4, noting that the maximum opening angle must be a smooth function of $t$
for all $t$ sufficiently large. By the argument above, the right hand side of
(4.10) is less than $-\delta$ for some $\delta>0$. But then there is only a
finite period of time when $P$ has positive area, a contradiction. ∎

###  5.5. Neck-to-neck surgery and the proof of the main theorem

In this section we analyse the behaviour of Lagrangian tori at the singular
time. The equivariant condition necessarily (and intentionally) restricts us
to two Hamiltonian isotopy classes of Lagrangian tori: the Clifford and
Chekanov tori. We have shown in Proposition 5.10 that an equivariant Chekanov
torus achieves a type II singularity at the origin with type II blow-up given
by a Lawlor neck. Resolving the Lawlor neck singularity by neck-to-neck
surgery gives a Clifford torus. We have also shown that almost-calibrated
Clifford tori have long-time existence and convergence to the equivariant
minimal Clifford torus. Our dream is that under Lagrangian mean curvature flow
with surgeries, exotic tori in $\mathbb{CP}^{2}$ flow towards a minimal
Clifford torus in infinite time, so we are led to ask the following question
in our symmetric case:

######  Question 5.11.

Does an equivariant Chekanov torus flow to a minimal Clifford torus after
neck-to-neck surgery?

The answer to this question is yes, with the caveat that the number of neck-
to-neck surgeries may be greater than one.

In fact, we will prove the following:

######  Theorem 5.12.

Let $L_{\gamma}$ be a equivariant Clifford or Chekanov torus in
$\mathbb{CP}^{2}$. Then, after a finite number of neck-to-neck surgeries,
$L_{\gamma}$ converges to the unique equivariant minimal Clifford torus
$L_{1}$ in infinite time.

######  Definition 5.13.

Let $L_{\gamma}$ be an equivariant Lagrangian mean curvature flow with a
finite-time singularity at $[0:0:1]$ at time $T<\infty$. Suppose the type I
blow-up is the cone $C^{0}_{\pi/2}$ and the type II blow-up is (independent of
rescaling) the Lawlor neck asympototic to $C^{0}_{\pi/2}$ intersecting the
real axis. For any $r>0$, we can find $\varepsilon>0$ and a least time
$t^{\prime}<T$ such that $L_{\gamma}\cap B_{r}$ intersects
$C^{0}_{\pi/2+2\varepsilon}$ as in Lemma 5.4. We define a new curve $\zeta$ in
$\mathbb{C}$ which will give a Lagrangian $L_{\zeta}$, which we will call the
scale $r$ surgery of $L_{\gamma}$.

Let $p^{\pm}=r^{\prime}e^{\pm i(\pi/4+\varepsilon)}$ be the smallest radius
points of intersection of $L_{\gamma}\cap B_{r}$ and
$C^{0}_{\pi/2+2\varepsilon}$. Similar to the proof of Lemma 5.4, we can find a
curve segment $\zeta^{\prime}$ (intersecting the imaginary axis this time),
smoothly tangent to $C^{0}_{\pi/2+\varepsilon}$ at points $q^{\pm}$,
$-q^{\pm}$ at radius $r^{\prime\prime}<r^{\prime}$. Define $\zeta$ to be the
union of $\zeta^{\prime}$, $\gamma\cap A(r^{\prime},\infty)$ and a smooth
curve interpolating between $p^{\pm}$ and $q^{\pm}$.

Rescale radially and perform Moser’s trick as in Vianna’s construction to
obtain from $L_{\zeta}$ a monotone surgery of $L_{\gamma}$. We call this
procedure neck-to-neck surgery.

######  Remark 5.14.

  1. (1)

On the level of Lagrangians, this construction is not canonical. Since we need
to use Moser’s trick to obtain a monotone torus, monotone surgery is never
going to be canonical unless you can flow directly through the singularity. In
this case however, there is a canonical way to perform Moser’s trick since the
equivariance means you can just rescale radially until you obtain a monotone
torus. This is a quirk of the equivariance and cannot be expected in general.

  2. (2)

The surgery procedure does not require that the type I blow-up is multiplicity
1 (even though we conjecture that all type I blow-ups in this situation are
multiplicity 1). In the case that the multiplicity is higher than 1, the
closest intersection point with the cone $C^{0}_{\pi/2+2\varepsilon}$ is
continuous in time for times $t$ sufficiently close to the singular time $T$.
Hence there is no ambiguity about the neck to be cut and rotated in any case.

  3. (3)

The surgery is canonical from a symplectic point of view since we always land
in the same Hamiltonian isotopy class.

  4. (4)

The surgery procedure is designed to reduce the number of intersections of
$L_{\gamma}$ with $C^{0}_{\pi/2}$. Since we rescale radially, applying Moser’s
trick does not alter the number of intersections.

###### Proof of Theorem 5.12.

As in the proof of Proposition 5.10, we restrict to the case where
$L_{\gamma}$ intersects the imaginary axis either twice (in the case of a
Clifford torus) or not at all (in the case of the Chekanov torus). Then by
Proposition 5.3, equivariant tori can only achieve finite-time singularities
at the origin $[0:0:1]$ with type I blow-up given by $C^{0}_{\pi/2}$, and type
II blow-up given by a Lawlor neck asymptotic to $C^{0}_{\pi/2}$.

Since $L_{\gamma}$ is compact, it has a finite number of intersections with
$C^{0}_{\pi/2}$. By Proposition 5.3, the number of intersections is a
decreasing function of time under mean curvature flow, and by definition neck-
to-neck surgery decreases the number of intersections with $C^{0}_{\pi/2}$.

If $L_{\gamma}$ is a Chekanov torus, Proposition 5.10 guarantees a finite-time
singularity, at which point $L_{\gamma}$ becomes a Clifford torus after
surgery. Similar to the proof of Proposition 5.10, if $L_{\gamma}$ is a
Clifford torus intersecting $C^{0}_{\pi/2}$ greater than 4 times, then
$L_{\gamma}$ has inflection points and hence cannot be minimal. So
$L_{\gamma}$ either has a finite-time singularity, or after a finite time has
only 4 intersections with $C^{0}_{\pi/2}$.

Since the number of intersections is strictly decreasing after surgery and
both the above cases end in either surgery or a reduction of the number of
intersections, after a finite time and a finite number of surgeries, we have
the minimum number of intersections. It has already been shown in Proposition
5.7 that if $L_{\gamma}$ has 4 intersections with $C^{0}_{\pi/2}$, then
$L_{\gamma}$ exists for all time and converges to $L_{1}$. The result follows.
∎

######  Remark 5.15.

As in the secondary proof of Proposition 5.10 we could work with the evolution
equations for $\int_{P}\omega$ directly, rather than just showing there are no
minimal objects and applying Proposition 2.1.

###  5.6. A Clifford torus with two singularities

Further to the result of the previous section, we also give a proof of the
following existence result:

######  Proposition 5.16.

Let $n\geq 0$ be an integer. Then

  1. (1)

There exists a Clifford torus that undergoes exactly $2n$ neck-to-neck
surgeries before converging to a minimal Clifford torus.

  2. (2)

There exists a Chekanov torus that undergoes exactly $2n+1$ neck-to-neck
surgeries before converging to a minimal Clifford torus.

We give an explicit construction for the $n=1$ case. The construction is
somewhat technical but the idea is fairly simple: construct a monotone
Clifford torus that has curvature sufficiently high in a neighbourhood of the
origin, and use a barrier to stop $\gamma$ from crossing the cone
$C^{0}_{2\pi/3}$ for long enough that a singularity is inevitable. The
constants chosen in the course of the proof are of no particular significance.

First, we prove a small lemma concerning the type I singular time of non-
monotone Chekanov tori.

######  Lemma 5.17.

Let $L_{\zeta}$ be a non-monotone equivariant torus bounding a Maslov 2 disc
$D$ of area $A=\int_{D}\omega<\pi/3$. If $\zeta$ has
$r>R(A)=\sqrt{A(\pi-2A)^{-1}}$ everywhere, then $L_{\zeta}$ has a type I
singularity away from the origin at time

$T=\frac{1}{6}\log\left(\frac{\pi}{3A-\pi}\right).$

###### Proof.

We have that

$\frac{d}{dt}\int_{D}\omega=\kappa\int_{D}\omega-2\pi,$

so denoting $f(t)=\int_{D}\omega$, we have

$f(t)=\left(A-\frac{\pi}{3}\right)e^{6t}+\frac{\pi}{3}.$

Hence the final existence time $T$ of $L_{\gamma}$ satisfies

$T\leq\frac{1}{6}\log\left(\frac{\pi}{3A-\pi}\right),$

with equality if the singularity is type I.

Now consider $L_{R(A)}$ given by $\gamma_{R(A)}(s)=R(A)e^{is}$. Equation (4.1)
reveals that $L_{R(A)}$ bounds a Maslov 4 disc of area $B=(2\pi
R(A)^{2})(1+2R(A)^{2})^{-1}$, so $B>2A$. Furthermore, a similar calculation to
above gives the final existence time $T^{\prime}$ of $L_{R(A)}$ as

$T^{\prime}=\frac{1}{6}\log\left(\frac{2\pi}{3B-2\pi}\right).$

Note that if $L_{\zeta}$ has a type II singularity, it must be at the origin
and since $L_{R(A)}$ is a barrier to $L_{\zeta}$, it must occur after
$T^{\prime}$. But $B>2A$ implies that $T^{\prime}>T$, and the result follows.
∎

###### Proof of Proposition 5.16.

We construct the $n=1$ case, i.e. an equivariant Clifford torus $L_{\gamma}$
with a finite-time singularity. The cases $n>1$ follow an iterated version of
the $n=1$ case, and the Chekanov case follows automatically from the Clifford
case.

Figure 12. The construction of a Clifford torus with a finite-time
singularity. Choosing the area of $P$ sufficiently small and the areas of $Q$
and $R$ sufficiently large guarantees a finite-time singularity.

See Figure 12 for the construction we now describe. Since $\gamma$ is
equivariant, it suffices to describe the construction only in the positive
real quadrant, i.e. the region

$Z=\\{re^{i\phi}\in\mathbb{C}:\phi\in[0,\pi/2]\\}.$

Let $\gamma$ be a equivariant curve such that $L_{\gamma}$ is a Clifford-type
torus and $\gamma$ intersects the cone $C^{0}_{2\pi/3}$ 3 times in $Z$.
Suppose $\gamma$ has the minimum of 2 inflection points, i.e. points with
$\langle\gamma,\nu\rangle=0$. Note that then there are two biangles $Q$ and
$R$ bounded by $\gamma$ and $C^{0}_{2\pi/3}$. Furthermore, there is a triangle
$P$ formed by $\gamma$ and the cone $C^{0}_{\psi}$ where $\psi$ is the widest
opening angle such that $C^{0}_{\psi}$ intersects $\gamma$ exactly 2 times in
$Z$.

We can choose $\gamma$ such that we can find two Euclidean circles
$\zeta\subset Q$ and $\eta\subset R$ each bounding discs with area $\pi/18$.
Furthermore, we can choose $\gamma$ such that $\zeta$ and $\eta$ both have
$r>R(\pi/18)$ as in the requirements of Lemma 5.17.

Furthermore, we can choose $\gamma$ such that the triangle $P$ has area
$\pi/216$. It is clear we can make these choices whilst also choosing $\gamma$
such that $L_{\gamma}$ is monotone.

The non-monotone Chekanov tori $L_{\zeta}$ and $L_{\eta}$ give lower bounds
for the final time $\int_{Q}\omega,\int_{R}\omega>0$ via direct application of
Lemma 5.17. We have that that $\int_{Q}\omega>0$ and $\int_{R}\omega>0$ (and
hence $\psi>2\pi/3$) for all times $t<T$ with

$T:=\frac{1}{6}\log\left(\frac{6}{5}\right).$

Suppose for a contradiction that the flow exists on the interval $[0,T]$. Then
the triangle $P$ exists for that period also, and the evolution of the
triangle $P$ is as in Lemma 4.4. We have that

$\frac{d}{dt}\int_{P}\omega\leq\kappa\int_{P}\omega+\left(\pi-2\psi\right)\leq\kappa\int_{P}\omega-\frac{\pi}{3},$

since $\psi$ is decreasing under the flow. Denoting
$u(t)=\kappa\int_{P}\omega-\pi/3$, we see that $u(t)$ satisfies the
differential inequality

$\frac{d}{dt}u(t)\leq\kappa u(t),$

to which we can apply Grönwall’s inequality. Since $\kappa=6$, we deduce that
$u(t)$ satisfies

$u(t)\leq u(0)e^{6t},$

and hence

$\int_{P}\omega\leq\left(\frac{\pi}{216}-\frac{\pi}{18}\right)e^{6t}+\frac{\pi}{18}=\left(1-\frac{11}{12}e^{6t}\right)\frac{\pi}{18}$

Hence the triangle $P$ has a maximum existence time of

$T^{\prime}:=\frac{1}{6}\log\left(\frac{12}{11}\right),$

which is strictly less than $T$, a contradiction. Hence a finite-time
singularity occurs in the period $[0,T]$. ∎

##  Appendix A Type II singularities and blow-ups in Lagrangian mean curvature
flow

Singularities in mean curvature flow are classified into two types based on
the rate of blow-up of the second fundamental form $A$. For a singularity at
time $T$, if

$\sup_{L_{t}}|A|^{2}\leq C(T-t)^{-1},$

for some constant $C$, we call the singularity type I, and if no such bound
exists, we call it type II. The primary reason for this distinction is the
following: for $\lambda>0$, $\tilde{x}=\lambda(x-x_{0})$,
$\tilde{t}=\lambda^{2}(t-t_{0})$,

$\tilde{F}^{\lambda}_{\tilde{t}}:=\lambda\left(F_{t_{0}+\lambda^{-2}\tilde{t}}-x_{0}\right)$

is a mean curvature flow, called a parabolic rescaling. Using Huisken’s
monotonicity formula, one can show that any sequence of parabolic rescalings
$F^{\lambda_{i}}_{\tilde{t}}$ with $\lambda_{i}\to\infty$ at a type I
singularity $(x_{0},t_{0})=(x,T)$ of the flow converges subsequentially to a
smooth limiting flow $F_{\tilde{t}}^{\infty}$, called a type I blow-up
(possibly not unique), with the property that

$\vec{H}=\frac{x^{\perp}}{2\tilde{t}}.$

Solitons of mean curvature flow of this type are called self-shrinkers since
they flow by homotheties. If the singularity is instead type II, one still can
find a weak limit to parabolic rescalings, though now the limiting flow is a
Brakke flow [Brakke1978]: a flow of rectifiable varifolds rather than smooth
manifolds. We also call this limit a type I blow-up, even though the
singularity is type II.

The most fundamental results on singularities in Lagrangian mean curvature
flow are the compactness results of Neves, originally established for zero-
Maslov Lagrangians in [Neves2007] but later extended to monotone Lagrangians
in [Neves2010]. We present them in the latter form since it is more applicable
to the subject matter of this paper. Compact monotone Lagrangians in
$\mathbb{C}^{n}$ have a maximal existence time strictly controlled by the
monotone constant. One can always normalise by homotheties of the ambient
space so that this maximal time of existence is $1/2$. Neves’ theorems concern
singularities happening before this time.

######  Theorem A.1 (Neves’ Theorem A).

Let $L$ be a normalised monotone Lagrangian in $\mathbb{C}^{n}$ developing a
singularity at $T<1/2$. For any sequence of rescaled flows $L^{j}_{s}$ at the
singularity with Lagrangian angles $\theta^{j}_{s}$, there exists a finite set
of angles $\\{\bar{\theta}_{1},\dots,\bar{\theta}_{N}\\}$ and special
Lagrangian cones $L_{1},\dots,L_{N}$ such that after passing to a subsequence
we have that for any smooth test function $\phi$ with compact support, every
$f\in C^{2}(S^{1})$ and $s<0$

$\lim_{j\to\infty}\int_{L^{j}_{s}}f(\exp(i\theta^{j}_{s}))\phi\,d\mathcal{H}^{n}=\sum_{k=1}^{N}m_{k}f(\exp(i\bar{\theta}_{k}))\mu_{k}(\phi),$

where $\mu_{k}$ and $m_{k}$ are the Radon measure of the support of $L_{k}$
and its multiplicity respectively.

Furthermore the set of angles is independent of the sequence of rescalings.

Theorem B applies for monotone Lagrangians in $\mathbb{C}^{2}$.

######  Theorem A.2 (Neves’ Theorem B).

Let $L$ be a normalised monotone Lagrangian in $\mathbb{C}^{2}$ developing a
singularity at $T<1/2$. For any sequence of rescaled flows $L^{j}_{s}$ at the
singularity with Lagrangian angles $\theta^{j}_{s}$, and for any sequence of
connected components $\Sigma_{j}$ of $L^{j}_{s}\cap B_{4R}(0)$ intersecting
$B_{R}(0)$, there exists a unique angle $\bar{\theta}$ and special Lagrangian
cone $\Sigma$ such that after passing to a subsequence we have that for any
smooth test function $\phi$ on $B_{2R}(0)$ with compact support, every $f\in
C^{2}(S^{1})$ and $s<0$

$\lim_{j\to\infty}\int_{\Sigma_{j}}f(\exp(i\theta^{j}_{s})\phi\,d\mathcal{H}^{n}=mf(\exp(i\bar{\theta}))\mu(\phi),$

where $\mu$ and $m$ are the Radon measure of the support of $\Sigma$ and its
multiplicity respectively.

Heuristically, these theorems give the type I blow-up models of type II
singularities of Lagrangian mean curvature flow as (unions of) special
Lagrangian cones. Consider the $n=2$ case: by considering the hyper-Kähler
rotation, one see that the only special Lagrangian cones are unions of special
Lagrangian planes with equal Lagrangian angle. Assuming all planes are
multiplicity 1, there is then only one blow-up model up to rotation, a union
of two transversely intersecting special Lagrangian planes with the same
Lagrangian angle.

We can further characterise singular behaviour by a procedure called the type
II blow-up. The precise details of this procedure are not important to this
paper, but we sketch the general principle. We refer the reader to Mantegazza
[mantegazza_2013] for additional details. Alternatively, the procedure is
described in depth in [wood_2020], where the first examples of Lawlor necks as
type II blow-ups were found. Instead of blowing up at a parabolic rate and at
a fixed point in time and space, we blow up at a sequence of space-time points
$(x_{i},t_{i})$ maximising the second fundamental form $|A|$ on the interval
$[0,T-1/i]$, at a rate dictated by the second fundamental form $|A|$. Thus we
guarantee convergence locally smoothly to an eternal mean curvature flow, i.e.
a mean curvature flow existing for all times $t\in(-\infty,\infty)$ (as
opposed to the self-shrinkers found by the type I procedure, which are ancient
but not eternal).

Note that the type II blow-up is not unique and doesn’t a priori have to
satisfy the same asymptotics as the type I blow-up. There are few results on
type II blow-ups for Lagrangian mean curvature flow so far, but the most
important appears in the work of Wood [Wood2019], where he shows that almost-
calibrated Lagrangian cylinders with prescribed asymptotic behaviour achieve
type II singularities in finite time, and the type II blow-up is given by a
special Lagrangian called a Lawlor neck asymptotic to the type I blow-up.

## References