# Hofer–Zehnder capacity of disc tangent bundles of projective spaces

Johanna Bimmermann

###### Abstract

We compute the Hofer–Zehnder capacity of disc tangent bundles of the complex
and real projective spaces of any dimension. The disc bundle is taken with
respect to the Fubini-Study resp. round metric, but we can obtain explicit
bounds for any other metric. In the case of the complex projective space we
also compute the Hofer–Zehnder capacity for the magnetically twisted case,
where the twist is proportional to the Fubini-Study form. For arbitrary twists
we can still give explicit upper bounds.

##  1 Introduction

The Hofer–Zehnder capacity [12, Chpt. 3] belongs to the class of symplectic
capacities, i.e. numerical symplectic invariants measuring the symplectic size
of a symplectic manifold. For a compact symplectic manifold $(M,\omega)$
possibly with boundary $\partial M$ the Hofer–Zehnder capacity is defined as
follows

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

where admissible means:

  * •

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

  * •

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

  * •

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

Here, $X_{H}$ denotes the Hamiltonian vector field defined imposing the
relation $\mathrm{d}H=-\iota_{X_{H}}\omega$. As all symplectic capacities,
also the Hofer–Zehnder capacity is fairly hard to compute and unknown in many
cases. A classic class of examples for symplectic manifolds are (co-) tangent
bundles. Recall that the cotangent bundle of any smooth manifold $N$ admits a
canonical symplectic structure $\mathrm{d}\lambda$, where
$\lambda\in\Omega^{1}(T^{*}N,\mathbb{R})$ is pointwise defined as follows

$\lambda_{(x,p)}:T_{(x,p)}TN\to\mathbb{R};\ \xi\mapsto
p(\mathrm{d}\pi_{(x,p)}(\xi))\ \ \ \forall\ (x,p)\in T^{*}N,$

where $\pi:TN\to N$ denotes the footpoint projection. Fixing a Riemannian
metric $g$ on $N$ we can pull back $\lambda$ to $TN$ via the metric
isomorphism. By abuse of notation this pullback will also be denoted by
$\lambda$. We will now study the disc subbundles with respect to the already
chosen metric $g$,

$D_{\rho}N:=\\{(x,v)\in TN\ |\ g_{x}(v,v)<\rho^{2}\\}$

and investigate the following question.

######  Question 1.1.

What is the value of $c_{HZ}(D_{\rho}N,\mathrm{d}\lambda)$?

To the authors knowledge this question is still wildly open and most of the
results in this direction concern finiteness. For example the Hofer–Zehnder
capacity is finite, if the base manifold carries a circle action with non-
contractible orbits [13], if the Hurewicz map $\pi_{2}(N)\to
H_{2}(N,\mathbb{Z})$ is non zero [2] or if $N$ is rationally inessential [10].
All these results rely on the observation by Irie [13, Cor. 3.5] that
vanishing of symplectic homology implies finiteness of the Hofer–Zehnder
capacity. Finiteness can also be shown by finding a Lagrangian embedding of
$N$ into a manifold with finite Hofer–Zehnder capacity, as the Lagrangian
neighborhood theorem assures that then also a small disc bundle
$(D_{\varepsilon}N,\mathrm{d}\lambda)\hookrightarrow(M,\omega)$ embeds
symplectically. This is also the easiest way to see that for the two examples
we will treat $c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)$ and
$c_{HZ}(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)$ are finite. Indeed
we have Lagrangian embeddings
$\mathbb{C}\mathrm{P}^{n}\hookrightarrow(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n},\omega_{FS}\ominus\omega_{FS})$
and
$\mathbb{R}\mathrm{P}^{n}\hookrightarrow(\mathbb{C}\mathrm{P}^{n},\omega_{FS})$.

However explicit values of the Hofer–Zehnder capacity of disc tangent bundles
are only given for certain flat tori [12, Ch.4.4, Prop. 4] and some convex
subsets of $\mathbb{R}^{n}$ [3], using that in these cases the tangent bundles
can be trivialized and then identified with subsets of $\mathbb{R}^{2n}$.

There is a variation of the Hofer–Zehnder capacity for which the analog of
Question 1.1 is partially answered. For a general compact symplectic manifold
$(M,\omega)$ possibly with boundary $\partial M$ consider a closed submanifold
$\Sigma\subset M$ not intersecting the boundary, i.e. $\Sigma\subset
M\setminus\partial M$. Then the Hofer–Zehnder capacity relative to $\Sigma$ is
defined as

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

For any homotopy class $\nu\in\pi_{1}(M)$ we say $H:M\to\mathbb{R}$ is
$\nu$-admissible if $H$ is smooth, $0\leq H$, vanishes on an open set
$U\subset M\setminus\partial M$, constantly attains its maximum on the
complement of a compact set $K\subset M\setminus\partial M$ and all non-
constant periodic solutions $\gamma:\mathbb{R}\to M$ of $\dot{\gamma}=X_{H}$
with $[\gamma]=\nu$ have period $T>1$. One defines the $\pi_{1}$-sensitive
Hofer–Zehnder capacity $c^{\nu}_{HZ}(M,\Sigma,\omega)$, replacing admissible
with $\nu$-admissible. In general we have the following chain of inequalities

$c_{HZ}(M,\Sigma,\omega)\leq c_{HZ}^{\nu}(M,\Sigma,\omega)\leq
c_{HZ}^{\nu}(M,\omega)$

and also

$c_{HZ}(M,\Sigma,\omega)\leq c_{HZ}(M,\omega)\leq c_{HZ}^{\nu}(M,\omega),$

simply because the class of admissible Hamiltonians is less and less
constrained. An important result by Weber [18] 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.

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

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 [4, Cor. 2.8]. Observe that restricting to the case
$\rho=1$ is fine as scaling the fibers yields a symplectomorphism identifying
$(D_{\rho}N,\mathrm{d}\lambda)\cong(D_{1}N,\rho\cdot\mathrm{d}\lambda)$.

In this article we derive the precise value of the Hofer–Zehnder capacity of
$(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)$ and
$(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)$ with respect to the
Fubini-Study metric respectively the round metric.

######  Theorem A.

Equip $\mathbb{C}\mathrm{P}^{n}$ with the Fubini-Study metric and
$\mathbb{R}\mathrm{P}^{n}$ with the round metric. Denote by $l$ the length of
the prime geodesics, then

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)=l,$

while

$c_{HZ}(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)=2l.$

Observe that in the case of $\mathbb{C}\mathrm{P}^{n}$ it coincides with the
relative capacity, while in the case of $\mathbb{R}\mathrm{P}^{n}$ there is a
factor of two. This factor of two is an example of the phenomenon of
Lagrangian barriers first described by Biran [6] and recently demonstrated
impressively by Brendel and Schlenk [7].

######  Remark 1.2.

We already made two choices here. First the metric and second the neighborhood
of the zero section. Both are in some sense related. Any two metrics $g$ and
$h$ are related by a symmetric positive definite bundle map $A:TN\to TN$ such
that for all $(x,v)\in TN$ we have

$h_{x}(v,v)=g_{x}(A_{x}v,v).$

Further denote $\lambda_{g}$ resp. $\lambda_{h}$ the pullback of the canonical
1-form with respect to $g$ resp. $h$. Then these two 1-forms can also be
identified via the bundle map $A$, i.e. $A^{*}\lambda_{g}=\lambda_{h}$. In
particular the unit disc subbundle $D_{1}N$ is identified with some ellipsoid
subbundle

$E_{A}N:=\\{(x,v)\in TN\ |\ g_{x}(v,A_{x}^{-1}v)\leq 1\\}.$

Therefore $E_{A}N\subset D_{a}N$ where $a:N\to\mathbb{R}_{\geq 0}$ is the
maximal eigenvalue of $A_{x}^{-1}$. In particular if $N$ is compact we can
obtain the following upper bound for the Hofer–Zehnder capacity

$c_{HZ}(D_{1}N,\mathrm{d}\lambda_{h})\leq\max_{x\in N}a(x)\cdot
c_{HZ}(D_{1}N,\mathrm{d}\lambda_{g}).$

We will also look at the magnetically twisted tangent bundle
$(D_{\rho}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})$ for
arbitrary constants $\rho>0,s\in\mathbb{R}$. We again have a scaling property,
in this case

$(D_{\rho}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})\cong\left(D_{1}\mathbb{C}\mathrm{P}^{n},\rho\left(\mathrm{d}\lambda-\frac{s}{\rho}\pi^{*}\omega_{FS}\right)\right).$

It is therefore enough to restrict to the case $\rho=1$ and $s\in\mathbb{R}$
arbitrary.

######  Theorem B.

Denote $\omega_{FS}$ the Fubini-Study form and $l$ the length of the prime
geodesics, then

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})=l\left(\sqrt{s^{2}+1}-|s|\right).$

######  Remark 1.3.

We can also say something about the Hofer–Zehnder capacity of non-constant
magnetic systems on $\mathbb{C}\mathrm{P}^{n}$. Let
$\nu\in\Omega^{2}(\mathbb{C}\mathrm{P}^{n})$ be an arbitrary closed 2-form. As
$H_{dR}^{2}(\mathbb{C}\mathrm{P}^{n})$ is generated by $[\omega_{FS}]$ we know
that $[\nu]=[s\omega_{FS}]$ in $H_{dR}^{2}(\mathbb{C}\mathrm{P}^{n})$ for some
$s\in\mathbb{R}$. Thus we find an exact form $\mathrm{d}\theta$ such that
$\nu+\mathrm{d}\theta=s\omega_{FS}$. But this means
$(T^{*}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-\pi^{*}\nu)$ is
symplectomorphic to
$(T^{*}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})$ via
the map that shifts the zero-section

$(x,p)\mapsto(x,p+\theta(x)).$

Clearly this map does not map disc bundles to disc bundles but we can still
get some inclusions depending on
$\theta\in\Omega^{1}(\mathbb{C}\mathrm{P}^{n})$. Denote

$\theta_{\max}=\max_{x}|\theta(x)|,$

then

$(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-\pi^{*}\nu)\hookrightarrow(D_{1+\theta_{\max}}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS}).$

There is a freedom of choosing the primitive $\theta$ that we can use to
optimize the bound. Set

$\rho=1+\inf_{\theta}\max_{x}|\theta(x)|,$

where the infimum is taken over all 1-forms $\theta\in\Omega^{1}(M)$
satisfying $\nu+\mathrm{d}\theta=s\omega_{FS}$. Then

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\omega_{\nu})\leq
l\left(\sqrt{s^{2}+\rho^{2}}-|s|\right).$

The proofs of Theorem A and Theorem B evolve in two steps: finding a lower
bound and finding an upper bound that coincides with the lower bound. For the
lower we will explicitly construct an admissible Hamiltonian using geodesic
billiards and the magnetic geodesic flow respectively. For the upper bound we
will need more abstract theory, the theory of pseudo-holomorphic curves. There
are not many cases where this theory becomes explicit, but we will construct
the following symplectomorphisms, realizing compactifications of the disc
bundles, that will put us in the perfect set up to study (pseudo-)holomorphic
curves. In this sense the following two theorems are maybe the key
contribution of this paper.

######  Theorem C.

Denote $\omega_{FS}\in\Omega_{2}(\mathbb{C}\mathrm{P}^{n})$ the Fubini-Study
form normalized such that the generator of
$H_{2}(\mathbb{C}\mathrm{P}^{n},\mathbb{Z})$ has area $4\pi$. Then there is a
symplectomorphism, which is equivariant with respect to the Hamiltonian
$\mathrm{SU}(n+1)$ actions,

$F:(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})\to(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta},R_{1}\omega_{FS}\ominus
R_{2}\omega_{FS}),$

where
$\bar{\Delta}\subset\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}$
denotes the anti-diagonal divisor

$\bar{\Delta}=\\{(p,q)\in\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\
|\ \mathrm{dist}(p,q)\ \text{maximal}\\}$

and

$R_{1}=\frac{1}{2}\left(\sqrt{s^{2}+1}+s\right),\ \
R_{2}=\frac{1}{2}\left(\sqrt{s^{2}+1}-s\right).$

Observe that the case $s=0$ is included in the Theorem and recovers the
untwisted case. For $\mathbb{R}\mathrm{P}^{n}$ we have a similar theorem.

######  Theorem D.

Denote $\omega_{FS}\in\Omega_{2}(\mathbb{C}\mathrm{P}^{n})$ the Fubini-Study
form normalized such that the generator of
$H_{2}(\mathbb{C}\mathrm{P}^{n},\mathbb{Z})$ has area $4\pi$. Then there is an
$SO(n+1)$-equivariant symplectomorphism

$F:(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)\to(\mathbb{C}\mathrm{P}^{n}\setminus
Q^{n-1},\omega_{FS}),$

where $Q^{n-1}\subset\mathbb{C}\mathrm{P}^{n}$ denotes the quadric

$Q^{n-1}:=\\{[z_{0}:\ldots:z_{n}]\ |\
z_{0}^{2}+\ldots+z_{n}^{2}=0\\}\subset\mathbb{C}\mathrm{P}^{n}.$

Note that the 2-dimensional version of this theorem was already proven by
Adaloglou in [1], but using a different approach.

Outline. In section 2 we construct the symplectic compactfications of
$(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)$ and
$(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)$, i.e. we prove Theorem C
in the case $s=0$ and Theorem D. Section 3 contains a recap on
pseudoholomorphic and uses them to obtain an upper bound on the Hofer–Zehnder
capacity. Section 4 is devoted to constructing admissible Hamiltonians to get
a lower bound. We will see that upper and lower bound agree and conclude that
Theorem A holds. The last section (section 5) contains the magnetic version,
i.e. the proofs of Theorem B and Theorem D.

Acknowledgments. I want to thank my advisor Gabriele Benedetti for his support
and suggestions on how to deal with numerous difficulties. Further I also want
to thank Peter Albers and Kai Zehmisch for helpful discussions regarding this
work. The author further acknowledges funding by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – 281869850 (RTG
2229) and 281071066 (TRR 191).

##  2 Compactifications

Let $N$ denote either $\mathbb{C}\mathrm{P}^{n}$ or
$\mathbb{R}\mathrm{P}^{n}$. We equip $\mathbb{C}\mathrm{P}^{n}$ with the
Fubini-Study metric and $\mathbb{R}\mathrm{P}^{n}$ with the round metric, both
denoted by $g$. Recall that the kinetic Hamiltonian

$E(x,v):=\frac{1}{2}g_{x}(v,v)\left(=\frac{1}{2}|v|_{x}^{2}\right)$

generates geodesic flow and since we chose Zoll metrics it is totally periodic
of period $l/|v|$, where $l$ denotes the length of the prime geodesics. It
follows that the Hamiltonian

$H:=l\sqrt{2E}:TN\setminus 0_{TN}\to\mathbb{R};\ (x,v)\mapsto l|v|_{x}$

generates a free circle action. One can now perform a Lerman cut (see [16]) at
$H=l$ to obtain symplectic compactifications

$\overline{(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)}\ \text{resp.}\
\overline{(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)}.$

These compactifications are per construction closed symplectic, but their
symplectic geometry is not very clear from the Lerman cut construction. This
section is concerned with identifying these ’abstract’ compactifications with
the well studied symplectic manifolds

$\left(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n},\frac{1}{2}(\omega_{FS}\ominus\omega_{FS})\right)\
\text{resp.}\ \left(\mathbb{C}\mathrm{P}^{n},\omega_{FS}\right),$

which puts us in the perfect set up to study (pseudo-) holomorphic spheres. To
find the identifying symplectomorphism, we will make use of the fact that the
action of the isometry groups $\mathrm{SU}(n+1)$ resp. $\mathrm{SO}(n+1)$ of
$\mathbb{C}\mathrm{P}^{n}$ resp. $\mathbb{R}\mathrm{P}^{n}$ lift to
Hamiltonian actions on the tangent bundle. We will see that indeed also the
spaces
$\left(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n},(\omega_{FS}\ominus\omega_{FS})\right)\
\text{resp.}\ \left(\mathbb{C}\mathrm{P}^{n},\omega_{FS}\right)$ admit
Hamiltonian $\mathrm{SU}(n+1)$\- resp. $\mathrm{SO}(n+1)$-actions. The map we
construct will be an equivariant smooth bijection and we will use the
following lemma to deduce it is a symplectomorphism.

######  Lemma 2.1.

Let $G$ be a Lie group and $(M_{i},\omega_{i})$, $i=1,2$ symplectic manifolds.
Assume we have a Hamiltonian G-action on $(M_{i},\omega_{i})$ with moment maps
$\mu_{i}:M_{i}\to\mathfrak{g}^{*}$ for $i=1,2$. If $\varphi:M_{1}\to M_{2}$ is
an equivariant smooth bijection such that

${M_{1}}$${M_{2}}$${\mathfrak{g}^{*}}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\varphi}$$\scriptstyle{\mu_{2}}$

commutes and the distribution $\mathcal{D}\subset TM_{1}$ tangent to the
$G$-orbits admits a complement $\Upsilon$ that is isotropic for both
symplectic forms $\omega_{1}$ and $\varphi^{*}\omega_{2}$, then $\varphi$ is
actually a symplectomorphism i.e. $\varphi^{*}\omega_{2}=\omega_{1}$.

###### Proof.

We need to show that

$\varphi^{*}\omega_{2}(v,w)=\omega_{1}(v,w)\ \ \forall v,w\in TM_{1}.$

By non-degeneracy of $\omega_{1}$ and $\omega_{2}$ it will then follow, that
also $\mathrm{d}\varphi$ must be non-degenerate hence a diffeomorphism. We can
decompose the tangent bundle $TM_{1}=\mathcal{D}\oplus\Upsilon$. The
distribution tangent to the $G$-orbits is given by
$\mathcal{D}_{x}=\\{(a_{1}^{\\#})_{x}:=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}e^{ta}(x)\
|\ a\in\mathfrak{g}\\}$. In particular $\mathrm{d}\varphi
a_{1}^{\\#}=a_{2}^{\\#}$ by equivariance, where the subscript indicates the
manifold the vector field induced by $a\in\mathfrak{g}$ lives on. So if either
$v=a_{1}^{\\#}\in\mathcal{D}$ or $w=a_{1}^{\\#}\in\mathcal{D}$ we get the
equality as follows

$\omega_{1}(a^{\\#}_{1},\cdot)=\mathrm{d}(\mu_{1},a)=\mathrm{d}(\mu_{2}\circ\varphi,a)=\omega_{2}(a^{\\#}_{2},\mathrm{d}\varphi\
\cdot)=\omega_{2}(\mathrm{d}\varphi\ a_{1}^{\\#},\mathrm{d}\varphi\
\cdot)=\varphi^{*}\omega_{2}(a^{\\#}_{1},\cdot).$

The second equality holds due to the moment map triangle and the fourth due to
equivariance of $\varphi$. If $v,w\in\Upsilon$ we get

$\omega_{1}(v,w)=0=\omega_{2}(\mathrm{d}\varphi(v),\mathrm{d}\varphi(w))$

as $\Upsilon$ is isotropic with respect to both symplectic forms by
assumption. ∎

###  2.1 Complex projective space

In order to give an accurate description of the Hamiltonian group actions,
including formulas for the moment maps, we will consider
$\mathbb{C}\mathrm{P}^{n}$ as (co-)adjoint orbit in $\mathfrak{su}(n+1)$ and
$\mathbb{R}\mathrm{P}^{n}\subset\mathbb{C}\mathrm{P}^{n}$ as a suborbit.
Observe that for compact Lie groups such as $\mathrm{SU}(n)$ we can use the
Killing form to identify $\mathfrak{g}^{*}\cong\mathfrak{g}$, which we will do
without further notice from now on. We follow [9] for the presentation of
$\mathbb{C}\mathrm{P}^{n}$ as adjoint orbit. The complex projective space can
be identified with the homogeneous space

$\mathbb{C}\mathrm{P}^{n}=\faktor{\mathrm{SU}(n+1)}{\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(1))},$

where $\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(1))$ denotes the matrices in
$\mathrm{U}(n)\times\mathrm{U}(1)$ of determinant one. Fix

$Z:=\frac{1}{n+1}\begin{pmatrix}-i_{n}&\quad\\\
\quad&ni\end{pmatrix}\in\mathfrak{su}(n+1).$

The stabilizer of $Z$ under the adjoint action is precisely
$\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(1))$, thus
$\mathbb{C}\mathrm{P}^{n}$ can be realized as the adjoint orbit
$O_{Z}\subset\mathfrak{su}(n+1)$. The stabilizer group
$\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(1))$ is isomorphic to
$\mathrm{U}(n)$ via the map

$\mathrm{U}(n)\to\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(1));\
A\mapsto\begin{pmatrix}A&\quad\\\ \quad&\det A^{-1}\end{pmatrix}.$

On Lie algebra level the inclusion looks like

$\mathfrak{u}(n)\hookrightarrow\mathfrak{su}(n+1);\
B\mapsto\begin{pmatrix}B&\quad\\\ \quad&-\mathrm{tr}B\end{pmatrix}.$

Thus the Lie-algebra splits as

$\mathfrak{su}(n+1)=\mathfrak{p}\oplus\mathfrak{u}(n),$

where

$\mathfrak{p}=\left\\{\begin{pmatrix}0_{n}&-\bar{z}\\\
z^{t}&0\end{pmatrix}\Big{|}\ z\in\mathbb{C}^{n}\right\\}$

is the subspace corresponding to the tangent space of
$\mathbb{C}\mathrm{P}^{n}$ at $Z$. Observe that for any

$v=\begin{pmatrix}0_{n}&-\bar{z}\\\ z^{t}&0\end{pmatrix}\in
T_{Z}\mathbb{C}\mathrm{P}^{n}=\mathfrak{p}$

we have that

$j_{Z}v:=[Z,v]=\frac{1}{n+1}\left(\begin{pmatrix}-i_{n}&\quad\\\
\quad&ni\end{pmatrix}\begin{pmatrix}0_{n}&-\bar{z}\\\
z^{t}&0\end{pmatrix}-\begin{pmatrix}0_{n}&-\bar{z}\\\
z^{t}&0\end{pmatrix}\begin{pmatrix}-i_{n}&\quad\\\
\quad&ni\end{pmatrix}\right)=\begin{pmatrix}0_{n}&-(\overline{iz})\\\
(iz)^{t}&0\end{pmatrix}$

recovers the standard complex structure on $\mathbb{C}\mathrm{P}^{n}$. Further
the negative of the Killing form

$(X,Y):=-2\mathrm{tr}(X\cdot Y),\ \ \ \forall\ X,Y\in\mathfrak{su}(n+1)$

yields an $\mathrm{Ad}$-invariant scalar product on $\mathfrak{su}(n+1)$.
Restricted to $\mathfrak{p}$ it is compatible with $j_{Z}$ and equivariantly
extended it yields the standard invariant Kähler structure of
$\mathbb{C}\mathrm{P}^{n}$, i.e. $g(j\cdot,\cdot)=\omega_{FS}(\cdot,\cdot)$.

######  Lemma 2.2.

The induced resp. diagonal group action of $\mathrm{SU}(n+1)$ on
$(T\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)$ resp.
$(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n},\omega_{FS}\ominus\omega_{FS})$
is Hamiltonian. The moment maps are given by

$\mu_{1}:T\mathbb{C}\mathrm{P}^{n}\to\mathfrak{su}(n+1);\ (x,v)\mapsto[x,v]$

resp.

$\mu_{2}:\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\to\mathfrak{su}(n+1);\
(a,b)\mapsto a-b.$

###### Proof.

For the first formula we compute for an arbitrary element
$a\in\mathfrak{su}(n+1)$

$\mathrm{d}(([x,v],a))=\mathrm{d}((v,[x,a]))=\mathrm{d}(g(v,\mathrm{d}\pi(a^{\\#})))=\mathrm{d}(\lambda(a^{\\#}))=\iota_{a^{\\#}}\mathrm{d}\lambda$

as $\lambda$ is invariant under the flow of $a^{\\#}$ and as a consequence
$\mathcal{L}_{a^{\\#}}\lambda=0$.

The formula for $\mu_{2}$ follows immediately from the fact that the inclusion
as adjoint orbit is a moment map for the group action on an adjoint orbit [15,
Ch. 1]. ∎

We can now prove the following theorem using the Lemmas above.

######  Theorem 2.3.

There is a symplectomorphism, which is equivariant with respect to the
Hamiltonian $\mathrm{SU}(n+1)$ actions,

$F:(D_{2}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)\to(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta},\omega_{FS}\ominus\omega_{FS}),$

where
$\bar{\Delta}\subset\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}$
denotes the anti-diagonal divisor

$\bar{\Delta}=\\{(a,b)\in\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\
|\ \mathrm{dist}(a,b)\ \text{maximal}\\}.$

###### Proof.

We make the ansatz

$F:D_{2}\mathbb{C}\mathrm{P}^{n}\to\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta};\
\ \ (x,v)\mapsto\left(\exp_{x}(-c(r)j_{x}v),\exp_{x}(c(r)j_{x}v)\right),$

for a function $c$ depending on $r:=|v|$. The map $F$ is depicted in Figure 1
for the two dimensional case of $\mathbb{C}\mathrm{P}^{1}$.

Figure 1: The Lie algebras $\mathfrak{su}(2)$ and $\mathbb{R}^{3}$ with cross
product as Lie bracket are isomorphic. This isomorphism identifies the adjoint
orbit $\mathbb{C}\mathrm{P}^{1}$ with the round 2-sphere. Now take a point
$(x,v)\in TS^{2}$ (in red). If we fix the radius of the sphere to be one the
complex structure is given by $jv=x\times v$ (in green). This is actually the
same as the moment map, i.e. $\mu_{1}(x,v)=x\times v=jv$. We follow the
geodesic (great circle) through $x$ in direction $jv$ resp. $-jv$ until the
points $a$ and $b$ satisfy $\mu_{2}(a,b)=a-b=jv$. The angle between $a$ and
$b$ uniquely determines the function $c$. That the higher dimensional case
works the same is no surprise as one can use the symmetry group
$\mathrm{SU}(n+1)$ to reduce everything to the two dimensional case.

For the proof we want to use Lemma 2.1, thus we need to show that the moment
map triangle commutes. We will determine $c$ imposing the relation of the
moment maps

$\mu_{1}(x,v)=\mu_{2}(F(x,v)).$

Per construction the moment maps and the map $F$ are equivariant. Hence, it is
enough to check the moment map triangle at some point

$x=Z=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&\quad&\quad\\\ \quad&-i&0\\\
\quad&0&ni\end{pmatrix},\ \ v=\frac{r}{2}\begin{pmatrix}0_{n-1}&\quad&\quad\\\
\quad&0&-i\\\ \quad&-i&0\end{pmatrix}\in T_{Z}\mathbb{C}\mathrm{P}^{n},\
r=|v|>0,$

as the group action is transitive on the sphere subbundle of
$T\mathbb{C}\mathrm{P}^{n}$. We compute

$\mu_{1}(x,v)=[x,v]=\frac{r}{2}\begin{pmatrix}0_{n}&\quad&\quad\\\
\quad&0&1\\\ \quad&-1&0\end{pmatrix}.$

The geodesic through $x$ in direction $j_{x}v$ is given by

$\gamma_{(x,j_{x}v)}(t)=\mathrm{Ad}_{\exp(-tv)}x=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&\quad&\quad\\\
\quad&-i(\cos(rt)^{2}-n\sin(rt)^{2})&(n+1)\sin(rt)\cos(rt)\\\
\quad&-(n+1)\sin(rt)\cos(rt)&i(n\cos(rt)^{2}-\sin(rt)^{2})\end{pmatrix}.$

It parametrizes an affine circle in $\mathfrak{su}(n+1)$ since

$V(t):=\frac{1}{2}\left(\gamma_{(x,j_{x}v)}(t)-\gamma_{(x,j_{x}v)}(t+\frac{\pi}{2r})\right)=\frac{1}{2}\begin{pmatrix}0_{n-1}&\quad&\quad\\\
\quad&-i\cos(2rt)&\sin(2rt)\\\ \quad&-\sin(2rt)&i\cos(2rt)\end{pmatrix}$

satisfies $|V|^{2}=-2\mathrm{tr}(V^{2})=1$. The circle is centered at

$y:=\frac{1}{2}\left(\gamma_{(x,j_{x}v)}(t)+\gamma_{(x,j_{x}v)}(t+\frac{\pi}{2r})\right)=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&0&0\\\
0&\frac{n-1}{2}i&0\\\ 0&0&\frac{n-1}{2}i\end{pmatrix}.$

Observe that $\gamma_{(x,j_{x}v)}(t)=V(t)+y$.

######  Remark 2.4.

The equality $|V|=1$ shows that the Fubini-Study form is normalized such that
prime geodesics have length $l=2\pi$.

Now the second moment map is given by

$\mu_{2}(F(x,v))=\gamma_{(x,j_{x}v)}(-c)-\gamma_{(x,j_{x}v)}(c)=V(-c)-V(c).$

Comparing the coefficients of the two matrices yields the following equation
for $c$:

$2\sin(2cr)=r.$

The moment map triangle commutes if and only if this equality holds. We
immediately see that the equation can be solved for $c$ if and only if
$r\in(-2,2)$, i.e.

$c:(-2,2)\to\mathbb{R};\ r\mapsto\frac{\sin^{-1}(r/2)}{2r}$

and is therefore smooth and even. This finishes the construction of the smooth
map
$F:D_{2}\mathbb{C}\mathrm{P}^{n}\to\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}$.
But is it a bijection? We can explicitly give an inverse. A point
$(a,b)\in\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta}$
defines a unique shortest geodesic $\gamma$ connecting $a=\gamma(0)$ to
$b=\gamma(1)$. Now we set $x=\gamma(1/2)$ and $v=-j(a-b)$ in total we obtain

$F^{-1}(a,b)=(\gamma(1/2),-j(a-b)).$

Thus $F$ is a smooth bijection intertwining the moment maps. The construction
of $F^{-1}$ in the 1-dimensional case is drawn in figure 2.

Figure 2: For any two points $a,b\in S^{2}$ (dark blue) that are not antipodal
there is a unique shortest geodesic $\gamma$ (light blue) such that
$\gamma(0)=a$ and $\gamma(1)=b$. We set $x:=\gamma(1/2)$ (red) and $jv=a-b$
(green). As $x$ is orthogonal to $a-b$ and thus orthogonal to $v=-j(jv)$ we
find that indeed $v\in T_{x}S^{2}$, thus $F^{-1}$ is well defined. In higher
dimensions the argument reduces to the 1-dimensional case as the symmetry
group acts transitively on the totally geodesic copies of
$\mathbb{C}\mathrm{P}^{1}\cong S^{2}$.

Finally observe that the codimension of the orbits of $SU(n+1)$ is one. Thus
any complement of $\mathcal{D}$ is 1-dimensional and therefore isotropic for
both symplectic forms $\mathrm{d}\lambda$ and
$F^{*}(\omega_{FS}\ominus\omega_{FS})$. It follows from Lemma 2.1 that $F$ is
indeed a symplectomorphism. ∎

######  Remark 2.5.

For $(x,v)\in T\mathbb{C}\mathrm{P}^{n}$ denote
$(a,b):=F(x,v)\in\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}$.
Observe now that

$F(H(x,v))=l|v|=l|jv|=l|\mu_{1}(x,v)|=l|\mu_{2}(F(x,v))=l|a-b|.$

We see that $H\circ F$ extends to a smooth map on
$\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\Delta$ that
is critical along $\bar{\Delta}$. We may therefore indeed interpret the map
$F$ as a realization of the Lerman cut with respect to $H$ at the value $2l$.

###  2.2 Real projective space

Using the description of $\mathbb{C}\mathrm{P}^{n}$ as adjoint orbit
$O_{Z}\subset\mathfrak{su}(n+1)$ explained before, we can identify
$\mathbb{R}\mathrm{P}^{n}$ as an $\mathrm{SO}(n+1)\subset\mathrm{SU}(n+1)$ sub
orbit

$\mathbb{R}\mathrm{P}^{n}\cong\left\\{\mathrm{Ad}_{g}(Z)|\ \
g\in\mathrm{SO}(n+1)\right\\}\subset O_{Z}\cong\mathbb{C}\mathrm{P}^{n}.$

It is fixed under the involution

$I:O_{Z}\to O_{Z};\ p\mapsto p^{T},$

as can be seen by direct computation

$(gZg^{-1})^{T}=gZ^{T}g^{T}=gZg^{-1}\ \ \forall g\in\mathrm{SO}(n+1).$

The involution $I$ is antiholomorphic, because for all $v\in\mathfrak{p}$ we
find

$\mathrm{d}I_{Z}(j_{Z}(v))=[Z,v]^{T}=(Zv-
vZ)^{T}=-(Zv^{T}-v^{T}Z)=-[Z,v^{T}]=-j_{Z}(\mathrm{d}I_{Z}(v)).$

######  Lemma 2.6.

The induced resp. subgroup action of $\mathrm{SO}(n+1)\subset\mathrm{SU}(n+1)$
on $(T\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)$ resp.
$(\mathbb{C}\mathrm{P}^{n},\omega_{FS})$ is Hamiltonian. The moment maps are
given by

$\mu_{1}:T\mathbb{R}\mathrm{P}^{n}\to\mathfrak{so}(n+1);\ (x,v)\mapsto[x,v]$

resp.

$\mu_{2}:\mathbb{C}\mathrm{P}^{n}\to\mathfrak{so}(n+1);\
a\mapsto\mathrm{Re}(a).$

###### Proof.

The map $\mu_{1}$ is the restriction of the moment map for
$(T\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)$. We therefore only need to
check that the map actually takes values in $\mathfrak{so}(n+1)$. From the
description as suborbit it is only clear that $[x,v]\in\mathfrak{su}(n+1)$.
The action of $\mathrm{SO(n+1)}$ on $T\mathbb{R}\mathrm{P}^{n}$ is transitive
on the sphere subbundle, thus it is fine to check this at some point

$x=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&\quad&\quad\\\ \quad&-i&0\\\
\quad&0&ni\end{pmatrix}\in\mathbb{R}\mathrm{P}^{n},\ \
v=\frac{r}{2}\begin{pmatrix}0_{n-1}&\quad&\quad\\\ \quad&0&-i\\\
\quad&-i&0\end{pmatrix}\in
T_{x}\mathbb{R}\mathrm{P}^{n}\cong[x,\mathfrak{so}(n+1)].$

Observe that $x=Z$ and $r=|v|$. Indeed for this choice we see

$\overline{[x,v]}=[\bar{x},\bar{v}]=[-x,-v]=[x,v]$

and therefore $[x,v]\in\mathfrak{so}(n+1)$.

Next we look at the map $\mu_{2}$. Equivariance is clear as conjugation with a
real matrix commutes with taking the real part. The rest of the proof is the
following straight forward computation. Any tangent vector $v\in
T_{p}\mathbb{C}\mathrm{P}^{n}$ is of the form $v=\xi^{\\#}_{p}$ for some
$\xi\in\mathfrak{su}(n+1)$, thus for every $a\in\mathfrak{so}(n+1)$ we find

$\displaystyle 0$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(\mathrm{Ad}_{e^{t\xi}}(p),\mathrm{Ad}_{e^{t\xi}}(a))$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(\mathrm{Ad}_{e^{t\xi}}(p),a)+(p,[\xi,a])$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(\mathrm{Re}(\mathrm{Ad}_{e^{t\xi}}(p)),a)+(p,[\xi,a])$
$\displaystyle=\mathrm{d}(\mathrm{Re},a)(\xi^{\\#})+\omega_{FS}(\xi^{\\#},a^{\\#})$
$\displaystyle=\mathrm{d}(\mathrm{Re},a)(v)+\omega_{FS}(v,a^{\\#}).$

For the third equality we used that real and imaginary part are orthogonal
with respect to the Killing form, i.e. for two real matrices $a,b$ such that
$a,ib\in\mathfrak{su}(n+1)$ we know that

$(a,ib)=-2\mathrm{tr}(a\cdot ib)=-2i\mathrm{tr}(a\cdot b)\in\mathbb{R}\ \
\Rightarrow\ \ (a,ib)=0.$

For the fourth equality we used that

$\omega_{FS}(\xi^{\\#},a^{\\#})=g(j\xi^{\\#},a^{\\#})=([p,\xi],a)=(p,[\xi,a]).$

∎

We can now prove the following theorem analogously to Theorem 2.3.

######  Theorem 2.7.

There is an $SO(n+1)$-equivariant symplectomorphism

$F:(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)\to(\mathbb{C}\mathrm{P}^{n}\setminus
Q^{n-1},\omega_{FS}),$

where $Q^{n-1}\subset\mathbb{C}\mathrm{P}^{n}$ denotes the quadric

$Q^{n-1}:=\\{[z_{0}:\ldots:z_{n}]\ |\
z_{0}^{2}+\ldots+z_{n}^{2}=0\\}\subset\mathbb{C}\mathrm{P}^{n}.$

###### Proof.

We make the following ansatz for the symplectomorphism realizing Theorem 2.7,

$F:D_{1}\mathbb{R}\mathrm{P}^{n}\to\mathbb{C}\mathrm{P}^{n};\ \
(x,v)\mapsto\exp_{x}(-f(|v|)j_{x}v)=e^{f(|v|)v}xe^{-f(|v|)v},$

for some smooth, even function $f:(-1,1)\to\mathbb{R}$ depending on $r:=|v|$.
This can be visualized nicely in dimension $n=1$ as shown in Figure 3.

Figure 3: The figure shows a sketch of the map
$F:D_{1}\mathbb{R}\mathrm{P}^{1}\to\mathbb{C}\mathrm{P}^{1}$. In dimension
$n=1$ the quadric is given by the two antipodal points furthest away from
$\mathbb{R}\mathrm{P}^{1}$. This can for example be seen in homogenous
coordinates, as $[1:\pm i]\in\mathbb{C}\mathrm{P}^{1}$ are the only points
solving the equation $z_{0}^{2}+z_{1}^{2}=0$. Now take a point $(x,v)\in
T\mathbb{R}\mathrm{P}^{1}$ (in blue) and follow the geodesic through $x$ in
direction $jv$ until $p$ satisfies $jv=\mathrm{Re}(p)$ (in green). The angle
between $p$ and $x$ determines $f$. In particular the image of $F$ coincides
with $\mathbb{C}\mathrm{P}^{1}\setminus Q^{0}$.

For the above choice of $(x,v)$ we find

$e^{f(r)v}=\begin{pmatrix}1_{n-1}&\quad&\quad\\\ \quad&\cos(fr)&i\sin(fr)\\\
\quad&i\sin(fr)&\cos(fr)\end{pmatrix}.$

It follows that

$e^{f(r)v}xe^{-f(r)v}=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&\quad&\quad\\\
\quad&i(n\sin(fr)^{2}-\cos(fr)^{2})&-\frac{1}{2}(n+1)\sin(2fr)\\\
\quad&\frac{1}{2}(n+1)\sin(2fr)&i(n\cos(fr)^{2}-\sin(fr)^{2})\end{pmatrix}$

and thus

$\mu_{2}(F(x,v))=\frac{1}{2}\begin{pmatrix}0_{n-1}&\quad&\quad\\\
\quad&0&-\sin(2fr)\\\ \quad&\sin(2fr)&0\end{pmatrix}.$

On the other hand

$\mu_{1}(x,v)=[x,v]=\frac{r}{2}\begin{pmatrix}0_{n-1}&\quad&\quad\\\
\quad&0&-1\\\ \quad&1&0\end{pmatrix}.$

Comparing the matrix entries we find that $f$ must satisfy

$\sin(2fr)=r,$ (2)

which is invertible for $r\in(-1,1)$ so that

$f(r)=\frac{\sin^{-1}(r)}{2r}$

is a smooth and even function. Thus it follows that $F$ is a smooth map
intertwining the moment maps. But is it a bijection? We can explicitly give an
inverse. A point $p\in\mathbb{C}\mathrm{P}^{n}\setminus Q^{n-1}$ and its
conjugate $I(p)$ are joined by a unique shortest geodesic
$\gamma:[0,1]\to\mathbb{C}\mathrm{P}^{n}$. This can be seen as follows. If
$p\notin\mathbb{R}\mathrm{P}^{n}$, $p$ and $I(p)$ lie in a unique copy of
$\mathbb{C}\mathrm{P}^{1}$. Now we look at Figure 4 to see that there is a
unique shortest geodesic joining $p$ and $I(p)$ if and only if $p\notin
Q^{n-1}$. Thus

$F^{-1}(p)=(\gamma(1/2),-j\mathrm{Re}(p))$

defines the inverse.

Figure 4: For any point $p\in\mathbb{C}\mathrm{P}^{1}\setminus Q^{0}$ there is
a unique shortest geodesic $\gamma$ (green) joining $p$ and its conjugate
$I(p)$ (blue). Observe that taking the real part corresponds to projection
onto the vertical axis (red). Now we set $x:=\gamma(1/2)$ and
$v=-j\mathrm{Re}(p)$. The higher dimensional case reduces to this case as
$\mathrm{SO}(n+1)$ acts transitively on the unit sphere bundle of
$\mathbb{R}\mathrm{P}^{n}$.

We see that $F$ is an equivariant smooth bijection that intertwines the moment
maps. Again any complement of $\mathcal{D}$ is 1-dimensional and thus
isotropic for any symplectic form. Thus $F$ fulfills the prerequisites of
Lemma 2.1 and therefore is the symplectomorphism realizing the symplectic
identification in Theorem 2.7. ∎

######  Remark 2.8.

For $(x,v)\in T\mathbb{R}\mathrm{P}^{n}$ denote
$p:=F(x,v)\in\mathbb{C}\mathrm{P}^{n}$. Observe now that

$H(F^{-1}(p))=l|v|=l|\mu_{1}(x,v)|=l|\mu_{2}(F(x,v))=l|\mathrm{Re}(p)|.$

We see that $H\circ F^{-1}$ extends to a smooth map on
$\mathbb{C}\mathrm{P}^{n}\setminus\mathbb{R}\mathrm{P}^{n}$ that is critical
along $Q^{n-1}$. We may thereforeindeed interpret the map $F$ as a realization
of the Lerman cut with respect to $H$ at the value $l$.

######  Remark 2.9.

A closed almost complex manifold $(W,J)$ is called (compact) complexification
of a real manifold $M$ if there exists an antiholomorphic involution

$I:W\to W;\ \ \ \mathrm{d}I\circ J=-J\circ\mathrm{d}I$

such that $M$ is isomorphic to the fixed point set of $I$. The
symplectomorphisms constructed above realizes these complexifications of
$\mathbb{C}\mathrm{P}^{n}$ resp. $\mathbb{R}\mathrm{P}^{n}$. The
antiholomorphic involution on
$(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n},j\ominus j)$ is given
by

$I:\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\to\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n};\
\ (a,b)\mapsto(b,a).$

The antiholomorphic involution on $(\mathbb{C}\mathrm{P}^{n},\omega_{FS})$
respresented as adjoint orbit is given by

$I:\mathbb{C}\mathrm{P}^{n}\subset\mathfrak{su}(n+1)\to\mathbb{C}\mathrm{P}^{n}\subset\mathfrak{su}(n+1);\
\ p\mapsto p^{T}.$

Observe that in both cases the symplectomorphism is of the form

$DM\to W,(x,v)\mapsto\exp_{x}^{W}(-Jc(v)v),$

where in our cases $c$ is a scalar depending only on the norm of $v$. It seems
likely that for general Kähler manifolds it is be possible to construct a
similar map, maybe promoting $c$ to be operator valued and depending also on
the direction of $v$ and possibly replacing the disc subbundle by a different
neighborhood of the zero section.

##  3 Upper bound

To find an upper bound of the Hofer–Zehnder capacity, we will use a theorem by
H. Hofer and C. Viterbo [11, Thm. 1.16] that shows existence of periodic
orbits in the presence of holomorphic spheres. Let $(M,\omega)$ be a closed
symplectic manifold and $J:TM\to TM$ a $\omega$-compatible almost complex
structure. Consider the moduli space $\mathcal{M}$ of $J$-holomorphic spheres
$u:S^{2}\to M$ satisfying:

$\displaystyle\begin{split}u(0)&\in\Sigma_{0},\\\
u(\infty)&\in\Sigma_{\infty},\\\ [u]&=A\\\ \frac{1}{2}\omega(A)&=\int_{|z|\leq
1}u^{*}\omega,\end{split}$ (3)

for two closed disjoint submanifolds $\Sigma_{0}$ and $\Sigma_{\infty}$ and
some homotopy class $A\in\pi_{2}(M)$. Observe that these conditions fix the
reparametrization symmetry up to an $S^{1}$ (rotation of the complex plane).
Further Hofer–Viterbo assume $A$ to be _minimal_ , i.e.

$\omega(A)=\inf\\{\omega(B)\ |B\in[S^{2},M],\ \omega(B)>0\\},$

which ensures that bubbling cannot occur. We replace this condition by
requiring $A$ to be $\omega$-indecomposable, i.e. $A$ can not be written as
sum $\sum_{i}B_{i}$ for free homotopy classes $B_{i}\in\pi_{2}(M)$ with
$\omega(B_{i})>0$. The reader my check that the proof of Theorem 1.16 in [11]
goes through without any changes also for $\omega$-indecomposable classes.

If $J$ is regular the moduli space $\mathcal{M}$ is a compact oriented
manifold without boundary and thus defines an oriented cobordism class. Since
the $S^{1}$ reparametrization symmetry of pseudoholomorphic curves is not
fixed, the moduli space inherits a free $S^{1}$-action and one considers
cobordisms of compact, oriented, free $S^{1}$-manifolds.

######  Theorem 3.1 (Hofer–Viterbo, Thm. 1.16 [11]).

Let $(M,\omega)$ be a closed symplectic manifold, $A\in\pi_{2}(M)$
$\omega$-indecomposable, $J$ a smooth, regular, $\omega$-compatible almost
complex structure and $\Sigma_{0},\Sigma_{\infty}$ two disjoint nonempty
closed submanifolds of $M$. Suppose $H:M\to\mathbb{R}$ is a Hamiltonian such
that

$H|_{\mathcal{U}(\Sigma_{0})}=\min(H)\ \ \text{and}\ \
H|_{\mathcal{U}(\Sigma_{\infty})}=\max(H),$

then, if the free $S^{1}$-cobordism class $[\mathcal{M}]\neq 0$, the
Hamiltonian system $\dot{x}=X_{H}(x)$ possesses a non constant contractible
$T$-periodic solution with

$0<T(\max(H)-\min(H))<\omega(A).$

In particular

$c_{HZ}(M\setminus\Sigma_{\infty},\Sigma_{0},\omega)\leq\omega(A)$

and if $\Sigma_{0}$ is a one point set we can conclude

$c_{HZ}(M\setminus\Sigma_{\infty},\omega)\leq\omega(A).$

We can now use this theorem and the compactifications constructed in the
previous section to determine an upper bound of the Hofer–Zehnder capacity of
$(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)$ resp.
$(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)$.

For the case
$(M,\omega)=\overline{(D_{2}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)}\cong(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n},\omega\ominus\omega)$
we pick $\Sigma_{0}=\\{(p,q)\\}$ a one point set and $\Sigma_{\infty}$ the
anti-diagonal divisor $\bar{\Delta}$. We fix
$A\in[S^{2},\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}]$ to be the
class of a degree one curve in the first factor. Further by [17, Prop. 7.4.3]
the split complex structure $j\ominus j$ is regular. Now every holomorphic
sphere through $(p,q)$ in class $A$ is of degree 1 in the first and constant
in the second factor. Further $\bar{\Delta}$ restricted to the fiber over $q$
is a copy of $\mathbb{C}\mathrm{P}^{n-1}$, in particular any $j\ominus
j$-holomorphic sphere through $(p,q)$ in class $A$ intersects $\bar{\Delta}$.
This means the moduli space of holomorphic curves $\mathcal{M}$ through
$(p,q)$ in class $A$ can be identified with the space of complex lines through
$p$. Recall that we left the $S^{1}$-reparametrization symmetry, therefore
$\mathcal{M}\cong S^{2n-1}$ and not
$\mathcal{M}\cong\mathbb{C}\mathrm{P}^{n-1}$ and the free circle action given
by rotating the holomorphic spheres parameterizes the Hopf fibration.

######  Lemma 3.2.

The oriented $S^{1}$-cobordism class of $S^{2n-1}$ does not vanish.

###### Proof.

We need to show that there is no oriented free $S^{1}$-manifold with boundary
$S^{2n-1}$. Assume there was, we could then glue in the standard ball
$B\subset\mathbb{C}^{n}$ of radius $1$ to obtain a closed orientable manifold
with a circle action with exactly one fixed point. This is because
multiplication with $e^{it}$ yields such an action on
$B\subset\mathbb{C}^{n}$. Now such a manifold does not exist as shown in [14,
Lem. 2.1]. ∎

We can use Theorem 3.1 to find

$c_{HZ}(D_{2}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)=c_{HZ}(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta},\omega\ominus\omega)\leq\omega(A)=4\pi=2l.$

It follows that

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)\leq l.$ (4)

For the case $(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)$ we need to
work less as Hofer–Viterbo already computed

$c_{HZ}(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)\leq
c_{HZ}(\mathbb{C}\mathrm{P}^{n},\omega_{FS})=4\pi=2l$ (5)

using Theorem 3.1. In the next section we will see that the bounds (4) and (5)
are actually good enough.

##  4 Lower bounds

Finding a lower bound is in some sense more elementary and much more hands on.
Indeed we will simply construct explicit admissible Hamiltonians. We observed
already that the Hamiltonian $H(x,v)=l|v|$ generates a Hamiltonian circle
action away from the zero-section. Composing $H$ with a smooth even function
$f:[-l,l]\to\mathbb{R}$ that satisfies $0\leq f^{\prime}<1$, $f\equiv 0$ near
$0$, $f\equiv l-\varepsilon$ near $\pm l$ for an arbitrary small
$\varepsilon>0$, yields an admissible function $\tilde{H}:=f\circ H$ that
extends smoothly to the zero-section. Further the oscillation of $\tilde{H}$
is $l-\varepsilon$ so that in the limit $\varepsilon\to 0$ we obtain $l$ as
lower bound. This finishes the computation of

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda)=l$

as the lower bound agrees with the upper bound derived in (4).

However in the case of $\mathbb{R}\mathrm{P}^{n}$ this lower bound is not good
enough. To improve it we look at the dynamics for a family of Hamiltonians

$H_{\varepsilon}(x,v)=\sqrt{\|v\|_{x}^{2}+V_{\varepsilon}(x)}$

with potentials $V_{\varepsilon}:\mathbb{R}\mathrm{P}^{n}\to\mathbb{R}$ that
approximate billiard dynamics on a hemisphere as $\varepsilon\to 0$. We may
lift the potential to $S^{n}$ and assume that
$V_{\varepsilon}(-x)=V_{\varepsilon}(x)$ so that the potential descends to a
function $V_{\varepsilon}$ on $\mathbb{R}\mathrm{P}^{n}$. We fix a point $N\in
S^{n}$. The isotropy subgroup of $SO(n+1)$ that fixes $N$ is isomorphic to
$SO(n)$ and leaves the equatorial copy of $S^{n-1}$ invariant. We want the
family $V_{\varepsilon}$ to be rotation invariant i.e.
$V_{\varepsilon}(R(x))=V_{\varepsilon}(x)$ for all $R\in\mathrm{SO(n)}$. This
implies that $V_{\varepsilon}(x)$ only depends on the distance of $x$ from
$N$. In particular the gradient $\nabla V_{\varepsilon}(x)$ points along the
geodesic connecting $x$ and $N$.

######  Lemma 4.1.

The Hamiltonian vector field for $H_{\varepsilon}$ with respect to
$\mathrm{d}\lambda$ is given by

$(X_{H_{\varepsilon}})_{(x,v)}=\frac{1}{H_{\varepsilon}}\left(X_{(x,v)}-\frac{1}{2}(\nabla
V_{\varepsilon}(x))^{\mathcal{V}}\right),$

where $X=X_{E}$ is the vector field generating geodesic flow and
$(\cdot)^{\mathcal{V}}:TS^{n}\to TTS^{n}$ denotes the vertical lift.

###### Proof.

The kernel of the differential of the footpoint projection $\pi:TS^{n}\to
S^{n}$ defines the so called vertical distribution $\mathcal{V}\subset
TTS^{n}$. Now the Levi-Civita connection determines a complement
$\mathcal{H}\subset TTS^{n}$ called the horizontal distribution. For any
$(x,v)\in TS^{n}$ we can identify $T_{x}S^{n}$ with $\mathcal{H}_{(x,v)}$ and
$\mathcal{V}_{(x,v)}$ using the so called horizontal and vertical lift
respectively. The horizontal lift is defined by

$T_{x}S^{n}\to\mathcal{H}_{(x,v)};\ \ w\mapsto
w^{\mathcal{H}}:=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(\gamma(t),P_{\gamma}v)$

where $\gamma:(-\varepsilon,\varepsilon)\subset\mathbb{R}\to M$ is a smooth
curve satisfying $\gamma(0)=x$ and $\dot{\gamma}(0)=w$ and $P_{\gamma}$
denotes parallel transport along $\gamma$. The vertical lift is defined by

$T_{x}S^{n}\to\mathcal{V}_{(x,v)};\ \ w\mapsto
w^{\mathcal{V}}:=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(x,v+tw).$

These lifts can be used to lift any vector field on $S^{n}$ to a vector field
on $TS^{n}$. Observe that per definition $\lambda$ vanishes on $\mathcal{V}$,
further as proven in [8, Lemma 2],
$[A^{\mathcal{V}},B^{\mathcal{H}}]\subset\mathcal{V}$ and
$[A^{\mathcal{V}},B^{\mathcal{V}}]=0$ for all vector fields $A,B$ on $S^{n}$.
We conclude that

$\displaystyle\mathrm{d}\lambda(A^{\mathcal{V}},B^{\mathcal{H}})$
$\displaystyle=A^{\mathcal{V}}(\lambda(B^{\mathcal{H}}))-B^{\mathcal{H}}(\lambda(A^{\mathcal{V}}))-\lambda([A^{\mathcal{V}},B^{\mathcal{H}}])$
$\displaystyle=A^{\mathcal{V}}(\lambda(B^{\mathcal{H}}))=A^{\mathcal{V}}(g(v,B))=g(A,B)$

and

$\mathrm{d}\lambda(A^{\mathcal{V}},B^{\mathcal{V}})=A^{\mathcal{V}}(\lambda(B^{\mathcal{V}}))-B^{\mathcal{V}}(\lambda(A^{\mathcal{V}}))-\lambda([A^{\mathcal{V}},B^{\mathcal{V}}])=0.$

Thus $\mathrm{d}\lambda(A^{\mathcal{V}},\cdot)=g(A,\mathrm{d}\pi\cdot)$. We
use this to verify the formula for $X_{H_{\varepsilon}}$,

$\displaystyle\mathrm{d}\lambda(X_{H_{\varepsilon}},\cdot)$
$\displaystyle=\frac{1}{H_{\varepsilon}}\left(\mathrm{d}\lambda(X,\cdot)-\frac{1}{2}\mathrm{d}\lambda((\nabla
V_{\varepsilon})^{\mathcal{V}},\cdot)\right)$
$\displaystyle=-\frac{1}{H_{\varepsilon}}\left(\mathrm{d}E+\frac{1}{2}g(\nabla
V_{\varepsilon},\mathrm{d}\pi\cdot)\right)=-\frac{1}{2H_{\varepsilon}}\left(2\mathrm{d}E+\mathrm{d}V_{\varepsilon}\right)=-\mathrm{d}H_{\varepsilon}.$

∎

The Lemma shows in particular that $(X_{H_{\varepsilon}})_{(x,v)}\in TTS^{2}$
for the totally geodesically embedded copy of $S^{2}$ through $x$ and $N$
tangent to $v$ and therefore its flow also preserves this copy of $TS^{2}$. We
reduced the problem to the situation in dimension $n=2$. Now we introduce
coordinates on this copy of $S^{2}$ to describe the dynamics explicitly. We
use $\theta\in[0,\pi]$ and $\varphi\in\mathbb{R}/2\pi\mathbb{Z}$. Where $N$
corresponds to $\theta=0$. Set

$V_{\varepsilon}(x)=\left\\{\begin{array}[]{ll}1,&\hbox{for}\
\theta(x)\in[\pi/2-\varepsilon/2,\pi/2]\\\ 0,&\hbox{for}\
\theta(x)\in[0,\pi/2-\varepsilon]\\\ \end{array}\right.$

and interpolate smoothly for
$\theta(x)\in[\pi/2-\varepsilon,\pi/2-\varepsilon/2]$ so that
$V_{\varepsilon}\geq 0$ and $V_{\varepsilon}^{\prime}\geq 0$. Now extend
$V_{\epsilon}$ according to the required symmetries, i.e.
$V_{\varepsilon}(x)=V_{\varepsilon}(-x)$ and
$V_{\varepsilon}(Rx)=V_{\varepsilon}(x)$ for all $R\in\mathrm{SO}(n)$. In this
coordinates the Hamiltonian reads

$H_{\varepsilon}(x,v)=\sqrt{\dot{\theta}^{2}+\sin(\theta)^{2}\dot{\varphi}^{2}+V_{\epsilon}(\theta)},$

where by abuse of notation we write $\dot{\varphi},\dot{\theta}$ instead of
$v_{\varphi},v_{\theta}$ and $V_{\varepsilon}(\theta)$ instead of
$V_{\epsilon}(x)$ as the potential only depends on $\theta$. Observe that
$H_{\varepsilon}$ does not depend on $\varphi$, this implies immediately that
$\sin(\theta)^{2}\dot{\varphi}$ is preserved. We conclude that the maximal
angular velocity $\dot{\varphi}_{\max}$ corresponds to the minimal radius
$\theta_{\min}$. We want to find upper bounds for the periods of all orbits
for the Hamiltonian $H_{\varepsilon}$ as $\varepsilon$ tends to zero.
Therefore we take an arbitrary orbit
$\gamma(t)=(\theta(t),\varphi(t),\dot{\theta}(t),\dot{\varphi}(t))$ with
energy $H_{\varepsilon}(\gamma)<1$. We split into two cases:
$\theta_{\min}\geq\frac{\pi}{2}-\sqrt{\varepsilon}$ and
$\theta_{\min}\leq\frac{\pi}{2}-\sqrt{\varepsilon}$.

Case $\theta_{\min}\geq\frac{\pi}{2}-\sqrt{\varepsilon}$ : At
$\theta(t)=\theta_{\min}$ we have $\dot{\theta}=0$, thus

$1>H_{\varepsilon}=\sqrt{\sin(\theta_{\min})^{2}\dot{\varphi}_{\max}^{2}+V_{\epsilon}(\theta_{\min})}\geq\sin(\theta_{\min})\dot{\varphi}_{\max}\Rightarrow\dot{\varphi}_{\max}\leq\frac{1}{\sin(\theta_{\min})}.$

Clearly the period $T$ must satisfy

$T>\frac{2\pi}{\dot{\varphi}_{\max}}\geq 2\pi\sin(\theta_{\min})\geq
2\pi\cos(\sqrt{\varepsilon})\xlongrightarrow{\varepsilon\longrightarrow
0}2\pi.$

Case $\theta_{\min}\leq\frac{\pi}{2}-\sqrt{\varepsilon}$ : For small
$\varepsilon$ certainly $\varepsilon<\sqrt{\varepsilon}$, so
$\theta_{\min}\leq\frac{\pi}{2}-\sqrt{\varepsilon}$ implies
$V(\theta_{\min})=0$. In particular segments of the orbit are geodesics
parametrized by unit speed. There are at least two of these segments as by
symmetry of the potential the orbit must leave the region where
$V_{\varepsilon}\neq 0$ with the same angle it enters the region. The length
of these segments is $\pi-2l_{\varepsilon}$, where $l_{\varepsilon}$ is the
length of the geodesic segment contained in the region
$\pi/2-\varepsilon\leq\theta\leq\pi/2$ where the potential $V_{\varepsilon}$
does not vanish as depicted in Figure 5. Now we can bound the period from
below as follows

$T>2\pi-4l_{\varepsilon}\geq
2\pi(1-\sin(l_{\varepsilon}))=2\pi\left(1-\frac{\sin(\varepsilon)}{\sin(\pi/2-\theta_{\min})}\right)\geq
2\pi\left(1-\frac{\sin(\varepsilon)}{\sin(\sqrt{\varepsilon})}\right)\xlongrightarrow{\varepsilon\longrightarrow
0}2\pi.$

The bound on $\sin(l_{\varepsilon})$ is obtained using the spherical law of
sines

$\frac{\sin(l_{\varepsilon})}{\sin(\pi/2)}=\frac{\sin(\varepsilon)}{\sin(\pi/2-\theta_{\min})}$

as explained in Figure 5.

Figure 5: On the left we see (in green undashed) a geodesic segment on the
hemisphere centered at $N$ (in red). Indeed it hits the equator with angle
$\frac{\pi}{2}-\theta_{\min}$ (in blue). On the right we see (in dark green) a
sketch of how the trajectory might continue when it enters the region where
the potential does not vanish. By conservation of angular momentum the picture
must be symmetric with respect to the grey dotted line. In particular the
trajectory leaves the region with the same angle entering it. Further we see
that the segment of the geodesic $l_{\varepsilon}$ is the hypotenuse of a
right spherical triangle. The length of the side opposite to
$\frac{\pi}{2}-\theta_{\min}$ is $\varepsilon$ and thus we can determine
$l_{\varepsilon}$ using the spherical law of sines.

In total we conclude that for small enough $\varepsilon>0$ the period is
bounded by

$T>2\pi(1-\sqrt{\varepsilon})$

and therefore the orbits for the scaled Hamiltonian
$(1-\sqrt{\varepsilon})2\pi H_{\varepsilon}$ have periods $T\geq 1$. Now we
can, analogously to the modifications for $H(x,v)=l|v|$, use a function
$f:[-1,1]\to[0,\infty)$ changing the oscillation arbitrary small so that
$f\circ H_{\varepsilon}$ is admissible. In particular

$c_{HZ}(D_{1}\mathbb{R}\mathrm{P}^{n},\mathrm{d}\lambda)\geq(1-\sqrt{\varepsilon})2\pi\mathrm{osc}(H_{\varepsilon})=(1-\sqrt{\varepsilon})2\pi\
\ \forall\varepsilon>0,$

which finishes the proof of Theorem A as $2\pi=2l$ for the normalization111As
$2\pi$ is the length of a prime geodesic on $S^{n}$, $\pi$ is the length of a
prime geodesic on $\mathbb{R}\mathrm{P}^{n}$. we worked with and this lower
bound agrees with the upper bound derived in (5).

##  5 A magnetic version

In this section we consider the tangent bundle of $\mathbb{C}\mathrm{P}^{n}$
with the twisted symplectic form $\mathrm{d}\lambda-s\pi^{*}\omega_{FS}$,
where $\omega_{FS}\in\Omega^{2}(\mathbb{C}\mathrm{P}^{n})$ denotes the Fubini-
Study form while $s\in\mathbb{R}$ is a parameter representing the strength of
the magnetic twist. We will compute the Hofer–Zehnder capacity for the unit
disc subbundle using the exact same strategy as in the proof of Theorem A.
Indeed the magnetic version is a generalization as the case $s=0$ recovers the
untwisted case.

###  5.1 Compactification

Observe that $\omega_{FS}$ is invariant under the $\mathrm{SU}(n+1)$-action
and therefore the induced action of $\mathrm{SU}(n+1)$ on
$(T\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})$ remains
symplectic. Indeed it is not too hard to check that

$\mu_{1}:T\mathbb{C}\mathrm{P}^{n}\to\mathfrak{su}(n+1);\
(x,v)\mapsto[x,v]-sx$ (6)

is a moment map. The term $-sx$ compensates the twist as the inclusion of
$\mathbb{C}\mathrm{P}^{n}\hookrightarrow\mathfrak{su}(n+1)$ as adjoint orbit
is a moment map for $\omega_{FS}$. We can use the same strategy as before to
show that the symplectomorphism from Theorem 2.3 can be adapted to the twisted
situation.

######  Theorem 5.1.

There is a symplectomorphism, which is equivariant with respect to the
Hamiltonian $\mathrm{SU}(n+1)$ actions,

$F:(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})\to(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta},R_{1}\omega_{FS}\ominus
R_{2}\omega_{FS}),$

where
$\bar{\Delta}\subset\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}$
denotes the anti-diagonal divisor

$\bar{\Delta}=\\{(p,q)\in\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\
|\ \mathrm{dist}(p,q)\ \text{maximal}\\}$

and

$R_{1}=\frac{1}{2}\left(\sqrt{s^{2}+1}+s\right),\ \
R_{2}=\frac{1}{2}\left(\sqrt{s^{2}+1}-s\right).$

###### Proof.

Compared to the proof of Theorem 2.3 we modify the Ansatz a little by
considering

$F:D_{1}\mathbb{C}\mathrm{P}^{n}\to\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta};\
\ \
(x,v)\mapsto\left(\exp_{x}(-c_{1}(r)j_{x}v),\exp_{x}(c_{2}(r)j_{x}v)\right)$

for two different functions $c_{1},c_{2}$ that will be determined imposing the
relation

$\mu_{1}(x,v)=\mu_{2}(F(x,v)).$

Per construction the moment maps and the map $F$ are equivariant. Hence it is
again enough to check the moment map triangle at some point

$x=Z=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&\quad&\quad\\\ \quad&-i&0\\\
\quad&0&ni\end{pmatrix},\ \ v=\frac{r}{2}\begin{pmatrix}0_{n-1}&\quad&\quad\\\
\quad&0&-i\\\ \quad&-i&0\end{pmatrix}\in T_{Z}\mathbb{C}\mathrm{P}^{n},\
r=|v|>0,$

as the group action is transitive on the sphere subbundle of
$T\mathbb{C}\mathrm{P}^{n}$. We compute

$\mu_{D_{\rho}\mathbb{C}\mathrm{P}^{n}}(x,v)=[x,v]-(R_{1}-R_{2})x=\frac{r}{2}\begin{pmatrix}0_{n}&\quad&\quad\\\
\quad&0&1\\\
\quad&-1&0\end{pmatrix}-\frac{R_{1}-R_{2}}{n+1}\begin{pmatrix}-i_{n-1}&\quad&\quad\\\
\quad&-i&0\\\ \quad&0&ni\end{pmatrix}.$

Recall that the geodesic through $x$ in direction $j_{x}v$ is given by

$\gamma_{(x,j_{x}v)}(t)=\mathrm{Ad}_{\exp(-tv)}x=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&\quad&\quad\\\
\quad&-i(\cos(rt)^{2}-n\sin(rt)^{2})&(n+1)\sin(rt)\cos(rt)\\\
\quad&-(n+1)\sin(rt)\cos(rt)&i(n\cos(rt)^{2}-\sin(rt)^{2})\end{pmatrix}.$

It parametrizes an affine circle in $\mathfrak{su}(n+1)$ since

$V(t):=\frac{1}{2}\left(\gamma_{(x,j_{x}v)}(t)-\gamma_{(x,j_{x}v)}(t+\frac{\pi}{2r})\right)=\frac{1}{2}\begin{pmatrix}0_{n-1}&\quad&\quad\\\
\quad&-i\cos(2rt)&\sin(2rt)\\\ \quad&-\sin(2rt)&i\cos(2rt)\end{pmatrix}$

satisfies $|V|^{2}=-2\mathrm{tr}(V^{2})=1$. The circle is centered at

$y:=\frac{1}{2}\left(\gamma_{(x,j_{x}v)}(t)+\gamma_{(x,j_{x}v)}(t+\frac{\pi}{2r})\right)=\frac{1}{n+1}\begin{pmatrix}-i_{n-1}&0&0\\\
0&\frac{n-1}{2}i&0\\\ 0&0&\frac{n-1}{2}i\end{pmatrix}.$

Observe that $\gamma_{(x,j_{x}v)}(t)=V(t)+y$. Now the second moment map is
given by

$\mu_{2}(F(x,v))=R_{1}\gamma_{(x,j_{x}v)}(-c_{1})-R_{2}\gamma_{(x,j_{x}v)}(c_{2})=(R_{1}-R_{2})y+R_{1}V(-c_{1})-R_{2}V(c_{2}).$

Comparing the coefficients of the two matrices yields two equations for
$c_{1}$ and $c_{2}$:

$\displaystyle R_{1}\sin(2c_{1}r)+R_{2}\sin(2c_{2}r)$ $\displaystyle=r,$ (7)
$\displaystyle R_{1}\cos(2c_{1}r)-R_{2}\cos(2c_{2}r)$
$\displaystyle=R_{1}-R_{2}.$

The moment map triangle commutes if and only if these two equations hold.
There is an elementary geometric interpretation of the equations (7) which
comes from the fact that geodesics in
$\mathbb{C}\mathrm{P}^{n}\subset\mathfrak{su}(n+1)$ restrict to round circles
in an affine copy of $\mathbb{R}^{2}\subset\mathfrak{su}(n+1)$ and is shown in
Figure 6 and Figure 7. From figure 8 we deduce that a solution exists if and
only if $r:=|v|\leq 1$.

Figure 6: The geodesic $\gamma_{(x,jv)}$ parametrizes a round circle. We draw
this circle twice (in black), one of radius $R_{1}$ and one of radius $R_{2}$.
The moment map $jv-(R_{1}-R_{2})x$ (in blue) must be equal to $R_{1}a-R_{2}b$.
Thus we need to decompose $jv-(R_{1}-R_{2})x$ into two vectors (in green), one
of length $R_{1}$ the other of length $R_{2}$. This is done with an auxiliary
circle (in orange) of radius $R_{2}$ centered at $jv-(R_{1}-R_{2})x$. In
principle there are two solutions (intersections of the orange and the outer
black circle). Counting counterclockwise from the $x$-axis we pick the first.
This is equivalent to demanding $2r(c_{1}+c_{2})\leq\pi$. Figure 7: To see
that the configuration drawn is a geometric interpretation of the equations
(7) denote the angle between the $x$-axis and the vectors by $2c_{1}r$ and
$2c_{2}r$ (in blue). Now projecting the difference $R_{1}a-R_{2}b$ to the
horizontal respectively the vertical axis yields the equations (7) (in
orange). Figure 8: The figure shows the boundary case where only one
intersection exists. Indeed it occurs when $r=1$ as
$(R_{1}+R_{2})^{2}=1+(R_{1}-R_{2})^{2}$ using that by the definition of
$R_{1}$ and $R_{2}$, the product $4R_{1}R_{2}=1$. It follows that there are
two solutions to the equations (7) if $0<r<1$. Demanding further that
$2r(c_{1}+c_{2})<\pi$ we fix one of the solutions.

######  Lemma 5.2.

The functions

$c_{1},c_{2}:\ (-1,1)\to(0,\infty)$

implicitly defined via the equation (7) are smooth and even.

###### Proof.

First observe that the defining equations (7) are invariant under $r\to-r$,
thus $c_{1}$ and $c_{2}$ must be even. We will prove smoothness applying the
implicit function formula to the map

$G:(-1,1)\times(0,\infty)^{2}\to\mathbb{R}^{2};\qquad(r,c_{1},c_{2})\to\begin{pmatrix}R_{1}\sin(2c_{1}r)+R_{2}\sin(2c_{2}r)-r,\\\
R_{1}(\cos(2c_{1}r)-1)-R_{2}(\cos(2c_{2}r)-1)\end{pmatrix}.$

Taking the derivative in $c_{1},c_{2}$ yields

$\mathrm{d}_{c_{1},c_{2}}G=\begin{pmatrix}2rR_{1}\cos(2c_{1}r)&2rR_{2}\cos(2c_{2}r)\\\
-2rR_{1}\sin(2c_{1}r)&2rR_{2}\sin(2c_{2}r)\end{pmatrix}.$

The determinant of $\mathrm{d}_{c_{1},c_{2}}G$ is equal to

$4r^{2}R_{1}R_{2}\left(\cos(2c_{1}r)\sin(2c_{2}r)-\sin(2c_{1}r)\cos(2c_{2}r)\right)=r^{2}\sin(2r(c_{1}+c_{2})).$

From Figure 8 we deduce that $0<2r(c_{1}+c_{2})<\pi$ for all $0<r<1$. Thus the
determinant of $\mathrm{d}_{c_{1},c_{2}}G$ does not vanish when $r\neq 0$. As
a consequence of the implicit function theorem $c_{1},c_{2}$ are smoothly
depending on $r$ whenever $r\neq 0$. For the case $r\to 0$ we rewrite the
equations (7) in terms of

$\tilde{G}:(-\rho,\rho)\times(0,\infty)^{2}\to\mathbb{R}^{2};\qquad(r,c_{1},c_{2})\to\begin{pmatrix}2R_{1}\tau(2c_{1}r)c_{1}+2R_{2}\tau(2c_{2}r)c_{2}-1\\\
4R_{1}\sigma(2c_{1}r)c_{1}^{2}-4R_{2}\sigma(2c_{2}r)c_{2}^{2}\end{pmatrix},$

where $\tau,\sigma:\mathbb{R}\to\mathbb{R}$ are the smooth functions given by

$\tau(x)=\frac{\sin(x)}{x},\qquad\sigma(x)=\frac{1-\cos(x)}{x^{2}}.$

The equations (7) are now equivalent to $\tilde{G}(r,c_{1},c_{2})=0$, taking
the derivative in $c_{1},c_{2}$ we obtain

$\mathrm{d}_{c_{1},c_{2}}\tilde{G}=\begin{pmatrix}4R_{1}r\tau^{\prime}(2c_{1}r)c_{1}+2R_{1}\tau(2c_{1}r)&4R_{2}r\tau^{\prime}(2c_{2}r)c_{2}+2R_{2}\tau(2c_{2}r)\\\
8R_{1}r\sigma^{\prime}(2c_{1}r)c_{1}^{2}+8R_{1}\sigma(2c_{1}r)c_{1}&-8R_{2}r\sigma^{\prime}(2c_{2}r)c_{2}^{2}-8R_{2}\sigma(2c_{2}r)c_{2}\end{pmatrix}.$

Observe that taking the limit $r\to 0$ in (7) yields

$R_{1}c_{1}(0)+R_{2}c_{2}(0)=1/2,\qquad
R_{1}c_{1}(0)^{2}-R_{2}c_{2}(0)^{2}=0,$

thus

$2c_{1}(0)=(R_{1}+\sqrt{R_{1}R_{2}})^{-1}=\left(R_{1}+\frac{1}{2}\right)^{-1},\qquad
2c_{2}(0)=(R_{2}+\sqrt{R_{1}R_{2}})^{-1}=\left(R_{2}+\frac{1}{2}\right)^{-1}.$

Further, $\tau(0)=1=2\sigma(0),\ \tau^{\prime}(0)=0=\sigma^{\prime}(0)$ and
therefore

$\mathrm{d}_{c_{1},c_{2}}\tilde{G}|_{r=0}=\begin{pmatrix}2R_{1}&2R_{2}\\\
4R_{1}c_{1}(0)&-4R_{2}c_{2}(0)\end{pmatrix}.$

We see that

$\det\left(\mathrm{d}_{c_{1},c_{2}}\tilde{G}|_{r=0}\right)=-8R_{1}R_{2}(c_{1}(0)+c_{2}(0))=-4\neq
0$

and it follows by the implicit function theorem that $c_{1},c_{2}$ are smooth
at $r=0$. ∎

This finishes the construction of the smooth map
$F:D_{\rho}\mathbb{C}\mathrm{P}^{n}\to\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}$.
But is it a bijection? We can explicitly give an inverse. A point
$(a,b)\in\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta}$
defines a unique shortest geodesic $\gamma$ connecting from $a$ to $b$. This
geodesic is again a segment of a round circle in an affine copy of
$\mathbb{R}^{2}\subset\mathfrak{su}(n+1)$. This situation is depicted in
figure 9. As shown in figure 9 there is a unique orthogonal decomposition of
the vector $R_{1}a-R_{2}b$ such that one of the vectors has length
$R_{1}-R_{2}$ and $x$ lies inside the angle $2r(c_{1}+c_{2})$. This
decomposition determines $jv$ and $(R_{1}-R_{2})x$ and thus $(x,v)\in
D_{1}\mathbb{C}\mathrm{P}^{n}$ and yields an inverse to $F$. So far we showed
that $F$ is a smooth, equivariant bijection.

Figure 9: The vector $R_{1}a-R_{2}b$ (in blue) is given and we need to
decompose it into two orthogonal vectors $jv$ and $-(R_{1}-R_{2})x$ (in red).
The second vector must be of length $R_{1}-R_{2}$, thus there are at most two
possible candidates. We pick the one, such that $x$ lies inside the angle
$2r(c_{1}+c_{2})$.

Observe that the codimension of the orbits of $SU(n+1)$ is one. Thus any
complement of $\mathcal{D}$ is 1-dimensional and thus isotropic for both
symplectic forms $\omega_{s}$ and $F^{*}(R_{1}\omega_{FS}\ominus
R_{2}\omega_{FS})$. It follows from Lemma 2.1 that $F$ is a symplectomorphism.
∎

###  5.2 Hofer–Zehnder capacity

The symplectomorphism $F$ can also be used to compute the Hofer–Zehnder
capacity of
$(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})$.

######  Theorem 5.3 (Thm. B).

Denote $\omega_{FS}$ the Fubini-Study form and $l$ the length of the
geodesics, then

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})=l\left(\sqrt{s^{2}+1}-|s|\right).$

###### Proof.

The upper bound works very similar to the untwisted case. We again pick
$A\in[S^{2},\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}]$ the
homotopy class of one of the two generators. As discussed for the untwisted
case, the moduli space of $j\ominus j$-holomorphic spheres in class $A$
through a fixed point and intersecting the anti-diagonal is a smooth, compact
$S^{1}$-manifold not cobordant to the empty set. It follows by Hofer–Viterbo’s
Theorem that $\omega(A)$ bounds the Hofer–Zehnder capacity from above. Observe
that in this case it matters which factor we choose, as both yield different
upper bounds. We obtain

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})\leq
4\pi\min\\{R_{1},R_{2}\\}=2\pi\left(\sqrt{s^{2}+1}-|s|\right).$ (8)

For the lower bound we need to work a little harder, as we can not use the
same Hamiltonian as in the untwisted case. Indeed the kinetic Hamiltonian
$E(x,v):=\frac{1}{2}g_{x}(v,v)$ does not generate geodesic flow anymore but
rather its magnetic version.

######  Lemma 5.4.

The magnetic geodesic flow (i.e. the flow generated by $X_{E}$) is totally
periodic.

###### Proof.

As shown in [5, Lemma 6.1] the vector field $X_{E}$ is pointwise given by

$(X_{E})_{(x,v)}=X_{(x,v)}+s(jv)^{\mathcal{V}},$

where $X$ denotes the vector field generating geodesic flow and
$(\cdot)^{\mathcal{V}}$ the vertical lift as introduced in the proof of Lemma
4.1. Now recall that for any point $(x,v)\in T\mathbb{C}\mathrm{P}^{n}$
outside the zero section there is a unique totally geodesic embedded copy of
$\mathbb{C}\mathrm{P}^{1}$ through $x$ tangent to $v$. In particular the
vector fields are $X+s(jv)^{\mathcal{V}}$ are tangent to these embedded copies
of $T\mathbb{C}\mathrm{P}^{1}$ and their flow will restrict to
$T\mathbb{C}\mathrm{P}^{1}$. It is therefore enough to study the one
dimensional case. So let $\gamma(t)=(x(t),v(t))$ be a magnetic geodesic, i.e.
$\dot{\gamma}=X+s(jv)^{\mathcal{V}}$. Projecting to the vertical distribution
we obtain $\dot{x}(t)=v(t)$. Now projecting to the horizontal distribution we
obtain

$\nabla_{\dot{x}}v=sjv.$

It follows that $\gamma(t)=(x(t),\dot{x}(t))$, where $x(t)$ is a curve of
constant geodesic curvature $\kappa_{g}=|\frac{s}{v}|$, i.e. it parametrizes a
geodesic circle and is therefore closed. ∎

If $R$ denotes the radius (with respect to the Riemannian metric $g$) of a
geodesic circle we know using normal polar coordinates that its circumference
$C$ and the geodesic curvature $\kappa_{g}$ are

$\displaystyle C$
$\displaystyle=\frac{2\pi}{\sqrt{\kappa}}\sin(\sqrt{\kappa}R)=\frac{2\pi\sqrt{\kappa}^{-1}\tan(\sqrt{\kappa}R)}{\sqrt{1+(\tan(\sqrt{\kappa}R))^{2}}},$
(9) $\displaystyle\kappa_{g}$
$\displaystyle=\frac{\sqrt{\kappa}}{\tan(\sqrt{\kappa}R)}.$ (10)

Inserting $\kappa_{g}$ into (9) yields

$C=\frac{2\pi}{\kappa_{g}\sqrt{1+\kappa/\kappa_{g}^{2}}}=\frac{2\pi|v|}{\sqrt{s^{2}+\kappa|v|^{2}}}.$

Now, we conclude that the period is given by

$T=\frac{C}{|v|}=\frac{2\pi}{\sqrt{s^{2}+\kappa|v|^{2}}}.$

In particular the reparametrization

$H:D_{1}\mathbb{C}\mathrm{P}^{n}\to\mathbb{R};\ \ (x,v)\mapsto
2\pi\left(\sqrt{s^{2}+|v|^{2}}-|s|\right)$

is the moment map of a circle action on
$(D_{1}\mathbb{C}\mathrm{P}^{n},\mathrm{d}\lambda-s\pi^{*}\omega_{FS})$.
Arguing as in the untwisted case at the beginning of section 4 we can modify
$H$ changing its oscillation only by an arbitrary small $\varepsilon>0$ to
make $H$ admissible. In particular we obtain the following lower bound

$c_{HZ}(D_{1}\mathbb{C}\mathrm{P}^{n})\geq\max H-\min
H=2\pi\left(\sqrt{s^{2}+1}-|s|\right).$

Comparing this to equation (8) we see that upper and lower bound agree,
further we choose the normalization of the metric such that $l=2\pi$,
therefore we just finished the proof of Theorem B. ∎

######  Remark 5.5.

We can interpret the symplectomorphism of Theorem 5.1 as a Lerman cut with
respect to $H$, i.e.

$\overline{\left(D_{\rho}\mathbb{C}\mathrm{P}^{n},\omega_{s}\right)}\cong\left(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n},R_{1}\omega_{FS}\ominus
R_{2}\omega_{FS}\right).$

###### Proof.

As $H$ generates a circle action also $H\circ F^{-1}$ must generate a
Hamiltonian circle action on
$(\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}\setminus\bar{\Delta},R_{1}\omega_{FS}\ominus
R_{2}\omega_{FS})$. Denote $(x,v):=F^{-1}(N,S)$. By construction
$\mu_{1}(x,v)=\mu_{2}(N,S)$ and therefore

$\displaystyle|\mu_{1}(x,v)|^{2}=|\mu_{2}(N,S)|^{2}\ $
$\displaystyle\Leftrightarrow\
|v|^{2}+(R_{2}-R_{1})^{2}=R_{1}^{2}+R_{2}^{2}-2R_{1}R_{2}(N,S)$
$\displaystyle\Leftrightarrow\ |v|^{2}=2R_{1}R_{2}(1-(N,S)).$

Thus

$H(F^{-1}(N,S))=2\pi\left(|R_{1}N-R_{2}S|-|R_{2}-R_{1}|\right),$

in particular $H\circ F^{-1}$ extends smoothly to
$\mathbb{C}\mathrm{P}^{n}\times\mathbb{C}\mathrm{P}^{n}$ and is critical on
the anti-diagonal divisor $\bar{\Delta}$. ∎

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