# An approach to Hamiltonian Floer theory for maps from surfaces

Ronen Brilleslijper and Oliver Fabert

###### Abstract

In $n$-dimensional classical field theory one studies maps from
$n$-dimensional manifolds in such a way that classical mechanics is recovered
for $n=1$. In previous papers we have shown that the standard polysymplectic
framework in which field theory is described, is not suitable for variational
techniques. In this paper, we introduce for $n=2$ a Lagrange-Hamilton
formalism that allows us to define a generalization of Hamiltonian Floer
theory. As an application, we prove a cuplength estimate for our Hamiltonian
equations that yields a lower bound on the number of solutions to Laplace
equations with nonlinearity. We also discuss the relation with holomorphic
Floer theory.

###### Contents

  1. 1 Introduction
    1. 1.1 Two-dimensional Bridges regularization
    2. 1.2 Outline of the article
  2. 2 Complex structures and the Lagrangian formalism
    1. 2.1 Relation to holomorphic Hamiltonian systems
    2. 2.2 Lagrangian approach
  3. 3 Cuplength proof
    1. 3.1 $C^{0}$-bounds
    2. 3.2 Floer curves
    3. 3.3 Proof of theorem 3.1
  4. 4 Fredholm theory
    1. 4.1 Linearization of the Floer map
    2. 4.2 Transversality
  5. 5 Compactness
  6. A Proof of lemma 5.3

##  1 Introduction

This articles continues in the lines of [BF23b, BF23a], developing a
Hamiltonian formalism for (respectively Minkowski and Euclidean) field
theories that is fit for variational calculus. In the current article, the
focus lies on the Laplace equation with non-linearity on the two-dimensional
torus. The aim of this article is twofold. First of all, we will provide a
connection between the Hamilton equations and their Lagrangian counterpart. In
this process a nice connection will emerge with complex structures and we will
highlight a relation to holomorphic Hamiltonian systems. Secondly, we will
prove a cuplength result similar to [BF23a] in the two-dimensional case using
Floer theory. The novel contributions of this paper are the Fredholm theory
and compactness results for the moduli space of Floer curves (see sections 4
and 5). Working out these aspects puts us a step closer to the development of
a full Floer theory. Furthermore the analysis should be useful also for
defining a holomorphic Floer theory.

The Hamiltonian formalism used in this article arises from what we call
_Bridges regularization_. This formalism can be found in [Bri06] and provides
a Hamiltonian framework for, amongst others, the Laplace equation that
pertains its nice analytical properties. To motivate its merits, let us start
by looking at Newton’s equation

$\displaystyle-\frac{d^{2}}{dt^{2}}q(t)=V^{\prime}_{t}(q(t)),$ (1)

for a particle $q\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\to Q$ subject
to some potential $V_{t}\mathrel{\mathop{\mathchar 58\relax}}Q\to\mathbb{R}$
on a manifold $Q$. To formulate this as a first-order equation, we can
introduce the momentum $p(t)=\frac{d}{dt}q(t)$ and the Hamiltonian
$H_{t}(u)=\frac{1}{2}|p|^{2}+V_{t}(q)$, for $u=(q,p)$ and write equation 1 as

$\displaystyle J\frac{d}{dt}u(t)=\nabla H_{t}(u(t)).$ (2)

Here $J=\begin{pmatrix}0&-\textup{Id}\\\ \textup{Id}&0\end{pmatrix}$ is the
standard complex structure on the cotangent bundle $T^{*}Q$. Note that this
so-called Hamiltonian equation is elliptic and that
$(J\frac{d}{dt})^{2}=-\frac{d^{2}}{dt^{2}}$. In Floer theory, one uses this
Hamiltonian formalism to rewrite solutions to equation 1 as critical points of
some action functional $\mathcal{A}_{H}$. One then studies gradient-lines of
$\mathcal{A}_{H}$ to conclude existence results for solutions to Newton’s
equation. A well-known result in Floer theory is that the number of critical
points of $\mathcal{A}_{H}$ is bounded below by the topology of $Q$. See
[AH16] for more explanation and [Cie94] for the specific case of cotangent
bundles. Crucial in the proofs of these results is the ellipticity of equation
2.

Note that equation 2 comes from a Lagrangian formalism. Periodic solutions to
equation 1 arise as critical points of the Lagrangian action functional

$\mathcal{S}[q]=\int_{S^{1}}L(q(t),q^{\prime}(t),t)\,dt,$

where $L(q,q^{\prime},t)=\frac{1}{2}(q^{\prime})^{2}-V_{t}(q)$. When we define
$p=\frac{\partial L}{\partial q^{\prime}}$ and $H=pq^{\prime}-L$, the Euler-
Lagrange equations for the Lagrangian action coincide with equation 2. Here,
in order to view $H_{t}$ as a function of $q$ and $p$, we need the Legendre
condition

$\det\frac{\partial^{2}L}{\partial q^{\prime 2}}\neq 0.$

We will see a similar condition when connecting the Lagrangian and Hamiltonian
approaches for the Laplace equation (see section 2.2).

###  1.1 Two-dimensional Bridges regularization

In this article, the starting point is the Laplace equation on the
2-dimensional torus. Let $q\mathrel{\mathop{\mathchar
58\relax}}\mathbb{T}^{2}\to\mathbb{T}^{d}$ satisfy the Laplace equation with
non-linearity

$\displaystyle-(\partial_{1}^{2}+\partial_{2}^{2})q(t)=\frac{dV}{dq}(q(t)),$
(3)

where $t=(t_{1},t_{2})\in\mathbb{T}^{2}$,
$\partial_{i}\mathrel{\mathop{\mathchar 58\relax}}=\partial_{t_{i}}$ and
$V\mathrel{\mathop{\mathchar 58\relax}}\mathbb{T}^{d}\to\mathbb{R}$ is a non-
linearity. We will prove a cuplength result for the number of solutions to
equation 3 and actually for more general equations. In comparison with
Newton’s equation we will first try to formulate 3 as a first-order equation.
The standard way of doing this, is by introducing ’momentum-like’ variables
for the different derivatives. The following set of equations follows this so-
called de Donder-Weyl approach:iiiNote that the de Donder-Weyl equations are
typically used as a first-order approach to the wave equation, so the signs
are different. See a.o. [Kan93, Gün87].

$\displaystyle\partial_{1}q$ $\displaystyle=p_{1}$
$\displaystyle\partial_{2}q$ $\displaystyle=p_{2}$ (4) $\displaystyle-$
$\displaystyle\partial_{1}p_{1}$ $\displaystyle-\partial_{2}p_{2}$
$\displaystyle=\frac{dV}{dq}.$

Even though equation 4 can be written using a Hamiltonian $H$ as
$K_{\partial}u(t)=\nabla H(u(t))$ with $u=(q,p_{1},p_{2})$ for a first-order
differential operator $K_{\partial}$, this equation on its own has some major
disadvantages. Most notably, the equation is not elliptic anymore, whereas
equation 3 is. For Floer theoretic purposes this is highly undesirable.
Moreover, $K_{\partial}$ has an infinite-dimensional kernel on the space of
maps on $\mathbb{T}^{2}$ and it does not square to the Laplacian in contrast
to the mechanical case, where $(J\frac{d}{dt})^{2}=-\frac{d^{2}}{dt^{2}}$. As
proven in [BF23b] for the case of the wave equation, the Floer curves
corresponding to equation 4 do not necessarily converge to solutions. Also see
[BF23a] for a similar problem for the Laplace equation on a 3-dimensional
torus.

Fortunately, there is a different way of writing equation 3 as a first-order
system that solves these problems. First proposed in [BD99] and later worked
out in [Bri06], this new set of equations follows from adding some extra terms
to equation 4, depicted in blue.

$\displaystyle\begin{split}\partial_{1}q_{1}{\color[rgb]{0,0,1}+\partial_{2}q_{2}}&=p_{1}\\\
{\color[rgb]{0,0,1}-\partial_{1}q_{2}+}\partial_{2}q_{1}&=p_{2}\\\
-\partial_{1}p_{1}-\partial_{2}p_{2}&=\frac{dV_{1}}{dq_{1}}\\\
{\color[rgb]{0,0,1}\partial_{1}p_{2}-\partial_{2}p_{1}}&{\color[rgb]{0,0,1}=\frac{dV_{2}}{dq_{2}}}\end{split}$
(5)

Note that we changed the variable $q$ from equation 4 into $q_{1}$ here, to
stress that this first-order system of equations actually describes two
Laplace equations

$\displaystyle-(\partial_{1}^{2}+\partial_{2}^{2})q_{1}(t)$
$\displaystyle=\frac{dV_{1}}{dq_{1}}(q_{1}(t))$
$\displaystyle-(\partial_{1}^{2}+\partial_{2}^{2})q_{2}(t)$
$\displaystyle=\frac{dV_{2}}{dq_{2}}(q_{2}(t)).$

The underlying geometric structure behind equation 5, which we call Bridges’
equations is explained in [Bri06]. A shorter version in the context of maps
between tori can also be found in [BF23b, BF23a], so we do not go into details
on this in the present article. However, we do want to stress again that
equation 5 is an elliptic first-order system. We can write it as

$\displaystyle J_{\partial}Z(t)=\nabla H(Z(t)),$

where $Z=(q_{1},q_{2},p_{1},p_{2})$,
$H(Z)=\frac{1}{2}p_{1}^{2}+\frac{1}{2}p_{2}^{2}+V_{1}(q_{1})+V_{2}(q_{2})$ and
$J_{\partial}$ is a first-order operator satisfying
$J_{\partial}^{2}=-(\partial_{1}^{2}+\partial_{2}^{2})$. Note that only
constant functions lie in the kernel of $J_{\partial}$.

######  Remark 1.1.

Note that there is a hyperkähler structure on the target space hidden in these
equations. When we write out
$J_{\partial}=J_{1}\partial_{1}+J_{2}\partial_{2}$,then $J_{1}$ and $J_{2}$
are both complex structures and their product is too. In remark 3.5 we will
note that this yields a relation between our Floer curves and the Fueter
equation. The complex structure $J_{1}$ pairs $q_{i}$ with $p_{i}$ whereas
$J_{2}$ pairs $q_{i}$ with $p_{j}$ for $i\neq j$. Their product
$J_{3}=J_{1}J_{2}$ therefore pairs $q_{1}$ with $q_{2}$ and $p_{1}$ with
$p_{2}$. In section 2 we will see that indeed equation 5 can be written using
some complex multiplication on $\mathbb{T}^{2d}$.

Equation 5 can also be described in terms of the polysymplectic form

$\displaystyle\Omega=\omega_{1}\otimes\partial_{1}+\omega_{2}\otimes\partial_{2},$
(6)

where $\omega_{1}=dp_{1}\wedge dq_{1}-dp_{2}\wedge dq_{2}$ and
$\omega_{2}=dp_{2}\wedge dq_{1}+dp_{1}\wedge dq_{2}$. This description will be
of use later, when we compare the equations to holomorphic Hamiltonian
systems. Note that both $\omega_{1}$ and $\omega_{2}$ are symplectic forms.
Heuristically, $\omega_{i}$ determines the derivative in the direction of
$t_{i}$. In other words, $Z=(q_{1},q_{2},p_{1},p_{2})$ is a solution to
equation 5 if and only if

$\displaystyle\omega_{1}(\cdot,\partial_{1}Z)+\omega_{2}(\cdot,\partial_{2}Z)=dH.$

See [Gün87] for an exposition of polysymplectic geometry.

###  1.2 Outline of the article

This paper is organized as follows. In section 2 we start by rewriting the
Bridges equations using language from complex analysis. After relating our
equations to holomorphic Hamiltonian systems, we discuss the underlying
Lagrangian formalism. In section 3 we state the main theorem 3.1 together with
its implication for Laplace equatinos with nonlinearity. The $C^{0}$-bounds
are given for both the orbits and the Floer curves. The section concludes with
a proof of the theorem, modulo the necessary Fredholm theory and compactness
results. These are treated in respectively sections 4 and 5. The proof of one
technical lemma needed for compactness is deferred to the appendix.

##  2 Complex structures and the Lagrangian formalism

Before motivating the Hamiltonian approach to Laplace’s equation mentioned
above from the corresponding Lagrangian system, we first want to note that we
can rewrite equation 5 in terms of complex structures. Note that for $H\equiv
0$ (i.e. a vanishing right-hand side in equation 5), Bridges’ equations reduce
to a set of two Cauchy Riemann equations for the pairs $(q_{1},q_{2})$ and
$(p_{1},p_{2})$. This leads us to define $t=t_{1}+it_{2}$ and correspondingly
$\bar{t}=t_{1}-it_{2}$. Also let $q=q_{1}+iq_{2}$, $p=p_{1}+ip_{2}$ and
correspondingly $\bar{q}=q_{1}-iq_{2}$, $\bar{p}=p_{1}-ip_{2}$. Then equation
5 is equivalent to the set of equations

$\displaystyle\begin{split}\partial_{t}q&=\frac{1}{2}\bar{p}=\frac{\partial
H}{\partial p}\\\ -\partial_{t}p&=\frac{\partial V}{\partial q}=\frac{\partial
H}{\partial q},\end{split}$ (7)

where $\partial_{t}=\frac{1}{2}(\partial_{1}-i\partial_{2})$,
$\frac{\partial}{\partial q}=\frac{1}{2}(\frac{\partial}{\partial
q_{1}}-i\frac{\partial}{\partial q_{2}})$ and
$V(q,\bar{q})=V_{1}(q_{1})+V_{2}(q_{2})\in\mathbb{R}$. Also
$H=H(q,\bar{q},p,\bar{p})=\frac{1}{2}p\bar{p}+V(q,\bar{q})$ is still the same
real-valued function as before, but now seen as a function of these new
coordinates.

While the Hamiltonian formalism for the Laplace equation might come across as
purely notation at this point, we will show in the section 2.2 that these
equations actually relate to a Lagrangian version of a minimization problem by
means of a Legendre transform. This will also provide us with a much bigger
class of examples then just Laplace equations. Before that, the next section
explains the relation to holomorphic Hamiltonian systems.

###  2.1 Relation to holomorphic Hamiltonian systems

Equation 7 looks very similar to the holomorphic Hamiltonian systems (HHS)
studied in [Wag23]. The main difference is that their Hamiltonians are
holomorphic functions with values in $\mathbb{C}$, whereas our Hamiltonian is
real-valued. We crucially need the derivative $\frac{\partial H}{\partial p}$
to actually depend on $\bar{p}$ in order to get the Laplace equation

$\displaystyle-\Delta(q_{1}+iq_{2})=-4\partial_{\bar{t}}\partial_{t}q=-2\partial_{\bar{t}}\bar{p}=2\frac{\partial
V}{\partial\bar{q}}=\frac{dV_{1}}{dq_{1}}+i\frac{dV_{2}}{dq_{2}}.$ (8)

Notice also that our Hamiltonians are not necessarily the real or imaginary
parts of a holomorphic Hamiltonian, since we do not require them to be
harmonic.

Interestingly, both the HHS’s from Wagner and the Bridges equations used in
this article can be described as special cases of a broader framework. Recall
the polysymplectic form $\Omega$ from equation 6. In the complex notation it
can be written as
$\Omega=\omega_{\mathbb{C}}\otimes\partial_{t}+\bar{\omega}_{\mathbb{C}}\otimes\partial_{\bar{t}}$,
where $\omega_{\mathbb{C}}=dp\wedge dq$ and
$\bar{\omega}_{\mathbb{C}}=d\bar{p}\wedge d\bar{q}$. Just as in [Wag23], the
form $\omega_{\mathbb{C}}$ (respectively $\bar{\omega}_{\mathbb{C}}$) induces
an isomorphisms between the holomorphic (respectively anti-holomorphic)
tangent and cotangent bundles

$\displaystyle\omega_{\mathbb{C}}^{\flat}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}T^{(1,0)}X\overset{\sim}{\to}T^{*(1,0)}X$
$\displaystyle\bar{\omega}_{\mathbb{C}}^{\flat}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}T^{(0,1)}X\overset{\sim}{\to}T^{*(0,1)}X,$

where $X=T^{*}\mathbb{T}^{2d}$. Thus, the polysymplectic form $\Omega$ induces
an isomorphism
$\omega_{\mathbb{C}}^{\flat}\oplus\bar{\omega}_{\mathbb{C}}^{\flat}\mathrel{\mathop{\mathchar
58\relax}}T_{\mathbb{C}}X\overset{\sim}{\to}T^{*}_{\mathbb{C}}X$, where the
complexified tangent and cotangent bundles are viewed as the direct sums of
their holomorphic and anti-holomorphic subbundles. In principle this means
that for any function $H\mathrel{\mathop{\mathchar 58\relax}}X\to\mathbb{C}$
we get a complex vector field by taking the inverse of $dH$ under this map. In
the study of HHS’s $H$ is taken to be a holomorphic function, which means that
$dH\in T^{*(1,0)}X$. Thus the Hamiltonian equations only take into account the
holomorphic vector field $(\omega_{\mathbb{C}}^{\flat})^{-1}(dH)$ and the
anti-holomorphic vector field vanishes. In our case the Hamiltonians are real-
valued meaning that its holomorphic and anti-holomorphic derivatives are
equal. Thus $dH$ lies in the diagonal of
$T^{*}_{\mathbb{C}}X=T^{*(1,0)}X\oplus T^{*(0,1)}X$ and we actually get two
sets of conjugate equations. The analysis in this article is aimed at the case
of real-valued Hamiltonians, but we expect similar techniques to work in the
holomorphic setting.

######  Remark 2.1.

Note that equation 7 is written in terms of local coordinates on the torus.
However, just as for HHS’s, the equations can be formulated on more general
manifolds. For equation 7 to make sense the domain should be a Riemann surface
and the target a complex manifold. From a dynamical point of view this could
be of particular interest for string theory, where the Riemann surface is
viewed as the closed world-sheet of a 1-dimensional string and the target
manifold is Calabi-Yau. The flatness of the Calabi-Yau manifold will be
crucial for the compactness result to hold, just like in [HNS09].

###  2.2 Lagrangian approach

We observe that solutions to the Laplace equations arise as critical points of
the functional

$\displaystyle\mathcal{S}[q]=-\int_{\mathbb{T}^{2}}\left(2\partial_{t}q\partial_{\bar{t}}\bar{q}-V(q,\bar{q})\right)\frac{dt\wedge
d\bar{t}}{2i},$

where we use the notation of equation 8 (for a proof see lemma 2.2 below). In
this section, we will develop a connection between the Lagrangian picture and
the Hamiltonian formalism sketched above.

Let $L=L(q,\bar{q},\partial_{t}q,\partial_{\bar{t}}\bar{q},t,\bar{t})$ be a
Lagrangian with values in $\mathbb{R}$ and

$\displaystyle\mathcal{S}[q]=-\int_{\mathbb{T}^{2}}L(q,\bar{q},\partial_{t}q,\partial_{\bar{t}}\bar{q},t,\bar{t})\frac{dt\wedge
d\bar{t}}{2i}$ (9)

the corresponding action. Note that $\mathcal{S}$ is real-valued as well. Our
main example is the Lagrangian
$L=2\partial_{t}q\partial_{\bar{t}}\bar{q}-V(q,\bar{q})$ with action given
above.

######  Lemma 2.2.

The Euler-Lagrange equations for the action (9) are given by

$\displaystyle\frac{\partial L}{\partial q}=\partial_{t}\frac{\partial
L}{\partial(\partial_{t}q)}$ $\displaystyle\frac{\partial
L}{\partial\bar{q}}=\partial_{\bar{t}}\frac{\partial
L}{\partial(\partial_{\bar{t}}\bar{q})}.$ (10)

Note that these two equations are each others conjugates as $L$ is real.

###### Proof.

Let us calculate the critical points of $\mathcal{S}$. To this extent we take
a family of maps $q_{s}$ where $s\in\mathbb{R}$ such that $q_{0}=q$ and write
$\frac{dq_{s}}{ds}|_{s=0}=\dot{q}$. Then

$\displaystyle\left.\frac{d}{ds}\right|_{s=0}\mathcal{S}[q_{s}]$
$\displaystyle=-\int_{\mathbb{T}^{2}}\left(\frac{\partial L}{\partial
q}\dot{q}+\frac{\partial L}{\partial\bar{q}}\dot{\bar{q}}+\frac{\partial
L}{\partial(\partial_{t}q)}\partial_{t}\dot{q}+\frac{\partial
L}{\partial(\partial_{\bar{t}}\bar{q})}\partial_{\bar{t}}\dot{\bar{q}}\right)\frac{dt\wedge
d\bar{t}}{2i}$
$\displaystyle=-\int_{\mathbb{T}^{2}}\left(\dot{q}\left(\frac{\partial
L}{\partial q}-\partial_{t}\frac{\partial
L}{\partial(\partial_{t}q)}\right)+\dot{\bar{q}}\left(\frac{\partial
L}{\partial\bar{q}}-\partial_{\bar{t}}\frac{\partial
L}{\partial(\partial_{\bar{t}}\bar{q})}\right)\right)\frac{dt\wedge
d\bar{t}}{2i}.$

We see that this derivative vanishes precisely when equation 10 is satisfied.
∎

In order to get back to a Hamiltonian formalism we define a momentum-like
variable $p=\frac{\partial L}{\partial(\partial_{t}q)}$ and correspondingly
$\bar{p}=\frac{\partial L}{\partial(\partial_{\bar{t}}\bar{q})}$. This yields

$\displaystyle dL$
$\displaystyle=d(p\partial_{t}q+\bar{p}\partial_{\bar{t}}\bar{q})-\partial_{t}qdp-\partial_{\bar{t}}\bar{q}d\bar{p}+\frac{\partial
L}{\partial q}dq+\frac{\partial L}{\partial\bar{q}}d\bar{q}+\frac{\partial
L}{\partial t}dt+\frac{\partial L}{\partial\bar{t}}d\bar{t},$ (11)

which motivates the definition

$H(q,\bar{q},p,\bar{p},t,\bar{t})=p\partial_{t}q+\bar{p}\partial_{\bar{t}}\bar{q}-L(q,\bar{q},\partial_{t}q,\partial_{\bar{t}}\bar{q},t,\bar{t}),$

where $(\partial_{t}q,\partial_{\bar{t}}\bar{q})$ is seen as a function of
$(q,\bar{q},p,\bar{p},t,\bar{t})$ by the inverse function theorem. This
requires the Legendre-like condition

$\det\left(\left(\frac{\partial^{2}L}{\partial(\partial_{t}q)\partial(\partial_{\bar{t}}\bar{q})}\right)^{2}-\frac{\partial^{2}L}{\partial(\partial_{t}q)^{2}}\frac{\partial^{2}L}{\partial(\partial_{\bar{t}}\bar{q})^{2}}\right)\neq
0.$

Note again that $H$ is real-valued.

######  Lemma 2.3.

The Euler-Lagrange equations (10) are equivalent to the Hamiltonian equations

$\displaystyle\begin{split}\partial_{t}q&=\frac{\partial H}{\partial p}\\\
-\partial_{t}p&=\frac{\partial H}{\partial q}.\end{split}$ (12)

###### Proof.

By the relation (11) above we get that

$dL=d(p\partial_{t}q+\bar{p}\partial_{\bar{t}}\bar{q})-dH.$

So

$\displaystyle\frac{\partial H}{\partial q}=-\frac{\partial L}{\partial q}$
$\displaystyle\frac{\partial H}{\partial p}=\partial_{t}q.$

Plugging in the Euler-Lagrange equations and the definition of $p$, the result
follows immediately. ∎

######  Example 2.4.

In the example $L=2\partial_{t}q\partial_{\bar{t}}\bar{q}-V(q,\bar{q})$ of the
Laplace equations, we get $H=\frac{1}{2}p\bar{p}+V(q,\bar{q})$ as before and
we recover equation 7.

##  3 Cuplength proof

In this section we will prove a cuplength result for solutions to the
Hamiltonian equations (12). Recall that we view $t=t_{1}+it_{2}$ and its
conjugate as the coordinates on $\mathbb{T}^{2}$ and $q=q_{1}+iq_{2}$ and its
conjugate as the coordinates on $\mathbb{T}^{2d}$. Consequently,
$Z\mathrel{\mathop{\mathchar 58\relax}}=(q,\bar{q},p,\bar{p})$ denote the
coordinates on $T^{*}\mathbb{T}^{2d}$, where $p=p_{1}+ip_{2}$.

######  Theorem 3.1.

Let $H\mathrel{\mathop{\mathchar 58\relax}}\mathbb{T}^{2}\times
T^{*}\mathbb{T}^{2d}\to\mathbb{R}$ be given by
$H_{t,\bar{t}}(Z)=\frac{1}{2}p\bar{p}+F_{t,\bar{t}}(Z)$, where $F$ is smooth,
real-valued and has finite $C^{3}$-norm. Then there exist at least $(2d+1)$
solutions to equation 12.

######  Corollary 3.2.

The set of Laplace equations (8) has at least $(2d+1)$ solutions, when $V_{1}$
and $V_{2}$ have finite $C^{3}$-norm.

Theorem 3.1 will be proven using a Floer theoretic argument along the lines of
[Sch98, AH16]. Note that similar statements have been proven in [HNS09, GH12,
GH13, AH16] for closed target manifolds and the proof for maps on a
3-dimensional torus was given in [BF23a]. In the latter article, the proof was
given as a corollary of results from [GH12] after establishing the necessary
$C^{0}$-bounds. This proof does not use Floer theoretic arguments. In the
present article we aim to establish a connection to the Floer theory of these
systems and will prove the necessary Fredholm and compactness results (see
sections 4 and 5 respectively). As pointed out in section 2.1, the techniques
used here should also be of use for studying holomorphic Hamiltonian systems
in the sense of [Wag23] using Floer theory.

###  3.1 $C^{0}$-bounds

First, we will prove the necessary $C^{0}$-bounds. The proofs are analogous to
the ones given in [BF23a], but as the notation with complex structures is
different in the present article and we treat the 2-dimensional torus here,
the proofs will be briefly be repeated below. First of all, define the action
by

$\displaystyle\mathcal{A}_{H}(Z)=\int_{\mathbb{T}^{2}}\left(p\partial_{t}q+\bar{p}\partial_{\bar{t}}\bar{q}-H_{t,\bar{t}}(Z)\right)\,d\mathcal{V},$
(13)

where $\,d\mathcal{V}=-\frac{dt\wedge d\bar{t}}{2i}$ is the standard volume
form on $\mathbb{T}^{2}$. In the case that $H$ actually comes from a
Lagrangian as in section 2.2, this action coincides with equation 9. We denote

$\displaystyle\mathcal{P}(H)$ $\displaystyle=\\{Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{T}^{2}\to T^{*}\mathbb{T}^{2d}\mid Z\textup{ is
nullhomotopic and satisfies \lx@cref{creftype~refnum}{eq:Ham}}\\}$
$\displaystyle\mathcal{P}_{a}(H)$ $\displaystyle=\\{Z\in\mathcal{P}(H)\mid
a\geq\mathcal{A}_{H}(Z)\\}.$

######  Lemma 3.3.

Let $H_{t,\bar{t}}(Z)=\frac{1}{2}p\bar{p}+F_{t,\bar{t}}(Z)$, where $F$ is
smooth, real-valued and has finite $C^{2}$-norm. Then for any $a\in\mathbb{R}$
the set $\mathcal{P}_{a}(H)$ is bounded in $L^{\infty}$.

###### Proof.

We start by proving that $\mathcal{P}_{a}(H)$ is bounded in $L^{2}$. Note that
for solutions of equation 12 we get

$\displaystyle\mathcal{A}_{H}(Z)$
$\displaystyle=\int_{\mathbb{T}^{2}}\left(p\frac{\partial H}{\partial
p}+\bar{p}\frac{\partial
H}{\partial\bar{p}}-H_{t,\bar{t}}(Z)\right)\,d\mathcal{V}$
$\displaystyle=\int_{\mathbb{T}^{2}}\left(\frac{1}{2}p\bar{p}+p\frac{\partial
F}{\partial p}+\bar{p}\frac{\partial
F}{\partial\bar{p}}-F_{t,\bar{t}}\right)\,d\mathcal{V}$ $\displaystyle\geq
h_{0}||p||_{L^{2}}^{2}-h_{1},$

for some $h_{0}>0$ and $h_{1}\geq 0$. The last inequality comes from the
boundedness of $F$ and its derivatives. The assumption
$a\geq\mathcal{A}_{H}(Z)$ now yields a uniform $L^{2}$-bound on $p$. As $q$
maps into $\mathbb{T}^{2d}$ it is $L^{2}$-bounded by construction. So indeed
$\mathcal{P}_{a}(H)$ is bounded in $L^{2}$.

We now proceed by a bootstrapping argument. For $Z\in\mathcal{P}_{a}(H)$ we
haveiiiiiiIn the first line we use partial integration, using that $Z$ is
nullhomotopic.

$\displaystyle||dZ||^{2}_{L^{2}}$
$\displaystyle=4\int_{\mathbb{T}^{2}}\left(|\partial_{t}q|^{2}+|\partial_{t}p|^{2}\right)\,d\mathcal{V}$
$\displaystyle=4\int_{\mathbb{T}^{2}}\left(\left|\frac{\partial H}{\partial
p}\right|^{2}+\left|\frac{\partial H}{\partial
q}\right|^{2}\right)\,d\mathcal{V}$
$\displaystyle\leq||p||_{L^{2}}^{2}+4\int_{\mathbb{T}^{2}}\left(\left|\frac{\partial
F}{\partial q}\right|^{2}+\left|\frac{\partial F}{\partial
p}\right|^{2}\right)\,d\mathcal{V}.$

As the first derivatives of $F$ are bounded and $p$ is uniformly bounded in
$L^{2}$ this induces a uniform $L^{2}$-bound on $dZ$. Therefore
$\mathcal{P}_{a}(H)$ is bounded in $H^{1}$. A similar argument shows that the
$H^{1}$-norm of $dZ$ is bounded by the $H^{1}$-norm of $Z$ and the
$C^{2}$-norm of $F$. Thus, we get that $\mathcal{P}_{a}(H)$ is bounded in
$H^{2}$. As $2\cdot 2>2$ we get boundedness in $L^{\infty}$ by Sobolev’s
embedding theorem. ∎

From lemma 3.3 it follows that we can actually use a cut-off version of $F$.
Let $\chi_{\rho}\mathrel{\mathop{\mathchar 58\relax}}[0,\infty)\to\mathbb{R}$
a smooth cut-off function such that $\chi_{\rho}\equiv 1$ on $[0,\rho-1]$ and
$\chi_{\rho}\equiv 0$ on $[\rho,\infty)$. We can define the cut-off
Hamiltonian as $\tilde{H}^{\rho}_{t,\bar{t}}(Z)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}p\bar{p}+\tilde{F}^{\rho}_{t,\bar{t}}(Z)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}p\bar{p}+\chi_{\rho}(|p|^{2})F_{t,\bar{t}}(Z)$.

######  Corollary 3.4.

For any $a\in\mathbb{R}$ there exists some $\rho>0$ such that
$\mathcal{P}_{a}(H)=\mathcal{P}_{a}(\tilde{H}^{\rho})$.

###  3.2 Floer curves

Now that we have the necessary bounds on the orbits, we turn to Floer curves.
A Floer curve is defined as a negative $L^{2}$-gradient flow line of the
action (13). Note that in the complex notation the $L^{2}$-inner product is
given by

$\langle
X,Y\rangle_{L^{2}}=\frac{1}{2}\int_{T^{2}}\left(X_{q}Y_{\bar{q}}+X_{\bar{q}}Y_{q}+X_{p}Y_{\bar{p}}+X_{\bar{p}}Y_{p}\right)\,d\mathcal{V}.$

It follows that the Floer curves are given by

$\displaystyle\frac{1}{2}\partial_{s}q-\partial_{\bar{t}}\bar{p}$
$\displaystyle=\frac{\partial\tilde{H}^{\rho}}{\partial\bar{q}}$
$\displaystyle\frac{1}{2}\partial_{s}\bar{q}-\partial_{t}p$
$\displaystyle=\frac{\partial\tilde{H}^{\rho}}{\partial q}$
$\displaystyle\frac{1}{2}\partial_{s}p+\partial_{\bar{t}}\bar{q}$
$\displaystyle=\frac{\partial\tilde{H}^{\rho}}{\partial\bar{p}}$
$\displaystyle\frac{1}{2}\partial_{s}\bar{p}+\partial_{t}q$
$\displaystyle=\frac{\partial\tilde{H}^{\rho}}{\partial p},$ (14)

where $Z=(q,\bar{q},p,\bar{p})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{T}^{2}\to T^{*}\mathbb{T}^{2d}$ and we have
taken the Hamiltonian to be $\tilde{H}^{\rho}$ in view of corollary 3.4. Note
that $s\in\mathbb{R}$ is a real coordinate, so that $\partial_{s}$ denotes the
ordinary derivative of functions on $\mathbb{R}$, whereas $\partial_{t}$ and
$\partial_{\bar{t}}$ still denote the complex derivative and its conjugate on
$\mathbb{T}^{2}$.

######  Remark 3.5.

Note that in the notation of remark 1.1 we can write the Floer equation with
vanishing Hamiltonian as
$\partial_{s}Z+J_{1}\partial_{1}Z+J_{2}\partial_{2}Z=0$. This is equivalent to
the Fueter equation

$J_{3}\partial_{s}Z+J_{2}\partial_{1}Z-J_{1}\partial_{2}Z=0,$

where $J_{3}=J_{1}J_{2}$ is the third complex structure from the hyperkähler
structure on the target space.

The following lemma shows that the image of a Floer curve actually sits within
a compact subset of $T^{*}\mathbb{T}^{2d}$ (compare with Lemma 3.4 from
[BF23a]).

######  Lemma 3.6.

Let $Z\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\times\mathbb{T}^{2}\to
T^{*}\mathbb{T}^{2d}$ be a solution to equation 14 which converges to critical
points of $\mathcal{A}_{\tilde{H}^{\rho}}$ for $|s|\to\infty$. Then
$|p(s,t)|^{2}\leq\rho$ for any $(s,t)\in\mathbb{R}\times\mathbb{T}^{2}$.

###### Proof.

Suppose $|p(s_{0},t_{0})|^{2}>\rho$ for some
$(s_{0},t_{0})\in\mathbb{R}\times\mathbb{T}^{2}$. Locally around
$(s_{0},t_{0})$ the Hamiltonian reduces to
$\tilde{H}^{\rho}(Z(s,t))=\frac{1}{2}p(s,t)\bar{p}(s,t)$. Therefore, on this
neighbourhood

$\displaystyle\partial_{s}^{2}p$
$\displaystyle=\partial_{s}p-2\partial_{s}\partial_{\bar{t}}\bar{q}$
$\displaystyle=\partial_{s}p-4\partial_{t}\partial_{\bar{t}}p.$

In the second equality, we use that $\partial_{s}\bar{q}=2\partial_{t}p$. It
follows that

$\displaystyle(\partial_{s}^{2}+\Delta)p=\partial_{s}^{2}p+4\partial_{t}\partial_{\bar{t}}p=\partial_{s}p.$

Define $\phi=\frac{1}{2}p\bar{p}$. Then

$\displaystyle(\partial_{s}^{2}+\Delta)\phi$
$\displaystyle=|\partial_{s}p|^{2}+2|\partial_{t}\bar{p}|^{2}+2|\partial_{t}p|^{2}+\frac{1}{2}p\partial_{s}\bar{p}+\frac{1}{2}\bar{p}\partial_{s}p$
$\displaystyle\geq\partial_{s}\phi.$

The maximum principle now tells us that $\phi$ cannot attain a maximum in the
neighbourhood of $(s_{0},t_{0})$, contradicting the assumption that the Floer
curve converges as $|s|\to\infty$. ∎

###  3.3 Proof of theorem 3.1

Having established the $C^{0}$-bounds we can prove theorem 3.1. The necessary
Fredholm and compactness results will be deferred to sections 4 and 5. For the
proof we will follow the lines of the paper [AH16].

To start the proof, assume there are finitely many solutions to equation 12,
since otherwise we would be done. Then their action is necessarily bounded.
Corollary 3.4 implies that we may replace $H$ by $\tilde{H}^{\rho}$ for some
$\rho$ and by lemma 3.6 also the Floer curves will take value in some compact
set $\mathbb{T}^{2d}\times B\subseteq T^{*}\mathbb{T}^{2d}$. Let $M\subseteq
C^{\infty}(\mathbb{T}^{2},\mathbb{T}^{2d}\times B)$ be the subset of
nullhomotopic maps. Just like in [AH16, §3.1] we define a smooth family of
functions $\beta_{r}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\to[0,1]$
for $r\geq 0$ such that

  * •

$\beta_{r}(s)=0$ for $s\leq-1$ and $s\geq(2d+1)r+1$ for all $r\geq 0$,

  * •

$\beta_{r}|_{[0,(2d+1)r]}\equiv 1$ for $r\geq 1$,

  * •

$0\leq\beta_{r}^{\prime}(s)\leq 2$ for $s\in(-1,0)$ and
$0\geq\beta_{r}^{\prime}(s)\geq-2$ for $s\in((2d+1)r,(2d+1)r+1)$,

  * •

$\lim_{r\to 0^{+}}\beta_{r}=0$ in the strong topology on $C^{\infty}$.

See [AH16] for the precise conditions on $\beta_{r}$ and figure 1 for an idea
of what they look like. We will use the functions $\beta_{r}$ to interpolate
between the quadratic Hamiltonian with and without non-linearity. To this
extent define

$\tilde{H}^{\rho,r}_{s,t,\bar{t}}(Z)=\frac{1}{2}p\bar{p}+\beta_{r}(s)\tilde{F}^{\rho}_{t,\bar{t}}(Z).$

One can see the Hamiltonian $\tilde{H}^{\rho,r}$ as the quadratic Hamiltonian
$H^{0}(Z)=\frac{1}{2}p\bar{p}$ with the non-linearity $\tilde{F}^{\rho}$
switched on just on an interval whose length is determined by $r$.
Correspondingly, we have the action $G_{r,s}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{A}_{\tilde{H}^{\rho,r}}$, interpolating between the
actions $\mathcal{A}_{H^{0}}$ and $\mathcal{A}_{\tilde{H}^{\rho}}$.

Figure 1: The functions $\beta_{r}$. Source: [AH16]. In our case we take
$k=\textup{cl}(\mathbb{T}^{2d})=2d$.

With this setup we can introduce the moduli space

$\displaystyle\mathcal{M}=\\{(r,Z)\mid r\geq 0,Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\to M,(\star 1)-(\star 3)\\},$ (15)

where $(\star 1)-(\star 3)$ are given below. Note that we can view $Z$ as a
contractible map $\mathbb{R}\times\mathbb{T}^{2}\to\mathbb{T}^{2d}\times B$.
The condition $(\star 1)$ is that $Z$ is a negative gradient line of
$G_{r,s}$, meaning a Floer curve for $\tilde{H}^{\rho,r}$. In formula this
means

$\displaystyle\frac{1}{2}\partial_{s}Z+\cancel{\partial}Z=\begin{pmatrix}\partial_{\bar{q}}\\\
\partial_{q}\\\ \partial_{\bar{p}}\\\
\partial_{p}\end{pmatrix}\tilde{H}^{\rho,r}(Z),$ ($\star 1$)

where

$\cancel{\partial}=\begin{pmatrix}0&0&0&-\partial_{\bar{t}}\\\
0&0&-\partial_{t}&0\\\ 0&\partial_{\bar{t}}&0&0\\\
\partial_{t}&0&0&0\end{pmatrix}$

and as before $Z=(q,\bar{q},p,\bar{p})$. Condition $(\star 2)$ is finiteness
of energy

$\displaystyle
E(Z)=\int_{-\infty}^{\infty}\int_{\mathbb{T}^{2}}|\partial_{s}Z(s,t)|^{2}\,d\mathcal{V}\,ds<+\infty$
($\star 2$)

and the final condition is convergence of the $p$-coordinates

$\displaystyle p(s,t)\to 0\textup{ as }|s|\to\infty.$ ($\star 3$)

We also introduce the subspaces

$\displaystyle\mathcal{M}_{[0,R]}(\mathbb{T}^{2d})$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=\\{(r,Z)\in\mathcal{M}\mid
0\leq r\leq R\textup{ and
}\lim_{s\to\pm\infty}Z(s)\in\mathbb{T}^{2d}\times\\{0\\}\\}$
$\displaystyle\mathcal{M}_{R}(\mathbb{T}^{2d})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\\{Z\mid(R,Z)\in\mathcal{M}_{[0,R]}(\mathbb{T}^{2d})\\},$

where $\mathbb{T}^{2d}\times\\{0\\}\subseteq M$ denotes the set of constant
maps. This coincides with the set of critical points of $\mathcal{A}_{H^{0}}$.
Note that for $R=0$ we get that $\mathcal{M}_{0}(\mathbb{T}^{2d})$ is equal to
this set of critical points.

It follows from theorem 4.1 and theorem 5.5 that $\mathcal{M}$ is a compact
1-dimensional manifold, possibly after a small perturbation of $F$. Also the
spaces $\mathcal{M}_{[0,R]}(\mathbb{T}^{2d})$ and
$\mathcal{M}_{R}(\mathbb{T}^{2d})$ will then be compact manifolds (again after
possibly perturbing $F$, see [AH16]). We have an evaluation map

$\displaystyle\textup{ev}_{R}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{M}_{R}(Z)$ $\displaystyle\to(\mathbb{T}^{2d}\times B)^{2d}$
$\displaystyle Z$
$\displaystyle\mapsto(Z(R,t_{0},\bar{t}_{0}),Z(2R,t_{0},\bar{t}_{0}),\ldots,Z(2dR,t_{0},\bar{t}_{0})),$

where $(t_{0},\bar{t}_{0})\in\mathbb{T}^{2}$ is some basepoint. For $R=0$ this
gives the diagonal embedding
$\mathbb{T}^{2d}\hookrightarrow(\mathbb{T}^{2d}\times B)^{2d}$.

The rest of the proof goes analogously to [AH16]. Take Morse functions
$f_{1},\ldots,f_{2d},f_{*}$ on $\mathbb{T}^{2d}$ and extend to Morse functions
$\bar{f}_{\alpha}$ for $\alpha=1,\ldots,2d$ on $\mathbb{T}^{2d}\times B$ using
a positive definite quadratic form on a tubular neighbourhood of
$\mathbb{T}^{2d}$ in the direction normal to $\mathbb{T}^{2d}$. We also choose
Riemannian metrics $g_{1},\ldots,g_{2d},g_{*}$ on $\mathbb{T}^{2d}\times B$.
For critical points $x_{\alpha}\in\mathbb{T}^{2d}$ of $f_{\alpha}$ and
$x_{*}^{\pm}\in\mathbb{T}^{2d}$ of $f_{*}$ we define

$\mathcal{M}(R,x_{1},\ldots,x_{2d},x_{*}^{-},x_{*}^{+})\mathrel{\mathop{\mathchar
58\relax}}=\left\\{Z\in\mathcal{M}_{R}(\mathbb{T}^{2d})\middle|(\star
4)-(\star 6)\right\\},$

where

$\displaystyle\textup{ev}_{R}$
$\displaystyle\in\prod_{\alpha}W^{s}(x_{\alpha},\bar{f}_{\alpha},g_{\alpha})$
($\star 4$) $\displaystyle\lim_{s\to-\infty}Z(s)$ $\displaystyle\in
W^{u}(x_{*}^{-},f_{*},g_{*})$ ($\star 5$)
$\displaystyle\lim_{s\to+\infty}Z(s)$ $\displaystyle\in
W^{s}(x_{*}^{+},f_{*},g_{*}).$ ($\star 6$)

Here, $W^{u}$ and $W^{s}$ denote the unstable and stable manifolds of the
critical points for the gradient flows of the indicated Morse functions.
Again, by choice of perturbations, we may assume these are all smooth
manifolds. Note that
$\mathcal{M}(0,x_{1},\ldots,x_{2d},x_{*}^{-},x_{*}^{+})=\bigcap_{\alpha}W^{s}(x_{\alpha},f_{\alpha},g_{\alpha})\cap
W^{u}(x_{*}^{-},f_{*},g_{*})\cap W^{s}(x_{*}^{+},f_{*},g_{*})$.

Define the following operation on Morse co-chain groups

$\displaystyle\theta_{R}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}\bigotimes_{\alpha}CM^{*}(f_{\alpha})\otimes CM_{*}(f_{*})\to
CM_{*}(f_{*})$ $\displaystyle x_{1}\otimes\cdots\otimes x_{2d}\otimes
x_{*}^{-}$
$\displaystyle\mapsto\sum_{x^{+}_{*}\in\textup{Crit}(f_{*})}\\#_{2}\mathcal{M}(R,x_{1},\ldots,x_{2d},x_{*}^{-},x_{*}^{+})\cdot
x^{+}_{*}.$

Here, $\\#_{2}$ denotes the parity of the set if it is discrete and 0 if it
not. For generic choices of Morse functions and Riemannian metrics, the map
$\theta_{0}$ realizes the usual cup-product in cohomology (see [AM10]). This
map does not vanish as the cuplength of $\mathbb{T}^{2d}$ is $2d$. Moreover,
$\theta_{R}$ is chain homotopic to $\theta_{0}$ and can therefore not vanish
either. Thus there have to be critical points $x_{\alpha}$ of $f_{\alpha}$ and
$x_{*}^{\pm}$ of $f_{*}$ of index at least 1 such that the moduli spaces
$\mathcal{M}(n,x_{1},\ldots,x_{2d},x_{*}^{-},x_{*}^{+})$ are non-empty for all
$n\in\mathbb{N}$.

Let $Z_{n}\in\mathcal{M}(n,x_{1},\ldots,x_{2d},x_{*}^{-},x_{*}^{+})$ and

$Z^{(\alpha)}=\lim_{n\to\infty}Z_{n}(\cdot+n\alpha),$

for $\alpha=1,\ldots,2d$. By the compactness discussed in section 5 these
limits exist in $C^{\infty}_{loc}(\mathbb{R},M)$. The $Z^{(\alpha)}$ satisfy
the Floer equation 14 for the Hamiltonian $\tilde{H}^{\rho}$ and they converge
to critical points $y^{\pm}_{\alpha}$ of $\mathcal{A}_{\tilde{H}^{\rho}}$ for
$s\to\pm\infty$ such that

$\mathcal{A}_{\tilde{H}^{\rho}}(y_{1}^{-})\geq\mathcal{A}_{\tilde{H}^{\rho}}(y_{1}^{+})\geq\mathcal{A}_{\tilde{H}^{\rho}}(y_{2}^{-})\geq\mathcal{A}_{\tilde{H}^{\rho}}(y_{2}^{+})\geq\ldots\geq\mathcal{A}_{\tilde{H}^{\rho}}(y_{2d}^{-})\geq\mathcal{A}_{\tilde{H}^{\rho}}(y_{2d}^{+}).$

The only thing left to do is show that at least $2d+1$ of these points are
different, by showing that the inequalities
$\mathcal{A}_{\tilde{H}^{\rho}}(y_{\alpha}^{-})>\mathcal{A}_{\tilde{H}^{\rho}}(y_{\alpha}^{+})$
are strict.

Note that since $\textup{Crit}(\mathcal{A}_{\tilde{H}^{\rho}})$ is finite by
assumption, we may assume critical points of $\mathcal{A}_{\tilde{H}^{\rho}}$
do not intersect the stable manifolds
$W^{s}(x_{\alpha},\bar{f}_{\alpha},g_{\alpha})$. If
$\mathcal{A}_{\tilde{H}^{\rho}}(y_{\alpha}^{-})=\mathcal{A}_{\tilde{H}^{\rho}}(y_{\alpha}^{+})$,
then $Z^{(\alpha)}$ is constant and therefore a critical point of
$\mathcal{A}_{\tilde{H}^{\rho}}$. Also note that by definition
$Z_{n}(n\alpha)\in W^{s}(x_{\alpha},\bar{f}_{\alpha},g_{\alpha})$ for any $n$.
Therefore $Z^{(\alpha)}(0)=\lim_{n\to\infty}Z_{n}(n\alpha)$ lies in the
closure of a stable manifold, which itself is a union of stable manifolds.
This is in contradiction with $Z^{(\alpha)}$ being a critical point of
$\mathcal{A}_{\tilde{H}^{\rho}}$. Thus indeed the inequalities
$\mathcal{A}_{\tilde{H}^{\rho}}(y_{\alpha}^{-})>\mathcal{A}_{\tilde{H}^{\rho}}(y_{\alpha}^{+})$
are strict and we get that $y_{1}^{-},y_{2}^{-},\ldots,y_{2d}^{-}$ and
$y_{2d}^{+}$ form $2d+1$ different critical points of
$\mathcal{A}_{\tilde{H}^{\rho}}$. These correspond to $2d+1$ distinct
solutions of equation 12 ∎

##  4 Fredholm theory

In this section we will prove the following theorem.

######  Theorem 4.1.

The moduli space $\mathcal{M}$ from equation 15 is a 1-dimensional manifold
for generic choice of $F$.

As is usual in Floer theory, this will be proven by describing $\mathcal{M}$
as the zero set of a Fredholm map. The results are standard, so we will not go
into full detail. However, we would like to give the proof of the
Fredholmness, to stress that in our setting elementary Fourier arguments can
be used.

We let $\mathcal{F}$ denote the Floer map

$\mathcal{F}(r,Z)=\mathcal{F}^{r}(Z)=\frac{1}{2}\partial_{s}Z+\cancel{\partial}Z-\begin{pmatrix}\partial_{\bar{q}}\\\
\partial_{q}\\\ \partial_{\bar{p}}\\\
\partial_{p}\end{pmatrix}\tilde{H}^{\rho,r}(Z),$

where as before $Z=(q,\bar{q},p,\bar{p})$. This means that
$\mathcal{M}=\mathcal{F}^{-1}(0)$. We start by proving $\mathcal{F}$ is a
nonlinear Fredholm map when acting on the appropriate spaces.

###  4.1 Linearization of the Floer map

Let $r$ be fixed. Then the linearization of $\mathcal{F}^{r}$ at $Z$ is given
by

$\displaystyle(d\mathcal{F}^{r})_{Z}=D-\beta_{r}(s)S_{Z},$ (16)

where $D=\frac{1}{2}\partial_{s}+\cancel{\partial}-\frac{1}{2}P$,

$\displaystyle P$ $\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{pmatrix}$ and $\displaystyle
S_{Z}=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}\textup{Hess}_{Z}\tilde{F}^{\rho}.$

By $H^{k}$ we denote the Sobolev space $W^{k,2}$. Note that for
$k>\frac{3}{2}$ we have an embedding
$H^{k}(\mathbb{R}\times\mathbb{T}^{2},\mathbb{R}^{4d})\hookrightarrow
C^{0}(\mathbb{R}\times\mathbb{T}^{2},\mathbb{R}^{4d})$.

######  Lemma 4.2.

The operator $D\mathrel{\mathop{\mathchar
58\relax}}H^{k}(\mathbb{R}\times\mathbb{T}^{2},\mathbb{R}^{4d})\to
H^{k-1}(\mathbb{R}\times\mathbb{T}^{2},\mathbb{R}^{4d})$ is a bijection for
$k>\frac{3}{2}$.

###### Proof.

For $Y\in H^{k}\mathrel{\mathop{\mathchar
58\relax}}=H^{k}(\mathbb{R}\times\mathbb{T}^{2},\mathbb{R}^{4d})$ consider the
Fourier expansion

$\displaystyle Y(s,t,\bar{t})$
$\displaystyle=\int\sum_{m_{1},m_{2}\in\mathbb{Z}}\hat{Y}(\xi,m_{1},m_{2})e^{i\xi
s}e^{im_{1}t_{1}}e^{im_{2}t_{2}}\,d\xi$
$\displaystyle=\int\sum_{m_{1},m_{2}\in\mathbb{Z}}\hat{Y}(\xi,m_{1},m_{2})e^{i\xi
s}e^{\frac{1}{2}t(im_{1}+m_{2})}e^{\frac{1}{2}\bar{t}(im_{1}-m_{2})}\,d\xi.$

Then on the Fourier coefficients the operator $D$ acts as

$\displaystyle\hat{D}(\xi,m_{1},m_{2})=\frac{1}{2}i\xi+\frac{1}{2}\begin{pmatrix}0&0&0&-im_{1}+m_{2}\\\
0&0&-im_{1}-m_{2}&0\\\ 0&im_{1}-m_{2}&0&0\\\
im_{1}+m_{2}&0&0&0\end{pmatrix}-\frac{1}{2}P.$

We compute
$\det(\hat{D}(\xi,m_{1},m_{2}))=\frac{1}{16}(m_{1}^{2}+m_{2}^{2}+\xi^{2}+i\xi)^{2}$.
This means only constant functions can be in the kernel of $D$, however the
only constant function that is square-integrable on
$\mathbb{R}\times\mathbb{T}^{2}$ is the zero function. So $\ker D=0$.

Now, for surjectivity let $W\in H^{k-1}$ and define

$\displaystyle\hat{Y}(\xi,m_{1},m_{2})=\begin{cases}\hat{D}(\xi,m_{1},m_{2})^{-1}\hat{W}(\xi,m_{1},m_{2})&(\xi,m_{1},m_{2})\neq(0,0,0)\\\
0&(\xi,m_{1},m_{2})=(0,0,0).\end{cases}$

Note that by the calculation of the determinant above
$\hat{D}(\xi,m_{1},m_{2})^{-1}$ exists. Now by definition we have that $DY=W$
provided $Y\in H^{k}$. To see this, note that the eigenvalues of
$\hat{D}(\xi,m_{1},m_{2})$ are

$\lambda^{\pm}(\xi,m_{1},m_{2})=\frac{1}{4}i\left(i+2\xi\pm
i\sqrt{1+4m_{1}^{2}+4m_{2}^{2}}\right).$

For $m_{1}^{2}+m_{2}^{2}$ large enough the norms of these eigenvalues can be
bounded from below. In other words, there exists some $N$ such that for
$m_{1}^{2}+m_{2}^{2}>N$, we have

$\displaystyle|\lambda^{\pm}(\xi,m_{1},m_{2})|^{2}\geq\frac{1}{4}\xi^{2}+\frac{1}{8}m_{1}^{2}+\frac{1}{8}m_{2}^{2}.$

Let pr denote the projection onto the subspace where $m_{1}^{2}+m_{2}^{2}>N$.
Then the $H^{k}$-norm of $\textup{pr}\,Y$ is

$\displaystyle||\textup{pr}\,Y||_{H^{k}}^{2}$
$\displaystyle=\int\sum_{m_{1}^{2}+m_{2}^{2}>N}(1+\xi^{2}+m_{1}^{2}+m_{2}^{2})^{k}|\hat{Y}(\xi,m_{1},m_{2})|^{2}\,d\xi$
$\displaystyle\leq\int\sum_{m_{1}^{2}+m_{2}^{2}>N}\frac{(1+\xi^{2}+m_{1}^{2}+m_{2}^{2})^{k}}{\frac{1}{4}\xi^{2}+\frac{1}{8}m_{1}^{2}+\frac{1}{8}m_{2}^{2}}|\hat{W}(\xi,m_{1},m_{2})|^{2}\,d\xi$
$\displaystyle\leq|||W||_{H^{k-1}}^{2}\sup_{m_{1}^{2}+m_{2}^{2}>N}\frac{1+\xi^{2}+m_{1}^{2}+m_{2}^{2}}{\frac{1}{4}\xi^{2}+\frac{1}{8}m_{1}^{2}+\frac{1}{8}m_{2}^{2}}<+\infty.$

Here, the third line follows from Hölders inequality. Thus $Y\in H^{k}$,
proving that $D$ is surjective. ∎

######  Corollary 4.3.

The Floer map $\mathcal{F}$ is a Fredholm map of index 1.

###### Proof.

Note that the operator $\beta_{r}(s)S_{Z}$ is compact as a map $H^{k}\to
H^{k-1}$. Thus, by equation 16 and lemma 4.2 the differential of
$\mathcal{F}^{r}$ is Fredholm of index 0. Varying $r$ as well, we see that
$(d\mathcal{F})_{(r,Z)}$ must be Fredholm of index 1. ∎

###  4.2 Transversality

In this section we will show that for generic choice of non-linearity the
differential of the Floer map is surjective. Combined with corollary 4.3 this
yields theorem 4.1.

Note that the moduli space $\mathcal{M}$ depends on the chosen non-linearity
$\beta_{r}(s)\tilde{F}^{\rho}_{t}$. By varying these non-linearities for each
$l$, we get the universal moduli space

$\widetilde{\mathcal{M}}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{F\in\mathcal{S}}\mathcal{M}(F_{s,t}),$

where
$\mathcal{S}=C_{c}^{l}(\mathbb{R}\times\mathbb{T}^{2}\times\mathbb{T}^{2d}\times
B,\mathbb{R})$ is the space of compactly supported smooth functions $F_{s,t}$
on $\mathbb{T}^{2d}\times B\subseteq T^{*}\mathbb{T}^{2d}$. We denote by
$\mathcal{M}(F_{s,t})$ the moduli space $\mathcal{M}$ of equation 15, where
the non-linearity $\beta_{r}(s)\tilde{F}^{\rho}_{t}$ is replaced by $F_{s,t}$.
Note that in this case $F_{s,t}$ is not necessarily the product of a function
in $s$, with a time-dependent non-linearity on $\mathbb{T}^{2d}\times B$.

This universal moduli space can be seen as the zero set of the map

$\widetilde{\mathcal{F}}(r,Z,F)=\widetilde{\mathcal{F}}^{r}(Z,F)=\frac{1}{2}\partial_{s}Z+\cancel{\partial}Z-\begin{pmatrix}\partial_{\bar{q}}\\\
\partial_{q}\\\ \partial_{\bar{p}}\\\
\partial_{p}\end{pmatrix}H^{F}_{s,t}(Z),$

where $H^{F}_{s,t}(Z)=\frac{1}{2}p\bar{p}+F_{s,t}(Z)$ and $F$ vanishes for
$s\notin[-1,(2d+1)r+1]$. The result would follow from Sard-Smale’s theorem, if
we know that the differential of $\widetilde{\mathcal{F}}^{r}$ is surjective
at every $(Z,F)\in\widetilde{\mathcal{M}}$. Let
$(Z,F)\in\widetilde{\mathcal{M}}$ and take a variation $(Y,G)\in
H^{k}(\mathbb{R}\times\mathbb{T}^{2},\mathbb{R}^{4d})\times
C^{l}_{c}(\mathbb{R}\times\mathbb{T}^{2}\times\mathbb{T}^{2d}\times
B,\mathbb{R})$. Then the linearization of $\widetilde{\mathcal{F}}^{r}$ at
$(Z,F)$ can be decomposed as

$\displaystyle(d\widetilde{\mathcal{F}}^{r})_{(Z,F)}\cdot(Y,G)=D^{1}_{F}G+D^{2}_{Z}Y.$

Assume we have some non-zero $W\in
H^{k-1}(\mathbb{R}\times\mathbb{T}^{2},\mathbb{R}^{4d})$ in the orthogonal
complement of the image of $(d\widetilde{\mathcal{F}}^{r})_{(Z,F)}$. Then
$\langle W,D^{2}_{Z}Y\rangle=0$ for all $Y$, so that
$W\in\ker\left((D^{2}_{Z})^{*}\right)$. Note that for $s\notin[-1,(2d+1)r+1]$
the non-linearity $F_{s,t}$ vanishes so that
$(D^{2}_{Z})^{*}=\frac{1}{2}\partial_{s}-\cancel{\partial}-\frac{1}{2}P$. This
operator satisfies a maximum principle, similar to the one used in lemma 3.6.
Therefore, by standard arguments $W(s,t)\neq 0$ for some
$(s,t)\in[-1,(2d+1)r+1]\times\mathbb{T}^{2}$. Thus, we can find some variation
$G$ with support for $s\in[-1,(2d+1)r+1]$, such that $\langle
W,D^{1}_{F}G\rangle\neq 0$. This contradicts the assumption that $\langle
W,(d\widetilde{\mathcal{F}}^{r})_{(Z,F)}\cdot(Y,G)\rangle=0$ for any $(Y,G)$.
We must conclude $(d\widetilde{\mathcal{F}}^{r})_{(Z,F)}$ is surjective. This
finishes the proof of theorem 4.1.∎

##  5 Compactness

In this section we will prove compactness of the moduli space (15) that is
needed in the proof of theorem 3.1. We start by proving a uniform energy
bound, using a standard argument.

######  Lemma 5.1.

For all $(r,Z)\in\mathcal{M}$ we have

$\displaystyle E(Z)\leq
2||\tilde{F}^{\rho}||_{\textup{Hofer}}\mathrel{\mathop{\mathchar
58\relax}}=2\int_{\mathbb{T}^{2}}\left(\sup_{Z^{\prime}\in
T^{*}\mathbb{T}^{2d}}\tilde{F}^{\rho}_{t}(Z^{\prime})-\inf_{Z^{\prime}\in
T^{*}\mathbb{T}^{2d}}\tilde{F}^{\rho}_{t}(Z^{\prime})\right)\,d\mathcal{V}.$

###### Proof.

By definition

$E(Z)=\int_{-\infty}^{\infty}\int_{\mathbb{T}^{2}}|\partial_{s}Z(s,t)|^{2}\,d\mathcal{V}\,ds=-\int_{-\infty}^{\infty}\langle\textup{grad}\,\mathcal{A}_{\tilde{H}^{\rho,r}}(Z),\partial_{s}Z\rangle\,ds,$

since $Z$ is a negative gradient line of $\mathcal{A}_{\tilde{H}^{\rho,r}}$.
Writing this out further we get

$\displaystyle E(Z)$
$\displaystyle=-\int_{-\infty}^{\infty}\left(d\mathcal{A}_{\tilde{H}^{\rho,r}}\right)_{Z_{s}}(\partial_{s}Z)\,ds$
$\displaystyle=-\int_{-\infty}^{\infty}\frac{d}{ds}(\mathcal{A}_{\tilde{H}^{\rho,r}}(Z_{s}))\,ds+\int_{-\infty}^{\infty}\frac{\partial\mathcal{A}_{\tilde{H}^{\rho,r}}}{\partial
s}(Z_{s})\,ds$
$\displaystyle=-\lim_{s\to\infty}\mathcal{A}_{\tilde{H}^{\rho,r}}(Z_{s})+\lim_{s\to-\infty}\mathcal{A}_{\tilde{H}^{\rho,r}}(Z_{s})-\int_{-\infty}^{\infty}\int_{\mathbb{T}^{2}}\beta_{r}^{\prime}(s)\tilde{F}^{\rho}_{t}(Z_{s})\,d\mathcal{V}\,ds.$

Note that the first two terms on the last line vanish. Also,
$\beta_{r}^{\prime}$ vanishes everywhere outside
$[-1,0]\cup[(2d+1)r,(2d+1)r+1]$ and is positive and negative respectively on
those two intervals. Therefore,

$\displaystyle E(Z)$
$\displaystyle=-\int_{-1}^{0}\int_{\mathbb{T}^{2}}\beta_{r}^{\prime}(s)\tilde{F}^{\rho}_{t}(Z_{s})\,d\mathcal{V}\,ds-\int_{(2d+1)r}^{(2d+1)r+1}\int_{\mathbb{T}^{2}}\beta_{r}^{\prime}(s)\tilde{F}^{\rho}_{t}(Z_{s})\,d\mathcal{V}\,ds$
$\displaystyle\leq 2\int_{\mathbb{T}^{2}}\left(\sup_{Z^{\prime}\in
T^{*}\mathbb{T}^{2d}}\tilde{F}^{\rho}_{t}(Z^{\prime})-\inf_{Z^{\prime}\in
T^{*}\mathbb{T}^{2d}}\tilde{F}^{\rho}_{t}(Z^{\prime})\right)\,d\mathcal{V}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}2||\tilde{F}^{\rho}||_{\textup{Hofer}}.$

∎

Now we want to relate the energy to the $L^{2}$-norm of the derivatives of
$Z$. To this end we define the energy density

$e_{Z}(s,t)=\frac{1}{2}|dZ(s,t)|^{2}=\frac{1}{2}|\partial_{s}Z(s,t)|^{2}+|\partial_{t}Z(s,t)|^{2}+|\partial_{\bar{t}}Z(s,t)|^{2}.$

######  Lemma 5.2.

Let $K=[-M,M]\times\mathbb{T}^{2}\subseteq\mathbb{R}\times\mathbb{T}^{2}$.
Then there exists a constant $C=C(K)$ such that for any $(r,Z)\in\mathcal{M}$

$\int_{K}e_{Z}(s,t)\,d\mathcal{V}\,ds\leq E(Z)+C(K).$

###### Proof.

We calculateiiiiiiiiiThe first equality follows from partial integration,
using that $Z$ is nullhomotopic.

$\displaystyle\int_{K}e_{Z}(s,t)\,d\mathcal{V}\,ds$
$\displaystyle=\frac{1}{2}\int_{K}|\partial_{s}Z|^{2}\,d\mathcal{V}\,ds+2\int_{K}\left(|\partial_{t}q|^{2}+|\partial_{t}p|^{2}\right)\,d\mathcal{V}\,ds$
$\displaystyle=\frac{1}{2}\int_{K}|\partial_{s}Z|^{2}\,d\mathcal{V}\,ds+2\int_{K}\left(\left|\frac{\partial
H}{\partial
p}-\frac{1}{2}\partial_{s}\bar{p}\right|^{2}+\left|\frac{1}{2}\partial_{s}\bar{q}-\frac{\partial
H}{\partial q}\right|^{2}\right)\,d\mathcal{V}\,ds$ $\displaystyle\leq
E(Z)+2\int_{K}\left(\left|\frac{\partial H}{\partial
q}\right|^{2}+\left|\frac{\partial H}{\partial
p}\right|^{2}\right)\,d\mathcal{V}\,ds$ $\displaystyle\leq
E(Z)+\frac{1}{2}\int_{K}|p|^{2}\,d\mathcal{V}\,ds+2\int_{K}\left(\left|\beta_{r}(s)\frac{\partial\tilde{F}^{\rho}}{\partial
q}\right|^{2}+\left|\beta_{r}(s)\frac{\partial\tilde{F}^{\rho}}{\partial
p}\right|^{2}\right)\,d\mathcal{V}\,ds.$

Now, as in lemma 3.6 we have that $|p(s,t)|^{2}<\rho$ for all $(s,t)\in K$.
Combined with the $C^{1}$-boundedness of $F$, we can estimate the right-hand
side purely in terms of $E(Z)$ and the volume of $K$. This proves the lemma. ∎

In order to get point-wise bounds on the derivatives, we want to use the Heinz
trick (theorem 5.4). We first need to show that the energy density satisfies
an appropriate inequality.

######  Lemma 5.3.

The energy density $e\mathrel{\mathop{\mathchar 58\relax}}=e_{Z}$ satisfies
the inequality

$\displaystyle(\partial_{s}^{2}+\Delta)e\geq-c(1+e^{\frac{3}{2}}),$ (17)

for some $c>0$.

###### Proof.

See appendix A. ∎

######  Theorem 5.4 (Heinz trick, see [HNS09]).

Let $M$ be an $(m+1)$-dimensional Riemannian manifold for
$\frac{3}{2}<\frac{m+3}{m+1}$ and $e\mathrel{\mathop{\mathchar
58\relax}}M\to\mathbb{R}_{\geq 0}$ a smooth function satisfying equation 17.
Then for $K\subseteq M$ compact $\sup_{K}e$ is bounded by $\int_{K}e$.

Now we are ready to prove the desired compactness theorem.

######  Theorem 5.5.

The moduli space $\mathcal{M}$ is relatively compact in the
$C^{\infty}_{\textit{loc}}$-topology.

###### Proof.

Let $(r_{n},Z_{n})\in\mathcal{M}$ for all $n\in\mathbb{N}$ and
$K=[-M,M]\times\mathbb{T}^{2}\subseteq\mathbb{R}\times\mathbb{T}^{2}$. By
lemma 3.6 the sequence $Z_{n}|_{K}$ is a bounded family of smooth functions on
$K$. By combining lemmas 5.1 and 5.2 we find a uniform bound on
$\int_{K}e_{Z_{n}}$. Then the Heinz trick (theorem 5.4) combined with the
density estimate in lemma 5.3 yields a uniform pointwise bound on
$e_{Z_{n}}(s,t)$ for all $(s,t)\in K$. By the mean-value theorem it follows
that the sequence $(Z_{n})$ is uniformly equicontinuous. The theorem of
Arzelà-Ascoli then yields a converging subsequence in the
$C^{0}_{\textit{loc}}$-topology. Finally, the ellipticity of the Floer
operatorivivivNote that
$(\frac{1}{2}\partial_{s}-\cancel{\partial})(\frac{1}{2}\partial_{s}+\cancel{\partial})=\frac{1}{4}\partial_{s}^{2}+\partial_{t}\partial_{\bar{t}}$,
so the Floer operator is a factor of the Laplacian on
$\mathbb{R}\times\mathbb{T}^{2}$. yields convergence of the subsequence in
$C^{\infty}_{\textit{loc}}$. ∎

##  Appendix A Proof of lemma 5.3

The proof of lemma 5.3 is easier to write out in the ’real’ notation using
$T=(t_{1},t_{2})\in\mathbb{T}^{2}$ and $Z=(q_{1},q_{2},p_{1},p_{2})$. Note
that it is just a matter of notation so that the proof still holds in the
complex notation as well. Recall that $J_{\partial}$ is an operator satisfying
$J_{\partial}^{2}=-(\partial_{1}^{2}+\partial_{2}^{2})$. We can write
$J_{\partial}=J_{1}\partial_{1}+J_{2}\partial_{2}$ for some matrices
$J_{1},J_{2}$ satisfying $J_{1}^{2}=J_{2}^{2}=-\textup{Id}$ and
$J_{1}J_{2}+J_{2}J_{1}=0$ (see [Bri06]). For any $(r,Z)$ in $\mathcal{M}$, it
holds that $Z$ satisfies the Floer equation

$\displaystyle\partial_{s}Z+J_{\partial}Z=\nabla\tilde{H}^{\rho,r}(Z),$ (18)

where
$\tilde{H}^{\rho,r}_{s,T}(Z)=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})+\beta_{r}(s)\tilde{F}^{\rho}_{T}(Z)$.
We define $t_{0}\mathrel{\mathop{\mathchar 58\relax}}=s$ and
$\xi_{i}\mathrel{\mathop{\mathchar 58\relax}}=\partial_{i}Z$ for $i=0,1,2$.
From the identity

$(\partial_{s}-J_{\partial})(\partial_{s}+J_{\partial})=\mathcal{L}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{i=0}^{2}\partial_{i}^{2}$

we see that the $\xi_{i}$ satisfy the equation

$\displaystyle\mathcal{L}\xi_{i}=(\partial_{s}-J_{\partial})(\sigma_{Z}\xi_{i}),$
(19)

where $\sigma_{Z}$ is the Hessian of $\tilde{H}^{\rho,r}$ at $Z$.

Note that in this notation the energy density is given by

$e(s,T)=\frac{1}{2}\sum_{i=0}^{2}|\xi_{i}(s,T)|^{2}.$

We calculate

$\displaystyle\mathcal{L}e$
$\displaystyle=\sum_{i,j=0}^{2}|\partial_{j}\xi_{i}|^{2}+\sum_{i=0}^{2}\langle\xi_{i},\mathcal{L}\xi_{i}\rangle$
$\displaystyle=\sum_{i,j=0}^{2}|\partial_{j}\xi_{i}|^{2}+\sum_{i=0}^{2}\langle\xi_{i},(\partial_{s}-J_{\partial})(\sigma_{Z}\xi_{i})\rangle,$

where we have used equation 19 in the second equality. Now note that the chain
rule gives that

$\displaystyle\partial_{j}(\sigma_{Z}\xi_{i})=T\sigma_{Z}\cdot\xi_{j}\cdot\xi_{i}+\sigma_{Z}\partial_{j}\xi_{i}.$

Denote $J_{0}=-\textup{Id}$. Then

$\displaystyle\mathcal{L}e$
$\displaystyle=\sum_{i,j=0}^{2}\left(|\partial_{j}\xi_{i}|^{2}-\langle(J_{j}\sigma_{Z})^{T}\xi_{i},\partial_{j}\xi_{i}\rangle-\langle\xi_{i},J_{j}T\sigma_{Z}\cdot\xi_{j}\cdot\xi_{i}\rangle\right)$
$\displaystyle=\sum_{i,j=0}^{2}\left(\left|\partial_{j}\xi_{i}-\frac{1}{2}(J_{j}\sigma_{Z})^{T}\xi_{i}\right|^{2}-\frac{1}{4}\left|(J_{j}\sigma_{Z})^{T}\xi_{i}\right|^{2}-\langle\xi_{i},J_{j}T\sigma_{Z}\cdot\xi_{j}\cdot\xi_{i}\rangle\right).$

Note that since $F$ is $C^{3}$-bounded, we have that both $\sigma_{Z}$ and
$T\sigma_{Z}$ are bounded. Thus

$\left|(J_{j}\sigma_{Z})^{T}\xi_{i}\right|^{2}\leq c_{1}|\xi_{i}|^{2}\leq
c_{2}e$

and

$\left|\langle\xi_{i},J_{j}T\sigma_{Z}\cdot\xi_{j}\cdot\xi_{i}\rangle\right|\leq
c_{3}|\xi_{i}|^{2}|\xi_{j}|\leq c_{4}e^{\frac{3}{2}}.$

Combined we get that

$\mathcal{L}e\geq-c_{5}\left(e+e^{\frac{3}{2}}\right).$

Finally, Young’s inequality with $p=3$, $q=\frac{3}{2}$ gives that

$e\leq\frac{1}{3}+\frac{2}{3}e^{\frac{3}{2}},$

so that

$\mathcal{L}e\geq-c(1+e^{\frac{3}{2}}).$

∎

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