# Chekanov torus and Gelfand–Zeitlin torus in $S^{2}\times S^{2}$

Yoosik Kim  Department of Mathematics, Pusan National University
<EMAIL_ADDRESS>
###### Abstract.

The Chekanov torus was the first known _exotic_ torus, a monotone Lagrangian
torus that is not Hamiltonian isotopic to the standard monotone Lagrangian
torus. We explore the relationship between the Chekanov torus in $S^{2}\times
S^{2}$ and a monotone Lagrangian torus that had been introduced before
Chekanov’s construction [Che96]. We prove that the monotone Lagrangian torus
fiber in a certain Gelfand–Zeitlin system is Hamiltonian isotopic to the
Chekanov torus in $S^{2}\times S^{2}$.

##  1\. Introduction

A symplectic manifold $(X,\omega)$ is called _monotone_ if the class of the
symplectic form $\omega$ and the first Chern class of the tangent bundle $TX$
in $H^{2}(X;\mathbb{R})$ are positively proportional, i.e. $[\omega]=r\cdot
c_{1}(TX)$ for a positive real number $r>0$. Each Lagrangian submanifold $L$
of $X$ carries two group homomorphisms$\colon$

  * •

$I_{\omega}\colon\pi_{2}(X,L)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{R}$
is given by the symplectic area,

  * •

$I_{\mu}\colon\pi_{2}(X,L)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{Z}$
is given by the Maslov index.

A Lagrangian submanifold $L$ is said to be _monotone_ if there exists a
positive real number $s>0$ such that $I_{\omega}(\beta)=s\cdot I_{\mu}(\beta)$
for every $\beta\in\pi_{2}(X,L)$. This constant $s$ is called the
_monotonicity_ of $L$. The notion of a monotone Lagrangian submanifold was
introduced by Y.-G. Oh to construct its Floer cohomology in [Oh93a, Oh93b].

As an attempt to classify the monotone Lagrangians in a symplectic manifold,
it has been an interesting problem to construct monotone Lagrangian tori that
are not related by any Hamiltonian isotopy. In the symplectic vector space
$\mathbb{R}^{2n}$, Chekanov [Che96] constructed a monotone Lagrangian torus
that is _not_ Hamiltonian isotopic to any standard product torus. Moreover, by
embedding a suitable subset of $\mathbb{R}^{2n}$ symplectically, one can
produce a monotone Lagrangian torus by transporting a Chekanov exotic torus of
$\mathbb{R}^{2n}$ into a closed symplectic manifold including the complex
projective space or the product of projective lines. Such an embedded monotone
Lagrangian torus is also called a _Chekanov torus_. A monotone torus not
Hamiltonian isotopic to the standard one is called an _exotic torus_. The
Chekanov torus is the first known exotic monotone Lagrangian torus. Recently,
there have been exciting developments constructing infinitely many exotic
Lagrangian tori, see [Via16, Aur15, Via17] for instance.

In this article, we are concerned with the Chekanov torus in
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$. To begin with, we briefly recall
several ways of constructing the Chekanov torus in
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}\simeq S^{2}\times S^{2}$.

  * •

(Chekanov–Schlenk torus $T_{\mathrm{CS}}$ in [CS10]) Let
$\mathbb{D}(\sqrt{2}):=\\{\zeta\in\mathbb{C}\mid|\zeta|<\sqrt{2}\\}$ be an
open disk in the complex plane with radius $\sqrt{2}$ and let
$\mathbb{H}(\sqrt{2}):=\\{\zeta\in\mathbb{D}(\sqrt{2})\mid\mathrm{Im}(\zeta)>0\\}$
be an open half-disk. Choose any simple closed curve $\Gamma$ in
$\mathbb{H}(\sqrt{2})$ which bounds the region having the area $\pi/2$. The
product space carries the Hamiltonian $S^{1}$-action given by

$e^{\sqrt{-1}\theta}\cdot(\zeta_{1},\zeta_{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\left(e^{-\sqrt{-1}\theta}\zeta_{1},e^{\sqrt{-1}\theta}\zeta_{2}\right).$

Consider the product space $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ of
projective planes equipped with the product symplectic form
$2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}})$ where
$([v_{0}:v_{1}],[w_{0}:w_{1}])$ is its homogeneous coordinate. Recall that
$2\omega_{\mathrm{FS}}=\frac{1}{2}\omega_{\mathrm{std}}$ and the symplectic
area of $\mathbb{C}P^{1}$ measured by $2\omega_{\mathrm{FS}}$ is $2\pi$. We
denote the diagonal map by

(1)
$\Delta_{\mathbb{D}}\colon\mathbb{D}(\sqrt{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{D}(\sqrt{2})\times\mathbb{D}(\sqrt{2}).$

Also, the composition of the inclusion
$\mathbb{H}(\sqrt{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{D}(\sqrt{2})$
and the diagonal map $\Delta_{\mathbb{D}}$ is denoted by

(2)
$\Delta_{\mathbb{H}}\colon\mathbb{H}(\sqrt{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{D}(\sqrt{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{D}(\sqrt{2})\times\mathbb{D}(\sqrt{2}).$

We have a symplectic embedding

(3)
$\rho\colon\left(\mathbb{D}(\sqrt{2})\times\mathbb{D}(\sqrt{2}),(\omega_{\mathrm{std}}\oplus\omega_{\mathrm{std}})|_{\mathbb{D}(\sqrt{2})\times\mathbb{D}(\sqrt{2})}\right)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\left(\mathbb{C}P^{1}\times\mathbb{C}P^{1},2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}})\right)$

such that the image of $\rho$ is the intersection of $v_{0}\neq 0$ and
$w_{0}\neq 0$. Chekanov and Schlenk in [CS10] constructed the monotone
Lagrangian torus $T_{\mathrm{CS}}$ as follows$\colon$

(4)
$T_{\mathrm{CS}}:=\left\\{\rho\left(e^{-\sqrt{-1}\theta}\zeta,e^{\sqrt{-1}\theta}\zeta\right)\in\mathbb{C}P^{1}\times\mathbb{C}P^{1}\mid\theta\in[0,2\pi],\zeta\in\Gamma\right\\}.$

  * •

(Entov–Polterovich torus $T_{\mathrm{EP}}$ in [EP09]) Let us regard the
product space $S^{2}\times S^{2}$ as an embedded submanifold of
$\mathbb{R}^{3}\times\mathbb{R}^{3}$ equipped with
$\frac{1}{2}(\omega_{\mathrm{std}}\oplus\omega_{\mathrm{std}})$. Entov and
Polterovich in [EP09] constructed a monotone Lagrangian torus as
follows$\colon$

$T_{\mathrm{EP}}:=\left\\{(\mathbf{a},\mathbf{b})\in S^{2}\times
S^{2}\mid(\mathbf{a}+\mathbf{b})\cdot\mathbf{e}_{1}=0,\mathbf{a}\cdot\mathbf{b}=-1/2\right\\}$

where $\mathbf{e}_{1}=(1,0,0)$.

  * •

(Fukaya–Oh–Ohta–Ono torus $T_{\mathrm{FOOO}}$ in [FOOO12]) Start with the
symplectic toric orbifold associated to the triangle whose vertices are
$(0,0),(0,1)$, and $(2,0)$ as the moment polytope. By replacing a neighborhood
of the singular point with the Milnor fiber to obtain a symplectic manifold
isomorphic to $S^{2}\times S^{2}$. Through this process, one obtains a semi-
toric system $\Phi_{\mathrm{semi}}$ whose image is the above polytope.
Fukaya–Oh–Ohta–Ono monotone Lagrangian torus is located at the center
$(1/2,1/2)$, that is,

$T_{\mathrm{FOOO}}:=\Phi^{-1}_{\mathrm{semi}}(1/2,1/2).$

  * •

(Albers–Frauenfelder torus $T_{\mathrm{AF}}$ in [AF08]) Let $\Delta$ be the
diagonal of $S^{2}\times S^{2}$. Choose a symplectomorphism

$\upsilon\colon(S^{2}\times
S^{2})\backslash\Delta\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}D_{1}^{*}S^{2}$

where $D_{1}^{*}S^{2}$ is the open unit disk bundle. Albers–Frauenfelder
monotone torus is defined by

$T_{\mathrm{AF}}:=\upsilon^{-1}\left(\left\\{(\mathbf{p},\mathbf{q})\in
D_{1}^{*}S^{2}\mid|\mathbf{p}|=1/2,(\mathbf{p}\times\mathbf{q})\cdot\mathbf{e}_{1}=0\right\\}\right).$

  * •

(Biran–Cornea torus $T_{\mathrm{BC}}$ in [BC12]) Let us start with the
diagonal $\Delta$ of $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$, a complex
hypersurface of $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$. Set

$P_{\Delta}=\left\\{(\mathbf{a},\mathbf{b})\in\mathbb{R}^{3}\times\mathbb{R}^{3}\colon|\mathbf{a}|=|\mathbf{b}|=1,\,\mathbf{a}\cdot\mathbf{b}=0\right\\}.$

with the circle action

$e^{\sqrt{-1}\theta}\cdot(\mathbf{a},\mathbf{b})=(\mathbf{a},(\cos\theta)\mathbf{b}+(\sin\theta)(\mathbf{a}\times\mathbf{b})).$

Identifying $\Delta$ with $S^{2}$, we then have the principal $S^{1}$-bundle
$\pi\colon
P_{\Delta}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Delta$
given by $\pi([(\mathbf{a},\mathbf{b})])=\mathbf{a}$, whose connection
$1$-from for $P_{\Delta}$ is defined by
$\alpha_{(\mathbf{a},\mathbf{b})}(\mathbf{p},\mathbf{q})=\mathbf{q}\cdot(\mathbf{a}\times\mathbf{b})$.
According to a work of Biran in [Bir01], there is a symplectomorphism from

$D_{\sqrt{2}}(P_{\Delta})=\frac{P_{\Delta}\times
D(\sqrt{2})}{(e^{\sqrt{-1}\theta}\cdot(\mathbf{a},\mathbf{b}),\zeta)\sim((\mathbf{a},\mathbf{b}),e^{\sqrt{-1}\theta}\cdot\zeta)}$

adorned with a certain symplectic form defined by using the connection
$1$-form to $(S^{2}\times S^{2})\backslash\overline{\Delta}$ where
$\overline{\Delta}$ is the anti-diagonal.

For each monotone Lagrangian submanifold $L$ in $\Delta$ and each $r$ with
$0<r<\sqrt{2}$, we have the submanifold

$L_{(r)}=\left\\{\left[((\mathbf{a},\mathbf{b}),\zeta)\right]\in
D_{\sqrt{2}}(P_{\Delta})\mid|\zeta|=r,\pi([((\mathbf{a},\mathbf{b}),\zeta)])\in
L\right\\}.$

According to [BC12], there is a unique value $r$ such that the embedding of
$L_{(r)}$ becomes a monotone Lagrangian submanifold in $S^{2}\times S^{2}$. In
this case, we choose a great circle of $S^{2}\simeq\Delta$ for $L$ and one for
$r$. The choice leads to Brian–Cornea monotone Lagrangian torus

$T_{\mathrm{BC}}=L_{(1)}$

in $S^{2}\times S^{2}$. For an explicit description for the embedding and the
embedded monotone Lagrangian, the reader is referred to [OU16, Section 4].

The relation between the above monotone tori had been explored. In [FOOO12],
Fukaya, Oh, Ohta, and Ono discussed the relation between $T_{\mathrm{FOOO}}$
and $T_{\mathrm{AF}}$. Gadbled [Gad13] showed that the $T_{\mathrm{BC}}$ and
$T_{\mathrm{CS}}$ are Hamiltonian isotopic. Oakley and Usher ultimately proved
that all five monotone Lagrangian tori are Hamiltonian isotopic to each other
by constructing detailed and explicit symplectomorphisms in [OU16].

######  Theorem 1 (Theorem 1.1 in [OU16]).

The five monotone Lagrangian tori $T_{\mathrm{CS}}$, $T_{\mathrm{EP}}$,
$T_{\mathrm{FOOO}}$, $T_{\mathrm{AF}}$, and $T_{\mathrm{BC}}$ listed above are
Hamiltonian isotopic to each other.

The main goal of this article is to add one more monotone Lagrangian torus to
the above list. This torus had been constructed even before the Chekanov’s
construction of “twist tori” in [Che96]. Specifically, we shall prove that the
monotone Gelfand–Zeitlin (GZ) Lagrangian torus fiber in the orthogonal
Grassmannian $\mathrm{OG}(1,\mathbb{C}^{4})$ is Hamiltonian isotopic to one
(and hence each) of the above listed tori.

Let us recall the quickest way of constructing the monotone GZ Lagrangian
torus.

  * •

(Gelfand–Zeitlin torus $T_{\mathrm{GZ}}$ in [GS83b]) Consider the set
$\mathcal{O}$ of $(4\times 4)$ skew-symmetric matrices with prescribed four
eigenvalues $\pm\lambda\sqrt{-1}$ and $0$ with multiplicity two for some real
number $\lambda$. In other words, the space $\mathcal{O}$ is the orbit of the
block diagonal matrix

(5) $\begin{bmatrix}0&\lambda&0&0\\\ -\lambda&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}$

under the adjoint $\mathrm{SO}(4)$-action. It carries the
Kirillov–Kostant–Souriau (KKS) symplectic form $\omega_{\mathrm{KKS}}$, which
is determined by $\lambda$. cf. Remark 2 below. To match up with the
convention on the symplectic form
$2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}})$ on
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$, we take $\lambda=1$.

For a matrix $A\in\mathcal{O}$, let $A^{(k)}$ be the leading principal
submatrix of the size $(k\times k)$. Since $A$ is skew-symmetric, $A^{(1)}$ is
the zero matrix. In order to define the desired GZ torus, we impose the
following conditions on $A^{(2)}$ and $A^{(3)}$.

    1. (1)

The eigenvalues of $A^{(2)}$ are required to be $0$ with multiplicity two.

    2. (2)

The eigenvalues of $A^{(3)}$ are required to be $\pm\frac{1}{2}\sqrt{-1}$, and
$0$.

The conditions $(\mathrm{1})$ and $(\mathrm{2})$ determine the GZ Lagrangian
torus $T_{\mathrm{GZ}}$ in $\mathcal{O}$. That is,

(6) $T_{\mathrm{GZ}}=\left\\{A\in\mathcal{O}~{}\Big{|}~{}A^{(2)}=O\mbox{ and
the eigenvalues of $A^{(3)}$ are $\pm\frac{1}{2}\sqrt{-1},0$}\right\\}.$

######  Remark 2.

When defining the KKS form $\omega_{\mathrm{KKS}}$, the symplectic form is
sometimes normalized by dividing the form by $2\pi$ so that the class of the
symplectic form agrees with an _integral_ multiple of the Chern class. In this
article, we do _not_ normalize the KKS form to facilitate comparison of the
KKS form with other symplectic forms. Namely, it is defined by

$\omega_{\mathrm{KKS}}\left(\mathrm{ad}_{\xi_{1}}(A),\mathrm{ad}_{\xi_{2}}(A)\right)=\langle
A,[\xi_{1},\xi_{2}]\rangle$

where $A\in\mathcal{O}$ and $\mathrm{ad}_{\xi_{i}}(A)$ is a tangent vector at
$A$ of $\mathcal{O}$ for $i=1,2$.

Additional explanation on $T_{\mathrm{GZ}}$ and $\mathcal{O}$ is in order. The
real special orthogonal group $\mathrm{SO}(4):=\mathrm{SO}(4;\mathbb{R})$
transitively acts on $\mathcal{O}$ by conjugation. A choice of an element of
$\mathcal{O}$ gives rise to a diffeomorphism
$\mathcal{O}\simeq{\mathrm{SO}(4)}/{\mathrm{S}(\mathrm{O}(2)\times\mathrm{O}(2))}$.
Therefore, the orbit $\mathcal{O}$ is diffeomorphic to the orthogonal
Grassmannian $\mathrm{OG}(1,\mathbb{C}^{4})$, which is the space of isotropic
subspaces of $\mathbb{C}^{4}$ with respect to a non-degenerate symmetric
bilinear form. Thus, $\mathrm{OG}(1,\mathbb{C}^{4})$ can be regarded as a
quadric hypersurface. As any two smooth quadric hypersurfaces are
symplectomorphic, the quadric hypersurface is symplectomorphic to the Segre
variety $\left\\{[\mathbf{z}]\in\mathbb{C}P^{3}\mid
z_{0}z_{1}-z_{2}z_{3}=0\right\\}$. Hence, the space $\mathcal{O}$ is regarded
as the product $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ of two complex
projective planes.

The min-max principle, a system of inequalities between eigenvalues of
submatrices of a skew-symmetric matrix, says that the eigenvalues of $A^{(2)}$
and $A^{(3)}$ are of the form$\colon$

  * •

the eigenvalues of $A^{(2)}$ are $\pm\lambda^{(2)}\sqrt{-1}$.

  * •

the eigenvalues of $A^{(3)}$ are $\pm\lambda^{(3)}\sqrt{-1},0$.

Each real-valued function determined by $\lambda^{(\bullet)}$ generates a
Hamiltonian circle action (on an open dense subset of $\mathcal{O}$) by the
result of Guillemin–Sternberg [GS83a]. Indeed, the fiber $T_{\mathrm{GZ}}$ is
a free Hamiltonian $T^{2}$-orbit so that it is a Lagrangian torus. Employing
the technique of the gradient Hamiltonian disks in [CK19], the torus
$T_{\mathrm{GZ}}$ was shown to be monotone.

######  Lemma 3 (Proposition 3.7 in [Kim21]).

The above Gelfand–Zeitlin torus fiber $T_{\mathrm{GZ}}$ is monotone.

The main theorem of this article is stated below.

######  Theorem 4.

There is a symplectomorphism from $\mathcal{O}$ to
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ taking the Gelfand–Zeitlin torus
$T_{\mathrm{GZ}}$ to the Chekanov–Schlenk torus $T_{\mathrm{CS}}$.
Consequently, all monotone Lagrangian tori $T_{\mathrm{CS}}$,
$T_{\mathrm{EP}},T_{\mathrm{FOOO}},T_{\mathrm{AF}}$, $T_{\mathrm{BC}}$, and
$T_{\mathrm{GZ}}$ are Hamiltonian isotopic to each other.

There had been decisive clues that $T_{\mathrm{GZ}}$ is Hamiltonian isotopic
to the Chekanov torus in literatures. First, Nishinou–Nohara–Ueda [NNU10]
described and constructed the toric degeneration of the Gelfand–Zeitlin
system. The image of the system on the quadric hypersurface of complex
dimension two agrees with the moment polytope of $\mathbb{C}P(1,1,2)$, which
is one of the starting points of this work. Second, the disk potential of the
GZ torus fiber $T_{\mathrm{GZ}}$ computed in [Kim21] agrees with that of
$T_{\mathrm{CS}}$ in [Aur07, CS10] and that of $T_{\mathrm{FOOO}}$ in [FOOO12]
up to some coordinate changes. Theorem 4 claims a stronger relationship
between $T_{\mathrm{GZ}}$ and $T_{\mathrm{CS}}$.

##  2\. Proof of Theorem 4

Consider the product space $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ of complex
planes equipped with the product symplectic form
$2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}})$ where
$\omega_{\mathrm{FS}}$ is the Fubini–Study form on $\mathbb{C}P^{1}$. It has
the $S^{1}$-action given by

(7)
$([v_{0}:v_{1}],[w_{0}:w_{1}])\mapsto\left(\left[v_{0}:e^{-\sqrt{-1}\theta}v_{1}\right],\left[w_{0}:e^{\sqrt{-1}\theta}w_{1}\right]\right).$

The product space $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ is embedded into
$\mathbb{C}P^{3}$ via the Segre embedding

$\sigma\colon\mathbb{C}P^{1}\times\mathbb{C}P^{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}P^{3}\quad([v_{0}:v_{1}],[w_{0}:w_{1}])\mapsto[v_{0}w_{1}:v_{1}w_{0}:v_{0}w_{0}:v_{1}w_{1}].$

The image under the Segre map $\sigma$ is a hypersurface of $\mathbb{C}P^{3}$
defined by $z_{0}z_{1}-z_{2}z_{3}=0$ where
$[\mathbf{z}]:=[z_{0}:z_{1}:z_{2}:z_{3}]$ is the homogeneous coordinates for
$\mathbb{C}P^{3}$. The hypersurface is denoted by

(8) $\mathcal{Q}^{\prime}:=\left\\{[\mathbf{z}]\in\mathbb{C}P^{3}\mid
z_{0}z_{1}-z_{2}z_{3}=0\right\\}.$

Let $S^{1}$ act on $\mathbb{C}P^{3}$ as follows.

(9)
$[z_{0}:z_{1}:z_{2}:z_{3}]\mapsto\left[e^{\sqrt{-1}\theta}z_{0}:e^{-\sqrt{-1}\theta}z_{1}:z_{2}:z_{3}\right].$

Adorning $\mathbb{C}P^{3}$ with twice Fubini–Study form
$2\omega_{\mathrm{FS}}$, the $S^{1}$-action becomes Hamiltonian. Setting
$\|\mathbf{z}\|^{2}:={|z_{0}|^{2}+|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}}$, we
choose a moment map
$\mu_{S^{1}}\colon\mathbb{C}P^{3}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{R}$
with respect to the symplectic form $2\omega_{\mathrm{FS}}$ for the action (9)
as

(10)
$\mu_{S^{1}}(\mathbf{z})=\frac{|z_{0}|^{2}-|z_{1}|^{2}}{\|\mathbf{z}\|^{2}}.$

Then the map $\sigma$ is $S^{1}$-equivariant. The embedded variety
$\mathcal{Q}^{\prime}$ is equipped with the Kähler form inherited from
$(\mathbb{C}P^{3},2\omega_{\mathrm{FS}})$. We denote the form restricted to
$\mathcal{Q}^{\prime}\subset\mathbb{C}P^{3}$ by
$\omega_{\mathcal{Q}^{\prime}}$.

To show that the map $\sigma$ preserves the symplectic form, it suffices to
show that it is a symplectomorphism on an affine chart, an open dense subset
of $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$. A straightforward computation shows
that $\sigma$ preserves the symplectic form.

######  Proposition 5.

The Segre embedding $\sigma$ is an $S^{1}$-equivariant symplectomorphism.

Under the coordinate change

(11) $\begin{bmatrix}z_{0}\\\ z_{1}\\\ z_{2}\\\
z_{3}\end{bmatrix}=\begin{bmatrix}1&\sqrt{-1}&0&0\\\ 1&-\sqrt{-1}&0&0\\\
0&0&1&\sqrt{-1}\\\ 0&0&-1&\sqrt{-1}\end{bmatrix}\begin{bmatrix}x_{0}\\\
x_{1}\\\ x_{2}\\\ x_{3}\end{bmatrix},$

the variety $\mathcal{Q}^{\prime}$ in (8) maps into the Fermat
hypersurface$\colon$

(12) $\mathcal{Q}=\left\\{[x_{0}:x_{1}:x_{2}:x_{3}]\in\mathbb{C}P^{3}\mid
x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0\right\\}.$

The linear transformation (11) is denoted by
$\Lambda\colon\mathcal{Q}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{Q}^{\prime}$.

Let
$G:=\mathrm{SO}(2)=\mathrm{SO}(2;\mathbb{R})\simeq\mathrm{diag}\left(\mathrm{SO}(2;\mathbb{R}),I_{2}\right)$
act on $\mathbb{C}^{4}$ linearly. The linear $\mathrm{SO}(2)$-action induces
the action on $\mathbb{C}P^{3}$ and the action on $\mathcal{Q}$. Let us fix
the following isomorphism

(13) $\iota\colon
S^{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{SO}(2;\mathbb{R}),\quad
e^{\sqrt{-1}\theta}\mapsto\begin{bmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{bmatrix}.$

The hypersurface $\mathcal{Q}$ in (12) admits the $S^{1}$-action via (13). We
then observe the following.

######  Lemma 6.

The map
$\Lambda\colon\mathcal{Q}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{Q}^{\prime}$
is $S^{1}$-equivariant.

###### Proof.

Using the identification (13) and the expression (11), one can directly verify
the $S^{1}$-equivalence of $\Lambda$. ∎

We now discuss symplectic forms on $\mathcal{Q}$. Two adorned symplectic forms
on $\mathcal{Q}$ are taken into account. The first one is the pull-backed
symplectic form $\Lambda^{*}\omega_{\mathcal{Q}^{\prime}}$. The second one is
the symplectic form coming from the twice Fubini–Study form
$2\omega_{\mathrm{FS}}$ on the ambient space $\mathbb{C}P^{3}$. Namely, we
take the reduction
$\mathbb{C}P^{3}\simeq(\mathbb{C}^{4}-\\{0\\})\mathbin{/\mkern-6.0mu/}S^{1}$
where the $S^{1}$-action is diagonal action with the stability condition
$\|\mathbf{x}\|=\sqrt{2}$ so that $\mathbb{C}P^{3}$ is equipped with twice
Fubini–Study form $2\omega_{\mathrm{FS}}$. The Kähler form restricted to
$\mathcal{Q}$ is denoted by $\omega_{\mathcal{Q}}$.

The Fermat hypersurface $\mathcal{Q}$ carries two circle actions. First, the
moment map of the corresponding $S^{1}$-action on $\mathcal{Q}$ with respect
to the first symplectic form $\Lambda^{*}\omega_{\mathcal{Q}^{\prime}}$ is
obtained by replacing $\mathbf{z}$ with $\Lambda(\mathbf{x})$ in (10) as
follows.

(14)
$\mu_{S^{1}}(\mathbf{x})=\frac{\sqrt{-1}\left(\overline{x}_{0}x_{1}-x_{0}\overline{x}_{1}\right)}{\|\mathbf{x}\|^{2}}.$

Second, by taking the Killing form

$\langle\xi_{1},\xi_{2}\rangle=-\frac{1}{2}\mathrm{Tr}(\xi_{1},\xi_{2}),$

we identify the Lie algebra $\mathfrak{g}$ and its dual Lie algebra
$\mathfrak{g}^{*}$. Under this identification, a moment map of the above
$\mathrm{SO}(2)$-action on $\mathcal{Q}$ is then of the form

(15)
$\mu_{G}\colon\mathcal{Q}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{g},\quad\mu_{G}(\mathbf{x})=\frac{\sqrt{-1}}{\|\mathbf{x}\|^{2}}\begin{bmatrix}0&(\overline{x}_{0}x_{1}-x_{0}\overline{x}_{1})\\\
-(\overline{x}_{0}x_{1}-x_{0}\overline{x}_{1})&0\end{bmatrix}.$

Via the isomorphism (13), the $\mathrm{SO}(2)$-action can be regarded as the
$S^{1}$-action. By the functoriality of moment maps, a moment map of the
$S^{1}$-action can be chosen as

(16)
$(\iota^{*}\circ\mu_{G})(\mathbf{x})=\frac{\sqrt{-1}\left(\overline{x}_{0}x_{1}-x_{0}\overline{x}_{1}\right)}{\|\mathbf{x}\|^{2}}\colon\mathcal{Q}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{R}.$

The following lemma compares those two circle actions.

######  Lemma 7.

The Hamiltonian function (14) on
$\left(\mathcal{Q},\Lambda^{*}\omega_{\mathcal{Q}^{\prime}}\right)$ and the
Hamiltonian function (16) on $\left(\mathcal{Q},\omega_{\mathcal{Q}}\right)$
generate the same $S^{1}$-action on $\mathcal{Q}$.

###### Proof.

We shall compare those two $S^{1}$-actions on $\mathcal{Q}$ after passing them
to $\mathcal{Q}^{\prime}$ via the isomorphism $\Lambda$ in (11). Recall that
the $S^{1}$-action on $\mathcal{Q}^{\prime}$ corresponding to the first
$S^{1}$-action is described in (9). By Lemma 6, the $S^{1}$-action on
$\mathcal{Q}^{\prime}$ corresponding to the second $S^{1}$-action is exactly
the action in (9) as desired. ∎

To interpolate these forms, we need an equivariant version of the Moser
theorem, which is stated below.

######  Theorem 8 (Equivariant Moser Theorem).

Let $X$ be a compact symplectic manifold. Suppose that $\omega_{0}$ and
$\omega_{1}$ are two symplectic forms in the same cohomology class, that is,
$[\omega_{0}]=[\omega_{1}]$ in $H^{2}(X;\mathbb{C})$. If
$\omega_{t}:=(1-t)\omega_{0}+t\omega_{1}$ is symplectic for each $t\in[0,1]$,
then there exists an isotopy

$\phi\colon[0,1]\times
X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$

such that $\phi_{t}^{*}\omega_{t}=\omega_{0}$ for each $t$.

If a compact Lie group $G$ acts on $X$ symplectically with respect to
$\omega_{t}$ for each $t$, then the map $\phi_{t}$ is $G$-equivariant for each
$t$.

Consider $\mathcal{Q}$ together with two symplectic forms
$\Lambda^{*}\omega_{\mathcal{Q}^{\prime}}$ and $\omega_{\mathcal{Q}}$. Since
two forms are Kähler forms, each form
$\omega_{t}:=(1-t)\Lambda^{*}\omega_{\mathcal{Q}^{\prime}}+t\omega_{\mathcal{Q}}$
in the linear interpolation is also symplectic. Since the $S^{1}$-action is
Hamiltonian (and hence symplectic) with respect to both $\omega_{0}$ and
$\omega_{1}$, the $S^{1}$-action is symplectic with respect to $\omega_{t}$
for each $t\in[0,1]$. By Theorem 8, there exists an $S^{1}$-equivariant
isotopy
$\phi_{t}\colon\mathcal{Q}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{Q}$
such that $\phi_{t}^{*}(\omega_{t})=\omega_{0}$. Set

$\Phi:=\phi_{1}\circ\Lambda^{-1}\circ\sigma\colon(\mathbb{C}P^{1}\times\mathbb{C}P^{1},2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathcal{Q}^{\prime},\omega_{\mathcal{Q}^{\prime}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathcal{Q},\Lambda^{*}\omega_{\mathcal{Q}^{\prime}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathcal{Q},\omega_{\mathcal{Q}}).$

######  Lemma 9.

The map $\Phi$ is an $S^{1}$-equivariant symplectomorphism

###### Proof.

By Proposition 5, the Segre map $\sigma$ is an $S^{1}$-equivariant
symplectomorphism. By Lemma 6, so is the second map $\Lambda^{-1}$. It follows
from Theorem 8 that $\phi_{1}$ is an $S^{1}$-equivariant symplectomorphism.
Therefore, Lemma 9 is established. ∎

Note that the quadric $\mathcal{Q}$ is the zero locus of the symmetric
bilinear form $\mathcal{B}$ corresponding to the identity matrix $I_{4}$ on
$\mathbb{C}P^{3}$. On the other hand, the vanishing condition imposes the
isotropy condition on the bilinear form $\mathcal{B}$ on
$\mathbb{C}^{4}=\mathbb{C}\langle x_{1},x_{2},x_{3},x_{4}\rangle$. Namely,
$\mathcal{Q}$ is isomorphic to the orthogonal Grassmannian
$\mathrm{OG}(1,\mathbb{C}^{4})$, which consists of one dimensional subspaces
$V$ of $\mathbb{C}^{4}$ satisfying $\mathcal{B}(v_{1},v_{2})=0$ for all
$v_{1},v_{2}\in V$. The orthogonal Grassmannian
$\mathrm{OG}(1,\mathbb{C}^{4})$ is adorned with the complex structure from the
identification with $\mathrm{SO}(4;\mathbb{C})/{P}$ where ${P}$ is a parabolic
subgroup of $\mathrm{SO}(4;\mathbb{C})$. We denote this isomorphism by
$\varrho\colon\mathrm{OG}(1,\mathbb{C}^{4})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{Q}$
as complex manifolds.

Regarding
$G=\mathrm{SO}(2)=\mathrm{SO}(2;\mathbb{R})\simeq\mathrm{diag}\left(\mathrm{SO}(2),I_{2}\right)$
as a subgroup of $\mathrm{SO}(4)=\mathrm{SO}(4;\mathbb{R})$, recall that the
quadric $\mathcal{Q}$ is acted by $G$ in (15). The linear $G$-action on
$\mathbb{C}^{4}$ induces the $G$-action on $\mathrm{OG}(1,\mathbb{C}^{4})$.
Note that the map
$\varrho\colon\mathrm{OG}(1,\mathbb{C}^{4})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{Q}$
is $G$-equivariant.

The group $\mathrm{SO}(4)$ acts on the moduli space of one dimensional
isotropic subspaces in $\mathbb{C}^{4}$ linearly and transitively. Also, the
adjoint $\mathrm{SO}(4)$-action on $\mathcal{O}$ is transitive. We then have

  * •

${\mathrm{SO}(4)}/{\mathrm{S}(\mathrm{O(2)}\times\mathrm{O(2)})}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{OG}(1,\mathbb{C}^{4})$

  * •

$\mathcal{O}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathrm{SO}(4)}/{\mathrm{S}(\mathrm{O(2)}\times\mathrm{O(2)})}$,

by choosing one flag and one element of $\mathcal{O}$ respectively. We then
have a diffeomorphism
$\Upsilon\colon\mathcal{O}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{OG}(1,\mathbb{C}^{4})$.
The orbit $\mathcal{O}$ has the adjoint action of the subgroup $G$ of
$\mathrm{SO}(4)$. Regarding
$G=\mathrm{SO}(2)\simeq\mathrm{diag}\left(\mathrm{SO}(2),I_{2}\right)$ as a
subgroup of $\mathrm{SO}(4)$, recall that the quadric $\mathcal{Q}$ is acted
by $G$ in (15). The linear $G$-action on $\mathbb{C}^{4}$ induces the
$G$-action on $\mathrm{OG}(1,\mathbb{C}^{4})$. The map $\Upsilon$ is
$G$-equivariant. Recall that $\omega_{\mathrm{KKS}}$ is a Kähler form with
respect to the complex structure from $\mathrm{SO}(4;\mathbb{C})/{P}$.

The orbit $\mathcal{O}$ carries two Kähler forms
$(\varrho\circ\Upsilon)^{*}\omega_{\mathcal{Q}}$ and $\omega_{\mathrm{KKS}}$.
Each form
$\omega^{\prime}_{t}:=(1-t)(\varrho\circ\Upsilon)^{*}\omega_{\mathcal{Q}}+t\omega_{\mathrm{KKS}}$
in the linear interpolation is also symplectic. Since the $G$-action is
Hamiltonian with respect to both $\omega^{\prime}_{0}$ and
$\omega^{\prime}_{1}$, there exists an $G$-equivariant isotopy
$\phi^{\prime}_{t}\colon\mathcal{O}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{O}$
such that $(\phi^{\prime}_{t})^{*}(\omega^{\prime}_{t})=\omega^{\prime}_{0}$
again by Theorem 8. Set

$\Phi^{\prime}:=\phi^{\prime}_{1}\circ\Upsilon^{-1}\circ\varrho^{-1}\colon(\mathcal{Q},\omega_{\mathcal{Q}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathrm{OG}(1,\mathbb{C}^{4}),\varrho^{*}\omega_{\mathcal{Q}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathcal{O},(\varrho\circ\Upsilon)^{*}\omega_{\mathcal{Q}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathcal{O},\omega_{\mathrm{KKS}}).$

In summary, we have derived the following lemma.

######  Lemma 10.

Under the identification (13), the map $\Phi^{\prime}$ is an
$S^{1}$-equivariant symplectomorphism.

We are now ready to transport the Gelfand–Zeitlin torus $T_{\mathrm{GZ}}$ into
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$. By Lemma 9 and Lemma 10, we obtain an
$S^{1}$-invariant Lagrangian torus
$(\Phi^{\prime}\circ\Phi)^{-1}(T_{\mathrm{GZ}})$ in the product space
$(\mathbb{C}P^{1}\times\mathbb{C}P^{1},2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}}))$.
Let

(17)
$T^{\prime}_{\mathrm{GZ}}:=(\Phi^{\prime}\circ\Phi)^{-1}(T_{\mathrm{GZ}}).$

The next proposition compares $T^{\prime}_{\mathrm{GZ}}$ with
$T_{\mathrm{CS}}$ in the same space
$(\mathbb{C}P^{1}\times\mathbb{C}P^{1},2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}}))$.

######  Proposition 11.

Two monotone Lagrangian tori $T^{\prime}_{\mathrm{GZ}}$ and $T_{\mathrm{CS}}$
are Hamiltonian isotopic.

Let us do preliminary work for verifying Proposition 11.

Let $\mathcal{U}$ be an affine chart of $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$
defined as

(18)
$\mathcal{U}:=(\mathbb{C}P^{1}\times\mathbb{C}P^{1})\backslash\\{v_{0}w_{0}=0\\}=\mathbb{C}P^{1}\backslash(\\{v_{0}=0\\})\times\mathbb{C}P^{1}\backslash(\\{w_{0}=0\\})\simeq\mathbb{C}\times\mathbb{C}.$

Recall that $\mathcal{U}$ is exactly the image of $\rho$ in (3). The
Chekanov–Schlenk monotone torus $T_{\mathrm{CS}}$ is contained in the level
set $\mu^{-1}_{S^{1}}(0)$ where the moment map $\mu_{S^{1}}$ is in (10). Note
that the $S^{1}$-action generated by $\mu_{S^{1}}$ induces that on
$\mathcal{U}\cap\mu^{-1}_{S^{1}}(0)$.

A fundamental domain of the $S^{1}$-action on
$\mathcal{U}\cap\mu^{-1}_{S^{1}}(0)$ can be chosen as the image of

(19)
$\mathcal{F}(\sqrt{2}):=\\{\zeta\in\mathbb{C}\mid\mathrm{Im}(\zeta)>0,\,0<|\zeta|<\sqrt{2}\\}\cup\\{\zeta\in\mathbb{C}\mid\mathrm{Im}(\zeta)=0,\,0\leq\zeta<\sqrt{2}\\}$

under the composition $\rho\circ\Delta_{\mathbb{H}}$ of (2) and (3) as
depicted in Figure 1. Since each pair of the antipodal points $\zeta$ and
$-\zeta$ in $\mathbb{D}(\sqrt{2}):=\\{\zeta\in\mathbb{C}\mid
0\leq|\zeta|<\sqrt{2}\\}$ lies on the same $S^{1}$-orbit,
$\mathcal{F}(\sqrt{2})$ can be regarded as $\mathbb{D}(\sqrt{2})$ via the map

$\Theta\colon\mathcal{F}(\sqrt{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{D}(\sqrt{2}),\quad\left(re^{\sqrt{-1}\theta}\mapsto
re^{2\sqrt{-1}\theta}\right).$

Figure 1.  The fundamental domain $\mathcal{F}(\sqrt{2})$ in (19)

######  Lemma 12.

Suppose that a symplectic manifold $(X,\omega)$ admits a Hamiltonian free
$S^{1}$-action. Choose a moment map $\mu_{S^{1}}$ of the $S^{1}$-action. Let
$L$ be an $S^{1}$-invariant connected Lagrangian submanifold. Then the
Lagrangian submanifold $L$ is contained in the set $\mu^{-1}_{S^{1}}(r)$ for
some $r\in\mathbb{R}\simeq\mathfrak{t}^{*}$.

###### Proof.

For each point $x\in X$, we know that

$\mathrm{ker}\left(d\mu_{S^{1},x}\right)=\left\\{v\in
T_{x}X\mid\omega_{x}(v,w)=0\mbox{ for every $w\in
T_{x}\mathscr{O}_{x}$}\right\\}$

where $\mathscr{O}_{x}$ is the orbit through $x$, see [MS17, Lemma 5.2.5].
Since $S^{1}$ acts freely on the level set $\mu_{S^{1}}^{-1}(r)$, each point
$x\in\mu_{S^{1}}^{-1}(r)$ is regular and hence $r\in\mathbb{R}$ is a regular
value of $\mu_{S^{1}}$. The dimension counting yields that
$\mathrm{ker}\left(d\mu_{S^{1},x}\right)=T_{x}\left(\mu^{-1}_{S^{1}}(r)\right)$.
Since $L$ is Lagrangian, we have

(20) $T_{x}L\subset\left\\{v\in T_{x}X\mid\omega_{x}(v,w)=0\mbox{ for every
$w\in T_{x}\mathscr{O}_{x}$}\right\\}=T_{x}\left(\mu^{-1}_{S^{1}}(r)\right)$

where $r:=\mu_{S^{1}}(x)$.

We claim that the image of a connected submanifold $L$ under $\mu_{S^{1}}$ is
a singleton set. Suppose on the contrary that the image of $L$ contains an
interval having positive measure. By Sard’s Theorem, there exist $x\in L$ and
$v\in T_{x}L$ such that $d\mu_{S^{1},x}(v)$ is non-zero. Then a non-zero
vector $w\in T_{x}\mathscr{O}_{x}$ satisfies $\omega_{x}(v,w)\neq 0$, which
contradicts to (20). ∎

######  Corollary 13.

The monotone $S^{1}$-invariant Lagrangian torus $T^{\prime}_{\mathrm{GZ}}$ in
(17) is contained in the zero level set $\mu^{-1}_{S^{1}}(0)$ where
$\mu_{S^{1}}$ is (10).

###### Proof.

Observe that $S^{1}$ acts on the complement of the four points

$([1:0],[1:0]),([0:1],[1:0]),([1:0],[0:1]),\mbox{ and }([0:1],[0:1])$

in $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ freely. Applying Lemma 12 to the
complement, the torus $T^{\prime}_{\mathrm{GZ}}$ is contained in
$\mu^{-1}_{S^{1}}(r)$ for some $r\in\mathbb{R}$.

We claim that $T^{\prime}_{\mathrm{GZ}}$ is contained in the level set of $0$,
that is, $r=0$. By Lemma 3, $T_{\mathrm{GZ}}$ is monotone and so is
$T^{\prime}_{\mathrm{GZ}}$ in $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$. Take a
point in $T^{\prime}_{\mathrm{GZ}}$ and choose an $S^{1}$-invariant almost
complex structure $J$. Let $\gamma$ be the integral curve starting from the
chosen point of the gradient vector field generated by the Riemannian metric
obtained by $\omega$ and $J$. The curve $\gamma$ converges to a certain fixed
point. The $S^{1}$-orbit of $\gamma$ is a $J$-holomorphic disk such that its
symplectic area is equal to $\pi/2$ times its Maslov index. Such a
$J$-holomorphic disk is bounded by an $S^{1}$-orbit in $T_{\mathrm{GZ}}$ and
called a _gradient holomorphic disk_ , see [CK19, Section 2] for more details.
As the level of $T^{\prime}_{\mathrm{GZ}}$ varies, the symplectic area of this
gradient disk changes due to Archimedes, while the Maslov index of the disk
does not change. It implies that the monotone torus must be contained in the
level zero. ∎

Corollary 13 says that the torus $T^{\prime}_{\mathrm{GZ}}$ is completely
contained in the $S^{1}$-orbit of the image of the open disk
$\mathbb{D}(\sqrt{2})$ under the composition map
$\rho\circ\Delta_{\mathbb{D}}$ in (1) and (3). By intersecting the fundamental
domain, we obtain a curve $\Gamma_{\mathrm{GZ}}$ determined by

$\Gamma_{\mathrm{GZ}}:=\Theta\left((\rho\circ\Delta_{\mathbb{D}})^{-1}(T^{\prime}_{GZ})\cap\mathcal{F}(\sqrt{2})\right).$

Since $T^{\prime}_{GZ}$ is a smooth torus, $\Gamma_{\mathrm{GZ}}$ is a simple
closed curve not passing through the origin. Then there are two
possibilities$\colon$

  1. (1)

$\Gamma_{\mathrm{GZ}}$ does not bound the closed region,

  2. (2)

$\Gamma_{\mathrm{GZ}}$ bounds the closed region.

We claim that $(2)$ holds.

######  Proposition 14.

The curve $\Gamma_{\mathrm{GZ}}\subset\mathbb{D}(\sqrt{2})\backslash\\{0\\}$
bounds the closed region.

To see Proposition 14, we collect some facts. Let $(X,\omega)$ be a closed
monotone symplectic manifold and $L$ a monotone Lagrangian submanifold. For
each homotopy class $\beta\in\pi_{2}(X,L)$ of Maslov index two, a generic
choice of $\omega$-compatible almost complex structure makes the moduli space
of stable maps in the class $\beta$ from $(\mathbb{D},\partial\mathbb{D})$ to
$(X,L)$ transversal. The virtual dimension of this moduli space is exactly
dimension of $L$. This monotonicity condition ensures that the moduli space is
closed. We then count the number of stable map passing through a generic point
of $L$ at the marking point on the boundary of the disk $\mathbb{D}$. Such a
number is called a _counting invariant_ of $L$ in $X$ and is denoted by
$n_{\beta}$. The sum of counting invariants is meant to be
$\sum_{\beta}n_{\beta}$ where the summation is taken over all homotopy class
$\beta\in\pi_{2}(X,L)$. By the dimension reason, almost all $n_{\beta}=0$ and
the sum is finite because of Gromov’s compactness theorem.

######  Lemma 15 ([EP97]).

Suppose that $\phi$ is a symplectomorphism from $(X,\omega)$ to
$(X^{\prime},\omega^{\prime})$. If $L$ and $L^{\prime}$ are monotone
Lagrangian submanifolds related by $\phi$, then the sum of counting invariants
bounded by $L$ in $X$ is equal to that of counting invariants by $L^{\prime}$
in $X^{\prime}$.

The sum of counting invariants of $T_{\mathrm{GZ}}$ in $\mathcal{O}$ can be
computed from the disk potential of $T_{\mathrm{GZ}}$ in [Kim21, Theorem A].

######  Lemma 16 ([Kim21]).

The sum of counting invariants of $T_{\mathrm{GZ}}$ in $\mathcal{O}$ is five.

###### Proof of Proposition 14.

Suppose that the curve $\Gamma_{\mathrm{GZ}}$ does not bound the closed
region. Then the torus $T^{\prime}_{\mathrm{GZ}}$ is Hamiltonian isotopic to
the product of equators of $S^{2}\times S^{2}$. If so, the sum of counting
invariants of $T^{\prime}_{\mathrm{GZ}}$ in $S^{2}\times S^{2}$ is four.
According to Lemma 15 and Lemma 16, the sum of counting invariants of
$T^{\prime}_{\mathrm{GZ}}$ has to be five. We have derived a contradiction. ∎

Recall that the chosen $\Gamma$ for (4) is contained in
$\mathbb{H}(\sqrt{2})$. Let

$\Gamma_{\mathrm{CS}}:=\Theta(\Gamma).$

Let us compare two simple closed curves $\Gamma_{\mathrm{CS}}$ and
$\Gamma_{\mathrm{GZ}}$. A priori, the simple closed curve
$\Gamma_{\mathrm{GZ}}$ might be complicated so that it may not be contained in
a branch, while $\Gamma_{\mathrm{CS}}$ is contained in the branch
$\mathbb{D}(\sqrt{2})\backslash\\{z\mid\mathrm{Im}(z)=0,z\geq 0\\}$.

We shall deform one to the other via a Hamiltonian isotopy. For this purpose,
we recall one lemma concerning Hamiltonian isotopy class of loops in a two
dimensional exact symplectic manifold. (Here, the dimension condition is
necessary to apply the Moser argument).

######  Lemma 17 (Lemma 2.3 in [LM14]).

Let $(\Sigma,\omega:=d\alpha)$ be an exact symplectic manifold of real
dimension two. Let $\Gamma_{1}$ and $\Gamma_{2}$ be simple closed curves in
$\Sigma$. Suppose that

  1. (1)

$\Gamma_{0}$ and $\Gamma_{1}$ are isotopic and

  2. (2)

$\int_{\partial\Gamma_{0}}\alpha=\int_{\partial\Gamma_{1}}\alpha$.

Then $\Gamma_{0}$ and $\Gamma_{1}$ are related by via a compactly supported
Hamiltonian isotopy.

In our case, two simple closed curves $\Gamma_{\mathrm{CS}}$ and
$\Gamma_{\mathrm{GZ}}$ in $\mathbb{D}(\sqrt{2})\backslash\\{0\\}$ bound
regions having the same area since the bounded regions can be lifted to a disk
bounded by $T_{\mathrm{CS}}$ and by $T^{\prime}_{\mathrm{GZ}}$ respectively.
The monotonicity of $T_{\mathrm{CS}}$ and $T^{\prime}_{\mathrm{GZ}}$ ensures
the bounded regions are same. By Stokes’ theorem, the second condition holds.
For the first condition, we can apply the following well-known fact from
differential topology.

######  Lemma 18 (Theorem 3.1 in Chapter 8 of [Hir94]).

Let $X$ be a connected $n$-manifold and
$f,g\colon\mathbb{D}^{k}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
embedding of the $k$-disk, $0\leq k\leq n$. If $k=n$ and $X$ is orientable,
assume that $f$ and $g$ both preserve orientation. Then $f$ and $g$ are
isotopic.

If $f(\mathbb{D}^{k})\cup g(\mathbb{D}^{k})$ is contained in $X-\partial X$,
an isotopy between them can be realized by a smooth isotopy of $X$ having
compact support.

We now discuss the relation between $T_{\mathrm{CS}}$ and by
$T^{\prime}_{\mathrm{GZ}}$. A choice of isotopy from $\Gamma_{\mathrm{CS}}$ to
$\Gamma_{\mathrm{GZ}}$ in $\mathbb{D}(\sqrt{2})\backslash\\{0\\}$ gives rise
to a Lagrangian isotopy between $T_{\mathrm{CS}}$ and by
$T^{\prime}_{\mathrm{GZ}}$. Proposition 11 further claim that the Lagrangian
isotopy arising from the Hamiltonian isotopy in
$\mathbb{D}(\sqrt{2})\backslash\\{0\\}$ can be extended to an ambient
Hamiltonian isotopy of $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$.

###### Proof of Proposition 11.

For simplicity of notation, let us set

$\left(X:=\mathbb{C}P^{1}\times\mathbb{C}P^{1},\omega:=2(\omega_{\mathrm{FS}}\oplus\omega_{\mathrm{FS}})\right),$

while presenting the proof. Suppose that $\ell\colon
T^{2}\times[0,1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{D}(\sqrt{2})\times\mathbb{D}(\sqrt{2})$
is a Lagrangian isotopy arising from the Hamiltonian isotopy from Lemma 17.
Since $\ell$ is a Lagrangian isotopy, the pull-backed form is of the following
form

$(\Delta_{\mathbb{D}}\circ\ell)^{*}\omega=\ell^{*}\omega_{\mathrm{std}}=\alpha_{s}\wedge
ds$

where $\\{\alpha_{s}\\}$ is a family of one forms.

We shall show that the Lagrangian isotopy $\Delta_{\mathbb{D}}\circ\ell$ is
exact. Then this Lagrangian isotopy can be extended to an ambient Hamiltonian
isotopy of $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ as desired, see [Pol01,
Section 6], [Oh15, Section 3.6] for instance.

It remain to show that for each $s$, $\alpha_{s}$ is exact. Note that
$\omega_{\mathrm{std}}=d\eta$. The pull-back of the primitive $\eta$ is

$\ell^{*}\eta=f(x,s)ds+\eta_{s}^{\prime}$

where $\\{\eta^{\prime}_{s}\\}$ is a family of one forms on
$T^{2}\times\\{s\\}$. To show that $\alpha_{s}$ is exact, we claim that the
integration of $\eta$ over any loop remains constant through the isotopy. Let
$\Gamma_{s}$ be an isotopy from $\Gamma_{\mathrm{CS}}$ to
$\Gamma_{\mathrm{GZ}}$ from Lemma 17. We then have an isotopy of loops

$\gamma_{s}:=(\rho\circ\Delta_{\mathbb{D}})\left(\Theta^{-1}(\Gamma_{s})\right).$

Then $\int_{\gamma_{s}}\eta$ is constant through the isotopy because of the
construction of the isotopy. We denote by $\gamma_{0}^{\prime}$ a Lefschetz
thimble bounded by $T_{\mathrm{CS}}$ in
$\mathbb{D}(\sqrt{2})\times\mathbb{D}(\sqrt{2})$. Let $\gamma_{s}^{\prime}$ be
the isotoped circle. In other words,
$\gamma_{s}^{\prime}:=\left(\ell_{s}\circ\ell^{-1}_{0}\right)\left(\gamma_{0}^{\prime}\right)$
where $\ell_{s}:=\ell(\cdot,s)\colon
T^{2}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{D}(\sqrt{2})\times\mathbb{D}(\sqrt{2})$.
Then $\int_{\gamma^{\prime}_{s}}\eta=0$ for all $s$. Hence, the claim is
verified.

The claim yields that the integration of $\eta^{\prime}_{s}$ over each loop in
$T^{2}$ is independent to $s$. Then $d_{x}f=\alpha_{s}$ as desired. ∎

###### Proof of Theorem 4.

It follows from Lemma 9, Lemma 10, and Proposition 11. ∎

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# Turán-type reverse Markov inequalities for polynomials with restricted zeros

Tamás Erdélyi  Department of Mathematics, Texas A&M University, College
Station, Texas 77843, College Station, Texas 77843  terdelyimath.tamu.edu

(September 21, 2019 )

Let ${\Cal{P}}_{n}^{c}$ denote the set of all algebraic polynomials of degree
at most $n$ with complex coefficients. Let

$D^{+}:=\\{z\in{\mathbb{C}}:|z|\leq 1,\enskip\text{\rm Im}(z)\geq 0\\}\,.$

For integers $0\leq k\leq n$ let ${\Cal{F}}_{n,k}^{c}$ be the set of all
polynomials $P\in{\Cal{P}}_{n}^{c}$ having at least $n-k$ zeros in $D^{+}$.
Let

$\|f\|_{A}:=\sup_{z\in A}{|f(z)|}$

for complex-valued functions defined on $A\subset{\mathbb{C}}$. We prove that
there are absolute constants $c_{1}>0$ and $c_{2}>0$ such that

$c_{1}\left(\frac{n}{k+1}\right)^{1/2}\leq\inf_{P}{\frac{\|P^{\prime}\|_{[-1,1]}}{\|P\|_{[-1,1]}}}\leq
c_{2}\left(\frac{n}{k+1}\right)^{1/2}$

for all integers $0\leq k\leq n$, where the infimum is taken for all
$0\not\equiv P\in{\Cal{F}}_{n,k}^{c}$ having at least one zero in $[-1,1]$.
This is an essentially sharp reverse Markov-type inequality for the classes
${\Cal{F}}_{n,k}^{c}$ extending earlier results of Turán and Komarov from the
case $k=0$ to the cases $0\leq k\leq n$.

Turán type reverse Markov inequalities, polynomials with restricted zeros

††support: 2010 Mathematics Subject Classifications. 41A17

## 1\. Introduction and Notation

Let ${\Cal{P}}_{n}$ denote the set of all algebraic polynomials of degree at
most $n$ with real coefficients Let ${\Cal{P}}_{n}^{c}$ denote the set of all
algebraic polynomials of degree at most $n$ with complex coefficients. Let

$\|f\|_{A}:=\sup_{z\in A}{|f(z)|}$

for complex-valued functions defined on $A\subset{\mathbb{C}}$. In 1939 Turán
[T-39] proved that

$\|P^{\prime}\|_{[-1,1]}\geq\frac{\sqrt{n}}{6}\,\|P\|_{[-1,1]}$ $None$

for all $P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all their zeros in the
interval $[-1,1]$. The examples $P(x)=(x^{2}-1)^{m}$ and
$P(x)=(x^{2}-1)^{m}(x+1)$ show that Turán’s reverse Markov-type inequality
(1.1) is essentially sharp, even though the multiplicative constant $1/6$ in
(1.1) is not the best possible. Note that the best possible multiplicative
constant $c=c_{n}$ in (1.1) had been found by Erőd [E-39]. Another simple
observation of Turán [T-39] is the inequality

$\|P^{\prime}\|_{D}\geq\frac{n}{2}\,\|P\|_{D}$ $None$

for all $P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all their zeros in the
closed unit disk $D\subset{\mathbb{C}}$. Malik [M-69], Govil [G-73], and Govil
and Mohapatra [GM-99, Section 4] established extensions of (1.2) proving that

$\|P^{\prime}\|_{D}\geq\frac{n}{1+R}\,\|P\|_{D}$

for all $P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all their zeros in the
disk $D(0,R)\subset{\mathbb{C}}$ of radius $R\leq 1$ centered at $0$, and

$\|P^{\prime}\|_{D}\geq\frac{n}{1+R^{n}}\,\|P\|_{D}$

for all $P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all its zeros in the disk
$D(0,R)\subset{\mathbb{C}}$ of radius $R\leq 1$ centered at $0$.

Let $\varepsilon\in[0,1]$ and let $D_{\varepsilon}$ be the ellipse of the
complex plane with large axis $[-1,1]$ and small axis
$[-i\varepsilon,i\varepsilon]$. Let ${\Cal{P}}_{n}^{c}(D_{\varepsilon})$
denote the collection of all $P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all
their zeros in $D_{\varepsilon}$. Extending Turán’s reverse Markov-type
inequality (1.1), Erőd [E-39, III. tétel] proved that

$c_{1}(n\varepsilon+\sqrt{n})\leq\inf_{P}{\frac{\|P^{\prime}\|_{D_{\varepsilon}}}{\|P\|_{D_{\varepsilon}}}}\leq
c_{2}(n\varepsilon+\sqrt{n})\,,$

where the infimum is taken for all $P\in{\Cal{P}}_{n}^{c}(D_{\varepsilon})$.
Levenberg and Poletcky [LP-02] rediscovered this beautiful result. In [LP-02]
they also proved that

$\frac{\sqrt{n}}{20\,\text{\rm
diam}K}\leq\inf_{P}{\frac{\|P^{\prime}\|_{K}}{\|P\|_{K}}}$

for all compact convex set $K\subset{\mathbb{C}}$, where the infimum is taken
for all $P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all their zeros in $K$.

Let $\varepsilon\in[0,1]$ and let $S_{\varepsilon}$ be the diamond of the
complex plane with diagonals $[-1,1]$ and $[-i\varepsilon,i\varepsilon]$. Let
${\Cal{P}}_{n}^{c}(S_{\varepsilon})$ denote the collection of all
$P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all their zeros in
$S_{\varepsilon}$. It has been proved in [E-07] that there are absolute
constants $c_{1}>0$ and $c_{2}>0$ such that

$c_{1}(n\varepsilon+\sqrt{n})\leq\inf_{P}{\frac{\|P^{\prime}\|_{S_{\varepsilon}}}{\|P\|_{S_{\varepsilon}}}}\leq
c_{2}(n\varepsilon+\sqrt{n})\,,$

where the infimum is taken for all $P\in{\Cal{P}}_{n}^{c}(S_{\varepsilon})$
with the property

$|P(z)|=|P(-z)|\,,\qquad z\in{\mathbb{C}}\,,$

or where the infimum is taken for all $P\in{\Cal{P}}_{n}^{c}(S_{\varepsilon})$
with real coefficients. It is an interesting question whether or not the lower
bound in the above inequality holds for all
$P\in{\Cal{P}}_{n}^{c}(S_{\varepsilon})$. Another result in [E-07] shows that
this is the case at least when $\varepsilon=1$, that is, there are absolute
constants $c_{1}>0$ and $c_{2}>0$ such that

$c_{1}n\leq\inf_{P}{\frac{\|P^{\prime}\|_{S_{1}}}{\|P\|_{S_{1}}}}\leq
c_{2}n\,,$

where the infimum is taken for all (complex) $P\in{\Cal{P}}_{n}^{c}(S_{1})$.
Motivated by the above results Révész [R-06b] established the right order
Turán-type reverse Markov inequalities on convex domains of the complex plane.
His main theorem contains the above mentioned results in [E-07] as special
cases. It states that

$\frac{\|P^{\prime}\|_{K}}{\|P\|_{K}}\geq c(K)n\qquad\text{\rm with}\qquad
c(K)=0.0003\frac{w(K)}{d(K)^{2}}\,,$

for all $P\in{\Cal{P}}_{n}^{c}$ of degree $n$ having all their zeros in a
bounded convex set $K\subset{\mathbb{C}}$, where $d(K)$ is the diameter of $K$
and

$w(K):=\min_{\gamma\in[-\pi,\pi]}{\left(\max_{z\in K}{\text{\rm
Re}(ze^{-i\gamma})}-\min_{z\in K}{\text{\rm Re}(ze^{-i\gamma})}\right)}$

is the minimal width of $K$. Révész’ proof is also elementary, but rather
subtle. Further reverse Markov-type inequalities may be found in the recent
papers [R-06a] and [R-13] by Révész and [GR-17a] and [GR-17b] by Glazyrina and
Révész.

G.G. Lorentz, M. von Golitschek, and Y. Makovoz devotes Chapter 3 of their
book [LG-96] to incomplete polynomials. E.B. Saff and R.S. Varga were among
the researchers having contributed significantly to this topic. See [BCS-78],
[SV-79], and [SV-81], for instance. Reverse Markov- and Bernstein type
inequalities were first studied by P. Turán [T-39] and J. Erőd [E-39] in 1939
(see also [E-06]). The research on Turán and Erőd type reverses of Markov- and
Bernstein-type inequalities got a new impulse suddenly in 2006 in large part
by the work of Sz. Révész [R-06b], and several results have been published on
such inequalities in recent years, see [E-07], [E-09], [EH-15], [GR-17a],
[GR-17b], [LP-02], [k-04], [MM-94], [R-06a], [R-13], [XZ-99], and [Z-95], for
example.

Let ${\Cal{P}}_{n,k}$ be the set of all algebraic polynomials, with real
coefficients, of degree at most $n+k$ having at least $n+1$ zeros at $0$. That
is, every $P\in{\Cal{P}}_{n,k}$ is of the form

$P(x)=x^{n+1}R(x)\,,\qquad R\in{\Cal{P}}_{k-1}\,.$

Let

$V_{a}^{b}(f):=\int_{a}^{b}{|f^{\prime}(x)|\,dx}$

denote the total variation of a continuously differentiable function $f$ on an
interval $[a,b]$. In [E-19] a question [EI-18] asked by A. Eskenazis and P.
Ivanisvili related to their paper [EI-19] is answered by proving that there
are absolute constants $c_{3}>0$ and $c_{4}>0$ such that

$c_{3}\frac{n}{k}\leq\min_{0\not\equiv
P\in{\Cal{P}}_{n,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{V_{0}^{1}(P)}}\leq\min_{0\not\equiv
P\in{\Cal{P}}_{n,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{|P(1)|}}\leq
c_{4}\left(\frac{n}{k}+1\right)$

for all integers $n\geq 1$ and $k\geq 1$. Here $c_{3}=1/12$ is a suitable
choice.

In [E-19] we also proved that there are absolute constants $c_{3}>0$ and
$c_{4}>0$ such that

$\begin{split}c_{3}\left(\frac{n}{k}\right)^{1/2}&\leq\min_{0\not\equiv
P\in{\Cal{P}}_{n,k}}{\frac{\|P^{\prime}(x)\sqrt{1-x^{2}}\|_{[0,1]}}{V_{0}^{1}(P)}}\cr&\leq\min_{0\not\equiv
P\in{\Cal{P}}_{n,k}}{\frac{\|P^{\prime}(x)\sqrt{1-x^{2}}\|_{[0,1]}}{|P(1)|}}\leq
c_{4}\left(\frac{n}{k}+1\right)^{1/2}\cr\end{split}$

for all integers $n\geq 1$ and $k\geq 1$. Here $c_{4}=1/8$ is a suitable
choice.

Let

$D^{+}:=\\{z\in{\mathbb{C}}:|z|\leq 1,\enskip\text{\rm Im}(z)\geq 0\\}\,.$

In [K-19] Komarov proved that

$\|P^{\prime}\|_{[-1,1]}\geq A\sqrt{n}\,\|P\|_{[-1,1]}\,,\qquad
A=\frac{2}{3\sqrt{210e}}=0.0279\ldots\,,$

for all polynomials of degree $n$ having all their zeros in the closed upper
half-disk $D^{+}$.

For integers $0\leq k\leq n$ let ${\Cal{F}}_{n,k}^{c}$ be the set of all
polynomials $P\in{\Cal{P}}_{n}^{c}$ having at least $n-k$ zeros in $D^{+}$. In
this paper we prove an essentially sharp reverse Markov-type inequality for
the classes ${\Cal{F}}_{n,k}^{c}$ extending the earlier results of Turán and
Komarov from the case $k=0$ to the cases $0\leq k\leq n$.

## 2\. New Results

Our main result in this paper is an essentially sharp reverse Markov-type
inequality for the classes ${\Cal{F}}_{n,k}^{c}$ extending the earlier results
of Turán and Komarov from the case $k=0$ to the cases $0\leq k\leq n$. The
upper bound of Theorem 2.1 below is quite a new result even in the case when
the infimum is taken for polynomials $P\in{\Cal{P}}_{n}^{c}$ having at least
$n-k$ zeros only in $[-1,1]$ rather than $D^{+}$.

###### Theorem 2.1

There are absolute constants $c_{1}>0$ and $c_{2}>0$ such that

$c_{1}\left(\frac{n}{k+1}\right)^{1/2}\leq\inf_{P}{\frac{\|P^{\prime}\|_{[-1,1]}}{\|P\|_{[-1,1]}}}\leq
c_{2}\left(\frac{n}{k+1}\right)^{1/2}$

for all integers $0\leq k\leq n$, where the infimum is taken for all
$0\not\equiv P\in{\Cal{F}}_{n,k}^{c}$ having at least one zero in $[-1,1]$.

Theorem 2.1 follows from the results below. In fact, Corollary 2.3 below
offers the explicit constant $c_{1}=1/808$ in a slightly modified form of the
lower bound in Theorem 2.1.

###### Theorem 2.2

Let $1\leq k\leq n/163000$. We have

$\|P^{\prime}\|_{[-1,1]}\geq\frac{1}{202}\left(\frac{n-k}{8k}\right)^{1/2}\|P\|_{[-1,1]}$

for all $P\in{\Cal{F}}_{n,k}^{c}$.

###### Corollary 2.3

Let $1\leq k\leq n$. We have

$\|P^{\prime}\|_{[-1,1]}\geq\max\left\\{\frac{1}{2},\frac{1}{808}\left(\frac{n-k}{k}\right)^{1/2}\right\\}\|P\|_{[-1,1]}$

for all $P\in{\Cal{F}}_{n,k}^{c}$ with at least one zero in $[-1,1]$.

###### Theorem 2.4

There is an absolute constant $c_{1}>0$ and there are polynomials $0\not\equiv
P=P_{n,k}\in{\Cal{F}}_{2n,2k}$ of the form

$P(x)=(x^{2}-1)^{n-k}R(x)\,,\qquad R\in{\Cal{P}}_{2k}\,,$

such that

$\frac{\|P^{\prime}\|_{[-1,1]}}{\|P\|_{[-1,1]}}\leq
c_{2}\left(\frac{n}{k}\right)^{1/2}$

for every $1\leq k\leq n/2$.

We remark that the upper bound of Theorem 2.1 does not remain valid if we
replace the closed upper half-disk $D^{+}$ with the closed unit disk $D$ in
the definition of ${\Cal{F}}_{n,k}^{c}$, not even in the case that $k=0$.
Given $\varepsilon>0$, let $m$ be the even integer for which
$1/\varepsilon<m\leq 1/\varepsilon+2$. We claim that for every $\varepsilon>0$
and for every integer $n\geq 1$ there is a $P_{n}\in{\Cal{P}}_{mn}^{c}$ of
degree $mn$ having all its zeros on the unit circle ${\partial D}$ such that

$\|P_{n}^{\prime}\|_{[-1,1]}\leq(1/\varepsilon+2)^{1-\varepsilon}(mn)^{\varepsilon}\|P_{n}\|_{[-1,1]}\,.$

To see this let $P_{n}\in{\Cal{P}}_{mn}^{c}$ be defined by
$P_{n}(z):=(z^{m}-1)^{n}$. Observe that $\|P_{n}\|_{[-1,1]}=1$ (as $m$ is
even), and the function

$|P_{n}^{\prime}(x)|=mn(1-x^{m})^{n-1}|x|^{m-1}$

achieves its maximum on $[-1,1]$ at the point $a\in(0,1)$, where i

$a^{m}=\frac{m-1}{mn-1}\leq\frac{1}{n}\,.$

Hence

$|P_{n}^{\prime}(a)|\leq mna^{m-1}\leq mnn^{1/m-1}\leq mn^{\varepsilon}\leq
m^{1-\varepsilon}(mn)^{\varepsilon}\leq(1/\varepsilon+2)^{1-\varepsilon}(mn)^{\varepsilon}\,.$

## 3\. Lemmas

Our proof of Theorem 2.2 is based on the following two non-trivial results.
Lemma 3.1 is proved in [GL-78] while the proof of Lemma 3.2 may be found in
Section 7.2 of [BE-95].

###### Lemma 3.1

If $Q\in{\Cal{F}}_{n,0}^{c}$ and

$E_{\delta}:=\left\\{x\in[-1,1]:\left|\frac{Q^{\prime}(x)}{Q(x)}\right|\leq
n\delta\right\\}\,,\qquad\delta>0\,,$

then

$m(E_{\delta})<A\delta\,,\qquad\delta>0\,,$

where $A:=70e$ is a suitable choice.

###### Lemma 3.2

If $R\in{\Cal{P}}_{k}^{c}$ and

$F_{\alpha}:=\left\\{x\in{\mathbb{R}}:\left|\frac{R^{\prime}(x)}{R(x)}\right|\geq\alpha\right\\}\,,\qquad\alpha>0\,,$

then

$m(F_{\alpha})\leq\frac{Bk}{\alpha}\,,\qquad\alpha>0\,,$

where $B:=8\sqrt{2}$ is a suitable choice.

Our third lemma is a simple consequence of the Mean Value Theorem.

###### Lemma 3.3

If a function $P$ is differentiable on $[-1,1]$,

$\|P^{\prime}\|_{[-1,1]}\leq M\|P\|_{[-1,1]}\,,$

and $x_{0}\in[-1,1]$ is such that

$|P(x_{0})|:=\|P\|_{[-1,1]}\,,$ $None$

then

$|P(y)|\geq\frac{1}{2}\,\|P\|_{[-1,1]}\,,\qquad
y\in[x_{0}-(2M)^{-1},x_{0}+(2M)^{-1}]\cap[-1,1]\,.$

To prove Theorem 2.4 we need the following two lemmas. Lemma 3.4 below is
stated and proved as Theorem 2.1 in [E-19] by using deep results from [B-85]
and [BE-94]. Recall that ${\Cal{P}}_{n-k,k}$, $0\leq k\leq n$, denotes the set
of all algebraic polynomials with real coefficients, of degree at most $n$
having at least $n-k+1$ zeros at $0$.

###### Lemma 3.4

There are absolute constants $c_{3}>0$ and $c_{4}>0$ such that

$c_{3}\frac{n-k}{k}\leq\min_{0\not\equiv
P\in{\Cal{P}}_{n-k,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{V_{0}^{1}(P)}}\leq\min_{0\not\equiv
P\in{\Cal{P}}_{n-k,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{|P(1)|}}\leq
c_{4}\frac{n}{k}$

for all integers $1\leq k\leq n-1$. Here $c_{3}=1/12$ is a suitable choice.

Lemma 3.5 below is stated and proved as Lemma 3.2 in [E-19].

###### Lemma 3.5

Let $1\leq k\leq n-1$ and let $S(x):=x^{n-k}R(x)$ with $R\in{\Cal{P}}_{k}$. We
have

$|S(x)|\leq x^{(n-k)/2}\|S\|_{[0,1]}\,,\qquad
x\in\left[0,1-\frac{10k}{n-k}\right]\,.$

Lemma 3.6 below follows simply from Lemma 3.5.

###### Lemma 3.6

Let $1\leq k\leq n/2$ and let $W(x):=(1-x)^{n-k}V(x)$ with
$V\in{\Cal{P}}_{k}$. We have

$|y^{1/2}W(y)|<\|u^{1/2}W(u)\|_{[0,1]}\,,\qquad
y\in\left[\frac{10(2k+1)}{n},1\right]\,.$

###### Demonstration Proof of Lemma 3.6

Replacing $n$ by $2n+1$ and $k$ by $2k+1$ in Lemma 3.6 we get obtain that

$|S(x)|\leq x^{n-k}\|S\|_{[0,1]}\,,\qquad
x\in\left[0,1-\frac{10(2k+1)}{n}\right]\subset\left[0,1-\frac{10(2k+1)}{2n-2k}\right]\,,$
$None$

whenever $1\leq k\leq n/2$ and $S(x):=x^{2n-2k}R(x)$ with
$R\in{\Cal{P}}_{2k+1}$. Replacing the variable $x$ by $1-x$ in (3.1) yields
that

$|S(x)|\leq(1-x)^{n-k}\|S\|_{[0,1]}\,,\qquad
x\in\left[\frac{10(2k+1)}{n},1\right]\,,$ $None$

whenever $1\leq k\leq n/2$ and $S(x):=(1-x)^{2n-2k}R(x)$ with
$R\in{\Cal{P}}_{2k+1}$. Now let $1\leq k\leq n/2$ and let
$W(x):=(1-x)^{n-k}V(x)$ with $V\in{\Cal{P}}_{k}$. Applying (3.2) to $S$
defined by

$S(x)=xW(x)^{2}=(1-x)^{2n-2k}(xV(x)^{2})\,,\qquad V\in{\Cal{P}}_{k}\,,$

we get the conclusion of the lemma. ∎

## 4\. Proof of the Theorems

###### Demonstration Proof of the Theorem 2.2

Let $0\not\equiv P\in{\Cal{F}}_{n,k}^{c}$, that is, $P=QR$, where
$P\in{\Cal{F}}_{n-k,0}^{c}$ and $R\in{\Cal{P}}_{k}^{c}$. We have

$\frac{P^{\prime}}{P}=\frac{Q^{\prime}}{Q}+\frac{R^{\prime}}{R}\,.$ $None$

By Lemma 3.1 we have

$m(E_{\delta})<A\delta,\qquad\delta>0,\qquad A:=70e\,,$ $None$

where

$E_{\delta}:=\left\\{x\in[-1,1]:\left|\frac{Q^{\prime}(x)}{Q(x)}\right|\leq(n-k)\delta\right\\}\,,\qquad\delta>0\,.$
$None$

By Lemma 3.2 we have

$m(F_{\delta})\leq B\delta,\qquad\delta>0,\quad B:=8\sqrt{2}\,,$ $None$

where

$F_{\delta}:=\left\\{x\in[-1,1]:\left|\frac{R^{\prime}(x)}{R(x)}\right|\geq\frac{k}{\delta}\right\\}\,,\qquad\delta>0\,.$
$None$

Now we choose $\delta>0$ such that

$\frac{k}{\delta}=\frac{1}{2}\,(n-k)\delta\,,$ $None$

that is,

$\delta:=\left(\frac{2k}{n-k}\right)^{1/2}\,.$ $None$

Then, combining (4.1)–(4.7), we can deduce that

$\begin{split}\left|\frac{P^{\prime}(x)}{P(x)}\right|\geq&\left|\frac{Q^{\prime}(x)}{Q(x)}\right|-\left|\frac{R^{\prime}(x)}{R(x)}\right|\cr\geq&(n-k)\delta-\frac{k}{\delta}\geq\frac{1}{2}\,(n-k)\delta\geq\frac{1}{2}\,(n-k)\left(\frac{2k}{n-k}\right)^{1/2}\cr\geq&\left(\frac{1}{2}\,(n-k)k\right)^{1/2}\,,\qquad
x\in[-1,1]\setminus H_{\delta}\,,\cr\end{split}$ $None$

where $H_{\delta}:=E_{\delta}\cup F_{\delta}$ with

$m(H_{\delta})\leq(A+B)\delta\,.$ $None$

Note that

$1\leq k\leq\frac{n}{163000}$

implies that

$(A+B)\delta=(A+B)\left(\frac{2k}{n-k}\right)^{1/2}\leq(70e+8\sqrt{2})\left(\frac{4k}{n}\right)^{1/2}<1\,.$
$None$

Assume now to the contrary of the theorem that

$\|P^{\prime}\|_{[-1,1]}\leq(2(A+B)\delta)^{-1}\|P\|_{[-1,1]}\,.$ $None$

Choose $x_{0}\in[-1,1]$ such that (3.1) holds. Observe that (4.9), (4.10), and
(4.11) imply that there is a

$y\in[x_{0}-(A+B)\delta,x_{0}+(A+B)\delta]\cap[-1,1]$ $None$

such that

$y\in[-1,1]\setminus H_{\delta}\,.$ $None$

Using Lemma 3.3 with $M:=(2(A+B)\delta)^{-1}$ and recalling (4.12) we obtain

$|P(y)|\geq\frac{1}{2}\,\|P\|_{[-1,1]}\,.$ $None$

Combining (4.13), (4.8) and (4.14) we obtain

$\begin{split}\|P^{\prime}\|_{[-1,1]}\geq&|P^{\prime}(y)|>\left(\frac{1}{2}\,(n-k)k\right)^{1/2}|P(y)|\cr\geq&\left(\frac{1}{2}\,(n-k)k\right)^{1/2}\frac{1}{2}\,\|P\|_{[-1,1]}\cr>&(2(A+B)\delta)^{-1}\|P\|_{[-1,1]}\,,\cr\end{split}$

which contradicts (4.11). Hence (4.11) is impossible and the proof is
finished. ∎

###### Demonstration Proof of Corollary 2.3

Let $1\leq k\leq n$. Suppose $0\not\equiv P\in{\Cal{F}}_{n,k}^{c}$ has at
least one zero in $[-1,1]$. Choose $a,b\in[-1,1]$ such that $P(a)=0$, and
$|P(b)|=\|P\|_{[-1,1]}$. By the Mean Value Theorem there is a $c\in(-1,1)$
between $a$ and $b$ such that

$\|P^{\prime}\|_{[-1,1]}\geq|P^{\prime}(c)|=\left|\frac{P(b)-P(a)}{b-a}\right|\geq\frac{1}{2}\|P\|_{[-1,1]}$
$None$

If $\displaystyle{1\leq k\leq\frac{n}{163000}}$, the result follows from
Theorem 2.2 and (4.15). If $\displaystyle{\frac{n}{163000}<k\leq n}$, then

$\frac{1}{808}\left(\frac{n-k}{k}\right)^{1/2}\leq\frac{1}{2}\,,$

and the result follows simply from (4.15). ∎

###### Demonstration Proof of Theorem 2.4

Let $1\leq k\leq n/2$. By (the upper bound of) Lemma 3.4 there is an absolute
constant $c_{4}>0$ and there are polynomials $0\not\equiv
Q=Q_{n,k}\in{\Cal{P}}_{n-k,k}$ such that

$\frac{\|Q^{\prime}\|_{[0,1]}}{\|Q\|_{[0,1]}}\leq c_{4}\frac{n}{k}\,.$ $None$

Let $0\not\equiv R(x)=R_{n,k}(x)=Q(1-x)$. Obviously $R$ is of the form

$R(x)=(1-x)^{n-k+1}U(x)\,,\qquad U\in{\Cal{P}}_{k-1}^{c}\,,$

$R^{\prime}$ is of the form

$R^{\prime}(x)=(1-x)^{n-k}V(x)\,,\qquad V\in{\Cal{P}}_{k-1}^{c}\,,$ $None$

Let $0\not\equiv P=P_{n,k}$ be defined by $P(x):=R(x^{2})$. Observe that $P$
is of the form

$P(x)=(1-x^{2})^{n-k+1}U(x)\,,\qquad U\in{\Cal{P}}_{2k-2}^{c}\,,$

hence $P\in{\Cal{F}}_{2n,2k}$. It is also clear that

$\|P\|_{[-1,1]}=\|R\|_{[0,1]}=\|Q\|_{[0,1]}$ $None$

and

$P^{\prime}(x)=2xR^{\prime}(x^{2})\,.$ $None$

Let $y:=x^{2}$. Using (4.19), (4.17), and Lemma 3.6, we obtain

$|P^{\prime}(x)|=|xR^{\prime}(x^{2})|=|2y^{1/2}R^{\prime}(y)|<\|2u^{1/2}R^{\prime}(u)\|_{[0,1]}=\|P^{\prime}\|_{[-1,1]}$

or every $y=x^{2}\in[10(2k+1)/n,1]$, and hence there is an

$a\in\left[0,\left(\frac{10(2k+1)}{n}\right)^{1/2}\right]$ $None$

such that

$|P^{\prime}(a)|=\|P^{\prime}\|_{[0,1]}\,.$ $None$

Combining (4.21), (4.19), (4.20), (4.16), and (4.18) we obtain

$\begin{split}\|P^{\prime}\|_{[-1,1]}&=\|P^{\prime}\|_{[0,1]}=|P^{\prime}(a)|=|2aR^{\prime}(a^{2})|\cr&\leq
2\left(\frac{10(2k+1)}{n}\right)^{1/2}\|R^{\prime}\|_{[0,1]}=2\left(\frac{10(2k+1)}{n}\right)^{1/2}\|Q^{\prime}\|_{[0,1]}\cr&\leq
2\left(\frac{10(2k+1)}{n}\right)^{1/2}c_{4}\frac{n}{k}\|Q\|_{[0,1]}\cr&\leq
c_{2}\left(\frac{n}{k}\right)^{1/2}\|Q\|_{[0,1]}=c_{2}\left(\frac{n}{k}\right)^{1/2}\|P\|_{[-1,1]}\,.\cr\end{split}$

∎

###### Demonstration Proof of Theorem 2.1

The case that $k=0$ is the result of Komarov [K-19] mentioned in the
Introduction, so we may assume that $1\leq k\leq n$. in which cases the lower
bound of the theorem follows immediately from Corollary 2.3. Let $f(n,k)$
defined by

$f(n,k):=\min_{0\not\equiv
P\in{\Cal{F}}_{n,k}^{c}}{\frac{\|P^{\prime}\|_{[-1,1]}}{\|P\|_{[-1,1]}}}\,.$

Observe that for a fixed positive integer $n$ the function $f(n,k)$ is
decreasing on the set of integers $0\leq k\leq n$, and for a fixed nonnegative
integer $k$ the function $f(n,k)$ is decreasing on the set of integers $n\geq
k$. So it is sufficient to show the upper bound of the theorem only for even
numbers $n=2\nu$ and $k=2\kappa$ satisfying $1\leq\kappa\leq\nu/2$ in which
cases the upper bound of the theorem follows from Theorem 2.4. ∎

## 5\. Acknowledgement

The author thanks Szilárd Révész for checking the details of the proofs in a
first draft of this paper and for helpful discussions.

## References

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