and $F_{t}=0$ for all $t\in\mathinner{\bigl{[}0,\frac{1}{2}\bigr{]}}$.

###### Lemma 3.9.

Let $\varphi\in\operatorname{Aut}(W,\lambda)$. For all
$H\in\mathscr{F}_{\varphi}(\Sigma)$ and
$\varphi_{F}\in\operatorname{Ham}(M,d\lambda)$ there exists a corresponding
twisted Moser pair $\mathfrak{M}$ such that the flow of $\chi X_{H}$ is a
time-reparametrisation of the flow of $X_{H}$.

###### Proof.

For constructing the Hamiltonian perturbation $\widetilde{F}$, fix $\rho\in
C^{\infty}(I,I)$ such that

$\rho(t)=\begin{cases}0&t\in\mathinner{\bigl{[}0,\frac{1}{2}\bigr{]}},\\\
1&t\in\mathinner{\bigl{[}\frac{2}{3},1\bigr{]}}.\end{cases}$

See Figure 9(a). Then define $\widetilde{F}\in
C^{\infty}_{\varphi}(M\times\mathbb{R})$ by

$\widetilde{F}(x,t)\mathrel{\mathop{\mathchar
58\relax}}=\dot{\rho}(t-k)F\mathinner{\bigl{(}\varphi^{-k}(x),\rho(t-k)\bigr{)}}\qquad\forall
t\in\mathinner{[k,k+1]},$

for $k\in\mathbb{Z}$. See Figure 9(b). Then $\widetilde{F}_{t}=0$ for all
$t\in\mathinner{\bigl{[}0,\frac{1}{2}\bigr{]}}$, and

$\phi^{X_{\widetilde{F}}}_{t}=\phi^{X_{F}}_{\rho(t)}\qquad\forall t\in I.$

Indeed, we compute

$\frac{d}{dt}\phi^{X_{F}}_{\rho(t)}=\dot{\rho}(t)\frac{d}{d\rho}\phi^{X_{F}}_{\rho(t)}=\dot{\rho}(t)\mathinner{\bigl{(}X_{F_{\rho(t)}}\circ\phi^{X_{F}}_{\rho(t)}\bigr{)}}=X_{\widetilde{F}_{t}}\circ\phi^{X_{F}}_{\rho(t)}.$

In particular

$\varphi_{\widetilde{F}}=\phi^{X_{\widetilde{F}}}_{1}=\phi^{X_{F}}_{\rho(1)}=\phi^{X_{F}}_{1}=\varphi_{F}.$

Finally, we have that

$\phi^{\chi
X_{H}}_{t}=\phi^{X_{H}}_{\tau(t)}\qquad\text{with}\qquad\tau(t)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{t}\chi,$

as we compute

$\frac{d}{dt}\phi^{X_{H}}_{\tau(t)}=\chi(t)\frac{d}{d\tau}\phi^{X_{H}}_{\tau(t)}=\chi(t)X_{H}\circ\phi^{X_{H}}_{\tau(t)},$

and thus we conclude by the uniqueness of integral curves. ∎

(a) The smooth function $\rho$.

(b) The derivative $\dot{\rho}$ of $\rho$.

Figure 9:

###### Lemma 3.10.

Let $\varphi\in\operatorname{Aut}(W,\lambda)$ and
$\varphi_{F}\in\operatorname{Ham}(M,d\lambda)$ a Hamiltonian
symplectomorphism. If
$(\gamma,\tau)\in\operatorname{Crit}\mathscr{A}^{\mathfrak{M}}_{\varphi}$,
then $x\mathrel{\mathop{\mathchar
58\relax}}=\gamma\mathinner{\bigl{(}\frac{1}{2}\bigr{)}}$ is a twisted leaf-
wise intersection point for $\varphi_{F}$.

###### Proof.

Let $\mathfrak{M}=(\chi H,F)$ denote the twisted Moser pair from Lemma 3.9.
Using Proposition 3.1 we compute

$\displaystyle\partial_{t}H(\gamma(t))$ $\displaystyle=dH(\dot{\gamma}(t))$
$\displaystyle=dH(\tau X_{\chi(t)H}(\gamma(t))+X_{F_{t}}(\gamma(t)))$
$\displaystyle=dH(\tau\chi(t)X_{H}(\gamma(t)))$
$\displaystyle=\tau\chi(t)dH(X_{H}(\gamma(t)))$ $\displaystyle=0$

for all $t\in\mathinner{\bigl{[}0,\frac{1}{2}\bigr{]}}$. Thus
$H\circ\gamma=c\in\mathbb{R}$ on $\mathinner{\bigl{[}0,\frac{1}{2}\bigr{]}}$
with

$0=\int_{0}^{1}\chi H(\gamma)=\int_{0}^{\frac{1}{2}}\chi
H(\gamma)=c\int_{0}^{\frac{1}{2}}\chi=c\int_{0}^{1}\chi=c.$

Consequently, $\gamma(0)\in L_{x}$ and $x\in\Sigma$. Moreover, also
$\gamma(1)=\varphi(\gamma(0))\in\Sigma$ by the $\varphi$-invariance of $H$.
For $t\in\mathinner{\bigl{[}\frac{1}{2},1\bigr{]}}$,
$\dot{\gamma}=X_{F_{t}}(\gamma)$ and so $\gamma(1)=\varphi_{F}(x)\in\Sigma$.
We conclude

$L_{\varphi(x)}=\\{\phi^{R}_{t}(\varphi(x))\mathrel{\mathop{\mathchar
58\relax}}t\in\mathbb{R}\\}=\\{\varphi(\phi^{R}_{t}(x))\mathrel{\mathop{\mathchar
58\relax}}t\in\mathbb{R}\\}=\varphi(L_{x}),$

and so $\varphi_{F}(x)=\gamma(1)=\varphi(\gamma(0))\in L_{\varphi(x)}$. ∎

###### Theorem 3.3.

Let $(W,\lambda)$ be a Liouville domain with displaceable boundary in the
completion $(M,\lambda)$ and $\varphi\in\operatorname{Aut}(W,\lambda)$. Then
$\operatorname{RFH}^{\varphi}(\Sigma,M)\cong 0$.

###### Proof.

Suppose that $\Sigma=\partial W$ is displaceable in $M$ via
$\varphi_{F}\in\operatorname{Ham}_{c}(M,d\lambda)$ and choose Rabinowitz–Floer
data $(H,J)$ for $\varphi$. Denote by $\mathfrak{M}=(\chi H,F)$ the associated
twisted Moser pair from Lemma 3.9. Then
$\operatorname{Crit}\mathscr{A}^{\mathfrak{M}}_{\varphi}=\varnothing$. Indeed,
if there exists
$(\gamma,\tau)\in\operatorname{Crit}\mathscr{A}^{\mathfrak{M}}_{\varphi}$,
then $\gamma\mathinner{\bigl{(}\frac{1}{2}\bigr{)}}$ is a twisted leaf-wise
intersection point for $\varphi_{F}$ by Lemma 3.10. However, this is
impossible as by displaceability we have that
$\varphi_{F}(\Sigma)\cap\Sigma=\varnothing$. Consequently, the perturbed
twisted Rabinowitz action functional $\mathscr{A}^{\mathfrak{M}}_{\varphi}$ is
a Morse function. By adapting the Fundamental Lemma to the current setting as
in [AF10, Theorem 2.9], the Floer homology
$\operatorname{HF}(\mathscr{A}^{\mathfrak{M}}_{\varphi})$ is well-defined. By
making use of continuation homomorphisms we have that

$0=\operatorname{HF}(\mathscr{A}^{\mathfrak{M}}_{\varphi})\cong\operatorname{HF}(\mathscr{A}^{(\chi
H,0)}_{\varphi})\cong\operatorname{RFH}^{\varphi}(\Sigma,M),$

where the last equation is the observation that twisted Rabinowitz–Floer
homology in the autonomous case extends to the weakly time-dependent case
without any issues. Crucial is, that the period–action equality (see Remark
3.5) is still valid. Indeed, we compute

$\mathscr{A}^{(\chi
H,0)}_{\varphi}(\gamma,\tau)=\int_{0}^{1}\gamma^{*}\lambda=\int_{0}^{1}\lambda(\dot{\gamma})=\tau\int_{0}^{1}\chi\lambda(R(\gamma))=\tau\int_{0}^{1}\chi=\tau$

for all $(\gamma,\tau)\in\operatorname{Crit}\mathscr{A}^{(\chi
H,0)}_{\varphi}$. ∎

##  4 Applications

In this chapter we give two applications of the abstract machinery developed
in the previous section and prove Theorem 1.1 as well as Theorem 1.5 (see
Theorem 4.3).

###  4.1 Existence of Noncontractible Periodic Reeb Orbits

We define an equivariant version of twisted Rabinowitz–Floer homology for the
discrete group $\mathbb{Z}_{m}$ following [AF12, p. 487]. In general, suppose
that a topological manifold $M$ admits a group action by a topological group
$G$. Then there exists a principal $G$-bundle $EG\to BG$, where $BG=EG/G$
denotes the classifying space of $G$ and $EG$ is weakly contractible. Then $G$
acts freely on $EG\times M$ via the diagonal action. Thus we can define the
$G$-equivariant homology of $M$ by

$\operatorname{H}_{\ast}^{G}(M;R)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{H}_{\ast}(EG\times M/G;R),$

for any coefficient ring $R$ and where $EG\times M/G$ is the homotopy quotient
of $M$ by $G$. See [Tu20, p. 30–31]. For example, if $G=\mathbb{Z}_{m}$, then
$E\mathbb{Z}_{m}=\mathbb{S}^{\infty}$ and
$B\mathbb{Z}_{m}=\mathbb{S}^{\infty}/\mathbb{Z}_{m}$ is a lens space. Since
$\mathbb{Z}_{m}$ acts freely on $\mathbb{S}^{2n-1}$, there is a fibre bundle

$EG\to EG\times M/G\to M/G,$

inducing an isomorphism

$\operatorname{H}_{*}^{G}(M)=\operatorname{H}_{*}(M/G)$

by [Tu20, Corollary 9.6] and [Tu20, Theorem 3.3]. This observation will be
crucial in widehat follows. Explicitly, consider the free smooth discrete
action on the odd-dimensional sphere generated by the rotation $\varphi$ from
Example 2.13. Define a twisted defining Hamiltonian function
$H\in\mathscr{F}_{\varphi}(\mathbb{S}^{2n-1})$ by

$H(z)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\mathinner{\bigl{(}\beta(\mathinner{\lvert
z\rvert}^{2})-1\bigr{)}}$

for some sufficiently small mollification of the piecewise linear function

$\beta(r)=\begin{cases}\frac{1}{2}&r\in\mathinner{\bigl{(}-\infty,\frac{1}{2}\bigr{]}},\\\
r&r\in\mathinner{\bigl{[}\frac{1}{2},\frac{3}{2}\bigr{]}},\\\
\frac{3}{2}&r\in\mathinner{\bigl{[}\frac{3}{2},+\infty\bigr{)}}.\end{cases}$

Fix a $\varphi$-invariant $\omega$-compatible almost complex structure on
$(\mathbb{C}^{n},\lambda)$, where $\lambda$ is given by (10). Then the
rotation $\varphi$ induces a free $\mathbb{Z}_{m}$-action on
$\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$ and on the moduli space of
twisted negative gradient flow lines with cascades of
$\mathscr{A}^{H}_{\varphi}$. Therefore, we can define the
$\mathbb{Z}_{m}$-equivariant twisted Rabinowitz-Floer homology

$\overline{\operatorname{RFH}}^{\varphi}_{k}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m})\mathrel{\mathop{\mathchar
58\relax}}=\frac{\ker\overline{\partial}_{k}}{\operatorname{im}\overline{\partial}_{k+1}}\qquad\forall
k\in\mathbb{Z},$

as the homology of the $\mathbb{Z}$-graded chain complex (see Remark 3.7)

$\overline{\partial}_{k}\colon\operatorname{RFC}^{\varphi}_{k}(\mathbb{S}^{2n-1},\mathbb{C}^{n})/\mathbb{Z}_{m}\to\operatorname{RFC}^{\varphi}_{k-1}(\mathbb{S}^{2n-1},\mathbb{C}^{n})/\mathbb{Z}_{m}$

given by

$\overline{\partial}_{k}[(\gamma,\tau)]\mathrel{\mathop{\mathchar
58\relax}}=[\partial_{k}(\gamma,\tau)]\qquad(\gamma,\tau)\in\operatorname{Crit}h,$

for some $\varphi$-invariant Morse function $h$ on
$\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$. More generally, if $G$ is a
finite symmetry of a Hamiltonian system which acts free on the displaceable
regular energy hypersurface, one can define the $G$-equivariant twisted
Rabinowitz–Floer homology as above if the twisted Rabinowitz–Floer homology is
defined. Under some mild index assumption on the Conley–Zehnder index, the
resulting $G$-equivariant twisted Rabinowitz–Floer homology is isomorphic to
the Tate homology of $G$ with coefficients in $\mathbb{Z}_{2}$. See [Ruc22,
Theorem 5.6] for a proof and [Wei94, Definition 6.2.4] as well as [Bro82, p.
135] for a definition of Tate homology.

###### Theorem 4.1.

Let $n\geq 2$. For $m\geq 1$ consider the rotation

$\varphi\colon\mathbb{C}^{n}\to\mathbb{C}^{n},\quad\varphi(z_{1},\dots,z_{n})\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\bigl{(}e^{2\pi ik_{1}/m}z_{1},\dots,e^{2\pi
ik_{n}/m}z_{n}\bigr{)}}$

for $k_{1},\dots,k_{n}\in\mathbb{Z}$ coprime to $m$. Then

$\overline{\operatorname{RFH}}^{\varphi}_{k}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m})\cong\begin{cases}\mathbb{Z}_{2}&m\text{
even},\\\ 0&m\text{ odd},\end{cases}\qquad\forall k\in\mathbb{Z},$

If $m$ is even, then
$\overline{\operatorname{RFH}}^{\varphi}_{k}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m})$
is generated by a noncontractible periodic Reeb orbit in the lens space
$\mathbb{S}^{2n-1}/\mathbb{Z}_{m}$ for all $k\in\mathbb{Z}$.

###### Proof.

First we consider the special case

$\varphi\colon\mathbb{C}^{n}\to\mathbb{C}^{n},\qquad\varphi(z)=e^{2\pi i/m}z.$

The hypersurface $\mathbb{S}^{2n-1}\subseteq(\mathbb{C}^{n},\lambda)$ is of
restricted contact type with contact form $\lambda|_{\mathbb{S}^{2n-1}}$ and
associated Reeb vector field

$R=2\sum_{j=1}^{n}\mathinner{\biggl{(}y_{j}\frac{\partial}{\partial
x_{j}}-x_{j}\frac{\partial}{\partial
y_{j}}\biggr{)}}\bigg{|}_{\mathbb{S}^{2n-1}}=2i\sum_{j=1}^{n}\mathinner{\biggl{(}\overline{z}_{j}\frac{\partial}{\partial\overline{z}_{j}}-z_{j}\frac{\partial}{\partial
z_{j}}\biggr{)}}\bigg{|}_{\mathbb{S}^{2n-1}}.$

Suppose $(\gamma,\tau)\in\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$. If
$\tau=0$, then $\gamma$ is constant. This cannot happen as
$\operatorname{Fix}(\varphi|_{\mathbb{S}^{2n-1}})=\varnothing$. So we assume
$\tau\neq 0$. Define a reparametrisation

$\gamma_{\tau}\colon\mathbb{R}\to\mathbb{S}^{2n-1},\qquad\gamma_{\tau}(t)\mathrel{\mathop{\mathchar
58\relax}}=\gamma(t/\tau).$

Then $\gamma_{\tau}$ is the unique integral curve of $R$ starting at
$z\mathrel{\mathop{\mathchar 58\relax}}=\gamma(0)$ and thus

$\gamma_{\tau}(t)=e^{-2it}z\qquad\forall t\in\mathbb{R}.$

From $\gamma(t)=\gamma_{\tau}(\tau t)$ and the requirement

$e^{-2i\tau}z=\gamma(1)=\varphi(\gamma(0))=\varphi(z)=e^{2\pi i/m}z,$

we conclude $\tau\in\frac{\pi}{m}(m\mathbb{Z}-1)$. Hence

$\operatorname{Crit}\mathscr{A}^{H}_{\varphi}=\mathinner{\bigl{\\{}\mathinner{\bigl{(}\phi^{\tau_{k}R}(z),\tau_{k}\bigr{)}}\mathrel{\mathop{\mathchar
58\relax}}k\in\mathbb{Z},z\in\mathbb{S}^{2n-1}\bigr{\\}}}\cong\mathbb{S}^{2n-1}\times\mathbb{Z},$

for any $H\in\mathscr{F}_{\varphi}(\mathbb{S}^{2n-1})$, where we define
$\tau_{k}\mathrel{\mathop{\mathchar 58\relax}}=\frac{\pi}{m}(mk-1)$. By
Proposition 3.2, $(z_{0},\eta)\in T_{z}\mathbb{S}^{2n-1}\times\mathbb{R}$
belongs to the kernel of the Hessian at
$(z,k)\in\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$ if and only if $\eta=0$
and

$z_{0}\in\ker\mathinner{\bigl{(}D(\phi^{R}_{-\tau_{k}}\circ\varphi)|_{z}-\operatorname{id}_{T_{z}\mathbb{S}^{2n-1}}\bigr{)}}.$

A direct computation yields
$D(\phi^{R}_{-\tau_{k}}\circ\varphi)|_{z}=\operatorname{id}_{T_{z}\mathbb{S}^{2n-1}}$
and thus the twisted Rabinowitz action functional $\mathscr{A}^{H}_{\varphi}$
is Morse–Bott with spheres.

Figure 10: The critical manifold $\mathbb{S}^{2n-1}\times\mathbb{Z}$ with the
standard height function, the Morse–Bott function $f$ and the resulting chain
complex.

The full Conley–Zehnder index [FK18, Definition 10.4.1] gives rise to a
locally constant function

$\widehat{\mu}_{\mathrm{CZ}}\colon\operatorname{Crit}\mathscr{A}^{H}_{\varphi}\to\mathbb{Z},\qquad\widehat{\mu}(z,k)=(2k-1)n.$

Note that the definition of the Conley–Zehnder index also applies in this
degenerate case, compare [FK18, Remark 10.4.2]. By the adapted proof of the
Hofer–Wysocki–Zehnder Theorem [FK18, Theorem 12.2.1] to the $n$-dimensional
setting, the full Conley–Zehnder index coincides with the transverse
Conley–Zehnder index $\mu_{\operatorname{CZ}}$. Indeed, for a critical point
$(\gamma,\tau)\in\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$ define a smooth
path

$\Psi\colon I\to\operatorname{Sp}(n),\qquad\Psi_{t}\mathrel{\mathop{\mathchar
58\relax}}=D\phi^{H}_{\tau
t}|_{\gamma(0)}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}.$

Adapting the proof of [FK18, Lemma 12.2.3 (iii)], we get that

$\Psi_{1}(R(\gamma(0)))=R(\gamma(1))\qquad\text{and}\qquad\Psi_{1}(\gamma(0))=\gamma(1).$

Arguing as in [FK18, p. 235–236] we conclude

$\mu_{\operatorname{CZ}}(\gamma,\tau)=\widehat{\mu}_{\operatorname{CZ}}(\gamma,\tau).$

Fix $z_{0}\in\mathbb{S}^{2n-1}$ and define $\eta\mathrel{\mathop{\mathchar
58\relax}}=\phi^{\tau_{0}R}(z_{0})$. Note that $\phi^{\tau_{k}R}(z)$ belongs
to the same equivalence class in
$\pi_{0}\mathscr{L}_{\varphi}\mathbb{S}^{2n-1}$ as $\eta$ for all
$z\in\mathbb{S}^{2n-1}$ and $k\in\mathbb{Z}$ because $\mathbb{S}^{2n-1}$ is
simply connected for $n\geq 2$. Let $h\in C^{\infty}(\mathbb{S}^{2n-1})$ be
the standard height function. By Remark 3.7,
$\operatorname{RFH}^{\varphi}(\mathbb{S}^{2n-1},\mathbb{C}^{n})$ carries the
$\mathbb{Z}$-grading

$\mu((z,k),(z_{0},0))+\operatorname{ind}_{h}(z)=2kn+\operatorname{ind}_{h}(z)\qquad\forall(z,k)\in\mathbb{S}^{2n-1}\times\mathbb{Z}.$

We claim that the number of twisted negative gradient flow lines between the
minimum of $\mathbb{S}^{2n-1}\times\\{k+1\\}$ and the maximum of
$\mathbb{S}^{2n-1}\times\\{k\\}$ must be odd, so that the critical manifold
$\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$ looks like a string of pearls,
see Figure 10. Indeed, if there is an even number of such negative gradient
flow lines, then
$\operatorname{RFH}^{\varphi}_{\ast}(\mathbb{S}^{2n-1},\mathbb{C}^{n})\neq 0$,
contradicting Theorem 3.3 as $\mathbb{S}^{2n-1}$ is displaceable in the
completion $\mathbb{C}^{n}$. To compute the $\mathbb{Z}_{m}$-equivariant
twisted Rabinowitz–Floer homology, choose the additional
$\mathbb{Z}_{m}$-invariant Morse–Bott function $f$ from Example 2.1.
Additionaly, choose a $\mathbb{Z}_{m}$-invariant Morse function on
$\operatorname{Crit}f$. For example, one can take

$h\colon\mathbb{T}\to\mathbb{R},\qquad h(t)\mathrel{\mathop{\mathchar
58\relax}}=\cos(2\pi mt).$

The resulting chain complex is given by

${\dots}$${\mathbb{Z}_{2}^{m}}$${\mathbb{Z}^{m}_{2}}$${\mathbb{Z}^{m}_{2}}$${\mathbb{Z}^{m}_{2}}$${\mathbb{Z}^{m}_{2}}$${\mathbb{Z}^{m}_{2}}$${\dots}$$\scriptstyle{\mathbbm{1}}$$\scriptstyle{A}$$\scriptstyle{\mathbbm{1}}$$\scriptstyle{A}$$\scriptstyle{\mathbbm{1}}$

where $\mathbbm{1}\in M_{m\times m}(\mathbb{Z}_{2})$ has every entry equal to
$1$ and $A\in M_{m\times m}(\mathbb{Z}_{2})$ is defined by

$A\mathrel{\mathop{\mathchar 58\relax}}=I_{m\times
m}+\sum_{j=1}^{m-1}e_{(j+1)j}+e_{1m},$

where $e_{ij}\in M_{m\times m}(\mathbb{Z}_{2})$ satisfies
$(e_{ij})_{kl}=\delta_{ik}\delta_{jl}$. Thus the resulting chain complex looks
like a rope ladder. Compare Figure 10. Passing to the quotient via the free
$\mathbb{Z}_{m}$-action, we get the acyclic chain complex

${\dots}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\dots}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{0}$

if $m$ is even and the alternating chain complex

${\dots}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\mathbb{Z}_{2}}$${\dots}$$\scriptstyle{1}$$\scriptstyle{0}$$\scriptstyle{1}$$\scriptstyle{0}$$\scriptstyle{1}$

if $m$ is odd.

For the general case, we note that
$\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$ is a disjoint union of spheres
and different copies of $\mathbb{Z}$. It is therefore rather hard to compute
the equivariant twisted Rabinowitz–Floer homology directly via analysing the
critical manifold. By Theorem 3.3, there is a canonical isomorphism

$\operatorname{RFH}^{\varphi}_{*}(\mathbb{S}^{2n-1},\mathbb{C}^{n})\cong\operatorname{RFH}_{*}(\mathbb{S}^{2n-1},\mathbb{C}^{n})$

inducing a canonical isomorphism

$\overline{\operatorname{RFH}}^{\varphi}_{*}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m})\cong\operatorname{RFH}^{\mathbb{Z}_{m}}_{*}(\mathbb{S}^{2n-1},\mathbb{C}^{n}),$

where
$\operatorname{RFH}^{\mathbb{Z}_{m}}_{*}(\mathbb{S}^{2n-1},\mathbb{C}^{n})$
denotes the $\mathbb{Z}_{m}$-equivariant Rabinowitz–Floer homology constructed
in [AF12, p. 487]. A computation similar as before shows

$\operatorname{RFH}^{\mathbb{Z}_{m}}_{k}(\mathbb{S}^{2n-1},\mathbb{C}^{n})\cong\begin{cases}\mathbb{Z}_{2}&m\text{
even},\\\ 0&m\text{ odd},\end{cases}\qquad\forall k\in\mathbb{Z}.$

The crucial observation is, that
$\operatorname{Crit}\mathscr{A}^{H}\cong\mathbb{S}^{2n-1}\times\mathbb{Z}$. In
particular, the string of pearls looks like in Figure 11.

Lastly,
$\overline{\operatorname{RFH}}^{\varphi}_{k}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m})$
is generated by a noncontractible periodic Reeb orbit in
$\mathbb{S}^{2n-1}/\mathbb{Z}_{m}$ for all $k\in\mathbb{Z}$ by Lemma 1.1. ∎

Figure 11: The critical manifold
$\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$ together with the standard
height function.

###### Remark 4.1 (Coefficients).

As
$\overline{\operatorname{RFH}}^{\varphi}_{*}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m})$
vanishes for odd $m$, one should rather consider twisted Rabinowitz–Floer
homology with coefficients in $\mathbb{Z}$ in this case. Using the polyfold
approach, it might be possible to invoke [HWZ21, Chapter 6] to define coherent
orientations on the moduli spaces. However, Lagrangian Floer homology admits
an abstract polyfold description, but it is not always possible to define
coherent orientations.

Using Theorem 4.1 we can finally prove Theorem 1.1.

###### of Theorem 1.1.

By assumption, $\Sigma$ bounds a star-shaped domain $D$ with respect to the
origin. Thus $(D\cup\Sigma,\lambda)$ is a Liouville domain with $\lambda$
given by (10). By rescaling we may assume that $\mathbb{S}^{2n-1}\subseteq D$.
Define a smooth function

$\delta\colon\Sigma\to\mathinner{(-\infty,0)}$

by requiring $\delta(x)$ to be the unique number such that
$\phi^{X}_{\delta(x)}(x)\in\mathbb{S}^{2n-1}$, $x\in\Sigma$, where
$X\in\mathfrak{X}(\mathbb{C}^{n})$ denotes the Liouville vector field (11). We
claim that $\delta\circ\varphi=\delta$. Indeed, $\delta(\varphi(x))$ is the
unique number such that
$\phi^{X}_{\delta(\varphi(x))}(\varphi(x))\in\mathbb{S}^{2n-1}$. As the flow
of $X$ and $\varphi$ commute by the proof of Lemma 3.6, we conclude that
$\phi^{X}_{\delta(\varphi(x))}(x)\in\mathbb{S}^{2n-1}$. Define a smooth family
of star-shaped hypersurfaces $(\Sigma_{\sigma})_{\sigma\in I}$

$\Sigma_{\sigma}\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\bigl{\\{}\phi^{X}_{\sigma\delta(x)}(x)\mathrel{\mathop{\mathchar
58\relax}}x\in\Sigma\bigr{\\}}}\subseteq\mathbb{C}^{n}.$

Then we compute

$\displaystyle\varphi(\Sigma_{\sigma})$
$\displaystyle=\mathinner{\bigl{\\{}\varphi\mathinner{\bigl{(}\phi^{X}_{\sigma\delta(x)}(x)\bigr{)}}\mathrel{\mathop{\mathchar
58\relax}}x\in\Sigma\bigr{\\}}}$
$\displaystyle=\mathinner{\bigl{\\{}\phi^{X}_{\sigma\delta(x)}(\varphi(x))\mathrel{\mathop{\mathchar
58\relax}}x\in\Sigma\bigr{\\}}}$
$\displaystyle=\mathinner{\bigl{\\{}\phi^{X}_{\sigma\delta(\varphi(x))}(\varphi(x))\mathrel{\mathop{\mathchar
58\relax}}x\in\Sigma\bigr{\\}}}$
$\displaystyle=\mathinner{\bigl{\\{}\phi^{X}_{\sigma\delta(y)}(y)\mathrel{\mathop{\mathchar
58\relax}}y\in\varphi(\Sigma)\bigr{\\}}}$
$\displaystyle=\mathinner{\bigl{\\{}\phi^{X}_{\sigma\delta(y)}(y)\mathrel{\mathop{\mathchar
58\relax}}y\in\Sigma\bigr{\\}}}$ $\displaystyle=\Sigma_{\sigma}$

for all $\sigma\in I$ and therefore we can find a twisted homotopy
$(H_{\sigma})_{\sigma\in I}$ of Liouville domains in $\mathbb{C}^{n}$. By
Theorem 3.2 we have that

$\operatorname{RFH}^{\varphi}_{*}(\Sigma,\mathbb{C}^{n})\cong\operatorname{RFH}_{*}^{\varphi}(\mathbb{S}^{2n-1},\mathbb{C}^{n}),$

giving rise to a canonical isomorphism of the associated
$\mathbb{Z}_{m}$-equivariant twisted Rabinowitz–Floer homology

$\overline{\operatorname{RFH}}^{\varphi}_{*}(\Sigma/\mathbb{Z}_{m})\cong\overline{\operatorname{RFH}}^{\varphi}_{*}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m}).$

However, by Theorem 4.1 the latter does not vanish as $m\geq 2$ is even. ∎

Using Theorem 4.1 it is also possible to generalise [AF12, Theorem 1.2].
Define the set of $\varphi$-invariant Hamiltonian symplectomorphisms by

$\operatorname{Ham}^{\varphi}(\mathbb{C}^{n},d\lambda)\mathrel{\mathop{\mathchar
58\relax}}=\\{\varphi_{F}\in\operatorname{Ham}(\mathbb{C}^{n},d\lambda)\mathrel{\mathop{\mathchar
58\relax}}F_{t}(x)=F_{t}(\varphi(x))\>\forall(x,t)\in\mathbb{C}^{n}\times
I\\}.$

If $\varphi_{F}\in\operatorname{Ham}^{\varphi}(\mathbb{C}^{n},d\lambda)$, then
$\varphi\circ\varphi_{F}=\varphi_{F}\circ\varphi$. In particular
$0\in\operatorname{Fix}(\varphi_{F})$, and thus no element in
$\operatorname{Ham}^{\varphi}(\mathbb{C}^{n},d\lambda)$ can displace a star-
shaped hypersurface with respect to the origin in $\mathbb{C}^{n}$. We have
the following result.

###### Theorem 4.2.

Let $\Sigma\subseteq\mathbb{C}^{n}$ be a compact connected star-shaped
hypersurface invariant under the rotation $\varphi$. Every element in
$\operatorname{Ham}^{\varphi}(\mathbb{C}^{n},d\lambda)$ admits infinitely many
leaf-wise intersection points on $\Sigma$ or there does exist a leaf-wise
intersection point on a closed leaf.

###### Proof.

We reproduce the proof in [AF12] for completeness with minor modifications.
Let $\varphi_{F}\in\operatorname{Ham}_{c}^{\varphi}(\mathbb{C}^{n},d\lambda)$
and for $r\in\mathinner{[0,1]}$ consider the smooth family of perturbed
Rabinowitz action functionals

$\mathscr{A}_{r}\colon\mathscr{L}\mathbb{C}^{n}\times\mathbb{R}\to\mathbb{R}$

defined by

$\mathscr{A}_{r}(\gamma,\tau)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}\gamma^{*}\lambda-\tau\int_{0}^{1}H_{r}(\gamma(t))dt-r\int_{0}^{1}F_{t}(\gamma(t))dt,$

where $(H_{r})_{r\in I}$ is a twisted homotopy of Liouville domains from
$\mathbb{S}^{2n-1}$ to $\Sigma$. Clearly, every $\mathscr{A}_{r}$ is
$\varphi$-invariant. As in [AF10a, Definition 5.1], we define the spectral
value $\sigma([\xi])$ of a homology class
$[\xi]\in\operatorname{RFH}^{\mathbb{Z}_{m}}_{\ast}(\mathbb{S}^{2n-1},\mathbb{C}^{n})$
by

$\sigma([\xi])\mathrel{\mathop{\mathchar
58\relax}}=\inf_{\eta\in[\xi]}\max_{\eta_{(\gamma,\tau)}\neq
0}\mathscr{A}_{0}(\gamma,\tau)\in\mathbb{R}\cup\\{-\infty\\}.$

Moreover, we define the set

$\mathfrak{S}\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\\{\sigma([\xi])\mathrel{\mathop{\mathchar
58\relax}}[\xi]\in\operatorname{RFH}^{\mathbb{Z}_{m}}_{\ast}(\mathbb{S}^{2n-1},\mathbb{C}^{n})\\}}.$

By Theorem 4.1, we conclude that $\mathfrak{S}=2\pi\mathbb{Z}$. Hence
$\mathscr{A}_{0}$ has critical values of arbitrarily large critical value and
so does $\mathscr{A}_{1}$ by [AF10a, Corollary 5.14]. Thus $\mathscr{A}_{1}$
has infinitely many critical points which give rise to leaf-wise intersection
points by Lemma 3.10. The map

$\operatorname{Crit}\mathscr{A}_{1}\to\mathinner{\\{\text{leaf-wise
intersection points}\\}}$

is injective unless there exists a leaf-wise intersection point on a closed
leaf. For the general case, use cut-off functions. ∎

###  4.2 A Forcing Theorem for Twisted Periodic Reeb Orbits

###### Definition 4.1 (Twisted Stable Hypersurface).

Let $(\Sigma,\omega|_{\Sigma},\lambda)$ be a stable hypersurface in a
connected symplectic manifold $(M,\omega)$ and
$\varphi\in\operatorname{Symp}(M,\omega)$. We say that $\Sigma$ is twisted by
$\varphi$, if $\varphi(\Sigma)=\Sigma$, $\varphi$ is of finite order and
$\varphi^{*}\lambda=\lambda$.

###### Example 4.1.

Consider the stable hypersurface
$\Sigma_{c}\subseteq(T^{*}\mathbb{T}^{n},\omega_{\sigma},H)$ for $c>0$ as in
Example 2.7. Let $\varphi\in\operatorname{Diff}(\mathbb{T}^{n})$ be an
isometry of finite order such that

$D\varphi\circ J=J\circ D\varphi$ (33)

holds and consider the cotangent lift (2.6)

$D\varphi^{\dagger}\colon\mathbb{T}^{n}\times\mathbb{R}^{n}\to\mathbb{T}^{n}\times\mathbb{R}^{n},\qquad
D\varphi^{\dagger}(q,p)=\mathinner{\bigl{(}\varphi(q),\mathinner{\bigl{(}D\varphi^{-1}(q)\bigr{)}}^{t}p\bigr{)}}.$

Then clearly $\varphi(\Sigma_{c})=\Sigma_{c}$ as $\varphi$ is an isometry and
$D\varphi^{\dagger}$ is of finite order as $\varphi$ is. Moreover,
$D\varphi^{\dagger}\in\operatorname{Symp}(T^{*}\mathbb{T}^{n},\omega_{\sigma})$,
as $D\varphi^{\dagger}\in\operatorname{Symp}(T^{*}\mathbb{T}^{n},\omega_{0})$
by Proposition 2.1 and $D\varphi^{\dagger}$ preserves $\sigma$ by assumption
(33). Lastly, we have that $\varphi^{*}\lambda=\lambda$ as one sees by
considering the formula (8) together with assumption (33).

Let $(\Sigma,\omega|_{\Sigma},\lambda)$ be a twisted stable hypersurface for
$\varphi\in\operatorname{Symp}(M,\omega)$ in a connected symplectically
aspherical symplectic manifold $(M,\omega)$, that is,
$[\omega]|_{\pi_{2}(M)}=0$. As $\varphi$ is of finite order by assumption, we
can define the set of twisted contractible loops, written
$\Lambda_{\varphi}M\subseteq\Lambda M$, as follows. We say that a contractible
free loop $v\in\Lambda M$ is in $\Lambda_{\varphi}M$, if there exists
$\gamma\in\mathscr{L}_{\varphi}M$ such that

$v(t)=\gamma(mt)\qquad\forall t\in\mathbb{T},$

where $m\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{ord}\varphi$. Then
we can define a generalisation of the twisted Rabinowitz action functional

$\mathscr{A}^{H}_{\varphi}\colon\Lambda_{\varphi}M\times\mathbb{R}\to\mathbb{R},\quad\mathscr{A}^{H}_{\varphi}(v,\tau)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{m}\int_{\mathbb{D}}\overline{v}^{*}\omega-\tau\int_{0}^{1}H(v(t))dt,$
(34)

where $\overline{v}\in C^{\infty}(\mathbb{D},M)$ is a filling of $v$ and $H$
is any twisted defining Hamiltonian function for $\Sigma$. Then
$(v,\tau)\in\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$ if and only if
$(\gamma,\tau)\in\mathscr{L}_{\varphi}\Sigma$ solves

$\dot{\gamma}(t)=\tau R(\gamma(t))\qquad\forall t\in\mathbb{R},$

where $R\in\mathfrak{X}(\Sigma)$ denotes the stable Reeb vector field 2.11. We
call the projection of the set of critical points of
$\mathscr{A}^{H}_{\varphi}$ to $\Lambda_{\varphi}M$ contractible twisted
closed characteristics and denote it by $\mathscr{C}_{\varphi}(\Sigma)$.
Define a function, called the $\omega$-energy, by

$\Omega\colon\mathscr{C}_{\varphi}(\Sigma)\to\mathbb{R},\qquad\Omega(v)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{m}\int_{\mathbb{D}}\overline{v}^{*}\omega.$

It follows that $\Omega(v)=\mathscr{A}^{H}_{\varphi}(v,\tau)$.

###### Example 4.2.

Consider the twisted stable hypersurface
$\Sigma_{c}\subseteq(T^{*}\mathbb{T}^{n},\omega_{\sigma},H)$ as in Example
4.1. By adapting Example 2.8, we have that $(q,p)\in\Sigma_{k}$ gives rise to
a contractible twisted closed characteristic if and only if

$\int_{0}^{\tau}e^{sJ}pds+q=\varphi(q),\quad e^{\tau
J}p=\mathinner{\bigl{(}D\varphi^{-1}(q)\bigr{)}}^{t}p,\quad\text{and}\quad\mathinner{\lVert
p\rVert}^{2}=2c.$

A computation similar to [CFP10, p. 1843] shows

$\Omega\colon\mathscr{C}_{\varphi}(\Sigma_{c})\to\mathbb{R},\qquad\Omega(v)=c\tau.$

In order to state the main result of this section, we need two additional
preliminary definitions.

###### Definition 4.2 (Morse–Bott Component, [AF10, p. 86]).

Let $M$ be a smooth manifold and $f\in C^{\infty}(M)$. A subset
$C\subseteq\operatorname{Crit}f$ is called a Morse–Bott component, if

  1. (i)

$C$ is a connected embedded submanifold of $M$.

  2. (ii)

$T_{x}C=\ker\operatorname{Hess}f(x)$ for all $x\in C$.

###### Example 4.3 ([AF10, Lemma 2.12]).

In the setting of Proposition 3.2, any connected component of
$\operatorname{Fix}(\varphi|_{\Sigma})\subseteq\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$
is a Morse–Bott component. Indeed, we have that

$\ker\operatorname{Hess}\mathscr{A}^{H}_{\varphi}|_{(x,0)}\cong\ker(D\varphi_{x}-\operatorname{id}_{T_{x}\Sigma})=T_{x}\operatorname{Fix}(\varphi|_{\Sigma})$

for all $x\in\operatorname{Fix}(\varphi|_{\Sigma})$.

###### Definition 4.3 ([CFP10, p. 1768]).

A symplectic manifold $(M,\omega)$ is called geometrically bounded, if there
exists an $\omega$-compatible almost complex structure $J$ and a complete
Riemannian metric such that the following conditions hold.

  1. (i)

There are constants $C_{0},C_{1}>0$ with

$\omega(Jv,v)\geq C_{0}\mathinner{\lVert
v\rVert}^{2}\qquad\text{and}\qquad\mathinner{\lvert\omega(u,v)\rvert}\leq
C_{1}\mathinner{\lVert u\rVert}\mathinner{\lVert v\rVert}$

for all $u,v\in T_{x}M$ and $x\in M$.

  2. (ii)

The sectional curvature of the metric is bounded above, and its injectivity
radius is bounded away from zero.

###### Example 4.4 ([CFP10, p. 1768]).

Twisted cotangent bundles are geometrically bounded.

###### Theorem 4.3 (Forcing).

Let $\Sigma$ be a twisted stable displaceable hypersurface in a symplectically
aspherical, geometrically bounded, symplectic manifold $(M,\omega)$ for some
$\varphi\in\operatorname{Symp}(M,\omega)$ and suppose that
$v^{-}\in\mathscr{C}_{\varphi}(\Sigma)$ belongs to a Morse–Bott component $C$
of the twisted Rabinowitz action functional (34). Then there exists a
contractible twisted closed characteristic
$v\in\mathscr{C}_{\varphi}(\Sigma)\setminus C$ such that

$\Omega(v)-\Omega(v^{-})\leq e(\Sigma).$

###### Corollary 4.1.

Let $\Sigma$ be a twisted stable displaceable hypersurface in a symplectically
aspherical, geometrically bounded, symplectic manifold $(M,\omega)$ for some
symplectomorphism $\varphi\in\operatorname{Symp}(M,\omega)$. If
$\operatorname{Fix}(\varphi|_{\Sigma})\neq\varnothing$, then there exists a
contractible twisted closed characteristic
$v\in\mathscr{C}_{\varphi}(\Sigma)\setminus\operatorname{Fix}(\varphi|_{\Sigma})$
such that

$\Omega(v)\leq e(\Sigma).$

###### Corollary 4.2 ([MS17, Theorem 12.3.4], [HZ94, p. 171]).

We have that

$e(\overline{B}_{r}^{2n}(0))=\pi r^{2}\qquad\forall r>0,$

where $\overline{B}^{2n}_{r}(0)\subseteq\mathbb{R}^{2n}$ denotes the closed
ball around the origin of radius $r$.

###### Proof.

By monotonicity and [MS17, Exercise 12.3.7] we have that

$e(\partial\overline{B}^{2n}_{r}(0))\leq e(\overline{B}^{2n}_{r}(0))\leq\pi
r^{2}\qquad\forall r>0.$

The Reeb flow on $\partial\overline{B}^{2n}_{r}(0)$ is given by

$\phi^{R_{r}}_{t}(z)=e^{-2it/r^{2}}z\qquad\forall
z\in\partial\overline{B}^{2n}_{r}(0).$

Hence the parametrised periodic Reeb orbits are $(\phi^{R_{r}}(z),\tau)$ with
$\tau\in\pi r^{2}\mathbb{Z}$. But Corollary 4.1 implies the existence of a
nonconstant closed characteristic $v$ on the hypersurface
$\partial\overline{B}^{2n}_{r}(0)$ such that

$0<\tau=\Omega(v)\leq e(\partial\overline{B}^{2n}_{r}(0))\leq\pi r^{2}.$

This is only possible for $\tau=\pi r^{2}$ and the statement follows. ∎

###### Proof of Theorem 4.3.

This proof uses a method called a “homotopy of homotopies argument”. Fix
$\varepsilon>0$ and choose a Hamiltonian function $F\in C^{\infty}_{c}(M\times
I)$ satisfying

$\mathinner{\\!\left\lVert
F\right\rVert}<e(\Sigma)+\varepsilon\qquad\text{and}\qquad\varphi_{F}(\Sigma)\cap\Sigma=\varnothing.$

For an appropriate twisted defining Hamiltonian function $H$ for $\Sigma$ we
denote by $\mathfrak{M}$ the associated twisted Moser pair. The actual
construction of $H$ is very cumbersome and is carried out in [CFP10]. The
crucial observation here is that [CFP10, Proposition 2.6] gives a
$\varphi$-invariant stable tubular neighbourhood of $\Sigma$ as
$\varphi^{*}\lambda=\lambda$ by invoking the equivariant Darboux–Weinstein
Theorem [VG84, Theorem 22.1]. Moreover, we choose a smooth family
$(\beta_{r})_{r\in\mathinner{[0,+\infty)}}$ of cutoff functions $\beta_{r}\in
C^{\infty}(\mathbb{R},I)$ such that

$\begin{cases}\beta_{r}(s)=0&\mathinner{\lvert s\rvert}\geq r,\\\
\beta_{r}(s)=1&\mathinner{\lvert s\rvert}\leq r-1,\\\
s\beta^{\prime}_{r}(s)\leq 0&\forall s\in\mathbb{R},\end{cases}$

for all $r\in\mathinner{[0,+\infty)}$. Define a family of twisted Rabinowitz
action functionals

$\mathscr{A}_{r}\colon\Lambda_{\varphi}M\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$

by

$\mathscr{A}_{r}(v,\tau,s)\mathrel{\mathop{\mathchar
58\relax}}=\mathscr{A}^{H}_{\varphi}(v,\tau)-\beta_{r}(s)\int_{0}^{1}F_{t}(v(t))dt$

for all $r\in\mathinner{[0,+\infty)}$. Note that
$\mathscr{A}_{0}=\mathscr{A}^{H}_{\varphi}$. For a suitable
$\varphi$-invariant $\omega$-compatible almost complex structure we consider
the moduli space

$\mathscr{M}\mathrel{\mathop{\mathchar 58\relax}}=\mathinner{\\{(u,\tau,r)\in
C^{\infty}(\mathbb{R},\mathscr{L}_{\varphi}M\times\mathbb{R})\times\mathinner{[0,+\infty)}\mathrel{\mathop{\mathchar
58\relax}}(u,\tau,r)\text{ solution of \eqref{eq:moduli_space}}\\}},$

where

$\begin{cases}\partial_{s}(u,\tau)=\operatorname{grad}\mathscr{A}_{r}|_{(u(s),\tau(s),s)}&\forall
s\in\mathbb{R},\\\
\displaystyle\lim_{s\to-\infty}(u(s),\tau(s))=(v^{-},\tau^{-}),\\\
\displaystyle\lim_{s\to+\infty}(u(s),\tau(s))\in C.\end{cases}$ (35)

Note that always $(v^{-},\tau^{-},0)\in\mathscr{M}$. The proof is now based on
the following observation. If

$\Omega(v)>\mathinner{\lVert F\rVert}+\Omega(v^{-})\qquad\forall
v\in\mathscr{C}_{\varphi}(\Sigma)\setminus C$ (36)

holds, then $\mathscr{M}$ is compact. This is absurd. Indeed, the moduli space
$\mathscr{M}$ is the zero level set of a Fredholm section of a bundle over a
Banach manifold. As $v^{-}$ belongs to a Morse–Bott component, the Fredholm
section is regular at the point $v^{-}$, that is, the linearisation of the
gradient flow equation is surjective there. By compactness, we can therefore
perturb the Fredholm section to make it transverse. Hence $\mathscr{M}$ is a
compact smooth manifold with boundary consisting precisely of the point
$v^{-}$. There do not exist such manifolds. Thus we conclude that there exists
$v\in\mathscr{C}_{\varphi}(\Sigma)\setminus C$ such that

$\Omega(v)-\Omega(v^{-})\leq\mathinner{\lVert F\rVert}<e(\Sigma)+\varepsilon.$

As $\varepsilon>0$ was arbitrary, the statement follows. We prove the
compactness of $\mathscr{M}$ under assumption (36) in four steps.

_Step 1: If $(u,\tau,r)\in\mathscr{M}$, then
$E(u,\tau)\leq\mathinner{\\!\left\lVert F\right\rVert}$._ We estimate

$\displaystyle E(u,\tau)$
$\displaystyle=\int_{-\infty}^{+\infty}\mathinner{\lVert\partial_{s}(u,\tau)\rVert}^{2}ds$
$\displaystyle=\int_{-\infty}^{+\infty}d\mathscr{A}_{r}(\partial_{s}(u,\tau),s)ds$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{d}{ds}\mathscr{A}_{r}(u,\tau,s)ds-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=\lim_{s\to+\infty}\mathscr{A}_{r}(u,\tau,s)-\lim_{s\to-\infty}\mathscr{A}_{r}(u,\tau,s)-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=\mathscr{A}_{0}(v^{+},\tau^{+})-\mathscr{A}_{0}(v^{-},\tau^{-})-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=\int_{-\infty}^{+\infty}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u(s,t))dtds$
$\displaystyle\leq\mathinner{\lVert
F\rVert}_{+}\int_{-\infty}^{0}\dot{\beta}_{r}(s)ds-\mathinner{\lVert
F\rVert}_{-}\int_{0}^{+\infty}\dot{\beta}_{r}(s)ds$
$\displaystyle=\beta_{r}(0)(\mathinner{\lVert F\rVert}_{-}+\mathinner{\lVert
F\rVert}_{+})$ $\displaystyle=\beta_{r}(0)\mathinner{\lVert F\rVert}$
$\displaystyle\leq\mathinner{\lVert F\rVert},$

as $\mathscr{A}_{0}(v^{+},\tau^{+})=\mathscr{A}_{0}(v^{-},\tau^{-})$ since $C$
is connected.

_Step 2: There exists $r_{0}\in\mathbb{R}$ such that $r\leq r_{0}$ for all
$(u,\tau,r)\in\mathscr{M}$._ Crucial is the existence of a constant $\delta>0$
such that

$\mathinner{\lVert\operatorname{grad}\mathscr{A}_{r}|_{(v,\tau,s)}\rVert}\geq\delta\qquad\forall(v,\tau,s)\in\Lambda_{\varphi}M\times\mathbb{R}\times\mathbb{R}.$

This is proven along the lines of [CF09, Lemma 3.9]. With the above inequality
and Step 1 we estimate

$\mathinner{\\!\left\lVert F\right\rVert}\geq
E(u,\tau)\geq\int_{-r}^{r}\mathinner{\lVert\operatorname{grad}\mathscr{A}_{r}|_{(u(s),\tau(s),s)}\rVert}^{2}ds\geq
2r\delta^{2},$

and thus we can set

$r_{0}\mathrel{\mathop{\mathchar 58\relax}}=\frac{\mathinner{\\!\left\lVert
F\right\rVert}}{2\delta^{2}}.$

_Step 3: There exists $C>0$ such that
$\mathinner{\\!\left\lVert\tau\right\rVert}_{\infty}\leq C$ for all
$(u,\tau,r)\in\mathscr{M}$._ This is a delicate estimate based on the
construction of the defining Hamiltonian $H$ for $\Sigma$ as well as an
extension of the stablising form and proceeds as in [CFP10]. Particularly
crucial is [CFP10, Proposition 4.1].

_Step 4: If ( 36) holds, then $\mathscr{M}$ is compact._ Let
$(u_{k},\tau_{k},r_{k})$ be a sequence in the moduli space $\mathscr{M}$. By
Step 2 and Step 3, the sequences $(r_{k})$ and $(\tau_{k})$ are uniformly
bounded. Thus $(u_{k},\tau_{k},r_{k})$ admits a
$C^{\infty}_{\operatorname{loc}}$-convergent subsequence by standard
arguments. Indeed, the uniform $L^{\infty}$-bound on the sequence $(u_{k})$
follows from the assumption that $(M,\omega)$ is geometrically bounded and the
uniform $L^{\infty}$-bound on the derivatives $(Du_{k})$ follows from
Corollary D.1 by the assumption that $(M,\omega)$ is symplectically
aspherical. Denote the limit of this subsequence by $(u,\tau,r)$. This limit
clearly satisfies the first equation in (35), thus one only needs to check the
asymptotic conditions in (35). Again by compactness, $(u,\tau)$ converges to
critical points $(w^{\pm},\tau^{\pm})$ of $\mathscr{A}_{0}$ at its asymptotic
ends. We claim that

$\mathscr{A}_{r}(u(s),\tau(s),s)\in\mathinner{[-\mathinner{\lVert
F\rVert}+\Omega(v^{-}),\mathinner{\lVert F\rVert}+\Omega(v^{-})]}\qquad\forall
s\in\mathbb{R}.$ (37)

In particular, $\Omega(w^{\pm})\in\mathinner{[-\mathinner{\lVert
F\rVert}+\Omega(v^{-}),\mathinner{\lVert F\rVert}+\Omega(v^{-})]}$. So if (37)
holds, then by assumption (36) we conclude $(w^{\pm},\tau^{\pm})\in C$ and
$\mathscr{M}$ is indeed compact. It remains to prove (37). It is enough to
prove

$\mathscr{A}_{r}(u_{k}(s),\tau_{k}(s),s)\in\mathinner{[-\mathinner{\lVert
F\rVert}+\Omega(v^{-}),\mathinner{\lVert F\rVert}+\Omega(v^{-})]}\qquad\forall
s\in\mathbb{R}$

for every $k\in\mathbb{N}$. As in the proof of [AF10, Lemma 2.8] we estimate

$\displaystyle 0$
$\displaystyle\leq\int^{+\infty}_{s_{0}}d\mathscr{A}_{r}(\partial_{s}(u_{k},\tau_{k}),s)ds$
$\displaystyle=\int^{+\infty}_{s_{0}}\frac{d}{ds}\mathscr{A}_{r}(u_{k},\tau_{k},s)ds-\int^{+\infty}_{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\lim_{s\to+\infty}\mathscr{A}_{r}(u_{k},\tau_{k},s)-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})-\int^{+\infty}_{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\mathscr{A}_{0}(v^{+},\tau^{+})-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\int^{+\infty}_{s_{0}}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u_{k}(s,t))dtds$
$\displaystyle\leq\mathscr{A}_{0}(v^{+},\tau^{+})-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\int^{+\infty}_{-\infty}\mathinner{\lVert\dot{\beta}_{r}(s)F\rVert}_{+}ds$
$\displaystyle\leq\mathscr{A}_{0}(v^{+},\tau^{+})-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\mathinner{\lVert
F\rVert}$
$\displaystyle=\Omega(v^{-})-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\mathinner{\lVert
F\rVert}$

for all $s_{0}\in\mathbb{R}$. Similarly, we compute

$\displaystyle 0$
$\displaystyle\leq\int_{-\infty}^{s_{0}}d\mathscr{A}_{r}(\partial_{s}(u_{k},\tau_{k}),s)ds$
$\displaystyle=\int_{-\infty}^{s_{0}}\frac{d}{ds}\mathscr{A}_{r}(u_{k},\tau_{k},s)ds-\int_{-\infty}^{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})-\lim_{s\to-\infty}\mathscr{A}_{r}(u_{k},\tau_{k},s)-\int_{-\infty}^{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})-\mathscr{A}_{0}(v^{-},\tau^{-})+\int_{-\infty}^{s_{0}}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u_{k}(s,t))dtds$

and thus we estimate

$\displaystyle\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})$
$\displaystyle\geq\mathscr{A}_{0}(v^{-},\tau^{-})-\int_{-\infty}^{s_{0}}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u_{k}(s,t))dtds$
$\displaystyle\geq\mathscr{A}_{0}(v^{-},\tau^{-})-\int_{-\infty}^{+\infty}\mathinner{\lVert\dot{\beta}(s)F\rVert}_{+}ds$
$\displaystyle\geq\Omega(v^{-})-\mathinner{\lVert F\rVert}.$

This completes the proof of the Forcing Theorem 4.3. ∎

We conlcude this section by applying the Forcing Theorem 4.3 to a displaceable
twisted stable hypersurface.

###### Example 4.5.

Consider the isometry

$\varphi\colon\mathbb{T}^{2}\times\mathbb{R}^{2}\to\mathbb{T}^{2}\times\mathbb{R}^{2},\qquad\varphi(q_{1},q_{2})\mathrel{\mathop{\mathchar
58\relax}}=(q_{2},-q_{1})$

and its cotangent lift

$D\varphi^{\dagger}\colon\mathbb{T}^{2}\times\mathbb{R}^{2}\to\mathbb{T}^{2}\times\mathbb{R}^{2},\quad
D\varphi^{\dagger}(q_{1},q_{2},p_{1},p_{2})=(q_{2},-q_{1},p_{2},-p_{1}).$

Then $\Sigma_{c}\subseteq(T^{*}\mathbb{T}^{2},\omega_{\sigma},H)$ is a
displaceable twisted stable hypersurface for the area form
$\sigma=dq_{1}\wedge dq_{2}$ by Example 4.1. By Example 4.2, we have that

$v\colon\mathbb{R}\to\Sigma_{c},\qquad v(t)\mathrel{\mathop{\mathchar
58\relax}}=\sqrt{2c}(\sin t,\cos t,\cos t,-\sin t)$

is a $\tau$-periodic twisted characteristic for all periods $\tau\in
2\pi\mathbb{Z}+\frac{\pi}{2}$ and $c>0$. Thus if we choose
$v^{-}\in\mathscr{C}_{\varphi}(\Sigma_{c})$ of period $\tau>0$, then we
compute for $v\in\mathscr{C}_{\varphi}(\Sigma_{c})$ of period $\tau+2\pi$

$\Omega(v)-\Omega(v^{-})=c(\tau+2\pi)-c\tau=2\pi c=e(\Sigma_{c})$

by Example 4.2 and Example 2.16. Hence, we have verified the statement of the
Forcing Theorem 4.3 for the displaceable twisted stable hypersurface
$\Sigma_{c}$ in the symplectically aspherical and geometrically bounded
symplectic manifold $(T^{*}\mathbb{T}^{2},\omega_{\sigma})$. Indeed,
$(T^{*}\mathbb{T}^{2},\omega_{\sigma})$ is geometrically bounded by Example
4.4, and symplectically aspherical as

$\pi_{2}(T^{*}\mathbb{T}^{n})\cong\pi_{2}(\mathbb{T}^{n})\times\pi_{2}(\mathbb{R}^{n})\cong
0.$

##  5 Further Steps in Twisted Rabinowitz–Floer Homology

In this final chapter we discuss some possible further research in twisted
Rabinowitz–Floer homology for future work. One can of course try to find a
twisted version of every result provided by standard Rabinowitz–Floer
homology. Following the survey article [AF12a], major results relate
Rabinowitz–Floer homology to symplectic homology.

In the first section we briefly outline a further computation of twisted
Rabinowitz–Floer homology, where the hypersurface is not displaceable.

In the second section, we explain an important physical setting where the
Forcing Theorem 4.3 and Theorem 1.1 might be applicable.

###  5.1 Cotangent Bundles

Let $(M,g)$ be a compact connected Riemannian manifold and let $(S^{*}M,pdq)$
be the spherisation of $M$ as in Example 2.11. By [AS09] or [CFO10, Theorem
1.10], we have

$\operatorname{RFH}_{k}(S^{*}M,T^{*}M)\cong\begin{cases}\operatorname{H}^{-k+1}(\mathscr{L}M)&k<0,\\\
\operatorname{H}_{k}(\mathscr{L}M)&k>1.\end{cases}$

In the degrees $k=0,1$ the answer is known and depends on the Euler class. The
proof uses a relation between Rabinowitz–Floer homology and symplectic
homology, respectively symplectic cohomology. If
$\varphi\in\operatorname{Aut}(D^{*}M,pdq|_{D^{*}M})$ is a Liouville
automorphism, then it is plausible to expect

$\operatorname{RFH}_{k}^{\varphi}(S^{*}M,T^{*}M)\cong\begin{cases}\operatorname{H}^{-k+1}(\mathscr{L}_{\varphi}M)&k<0,\\\
\operatorname{H}_{k}(\mathscr{L}_{\varphi}M)&k>1.\end{cases}$

However, it would be also interesting to study the loop space homology
$\operatorname{H}_{\ast}(\mathscr{L}_{\varphi}M)$ itself, because usually one
computes the free loop space homology via Morse theory. For details see [LO15,
Chapter 2].

###  5.2 Stark–Zeeman Systems

Following [CFZ19] we introduce Stark–Zeeman systems. For $\mu_{\pm}>0$ define
the potential functions

$V_{\pm}\colon\mathbb{C}\setminus\\{\pm 1\\}\to\mathbb{R},\qquad
V_{\pm}(z)\mathrel{\mathop{\mathchar
58\relax}}=-\frac{\mu_{\pm}}{\mathinner{\lvert z\pm 1\rvert}}.$

Let $U_{0}\subseteq\mathbb{C}$ be open and star-shaped with respect to the
origin such that $\pm 1\in U_{0}$. Choose $V_{0}\in C^{\infty}(U_{0})$.
Moreover, set

$V\mathrel{\mathop{\mathchar 58\relax}}=V_{+}+V_{-}+V_{0}\in C^{\infty}(U)$

for $U\mathrel{\mathop{\mathchar 58\relax}}=U_{0}\setminus\\{\pm 1\\}$. For a
function $B\in C^{\infty}(U_{0})$, let $\sigma_{B}\mathrel{\mathop{\mathchar
58\relax}}=Bdq_{1}\wedge dq_{2}$ and abbreviate by $(T^{*}U,\omega_{B})$ the
associated magnetic cotangent bundle. Fix a Riemannian metric $g$ on $U_{0}$
which is conformal to the standard metric. A Stark–Zeeman system is the
magnetic Hamiltonian system $(T^{*}U,\omega_{B},H)$ with

$H(q,p)\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{2}\mathinner{\lVert
p\rVert}^{2}_{g^{*}}+V(q)\qquad\forall(q,p)\in T^{*}U.$

For $c\in\mathbb{R}$ a regular value of $H$, we consider a connected component
$\Sigma_{c}\subseteq H^{-1}(c)$ such that $\mathfrak{K}_{c}\cup\\{\pm 1\\}$ is
bounded and simply connected, where the Hill’s region $\mathfrak{K}_{c}$ is
defined by

$\mathfrak{K}_{c}\mathrel{\mathop{\mathchar
58\relax}}=\pi(\Sigma_{c})\subseteq\\{q\in U\mathrel{\mathop{\mathchar
58\relax}}V(q)\leq c\\}.$

For example, the planar circular restricted three-body problem is a
Stark–Zeeman system. In order to deal with collisions, we regularise
$\Sigma_{c}$.

###### Definition 5.1 (Regularisation, [FK18, p. 48]).

Let $(\Sigma,\omega)$ be a noncompact Hamiltonian manifold. A regularisation
of $(\Sigma,\omega)$ is defined to be a compact Hamiltonian manifold
$(\overline{\Sigma},\overline{\omega})$ such that there exists an embedding
$\iota\colon\Sigma\hookrightarrow\overline{\Sigma}$ with
$\iota^{*}\overline{\omega}=\omega$.

Consider the Birkhoff regularisation map

$\varphi\colon\mathbb{C}^{*}\to\mathbb{C},\qquad
B(z)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\mathinner{\biggl{(}z+\frac{1}{z}\biggr{)}}.$

By Example 2.4, the cotangent lift $D\varphi^{\dagger}$ of $\varphi$ is given
by

$D\varphi^{\dagger}\colon T^{*}\mathbb{C}^{*}\to T^{*}\mathbb{C},\qquad
D\varphi^{\dagger}(z,w)=\mathinner{\biggl{(}\frac{z^{2}+1}{2z},\frac{2\overline{z}^{2}w}{\overline{z}^{2}-1}\biggr{)}}=(q,p).$

The regular energy surface $\Sigma_{c}$ gives rise to a regular energy surface
$\Sigma^{B}_{c}\subseteq K^{-1}(0)$, where the rescaled Hamiltonian
$K\mathrel{\mathop{\mathchar 58\relax}}=H\circ D\varphi^{\dagger}$ is given by

$K(z,w)=\frac{\mathinner{\lVert
w\rVert}^{2}_{g^{*}}}{2}-\frac{\mu_{+}\mathinner{\lvert
z+1\rvert}^{2}}{2\mathinner{\lvert
z\rvert}^{3}}-\frac{\mu_{-}\mathinner{\lvert
z-1\rvert}^{2}}{2\mathinner{\lvert
z\rvert}^{3}}+\frac{(V_{0}(q)-c)\mathinner{\lvert
z^{2}-1\rvert}^{2}}{4\mathinner{\lvert z\rvert}^{4}}.$

The compact regular energy surface $\Sigma^{B}_{c}$ is called the Birkhoff
regularisaton of $\Sigma_{c}$. This regularisation is invariant under the
cotangent lift

$\Phi\colon T^{*}\mathbb{C}^{*}\to
T^{*}\mathbb{C}^{*},\qquad\Phi(z,w)\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\biggl{(}\frac{1}{z},-\overline{z}^{2}w\biggr{)}}$

As the induced action of $\Phi$ on $\Sigma_{c}^{B}$ is free, we obtain the
cover

$\Sigma^{B}_{c}\to\Sigma^{B}_{c}/\mathbb{Z}_{2}.$

Explicitly, there exist diffeomorphisms such that

$\Sigma^{B}_{c}\cong\mathbb{S}^{1}\times\mathbb{S}^{2}\qquad\text{and}\qquad\Sigma^{B}_{c}/\mathbb{Z}_{2}\cong\mathbb{RP}^{3}\\#\mathbb{RP}^{3}.$

Therefore, it may be possible to apply the ideas developed in the proof of
Theorem 1.1 or the Forcing Theorem 4.3 to these hypersurfaces. The analysis of
these hypersurfaces is already quite delicate in the special case of the
planar circular restricted three-body problem. Indeed, it requires some work
to show that the regularised energy hypersurface is fibrewise star-shaped for
energy values below the first critical value. For details, see [FK18, Theorem
5.7.2]. Hence we cannot expect stability of the hypersurfaces in a general
Stark–Zeeman system.

## Acknowledgements

First of all I would like to thank my supervisor Urs Frauenfelder for his
inspiring guidance. I owe thanks to Kai Cieliebak and Igor Uljarevic for many
helpful discussions. Moreover, I cordially thank Felix Schlenk for the
pleasant stay at the University of Neuchâtel in April 2022. I also thank Will
J. Merry for his constant encouragement during my Master’s thesis as well as
the doctoral students of the Mathematics institute at the university of
Augsburg for enduring my numerous talks. Lastly, I thank Springer National for
providing a freely available template for monographs. Without it, this thesis
would not be what it is. This thesis is dedicated to my family, most
importantly, my two loves Jil and Neil Kern.

##  Appendix A Twisted Loops in Universal Covering Manifolds

In this Appendix, we will consider the category of topological manifolds
rather than the category of smooth manifolds, because smoothness does not add
much to the discussion. Free and based loop spaces are fundamental objects in
Algebraic Topology, for a vast treatment of the geometry and topology of based
as well as free loop spaces see for example [LO15]. But so-called twisted loop
spaces are not considered that much.

###### Theorem A.1 (Twisted Loops in Universal Covering Manifolds).

Let $(M,x)$ be a connected pointed topological manifold and
$\pi\colon\tilde{M}\to M$ the universal covering.

  1. (a)

Fix $[\eta]\in\pi_{1}(M,x)$ and denote by $U_{\eta}\subseteq\mathscr{L}(M,x)$
the path component corresponding to $[\eta]$ via the bijection
$\pi_{0}(\mathscr{L}(M,x))\cong\pi_{1}(M,x)$. For every
$e,e^{\prime}\in\pi^{-1}(x)$ and
$\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})$ such that
$\varphi(e)=\tilde{\eta}_{e}(1)$, where $\tilde{\eta}_{e}$ denotes the unique
lift of $\eta$ with $\tilde{\eta}_{e}(0)=e$, we have a commutative diagram of
homeomorphisms

${\mathscr{L}_{\varphi}(\tilde{M},e)}$${\mathscr{L}_{\psi\circ\varphi\circ\psi^{-1}}(\tilde{M},e^{\prime})}$${U_{\eta},}$$\scriptstyle{L_{\psi}}$$\scriptstyle{\Psi_{e}}$$\scriptstyle{\Psi_{e^{\prime}}}$
(38)

where $\psi\in\operatorname{Aut}_{\pi}(\tilde{M})$ is such that
$\psi(e)=e^{\prime}$,

$\qquad
L_{\psi}\colon\mathscr{L}_{\varphi}(\tilde{M},e)\to\mathscr{L}_{\psi\circ\varphi\circ\psi^{-1}}(\tilde{M},e^{\prime}),\qquad
L_{\psi}(\gamma)\mathrel{\mathop{\mathchar 58\relax}}=\psi\circ\gamma,$

and

$\displaystyle\Psi_{e}\colon U_{\eta}\to\mathscr{L}_{\varphi}(\tilde{M},e),$
$\displaystyle\Psi_{e}(\gamma)\mathrel{\mathop{\mathchar
58\relax}}=\tilde{\gamma}_{e},$ $\displaystyle\Psi_{e^{\prime}}\colon
U_{\eta}\to\mathscr{L}_{\psi\circ\varphi\circ\psi^{-1}}(\tilde{M},e^{\prime}),$
$\displaystyle\Psi_{e^{\prime}}(\gamma)\mathrel{\mathop{\mathchar
58\relax}}=\tilde{\gamma}_{e^{\prime}}.$

Moreover, $U_{c_{x}}\cong\mathscr{L}_{\varphi}(\tilde{M},e)$ via $\Psi_{e}$ if
and only if $\varphi=\operatorname{id}_{\tilde{M}}$, where $c_{x}$ denotes the
constant loop at $x$.

  2. (b)

For every $\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})$ and
$e,e^{\prime}\in\pi^{-1}(x)$ we have a commutative diagram of isomorphisms

${\operatorname{Aut}_{\pi}(\tilde{M})}$${\operatorname{Aut}_{\pi}(\tilde{M})}$${\pi_{1}(M,x),}$$\scriptstyle{C_{\psi}}$$\scriptstyle{\Phi_{e}}$$\scriptstyle{\Phi_{e^{\prime}}}$

where for $\psi\in\operatorname{Aut}_{\pi}(\tilde{M})$ sucht that
$\psi(e)=e^{\prime}$

$\qquad
C_{\psi}\colon\operatorname{Aut}_{\pi}(\tilde{M})\to\operatorname{Aut}_{\pi}(\tilde{M}),\qquad
C_{\psi}(\varphi)\mathrel{\mathop{\mathchar
58\relax}}=\psi\circ\varphi\circ\psi^{-1},$

and

$\displaystyle\Phi_{e}\colon\pi_{1}(M,x)\to\operatorname{Aut}_{\pi}(\tilde{M}),$
$\displaystyle\Phi_{e}([\gamma])\mathrel{\mathop{\mathchar
58\relax}}=\varphi^{e}_{[\gamma]},$
$\displaystyle\Phi_{e^{\prime}}\colon\pi_{1}(M,x)\to\operatorname{Aut}_{\pi}(\tilde{M}),$
$\displaystyle\Phi_{e^{\prime}}([\gamma])\mathrel{\mathop{\mathchar
58\relax}}=\varphi^{e^{\prime}}_{[\gamma]},$

with $\varphi^{e}_{[\gamma]}(e)=\tilde{\gamma}_{e}(1)$ and
$\varphi^{e^{\prime}}_{[\gamma]}(e^{\prime})=\tilde{\gamma}_{e^{\prime}}(1)$.

  3. (c)

The projection

$\qquad\tilde{\pi}_{x}\colon\coprod_{\begin{subarray}{c}\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})\\\
e\in\pi^{-1}(x)\end{subarray}}\mathscr{L}_{\varphi}(\tilde{M},e)\to\mathscr{L}(M,x)$

defined by $\tilde{\pi}_{x}(\gamma)\mathrel{\mathop{\mathchar
58\relax}}=\pi\circ\gamma$ is a covering map with number of sheets coinciding
with the cardinality of $\pi_{1}(M,x)$. Moreover, $\tilde{\pi}_{x}$ restricts
to define a covering map

$\qquad\tilde{\pi}_{x}|_{\operatorname{id}_{\tilde{M}}}\colon\coprod_{e\in\pi^{-1}(x)}\mathscr{L}(\tilde{M},e)\to
U_{c_{x}},$

and $\tilde{\pi}_{x}$ gives rise to a principal
$\operatorname{Aut}_{\pi}(\tilde{M})$-bundle. If $M$ admits a smooth
structure, then this bundle is additionally a bundle of smooth Banach
manifolds.

###### Proof.

For proving part (a), fix a path class $[\gamma]\in\pi_{1}(M,x)$. As any
topological manifold is Hausdorff, paracompact and locally metrisable by
definition, the Smirnov Metrisation Theorem [Mun00, Theorem 42.1] implies that
$M$ is metrisable. Let $d$ be a metric on $M$ and $\bar{d}$ be the standard
bounded metric corresponding to $d$, that is,

$\bar{d}(x,y)=\min\mathinner{\\{d(x,y),1\\}}\qquad\forall x,y\in M.$

The metric $\bar{d}$ induces the same topology on $M$ as $d$ by [Mun00,
Theorem 20.1]. Topologise the based loop space
$\mathscr{L}(M,x)\subseteq\mathscr{L}M$ as a subspace of the free loop space
on $M$, where $\mathscr{L}M$ is equipped with the topology of uniform
convergence, that is, with the supremum metric

$\bar{d}_{\infty}(\gamma,\gamma^{\prime})=\sup_{t\in\mathbb{S}^{1}}\bar{d}\mathinner{\bigl{(}\gamma(t),\gamma^{\prime}(t)\bigr{)}}\qquad\forall\gamma,\gamma^{\prime}\in\mathscr{L}M.$

There is a canonical pseudometric on the universal covering manifold
$\tilde{M}$ induced by $\bar{d}$ given by $\bar{d}\circ\pi$. As every
pseudometric generates a topology, we topologise the based twisted loop space
$\mathscr{L}_{\varphi}(\tilde{M},e)\subseteq\mathscr{P}\tilde{M}$ as a
subspace of the free path space on $\tilde{M}$ for every $e\in\pi^{-1}(x)$ via
the supremum metric $\tilde{d}_{\infty}$ corresponding to $\bar{d}\circ\pi$.
In fact, $\tilde{d}_{\infty}$ is a metric as if
$\tilde{d}_{\infty}(\gamma,\gamma^{\prime})=0$, then by definition of
$\tilde{d}_{\infty}$ we have that $\pi(\gamma)=\pi(\gamma^{\prime})$. But as
$\gamma(0)=e=\gamma^{\prime}(0)$, we conclude $\gamma=\gamma^{\prime}$ by the
unique lifting property of paths [Lee11, Corollary 11.14]. Note that the
resulting topology of uniform convergence on
$\mathscr{L}_{\varphi}(\tilde{M},e)$ coincides with the compact-open topology
by [Mun00, Theorem 46.8] or [Hat01, Proposition A.13]. In particular, the
topology of uniform convergence does not depend on the choice of a metric (see
[Mun00, Corollary 46.9]). It follows from [Lee11, Theorem 11.15 (b)], that
$\Psi_{e}$ and $\Psi_{e^{\prime}}$ are well-defined. Moreover, it is immediate
by the fact that the projection $\pi\colon\tilde{M}\to M$ is an isometry with
respect to the above metric, that $\Psi_{e}$ and $\Psi_{e^{\prime}}$ are
continuous with continuous inverse given by the composition with $\pi$. It is
also immediate that $L_{\psi}$ is continuous with continuous inverse
$L_{\psi^{-1}}$.

Next we show that the diagram (38) commutes. Note that

$\pi\circ
L_{\psi}\circ\Psi_{e}=\pi\circ\Psi_{e}=\operatorname{id}_{U_{\eta}}=\pi\circ\Psi_{e^{\prime}},$

thus by

$(L_{\psi}\circ\Psi_{e}(\gamma))(0)=\psi(\tilde{\gamma}_{e}(0))=\psi(e)=e^{\prime}=\tilde{\gamma}_{e^{\prime}}(0)=\Psi_{e^{\prime}}(\gamma)(0)$

and by uniqueness it follows that

$L_{\psi}\circ\Psi_{e}=\Psi_{e^{\prime}}.$

In particular

$\Psi_{e^{\prime}}(1)=(L_{\psi}\circ\Psi_{e})(1)=\psi(\varphi(e))=(\psi\circ\varphi\circ\psi^{-1})(e^{\prime}),$

and thus
$\Psi_{e^{\prime}}(\gamma)\in\mathscr{L}_{\psi\circ\varphi\circ\psi^{-1}}(\tilde{M},e^{\prime})$.
Consequently, the homeomorphism $\Psi_{e^{\prime}}$ is well-defined.

Recall, that by the Monodromy Theorem [Lee11, Theorem 11.15 (b)]

$\gamma\simeq\gamma^{\prime}\qquad\Leftrightarrow\qquad\Psi_{e}(\gamma)(1)=\Psi_{e}(\gamma^{\prime})(1)$

for all paths $\gamma$ and $\gamma^{\prime}$ in $M$ starting at $x$ and ending
at the same point. Note that the statement of the the Monodromy Theorem is an
if-and-only-if statement since $\tilde{M}$ is simply connected.

Suppose $\gamma\in\mathscr{L}(M,x)$ is contractible. Then $\gamma\simeq
c_{x}$, implying $e\in\operatorname{Fix}(\varphi)$. But the only deck
transformation of $\pi$ fixing any point of $\tilde{M}$ is
$\operatorname{id}_{\tilde{M}}$ by [Lee11, Proposition 12.1 (a)].

Conversely, assume that $\gamma\in\mathscr{L}(M,x)$ is not contractible. Then
we have that $\Psi_{e}(\gamma)(1)\neq e$. Indeed, if $\Psi_{e}(\gamma)(1)=e$,
then $\gamma\simeq c_{x}$ and consequently, $\gamma$ would be contractible. As
normal covering maps have transitive automorphism groups by [Lee11, Corollary
12.5], there exists
$\psi\in\operatorname{Aut}_{\pi}(\tilde{M})\setminus\mathinner{\\{\operatorname{id}_{\tilde{M}}\\}}$
such that $\Psi_{e}(\gamma)(1)=\psi(e)$.

For proving part (b), observe that $\Phi_{e}$ and $\Phi_{e^{\prime}}$ are
isomorphisms follows from [Lee11, Corollary 12.9]. Moreover, it is also clear
that $C_{\psi}$ is an isomorphism with inverse $C_{\psi^{-1}}$. Let
$[\gamma]\in\pi_{1}(M,x)$. Then using part (a) we compute

$\displaystyle(C_{\psi}\circ\Phi_{e})[\gamma](e^{\prime})$
$\displaystyle=(\psi\circ\Phi_{e}[\gamma]\circ\psi^{-1})(e^{\prime})$
$\displaystyle=\psi\mathinner{\bigl{(}\varphi_{[\gamma]}^{e}(e)\bigr{)}}$
$\displaystyle=\psi(\tilde{\gamma}_{e}(1))$
$\displaystyle=(L_{\psi}\circ\Psi_{e})(\gamma)(1)$
$\displaystyle=\Psi_{e^{\prime}}(\gamma)(1)$
$\displaystyle=\tilde{\gamma}_{e^{\prime}}(1)$
$\displaystyle=\varphi^{e^{\prime}}_{[\gamma]}(e^{\prime})$
$\displaystyle=\Phi_{e^{\prime}}[\gamma](e^{\prime}).$

Thus by uniqueness [Lee11, Proposition 12.1 (a)], we conclude

$C_{\psi}\circ\Phi_{e}=\Phi_{e^{\prime}}.$

Finally for proving (c), define a metric $\tilde{d}_{\infty}$ on

$E\mathrel{\mathop{\mathchar
58\relax}}=\coprod_{\begin{subarray}{c}\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})\\\
e\in\pi^{-1}(x)\end{subarray}}\mathscr{L}_{\varphi}(\tilde{M},e)$

by

$\tilde{d}_{\infty}(\gamma,\gamma^{\prime})\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\bar{d}_{\infty}\mathinner{\bigl{(}\pi(\gamma),\pi(\gamma^{\prime})\bigr{)}}&\gamma,\gamma^{\prime}\in\mathscr{L}_{\varphi}(\tilde{M},e),\\\
1&\text{else}.\end{cases}$

Then the induced topology coincides with the disjoint union topology and with
respect to this topology, $\tilde{\pi}_{x}$ is continuous. So left to show is
that $\tilde{\pi}_{x}$ is a covering map. Surjectivity is clear. So let
$\gamma\in\mathscr{L}(M,x)$. Then $\gamma\in U_{\eta}$ for some
$[\eta]\in\pi_{1}(M,x)$. Now note that $U_{\eta}$ is open in
$\mathscr{L}(M,x)$ and by part (a) we conclude

$\tilde{\pi}_{x}^{-1}(U_{\eta})=\coprod_{\psi\in\operatorname{Aut}_{\pi}(\tilde{M})}\mathscr{L}_{\psi\circ\varphi\circ\psi^{-1}}(\tilde{M},\psi(e))$
(39)

for some fixed $e\in\pi^{-1}(x)$ and
$\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})$ such that
$\varphi(e)=\tilde{\eta}_{e}(1)$.

As the cardinality of the fibre $\pi^{-1}(x)$ and of
$\operatorname{Aut}_{\pi}(\tilde{M})$ coincides with the cardinality of the
fundamental group $\pi_{1}(M,x)$ by [Lee11, Corollary 11.31] and part (b), we
conclude that the number of sheets is equal to the cardinality of the
fundamental group $\pi_{1}(M,x)$.

Equip $\operatorname{Aut}_{\pi}(\tilde{M})$ with the discrete topology. As the
fundamental group of every topological manifold is countable by [Lee11,
Theorem 7.21], we have that $\operatorname{Aut}_{\pi}(\tilde{M})$ is a
discrete topological Lie group. Now label the distinct path classes in
$\pi_{1}(M,x)$ by $\beta\in B$ and for fixed $e\in\pi^{-1}(x)$ define local
trivialisations

$(\tilde{\pi}_{x},\alpha_{\beta})\colon\tilde{\pi}_{x}^{-1}(U_{\beta})\xrightarrow{\cong}U_{\beta}\times\operatorname{Aut}_{\pi}(\tilde{M}),$

making use of (39) by

$\alpha_{\beta}(\gamma)\mathrel{\mathop{\mathchar 58\relax}}=\psi^{-1},$

whenever
$\gamma\in\mathscr{L}_{\psi\circ\varphi\circ\psi^{-1}}(\tilde{M},\psi(e))$.
Consequently, $\tilde{\pi}_{x}$ is a fibre bundle with discrete fibre
$\operatorname{Aut}_{\pi}(\tilde{M})$ and bundle atlas
$(U_{\beta},\alpha_{\beta})_{\beta\in B}$. Define a free right action

$E\times\operatorname{Aut}_{\pi}(\tilde{M})\to
E,\qquad\gamma\cdot\xi\mathrel{\mathop{\mathchar
58\relax}}=\xi^{-1}\circ\gamma.$

Then $\alpha_{\beta}$ is $\operatorname{Aut}_{\pi}(\tilde{M})$-equivariant
with respect to this action for all $\beta\in B$. Indeed, using again the
commutative diagram (38) we compute

$\alpha_{\beta}(\gamma\cdot\xi)=\alpha_{\beta}(\xi^{-1}\circ\gamma)=\mathinner{\bigl{(}\xi^{-1}\circ\psi\bigr{)}}^{-1}=\psi^{-1}\circ\xi=\alpha_{\beta}(\gamma)\circ\xi$

for all $\xi\in\operatorname{Aut}_{\pi}(\tilde{M})$ and
$\gamma\in\mathscr{L}_{\psi\circ\varphi\circ\psi^{-1}}(\tilde{M},\psi(e))$.
Note, that here we use again the fact that
$\operatorname{Aut}_{\pi}(\tilde{M})$ acts transitively on the fibre
$\pi^{-1}(x)$.

Suppose that $M$ admits a smooth structure. Then for every compact smooth
manifold $N$ we have that the mapping space $C(N,M)$ admits the structure of a
smooth Banach manifold by [Wit19]. By [LO15, Theorem 1.1 p. 24], there is a
smooth fibre bundle, called the loop-loop fibre bundle,

$\mathscr{L}(M,x)\hookrightarrow\mathscr{L}M\xrightarrow{\operatorname{ev}_{0}}M$

where

$\operatorname{ev}_{0}\colon\mathscr{L}M\to
M,\qquad\operatorname{ev}_{0}(\gamma)\mathrel{\mathop{\mathchar
58\relax}}=\gamma(0).$

Thus the based loop space $\mathscr{L}(M,x)=\operatorname{ev}_{0}^{-1}(x)$ on
$M$ is a smooth Banach manifold by the implicit function theorem [MS12,
Theorem A.3.3] for all $x\in M$. Likewise, by [LO15, Theorem 1.2 p. 25], there
is a smooth fibre bundle, called the path-loop fibre bundle,

$\mathscr{L}(\tilde{M},e)\hookrightarrow\mathscr{P}(\tilde{M},e)\xrightarrow{\operatorname{ev}_{1}}\tilde{M},$

where

$\mathscr{P}(\tilde{M},e)\mathrel{\mathop{\mathchar 58\relax}}=\\{\gamma\in
C(I,\tilde{M})\mathrel{\mathop{\mathchar 58\relax}}\gamma(0)=e\\}$

denotes the based path space and

$\operatorname{ev}_{1}\colon\mathscr{P}(\tilde{M},e)\to\tilde{M},\qquad\operatorname{ev}_{1}(\gamma)\mathrel{\mathop{\mathchar
58\relax}}=\gamma(1).$

Therefore, the twisted loop space
$\mathscr{L}_{\varphi}(\tilde{M},e)=\operatorname{ev}_{1}^{-1}(\varphi(e))$ is
also a smooth Banach manifold for all
$\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})$ and $e\in\pi^{-1}(x)$ by the
implicit function theorem [MS12, Theorem A.3.3]. As the fundamental group
$\pi_{1}(M,x)$ is countable, the topological space $E$ has only countably many
connected components being smooth Banach manifolds and thus the total space
itself is a smooth Banach manifold. Finally,
$\operatorname{Aut}_{\pi}(\tilde{M})$ is trivially a Banach manifold with
$\dim\operatorname{Aut}_{\pi}(\tilde{M})=0$ as a discrete Lie group. ∎

###### Corollary A.1.

Let $(M,x)$ be a connected pointed topological manifold and denote by
$\pi\colon\tilde{M}\to M$ the universal covering of $M$. Assume that
$\pi_{1}(M,x)$ is abelian.

  1. (a)

Fix a path class $[\eta]\in\pi_{1}(M,x)$. For every
$e,e^{\prime}\in\pi^{-1}(x)$ and deck transformation
$\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})$ such that
$\varphi(e)=\tilde{\eta}_{e}(1)$, we have a commutative diagram of
homeomorphisms

${\mathscr{L}_{\varphi}(\tilde{M},e)}$${\mathscr{L}_{\varphi}(\tilde{M},e^{\prime})}$${U_{\eta},}$$\scriptstyle{L_{\psi}}$$\scriptstyle{\Psi_{e}}$$\scriptstyle{\Psi_{e^{\prime}}}$

where $\psi\in\operatorname{Aut}_{\pi}(\tilde{M})$ is such that
$\psi(e)=e^{\prime}$.

  2. (b)

For every $\varphi\in\operatorname{Aut}_{\pi}(\tilde{M})$ we have that
$\Phi_{e}=\Phi_{e^{\prime}}$ for all $e,e^{\prime}\in\pi^{-1}(x)$.

Lemma 1.1 now follows from part (a) of Theorem A.1. Indeed, by assumption
$\varphi\in\operatorname{Aut}_{\pi}(\Sigma)\setminus\\{\operatorname{id}_{\Sigma}\\}$
and using the long exact sequence of homotopy groups of a fibration [Hat01,
Theorem 4.41], there is a short exact sequence

${0}$${\pi_{1}(\Sigma,x)}$${\pi_{1}(\Sigma/\mathbb{Z}_{m},\pi(x))}$${\pi_{0}(\mathbb{Z}_{m})}$${0.}$

In particular, by [Lee11, Corollary 12.9] we conclude

$\operatorname{Aut}_{\pi}(\Sigma)\cong\pi_{1}(\Sigma/\mathbb{Z}_{m},\pi(x))\cong\mathbb{Z}_{m}\cong\\{\operatorname{id}_{\Sigma},\varphi,\dots,\varphi^{m-1}\\}.$

Finally, we discuss a smooth structure on the continuous free twisted loop
space of a smooth manifold.

###### Lemma A.1.

Let $M$ be a smooth manifold and $\varphi\in\operatorname{Diff}(M)$. Then the
continuous free twisted loop space $\mathscr{L}_{\varphi}M$ is the pullback of

$(\operatorname{ev}_{0},\operatorname{ev}_{1})\colon\mathscr{P}M\to M\times
M,\qquad\gamma\mapsto(\gamma(0),\gamma(1)),$

where we abbreviate $\mathscr{P}M\mathrel{\mathop{\mathchar
58\relax}}=C(I,M)$, along the graph of $\varphi$

$\Gamma_{\varphi}\colon M\to M\times
M,\qquad\Gamma_{\varphi}(x)\mathrel{\mathop{\mathchar
58\relax}}=(x,\varphi(x)),$

in the category of smooth Banach manifolds. Moreover, we have that

$T_{\gamma}\mathscr{L}_{\varphi}M=\\{X\in\Gamma^{0}(\gamma^{*}TM)\mathrel{\mathop{\mathchar
58\relax}}X(1)=D\varphi(X(0))\\}$

for all $\gamma\in\mathscr{L}_{\varphi}M$.

###### Proof.

Write $f\mathrel{\mathop{\mathchar
58\relax}}=(\operatorname{ev}_{0},\operatorname{ev}_{1})$. Then

$\mathscr{L}_{\varphi}M=f^{-1}(\Gamma_{\varphi}(M)).$

Thus in order to show that the free twisted loop space
$\mathscr{L}_{\varphi}M$ is a smooth Banach manifold, it is enough to show
that $f$ is transverse to the properly embedded smooth submanifold
$\Gamma_{\varphi}(M)\subseteq M\times M$. By [Lan99, Proposition 2.4] we need
to show that the composition

$\Phi_{\gamma}\colon
T_{\gamma}\mathscr{P}M\xrightarrow{Df_{\gamma}}T_{(x,\varphi(x))}(M\times
M)\to T_{(x,\varphi(x))}(M\times M)/T_{(x,\varphi(x))}\Gamma_{\varphi}(M)$

is surjective and $\ker\Phi_{\gamma}$ is complemented for all $\gamma\in
f^{-1}(\Gamma_{\varphi}(M))$, where we abbreviate $x\mathrel{\mathop{\mathchar
58\relax}}=\gamma(0)$. Note that we have a canonical isomorphism

$T_{(x,\varphi(x))}(M\times M)/T_{(x,\varphi(x))}\Gamma_{\varphi}(M)\to
T_{\varphi(x)}M,\quad[(v,u)]\mathrel{\mathop{\mathchar
58\relax}}=u-D\varphi(v).$

Under this canonical isomorphism, the linear map $\Phi_{\gamma}$ is given by

$\Phi_{\gamma}(X)=X(1)-D\varphi(X(0)),\qquad\forall
X\in\Gamma^{0}(\gamma^{*}TM).$

Fix a Riemannian metric on $M$ and let $X_{v}\in\Gamma(\gamma^{*}TM)$ be the
unique parallel vector field with $X_{v}(1)=v\in T_{\varphi(x)}M$. Fix a
cutoff function $\beta\in C^{\infty}(I)$ such that
$\operatorname{supp}\beta\subseteq\mathinner{\bigl{[}\frac{1}{2},1\bigr{]}}$
and $\beta=1$ in a neighbourhood of $1$. Then $\Phi_{\gamma}(\beta X_{v})=v$
and consequently, $\Phi_{\gamma}$ is surjective. Moreover

$\ker\Phi_{\gamma}=\mathinner{\\{X\in\Gamma^{0}(\gamma^{*}TM)\mathrel{\mathop{\mathchar
58\relax}}X(1)=D\varphi(X(0))\\}}$

is complemented by the finite-dimensional vector space

$V\mathrel{\mathop{\mathchar 58\relax}}=\mathinner{\\{\beta
X_{v}\in\Gamma(\gamma^{*}TM)\mathrel{\mathop{\mathchar 58\relax}}v\in
T_{\varphi(x)}M\\}}.$

Indeed, any $X\in\Gamma^{0}(\gamma^{*}TM)$ can be decomposed uniquely as

$X=X-\beta X_{v}+\beta X_{v},\qquad v\mathrel{\mathop{\mathchar
58\relax}}=X(1)-D\varphi(X(0)).$

Abbreviating $Y\mathrel{\mathop{\mathchar 58\relax}}=X-\beta
X_{v}\in\Gamma^{0}(\gamma^{*}TM)$, we have that

$Y(1)=D\varphi(X(0))=D\varphi(Y(0)),$

implying $Y\in\ker\Phi_{\gamma}$. Thus $\mathscr{L}_{\varphi}M$ is a smooth
Banach manifold.

Now note that $\mathscr{L}_{\varphi}M$ can be identified with the pullback

$f^{*}\mathscr{P}M=\\{(x,\gamma)\in
M\times\mathscr{P}M\mathrel{\mathop{\mathchar
58\relax}}(\gamma(0),\gamma(1))=\mathinner{(x,\varphi(x))}\\},$

making the diagram

${f^{*}\mathscr{P}M}$${\mathscr{P}M}$${M}$${M\times
M}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{f}$$\scriptstyle{\Gamma_{\varphi}}$

commute, via the homeomorphism

$\mathscr{L}_{\varphi}M\to
f^{*}\mathscr{P}M,\qquad\gamma\mapsto(\gamma(0),\gamma).$

Finally, one computes

$T_{(x,\gamma)}f^{*}\mathscr{P}M=\\{(v,X)\in T_{x}M\times
T_{\gamma}\mathscr{P}M\mathrel{\mathop{\mathchar
58\relax}}Df_{\gamma}X=D\Gamma_{\varphi}|_{x}(v)\\}$

for all $(x,\gamma)\in f^{*}\mathscr{P}M$. ∎

###### Remark A.1.

Using Lemma A.1 one should be able to prove similar results as in Theorem A.1
in the case of free twisted loop spaces. However, in the non-abelian case the
situation gets much more complicated as in general it is not true, that lifts
of conjugated elements of the fundamental group lie in the same free twisted
loop space by [LO15, Theorem 1.6 (i)].

##  Appendix B On the Nonexistence of the Gradient of the Twisted Rabinowitz
Action Functional

Let $(M,\lambda)$ be an exact symplectic manifold and
$\varphi\in\operatorname{Symp}(M,d\lambda)$ a symplectomorphism of finite
order. For $H\in C^{\infty}(M)$ such that $H\circ\varphi=H$, one can define
the twisted Rabinowitz action functional

$\mathscr{A}^{H}_{\varphi}\colon\mathscr{L}_{\varphi}M\times\mathbb{R}\to\mathbb{R},\qquad\mathscr{A}^{H}_{\varphi}(\gamma,\tau)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}\gamma^{*}\lambda-\tau\int_{0}^{1}H(\gamma(t))dt.$ (40)

Let $J$ be a $d\lambda$-compatible almost complex structure such that
$\varphi^{*}J=J$. Then one can consider the gradient of
$\mathscr{A}^{H}_{\varphi}$ with respect to the $L^{2}$-metric

$\langle(X,\eta),(Y,\sigma)\rangle_{J}\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}d\lambda(JX(t),Y(t))dt+\eta\sigma$ (41)

for all $(X,\eta),(Y,\sigma)\in
T_{\gamma}\mathscr{L}_{\varphi}M\times\mathbb{R}$ and
$(\gamma,\tau)\in\mathscr{L}_{\varphi}M\times\mathbb{R}$.

###### Theorem B.1 (Nonexistence Gradient).

Let $(M,\lambda)$ be a connected exact symplectic manifold,
$\varphi\in\operatorname{Symp}(M,d\lambda)$ of finite order and $H\in
C^{\infty}(M)$ such that $H\circ\varphi=H$. If
$\varphi^{*}\lambda\neq\lambda$, then the gradient of the twisted Rabinowitz
action functional (40) with respect to the $L^{2}$-metric (41) does not exist.

###### Proof.

Assume that the gradient
$\operatorname{grad}\mathscr{A}^{H}_{\varphi}\in\mathfrak{X}(\mathscr{L}_{\varphi}M\times\mathbb{R})$
exists. We write

$\operatorname{grad}\mathscr{A}^{H}_{\varphi}=\operatorname{grad}\mathscr{A}^{H}+V$

for some $V\in\mathfrak{X}(\mathscr{L}_{\varphi}M)$ and

$\operatorname{grad}\mathscr{A}^{H}|_{(\gamma,\tau)}=\begin{pmatrix}\displaystyle
J(\dot{\gamma}-\tau X_{H}(\gamma))\\\
\displaystyle-\int_{0}^{1}H(\gamma(t))dt\end{pmatrix}$

for all $(\gamma,\tau)\in\mathscr{L}_{\varphi}M\times\mathbb{R}$. Indeed, this
follows from

$d\mathscr{A}^{H}_{\varphi}|_{(\gamma,\tau)}(X,\eta)=d\mathscr{A}^{H}|_{(\gamma,\tau)}(X,\eta)+(\varphi^{*}\lambda-\lambda)(X(0))$
(42)

for all $(X,\eta)\in T_{\gamma}\mathscr{L}_{\varphi}M\times\mathbb{R}$, where

$d\mathscr{A}^{H}|_{(\gamma,\tau)}(X,\eta)=\int_{0}^{1}d\lambda(X,\dot{\gamma}(t)-\tau
X_{H}(\gamma(t)))dt-\eta\int_{0}^{1}H(\gamma(t))dt.$

By assumption, there exists $x\in M$ and $v\in T_{x}M$ with
$(\varphi^{*}\lambda)_{x}(v)\neq\lambda_{x}(v)$. As by assumption $M$ is
connected, there exists a smooth path $u\in C^{\infty}(\mathinner{[0,1]},M)$
from $x$ to $\varphi(x)$. Fix a smooth cutoff function $\beta\in
C^{\infty}(\mathinner{[0,1]},\mathinner{[0,1]})$ such that $\beta=0$ in a
neighbourhood of $0$ and $\beta=1$ in a neighbourhood of $1$. Then we can
extend $u$ by

$\gamma(t)\mathrel{\mathop{\mathchar
58\relax}}=\varphi^{k}(u(\beta(t-k)))\qquad\forall
t\in\mathinner{[k,k+1]},k\in\mathbb{Z}.$

Clearly, $\gamma\in\mathscr{L}_{\varphi}M$ by construction. Extend $v\in
T_{\gamma(0)}M$ to $X_{v}\in T_{\gamma}\mathscr{L}_{\varphi}M$ by

$X_{v}(t)\mathrel{\mathop{\mathchar
58\relax}}=(1-\beta(t-k))P_{0,\beta(t-k)}^{\varphi^{k}\circ
u}(D\varphi^{k}(v))+\beta(t-k)P_{1,\beta(t-k)}^{\varphi^{k}\circ
u}(D\varphi^{k+1}(v)),$

for all $t\in\mathinner{[k,k+1]}$ and $k\in\mathbb{Z}$, where $P$ denotes the
parallel transport system induced by the Levi–Civita connection associated
with the metric $m_{J}$. Choose a sequence $(\beta_{j})\subseteq
C^{\infty}(\mathbb{S}^{1},\mathinner{[0,1]})$ with $\beta_{j}=1$ on
$\mathinner{\bigl{[}0,\frac{1}{2j}\bigr{]}}\cup\mathinner{\bigl{[}1-\frac{1}{2j},1\bigr{]}}$
and such that
$\operatorname{supp}\beta_{j}\subseteq\mathinner{\bigl{[}0,\frac{1}{j}\bigr{]}}\cup\mathinner{\bigl{[}1-\frac{1}{j},1\bigr{]}}$
for all $j\in\mathbb{N}$. Using (42) we compute

$\displaystyle\langle V,\beta_{j}X_{v}\rangle_{J}$
$\displaystyle=\langle\operatorname{grad}\mathscr{A}^{H}_{\varphi},\beta_{j}X_{v}\rangle_{J}-\langle\operatorname{grad}\mathscr{A}^{H},\beta_{j}X_{v}\rangle_{J}$
$\displaystyle=d\mathscr{A}^{H}_{\varphi}(\beta_{j}X_{v})-d\mathscr{A}^{H}(\beta_{j}X_{v})$
$\displaystyle=(\varphi^{*}\lambda-\lambda)(\beta_{j}(0)X_{v}(0))$
$\displaystyle=(\varphi^{*}\lambda-\lambda)(v)$

for all $j\in\mathbb{N}$, implying

$\displaystyle(\varphi^{*}\lambda-\lambda)(v)$
$\displaystyle=\lim_{j\to\infty}\langle V,\beta_{j}X_{v}\rangle_{J}$
$\displaystyle=\lim_{j\to\infty}\int_{0}^{1}d\lambda(JV(t),\beta_{j}(t)X_{v}(t))dt$
$\displaystyle=0$

by dominated convergence. ∎

##  Appendix C M-Polyfolds

The classical approach for establishing generic transversality results in
Floer theories is via a suitable version of the Sard–Smale Theorem [MS12,
Theorem A.5.1]. The idea is to represent the moduli space of negative gradient
flow lines as the zero set of an appropriate Fredholm section. Unfortunately,
this does not work for the moduli space of unparametrised negative gradient
flow lines as the reparametrisation action is not smooth. Moreover, the
transversality results usually require perturbing the given metric to a
generic one. There is a more abstract approach for proving transversality
results via polyfold theory. This theory was and is still developed by
Hofer–Wysocki–Zehnder [HWZ21] primarily having symplectic field theory in
mind. Another more algebraic approach to abstract perturbations is via
Kuranishi structures developed by Fukaya–Oh–Ohta–Ono [Fuk+20].

In the first section we introduce the basic terminology of polyfold theory,
namely the notion of scale smoothness on scale Banach spaces.

In the second section we formulate a prototypical result for Morse–Bott
homology following the brilliant lecture notes [Cie18].

###  C.1 Scale Calculus

###### Definition C.1 (Scale Structure, [HWZ21, Definition 1.1.1]).

A scale structure on a Banach space $E$ is a decreasing sequence

$E=\mathrel{\mathop{\mathchar 58\relax}}E_{0}\supseteq E_{1}\supseteq
E_{2}\supseteq\dots$

of Banach spaces such that the inclusion $E_{k+1}\hookrightarrow E_{k}$ is
compact for every $k\in\mathbb{N}_{0}$ and such that
$E_{\infty}\mathrel{\mathop{\mathchar 58\relax}}=\cap_{k=0}^{\infty}E_{k}$ is
dense in every $E_{k}$. A Banach space $E$ together with a scale structure
$(E_{k})$ is called a scale Banach space.

###### Example C.1 (Shifted Scale Banach Space, [FW21, Definition 3.3]).

Let $(E,(E_{k}))$ be a scale Banach space and $m\in\mathbb{N}_{0}$. Then
$\mathinner{\bigl{(}E^{k},(E^{m}_{k})\bigr{)}}$ is a scale Banach space where
$E^{m}_{k}\mathrel{\mathop{\mathchar 58\relax}}=E_{m+k}$ for all
$k\in\mathbb{N}_{0}$.

###### Example C.2 (Scale Direct Sum, [FW21, Definition 3.4]).

Let $(E,(E_{k}))$ and $(F,(F_{k}))$ be scale Banach spaces. Then $(E\oplus
F,(E_{k}\oplus F_{k}))$ is also a scale Banach space.

The following example underlies Morse and Floer theory.

###### Example C.3 (Weighted Sobolev Spaces, [FW21, Example 3.9]).

Fix a monotone cutoff function $\beta\in
C^{\infty}(\mathbb{R},\mathinner{[-1,1]})$ with

$\beta(s)=\begin{cases}1&s\geq 1,\\\ -1&s\leq-1,\end{cases}$

and $\delta>0$. Define

$\gamma_{\delta}\colon\mathbb{R}\to\mathbb{R},\qquad\gamma_{\delta}(s)\mathrel{\mathop{\mathchar
58\relax}}=e^{\delta\beta(s)s}.$

For $p\in\mathinner{(1,+\infty)}$ and $k\in\mathbb{N}_{0}$ define the Sobolev
spaces with weight $\delta$ by

$W^{k,p}_{\delta}(\mathbb{R},\mathbb{R}^{n})\mathrel{\mathop{\mathchar
58\relax}}=\\{u\in
W^{k,p}(\mathbb{R},\mathbb{R}^{n})\mathrel{\mathop{\mathchar
58\relax}}\gamma_{\delta}u\in W^{k,p}(\mathbb{R},\mathbb{R}^{n})\\}.$

Choose a strictly increasing sequence $(\delta_{k})$ with $\delta_{0}=0$. Then
$(E,(E_{k}))$ is a scale Banach space with

$E_{k}\mathrel{\mathop{\mathchar
58\relax}}=W^{k,p}_{\delta_{k}}(\mathbb{R},\mathbb{R}^{n})\qquad\forall
k\in\mathbb{N}_{0}.$

###### Definition C.2 (Scale Continuity, [FW21, Definition 4.1]).

Let $(E,(E_{k}))$ and $(F,(F_{k}))$ be two scale Banach spaces. A map $f\colon
E\to F$ is called scale continuous, iff $f(E_{k})\subseteq F_{k}$ for all
$k\in\mathbb{N}_{0}$ and the restriction $f|_{E_{k}}\colon E_{k}\to F_{k}$ is
continuous.

###### Definition C.3 (Tangent Bundle, [HWZ21, Definition 1.1.14]).

For a scale Banach space $(E,(E_{k}))$ define its tangent bundle by
$TE\mathrel{\mathop{\mathchar 58\relax}}=E^{1}\oplus E$.

###### Definition C.4 (Scale Differentiability, [FW21, Definition 4.2]).

A scale continuous map $f\colon E\to F$ between scale Banach spaces
$(E,(E_{k}))$ and $(F,(F_{k}))$ is called scale differentiable, iff for every
$x\in E_{1}$ there exists a bounded linear operator

$Df(x)\colon E_{0}\to F_{0},$

called the scale differential of $f$, such that the restriction
$f|_{E_{1}}\colon E_{1}\to F_{0}$ is Fréchet differentiable with derivative
$Df|_{E_{1}}$ and the tangent map

$Tf\colon TE\to TF,\qquad Tf(x,h)\mathrel{\mathop{\mathchar
58\relax}}=(f(x),Df(x)h)$

is scale continuous.

Using the iterated notion of scale differentiability one can define higher
scale regularity. We say that a scale differentiable map is scale smooth, iff
its tangent map is infinitely scale differentiable. The following example
motivated the development of scale calculus.

###### Example C.4 (Weighted Sobolev Spaces, [FW21, Theorem 6.2]).

Let $E_{k}$ be the scale of weighted Sobolev spaces introduced in Example C.3
on the scale Banach space $E=L^{p}(\mathbb{R},\mathbb{R}^{n})$. Then the shift
map

$\mathbb{R}\oplus E\to E,\qquad(r,x)\mapsto x(\cdot+r)$

is scale smooth.

One important property of scale differentiability is that the chain rule
remains valid in this general setting.

###### Proposition C.1 (Chain Rule, [HWZ21, Theorem 1.3.1]).

Uppose that $f\colon E\to F$ and $g\colon F\to G$ are scale differentiable
maps for scale Banach spaces $E$, $F$ and $G$. Then the composition $g\circ f$
is scale differentiable with tangent map

$T(g\circ f)=Tg\circ Tf\colon TE\to TG.$

The proof of the chain rule heavily relies on the compactness of the
embeddings of the scales. Using the chain rule one is able to define the
notion of scale manifolds and scale differential geometry in analogy to the
finite-dimensional case. For details see [Cie18, Chapter 5]. Thus with the
theory developed so far, one can make sense of the smooth moduli space of
unparametrised negative gradient flow lines. However, for broken negative
gradient flow lines one needs an even more general notion, including scale
manifolds.

###### Definition C.5 (Retraction, [HWZ21, Definition 2.1.1]).

Let $E$ be scale Banach space. A retraction on $E$ is a scale smooth map
$r\colon E\to E$ such that $r^{2}=r$.

###### Remark C.1.

If $X$ is a smooth Banach manifold, then $\operatorname{Fix}(r)=r(M)$ is a
smooth submanifold for every smooth retraction $r\colon X\to X$ by a result of
Cartan [HWZ21, Proposition 2.1.2].

It is in general not true that the fixed point set of a scale smooth
retraction of a scale manifold is a scale submanifold. This lead
Hofer–Wysocki–Zehnder to the generalised notion of an M-polyfold, where the
“M” stands for “manifold flavoured”. Heuristically, an M-polyfold is locally
the fixed point set of a scale smooth retraction of a scale smooth manifold.
The main aspect for us is that Fredholm theory still is valid in M-polyfolds
in some sense. For an extensive treatment see [HWZ21, Part I].

###  C.2 M-Polyfold Setup for Morse–Bott Homology

In this section we explain how an M-polyfold Fredholm setup for Morse–Bott
homology can be defined. We follow [Cie18, Section 8.4]. The essential
arguments are contained in [Fra04, Appendix A]. We assume the following setup.
Let $(M,g)$ be a compact Riemannian manifold and $f\in C^{\infty}(M)$ a
Morse–Bott function. Choose an additional Morse function $h\in
C^{\infty}(\operatorname{Crit}f)$ and a Riemannian metric $g_{0}$ on
$\operatorname{Crit}f$ such that $(h,g_{0})$ is a Morse–Smale pair, that is,
the stable and unstable manifolds intersect transversally. Pick two connected
critical components $C^{\pm}\subseteq\operatorname{Crit}f$. For

$0<\delta<\min\\{\mathinner{\lvert\lambda\rvert}\mathrel{\mathop{\mathchar
58\relax}}\lambda\in\sigma(\operatorname{Hess}_{x}f))\setminus\\{0\\},x\in\operatorname{Crit}f\\}$

and a positive strictly increasing sequence
$0<\delta_{0}<\delta_{1}<\dots<\delta$, consider the Banach manifold
$\widetilde{\mathscr{E}}_{k}(C^{-},C^{+})$ consisting of all maps $u\in
H^{k+2}_{\delta_{k}}(\mathbb{R},M)$ converging exponentially to $C^{\pm}$,
that is, there exist $x^{\pm}\in C^{\pm}$ as well as

$\xi^{-}\in
H^{k+2}_{\delta_{k}}(\mathinner{(-\infty,-T]},T_{x^{-}}M)\quad\text{and}\quad\xi^{+}\in
H^{k+2}_{\delta_{k}}(\mathinner{[T,+\infty)},T_{x^{+}}M)$

for some $T\in\mathbb{R}$ with

$u(s)=\exp_{x^{\pm}}(\xi^{\pm}(s))\qquad\forall\pm s\geq T.$

See Figure 12. Local charts on $\widetilde{\mathscr{E}}_{k}(C^{-},C^{+})$ are
constructed by exponential neighbourhoods around smooth paths [Sch93, Appendix
A]. Similar to [AW13], one can construct scale smooth charts using those
exponential neighbourhoods, equipping the Banach manifold
$\widetilde{\mathscr{E}}(C_{-},C_{+})\mathrel{\mathop{\mathchar
58\relax}}=\widetilde{\mathscr{E}}_{0}(C_{-},C_{+})$ with a scale smooth
structure. Moreover, there are two natural scale smooth evaluation maps

$\operatorname{ev}^{\pm}\colon\widetilde{\mathscr{E}}(C^{-},C^{+})\to\operatorname{Crit}f,\qquad\operatorname{ev}^{\pm}(u)=x^{\pm}.$

Figure 12: An asymptotically exponential Sobolev path connecting two critical
components of a Morse–Bott function.

Fix $x^{\pm}\in C^{\pm}\cap\operatorname{Crit}h$ and define the stable and
unstable manifolds by

$W^{\pm}(x^{\pm})\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\biggl{\\{}x\in\operatorname{Crit}f\mathrel{\mathop{\mathchar
58\relax}}\lim_{s\to\pm\infty}\phi_{s}(x)=x^{\pm}\biggr{\\}}},$

where $\phi\colon\mathbb{R}\times\operatorname{Crit}f\to\operatorname{Crit}f$
denotes the negative gradient flow of $h$ with respect to the Morse–Smale
metric $m_{0}$. Define

$\widetilde{\mathscr{E}}_{k}^{1}(x^{-},x^{+})\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\bigl{\\{}u\in\widetilde{\mathscr{E}}_{k}(C^{-},C^{+})\mathrel{\mathop{\mathchar
58\relax}}\operatorname{ev}^{\pm}(u)\in W^{\pm}(x^{\pm})\bigr{\\}}}$

for all $k\in\mathbb{N}$. See Figure 13. Again,
$\widetilde{\mathscr{E}}^{1}(x_{-},x_{+})\mathrel{\mathop{\mathchar
58\relax}}=\widetilde{\mathscr{E}}^{1}_{0}(x_{-},x_{+})$ is a scale manifold.
For details, see the proof of [Fra04, Theorem A.12] and [Fra04, Theorem A.14].

Figure 13: A weighted Sobolev path with one cascade.

There exists a canonical scale smooth bundle

$\widetilde{\mathscr{F}}^{1}(x^{-},x^{+})\to\widetilde{\mathscr{E}}^{1}(x^{-},x^{+}),$

where the fibre over $u\in\widetilde{\mathscr{E}}_{k}^{1}(C^{-},C^{+})$ is
given by

$\widetilde{\mathscr{F}}_{k}^{1}(x^{-},x^{+})|_{u}=H^{k+1}_{\delta_{k}}(\mathbb{R},u^{*}TM)$

for all $k\in\mathbb{N}$. Then

$\widetilde{\partial}\colon\widetilde{\mathscr{E}}^{1}(x^{-},x^{+})\to\widetilde{\mathscr{F}}^{1}(x^{-},x^{+}),\qquad\widetilde{\partial}(u)\mathrel{\mathop{\mathchar
58\relax}}=\dot{u}+\operatorname{grad}_{m}f(u)$

is a scale Fredholm operator. If $x^{-}\neq x^{+}$, the scale smooth
reparametrisation action

$\mathbb{R}\oplus\widetilde{\mathscr{E}}^{1}(x^{-},x^{+})\to\widetilde{\mathscr{E}}^{1}(x^{-},x^{+}),\qquad(r,u)\mapsto
u(\cdot+r),$

as in Example C.4 is free, giving rise to a scale smooth bundle

$\widetilde{\mathscr{F}}^{1}(x^{-},x^{+})/\mathbb{R}\to\widetilde{\mathscr{E}}^{1}(x^{-},x^{+})/\mathbb{R}$

to which $\widetilde{\partial}$ descends to a scale Fredholm section
$\partial$. This construction can be generalised to $m$ cascades and thus
yields the scale smooth bundle

$\mathscr{F}^{m}(x^{-},x^{+})\to\mathscr{E}^{m}(x^{-},x^{+})$

of unparametrised paths from $x^{-}$ to $x^{+}$ with $m$ cascades. Now we
consider broken paths. For $x^{0},\dots,x^{l}\in\operatorname{Crit}h$ with

$f(x^{0})\geq f(x^{1})\geq\dots\geq f(x^{l-1})\geq f(x^{l}),$

we define the space of broken paths from $x^{0}$ to $x^{l}$ by

$\mathscr{E}(x^{0},\dots,x^{l})\mathrel{\mathop{\mathchar
58\relax}}=\coprod_{(\alpha_{0},\dots,\alpha_{l})\in\mathbb{N}^{l+1}_{0}}\mathscr{E}^{\alpha_{0}}(x^{0},x^{1})\times\dots\times\mathscr{E}^{\alpha_{l}}(x^{l-1},x^{l}).$

Then $\mathscr{E}(x^{0},\dots,x^{l})$ is a scale manifold over which we have
the scale bundle

$\mathscr{F}(x^{0},\dots,x^{l})\mathrel{\mathop{\mathchar
58\relax}}=\coprod_{(\alpha_{0},\dots,\alpha_{l})\in\mathbb{N}^{l+1}_{0}}\mathscr{F}^{\alpha_{0}}(x^{0},x^{1})\times\dots\times\mathscr{F}^{\alpha_{l}}(x^{l-1},x^{l})$

and the scale Fredholm section

$\partial\times\dots\times\partial\colon\mathscr{E}(x^{0},\dots,x^{l})\to\mathscr{F}(x^{0},\dots,x^{l}).$

For $x^{\pm}\in\operatorname{Crit}h$ with $f(x^{-})\geq f(x^{+})$ we define
the space of broken paths from $x^{-}$ to $x^{+}$ by

$X(x^{-},x^{+})\mathrel{\mathop{\mathchar
58\relax}}=\coprod_{l\in\mathbb{N}}\coprod_{\begin{subarray}{c}x^{1},\dots,x^{l-1}\in\operatorname{Crit}h\\\
f(x^{-})\geq f(x^{1})\geq\dots\geq f(x^{l-1})\geq
f(x^{+})\end{subarray}}\mathscr{E}(x^{-},x^{1},\dots,x^{l-1},x^{+}).$

Similarly, one defines

$Y(x^{-},x^{+})\mathrel{\mathop{\mathchar
58\relax}}=\coprod_{l\in\mathbb{N}}\coprod_{\begin{subarray}{c}x^{1},\dots,x^{l-1}\in\operatorname{Crit}h\\\
f(x^{-})\geq f(x^{1})\geq\dots\geq f(x^{l-1})\geq
f(x^{+})\end{subarray}}\mathscr{F}(x^{-},x^{1},\dots,x^{l-1},x^{+}).$

One can show that $X(x^{-},x^{+})$ carries the natural structure of an
M-polyfold and

$Y(x^{-},x^{+})\to X(x^{-},x^{+})$

is a bundle. Moreover, the scale Fredholm section above induces a scale
Fredholm section

$\partial\colon X(x^{-},x^{+})\to Y(x^{-},x^{+})$

of index

$\operatorname{ind}\partial=\operatorname{ind}(x^{-})-\operatorname{ind}(x^{+})-1,$

where either

$\operatorname{ind}=\operatorname{ind}_{f}+\operatorname{ind}_{h}\qquad\text{or}\qquad\operatorname{ind}=-\frac{1}{2}\operatorname{sgn}\operatorname{Hess}f-\frac{1}{2}\operatorname{sgn}\operatorname{Hess}h.$

For details see [CF09, Appendix A]. One can show that the level set
$\partial^{-1}(0)$ of unparametrised broken negative gradient flow lines is
compact and that there exists an abstract perturbation of $\partial$, that is,
a scale smooth section $\sigma\colon X\to Y$, where

$X\mathrel{\mathop{\mathchar
58\relax}}=\coprod_{\begin{subarray}{c}x^{\pm}\in\operatorname{Crit}h\\\
f(x^{-})\geq f(x^{+})\end{subarray}}X(x^{-},x^{+})\qquad\text{and}\qquad
Y\mathrel{\mathop{\mathchar
58\relax}}=\coprod_{\begin{subarray}{c}x^{\pm}\in\operatorname{Crit}h\\\
f(x^{-})\geq f(x^{+})\end{subarray}}Y(x^{-},x^{+}),$

with the property that for all $x^{\pm}\in\operatorname{Crit}h$ the moduli
space

$\mathscr{M}(x^{-},x^{+})\mathrel{\mathop{\mathchar 58\relax}}=\\{u\in
X(x^{-},x^{+})\mathrel{\mathop{\mathchar 58\relax}}(\partial+\sigma)(u)=0\\}$

is compact and $\partial+\sigma$ is transverse to the zero section. Hence we
arrive at a prototypical result for Morse–Bott homology, compare [Cie18,
Corollary 8.9].

###### Theorem C.1 (M-Polyfold Setup for Morse–Bott Homology).

Let $(M,g)$ be a compact Riemannian manifold without boundary and $f\in
C^{\infty}(M)$ a Morse–Bott function. Choose an additional Morse function
$h\in C^{\infty}(\operatorname{Crit}f)$ and an additional Riemannian metric
$g_{0}$ on $\operatorname{Crit}f$ such that $(h,g_{0})$ is a Morse–Smale pair.
For all critical points $x^{\pm}\in\operatorname{Crit}h$, the moduli space

$\mathscr{M}(x^{-},x^{+})\mathrel{\mathop{\mathchar 58\relax}}=\\{u\in
X(x^{-},x^{+})\mathrel{\mathop{\mathchar 58\relax}}(\partial+\sigma)(u)=0\\}$

is a smooth compact manifold with corners of dimension

$\dim\mathscr{M}(x^{-},x^{+})=\operatorname{ind}(x^{-})-\operatorname{ind}(x^{+})-1.$

Moreover, there is a canoncial diffeomorphism

$\partial\mathscr{M}(x^{-},x^{+})=\mathscr{M}(x^{-},x^{+})\cap\partial
X\cong\coprod_{x\in\operatorname{Crit}h}\mathscr{M}(x^{-},x)\times\mathscr{M}(x,x^{+}).$

##  Appendix D Bubbling Analysis

In this section we prove the main result about the compactness of the moduli
space of negative gradient flow lines of the symplectic action functional.

###### Definition D.1 (Symplectic Asphericity).

A connected symplectic manifold $(M,\omega)$ is said to be symplectically
aspherical, if

$\int_{\mathbb{S}^{2}}f^{*}\omega=0\qquad\forall f\in
C^{\infty}(\mathbb{S}^{2},M).$

###### Remark D.1.

A symplectic manifold $(M,\omega)$ is symplectically aspherical, if and only
if $[\omega]|_{\pi_{2}(M)}=0$, where
$[\omega]\in\operatorname{H}_{\mathrm{dR}}^{2}(M;\mathbb{R})$ denotes the
cohomology class of the closed form $\omega$.

###### Theorem D.1 (Bubbling).

Let $(M,\omega)$ be a compact symplectically aspherical symplectic manifold
and let $(u_{k})$ be a sequence of negative gradient flow lines of the
symplectic action functional $\mathcal{A}_{H}$ for some $H\in
C^{\infty}(M\times\mathbb{T})$ with uniformly bounded energy

$E_{J}(u_{k})\mathrel{\mathop{\mathchar
58\relax}}=\int_{-\infty}^{+\infty}\mathinner{\lVert\partial_{s}u_{k}\rVert}^{2}_{J}$

for some, and hence every, $\omega$-compatible almost complex structure $J$.
Then the derivatives of $(u_{k})$ are uniformly bounded.

The main idea of the proof is to assume that the derivatives $(Du_{k})$
explode and then to construct a nonconstant $J$-holomorphic sphere. Indeed,
assume that there exists a sequence $(s_{k},t_{k})$ in
$\mathbb{R}\times\mathbb{T}$ such that

$\lim_{k\to\infty}\mathinner{\lVert\partial_{s}u_{k}(s_{k},t_{k})\rVert}\to+\infty.$

Then we rescale the sequence $(u_{k})$, see Figure 14. Set

$m_{k}\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\lVert\partial_{s}u_{k}(s_{k},t_{k})\rVert}\qquad\text{and}\qquad
v_{k}(\sigma,\tau)\mathrel{\mathop{\mathchar
58\relax}}=u_{k}\mathinner{\biggl{(}\frac{\sigma}{m_{k}}+s_{k},\frac{\tau}{m_{k}}+t_{k}\biggr{)}}$

for all $(\sigma,\tau)\in\mathbb{C}$.

Then we compute

$\displaystyle m_{k}\partial_{\sigma}v_{k}(\sigma,\tau)$
$\displaystyle=\partial_{s}u_{k}\mathinner{\biggl{(}\frac{\sigma}{m_{k}}+s_{k},\frac{\tau}{m_{k}}+t_{k}\biggr{)}},$
$\displaystyle m_{k}\partial_{\tau}v_{k}(\sigma,\tau)$
$\displaystyle=\partial_{t}u_{k}\mathinner{\biggl{(}\frac{\sigma}{m_{k}}+s_{k},\frac{\tau}{m_{k}}+t_{k}\biggr{)}}.$

In particular $\mathinner{\lVert\partial_{\sigma}v_{k}(0,0)\rVert}=1$ for all
$k\in\mathbb{N}$. Moreover, every $v_{k}$ solves

$\partial_{\sigma}v_{k}(\sigma,\tau)+J\partial_{\tau}v_{k}(\sigma,\tau)=\frac{1}{m_{k}}JX_{H_{\frac{\tau}{m_{k}}+t_{k}}}(v_{k}(\sigma,\tau))\qquad\forall(\sigma,\tau)\in\mathbb{C}$

as $u_{k}$ satisfies the Floer equation (5). If there exists $v_{\infty}\in
C^{\infty}(\mathbb{C},M)$ such that

$v_{k}\xrightarrow{C^{\infty}_{\operatorname{loc}}}v_{\infty},\qquad
k\to\infty,$

modulo subsequences, then $v_{\infty}$ satisfies

$\partial_{\sigma}v_{\infty}(\sigma,\tau)+J\partial_{\tau}v_{\infty}(\sigma,\tau)=0\qquad\forall(\sigma,\tau)\in\mathbb{C}.$

Consequently, $v_{\infty}$ is a $J$-holomorphic plane. Using the assumption
that the energy of the sequence $(u_{k})$ is uniformly bounded, one can extend
$v_{\infty}$ to a $J$-holomorphic sphere $v\in C^{\infty}(\mathbb{S}^{2},M)$
such that $v|_{\mathbb{C}}=v_{\infty}$ via the identification
$\mathbb{S}^{2}\cong\mathbb{C}\cup\\{\infty\\}$.

Figure 14: Looking at the sequence $(u_{k})$ of negative gradient flow lines
of the symplectic action functional with a magnifying glass via rescaling.

###  D.1 Rescaling

###### Lemma D.1 (Hofer, [AH19, Lemma 4.4.4]).

Let $(X,d)$ be a complete metric space and

$f\colon X\to\mathinner{[0,+\infty)}$

continuous. Given $\varepsilon>0$ and $x\in X$, there exist
$0<\delta\leq\varepsilon$ and $y\in X$ such that

$d(x,y)\leq 2\delta,\qquad\delta f(y)\geq\varepsilon
f(x),\qquad\text{and}\qquad\sup_{z\in\overline{B}_{\delta}(y)}f(z)\leq 2f(y).$

###### Lemma D.2 (Rescaling).

Let $(u_{k})$ be a sequence of negative gradient flow lines of the symplectic
action functional $\mathcal{A}_{H}$ for some $H\in
C^{\infty}(M\times\mathbb{T})$ and $J$ some $\omega$-compatible almost complex
structure. Then there exists a sequence $(R_{k})$ in $\mathinner{(0,+\infty)}$
with $R_{k}\to+\infty$ and a sequence $v_{k}\in C^{\infty}(B_{R_{k}}(0),M)$
such that

$\lim_{k\to\infty}\mathinner{\lVert\partial_{\sigma}v_{k}+J\partial_{\tau}v_{k}\rVert}_{\infty}=0\qquad\text{and}\qquad
1\leq\mathinner{\lVert\partial_{\sigma}v_{k}\rVert}_{\infty}\leq 2.$

###### Proof.

Abbreviate $z_{k}\mathrel{\mathop{\mathchar
58\relax}}=(s_{k},t_{k})\in\mathbb{R}\times\mathbb{T}$. Apply Hofer’s Lemma
D.1 with

$f\colon\mathbb{R}\times\mathbb{T}\to\mathinner{[0,+\infty)},\qquad
f(s,t)\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\lVert\partial_{s}u_{k}(s,t)\rVert},$

as well as

$\varepsilon_{k}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sqrt{m_{k}}},\qquad\text{and}\qquad
x_{k}\mathrel{\mathop{\mathchar 58\relax}}=z_{k}.$

Thus there exists a nearby sequence
$z^{\prime}_{k}=(s^{\prime}_{k},t^{\prime}_{k})\in\mathbb{R}\times\mathbb{T}$
and $0<\delta_{k}\leq\varepsilon_{k}$ with

$\delta_{k}\mathinner{\lVert\partial_{s}u_{k}(z^{\prime}_{k})\rVert}\geq\varepsilon_{k}\mathinner{\lVert\partial_{s}u_{k}(z_{k})\rVert}=\sqrt{m}_{k}$

and

$\sup_{z\in
B_{\delta_{k}}(z_{k}^{\prime})}\mathinner{\lVert\partial_{s}u_{k}(z)\rVert}\leq
2\mathinner{\lVert\partial_{s}u_{k}(z^{\prime}_{k})\rVert}.$

Rescale $m_{k}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\lVert\partial_{s}u_{k}(z_{k}^{\prime})\rVert}$, set
$R_{k}\mathrel{\mathop{\mathchar 58\relax}}=\delta_{k}m_{k}^{\prime}$ and

$v_{k}(\sigma,\tau)\mathrel{\mathop{\mathchar
58\relax}}=u_{k}\mathinner{\biggl{(}\frac{\sigma}{m_{k}^{\prime}}+s_{k}^{\prime},\frac{\tau}{m_{k}^{\prime}}+t_{k}^{\prime}\biggr{)}}\qquad\forall(\sigma,\tau)\in
B_{R_{k}}(0).$

As $u_{k}$ is a solution to the Floer equation (5), $v_{k}$ satisfies the
equation

$\partial_{\sigma}v_{k}(\sigma,\tau)+J\partial_{\tau}v_{k}(\sigma,\tau)=\frac{1}{m_{k}^{\prime}}JX_{H_{\frac{\tau}{m_{k}^{\prime}}+t_{k}^{\prime}}}(v_{k}(\sigma,\tau))$
(43)

for all $(\sigma,\tau)\in B_{R_{k}}(0)$. As $m^{\prime}_{k}\geq
m_{k}\to+\infty$, we conclude

$\lim_{k\to\infty}\mathinner{\lVert\partial_{\sigma}v_{k}+J\partial_{\tau}v_{k}\rVert}_{\infty}=0.$

Finally, we compute

$\displaystyle\mathinner{\lVert\partial_{\sigma}v_{k}\rVert}_{\infty}$
$\displaystyle=\sup_{z\in
B_{R_{k}}(0)}\mathinner{\lVert\partial_{\sigma}v_{k}(z)\rVert}$
$\displaystyle=\frac{1}{m_{k}^{\prime}}\sup_{z\in
B_{\delta_{k}}(z_{k}^{\prime})}\mathinner{\lVert\partial_{s}u_{k}(z)\rVert}$
$\displaystyle\leq\frac{2}{m_{k}^{\prime}}\mathinner{\lVert\partial_{s}u_{k}(z^{\prime}_{k})\rVert}$
$\displaystyle=2,$

and

$m_{k}^{\prime}\mathinner{\lVert\partial_{\sigma}v_{k}(0,0)\rVert}=\mathinner{\lVert\partial_{s}u_{k}(z_{k}^{\prime})\rVert}=m_{k}^{\prime}.$

This concludes the proof of the lemma. ∎

###### Lemma D.3.

Let $(v_{k})$ be the sequence constructed in the Rescaling Lemma D.2. Then
there exists $v_{\infty}\in C^{0}(\mathbb{C},M)$ such that

$v_{k}\xrightarrow{C^{0}_{\mathrm{loc}}}v_{\infty},\qquad k\to\infty,$

up to a subsequence.

###### Proof.

By the Rescaling Lemma D.2 and compactness of $M$ there exists $C>0$ such that

$\mathinner{\lVert\partial_{\tau}v_{k}\rVert}_{\infty}\leq C\qquad\forall
k\in\mathbb{N}.$

Fix $R>0$ and choose $K_{R}\in\mathbb{N}$ such that $R_{k}\geq R$ for all
$k\geq K_{R}$ and consider the restrictions $v_{k}|_{B_{R}(0)}$ for all $k\geq
K_{R}$. As the derivatives of $v_{k}$ are uniformly bounded, we conclude that
$v_{k}|_{B_{R}(0)}$ is of Sobolev class

$W^{1,\infty}(B_{R}(0))\mathrel{\mathop{\mathchar 58\relax}}=\\{f\in
W^{1,\infty}(B_{R}(0),\mathbb{R}^{4n+1})\mathrel{\mathop{\mathchar
58\relax}}f(B_{R}(0))\subseteq M\\}$

for all $k\geq K_{k}$ where we consider
$M^{2n}\hookrightarrow\mathbb{R}^{4n+1}$ via the Whitney embedding Theorem.
Thus by Morrey’s inequality [Bre11, Corollary 9.14], every $v_{k}|_{B_{R}(0)}$
is of Hölder class $C^{0,1}(B_{R}(0))$ , and hence $v_{k}|_{B_{R}(0)}$ is
equicontinuous. As $M$ is compact, Ascolis Theorem implies the existence of
$v^{R}\in C^{0}(B_{R}(0),M)$ with

$v_{k}|_{B_{R}(0)}\xrightarrow{C^{0}}v^{R},\qquad k\to\infty,$

up to a subsequence. Choose a subsequence $(k_{j}^{1})$ with $k_{j}^{1}\geq
K_{1}$ for all $j\in\mathbb{N}$ and such that there exists $v^{1}\in
C^{0}(B_{1}(0),M)$ with

$v_{k_{j}^{1}}|_{B_{1}(0)}\xrightarrow{C^{0}}v^{1},\qquad j\to\infty.$

Inductively, choose a subsequence $(k_{j}^{\mu+1})$ of $(k_{j}^{\mu})$ for all
$\mu\in\mathbb{N}$ with $k_{j}^{\mu+1}\geq K_{\mu+1}$ for all $j\in\mathbb{N}$
and such that there exists $v^{\mu+1}\in C^{0}(B_{\mu+1}(0),M)$ with

$v_{k^{\mu+1}_{j}}|_{B_{\mu+1}(0)}\xrightarrow{C^{0}}v^{\mu+1},\qquad
j\to\infty.$

Finally, taking the diagonal subsequence yields

$v_{k^{j}_{j}}\xrightarrow{C^{0}_{\operatorname{loc}}}v_{\infty}\in
C^{0}(\mathbb{C},M),\qquad j\to\infty.$

∎

###  D.2 Elliptic Bootstrapping

###### Lemma D.4 (Elliptic Bootstrapping).

Denote by $v_{\infty}\in C^{0}(\mathbb{C},M)$ the map constructed in Lemma
D.3. Then $v_{\infty}\in C^{\infty}(\mathbb{C},M)$ is a nonconstant
$J$-holomorphic plane.

###### Proof.

Fix sequences $(z_{j})\subseteq\mathbb{C}$ and
$r_{j}\subseteq\mathinner{(0,+\infty)}$ such that

  * •

$v_{\infty}|_{B_{4r_{j}}(z_{j})}$ is contained in a chart $U_{j}$ of $M$ for
all $j\in\mathbb{N}$.

  * •

$\bigcup_{j\in\mathbb{N}}B_{r_{j}}(z_{j})=\mathbb{C}$.

  * •

for all $j\in\mathbb{N}$ there exists $K_{j}$ such that
$v_{k}|_{B_{2r_{j}}(z_{j})}\subseteq U_{j}$ for all $k\geq K_{j}$.

Fix smooth bump functions $\beta_{j}\in
C^{\infty}(\mathbb{C},\mathinner{[0,1]})$ for $\overline{B}_{r_{j}}(z_{j})$
supported in $B_{2r_{j}}(z_{j})$ and define

$\overline{v}_{k}^{j}\mathrel{\mathop{\mathchar 58\relax}}=\beta_{j}v_{k}\in
C^{\infty}_{c}(\mathbb{C},\mathbb{R}^{2n})$

for all $k,j\in\mathbb{N}$. We compute

$\Delta\overline{v}^{j}_{k}=(\Delta\beta_{j})v_{k}+2\partial_{\sigma}\beta_{j}\partial_{\sigma}v_{k}+2\partial_{\tau}\beta_{j}\partial_{\tau}v_{k}+\beta_{j}\Delta
v_{k}.$ (44)

To compute $\Delta v_{k}$, we differentiate (43) in charts. Applying
$\partial_{\sigma}$ we get that

$\partial_{\sigma}^{2}v_{k}(\sigma,\tau)+DJ\partial_{\sigma}v_{k}(\sigma,\tau)\partial_{\tau}v_{k}(\sigma,\tau)+J\partial_{\sigma}\partial_{\tau}v_{k}(\sigma,\tau)$

is equal to

$\frac{1}{m_{k}^{\prime}}D\operatorname{grad}_{J}H_{\frac{\tau}{m_{k}^{\prime}}+t_{k}^{\prime}}(v_{k}(\sigma,\tau))\partial_{\sigma}v_{k}(\sigma,\tau),$

where $\operatorname{grad}_{J}H_{t}$ denotes the gradient of $H_{t}$ with
respect to the Riemannian metric induced by $J$, that is, the Riemannian
metric $\omega(J\cdot,\cdot)$. Applying $\partial_{\tau}$ to (43) we get that

$\partial_{\tau}\partial_{\sigma}v_{k}(\sigma,\tau)+J\partial_{\tau}^{2}v_{k}(\sigma,\tau)+DJ(\partial_{\tau}v_{k}(\sigma,\tau))^{2}$

is equal to

$\frac{1}{m_{k}^{\prime}}D\operatorname{grad}_{J}H_{\frac{\tau}{m_{k}^{\prime}}+t_{k}^{\prime}}(v_{k}(\sigma,\tau))\partial_{\tau}v_{k}(\sigma,\tau)+\frac{1}{m_{k}^{\prime}}\partial_{\tau}\operatorname{grad}_{J}H_{\frac{\tau}{m_{k}^{\prime}}+t_{k}^{\prime}}(v_{k}(\sigma,\tau)).$

Hence

$\Delta v_{k}=P(\partial_{\sigma}v_{k},\partial_{\tau}v_{k}),$

where $P$ is a polynomial of degree $2$ with $C^{\infty}$-coefficients and so

$\Delta\overline{v}_{k}^{j}=P(\partial_{\sigma}\overline{v}_{k}^{j},\partial\overline{v}_{k}^{j})$

by (44). As the derivatives $\partial_{\sigma}v_{k}$ and
$\partial_{\tau}v_{k}$ are uniformly bounded, we conclude that
$\Delta\overline{v}_{k}^{j}$ is also uniformly bounded. Thus
$\mathinner{\lVert\Delta\overline{v}_{k}^{j}\rVert}_{L^{p}}$ is uniformly
bounded for all exponents $p\in\mathinner{[1,+\infty]}$ as
$\Delta\overline{v}_{k}^{j}$ is compactly supported by construction.
Therefore, the Calderon–Zygmund inequality [MS12, Corollary B.2.7] implies
that $\mathinner{\lVert\overline{v}_{k}^{j}\rVert}_{W^{2,p}}$ is uniformly
bounded for all $1<p<+\infty$. In particular,

$\mathinner{\lVert\partial_{\sigma}\overline{v}_{k}^{j}\rVert}_{W^{1,p}}\qquad\text{and}\qquad\mathinner{\lVert\partial_{\tau}\overline{v}_{k}^{j}\rVert}_{W^{1,p}}$

are uniformly bounded. Thus again by Morrey’s inequality,
$\partial_{\sigma}\overline{v}_{k}^{j}$ and
$\partial_{\tau}\overline{v}_{k}^{j}$ belong to the Hölder class
$C^{0,\alpha}$ for all $0<\alpha<1$. By Ascolis Theorem,
$\partial_{\sigma}\overline{v}_{k}^{j}$ and
$\partial_{\tau}\overline{v}_{k}^{j}$ admit convergent subsequences and

$\overline{v}_{\infty}^{j}\mathrel{\mathop{\mathchar
58\relax}}=\beta_{j}v_{\infty}\in C^{1}_{c}(\mathbb{C},\mathbb{R}^{2n})$

satisfies

$v_{k}|_{B_{r_{j}}(z_{j})}\xrightarrow{C^{1}}\overline{v}_{\infty}^{j}$

for all $j\in\mathbb{N}$. Hence we have showed that $v_{\infty}\in
C^{1}(\mathbb{C},M)$ and

$v_{k}\xrightarrow{C^{1}_{\operatorname{loc}}}v_{\infty}$

up to subsequences. This procedure can be generalised to higher derivatives
and is referred to as elliptic bootstrapping. Note that taking higher order
derivatives makes only sense locally in a chart if we do not refer to a
particular connection. Thus we get $v_{\infty}\in C^{\infty}(\mathbb{C},M)$
and

$v_{k}\xrightarrow{C^{\infty}_{\operatorname{loc}}}v_{\infty},\qquad
k\to\infty,$

up to subsequences. Finally, $v_{\infty}$ is nonconstant as

$\mathinner{\lVert\partial_{\sigma}v_{\infty}\rVert}_{\infty}=\lim_{k\to\infty}\mathinner{\lVert\partial_{\sigma}v_{k}\rVert}_{\infty}\geq
1$

and satisfies

$\partial_{\sigma}v_{\infty}+J\partial_{\tau}v_{\infty}=0$

by the Rescaling Lemma D.2. ∎

###  D.3 Removal of Singularities

Consider the conformal diffeomorphism

$\varphi\colon\mathbb{C}\setminus\\{0\\}\to\mathbb{R}\times\mathbb{T},\qquad\varphi(re^{2\pi
i\theta})\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Log}(re^{2\pi
i\theta}),$

and define

$w\colon\mathbb{R}\times\mathbb{T}\to M,\qquad w\mathrel{\mathop{\mathchar
58\relax}}=v_{\infty}\circ\varphi^{-1},$

where $v_{\infty}\in C^{\infty}(\mathbb{C},M)$ denotes the nonconstant
$J$-holomorphic plane constructed in Lemma D.3. Then

$\lim_{s\to-\infty}w(s,\cdot)=v_{\infty}(0)$

and $w$ is of finite energy. Indeed, we compute

$E_{J}(w)=\int_{\mathbb{R}\times\mathbb{T}}w^{*}\omega=\int_{-\infty}^{+\infty}\int_{0}^{1}\omega(J\partial_{s}w(s,t),\partial_{s}w(s,t))dtds\leq\sup_{k\in\mathbb{N}}E_{J}(u_{k})$

by [MS12, Lemma 2.2.1]. Moreover, $w$ is a negative gradient flow line of the
symplectic area functional

$\mathscr{A}\colon\Lambda
M\to\mathbb{R},\qquad\mathscr{A}(\gamma)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\mathbb{D}}\overline{\gamma}^{*}\omega.$

###### Lemma D.5.

For every sequence $(r_{k})\subseteq\mathinner{(0,+\infty)}$ with
$r_{k}\to+\infty$ as $k\to\infty$ define a sequence $(w_{k})$ by

$w_{k}(s,t)\mathrel{\mathop{\mathchar
58\relax}}=w(s+r_{k},t)\qquad\forall(s,t)\in\mathbb{R}\times\mathbb{T}.$

Then there exists a point $w_{\infty}\in M$ such that

$w_{k}\xrightarrow{C^{\infty}_{\operatorname{loc}}}w_{\infty},\qquad
k\to\infty,$

up to a subsequence.

###### Proof.

By [MS12, p. 89–90] there exists a constant $a\geq 0$ such that

$\Delta e\geq-ae^{2},$

where

$e\colon\mathbb{R}\times\mathbb{T}\to\mathinner{[0,+\infty)},\qquad
e(s,t)\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\lVert\partial_{s}w(s,t)\rVert}^{2}$

denotes the energy density. As $E_{J}(w)<+\infty$, there exists $R>0$ such
that

$\int_{R}^{+\infty}\int_{0}^{1}e(s,t)dtds\leq\min\mathinner{\biggl{\\{}\frac{\pi}{8a},1\biggr{\\}}}$

and

$\int_{-\infty}^{-R}\int_{0}^{1}e(s,t)dtds\leq\min\mathinner{\biggl{\\{}\frac{\pi}{8a},1\biggr{\\}}}.$

Let $z=(s,t)\in\mathbb{R}\times\mathbb{T}$ such that $\mathinner{\lvert
s\rvert}\geq R+1$. Then

$\int_{B_{1}(z)}e<\frac{\pi}{8a}.$

By [MS12, Lemma 4.3.3] we conclude

$e(z)\leq\frac{8}{\pi}\int_{B_{1}(z)}e\leq\frac{8}{\pi}.$

Consequently, $e$ is uniformly bounded, as
$e|_{\mathinner{[-R-1,R+1]}\times\mathbb{T}}$ is uniformly bounded by
continuity of $e$. Hence $\mathinner{\lVert\partial_{s}w\rVert}$ is uniformly
bounded by definition of $e$ and $\mathinner{\lVert\partial_{t}w\rVert}$ is
uniformly bounded as $w$ is a $J$-holomorphic curve. By an eliptic
bootrstrapping argument as in Lemma D.4, we conclude

$w_{k}\xrightarrow{C^{\infty}_{\operatorname{loc}}}w_{\infty},\qquad
k\to\infty,$

up to a subsequence, where $w_{\infty}$ is a negative gradient flow line of
the symplectic area functional $\mathscr{A}$. Then $w_{\infty}$ is constant.
Indeed, assume that $w_{\infty}$ is not constant. Then there exists
$s<s^{\prime}$ such that

$\varepsilon\mathrel{\mathop{\mathchar
58\relax}}=\mathscr{A}(w_{\infty}(s))-\mathscr{A}(w_{\infty}(s^{\prime}))>0.$

Moreover, there exists $K\in\mathbb{N}$ such that

$\mathscr{A}(w_{k}(s))-\mathscr{A}(w_{k}(s^{\prime}))=\mathscr{A}(w(s+r_{k}))-\mathscr{A}(w(s^{\prime}+r_{k}))\geq\frac{\varepsilon}{2}$

for all $k\geq K$. Define a subsequence $(r_{k_{j}})$ of $(r_{k})$ recursively
by

$k_{0}\mathrel{\mathop{\mathchar 58\relax}}=K\qquad\text{and}\qquad
k_{j}\mathrel{\mathop{\mathchar
58\relax}}=\min\\{l\in\mathbb{N}\mathrel{\mathop{\mathchar
58\relax}}s^{\prime}+r_{k_{j-1}}\leq s+r_{l}\\}.$

This works as $r_{k}\to+\infty$ as $k\to\infty$. Fix $l\in\mathbb{N}$. Then we
compute

$\displaystyle E(w)$
$\displaystyle=\sup_{s\in\mathbb{R}}\mathscr{A}(w(s))-\inf_{s\in\mathbb{R}}\mathscr{A}(w(s))$
$\displaystyle\geq\mathscr{A}(w(s+r_{k_{0}}))-\mathscr{A}(w(s+r_{k_{l}}))$
$\displaystyle=\sum_{\nu=1}^{l}\mathinner{\bigl{(}\mathscr{A}(w(s+r_{k_{\nu-1}}))-\mathscr{A}(w(s+r_{k_{\nu}}))\bigr{)}}$
$\displaystyle\geq\sum_{\nu=1}^{l}\mathinner{\bigl{(}\mathscr{A}(w(s+r_{k_{\nu-1}}))-\mathscr{A}(w(s^{\prime}+r_{k_{\nu-1}}))\bigr{)}}$
$\displaystyle\geq\frac{\varepsilon l}{2}.$

As $l\in\mathbb{N}$ was arbitrary, we conlcude that $E(w)=+\infty$. ∎

###### Proof of Theorem D.1.

As $M$ is compact, there exists a finite open cover $U_{1},\dots,U_{m}$ of $M$
such that $\overline{U}_{j}$ is contained in a Darboux chart. Define

$\mathscr{U}\mathrel{\mathop{\mathchar 58\relax}}=\\{\gamma\in\Lambda
M\mathrel{\mathop{\mathchar 58\relax}}\gamma(\mathbb{T})\subseteq U_{j}\text{
for some $j=1,\dots,m$}\\}.$

_Step 1: There exists $s_{0}\in\mathbb{R}$ such that $w(s)\in\mathscr{U}$ for
all $s\geq s_{0}$._ Assume that there exists a sequence
$(r_{k})\subseteq\mathinner{(0,+\infty)}$ with $r_{k}\to+\infty$ as
$k\to\infty$ and $w(r_{k})\notin\mathscr{U}$ for all $k\in\mathbb{N}$. By
Lemma D.5, there exists $w_{\infty}\in M$ such that

$w_{k}\xrightarrow{C^{\infty}_{\operatorname{loc}}}w_{\infty},\qquad
k\to\infty,$

up to a subsequence. But $w_{\infty}\in U_{j}$ for some $j$ and thus there
exists $K\in\mathbb{N}$ such that $w(r_{k},t)\in U_{j}$ for all
$t\in\mathbb{T}$ and $k\geq K$. Consequently, $w(r_{k})\in\mathscr{U}$ for all
$k\geq K$.

_Step 2: There exist constants $C>0$ and $\kappa>0$ such that_

$d_{L^{2}}(w(s),w_{\infty})\leq Ce^{-\kappa s}\qquad\forall s\geq s_{0},$

_where $w_{\infty}$ is the limit of $w(k)=w_{k}(0)$ as $k\to\infty$ up to a
subsequence $(k_{j})$._ By [FN20, Proposition 6.4] and [FN20, Lemma 6.3], we
have the action-energy inequality

$\mathinner{\lvert\mathscr{A}(\gamma)\rvert}\leq
C_{0}\mathinner{\lVert\operatorname{grad}_{J}\mathscr{A}(\gamma)\rVert}_{J}^{2}\qquad\forall\gamma\in\mathscr{U}$

for some constant $C_{0}>0$ as the symplectic area functional $\mathscr{A}$ is
Morse–Bott. See Remark 2.1. Let $s\geq s_{0}$. Choose $j_{0}$ such that
$k_{j_{0}}>s$. We estimate

$\mathscr{A}(w(s))>\mathscr{A}(w(k_{j_{0}}))\geq\lim_{j\to\infty}\mathscr{A}(w(k_{j}))=\lim_{j\to\infty}\mathscr{A}(w_{k_{j}}(0))=\mathscr{A}(w_{\infty})=0.$

Let $s_{2}>s_{1}>s_{0}$. Using [FN20, Lemma 6.5] and [FN20, Lemma 6.6] we
compute

$\displaystyle d(w(s_{1}),w(s_{2}))$
$\displaystyle\leq\frac{2}{\sqrt{C_{0}}}\mathinner{\bigl{(}\sqrt{\mathscr{A}(w(s_{1}))}-\sqrt{\mathscr{A}(w(s_{2}))}\bigr{)}}$
$\displaystyle\leq\frac{2}{\sqrt{C_{0}}}\sqrt{\mathscr{A}(w(s_{1}))}$
$\displaystyle\leq\frac{2}{\sqrt{C_{0}}}\sqrt{\mathscr{A}(w(s_{0}))}e^{\frac{1}{2C_{0}}(s_{0}-s_{1})}$

Choose $j_{1}$ such that $k_{j_{1}}\geq s_{1}$. Then for all $j\geq j_{1}$, we
have that

$d_{L^{2}}(w(s_{1}),w(k_{j}))\leq Ce^{-\kappa s_{1}},$

where

$C\mathrel{\mathop{\mathchar
58\relax}}=\frac{2}{\sqrt{C_{0}}}\sqrt{\mathscr{A}(w(s_{0}))}e^{\frac{1}{2C_{0}}s_{0}}\qquad\text{and}\qquad\kappa\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2C_{0}}.$

Thus

$d_{L^{2}}(w(s_{1}),w_{\infty})=\lim_{j\to\infty}d_{L^{2}}(w(s_{1}),w(k_{j}))\leq
Ce^{-\kappa s_{1}}.$

_Step 3: There exists a unique point $w_{\infty}\in M$ such that_

$w(s)\xrightarrow{C^{\infty}}w_{\infty},\qquad s\to+\infty.$

Uniqueness follows immediately from the exponential decay established in Step
2. Indeed, a priori, $w_{\infty}$ does depend on the choice of subsequence
$(k_{j})$. Let $w_{\infty}^{\prime}\in M$ be the limit of a different
subsequence. Then we compute

$d_{L^{2}}(w_{\infty},w_{\infty}^{\prime})\leq
d_{L^{2}}(w(s),w_{\infty})+d_{L^{2}}(w(s),w_{\infty}^{\prime})\leq Ce^{-\kappa
s}+C^{\prime}e^{-\kappa^{\prime}s}\to 0$

as $s\to+\infty$. That the limit can also be taken with respect to the
$C^{\infty}$-topology follows from [AD14, Proposition 6.5.3]. ∎

###### Corollary D.1.

Let $(M,\omega)$ be a symplectically aspherical symplectic manifold and let
$(u_{k},\tau_{k})$ be a sequence of solutions $(u_{k},\tau_{k})\in
C^{\infty}(\mathbb{R}\times\mathbb{T},M)\times
C^{\infty}(\mathbb{R},\mathbb{R})$ of

$\begin{cases}\partial_{s}u_{k}(s,t)+J(\partial_{t}u_{k}(s,t)-\tau_{k}(s)X_{H}(u_{k}(s,t)))=0,\\\
\displaystyle\partial_{s}\tau_{k}(s)=\int_{0}^{1}H(u_{k}(s,t))dt,\end{cases}$

for all $s\in\mathbb{R}$ and $k\in\mathbb{N}$ for some $H\in C^{\infty}(M)$,
with uniformly bounded energy. If there exists a compact subset $K\subseteq
M\times\mathbb{R}$ such that

$\operatorname{im}(u_{k},\tau_{k})\subseteq K\qquad\forall k\in\mathbb{N},$

then the derivatives of $(u_{k})$ are uniformly bounded.

###### Proof.

Crucial is the observation that under the above assumptions, the limit of (43)
is still a $J$-holomorphic curve. ∎

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