# Local exotic tori

Joé Brendel  Joé Brendel, School of Mathematical Sciences, Tel Aviv University
Ramat Aviv, Tel Aviv 69978 Israel<EMAIL_ADDRESS>
###### Abstract.

Let $(X,\omega)$ be a geometrically bounded symplectic manifold of dimension
at least six. We show that every open subset of $X$ contains infinitely many
Lagrangian tori which are distinct up to symplectomorphisms of $X$ while being
Lagrangian isotopic and having the same area class.

The existence of infinitely many monotone Lagrangian tori in $\mathbb{R}^{6}$
is due to Auroux. We give a different construction of such tori and
distinguish them using the _displacement energy germ_. Our tori in
$\mathbb{R}^{6}$ come in countably many two-dimensional families, each of
which has monotone and non-monotone members. We show that no two tori from
different families can be mapped to one another by a symplectomorphism of
$\mathbb{R}^{6}$. For small enough tori (in terms of their area class) in
these families, we show that the displacement energy germ is preserved under
Darboux embeddings into $X$. We call this the _locality property_ of the
displacement energy germ. The locality property implies that the resulting
embedded tori are distinct up to symplectomorphisms of $X$.

##  1\. Introduction

This paper is motivated by the classification question of Lagrangian
submanifolds up to symplectomorphisms of the ambient symplectic manifold. The
full classification is open in most cases, for example for Lagrangian tori in
$\mathbb{R}^{4}$. In this paper we show that infinitely many Lagrangian tori,
which cannot be mapped to one another by an ambient symplectomorphism, occur
in every open set of every geometrically bounded symplectic manifold of
dimension at least six.

###  1.1. Lagrangian tori in symplectic vector spaces

Equip $\mathbb{R}^{2}=\mathbb{C}$ with the standard area form
$\omega_{0}=d\lambda_{0}$. In $\mathbb{R}^{2}$, the only closed Lagrangians
are closed embedded curves and Hamiltonian isotopies are just smooth area-
preserving isotopies with compact support. Every closed embedded curve $L$ is
isotopic to a circle $S^{1}(a)=\\{\pi|z|^{2}=a\\}$ by an area-preserving
isotopy which has compact support, where $a=\int_{L}\lambda_{0}$ is the area
enclosed by $L$. This settles the classification question of Lagrangian tori
in $(\mathbb{R}^{2},\omega_{0})$.

Let $\mathbb{R}^{2n}=\mathbb{C}^{n}$ be equipped with the product symplectic
form $\omega_{0}\oplus\ldots\oplus\omega_{0}=d\lambda$. Then we obtain a
family of Lagrangian tori, called _product tori_ (or _split tori_), by setting

$T(a_{1},\ldots,a_{n})=S^{1}(a_{1})\times\ldots\times S^{1}(a_{n}),\quad
a_{i}>0.$ (1)

Inspired by the case $n=1$, it is natural to ask if every Lagrangian torus in
$\mathbb{C}^{n}$ is Hamiltonian isotopic to a product torus.

In $\mathbb{C}^{2}$ this question was answered negatively by Chekanov [13],
who constructed the so-called _Chekanov torus_ , which cannot be mapped by a
symplectomorphism to a product torus. A Lagrangian torus in a symplectic
vector space $\mathbb{R}^{2n}=\mathbb{C}^{n}$ which cannot be mapped by a
symplectomorphism to a product torus is called _exotic_. An alternate
construction of the Chekanov torus, which heavily influenced the present
paper, was given by Eliashberg–Polterovich [20]. See also [3, 7, 15] for
different points of view on the same construction. Despite significant
progress made by Dimitroglou–Rizell [18] towards a classification, it is
unknown as of yet if there are other tori in $\mathbb{R}^{4}$ besides product
and Chekanov tori.

In $\mathbb{C}^{3}$, on the other hand, it is known that there are infinitely
many Lagrangian tori which are exotic and pairwise distinct up to
symplectomorphisms. This was proved by Auroux [5]. The Chekanov torus as well
as Auroux’s examples are _monotone_ (meaning their area and Maslov classes are
positively proportional) and appear in one-parameter families. For every
$k\in\mathbb{Z}_{\geqslant 2}$, set

$U_{k}=\left\\{(a_{1},a_{2})\in\mathbb{R}_{>0}^{2}\,\left|\,\frac{a_{1}}{k+1}\leqslant
a_{2}<\frac{a_{1}}{k}\right.\right\\}\subset\mathbb{R}^{2}.$ (2)

In §§4.1-4.2, we construct Lagrangian tori,

$\Upsilon_{k}(a)\subset\mathbb{C}^{3},\quad a\in U_{k}.$ (3)

The parameter $a=(a_{1},a_{2})$ has the following geometric significance.
There is a basis $e_{1},e_{2},e_{3}\in H_{1}(\Upsilon_{k}(a))$ of Maslov two
classes with

$\int_{e_{1}}\lambda|_{\Upsilon_{k}(a)}=a_{1}-ka_{2},\quad\int_{e_{2}}\lambda|_{\Upsilon_{k}(a)}=\int_{e_{3}}\lambda|_{\Upsilon_{k}(a)}=a_{2}.$
(4)

This means that $a$ determines the area class as defined in (5). In
particular, the torus $\Upsilon_{k}(a)$ is monotone if and only if
$a_{1}=(k+1)a_{2}$. In §4.3, we prove the following.

######  Theorem A.

For every $k\in\mathbb{Z}_{\geqslant 2}$, the torus $\Upsilon_{k}(a)$ is
exotic. Let $k\neq k^{\prime}$ and $a\in U_{k},a^{\prime}\in U_{k^{\prime}}$,
then $\Upsilon_{k}(a)$ cannot be mapped to $\Upsilon_{k^{\prime}}(a^{\prime})$
by a symplectomorphism of $\mathbb{C}^{3}$.

Our construction, as well as our method of distinguishing Lagrangians differs
from [5]. Due to the occurence of non-monotone exotic tori, Theorem A
strenghtens Auroux’s result.

######  Remark 1.1.

We do not know for which $a,a^{\prime}\in U_{k}$, the tori $\Upsilon_{k}(a)$
and $\Upsilon_{k}(a^{\prime})$ are equivalent up to a symplectomorphism of
$\mathbb{C}^{3}$. Note that even the classification of product tori up to
ambient symplectomorphisms is a subtle question, see [13, Theorem A].

######  Remark 1.2.

The monotone members of (3) were independently constructed and studied in
unpublished work by Dmitry Tonkonog and Renato Vianna; Jeff Hicks was also
aware of the construction. Furthermore, R. Vianna informed the author that it
is likely, based on computing the superpotential, that the monotone subfamily
of the $\Upsilon_{k}$ corresponds to Auroux’s examples [5].

###  1.2. Lagrangian tori in general symplectic manifolds

We consider the question of exotic tori in more general ambient spaces. In
sharp contrast to what is known in the case of $\mathbb{C}^{2}$, Vianna [51,
50] proved that there are infinitely many monotone Lagrangian tori in
$\mathbb{C}P^{2}$ and other Del Pezzo surfaces. We will discuss this in detail
in §3.3. This result was conjectured earlier by Galkin–Usnich [32] and worked
out independently by Galkin–Mikhalkin [31]. Earlier work producing finitely
many exotic Lagrangian tori in different ambient spaces includes [2, 6, 15,
22, 29, 30, 43, 52]. We refer to the introduction of [43] for an overview.

The main aim of this paper is to construct _exotic tori_ in general
(geometrically bounded) symplectic manifolds of dimension greater or equal to
six. The notion of _exoticity_ only makes sense in spaces in which there is a
natural family of Lagrangian tori, from which the _exotic_ examples are
distinguished. For example, in $\mathbb{C}^{n}$, the standard tori are product
tori; in more general toric manifolds, standard tori are toric fibres.

For a general symplectic manifold $(X,\omega)$, let us clarify the question we
are interested in.

######  Definition 1.3.

We say that two Lagrangians $L,L^{\prime}\subset(X,\omega)$ are _equivalent_
if there is $\phi\in\operatorname{Ham}(X,\omega)$ with $\phi(L)=L^{\prime}$.
This is denoted by $L\cong L^{\prime}$. We say that $L,L^{\prime}$ are
_inequivalent_ or _distinct_ if there is no
$\phi\in\operatorname{Symp}(X,\omega)$ with $\phi(L)=L^{\prime}$ and denote
this by $L\ncong L^{\prime}$.

The mix of Hamiltonian diffeomorphisms and symplectomorphisms in the
definition is justified by the fact that in all the examples we treat,
Lagrangians are distinguished in the strongest sense (up to
symplectomorphisms) or shown to be equivalent in the strongest sense (up to
Hamiltonian diffeomorphisms). We are thus looking for sets of _distinct_
Lagrangian tori.

Lagrangian tori may be distinct because they have different _soft invariants_.
By soft invariants, we mean Maslov and area class,

$m_{L}\colon H_{2}(X,L)\rightarrow\mathbb{Z},\quad\sigma_{L}\colon
H_{2}(X,L)\rightarrow\mathbb{R}.$ (5)

We have $\phi^{*}m_{\phi(L)}=m_{L}$ and $\phi^{*}\sigma_{\phi(L)}=\sigma_{L}$
for all $\phi\in\operatorname{Symp}(X,\omega)$. For example if
$\operatorname{im}\sigma_{L}\neq\operatorname{im}\sigma_{L^{\prime}}\subset\mathbb{R}$,
then $L\ncong L^{\prime}$. In light of this, let us make the following
definition.

######  Definition 1.4.

We say that $L,L^{\prime}\subset X$ are _area-equivalent (in $X$)_ if there is
a Lagrangian isotopy with endpoints $L$ and $L^{\prime}$ preserving the area
class in the sense that $\Phi^{*}\sigma_{L^{\prime}}=\sigma_{L}$, where
$\Phi\colon H_{2}(X,L)\rightarrow H_{2}(X,L^{\prime})$ is the isomorphism
induced by the Lagrangian isotopy. We write $L\sim L^{\prime}$ in that case.

The Maslov class is preserved by Lagrangian isotopies, meaning that both soft
invariants of $L$ and $L^{\prime}$ agree if $L$ and $L^{\prime}$ are area-
equivalent. Both the area and the Maslov class are preserved by Hamiltonian
isotopies (and symplectomorphisms more generally), meaning that $L\cong
L^{\prime}$ implies $L\sim L^{\prime}$. The converse gives rise to the main
question addressed in this paper.

######  Question 1.5.

For a given symplectic manifold $X$, are there Lagrangians
$L,L^{\prime}\subset X$ such that $L\sim L^{\prime}$, but $L\ncong
L^{\prime}$?

All of the exotic tori from [5, 13, 15, 51, 50] discussed so far (and all
other examples of exotic tori known to the author) yield examples answering
Question 1.5 positively. For example the Chekanov torus in $\mathbb{C}^{2}$ is
area-equivalent to a monotone product torus $T(a,a)$ for a suitable $a>0$,
depending on the size of the chosen Chekanov torus. Similarly, the tori
$\Upsilon_{k}(a)$ appearing in Theorem A are area-equivalent for different
$k$. More precisely, we prove that $\Upsilon_{k}(a_{1},a_{2})\sim
T(a_{1}-ka_{2},a_{2},a_{2})$, where $T(\cdot)$ denotes a product torus as in
(1). See also Remark 4.7.

We prove the following Theorem in §4.4. See [38, Chapter X, Definition 2.2.1]
for a definition of geometrically bounded symplectic manifolds.

######  Theorem B.

Let $(X^{6},\omega)$ be a geometrically bounded symplectic manifold and
$\psi\colon B^{6}(R)\rightarrow X$ a Darboux chart. Then there is
$\varepsilon>0$ such that

$\psi(\Upsilon_{k}(a))\ncong\psi(\Upsilon_{k^{\prime}}(a^{\prime}))$ (6)

for all $k\neq k^{\prime}$ and $a=(a_{1},a_{2})\in
U_{k},a^{\prime}=(a_{1}^{\prime},a_{2}^{\prime})\in U_{k^{\prime}}$ with
$a_{1},a_{1}^{\prime}<\varepsilon$.

Again, we have $\psi(\Upsilon_{k}(a))\sim\psi(T(a_{1}-ka_{2},a_{2},a_{2}))$ in
$X$, meaning that we find tori answering Question 1.5 in the affirmative.

The condition $a_{1}<\varepsilon$ means that the tori to which Theorem B
applies have a sufficiently small area class. The threshold $\varepsilon>0$ is
discussed in Remark 1.16. A significant drawback of this construction is the
following. For fixed $\varepsilon>0$, the area class of tori to which the
statement of Theorem B applies become smaller as $k$ increases. Indeed, this
follows from (4), which yields the area class of $\Upsilon_{k}(a)$ in terms of
$a\in U_{k}$, togther with (2). This means that in a fixed area class, Theorem
B produces only finitely many inequivalent Lagrangian tori in $(X,\omega)$.

To remedy this drawback, we construct another family of exotic tori in
$\mathbb{C}^{3}$ by lifting Vianna’s tori in $\mathbb{C}P^{2}$ to tori in
$\mathbb{C}^{3}$ via the Hopf reduction map. For every Markov triple
${\mathfrak{m}}$, meaning every triple ${\mathfrak{m}}=(\alpha,\beta,\gamma)$
of natural numbers satisfying the Diophantine equation

$\alpha^{2}+\beta^{2}+\gamma^{2}=3\alpha\beta\gamma,$ (7)

there is a monotone Lagrangian torus
$T_{{\mathfrak{m}}}\subset\mathbb{C}P^{2}$ appearing as the monotone fibre of
a certain almost toric fibration of $\mathbb{C}P^{2}$ with
$T_{{\mathfrak{m}}}\cong T_{{\mathfrak{m}}^{\prime}}$ if and only if
${\mathfrak{m}}={\mathfrak{m}}^{\prime}$, see §§3.2-3.3 or [31, 50]. In §4.5,
we show that the lifted tori
$\Theta_{{\mathfrak{m}}}(a)=p_{a}^{-1}(T_{{\mathfrak{m}}})\subset\mathbb{C}^{3}$
by the reduction map $p_{a}\colon\mathbb{C}^{3}\supset
S^{5}(a)\rightarrow(\mathbb{C}P^{2},a\omega_{\mathbb{C}P^{2}})$ have the same
property. Note that these tori are monotone, as well.

######  Proposition 1.6.

Let ${\mathfrak{m}},{\mathfrak{m}}^{\prime}$ be Markov triples, then
$\Theta_{{\mathfrak{m}}}(a)\cong\Theta_{{\mathfrak{m}}^{\prime}}(a)$ if and
only if ${\mathfrak{m}}={\mathfrak{m}}^{\prime}$.

This result was expected and may have been known to experts. What is
interesting in the context of the present paper is that this property persists
under Darboux embeddings into geometrically bounded symplectic manifolds,
provided $a>0$ is small enough.

######  Theorem C.

Let $(X^{6},\omega)$ be a geometrically bounded symplectic manifold and
$\psi\colon B^{6}(R)\rightarrow X$ a Darboux chart. Then there is
$\varepsilon>0$ such that

$\psi(\Theta_{{\mathfrak{m}}}(a))\ncong\psi(\Theta_{{\mathfrak{m}}^{\prime}}(a)),\quad\psi(\Theta_{{\mathfrak{m}}}(a))\sim\psi(\Theta_{{\mathfrak{m}}^{\prime}}(a))$
(8)

for all $a<\varepsilon$ and ${\mathfrak{m}}\neq{\mathfrak{m}}^{\prime}$.

See §4.6 for a proof. Since the set of Markov triples is infinite, Theorem C
shows that $(X,\omega)$ contains infinitely many distinct Lagrangian tori in a
(small enough) fixed area class. Thus Theorem C remedies the drawback of
Theorem B pointed out above. The idea used to upgrade Proposition 1.6 to
Theorem C is the same as the one used to upgrade Theorem A to Theorem B, and
will be discussed in detail in §1.6. Again, we refer to Remark 1.16 for a
discussion of the size $\varepsilon>0$ of tori to which Theorem C applies.

######  Remark 1.7.

Note that, although we will not consider these in detail, the lifted Vianna
tori $\Theta_{{\mathfrak{m}}}$ can be extended to natural two-parameter
families, by lifting non-monotone Vianna tori from $\mathbb{C}P^{2}$ to
$\mathbb{C}^{3}$. These were for example considered in a paper by
Shelukhin–Tonkonog–Vianna [47], see also Example 3.7. Although these two-
parameter families can be distinguished by their displacement energy germ by
the methods used in this paper, we restrict our attention to the one-parameter
families $\\{\Theta_{{\mathfrak{m}}}(a)\\}_{a>0}$ coming from lifts of
monotone Vianna tori to simplify the exposition.

######  Remark 1.8.

The families $\Upsilon_{k}$ and $\Theta_{{\mathfrak{m}}}$ are disjoint up to
one exception in the sense that $\Upsilon_{k}\ncong\Theta_{{\mathfrak{m}}}$
for all $k\geqslant 3$ and all Markov triples ${\mathfrak{m}}\neq(2,1,1)$. The
exception is $k=2$ and ${\mathfrak{m}}=(2,1,1)$, in which case
$\Upsilon_{2}(3a,a)\cong\Theta_{(2,1,1)}(3a)$ for all $a>0$. The same
statements hold for the embeddings of the tori into geometrically bounded
symplectic manifolds by Darboux charts. The distinction of the tori follows
from the fact that they have inequivalent displacement energy germs, see §4.3
and §4.5 for $\Upsilon_{k},\Theta_{{\mathfrak{m}}}\subset\mathbb{C}^{3}$ and
§4.4 and §4.6 for their embeddings into tame symplectic manifolds.

###  1.3. Higher dimensions

For simplicity, we have restricted our attention to dimension six in the above
discussion. Since our methods behave well when taking products, one obtains
the statements of Theorems A, B and C in higher dimensions by considering tori
of the form $L\times T(c_{1},\ldots,c_{m})\subset\mathbb{C}^{m+3}$, where
$L\subset\mathbb{C}^{3}$ is a $\Upsilon_{k}$\- or
$\Theta_{{\mathfrak{m}}}$-torus. The so-obtained tori can be distinguished
from one another, provided the $c_{i}$ are large enough for the displacement
energy germ of the $L$-factor to be visible in the displacement energy germ of
the product, see Remark 4.13 for details. Furthermore, similar methods can be
used to distinguish products of exotic tori of either $\Upsilon_{k}$\- or
$\Theta_{{\mathfrak{m}}}$-type. Using products, we prove the following.

######  Theorem D (Local exotic tori).

Let $X$ be a geometrically bounded symplectic manifold of dimension at least
six and $U\subset X$ an open subset. Then $U$ contains infinitely many area-
equivalent tori which are distinct up to symplectomorphisms of $X$.

These local exotic tori naturally appear in $(n-1)$-dimensional families,
where $\dim X=2n$, which seems to be a general feature of exotic tori, see
Remark 4.12 and Question 5.2. Beyond Theorem D, we do not give a comprehensive
treatment of tori obtained as products from known tori. All proofs in that
direction follow the same pattern as the proof of Theorem D given in §4.7.

###  1.4. Key ideas: construction

The idea for the construction of the tori $\Upsilon_{k}$ is inspired by the
Eliashberg–Polterovich [20] construction of the Chekanov torus in
$\mathbb{C}^{2}$ and its subsequent interpretations by Auroux [3, 4] and
Chekanov–Schlenk [15]. Let us review this construction of the Chekanov torus
in $\mathbb{C}^{2}$ using the language of toric geometry and symplectic
reduction and illustrate how it inspires the construction of the
$\Upsilon_{k}$-tori. For more details on the point of view discussed here, we
refer to the expository paper [7].

Let $\ell=\\{(x,x)\\}\subset\mathbb{R}^{2}_{\geqslant 0}$ be the diagonal in
the moment map image $\mathbb{R}^{2}_{\geqslant 0}$ of $\mathbb{C}^{2}$ under
the standard moment map $\mu(z_{1},z_{2})=(\pi|z_{1}|^{2},\pi|z_{2}|^{2})$.
Note that the regular fibres of this moment map are precisely the product tori
$T(a_{1},a_{2})\subset\mathbb{C}^{2}$. To the ray $\ell$, we can naturally
associate a symplectic quotient space111In general, every rational affine
subspace $V\subset\mathbb{R}^{n}$ intersected with a toric moment polytope
$\Delta\subset\mathbb{R}^{n}$ gives rise to a (possibly singular) symplectic
reduction by the action of the orthogonal torus $T_{V}=\exp(V^{\perp})\subset
T^{n}$ and has as reduced space a (possibly singular) toric symplectic
manifold. We refer to [8] for details on _toric reduction_. by viewing $\ell$
as the projection of the level set $\\{H=0\\}$ of the Hamiltonian
$H=\pi|z_{1}|^{2}-\pi|z_{2}|^{2}$ generating the Hamiltonian $S^{1}$-action
_orthogonal_ to $\ell$. Note that the level set $\\{H=0\\}$ contains the fixed
point $0\in\mathbb{C}^{2}$, which we remove to obtain the reduced space
$(\\{H=0\\}\setminus\\{0\\})/S^{1}$ symplectomorphic to the punctured standard
plane $(\mathbb{C}^{\times},\omega_{0}|_{\mathbb{C}^{\times}})$. On one hand,
for $a>0$ the product torus $T(a,a)\subset\\{H=0\\}$ projects to the circle
$S^{1}(a)\subset\mathbb{C}^{\times}$. On the other hand one can lift closed
embedded curves from the reduced space to obtain tori in $\mathbb{C}^{2}$. The
Chekanov torus $T_{\rm Ch}^{2}(a)$ is obtained by lifting a curve in
$\mathbb{C}^{\times}$ bounding a disk of symplectic area $a>0$ in
$\mathbb{C}^{\times}$. Note that such circles are not isotopic to standard
circles $S^{1}(a)$ in $\mathbb{C}^{\times}$, since the latter enclose the
puncture and the former do not.

The tori $\Upsilon_{k}(a)$ are similarly defined by choosing suitable families
of rational segments

$\ell_{k}(a_{1})=\\{(a_{1},0,0)+t(-k,1,1)\,|\,t\in\mathbb{R}\\}\cap\mathbb{R}^{3}_{\geqslant
0},\quad k\in\mathbb{Z}_{\geqslant 2},a_{1}>0$ (9)

in the moment polytope $\mathbb{R}^{3}_{\geqslant 0}$ of $\mathbb{C}^{3}$ and
performing $T^{2}$-reduction on their preimages. The reduced space is a sphere
of area $\frac{a_{1}}{k}$ with a puncture and an orbifold point of order $k$
and the torus $\Upsilon_{k}(a_{1},a_{2})$ is defined as the lift of a closed
embedded curve bounding a disk of area $0<a_{2}<\frac{a_{1}}{k}$ which does
not contain the orbifold point (nor the puncture). We refer to §§4.1-4.2 for
more details.

######  Remark 1.9.

The segments $\ell_{k}(a_{1})$ intersect the edge of $\mathbb{R}_{\geqslant
0}^{3}$ given by the first coordinate axis. The same construction can be
carried out for any face of codimension $\geqslant 2$ in the moment polytope
$\mathbb{R}_{\geqslant 0}^{n}$ of $\mathbb{C}^{n}$. We do not discuss the
general construction here, but instead we rely on taking products to produce
tori in higher dimensions, see §1.3 and §4.7.

###  1.5. The invariant

The invariant we use to distinguish Lagrangian tori is the so-called
_displacement energy germ_. It is a strengthened symplectic invariant of
Lagrangian submanifolds derived from displacement energy via _versal
deformations_. Versal deformations of Lagrangian submanifolds were first
introduced by Chekanov [13]. See §2.4 for a detailed introduction to the
displacement energy germ and [9, 7, 15, 16] for further applications. Here we
give a brief discussion.

The _displacement energy germ_ of a closed Lagrangian $L\subset(X,\omega)$ is
the function

${\mathcal{E}}_{L}\colon H^{1}(L;\mathbb{R})\rightarrow\mathbb{R}_{\geqslant
0}\cup\\{+\infty\\},$ (10)

defined as a germ at $0\in H^{1}(L;\mathbb{R})$ of the displacement energy
$e(\cdot)\in\mathbb{R}\cup\\{+\infty\\}$ viewed as a function on the space of
Lagrangians. By Weinstein’s neighbourhood theorem, a neighbourhood of $L$ in
the space of Lagrangians is parametrized by $C^{1}$-small closed differential
one-forms on $L$. Furthermore, the graphs of two such forms are Hamiltonian
isotopic in the Weinstein neighbourhood if and only if the corresponding forms
are cohomologuous. The germ of Lagrangian neighbours of $L$ up to the germ of
Hamiltonian isotopies is thus in natural bijection with $H^{1}(L;\mathbb{R})$.
Since $e(\cdot)$ is invariant under symplectomorphisms ${\mathcal{E}}_{L}$ is
well defined. The displacement energy germ is a symplectic invariant in the
sense that

${\mathcal{E}}_{\phi(L)}={\mathcal{E}}_{L}\circ(\phi|_{L})^{*},\quad\phi\in\operatorname{Symp}(X,\omega).$
(11)

In practice, one picks a _versal deformation_ of $L$, meaning a continuous
family of Lagrangians $b\mapsto v_{L}(b)$ with $b\in U\subset
H^{1}(L;\mathbb{R})$ which are parametrized by their Lagrangian flux, see
Definition 2.12. Versal deformations induce the bijection discussed above and
thus, to compute ${\mathcal{E}}_{L}$, one can take the germ at the origin
$0\in H^{1}(L;\mathbb{R})$ of the function $b\mapsto e(v_{L}(b))$. This is
independent of the choice of versal deformation.

In our case, $L$ is diffeomorphic to a torus and thus
$H^{1}(L;\mathbb{R})\cong\mathbb{R}^{n}$ and we obtain an invariant up to
transformations $(\phi|_{L})^{*}\in\operatorname{GL}(n;\mathbb{Z})$.
Furthermore, in applications, we usually determine $b\mapsto e(v_{L}(b))$ on
an open dense subset of a neighbourhood of the origin $0\in\mathbb{R}^{n}$.
This yields ${\mathcal{E}}_{L}$ on an open dense subset of $\mathbb{R}^{n}$,
which is sufficient to distinguish Lagrangians. We make the following
definition.

######  Definition 1.10.

Let $L,L^{\prime}\subset X$ be Lagrangian tori. We say that
${\mathcal{E}}_{L},{\mathcal{E}}_{L^{\prime}}\colon\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\\{+\infty\\}$
are _equivalent_ if there is $\Phi\in\operatorname{GL}(n;\mathbb{Z})$ such
that ${\mathcal{E}}_{L^{\prime}}={\mathcal{E}}_{L}\circ\Phi$ on an open dense
subset of $\mathbb{R}^{n}$. We write
${\mathcal{E}}_{L}\sim{\mathcal{E}}_{L^{\prime}}$.

###  1.6. Key ideas: distinguishing tori

Let us first discuss how to distinguish tori in $\mathbb{C}^{n}$. Let
$L=\Upsilon_{k}\subset\mathbb{C}^{3}$ (see Theorem A),
$L=\Theta_{\mathfrak{m}}\subset\mathbb{C}^{3}$ (see Proposition 1.6), or
$L=\Theta_{{\mathfrak{m}}}\times T(c_{4},\ldots,c_{n})\subset\mathbb{C}^{n}$
(see the proof of Theorem D in §4.7). To compute ${\mathcal{E}}_{L}$, we use
the _instability of exotic tori_ , by which we mean that most tori which are
$C^{1}$-close to $L$ are equivalent to product tori in $\mathbb{C}^{n}$ and
thus not exotic themselves. We note that this remarkable property is shared by
all exotic tori known to the author. More precisely, for every neighbourhood
$U\subset H^{1}(L;\mathbb{R})\cong\mathbb{R}^{n}$ of the origin, there is an
open dense subset $U^{\prime}\subset U$ such that the versal deformation
$v_{L}(b)$ is equivalent to a product torus for all $b\in U^{\prime}$. The
displacement energy of product tori is known, see Example 2.7,

$e(T(a))=\underline{a},\quad a=(a_{1},\ldots,a_{n})\in\mathbb{R}_{>0}^{n},$
(12)

where $\underline{a}=\min\\{a_{1},\ldots,a_{n}\\}$ denotes the minimum
throughout the paper. We can compute ${\mathcal{E}}_{L}$ on an open and dense
subset by determining the bijection $\sigma\colon
U^{\prime}\rightarrow\sigma(U^{\prime})\subset\mathbb{R}^{n}$ for which

$v_{L}(b)\cong T(\sigma(b)),\quad b\in U^{\prime}.$ (13)

Interestingly enough, in all examples we consider, the map $\sigma$ is
piecewise linear and the linear components are all in
$\operatorname{GL}(n;\mathbb{Z})$. Note that it is sufficient to have
${\mathcal{E}}_{L}\nsim{\mathcal{E}}_{L^{\prime}}$ to conclude $L\ncong
L^{\prime}$. See (76) for the displacement energy germ of the
$\Upsilon_{k}$-tori and (84) for the case of the
$\Theta_{{\mathfrak{m}}}$-tori.

######  Remark 1.11.

The idea of using the instability of exotic tori to compute invariants coming
from versal deformations is due to Chekanov [13]. See [15, Section 3] for an
explicit computation for the Chekanov torus in $S^{2}\times S^{2}$ and see [7,
Section 5] for the case of the Chekanov torus in $\mathbb{R}^{4}$.

To distinguish embeddings of $L$ into a geometrically bounded symplectic
manifold $(X,\omega)$ by a Darboux chart $\psi\colon B^{2n}(R)\rightarrow X$
(see Theorems B, C and D) we show that

${\mathcal{E}}_{\psi(L)}\sim{\mathcal{E}}_{L},$ (14)

provided $L$ is small enough. We stress that ${\mathcal{E}}_{\psi(L)}$ is
computed from displacement energy in $X$, whereas ${\mathcal{E}}_{L}$ is
computed from displacement energy in $\mathbb{C}^{n}$. If (14) holds, then the
image under Darboux embeddings of the tori
$\Upsilon_{k},\Theta_{{\mathfrak{m}}},\Theta_{{\mathfrak{m}}}\times T(c)$ are
still distinguished by the displacement energy germ.

######  Definition 1.12.

Let $L\subset\mathbb{C}^{n}$ be a Lagrangian. We say that an embedding
$\psi(L)$ under a Darboux chart $\psi$ has the _locality property (for the
displacement energy germ)_ if (14) holds.

The main idea to prove the locality property is to combine the _instability of
exotic tori_ with the fact that, under certain conditions, _the displacement
energy of product tori is preserved under Darboux embeddings_. This goes back
to Chekanov–Schlenk [16]. Let us make some definitions first. Let
$J\in{\mathcal{J}}_{\omega}$ be an $\omega$-tame almost complex structure. Let
$\lambda_{S}(X,J)$ be the infimum over areas of non-constant $J$-holomorphic
spheres in $X$. If $\lambda_{S}$ is finite, then, by Gromov compactness, it is
positive and realized by a $J$-holomorphic sphere. Let $\psi\colon
B(R)\rightarrow(X,\omega)$ be a Darboux chart. We will work with $\omega$-tame
almost complex structures which agree with the push-forward of the standard
complex structure $J_{0}$ on $\operatorname{im}\psi=\psi(B(R))$. The following
quantity will play a crucial role in the proofs of Theorems B and C,

$\lambda_{S}(X,\psi)=\sup\\{\lambda_{S}(X,J)\,|\,J\in{\mathcal{J}}_{\omega},J|_{\operatorname{im}\psi}=\psi_{*}J_{0}\\}.$
(15)

######  Definition 1.13.

We say that a Lagrangian torus $L\subset(X,\omega)$ has _property (CS)_ if
there is a Darboux chart $\psi\colon B(R)\rightarrow(X,\omega)$ with
$L=\psi(T(a))$ such that

$a_{1}+\ldots+a_{n}+\underline{a}<R,$ (16)

and

$\underline{a}<\lambda_{S}(X,\psi).$ (17)

In the aspherical case, the following proposition goes back to [16,
Proposition 2.1].

######  Proposition 1.14 (Chekanov–Schlenk).

Let $L=\psi(T(a))\subset X$ be a Lagrangian torus with property (CS), then

$e_{X}(L)=e_{\mathbb{C}^{n}}(T(a))=\underline{a}.$ (18)

See §2.3 for a proof. The following is a general criterion for when the
locality property (14) holds.

######  Lemma 1.15.

Let $L^{\prime}\subset B(R)$ be a Lagrangian torus and $\psi\colon
B(R)\rightarrow(X,\omega)$ a Darboux chart. If there is a versal deformation
$v_{L^{\prime}}$ defined on $U$ such that $v_{L^{\prime}}(b)\cong
T(\sigma(b))$ inside $B(R)$ for all $b$ in an open dense subset
$U^{\prime}\subset U$ and such that all $\psi(T(\sigma(b)))$ have property
(CS), then the image $L=\psi(L^{\prime})$ has the locality property.

Proof. We identify $H^{1}(L;\mathbb{R})$ and $H^{1}(L^{\prime};\mathbb{R})$
via $\psi$. The versal deformation $v_{L^{\prime}}$ induces a natural versal
deformation $v_{L}=\psi\circ v_{L^{\prime}}$ of $L=\psi(L^{\prime})$ and
${\mathcal{E}}_{L}$ is the germ of $b\mapsto e_{X}(v_{L}(b))$. For all $b\in
U^{\prime}$, we find

$e_{X}(v_{L}(b))=e_{X}(\psi(v_{L^{\prime}}(b)))=e_{X}(\psi(T(\sigma(b))))=e_{\mathbb{C}^{n}}(T(\sigma(b)))=e_{\mathbb{C}^{n}}(v_{L^{\prime}}(b)).$
(19)

The third equality follows from Proposition 1.14. Since
${\mathcal{E}}_{L^{\prime}}$ is the germ of $b\mapsto
e_{\mathbb{C}^{n}}(v_{L^{\prime}}(b))$, this proves
${\mathcal{E}}_{L}\sim{\mathcal{E}}_{L^{\prime}}$ and therefore the locality
property. $\Box$

For a fixed Darboux chart $\psi\colon B(R)\rightarrow X$, we prove Theorems B,
C and D by showing that the respective families of tori contain members which
are small enough such that their image under $\psi$ satisfies the hypotheses
of Lemma 1.15.

######  Remark 1.16.

In Theorems B and C, the constant $\varepsilon>0$, determining the size of the
tori which can be distinguished by our methods, is the minimum of a linear
function of the size $R$ of the Darboux chart and $\lambda_{S}(X,\psi)$. These
two terms in the minimum reflect the conditions (16) and (17) for the versal
deformations to have property (CS). Note, that for fixed $\psi$, the only
input coming from the ambient manifold in the statements of Theorems B and C
is the constant $\lambda_{S}(X,\psi)$.

######  Remark 1.17.

If $(X,\omega)$ is symplectically aspherical, we can refine the statements of
Theorems B, C and D. In this case $\lambda_{S}(X,\psi)=\infty$ and hence we
can pick a sequence of ball embeddings whose sizes tend towards the Gromov
width $w=w_{G}(X,\omega)$ to obtain $\varepsilon=\frac{w}{2}$ in Theorem B and
$\varepsilon=\frac{3w}{4}$ in Theorem C. Similarly, the statement of Theorem D
applies to all tori $\psi(\Theta_{{\mathfrak{m}}}^{n}(a,a_{4},\ldots,a_{n}))$
with $\frac{4a}{3}+a_{4}+\ldots+a_{n}<w$. For details, see the proofs in §4.4,
§4.6 and §4.7, respectively.

######  Remark 1.18.

Let $X$ be aspherical. If $L\subset\mathbb{C}^{n}$ is monotone, then so is its
image $\psi(L)\subset X$ under the Darboux chart. In that case we obtain one-
parameter families of monotone exotic tori in $X$ from Theorems B, C and D.

######  Remark 1.19.

If $X$ is toric, it contains a natural family of Lagrangian tori appearing as
free orbits of the Hamiltonian $T^{n}$-action, or equivalently, regular fibres
of the moment map. These tori are called _toric fibres_. We call a Lagrangian
torus $L\subset X$ _exotic_ if it is not equivalent to any toric fibre. In
case $X$ is monotone toric and the displacement energy of its toric fibres is,
on an open and dense subset of the moment polytope, given by the integral
distance to the boundary of its moment polytope, then $X$ contains infinitely
many distinct exotic tori, see Corollary 2.20. The condition on displacement
energies is conjecturally true for all monotone toric manifolds and has been
checked for all $X$ with $\dim X\leqslant 18$, see Example 2.8 or [9, §3.4]
for more details

######  Remark 1.20.

As discussed in Example 2.16, every Lagrangian torus $L=\psi(T(a))\subset X$
which has property (CS) has displacement energy germ
${\mathcal{E}}_{L}\sim{\mathcal{E}}_{T(a)}$. Having the same displacement
energy germ as product tori, we may consider (CS)-tori to be the _standard
tori_ in general geometrically bounded symplectic manifolds against which to
compare exotic tori. Indeed, as a corollary of the proofs of Theorems B and C,
we obtain that neither the $\Upsilon_{k}$-, nor the
$\Theta_{{\mathfrak{m}}}$-tori are equivalent to a torus having property (CS).
Thus we recover an absolute notion of what exoticity means.

###  1.7. Discussion and further questions

In geometrically bounded symplectic manifolds of dimension at least six, the
existence of an infinite collection of tori which are area-equivalent but
pairwise distinct is a _local_ phenomenon, see Theorem D. Relative to the
current state of knowledge, this is in contrast to the situation in dimension
four. As of now – meaning as long as no other tori besides product and
Chekanov are found in $\mathbb{C}^{2}$ – the existence of infinitely many
exotic tori in four-dimensional symplectic manifolds is a _global_ phenomenon
in the sense that it is highly dependent on the geometry of the ambient space.
As illustrated in Example 5.6, even embedding the Chekanov torus into a
manifold by a Darboux chart does not yield a torus which is exotic in a robust
way. However, in dimension four there is a semi-local analogue of the
constructions discussed in this paper. Instead of using Darboux charts, i.e.
ball embeddings, one can use symplectic embeddings of more complicated domains
into $X^{4}$ in order to construct exotic Lagrangian tori. This is ongoing
work joint with Johannes Hauber and Joel Schmitz. See §5.2 for more details.

The second main point of this paper is that it sheds some light on the
(un-)importance of monotonicity when it comes to the study of exotic tori.
Most exotic tori which have been studied are monotone, see [5, 9, 13, 15, 31,
51, 50], with the notable exceptions of [29] and [47]. However, we make the
case that this circumstance is mainly an artifact of the methods used to
distinguish Lagrangians. In contrast to the superpotential counting Maslov-two
$J$-holomorphic disks, the displacement energy germ does not require any
adaptations in order to yield a well-defined invariant in the case of non-
monotone Lagrangians. In some sense, the tori studied in Theorems B, C and D
are the very opposite of monotone, since they bound very small Maslov two
disks inside the Darboux chart as opposed to potentially very large disks
coming from the ambient space. The fact that the embedded tori in Theorems B,
C and D are chosen small relative to the size of the ball embedding contrasts
with the phenomenon of exotic tori _untwisting_ when they are given enough
space, see for example [15, §4].

An interesting direction of inquiry raised in §5.3 is that of interpreting the
$\Upsilon_{k}$-tori as fibres of Lagrangian torus fibrations on
$\mathbb{C}^{3}$ arising from mutations of its toric structure, just as the
Vianna tori in $\mathbb{C}P^{2}$ appear as fibres of almost toric mutations of
the toric structure on $\mathbb{C}P^{2}$. This seems to lead to an analogue of
almost toric fibrations in higher dimensions which would provide a framework
to carry out higher mutations in arbitrary dimensions to produce more exotic
tori. Taking fibres of such mutated Lagrangian torus fibrations should provide
a general framework in which both the $\Upsilon_{k}$\- and the
$\Theta_{{\mathfrak{m}}}$-tori appear as special cases.

###  1.8. Structure of the paper

Section 2 provides background on symplectic reduction, integral geometry,
displacement energy and the displacement energy germ which is used in later
sections. Section 3 gives an overview of _almost toric fibrations_ , mutations
of ATFs on $\mathbb{C}P^{2}$ and a computation of the displacement energy germ
of certain almost toric fibres. The approach we follow in §3.3 was suggested
to us by Felix Schlenk based on discussions with Yuri Chekanov. The
displacement energy germ of almost toric fibres in $\mathbb{C}P^{2}$ is used
to distiguish the tori $\Theta_{{\mathfrak{m}}}(a)\subset\mathbb{C}^{3}$ and
their embeddings, see Proposition 1.6 and Theorems C, D. The heart of the
paper can be found in Section 4, which contains the construction of the
$\Upsilon_{k}$-tori, as well as all proofs. In Section 5, we discuss open
questions, we give an outlook on future work in dimension four and speculate
on mutations of Lagrangian torus fibrations in dimensions greater than or
equal to six, which may lead to a theory of almost toric fibrations in these
dimensions.

### Acknowledgements

We thank Denis Auroux, Jonny Evans, Jeff Hicks, Yael Karshon, Leonid
Polterovich and Joel Schmitz for interesting discussions, encouragement and
insightful comments on an earlier draft. We thank Renato Vianna for sharing
insights into his discussions joint with Jeff Hicks and Dmitry Tonkonog, see
Remark 1.2. We thank Felix Schlenk for detailed comments on an earlier draft
and for generously sharing his ideas joint with Yuri Chekanov, in particular
those surrounding §3.3. We are endebted to Paul Biran for a crucial remark
pointing out a mistake after the author’s talk at ETH in October 2022. We
acknowledge the support of the Israel Science Foundation grant 1102/20 and the
ERC Starting Grant 757585.

##  2\. Background

###  2.1. Symplectic reduction

In this subsection we introduce some notions surrounding symplectic reduction
which will be used in subsequent sections. Although most properties hold in
the more general context of Hamiltonian actions of compact Lie groups, we
restrict our attention to Hamiltonian torus actions. We refer to [12, 35, 41,
42] for more details on this classical topic. Let $(X,\omega)$ be a symplectic
manifold equipped with a Hamiltonian $T^{k}$-action, generated by the moment
map $\mu\colon X\rightarrow\mathbb{R}^{k}$. If the restriction of the
$T^{k}$-action to a level set $\mu^{-1}(c)$ is free, then the corresponding
level set is smooth and its quotient $X_{c}=\mu^{-1}(c)/T^{k}$ is a smooth
manifold equipped with a naturally induced symplectic form $\omega_{c}$
satisfying (21). This procedure is called _symplectic reduction_ and the
symplectic manifold $(X_{c},\omega_{c})$ is called _symplectic quotient_. We
will use a slightly generalized version of symplectic reduction, where we
allow for group actions having finite stabilizers. In that case, the reduced
space is a symplectic orbifold, see for example [39] for details. Symplectic
reduction is best summarized by a diagram,

${\mu^{-1}(c)}$${(X,\omega)}$${´(X_{c},\omega_{c}),}$$\scriptstyle{\pi}$ (20)

with

$\pi^{*}\omega_{c}=\omega|_{\mu^{-1}(c)}.$ (21)

A crucial feature of symplectic reduction we will use repeatedly in this paper
is the fact that one can lift Lagrangian submanifolds and Hamiltonian
isotopies in a natural way from the reduced space $X_{c}$ to $X$, see for
example [1, 9, 8].

######  Proposition 2.1.

Let $L\subset X_{c}$ be a Lagrangian, then
$\pi^{-1}(L)\subset\mu^{-1}(c)\subset X$ is a $T^{k}$-invariant Lagrangian.
Let ${\varphi_{t}}$ be a Hamiltonian isotopy of $X_{c}$, then there is a
$T^{k}$-equivariant Hamiltonian isotopy $\\{\phi_{t}\\}$ satisfying
$\pi\circ\phi_{t}=\varphi_{t}\circ\pi$.

Similarly, invariant Lagrangians and equivariant Hamiltonian isotopies in $X$
project to the symplectic quotient $X_{c}$. For Hamiltonian group actions we
have the following.

######  Proposition 2.2.

Let $\nu\colon X\rightarrow\mathbb{R}^{l}$ be another Hamiltonian
$T^{l}$-action which commutes with the $T^{k}$-action, i.e.
$\\{\mu_{i},\nu_{j}\\}=0$ for all $i,j$. Then there is an induced Hamiltonian
$T^{l}$-action on the symplectic quotient $X_{c}$ generated by $\nu_{c}\colon
X_{c}\rightarrow\mathbb{R}^{l}$ with $\pi^{*}\nu_{c}=\nu|_{\mu^{-1}(c)}$ for
which $\pi$ is $T^{l}$-equivariant.

A special case of this is _toric reduction_. A toric manifold is a symplectic
$2n$-manifold $(X,\omega)$ equipped with an effective Hamiltonian
$T^{n}$-aciton. The image of the moment map
$\Delta=\mu(X)\subset\mathbb{R}^{n}$ is a polytope with special properties, a
so-called _Delzant polytope_ , whose integral affine geometry classifies
$(X,\omega,\mu)$ up to equivariant symplectomorphisms by the Delzant Theorem
[17]. We refer to [23, 34] for more details. Performing symplectic reduction
with respect to a subtorus $K\subset T^{n}$ of the toric action, we are in the
context of Proposition 2.2, since the actions of $K$ and $T^{n}$ commute. The
quotient $X_{c}$ carries an effective $T^{n}/K$-action and is thus a toric
manifold in its own right. On the level of moment polytopes, toric reduction
is encoded by an integral affine embedding of the Delzant polytope of $X_{c}$
into $\Delta$. The Delzant construction of toric manifolds [34, Chapter 1] and
McDuff’s probes [40] are special cases of toric reduction. The symplectic
reductions in §4.1 are of this type, too, and can be seen as performing toric
reduction on a family of segments in the moment polytope
$\mathbb{R}^{3}_{\geqslant 0}$ of $\mathbb{C}^{3}$. We refer to [8, §2.2] for
more details.

Regular fibres $\mu^{-1}(x)$ of toric moment maps are Lagrangian tori, see
also Example 2.17. As a special case of Proposition 2.1, recall from [8,
Proposition 2.8] that toric fibres lift to toric fibres under the toric
reduction.

######  Proposition 2.3.

Let $(X_{c},\omega_{c},\mu_{c})$ be the toric reduction of a toric
$(X,\omega,\mu)$ with $\iota\colon\Delta_{c}\hookrightarrow\Delta$ the
corresponding integral affine embedding of Delzant polytopes. Then

$\mu^{-1}(\iota(x))=\pi^{-1}(\mu_{c}^{-1}(x)),\quad
x\in\operatorname{int}\Delta_{c}.$ (22)

###  2.2. Some integral geometry

Integral geometry is the geometry associated to transformations preserving a
lattice of maximal rank in a finite-dimensional vector space. We restrict our
attention to the lattice $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$ which has
symmetry group $\operatorname{GL}(n;\mathbb{Z})$. Our interest in integral
geometry comes from the fact that, as discussed in §1.5 and §2.4, the
displacement energy germ of Lagrangian tori is an invariant up to
$\operatorname{GL}(n;\mathbb{Z})$. For related reasons, the regular locus of
the base of any Lagrangian torus fibration has an _integral affine structure_.
This means that the Lagrangian torus fibration induces a manifold atlas with
transition charts in $\operatorname{GL}(n;\mathbb{Z})\ltimes\mathbb{R}^{n}$,
which essentially comes from action coordinates and the Arnold–Liouville
theorem, see [23, §2.3]. The integral affine structure of Lagrangian torus
fibrations is intimately related to the versal deformations of its regular
fibres, see Proposition 2.14.

A vector $v\in\mathbb{Z}^{n}$ is called _primitive_ if its entries are
coprime. Let $w\in\mathbb{R}^{n}$ be a vector with a rational direction. Then
there is a unique $\alpha>0$ such that $w=\alpha v$ for a primitive
$v\in\mathbb{Z}^{n}$ and $\alpha$ is called _integral affine length of $w$_
and denoted by $\ell_{\rm IA}(w)=\alpha$. Note that this corresponds to the
Euclidean length of the image of $w$ under a map in
$\operatorname{GL}(n;\mathbb{Z})$ mapping $w$ into a coordinate axis of
$\mathbb{R}^{n}$. Let $P\subset\mathbb{R}^{n}$ be a rational affine
hyperplane. Then the _integral affine distance_ $d_{\operatorname{IA}}(x,P)$
of a point $x$ to $P$ is defined as the Euclidean distance of $\phi(x)$ to
$\\{x_{n}=0\\}$, where
$\phi\in\operatorname{GL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ is an integral
affine transformation with $\phi(P)=\\{x_{n}=0\\}$. This is independent of the
choice of $\phi$. Write $P=\\{y\in\mathbb{R}^{n}\,|\,\langle
y,v\rangle+\lambda=0\\}$ for a primitive $v\in\mathbb{Z}^{n}$ and
$\lambda\in\mathbb{R}$, then $d_{\rm IA}(x,P)=|\langle x,v\rangle+\lambda|$.
Now let

$\Delta=\\{x\in\mathbb{R}^{n}\;|\;\ell_{i}(x)\geqslant
0\\},\quad\ell_{i}(x)=\langle x,v_{i}\rangle+\lambda_{i}.$ (23)

be a convex polytopes defined by the intersection of half-spaces delimited by
rational affine hyperplanes $P_{i}=\\{\ell_{i}(y)=0\\}$ for
$i\in\\{1,\ldots,N\\}$. Again, the $v_{i}\in\mathbb{Z}^{n}$ are chosen to be
primitive. They can be thought of as the inwards pointing normal vectors to
the facets of $\Delta$. Then the _integral affine distance_ $d_{\rm
IA}(x,\partial\Delta)$ of $x\in\Delta$ to the boundary $\partial\Delta$ is
defined as

$d_{\rm IA}(x,\partial\Delta)=\min_{1\leqslant i\leqslant N}\\{d_{\rm
IA}(x,P_{i})\\}=\min\\{\ell_{1}(x),\ldots,\ell_{N}(x)\\}.$ (24)

######  Definition 2.4.

Let $v_{1},\ldots,v_{k}\in\mathbb{Z}^{n}$ for $k\leqslant n$. The _integral
index_ $D_{\rm IA}=D_{\rm IA}(v_{1},\ldots,v_{k})$ is defined as the greatest
common divisor of

$\left\\{\det(v_{1}|\ldots|v_{k}|w_{k+1}|\ldots|w_{n})\in\mathbb{Z}\,|\,w_{k+1},\ldots,w_{n}\in\mathbb{Z}^{n}\right\\}.$
(25)

This is an integral invariant of the unordered $k$-tuple $v_{1},\ldots,v_{k}$.
Its geometric interpretation is that it is equal to the index of the
sublattice $\operatorname{span}_{\mathbb{Z}}\\{v_{1},\ldots,v_{k}\\}$ of the
lattice
$\mathbb{Z}^{n}\cap\operatorname{span}_{\mathbb{R}}\\{v_{1},\ldots,v_{k}\\}$.
For computations, the following characterization is the most convenient, see
[37, Chapter 18] for a proof and more details.

######  Proposition 2.5.

The integral index $D_{\rm IA}(v_{1},\ldots,v_{k})\in\mathbb{Z}$ is equal to
the greatest common divisor of all $k$-minors (meaning determinants of
$k\times k$ submatrices) of the matrix $(v_{1}|\ldots|v_{k})$.

###  2.3. Displacement energy

Let $A$ be a compact subset of a symplectic manifold $(X,\omega)$. Recall that
the _displacement energy of $A$_ is defined as

$e(A)=\inf\\{\|H\|\,|\,\phi^{H}_{1}(A)\cap A=\varnothing\\},$ (26)

where $\phi^{H}_{1}$ denotes the time-one flow of a compactly supported, time-
dependent Hamiltonian $H$. We set $e(A)=\infty$ if the infimum in (26) is
taken over the empty set and by $\|\cdot\|$ we denote the Hofer norm,

$\|H\|=\int_{0}^{1}\left(\max_{x\in X}H_{t}(x)-\min_{x\in
X}H_{t}(x)\right)dt.$ (27)

We refer to the classic [46] for details on Hofer geometry. The quantity
$e(A)$ is a symplectic invariant of the set $A$, in the sense that if there is
a symplectomorphism of $(X,\omega)$ mapping $A$ to another subset $B\subset
X$, then the displacement energies of $A$ and $B$ agree. In this article, we
are exclusively interested in the displacement energy of Lagrangian tori.
Chekanov [14] (see also [44]) gave a lower bound for displacement energy of
compact Lagrangian submanifolds in geometrically bounded symplectic manifolds
in terms of $J$-holomorphic spheres in $X$ and $J$-holomorphic disks with
boundary on the Lagrangian in question. Denote by
${\mathcal{J}}={\mathcal{J}}(X,\omega)$ the set of $\omega$-tame almost
complex structures. Let $\lambda_{D}(X,L,J)$ denote the infimum of symplectic
areas realized by non-trivial $J$-disks with boundary on $L$. If such disks
exist, then the infimum is positive and is attained by at least one disk by
Gromov compactness. If not, then set $\lambda_{D}(X,L,J)=\infty$. Similarly,
denote by $\lambda_{S}(X,J)$ the infimum of areas realized by non-trivial
$J$-spheres.

######  Theorem 2.6 (Chekanov).

Let $L\subset X$ be a compact Lagrangian in a geometrically bounded symplectic
manifold and $J$ an $\omega$-tame almost complex structure. Then

$\min\\{\lambda_{D}(X,L,J),\lambda_{S}(X,J)\\}\leqslant e(L).$ (28)

######  Example 2.7 (Product tori in $\mathbb{C}^{n}$).

Let $\mathbb{C}$ be equipped with the standard symplectic form and let
$S^{1}(x)\subset\mathbb{C}$ be the circle enclosing symplectic area $x>0$.
Then $e(S^{1}(x))=x$. The upper bound comes from an explicit construction and
the lower bound comes from the obvious holomorphic disk with boundary on
$S^{1}(x)$ together with Theorem 2.6. Taking products of this example, we
obtain the _product tori_ in $\mathbb{C}^{n}$, which we will denote by

$T(x)=S^{1}(x_{1})\times\ldots\times S^{1}(x_{n})\subset\mathbb{C}^{n},\quad
x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}_{>0}.$ (29)

Product tori have displacement energy

$e(T(x))=\min\\{x_{1},\ldots,x_{n}\\}.$ (30)

This was proved by Sikorav [48] without using $J$-holomorphic curves. It can
also be proved using Theorem 2.6 and the standard holomorphic structure on
$\mathbb{C}^{n}$.

######  Example 2.8 (Toric fibres).

Let $(X,\omega)$ be a toric manifold with moment map $\mu$ and moment polytope
$\Delta=\\{x\in\mathbb{R}^{n}\,|\,\ell_{i}(x)\geqslant 0\\}$, with the same
conventions as in (23). Every fibre $T(x)=\mu^{-1}(x)$ over a point
$x\in\operatorname{int}\Delta$ in the interior of $\Delta$ is a Lagrangian
torus, called _toric fibre_. In all examples known to the author,

$e(T(x))=d_{\rm IA}(x,\partial\Delta)=\min_{1\leqslant i\leqslant
N}\\{\ell_{i}(x)\\},$ (31)

for all $x$ in an open dense subset of $\operatorname{int}\Delta$, where
$d_{\rm IA}(\cdot,\partial\Delta)$ denotes the integral affine distance to the
boundary of $\Delta$, see (24). Note that Example 2.7 is a special case of
this, since product tori are the fibres of the standard toric moment map on
$\mathbb{C}^{n}$. There are toric fibres where (31) does not hold; e.g.
Entov–Polterovich [21] proved that every compact toric manifold contains at
least one non-displaceable fibre, see also [26, 27, 28]. Note that for example
in $\mathbb{C}P^{2}$, the non-displaceable fibre is the only point where (31)
fails. The inequality $e(T(x))\geqslant d_{\rm IA}(x,\partial\Delta)$ was
shown in [9, Proposition 3.2] for _all_ toric fibres in compact toric
manifolds. The upper bound on displacement energy relies on probes [40] and
therefore on delicate properties of the moment polytope. If $(X^{2n},\omega)$
is monotone toric, the upper bound follows from a conjecture (related to the
Ewald conjecture, see [40, §3.1]) about lattice polytopes, which has been
checked in dimensions $2n\leqslant 18$, see [9, §3.4] for more details.

Let us prove Proposition 1.14 along the lines of [16].

Proof of Proposition 1.14.  Let $L=\psi(T(a))\subset X$ be a Lagrangian torus
which has property (CS). For the inequality $e_{X}(L)\leqslant\underline{a}$,
see [16, Proposition 2.1] or consider the standard moment map
$\mu\colon\mathbb{C}^{n}\rightarrow\mathbb{R}_{>0}^{n}$ to view $B(R)$ as an
open toric manifold and use McDuff’s method of probes [40]. The condition (16)
precisely guarantees that there is a sufficiently long probe contained in the
moment map image of $B(R)$ to displace $T(a)$. For the inequality
$e_{X}(L)\geqslant\underline{a}$, let $\\{J_{k}\\}_{k\in\mathbb{N}}$ be a
sequence of $\omega$-tame almost complex structures satisfying
$J_{k}|_{\operatorname{im}\psi}=\psi_{*}J_{0}$ such that
$\lambda_{S}(X,J_{k})\rightarrow\lambda_{S}(X,\psi)$ for
$k\rightarrow+\infty$. By Theorem 2.6, we have

$\min\\{\lambda_{D}(X,L,J_{k}),\lambda_{S}(X,J_{k})\\}\leqslant e(L),\quad
k\in\mathbb{N}.$ (32)

It follows from the proof of [16, Proposition 2.1] that
$\lambda_{D}(X,L,J_{k})=\underline{a}$. The essential ingredient is the fact
that the minimal $\psi_{*}J_{0}$-holomorphic disk inside
$\operatorname{im}\psi$ has area $\underline{a}$ and that all disks leaving
$\operatorname{im}\psi$ have area $>\underline{a}$, which follows from the
area estimate [16, Appendix A] together with (16). Since
$\lambda_{S}(X,J_{k})\rightarrow\lambda_{S}(X,\psi)$ and
$\underline{a}<\lambda_{S}(X,\psi)$ by (17), there is $k_{0}\in\mathbb{N}$
such that the left hand side of (32) is equal to $\underline{a}$. $\Box$

###  2.4. Versal deformations and the displacement energy germ

We give a brief overview of the method of versal deformations here. When
applied to displacement energy, it yields the _displacement energy germ_ ,
which is the invariant used in this paper to distinguish Lagrangian
submanifolds. See also [13, 16] and [9, 7, 15] for further applications.

Let $L\subset X$ be a compact Lagrangian and let
${\mathcal{L}}={\mathcal{L}}_{L}$ be the space of Lagrangians which are
Lagrangian isotopic to $L$, equipped with the $C^{1}$-topology, see for
example [45] for details. In this topology, a small enough neighbourhood
${\mathcal{U}}$ of $L\in{\mathcal{L}}$ is in bijection with a neighbourhood of
the zero $1$-form in the set of closed differential one-forms on $L$. This
follows from Weinstein’s neighbourhood theorem applied to $L$ together with
the fact that a graph $\Gamma_{\alpha}\subset(T^{*}L,d\lambda)$ of a
differential $1$-form $\alpha\in\Omega^{1}(L)$ is Lagrangian if and only if
$\alpha$ is closed. Furthermore, the graphs $\Gamma_{\alpha},\Gamma_{\beta}$
are Hamiltonian isotopic in $T^{*}L$ if and only if the corresponding forms
are cohomologous, $[\alpha]=[\beta]\in H_{\rm dR}^{1}(L;\mathbb{R})$. In other
words, there is a continuous bijection between $C^{1}$-small Lagrangian
perturbations of $L$ up to locally supported Hamiltonian isotopies and small
elements in $H^{1}(L;\mathbb{R})$. In terms of a Weinstein chart
$\varphi\colon T^{*}L\supset V\rightarrow X$, this bijection can be realized
by taking the class up to Hamiltonian isotopies supported in
$\operatorname{im}\varphi$ of $v_{L}^{\varphi}(b)=\varphi(\Gamma_{\beta(b)})$.
Here $b\mapsto\beta(b)$ is a continuous family of closed $1$-forms satisfying
$[\beta(b)]=b$.

######  Example 2.9.

Let $L\subset X$ be a Lagrangian torus and $\varphi\colon T^{*}T^{n}\supset
V\rightarrow X$ be a Weinstein chart. Identifying
$T^{n}=(\mathbb{R}/\mathbb{Z})^{n}$, we obtain induced identifications
$T^{*}T^{n}\cong(\mathbb{R}/\mathbb{Z})^{n}\times\mathbb{R}^{n}=\\{(\theta_{1},\ldots,\theta_{n},p_{1},\ldots,p_{n})\\}$
and $H^{1}(T^{n};\mathbb{R})\cong\mathbb{R}^{n}$. Set
$\beta(b)=\sum_{i}b_{i}d\theta_{i}$ to obtain

$v^{\varphi}_{L}(b)=\varphi(T^{n}\times\\{b\\}).$ (33)

The above discussion is closely related to the Lagrangian flux of Lagrangian
isotopies contained in the image of $\varphi$. We recall the definition of
Lagrangian flux for the reader’s convenience, see also [45]. Let
$\\{i_{t}\\}_{t\in[0,1]}$ be a Lagrangian isotopy, meaning that there is a
smooth map $i\colon[0,1]\times L\rightarrow X$ such that $i_{t}=i(t,\cdot)$ is
a Lagrangian embedding for every $t\in[0,1]$. The embeddings being Lagrangian
is equivalent to the forms $i_{t}^{*}\iota(X_{t})\omega\in\Omega^{1}(L)$ being
closed for all $t\in[0,1]$, where $X_{t}\in\Gamma(i_{t}^{*}TX)$ is the vector
field generating $i_{t}$. The _Lagrangian flux of $\\{i_{t}\\}$_ is defined as
the average cohomology class of this family of forms,

$\operatorname{Flux}\\{i_{t}\\}=\int_{0}^{1}[i_{t}^{*}\iota(X_{t})\omega]\in
H^{1}(L;\mathbb{R}).$ (34)

As in the case of the flux of symplectic isotopies, see [46, Chapter 14] or
[41, §10.2], the flux of Lagrangian isotopies can be geometrically interpreted
as a measure of the symplectic area of cylinders swept out by one-cycles. Let
$c$ be a one-cycle on $L$ with homology class $\xi=[c]\in H_{1}(L)$. We obtain
a two-chain $C_{\xi}=\cup_{t\in[0,1]}i_{t}(c)$ in $X$, defining a relative
homology class $[C_{\xi}]\in H_{2}(X,\iota_{0}(L)\cup\iota_{1}(L))$. The
assignment $\xi\mapsto[C_{\xi}]$ yields a well-defined, linear map
$H_{1}(L)\rightarrow H_{2}(X,\iota_{0}(L)\cup\iota_{1}(L))$. Since
$\iota_{0},\iota_{1}$ are Lagrangian embeddings, the symplectic form yields a
well-defined functional $[\omega]\colon
H_{2}(X,\iota_{0}(L)\cup\iota_{1}(L))\rightarrow\mathbb{R}$. From this point
of view, Lagrangian flux can be viewed as

$\langle\operatorname{Flux}\\{i_{t}\\},\xi\rangle=\langle[\omega],[C_{\xi}]\rangle,\quad\xi\in
H_{1}(L).$ (35)

In this paper, we work with unparametrized Lagrangian isotopies, i.e. families
$\\{L_{t}\\}_{t\in[0,1]}$ of Lagrangian submanifolds. One can always choose a
parametrization $i_{t}(L)=L_{t}$ and prove that the above discussion is
independent of this choice up to picking an identification $L=L_{0}$.
Furthermore, we sometimes consider _continuous_ families of Lagrangians, i.e.
continuous maps $[0,1]\rightarrow{\mathcal{L}}$. By its homological
interpretation (35), the Lagrangian flux is well-defined. Furthermore since
$\omega$ is closed, Stokes’ theorem and (35) show that the Lagrangian flux is
invariant under homotopies of Lagrangian isotopies with fixed endpoints.

######  Remark 2.10.

A Lagrangian isotopy $\\{i_{t}\\}$ is induced by an ambient Hamiltonian
isotopy if and only if the forms $i_{t}^{*}\iota(X_{t})\omega$ are exact, i.e.
the Lagrangian flux $\operatorname{Flux}\\{i_{t}\\}_{t\in[0,t_{0}]}$ is zero
for all $t_{0}\in[0,1]$.

######  Remark 2.11.

The Lagrangian flux measures the difference between the area classes
$\sigma_{L_{0}},\sigma_{L_{1}}$ (see (5)) of endpoints of a Lagrangian isotopy
$\\{L_{t}\\}$, in the sense that

$\langle\Phi^{*}\sigma_{L_{1}}-\sigma_{L_{0}},D\rangle=\langle\operatorname{Flux}\\{L_{t}\\},\partial
D\rangle,\quad D\in H_{2}(X,L_{0}),$ (36)

where $\Phi\colon H_{2}(X,L_{0})\rightarrow H_{2}(X,L_{1})$ is the isomorphism
induced by $\\{L_{t}\\}$ and $\partial\colon H_{2}(X,L_{0})\rightarrow
H_{1}(L_{0})$ is the boundary map of the long exact sequence of the pair
$(X,L_{0})$. In particular, a Lagrangian isotopy with vanishing flux and
endpoints on non-equivalent Lagrangians yields an example answering Question
1.5 positively. The existence of such Lagrangian isotopies (see for example
Remark 4.7) is in contrast with the case of symplectic flux, where every
isotopy with vanishing symplectic flux is homotopic to a Hamiltonian isotopy,
see [41, Theorem 10.2.5].

Let us return to the map $v_{L}^{\varphi}$ defined above. Since
$\varphi^{*}\omega=d\lambda$ on $\operatorname{im}\varphi$, Stokes’ theorem
shows that the flux of a Lagrangian isotopy $\\{L_{t}\\}$ contained in
$\operatorname{im}\varphi$ depends only on its endpoints. Furthermore, if
$L_{1}=\varphi(\Gamma_{\beta})$, then
$\operatorname{Flux}\\{L_{t}\\}=[\beta]\in H^{1}(L;\mathbb{R})$. This means
that the Lagrangian flux induces an inverse map to the bijection
$v_{L}^{\varphi}$. The map $v_{L}^{\varphi}$ is an example of a versal
deformation. However, we sometimes work with versal deformations without
considering an underlying Weinstein chart $\varphi$. In general, a versal
deformation is defined as a right-inverse to the Lagrangian flux map.

######  Definition 2.12.

A continuous map $v_{L}\colon H^{1}(L;\mathbb{R})\supset
U\rightarrow{\mathcal{L}}_{L}$ defined on a simply-connected neighbourhood $U$
of the origin is called a _versal deformation_ of $L$ if $v_{L}(0)=L$ and if
the Lagrangians in its image are parametrized by their flux. By this we mean
that

$\operatorname{Flux}\\{v_{L}(\gamma(t))\\}_{t\in[0,1]}=b,$ (37)

for any $b\in U$ and any continuous curve $\gamma\colon[0,1]\rightarrow U$
with $\gamma(0)=0$ and $\gamma(1)=b$.

Although versal deformations are not unique, they are locally unique up to
Hamiltonian isotopy. More precisely, let $v_{L}\colon
U\rightarrow{\mathcal{L}}_{L}$ and $v_{L}^{\prime}\colon
U^{\prime}\rightarrow{\mathcal{L}}_{L}$ be two versal deformations of $L$.
Then there is a neighbourhood of the origin $V\subset U\cap U^{\prime}$ such
that

$v_{L}(b)\cong v_{L}^{\prime}(b),\quad b\in V.$ (38)

This follows from the fact that, given a Weinstein chart $\varphi$ of $L$,
there is a small enough neighbourhood $V$ of the origin $0\in
H^{1}(L;\mathbb{R})$ such that every Lagrangian in the image of $v_{L}|_{V}$
and $v_{L}^{\prime}|_{V}$ is the image of a graph of a closed $1$-form under
$\varphi$, and closed $1$-forms are classified up to Hamiltonian isotopy by
their cohomology class. Furthermore, let
$\phi\in\operatorname{Symp}(X,\omega)$ and $v_{\phi(L)}$ be a versal
deformation of $\phi(L)$, then there is a neighbourhood of the origin
$V^{\prime}\subset H^{1}(\phi(L);\mathbb{R})$ such that

$v_{\phi(L)}(b)\cong(\phi\circ v_{L}\circ\phi|_{L}^{*})(b),\quad b\in
V^{\prime}.$ (39)

Chekanov’s idea is to evaluate a symplectic invariant of Lagrangian
submanifolds on the image of a versal deformation and then take its germ to
extract an invariant. In our case, we take displacement energy as symplectic
invariant and consider the function

$H^{1}(L;\mathbb{R})\supset U\rightarrow\mathbb{R}\cup\\{+\infty\\},\quad
b\mapsto e(v_{L}(b)),$ (40)

where $v_{L}$ is a versal deformation of $L$.

######  Definition 2.13.

Let $L\subset(X,\omega)$ be a compact Lagrangian. The germ
${\mathcal{E}}_{L}\colon
H^{1}(L;\mathbb{R})\rightarrow\mathbb{R}\cup\\{+\infty\\}$ at $0\in
H^{1}(L;\mathbb{R})$ of the function (40) is called _displacement energy germ
of $L$_.

By (38) and the invariance of $e(\cdot)$ under symplectomorphisms, the
displacement energy germ is independent of the choice of versal deformation.
The symplectic invariance property (11) of the displacement energy germ
similarly follows from (39). We are interested in versal deformations of
Lagrangian tori $L\approx T^{n}$. It is useful to note that if $L$ is an orbit
of a Hamiltonian $T^{n}$-actions, then the moment map generating the
Hamiltonian $T^{n}$-action yields a natural versal deformation of $L$. Indeed,
let $\varphi\colon T^{n}\times U\rightarrow X$ be a $T^{n}$-equivariant
Weinstein chart of $L$ and consider the versal deformation
$v_{L}^{\varphi}\colon U\rightarrow{\mathcal{L}}_{L}$ defined by
$v_{L}^{\varphi}(b)=\varphi(T^{n}\times\\{b\\})$. By $T^{n}$-equivariance,
this proves the following.

######  Proposition 2.14.

Let $L\subset X$ be a Lagrangian torus and $\mu\colon
U\rightarrow\mathbb{R}^{n}$ the moment map of a (locally defined) Hamiltonian
$T^{n}$-action having $L=\mu^{-1}(a)$ as orbit. Then $v_{L}(b)=\mu^{-1}(a+b)$
is a versal deformation.

In particular, a versal deformation of a regular fibre of a Lagrangian torus
fibration (or integrable system) can be computed by computing local action-
coordinates on the basis. Recall that this is equivalent to computing the flux
map, see [23, Lemma 2.15], which connects back to the appearance of Lagrangian
flux in the discussion of versal deformations.

######  Example 2.15 (Product tori in $\mathbb{C}^{n}$).

Let $T(a)\subset\mathbb{C}^{n}$ be a product torus. It is easy to see that
$v_{T(a)}(b)=T(a+b)$ is a versal deformation, e.g. as a special case of
Proposition 2.14, since the standard moment map $\mu$ on $\mathbb{C}^{n}$
satisfies $\mu^{-1}(a)=T(a)$. By Example 2.7, its displacement energy germ is
the germ of $b\mapsto\min_{i}\\{a_{i}+b_{i}\\}$ and is thus given by

${\mathcal{E}}_{T(a)}(b)=\underline{a}+\min_{i\in I(a)}\\{b_{i}\\},$ (41)

where $I(a)$ is defined as
$I(a)=\\{i\in\\{1,\ldots,n\\}\,|\,a_{i}=\underline{a}\\}$.

######  Example 2.16 (Tori with property (CS)).

Let $L=\psi(T(a))\subset X$ be a Lagrangian torus in a geometrically bounded
symplectic manifold having property (CS) as in Definition 1.13. Then
${\mathcal{E}}_{L}\sim{\mathcal{E}}_{T(a)}$. Indeed, take the versal
deformation $b\mapsto\psi(T(a+b))$ and note that, for small enough $b$, all of
its members have property (CS), too. By Proposition 1.14, the claim follows.

######  Example 2.17 (Toric fibres).

As in Example 2.8, let $X$ be a toric manifold with moment polytope $\Delta$
defined by vectors $v_{i}$. Suppose that (31) holds on an open dense subset of
$\operatorname{int}\Delta$. For every toric fibre $T(x)=\mu^{-1}(x)$, the map
$v_{T(x)}(b)=T(x+b)$ is a versal deformation by Proposition 2.14. We find

${\mathcal{E}}_{T(x)}(b)\sim d_{\rm IA}(x,\partial\Delta)+\min_{i\in
I(x)}\\{\langle b,v_{i}\rangle\\},$ (42)

where $\sim$ denotes equality on an open dense subset, see Definition 1.10,
and where $I(x)=\\{i\,|\,\ell_{i}(x)=d_{\rm IA}(x,\partial\Delta)\\}$. Note
that (42) holds for all _all_ toric fibres in $X$, even those for which (31)
fails. Example 2.15 is a special case of this example.

In all of the examples known to the author, the displacement energy germ of a
Lagrangian torus $L$ has the form

${\mathcal{E}}_{L}(b)\sim a+\min\\{\langle b,v_{1}\rangle,\ldots,\langle
b,v_{k}\rangle\\},\quad a>0,v_{i}\in H_{1}(L)\cong\mathbb{Z}^{n}.$ (43)

Now let $L^{\prime}$ be another torus whose displacement energy germ has the
form (43) with $a^{\prime}>0$ and $v_{1}^{\prime},\ldots
v_{k^{\prime}}^{\prime}\in\mathbb{Z}^{n}$. If $L\cong L^{\prime}$, then by
(11) we find $a=a^{\prime}$, $k=k^{\prime}$ and that there is
$\Phi\in\operatorname{GL}(n;\mathbb{Z})$ such that
$\Phi(v_{i})=v_{i}^{\prime}$ up to relabelling. This explains our interest in
the integral geometry of sets of vectors
$v_{1},\ldots,v_{k}\in\mathbb{Z}^{n}$, see Definition 2.4.

######  Remark 2.18.

It would be interesting to interpret (43) in terms of minimal area Maslov two
$J$-holomorphic disks with boundary on $L$. The quantity $a>0$ should
correspond to the minimal area and the classes $v_{i}\in
H_{1}(L;\mathbb{Z})\cong\mathbb{Z}^{n}$ to the boundaries of disks realizing
the minimal area $a$. See [47] for related ideas.

######  Remark 2.19 (Virtual displacement energy).

Let $L$ be a Lagrangian torus whose displacement energy germ has the form
(43). Then $ve(L)=a>0$ is an invariant of $L$ under symplectomorphisms of $X$,
which we call _virtual displacement energy_. In many examples $ve(L)=e(L)$,
but sometimes this is not true. Take for example the monotone torus $T_{\rm
Cl}$ in $(\mathbb{C}P^{2},\omega_{\mathbb{C}P^{2}})$ with normalization
$\int_{\mathbb{C}P^{1}}\omega_{\mathbb{C}P^{2}}=1$. Then $e(T_{\rm
Cl})=+\infty$, and

${\mathcal{E}}_{T_{\rm
Cl}}(b)\sim\frac{1}{3}+\min\left\\{b_{1},b_{2},-b_{1}-b_{2}\right\\},$ (44)

meaning that $ve(T_{\rm Cl})=\frac{1}{3}$. More generally, let $X$ be toric
such that (31) holds on an open dense subset of $\operatorname{int}\Delta$,
then $ve(x)=d_{\operatorname{IA}}(x,\partial\Delta)$ everywhere. Assuming
every $L^{\prime}\in{\mathcal{L}}_{L}$ has displacement energy germ of the
form (43), we find that $ve\colon{\mathcal{L}}\rightarrow\mathbb{R}$ is
continuous, as opposed to $e(\cdot)$. This is reminiscent of the invariant
$\Psi(\cdot)$ from [47].

As an application of this discussion and Example 2.17, let us focus on toric
manifolds. Note that toric fibres in a toric manifold are the analogue of
product tori in $\mathbb{C}^{n}$. Therefore, we can call a torus in a toric
manifold _exotic_ if it is not equivalent to a toric fibre.

######  Corollary 2.20.

Let $X$ be a monotone toric manifold satisfying222Recall from the discussion
in Example 2.8, that this condition conjecturally holds for all monotone toric
manifolds and has been checked for $\dim X\leqslant 18$. (31) on an open dense
subset of $\operatorname{int}\Delta$. Then $X$ contains infinitely many exotic
tori, meaning tori distinct from all toric fibres.

Proof. Let $T(x)\subset X$ be a toric fibre. We can assume that $T(x)$ is non-
monotone, since the tori embedded in the proof of Theorem D are not montone in
$X$. Recall from Example 2.17 that the displacement energy germ of $T(x)$ is
of the form (42). Let $v_{i},v_{j}$ be a pair of vectors appearing in (42). We
claim that $D_{\rm IA}(v_{i},v_{j})=1$, where $D_{\rm IA}$ is the integral
index, see Definition 2.4. Indeed, by monotonicity, the facets associated to
$v_{i}$ and $v_{j}$ have non-empty intersection and thus the claim follows
from the Delzant property of the moment polytope $\Delta$ of $X$. For the
local exotic tori constructed in Theorem D on the other hand, the displacement
energy germ is given by (102), where the minimum is taken over a set
containing a pair of vectors for which $D_{\rm IA}\neq 1$ whenever the
corresponding Markov triple ${\mathfrak{m}}$ is non-trivial. $\Box$

##  3\. Almost toric fibrations

###  3.1. Overview

Almost toric fibrations (ATFs) are a type of Lagrangian torus fibrations on
four-dimensional symplectic manifolds introduced by Symington [49], which are
slightly more general than toric fibrations. We give a brief outline of ATFs
here with an emphasis on the properties used in the context of this paper. For
more details on almost toric fibrations, we recommend [23].

Toric fibrations lie at the intersection of Hamiltonian group actions and
Lagrangian torus fibrations (or integrable systems, the two can be used
interchangeably for the present purposes). Indeed the moment map $\mu\colon
X\rightarrow\mathbb{R}^{n}$ of a toric action on a symplectic manifold
$(X,\omega)$ can be viewed as a special type of Lagrangian torus fibration. It
is special in the sense that the moment map yields _global_ action
coordinates, i.e. action coordinates on the regular locus
$\mu^{-1}(\operatorname{int}\Delta)$ of $\mu$. In the vicinity of a compact
regular fibre, every Lagrangian torus fibration can be equipped with _local_
action-angle coordinates compatible with the fibration; this is the classical
Arnold–Liouville Theorem [23, Theorem 1.42]. However, in general, action-angle
coordinates do not extend globally. See Duistermaat [19] for a study of
obstructions to this extension. One obstruction that we encounter when
discussing ATFs is the topological monodromy of the torus bundle over the
regular locus. If the regular locus is simply-connected, as is the case for
toric fibrations, then the topological monodromy is trivial.

To move from toric to almost toric fibrations, we lose the Hamiltonian group
action. In terms of Lagrangian torus fibrations/integrable systems, four-
dimensional toric fibrations have a very restricted set of allowable
singularities, namely only elliptic-regular and elliptic-elliptic
singularities appear. We refer to [53, 55] for details. In the almost toric
case, the set of permitted singularities is extended by adding the focus-focus
case, see [54]. An almost toric fibration is thus a (proper) Lagrangian torus
fibration $F\colon X^{4}\rightarrow B^{2}$ with singularities of the three
types discussed above. In the context of this paper, we have
$B^{2}\subset\mathbb{R}^{2}$. Note that the focus-focus singularity does not
have $T^{2}$-symmetry (it only has $S^{1}$-symmetry), meaning that we lose the
globally defined Hamiltonian $T^{2}$-action present in the toric case.
Furthermore, the focus-focus singularities map to singular values (called
_nodes_ or _nodal points_) in the interior of $B$, meaning that the regular
locus $B_{\rm reg}\subset B$ is not simply connected (as opposed to the toric
case) and the torus bundle $F|_{F^{-1}(B_{\rm reg})}\colon F^{-1}(B_{\rm
reg})\rightarrow B_{\rm reg}$ has non-trivial topological monodromy. This
means that whenever $F$ has at least one focus-focus singularity, there are no
global action-angle coordinates on $F^{-1}(B_{\rm reg})$, meaning that a
priori it is not obvious what the analogue of the moment map $\mu$ and the
moment polytope $\Delta$ should be in the almost toric case. However, one can
find curves $C_{i}$ in $B$, such that their complement $B_{\rm
reg}\setminus\cup_{i}C_{i}\subset B_{\rm reg}$ is simply-connected. In the
cases relevant to this paper, every $C_{i}$ connects a node to the toric
boundary of $B$. For simplicity, we assume the set of curves $C_{i}$ to be
mutually disjoint. On $B_{\rm reg}\setminus\cup_{i}C_{i}$, there are global
action coordinates. Thus one gets a reasonable analogue of the moment map as
follows. First, post-compose $F|_{F^{-1}(B_{\rm reg}\setminus\cup_{i}C_{i})}$
with the diffeomorphism yielding action coordinates. Denote the resulting map
by $\pi_{0}$. Second, as it turns out, one can choose the curves $C_{i}$ in
such a way that there is a continuous extension $\pi$ of $\pi_{0}$ mapping $X$
to the closure
$\Delta_{\pi}=\overline{\operatorname{im}\pi_{0}}\subset\mathbb{R}^{2}$ such
that $\Delta_{\pi}$ is a polytope and such that the image of every
$F^{-1}(C_{i})$ in $\Delta_{\pi}$ is a line segment, which we call _branch
cut_. Decorating $\Delta_{\pi}$ with branch cuts (represented by dashed lines
in the conventions of [49]) indicating the image of the curves $C_{i}$ and
with nodes (represented by crosses) indicating which fibres contain focus-
focus singularities, we obtain a so-called _ATF base diagram_ of the
Lagrangian torus fibration $F\colon X\rightarrow B$ we started with. Abusing
notation, we denote by $\Delta_{\pi}$ both the polytope
$\Delta_{\pi}\subset\mathbb{R}^{2}$ obtained as the image of $\pi$ and the
base diagram, i.e. the polytope decorated with branch cuts and nodes. A
crucial point in the ATF-framework is that there is a Delzant-type
theorem333The classical Delzant Theorem [17] states that a toric manifold is
determined, up to equivariant symplectomorphism, by the integral affine
equivalence class of its moment polytope. for ATF base diagrams. More
precisely, the ATF base diagram $\Delta_{\pi}$ determines the symplectic
manifold $X$ equipped with an almost toric fibration that is unique up to
symplectomorphisms which are fibre-preserving up to a union of arbitrarily
small neighbourhoods of the nodal points in $B$, see [23, Theorem 8.5].

The almost toric base diagram $\Delta_{\pi}$ should be thought of as a
geometric-combinatorial object encoding an almost toric fibration $F\colon
X\rightarrow B$ having $\Delta_{\pi}$ as its ATF base diagram in the sense
described above, i.e. by removing suitable curves $C_{i}\subset B$ and
computing action coordinates on their complement. In practice, these
subtleties are often glossed over, and, by abuse of terminology, the map
$\pi\colon X\rightarrow\Delta_{\pi}$ is called _almost toric fibration_. When
working with the map $\pi$ explicitely, we keep in mind that it is not smooth
over the branch cuts, but that it yields honest action coordinates when
restricted to the complement of branch cuts.

Note that, since it involves a choice of curves $C_{i}$, the ATF base diagram
is _not_ uniquely defined up to integral affine transformations. However, the
choice of curves $C_{i}$ for which the outcome is as described above is very
restricted. In fact, there are only two possible choices of $C_{i}$ (pointing
in opposite directions) for the $i$-th node. This means that, still assuming
that all possible choices of $C_{i}$ are mutually disjoint, there are
$2^{\\#\\{\text{nodes}\\}}$ possible ATF base diagrams associated to $F$.
Switching at the $i$-th node from a given choice of $C_{i}$ to the one in the
opposite direction is called _changing the branch cut_. Note that changing a
branch cut does not change the underlying almost toric fibration $F\colon
X\rightarrow B$, merely the way of representing it by a base diagram. Let
$l_{i}$ be the line containing the branch cut emanating from the $i$-th node
in $\Delta_{\pi}$. In terms of base diagrams, changing the branch cut at the
$i$-th node corresponds to passing from $\Delta_{\pi}$ to
$\Delta_{\pi^{\prime}}$ by the branch cut switching sides in $l_{i}$ at the
cross representing the $i$-th node. A computation of action coordinates on the
so-obtained simply-connected domain yields that passing from $\Delta_{\pi}$ to
$\Delta_{\pi^{\prime}}$ corresponds to applying a piece-wise linear
transformation $\tau\colon\Delta_{\pi}\rightarrow\Delta_{\pi^{\prime}}$. The
map $\tau$ is continuous, has eigenline $l_{i}$ and acts by integral affine
transformations on the two halves of $\mathbb{R}^{2}\setminus l_{i}$. By
convention, since everything is up to integral affine transformations, one
chooses $\tau$ to be the identity on one of the halves. Then it is explicitely
given by

$\tau(x)=\begin{cases}\sigma_{w}(x)=x-\det(w|x)w,&\det(w|x)\geqslant 0\\\
x,&\det(w|x)<0,\end{cases}$ (45)

where $w\in\mathbb{Z}^{2}$ is the primitive vector contained in $l_{i}$
pointing in the direction of the new branch cut. We call $\tau$ a _mutation of
ATF base diagrams_.

Another crucial feature of almost toric fibrations is that, as opposed to
toric fibrations, one can apply certain operations to them which produce new
ATFs from given ones. These operations are called _nodal trades_ and _nodal
slides_. Nodal trades exchange an elliptic-elliptic singularity for a focus-
focus singularity, merging the two one-parameter families of elliptic-regular
singularities emanating from the original elliptic-elliptic singularity. Nodal
slides change the position of the node in the ATF base diagram $\Delta_{\pi}$
by sliding along the line $l_{i}$ containing the corresponding node and its
branch cut. See [23, Sections 8.2/8.3] for details. Thus, given a toric
fibration, one can first apply nodal trades to obtain an ATF and then nodal
slides to obtain new ATFs on a fixed symplectic manifold. Usually, one changes
the branch cut along the way in order for the different branch cuts not to
intersect. This process is called _mutation_ of ATFs, see [23, Section 8.4].
As we shall see, mutations are an effective method, pioneered by Vianna [51,
50], to come up with exotic Lagrangian tori.

###  3.2. Mutations of the complex projective plane

For the reader’s convenience and to set up notation, let us briefly recall the
almost toric fibrations obtained on $\mathbb{C}P^{2}$ by applying mutations to
the standard toric fibration. For details, we refer to [50]. Let
$\mathbb{C}P^{2}$ be equipped with the symplectic form
$\omega_{\mathbb{C}P^{2}}$ with normalization
$\int_{\mathbb{C}P^{1}}\omega_{\mathbb{C}P^{1}}=\frac{1}{2}$ and the toric
system having moment polytope

$\Delta_{\mathbb{C}P^{2}}=\left\\{x\in\mathbb{R}^{2}\,\left|\,\langle
x,v_{i}\rangle\geqslant-\frac{1}{3}\right.\right\\},\quad
v_{1}=\begin{pmatrix}1\\\ 0\end{pmatrix},v_{2}=\begin{pmatrix}0\\\
1\end{pmatrix},v_{3}=\begin{pmatrix}-1\\\ -1\end{pmatrix}.$ (46)

By applying mutations, one obtains a set of almost toric fibrations which is
in bijection with the so-called _Markov triples_ , i.e. triples
${\mathfrak{m}}=(\alpha,\beta,\gamma)$ of natural numbers which satisfy the
_Markov equation_

$\alpha^{2}+\beta^{2}+\gamma^{2}=3\alpha\beta\gamma.$ (47)

These triples appear in a tree rooted in the solution $(1,1,1)$ (corresponding
to the _toric_ fibration) which is trivalent except at the first two triples.
The edges of the tree correspond to mutations of Markov triples, each of which
replaces one of the numbers in the triple by a new Markov number. This comes
from the observation that keeping two out of three Markov numbers fixed turns
(47) into a quadratic equation having two solutions. On the ATF side, mutating
a Markov triple corresponds to mutating the associated almost toric fibration.
We denote the polytope of the ATF base diagram corresponding to a Markov
triple ${\mathfrak{m}}$ by

$\Delta_{{\mathfrak{m}}}=\left\\{x\in\mathbb{R}^{2}\,\left|\,\langle
x,u\rangle,\,\langle x,v\rangle,\,\langle
x,w\rangle\geqslant-\frac{1}{3}\right.\right\\},$ (48)

with $\det(u|v)=\alpha^{2}$, $\det(v|w)=\beta^{2}$ and $\det(w|u)=\gamma^{2}$.
More precisely, every pair of vectors in $\\{u,v,w\\}$ can be mapped by an
integral affine transformation to a normal form given by $(1,0)$ and
$(1-pq,p^{2})$, where $p$ is the corresponding Markov number and $1\leqslant
q<p$ is a natural number coprime to $p$. See for example [50, Figure 2] or
[25, Figure 2] for details. From this normal form it follows that

$u-v=\alpha y_{1},\quad v-w=\beta y_{2},\quad w-u=\gamma y_{3},$ (49)

where $y_{i}\in\mathbb{Z}^{2}$ are primitive vectors. This will be used in
§4.5. For every ${\mathfrak{m}}$, there are different ATF base diagrams having
the same underlying polytope $\Delta_{{\mathfrak{m}}}$, all of which are
related by nodal slides. We choose one such base diagram such that all branch
cuts are disjoint from the central point $0\in\Delta_{{\mathfrak{m}}}$ and
denote the corresponding projection map by
$\pi_{{\mathfrak{m}}}\colon\mathbb{C}P^{2}\rightarrow\Delta_{{\mathfrak{m}}}$.
Recall that this projection map is an honest $T^{2}$-moment map on the
complement of the branch cuts. Then the almost toric fibre

$T_{{\mathfrak{m}}}=\pi_{{\mathfrak{m}}}^{-1}(0)\subset\mathbb{C}P^{2},\quad{\mathfrak{m}}=(\alpha,\beta,\gamma),$
(50)

is a monotone Lagrangian torus called the _Vianna torus_ associated to the
Markov triple ${\mathfrak{m}}$. Vianna [50] proved that one can recover
$\Delta_{{\mathfrak{m}}}$ up to a $\operatorname{GL}(2;\mathbb{Z})$-map from
the Maslov two $J$-holomorphic disks with boundary on $T_{{\mathfrak{m}}}$.
Since no $\Delta_{{\mathfrak{m}}}$ can be mapped to
$\Delta_{{\mathfrak{m}}^{\prime}}$ for
${\mathfrak{m}}\neq{\mathfrak{m}}^{\prime}$ by a
$\operatorname{GL}(2;\mathbb{Z})$-map (the integral affine angles are
$\operatorname{GL}(2;\mathbb{Z})$-invariants), one can conclude the following.

######  Theorem 3.1 (Vianna).

The tori
$T_{{\mathfrak{m}}},T_{{\mathfrak{m}}^{\prime}}\subset\mathbb{C}P^{2}$ are
equivalent if and only if ${\mathfrak{m}}={\mathfrak{m}}^{\prime}$.

This result was conjectured earlier by Galkin–Usnich [32] and proved
independently by Galkin–Mikhalkin [31]. From their point of view, the tori
$T_{{\mathfrak{m}}}$ are obtained by considering certain degenerations of
$\mathbb{C}P^{2}$ to weighted projective spaces
$\mathbb{C}P(\alpha^{2},\beta^{2},\gamma^{2})$. The torus $T_{{\mathfrak{m}}}$
can be defined as the preimage under the parallel transport map of the
degeneration of the monotone fibre in the weighted projective space. Such
degenerations are in bijection with Markov triples by work of
Hacking–Prokhorov [36]. Note that this viewpoint is closely related to the
approach based on ATFs, since one should be able to obtain ATFs by a
perturbation of the pull-back to $\mathbb{C}P^{2}$ of the orbifold toric
system on $\mathbb{C}P(\alpha^{2},\beta^{2},\gamma^{2})$ under the parallel
transport map of the degeneration.

###  3.3. Almost toric fibres

In this subsection, we compute the displacement energy of Lagrangian tori
appearing as regular fibres of certain almost toric fibrations, which allows
us to determine their displacement energy germ. The idea to use the
displacement energy germ to distinguish almost toric fibres to obtain results
similar to [51, 50] was suggested to us by Felix Schlenk based on joint work
with Yuri Chekanov. Let $x\in\Delta_{\pi}$ be a point in the complement of the
set of nodes in the base diagram of an almost toric fibration $\pi\colon
X\rightarrow\Delta_{\pi}$. Then $\pi^{-1}(x)$ is a Lagrangian torus. We study
the displacement energy of these Lagrangian tori in the special case of almost
toric fibrations coming, as for example in §3.2, from mutations of an honest
toric fibration. The crucial observation is the fact that ATF-fibres which do
not lie on a branch cut line are untouched by mutations, see Lemma 3.3.

######  Theorem 3.2.

Let $\pi\colon X\rightarrow\Delta_{\pi}$ be an ATF on a compact monotone
symplectic four-manifold, which is obtained by mutations from a toric
fibration $\mu\colon X\rightarrow\Delta$. Then

$e(\pi^{-1}(x))=d_{\rm IA}(x,\partial\Delta_{\pi})$ (51)

for all $x$ in an open dense subset of $\Delta_{\pi}$.

Before turning to the proof of the theorem, we prove two lemmata.

######  Lemma 3.3.

Let $\tau^{-1}\colon\Delta_{\pi}\rightarrow\Delta_{\pi^{\prime}}$ be a
mutation of ATF base diagrams as in (45) and let $\pi\colon
X\rightarrow\Delta_{\pi}$ be an ATF. Let $x\in\Delta_{\pi^{\prime}}\setminus
l$ be a point which is not contained in the branch cut line $l$ of the
mutation $\tau^{-1}$. Then there is an ATF $\pi^{\prime}\colon
X\rightarrow\Delta_{\pi^{\prime}}$ such that

$(\pi^{\prime})^{-1}(x)=\pi^{-1}(\tau(x)).$ (52)

Proof. Nodal slides and nodal trades are supported in arbitrary small
neighbourhoods of $l$ and thus, we can choose $\pi^{\prime}$ to differ from
$\pi$ by a change in fibration supported away from $x\in\Delta_{\pi}\setminus
l$ (see for example the proof of [23, Theorem 8.10]) and a change of branch
cut. Recall that a change of branch cut does not change the underlying
fibration, merely its ATF base diagram by applying $\tau^{-1}$, therefore the
claim follows. $\Box$

######  Remark 3.4.

Actually, a more interesting statement is true. Namely, let $\pi,\pi^{\prime}$
be two ATFs related by a nodal trade or nodal slide contained in the branch
cut line $l$, then the almost toric fibres are Hamiltonian isotopic,

$(\pi^{\prime})^{-1}(x)\cong\pi^{-1}(x),\quad x\in\Delta_{\pi}\setminus l.$
(53)

Here, both almost toric fibrations are fixed a priori, as opposed to the
statement in Lemma 3.3, which comes at the cost of having Hamiltonian
isotopies instead of equalities as sets. This statement can be proved by
performing toric reduction on segments which are parallel to the branch cut
line in question. ATF-fibres project to circles under this reduction and nodal
trades and slides correspond to perturbing these reduced fibres, see for
example [7] or [33] for related ideas. Area arguments in the two-dimensionsal
reduced space allow to deduce (53). Although more broadly applicable than
Lemma 3.3, the statement (53) does not appear in the literature and its proof
is more involved, which is why we rely on Lemma 3.3 instead.

######  Lemma 3.5.

Let $\tau^{-1}\colon\Delta_{\pi}\rightarrow\Delta_{\pi^{\prime}}$ be a
mutation of ATF base diagrams which come, via mutations, from the Delzant
polytope $\Delta$ of a compact monotone toric manifold. Then for all
$x\in\Delta_{\pi}$,

$d_{\rm IA}(x,\partial\Delta_{\pi})=d_{\rm
IA}(\tau^{-1}(x),\tau^{-1}(\partial\Delta_{\pi}))=d_{\rm
IA}(\tau^{-1}(x),\partial\Delta_{\pi^{\prime}}).$ (54)

Proof. Let $\Delta_{\pi}=\\{x\in\mathbb{R}^{2}\,|\,\langle
x,v_{i}\rangle+\lambda\geqslant 0\\}$ and
$\Delta_{\pi^{\prime}}=\\{x\in\mathbb{R}^{2}\,|\,\langle
x,v_{i}^{\prime}\rangle+\lambda\geqslant 0\\}$ with the same conventions as in
(23). The defining vectors $v_{i}$ are labelled starting with the edge
adjacent to the branch cut line $l$ with respect to which the mutation is
performed and in the mathematically positive sense. The labelling conventions
for the $v_{i}^{\prime}$ is determined below. We denote the edge corresponding
to $v_{i}$ by $E_{i}\subset\Delta_{\pi}$ and the one corresponding to
$v_{i}^{\prime}$ by $E_{i}^{\prime}\subset\Delta_{\pi^{\prime}}$. The fact
that the constant $\lambda>0$ is the same in every defining equation of
$\Delta_{\pi}$ and $\Delta_{\pi^{\prime}}$ follows from monotonicity. Here we
suppose that the branch cut line $l$ of the mutation $\tau^{-1}$ intersects
the interior of an edge opposite to the vertex of the branch cut at which the
mutation is performed. In that case the number of edges of $\Delta_{\pi}$ and
of $\Delta_{\pi^{\prime}}$ coincides and we denote it by $N$. The case where
$l$ intersects a vertex is similar up to relabelling some indices. For the
mutation $\tau^{-1}\colon\Delta_{\pi}\rightarrow\Delta_{\pi^{\prime}}$, where
$\tau$ is given by (45), we find that there is $i_{0}\in\\{2,\ldots,N-1\\}$
and a labelling of the defining vectors $v_{i}^{\prime}$ of
$\Delta_{\pi^{\prime}}$ such that

$v_{1}^{\prime}=\sigma_{w}^{*}v_{1}\mskip 1.0mu,\ldots,\mskip
1.0muv_{i_{0}}^{\prime}=\sigma_{w}^{*}v_{i_{0}},\mskip 1.0mu\mskip
1.0muv_{i_{0}+1}^{\prime}=v_{i_{0}}\mskip 1.0mu,\ldots,\mskip
1.0muv_{N}^{\prime}=v_{N-1}.$ (55)

Furthermore, since the two edges $E_{1},E_{N}\subset\Delta_{\pi}$ adjacent to
the vertex at the branch cut at which the mutation is performed map to the
same edge $E_{1}^{\prime}\subset\Delta_{\pi^{\prime}}$ under $\tau^{-1}$, we
obtain $v_{1}^{\prime}=v_{N}$. For $i\in\\{1,\ldots,N\\}$ let

$K_{i}=\\{x\in\Delta_{\pi}\,|\,d_{\rm IA}(x,\partial\Delta_{\pi})=d_{\rm
IA}(x,E_{i})=\langle x,v_{i}\rangle+\lambda\\}$ (56)

be the chamber of $\Delta_{\pi}$ in which the distance to the $i$-th facet is
minimal. The chambers $K_{i}^{\prime}\subset\Delta_{\pi^{\prime}}$ are defined
similarly. We claim that the chamber $K_{i}$ is the triangle with vertices
$0,V_{i},V_{i+1}$, where $V_{i},V_{i+i}$ are the vertices adjacent to the edge
$E_{i}$. Indeed, since $d_{\rm
IA}(x,\partial\Delta_{\pi})=\min_{i}\left\\{d_{\rm IA}(x,E_{i})\right\\}$ the
subset $\partial K_{i}\cap\operatorname{int}\Delta_{\pi}$ of the boundary of
$K_{i}$ is formed by the sets satisfying $d_{\rm IA}(x,E_{i})=d_{\rm
IA}(x,E_{j})$, where $j$ is the index of an adjacent edge. This is equivalent
to $\langle x,v_{i}-v_{j}\rangle=0$, and the latter equation defines a line
containing $0$ and a vertex of $\Delta_{\pi}$ adjacent to $E_{i}$. The
corresponding statements hold for $\Delta_{\pi^{\prime}}$. We have

$\displaystyle\tau^{-1}(K_{1})\cup\tau^{-1}(K_{N})$ $\displaystyle=$
$\displaystyle K_{1}^{\prime},$ $\displaystyle\tau^{-1}(K_{i_{0}})$
$\displaystyle=$ $\displaystyle K_{i_{0}}^{\prime}\cup K_{i_{0}+1}^{\prime},$
$\displaystyle\tau^{-1}(K_{i})$ $\displaystyle=$ $\displaystyle
K_{i}^{\prime},\quad 1<i<i_{0},$ $\displaystyle\tau^{-1}(K_{i})$
$\displaystyle=$ $\displaystyle K_{i+1}^{\prime}\quad i_{0}<i<N,$

which can be deduced from $\tau^{-1}(0)=0$ together with the fact that
$\tau^{-1}$ induces a bijection of vertices of $\Delta_{\pi}$ to vertices of
$\Delta_{\pi^{\prime}}$, except for the vertex $V_{1}\subset\Delta_{\pi}$,
which maps to the interior of $E_{1}^{\prime}$ under $\tau^{-1}$, and the
vertex $V_{i_{0}}^{\prime}$ whose preimage lies in the interior of
$E_{i_{0}}$. Let $x\in K_{i}$ with $1<i<i_{0}$. Then

$\displaystyle d_{\rm IA}(x,\partial\Delta_{\pi})$ $\displaystyle=$
$\displaystyle\langle x,v_{i}\rangle+\lambda$ $\displaystyle=$
$\displaystyle\langle
x,(\sigma_{w}^{-1})^{*}\sigma_{w}^{*}v_{i}\rangle+\lambda$ $\displaystyle=$
$\displaystyle\langle\sigma_{w}^{-1}x,\sigma_{w}^{*}v_{i}\rangle+\lambda$
$\displaystyle=$ $\displaystyle d_{\rm
IA}(\sigma_{w}^{-1}x,\partial\Delta_{\pi^{\prime}})$ $\displaystyle=$
$\displaystyle d_{\rm IA}(\tau^{-1}(x),\partial\Delta_{\pi^{\prime}}).$

We have used the definition (56) of the chambers $K_{i}$, equation (55), and
the fact that $\tau^{-1}(K_{i})=K_{i}^{\prime}$. For $x\in K_{i}$ in another
chamber, the claim follows similarly. Since $\Delta_{\pi}=\cup_{i}K_{i}$, the
claim of the lemma follows. $\Box$

Proof of Theorem 3.2.  Let $S\subset\Delta_{\pi}$ be the union of branch cut
lines used in the mutations $\tau_{k},\ldots,\tau_{1}$ bringing $\Delta_{\pi}$
to the toric base $\Delta$, i.e. for which we have
$\Delta=(\tau_{k}\circ\ldots\circ\tau_{1})(\Delta_{\pi})$. The set $S$ is a
finite union of line segments. Since $\mu\colon X\rightarrow\Delta$ is a
monotone toric four-manifold, (31) holds on an open dense subset
$U\subset\Delta$, see Example 2.8. Let $x$ be an element in the open dense
subset $\tau^{-1}(U)\setminus S\subset\Delta_{\pi}$, where
$\tau=\tau_{k}\circ\ldots\circ\tau_{1}$. By Lemma 3.3, the mutations of
polytopes can be realized by ATFs such that

$\pi^{-1}(x)=\mu^{-1}(\tau(x)).$ (57)

By Lemma 3.5, we find

$e(\pi^{-1}(x))=d_{\rm IA}(\tau(x),\partial\Delta)=d_{\rm
IA}(x,\tau^{-1}(\partial\Delta))=d_{\rm IA}(x,\partial\Delta_{\pi}).$ (58)

$\Box$

Let $\Delta_{\pi}=\\{\ell_{i}(x)\geqslant 0\\}$, where the functionals
$\ell_{i}(x)=\langle x,v_{i}\rangle+\lambda$ are chosen by the same
conventions as in (23). Then we deduce the following.

######  Corollary 3.6.

Let $\Delta_{\pi}=\\{\ell_{i}(x)\geqslant 0\\}$ be the base of an ATF
$\pi\colon X\rightarrow\Delta_{\pi}$ satisfying the same hypotheses as in
Theorem 3.2. Then

${\mathcal{E}}_{\pi^{-1}(x)}(b)\sim d_{\rm
IA}(x,\partial\Delta_{\pi})+\min_{i\in I(x)}\\{\langle b,v_{i}\rangle\\},\quad
I(x)=\\{i\,|\,\ell_{i}(x)=d_{\rm IA}(x,\partial\Delta_{\pi})\\},$ (59)

for all regular fibres $\pi^{-1}(x)\subset X$.

Proof. Let $\pi^{-1}(x)$ be a regular fibre. We can assume, possibly after
performing changes in branch cuts, that $x$ does not lie on a branch cut.
Recall that the map $\pi\colon X\rightarrow\Delta_{\pi}$ yields honest action
coordinates on the complement of branch cuts. Therefore, for every
neighbourhood $U\subset\Delta_{\pi}$ of $x$ contained in the complement of
branch cuts, $\pi$ is a $T^{2}$-moment map and we obtain a versal deformation
$v_{\pi^{-1}(x)}(b)=\pi^{-1}(x+b)$. This follows from Proposition 2.14. The
claim now follows from Theorem 3.2. $\Box$

This implies Vianna’s Theorem 3.1. Indeed, for the monotone Vianna tori
$T_{{\mathfrak{m}}}\subset\mathbb{C}P^{2}$, the displacement energy germ (59)
recovers the integral affine type of the full polytope
$\Delta_{{\mathfrak{m}}}$ and thus Theorem 3.1 follows.

######  Example 3.7 (Exotic tori in $\mathbb{C}P^{2}$).

Furthermore, certain non-monotone almost toric fibres can be distinguished by
the same method. Indeed, let $x$ be a point such that $d_{\rm
IA}(x,\partial\Delta)$ is realized by the integral affine distance to two
different edges. Then its displacement energy germ recovers the integral
affine geometry of the vertex formed by these two edges. The determinant of
the primitive vectors associated to the edges forming this vertex is, by the
discussion surrounding (48), equal to the square of the associated Markov
number. We conclude that $\mathbb{C}P^{2}$ contains infinitely many distinct
one-parameter families of non-monotone tori. These families are in bijeciton
with the Markov numbers and a triple of such families intersects at a monotone
Vianna torus $T_{{\mathfrak{m}}}$ if the three Markov numbers form a Markov
triple. This implies some of the results in [47, Section 7].

######  Remark 3.8.

In [51], the results in $\mathbb{C}P^{2}$ were extended to other monotone
toric four-manifolds. These also follow from the methods discussed here.
Although an analogue of Theorem 3.2 in general does not hold for ATFs obtained
by mutation from _non-monotone_ toric manifolds, the displacement energy of
almost toric fibres can still be determined by induction on mutations. This
way, one can find exotic tori in non-monotone toric four-manifolds. See for
example the forthcoming [11], where all non-monotone $S^{2}\times S^{2}$ will
be treated.

######  Remark 3.9.

For every Markov triple ${\mathfrak{m}}$ the torus $T_{{\mathfrak{m}}}$ is
area-equivalent to the monotone toric fibre $T_{\rm Cl}=T_{(1,1,1)}$, meaning
that every pair of Vianna tori yields a positive answer to Question 1.5.
Indeed, as on an open dense subset of its domain, the tori in the versal
deformation of $T_{{\mathfrak{m}}}$ are Hamiltonian isotopic to toric fibres,
we can pick a Lagrangian isotopy from $T_{{\mathfrak{m}}}$ to a toric fibre
which can be concatenated with a Lagrangian isotopy from that toric fibre to
$T_{\rm Cl}$. Since both endpoints of the resulting Lagrangian isotopy are
monotone, we find $T_{{\mathfrak{m}}}\sim T_{\rm Cl}$. See also Remark 4.7,
where the same idea is applied to the tori
$\Upsilon_{k}\subset\mathbb{C}^{3}$.

##  4\. Constructions and proofs

###  4.1. A family of symplectic reductions

In this subsection we prepare the construction of the $\Upsilon_{k}$-tori
given in §4.2. The $T^{3}$-action on $\mathbb{C}^{3}$ by coordinate-wise
rotation is Hamiltonian and generated by the standard moment map,

$\mu\colon\mathbb{C}^{3}\rightarrow\mathbb{R}_{\geqslant
0}^{3},\quad\mu(z_{1},z_{2},z_{3})=(\pi|z_{1}|^{2},\pi|z_{2}|^{2},\pi|z_{3}|^{2})$
(60)

Let $k\in\mathbb{Z}_{\geqslant 2}$ and define the moment map

$\nu_{k}\colon\mathbb{C}^{3}\rightarrow\Delta_{k},\quad\nu_{k}=(\mu_{1}+k\mu_{3},\mu_{2}-\mu_{3}).$
(61)

This moment map generates a Hamiltonian $T^{2}$-action induced by the standard
$T^{3}$-action via the inclusion

$T^{2}\hookrightarrow
T^{3},\quad\exp(\theta_{1},\theta_{2})\mapsto\exp(\theta_{1},\theta_{2},k\theta_{1}-\theta_{2}).$
(62)

Here, $\Delta_{k}=\\{(x,y)\in\mathbb{R}^{2}\,|\,x\geqslant 0,x+ky\geqslant
0\\}$ denotes the image of $\nu_{k}$. The set of critical points is given by
$\operatorname{Crit}\nu_{k}=\cup_{i\neq j}\\{z_{i}=z_{j}=0\\}$ and the
critical values are given by $\partial\Delta_{k}\cup(\mathbb{R}_{\geqslant
0}\times\\{0\\})$. Denote the subtorus obtained as image of the inclusion (62)
by $T_{k}\subset T^{3}$.

We discuss symplectic reduction with respect to $\nu_{k}$ for
$c=(c_{1},c_{2})\in\operatorname{int}\Delta_{k}$, paying special attention to
the critical values of the form $c=(c_{1},0)$. Every level set
$\nu_{k}^{-1}(c)$ contains one orbit with stabilizer isomorphic to
$\mathbb{Z}/k\mathbb{Z}$, namely the orbit obtained by intersecting
$\nu_{k}^{-1}(c)$ with $\\{z_{1}=0\\}$. This orbit maps to an orbifold point
of order $k$ in the reduced space $\Sigma_{k,c}$. For $c_{2}\neq 0$, the group
$T_{k}$ acts freely on the complement in $\nu_{k}^{-1}(c)$ of that orbit. For
$c=(c_{1},0)$, the level set $\nu_{k}^{-1}(c)$ intersects
$\\{z_{2}=z_{3}=0\\}$ in the orbit $\mu^{-1}(c_{1},0,0)$ and thus for
$c_{1}>0$, the stabilizer of that orbit is isomorphic to $S^{1}$. Removing
this exceptional orbit, we can perform symplectic reduction to obtain an
orbifold reduced space with one puncture. To summarize, for
$c\in\operatorname{int}\Delta_{k}$ we have the following reduction diagram,

${Z_{k,c}}$${\mathbb{C}^{3}}$${´(\Sigma_{k,c},\omega_{k,c}),}$$\scriptstyle{\pi_{k,c}}$
(63)

where $Z_{k,c}=\nu_{k}^{-1}(c)$ in case $c_{2}\neq 0$, and
$Z_{k,c}=\nu_{k}^{-1}(c)\setminus\mu^{-1}(c_{1},0,0)$ in case $c_{2}=0$ and
where $(\Sigma_{k,c},\omega_{k,c})$ is the corresponding reduced space.

######  Proposition 4.1.

For $c_{2}\neq 0$, the reduced space $\Sigma_{k,c}$ is a symplectic orbifold
sphere with one orbifold point of order $k$. For $c_{2}=0$, the reduced space
is an orbifold sphere with one orbifold point of order $k$ and one puncture.
The quotient space carries an induced effective Hamiltonian $S^{1}$-action
generated by a moment map
$H_{k,c}\colon\Sigma_{k,c}\rightarrow[0,\alpha_{k}(c)]$, where

$\alpha_{k}(c)=\int_{\Sigma_{k,c}}\omega_{k,c}=\begin{cases}\frac{c_{1}}{k},&c_{2}\geqslant
0\\\ \frac{c_{1}}{k}+c_{2},&c_{2}<0\end{cases}$ (64)

is the symplectic area of the reduced space. Furthermore, product tori in
$\mathbb{C}^{3}$ map to orbits of the $H_{k,c}$-action on $\Sigma_{k,c}$. More
precisely, we have

$\pi_{k,c}^{-1}(H_{k,c}^{-1}(c_{3}))=\begin{cases}T(c_{1}-kc_{3},c_{3}+c_{2},c_{3}),&c_{2}\geqslant
0\\\ T(c_{1}-kc_{3}+kc_{2},c_{3},c_{3}-c_{2}),&c_{2}<0.\end{cases}$ (65)

Proof. The topology of the reduced spaces has been discussed above. Away from
the orbifold point and the puncture in the case $c_{2}=0$, the reduction (63)
is a special case of toric reduction (see the discussion in §2.1), meaning
that there is an induced Hamiltonian circle action on $\Sigma_{k,c}$ which is
toric. This can be used to prove (64) and (65), see Proposition 2.3.

For the reader’s convenience, let us nonetheless discuss the details. The
Hamiltonian $T^{3}$-action on $\mathbb{C}^{3}$ descends to the quotients
$\Sigma_{k,c}$ since $\nu_{k}$ (generating a subaction of the standard
$T^{3}$-action) commutes with $\mu$, see Proposition 2.2. The induced
$T^{3}$-action on $\Sigma_{k,c}$ has stabilizer $T_{k}$, meaning that it
yields an effective Hamiltonian action by $S^{1}\cong T^{3}/T_{k}$ on the
quotient. A moment map for this induced action can be defined by picking a
circle which is complementary to $T_{k}\subset T^{3}$, e.g. the circle
$\\{1\\}\times\\{1\\}\times S^{1}\subset T^{3}$, whose action is generated by
$\mu_{3}$. Thus $\mu_{3}|_{Z_{k,c}}\colon Z_{k,c}\rightarrow\mathbb{R}$
descends to the Hamiltonian $H_{k,c}\colon\Sigma_{k,c}\rightarrow\mathbb{R}$.
By computing the image $\mu_{3}(Z_{k,c})$, we find that $H_{k,c}$ takes values
in $[0,\alpha_{k}(c)]$ (respectively $(0,\alpha_{k}(c)]$ for $c=(c_{1},0)$)
where $\alpha_{k}(c)$ is defined as in (64). Since this Hamiltonian circle
action is effective, it is toric and thus (as an action coordinate) its range
computes the area of $\Sigma_{k,c}$, proving (64).

To prove (65), note that $H^{-1}_{k,c}(c_{3})$ is an orbit of the residual
action induced by the $T^{3}$-action on $\mathbb{C}^{3}$. By
$T^{3}$-equivariance of $\pi_{k,c}$, see Proposition 2.2, this orbit lifts to
an orbit of the $T^{3}$-action on $\mathbb{C}^{3}$, i.e. a product torus
$T(a_{1},a_{2},a_{3})\subset\mathbb{C}^{3}$. To determine
$a_{1},a_{2},a_{3}\in\mathbb{R}_{>0}$, recall that on $Z_{k,c}$, we have
$\nu_{k}=c$ and thus $a_{1}+ka_{3}=c_{1}$ and $a_{2}-a_{3}=c_{2}$ by
definition of $\nu_{k}$, see (61). Furthermore, the $c_{3}>0$ is equal to the
area of the sublevel set $H_{k,c}^{-1}(\leqslant c_{3})$, since $H_{k,c}$
generates a Hamiltonian circle action; therefore $c_{3}$ is the integral
affine distance of the point $(a_{1},a_{2},a_{3})$ to the union of facets
$\\{x_{1}=0\\}\cup\\{x_{2}=0\\}$. This yields (65). $\Box$

###  4.2. Tori and their versal deformations

We use the same set-up and notation as in §4.1.

######  Definition 4.2.

Let $a_{1},a_{2}>0$ with $a_{2}<\alpha_{k}(a_{1},0)$ and set

$\Upsilon_{k}(a_{1},a_{2})=\pi_{k,(a_{1},0)}^{-1}(C(a_{2})),$ (66)

where $C(a_{2})\subset\Sigma_{k,(a_{1},0)}$ is a closed embedded curve in
$\Sigma_{k,(a_{1},0)}$ bounding a disk (not containing the puncture) of
symplectic area $a_{2}$ in the punctured orbifold $\Sigma_{k,(a_{1},0)}$.

Note that $C(a_{2})\subset\Sigma_{k,(a_{1},0)}$ is unique up to Hamiltonian
isotopy and thus so is $\Upsilon_{k}(a_{1},a_{2})$, since Hamiltonian
isotopies lift by Proposition 2.1.

######  Remark 4.3.

The same construction can also be carried out, with small modifications at
various points, for $k\in\\{-1,0,1\\}$. Note that in the cases
$k\in\\{-1,0\\}$, the reduced space is non-compact and in the case $k=1$ it
does not have an orbifold point. To simplify the exposition, we do not treat
these cases in detail. Furthermore, the tori we obtain in these cases are
already known, namely, $\Theta_{-1}\cong T^{3}_{\rm Ch}$ is the
$3$-dimensional version of the Chekanov torus, $\Theta_{0}\cong S^{1}\times
T^{2}_{\rm Ch}$ is the product of the Chekanov torus with a circle and
$\Theta_{1}$ is Hamiltonian isotopic to a monotone product torus and thus not
exotic.

Let $a_{1},a_{2}>0$ with $a_{2}<\alpha_{k}(a_{1},0)$. From now on, we denote
$\Upsilon_{k}=\Upsilon_{k}(a_{1},a_{2})$ to simplify notation. To construct a
versal deformation of $\Upsilon_{k}$, we choose a $T^{2}$-equivariant
Weinstein chart, $\varphi\colon T^{*}T^{3}\supset
V\hookrightarrow\mathbb{C}^{3}$, which intertwines rotation of the first two
components of $T^{3}$ in $T^{*}T^{3}$ with the $T_{k}$-action on
$\mathbb{C}^{3}$. Rotation of the first two coordinates of
$T^{*}T^{3}=(\mathbb{R}/\mathbb{Z})^{3}\times\mathbb{R}^{3}=\\{(\theta,p)=(\theta_{1},\theta_{2},\theta_{3},p_{1},p_{2},p_{3})\\}$
equipped with $\omega_{T^{*}T^{3}}=\sum_{i}d\theta_{i}\wedge dp_{i}$ is
generated by the moment map $\eta(\theta,p)=(p_{1},p_{2})$. Since $\varphi$ is
equivariant, we find that $\varphi^{*}\nu_{k}=\eta+(a_{1},0)$ on $U$. As in
(33), we choose the versal deformation

$v_{\Upsilon_{k}}^{\varphi}\colon
H^{1}(T^{3};\mathbb{R})\cong\mathbb{R}^{3}\supset
U\rightarrow{\mathcal{L}}_{\Upsilon_{k}},\quad
v_{\Upsilon_{k}}^{\varphi}(b)=\varphi(T^{3}\times\\{b\\}),$ (67)

where we have identified $H^{1}(T^{3},\mathbb{R})\cong\mathbb{R}^{3}$ with the
fibre part of $T^{*}T^{3}=T^{3}\times\mathbb{R}^{3}$ and $U$ is a
neighbourhood of $0\in\mathbb{R}^{3}$ such that $T^{3}\times U$ is contained
in the domain $V$ of $\varphi$.

######  Proposition 4.4.

Let $b\in U$, then

$v_{\Upsilon_{k}}^{\varphi}(b)=\pi_{k,(a_{1}+b_{1},b_{2})}^{-1}(C),$ (68)

for some closed embedded curve $C\subset\Sigma_{k,(a_{1}+b_{1},b_{2})}$,
bounding a disk of area $a_{2}+b_{3}$.

Proof. By equivariance of $\varphi$, the torus
$v_{\Upsilon_{k}}^{\varphi}(b)=\varphi(T^{3}\times\\{b\\})$ is
$T_{k}$-invariant. Furthermore, since $\varphi^{*}\nu_{k}=\eta+(a_{1},0)$, we
find that $v^{\varphi}_{\Upsilon_{k}}(b)$ is contained in the level set
$\nu_{k}^{-1}(a_{1}+b_{1},b_{2})$. Therefore, $v^{\varphi}_{\Upsilon_{k}}(b)$
projects to some compact embedded Lagrangian
$C\subset\Sigma_{k,(a_{1}+b_{1},b_{2})}$. Let us now prove that $C$ bounds a
disk of area $a_{2}+b_{3}$. Set

$\phi_{b}=(\varphi|_{T^{3}\times\\{b\\}})_{*}\colon H_{1}(T^{3})\rightarrow
H_{1}(v^{\varphi}_{\Upsilon_{k}}(b)),$ (69)

and let $e_{1},e_{2},e_{3}\in H_{1}(T^{3})$ be the standard basis of the first
homology of $T^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$. The classes
$\phi_{b}e_{1},\phi_{b}e_{2}$ are generated by orbits of the $T_{k}$-action
and hence
$(\pi_{k,(a_{1}+b_{1},b_{2})}|_{v_{\Upsilon_{k}}^{\varphi}(b)})_{*}\phi_{b}e_{i}=0$
for $i=1,2$. Furthermore, we find that

$\left(\pi_{k,(a_{1}+b_{1},b_{2})}|_{v_{\Upsilon_{k}}^{\varphi}(b)}\right)_{*}\phi_{b}e_{3}=[C]\in
H_{1}(C),$ (70)

where $[C]$ is the fundamental class with appropriate orientation. Indeed, any
basis of $H_{1}(v_{\Upsilon_{k}}^{\varphi}(b))$ maps to a basis of $H_{1}(C)$
under the reduction map. Recall that $C$ bounds a disk in
$\Sigma_{k,(a_{1}+b_{1},b_{2})}$. Let $d_{b}\in
H_{2}(\Sigma_{k,(a_{1}+b_{1},b_{2})},C)$ be the class of this disk with
$\partial d_{b}=[C]$. Since $C$ is contractible, the disk $d_{b}$ has a unique
lift $D_{b}\in
H_{2}(\nu_{k}^{-1}(a_{1}+b_{1},b_{2}),v^{\varphi}_{\Upsilon_{k}}(b))$ with
boundary $\partial D_{b}=\phi_{b}e_{3}$. The cylinder $C_{e_{3}}$ swept out by
$\phi_{0}e_{3}$ under a Lagrangian isotopy $i_{t}$ (contained in the image of
$\varphi$) with endpoints $v_{\Upsilon_{k}}^{\varphi}(0)$ and
$v_{\Upsilon_{k}}^{\varphi}(b)$ has symplectic area
$\langle[\omega],C_{e_{3}}\rangle=b_{3}$. Indeed, this follows from the fact
that $v_{\Upsilon_{k}}^{\varphi}$ is a versal deformation, see Definition
2.12, together with (35). Furthermore $\partial
C_{e_{3}}=\phi_{b}e_{3}-\phi_{0}e_{3}$ and therefore it can be compactified to
a sphere $S=D_{0}+C_{e_{3}}-D_{b}$, by attaching two disks at its boundary.
Using the property (21) of symplectic reduction, we obtain

$\langle[\omega],D_{b}\rangle=\langle[\pi_{k,(a_{1}+b_{1},b_{2})}^{*}\omega_{k,(a_{1}+b_{1},b_{2})}],D_{b}\rangle=\langle[\omega_{k,(a_{1}+b_{1},b_{2})}],d_{b}\rangle.$
(71)

For $b=0$, i.e. for the torus $v_{\Upsilon_{k}}^{\varphi}(0)=\Upsilon_{k}$,
this quantity is equal to $a_{2}$, by the definition of
$\Upsilon_{k}=\Upsilon_{k}(a_{1},a_{2})$. Since $S\in
H_{2}(\mathbb{C}^{3})=0$, we find

$0=\langle[\omega],S\rangle=\langle[\omega],D_{0}\rangle+\langle[\omega],C_{e_{3}}\rangle-\langle[\omega],D_{b}\rangle=a_{2}+b_{3}-\langle[\omega_{k,(a_{1}+b_{1},b_{2})}],d_{b}\rangle,$
(72)

which proves the claim. $\Box$

In other words, all tori in the versal deformation
$v_{\Upsilon_{k}}^{\varphi}$ are lifts of curves in the respective reduced
spaces of the action generated by $\nu_{k}$ and furthermore, the parameters
$b_{1},b_{2}$ determine the reduced space and $b_{3}$ the area of the curve.

Note that whenever $b_{2}\neq 0$, the reduced space
$\Sigma_{k,(a_{1}+b_{1},b_{2})}$ is not punctured and hence any two curves
$C,C^{\prime}\subset\Sigma_{k,(a_{1}+b_{1},b_{2})}$ enclosing the same
symplectic area are Hamiltonian isotopic. In particular, the corresponding
versal deformations are Hamiltonian isotopic to certain product tori. This is
a typical example of the _instability of exotic tori_ discussed in §1.6. More
precisely, combining Propositions 4.4 and 4.1, find the following.

######  Corollary 4.5.

For all $b\in U$ with $b_{2}\neq 0$, the tori in the versal deformation of
$\Upsilon_{k}$ are Hamiltonian isotopic to the product tori,

$v_{\Upsilon_{k}}^{\varphi}(b)\cong\begin{cases}T(a_{1}-ka_{2}+b_{1}-kb_{3},a_{2}+b_{2}+b_{3},a_{2}+b_{3}),&b_{2}>0\\\
T(a_{1}-ka_{2}+b_{1}+kb_{2}-kb_{3},a_{2}+b_{3},a_{2}-b_{2}+b_{3}),&b_{2}<0.\end{cases}$
(73)

Proof. Let $C\subset\Sigma_{k,(a_{1}+b_{1},b_{2})}$ be the projection of
$v_{\Upsilon_{k}}^{\varphi}(b)$ as in Proposition 4.4. For $b_{2}\neq 0$, the
reduced space is a orbifold sphere with one orbifold point of order $k$ by
Proposition 4.1. Since $C$ bounds a disk of area $a_{2}+b_{3}$, it is
Hamiltonian isotopic to the orbit
$H_{k,(a_{1}+b_{1},b_{2})}^{-1}(a_{2}+b_{3})$ of the residual $S^{1}$-action
on the reduced space. By Proposition 2.1, the Hamiltonian isotopy can be
lifted to yield,

$v_{\Upsilon_{k}}^{\varphi}(b)\cong\pi^{-1}_{k,(a_{1}+b_{1},b_{2})}(H_{k,(a_{1}+b_{1},b_{2})}^{-1}(a_{2}+b_{3})).$
(74)

The claim follows from applying (65). $\Box$

######  Remark 4.6.

Note that the support of the Hamiltonian isotopy in Corollary 4.5 mapping
$v_{\Upsilon_{k}}^{\varphi}(b)$ to the corresponding product torus can be
chosen to lie arbitrarily close to the set $\nu^{-1}_{k}(a_{1}+b_{1},b_{2})$
on which we perform symplectic reduction. In particular, since $k\geqslant 2$,
its support lies in the ball $B(a_{1}+b_{1}+b_{2})$. This will be used in
§4.4.

###  4.3. Proof of Theorem A

We compute the displacement energy germ of
$\Upsilon_{k}=\Upsilon_{k}(a_{1},a_{2})$ on the subset $\\{b_{2}\neq
0\\}\subset H^{1}(T^{3};\mathbb{R})\cong\mathbb{R}^{3}$. Note that this is the
subset of $H^{1}(T^{3};\mathbb{R})$ corresponding to tori which are
Hamiltonian isotopic to product tori by Corollary 4.5. This means that we do
not compute the displacement energy of the tori $\Upsilon_{k}(a_{1},a_{2})$
(corresponding to versal deformations with $b_{2}=0$) themselves, see Remark
4.8. By (73) and (30), we find

$e(v_{\Upsilon_{k}}^{\varphi}(b))=\begin{cases}\min\\{a_{1}-ka_{2}+b_{1}-kb_{3},a_{2}+b_{2}+b_{3},a_{2}+b_{3}\\}´,&b_{2}>0,\\\
\min\\{a_{1}-ka_{2}+b_{1}+kb_{2}-kb_{3},a_{2}+b_{3},a_{2}-b_{2}+b_{3}\\},&b_{2}<0\end{cases}$
(75)

for $b\in U$. The germ at the origin of $b\mapsto
e(v^{\varphi}_{\Upsilon_{k}}(b))$ depends on which terms in the minima in (75)
are dominant for $b=0$. In case $a_{1}-ka_{2}<a_{2}$, the first terms in the
respective minima are dominant. For $a_{1}-ka_{2}=a_{2}$, all terms are
dominant. Note however that $b_{3}<b_{3}-b_{2}$ for $b_{2}<0$ and
$b_{3}<b_{3}+b_{2}$ for $b_{2}>0$, meaning that there are only three dominant
terms in total. By the same token, there is only one dominant term in the case
$a_{1}-ka_{2}>a_{2}$. We find

$\mathcal{E}_{\Upsilon_{k}}(b)=\begin{cases}a_{1}-ka_{2}+\min\\{\langle
b,v_{1}\rangle,\langle b,v_{2}\rangle\\},&a_{1}-ka_{2}<a_{2}\\\
a_{1}-ka_{2}+\min\\{\langle b,v_{1}\rangle,\langle b,v_{2}\rangle,\langle
b,v_{3}\rangle\\},&a_{1}-ka_{2}=a_{2}\\\
a_{2}+b_{3},&a_{1}-ka_{2}>a_{2},\end{cases}$ (76)

with

$v_{1}=\begin{pmatrix}1\\\ k\\\ -k\end{pmatrix},v_{2}=\begin{pmatrix}1\\\ 0\\\
-k\end{pmatrix},v_{3}=\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix},$ (77)

for all $b\in\mathbb{R}^{3}$ with $b_{2}\neq 0$. Let $a=(a_{1},a_{2})\in
U_{k}$ and $a^{\prime}=(a_{1}^{\prime},a_{2}^{\prime})\in U_{k^{\prime}}$,
with $U_{k},U_{k^{\prime}}$ defined as in (2). This means we are only
interested in the first two cases in (76). Indeed, note that the displacement
energy germ in the case $a_{1}-ka_{2}>a_{2}$ does not depend on $k$ and thus
cannot distinguish the corresponding $\Upsilon_{k}$. If
$\Upsilon_{k}(a)\cong\Upsilon_{k^{\prime}}(a^{\prime})$, then
${\mathcal{E}}_{\Upsilon_{k}}\sim{\mathcal{E}}_{\Upsilon_{k^{\prime}}}$ and,
since (76) is of the form (43), there is a
$\Phi\in\operatorname{GL}(3;\mathbb{Z})$ mapping $\\{v_{1},v_{2}\\}$ to
$\\{v_{1}^{\prime},v_{2}^{\prime}\\}$ in the non-monotone and
$\\{v_{1},v_{2},v_{3}\\}$ to
$\\{v_{1}^{\prime},v_{2}^{\prime},v_{3}^{\prime}\\}$ in the monotone case.
However since $D_{\rm IA}(v_{1},v_{2})=k$ by Proposition 2.5, this implies
$k=k^{\prime}$. This proves Theorem A. $\Box$

######  Remark 4.7.

It follows from the proof of Theorem A that $\Upsilon_{k}(a_{1},a_{2})\sim
T(a_{1}-ka_{2},a_{2},a_{2})$, i.e. that every $\Upsilon_{k}$ is area-
equivalent in the sense of Definition 1.4. Again, we use the fact that small
versal deformations of $\Upsilon_{k}$ are Hamiltonian isotopic to product
tori. The desired Lagrangian isotopy is obtained by concatenation of a
deformation of $\Upsilon_{k}(a_{1},a_{2})$ to a suitable torus in its versal
deformation with its Hamiltonian isotopy to a product torus and finally with a
deformation of the product torus to $T(a_{1}-ka_{2},a_{2},a_{2})$. Such a
Lagrangian isotopy has the desired endpoints and vanishing Lagrangian flux. By
Remark 2.11, this proves the claim.

######  Remark 4.8.

Recall that $e(\cdot)$ is an upper semicontinuous function on the space of
Lagrangians ${\mathcal{L}}$ equipped with the $C^{1}$-topology. Since we have
computed the displacement energy on an open dense subset of a neighbourhood of
$\Upsilon_{k}\in{\mathcal{L}}$ in the proof of Theorem A above, we deduce
$e(\Upsilon_{k}(a_{1},a_{2}))\geqslant a_{1}-ka_{2}$ in case
$a_{1}-ka_{2}\leqslant a_{2}$ and $e(\Upsilon_{k}(a_{1},a_{2}))\geqslant
a_{2}$ otherwise. We do not know if these inequalities are sharp.

###  4.4. Proof of Theorem B

Let $(X,\omega)$ be a geometrically bounded symplectic manifold and
$\psi\colon B(R)\rightarrow X$ a symplectic ball embedding. As outlined in
§1.6, we prove that there is $\varepsilon>0$ such that $\psi(\Upsilon_{k})$
has the _locality property_ , i.e. that
${\mathcal{E}}_{\Upsilon_{k}}\sim{\mathcal{E}}_{\psi(\Upsilon_{k})}$ (see
Definition 1.12) where $\Upsilon_{k}=\Upsilon_{k}(a_{1},a_{2})$ for all
$a_{1}<\varepsilon$. As in the proof of Theorem A in §4.3, this allows us to
conclude that
$\psi(\Upsilon_{k}(a))\cong\psi(\Upsilon_{k^{\prime}}(a^{\prime}))$ implies
$k=k^{\prime}$ whenever $a\in U_{k}$ and $a^{\prime}\in U_{k^{\prime}}$.

Set $\varepsilon=\min\left\\{\frac{R}{2},\lambda_{S}(X,\psi)\right\\}$, where
$\lambda_{S}(X,\psi)$ is defined in (15). Let $a_{1}<\varepsilon$ and
$\Upsilon_{k}=\Upsilon_{k}(a_{1},a_{2})$. Note that $\Upsilon_{k}\subset
B(\varepsilon)$. We apply Lemma 1.15 to the versal deformation
$v_{\Upsilon_{k}}^{\varphi}\colon U\rightarrow{\mathcal{L}}_{\Upsilon_{k}}$ of
$\Upsilon_{k}$ constructed in §4.2. Here we have restricted, if necessary, the
domain $U$, such that $a_{1}+b_{1}+b_{2}<\varepsilon$. By Remark 4.6, this
implies that for all $b\in U$ with $b_{2}\neq 0$,

$v_{\Upsilon_{k}}^{\varphi}(b)\cong T(\sigma(b)),$ (78)

by a Hamiltonian isotopy supported in $B(\varepsilon)$. The piecewise affine
linear map $\sigma$ determining $T(\sigma(b))$ is given as in Corollary 4.5.
By Lemma 1.15, it is sufficient to check that the images $\psi(T(\sigma(b)))$
of the tori appearing in (78) have property (CS), see Definition 1.13. By
(73), the condition $a_{1}+b_{1}+b_{2}<\varepsilon$ implies
$T(\sigma(b))\subset B(\varepsilon)$. Let $x=\sigma(b)$. By our choice
$\varepsilon\leqslant\frac{R}{2}$, we find that

$x_{1}+\ldots+x_{n}+\underline{x}<\varepsilon+\varepsilon\leqslant\frac{R}{2}+\frac{R}{2}=R$
(79)

and thus (16) holds. Furthermore,
$\underline{x}<\varepsilon\leqslant\lambda_{S}(X,\psi)$, and thus (17) holds.
This proves that $\psi(\Upsilon_{k})$ has the locality property and thus that
the claim of Theorem B follows. $\Box$

###  4.5. Proof of Proposition 1.6

Let $H\colon\mathbb{C}^{3}\rightarrow\mathbb{R}$ be the Hamiltonian defined by

$H(z_{1},z_{2},z_{3})=\pi(|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}),$ (80)

which generates the diagonal $S^{1}$-action on $\mathbb{C}^{3}$. Its
symplectic quotient at the level $c>0$ is

${H^{-1}(c)=S^{5}(c)}$${\mathbb{C}^{3}}$${(\mathbb{C}P^{2},c\omega_{\mathbb{C}P^{2}}).}$$\scriptstyle{p_{c}}$
(81)

Let $T_{{\mathfrak{m}}}\subset\mathbb{C}P^{2}$ be the Vianna torus associated
with the Markov triple ${\mathfrak{m}}$ as in §3.2 and set

$\Theta_{{\mathfrak{m}}}(a)=p_{a}^{-1}(T_{{\mathfrak{m}}})\subset
S^{5}(a)\subset\mathbb{C}^{3}.$ (82)

Note that, technically speaking, the identification in (81) of the reduced
spaces $X_{c}=H^{-1}(c)/S^{1}$ with $\mathbb{C}P^{2}$ involves a choice of
diffeomorphism. Since this will be used in the proof of Lemma 4.9, we make it
explicit here. Our definition of $\mathbb{C}P^{2}$ is
$\mathbb{C}P^{2}=X_{1}=H^{-1}(1)/S^{1}$ equipped with the symplectic form
$\omega_{\mathbb{C}P^{2}}$ coming from reduction, i.e. satisfying
$p_{1}^{*}\omega_{\mathbb{C}P^{2}}=\omega$, where $\omega$ is the canonical
form on $\mathbb{C}^{3}$. Note that this coincides with the conventions in
§3.2, yielding $\Delta=\Delta_{\mathbb{C}P^{2}}$ as in (46) as moment
polytope. For $c_{1},c_{2}>0$, consider the natural diffeomorphism
$S_{c_{1},c_{2}}\colon H^{-1}(c_{1})\rightarrow H^{-1}(c_{2})$ defined by
scaling $z\mapsto\sqrt{\frac{c_{2}}{c_{1}}}z$. This induces a diffeomorphism

$s_{c_{1},c_{2}}\colon X_{c_{1}}\rightarrow X_{c_{2}},\quad
s_{c_{1},c_{2}}^{*}\omega_{c_{2}}=\frac{c_{2}}{c_{1}}\,\omega_{c_{1}}.$ (83)

In (81), we have implicitely made use of the diffeomorphism
$s_{1,c}\colon\mathbb{C}P^{2}\rightarrow X_{c}$ with
$s_{1,c}^{*}\omega_{c}=c\omega_{\mathbb{C}P^{2}}$.

######  Lemma 4.9.

${\mathcal{E}}_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})\sim d_{\rm
IA}(b,(a+b_{3})\partial\Delta_{{\mathfrak{m}}}),\quad\overline{b}=(b,b_{3}).$
(84)

Let us first prove that Proposition 1.6 follows.

Proof of Proposition 1.6.  Let ${\mathfrak{m}}=(\alpha,\beta,\gamma)$ be a
Markov triple and
$u=(u_{1},u_{2}),v=(v_{1},v_{2}),w=(w_{1},w_{2})\in\mathbb{Z}^{2}$ be the
defining vectors of the Markov triangle $\Delta_{{\mathfrak{m}}}$ as in (48).
Lemma 4.9 yields,

${\mathcal{E}}_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})\sim\frac{a}{3}+\min\\{\langle\overline{b},\overline{u}\rangle,\langle\overline{b},\overline{v}\rangle,\langle\overline{b},\overline{w}\rangle\\},$
(85)

where $\overline{u}=(u_{1},u_{2},1)$, $\overline{v}=(v_{1},v_{2},1)$ and
$\overline{w}=(w_{1},w_{2},1)$. We show that by (85), the
$\operatorname{GL}(3;\mathbb{Z})$-equivalence class of the displacement energy
germ ${\mathcal{E}}_{\Theta_{{\mathfrak{m}}}}$ determines the Markov triple
${\mathfrak{m}}=(\alpha,\beta,\gamma)$ which implies the claim. Indeed, the
minors of e.g. the matrix $(\overline{u}|\overline{v})$ are
$\det(u|v)=\alpha^{2}$, $u_{1}-v_{1}$ and $u_{2}-v_{2}$. Therefore Proposition
2.5 and (49) allow us to conclude that the integral index is $D_{\rm
IA}(\overline{u},\overline{v})=\alpha$. Similarly, we find $D_{\rm
IA}=\beta,\gamma$ for the other two pairs of vectors. $\Box$

Proof of Lemma 4.9.  To construct a versal deformation of
$\Theta_{{\mathfrak{m}}}(a)\subset\mathbb{C}^{3}$, we choose a Weinstein chart
$\varphi\colon T^{*}T^{3}\supset U\rightarrow\mathbb{C}^{3}$ of
$\Theta_{{\mathfrak{m}}}(a)$ which is $S^{1}$-equivariant with respect to the
action by rotation of the last component on
$T^{3}=(\mathbb{R}/\mathbb{Z})^{3}$ and the action generated by $H$ on
$\mathbb{C}^{3}$. Then $\varphi^{*}H=p_{3}+a$, where $p_{3}$ is the third
component in $\mathbb{R}^{3}$, in the identification
$T^{*}T^{3}=T^{3}\times\mathbb{R}^{3}$. Recall that the associated versal
deformation is given by

$v_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})=\varphi\left(T^{3}\times\\{\overline{b}\\}\right),\quad\overline{b}=(b,b_{3}).$
(86)

It follows that every torus
$v_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})\subset H^{-1}(a+b_{3})$ in the
versal deformation is invariant under the action generated by $H$ and thus
projects to a Lagrangian torus
$p_{a+b_{3}}\left(v_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})\right)\subset
X_{a+b_{3}}$. To compute its displacement energies, we will relate
$v_{\Theta_{{\mathfrak{m}}}(a)}$ to the versal deformation
$v_{T_{{\mathfrak{m}}}}(b)=\pi_{{\mathfrak{m}}}^{-1}(b)$ of
$T_{{\mathfrak{m}}}\subset X_{a}\approx\mathbb{C}P^{2}$ coming from the almost
toric fibration $\pi_{{\mathfrak{m}}}\colon X_{a}\rightarrow
a\Delta_{{\mathfrak{m}}}$, discussed (in a more general setting) in §3.3, see
in particular the proof of Theorem 3.2. The ATF base diagram is scaled by a
factor of $a>0$ since we consider the reduced space
$X_{a}\approx\mathbb{C}P^{2}$. We will see for example that $b\mapsto
p_{a}\left(v_{\Theta_{{\mathfrak{m}}}(a)}(b,0)\right)$ is a versal deformation
of $T_{{\mathfrak{m}}}=\left(v_{\Theta_{{\mathfrak{m}}}(a)}(0,0)\right)\subset
X_{a}$. In particular, by the uniqueness of versal deformations up to
Hamiltonian isotopy (38), we find

$p_{a}\left(v_{\Theta_{{\mathfrak{m}}}(a)}(b,0)\right)\cong
v_{T_{{\mathfrak{m}}}}(\Psi(b)),$ (87)

for some $\Psi\in\operatorname{GL}(2;\mathbb{Z})$ and small enough
$b\in\mathbb{R}^{2}$. More generally, we claim that

$w_{b_{3}}(b)=s_{a+b_{3},a}\left(p_{a+b_{3}}\left(v_{\Theta_{{\mathfrak{m}}}(a)}\left(\frac{a+b_{3}}{a}\,b,b_{3}\right)\right)\right)\subset
X_{a}$ (88)

is a versal deformation of (a torus equivalent to) $T_{{\mathfrak{m}}}\subset
X_{a}$. This is defined for all small enough $b$. Here $s_{a+b_{3},a}\colon
X_{a+b_{3}}\rightarrow X_{a}$ denotes the diffeomorphism (83) satisfying
$s_{a+b_{3},a}^{*}\omega_{a}=\frac{a}{a+b_{3}}\omega_{a+b_{3}}$. As becomes
clear below, the scaling by $\frac{a+b_{3}}{a}$ in (88) compensates for the
scaling of the symplectic form when changing level sets. First, note that
$w_{b_{3}}(0)\cong T_{{\mathfrak{m}}}\subset X_{a}$. Indeed, $b_{3}\mapsto
w_{b_{3}}(0)$ is a Lagrangian isotopy of monotone tori, implying that it has
vanishing Lagrangian flux and is therefore induced by a Hamiltonian isotopy.
We use (35) to prove that $w_{b_{3}}$ is a versal deformation in the sense of
Definition 2.12. Let $E_{1},E_{2},E_{3}\in H_{1}(\Theta_{{\mathfrak{m}}}(a))$
be the basis induced by the standard basis on
$T^{3}=(\mathbb{R}/\mathbb{Z})^{3}$ via the Weinstein chart $\varphi$. Note
that $E_{3}$ is the class of the $H$-action and that
$e_{1}=\left(p_{a}|_{\Theta_{{\mathfrak{m}}}(a)}\right)_{*}E_{1},e_{2}=\left(p_{a}|_{\Theta_{{\mathfrak{m}}}(a)}\right)_{*}E_{2}\in
H_{1}(T_{{\mathfrak{m}}})$ form a basis. Since
$v_{\Theta_{{\mathfrak{m}}}(a)}$ is a versal deformation, we have

$\langle\operatorname{Flux}\widehat{\Lambda}_{t},E_{i}\rangle=\langle[\omega],C_{E_{i}}\rangle=b_{i},$
(89)

for every Lagrangian isotopy $\widehat{\Lambda}_{t}$ contained in the image of
the versal deformation with $\widehat{\Lambda}_{0}=\Theta_{{\mathfrak{m}}}(a)$
and $\widehat{\Lambda}_{1}=v_{\Theta_{{\mathfrak{m}}}(a)}(b)$ and for every
relative cycle $C_{E_{i}}\in
H_{2}(\mathbb{C}^{3},\widehat{\Lambda}_{0}\sqcup\widehat{\Lambda}_{1})$
contained in the image of $\varphi$ bounding $E_{i}\in
H_{1}(\widehat{\Lambda}_{0})$ on $\Lambda_{0}$. Now let $b\in\mathbb{R}^{2}$
be small enough and let $\Lambda_{t}=w_{b_{3}}(b(t))$ be a Lagrangian isotopy
with $b(0)=0$ and $b(1)=b$. Since $(p_{a+b_{3}})_{*}E_{i}=e_{i}$, every class
$C_{e_{i}}\in H_{2}(X_{a},\Lambda_{0}\sqcup\Lambda_{1})$ is the image
$(s_{a+b_{3},a}\circ p_{a+b_{3}})_{*}C_{E_{i}}$ of a class

$C_{E_{i}}\in
H_{2}\left(\mathbb{C}^{3},v_{\Theta_{{\mathfrak{m}}}(a)}\left(0,b_{3}\right)\sqcup
v_{\Theta_{{\mathfrak{m}}}(a)}\left(\frac{a+b_{3}}{a}b,b_{3}\right)\right),$
(90)

contained in $H^{-1}(a+b_{3})$. Since $v_{\Theta_{{\mathfrak{m}}}(a)}$ is a
versal deformation, we have
$\langle[\omega],C_{E_{i}}\rangle=\frac{a+b_{3}}{a}b_{i}$ and thus we find,

$\displaystyle\langle\operatorname{Flux}\Lambda_{t},e_{i}\rangle$
$\displaystyle=$ $\displaystyle\langle[\omega_{a}],C_{e_{i}}\rangle$
$\displaystyle=$
$\displaystyle\langle[p_{a+b_{3}}^{*}s_{a+b_{3},a}^{*}\omega_{a}],C_{E_{i}}\rangle$
$\displaystyle=$
$\displaystyle\frac{a}{a+b_{3}}\langle[p_{a+b_{3}}^{*}\omega_{a+b_{3}}],C_{E_{i}}\rangle$
$\displaystyle=$
$\displaystyle\frac{a}{a+b_{3}}\langle[\omega],C_{E_{i}}\rangle$
$\displaystyle=$ $\displaystyle\frac{a}{a+b_{3}}\frac{a+b_{3}}{a}b_{i}=b_{i},$

where we have used (21) in the fourth equality. Thus we have proved that (88)
is a versal deformation of $w_{b_{3}}(0)\cong T_{{\mathfrak{m}}}$. By the
uniqueness of versal deformations (38), we deduce that there is
$\Psi\in\operatorname{GL}(2;\mathbb{Z})$ such that

$s_{a+b_{3},a}\left(p_{a+b_{3}}\left(v_{\Theta_{{\mathfrak{m}}}(a)}\left(\frac{a+b_{3}}{a}\,b,b_{3}\right)\right)\right)\cong
v_{T_{{\mathfrak{m}}}}(\Psi(b))$ (91)

Using $s_{a+b_{3},a}\circ p_{a+b_{3}}=p_{a}\circ S_{a+b_{3},a}$, we find

$v_{\Theta_{{\mathfrak{m}}}(a)}\left(\Psi^{-1}(b),b_{3}\right)\cong
S_{a+b_{3},a}^{-1}\left(p_{a}^{-1}\left(v_{T_{{\mathfrak{m}}}}\left(\frac{a}{a+b_{3}}\,b\right)\right)\right).$
(92)

As discussed in §3.3, there is a piece-wise linear map
$\tau=(\tau_{1},\tau_{2})\colon\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$, the
linear pieces of which are in $\operatorname{GL}(2;\mathbb{Z})$, such that

$v_{T_{{\mathfrak{m}}}}(b)\cong\mu^{-1}(\tau(b)),\quad b\in V,$ (93)

where $\mu^{-1}(\cdot)$ denotes toric fibres of the standard moment map $\mu$
and $V$ is an open dense subset of a neighbourhood of the origin. See the
proof of Theorem 3.2 for details. Under $p_{a}$, toric fibres lift to product
tori in $\mathbb{C}^{3}$,

$p_{a}^{-1}(\mu^{-1}(y_{1},y_{2}))=T\left(\frac{a}{3}+y_{1},\frac{a}{3}+y_{2},\frac{a}{3}-y_{1}-y_{2}\right),\quad(y_{1},y_{2})\in
a\Delta,$ (94)

see Proposition 2.3. Product tori are mapped to product tori under the scaling
map $S_{a+b_{3},a}$,

$S_{a+b_{3},a}(T(x))=T\left(\frac{a}{a+b_{3}}x\right),\quad
x\in\mathbb{R}_{\geqslant 0}^{3}.$ (95)

Applying (93), (94) and (95) to (92), we find, for every $b\in V$,

$v_{\Theta_{{\mathfrak{m}}}(a)}\left(\Psi^{-1}(b),b_{3}\right)\cong
T\left(\frac{a+b_{3}}{3}+\tau_{1}(b),\frac{a+b_{3}}{3}+\tau_{2}(b),\frac{a+b_{3}}{3}-\tau_{1}(b)-\tau_{2}(b)\right).$
(96)

Therefore, using Lemma 3.5 we find (recall Definition 1.10) that

$\displaystyle{\mathcal{E}}_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})$
$\displaystyle\sim$
$\displaystyle\frac{a+b_{3}}{3}+\min\\{\tau_{1}(b),\tau_{2}(b),-\tau_{1}(b)-\tau_{2}(b)\\}$
$\displaystyle=$ $\displaystyle d_{\rm IA}(\tau(b),(a+b_{3})\partial\Delta)$
$\displaystyle=$ $\displaystyle d_{\rm
IA}(b,(a+b_{3})\tau^{-1}(\partial\Delta))$ $\displaystyle=$ $\displaystyle
d_{\rm IA}(b,(a+b_{3})\partial\Delta_{{\mathfrak{m}}})$

$\Box$

######  Remark 4.10.

One can consider constructing the analogue of an almost toric fibration for
$\mathbb{C}^{3}\setminus\\{0\\}$, starting from a given ATF
$\pi_{{\mathfrak{m}}}\colon\mathbb{C}P^{2}\rightarrow\Delta_{{\mathfrak{m}}}$,
by considering the projection map

$\overline{\pi}_{{\mathfrak{m}}}\colon\mathbb{C}^{3}\setminus\\{0\\}\rightarrow\mathbb{R}^{3},\quad\overline{\pi}_{{\mathfrak{m}}}(z)=(h\cdot\pi_{{\mathfrak{m}}}(p_{h}(z)),h),\quad
h=H(z)=\pi(|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}).$ (97)

The image of $\overline{\pi}_{{\mathfrak{m}}}$ is the infinite cone over
$\Delta_{{\mathfrak{m}}}$,

$C(\Delta_{{\mathfrak{m}}})=\\{(hx_{1},hx_{2},h)\in\mathbb{R}^{3}\,|\,h>0,(x_{1},x_{2})\in\Delta_{{\mathfrak{m}}}\\}.$
(98)

The associated Lagragian torus fibration has three one-parameter families of
focus-focus-regular singularities, projecting to three rays
$C(\Delta_{{\mathfrak{m}}})$. Note that the branch cuts are two dimensional
wedges obtained by taking the cones over the three branch cuts in
$\Delta_{{\mathfrak{m}}}$. On the complement of these, the map
$\overline{\pi}_{{\mathfrak{m}}}$ generates an honest $T^{3}$-action. The
torus $\Theta_{{\mathfrak{m}}}(a)$ is the fibre over the point $(0,0,a)$. As
in the case of (almost) toric fibres in dimension four, the displacement
energy germ of ${\mathcal{E}}_{\Theta_{{\mathfrak{m}}}(a)}$ is given by the
integral affine distance to the boundary of the base of the Lagrangian torus
fibration,

${\mathcal{E}}_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})\sim d_{\rm
IA}(\overline{b},\partial C(\Delta_{{\mathfrak{m}}})).$ (99)

See also the discussion in §5.3.

###  4.6. Proof of Theorem C

Let $(X^{6},\omega)$ be a geometrically bounded symplectic manifold and
$\psi\colon B(R)\rightarrow X$ a Darboux chart. Let
$\varepsilon=\min\left\\{\frac{3R}{4},\lambda_{S}(X,\psi)\right\\}$. As in the
proof of Theorem B in §4.4, we show that embeddings
$\psi(\Theta_{{\mathfrak{m}}})$ of lifted Vianna tori
$\Theta_{{\mathfrak{m}}}=\Theta_{{\mathfrak{m}}}(a)$ with $a<\varepsilon$ have
the locality property,
${\mathcal{E}}_{\psi(\Theta_{{\mathfrak{m}}})}\sim{\mathcal{E}}_{\Theta_{{\mathfrak{m}}}}$.
We apply Lemma 1.15 to the versal deformation
$v_{\Theta_{{\mathfrak{m}}}}\colon
V\rightarrow{\mathcal{L}}_{\Theta_{{\mathfrak{m}}}}$ constructed in §4.5.
Since $\Theta_{{\mathfrak{m}}}\subset B(\varepsilon)$, we restrict the domain
$V$ such that the Lagrangian tori in its image contained in $B(\varepsilon)$.
Recall from the proof of Proposition 1.6 in §4.5 that
$v_{\Theta_{{\mathfrak{m}}}}(\overline{b})\cong T(\sigma(\overline{b}))$ for
$\overline{b}$ in an open dense subset of $V$. Since this Hamiltonian isotopy
is lifted from the corresponding reduced space
$H^{-1}(c)/S^{1}\cong(\mathbb{C}P^{2},c\omega_{\mathbb{C}P^{2}})$ with
$c<\varepsilon$, it has support in $B(\varepsilon)$ and thus it suffices to
check that the image $\psi(T(\sigma(x)))$ of the tori appearing the versal
deformation have property (CS) to apply Lemma 1.15. Since