# On the Index of Bipolar Surfaces to Otsuki Tori

Еgor Morozov  The author is partially supported by the Theoretical Physics and
Mathematics Advancement Foundation ‘‘BASIS’’.

###### Abstract

We obtain estimates on the Morse index and nullity of bipolar surfaces to
Otsuki tori $\tilde{O}_{p/q}$ for $p/q$ sufficiently close to $\sqrt{2}/2$.

##  1 Introduction

A minimal surface is a critical point of the area functional and the index of
a minimal surface is the Morse index of the corresponding critical point.
Despite the naturalness and importance of this notion, the index is known only
for a few examples of minimal surfaces. Unfortunately, there is no universal
method for computing the index. In fact, for each case one should invent a new
approach.

In the present paper we consider minimal surfaces in $\mathbb{S}^{4}$ called
_bipolar surfaces to Otsuki tori_. Otsuki tori are minimal surfaces in
$\mathbb{S}^{3}$. They first appeared in [O70] but the original definition was
implicit and very complicated. Later the definition was simplified in works
[HsL71] and [Pen13]. The second work contains also the study of extremal
spectral properties of Otsuki tori.

Bipolar surfaces to Otsuki tori are obtained by applying the construction from
[Law70, §11] to Otsuki tori. Generally, this construction gives a minimal
surface in $\mathbb{S}^{5}$ but in the case of Otsuki tori this surface always
lies in some equatorial sphere $\mathbb{S}^{4}\subset\mathbb{S}^{5}$. The
study of extremal spectral properties of bipolar surfaces to Otsuki tori was
done in [Kar14]. Moreover, this work contains a convenient parametrization of
bipolar surfaces to Otsuki tori (see §2.2 and also Remark 1).

Our main result is Theorem 1 where we obtain estimates on the index and
nullity of a bipolar surface to Otsuki tori under the condition that this
surface is ‘‘close enough to the Clifford torus’’ (these words are clarified
in §2.2). Unfortunately, our method allows neither compute the index exactly
nor at least give an example of surface for which our estimates holds. However
we hope that the method provides some intuition and can be easily adopted for
numerical computations. For example, we are able to compute numerically the
index of $\tilde{O}_{2/3}$, which is one of the surfaces of the family. We
suppose that finding the exact value of index is a very difficult problem. See
Remark 3 for the discussion of our main result.

The plan of the proof of Theorem 1 is the following. First we compute the
Jacobi stability operator in appropriate local coordinates. Then, using the
separation of variables, we reduce the spectral problem for Jacobi operator to
the spectral problem for some periodic matrix Sturm-Liouville problem.
Finally, we attack this problem using methods of the classical work [Edw64].
Thus our approach is a direct generalization of one used in [Pen12, Pen13,
Kar14]. Note however that in our case the situation is complicated by the fact
that bipolar surfaces to Otsuki tori are of codimension 2 in $\mathbb{S}^{4}$.
This leads to multidimensional (matrix) Sturm-Liouville problem, which is much
more complicated than the one-dimensional (scalar) one appeared in the works
cited above. For an alternative approach to the codimension 2 case based on
complex geometry see [KW18, Med21, Med22].

##  2 Preliminaries

###  2.1 Index and nullity of a minimal submanifold

In this section we fix our notation and recall the definitions of index and
nullity of a minimal submanifold.

Let $E$ be a vector bundle equipped with a Riemannian metric. We denote by
$\Gamma(E)$ the set of all global sections of $E$. If $F$ is a subbundle of
$E$ with the induced metric and $s\in\Gamma(E)$, then $s^{F}\in\Gamma(F)$
denotes the orthogonal projection of $s$ on $F$. For a smooth manifold $M$ we
denote by $TM$ the tangent bundle of $M$.

Now let $\Phi\colon M\looparrowright\bar{M}$ be an immersion of a
$p$-dimensional manifold $M$ in an $n$-dimensional Riemannian manifold
$(\bar{M},\bar{g})$. Denote by $NM$ the normal bundle to $M$ in $\bar{M}$. Let
$\bar{R}(\cdot,\cdot)\cdot$ denote the Riemann curvature tensor on $\bar{M}$.
For each $V\in\Gamma(NM)$ define $\mathcal{R}(V)\in\Gamma(NM)$ as the trace of
the bilinear form

$X,Y\mapsto(\bar{R}(X,V)Y)^{NM}.$

Recall that for each $V\in\Gamma(NM)$ the Weignharten operator

$A^{V}\colon\Gamma(TM)\to\Gamma(TM),\quad
A^{V}(X)=-(\nabla_{X}^{T\bar{M}}V)^{TM}$

is defined. Let $SM\subset T^{*}M\otimes TM$ be the bundle of symmetric linear
operators on $TM$. Consider the operator
$A\colon\Gamma(NM)\to\Gamma(SM),V\mapsto A^{V}$. Let
$A^{*}\colon\Gamma(SM)\to\Gamma(NM)$ be the adjoint operator and
$\tilde{A}=A^{*}A$.

Finally, for each $V\in\Gamma(NM)$ consider the operator

$\nabla^{NM}V\colon\Gamma(TM)\to\Gamma(NM),\quad X\mapsto\nabla_{X}^{NM}V$

and the bilinear form

$X,Y\mapsto-(\nabla_{X}^{T^{*}M\otimes
NM}(\nabla^{NM}V))(Y)=\nabla_{\nabla_{X}^{TM}Y}^{NM}V-\nabla_{X}^{NM}(\nabla_{Y}^{NM}V)$

with values in $\Gamma(NM)$. Denote the trace of this bilinear form by
$\Delta^{NM}V$ (this is so-called _Laplace-Beltrami operator in normal
bundle_). If $e_{1},\ldots,e_{p}$ is a local orthonormal frame in
$\Gamma(TM)$, then

$\Delta^{NM}V=\sum_{i=1}^{p}\nabla_{\nabla_{e_{i}}^{TM}e_{i}}^{NM}V-\nabla_{e_{i}}^{NM}(\nabla_{e_{i}}^{NM}V).$
(1)

Now suppose that the immersion $\Phi$ is minimal and $g=\Phi^{*}\bar{g}$ is
the induced metric on $M$. Let $\Phi_{t}$ be a variation of $\Phi$ and
$V:=\left.\frac{\partial}{\partial
t}\right|_{t=0}\Phi_{t}\in\Gamma(\Phi^{*}T\bar{M})$. Then since $\Phi$ is
minimal, the first variation of the volume functional vanishes:
$\left.\frac{d}{dt}\right|_{t=0}\mathrm{Vol}(M,\Phi_{t}^{*}\bar{g})=0$. The
second variation of this functional has the form

$\left.\frac{d^{2}}{dt^{2}}\right|_{t=0}\mathrm{Vol}(M,\Phi_{t}^{*}\bar{g})=\int_{M}\langle\Delta^{NM}(V^{NM})+\mathcal{R}(V^{NM})-\tilde{A}(V^{NM}),V^{NM}\rangle_{g}\,dv_{g},$

where $dv_{g}$ stands for the measure associated with the metric $g$ (see
[S68, Theorems 3.2.1—3.2.2]).

######  Definition 1.

The operator $J\colon\Gamma(NM)\to\Gamma(NM)$, defined by the formula

$J(V)=\Delta^{NM}V+\mathcal{R}(V)-\tilde{A}(V),$

is called the _Jacobi (stability) operator_ on $M$.

It is well-known that Jacobi operator is an elliptic differential operator and
its spectrum has the form

$\lambda_{1}\leqslant\lambda_{2}\leqslant\dots\leqslant\lambda_{n}\leqslant\dots\nearrow+\infty,$

where each $\lambda_{i}$ is listed as many times as its multiplicity. In
particular, the number of indices $i$ with $\lambda_{i}<0$ is finite.

######  Definition 2.

The number of negative eigenvalues of $J$ counted with multiplicity is called
the _index_ of $M$ and is denoted by $\operatorname{ind}M$. The multiplicity
of the zero eigenvalue of $J$ is called the _nullity_ of $M$ and is denoted by
$\operatorname{null}M$.

###  2.2 Bipolar surfaces to Otsuki tori

In this section we shortly describe the construction of bipolar surfaces to
Otsuki tori. This construction is based on the Hsiang-Lawson Reduction Theorem
[HsL71, Theorem 2] and is similar to the construction of ordinary Otsuki tori
from [Pen13]. Here we give only the most necessary definitions. For more
detailed exposition of the construction and its motivation see [Kar14, §2.4].
See also Remark 1.

Let $\mathbb{S}^{2}$ be the standard unit sphere in $\mathbb{R}^{3}$ with
spherical coordinates $(\varphi,\theta)$ so that

$\mathbb{S}^{2}=\\{(\cos\varphi\sin\theta,\cos\varphi\cos\theta,\sin\varphi)\in\mathbb{R}^{3}\mid-\frac{\pi}{2}\leqslant\varphi\leqslant\frac{\pi}{2},0\leqslant\theta<2\pi\\},$

Denote $N=(0,0,1),S=(0,0,-1)\in\mathbb{S}^{2}$. Consider the metric

$\tilde{g}_{1}=E\,d\varphi^{2}+G\,d\theta^{2},\quad
E=4\pi^{2}\cos^{2}\varphi,\,G=4\pi^{2}\cos^{4}\varphi$

on $\mathbb{S}^{2}\setminus\\{N,S\\}$. Let

$\gamma\colon[0,t_{0}]\to\mathbb{S}^{2},\quad\gamma(t)=(\varphi(t),\theta(t))$

be a closed geodesic on $(\mathbb{S}^{2}\setminus\\{N,S\\},\tilde{g}_{1})$
parametrized by arc length (so that $t_{0}$ is the full length of the
geodesic). Here we consider $\theta(t)$ modulo $2\pi$.

######  Proposition 2.2.1.

The surface in $\mathbb{R}^{5}$ given parametrically by

$\begin{cases}x^{1}(\alpha,t)=\cos\alpha\cos\varphi(t)\sin\theta(t),\\\
x^{2}(\alpha,t)=\sin\alpha\cos\varphi(t)\sin\theta(t),\\\
x^{3}(\alpha,t)=\cos\alpha\cos\varphi(t)\cos\theta(t),\\\
x^{4}(\alpha,t)=\sin\alpha\cos\varphi(t)\cos\theta(t),\\\
x^{5}(\alpha,t)=\sin\varphi(t),\end{cases}\quad\alpha\in[0,2\pi),t\in[0,t_{0}),$
(2)

is a minimal torus in
$\mathbb{S}^{4}=\\{(x^{1},x^{2},x^{3},x^{4},x^{5})\in\mathbb{R}^{5}\mid\sum_{i=1}^{5}(x^{i})^{2}=1\\}$.

The surface given by (2) is a _bipolar surface to Otsuki tori_.

Now let us describe closed geodesics on
$(\mathbb{S}^{2}\setminus\\{N,S\\},\tilde{g}_{1})$. To this end, for each
$b\in(-\frac{\pi}{2},\frac{\pi}{2})$ consider the geodesic
$\gamma_{b}(t)=(\varphi(t),\theta(t))$ defined by the following initial
conditions:

$\varphi(0)=b,\quad\dot{\varphi}(0)=0,\quad\theta(0)=0,\quad\dot{\theta}(0)>0.$
(3)

We want to find out for which values of $b$ this geodesic closes. Note that
any closed geodesic on $(\mathbb{S}^{2}\setminus\\{N,S\\},\tilde{g}_{1})$ can
be parametrized in such a way that conditions (3) hold for some $b$. Indeed,
it suffices to take $b$ equal to the lowest value of the coordinate $\varphi$
on the geodesic. Thus in such a way we describe all closed geodesics.
Moreover, since the metric $\tilde{g}_{1}$ is invariant under the
transformation $\varphi\mapsto-\varphi$, we may assume that $b\leqslant 0$.

The equation of geodesics for $\ddot{\theta}$ reads

$\ddot{\theta}+\frac{1}{G}\frac{\partial
G}{\partial\varphi}\dot{\varphi}\dot{\theta}=0\quad\Leftrightarrow\quad\frac{d}{dt}(G\dot{\theta})=0.$

Hence, $\dot{\theta}=\frac{c}{G}$ for some constant $c$. Since the geodesic is
naturally parametrized, we get

$E\dot{\varphi}^{2}+G\dot{\theta}^{2}=1\quad\Leftrightarrow\quad\dot{\varphi}^{2}=\frac{G-c^{2}}{EG}.$

Substituting $t=0$, we obtain $c=\sqrt{G}|_{\varphi=b}=\pm 2\pi\cos^{2}b$, and
since $\dot{\theta}(0)>0$, we have $c>0$. Hence,

$\dot{\theta}=\frac{\cos^{2}b}{2\pi\cos^{4}\varphi}$ (4)

and

$\dot{\varphi}^{2}=\frac{\cos^{4}\varphi-\cos^{4}b}{4\pi^{2}\cos^{6}\varphi}.$
(5)

From the last equation it follows that $\dot{\varphi}$ vanishes exactly when
$\varphi=\pm b$ and that the geodesic $\gamma_{b}$ lies in the spherical
segment $b\leqslant\varphi\leqslant-b$.

Note that there is a special case $b=0$. In this case we have
$\varphi(t)\equiv 0,\theta(t)=\frac{t}{2\pi}$ and the surface (2) is just the
Clifford torus in equatorial $\mathbb{S}^{3}\subset\mathbb{S}^{5}$. Further we
assume that $b\neq 0$.

######  Proposition 2.2.2.

1) The function $\varphi(t)$ is $T$-antiperiodic where

$T=T(b):=\int_{b}^{-b}\frac{2\pi\cos^{3}\varphi\,d\varphi}{\sqrt{\cos^{4}\varphi-\cos^{4}b}}<+\infty.$

2) The function $\varphi(t)$ is odd w.r.t. the transformation $t\mapsto T-t$.

3) The function $\theta(t)$ satisfies $\theta(t+T)\equiv\theta(t)+\theta(T)$.

###### Proof.

The number $T$ is nothing but the difference between the value of $t$,
corresponding to $\varphi=b$, and the nearest value of $t$, corresponding to
$\varphi=-b$. In [Kar14, Corollary 4] it is shown that
$T(b)<\sqrt{2}\pi^{2}<+\infty$.

A solution of the ODE (5) has two branches. The functions $\varphi(t)$ and
$-\varphi(t+T)$ correspond to the same branch of the same solution of (5)
(this solution satisfies the initial condition $\varphi(0)=b$). Thus part 1)
follows. Part 2) can be proved similarly and part 3) follows easily from part
1) and (4). ∎

Finally, consider the issue of closeness of the geodesic $\gamma_{b}$. It
follows from Proposition 2.2.2 that the geodesic $\gamma_{b}$ is closed
exactly when the number $\Xi(b):=\theta(T(b))$ is a rational multiple of
$\pi$. From equations (4),(5) we obtain

$\Xi(b)=\cos^{2}b\int_{b}^{-b}\frac{d\varphi}{\cos\varphi\sqrt{\cos^{4}\varphi-\cos^{4}b}}.$

The function $\Xi(b)$ (as well as $T(b)$) is not expressible in terms of
elementary functions. However one can prove the following assertions (see
[Kar14, Proposition 5]):

  * (i)

$\Xi(b)$ is continuous on $(-\frac{\pi}{2},0)$ and strictly increasing;

  * (ii)

$\lim\limits_{b\to
0-}\Xi(b)=\frac{\sqrt{2}}{2}\pi,\quad\lim\limits_{b\to(-\frac{\pi}{2})+}\Xi(b)=\frac{\pi}{2}$.

It follows that closed geodesics on
$(\mathbb{S}^{2}\setminus\\{N,S\\},\tilde{g}_{1})$ (and so the bipolar
surfaces to Otsuki tori) are in 1-1 correspondence with rational numbers
$\frac{p}{q}$ such that

$p,q>0,\quad(p,q)=1,\quad\text{and}\quad\frac{1}{2}<\frac{p}{q}<\frac{\sqrt{2}}{2}.$

We denote the bipolar surface to the Otsuki torus corresponding to the number
$\frac{p}{q}$ by $\tilde{O}_{p/q}$. The parameter $b$ of the corresponding
closed geodesic $\gamma_{b}$ satisfies $\Xi(b)=\frac{p}{q}\pi$ and the full
length of this geodesic equals $t_{0}=2qT(b)$.

######  Remark 1.

The above construction _is not_ the standard definition of bipolar surfaces to
Otsuki tori. According to the standard definition, the surface
$\tilde{O}_{p/q}$ is obtained as the _bipolar surface_ to some minimal torus
in $\mathbb{S}^{3}$ called _Otsuki torus_ and denoted by $O_{p/q}$, see
[Kar14, §2.2-2.3]. For details on bipolar surfaces and Otsuki tori see [Law70,
§11] and [Pen13] respectively.

The following proposition contains some important properties of the functions
$\varphi(t),\theta(t)$ and immersion (2).

######  Proposition 2.2.3.

1) The function $\varphi(t)$ has exactly $2q$ zeros on $[0,t_{0})$, located at
the points $\frac{2d+1}{2}T$ where $d=0,\ldots,2q-1$. The function
$\dot{\varphi}(t)$ has exactly $2q$ zeros on $[0,t_{0})$, located at the
points $dT$ where $d=0,\ldots,2q-1$.

2) The functions $\cos\theta(t)$ and $\sin\theta(t)$ each have exactly $2p$
zeros on $[0,t_{0})$.

3) For even $q$ immersion (2) is invariant under the transformation

$(\alpha,t)\mapsto\left(\alpha+\pi,t+\frac{t_{0}}{2}\right).$

The immersion (2) is not invariant under any other transformations.

###### Proof.

Part 1) follows from Proposition 2.2.2 and equation (5). Parts 2) and 3) are
proved in [Kar14, Proposition 4]. ∎

The aim of this paper is to obtain estimates on the index and nullity of the
surfaces $\tilde{O}_{p/q}$. A rough upper bound can be obtained as follows. It
follows from [EM08, Theorem 1.1] and [Kar21, Proposition 1.6] that

$\operatorname{ind}\tilde{O}_{p/q}\leqslant
5\operatorname{ind}_{S}\tilde{O}_{p/q}+2,$

where $\operatorname{ind}_{S}\tilde{O}_{p/q}$ denotes so-called _spectral
index_ of $\tilde{O}_{p/q}$, i.e. the number of eigenvalues of the Laplace-
Beltrami operator on $\tilde{O}_{p/q}$ less than 2. On the other hand, in
[Kar14] it is shown that

$\operatorname{ind}_{S}\tilde{O}_{p/q}=\begin{cases}2q+4p-2,&\text{$q$ is
odd;}\\\ q+2p-2,&\text{$q$ is even.}\end{cases}$

It follows immediately that

$\operatorname{ind}\tilde{O}_{p/q}\leqslant\begin{cases}10q+20p-8,&\text{$q$
is odd;}\\\ 5q+10p-8,&\text{$q$ is even.}\end{cases}$

Our main result reads as that if $\frac{p}{q}$ is sufficiently close to
$\frac{\sqrt{2}}{2}$ (i.e., speaking informally, if $\tilde{O}_{p/q}$ is
‘‘close to the Clifford torus’’), then one can improve the rough upper bound
mentioned above and also obtain a lower bound.

######  Theorem 1.

There exists $\delta>0$ such that if $\frac{\sqrt{2}}{2}-\frac{p}{q}<\delta$,
then for odd $q$ the inequalities

$6q+8p-3\leqslant\operatorname{ind}\tilde{O}_{p/q}\leqslant 10q+4p-5$

hold, and for even $q$ the inequalities

$3q+4p-3\leqslant\operatorname{ind}\tilde{O}_{p/q}\leqslant 5q+2p-5$

hold. Moreover, for such $\frac{p}{q}$ the inequalities

$9\leqslant\operatorname{null}\tilde{O}_{p/q}\leqslant 13$

hold independently of the parity of $q$.

###  2.3 Sturm-Liouville equation

In this section we recall certain properties of Sturm-Liouville equation,
which are used in the sequel.

######  Theorem 2 (Sturm Oscillation Theorem for periodic Sturm-Liouville
equation, [CL, Theorem 3.1 in Chapter 8]).

Consider a periodic Sturm-Liouville problem

$-(p(t)h^{\prime}(t))^{\prime}+q(t)h(t)=\lambda h(t),$ (6)

where

$p(t)>0,\quad p(t+T)\equiv p(t),\quad q(t+T)\equiv q(t).$

Denote by $\lambda_{i}$ and $h_{i}(t)$ (where $i=0,1,2,\ldots$) eigenvalues
and eigenfunctions of problem (6) with periodic boundary conditions

$h(t+T)\equiv h(t).$

Denote by $\tilde{\lambda}_{i}$ и $\tilde{h}_{i}(t)$ (where $i=1,2,\ldots$)
eigenvalues and eigenfunctions of problem (6) with antiperiodic boundary
conditions

$h(t+T)\equiv-h(t).$

Then the following inequalities hold:

$\lambda_{0}<\tilde{\lambda}_{1}\leqslant\tilde{\lambda}_{2}<\lambda_{1}\leqslant\lambda_{2}<\tilde{\lambda}_{3}\leqslant\tilde{\lambda}_{4}<\lambda_{3}\leqslant\lambda_{4}<\ldots.$

For $\lambda=\lambda_{0}$ there exists a unique (up to multiplication by a
non-zero constant) eigenfunction $h_{0}(t)$. If
$\lambda_{2i+1}<\lambda_{2i+2}$ for some $i\geqslant 0$, then there is a
unique (up to multiplication by a non-zero constant) eigenfunction
$h_{2i+1}(t)$ with eigenvalue $\lambda_{2i+1}$ of multiplicity 1 and there is
a unique (up to multiplication by a non-zero constant) eigenfunction
$h_{2i+2}(t)$ with eigenvalue $\lambda_{2i+2}$ of multiplicity 1. If
$\lambda_{2i+1}=\lambda_{2i+2}$, then there exist two linearly independent
eigenfunctions $h_{2i+1}(t)$ and $h_{2i+2}(t)$ with eigenvalue
$\lambda=\lambda_{2i+1}=\lambda_{2i+2}$ of multiplicity 2. The same holds in
cases $\tilde{\lambda}_{2i+1}<\tilde{\lambda}_{2i+2}$ and
$\tilde{\lambda}_{2i+1}=\tilde{\lambda}_{2i+2}$.

The eigenfunction $h_{0}(t)$ has no zeros on $[0,T)$. The eigenfunctions
$h_{2i+1}(t)$ and $h_{2i+2}(t)$ each have exactly $2i+2$ zeros on $[0,T)$. The
eigenfunctions $\tilde{h}_{2i+1}(t)$ and $\tilde{h}_{2i+2}(t)$ each have
exactly $2i+1$ zeros on $[0,T)$.

One more useful fact concerns the case when the coefficients of the equation
(6) are periodic with period less than $t_{0}$.

######  Claim 1 ([Kar14, Propositions 11 and 12]).

1) Suppose that in the conditions of Theorem 2 the coefficients of the
equation (6) are periodic with period $\frac{t_{0}}{2n}$ for some
$n\in\mathbb{Z},n\geqslant 1$. Then $\frac{t_{0}}{2n}$-antiperiodic
eigenfunctions of the problem (6) are $h_{2n(2k+1)}(t)$ and
$h_{2n(2k+1)-1}(t)$, where $k\geqslant 0$.

2) Suppose that in the conditions of Theorem 2 the coefficients of the
equation (6) are periodic with period $\frac{t_{0}}{n}$ for some
$n\in\mathbb{Z},n\geqslant 2$. Then $\frac{t_{0}}{n}$-periodic eigenfunctions
of the problem (6) are $h_{0}(t),h_{2nk-1}(t)$ and $h_{2nk}(t)$, where
$k\geqslant 1$.

##  3 Proof of the main theorem

###  3.1 Jacobi operator on $\tilde{O}_{p/q}$

In this section we compute the Jacobi stability operator $J$ on
$\tilde{O}_{p/q}$ in local coordinates $\alpha,t$.

######  Proposition 3.1.1.

Let $x\in\tilde{O}_{p/q}$. The following vectors form an orthonormal basis of
$T_{x}\mathbb{R}^{5}$:

$\displaystyle N=\begin{pmatrix}\cos\alpha\cos\varphi\sin\theta\\\
\sin\alpha\cos\varphi\sin\theta\\\ \cos\alpha\cos\varphi\cos\theta\\\
\sin\alpha\cos\varphi\cos\theta\\\ \sin\varphi\end{pmatrix},$ $\displaystyle
e_{1}=\frac{1}{\cos\varphi}\frac{\partial}{\partial\alpha}=\begin{pmatrix}-\sin\alpha\sin\theta\\\
\cos\alpha\sin\theta\\\ -\sin\alpha\cos\theta\\\ \cos\alpha\cos\theta\\\
0\end{pmatrix},e_{2}=2\pi\cos\varphi\frac{\partial}{\partial
t}=2\pi\cos\varphi\begin{pmatrix}\cos\alpha(-\sin\varphi\sin\theta\,\dot{\varphi}+\cos\varphi\cos\theta\,\dot{\theta})\\\
\sin\alpha(-\sin\varphi\sin\theta\,\dot{\varphi}+\cos\varphi\cos\theta\,\dot{\theta})\\\
\cos\alpha(-\sin\varphi\cos\theta\,\dot{\varphi}-\cos\varphi\sin\theta\,\dot{\theta})\\\
\sin\alpha(-\sin\varphi\cos\theta\,\dot{\varphi}-\cos\varphi\sin\theta\,\dot{\theta})\\\
\cos\varphi\,\dot{\varphi}\end{pmatrix},$ $\displaystyle
n_{1}=\begin{pmatrix}\sin\alpha\cos\theta\\\ -\cos\alpha\cos\theta\\\
-\sin\alpha\sin\theta\\\ \cos\alpha\sin\theta\\\
0\end{pmatrix},n_{2}=2\pi\cos\varphi\begin{pmatrix}-\cos\alpha(\cos\theta\,\dot{\varphi}+\sin\theta\sin\varphi\cos\varphi\,\dot{\theta})\\\
-\sin\alpha(\cos\theta\,\dot{\varphi}+\sin\theta\sin\varphi\cos\varphi\,\dot{\theta})\\\
\cos\alpha(\sin\theta\,\dot{\varphi}-\cos\theta\sin\varphi\cos\varphi\,\dot{\theta})\\\
\sin\alpha(\sin\theta\,\dot{\varphi}-\cos\theta\sin\varphi\cos\varphi\,\dot{\theta})\\\
\cos^{2}\varphi\,\dot{\theta}\end{pmatrix}.$

Moreover, $e_{1},e_{2}$ is a basis of $T_{x}\tilde{O}_{p/q}$ and $n_{1},n_{2}$
is a basis of $N_{x}\tilde{O}_{p/q}$.

###### Proof.

The first assertion is easily seen from a direct computation. The vectors
$e_{1}$ and $e_{2}$ lie in $T_{x}\tilde{O}_{p/q}$ because they are
proportional to $\frac{\partial}{\partial\alpha}$ and
$\frac{\partial}{\partial t}$ respectively. It remains to note that $N$ is
just a unit normal vector to $\mathbb{S}^{4}$. ∎

In the sequel, all computations are made w.r.t. the basis from Proposition
3.1.1 and the elements of $\Gamma(N\tilde{O}_{p/q})$ are identified with
vector-functions:
$f_{1}n_{1}+f_{2}n_{2}\leftrightarrow\left(\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right)$.

######  Proposition 3.1.2.

The matrix of the operator
$\tilde{A}\colon\Gamma(N\tilde{O}_{p/q})\to\Gamma(N\tilde{O}_{p,q})$ in the
basis $n_{1},n_{2}$ is

$\tilde{A}=\begin{pmatrix}8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2}&0\\\
0&8\pi^{2}\sin^{2}\varphi\cos^{2}\varphi\,\dot{\theta}^{2}\end{pmatrix}.$

###### Proof.

We have

$\langle
A^{n_{i}}e_{j},e_{k}\rangle=-\langle\nabla_{e_{j}}^{T\tilde{O}_{p/q}}n_{i},e_{k}\rangle=-\langle\nabla_{e_{j}}^{\mathbb{R}^{5}}n_{i},e_{k}\rangle=\langle
n_{i},\nabla_{e_{j}}^{\mathbb{R}^{5}}e_{k}\rangle$

for all $i,j,k=1,2$. Then,

$\displaystyle\nabla_{e_{1}}^{\mathbb{R}^{5}}e_{1}=\frac{1}{\cos\varphi}\nabla_{\frac{\partial}{\partial\alpha}}^{\mathbb{R}^{5}}e_{1}=\frac{1}{\cos\varphi}\begin{pmatrix}-\cos\alpha\sin\theta\\\
-\sin\alpha\sin\theta\\\ -\cos\alpha\cos\theta\\\ -\sin\alpha\cos\theta\\\
0\end{pmatrix},\nabla_{e_{2}}^{\mathbb{R}^{5}}e_{1}=2\pi\cos\varphi\nabla_{\frac{\partial}{\partial
t}}^{\mathbb{R}^{5}}e_{1}=2\pi\cos\varphi\begin{pmatrix}-\sin\alpha\cos\theta\,\dot{\theta}\\\
\cos\alpha\cos\theta\,\dot{\theta}\\\ \sin\alpha\sin\theta\,\dot{\theta}\\\
-\cos\alpha\sin\theta\,\dot{\theta}\\\ 0\end{pmatrix}.$ (7)

From this it is easy to see that the matrices of the operators
$A^{n_{1}},A^{n_{2}}$ in the basis $e_{1},e_{2}$ are

$A^{n_{1}}=\begin{pmatrix}0&-2\pi\cos\varphi\,\dot{\theta}\\\
-2\pi\cos\varphi\,\dot{\theta}&0\end{pmatrix},\quad
A^{n_{2}}=\begin{pmatrix}2\pi\sin\varphi\cos\varphi\,\dot{\theta}&0\\\
0&-2\pi\sin\varphi\cos\varphi\,\dot{\theta}\end{pmatrix}.$

Here we used that $A^{n_{1}},A^{n_{2}}$ are symmetric and that
$\operatorname{tr}A^{n_{1}}=\operatorname{tr}A^{n_{2}}=0$ because the surface
$\tilde{O}_{p/q}$ is minimal. Hence, the matrix of the operator $A$ is

$A=\begin{pmatrix}0&2\pi\sin\varphi\cos\varphi\,\dot{\theta}\\\
-2\pi\cos\varphi\,\dot{\theta}&0\\\ -2\pi\cos\varphi\,\dot{\theta}&0\\\
0&-2\pi\sin\varphi\cos\varphi\,\dot{\theta}\end{pmatrix},$

and since $\tilde{A}=A^{*}A$, the result follows. ∎

######  Proposition 3.1.3.

The operator
$\Delta^{N\tilde{O}_{p/q}}\colon\Gamma(N\tilde{O}_{p/q})\to\Gamma(N\tilde{O}_{p/q})$
in the basis $n_{1},n_{2}$ has the form

$\Delta^{N\tilde{O}_{p/q}}\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}=\begin{pmatrix}\Delta
f_{1}+4\pi^{2}\dot{\varphi}^{2}f_{1}-\frac{4\pi\dot{\varphi}}{\cos\varphi}\frac{\partial
f_{2}}{\partial\alpha}\\\ \Delta
f_{2}+4\pi^{2}\dot{\varphi}^{2}f_{2}+\frac{4\pi\dot{\varphi}}{\cos\varphi}\frac{\partial
f_{1}}{\partial\alpha}\end{pmatrix}.$

###### Proof.

It easily follows from (1) that for any $n\in\Gamma(N\tilde{O}_{p/q})$ and
$f\in C^{\infty}(\tilde{O}_{p/q})$ we have

$\Delta^{N\tilde{O}_{p/q}}(fn)=f\Delta^{N\tilde{O}_{p/q}}n+(\Delta
f)n-2\sum_{i=1}^{2}(e_{i}f)\nabla_{e_{i}}^{N\tilde{O}_{p/q}}n.$ (8)

Therefore it suffices to calculate $\Delta^{N\tilde{O}_{p/q}}n_{i}$ and
$\nabla_{e_{j}}^{N\tilde{O}_{p/q}}n_{i}$ for $i,j=1,2$. We have

$\nabla_{e_{1}}^{\mathbb{R}^{5}}n_{1}=\frac{1}{\cos\varphi}\nabla_{\frac{\partial}{\partial\alpha}}^{\mathbb{R}^{5}}n_{1}=\begin{pmatrix}\cos\alpha\cos\theta\\\
\sin\alpha\cos\theta\\\ -\cos\alpha\sin\theta\\\ -\sin\alpha\sin\theta\\\
0\end{pmatrix},\quad\nabla_{e_{2}}^{\mathbb{R}^{5}}n_{1}=2\pi\cos\varphi\nabla_{\frac{\partial}{\partial
t}}^{\mathbb{R}^{5}}n_{1}=\begin{pmatrix}-\sin\alpha\sin\theta\,\dot{\theta}\\\
\cos\alpha\sin\theta\,\dot{\theta}\\\ -\sin\alpha\cos\theta\,\dot{\theta}\\\
\cos\alpha\cos\theta\,\dot{\theta}\\\ 0\end{pmatrix}.$

From this it is easy to see that

$\langle\nabla_{e_{1}}^{\mathbb{R}^{5}}n_{2},n_{1}\rangle=-\langle\nabla_{e_{1}}^{\mathbb{R}^{5}}n_{1},n_{2}\rangle=2\pi\dot{\varphi}\quad\text{and}\quad\langle\nabla_{e_{i}}^{\mathbb{R}^{5}}n_{j},n_{k}\rangle=0\text{
for all other $i,j,k=1,2$.}$

Hence,

$\begin{gathered}\nabla_{e_{1}}^{N\tilde{O}_{p/q}}n_{1}=\langle\nabla_{e_{1}}^{\mathbb{R}^{5}}n_{1},n_{2}\rangle
n_{2}=-2\pi\dot{\varphi}\,n_{2},\quad\nabla_{e_{1}}^{N\tilde{O}_{p/q}}n_{2}=\langle\nabla_{e_{1}}^{\mathbb{R}^{5}}n_{2},n_{1}\rangle
n_{1}=2\pi\dot{\varphi}\,n_{1},\\\
\nabla_{e_{2}}^{N\tilde{O}_{p/q}}n_{1}=\nabla_{e_{2}}^{N\tilde{O}_{p/q}}n_{2}=0.\end{gathered}$
(9)

Further, from (7) we have

$\nabla_{e_{1}}^{T\tilde{O}_{p/q}}e_{1}=\langle\nabla_{e_{1}}^{\mathbb{R}^{5}}e_{1},e_{2}\rangle
e_{2}=2\pi\sin\varphi\,\dot{\varphi}\,e_{2},\quad\nabla_{e_{2}}^{T\tilde{O}_{p/q}}e_{2}=-\langle\nabla_{e_{2}}^{\mathbb{R}^{5}}e_{1},e_{2}\rangle
e_{1}=0,$ (10)

and using (1), we get

$\Delta^{N\tilde{O}_{p/q}}n_{1}=-\nabla_{e_{1}}^{N\tilde{O}_{p/q}}\nabla_{e_{1}}^{N\tilde{O}_{p/q}}n_{1}=4\pi^{2}\dot{\varphi}^{2}n_{1},\quad\Delta^{N\tilde{O}_{p/q}}n_{2}=-\nabla_{e_{1}}^{N\tilde{O}_{p/q}}\nabla_{e_{1}}^{N\tilde{O}_{p/q}}n_{2}=4\pi^{2}\dot{\varphi}^{2}n_{2},$
(11)

where all other summands from (1) vanish because of (9) and (10). The
proposition now follows from (8), (9), (11). ∎

######  Proposition 3.1.4 ([Kar14, Proposition 6]).

The Laplace-Beltrami operator on $\tilde{O}_{p/q}$ is given by the formula

$\Delta
f=-\frac{1}{\cos^{2}\varphi}\frac{\partial^{2}f}{\partial\alpha^{2}}-\frac{\partial}{\partial
t}\left(4\pi^{2}\cos^{2}\varphi\frac{\partial f}{\partial t}\right).$

######  Proposition 3.1.5 ([S68, equation (5.1.1)]).

The operator
$\mathcal{R}\colon\Gamma(N\tilde{O}_{p/q})\to\Gamma(N\tilde{O}_{p/q})$ is the
multiplication by $-2$.

######  Proposition 3.1.6.

The Jacobi stability operator
$J\colon\Gamma(N\tilde{O}_{p/q})\to\Gamma(N\tilde{O}_{p/q})$ in the basis
$n_{1},n_{2}$ has the form

$J\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}=\begin{pmatrix}\Delta
f_{1}+4\pi^{2}\dot{\varphi}^{2}f_{1}-\frac{4\pi\dot{\varphi}}{\cos\varphi}\frac{\partial
f_{2}}{\partial\alpha}-2f_{1}-8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2}f_{1}\\\
\Delta
f_{2}+4\pi^{2}\dot{\varphi}^{2}f_{2}+\frac{4\pi\dot{\varphi}}{\cos\varphi}\frac{\partial
f_{1}}{\partial\alpha}-2f_{2}-8\pi^{2}\cos^{2}\varphi\sin^{2}\varphi\,\dot{\theta}^{2}f_{1}\end{pmatrix}.$

###### Proof.

This follows from Propositions 3.1.2–3.1.5. ∎

###  3.2 Some elements from the kernel of $J$

It turns out that some elements of the kernel of $J$ can be found from
geometrical considerations. We use the following fact.

######  Proposition 3.2.1 ([S68, Lemmas 5.1.7—5.1.9]).

The image of any Killing vector field on $\mathbb{S}^{4}$ under the orthogonal
projection on $N\tilde{O}_{p/q}$ lies in the kernel of $J$.

It is well-known that the space of Killing vector fields on $\mathbb{S}^{4}$
has dimension 10 and is spanned by the fields

$x^{j}\frac{\partial}{\partial x^{i}}-x^{i}\frac{\partial}{\partial
x^{j}},\quad 1\leqslant i,j\leqslant 5.$

The following proposition can be proved by a direct computation.

######  Proposition 3.2.2.

The image of the space of Killing vector fields on $\mathbb{S}^{4}$ under the
orthogonal projection on $N\tilde{O}_{p/q}$ is spanned by the following
vector-functions

$\displaystyle\begin{pmatrix}\cos\varphi\sin 2\theta\\\
0\end{pmatrix},\begin{pmatrix}\cos\varphi\cos 2\theta\\\
0\end{pmatrix},\begin{pmatrix}0\\\
2\pi\cos^{2}\varphi\,\dot{\varphi}\end{pmatrix},\begin{pmatrix}-\sin\alpha\sin\varphi\cos\theta\\\
2\pi\cos\alpha\cos\varphi(\sin\varphi\cos\theta\,\dot{\varphi}+\sin\theta\cos\varphi\,\dot{\theta})\end{pmatrix},$
$\displaystyle\begin{pmatrix}\cos\alpha\sin\varphi\cos\theta\\\
2\pi\sin\alpha\cos\varphi(\sin\varphi\cos\theta\,\dot{\varphi}+\sin\theta\cos\varphi\,\dot{\theta})\end{pmatrix},\begin{pmatrix}-\sin\alpha\sin\varphi\sin\theta\\\
2\pi\cos\alpha\cos\varphi(\sin\varphi\sin\theta\,\dot{\varphi}-\cos\theta\cos\varphi\,\dot{\theta})\end{pmatrix},$
$\displaystyle\begin{pmatrix}\cos\alpha\sin\varphi\sin\theta\\\
2\pi\sin\alpha\cos\varphi(\sin\varphi\sin\theta\,\dot{\varphi}-\cos\theta\cos\varphi\,\dot{\theta})\end{pmatrix},\begin{pmatrix}-\sin
2\alpha\cos\varphi\\\ 2\pi\cos
2\alpha\cos^{2}\varphi\,\dot{\varphi}\end{pmatrix},\begin{pmatrix}\cos
2\alpha\cos\varphi\\\ 2\pi\sin
2\alpha\cos^{2}\varphi\,\dot{\varphi}\end{pmatrix}.$

###  3.3 Separation of variables

######  Proposition 3.3.1.

Let $h(l,t)=\left(\begin{smallmatrix}h_{1}(l,t)\\\
h_{2}(l,t)\end{smallmatrix}\right)$ be a solution of the following periodic
matrix Sturm-Liouville problem

$\displaystyle\left\\{\begin{aligned}
-(4\pi^{2}\cos^{2}\varphi\,h_{1}^{\prime})^{\prime}+\left(\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2}\right)h_{1}-\frac{4\pi
l\dot{\varphi}}{\cos\varphi}h_{2}=\lambda h_{1},\\\
-(4\pi^{2}\cos^{2}\varphi\,h_{2}^{\prime})^{\prime}+\left(\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\sin^{2}\varphi\cos^{2}\varphi\,\dot{\theta}^{2}\right)h_{2}-\frac{4\pi
l\dot{\varphi}}{\cos\varphi}h_{1}=\lambda h_{2},\end{aligned}\right.$ (12)
$\displaystyle h(l,t)\equiv h(l,t+t_{0}).$ (13)

Then the vector-functions

$\begin{pmatrix}h_{1}(l,t)\cos l\alpha\\\ h_{2}(l,t)\sin
l\alpha\end{pmatrix}\quad\text{and}\quad\begin{pmatrix}-h_{1}(l,t)\sin
l\alpha\\\ h_{2}(l,t)\cos l\alpha\end{pmatrix},\quad l=0,1,2,\ldots$

form a basis in the space of eigenfunctions of the operator $J$ with
eigenvalue $\lambda$. If $q$ is even, then one can chose this basis in such a
way that each basis eigenfunction is either $\frac{t_{0}}{2}$-periodic or
$\frac{t_{0}}{2}$-antiperiodic.

###### Proof.

It suffices to note that the operator $J$ commutes with
$\frac{\partial}{\partial\alpha}$ and that for even $q$ the coefficients of
the system (12) are $\frac{t_{0}}{2}$-periodic. ∎

Denote by $\lambda_{k}(l)$ the $k$-th ($k\geqslant 1$) eigenvalue of the
problem (12,13). For even $q$ denote by $\lambda_{k}^{+}(l)$ (respectively, by
$\lambda_{k}^{-}(l)$) the $k$-th ($k\geqslant 1$) eigenvalue of the problem
(12,13) for which the corresponding eigenfunction is
$\frac{t_{0}}{2}$-periodic (respectively, $\frac{t_{0}}{2}$-antiperiodic).
Also let us fix the following notation

$\displaystyle p(t)=4\pi^{2}\cos^{2}\varphi(t),$ $\displaystyle
Q_{l}(t)=\begin{pmatrix}\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2}&-\frac{4\pi
l\dot{\varphi}}{\cos\varphi}\\\ -\frac{4\pi
l\dot{\varphi}}{\cos\varphi}&\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\sin^{2}\varphi\cos^{2}\varphi\,\dot{\theta}^{2}\end{pmatrix}.$

######  Proposition 3.3.2.

For $l\geqslant 3$ the inequality $\lambda_{1}(l)>0$ holds.

###### Proof.

Fix arbitrary $l\geqslant 3$. Since

$\lambda_{1}(l)=\inf_{\begin{subarray}{c}h\in
C^{\infty}(\mathbb{R})\setminus\\{0\\}\\\ h(t)\equiv
h(t+t_{0})\end{subarray}}\frac{\int_{0}^{t_{0}}(p(t)|h^{\prime}(t)|^{2}+\langle
h(t),Q_{l}(t)h(t)\rangle)\,dt}{\int_{0}^{t_{0}}|h(t)|^{2}\,dt}\geqslant\inf_{\begin{subarray}{c}h\in
C^{\infty}(\mathbb{R})\setminus\\{0\\}\\\ h(t)\equiv
h(t+t_{0})\end{subarray}}\frac{\int_{0}^{t_{0}}\langle
h(t),Q_{l}(t)h(t)\rangle\,dt}{\int_{0}^{t_{0}}|h(t)|^{2}\,dt},$

it suffices to show that the matrix $Q_{l}(t)$ is positive definite for each
$t$. This last assertion holds as soon as the inequality

$\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2}>\left|\frac{4\pi
l\dot{\varphi}}{\cos\varphi}\right|$

holds for each $t$. Using (4) and (5), one can rewrite this inequality in the
form

$\left(l-\sqrt{1-\frac{\cos^{4}b}{\cos^{4}\varphi}}\right)^{2}>2\left(\cos^{2}\varphi+\frac{\cos^{4}b}{\cos^{4}\varphi}\right).$

This last inequality holds for $l\geqslant 3$ because in this case we have
$\mathrm{LHS}\geqslant 4>\mathrm{RHS}$. ∎

Thus, the problem (12,13) admits nonpositive eigenvalues only for $l=0,1,2$.
In the following sections the cases $l=0$ and $l=1,2$ are considered
separately.

###  3.4 Case $l=0$

This case is the simplest one because in this case the problem (12,13) is
decoupled. We arrive at two scalar periodic Sturm-Liouville problems

$\displaystyle-(4\pi^{2}\cos^{2}\varphi\,h_{1}^{\prime})^{\prime}+(4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2})h_{1}=\lambda
h_{1},$ (14) $\displaystyle h_{1}(t+t_{0})\equiv h_{1}(t),$ (15)

and

$\displaystyle-(4\pi^{2}\cos^{2}\varphi\,h_{2}^{\prime})^{\prime}+(4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\sin^{2}\varphi\cos^{2}\varphi\,\dot{\theta}^{2})h_{2}=\lambda
h_{2},$ (16) $\displaystyle h_{2}(t+t_{0})\equiv h_{2}(t).$ (17)

Our aim is the following

######  Proposition 3.4.1.

The following inequalities hold

$\\#\\{k\mid\lambda_{k}(0)<0\\}=2q+4p-1,\quad\\#\\{k\mid\lambda_{k}(0)=0\\}=3.$

If, in addition, $q$ is even, then

$\\#\\{k\mid\lambda_{k}^{+}(0)<0\\}=q+2p-1,\quad\\#\\{k\mid\lambda_{k}^{+}(0)=0\\}=3.$

Denote the $k$-th ($k\geqslant 0$) eigenvalues of the problems (14,15) and
(16,17) by $\lambda_{k}^{(1)}$ and $\lambda_{k}^{(2)}$ respectively.

######  Proposition 3.4.2.

$\lambda_{4p-1}^{(1)}=\lambda_{4p}^{(1)}=0$.

###### Proof.

It follows from Proposition 3.2.2 that the functions $\cos\varphi\cos 2\theta$
and $\cos\varphi\sin 2\theta$ are the eigenfunctions of the problem (14,15)
with eigenvalue 0. These functions are linearly independent and each have
exactly $4p$ zeros on $[0,t_{0})$ (see Proposition 2.2.3), hence the
proposition follows from Theorem 2. ∎

Now consider the problem (16,17). Unfortunately, for this problem Proposition
3.2.2 gives only one eigenfunction $2\pi\cos^{2}\varphi\,\dot{\varphi}$ and we
cannot determine the number of the eigenvalue 0 using only Theorem 2. However,
this eigenfunction is $T$-antiperiodic and the coefficients of the equation
(16) are periodic with the same period. Therefore it is natural to consider
the following antiperiodic Sturm-Liouville problem

$\displaystyle-(4\pi^{2}\cos^{2}\varphi\,h_{2}^{\prime})^{\prime}+(4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\sin^{2}\varphi\cos^{2}\varphi\,\dot{\theta}^{2})h_{2}=\lambda
h_{2},$ (18) $\displaystyle h_{2}(t+T)\equiv-h_{2}(t).$ (19)

Denote the $k$-th ($k\geqslant 1$) eigenvalue of the problem (18,19) by
$\tilde{\lambda}_{k}^{(2)}$.

######  Proposition 3.4.3.

$\tilde{\lambda}_{1}^{(2)}<0,\tilde{\lambda}_{2}^{(2)}=0$.

###### Proof.

It follows from Proposition 3.2.2 that the function
$2\pi\cos^{2}\varphi\,\dot{\varphi}$ is the eigenfunction of the problem
(18,19) with eigenvalue 0. Since on the segment $[0,T]$ this function vanishes
only at the points $t=0,T$, it follows from Theorem 2 that either
$\tilde{\lambda}_{1}^{(2)}=0$ or $\tilde{\lambda}_{2}^{(2)}=0$.

Let us show that in fact $\tilde{\lambda}_{2}^{(2)}=0$. One can verify by a
direct computation that the substitution
$z(t)=\sqrt{p(t)}h_{2}(t)=2\pi\cos\varphi(t)\,h_{2}(t)$ transforms the problem
(16,17) into the problem

$\displaystyle-z^{\prime\prime}+\frac{\lambda+2}{p(t)}z=0,$ (20)
$\displaystyle z(t+T)\equiv-z(t).$ (21)

We need the following claim, which one can consider as the integral version of
the permutation inequality.

######  Claim 2.

Let $f(t)$ and $g(t)$ be two (non-strictly) monotonic functions on $[0,a]$
with different kinds of monotonicity. Then

$\int_{0}^{a}f(t)g(t)\,dt\leqslant\int_{0}^{a}f(t)g(a-t)\,dt,$

where the equality is attained exactly when at least one of the functions
$f,g$ is a constant function.

###### Proof.

$\int_{0}^{a}f(t)(g(t)-g(a-t))\,dt=\int_{0}^{\frac{a}{2}}f(t)(g(t)-g(a-t))\,dt+\int_{\frac{a}{2}}^{a}f(t)(g(t)-g(a-t))\,dt=\\\
\int_{0}^{\frac{a}{2}}f(t)(g(t)-g(a-t))\,dt-\int_{0}^{\frac{T}{2}}f(a-t)(g(a-t)-g(t))\,dt=\int_{0}^{\frac{a}{2}}(f(t)-f(a-t))(g(t)-g(a-t))\,dt\leqslant
0,$

where the last inequality follows from the monotonicity of $f$ and $g$. The
equality happens exactly when the integrand is identically zero. In such a
case we have $f(0)=f(a)$ or $g(0)=g(a)$ and from the monotonicity we have
$f=\mathrm{const}$ or $g=\mathrm{const}$. ∎

Let us continue the proof of Proposition 3.4.3. The function
$\zeta(t)=2\pi\cos^{3}\varphi(t)\,\dot{\varphi}(t)$ solves the problem (20,21)
with $\lambda=0$. Note that the functions $p(t)$ and $\zeta(t)$ are strictly
increasing on $[0,\frac{T}{2}]$. Using $\zeta(\frac{T}{2}-t)$ as a test
function, we find

$\tilde{\lambda}_{1}^{(2)}+2\leqslant\frac{\int_{0}^{T}\zeta^{\prime}(\frac{T}{2}-t)^{2}\,dt}{\int_{0}^{T}p(t)^{-1}\zeta(\frac{T}{2}-t)^{2}\,dt}=\frac{\int_{0}^{\frac{T}{2}}\zeta^{\prime}(t)^{2}\,dt}{\int_{0}^{\frac{T}{2}}p(t)^{-1}\zeta(\frac{T}{2}-t)^{2}\,dt}<\frac{\int_{0}^{\frac{T}{2}}\zeta^{\prime}(t)^{2}\,dt}{\int_{0}^{\frac{T}{2}}p(t)^{-1}\zeta(t)^{2}\,dt}=\frac{\int_{0}^{T}\zeta^{\prime}(t)^{2}\,dt}{\int_{0}^{T}p(t)^{-1}\zeta(t)^{2}\,dt}=2.$

Here we used Claim 2 and the fact that $\zeta$ is even w.r.t. the
transformation $t\mapsto T-t$. ∎

######  Proposition 3.4.4.

$\lambda_{2q-1}^{(2)}=\tilde{\lambda}_{1}^{(2)}<0,\lambda_{2q}^{(2)}=\tilde{\lambda}_{2}^{(2)}=0$.

###### Proof.

This follows from Proposition 3.4.3 and assertion 1) of Proposition 1 for
$n=q$. ∎

###### Proof of Proposition 3.4.1.

This follows from Propositions 3.4.2, 3.4.4 and Claim 1 for $n=2$. ∎

######  Remark 2.

The trick, used in the proof of Proposition 3.4.3, can be used to simplify the
proof of [Kar14, §3.4].

###  3.5 Preparation for the cases $l=1$ and $l=2$

In this section we describe our approach for more complicated cases $l=1$ and
$l=2$, in which the system (12) is not decoupled.

Consider the following matrix Sturm-Liouville problem

$\displaystyle\left\\{\begin{aligned}
-(4\pi^{2}\cos^{2}\varphi\,h_{1}^{\prime})^{\prime}+\left(\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2}\right)h_{1}-\frac{4\pi
l\dot{\varphi}}{\cos\varphi}h_{2}=\lambda h_{1},\\\
-(4\pi^{2}\cos^{2}\varphi\,h_{2}^{\prime})^{\prime}+\left(\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\sin^{2}\varphi\cos^{2}\varphi\,\dot{\theta}^{2}\right)h_{2}-\frac{4\pi
l\dot{\varphi}}{\cos\varphi}h_{1}=\lambda h_{2},\end{aligned}\right.$ (22)
$\displaystyle h_{1}(t+T)\equiv\omega h_{1}(t),\quad h_{2}(t+T)\equiv-\omega
h_{2}(t),$ (23)

where $\omega$ is some $2q$-th root of unity and $h(t)$ is a complex-valued
vector-function. Denote the $k$-th ($k\geqslant 1$) eigenvalue of the problem
(22,23) by $\lambda_{k}^{[\omega]}(l)$.

######  Proposition 3.5.1.

Denote $\varepsilon=e^{\frac{1}{q}\pi i}$. Then

$\\{\lambda_{k}(l)\mid k\geqslant
1\\}=\bigcup_{r=0}^{2q-1}\\{\lambda_{k}^{[\varepsilon^{r}]}(l)\mid k\geqslant
1\\},$

and if $q$ is even, then

$\\{\lambda_{k}^{+}(l)\mid k\geqslant
1\\}=\bigcup_{r=0}^{q-1}\\{\lambda_{k}^{[\varepsilon^{2r}]}(l)\mid k\geqslant
1\\},\quad\\{\lambda_{k}^{-}(l)\mid k\geqslant
1\\}=\bigcup_{r=0}^{q-1}\\{\lambda_{k}^{[\varepsilon^{2r+1}]}(l)\mid
k\geqslant 1\\}.$

Here we mean the equalities of multisets, i.e. each eigenvalue is counted with
multiplicity.

###### Proof.

This is standard. It suffices to note that the cyclic group of order $2q$ acts
on the solution space of the problem (12,13) by the rule

$\sigma*\begin{pmatrix}h_{1}(t)\\\
h_{2}(t)\end{pmatrix}=\varepsilon\begin{pmatrix}h_{1}(t)\\\
-h_{2}(t)\end{pmatrix},$ (24)

where $\sigma$ is a generator of the cyclic group. Thus the first assertion of
the proposition follows easily from the existence of a joint basis of
eigenfunctions. For the second assertion just note that an eigenfunction of
(22,23) is $\frac{t_{0}}{2}$-antiperiodic exactly when $\omega^{q}=1$. ∎

Proposition 3.5.1 allows us to reduce the counting of negative eigenvalues of
the problem (12,13) to the counting of negative eigenvalues of the problems
(22,23) where $\omega$ runs through all $2q$-th roots of unity. For the
problems (22,23) we use an approach based on the work of Edwards [Edw64].

Denote by $\mu_{k}(l)$ the $k$-th eigenvalue of the Dirichlet problem

$\displaystyle\left\\{\begin{aligned}
-(4\pi^{2}\cos^{2}\varphi\,h_{1}^{\prime})^{\prime}+\left(\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\cos^{2}\varphi\,\dot{\theta}^{2}\right)h_{1}-\frac{4\pi
l\dot{\varphi}}{\cos\varphi}h_{2}=\mu h_{1},\\\
-(4\pi^{2}\cos^{2}\varphi\,h_{2}^{\prime})^{\prime}+\left(\frac{l^{2}}{\cos^{2}\varphi}+4\pi^{2}\dot{\varphi}^{2}-2-8\pi^{2}\sin^{2}\varphi\cos^{2}\varphi\,\dot{\theta}^{2}\right)h_{2}-\frac{4\pi
l\dot{\varphi}}{\cos\varphi}h_{1}=\mu h_{2},\end{aligned}\right.$
$\displaystyle h(0)=h(T)=0.$

Also denote

$\Omega_{l}[g,h]=\int_{0}^{T}(p(t)\langle
g^{\prime}(t),h^{\prime}(t)\rangle+\langle g(t),Q_{l}(t)h(t)\rangle)dt=\\\
\langle g(t),p(t)h^{\prime}(t)\rangle\Bigr{|}_{0}^{T}+\int_{0}^{T}\langle
g(t),-(p(t)h^{\prime}(t))^{\prime}+Q_{l}(t)h(t)\rangle dt,$

where $\langle\cdot,\cdot\rangle$ stands for the standard Hermitian product in
$\mathbb{C}^{2}$.

Let $S_{l}$ be the solution space of the system (22) on the segment $[0,T]$.
Consider the map

$S_{l}\to\mathbb{C}^{2}\oplus\mathbb{C}^{2},\quad h(t)\mapsto(h(0),h(T)).$
(25)

Suppose that $\mu_{k}(l)\neq 0$ for each $k\geqslant 1$ (we will see that this
is the case at least for $b$ close to zero). Then the kernel of the map (25)
is trivial and since $\dim S_{l}=4$, we see that this map is a bijection.
Define the Hermitian form $\alpha_{l}[\cdot]$ on
$\mathbb{C}^{2}\oplus\mathbb{C}^{2}$ by the formula

$\alpha[(v_{0},v_{T})]=\Omega_{l}[h,h],$

where $h$ is a unique element of $S_{l}$ such that $h(0)=v_{0},h(T)=v_{T}$. We
denote the corresponding sesquilinear form by $\alpha_{l}[\cdot,\cdot]$.
Finally, let us fix the following basis of
$\mathbb{C}^{2}\oplus\mathbb{C}^{2}$

$e_{1}=\left(\begin{pmatrix}1\\\ 0\end{pmatrix},\begin{pmatrix}0\\\
0\end{pmatrix}\right),\quad e_{2}=\left(\begin{pmatrix}0\\\
1\end{pmatrix},\begin{pmatrix}0\\\ 0\end{pmatrix}\right),\quad
e_{3}=\left(\begin{pmatrix}0\\\ 0\end{pmatrix},\begin{pmatrix}1\\\
0\end{pmatrix}\right),\quad e_{4}=\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}0\\\ 1\end{pmatrix}\right).$

######  Proposition 3.5.2 (see [Edw64, Proposition 2.6]).

Suppose that $\mu_{k}(l)\neq 0$ for each $k\geqslant 1$. Then

$\displaystyle\\#\\{k\mid\lambda_{k}^{[\omega]}(l)<0\\}=\\#\\{k\mid\mu_{k}(l)<0\\}+\operatorname{ind}(\alpha_{l}|\Pi_{\omega}),$
$\displaystyle\\#\\{k\mid\lambda_{k}^{[\omega]}(l)=0\\}=\operatorname{null}(\alpha_{l}|\Pi_{\omega}),$

where $\Pi_{\omega}=\mathrm{span}\\{e_{1}+\omega e_{3},e_{2}-\omega e_{4}\\}$,
vertical bar denotes restriction, and $\operatorname{ind},\operatorname{null}$
denote the index and the nullity of a Hermitian form on a subspace
respectively.

Denote $a_{ij}=\alpha_{l}[e_{i},e_{j}]$. Since the coefficients of the system
(22) are real-valued functions, we have $a_{ij}\in\mathbb{R}$, in particular,
$a_{ij}=a_{ji}$. Moreover, since the coefficients of the system (22) are even
functions w.r.t. the transformation $t\mapsto T-t$, we obtain

$a_{ij}=a_{\sigma(i)\sigma(j)},\text{ where }\sigma=\begin{pmatrix}1&2&3&4\\\
3&4&1&2\end{pmatrix}.$

Therefore the Gram matrix of the restriction $\alpha_{l}|\Pi_{\omega}$ has the
form

$A_{l}(\omega)=\begin{pmatrix}a_{11}+(\omega+\bar{\omega})a_{13}+a_{33}&a_{12}+\omega
a_{23}-\bar{\omega}a_{14}-a_{34}\\\ a_{12}+\bar{\omega}a_{23}-\omega
a_{14}-a_{34}&a_{22}-(\omega+\bar{\omega})a_{24}+a_{44}\end{pmatrix}=\\\
=\begin{pmatrix}2a_{11}+(\omega+\bar{\omega})a_{13}&(\omega-\bar{\omega})a_{14}\\\
(\bar{\omega}-\omega)a_{14}&2a_{22}-(\omega+\bar{\omega})a_{24}\end{pmatrix}.$
(26)

In particular, $\det A_{l}(\omega)$ is a polynomial in
$s=\mathop{\mathrm{Re}}\nolimits\omega$ of degree at most 2. We denote this
polynomial by $P_{l}(s)$.

Now consider the case of Clifford torus $b=0$. In this case we have

$\varphi(t)\equiv 0,\quad\theta(t)=\frac{t}{2\pi},\quad T=\lim_{b\to
0}T(b)=\sqrt{2}\pi^{2}.$

The problem (22,23) becomes

$\displaystyle\left\\{\begin{aligned}
-4\pi^{2}h_{1}^{\prime\prime}+(l^{2}-4)h_{1}=\lambda h_{1},\\\
-4\pi^{2}h_{2}^{\prime\prime}+(l^{2}-2)h_{2}=\lambda
h_{2},\end{aligned}\right.$ (27) $\displaystyle h_{1}(t+T)\equiv\omega
h_{1}(t),\quad h_{2}(t+T)\equiv-\omega h_{2}(t).$ (28)

Since the problem (27,28) is decoupled, it is easy to see that

$\mu_{1}(1)=-1,\,\mu_{2}(1)=1>0,\quad\mu_{1}(2)=2>0.$ (29)

In particular, $\mu_{k}(l)\neq 0$ for each $k\geqslant 0$. Therefore the
matrices $A_{l}(\omega),l=1,2$ are defined and can be computed.

######  Proposition 3.5.3.

For $b=0$ we have

$\displaystyle A_{1}(\omega)$
$\displaystyle=\begin{pmatrix}\frac{4\sqrt{3}\pi}{\sin\frac{\sqrt{6}}{2}\pi}(\cos\frac{\sqrt{6}}{2}\pi-\mathop{\mathrm{Re}}\nolimits\omega)&0\\\
0&\frac{4\pi}{\sin\frac{\sqrt{2}}{2}\pi}(\cos\frac{\sqrt{2}}{2}\pi+\mathop{\mathrm{Re}}\nolimits\omega)\end{pmatrix},$
$\displaystyle A_{2}(\omega)$
$\displaystyle=\begin{pmatrix}4\sqrt{2}(1-\mathop{\mathrm{Re}}\nolimits\omega)&0\\\
0&\frac{4\sqrt{2}\pi}{\sinh\pi}(\cosh\pi+\mathop{\mathrm{Re}}\nolimits\omega)\end{pmatrix}.$

###### Proof.

We consider only the case $l=1$ because the case $l=2$ is similar. Let
$\psi_{i}(t)$ be the solution of the system (27) satisfying
$(\psi_{i}(0),\psi_{i}(T))=e_{i}\,(i=1,2,3,4)$. On can verify by a direct
computation that

$\displaystyle\psi_{1}$
$\displaystyle=\frac{1}{\sin\frac{\sqrt{6}}{2}\pi}\begin{pmatrix}\sin\frac{\sqrt{3}(T-t)}{2\pi}\\\
0\end{pmatrix},$ $\displaystyle\psi_{2}$
$\displaystyle=\frac{1}{\sin\frac{\sqrt{2}}{2}\pi}\begin{pmatrix}0\\\
\sin\frac{T-t}{2\pi}\end{pmatrix},$ $\displaystyle\psi_{3}$
$\displaystyle=\frac{1}{\sin\frac{\sqrt{6}}{2}\pi}\begin{pmatrix}\sin\frac{\sqrt{3}t}{2\pi}\\\
0\end{pmatrix},$ $\displaystyle\psi_{4}$
$\displaystyle=\frac{1}{\sin\frac{\sqrt{2}}{2}\pi}\begin{pmatrix}0\\\
\sin\frac{t}{2\pi}\end{pmatrix}.$

Since

$a_{ij}=\alpha_{1}[e_{i},e_{j}]=\Omega_{1}[\psi_{i},\psi_{j}]=\langle\psi_{i},p\psi_{j}^{\prime}\rangle\Bigr{|}_{0}^{T},$

we see that all the values $a_{ij}$ can be easily computed. ∎

Hence for $b=0$ we have

$\displaystyle\operatorname{ind}(\alpha_{1}|\Pi_{\omega})$
$\displaystyle=\begin{cases}2,&\mathop{\mathrm{Re}}\nolimits\omega\in[-1,\cos\frac{\sqrt{6}}{2}\pi),\\\
1,&\mathop{\mathrm{Re}}\nolimits\omega\in[\cos\frac{\sqrt{6}}{2}\pi,-\cos\frac{\sqrt{2}}{2}\pi),\\\
0,&\mathop{\mathrm{Re}}\nolimits\omega\in[-\cos\frac{\sqrt{2}}{2}\pi,1],\end{cases}$
$\displaystyle\operatorname{null}(\alpha_{1}|\Pi_{\omega})$
$\displaystyle=\begin{cases}1,&\mathop{\mathrm{Re}}\nolimits\omega=\cos\frac{\sqrt{6}}{2}\pi,-\cos\frac{\sqrt{2}}{2}\pi,\\\
0,&\text{otherwise}.\end{cases}$
$\displaystyle\operatorname{ind}(\alpha_{2}|\Pi_{\omega})$ $\displaystyle=0,$
$\displaystyle\operatorname{null}(\alpha_{2}|\Pi_{\omega})$
$\displaystyle=\begin{cases}1,&\omega=1,\\\ 0,&\text{otherwise}.\end{cases}$

We need the following technical proposition.

######  Proposition 3.5.4.

The values $a_{ij}$ and $\mu_{k}(l)$ depend continuously on $b$.

###### Proof.

The substitution $t\mapsto\frac{t}{T}$ allows us to consider the system (22)
as a system on the segment $[-1,1]$ independent of $b$. Then the continuity of
$a_{ij}$ as a function of $b$ follows easily from the continuous dependence of
the solution of a system of ODEs on the coefficients. The continuity of the
Dirichlet eigenvalues $\mu_{k}(l)$ follows, for instance, from [HLWZ19,
Theorem 6]. ∎

###  3.6 Estimates for the cases $l=1$ и $l=2$

######  Proposition 3.6.1.

For each $b$ sufficiently close to zero we have

$\displaystyle 2q+2p-1\leqslant$
$\displaystyle\\#\\{k\mid\lambda_{k}(1)<0\\}\leqslant 4q-2,$ (30)
$\displaystyle 2\leqslant$
$\displaystyle\\#\\{k\mid\lambda_{k}(1)=0\\}\leqslant 4.$

If, in addition, $q$ is even, then

$\displaystyle q+p-1\leqslant$
$\displaystyle\\#\\{k\mid\lambda_{k}^{-}(1)<0\\}\leqslant 2q-2,$ (31)
$\displaystyle 2\leqslant$
$\displaystyle\\#\\{k\mid\lambda_{k}^{-}(1)=0\\}\leqslant 4.$

###### Proof.

First note that it follows from (29) and Proposition 3.5.4 that for $b$
sufficiently close to zero we have

$\\#\\{k\mid\mu_{k}(1)<0\\}=1.$ (32)

When $b=0$ the roots of the polynomial $P_{1}(s)$ are
$\cos\frac{\sqrt{6}}{2}\pi$ and $-\cos\frac{\sqrt{2}}{2}\pi$. It follows from
Proposition 3.5.4 that for $b$ sufficiently close to zero the roots $s_{1}$
and $s_{2}$ of the polynomial $P_{1}(s)$ are close to
$\cos\frac{\sqrt{6}}{2}\pi$ and $-\cos\frac{\sqrt{2}}{2}\pi$ respectively and
we have

$\operatorname{ind}(\alpha_{1}|\Pi_{\omega})=\begin{cases}2,&\mathop{\mathrm{Re}}\nolimits\omega\in[-1,s_{1}),\\\
1,&\mathop{\mathrm{Re}}\nolimits\omega\in[s_{1},s_{2}),\\\
0,&\mathop{\mathrm{Re}}\nolimits\omega\in[s_{2},1],\end{cases}\quad\operatorname{null}(\alpha_{1}|\Pi_{\omega})=\begin{cases}1,&\mathop{\mathrm{Re}}\nolimits\omega=s_{1},s_{2},\\\
0,&\text{otherwise}.\end{cases}$ (33)

At the same time, it follows from Proposition 3.2.2 that the vector-function

$e^{i\theta}\begin{pmatrix}\sin\varphi\\\
2\pi\cos\varphi(\sin\varphi\,\dot{\varphi}-i\cos\varphi\,\dot{\theta})\end{pmatrix}=\begin{pmatrix}\sin\varphi\cos\theta\\\
2\pi\cos\varphi(\sin\varphi\cos\theta\,\dot{\varphi}+\sin\theta\cos\varphi\,\dot{\theta})\end{pmatrix}+\\\
+i\begin{pmatrix}\sin\varphi\sin\theta\\\
2\pi\cos\varphi(\sin\varphi\sin\theta\,\dot{\varphi}-\cos\theta\cos\varphi\,\dot{\theta})\end{pmatrix}$

solves the problem (22,23) with $l=1,\lambda=0,\omega=-e^{\frac{p}{q}\pi i}$.
Therefore it follows from Proposition 3.5.2 that $\det
A_{1}(-e^{\frac{p}{q}\pi i})=0$, i.e.

$s_{2}=-\cos\frac{p}{q}\pi.$ (34)

Note that for $b$ sufficiently close to zero we have $|s_{1}|>s_{2}$. Thus
from (33), (34) we have

$2p-1\leqslant\sum_{r=0}^{2q}\operatorname{ind}(\alpha_{1}|\Pi_{\varepsilon^{r}})\leqslant
2q-2,\quad
2\leqslant\sum_{r=0}^{2q}\operatorname{null}(\alpha_{1}|\Pi_{\varepsilon^{r}})\leqslant
4,$ (35)

and for even $q$ we have

$p-1\leqslant\sum_{r=0}^{q-1}\operatorname{ind}(\alpha_{1}|\Pi_{\varepsilon^{2r+1}})\leqslant
q-2,\quad
2\leqslant\sum_{r=0}^{q-1}\operatorname{null}(\alpha_{1}|\Pi_{\varepsilon^{2r+1}})\leqslant
4.$ (36)

The required inequalities (30) and (31) follow now from (32), (35), (36) and
Propositions 3.5.1, 3.5.2. ∎

######  Proposition 3.6.2.

For any $b$ sufficiently close to zero we have

$\lambda_{1}(2)=0,\quad\lambda_{2}(2)>0.$

If, in addition, $q$ is even, then

$\lambda_{1}^{+}(2)=0,\quad\lambda_{2}^{+}(2)>0.$

###### Proof.

This is similar to the proof of Proposition 3.6.1 but easier. Again we note
that it follows from (29) and Proposition 3.5.4 that for $b$ sufficiently
close to zero we have

$\\#\\{k\mid\mu_{k}(2)<0\\}=0.$ (37)

For $b=0$ the roots of the polynomial $P_{2}(s)$ are $1$ and $-\cosh\pi$. At
the same time, it follows from Proposition 3.2.2 that the vector-function
$\bigl{(}\begin{smallmatrix}\cos\varphi\\\
2\pi\cos^{2}\varphi\,\dot{\varphi}\end{smallmatrix}\bigr{)}$ solves the
problem (22,23) with $l=2,\lambda=0,\omega=1$. Therefore for each $b$ the
polynomial $P_{2}(s)$ has the root $1$ and for $b$ sufficiently close to zero
the second root of $P_{2}(s)$ is close to $-\cosh\pi<-1$. Thus for $b$
sufficiently close to zero we have

$\operatorname{ind}(\alpha_{2}|\Pi_{\omega})=0,\quad\operatorname{null}(\alpha_{2}|\Pi_{\omega})=\begin{cases}1,\quad\omega=1,\\\
0,\quad\text{otherwise}.\end{cases}$ (38)

The proposition now follows from (37), (38) and Propositions 3.5.1, 3.5.2. ∎

###  3.7 Estimates on index and nullity

###### Proof of Theorem 1.

Let us put together the results of §3.4 and §3.6. Suppose that $q$ is odd.
According to Proposition 3.3.1 each eigenfunction of the problem (12,13) with
$l=0$ and $\lambda<0$ ($\lambda=0$) contributes $1$ to
$\operatorname{ind}\tilde{O}_{p/q}$ ($\operatorname{null}\tilde{O}_{p/q}$),
and each eigenfunction of the problem (12,13) with $l=1,2$ and $\lambda<0$
($\lambda=0$) contributes $2$ to $\operatorname{ind}\tilde{O}_{p/q}$
($\operatorname{null}\tilde{O}_{p/q}$). Hence,

$\displaystyle\operatorname{ind}\tilde{O}_{p/q}$
$\displaystyle=\\#\\{k\mid\lambda_{k}(0)<0\\}+2\\#\\{k\mid\lambda_{k}(1)<0\\}+2\\#\\{k\mid\lambda_{k}(2)<0\\},$
$\displaystyle\operatorname{null}\tilde{O}_{p/q}$
$\displaystyle=\\#\\{k\mid\lambda_{k}(0)=0\\}+2\\#\\{k\mid\lambda_{k}(1)=0\\}+2\\#\\{k\mid\lambda_{k}(2)=0\\},$

and from Propositions 3.4.1, 3.6.1, 3.6.2 we have

$\displaystyle 6q+8p-3=(2q+4p-1)+2(2q+2p-1)\leqslant$
$\displaystyle\operatorname{ind}\tilde{O}_{p/q}\leqslant(2q+4p-1)+2(4q-2)=10q+4p-5,$
$\displaystyle 9=3+2\cdot 2+2\cdot 1\leqslant$
$\displaystyle\operatorname{null}\tilde{O}_{p/q}\leqslant 3+2\cdot 4+2\cdot
1=13.$

Suppose that $q$ is even. Then the immersion (2) is invariant under the
transformation

$(\alpha,t)\mapsto\left(\alpha+\pi,t+\frac{t_{0}}{2}\right).$

It follows that for even (respectively, odd) $l$ only
$\frac{t_{0}}{2}$-periodic (respectively, $\frac{t_{0}}{2}$-antiperiodic)
eigenfunctions of the problem (12,13) contribute to index and nullity. Hence,

$\displaystyle\operatorname{ind}\tilde{O}_{p/q}$
$\displaystyle=\\#\\{k\mid\lambda_{k}^{+}(0)<0\\}+2\\#\\{k\mid\lambda_{k}^{-}(1)<0\\}+2\\#\\{k\mid\lambda_{k}^{+}(2)<0\\},$
$\displaystyle\operatorname{null}\tilde{O}_{p/q}$
$\displaystyle=\\#\\{k\mid\lambda_{k}^{+}(0)=0\\}+2\\#\\{k\mid\lambda_{k}^{-}(1)=0\\}+2\\#\\{k\mid\lambda_{k}^{+}(2)=0\\},$

and from Propositions 3.4.1, 3.6.1, 3.6.2 we have

$\displaystyle 3q+4p-3=(q+2p-1)+2(q+p-1)\leqslant$
$\displaystyle\operatorname{ind}\tilde{O}_{p/q}\leqslant(q+2p-1)+2(2q-2)=5q+2p-5,$
$\displaystyle 9=3+2\cdot 2+2\cdot 1\leqslant$
$\displaystyle\operatorname{null}\tilde{O}_{p/q}\leqslant 3+2\cdot 4+2\cdot
1=13.$

∎

######  Remark 3.

1) Unfortunately, our method does not allow us to obtain explicit estimates on
$\delta$.

2) We suppose that finding the _exact_ value of
$\operatorname{ind}\tilde{O}_{p/q}$ for each $\frac{p}{q}$ is a very difficult
problem. Indeed, such a computation at least should contain a very accurate
control over the root $s_{1}$ from the proof of Proposition 3.6.1, while this
root hardly can be calculated exactly.

3) A numerical experiment in Wolfram Mathematica allows us to conjecture that

$\operatorname{ind}\tilde{O}_{2/3}=31,\quad\operatorname{null}\tilde{O}_{2/3}=9,$

which coincides with the lower bounds from Theorem 1. However it is easy to
see from the proof of Theorem 1 that for some of the surfaces
$\tilde{O}_{p/q}$ this is not the case. Indeed, analyzing the proof of
Proposition 3.6.1, one can see that the lower bound on
$\operatorname{ind}\tilde{O}_{p/q}$ can be attained only if $s_{1}<-1$ while
for $b$ sufficiently close to zero the inequality $s_{1}>-1$ holds.

## Acknowledgments

I am grateful to A. V. Penskoi for stating the problem and constant attention
to this work. Also thanks to V. O. Medvedev for useful discussions.

## References

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Faculty of Mathematics, National Research University Higher School of
Economics,
---
6 Usacheva Str., 119048, Moscow, Russia
E-mail address<EMAIL_ADDRESS>


#  A Data-Centric Framework for Composable NLP Workflows

Zhengzhong Liu  Carnegie Mellon University  Petuum Inc<EMAIL_ADDRESS>Guanxiong Ding  Petuum Inc.  Avinash Bukkittu  Petuum Inc.  Mansi Gupta
Petuum Inc.  Pengzhi Gao  Petuum Inc.  Atif Ahmed  Petuum Inc.
Shikun Zhang  Carnegie Mellon University  Xin Gao  Carnegie Mellon University
Petuum Inc.  Swapnil Singhavi  Petuum Inc.  Linwei Li  Carnegie Mellon
University  Wei Wei  Carnegie Mellon University  Zecong Hu  Carnegie Mellon
University  Haoran Shi  Carnegie Mellon University  Xiaodan Liang  Sun Yat-sen
University
Teruko Mitamura  Carnegie Mellon University  Eric P. Xing  Carnegie Mellon
University  Petuum Inc.  Zhiting Hu  Carnegie Mellon University  UC San Diego

###### Abstract

Empirical natural language processing (NLP) systems in application domains
(e.g., healthcare, finance, education) involve interoperation among multiple
components, ranging from data ingestion, human annotation, to text retrieval,
analysis, generation, and visualization. We establish a unified open-source
framework to support fast development of such sophisticated NLP workflows in a
composable manner. The framework introduces a uniform data representation to
encode heterogeneous results by a wide range of NLP tasks. It offers a large
repository of processors for NLP tasks, visualization, and annotation, which
can be easily assembled with full interoperability under the unified
representation. The highly extensible framework allows plugging in custom
processors from external off-the-shelf NLP and deep learning libraries. The
whole framework is delivered through two modularized yet integratable open-
source projects, namely _Forte_ 111https://github.com/asyml/forte (for
workflow infrastructure and NLP function processors) and _Stave_
222https://github.com/asyml/stave (for user interaction, visualization, and
annotation).

Figure 1:  Stack of the data-centric framework for NLP workflows, including
workflow _infrastructure_ , and _processors_ for NLP tasks and interactions
(e.g., visualization, annotation). Different processors are composed together
with the infrastructure APIs to form an arbitrary complex workflow. The
example workflow transforms an unstructured text corpus into a knowledge graph
through a series of NLP functions.

##  1 Introduction

Natural language processing (NLP) techniques are playing an increasingly
central role in industrial applications. A real-world NLP system involves a
wide range of NLP tasks that interoperate with each other and interact with
users to accomplish complex workflows. For example, in an assistive medical
system for diagnosis (Figure 4), diverse text _analysis_ tasks (e.g., named
entity recognition, relation extraction, entity coreference) are performed to
extract key information (e.g., symptoms, treatment history) from clinical
notes and link to knowledge bases; a medical practitioner could select any
extracted entity to _retrieve_ similar past cases for reference; text
_generation_ techniques are used to produce summaries from diverse sources.

To develop domain-specific NLP systems fast, it is highly desirable to have a
unified open-source framework that supports: (1) seamless integration and
interoperation across NLP functions ranging from text analysis to retrieval to
generation; (2) rich user interaction for data visualization and annotation;
(3) extensible plug-ins for customized components; and (4) highly reusable
components.

A wealth of NLP toolkits exist (§4), such as spaCy (Honnibal and Montani,
2017), DKPro (Eckart de Castilho and Gurevych, 2014), CoreNLP (manning-
etal-2014-stanford), for pipelining multiple NLP functions; BRAT (Stenetorp et
al., 2012) and YEDDA (Yang et al., 2018) for annotating certain types of data.
None of them have addressed all the desiderata uniformly. Combining them for a
complete workflow requires non-trivial effort and expertise (e.g., ad-hoc
gluing code), posing challenges for maintenance and upgrading.

We introduce a new unified framework to support complex NLP workflows that
involve text data ingestion, analysis, retrieval, generation, visualization,
and annotation. The framework provides an infrastructure to simply plug in
arbitrary NLP functions and offers pre-built and reusable components to build
desired workflows. Importantly, the framework is designed to be extensible,
allowing users to write custom components (e.g., specialized annotation
interfaces) or wrap other existing libraries (e.g., Hu et al., 2019; Wolf et
al., 2019) easily.

The framework’s design is founded on a data-centric perspective. We design a
universal text data representation that can encode diverse input/output
formats of various NLP tasks uniformly. Each component (“ _processor_ ”) in
the workflow fetches relevant parts of data as inputs, and passes its results
to subsequent processors by adding the results to the data flow (Figure 1). In
this way, different processors are properly decoupled, and each is implemented
with a uniform interface without the need of accommodating other processors.
Visualization and annotation are also abstracted as standalone components
based on the data representation.

We demonstrate two case studies on using the framework to build a
sophisticated assistive medical workflow and a neural-symbolic hybrid chatbot.

Figure 2: Top Left: A dependency parser processor that calls a neural model
and save the results; Top Right: A relation extractor can use the same model
architecture. Bottom Left: A pipeline can be constructed by simply adding
processors. Bottom Right: Example data types offered by the framework or
customized by users. Relation and Dependency both extends Link. Definition of
dependency is done through a simple JSON configuration.

##  2 Data-Centric NLP Framework

Figure 1 shows the stack of the framework, consisting of several major parts:
(1) We first introduce the underlying infrastructure (§2.1), in particular, a
universal representation scheme for heterogeneous NLP data. The highly-
organized unified representation plays a key role in supporting composable NLP
workflows, which differentiates our framework from prominent toolkits such as
CoreNLP (manning-etal-2014-stanford), spaCy (Honnibal and Montani, 2017), and
AllenNLP (Gardner et al., 2018). We then introduce a range of functionalities
that enable the convenient use of the symbolic data/features in neural
modeling, which are not available in traditional NLP workflow toolkits such as
DKPro (Eckart de Castilho and Gurevych, 2014). (2) §2.2 describes how
processors for various NLP tasks can be developed with a uniform interface,
and can be simply plugged into a complex workflow. (3) finally, the human
interaction part offers rich composable processors for visualization,
annotation, and other forms of interactions.

###  2.1 Infrastructure

####  2.1.1 Universal Data Representation

NLP data primarily consists of two parts: the raw _text source_ and the
structured _markups_ on top of it (see Figure 3 for an example). The markups
represent the information overlaid on the text, such as part-of-speech tags,
named entity mentions, dependency links, and so forth. NLP tasks are to
produce desired text or markups as output, based on vastly different input
information and structures,

To enable full interoperation among distinct tasks, we summarize the
underlying commonalities between the myriad formats across different NLP
tasks, and develop a universal data representation encapsulating information
uniformly. The representation scheme defines a small number of _template_ data
types with high-level abstraction, which can be further extended to encode
domain-specific data.

##### Template data types:

We generalize the previous UIMA representation scheme (Götz and Suhre, 2004)
to cover the majority of common NLP markups. This results in three template
data types, each of which contains a couple of attributes.

  * •

Span contains two integer attributes, begin and end, to denote the offsets of
a piece of text. This type can mark tokens, entity mentions, and etc.

  * •

Link defines a pair of (parent, child) which are pointers to other markups, to
mark dependency arcs, semantic roles, entity relations, etc.

  * •

Group has a members attribute, which is a collection of markups. This type can
mark coreference clusters, topical clusters, etc.

##### Extended data types:

In order to encode more specific information, each of the template data types
can be extended by adding new attributes. For example, the framework offers
over 20 extended types for commonly used NLP concepts, such as Token and
EntityMention. Moreover, users can easily add _custom_ data types through
simple JSON definitions (Figure 2) to fulfill specific needs, such as
MedicalEntity that extends EntityMention with more attributes like patient
IDs. Once a new data type is defined, rich data operations (e.g., structured
access) as below are automatically enabled for the new type.

##### Flexible Data Sources:

Modern NLP systems face challenges imposed by the volume, veracity and
velocity of data. To cope with these, the system is designed with customizable
and flexible data sources that embrace technologies such as Indexing (e.g.
Elastic Search Elastic.co ), Databases (e.g. Sqlite), Vector Storage (e.g.
Faiss Johnson et al. (2017)). Users are free to implement flexible “Reader”
interface to ingest any source of data.

####  2.1.2 Facilitation for Neural Modeling

The framework provides extensive functionalities for effortless integration of
the above symbolic data representation with tensor-based neural modeling.

Neural representations. All data types are associated with an optional
_embedding_ attribute to store continuous neural representations. Hence, users
can easily access and manipulate the embeddings of arbitrary markups (e.g.,
entity, relation) extracted from neural models like word2vec Mikolov et al.
(2013) and BERT Devlin et al. (2019). The system also supports fast embedding
indexing and lookup with embedding storage systems such as Faiss Johnson et
al. (2017).

Rich data operations: auto-batching, structured access, etc.  Unified data
representation enables a rich array of operations to support different data
usage, allowing users to access any information in a structured manner. Figure
2 (top left) shows API calls that get all dependency links in a sentence.
Utilities such as _auto-batching_ and _auto-padding_ help aggregates relevant
information (e.g., event embeddings) from individual data instances into
tensors, which are particularly useful for neural modeling on GPUs.

Neural-symbolic hybrid modeling. Unified data representations and rich data
operations make it convenient to support hybrid modeling using both neural and
symbolic features. Take retrieval for example, the framework offers retrieval
processors (§2.2) that retrieve a coarse-grained candidate set with symbolic
features (e.g., TF.IDF) first, and then refine the results with more expensive
embedding-based re-ranking Nogueira and Cho (2019). Likewise, fast embedding
based search is facilitated with the Faiss library Johnson et al. (2017).

Shared modeling approaches.  The uniform input/output representation for NLP
tasks makes it easy to share the modeling approaches across diverse tasks. For
example, similar to Jiang et al. (2020), all tasks involving the Span and Link
data types as outputs (e.g., dependency parsing, relation extraction,
coreference resolution) can potentially use the exact same neural network
architecture for modeling. Further with the standardized APIs of our
framework, users can spawn models for all such tasks using the same code with
minimal edits. Top right of Figure 2 shows an example where the same relation
extractor is implemented with dependency parser for a new task, and the only
difference lies in accessing different data features.

Figure 3: Top: Screenshot of the single doc interface shows predicates, entity
mentions and semantic role links of one sentence. A new link is being created
from “bring” to “Sharm Ei-eikh”. Bottom: Screenshot of the two document
interface for annotating event coreference. The system is suggesting a
potential coreference pair. The interfaces are rendered based on the data
types. Users can customize the interface to use different UI components.

###  2.2 Processors

Universal data representation enables a uniform interface to build processors
for different NLP tasks. Most notably, _interoperation_ across processors
supported by the system abstraction allows each processor to interacts with
each other via the data flow.

Each processor takes uniformly represented data as inputs and performs
arbitrary actions on them. A processor can edit text source (e.g., language
generation), add additional markups (e.g., entity detection), or produce side
effects (e.g., writing data to disk). Top left of Figure 2 shows the common
structure of a processor, that fetches relevant information from the data pack
with high-level APIs, performs operations such as neural model inference,
writes results back to the data pack and pass them over to subsequent
processors. Top right shows the simple API used for plugging processors into
the workflow.

A comprehensive repository of pre-built processors. With the standardized
concept-level APIs for NLP data management, users can easily develop any
desired processors. One can wrap existing models from external libraries by
conforming to the simple interfaces. Moreover, we offer a large set of pre-
built processors for various NLP tasks, ranging from text retrieval, to
analysis and generation. Figure 1 lists a subset of processors.

###  2.3 Visualization, Annotation, & Interaction

The interfaces for visualization and annotation are implemented as standalone
components and designed for different data types.

Single document viewer.  We provide a single document interface (Figure 3)
that renders the template types. For example, Spans are shown by colored
highlights, Links are shown as connectors between the spans. A user can create
new spans, add links, create groups, or edit the attributes through intuitive
interfaces.

Figure 4: A system for diagnosis analysis and retrieval from clinical notes.
The data-centric approach makes it easy to assemble a variety of components
and UI elements. Example text was obtained from UNC School of Medicine .

Multi document viewer.  _Stave_ currently supports a two-document interface.
Users can create links across documents. The bottom of Figure 3 shows an
example of annotating event coreference. The system is suggesting an event
coreference pair and asking for the annotator’s decision.

Customization with plugins.  While default interfaces support a wide range of
tasks, users can create customized interfaces to meet more specific needs. We
build a system that can quickly incorporate independently-developed plugins,
such as a plugin for answering multiple-choice questions, or a dialogue
interface for chat-bots. Some pre-built plugins are showcased in Figure 4.
Additionally, the layout can be customized to display specific UI elements,
allowing for greater flexibility to use plugins together.

Human-machine collaboration.  Universal data representation across all modules
not only enhances interoperation between components, but also allows machines
and humans to collaborate in a workflow. Human-interactive components can be
integrated at any part of the workflow for visualization/reviewing to produce
a downstream system that combines the advantages of humans and machines.
Machine-assisted annotation can be undertaken straightforwardly: the
annotation system simply ingests the data produced by a back-end processor
(Figure 3).

##  3 Case Studies

###  3.1 A Clinical Information Workflow

We demonstrate an information system for clinical diagnosis analysis,
retrieval, and user interaction. Figure 4 shows an overview of the system. To
build the workflow, we first define domain-specific data types, such as
Clinical Entity Mention, via JSON config files as shown in Figure 2. We then
develop processors for text processing: (1) we create an LSTM-based clinical
NER processor Boag et al. (2015), a Span-Relation model based relation
extraction processor He et al. (2018), and a coreference processor with the
End-to-End model Lee et al. (2017) to extract key information; (2) we build a
report generation processor following Li et al. (2019) with extracted mentions
and relations; (3) we build a simple keyword based dialogue system for user to
interact using natural languages. The whole workflow is implemented with
minimal engineering effort. For example, the workflow is assembled with just
20 lines of code; and the IE processors are implemented with around 50 lines
of code by reusing libraries and models.

###  3.2 A ChatBot Workflow

The case study considers the scenario where we have a corpus of movie reviews
in English to answer complex queries (e.g., “movies with a positive sentiment
starring by a certain actor”) by a German user. The iterative workflow
consists of a review retrieval processor based on the hybrid symbolic-neural
feature modeling (§2.1.2), an NER processor Gardner et al. (2018) to find
actors and movies from the retrieved reviews, a sentiment processor Hutto and
Gilbert (2014) for sentence polarity, and an English-German translation
processor.

##  4 Related Work

The framework shares some characteristics with UIMA Götz and Suhre (2004)
backed systems, such as DKPro Eckart de Castilho and Gurevych (2014), ClearTK
Bethard et al. (2014) and cTakes Khalifa and Meystre (2015). There are NLP
toolboxes like NLTK Bird and Loper (2016) and AllenNLP Gardner et al. (2018),
GluonNLP Guo et al. (2019), NLP pipelines like Stanford CoreNLP manning-
etal-2014-stanford, SpaCy Honnibal and Montani (2017), and Illinois Curator
Clarke et al. (2012). As in §2.2, our system develops a convenient scaffold
and provides a rich set of utilities to reconcile the benefits of symbolic
data system, neural modeling, and human interaction, making it suit for
building complex workflows.

Compared to open-source text annotation toolkits, such as Protégé Knowtator
Ogren (2006), BRAT Stenetorp et al. (2012), Anafora Chen and Styler (2013),
GATE Cunningham et al. (2013), WebAnno Castilho (2016), and YEDDA Yang et al.
(2018), our system provides a more flexible experience with customizable plug-
ins, extendable data types, and full-fledged NLP support. The Prodigy tool by
spaCy is not open-source and supports only pre-defined annotation tasks like
NER.

##  5 Conclusions and Future Work

We present a data-centric framework for building complex NLP workflows with
heterogeneous modules. We will continue to improve the framework on other
advanced functionalities, such as multi-task learning, joint inference, data
augmentation, and provide a broader arsenal of processors to help build better
NLP solutions and other data science workflows. We also plan to further
facilitate workflow development by providing more flexible and robust data
management processors.

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