# Magnetic billiards and the Hofer-Zehnder capacity of disk tangent bundles of
lens spaces

Johanna Bimmermann111Fakultät für Mathematik, Ruhr-Universität Bochum,
Universitätsstraße 150, 44801 Bochum, Germany<EMAIL_ADDRESS>&
Levin Maier222Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg,
Im Neuenheimer Feld 206, 69120 Heidelberg, Germany<EMAIL_ADDRESS>heidelberg.de

###### Abstract

We compute the Hofer–Zehnder capacity of disk tangent bundles of certain lens
spaces with respect to the round metric. Interestingly we find that the
Hofer–Zehnder capacity does not see the covering, i.e. the capacity of the
disk tangent bundle of the lens space coincides with the capacity of the disk
tangent bundle of the 3-sphere covering it. In particular, this gives a first
example, where Gromov width and Hofer–Zehnder capacity of a disk tangent
bundle disagree. Techniques we use include for the lower bound magnetic
billiards and for the upper bound Gromov–Witten invariants.

##  1 Introduction

Since Gromov’s famous non-squeezing theorem symplectic embedding problems lie
at the heart of symplectic geometry. To obtain obstructions for symplectic
embeddings, symplectic capacities were introduced as numerical invariants of
symplectic manifolds. A normalized symplectic capacity is a function $c$ that
assigns a number $c(M,\omega)\in[0,\infty]$ to a symplectic manifold
$(M^{2n},\omega)$ of certain dimension $2n$, such that the following axioms
hold

  1. 1.

(Monotonicity) if $(M,\omega)$ symplectically embeds into
$(\tilde{M},\tilde{\omega})$ then $c\left(M,\omega\right)\leq
c\left(\tilde{M},\tilde{\omega}\right)$,

  2. 2.

(Conformality) for all $r>0$, we have
$c\left(M,r\omega\right)=r^{2}c\left(M,\omega\right)$,

  3. 3.

(Normalization)
$c\left(B^{2n},\omega_{0}\right)=\pi=c\left(Z,\omega_{0}\right)$.

Here $\left(B^{2n},\omega_{0}\right)$ denotes a ball of radius $1$ in
$\mathbb{R}^{2n}$ and $\left(Z=B^{2}\times\mathbb{R}^{2n-2},\omega_{0}\right)$
denotes a the cylinder of radius $1$ in $\mathbb{R}^{2n}$, where each is
equipped with the standard symplectic structure
$\omega_{0}=\sum_{j=1}^{n}\mathrm{d}x_{i}\wedge\mathrm{d}y_{j}$ . Before
moving on we would like to mention that normalized symplectic capacities are
intrinsically hard to compute since the existence of a normalized symplectic
capacity automatically leads to a proof of Gromov’s non squeezing theorem.
Nowadays there are several normalized symplectic capacities, see for an
overview for example [11], so it’s quite natural to ask on which class of
domains which symplectic capacities agree? This question is closely related to
two of the biggest unsolved problems of symplectic geometry: the _Viterbo
conjecture_ and the _strong Viterbo conjecture_.

######  Conjecture 1.1 (Strong Viterbo conjecture).

Let $X\subseteq\mathbb{R}^{2n}$ be a convex domain. Then all normalized
symplectic capacities agree on $X$.

Recent work of Abbondandolo-Benedetti-Edtmair [1] proves that the strong
Viterbo conjecture holds true in a $C^{2}$-neighborhood of the ball in
$\mathbb{R}^{2n}$. This extends earlier work of Edtmair [13] about the strong
Viterbor conjecture in a $C^{3}$-neighborhood of the ball in $\mathbb{R}^{4}$.
Let us also mention the recent work of Cristofaro-Gardiner-Hind [12] that
provide a positve answer to the strong Viterbo conjeture for all monotone
toric domains of any dimension. For a general overview we refer to Gutt-
Hutchings-Ramos [15]. Furthermore note that the strong Viterbo conjecture
implies:

######  Conjecture 1.2 (Viterbo conjecture).

Let $X\subseteq\mathbb{R}^{2n}$ be a convex domain. Then any normalized
symplectic capacity $c$ satisfies the inequality

$c(X)\leq\left(n!\ \mathrm{vol}(X)\right)^{\frac{1}{n}}.$

Note that by the seminal work of Artstein-Avidan-Karasev-Ostrover [5] the
_Viterbo conjecture_ implies the famous _Mahler conjecture_ , an old
conjecture in convex geometry. In addition, we note at this point that the
Gromov width $c_{G}$ fulfills the Viterbo conjecture, as it is volume-
sensitive. From a topological point the next natural class of symplectic
manifolds to investigate the behavior of normalized symplectic capacities are
unit disk bundles $D^{*}M$!
In this paper, we will study two classical normalized symplectic capacities
the Gromov width $c_{G}$ and the Hofer-Zehnder capacity $c_{\mathrm{HZ}}$ on
unit disk bundles over Lens spaces $L(p,1)$. Where the Gromov width $c_{G}$ of
a symplectic manifold $(M,\omega)$ measures the size of largest symplectically
embedded ball into $(M,\omega)$ i.e.,

$c_{G}(M,\omega):=\sup\\{\pi r^{2}\mid\text{ there exists a symplectic
embedding of }rB^{2n}\text{ into }(M,\omega)\\}.$

In contrast, the Hofer–Zehnder capacity [17, Ch. 3] measures the size of a
symplectic manifold in terms of the Hamiltonian dynamics on it. It is defined
as follows:

$c_{\mathrm{HZ}}(M,\omega):=\sup\left\\{\mathrm{max}(H)\ |\ H:M\to\mathbb{R}\
\ \text{smooth, admissible}\right\\},$

where admissible means:

  * (1)

$0\leq H$ and there exists an open set $U\subset M\setminus\partial M$ such
that $H|_{U}\equiv 0$,

  * (2)

there exists a compact set $K\subset M\setminus\partial M$ such that
$H|_{M\setminus K}\equiv\max(H)$,

  * (3)

all non-constant periodic solutions $\gamma:\mathbb{R}\to M$ of
$\dot{\gamma}=X_{H}$ have period $T>1$.

We are interested in the following set up. Let $(N,g)$ be a Riemannian
manifold and denote

$D_{1}N:=\\{(x,v)\in TN\ |\ g_{x}(v,v)<1\\}$

the unit disk bundle over $(N,g)$. As symplectic structure we consider the
metric pullback of the canonical symplectic structure on $T^{*}N$ and denote
it by $\mathrm{d}\lambda$, i.e.
$\lambda_{(x,v)}(\xi)=g_{x}(v,\mathrm{d}\pi(\xi))$ for all $\xi\in
T_{(x,v)}TN$. Although, (co-)tangent bundles are a very classical class of
symplectic manifolds, the Hofer-Zehnder capacity is only known in a few
examples namely real and complex projective space [9], flat tori [7] and some
convex subsets of $\mathbb{R}^{n}$ [5]. In this paper we will add (some) lens
spaces to the list.

######  Theorem 1.3.

For any odd $p\in\mathbb{N}$ look at $L(p;1)$ equipped with the metric induced
from the round metric on $S^{3}$, denote by $l$ the length of the shortest
closed geodesics. Then the Hofer-Zehnder capacity of $D_{1}L(p;1)$ equipped
with the canonical symplectic form is given by

$c_{\mathrm{HZ}}(D_{1}L(p;1),\mathrm{d}\lambda)=pl=2\pi.$

So far, in all examples where both Gromov width and Hofer–Zehnder capacity of
disk tangent bundles are known (e.g. $S^{2},\mathbb{R}\mathrm{P}^{2}$ and flat
tori, see [7, 9, 14]), they agreed. This theorem however shows that for $p$
large enough the Hofer–Zehnder capacity can not agree with the Gromov width.
This is easy to see as the Gromov width is volume sensitive i.e. must satisfy

$c_{G}(D_{1}L(p;1))\leq\left(3!\mathrm{vol}(D_{1}L(p;1))\right)^{\frac{1}{3}}=\left(3!\mathrm{vol}(D_{1}S^{3})/p)\right)^{\frac{1}{3}}$

while the Hofer–Zehnder capacity of $D_{1}L(p,1)$ is by Theorem 1.3 constant
in $p$.333At first it might not look like it, but the length of the shortest
geodesics on $L(p;1)$ is the length of prime geodesics on $S^{3}$ divided by
$p$. So, in particular, as a corollary of Theorem 1.3, we proved that neither
the analog of the Viterbo conjecture nor the analog of the strong Viterbo
conjecture can hold for disk bundles:

######  Corollary 1.4.

Let $(D_{1}N,\mathrm{d}\lambda)$ be the unit disk bundle over the
n-dimensional Riemannian manifold $(N,g)$ equipped with the canonical
symplectic structure. Then

  1. 1.

not all normalized symplectic capacities have to coincide on
$(D_{1}N,\mathrm{d}\lambda)$,

  2. 2.

not all normalized symplectic capacities $c$ have to satisfy the inequality

$c\left(D_{1}N,\mathrm{d}\lambda\right)\leq\left(n!\
\mathrm{vol}(D_{1}N,\mathrm{d}\lambda)\right)^{\frac{1}{n}}.$

######  Remark 1.5.

In particular the Hofer–Zehnder capacity of $D_{1}L(p,1)$ does not obey any
volume constraint, which is not too surprising as Usher showed that many
closed symplectic manifolds have infinite Hofer–Zehnder capacity [22].

The Theorem 1.3 is also interesting, when compared to a relative version of
the Hofer–Zehnder capacity. For a closed submanifold $\Sigma\subset M$ not
intersecting the boundary, i.e. $\Sigma\subset M\setminus\partial M$, the
Hofer–Zehnder capacity relative to $\Sigma$ is defined as

$c_{\mathrm{HZ}}(M,\Sigma,\omega):=\sup\left\\{\mathrm{max}(H)\ |\
H:M\to\mathbb{R}\ \ \text{smooth, admissible and vanishing on
}\Sigma\right\\}.$

Further for any homotopy class $\nu\in\pi_{1}(M)$ we define
$c_{\mathrm{HZ}}^{\nu}(M,\Sigma,\omega)$ by replacing condition (3) with
condition

  * (3)’

all non-constant periodic solutions $\gamma:\mathbb{R}\to M$ with
$[\gamma]=\nu$ of $\dot{\gamma}=X_{H}$ have period $T>1$.

An important result by Weber [25] shows that for closed $N$ and a non-zero
class $\nu\in\pi_{1}(N)$, the Hofer-Zehnder capacity relative to the zero
section is given by the length $l_{\nu}$ of the shortest closed geodesic in
the class $\nu\in\pi_{1}(M)$, i.e.

$c_{\mathrm{HZ}}^{\nu}(D_{1}N,N,\mathrm{d}\lambda)=l_{\nu}.$

For non-aspherical, homogeneous spaces and positive curvature metrics on the
2-sphere the result was extended to the class of contractible loops ($\nu=0$)
by Benedetti and Kang [8, Cor. 2.8]. In particular if either the shortest
geodesic is non-contractible or $N$ is non-aspherical, homogeneous spaces or a
positive curvature 2-sphere we can deduce

$c_{\mathrm{HZ}}(D_{1}N,N,\mathrm{d}\lambda)=l,$ (1)

where $l$ denotes the length of the shortest closed geodesic. This is because
$c_{\mathrm{HZ}}^{\nu}(D_{1}N,N,\mathrm{d}\lambda)$ yields an upper bound for
any homotopy class $\nu$ and the Hamiltonian $H(x,v):=l|v|$ can easily be
modified, without changing the oscillation much, so that it becomes
admissible.
In [9] it was shown that for $N=\mathbb{C}\mathrm{P}^{n}$ the absolute
capacity coincides with relative one, while for $N=\mathbb{R}\mathrm{P}^{n}$
there is a factor of 2. Using theorem 1.3 and eq. 1 we can extend the result
for $N=\mathbb{R}\mathrm{P}^{3}$ to the following class of lens spaces
$L(p,1)$:

######  Corollary 1.6.

For odd $p\in\mathbb{N}$, the Hofer-Zehnder capacity and the relative Hofer-
Zehnder capacity of $D_{1}L(p,1)$ are in the following ratio to each other:

$c_{\mathrm{HZ}}(D_{1}L(p;1),\mathrm{d}\lambda)=p\cdot
c_{\mathrm{HZ}}(D_{1}L(p;1),L(p;1),\mathrm{d}\lambda).$

Outline: The proof of Theorem 1.3 has two steps. The first is finding a lower
bound. For this we construct a nice Hamiltonian and determine its flow and in
particular its periodic orbits explicitly. The dynamics that will give the
correct lower bound lift to magnetic billiard dynamics on the 3-sphere with
the Hopf-link removed. The construction of this flow is carried out in section
2. The second step is finding an upper bound that coincides with the lower
bound. For the upper bound we use that

$c_{\mathrm{HZ}}(DL(p;1),\mathrm{d}\lambda)\leq
c_{\mathrm{HZ}}(DS^{3},\mathrm{d}\lambda),$

as admissible Hamiltonians on $DL(p;1)$ lift to admissible Hamiltonians on
$DS^{3}$. An upper bound for the capacity of $DS^{3}$ is then obtained from
results by Hofer–Viterbo [16] and Lu [20], which heavily relies on
pseudoholomorphic curve techniques. This works because the Lerman cut [18]
with respect to the geodesic flow $\overline{(DS^{3},\mathrm{d}\lambda)}$ is
symplectomorphic to the complex quadric $Q^{3}:=\\{[z_{0}:\ldots:z_{4}]\ |\
z_{0}^{2}+\ldots+z_{4}^{2}=0\\}\subset\mathbb{C}\mathrm{P}^{4}$ with Fubini-
Study form. Hence Gromov-Witten invariants are explicitly computable.

Acknowledgments.
The authors are grateful to G. Benedetti, D. Cristofaro-Gardiner, R. Hind and
S. Tabachnikov for helpful comments on a earlier draft of this paper. The
authors are also grateful to their old friend G. Mogol, who helped them out
with his Python skills. The authors further acknowledges funding by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 281869850
(RTG 2229), 390900948 (EXC-2181/1) and 281071066 (TRR 191). L.M. gratefully
acknowledges support from the Simons Center for Geometry and Physics, Stony
Brook University at which some of the research for this paper was performed
during the program Mathematical Billiards: at the Crossroads of Dynamics,
Geometry, Analysis, and Mathematical Physics. L.M. thanks also V. Assenza and
V. Ramos for their warm hospitality at IMPA in Rio de Janeiro in February 2024
where some of the research for this paper was performed. L.M. thanks the
participants of the conferences Symplectic Dynamics in Montevideo in
Montevideo in February 2024 for helpful feedback on preliminary versions of
this work.

##  2 Lower bound: magnetic billiards on lens spaces

Consider $S^{3}:=\\{|x|^{2}=1\\}\subset\mathbb{R}^{4}$, where $|\cdot|$
denotes the standard norm on $\mathbb{R}^{4}$. In order to obtain the correct
lower bound of the Hofer–Zehnder capacity for $D_{1}S^{3}$ one can use the
Hamiltonian $H(x,v):=2\pi|v|$. This reparametrizes the geodesic flow, so that
all orbits have period one. Now modifying $H$ a little near the zero-section
and the boundary one obtains an admissible Hamiltonian that has ossicillation
arbitrarily close to $1\cdot 2\pi$, hence gives the correct lower bound to
prove Theorem 1.3 in the case of $p=1$. For lens spaces the situation will be
more complicated as prime geodesics can have different lengths. Let us look at
the set up more closely. Fix an odd number $p\in\mathbb{N}$ and denote

$L(p;1)=S^{3}/\mathbb{Z}_{p}$

the lens space obtained from the 3-sphere by dividing out the following
$\mathbb{Z}_{p}$-action

$x\mapsto e^{\frac{2\pi}{p}j}x,$

where $j=i\ominus i=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}$ and we identified
$\mathbb{R}^{4}\cong\mathbb{C}\times\mathbb{C}$.

######  Lemma 2.1.

All prime geodesics, but the ones in direction $jx$, have length $2\pi$. The
prime geodesics in direction $jx$ have length $2\pi/p$.

###### Proof.

Take a point $x\in S^{3}$, now all points $e^{2\pi jq/p}x$ identified with $x$
under the $\mathbb{Z}_{p}$ action lie on the geodesic in direction $jx$. As
$p$ is odd for all $q$ the geodesic in direction $jx$ is the unique geodesic
through $x$ and $e^{2\pi jq/p}x$. ∎

Observe that this means simply reparametrizing the geodesic flow, as in the
case of $S^{3}$, is not enough to obtain the correct lower bound. Instead, we
want to introduce a small magnetic perturbation to destroy almost all of the
short geodesics. By abuse of notation, we write $i=i\oplus i$ for the standard
complex structure on $\mathbb{C}^{2}$. This complex structure induces the
standard contact form on $S^{3}$, namely

$\alpha_{x}:T_{x}S^{3}\to\mathbb{R};\ v\mapsto\frac{1}{2}\mathrm{Re}\langle
ix,v\rangle.$

Here $\langle\cdot,\cdot\rangle$ denotes the standard Hermitian product on
$\mathbb{C}^{2}$ with respect to $i$. To find a lower bound of the
Hofer–Zehnder capacity we need to explicitly construct an admissible
Hamiltonian. The strategy for constructing such a Hamiltonian is as follows:

  1. 1.

We will use the magnetic geodesic flow on
$\left(S^{3},g,\varepsilon\cdot\mathrm{d}\alpha\right)$ to destroy almost all
short geodesics except two, ${\gamma}_{\pm}$ parametrizing the Hopf link (see
Lemma 2.5).

  2. 2.

After removing the Hopf link $L={\gamma}_{+}\cup{\gamma}_{-}$ in $S^{3}$ and
adding a $\mathbb{Z}_{p}$-invariant potential, which tends to infinity near
${\gamma}_{\pm}$, to the kinetic Hamiltonian the induced Hamiltonian flow on
$L(p,1)$ has no fast periodic orbits (see Corollary 2.10).

  3. 3.

The Hamiltonian can be modified to be admissible, and its oscillation is given
by $pl$, which yields a lower bound on the Hofer-Zehnder capacity (see
Corollary 2.13).

Intro: Magnetic systems
Before we start with the construction of the lower bound we recall basic facts
about magnetic systems, for a more detailed introduction look at [6] and the
references there in. The triple $(M,g,\sigma)$ is called a _magnetic system_ ,
where $M$ is a closed manifold, $g$ a Riemannian metric on $M$ and
$\sigma\in\Omega^{2}(M)$ a closed $2$-form on $M$. This data gives rise to a
twisted symplectic form $\omega_{\sigma}=\mathrm{d}\lambda-\pi^{*}\sigma$ on
the tangent bundle $TM$, where $\pi$ denotes the foot point projection of $TM$
onto $M$ and $\lambda$ the pullback of the canonical symplectic form on
$T^{*}M$ to $TM$ via the duality isomorphism given by $g$. The symplectic form
$\omega_{\sigma}$ is nondegenerate so there exist a unique vector field
$X_{E}^{\sigma}$ on $TM$, called the Hamiltonian vector field of
$(E,\omega_{\sigma})$, such that

$\omega_{\sigma}\left(X_{E}^{\sigma},\cdot\right)=\mathrm{d}E(\cdot),$

where $E(x,v)=\frac{1}{2}g_{x}(v,v)$ denotes the kinetic Hamiltonian on $TM$.
The flow associated to $X_{E}^{\sigma}$ is called the Hamiltonian flow of $E$
and is given by

$\varPhi_{E,\sigma}^{t}:TM\longrightarrow TM:(x,v)\mapsto\Gamma_{(x,v)}(t)$

where $\Gamma_{(x,v)}(t)$ is the unique solution of
$\dot{\Gamma}=X_{E}^{\sigma}$ with $\Gamma(0)=(x,v)$. Since the derivative of
$E$ vanishes on the energy surface

$\Sigma_{k}=\\{(x,v)\in TM:E(x,v)=k\\}=E^{-1}(k)\subset TM,$

with $k\in[0,\infty)$, these submanifolds are invariant by the Hamiltonian
flow $\varPhi_{E,\sigma}$. It is well known that flow lines are of the form
$\Gamma=(\gamma,\dot{\gamma})$, where $\gamma:\mathbb{R}\to M$ is a magnetic
geodesic on $(M,g,\sigma)$ i.e. a solution to

$\nabla_{\dot{{\gamma}}}\dot{{\gamma}}=Y_{{\gamma}}\dot{{\gamma}},$ (2)

where $Y:TM\longrightarrow TM$, called the _Lorenz force_ , is the unique
bundle endomorphism which satisfies

$g_{p}\left(Y_{p}(v),(w)\right)=\sigma_{p}(v,w)\ \forall(p,v)\in TM.$

Using this duality between magnetic geodesic on $(M,g,\sigma)$ and flow lines
of $X_{E}^{\sigma}$ and the fact that the energy surfaces $\Sigma_{k}\subset
TM$ are invariant by Hamiltonian flow $\varPhi_{E,\sigma}$ we see that if
${\gamma}$ is a magnetic geodesic on $(M,g,\sigma)$, then its kinetic energy
is a conserved quantity i.e.

$\frac{1}{2}g_{{\gamma}(t)}(\dot{{\gamma}}(t),\dot{{\gamma}}(t))=E({\gamma}(t),\dot{{\gamma}}(t))=E({\gamma}(0),\dot{{\gamma}}(0))=\frac{1}{2}g_{{\gamma}(0)}(\dot{{\gamma}}(0),\dot{{\gamma}}(0))\
\forall t\in\mathbb{R}.$ (3)

In addition, it is quite natural to ask how the dynamics of the Hamiltonian
flow $\varPhi_{E,\sigma}$ changes if we deform the twisted symplectic form as
follows

$\left(\omega_{\sigma}\right)_{\varepsilon}=\mathrm{d}\lambda-\varepsilon\pi^{*}\sigma\textit{
for }\varepsilon\in\mathbb{R}.$

Thus a deformation of $\varepsilon$ induce a deformation of the Hamiltonian
vector field $\left(X_{E,\sigma}\right)_{\varepsilon}$, where
$\left(X_{E,\sigma}\right)_{\varepsilon}$ is defined through

$\left(\omega_{\sigma}\right)_{\varepsilon}\left(\left(X_{E,\sigma}\right)_{\varepsilon},(\cdot)\right)=\mathrm{d}E(\cdot).$

Note also that for $\varepsilon=0$, the vector field
$\left(X_{E,\sigma}\right)_{0}$ is exactly the geodesic vector field, thus is
associated Hamiltonian flow is the geodesic flow of $(M,g)$. It is another
exercise in logic to check that if $\Gamma_{\varepsilon}$ is a flow line of
Hamiltonian vector field $\left(X_{E,\sigma}\right)_{\varepsilon}$ then its
foot point projection $\pi\left(\Gamma_{\varepsilon}\right)$ is a magnetic
geodesic on $(M,g,\sigma)$ of strength $\varepsilon$ i.e. it is a solution of

$\nabla_{\dot{{\gamma}}}\dot{{\gamma}}=\varepsilon
Y_{{\gamma}}\dot{{\gamma}}.$ (4)

If we make use of eq. 3 it follows directly that if ${\gamma}$ is a magnetic
geodesic of strength $\varepsilon$, then its kinetic energy is an integral of
motion. Furthermore, if the magnetic field $\sigma\in\Omega^{2}(M)$ is exact,
i.e. there exists a $\beta\in\Omega^{1}(M)$ such that
$\mathrm{d}\beta=\sigma$, there is another point of view, the _Lagrangian
point of view_. To be precise the curve ${\gamma}$ is a magnetic geodesic on
$(M,g,\sigma)$ of strength $\varepsilon$, if and only if ${\gamma}$ is a
solution of the Euler–Lagrange equations associated to the Lagrangian

$L:TM\longrightarrow\mathbb{R}:(x,v)\mapsto\frac{1}{2}g_{x}(v,v)+\varepsilon\beta_{x}(v).$

Step 1: Magnetic geodesic flow on $L(p;1)$
Now we will focus on a more specific magnetic system:
$\left(S^{3},g,\varepsilon\cdot\mathrm{d}\alpha\right)$, the 3-sphere equipped
with the round metric $g$ and the standard contact form $\alpha$ as a magnetic
field. To construct the lower bound of the Hofer-Zehnder capacity we will use
this magnetic system. This dynamical system was studied in full detail by
Albers-Benedetti and the second author in [3]. Recall from [3] that a magnetic
geodesic $\gamma$ starting at $\gamma(0)=x\in S^{3}$ in direction
$\dot{\gamma}(0)=v\in T_{x}S^{3}$ is a curve which is a solution of the
following differential equation

$\ddot{\gamma}-i\varepsilon\dot{\gamma}+(c^{2}-\varepsilon\delta)\gamma=0,$
(5)

in the ambient space $\mathbb{C}^{2}$, where we used that
$c=|\dot{\gamma}|=|v|$ and $\delta=\mathrm{Re}\langle
i\gamma,\dot{\gamma}\rangle=\mathrm{Re}\langle ix,v\rangle$ are integrals of
motion. By the previous discussion we see that ${\gamma}$ is magnetic geodesic
on $(S^{3},g,\varepsilon\mathrm{d}\alpha)$ if and only if
$({\gamma},\dot{{\gamma}})$ is a flow line of the Hamiltonian system
$(TS^{3},\mathrm{d}\lambda-\varepsilon\pi^{*}\mathrm{d}\alpha,E)$. As the
magnetic form is exact we can shift the zero-section to symplectically embed
the twisted into the standard tangent bundle

$\iota:(D_{1-2\varepsilon}S^{3},\mathrm{d}\lambda-\varepsilon\pi^{*}\mathrm{d}\alpha)\hookrightarrow(D_{1}S^{3},\mathrm{d}\lambda):\
(x,v)\mapsto(x,v+\varepsilon ix).$ (6)

To give a lower bound on the Hofer–Zehnder capacity of
$(D_{1}S^{3},\mathrm{d}\lambda)$ we will first give a lower bound on the
periods of orbits of the Hamiltonian system
$(D_{1-2\varepsilon}S^{3},\mathrm{d}\lambda-\varepsilon\pi^{*}\mathrm{d}\alpha,E)$:

######  Lemma 2.2.

All non-constant periodic orbits of the kinetic Hamiltonian

$E:D_{1-2\varepsilon}S^{3}\to\mathbb{R};\ (x,v)\mapsto\frac{1}{2}|v|_{x}^{2}$

have period

$T_{E}\geq\begin{cases}\frac{2\pi}{|v|}\ \ \text{for}\ |v|\geq\varepsilon\\\
\frac{2\pi}{\varepsilon}\ \ \text{for}\ |v|\leq\varepsilon.\end{cases}$

###### Proof.

In summary in [3] it is shown that the orbits stay on Clifford tori, spiraling
around the Reeb direction. Indeed there is an explicit formula for general
magnetic geodesics

$\gamma(t)=e^{i\theta_{+}t}p_{+}+e^{i\theta_{-}t}p_{-},$

where

$\theta_{\pm}=\frac{\varepsilon\pm\sqrt{\varepsilon^{2}+4(c^{2}-\varepsilon\delta)}}{2}\
\ \&\ \ p_{\pm}=\mp\frac{\theta_{\mp}x+iv}{\theta_{+}-\theta_{-}}.$

Further $c:=|\dot{\gamma}|=|v|$ and $\delta:=\mathrm{Re}\langle
i\gamma,\dot{\gamma}\rangle=\mathrm{Re}\langle ix,v\rangle$ are integrals of
motion. If $\theta_{-}\neq 0$ it follows immediately that

$T_{E}\geq\frac{2\pi}{|\theta_{-}|}\geq\frac{2\pi}{c},$

where we used that

$c=\sqrt{\theta_{+}^{2}|p_{+}|^{2}+\theta_{-}^{2}|p_{-}|^{2}}\geq\sqrt{\theta_{-}^{2}\left(|p_{+}|^{2}+|p_{-}|^{2}\right)}=|\theta_{-}|.$

If on the other hand $\theta_{-}=0$, we find $\gamma(t)=e^{i\varepsilon
t}p_{+}+p_{-}$. It easily follows that

$T_{E}=\frac{2\pi}{\varepsilon}.$

As $\theta_{-}=0$ implies $c^{2}=\varepsilon\delta$ and $\delta<c$ we have
$c<\varepsilon$. ∎

The advantage of the magnetic geodesic flow becomes clear when studying the
lens space. The idea is that the magnetic term bends the short flow lines
(just an $\varepsilon$-bit) away from the geodesic that is contracted by the
quotient $S^{3}\to L(p;1)$. Now lets look at the details. First, we need to
make sure that the magnetic term $\alpha$ actually descends to the quotient.
Hence, we can think of $\alpha\in\Omega_{1}(L(p;1))$.

######  Lemma 2.3.

The 1-form $\alpha\in\Omega_{1}(S^{3})$ is $\mathbb{Z}_{p}$-invariant, i.e.

$\left(e^{\frac{2\pi}{p}j}\right)^{*}\alpha_{x}=\alpha_{e^{-\frac{2\pi}{p}j}x}.$

###### Proof.

Observe that $ji=ij$, hence

$\left(e^{\frac{2\pi}{p}j}\right)^{*}\alpha_{x}(\cdot)=\frac{1}{2}\langle
ix,e^{\frac{2\pi}{p}j}\cdot\rangle=\frac{1}{2}\langle
e^{-\frac{2\pi}{p}j}ix,\cdot\rangle=\frac{1}{2}\langle
e^{-\frac{2\pi}{p}j}x,\cdot\rangle=\alpha_{e^{-\frac{2\pi}{p}j}x}.$

∎

This means that the magnetic twist descends to the lens space, but the
magnetic geodesic flow on $L(p;1)$ has (at least) two short periodic orbits
$\gamma_{+}(t)=(e^{2\pi it},0)$ and $\gamma_{-}(t)=(0,e^{2\pi it})$, as
$j\gamma_{\pm}=\pm i\gamma_{\pm}$ and magnetic geodesics in Reeb direction
$ix$ are ordinary geodesics.

Alternative description of magnetic geodesics on $S^{3}$
In order to finish Step 1 we need to prove that these two are the only short
periodic orbits of the magnetic flow on $L(p;1)$. The following alternative
description of magnetic geodesics will be useful. It is based on the geometric
intuition that one can think of a magnetic geodesic as a composition of
rotation in Reeb direction and a rotation of some special complex structure,
to be precise:

######  Proposition 2.4.

A solution $\gamma$ of (5) with initial conditions $\gamma(0)=x\in S^{3}$ and
$\dot{\gamma}(0)=v\in T_{x}S^{3}$ can be written as

$\gamma(t)=e^{\frac{i\varepsilon}{2}t}e^{J(x,v,\varepsilon)at}x$

where $a=\frac{1}{2}\sqrt{\varepsilon^{2}+4(c^{2}-\varepsilon\delta)}$ and
$J(x,v,\varepsilon)$ is the unique complex structure, preserving the standard
orientation and round metric on $S^{3}$ that satisfies

$J(x,v,\varepsilon)x=\frac{1}{a}\left(v-i\frac{\varepsilon}{2}x\right).$

Before we prove the proposition, observe that $J(x,v,\varepsilon)$ is indeed
unique, as orientation determines it on the orthogonal complement of
$\mathrm{span}\\{x,v-i\frac{\varepsilon}{2}x\\}$. Explicitly we set
$J(x,v,\varepsilon)e_{1}=e_{2},J(x,v,\varepsilon)e_{2}=-e_{1}$ for
$(e_{1},e_{2})$ any ordered (hence oriented) orthonormal bases of the
orthogonal complement of $\mathrm{span}\\{x,v-i\frac{\varepsilon}{2}x\\}$.
Further, $J(x,v,\varepsilon)$ belongs to the 2-sphere of complex structures
(commuting with $i$) that comes from identifying $\mathbb{C}^{2}$ as the
quaternionic space.

###### Proof.

We simply check that
$\gamma(t)=e^{\frac{i\varepsilon}{2}t}e^{J(x,v,\varepsilon)at}x$ satisfies
(5). We abbreviate $J=J(x,v,\varepsilon)$ and compute

$\dot{\gamma}=\left(i\frac{\varepsilon}{2}+Ja\right)\gamma$

and

$\ddot{\gamma}=\left(i\frac{\varepsilon}{2}+Ja\right)^{2}\gamma=\left(-\frac{\varepsilon^{2}}{4}+iJ\varepsilon
a-a^{2}\right)\gamma.$

This indeed implies

$\displaystyle\ddot{\gamma}-i\varepsilon\dot{\gamma}+(c^{2}-\varepsilon\delta)\gamma$
$\displaystyle=\left(-\frac{\varepsilon^{2}}{4}+iJ\varepsilon
a-a^{2}-i\varepsilon\left(i\frac{\varepsilon}{2}+Ja\right)+c^{2}-\varepsilon\delta\right)\gamma$
$\displaystyle=\left(\frac{\varepsilon^{2}}{4}-a^{2}+c^{2}-\varepsilon\delta\right)\gamma=0.$

∎

This description of the magnetic geodesics yields a nice visualization (Figure
1) in terms of the Hopf-fibration $\pi:S^{3}\to S^{2}$ (for $i$). The factor
$e^{\frac{i\varepsilon}{2}t}$ is a rotation in the fibers, while the factor
$e^{Jat}$ projects down to a rotation of $S^{2}$ with rotation axis determined
by $J$. Explicitly, the two fixed antipodal points $\pi(x_{\pm})$ of the
rotation are the projections of solutions to the equation $Jx_{\pm}=\pm
ix_{\pm}$.

Figure 1: The figure shows the Hopf projection $\pi:S^{3}\to S^{2}$ with
respect to $i$. The orbits $\gamma_{\pm}$ that satisfy $j\gamma_{\pm}=\pm
i\gamma_{\pm}$ are projected to antipodal points we call $N$ and $S$. The
rotation $e^{jt}$ is projected to the rotation that fixes $N$ and $S$.
Similarly any rotation of the form $e^{Jt}$ (for a complex structure $J$
commuting with $i$) projects to a rotation fixing the points with fibers
determined by $J\gamma_{\pm}=\pm i\gamma_{\pm}$.

We can now prove that the only fast periodic magnetic geodesics on $L(p;1)$
are the quotients of the Reeb orbits over the antipodal points $N,S$.

######  Lemma 2.5.

All non-constant periodic magnetic geodesics on $L(p;1)$ of period $T$ except
for the Reeb orbits $\gamma_{\pm}$ satisfy

$T_{E}\geq\begin{cases}\frac{2\pi}{|v|}\ \ \text{for}\ |v|\geq\varepsilon\\\
\frac{2\pi}{\varepsilon}\ \ \text{for}\ |v|\leq\varepsilon.\end{cases}$

###### Proof.

All magnetic geodesics on $L(p;1)$ that lift to closed magnetic geodesic on
$S^{3}$ satisfy the bound by Lemma 2.2. Therefor let $\gamma$ be a magnetic
geodesic of period $T$ on $S^{3}$ that satisfies

$e^{j\frac{2\pi}{p}}\gamma(t)=\gamma(t+T_{E}/p),$ (7)

which is equivalent to projecting to a short magnetic geodesic on $L(p;1)$. As
$[j,i]=0$ this implies that $e^{j\frac{2\pi}{p}}$ projects under the Hopf map
to a rotation of $S^{2}$ that preserves the geodesic circle $\pi(\gamma)$,
which implies $J(x,v,\varepsilon)=j$. Evaluating equation (7) at $t=0$ we find

$(e^{i\frac{2\pi}{p}}z_{1},e^{-i\frac{2\pi}{p}}z_{2})=e^{i\frac{\varepsilon}{2}\frac{T_{E}}{p}}(e^{\frac{iaT_{E}}{p}}z_{1},e^{\frac{-iaT_{E}}{p}}z_{2}),$

where $\gamma(0)=:(z_{1},z_{2})$. Multiplying both entries yields
$e^{i\varepsilon\frac{T_{E}}{p}}z_{1}z_{2}=z_{1}z_{2}$ which implies

$\frac{T_{E}}{p}=\frac{2\pi k}{\varepsilon}\geq\frac{2\pi}{\varepsilon}$

unless $z_{1}=0$ or $z_{2}=0$ which are precisely the orbits $\gamma_{\pm}$. ∎

Step 2: Adding a potential – magnetic billiards
In order to construct an admissible Hamiltonian on (an open set of)
$D_{1}L(p;1)$ we need to modify $E$, so that all non-constant periodic orbits
have period $T>1$. Recall that $\gamma_{\pm}$ are the only periodic orbits
that will cause problems, so we will try to cut them out using a potential.
Again we first construct the potential on $S^{3}$ and later argue that it
descends to $L(p;1)$. If

$\pi:S^{3}(1)\to S^{2}\left(\frac{1}{2}\right)$

denotes the Hopf-map with respect to $i$, these two orbits ${\gamma}_{\pm}$
project to antipodal points. We call them $N$ and $S$. Now the rotation
$e^{jt}$ projects under $\pi$ to the rotation of $S^{2}(\frac{1}{2})$ fixing
$N$ and $S$ (see Figure 1). We want to modify $E$ by adding a potential
$V_{\varepsilon}:L(p;1)\setminus\gamma_{\pm}\to\mathbb{R}$ that only depends
on the distance $d(x,\gamma_{\pm})$, goes to infinity near $\gamma_{\pm}$ and
is zero outside the $\varepsilon$ neighborhood of $\gamma_{\pm}$.

######  Proposition 2.6.

All periodic orbits of $E+V_{\varepsilon}$ on $TS^{3}$ have period

$T_{E+V_{\varepsilon}}\geq\begin{cases}\frac{2\pi}{\max|v|}\ \ &\text{for}\
\max|v|\geq\varepsilon\\\ \frac{2\pi}{\varepsilon}\ \ &\text{for}\
\max|v|\leq\varepsilon\end{cases}-\mathcal{O}(\varepsilon).$

Here $\max|v|:=\max_{t}|\dot{\gamma}(t)|$ refers to the maximal velocity along
the periodic orbit. Observe that
$E(\gamma,\dot{\gamma})+V_{\varepsilon}(\gamma)=\frac{1}{2}\max|v|^{2}$ as the
potential vanishes where the velocity is maximal.

######  Notation 2.7.

From now on we will use the term $\mathcal{O}(\varepsilon)$ by abuse of
notation as a term that tends to zero as $\varepsilon\to 0$, but is not
necessarily of order $\varepsilon$.

The idea of proof is very geometric: If $\gamma_{\varepsilon}$ is a sequence
of periodic orbits of $E+V_{\varepsilon}$, then the $H^{1}$-limit of
${\gamma}_{\varepsilon}$ as $\varepsilon\to 0$ is either a magnetic bounce
orbit on the billiard table $S^{3}\setminus\gamma_{\pm}$ or an orbit trapped
near the boundary of the billiard table $\gamma_{\pm}$. (For a precise
definition we refer to Appendix B.) In particular, when the periods of
$\gamma_{\varepsilon}$ are bounded uniformly in $\varepsilon$, their periods
converge to the periods of the bounce orbits.

Figure 2: There are three types of bounce orbits: (1) no bounce, (2) bounce at
exactly one pole, (3) bounce at both poles. At every bounce the velocity in
contact direction is reversed, while $\delta$, the velocity in Reeb direction,
is preserved. In particular as depicted on the right, every periodic bounce
orbit corresponds by symmetry to a periodic magnetic geodesic (without
bounces). Here symmetry means a reflection at a suitable plane.

######  Lemma 2.8.

All periodic magnetic bounce orbits on $S^{3}$ have period
$T_{\text{bounce}}=T_{E}.$

###### Proof.

As shown in Figure 2 there are three types of bounce orbits. Type (1) is a
smooth orbit and thus satisfies the period bound by Lemma 2.2. Type (2)
bounces exactly at one of the poles. If it is periodic, its period is by
symmetry the same as the orbit of the corresponding smooth orbit. Type (3)
bounces at both poles and again using symmetries one can sees that its period
is the same as the period of the corresponding smooth orbit. ∎

The detailed proof of Proposition 2.6 is written in Appendix B. It follows the
strategy above arguing that for small $\varepsilon$ any periodic orbit must
either be close to a periodic bounce orbit or trapped near the poles. In the
first case one obtains the bound from Lemma 2.8, in the second case from using
some integrals of motion that are derived in Appendix A.

######  Remark 2.9.

At this point we also remark that in principle it would be also possible to
construct the magnetic bounce orbits with the approximating scheme of magnetic
bounce orbits developed by Vocke in her thesis [23], which is a generalization
of earlier work of Albers-Mazzucchelli [4].

The potential $V_{\varepsilon}$ is $\mathbb{Z}_{p}$-invariant, therefor the
Hamiltonian $E+V_{\varepsilon}$ descends to the lens space.

######  Corollary 2.10.

All periodic orbits of $E+V_{\varepsilon}$ on $TL(p;1)$ have period

$T_{E+V_{\varepsilon}}\geq\begin{cases}\frac{2\pi}{\max|v|}\ \ &\text{for}\
\max|v|\geq\varepsilon\\\ \frac{2\pi}{\varepsilon}\ \ &\text{for}\
\max|v|\leq\varepsilon\end{cases}-\mathcal{O}(\varepsilon).$

###### Proof.

We need to check that $\mathbb{Z}_{p}$-invariant orbits on $S^{3}$ have period
at least $p$ times the above bound. A $\mathbb{Z}_{p}$-invariant orbit is
either a magnetic geodesic as in Lemma 2.5 and therefor its period satisfies
Proposition 2.6 or it is a $\mathbb{Z}_{p}$-symmetric configuration of
magnetic geodesic arcs (see Figure 3). If it is not trapped near the caps this
means it must have length tending to infinity as the bounce orbits are not
$\mathbb{Z}_{p}$-symmetric. If the orbit is trapped near a cap, it also must
consists of at least $p$ magnetic geodesic arcs. This means it has winding
number at least $p$ in Reeb direction and therefor $T_{E+V_{\varepsilon}}\geq
p\cdot\frac{2\pi}{\max|v|}$ on $S^{3}$. ∎

Figure 3: The vertical grey lines indicate the fundamental domain. A
$\mathbb{Z}_{p}$-symmetric periodic orbit needs to consist of at least $p$
magnetic geodesic arcs, this is what we refer to as $\mathbb{Z}_{p}$-symmetric
configuration. Two examples are illustrated above.

Step 3: Constructing an admissible Hamiltonian
We can now reparametrize the magnetic billiard dynamics to write down a
Hamiltonian with no fast periodic orbits.

######  Proposition 2.11.

We define the following Hamiltonian to reparametrize the dynamics induced by
$E+V_{\varepsilon}$,

$H_{\varepsilon}:D_{1-2\varepsilon}L(1,p)\to\mathbb{R};\ (x,v)\mapsto
h_{\varepsilon}\left(E(x,v)+V_{\varepsilon}(x)\right)$

where $h_{\varepsilon}(y)$ is given by

$h_{\varepsilon}:[0,\infty)\longrightarrow\mathbb{R}:y\mapsto
2\pi\begin{cases}\frac{y}{\varepsilon}+\frac{\varepsilon}{2},&\textit{ for
}y\leq\frac{\varepsilon^{2}}{2}\\\ \sqrt{2y},&\textit{ for
}y\geq\frac{\varepsilon^{2}}{2}\end{cases}.$

Then periodic orbits $(\gamma,\dot{{\gamma}})$ with respect to
$\mathrm{d}\lambda-\varepsilon\pi^{*}\mathrm{d}\alpha$ have periods

$T_{H_{\varepsilon}}\geq 1-\mathcal{O}(\varepsilon).$

######  Remark 2.12.

Note that $h_{\varepsilon}$ is differentiable in particular at
$y=\frac{\varepsilon^{2}}{2}$. The function $h_{\varepsilon}$ is not smooth,
i.e. of class $C^{\infty}$, but one can approximate it by a smooth function,
and as the period only sees the first derivative this will not change the
bound on the period. In addition, for the reader’s convenience, we add an
example of a possible graph of $h_{\varepsilon}$.

Figure 4: The figure shows a plot of the function $h_{\varepsilon}$ for
$\varepsilon=\frac{1}{10}$.

###### Proof.

Let $({\gamma},\dot{{\gamma}})$ be a periodic orbit of $H_{\varepsilon}$.
There are two cases that we have to investigate. We start with
$E({\gamma},\dot{{\gamma}})+V_{\varepsilon}({\gamma})\leq\frac{\varepsilon^{2}}{2}$.
Using that
$H_{\varepsilon}=h_{\varepsilon}\circ(E({\gamma},\dot{{\gamma}})+V_{\varepsilon}({\gamma}))$
for $h_{\varepsilon}^{\prime}(y)=\frac{2\pi}{\varepsilon}$ if
$y\leq\frac{\varepsilon^{2}}{2}$, we find

$T_{H_{\varepsilon}}=\frac{T_{E+V_{\varepsilon}}}{h_{\varepsilon}^{\prime}(E({\gamma},\dot{{\gamma}})+V_{\varepsilon}({\gamma}))}=\frac{\varepsilon}{2\pi}T_{E+V_{\varepsilon}}\overset{\ref{prop
potential}}{\geq}1-\mathcal{O}(\varepsilon).$

Now if $E({\gamma},\dot{{\gamma}})+V_{\varepsilon}({\gamma})\geq\varepsilon$,
recall that $H_{\varepsilon}=h_{\varepsilon}\circ(E+V_{\varepsilon})$ with
$h_{\varepsilon}(y)=2\pi\sqrt{2y}$. Thus we find

$T_{H_{\varepsilon}}=\frac{T_{E+V_{\varepsilon}}}{h^{\prime}(E({\gamma},\dot{{\gamma}})+V_{\varepsilon}({\gamma}))}=\frac{\sqrt{2(E({\gamma},\dot{{\gamma}})+V_{\varepsilon}({\gamma}))}}{2\pi}T_{E+V_{\varepsilon}}=\frac{\max|v|}{2\pi}T_{E+V_{\varepsilon}}\overset{\ref{prop
potential}}{\geq}1-\mathcal{O}(\varepsilon).$

∎

We are finally in the position to prove the lower bound of the Hofer–Zehnder
capacity.

######  Corollary 2.13.

The Hofer-Zehnder capacity $c_{\mathrm{HZ}}(D_{1}L(p;1),\mathrm{d}\lambda)$ is
bounded from below by

$c_{\mathrm{HZ}}(D_{1}L(p;1),\mathrm{d}\lambda)\geq pl=2\pi,$

where $l$ denotes the length of the shortest prime geodesics on $L(p,1)$.

###### Proof.

By monotonicity of capacities and using the shift of the zero-section (6) we
find

$c_{HZ}(D_{1-2\varepsilon}L(p;1),\mathrm{d}\lambda-\varepsilon\pi^{*}\mathrm{d}\alpha)\leq
c_{HZ}(D_{1}L(p;1),\mathrm{d}\lambda).$

Denote the sub level set of $H_{\varepsilon}$ by

$U_{\varepsilon}=\left\\{(x,v)\in TL(p,1)\ \Big{|}\ \ H_{\varepsilon}(x,v)\leq
2\pi\sqrt{\frac{1}{2}(1-2\varepsilon)^{2}}\right\\}.$

Observe that $U_{\varepsilon}\subset D_{1-2\varepsilon}L(p;1)$. Now we can
modify

$H_{\varepsilon}:U_{\varepsilon}\to\left[\frac{\varepsilon}{2},\max(H_{\varepsilon})\right]$

by composing it with another function

$f_{\varepsilon}:\left[\frac{\varepsilon}{2},\max(H_{\varepsilon})\right]\to[0,\infty),$

with $0\leq f_{\varepsilon}^{\prime}<1-2\mathcal{O}(\varepsilon)$,
$\min(f_{\varepsilon}\circ H_{\varepsilon})=0$ and $\max(f_{\varepsilon}\circ
H_{\varepsilon})=\max(H_{\varepsilon})-\mathcal{O}(\varepsilon)$, so that
$f_{\varepsilon}\circ H_{\varepsilon}$ is admissible. Indeed the periods then
satisfy

$T_{f_{\varepsilon}\circ
H_{\varepsilon}}=\frac{T_{H_{\varepsilon}}}{f^{\prime}_{\varepsilon}(H_{\varepsilon})}>1.$

In particular, we obtain

$c_{HZ}(D_{1}L(p;1),\mathrm{d}\lambda)\geq
c_{HZ}(U_{\varepsilon},\mathrm{d}\lambda-\varepsilon\pi^{*}\mathrm{d}\alpha)\geq\max(f_{\varepsilon}\circ
H_{\varepsilon})=\max(H_{\varepsilon})-\mathcal{O}(\varepsilon)\geq
2\pi-\mathcal{O}(\varepsilon).$

Taking $\varepsilon\to 0$ yields the corollary as the sphere covering the lens
space was of radius $1$ and therefore its prime geodesic have length $2\pi$,
while the shortest prime geodesics on $L(p;1)$ have length $l=2\pi/p$. ∎

##  3 Upper bound: Capacity of $(D_{1}S^{3},\mathrm{d}\lambda)$

We will deduce the upper bound for disk tangent bundles of the lens spaces
from the fact that they are all covered by the disk tangent bundle of the
3-sphere. To do so we first need to compute the Hofer–Zehnder capacity of
$(D_{1}S^{3},\mathrm{d}\lambda)$.

######  Theorem 3.1.

Take the disk bundle with respect to the round metric and denote by $l$ the
length of all prime geodesics, then

$c_{\mathrm{HZ}}(D_{1}S^{3},\mathrm{d}\lambda)=l.$

###### Proof.

We prove the Theorem by finding a lower and an upper bound that coincide.
Further we normalize the metric such that $l=2\pi$, the general case follows
by scaling.

Lower bound: All orbits of $H:D_{1}S^{3}\setminus 0_{TS^{3}}\to\mathbb{R},\
(x,v)\mapsto 2\pi|v|$ are periodic of period one. Composing the Hamiltonian
$H$ with a suitable function $f:[0,2\pi]\to\mathbb{R}$ we can extend $f\circ
H$ smoothly to the zero-section such that $f\circ H$ is admissible and
$\max(f\circ H)=\max(H)-\mathcal{O}(\varepsilon)$. Now the lower bound follows
from $\varepsilon\to 0$.

######  Remark 3.2.

For the upper bound we will use results by Hofer–Viterbo [16] and Lu [20],
using the theory of pseudoholomorphic curves or more precisely Gromov–Witten
invariants to obtain an upper bound for the Hofer–Zehnder capacity. For a
detailed introduction to Gromov-Witten invariants in symplectic topology we
refer to [21]. We use the notation of Wendl [26] and denote by
$\mathrm{GW}_{A}([pt.],\alpha,\beta)$ the count of pseudoholomorphic spheres
$u:\mathbb{C}\mathrm{P}^{1}\to Q^{3}$ representing the homology class $A\in
H_{2}(Q^{3},\mathbb{Z})$ with three marked points mapped to representatives of
the homology classes $[pt.],\alpha,\beta\in H_{2}(Q^{3},\mathbb{Z})$.

Upper bound: As shown for example in [2, Lem. 3.2] the disk tangent bundle is
symplectomorphic to the complement of 2-quadric in the 3-quadric, i.e.

$(D_{1}S^{3},\mathrm{d}\lambda)\cong(Q^{3}\setminus Q^{2},2\sigma).$

Here $Q^{n}:=\\{[z_{0}:\ldots:z_{n+1}]\in\mathbb{C}\mathrm{P}^{n+1}|\
z_{0}^{2}+\ldots+z_{n+1}^{2}=0\\}$ and $\sigma$ denotes the restriction of the
Fubini-Study to $Q^{3}\subset\mathbb{C}\mathrm{P}^{4}$ normalized to take
value $\pi$ on the generator of $H_{2}(\mathbb{C}\mathrm{P}^{n},\mathbb{Z})$.

Note that $Q^{3}\cong\mathrm{SO}(5)/(\mathrm{SO}(3)\times\mathrm{SO}(2))$ is
actually an irreducible Hermitian symmetric space of rank 2. By [19, Lem. 15]
a Gromov–Witten invariant of the form $\mathrm{GW}_{A}([pt.],\alpha,\beta)$
for $A$ the generator of $H_{2}(Q^{3},\mathbb{Z})$ does not vanish. As
$[Q^{2}]\neq 0$ by Lemma C.1 we have $[Q^{2}]\cdot[A]\neq 0$ and therefor we
find

$\mathrm{GW}_{A}\left([pt.],\alpha,\beta,[Q^{2}]\right)=\mathrm{GW}_{A}\left([pt.],\alpha,\beta\right)\left(A\cdot[Q^{2}]\right)\neq
0.$

Finally we can use a corollary [10, Cor. 3.2.11] of a theorem by Lu [20, Thm.
1.10] to find

$c_{\mathrm{HZ}}(D_{1}S^{3},\mathrm{d}\lambda)=c_{\mathrm{HZ}}(Q^{3}\setminus
Q^{2},2\sigma)\leq 2\sigma(A)=2\pi.$

∎

######  Remark 3.3.

Observe that $A$ is the generator of $H_{2}(Q^{3},\mathbb{Z})\cong\mathbb{Z}$,
therefor it is a minimal class and the moduli space of pseudoholomorphic
curves in class $A$ is compact. The Gromov-Witten invariant we use here is
thus defined without using any virtual techniques.

We can now deduce an upper bound for the Hofer–Zehnder capacity of
$(D_{1}L(p;1),\mathrm{d}\lambda)$.

######  Corollary 3.4.

The Hofer-Zehnder capacity $c_{\mathrm{HZ}}(D_{1}L(p;1),\mathrm{d}\lambda)$ is
bounded from above by

$c_{\mathrm{HZ}}(D_{1}L(p;1),\mathrm{d}\lambda)\leq pl=2\pi,$

where $l$ denotes the length of the shortest prime geodesics.

###### Proof.

Let $H:D_{1}L(p;1)\to\mathbb{R}$ be an admissible Hamiltonian, then
$H\circ\mathrm{d}\mathrm{pr}:D_{1}S^{3}\to\mathbb{R}$ is admissible for
$(DS^{3},\mathrm{d}\lambda)$. In particular

$\max(H)=\max(H\circ\mathrm{d}\mathrm{pr})\leq
c_{\mathrm{HZ}}(DS^{3},\mathrm{d}\lambda)=pl=2\pi,$

as $pl=2\pi$ is the length of the prime geodesics on the sphere covering
$L(p;1)$ isometrically. ∎

As the upper bound of the Hofer-Zehnder capacity
$(D_{1}L(p;1),\mathrm{d}\lambda)$ in Corollary 3.4 coincides with the lower
bound constructed in the previous section in Corollary 2.13 this finishes the
proof of Theorem 1.3.

##  4 Further directions

We will finish the article by discussing open problems which are in our
opinion closely related to Theorem 1.3. The first question one has is
probably, what about other lens spaces? First note that when considering lens
spaces $L(p;1)$ for even $p$ 2.1 does not hold, as every geodesic on $S^{3}$
covers a geodesic of $L(p;1)$ at least twice. Still, as shown by the first
author [9], for $\mathbb{R}\mathrm{P}^{2}$ (e.g. $p=2$) Theorem 1.3 continues
to hold. The lower bound in this case was obtained using geodesic billiards on
one of the hemispheres of $S^{3}$. For general odd $p$ the authors are
optimistic that one could combine the two dynamics in order to find a lower
bound. As the upper bound comes from $S^{3}$ it actually holds for any lens
space $L(p;q)$.

######  Question 4.1.

Let $p,q\in\mathbb{Z}$ co-prime and odd. Can one extend the proof of the lower
bound in theorem 1.3 to compute the Hofer-Zehnder capacity $D_{1}L(p,q)$?

The guess of the authors would be yes, with the following modification one
should instead of studying the magnetic flow $S^{3}$ equipped with it’s
standard contact form the magnetic flow on the ellipsoid $E\left(1,q\right)$
equipped with it’s standard contact form. This can be done as follows: we
equip $S^{3}$ with the contact form
$\alpha_{q}=\frac{\mathrm{i}}{2}\left(z_{1}\mathrm{d}\bar{z_{1}}-\bar{z}_{1}\mathrm{d}z_{1}+\frac{1}{q}\left(z_{2}\mathrm{d}\bar{z}_{2}-\bar{z}_{2}\mathrm{d}z_{2}\right)\right)$.
Note that the Reeb vector field is given by
$R_{(z_{1},z_{2})}=2(\mathrm{i}z_{1},\mathrm{i}qz_{2})$ thus the Reeb flow
w.r.t. $\alpha_{q}$ on $S^{3}$ is given by
$\varPhi_{t}(z_{1},z_{2})=\left(e^{2\mathrm{i}t}z_{1},e^{2q\mathrm{i}t}z_{2}\right)$.
If we replace $\mathrm{i}$ by $\mathrm{j}$ again this Reeb flow can used to
define a $\mathbb{Z}_{p}$ action on $S^{3}$ as follows

$(z_{1},z_{2})\mapsto\left(e^{2\pi\frac{1}{p}j}z_{1},e^{2\pi\frac{q}{p}j}z_{2}\right).$

Dividing out this $\mathbb{Z}_{p}$ action on $S^{3}$ we obtain the Lens space
$L(p,q)=S^{3}/\mathbb{Z}_{p}$.

######  Conjecture 4.2.

For all $p,q\in\mathbb{N}$ co-prime the Hofer-Zehnder capacity of the disk
bundle of $L(p,q)$ is given by

$c_{HZ}\left(D_{1}L(p,q),\mathrm{d}\lambda\right)=2\pi.$

Lastly we also want to say a few words about higher dimensions. Again the
upper bound continues to hold as also in higher dimensions, since we have

$(D_{1}S^{2n+1},\mathrm{d}\lambda)\cong(Q^{2n+1}\setminus Q^{2n},2\sigma).$

For the lower bound one could try to use the fact that the magnetic geodesic
flow on $S^{2n+1}$ actually stays on totally geodesic copies of $S^{3}$ which
are invariant under the magnetic flow, see for reference [3]. Certainly the
potential $U$ one uses can be extended to $S^{2n+1}$ such that $U$ tends to
infinity in a neighborhood of the Reeb orbits ${\gamma}_{k}$ where
${\gamma}_{k}(t)$ is a rotation in the $k$-th coordinate by the angle $2t$.
So, we end up with the following:

######  Conjecture 4.3.

For all $p\in\mathbb{N}$ odd the Hofer-Zehnder capacity of the disk bundle of
$L(p,1,\dots,1)=S^{2n+1}/\mathbb{Z}_{p}$ is given by

$c_{HZ}\left(D_{1}L(p,1,\dots,1),\mathrm{d}\lambda\right)=2\pi.$

##  Appendix A Integrals of motion

We introduce the following coordinates on $S^{3}\subset\mathbb{C}^{2}$,

$\displaystyle z_{1}=e^{i\varphi_{1}}\sin(\theta)\ \&\
z_{2}=e^{i\varphi_{2}}\cos(\theta)$

for $\varphi_{1},\varphi_{2}\in[0,2\pi)$ and $\theta\in[0,\pi/2]$. Observe
that $\gamma_{+}$ corresponds to $\theta=\pi/2$ and $\gamma_{-}$ corresponds
to $\theta=0$ and that

$\mathrm{dist}_{\partial\Omega_{\varepsilon}}(x)=\left\\{\begin{array}[]{ll}\theta(x),&\theta(x)\leq\pi/4\\\
\pi/2-\theta(x),&\theta(x)\geq\pi/4\end{array}\right..$

Hence, $V_{\varepsilon}$ depends only on $\theta(x)$. The metric in these
coordinates is

$\mathrm{d}s^{2}=\mathrm{d}\theta^{2}+\sin(\theta)^{2}\mathrm{d}\varphi_{1}^{2}+\cos(\theta)^{2}\mathrm{d}\varphi_{2}^{2}.$

We plug
$v=\dot{\theta}\partial_{\theta}+\dot{\varphi}_{1}\partial_{\varphi_{1}}+\dot{\varphi}_{2}\partial_{\varphi_{2}}$
into $\mathrm{d}s^{2}$ to obtain

$|v|^{2}=\dot{\theta}^{2}+\sin(\theta)^{2}\dot{\varphi}_{1}^{2}+\cos(\theta)^{2}\dot{\varphi}_{2}^{2}.$

Using that $ix=\partial_{\varphi_{1}}+\partial_{\varphi_{2}}$ we can express
the Lagrangian in coordinates

$L\left(\varphi_{1},\varphi_{2},\theta,\dot{\varphi_{1}},\dot{\varphi}_{2},\dot{\theta}\right)=\frac{1}{2}\left(\dot{\theta}^{2}+\sin(\theta)^{2}\dot{\varphi}_{1}^{2}+\cos(\theta)^{2}\dot{\varphi}_{2}^{2}\right)+\varepsilon\left(\sin(\theta)^{2}\dot{\varphi}_{1}+\cos{\theta}^{2}\dot{\varphi}_{2}\right)-V_{\varepsilon}(\theta).$

In particular, the Lagrangian $L$ does not depend on $\varphi_{1}$ and
$\varphi_{2}$. Therefore the conjugate momenta

$\displaystyle c_{1}:=\frac{\partial
L}{\partial\dot{\varphi}_{1}}=\sin^{2}(\theta)(\dot{\varphi}_{1}+\varepsilon)\
\ \&\ \ \ c_{2}:=\frac{\partial
L}{\partial\dot{\varphi}_{2}}=\cos^{2}(\theta)(\dot{\varphi}_{2}+\varepsilon)$
(8)

are preserved. Since the kinetic energy $E$ is also an integral of motion the
Lagrangian system $L_{\varepsilon}$ has three independent integrals of motion.
So the corresponding Hamiltonian system $H_{\varepsilon}$ on $TS^{3}$ is
integrable, so we can conclude:

######  Corollary A.1.

The Hamiltonian system
$\left(TS^{3},E+V_{\varepsilon},\mathrm{d}\lambda-\varepsilon\pi^{*}\mathrm{d}\alpha\right)$
is integrable.

##  Appendix B Bound on periods

Let $(\gamma_{\varepsilon},\dot{{\gamma}}_{\varepsilon})$ be a periodic orbit
of $E+V_{\varepsilon}$. We consider the following cases.

$\gamma_{\varepsilon}:\ \ \begin{cases}\text{avoids both caps},\\\
\text{enters a cap}:\ \ \begin{cases}\text{trapped},\\\ \text{not trapped}:\ \
\begin{cases}\text{enters only one cap},\\\ \text{enters both
caps}.\end{cases}\end{cases}\end{cases}$

As we added the potential $V_{\varepsilon}$, the norm of the velocity $|v|$ is
not a constant of motion. However, every orbit leaves the region where the
potential is non-vanishing and by conservation of energy $E+V_{\varepsilon}$
the velocity $|v|$ is maximal there. From now on we denote $c:=\max|v|$ the
maximal velocity.

Avoids both caps: If $\gamma_{\varepsilon}$ avoids both caps it is a magnetic
geodesic and therefor satisfies the bound in Lemma 2.2.

Trapped: The periodic orbit is ’trapped’ if it stays near the cap in the
following sense

$d(\gamma(t),\gamma_{\pm})\leq\sqrt{\varepsilon}\ \ \forall t.$

In coordinates this means either $\theta\geq\frac{\pi}{2}-\sqrt{\varepsilon}$
or $\theta\leq\sqrt{\varepsilon}$. Both cases work analogously, so we only
consider the case $\theta\leq\sqrt{\varepsilon}$. As
$c^{2}=\dot{\theta}^{2}+\sin\theta^{2}\dot{\varphi}_{1}^{2}+\cos\theta^{2}\dot{\varphi}_{2}^{2}$
we immediately find

$|\dot{\varphi}_{1}|\leq\frac{c}{\sin\theta}\ \ \&\ \
|\dot{\varphi}_{2}|\leq\frac{c}{\cos\theta}.$

In particular as $\theta\leq\sqrt{\varepsilon}$ we have

$|\dot{\varphi}_{2}|\leq\frac{c}{\cos(\sqrt{\varepsilon})}.$

If we could assume that $\dot{\varphi}_{2}\neq 0$, we could conclude

$T_{E+V_{\varepsilon}}\geq\frac{2\pi}{\max|\dot{\varphi}_{2}|}\geq\frac{2\pi}{c}\cos(\sqrt{\varepsilon}).$

Indeed $\dot{\varphi}_{2}\neq 0$ by the following argument. First, recall from
[3] that if ${\gamma}$ is a magnetic geodesic of strength $\varepsilon$, speed
$c$ and enclose the angle $\delta$ with the Reeb vector field $R$ on $S^{3}$,
then the projection $\pi\circ{\gamma}$ of ${\gamma}$ onto
$S^{2}\left(\frac{1}{2}\right)$ for the Hopf fibration is a closed curve of
radius

$R_{c,\delta,\varepsilon}=\sqrt{\frac{(c-\delta)(c+\delta)}{\varepsilon^{2}+4(c^{2}-\varepsilon\delta)}}.$

That might be surprising since one would expect that the radius of projections
of magnetic geodesics only depends on the speed and the velocity. Now on the
one hand we know that

$\displaystyle\frac{1}{2}\sin(\sqrt{\varepsilon})\geq
R=\sqrt{\frac{c^{2}-\delta^{2}}{\varepsilon^{2}+4(c^{2}-\varepsilon\delta)}}\geq\sqrt{\frac{c^{2}-\delta^{2}}{(\varepsilon-2c)^{2}}}$
$\displaystyle\Rightarrow|\delta|\geq\sqrt{c^{2}-\frac{1}{4}(\varepsilon-2c)^{2}\sin(\sqrt{\varepsilon})^{2}}\longrightarrow
c$

as $\varepsilon$ goes to zero. On the other hand if $\dot{\varphi}_{2}=0$
somewhere, it follows that
$|c_{2}|=\cos(\theta)^{2}|\dot{\varphi}_{2}+\varepsilon|\leq\varepsilon$.
Using the energy bound $|\dot{\varphi}_{1}|\leq\frac{c}{\sin(\theta)}$ we find

$|c_{1}|=\sin(\theta)^{2}|\dot{\varphi}_{1}+\varepsilon|\leq\sin(\sqrt{\varepsilon})c+\sin(\sqrt{\varepsilon})^{2}\varepsilon.$

All together this implies

$|\delta|=|c_{1}+c_{2}-\varepsilon|\leq\sin(\sqrt{\varepsilon})c+\sin(\sqrt{\varepsilon})^{2}\varepsilon+2\varepsilon\longrightarrow
0$

as $\varepsilon\to 0$. This contradicts $|\delta|\to c$ and we conclude that
$\dot{\varphi}_{2}\neq 0$.

Enters only one cap: If $\gamma_{\varepsilon}$ enters only one cap and is not
trapped, we know that the magnetic geodesic it follows outside the caps is of
(euclidean) radius
$R>\frac{1}{2}\sin\left(\frac{\sqrt{\varepsilon}+\varepsilon}{2}\right)$.
Denote $d$ the (euclidean) distance, between the center of the cap and the
center of the magnetic geodesic and $r=\frac{1}{2}\sin(\varepsilon)$ the
radius of the cap. Now

$\cos(\alpha_{\varepsilon})=\frac{d-x}{R};\ \ x=\frac{r^{2}+d^{2}-R^{2}}{2d},$

thus using the estimates $R-r<d<R+r$ we find

$\cos(\alpha_{\varepsilon})=\frac{d^{2}-r^{2}+R^{2}}{2dR}\geq\frac{R^{2}+(R-r)^{2}-r^{2}}{2(R+r)R}=\frac{R-r}{R+r}.$

We may assume $\varepsilon$ to be small so that we can approximate
$\sin(\varepsilon)=\varepsilon,\
\sin\left(\frac{\sqrt{\varepsilon}+\varepsilon}{2}\right)=\frac{\sqrt{\varepsilon}+\varepsilon}{2}$
to find

$\cos{\alpha_{\varepsilon}}\geq\frac{1+\sqrt{\varepsilon}}{1+3\sqrt{\varepsilon}}=1-\mathcal{O}(\varepsilon).$

It follows that

$\alpha_{\varepsilon}=\mathcal{O}(\varepsilon).$

This means as $\varepsilon\to 0$ these orbits either approximate periodic
bounce orbits of type (2) or have length tending to infinity. In particular

$T_{E+V_{\varepsilon}}=T_{E}-\mathcal{O}(\varepsilon).$

Enters both caps: Periodic orbits that enter both caps either approximate
orbits of type (3) or their length tends to infinity as $\varepsilon\to 0$. In
particular

$T_{E+V_{\varepsilon}}>T_{E}-\mathcal{O}(\varepsilon).$

##  Appendix C Cohomology of the quadric

The cohomology of $Q^{3}$ is well known to experts, but for the sake of
completeness, we add its computation in this article. The main tools are the
Lefschetz hyperplane theorem and the Mayer-Vietoris sequence.

######  Lemma C.1.

The cohomology of the complex quadric $Q^{3}$ is given by

$H^{*}(Q^{3},\mathbb{Z})=\left\\{\begin{array}[]{ll}\mathbb{Z},&*=0,2,4,6\\\
0,&*=1,3,5\end{array}\right.$

Further the generators $A\in H_{2}(Q^{3},\mathbb{Z})$ and $B\in
H_{4}(Q^{3},\mathbb{Z})$ have intersection equal to 1, i.e. $A\cdot B=1$.

###### Proof.

Using the direct corollaries [24, Cor. 1.24 & 1.25] of the Lefschetz
hyperplane theorem for hypersurfaces, we immediately find
$H^{*}(\mathbb{C}\mathrm{P}^{3},\mathbb{Z})\cong H^{*}(Q^{3},\mathbb{Z})$ for
$*\neq 3$. To find $H^{3}(Q^{3},\mathbb{Z})\cong H_{3}(Q^{3},\mathbb{Z})$ one
can for example apply the Mayer–Vietoris sequences to the decomposition
$Q^{3}\cong DS^{3}\cup NQ^{2},$ where $NQ^{2}$ denotes the disk normal bundle
of $Q^{2}$. Observe that $DS^{3}\simeq S^{3}$, $NQ^{2}\simeq Q^{2}\simeq
S^{2}\times S^{2}$ and $DS^{3}\cap NQ^{2}\simeq S^{3}\times S^{2}$ as the
tangent bundle of $S^{3}$ is trivializable. Therefor all terms but
$H_{3}(Q^{3},\mathbb{Z})$ are known. The statement about the intersection
again follows directly from [24, Cor. 1.25]. ∎

Figure 5: The circle of radius $r$ represents the cap. The circle of radius
$R$ the magnetic geodesic. We need to show that the angle
$\alpha_{\varepsilon}$ tends to zero, as this also means that most of the time
the trajectory stays outside the cap where the potential does not vanish.

## References

  * [1] A. Abbondandolo, G. Benedetti, and O. Edtmair.  Symplectic capacities of domains close to the ball and Banach–Mazur geodesics in the space of contact forms.  arXiv preprint arxiv:2312.07363, 2023.
  * [2] N. Adaloglou.  Uniqueness of Lagrangians in ${T}^{*}\mathbb{R}\mathrm{P}^{2}$.  arXiv preprint arXiv:2201.09299, 2022.
  * [3] P. Albers, G. Benedetti, and L. Maier.  Magnetic geodesics on odd dimensional spheres.  To appear.
  * [4] P. Albers and M. Mazzucchelli.  Periodic Bounce Orbits of Prescribed Energy.  International Mathematics Research Notices, 2011(14):3289–3314, 01 2011.
  * [5] S. Artstein-Avidan, R. Karasev, and Y. Ostrover.  From symplectic measurements to the Mahler conjecture.  Duke Math. J., 163(11), 2014.
  * [6] G. Benedetti.  The contact property for magnetic flows on surfaces.  PhD Thesis, University of Cambridge, (see https://doi.org/10.17863/CAM.16235, or arXiv:1805.04916), 2014.
  * [7] G. Benedetti, J. Bimmermann, and K. Zehmisch.  Symplectic capacities of disc cotangent bundles of flat tori.  arXiv preprint arXiv:2311.07351, 2023.
  * [8] G. Benedetti and J. Kang.  Relative Hofer–Zehnder capacity and positive symplectic homology.  Journal of Fixed Point Theory and Applications, 24(2):44, 2022.
  * [9] J. Bimmermann.  Hofer-Zehnder capacity of disc tangent bundles of projective spaces.  arXiv preprint arXiv:2306.11382, 2023.
  * [10] J. Bimmermann.  On the Hofer–Zehnder Capacity of Twisted Tangent Bundles.  PhD thesis, 2023.
  * [11] K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk.  Quantitative symplectic geometry, page 1–44.  Mathematical Sciences Research Institute Publications. Cambridge University Press, 2007.
  * [12] D. Cristofaro-Gardiner and R. Hind.  On the agreement of symplectic capacities in high dimension.  arXiv preprint arxiv:2307.12125, 2023.
  * [13] O. Edtmair.  Disk-like surfaces of section and symplectic capacities.  arXv preprint arxiv:2206.07847, 2023.
  * [14] B. Ferreira and V. G. Ramos.  Symplectic embeddings into disk cotangent bundles.  Journal of Fixed Point Theory and Applications, 24(3):62, 2022.
  * [15] J. Gutt, M. Hutchings, and V. G. B. Ramos.  Examples around the strong Viterbo conjecture.  Journal of Fixed Point Theory and Applications, 41, 2022.
  * [16] H. Hofer and C. Viterbo.  The Weinstein conjecture in the presence of holomorphic spheres.  Communications on Pure and Applied Mathematics, 45(5):583–622, 1992.
  * [17] H. Hofer and E. Zehnder.  Symplectic invariants and Hamiltonian dynamics.  Birkhäuser, 2011.
  * [18] E. Lerman.  Symplectic cuts.  Math. Res. Lett., 2(3):247–258, 1995.
  * [19] A. Loi, R. Mossa, and F. Zuddas.  Symplectic capacities of Hermitian symmetric spaces.  arXiv preprint arXiv:1302.1984, 2013.
  * [20] G. Lu.  Gromov-Witten invariants and pseudo symplectic capacities.  Israel Journal of Mathematics, 156(1):1–63, 2006.
  * [21] D. McDuff and D. Salamon.  J-holomorphic curves and symplectic topology, volume 52.  American Mathematical Soc., 2012.
  * [22] M. Usher.  Many closed symplectic manifolds have infinite Hofer–Zehnder capacity.  Transactions of the American Mathematical Society, 364(11):5913–5943, 2012.
  * [23] A.-M. Vocke.  Periodic bounce orbits in magnetic billiard systems.  PhD thesis, 2021.
  * [24] C. Voisin.  Hodge Theory and Complex Algebraic Geometry II: Volume 2, volume 77.  Cambridge University Press, 2003.
  * [25] J. Weber.  Noncontractible periodic orbits in cotangent bundles and Floer homology.  Duke Mathematical Journal, 133(3):527–566, 2006.
  * [26] C. Wendl.  Holomorphic curves in low dimensions.  Springer, 2018.


# Massive white dwarfs in Rastall-Rainbow gravity

Jie Li  Bo Yang  Wenbin Lin,11footnotetext: Corresponding author.

###### Abstract

We investigate the hydrostatic equilibrium of white dwarfs within the
framework of Rastall-Rainbow gravity, aiming to explore the effects of this
modified gravitational theory on their properties. By employing the
Chandrasekhar equation of state in conjunction with the modified Tolman-
Oppenheimer-Volkoff equation, we derive the mass-radius relations for white
dwarfs. Our results show that the maximum mass of white dwarfs deviates
significantly from the predictions of general relativity, potentially
exceeding the Chandrasekhar limit. Furthermore, we discuss other properties of
white dwarfs, such as the gravitational redshift, compactness and dynamical
stability, shedding light on their behavior within the context of this
modified gravitational framework.

##  1 Introduction

Einstein’s general relativity (GR), proposed a century ago, is a successful
theory of gravity which has helped us understand many aspects of the universe.
In particular, some experimental tests from weak gravity, as well as recent
observations like the shadows of black holes and gravitational waves by binary
stars mergers, have strongly supported GR. Nonetheless, there are some
phenomena at the larger (cosmological) scale that are not well described in
the context of GR. For example, GR fails to explain the accelerated expansion
of the universe without further refinement. To overcome this limitation, two
possibilities were proposed. One is that there is a large amount of dark
energy with negative pressure in the universe. The other is that GR’s
predictions may be biased on the cosmological scale, so alternative theories
of gravity were proposed, such as massive gravity [1], Brans-Dicke gravity [2]
and $f(R)$ gravity [3].

Additionally, Rastall [4] proposed a possible modified theory of gravity.
Rastall argued that current experiments merely verified the vanishing of the
covariant derivative of the energy-momentum tensor in flat spacetimes, without
necessarily holding true in curved spacetime. Thus, Rastall replaced the
energy-momentum tensor conservation law in GR by assuming that the covariant
divergence of the energy-momentum tensor varies proportionally to the gradient
of the Ricci scalar. In the past few years, Rastall gravity has attracted
attention from researchers due to its potential to explain some phenomena in
the universe [5, 6, 7, 8]. Recently, several works have been done to test
Rastall gravity by studying compact stars with strong gravitational fields.
Some solutions of black hole in the Rastall gravity are obtained in Refs. [9,
10, 11, 12, 13, 14], as well as various physical properties of these black
holes have been extensively studied [15, 16]. Oliveira et al., [17] first
obtained the modified equilibrium equation in this gravity, and discussed the
effects of Rastall parameter on the structure of neutron stars. Ghosh et al.,
[18] investigated gravastars in Rastall gravity. In the literatures [19, 20],
the authors studied anisotropic compact stars in Rastall gravitational
framework using Karori-Barua metric or Tolman-Kuchowicz metric. Quark stars
and quintessence compact stars were discussed in Refs. [21, 22].

Rainbow gravity was proposed by Magueijo and Smolin [23], who incorporate
double special relativity into the framework of GR. Recently, Mota et al.
[24]combined Rastall gravity with Rainbow theory and employed the modern
equation of state (EoS) of nuclear matter to derive models of neutron stars.
They concluded that slight changes in both parameters of this theory of
gravity have a substantial impact on the structure of the neutron star.
Subsequently, anisotropic neutron stars in this gravitational background was
studied [25]. Charged anisotropic strange stars and charged gravastars in
Rastall-Rainbow gravity were investigated in Refs. [26, 27].

White dwarfs are extraordinarily dense celestial bodies that formed from the
collapse of stellar stars following the complete depletion of their nuclear
fuel within the central region. The study of white dwarfs has been crucial in
advancing our understanding of stellar evolution and the physics of compact
objects. In the past few years, a lot of researches have been conducted on the
relationship between the mass and radius of white dwarfs, revealing that the
existence of an upper mass limit known as the Chandrasekhar mass limit [28,
29]. However, some recently observed peculiar over-luminous type-Ia
supernovae, such as SN 2007if, SN 2006gz, SN 2003fg and SN 2009DC [30, 31],
predicted the existence of white dwarfs with masses ranging from 2.1
$M_{\odot}$ to 2.8 $M_{\odot}$ ($M_{\odot}$ is the mass of the Sun) [32, 33,
34, 35]. In order to explain the formation mechanism of super-Chandrasekhar
white dwarfs, extensive research investigated white dwarfs in various
contexts, such as super strong uniform magnetized white dwarfs [36, 37, 38,
39, 40], white dwarfs in modified theories of gravity [41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51], electrical charged white dwarfs [52, 53] and rotating
white dwarfs [54, 55]. In this work, we explore the behavior of white dwarfs
within the framework of Rastall-Rainbow gravity, specifically investigating
the potential existence of super-Chandrasekhar white dwarfs within this
modified gravitational theory.

This article is organized as follows. In Sec. II, we briefly review Rastall
gravity and Rainbow gravity, and proceed to derive the modified Tolman-
Oppenheimer-Volkoff (TOV) equation within the framework of Rastall-Rainbow
gravity. In Sec. III, we recall the Chandrasekhar EoS. In Sec. IV, we solve
modified TOV equation numerically and present the mass-radius relations for
white dwarfs in Rastall-Rainbow gravity. In Sec. V, some properties of white
stars are investigated, including the gravitational redshift, compactness and
dynamic stability. Finally, summary is given in Sec. VI.

##  2 Basic equations of Rastall-Rainbow gravity

###  2.1 Rastall-Rainbow gravity

Double special relativity was proposed due to the existence of the minimum
observable length, which requires the speed of light and the Planck energy
remain constant in the low energy limit. Consequently, the energy-momentum
dispersion relation for a particle of mass $m$ can be written in the following
general form

$\displaystyle
E^{2}\Delta^{2}(\varepsilon)-p^{2}\Theta^{2}(\varepsilon)=m^{2}~{},$ (2.1)

where $\varepsilon=E/E_{p}$ is the ratio of the particle’s energy $E$ to the
Planck constant $E_{p}$. $p$ is the particle’s momentum. $\Delta(\varepsilon)$
and $\Theta(\varepsilon)$ are known as the rainbow functions, and they satisfy
the conditions

$\displaystyle\lim_{\varepsilon\rightarrow
0}~{}\Delta(\varepsilon)=1,~{}~{}~{}~{}\lim_{\varepsilon\rightarrow
0}~{}\Theta(\varepsilon)=1~{};$ (2.2)

Magueijo and Smolin integrated the double special relativity into the
framework of general relativity to propose Rainbow gravity. In this proposal,
the modified metric can be written as

$\displaystyle g(\varepsilon)=\eta^{ab}e_{a}(\varepsilon)\tiny\otimes
e_{b}(\varepsilon)~{},$ (2.3)

where $e_{a}(\varepsilon)$ and $e_{b}(\varepsilon)$ are the energy-dependent
orthonormal frame fields, and their expressions are as follows

$\displaystyle
e_{0}(\varepsilon)=\frac{1}{\Delta(\varepsilon)}~{}{e_{0}},~{}~{}~{}e_{i}(x)=\frac{1}{\Theta(\varepsilon)}~{}{e_{i}}~{};$
(2.4)

According to the above description, the spherical symmetry metric in Rainbow
gravity can be written as

$\displaystyle
ds^{2}=-\frac{e^{2\nu}}{\Delta(\varepsilon)^{2}}~{}dt^{2}+\frac{e^{2\lambda}}{\Theta(\varepsilon)^{2}}~{}dr^{2}+\frac{r^{2}}{\Theta(\varepsilon)^{2}}~{}(d\theta^{2}+\sin\theta^{2}d\phi^{2})~{},$
(2.5)

where $e^{2\nu}$ and $e^{2\lambda}$ are radial functions. Obviously, the
spherically symmetric metric is related to the rainbow function, so it depends
on the energy of the probe particle.

According to Rastall’s proposal, the conservation law for the energy-momentum
tensor may not hold true in curved spacetime, and it should be rewritten as

$\displaystyle\nabla_{\mu}T^{\mu\nu}=\frac{1-\kappa}{16\pi
G}\nabla_{\nu}R~{},$ (2.6)

where $\kappa$ is a constant called Rastall parameter, which measures the
deviation from GR and describes the affinity of the matter field to couple
with geometry. The field equation in Rastall’s gravity can be written as

$\displaystyle R_{\mu\nu}-\frac{\kappa}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}~{};$
(2.7)

###  2.2 Hydrostatic equilibrium equation in Rastall-Rainbow gravity

Mota et al [8] first combined the above two modified gravitational theories
and derived the Hydrostatic equilibrium equations. Next, we introduce the
modified TOV equation in this gravitational theory. The field equations in
Rastall-Rainbow is

$\displaystyle
R_{\mu\nu}(\varepsilon)-\frac{\kappa}{2}g_{\mu\nu}R(\varepsilon)=8\pi
G(\varepsilon)T_{\mu\nu}(\varepsilon)~{};$ (2.8)

In this investigation, we consider the isotropic perfect fluid whose energy-
momentum tensor is

$\displaystyle T_{\mu\nu}=(P+c^{2}\rho)U_{\mu}U_{\nu}-Pg_{\mu\nu}~{},$ (2.9)

where $P$ and $\rho$ denote the pressure and the energy density of the fluid,
respectively, while $U_{\mu}$ corresponds to the four-velocity.

By employing Eqs. (2.5) and (2.9), we can express the components of the
energy-momentum tensor in the four-dimensional spacetime as

$\displaystyle T^{0}_{0}=\rho
c^{2},~{}~{}~{}T^{1}_{1}=T^{2}_{2}=T^{3}_{3}=-P~{};$ (2.10)

Using the energy-momentum tensor Eq. (2.10) and the spherical symmetry metric
Eq. (2.5), the non-vanishing components of the field equations are given as

$\displaystyle\frac{e^{-2\lambda}}{r^{2}}(2r\lambda^{{}^{\prime}}-1)+\frac{1}{r^{2}}=\frac{8\pi\rho_{eff}(r)}{c^{2}}~{},$
(2.11)
$\displaystyle\frac{e^{-2\lambda}}{r^{2}}(2r\upsilon^{{}^{\prime}}+1)-\frac{1}{r^{2}}=\frac{8\pi
P_{eff}(r)}{c^{4}}~{},$ (2.12)
$\displaystyle\frac{e^{-2\lambda}}{r}\big{(}\upsilon^{{}^{\prime}}+r\upsilon^{{}^{\prime}2}-\lambda^{{}^{\prime}}-r\lambda^{{}^{\prime}}\upsilon^{{}^{\prime}}+r\upsilon^{{}^{\prime\prime}}\big{)}=\frac{8\pi
P_{eff}(r)}{c^{4}}~{},$ (2.13)

where prime denotes the differentiation with $r$. $\rho_{eff}(r)$ and
$p_{eff}(r)$ are the effective density and pressure, the expressions are

$\displaystyle\rho_{eff}(r)=\frac{1}{\Delta(x)^{2}}\Big{(}\alpha
c^{2}\rho+3\beta P\Big{)}~{},$ (2.14) $\displaystyle
P_{eff}(r)=\frac{1}{\Delta(x)^{2}}\Big{(}\beta
c^{2}\rho+(1-3\beta)P\Big{)}~{},$ (2.15)

where

$\displaystyle\alpha=\frac{2+3\chi}{2~{}(2+2\chi)},~{}~{}~{}\beta=\frac{\chi}{2~{}(2+2\chi)}~{};$
(2.16)

Here we assumed that $\chi=\kappa-1$. The differential equations Eqs.
(2.11)-(2.13) are similar to those obtained in GR. Thus, the expression of the
function $e^{-2\lambda}$ can be obtained by directly integrating Eq. (2.11),
as

$\displaystyle e^{-2\lambda}=1-\frac{2Gm(r)}{c^{2}r}~{},$ (2.17)

where $m(r)$ is the gravitational mass within the sphere of radius $r$. An
additional equation related to the metric function $\nu(r)$ is obtained from
Eq. (2.6) and reads

$\displaystyle\frac{d\nu}{dr}=-\Big{[}\beta\frac{d\rho}{dP}+1-3\beta\Big{]}\Big{[}(\alpha+\beta)c^{2}\rho+P\Big{]}^{-1}\frac{dP}{dr}~{};$
(2.18)

The expression for $d\nu/dr$ is derived from Eqs. (2.11) and (2.17), and it is
brought into Eq. (2.18), one can obtain the modified TOV equations in Rastall-
Rainbow gravity as

$\displaystyle\frac{dP}{dr}=-\frac{Gm}{c^{2}r^{2}}\Big{[}\Big{(}\beta\frac{d\rho}{dP}+1-3\beta\Big{)}\Big{(}1-\frac{2m}{c^{2}r}\Big{)}\Big{]}^{-1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Big{[}\big{(}\alpha+\beta\big{)}c^{2}\rho+P\Big{]}\Big{[}1+\frac{4\pi
r^{3}}{m\Delta^{2}}\Big{(}\beta c^{2}\rho+P-3\beta P\Big{)}\Big{]}~{},$ (2.19)

and

$\displaystyle\frac{dm(r)}{dr}=\frac{4\pi r^{2}}{\Delta^{2}}\Big{(}\alpha
c^{2}\rho+3\beta P\Big{)}~{};$ (2.20)

Naturally, by setting the parameter $\chi=0$, that is, $\alpha=1$ and
$\beta=0$, the modified TOV equation in Rainbow is recovered. When $\chi=0$
and $\Delta=0$, the standard TOV equation in GR is recovered.

##  3 Equation of state

Solving the modified TOV equation necessitates the adoption of a specific EoS
that establishes the interrelation between the pressure and density within the
star. White dwarfs, as compact celestial bodies primarily composed of
degenerate electron gas, have been extensively studied using the Chandrasekhar
EoS [29], which can be expressed in the following form

$\displaystyle
P(k_{F})=\frac{1}{3\pi^{2}\hbar^{3}}\int^{k_{F}}_{0}\frac{k^{4}}{\sqrt{k^{2}+m_{e}^{2}}}dk$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{\pi
m_{e}^{4}}{3h^{3}}\Big{[}x_{F}(2x_{F}^{2}-3)\sqrt{x_{F}^{2}+1}+3\sinh^{-1}x_{F}\Big{]}~{},$
(3.1)
$\displaystyle~{}~{}~{}\rho=\frac{8\pi\mu_{e}m_{H}m_{e}^{3}}{3h^{3}}x_{F}^{3}~{},$
(3.2)

where $x_{F}\equiv p_{F}/m_{e}c$, $p_{F}$ being the Fermi momentum, $k$ is the
momentum of electrons, $m_{e}$ represents the electron mass, $m_{H}$ is the
nucleon mass, $h$ denotes the Planck constant. $\mu_{e}$ is understood as the
mean molecular weight per electron. For carbon-oxygen white dwarfs,
$\mu_{e}=2$.

##  4 Numerical results

Some recently observed peculiar over-luminous SNIa predict the mass of white
dwarfs to be between 2.1$M_{\odot}$ and 2.8 $M_{\odot}$. Therefore, the upper
limit of the white dwarf’s mass is still an open question. In the following,
we delve into the properties of white dwarfs within the context of Rastall-
Rainbow gravity. The objective is to determine whether this gravitational
model can account for the existence of super-Chandrasekhar white dwarfs. In
this section, the results for white dwarfs are presented using the
Chandrasekhar EoS given in the previous subsection. The mass and radius of
white stars are obtained by numerically integrating the modified TOV equations
Eqs. (2.19) and (2.20). Here, we set the boundary conditions at the center of
the star $(r=0)$ as

$\displaystyle\rho(0)=\rho_{c},~{}~{}and~{}~{}M(0)=0~{},$ (4.1)

where $\rho_{c}$ is the central energy density. The integral is interrupted on
the surface of the white dwarf, where the pressure disappears ($P(R)=0$), the
corresponding $R$ and $M(R)$ are the radius and total mass of the white dwarf.

Figure 1: Mass function $m(r)$ from the center of white dwarfs to the surface
for different parameters $\chi$ (left panel) and $\Delta$ (right panel), where
we set $\rho_{c}=1\times 10^{12}~{}$kg/m3.

Figure 2: The mass-radius (left) and mass-central density $\rho_{c}$ relations
(right) for the white stars in Rastall gravity for different values of $\chi$.

Figure 3: The mass-radius (left) and mass-central density $\rho_{c}$ relations
(right) for the white stars in Rastall-Rainbow gravity for different values of
$\Delta$.

First, Fig. 1 shows the behavior of the mass against the radius from the
center to the surface of the star for $\rho_{c}=1\times 10^{12}~{}$kg/m3. Our
results indicate that as the radius increases, the mass of the white dwarf
increases. Furthermore, the mass obtained in the case of GR is the smallest
compared to other scenarios.

Next, we present the mass-radius relations for white dwarfs within the
framework of Rastall gravity, depicted in the left panel of Fig. 2. Here, we
consider a fixed rainbow parameter $\Delta=0$, while varying the parameter
$\chi$ in the range of $0$ to $6\times 10^{-5}$. Notably, for $\chi=0$ (the
blue solid line), the mass and radius of the white dwarf return to the results
obtained in the general relativity, where the maximum mass $M=1.41M_{\odot}$
is close to the Chandrasekhar limit. The other three curves represent the
Rastall gravity case, which demonstrates that the maximum mass of white dwarfs
exceeds the Chandrasekhar limit and increases with an increase in $\chi$. For
example, in the case of $\chi=4\times 10^{-5}$, the maximum mass increases to
$1.48M_{\odot}$. The right panel of Fig. 2 shows the mass-center density
relations, and the black dots in the figure mark the position of the maximum
mass. According to the Harrison-Zeldovich Novikov criterion,
$dM/d\epsilon_{c}>0$ corresponds to a stable configuration, while
$dM/d\epsilon_{c}<0$ is unstable one. The results show that in the scenario
where $\chi<6\times 10^{-5}$, the mass of the white dwarf increases with the
center density until reaching the maximum mass, indicating a state of
stability. Beyond this point, the mass exhibits a decreasing trend with the
center density. It is notable that, for $\chi=6\times 10^{-5}$, the mass
decreases monotonically with the central density, implying that these white
dwarf models are unstable. Consequently, the Chandrasekhar limit in Rastall
gravity is estimated to be $M\approx 1.51M_{\odot}$, with detailed data
provided in Table 1.

Finally, we investigate the effect of the rainbow parameter $\Delta$ on the
properties of white dwarfs, and the results are shown in Fig. 3. Here we fix
$\chi=2\times 10^{-5}$ and vary the value of $\Delta$. The results indicate
that when considering the effect of the Rainbow parameter $\Delta$, the
maximum mass of white dwarfs is larger compared to that in Rastall gravity. It
can be seen that super-Chandrasekhar white dwarfs can be obtained for some
specific values of the Rainbow parameters. For example, in the case of
$\Delta=1.5$, the maximum mass of the white dwarf is $2.14M_{\odot}$.
Meanwhile, considering the upper limit of the super-Chandrasekhar white dwarf
mass as $2.8M_{\odot}$, we can constrain the value of the rainbow parameter to
$\Delta_{upper}=2$. In this scenario, the maximum mass reaches
$2.86M_{\odot}$. For more detailed analysis, Fig. 4 shows the $equi-M_{max}$
contours in $\Delta-\chi$ plane. It can be observed that larger values of
$\chi$ and $\Delta$ facilitate the attainment of super-Chandrasekhar white
dwarfs, and $\Delta_{upper}$ exhibiting a decrease as the Rastall parameter
$\chi$ increases.

Figure 4: $\Delta-\chi$ plane for $equi-M_{max}$ contours. $\chi$ | $M_{max}~{}(M_{\odot})$ | R (km) | $\sigma~{}(10^{-2})$ | $z_{s}~{}(10^{-2})$
---|---|---|---|---
$0.0\times 10^{-5}$ | 1.41 | 898 | 0.46 | 0.23
$1.0\times 10^{-5}$ | 1.42 | 1131 | 0.37 | 0.19
$2.0\times 10^{-5}$ | 1.43 | 1508 | 0.27 | 0.14
$3.0\times 10^{-5}$ | 1.45 | 2166 | 0.19 | 0.10
$4.0\times 10^{-5}$ | 1.48 | 3387 | 0.12 | 0.06
$5.0\times 10^{-5}$ | 1.51 | 6908 | 0.06 | 0.03
Table 1: Structure properties of white dwarfs for different values of $\chi$. Here $\Delta=0$. $\Delta$ | $M_{max}~{}(M_{\odot})$ | R (km) | $\sigma~{}(10^{-2})$ | $z_{s}~{}(10^{-2})$
---|---|---|---|---
1.00 | 1.43 | 1508 | 0.27 | 0.14
1.25 | 1.79 | 1885 | 0.28 | 0.14
1.50 | 2.14 | 2263 | 0.28 | 0.14
1.75 | 2.50 | 2639 | 0.28 | 0.14
2.00 | 2.86 | 3016 | 0.28 | 0.14
Table 2: Structure properties of white dwarfs for different values of
$\Delta$. Here $\chi=2.0\times 10^{-5}$.

##  5 Structure properties of white dwarfs

###  5.1  Gravitational redshift and compactness

The gravitational redshift provides a direct measurement of the gravitational
field strength near the compact star, aiding in a understanding of
gravitational properties under extreme conditions. Moreover, it serves as a
significant probe for investigating the EoS of compact stars, and its
expression is

$\displaystyle z_{s}=\frac{1}{\sqrt{1-\sigma}}-1~{}.$ (5.1)

where $\sigma$ is the compactness of compact stars, which is defined as

$\displaystyle\sigma=\frac{2GM}{c^{2}R}~{},$ (5.2)

The compactness and gravitational redshift of white dwarfs are given in the
last two columns of Table 1 and 2. As can be seen from the above two tables,
the gravitational redshift and compactness decrease with the increase of
parameter $\chi$, while the rainbow parameter $\Delta$ has no significant
impact on these properties.

###  5.2 Dynamical stability

The adiabatic index $\Gamma$, defined as the ratio of two specific heats,
plays a crucial role in assessing the stability of white dwarfs. Chandrasekhar
first studied the dynamical stability of stellar models against infinitesimal
radial adiabatic perturbation [56]. Heintzmann and Hillebrandt conducted
calculations [57] and established that for compact objects to be dynamically
stable at any position, the adiabatic index must exceed $\frac{4}{3}$. The
expression for $\Gamma$ can be written as

$\displaystyle\Gamma=\Big{(}\frac{c^{2}\rho+P}{c^{2}P}\Big{)}\frac{dP}{d\rho}~{},$
(5.3)

Fig. 5 shows the adiabatic index versus the density for white stars in
Rastall-Raindow gravity. The grey solid line represents the critical adiabatic
index, $\Gamma=1.33$. The result shows that for the different values of the
parameters $\chi$ and $\Delta$ are over 1.33, which means that these white
dwarf models are dynamically stable.

Figure 5: Adiabatic index - density relations for different values of $\chi$
and $\Delta$. Here the mass of white dwarfs is $1.4M_{\odot}$.

##  6 Summary

Super-Chandrasekhar white dwarfs are compact stellar remnants that exceed the
standard mass limit. The supernova explosions triggered by these unusually
massive white dwarfs exhibit distinct spectral and luminosity characteristics,
making them significant for studying their properties. In Refs [41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51], authors obtained massive white dwarfs by
considering various modified gravitational theories. Motivated by this, we
investigate white dwarfs in Rastall-Rainbow gravity.

We numerically solve the modified TOV equation with Chandrasekhar EoS,
resulting in the derivation of the mass-radius relations for white dwarfs
under varying parameters of $\chi$ and $\Delta$. It was demostrated that the
parameters of Rastall-Rainbow gravity have significant effects on the
structure of white dwarfs. In the realm of Rastall gravity, when $\chi>1\times
10^{-5}$, the maximum mass of white dwarfs exhibits a substantial deviation
from the predictions of GR, with a noticeable upward trend as $\chi$
increases. For the case of $\chi=4\times 10^{-5}$, the maximum mass increases
to $1.48M_{\odot}$, which is larger than the Chandrasekhar limit. Based on the
fact that the structure of white dwarfs is unstable at $\chi>6\times 10^{-5}$,
we can conclude that the Chandrasekhar limit in Rastall gravity is
$M_{max}=1.51M_{\odot}$. Futhermore, we fix $\chi=2\times 10^{-5}$ and vary
the value of rainbow parameter $\Delta$. the results show that super-
Chandrasekhar white dwarfs can be obtained by increasing the value of rainbow
parameter $\Delta$. We can constrain $\Delta<2$ due to that the upper limit of
the super-Chandrasekhar white dwarf’s mass is $2.8M_{\odot}$.

Next, we investigate the gravitational redshift and compactness of white
dwarfs and find that they decrease with the increase of the parameter $\chi$,
but have no significant dependence on the rainbow parameter $\Delta$. Finally,
we confirm that these white dwarf models are dynamically stable by analyzing
the adiabatic index.

## Acknowledgments

We would like to thank Prof. Remo Riffini for helpful discussions. This work
was supported by the National Natural Science Foundation of China (Grant Nos.
11973025 and 12247157), and Scientific Research Fund of Hunan Provincial
Education Department (Grant No. 22B0446).

## References

  * [1] C. de Rham, G. Gabadadze and A. J. Tolle, _Resummation of Massive gravity_ , _Phys. Rev. Lett._ 106 (2011) 231101.
  * [2] C. Brans and R.H. Dicke, _Mach’s principle and a relativistic theory of gravitation_ , _Phys. Rev._ 124 (1961) 925.
  * [3] A. D. Felice and S. Tsujikawa, _f(R) theories_ , _Living Rev. Relativ._ 13 (2010) 3.
  * [4] P. Rastall, _Generalization of the Einstein theory_ , _Phys. Rev. D_ 6 (1972) 3357.
  * [5] C. E. M. Batista, M. H. Daouda, J. C. Fabris, O. F. Piattella and D. C. Rodrigues, _Rastall cosmology and the $\Lambda$CDM model_, _Phys. Rev. D_ 85 (2012) 084008.
  * [6] J. C. Fabris, M. H. Daouda and O. F. Piattella, _Note on the evolution of the gravitational potential in Rastall scalar field theories_ , _Phys. Lett. B_ 711 (2012) 232.
  * [7] H. Moradpour, Y. Heydarzade, F. Darabi and Ines G. Salako, _A generalization to the Rastall theory and cosmic eras_ , _Eur. Phys. J. C_ 78 (2017) 810.
  * [8] F. Darabi, H. Moradpou, I. Licata, Y. Heydarzade and C. Corda, _Einstein and Rastall theories of gravitation in comparison_ , _Eur. Phys. J. C_ 78 (2018) 25.
  * [9] K. A. Bronnikov, J. C. Fabris, O. F. Piattella and E. C. Santos, _Static, spherically symmetric solutions with a scalar field in Rastall gravity_ , _Gen. Rel. Grav._ 48 (2016) 162.
  * [10] Y. Heydarzade and H. Moradpour and F. Darabi, _Black hole solutions in Rastall theory_ , _Can. J. Phys._ 95 (2017) 1253.
  * [11] M. S. Ma and R. Zhao, _Noncommutative geometry inspired black holes in Rastall gravity_ , _Eur. Phys. J. C_ 77 (2017) 629.
  * [12] Y. Heydarzade and F. Darabi, _Black hole solutions surrounded by perfect fluid in Rastall theory_ , _Phys. Lett. B_ 771 (2017) 365.
  * [13] J. P. M. Graca and I. P. Lobo, _Scalar QNMs for higher dimensional black holes surrounded by quintessence in Rastall gravity_ , _Eur. Phys. J. C_ 78 (2018) 101.
  * [14] Z. Xu and X. Hou and X. Gong and J. Wang, _Kerr-Newman-AdS black hole surrounded by perfect fluid matter in Rastall gravity_ , _Eur. Phys. J. C_ 78 (2018) 513.
  * [15] K. Bamba, A. Jawad, S. Rafique and H. Moradpour, _Thermodynamics in Rastall gravity with entropy corrections_ , _Eur. Phys. J. C_ 78 (2018) 986.
  * [16] I. Lobo, P. Moradpour, J. P. M. Graca and I. G. Salako, _Thermodynamics of black holes in Rastall gravity_ , _Int. J. Mod. Phys. D_ 27 (2018) 1850069.
  * [17] A. Oliveira, H. Velten, J. Fabris and L. Casarini, _Neutron stars in Rastall gravity_ , _Phys. Rev. D_ 92 (2015) 044020.
  * [18] S. Ghosh, S. Dey, A. Das, A. Chanda and B. C. Paul, _Study of Gravastars in Rastall Gravity_ , _JCAP_ 07 (2021) 004.
  * [19] P. Bhar, F. Tello-Ortiz, Á. Rincón and Y. Gomez-Leyton, _Study on anisotropic stars in the framework of Rastall gravity_ , _Astrophys. Space Sci._ 365 (2020) 1.
  * [20] G. Abbas and M. Shahzad, _Models of anisotropic compact stars in the Rastall theory of gravity_ , _Astrophys. Space Sci._ 364 (2019) 50.
  * [21] G. Abbas and M. Shahzad, _A new model of quintessence compact stars in the Rastall theory of gravity_ , _Eur. Phys. J. A_ 54 (2018) 211.
  * [22] M. R. Shahzad and G. Abbas, _Strange stars with MIT bag model in the Rastall theory of gravity_ , _Int. J. Geom. Meth. Mod. Phys._ 16 (2019) 1950132.
  * [23] J. Magueijo and L. Smolin, _Gravity’s rainbow_ , _Class. Quant. Grav._ 21 (2004) 1725.
  * [24] C. E. Mota, L. C. N. Santos, G. Grams, F. M. da Silva and D. P. Menezes, _Combined Rastall and Rainbow theories of gravity with applications to neutron stars_ , _Phys. Rev. D_ 100 (2019) 024043.
  * [25] C. E. Mota, L. C. N. Santos, G. Grams and F. M. da Silva, _Anisotropic compact stars in Rastall-Rainbow Gravity_ , _Class. Quant. Grav._ 39 (2022) 085008.
  * [26] U. Debnath, _Charged gravastars in Rastall-Rainbow Gravity_ , _Eur. Phys. J. P_ 136 (2021) 442.
  * [27] K. P. Das, S. Maity, P. Saha and U. Debnath, _Charged anisotropic strange star in Rastall-Rainbow gravity_ , _Mod. Phys. Lett. A_ 37 (2022) 2250201.
  * [28] S. Chandrasekhar, _The maximum mass of ideal white dwarfs_ , _Astrophys. J._ 74 (1931) 81.
  * [29] S. Chandrasekhar, _The highly collapsed configurations of a stellar mass_ , _Mon. Not. R. Astron. Soc._ 95 (1935) 207.
  * [30] D. A. Howell et al., _The type Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star_ , _Nature_ 443 (2006) 308.
  * [31] R. A. Scalzo et al., _Nearby supernova factory observations of SN 2007if: First total mass measurement of a super-Chandrasekhar-mass progenitor_ , _Astrophys. J._ 713 (2010) 1073.
  * [32] Hicke2007 et al., _The luminous and Carbon-Rich Supernova 2006gz: a double degenerate merger?_ , _Astrophys. J._ 669 (2007) L17.
  * [33] M. Yamanaka et al., _Early phase observations of extremely luminous Type Ia Supernova 2009dc_ , _Astrophys. J._ 707 (2009) L118.
  * [34] J. M. Silverman, M. Ganeshalingam, W. Li, A.V. Filippenko, A.A. Miller and D. Poznanski, _Fourteen months of observations of the possible super-Chandrasekhar mass Type Ia Supernova 2009dc_ , _Mon. Not. R. Astron. Soc._ 410 (2011) 585.
  * [35] S. Taubeberger et al., _Nearby supernova factory observations of SN 2007if: First total mass measurement of a super-Chandrasekhar-mass progenitor_ , _Mon. Not. R. Astron. Soc._ 412 (2011) 2735.
  * [36] U. Das and B. Mukhopadhyay, _Strongly magnetized cold degenerate electron gas: Mass-radius relation of the magnetized white dwarf_ , _Phys. Rev. D_ 86 (2012) 042001.
  * [37] U. Das and B. Mukhopadhyay, _New Mass Limit for White Dwarfs: Super-Chandrasekhar Type Ia Supernova as a New Standard Candle_ , _Phys. Rev. Lett._ 110 (2013) 071102.
  * [38] U. Das and B. Mukhopadhyay, _Maximum mass of stable magnetized highly super-Chandrasekhar white dwarfs: stable solutions with varying magnetic fields_ , _JCAP_ 06 (2014) 050.
  * [39] B. Franzon and S. Schramm, _Effects of strong magnetic fields and rotation on white dwarf structure_ , _Phys. Rev. D_ 92 (2015) 083006.
  * [40] D. Deb, B. Mukhopadhyay and F. Weber, _Anisotropic magnetized white dwarfs: unifying under and overluminous peculiar and standard Type Ia supernovae_ , _Astrophys. J._ 926 (2022) 66.
  * [41] U. Das and B. Mukhopadhyay, _Modified Einstein’s gravity as a possible missing link between sub-and super-Chandrasekhar type Ia supernovae_ , _JCAP_ 05 (2015) 045.
  * [42] Z. Z. Jing and D. H. Wen, _A new solution in understanding massive white dwarfs_ , _Chin. Phys. Lett._ 33 (2016) 050401.
  * [43] S. Banerjee and S. Shankarb and T. P. Singh, _Constraints on modified gravity models from white dwarfs_ , _JCAP_ 10 (2017) 004.
  * [44] G. A. Carvalho, R. V. Lobato, P. H. R. S. Morae et al., _Stellar equilibrium configurations of white dwarfs in the f(R, T) gravity_ , _Eur. Phys. J. C_ 77 (2017) 817.
  * [45] S. Kalita and B. Mukhopadhyay, _Modified Einstein’s gravity to probe the sub-and super-Chandrasekhar limiting mass white dwarfs: a new perspective to unify under-and over-luminous type Ia supernovae_ , _JCAP_ 09 (2018) 007.
  * [46] B. Eslam Panah and H. L. Liu, _White dwarfs in de Rham-Gabadadze-Tolley like massive gravity_ , _Phys. Rev. D_ 99 (2019) 104074.
  * [47] H. L. Liu and G. L. Lü, _Properties of white dwarfs in Einstein- $\Lambda$ gravity_, _JCAP_ 02 (2019) 040.
  * [48] F. Rocha, G. A. Carvalho, D. Deb and M. Malheiro, _Study of the charged super-Chandrasekhar limiting mass white dwarfs in the f(R,T) gravity_ , _Phys. Rev. D_ 101 (2020) 104008.
  * [49] A. Wojnar, _White dwarf stars in modified gravity_ , _Int. J. Geom. Meth. Mod. Phys._ 18 (2021) 2140006.
  * [50] S. Kalita, B. Mukhopadhyay and T. R. Govindarajan, _Significantly super-Chandrasekhar mass-limit of white dwarfs in noncommutative geometry_ , _Int. J. Mod. Phys. D_ 30 (2021) 2150034.
  * [51] S. Kalita and L. Sarmah, _Weak-field limit of f(R) gravity to unify peculiar white dwarfs_ , _Phys. Lett. B_ 827 (2022) 136942.
  * [52] H. L. Liu, X. D. Zhanga and D. H . Wen, _One possible solution of peculiar Type Ia Supernovae explosions caused by a charged white dwarf_ , _Phys. Rev. D_ 89 (2014) 104043.
  * [53] G. A. Carvalho, J. D. V. Arba$\tilde{n}$il, R. M. Marinho and M. Malheiro, _White dwarfs with a surface electrical charge distribution: equilibrium and stability_ , _Eur. Phys. J. C_ 78 (2018) 411.
  * [54] K. Boshkayev, J. Rueda and R. Ruffini, _On the maximum mass of general relativistic uniformly rotating white dwarfs_ , _Int. J. Mod. Phys. E_ 20 (2011) 136.
  * [55] K. Boshkayev, J. A. Rueda, R. Ruffini and I. Siutsou, _On general relativistic uniformly rotating white dwarfs_ , _Astrophys. J._ 762 (2012) 117.
  * [56] S. Chandrasekhar, _The dynamical instability of gaseous masses approaching the Schwarzschild limit in general relativity_ , _Astrophys. J._ 140 (1964) 417.
  * [57] H. Heintzmann and W. Hillebrandt, _Neutron stars with an anisotropic equation of state-mass, redshift and stability_ , _Astrophys. J._ 38 (1975) 51.