# Fat equator effect and Minimality in immersions and submersions of the
Sphere

Vicent Gimeno i Garcia  Department of Mathematics, Universitat Jaume I-IMAC,
E-12071, Castelló, Spain<EMAIL_ADDRESS>and  Vicente Palmer  Department
of Mathematics, Universitat Jaume I-INIT, E-12071, Castelló, Spain
<EMAIL_ADDRESS>
###### Abstract.

Inspired by the equatorial concentration of measure phenomenon in the sphere,
a result which is deduced from the general, (and intrinsic), concentration of
measure in $\mathbb{S}^{n}(1)$, we describe in this paper an equatorial
concentration of measure satisfied by the closed, (compact without boundary),
isometric and minimal immersions $x:\Sigma^{m}\rightarrow\mathbb{S}^{n}(1)$,
($m\leq n$), and by the minimal Riemannian submersions
$\pi:\Sigma^{m}\rightarrow\mathbb{S}^{n}(1)$, ($m\geq n$).

###### Key words and phrases:

Concentration of measure, isometric immersion, Riemannian submersion, mean
curvature, minimal submanifold, minimal fiber

###### 2020 Mathematics Subject Classification:

Primary 58C35, 49Q15, 53A10

Research partially supported by the Research Program of University Jaume I
Project UJI-B2021-08, and and Research grant PID2020-115930GA-100 funded by
MCIN/ AEI /10.13039/501100011033

##  1\. Introduction

The deep connection between the notions of measure of sets, (in its meaning of
Riemannian volume), distance between points and dimension in a Riemannian
manifold is illustrated by the concentration of (intrinsic) measure phenomenon
in the sphere $\mathbb{S}^{n}(1)$, (that we consider in this paper equipped
with a metric of constant sectional curvature $1$), as we can find it in the
lecture notes [7], (see Theorem A in this Introduction).

From this concentration of measure it can be deduced a particular result,
namely, an intrinsic equatorial measure concentration property satisfied by
the strips around the equators of the sphere, also known as “fat equator
effect”, which can be roughly described by saying that for “very large”
dimension, almost all measure in the sphere concentrates in these strips
surrounding its equators, independently of its width, (see Theorems C and
Theorem 1.2 in this Section).

Now, an interesting question that can be raised from this last result appears
when considering it from an extrinsic point of view, that is, from the point
of view of the submanifold theory. We introduce at this point the concept of
“extrinsic equatorial measure concentration” for submanifolds in the sphere:
we shall study then the relative volume of the intersection of a closed and
minimal submanifold with the above mentioned strips around the equators of the
ambient sphere, concluding, as in the intrinsic case, that for “very large”
dimension, almost all measure in the submanifold concentrates in their
intersection with the strips of any width surrounding these equators, (see
Theorem 1.4 in this Introduction).

On the other hand, there is a concentration of measure in the dual geometric
setting with respect the isometric immersions given by the Riemmanian
submersions. Namely, we shall prove a concentration of measure satisfied by
the compact manifolds which admits a Riemannian submersion onto the sphere
with minimal fibers. In this vein, such compact manifolds accumulates almost
the whole measure on the points which project to the strips of any width
surrounding the equator of the sphere (see Theorem 1.5).

In the lecture notes by V. Milman and G. Schechtman [7] we can find the
following theorem:

######  Theorem A (Corollary 2.2 in [7]).

If $A\subset\mathbb{S}^{n}(1)$ is a domain with ${\rm
vol}(A)\geq\frac{1}{2}{\rm vol}(\mathbb{S}^{n}(1))$ then, for all
$\epsilon\in[0,\frac{\pi}{2}]$, the $\epsilon$-fattening of $A$ satisfies the
inequality

(1) $1\geq\frac{{\rm vol}(A_{\epsilon})}{{\rm vol}(\mathbb{S}^{n}(1))}\geq
1-\sqrt{\frac{\pi}{8}}e^{-\epsilon^{2}(n-1)/2}$

and hence, for all $\epsilon\in[0,\frac{\pi}{2}]$,

(2) $\lim_{n\to\infty}\frac{{\rm vol}(A_{\epsilon})}{{\rm
vol}(\mathbb{S}^{n}(1))}=1.$

######  Remark 1.1.

Here, the $\epsilon$-fattening of $A$ is defined, using the intrinsic distance
in the sphere $\mathbb{S}^{n}(1)$, as

$A_{\epsilon}:=\left\\{x\in\mathbb{S}^{n}(1)\,:\,{\rm
dist}^{\mathbb{S}^{n}(1)}(x,A)<\epsilon\,\right\\}$

Theorem A is a consequence of the Levy’s isoperimetric inequality in the
sphere, (see [7]). To see the idea behind the proof of assertions (1) and (2),
let us consider a domain $A$ with ${\rm vol}(A)=a\geq\frac{1}{2}{\rm
vol}(\mathbb{S}^{n}(1))$ and let us fix $\epsilon\in[0,\frac{\pi}{2}]$. Then,
applying Levy’s isoperimetric inequality, and having into account that, fixed
$\epsilon\in[0,\frac{\pi}{2}]$, the $\epsilon$-fattening of a geodesic ball
$B^{n,1}_{r}$ is the ball $(B^{n,1}_{r})_{\epsilon}=B^{n,1}_{r+\epsilon}$, for
all $r\in[0,\pi/2-\epsilon)$, we have the inequality

(3) $\displaystyle\frac{{\rm vol}\left(A_{\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}$ $\displaystyle\geq\frac{{\rm
vol}\left(B^{n,1}_{\pi/2+\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}=\frac{\int_{0}^{\frac{\pi}{2}+\epsilon}\sin^{n}tdt}{\int_{0}^{\pi}\sin^{n}tdt}$
$\displaystyle=1-\frac{\int_{\frac{\pi}{2}+\epsilon}^{\pi}\sin^{n}tdt}{\int_{0}^{\pi}\sin^{n}tdt}$

To obtain equality (2) from this point, we remark that , given
$\epsilon\in[0,\frac{\pi}{2}]$, in the intervals
$[0,\frac{\pi}{2}+\epsilon]\subseteq[0,\pi]$, the function $\sin t$ attains
its maximum of $1$ in $t=\frac{\pi}{2}$, so, when $n\to\infty$, the function
$\sin^{n}t$ has a “peak” at $\frac{\pi}{2}$ and is negligible out of this
value. Hence, we have that

(4)
$\lim_{n\to\infty}\frac{\int_{\frac{\pi}{2}+\epsilon}^{\pi}\sin^{n}tdt}{\int_{0}^{\pi}\sin^{n}tdt}=0$

and

(5) $1\geq\lim_{n\to\infty}\frac{{\rm vol}\left(A_{\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}\geq\lim_{n\to\infty}\frac{\int_{0}^{\frac{\pi}{2}+\epsilon}(\sin
t)^{n}dt}{\int_{0}^{\pi}(\sin t)^{n}dt}=1$

Now, we are going to focus on a particular concentration of measure phenomenon
in the sphere, that we call equatorial, what happens in this case around any
totally geodesic equator $\mathbb{S}^{n-1}(1)$ of the sphere
$\mathbb{S}^{n}(1)$, and which can be deduced from the general concentration
of measure described above. We will take it as a starting point for our
definition of the extrinsic concentration of measure of an immersed
submanifold in the sphere.

In order to describe it and to give an idea of its proof, we are going to
consider the particular case constituted by the equator

$E=\\{(x_{1},\ldots,x_{n+1})\in\mathbb{S}^{n}(1)\,:\,x_{n+1}=0\\}$

together the domain $\Omega_{\epsilon}\subseteq\mathbb{S}^{n}(1)$, (called
$\epsilon$-strip around the equator $E$, see also figure 1), given by

$\Omega_{\epsilon}:=\left\\{(x_{1},\ldots,x_{n+1})\in\mathbb{S}^{n}(1)\,:\,-\sin(\epsilon)<x_{n+1}<\sin(\epsilon)\right\\}$

where $\epsilon\in[0,\frac{\pi}{2}]$

We must remark at this point that any equator in the sphere can be placed in
the position of $E$ using the accurated rotation, (an isometry which preserves
volume and distances), so the argument we are going to show applies in fact
for any equator.

Figure 1. The $\epsilon$-strip of $\mathbb{S}^{2}(1)$ is the
$\epsilon$-fattening of the equator of $\mathbb{S}^{2}(1)$

Then, we can prove the following

######  Theorem B.

Given the $\epsilon$-strip $\Omega_{\epsilon}$ around the equator $E$,

(6) $\frac{{\rm vol}\left(\Omega_{\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}\geq
1-\sqrt{\frac{\pi}{2}}e^{-\epsilon^{2}\frac{n-1}{2}}$

and hence, for $0<\epsilon<\frac{\pi}{2}$ we have

(7) $\lim_{n\to\infty}\frac{{\rm vol}\left(\Omega_{\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}=1.$

This property comes from the concentration of measure stated in Theorem A in
the following way: let us denote as $\mathbb{S}^{n+}$ the half sphere centered
at the north pole and as $\mathbb{S}^{n-}$ the half sphere centered at the
south pole. Then, the $\epsilon$-strip around the equator can be expressed, in
terms of the $\epsilon$-fattenings $\mathbb{S}^{n+}_{\epsilon}$ and
$\mathbb{S}^{n-}_{\epsilon}$ as the intersection,

$\Omega_{\epsilon}=\mathbb{S}^{n+}_{\epsilon}\cap\mathbb{S}^{n-}_{\epsilon}$

Now, inequality (6) follows applying Theorem A to the $\epsilon$-fattenings
$\mathbb{S}^{n+}_{\epsilon}$ and $\mathbb{S}^{n-}_{\epsilon}$ and using the
fact that ${\rm vol}(\mathbb{S}^{n+}_{\epsilon})={\rm
vol}(\mathbb{S}^{n-}_{\epsilon})$.The conclusion is that for “very large”
dimension, almost all measure in the Sphere is concentrated surrounding the
equator.

This equatorial concentration of measure just described above is also known as
“fat equator effect” (see [2]) and can be alternatively deduced by using
another concentration of measure, namely, the concentration of measure around
the mean/averaged value of a Lipschitz function defined on the sphere
$\mathbb{S}^{n}(1)$, as it is stated in Corollary V.2 of [7]:

######  Theorem C (Corollary V.2 in [7]).

Let $f:\mathbb{S}^{n}(1)\rightarrow\mathbb{R}$ be a Lipschitz function, with
Lipschitz constant $\sigma$. Then there exists an absolute constant $\delta$
such that

(8) $\frac{{\rm
vol}(\\{x\in\mathbb{S}^{n}(1)/|f(x)-\frac{\int_{\mathbb{S}^{n}}fd\mu}{{\rm
vol}(\mathbb{S}^{n})}|\geq C\\}}{{\rm vol}(\mathbb{S}^{n})}\geq 4e^{-\delta
C^{2}\frac{n+1}{\sigma^{2}}}$

We must remark that Theorem C is deduced from a previous result due to B.
Maurey and G. Pisier, (see Theorem V.1 in [7]). In this theorem it is proved a
concentration of measure of a multivariate and Lipschitz function which
depends on several independent Gaussian variables and it is stated and proved
using probabilistic arguments. The particularization of this result to
functions defined on the sphere $\mathbb{S}^{n}(1)$ is equivalent to the
concentration of measure of the values of any continuous function defined on
this sphere $\mathbb{S}^{n}(1)$ around its Levy mean, (see Lemma 2.3 in [7]),
which in its turn is a corollary deduced from Theorem A.

Then, using the cosinus of the distance function to a fixed point in the
sphere as our Lipschitz function and applying Theorem C we obtain:

######  Theorem 1.2.

Given the $\epsilon$-strip $\Omega_{\epsilon}$ around the equator $E$,
($\epsilon\in(0,\pi/2)$), there exists an absolute constant, (in the sense
that do not depends on $n$), $\delta>0$ such that

(9) $\frac{{\rm vol}\left(\Omega_{\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}\geq 1-4e^{-\delta\bar{\epsilon}^{2}(n+1)}$

where $\bar{\epsilon}=\sin(\epsilon)\in(0,1)$ and hence,

(10) $\lim_{n\to\infty}\frac{{\rm vol}\left(\Omega_{\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}=1.$

As the statement of our main result is directly inspired in Theorem 1.2 and
for the sake of completeness we will prove this theorem in the Appendix
Appendix I: Proof of theorem 1.2.

######  Remark 1.3.

We would like to draw attention to the following observation, which only
apparently constitutes a paradox that would refute the last property: let us
consider small $\epsilon>0$. Then, by virtue of the limit (10), as the
dimension increases, the equatorial band occupies more and more area of the
sphere, so that the two complementary big caps to the band have a small volume
compared to the total volume of the sphere.

Let us now consider a rotation of the equatorial strip, (an isometry of the
sphere acting on the strip), in such a way that the image due to this rotation
of the strip is a domain in the sphere that cuts the two previous big caps.
Since the image by the isometry of the band is a domain with the same area as
the band and it occupies a large amount of area of the sphere, all this area
will be concentrated outside the intersection with the caps, (which have small
area). The conclusion is that the entire area of the band will be concentrated
in two small squares, antipodal and disconnected, which are the result of
intersecting the old band prior to the rotation with the new one, (see picture
below).

Figure 2. Two orthogonal equatorial bands which intersects in two small
antipodal squares

This fact not only does not contradict the corollary, but we must observe
that, if we take, within one of these small squares, a spherical cap centered
on it and circumscribed in it, we have that, when the dimension tends to
infinity, the volume of this cap tends to zero while the area of the square
tends to infinity.

As we have mentioned before, we use the equatorial concentration of measure or
“fat equator effect”, as it has been described in Theorem 1.2 to introduce the
notion of extrinsic equatorial concentration of measure of a submanifold in
the sphere. Let us consider $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be a complete
isometric immersion of the $m$-dimensional manifold $\Sigma^{m}$ in the sphere
$\mathbb{S}^{n}(1)$. Any totally geodesic equator $\mathbb{S}^{n-1}(1)$ on a
sphere $\mathbb{S}^{n}(1)$ can be described in terms of the intrinsic distance
of its points to a given fixed point $p$, (and, in this sense, the equator $E$
that has served above to exemplify the phenomenon of concentration of the
equatorial measure, is the set of points that are distant from the north pole
by a distance equal to $\frac{\pi}{2}$). This description is given in the
following way: given a point $p\in\mathbb{S}^{n}(1)$, _the equator of
$\mathbb{S}^{n}(1)$ with respect to $p$_ is defined as

$E(p):=\left\\{x=(x_{1},\cdots,x_{n+1})\in\mathbb{S}^{n}(1)\,:\,\,{\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)=\frac{\pi}{2}\right\\},$

where

${\rm dist}^{\mathbb{S}^{n}(1)}(p,x)=\angle{(p,x)}=\arccos(p\cdot x)$

denotes the intrinsic distance in $\mathbb{S}^{n}(1)$ between the points $p$
and $x$. Then, _the strip around the equator $E(p)$_ is defined as the set:

$\displaystyle\Omega(p,\epsilon)$
$\displaystyle=\left\\{x\in\mathbb{S}^{n}(1)\,:\,\frac{\pi}{2}-\epsilon<{\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)=\angle{(p,x)}<\frac{\pi}{2}+\epsilon\right\\}$
$\displaystyle=\left\\{x\in\mathbb{S}^{n}(1)\,:\,\cos\left({\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)\right)\in(-\sin\epsilon,\sin\epsilon)\right\\}.$

Note that the strip around the equator $E$, $\Omega_{\epsilon}$, used in the
estimations (6) and (9), is the strip $\Omega(p_{N},\epsilon)$ where $p_{N}$
is the north pole of the sphere $\mathbb{S}^{n}(1)$.

Our main result can be summarized asserting that there is an extrinsic
equatorial concentration of measure when we consider minimal and compact
submanifolds $\Sigma^{m}$ of any co-dimension in the Sphere
$\mathbb{S}^{n}(1)$ and it is stated as follows:

######  Theorem 1.4.

Let $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be an isometric and minimal immersion
of the closed $m$-dimensional manifold $\Sigma$ in the sphere
$\mathbb{S}^{n}(1)$. Then, for any point $p\in\mathbb{S}^{n}(1)$ , any
$\epsilon\in(0,\pi/2)$ and any absolute $\delta\in[0,\frac{1}{2})$, (in the
sense that do not depends on $n$), we have

(11) $1\geq\frac{{\rm vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}\geq
1-\sqrt{\frac{1}{1-2\delta}}e^{-\delta\bar{\epsilon}^{2}(m+1)}$

where $\bar{\epsilon}=\sin\epsilon\in(0,1)$. Hence, for each
$\epsilon\in(0,\pi/2)$,

(12) $\lim_{m\to\infty}\frac{{\rm
vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm vol}(\Sigma)}=1.$

We are going to finish this Introduction presenting a concentration of measure
result of the fiber bundle projecting on the strip $\Omega(p,\epsilon)$ of a
compact without boundary Riemannian submersion
$\pi:\Sigma\rightarrow\mathbb{S}^{n}(1)$ when this submersion is minimal,
namely, when the (compact) fibers $\pi^{-1}(q)\subseteq\Sigma$ with
$q\in\mathbb{S}^{n}(1)$ are minimally immersed in $\Sigma$. This result, which
can be considered a dual property of the extrinsic equatorial concentration of
measure given by Theorem 1.4, comes from both the structural duality of the
equations that govern the geometry of the submersion with respect to the
corresponding equations of the immersions and from the fact that the
projection $\pi$ commutes with the Laplacians $\Delta^{\Sigma}$ and
$\Delta^{\mathbb{S}^{n}(1)}$ when the fibers of
$\pi:\Sigma\rightarrow\mathbb{S}^{n}(1)$ are minimal submanifolds of $\Sigma$.

######  Theorem 1.5.

Let $\pi:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be a Riemannian submersion with
minimal fibers of the closed $m$-dimensional manifold $\Sigma$ on to the
sphere $\mathbb{S}^{n}(1)$. Then, for any point $p\in\mathbb{S}^{n}(1)$ , any
$\epsilon\in(0,\pi/2)$ and any absolute $\delta\in[0,\frac{1}{2})$ we have

(13) $1\geq\frac{{\rm vol}\left(\pi^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}=\frac{{\rm vol}\left(\Omega(p,\epsilon)\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}\geq
1-\sqrt{\frac{\pi}{2}}e^{-\epsilon^{2}\frac{n-1}{2}}.$

Hence, for each $\epsilon\in(0,\pi/2)$,

(14) $\lim_{n\to\infty}\frac{{\rm
vol}\left(\pi^{-1}(\Omega(p,\epsilon))\right)}{{\rm vol}(\Sigma)}=1.$

We shall finish this Introduction with some general observations concerning
the statements of Theorems 1.4 and 1.5.

######  Remark 1.6.

  1. (1)

In contrast to the intrinsic approach, in the extrinsic case given by Theorem
1.4, the proof of inequality (11) and equality (12) is based on the divergence
theorem, rather than on the isoperimetric inequality, (as it is the case of
the proof of inequality (6) and equality (7) in Theorem B, proved as a
corollary of Theorem A) or rather than on probabilistic arguments using the
Gaussian distribution, (as it is done in Theorem 1.2, deduced as a direct
application of Theorem C).

  2. (2)

We can recover the intrinsic equality (10) in Theorem 1.2 as a corollary of
the extrinsic equality (12) in Theorem 1.4 , taking
$\Sigma=\mathbb{S}^{n}(1)$. In fact, if $\Sigma=\mathbb{S}^{n}(1)$, then we
can use $x$ as the identity map and hence,

$x^{-1}(\Omega(p,\epsilon))=x(\Sigma)\cap\Omega(p,\epsilon)=\Omega(p,\epsilon)$

so, from equality (12), we obtain

$\lim_{n\to\infty}\frac{{\rm vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}=\lim_{n\to\infty}\frac{{\rm
vol}\left(\Omega(p,\epsilon)\right)}{{\rm vol}(\mathbb{S}^{n}(1))}=1$

But it is important to remark that inequality (9) in Theorem 1.2 is not the
same nor it is a consequence of inequality (11) in Theorem 1.4 nor, on the
other hand, does inequality (6) can be derived directly from (9). Note that
the constants $\delta$ are not the same in Theorem 1.4 and Theorem 1.2.

  3. (3)

When we consider $\Sigma=\mathbb{S}^{n}(1)$ in Theorem 1.4, we can compare
inequality (11) with inequality (9) in Theorem 1.2. To do this comparison, we
must have into account that the constants $\delta$ are not the same in both
theorems, (which, on the other hand, limits the scope of this comparison),
and, moreover, that while $\delta$ is a fixed constant in Theorem 1.2, in
Theorem 1.4 the values of the constant $\delta$ ranges over the interval
$[0,\frac{1}{2})$. If we start our comparison considering the bound (9) in
Theorem 1.2, namely, the bound

$\frac{{\rm vol}\left(\Omega_{\epsilon}\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}\geq 1-4e^{-\delta\bar{\epsilon}^{2}(n+1)}$

we can to choose the accurate value of $\delta=\delta_{0}$ in Theorem 1.4 in
order to have the bound
$1-\sqrt{\frac{1}{1-2\delta_{0}}}e^{-\delta_{0}\bar{\epsilon}^{2}(n+1)}=1-4e^{-\delta_{0}\bar{\epsilon}^{2}(n+1)}$.
This value is $\delta_{0}=\frac{15}{32}$ and inequality (11) will be

$\frac{{\rm vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\mathbb{S}^{n}(1))}\geq 1-4e^{-\frac{15}{32}\bar{\epsilon}^{2}(n+1)}.$

  4. (4)

It should rest open to obtain, using our techniques, the statement of an
intrinsic concentration of measure for compact and minimal submanifolds of the
sphere, namely, if $A\subseteq\Sigma$ is a domain with ${\rm
vol}(A)\geq\frac{1}{2}{\rm vol}(\Sigma)$, then the $\epsilon$-fattening of $A$
in $\Sigma$ satisfies the inequality

$1\geq\frac{{\rm vol}(A_{\epsilon})}{{\rm vol}(\Sigma)}\geq
1-\sqrt{\frac{\pi}{8}}e^{-\epsilon^{2}(m-1)/2}$

and hence, for all $\epsilon\in[0,\frac{\pi}{2}]$,

$\lim_{m\to\infty}\frac{{\rm vol}(A_{\epsilon})}{{\rm vol}(\Sigma)}=1$

Here, the $\epsilon$-fattening of $A$ in $\Sigma$ should be defined as

$A_{\epsilon}:=\left\\{x\in\Sigma\,:\,{\rm
dist}^{\Sigma}(x,A)<\epsilon\,\right\\}.$

In fact, there is a result in this vein given as a Corollary of Theorem 6.9 in
[7], where an intrinsic concentration of measure in the line of Theorem A is
stated in terms of the first Dirichlet eigenvalue of a compact and connected
Riemannian manifold. In this sense, and if we consider a compact and minimal
submanifold in the sphere $x:\Sigma^{m}\rightarrow\mathbb{S}^{n}(1)$ as a
manifold in itself, then, applying Theorem 6.9 we obtain, for any domain
$A\subset\Sigma$ with $\frac{{\rm vol}(A)}{{\rm vol}(\Sigma)}=a$, the
following concentration of measure

(15) $\frac{{\rm vol}\left(A_{\epsilon}\right)}{{\rm vol}(\Sigma)}\geq
1-(1-a^{2})e^{-\epsilon\sqrt{\lambda_{1}(\Sigma)}\ln(1+a)},$

where $\lambda_{1}(\Sigma)$ is the first (non-zero) eigenvalue for the
Laplacian of $\Sigma$.

  5. (5)

We should note at this point that the concentration of measure (15) is not
related, a priori, with the dimension of the submanifold, so we can’t conclude
the description of the behaviour at infinity of the volume of the fattenings
or the strips around the equators given by equalities (2), (7), (10) and (11).
However, looking again to inequality (15), the things should change if we
could obtain estimations for $\lambda_{1}(\Sigma)$ in terms of the dimension
of the submanifold. In this vein, in problem 100 of [14], S.T Yau conjectured
that the first eigenvalue of a compact minimal embedded hypersurface
$\Sigma\subset\mathbb{S}^{n}$ is given by the dimension of $\Sigma$, namely

$\lambda_{1}(\Sigma)=n-1.$

If Yau’s conjecture is true it would imply, by using inequality (15) that for
any domain $A\subset\Sigma\subset\mathbb{S}^{n}$ of a minimal
$n-1$-dimensional hypersurface, with $\frac{{\rm vol}(A)}{{\rm
vol}(\Sigma)}\geq\frac{1}{2}$ there will be two constants
$c_{1},c_{2}\in\mathbb{R}_{>0}$ such that

(16) $\frac{{\rm vol}\left(A_{\epsilon}\right)}{{\rm vol}(\Sigma)}\geq
1-c_{1}e^{-c_{2}\epsilon\sqrt{n-1}}.$

so we should have an intrinsic concentration of measure for $\Sigma$ which
could be considered a Wirtinger’s type inequality for compact and minimal
embedded hypersurfaces of the sphere, (see [5]).

The rate of decline with respect to the dimension in this result should be
slower than our rate of decrease on the dimension given by (11) but it should
be true for other domains than any extrinsic fattening of the equator.

  6. (6)

The Yau’s conjecture remains unsolved but it is known to be true in several
cases: for instance for homogeneous hypersurfaces (see [8], [6]) and for
certain isoparametric minimal hypersurfaces (see [15]). On the other hand,
Choi and Wang proved in [3] that for compact, embedded and minimal
hypersurfaces $\Sigma\subset\mathbb{S}^{n+1}$ there exists the following lower
bound in the first eigenvalue

$\lambda_{1}(\Sigma)\geq\frac{n}{2}.$

This lower bound is enough to state the concentration of measure provided by
inequality (16) in this particular context, namely, when we consider compact,
embedded and minimal hypersurfaces $\Sigma\subset\mathbb{S}^{n+1}$. Observe
that this concentration phenomena would be attained for fattenings of domains
of any kind in compact, embedded and minimal hypersurfaces of the sphere, in
contrast to the results presented in this paper which would be valid for a
specific type of equatorial domains in the otherwise broader context of
minimal and compact immersions of any codimension in the sphere. Observe
moreover that, as we have noted before, the rate of decline in the
concentration of measure provided by inequality (16) is slower than the rate
of decrease provided by inequality (11).

  7. (7)

Note that, when the case of Riemannian submersions
$\pi:\Sigma\rightarrow\mathbb{S}^{n}(1)$ with minimal fibers is considered,
and unlike what happens with minimal and embedded immersions of codimension 1
in the sphere, (see observation (6) in Remark 1.6), the first eigenvalue of
the Laplacian is not bounded from below in general. Let us take for instance
the family of Riemannian manifolds
$\Sigma_{\delta}:=(\mathbb{S}^{n}(1)\times\mathbb{S}^{1},g_{\mathbb{S}^{n}(1)}+\delta^{2}g_{\mathbb{S}^{1}(1)})$,
and the family of Riemannian submersions
$\pi_{\delta}:\Sigma_{\delta}\rightarrow\mathbb{S}^{n}(1)$ given by

$(p,q)\mapsto\pi_{\delta}(p,q)=p\,\,\forall\delta\in\mathbb{R}_{>0}$

All these Riemannian submersions are minimal, indeed totally geodesic ones,
and, as
$\Sigma_{\delta}:=(\mathbb{S}^{n}(1),g_{\mathbb{S}^{n}(1)})\times(\mathbb{S}^{1},\delta^{2}g_{\mathbb{S}^{1}(1)})$
for any $\delta\in\mathbb{R}$ we have that

$\lambda_{1}(\Sigma_{\delta})\leq{\rm
min}\\{\lambda_{1}(\mathbb{S}^{n}(1),g_{\mathbb{S}^{n}(1)}),\lambda_{1}(\mathbb{S}^{1},\delta^{2}g_{\mathbb{S}^{1}(1)})\\}={\rm
min}\\{n,\frac{1}{\delta^{2}}\\}$

so, for any $a\in(0,n]$ there is a value of $\delta$ such that

$\lambda_{1}(\Sigma_{\delta})<a.$

Thence, in contrast to what happens for minimal hypersurfaces embedded in the
sphere, there are no lower bounds for the first eigenvalue of a compact
manifold which admits a Riemannian submersion to the sphere with minimal
fibers.

###  1.1. Outline

The structure of the paper can be outlined as follows: after this Introduction
where the notions related with the intrinsic and extrinsic equatorial
concentration in the sphere has been presented, as well as a first glimpse of
our main results, (Theorems 1.4 and 1.5), we shall follow in the following
Section §.2 with the study of the relative position of the closed and minimal
submanifolds in the sphere with respect the equators of this sphere, proving a
two-piece property which is satisfied by these closed and minimal
submanifolds. We have proved, using the same techniques, the corresponding
result which is satisfied by the total space $\Sigma$ of a Riemannian
submersion. In Section §.3, we shall state and prove our main results, and we
have added an Appendix I with a proof of Theorem 1.2, because although we
think that it is a folk result, we have not found a proof of it in the
literature of the field.

##  2\. A high-dimensional two-piece property for parabolic and minimal
submanifolds of the sphere $\mathbb{S}^{n}(1)$

We start with the definition of the equator with respect a point in the sphere
$\mathbb{S}^{n}(1)$, and the strip around an equator .

######  Definition 2.1.

Given a point $p\in\mathbb{S}^{n}(1)$, _the equator of $\mathbb{S}^{n}(1)$
with respect to $p$_ is defined as

$\displaystyle E(p):$
$\displaystyle=\left\\{x=(x_{1},\cdots,x_{n+1})\in\mathbb{S}^{n}(1)\,:\,\,{\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)=\frac{\pi}{2}\right\\}$
$\displaystyle=\left\\{(x_{1},\cdots,x_{n+1})\in\mathbb{S}^{n}(1)\,:\,\,\,p_{1}x_{1}+\cdots+p_{n+1}x_{n+1}=0\right\\}$

In its turn, the strip around an equator  is defined as:

######  Definition 2.2.

The strip around the equator $E(p)$ is defined as the set:

$\displaystyle\Omega(p,\epsilon)$
$\displaystyle=\left\\{x\in\mathbb{S}^{n}(1)\,:\,\frac{\pi}{2}-\epsilon<{\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)=\angle{(p,x)}<\frac{\pi}{2}+\epsilon\right\\}$
$\displaystyle=\left\\{x\in\mathbb{S}^{n}(1)\,:\,\cos\left({\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)\right)\in(-\sin\epsilon,\sin\epsilon)\right\\}$

Hence, as $\epsilon\in(0,\frac{\pi}{2})$, if we put
$\bar{\epsilon}=\sin\epsilon\in(0,1)$, then

$\Omega(p,\epsilon)=\left\\{x\in\mathbb{S}^{n}(1)\,:\,-\bar{\epsilon}<\cos\left({\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)\right)<\bar{\epsilon}\right\\}.$

With these definitions at hand, it is evident that for any point $q\in E(p)$,

${\rm dist}^{\mathbb{S}^{n}(1)}(p,q)=\frac{\pi}{2}={\rm
dist}^{\mathbb{S}^{n}(1)}(-p,q)$

where $-p$ denotes the antipodal point of $p\in\mathbb{S}^{n}(1)$. It is also
evident that any equator in the sphere $\mathbb{S}^{n}(1)$ is an equator with
respect to some $p$.

Let us consider now $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ a complete isometric
immersion of the $m$-dimensional manifold $\Sigma^{m}$ in the sphere. A
natural question that arises in this context is to ask ourselves about the
position of the submanifold with respect to the equators of the ambient
sphere.

In fact, when $\Sigma=\mathbb{S}^{1}(1)$ is a great circle isometrically
immersed in the sphere $\mathbb{S}^{2}(1)$, it is not hard to show that then
$\Sigma$ intersects all the equators of $\mathbb{S}^{2}(1)$, (which are the
great circles of the $2$-dimensional sphere).

When the dimension of the ambient sphere is $3$, A. Ros showed in [10] a Two-
piece property, which implies a more general intersection result to that
described above for the two-dimensional case. Namely, every equator of the
3-sphere $\mathbb{S}^{3}(1)$ divides each embedded closed (compact, without
boundary) non-totally geodesic minimal surface in exactly two open connected
pieces.

Therefore, a natural question which arises in this context is following: under
what conditions, (topological, analytical, curvature restrictions, etc), it is
guaranteed, (for any dimension of the ambient sphere and any co-dimension of
the immersed submanifold), that the submanifold cuts all the equators of the
ambient sphere, _i.e._ ,

$x(\Sigma)\cap E(p)\neq\emptyset\,\,\,\forall p\in\mathbb{S}^{n}(1)$

and hence, will intersect all the $\epsilon$-strips

$x(\Sigma)\cap\Omega(p,\epsilon)\neq\emptyset,(\text{i.e.},x^{-1}(\Omega(p,\epsilon))\neq\emptyset\,\,\,\forall
p\in\mathbb{S}^{n}(1))$

Our first result takes the form of a kind of a “high dimensional two-piece
property”, and describes the position of the submanifold with respect the
equators of $\mathbb{S}^{n}(1)$ and their strips, when this submanifold is
minimal and parabolic:

######  Theorem 2.3.

Let $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be a complete, minimal and isometric
immersion, and let $p$ be any point of $\mathbb{S}^{n}(1)$. Suppose that
$\Sigma$ is parabolic. Then

  * •

Either $x(\Sigma)\subseteq E(p)$,

  * •

or, $x(\Sigma)\cap E(p)\neq\emptyset$

and hence, in any case, $x(\Sigma)\cap\Omega(p,\epsilon)\neq\emptyset$, (i.e.
$x^{-1}(\Omega(p,\epsilon))\neq\emptyset$).

###### Proof.

Let $K$ be a connected component of $\Sigma$. Since $\Sigma$ is parabolic, $K$
is a parabolic manifold as well. Given $p\in\mathbb{S}^{n}(1)$, the _extrinsic
distance function_ with respect to $p$ is the function given by

$\displaystyle r_{p}:K\to\mathbb{R},\quad q\mapsto r_{p}(q):$
$\displaystyle={\rm dist}^{\mathbb{S}^{n}(1)}(p,x(q))$
$\displaystyle=\angle{(p,x(q))}=\arccos(p\cdot x(q))$

so, given $q\in K$, with $x(q)=(x_{1},...,x_{n+1})\in\mathbb{S}^{n}(1)$, we
have that

$\cos(r_{p}(q))=p\cdot x(q)=p_{1}x_{1}+\cdots+p_{n+1}x_{n+1}$

and hence, $\cos\circ r_{p}$ is smooth on $K$. We need the following

######  Lemma 2.4.

Let $x:K^{m}\to\mathbb{S}^{n}(1)$ be a complete and minimal isometric
immersion, and let $p$ be any point of $\mathbb{S}^{n}(1)$. Then, for all
$q\in K$,

(17) $\Delta^{K}\cos(r_{p}(q))=-m\cos(r_{p}(q))$

###### Proof.

Applying Takahasi’s Lemma, (see [12]), we have, as $x$ is minimal, that

$\Delta^{K}x_{i}=-mx_{i}\,\,\forall i=1,...,n+1$

so we conclude

$\displaystyle\Delta^{K}\cos(r_{p}(q))=$ $\displaystyle
p_{1}\Delta^{\Sigma}x_{1}+\cdots+p_{n+1}\Delta^{\Sigma}x_{n+1}$
$\displaystyle=$
$\displaystyle-m\left(p_{1}x_{1}+\cdots+p_{n+1}x_{n+1}\right)=-m\cos(r_{p}(q))$

∎

To prove the Theorem, first, let us observe that, if the extrinsic distance
function is constant, $\cos\circ r_{p}(q)=c\,\,\forall q\in\Sigma$, then

$0=\Delta^{K}\cos(r_{p}(q))=-m\cos\circ r_{p}(q)\,\forall q\in K,$

so $r_{p}(q)=\frac{\pi}{2}\,\,\forall q\in\Sigma$ and hence, in this case we
have that

$x(K)\subseteq E(p).$

This implies that the extrinsic distance function is a constant function in
$K$ if and only if $x(K)\subseteq E(p)$. Now, let us suppose that
$x(K)\not\subseteq E(p)$ and that $x(K)\cap E(p)=\emptyset$. Then $\cos\circ
r_{p}$ is not constant on $K$ and, moreover, as $K$ is connected, if $x(K)\cap
E(p)=\emptyset$ then $r_{p}(q)\in[0,\frac{\pi}{2})\,\,\forall q\in K$ or
$r_{p}(q)\in(\frac{\pi}{2},\pi]\,\,\forall q\in K$. To see this last
assertion, let us note that, in case there are points $q_{1},q_{2}\in K$ such
that $r_{p}(q_{1})\in[0,\frac{\pi}{2})$ and
$r_{p}(q_{2})\in(\frac{\pi}{2},\pi]$, then we can conclude that there are
points $q_{1},q_{2}\in K$ such that $\cos(r_{p}(q_{1}))>0$ and
$\cos(r_{p}(q_{2}))<0$, and therefore, $K$ is not connected because $K=\\{q\in
K:\cos(r_{p}(q))>0\\}\cup\\{q\in K:\cos(r_{p}(q))<0\\}$. We can conclude
therefore that $\cos\circ r_{p}$ is a bounded non-constant subharmonic,
(superharmonic), function defined on $K$, which is a contradiction with the
parabolicity of $K$. Hence, if $x(K)\not\subseteq E(p)$, then $x(K)\cap
E(p)\neq\emptyset$, which implies that $x(\Sigma)\cap E(p)\neq\emptyset$.∎

Note that, given the compact, (and hence, parabolic), and minimal submanifold
$\Sigma$, we can apply the above theorem 2.3 to conclude the following
corollary:

######  Corollary 2.5.

Let $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be an isometric and minimal immersion
of a closed, (compact without boundary), $m$-dimensional manifold $\Sigma$ in
the sphere $\mathbb{S}^{n}(1)$, and let $p$ be any point of
$\mathbb{S}^{n}(1)$. Then

  * •

Either $x(\Sigma)\subseteq E(p)$,

  * •

or, $x(\Sigma)\cap E(p)\neq\emptyset$

and hence, in any case, $x(\Sigma)\cap\Omega(p,\epsilon)\neq\emptyset$, (i.e.
$x^{-1}(\Omega(p,\epsilon))\neq\emptyset$).

We can show an analogous result for Riemannian submersions on the sphere,
using the fact, alluded in the introduction, that the projection $\pi$
commutes with the Laplacians $\Delta^{\Sigma}$ and
$\Delta^{\mathbb{S}^{n}(1)}$ when the fibers of
$\pi:\Sigma\rightarrow\mathbb{S}^{n}(1)$ are minimal submanifolds of $\Sigma$.

In this case, the set
$\pi^{-1}(\Omega(p,\epsilon))=\cup_{q\in\Omega(p,\epsilon)}\pi^{-1}(q)$ is the
fiber bundle projecting on the equatorial strip $\Omega(p,\epsilon)$, and we
want to know if this set is not empty for every $p\in\mathbb{S}^{n}(1)$ and
$\epsilon>0$. To show this fact, we will prove that the “shadow” of the total
manifold $\Sigma$ projected by $\pi$ on the sphere cuts all the equators
$E(p)$ in the sphere, namely, that

$\pi(\Sigma)\cap E(p)\neq\emptyset\,\,\,\forall p\in\mathbb{S}^{n}(1)$

and hence, will intersect all the $\epsilon$-strips

$\pi(\Sigma)\cap\Omega(p,\epsilon)\neq\emptyset$

so we eventually have

$\pi^{-1}(\Omega(p,\epsilon))\neq\emptyset\,\,\,\forall p\in\mathbb{S}^{n}(1)$

These questions are answered in the following

######  Theorem 2.6.

Let $\pi:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be a complete, minimal and parabolic
Riemannian submersion, and let $p$ be any point of $\mathbb{S}^{n}(1)$. Then

  * •

Either $\pi(\Sigma)\subseteq E(p)$,

  * •

or, $\pi(\Sigma)\cap E(p)\neq\emptyset$

and hence, in any case, $\pi(\Sigma)\cap\Omega(p,\epsilon)\neq\emptyset$,
(i.e. $\pi^{-1}(\Omega(p,\epsilon))\neq\emptyset$).

The proof of Theorem 2.6 follows the lines of the proof of Theorem 2.3, when
we consider an isometric immersion but now using the function $\cos(r_{p})$
with $r_{p}:=d_{p}\circ\pi$ and $d_{p}$ the geodesic distance in
$\mathbb{S}^{n}(1)$ to $p\in\mathbb{S}^{n}$ . We take into account in this
context that since the fibers of $\pi$ are minimal submanifolds, then, (see
[13], [4])

$\Delta^{\Sigma}(f\circ\pi)(q)=\bigg{(}\big{(}\Delta^{\mathbb{S}^{n}(1)}f\big{)}\circ\pi\bigg{)}(q)$

for any smooth function $f:\mathbb{S}^{n}\to\mathbb{R}$. Choosing $f$ as the
geodesic distance $d_{p}$ to $p\in\mathbb{S}^{n}(1)$ we can state, see Theorem
27 of [9] for instance,

$\Delta^{\Sigma}r_{p}=\Delta^{\Sigma}(d_{p}\circ\pi)=\big{(}\Delta^{\mathbb{S}^{n}(1)}d_{p}\big{)}\circ\pi=(n-1)\frac{\cos(d_{p})}{\sin(d_{p})}\circ\pi=(n-1)\frac{\cos(r_{p})}{\sin(r_{p})}$

Hence for any smooth function $F:\mathbb{R}\to\mathbb{R}$,

$\Delta^{\Sigma}F(r_{p})={\rm
div}(F^{\prime}(r_{p})\nabla^{\Sigma}r_{p})=F^{\prime\prime}(r_{p})\|\nabla^{\Sigma}r_{p}\|^{2}+(n-1)\frac{\cos(r_{p})}{\sin(r_{p})}F^{\prime}(r_{p}).$

By expressing $\nabla^{\Sigma}r$ in terms an orthonormal basis of the
horizontal distribution and then projecting on the base space using the fact
that $r_{p}=d_{p}\circ\pi$ we conclude that $\|\nabla^{\Sigma}r_{p}\|^{2}=1$
and hence

(18)
$\Delta^{\Sigma}F(r_{p})=F^{\prime\prime}(r_{p})+(n-1)\frac{\cos(r_{p})}{\sin(r_{p})}F^{\prime}(r_{p}).$

In particular

$\Delta^{\Sigma}\cos(r_{p})=-n\cos(r_{p}),$

what is everything needed for the proof of the above theorem.

As in the case of immersions, where we obtain Corollary 2.5 for the compact
case as a direct consequence of Theorem 2.3, we have the following
corresponding corollary for compact submersions deduced as a direct
application of Theorem 2.6:

######  Corollary 2.7.

Let $\pi:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be a complete, minimal and compact
Riemannian submersion, and let $p$ be any point of $\mathbb{S}^{n}(1)$. Then

  * •

Either $\pi(\Sigma)\subseteq E(p)$,

  * •

or, $\pi(\Sigma)\cap E(p)\neq\emptyset$

and hence, in any case, $\pi(\Sigma)\cap\Omega(p,\epsilon)\neq\emptyset$,
(i.e. $\pi^{-1}(\Omega(p,\epsilon))\neq\emptyset$).

##  3\. Main results

In the above Section, we have proved in Corollaries 2.5 and 2.7 that, when the
immersion $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ is compact and minimal, (and
respectively, the submersion $\pi:\Sigma^{m}\to\mathbb{S}^{n}(1)$ is compact
and minimal), then _always_ there are points of $x(\Sigma)$, (or
$\pi(\Sigma)$), included in any equatorial strip, some in one part of the
equator, some in the other part. The question now is to estimate the “amount”
of such points (in $x(\Sigma)$ or $\pi(\Sigma)$), which are included in these
equatorial strips.

We shall see that, as occurs in the intrinsic case, as the dimension
increases, the “greater” the number of points of the sets $x(\Sigma)$ or
$\pi(\Sigma)$ within the equatorial strips. Namely, we observe a _measure
concentration phenomenon_ satisfied by $x:\Sigma\to\mathbb{S}^{n}(1)$ and
$\pi:\Sigma^{m}\to\mathbb{S}^{n}(1)$.

The following first two theorems, (Theorem 3.1 and Theorem 3.2), describes a
“Fat equator effect” in a closed and minimal submanifold in the sphere, the
latter being a refinement of the first with respect to the rate of measure
concentration. Indeed, while in the first theorem this rate is polynomial, in
the second theorem it will be shown that the concentration of measure grows at
an exponential rate, as in Theorem A and its corollary Theorem B. Although the
second theorem improves clearly the first one, assuming the same hypotheses,
we show both in order to make visible the techniques used in both results.

We shall finish this Section stating and proving in Theorem 3.5 that, when the
total space $\Sigma$ in a minimal Riemannian submersion
$\pi:\Sigma\rightarrow\mathbb{S}^{n}(1)$ is compact, almost all measure in
$\Sigma$ concentrates in the the fiber bundle of the fibers projecting on the
equatorial strips $\Omega(p,\epsilon)$ of any width $\epsilon$ in the sphere
$\mathbb{S}^{n}(1)$. As we have mentioned in the Introduction, its proof is
based in the same techniques than we have used for the immersions.

######  Theorem 3.1 (Main I).

Let $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be an isometric and minimal immersion
of the closed $m$-dimensional manifold $\Sigma$ in the sphere
$\mathbb{S}^{n}(1)$. Then, for any point $p\in\mathbb{S}^{n}(1)$ and any
$\epsilon\in(0,\pi/2)$ we have

$1\geq\frac{{\rm vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}\geq 1-\frac{1}{(m+1)\sin^{2}\epsilon}.$

Hence, for each $p$ and $\epsilon$,

$\lim_{m\to\infty}\frac{{\rm vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}=1.$

###### Proof.

Given the extrisic distance function $r_{p}:\Sigma\rightarrow\mathbb{R}$, we
know that

(19)
$\displaystyle\operatorname{div}^{\Sigma}\big{(}\cos(r_{p}(q))\,\nabla^{\Sigma}\cos(r_{p}(q))\big{)}=$
$\displaystyle\cos(r_{p}(q))\big{(}\Delta^{\Sigma}\cos(r_{p}(q))\big{)}$
$\displaystyle+\|\nabla^{\Sigma}\cos(r_{p}(q))\|^{2}.$

Then, since $\Delta^{\Sigma}\cos(r_{p}(q))=-m\cos(r_{p}(q))$, and integrating
equation (19) along $\Sigma$, we have, using that $\Sigma$ is compact, that
$\cos\circ r_{p}$ is smooth in $\Sigma$ and the divergence theorem:

(20) $\displaystyle m\int_{\Sigma}\cos^{2}(r_{p})=$
$\displaystyle\int_{\Sigma}\|\nabla^{\Sigma}\cos(r_{p})\|^{2}-\int_{\Sigma}\operatorname{div}^{\Sigma}\big{(}\cos(r_{p})\nabla^{\Sigma}\cos(r_{p})\big{)}$
$\displaystyle=$
$\displaystyle\int_{\Sigma}\sin^{2}(r_{p})\|\nabla^{\Sigma}r_{p}\|^{2}$
$\displaystyle\leq$ $\displaystyle\int_{\Sigma}\sin^{2}(r_{p})={\rm
vol}(\Sigma)-\int_{\Sigma}\cos^{2}(r_{p}).$

Then

(21) $(m+1)\int_{\Sigma}\cos^{2}(r_{p})\leq{\rm vol}(\Sigma).$

Which implies, as $\Sigma-x^{-1}(\Omega(p,\epsilon))\subseteq\Sigma$,

(22) $(m+1)\int_{\Sigma-x^{-1}(\Omega(p,\epsilon))}\cos^{2}(r_{p})\leq{\rm
vol}(\Sigma)$

The strip around the equator $E(p)$ is ($\bar{\epsilon}=\sin\epsilon$):

$\Omega(p,\epsilon)=\left\\{x\in\mathbb{S}^{n}(1)\,:\,-\bar{\epsilon}<\cos\left({\rm
dist}^{\mathbb{S}^{n}(1)}(p,x)\right)<\bar{\epsilon}\right\\}.$

Since $r_{p}(q):={\rm dist}^{\mathbb{S}^{n}(1)}(p,x(q))$, the set
$x^{-1}(\Omega(p,\epsilon))$ can be written as

$x^{-1}(\Omega(p,\epsilon))=\left\\{q\in\Sigma\,:\,-\bar{\epsilon}<\cos\left(r_{p}(q))\right)<\bar{\epsilon}\right\\}.$

Then for all $q\in\Sigma-x^{-1}(\Omega(p,\epsilon))$, we have
$\bar{\epsilon}^{2}\leq\cos^{2}(r_{p}(q))$ and by using inequality (22) we
conclude that

$(m+1)\bar{\epsilon}^{2}\left({\rm vol}(\Sigma)-{\rm
vol}(x^{-1}(\Omega(p,\epsilon)))\right)\leq{\rm vol}(\Sigma),$

which finally implies the statement of the Theorem:

$\frac{{\rm vol}(x^{-1}(\Omega(p,\epsilon)))}{{\rm vol}(\Sigma)}\geq
1-\frac{1}{(m+1)\bar{\epsilon}^{2}}.$

∎

######  Theorem 3.2 (Main II).

Let $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be an isometric and minimal immersion
of the closed $m$-dimensional manifold $\Sigma$ in the sphere
$\mathbb{S}^{n}(1)$. Then, for any point $p\in\mathbb{S}^{n}(1)$ , any
$\epsilon\in(0,\pi/2)$ and any $\delta\in[0,\frac{1}{2})$ we have

(23) $1\geq\frac{{\rm vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}\geq
1-\sqrt{\frac{1}{1-2\delta}}e^{-\delta\bar{\epsilon}^{2}(m+1)}$

where $\bar{\epsilon}=\sin\epsilon$. Hence, for each $p\in\mathbb{S}^{n}(1)$
and $\epsilon>0$,

$\lim_{m\to\infty}\frac{{\rm vol}\left(x^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}=1$

###### Proof.

We need the following

######  Lemma 3.3.

Let $x:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be an isometric and minimal immersion
of the closed $m$-dimensional manifold $\Sigma$ in the sphere
$\mathbb{S}^{n}(1)$ and let $p\in\mathbb{S}^{n}(1)$. Then

(24)
$\int_{\Sigma}\cos^{2k}(r_{p})\leq\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2k}(\widetilde{r}_{p})}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

where $\widetilde{x}:\mathbb{S}^{m}(1)\rightarrow\mathbb{S}^{n}(1)$ is a
totally geodesic immersion of $\mathbb{S}^{m}(1)$ in $\mathbb{S}^{n}(1)$,
$r_{p}$ is the extrinsic distance function to $p$ given by the immersion $x$,
and $\widetilde{r}_{p}$ is the extrinsic distance to $p$ on
$\mathbb{S}^{n}(1)$ given by the totally geodesic immersion $\widetilde{x}$.
As a consequence, we have the following inequality, for a given $t>0$:

(25)
$\int_{\Sigma}e^{t\cos^{2}r_{p}}\leq\frac{\int_{\mathbb{S}^{m}(1)}e^{t\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

###### Proof.

Let us apply an inductive argument to prove inequality (24): we are going to
see first that this inequality is true for $k=1$. From (20) using that
$\|\nabla^{\Sigma}r_{p}\|\leq 1$ we conclude inequality (21):

$\int_{\Sigma}\cos^{2}(r_{p})\leq\frac{1}{m+1}{\rm vol}(\Sigma)$

Now, if we consider the totally geodesic submanifold
$\widetilde{x}:\mathbb{S}^{m}(1)\rightarrow\mathbb{S}^{n}(1)$, we conclude the
following equality, (in an analogous way to (20), but using now that
$\|\nabla^{\mathbb{S}^{m}(1)}\widetilde{r}_{p}\|=1$, because $\widetilde{x}$
is a totally geodesic immersion),

(26) $\int_{\mathbb{S}^{m}(1)}\cos^{2}(\widetilde{r}_{p})=\frac{1}{m+1}{\rm
vol}(\mathbb{S}^{m}(1)).$

Hence, we have

(27)
$\int_{\Sigma}\cos^{2}(r_{p})\leq\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2}(\widetilde{r}_{p})}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

and we have proved that the inequality (24) is true for $k=1$.

Assuming that it is true for $k>1$, namely, that

(28)
$\int_{\Sigma}\cos^{2k}(r_{p})\leq\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2k}(\widetilde{r}_{p})}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma),$

we see that it holds for $k+1$ in the following way: by using lemma 2.4 we
have for any $q\in\Sigma$ that

(29)
$\displaystyle\operatorname{div}^{\Sigma}\left(\cos^{2k+1}(r_{p}(q))\,\nabla^{\Sigma}\cos(r_{p}(q)\right)=\cos^{2k+1}(r_{p}(q))\Delta^{\Sigma}\cos(r_{p}(q)$
$\displaystyle+(2k+1)\cos^{2k}(r_{p}(q))\|\nabla^{\Sigma}\cos(r_{p}(q))\|^{2}$
$\displaystyle=-m\cos^{2k+2}(r_{p}(q))$
$\displaystyle+(2k+1)\cos^{2k}(r_{p}(q))\|\nabla^{\Sigma}\cos(r_{p}(q))\|^{2}.$

Then, integrating the above equation along $\Sigma$, we have, using that
$\Sigma$ is compact and the divergence theorem:

(30) $\displaystyle m\int_{\Sigma}\cos^{2(k+1)}(r_{p})=$
$\displaystyle(2k+1)\int_{\Sigma}\cos^{2k}(r_{p})\|\nabla^{\Sigma}\cos(r_{p})\|^{2}$
$\displaystyle-\int_{\Sigma}\operatorname{div}^{\Sigma}\left(\cos^{2k+1}(r_{p}))\nabla^{\Sigma}\cos(r_{p})\right)$
$\displaystyle=$
$\displaystyle(2k+1)\int_{\Sigma}\cos^{2k}(r_{p})\sin^{2}(r_{p})\|\nabla^{\Sigma}r_{p}\|^{2}$
$\displaystyle=$
$\displaystyle(2k+1)\int_{\Sigma}\cos^{2k}(r_{p})\left(1-\cos^{2}(r_{p})\right)\|\nabla^{\Sigma}r_{p}\|^{2}$
$\displaystyle\leq$
$\displaystyle(2k+1)\int_{\Sigma}\cos^{2k}(r_{p})-(2k+1)\int_{\Sigma}\cos^{2k+2}(r_{p}).$

Thence

(31) $\displaystyle\int_{\Sigma}\cos^{2(k+1)}(r_{p})\leq$
$\displaystyle\frac{2k+1}{m+2k+1}\int_{\Sigma}\cos^{2k}(r_{p})$
$\displaystyle\leq$
$\displaystyle\frac{2k+1}{m+2k+1}\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2k}(\widetilde{r}_{p})}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

On the other hand, same computations than in (30), but when we consider the
totally geodesic immersion of $\mathbb{S}^{m}(1)$ in $\mathbb{S}^{n}(1)$ gives
the formula

$\displaystyle m\int_{\mathbb{S}^{m}(1)}\cos^{2(k+1)}(\widetilde{r}_{p})=$
$\displaystyle(2k+1)\int_{\mathbb{S}^{m}(1)}\cos^{2k}(\widetilde{r}_{p})-(2k+1)\int_{\mathbb{S}^{m}(1)}\cos^{2k+2}(\widetilde{r}_{p})$

and hence

(32)
$\int_{\mathbb{S}^{m}(1)}\cos^{2(k+1)}(\widetilde{r}_{p})=\frac{2k+1}{m+2k+1}\int_{\mathbb{S}^{m}(1)}\cos^{2k}(\widetilde{r}_{p})$

Finally, using (31) and (32) we obtain

(33) $\displaystyle\int_{\Sigma}\cos^{2(k+1)}(r_{p})\leq$
$\displaystyle\frac{2k+1}{m+2k+1}\int_{\Sigma}\cos^{2k}(r_{p})$
$\displaystyle\leq$
$\displaystyle\frac{2k+1}{m+2k+1}\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2k}(\widetilde{r}_{p})}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$ $\displaystyle=$
$\displaystyle\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2(k+1)}(\widetilde{r}_{p})}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

Now, we are going to prove inequality (25): given a fixed $t>0$, we have,
applying inequality (24) that

$\int_{\Sigma}\frac{t^{k}\cos^{2k}(r_{p})}{k!}\leq\frac{\int_{\mathbb{S}^{m}(1)}\frac{t^{k}\cos^{2k}(\widetilde{r}_{p})}{k!}}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

so, applying dominated convergence theorem and the power series expansion of
the exponential function, we have, for a given $t>0$:

$\displaystyle\int_{\Sigma}e^{t\cos^{2}r_{p}}=$
$\displaystyle\sum_{k=0}^{\infty}\int_{\Sigma}\frac{t^{k}\cos^{2k}(r_{p})}{k!}$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}\frac{\int_{\mathbb{S}^{m}(1)}\frac{t^{k}\cos^{2k}(\widetilde{r}_{p})}{k!}}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm
vol}(\Sigma)=\frac{\int_{\mathbb{S}^{m}(1)}e^{t\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

∎

Now, as we have seen in the proof of Theorem 3.1 we have that for all
$q\in\Sigma-x^{-1}(\Omega(p,\epsilon))$,

$\bar{\epsilon}^{2}\leq\cos^{2}(r_{p}(q))$

and hence, for every $t>0$, and for all
$q\in\Sigma-x^{-1}(\Omega(p,\epsilon))$,

$e^{t\bar{\epsilon}^{2}}\leq e^{t\cos^{2}(r_{p}(q))}$

Therefore, using inequality (27), we conclude

(34) $\displaystyle e^{t\bar{\epsilon}^{2}}$
$\displaystyle\int_{\Sigma-x^{-1}(\Omega(p,\epsilon))}1\leq\int_{\Sigma-x^{-1}(\Omega(p,\epsilon))}e^{t\cos^{2}r_{p}}$
$\displaystyle\leq\int_{\Sigma}e^{t\cos^{2}r_{p}}\leq\frac{\int_{\mathbb{S}^{m}(1)}e^{t\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}{\rm vol}(\Sigma)$

Then we can state that

(35) $1-\frac{{\rm vol}(x^{-1}(\Omega(p,\epsilon))}{{\rm vol}(\Sigma)}\leq
e^{-t\bar{\epsilon}^{2}}\frac{\int_{\mathbb{S}^{m}(1)}e^{t\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}.$

Now we need the following

######  Lemma 3.4.

For any $0<t<\frac{m+1}{2}$,

$\frac{\int_{\mathbb{S}^{m}(1)}e^{t\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}\leq\sqrt{\frac{m+1}{m+1-2t}}.$

###### Proof.

Let us define the function

$s\mapsto
F(s):=\frac{\int_{\mathbb{S}^{m}(1)}e^{s\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}.$

By the dominated convergence theorem we know that $F(0)=1$ and

$F^{\prime}(s)=\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2}\widetilde{r}_{p}e^{s\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}$

By using lemma 2.4, the divergence theorem, and that since the immersion is
totally geodesic $\|\nabla\widetilde{r}_{p}\|^{2}=1$ we have

$\displaystyle mF^{\prime}(s)=$
$\displaystyle\frac{-\int_{\mathbb{S}^{m}(1)}\cos\widetilde{r}_{p}e^{s\cos^{2}\widetilde{r}_{p}}\Delta\cos\widetilde{r}_{p}}{{\rm
vol}(\mathbb{S}^{m}(1))}=\frac{-\int_{\mathbb{S}^{m}(1)}{\rm
div}\left(\cos\widetilde{r}_{p}e^{s\cos^{2}\widetilde{r}_{p}}\nabla\cos\widetilde{r}_{p}\right)}{{\rm
vol}(\mathbb{S}^{m}(1))}$
$\displaystyle+\frac{\int_{\mathbb{S}^{m}(1)}\left\langle\nabla\left(\cos\widetilde{r}_{p}e^{s\cos^{2}\widetilde{r}_{p}}\right),\nabla\cos\widetilde{r}_{p}\right\rangle}{{\rm
vol}(\mathbb{S}^{m}(1))}$ $\displaystyle=$
$\displaystyle\frac{\int_{\mathbb{S}^{m}(1)}\sin^{2}\widetilde{r}_{p}e^{s\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}+2s\frac{\int_{\mathbb{S}^{m}(1)}\sin^{2}\widetilde{r}_{p}{\cos^{2}\widetilde{r}_{p}}e^{s\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}$ $\displaystyle=$
$\displaystyle\frac{\int_{\mathbb{S}^{m}(1)}e^{s\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}-\frac{\int_{\mathbb{S}^{m}(1)}\cos^{2}\widetilde{r}_{p}e^{s\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}+2s\frac{\int_{\mathbb{S}^{m}(1)}\sin^{2}\widetilde{r}_{p}{\cos^{2}\widetilde{r}_{p}}e^{s\cos^{2}\widetilde{r}_{p}}}{{\rm
vol}(\mathbb{S}^{m}(1))}$ $\displaystyle\leq$ $\displaystyle
F(s)-F^{\prime}(s)+2sF^{\prime}(s).$

Thence, for $s<\frac{m+1}{2}$,

$\frac{F^{\prime}(s)}{F(s)}\leq\frac{1}{m+1-2s}$

Integrating the above inequality between $0$ and $t$ we conclude that

$\ln F(t)\leq\ln\sqrt{\frac{m+1}{m+1-2t}}.$

and the lemma is proved. ∎

Taking now $t=\delta(m+1)$ with $0\leq\delta<\frac{1}{2}$, the above lemma and
inequality (35) we conclude that

(36) $1-\frac{{\rm vol}(x^{-1}(\Omega(p,\epsilon))}{{\rm vol}(\Sigma)}\leq
e^{-\delta\bar{\epsilon}^{2}(m+1)}\sqrt{\frac{1}{1-2\delta}}.$

and the theorem is proved.∎

Finally, we prove that, in a Riemannian submersion of the sphere, the measure
concentrates in the fiber bundles which projects inside its equatorial strips:

######  Theorem 3.5.

Let $\pi:\Sigma^{m}\to\mathbb{S}^{n}(1)$ be a Riemannian submersion with
minimal fibers of the closed $m$-dimensional manifold $\Sigma$ on to the
sphere $\mathbb{S}^{n}(1)$. Then, for any point $p\in\mathbb{S}^{n}(1)$ , any
$\epsilon\in(0,\pi/2)$ and any $\delta\in[0,\frac{1}{2})$ we have

(37) $1\geq\frac{{\rm vol}\left(\pi^{-1}(\Omega(p,\epsilon))\right)}{{\rm
vol}(\Sigma)}=\frac{{\rm vol}\left(\Omega(p,\epsilon)\right)}{{\rm
vol}\left(\mathbb{S}^{n}(1)\right)}\geq
1-\sqrt{\frac{\pi}{2}}e^{-\epsilon^{2}\frac{n-1}{2}}.$

Hence, for each $\epsilon$,

(38) $\lim_{n\to\infty}\frac{{\rm
vol}\left(\pi^{-1}(\Omega(p,\epsilon))\right)}{{\rm vol}(\Sigma)}=1.$

###### Proof.

Since $\pi:\Sigma\rightarrow\mathbb{S}^{n}(1)$ is a Riemmanian submersion we
have that

$\|\nabla^{\Sigma}r_{p}(x)\|=\|\nabla\Sigma\left(d_{p}\circ\pi\right)\|=1,\quad
x\in\Sigma-\\{\pi^{-1}(p)\\}\cup\\{\pi^{-1}(-p)\\}$

where $d_{p}$ is the distance function on $\mathbb{S}^{n}(1)$ to $p$. Let us
denote by $D_{R}$ the sublevel sets of $r_{p}$, namely

$D_{R}:=\\{x\in\Sigma\,:\,r_{p}(x)<R\\}=\pi^{-1}(B_{R}(p))$

where $B_{R}(p)$ is the geodesic ball of radius $R$ of $\mathbb{S}^{n}(1)$
centered at $p$. By the coarea formula the function $t\mapsto U(t)={\rm
vol}(D_{t})$ has derivative

$U^{\prime}(t)=\int_{\partial D_{t}}\frac{1}{\|\nabla^{\Sigma}r_{p}\|}={\rm
vol}(\partial D_{t})$

for any $t\in(0,\pi)$. Now, we consider the function $E:\Sigma\to\mathbb{R}$
given by

$q\in\Sigma\mapsto E(q)=\int_{0}^{r_{p}(q)}\frac{V(t)}{V^{\prime}(t)}dt$

where $V(t)={\rm vol}(B_{t}(p))$, and $V^{\prime}(t)=\frac{d}{dt}V(t)={\rm
vol}(\partial B_{t}(p))$. As $E=E(r_{p})$ is a radial function, we have, using
equation (18), that

$\Delta^{\Sigma}E=1.$

Hence by using the divergence theorem we can deduce that

$\displaystyle
U(t)=\int_{D_{t}}\Delta^{\Sigma}E=\frac{V(t)}{V^{\prime}(t)}\int_{\partial
D_{t}}\|\nabla r_{p}\|=\frac{V(t)}{V^{\prime}(t)}U^{\prime}(t).$

Thence,

$\frac{d}{dt}\left(\frac{U(t)}{V(t)}\right)=0$

for all $t\in(0,\pi)$. This implies that for any $R\in(0,\pi)$

$\frac{{\rm vol}(\pi^{-1}(B_{R}(p)))}{{\rm
vol}(B_{R}(p))}=\lim_{R\to\pi}\frac{{\rm vol}(\pi^{-1}(B_{R}(p)))}{{\rm
vol}(B_{R}(p))}=\frac{{\rm vol}(\Sigma)}{{\rm vol}(\mathbb{S}^{n}(1))}.$

Finally the theorem follows by using

$\displaystyle\frac{{\rm vol}(\pi^{-1}(\Omega(p,\epsilon)))}{{\rm
vol}(\Sigma)}=$ $\displaystyle\frac{{\rm
vol}(\pi^{-1}(B_{\frac{\pi}{2}+\epsilon}(p)))}{{\rm vol}(\Sigma)}-\frac{{\rm
vol}(\pi^{-1}(B_{\frac{\pi}{2}-\epsilon}(p)))}{{\rm vol}(\Sigma)}$
$\displaystyle=$ $\displaystyle\frac{{\rm
vol}(B_{\frac{\pi}{2}+\epsilon}(p)))\frac{{\rm
vol}(\pi^{-1}(B_{\frac{\pi}{2}+\epsilon}(p)))}{{\rm
vol}(B_{\frac{\pi}{2}+\epsilon}(p)))}}{{\rm vol}(\Sigma)}-\frac{{\rm
vol}(B_{\frac{\pi}{2}-\epsilon}(p)))\frac{{\rm
vol}(\pi^{-1}(B_{\frac{\pi}{2}-\epsilon}(p)))}{{\rm
vol}(B_{\frac{\pi}{2}-\epsilon}(p)))}}{{\rm vol}(\Sigma)}$ $\displaystyle=$
$\displaystyle\frac{{\rm vol}(B_{\frac{\pi}{2}+\epsilon}(p))}{{\rm
vol}(\mathbb{S}^{n}(1))}-\frac{{\rm vol}(B_{\frac{\pi}{2}-\epsilon}(p))}{{\rm
vol}(\mathbb{S}^{n}(1))}$ $\displaystyle=$ $\displaystyle\frac{{\rm
vol}(\Omega(p,\epsilon))}{{\rm vol}(\mathbb{S}^{n}(1))},$

and the Theorem follows by using Theorem B.

∎

## Appendix I: Proof of theorem 1.2

Our theorem 1.2 is a consequence of Corollary V.2 in [7] where it is proved a
concentration of measure around the averaged/mean value of a Lipschitz
function defined on the sphere $\mathbb{S}^{n}(1)$. To apply this result in
our present context, we shall define, given a point $p\in\mathbb{S}^{n}$, the
function $f_{p}:\mathbb{S}^{n}(1)\rightarrow\mathbb{R}^{+}$ as

$q\mapsto f_{p}(q):=\cos(d_{p}(q))=\langle p,q\rangle$

where $d_{p}(q):={\rm dist}^{\mathbb{S}^{n}(1)}(p,q)={\rm arccos}(\langle
p,q\rangle)$ is the distance in the sphere from $p$ to $q$. The function
$f_{p}$ is a Lipschitz function with Lipschitz constant $1$ because given two
points $q,s\in\mathbb{S}^{n}$, we have that

$\displaystyle\left\|f_{p}(q)-f_{p}(s)\right\|$
$\displaystyle=\left\|\cos(d_{p}(q))-\cos(d_{p}(s))\right\|=\|\langle
p,q\rangle-\langle p,s\rangle\|$ $\displaystyle=\|\langle
p,q-s\rangle\|\leq\|q-s\|={\rm dist}^{\mathbb{R}^{n+1}}(q,s)\leq{\rm
dist}^{\mathbb{S}^{n}}(p,q).$

Moreover since $\Delta_{\mathbb{S}^{n}}\cos(d_{p})=-n\cos(d_{p})$ by using the
divergence theorem we conclude that the mean of $f_{p}$ in $\mathbb{S}^{n}(1)$
is

$\langle
f_{p}\rangle=\langle\cos(d_{p})\rangle:=\frac{\int_{\mathbb{S}^{n}}\cos(d_{p})}{{\rm
vol}(\mathbb{S}^{n})}=0.$

Now, let us consider the strip, (putting $\bar{\epsilon}:=\sin\epsilon$),

$\Omega(p,\epsilon)=\left\\{q\in\mathbb{S}^{n}(1)\,:\,\cos\left(d_{p}(q)\right)\in(-\sin\epsilon,\sin\epsilon)\right\\}=\\{q\in\mathbb{S}^{n}\,:\,|\cos(d_{p}(q))|<\bar{\epsilon}\\}$

so

$\displaystyle{\rm vol}(\Omega(p,\epsilon))$ $\displaystyle={\rm
vol}(\\{q\in\mathbb{S}^{n}\,:\,|\cos(d_{p}(q))|<\bar{\epsilon}\\})$
$\displaystyle={\rm vol}(\mathbb{S}^{n}(1))-{\rm
vol}(\\{q\in\mathbb{S}^{n}\,:\,|\cos(d_{p}(q))|\geq\bar{\epsilon}\\})$

Now, applying corollary V.2 of [7] to the Lipschitz function $f_{p}=\cos\circ
d_{p}$ with Lispchitz constant $\sigma=1$, we have that

$\frac{{\rm
vol}(\\{q\in\mathbb{S}^{n}\,:\,|\cos(d_{p}(q))|\geq\bar{\epsilon}\\})}{{\rm
vol}(\mathbb{S}^{n})}\leq 4e^{-\delta\bar{\epsilon}^{2}(n+1)}$

and hence

$\displaystyle\frac{{\rm vol}(\Omega(p,\epsilon))}{{\rm vol}(\mathbb{S}^{n})}$
$\displaystyle=\frac{{\rm vol}(\mathbb{S}^{n}(1))-{\rm
vol}(\\{q\in\mathbb{S}^{n}\,:\,|\cos(d_{p}(q))|\geq\bar{\epsilon}\\})}{{\rm
vol}(\mathbb{S}^{n})}$ $\displaystyle=1-\frac{{\rm
vol}(\\{q\in\mathbb{S}^{n}\,:\,|\cos(d_{p}(q))|\geq\bar{\epsilon}\\})}{{\rm
vol}(\mathbb{S}^{n}(1))}\geq 1-4e^{-\delta\bar{\epsilon}^{2}(n+1)}$

and the theorem is proved.

## References

  * [1]
  * [2] I. Bengtsson and K. Życzkowski.  Geometry of quantum states.  Cambridge University Press, Cambridge, 2006.  An introduction to quantum entanglement.
  * [3] Hyeong In Choi and Ai Nung Wang.  A first eigenvalue estimate for minimal hypersurfaces.  Journal of Differential Geometry, 18(3):559 – 562, 1983.
  * [4] P.B. Gilkey and J.H. Park  Riemannian submersions which preserve the eigenforms of the Laplacian,  Illinois J. of Mathematics, 40, no. 2, (1996), 194–201
  * [5] Mikhael Gromov, Filling Riemannian manifolds,  J. Differential Geometry, 18:1–147, 1983.
  * [6] Motoko Kotani.  The first eigenvalue of homogeneous minimal hypersurfaces in a unit sphere $S^{n+1}(1)$.  Tohoku Math. J. (2), 37(4):523–532, 1985.
  * [7] Vitali D. Milman and Gideon Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, with and Appendix by M. Gromov, 2nd Printing, Lecture Notes in Mathematics, 1200, Springer Verlag, Berlin (2001).
  * [8] Hideo Muto, Yoshihiro Ohnita, and Hajime Urakawa.  Homogeneous minimal hypersurfaces in the unit spheres and the first eigenvalues of their Laplacian.  Tohoku Math. J. (2), 36(2):253–267, 1984.
  * [9] P. Petersen.  Riemannian geometry, volume 171 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1998.
  * [10] Antonio Ros, A Two-Piece Property for Compact Minimal Surfaces in a Three-Sphere, Indiana University Mathematics Journal, 44(3) (1995), 841–849
  * [11] Takashi Sakai, Riemannian Geometry, Translations of Mathematical Monographs Volume 149, American Mathematical Society, 1996.
  * [12] T. Takahasi,  Minimal immersions of Riemannian manifolds,  J. Math. Soc. Japan, 18, n. 4, (1966), 380–385
  * [13] W. Watson  Manifold maps commuting with the Laplacian,  J. Differential Geometry, 8, (1973), 85–94
  * [14] Shing Tung Yau, editor.  Seminar on Differential Geometry, volume No. 102 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982.  Papers presented at seminars held during the academic year 1979–1980.
  * [15] Zizhou Tang and Wenjiao Yan.  Isoparametric foliation and Yau conjecture on the first eigenvalue.  J. Differential Geom., 94(3):521–540, 2013.


# To the Moon: Analyzing Collective Trading Events on the Wings of Sentiment
Analysis

Tim<EMAIL_ADDRESS>Universität Hamburg, Department
of Physics, 20355 Hamburg, Germany  Thomas Löhden  Universität Hamburg,
Department of Informatics, 20355 Hamburg, Germany  Stephan Leible  Universität
Hamburg, Department of Informatics, 20355 Hamburg, Germany  Jun-Patrick Raabe
Universität Hamburg, Department of Informatics, 20355 Hamburg, Germany

###### Abstract

This research investigates the growing trend of retail investors participating
in certain stocks by organizing themselves on social media platforms,
particularly Reddit. Previous studies have highlighted a notable association
between Reddit activity and the volatility of affected stocks. This study
seeks to expand the analysis to Twitter, which is among the most impactful
social media platforms. To achieve this, we collected relevant tweets and
analyzed their sentiment to explore the correlation between Twitter activity,
sentiment, and stock volatility. The results reveal a significant relationship
between Twitter activity and stock volatility but a weak link between tweet
sentiment and stock performance. In general, Twitter activity and sentiment
appear to play a less critical role in these events than Reddit activity.
These findings offer new theoretical insights into the impact of social media
platforms on stock market dynamics, and they may practically assist investors
and regulators in comprehending these phenomena better.

Keywords: Twitter, Sentiment, Stock market, GameStop, Short squeeze

##  1 Introduction

In 2021, the stock market witnessed several notable events that appear to be
linked to social media. Among these, the GameStop short squeeze gained the
most media attention. During this time, retail investors collectively
purchased large amounts of GameStop stock, originating from the social media
platform Reddit. Despite the lack of exceptional earnings or news, GameStop’s
stock price increased by over 20 times while retail investors coordinated and
shared their buying activity on various social media platforms, as described
by Chohan (2021). This period was characterized by an anti-establishment
attitude, community, and coordinated mobilization of capital, which we find
particularly intriguing.

Given the critical role of sentiment in motivating investment decisions during
these events, including this factor in our analysis of their impact is
essential. Although a direct link between the stock price increase and social
media attention may exist, we must further explore the exact nature of this
link. Lyócsa et al. (2022) made a good start by showing a significant
relationship between Reddit activity and stock price volatility. They used the
term "YOLO-Trading" to describe the investment decisions made during these
events, often driven by a fear of missing out without fully considering the
risks and consequences.

The observed decentralized, yet collective behavior strongly resembles herd
behavior in financial markets, which we find fascinating. The popular meme "To
the moon" expressed the hope that the stock would continue to rise, and we
coined the "MOON" variable as a reference to this sentiment. Although the
movement originated on Reddit, it is unclear whether Reddit remained the
center of influence on the stock market during the development of this event.
Therefore, our study aims to extend this analysis by focusing on Twitter,
which has a broader reach and a more prominent media presence than Reddit.

We believe that Twitter as a social media platform which is particularly
present in the media is important in this case. The nature of these events
requires a large number of people to participate, and Twitter is an excellent
platform for users to broadcast their posts to the world. The importance of
Twitter data for financial markets has been studied by Sul et al. (2014),
Tafti et al. (2016), and Abendschein & Watfe (2022). Therefore, we analyzed
the general sentiment on Twitter during these events by gathering 2 million
tweets from October 2020 to June 2021 using the Twitter API and the sentiment
analysis tool VADER (Valence Aware Dictionary for sEntiment Reasoning) (Hutto
& Gilbert (2014)) to calculate the sentiment of these tweets. We collected
tweets from four stocks: GameStop, AMC, Nokia, and BlackBerry. All of them
experienced unusually high price fluctuation in the time frame mentioned
above.

We suggest a significant relationship between a stock’s price fluctuations and
the activity or sentiment on Twitter regarding that stock experiencing such an
event. Therefore, we aim to answer the following research question: What is
the relationship between the sentiment analysis of Twitter data and stock
price variation? The remainder of this paper is structured as follows: In
chapter 2, we discuss relevant literature for our study, particularly the work
of Lyócsa et al. (2022). In chapter 3, we describe our research method. In
chapter 4, we state our results. In chapter 5, we discuss their impact on
relevant actors such as brokers, stock traders, and market regulators, and in
chapter 6, we present our concluding remarks.

##  2 Related Work

###  2.1 Sentiment analysis for financial interest

Recent studies have shown that sentiment analysis can be an effective tool for
predicting stock market movements and prices. Professional market
participants, as well as general sentiment expressed through social media
platforms like Twitter, have been found to predict future stock market
behavior strongly (Chen et al. (2014); Zheludev et al. (2014); Subramanian et
al. (2022); Bollen et al. (2011)). However, the volume of social media posts
can limit the usefulness of sentiment analysis, as there may not be enough
data to make a broad analysis. In our study, we address this problem by
focusing on social media-driven events, which generate an abundance of tweets.
We use Twitter sentiment analysis to explore the relationship between Twitter
sentiment and the stock market, a topic that has been extensively researched
(Souza et al. (2015)). To conduct our sentiment analysis on a large scale, we
used VADER, which has been shown to be especially well-suited for analyzing
the sentiment of Twitter data (Pano & Kashef (2020); Elbagir & Yang (2019)).
VADER outperforms other commonly used sentiment analyzers, particularly in the
social media domain (Bonta & Janardhan (2019)). The sentiment of a tweet can
be used to draw conclusions about volatility.

###  2.2 Volatility

Financial risks are often measured using volatility, the standard deviation of
a stock’s returns over a period. If the volatility is predictable, it can be a
valuable tool for risk management, as demonstrated by Christoffersen & Diebold
(2000). However, the predictability of volatility decreases as the forecast
horizon increases. According to Dumas et al. (2009), publicly available
information can be used effectively to improve the prediction of stock price
volatility. Behrendt & Schmidt (2018) say that the intraday volatility of
individual stocks can be affected by Twitter sentiment. Audrino et al. (2020)
also discovered that sentiment and attention variables could have significant
predictive power for future volatility.

###  2.3 Reddit as the primary source

The primary source of information for YOLO-Events is the website Reddit and
its subreddit "r/WallStreetBets." Lyócsa et al. (2022) analyzed the
correlation between Reddit activity and market price variation of stocks
experiencing YOLO-Events. The study found that increased activity on the
subreddit "r/WallStreetBets" led to increased stock price volatility for a
stock undergoing a YOLO-Event, as measured by the volume of comments
indicating an intensifying discussion. Long et al. (2023) extended this
analysis by exploring the sentiment of the Reddit discussion about GameStop.
They found that the links between the discussion were stronger during bullish
periods and weaker during bearish periods.

As we proceed with this research, our analysis will focus on the sentiment
expressed on Twitter, which is one of the most influential social media
platforms. The relationship between Twitter sentiment and stock price
volatility will be explored and compared with previous studies.

##  3 Research Method

###  3.1 Twitter data and sentiment analysis

For our research, it is necessary to gather a large number of tweets from the
period surrounding the GameStop event in order to study overall sentiment on
Twitter. To access the Twitter API V2, we utilize the Python library Tweepy
Roesslein (n.d.). We collect tweets from October 2020 to June 2021, with a
maximum of 100 tweets every 15 minutes, resulting in a potential total of 2
million tweets. However, not every 15-minute interval contains 100 tweets.
Additionally, we filter out tweets with images, videos, or retweets to ensure
that we include only the most relevant information. The final number of tweets
for each stock can be found in Table 1. table 1.

We use the following queries for each stock: “GameStop OR GME,” “AMC,”
“Blackberry OR $BB,” and “Nokia OR NOK.” For BlackBerry, we use the cashtag
“$BB” instead of the stock symbol “BB” to filter out irrelevant tweets.
Cashtags have been studied as a predictive tool in previous research (Cresci
et al. (2019); Bujari et al. (2017); Hentschel & Alonso (2014)). We sort the
tweets by relevancy to obtain the most relevant tweets for each 15-minute
interval. After collecting the tweets, we conduct sentiment analysis using the
VADER by Hutto & Gilbert (2014), exclusively using the compound score. We
calculate the average sentiment for each trading day and combine sentiment
data for non-trading days with the previous trading day.

| GameStop | AMC | Nokia | BlackBerry
---|---|---|---|---
Number of Tweets | $746\,560$ | $680\,872$ | $294\,608$ | $209\,690$
Table 1: Amount of tweets collected that were tweeted from October 2020 to
June 2021.

After collecting the tweets, we proceeded with the sentiment analysis. Using
VADER’s (Hutto & Gilbert (2014)) compound score, we calculate the average of
the sentiments for each trading day. All the sentiment data for non-trading
days will be combined with the previous trading day.

###  3.2 Volatility data

In order to investigate the price variations, we proceed similarly to Lyócsa
et al. (2022). The same approximation for the volatility is used, which
consists of three averages of different volatility estimations. This
approximation uses only daily data, which is an advantage over the
conventional volatility calculations that require intraday data. The daily
data can be obtained by different services. We use Yahoo Finance - Stock
Market Live, quotes, Business & Finance News (n.d.) for retrieving the daily
stock data. Our daily data consists of the opening price $O_{t}$, the price
high $H_{t}$, the price low $L_{t}$, and the closing price $C_{t}$ for a given
trading day $t$. With this, we define the following four quantities:

$\displaystyle h_{t}$ $\displaystyle=\ln{\frac{H_{t}}{O_{t}}},$ $\displaystyle
l_{t}$ $\displaystyle=\ln{\frac{L_{t}}{O_{t}}},$ $\displaystyle c_{t}$
$\displaystyle=\ln{\frac{C_{t}}{O_{t}}},$ $\displaystyle J_{t}$
$\displaystyle=\left(\ln{\frac{O_{t}}{C_{t-1}}}\right)^{2}.$ (1)

$h_{t}$, $l_{t}$ & $c_{t}$ are the logarithm of $H_{t}$, $L_{t}$ & $C_{t}$ in
relation to the opening price $O_{t}$. The variable $J_{t}$ is overnight price
variation. We will now explain the three estimators:

$\displaystyle P_{t}$
$\displaystyle=\frac{\left(h_{t}-l_{t}\right)^{2}}{4\cdot\ln{2}},$ (2)
$\displaystyle G_{t}$
$\displaystyle=0.511\left(h_{t}-l_{t}\right)^{2}-0.383\left(c_{t}\right)^{2}-0.019\left(c_{t}\left(h_{t}+l_{t}\right)-2h_{t}l_{t}\right),$
(3) $\displaystyle R_{t}$
$\displaystyle=h_{t}\left(h_{t}-c_{t}\right)-l_{t}\left(l_{t}-c_{t}\right).$
(4)

$P_{t}$ is the estimator of Parkinson (1980), $G_{t}$ is the estimator of
Garman & Klass (1980), and $R_{t}$ is the estimator of Rogers & Satchell
(1991). Finally, we combine these, adjust for the overnight price variation
$J_{t}$, and take the logarithm to approximate the logarithmic volatility:

$V_{t}=\ln\left(100^{2}\cdot
252\left(\frac{1}{3}\left(P_{t}+G_{t}+R_{t}\right)\right)\right)$ (5)

###  3.3 Regression models

We investigate different models to predict the next trading day’s volatility
using only the previous trading day’s data. The models M1, M2, M3, & M4 are
identical to that created by Lyócsa et al. (2022). They proposed two
parameters: $\text{YOLO}_{1,t}$ and $\text{YOLO}_{2,t}$. $\text{YOLO}_{1,t}$
corresponds to the Reddit activity for a given stock, and $\text{YOLO}_{2,t}$
captures a more general activity of the specific stock using Google Trends.
The models are linear regression models where the parameters $\beta_{i}$ are
obtained by Ordinary Least Squares (OLS). The four models are:

$\displaystyle\textbf{M1: }V_{t}$
$\displaystyle=\beta_{0}+\beta_{1}V_{t-1}+\beta_{2}M_{t-1}+\beta_{3}VIX_{t-1}+\epsilon_{t},$
(6) $\displaystyle\textbf{M2: }V_{t}$
$\displaystyle=\textbf{M1}_{t-1}+\beta_{4}\text{YOLO}_{1,t-1}+\epsilon_{t},$
(7) $\displaystyle\textbf{M3: }V_{t}$
$\displaystyle=\textbf{M1}_{t-1}+\beta_{5}\text{YOLO}_{2,t-1}+\epsilon_{t},$
(8) $\displaystyle\textbf{M4: }V_{t}$
$\displaystyle=\textbf{M1}_{t-1}+\beta_{4}\text{YOLO}_{1,t-1}+\beta_{5}\text{YOLO}_{2,t-1}+\epsilon_{t}.$
(9)

$V_{t-1}$ is the volatility of the previous trading day, $M_{t-1}$ is the
Google Trends activity for general terms linked to the GameStop short squeeze
(Lyócsa et al. (2022)), and $\text{VIX}_{t-1}$ is the Chicago Board Options
Exchange’s (CBOE) Volatility Index. M1 is the baseline model without any
Reddit or Twitter data. The models M2, M3, & M4 use all the parameters of M1,
hence the $\textbf{M1}_{t-1}$ term. In section 4, we will extend these models
with our parameters. M1-4 will be used for comparison.

##  4 Results

###  4.1 MOON parameter

We introduce two MOON parameters. The first MOON parameter
$\text{MOON}_{1,t}$, doesn’t use the sentiment. It is intended to correspond
to the general activity on Twitter for a given day. For each day, we collect
the tweet count for our queries for a given day in $N_{t}$. For all non-
trading days, we averaged the tweet count with the previous trading day. In
contrast to the sentiment analysis, we included tweets with video, images, and
other media attachments. As we will look at the logarithmic volatility, we
took the logarithm of this quantity. To summarize, we write
$\text{MOON}_{1,t}$ as:

$\text{MOON}_{1,t}=\log N_{t}$ (10)

The second parameter $\text{MOON}_{2,t}$, includes the VADER compound
sentiment value $S(T)$. Here, $T$ stands for a given tweet. $S(T)=1$ if the
tweet $T$ has positive sentiment, $S(T)=0$ for a neutral tweet, and $S(T)=-1$
for a negative tweet. We will take the standard deviation to get a sense of
the polarity for a given day:

$\text{MOON}_{2,t}=\text{std}\left(S(T)\right)_{t}$ (11)

We tried multiple ways of combining the sentiment, but the standard deviation
is one of the simplest and yielded consistent results. Other approaches we
tried were: the mean, the weighted mean by public metrics (likes, retweets,
quotes, and comments), and the ratio of positive or negative tweets.

In figs. 1, 2, 4 and 3, we visualize both of our MOON variables,
$\text{MOON}_{1,t}$ in red & $\text{MOON}_{2,t}$ in blue, together with the
volatility $V_{t}$ in black. The x-axis notes the observed time frame. While
the left y-axis shows the normalized MOON variables. $\text{MOON}_{1,t}$ and
$\text{MOON}_{2,t}$ were mapped linearly to the interval $[0,1]$ to improve
readability. The right y-axis shows the values of the logarithmic volatility,
as described in section 3.2.

The correlations in table 2 seem to agree with the visual representation in
fig. 1. There appears to be a correlation between the general Twitter activity
$\text{MOON}_{1,t}$ and the volatility $V_{t}$, but less so for the sentiment
on Twitter $\text{MOON}_{2,t}$. This pattern continues for the three other
stocks in figs. 2, 4 and 3.

Figure 1: Visualization of $\text{MOON}_{1,t}$, $\text{MOON}_{2,t}$, and
$V_{t}$ for GameStop.

Figure 2: Visualization of $\text{MOON}_{1,t}$, $\text{MOON}_{2,t}$, and
$V_{t}$ for AMC.

Figure 3: Visualization of $\text{MOON}_{1,t}$, $\text{MOON}_{2,t}$, and
$V_{t}$ for Nokia.

Figure 4: Visualization of $\text{MOON}_{1,t}$, $\text{MOON}_{2,t}$, and
$V_{t}$ for BlackBerry.

###  4.2 Correlations

We investigated the Pearson correlation coefficient between $V_{t}$,
$\text{MOON}_{1,t}$, $\text{MOON}_{2,t}$, $\text{YOLO}_{1,t}$, and
$\text{YOLO}_{2,t}$. The specific values for GameStop can be seen in table 2.
The results for the other stocks are in the Appendix.

GameStop | $\text{MOON}_{1,t}$ | $\text{MOON}_{2,t}$ | $\text{YOLO}_{1,t}$ | $\text{YOLO}_{2,t}$
---|---|---|---|---
$V_{t}$ | 0.699 | -0.119 | 0.762 | 0.527
$\text{MOON}_{1,t}$ |  | 0.088 | 0.628 | 0.753
$\text{MOON}_{2,t}$ |  |  | -0.104 | 0.271
$\text{YOLO}_{1,t}$ |  |  |  | 0.470
Table 2: GME Correlation

###  4.3 Regression models

We extend the models from section 3.3 by M5, M6, M7, and M8:

$\displaystyle\textbf{M5: }V_{t}$
$\displaystyle=\textbf{M1}_{t-1}+\beta_{6}\text{MOON}_{1,t-1}+\epsilon_{t},$
(12) $\displaystyle\textbf{M6: }V_{t}$
$\displaystyle=\textbf{M1}_{t-1}+\beta_{7}\text{MOON}_{2,t-1}+\epsilon_{t},$
(13) $\displaystyle\textbf{M7: }V_{t}$
$\displaystyle=\textbf{M1}_{t-1}+\beta_{6}\text{MOON}_{1,t-1}+\beta_{7}\text{MOON}_{2,t-1}+\epsilon_{t},$
(14) $\displaystyle\textbf{M8: }V_{t}$
$\displaystyle=\textbf{M1}_{t-1}+\beta_{6}\text{MOON}_{1,t-1}+\beta_{7}\text{MOON}_{2,t-1}$
$\displaystyle\quad\quad\quad\quad\,+\beta_{8}\text{YOLO}_{1,t-1}\;+\beta_{9}\text{YOLO}_{2,t-1}\;+\epsilon_{t}.$
(15)

The quality of the models, as expressed through the $R^{2}$ value, can be seen
in table 3. The models M1-4 use Reddit data, and M5-7 use Twitter data. M8
always performs the best because it uses all variables. Model M6
($\text{MOON}_{2,t}$) always performs worse than M5 ($\text{MOON}_{1,t}$), and
model M4 (Reddit data) outperforms M7 (Twitter data). A notable exception is
GME, where both models’ $R^{2}$ values are similar.

| GME |
---|---|---
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8
$R^{2}$ value | 0.429 | 0.499 | 0.463 | 0.520 | 0.512 | 0.435 | 0.512 | 0.560
| AMC |
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8
$R^{2}$ value | 0.335 | 0.459 | 0.373 | 0.460 | 0.366 | 0.343 | 0.384 | 0.468
| NOK |
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8
$R^{2}$ value | 0.304 | 0.451 | 0.347 | 0.463 | 0.335 | 0.319 | 0.346 | 0.479
| BB |
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8
$R^{2}$ value | 0.453 | 0.529 | 0.481 | 0.544 | 0.492 | 0.456 | 0.499 | 0.547
Table 3: Linear Regression Models

##  5 Discussion

In our study, we examined the relationship between Twitter activity, the
sentiment of this activity, and the price variation of the stocks of several
companies, namely GameStop, AMC, BlackBerry, and Nokia, during time frames of
special social media attention. Thereby, we have indeed found correlations
between sentiment and volatility. However, a rise in Twitter activity seems
more correlated to price variation. The sentiment parameter
$\text{MOON}_{1,t}$ can still improve the overall model performance indicating
that the sentiment also has insight into the volatility prediction. While a
lot of this price variation is linked to general stock market factors, our
findings conclude that Twitter activity can motivate more people to
participate in the event and become retail investors in these stocks.

Our findings extend the current body of knowledge on YOLO Events by making a
useful comparison with the findings of Lyócsa et al. (2022) and extending it
to Twitter. The study and methodology deliberately closely resemble theirs to
allow for comparison. We found that while the more significant part of the
price variation depends on other variables, increased attention on Twitter
seems to impact it. When comparing our results, we see that overall, Twitter
activity is a worse predictor of stock volatility than Reddit activity.
Notably, GME had the best performance of all the examined stocks and
approached Reddit data in predictability. Even though the event originated
from Reddit, Twitter was almost as good a predictor in the most attention-
grabbing of the stocks: GME. For the others, Twitter performed notably worse.
There seems to be a hurdle of public attention for Twitter to become a
predictive factor. Otherwise, the importance of Reddit as the origin of these
events remains. As seen in table 2, there is also a high correlation between
Twitter activity $\text{MOON}_{1,t}$, and the volatility $V_{t}$ compared to
the correlation between the sentiment $\text{MOON}_{2,t}$, and the volatility
$V_{t}$ for GME. In this case, the correlation with Twitter activity
$\text{MOON}_{1,t}$ is also higher than the factor corresponding to Google
activity $\text{YOLO}_{2,t}$.

Brokers were in an especially awkward position during the course of these
events. They are the main gateway to facilitate the trade of retail investors.
Since facilitating a transaction usually takes about two business days, they
were occasionally forced to stop and restrict these trades to continue
fulfilling their legal security obligations as brokers. These restrictions
greatly impact the stocks and their customers (Jones et al. (2021)). The
online broker “Robinhood Markets, Inc.” played a special role that was
especially popular among retail investors. On 28 January 2021, they restricted
the trading of stocks with high volatility, including the four stocks examined
by Kelleher et al. (2022). The role that Robinhood played during these events
was investigated by Pasztor (2021), Tan (2021), and Barber et al. (2022). Our
findings show that by observing Twitter activity and sentiment, these brokers
can’t prepare for these occasions as well as if they observed Reddit activity
directly. Only for the most-attention grabbing events, like the GameStop short
squeeze, can brokers utilize Twitter to anticipate these kinds of events to
better prepare for these occasions and avoid implementing trading
restrictions. On another note, brokers can also gain a competitive advantage
by offering their customers this real-time analysis of stocks. Sentiment
analysis is already recognized as beneficial for stock prediction (Ho et al.
(2019)). Our findings can be used to signal the onset and occurrence of such
special equity rallies. For stock traders, the knowledge that more extensive
attention on Twitter during these events can translate to increased
predictability of the volatility of the stock is also valuable when
participating in these events. The finding that sentiment has a negligible
impact on the predictability of price variation also opens up additional
questions regarding how it can be instrumentalized during these events for
stock traders.

For market regulators, these kinds of events pose an extraordinary challenge.
The question of whether and how to regulate these events and manage their
impact remains to be resolved and requires a careful balance between
protection and freedom (Campbell (2016)). But the fact that these events can
have such an extraordinary impact on the stock market means that there has to
be careful consideration about the way they are managed. Social media analysis
can be useful for diagnosing whether such an event is taking place. If they
generally want to foresee these events, we find that Reddit is the better
place to pay attention. But if the goal is to spot more considerable market
disruptions to react to them, we find that Twitter can be almost as good at
signaling them. In combination, the two sources can even lead to better
predictability in general. For market regulators that want to anticipate major
events to react to them, we find that a combined observation of both Reddit
and Twitter has the best results.

In recent years, the number of bots on Twitter has increased Chu et al.
(2010), so the question naturally arises: Can bots be used to manipulate the
overall sentiment on Twitter for a given stock? This was also investigated by
Fan et al. (2020) and Cresci et al. (2019). Therefore, the detection and
filtering of bot messages could improve the proposed method here (Alothali et
al. (2018); Chavoshi et al. (2016)). Especially since Twitter, one of the most
influential social media platforms was acquired and privatized, leaving the
potential for the instrumentalization of these platforms for stock market
manipulation. Generating such an event artificially could have a similar
impact to what we showed, and therefore this avenue of research should be
explored. Can such an event be manufactured by, e.g., bots and still have a
similar impact, or are the effects caused entirely by the actual retail
investors?

This study has some potential limitations that should be considered in terms
of the validity of the results. We based our analysis on tweets collected via
the Twitter API. Due to our dependence on this API and its black-box nature,
we may have introduced a selection bias. By choosing a different query, we may
obtain other data and, therefore, different results. Additionally, since our
tweet selection was rudimentary, it stands to reason that the actual sentiment
regarding a stock, when including memes, videos, and other media, can be
different. We chose this tweet selection because we only focused on the
sentiment of text data. An interesting question for future research is whether
the value of observed sentiment can be substantially increased through a
better sentiment calculation process, e.g., finding a way to meaningfully
include memes and pictures or a better Tweet selection process. Future
research may try to solve these problems by having approaches to include memes
and other media in this analysis, as done by French (2017). The change in
observed sentiment could alter our analyzed connections between our parameters
and stock price volatility, which could lead to different conclusions from
this research.

##  6 Conclusion

In recent years we observed several occasions where certain stocks seem to be
strongly influenced by coordinated movements on social media, called YOLO-
Events. This study investigates the link between stock volatility, Twitter
activity, and sentiment regarding stocks during YOLO-Events. We do this by
constructing parameters from the observed sentiment and activity and
correlating them to approximated stock volatility. Additionally, we created
linear regression models with these parameters.

Regarding our research question, we find that the sentiment doesn’t improve
the performance of these models significantly most of the time. During the
biggest of the events, sentiment didn’t improve the model performance at all.
Generally, the activity on Twitter and stock volatility have a high
correlation, but we also find that Reddit is the better predictor for these
events. Our findings point to an existing link between especially Twitter
activity and stock volatility and a weak link to Twitter sentiment.

##  7 Acknowledgements

We thank the authors of Lyócsa et al. (2022) for corespondents and for making
their data available to us. We thank the Digital and Data Literacy in Teaching
Lab and the Stiftung Innovation in der Hochschullehre for their expertise and
financial support.

##  8 Appendix

AMC | $\text{MOON}_{1,t}$ | $\text{MOON}_{2,t}$ | $\text{YOLO}_{1,t}$ | $\text{YOLO}_{2,t}$
---|---|---|---|---
$V_{t}$ | 0.459 | -0.363 | 0.618 | 0.417
$\text{MOON}_{1,t}$ |  | 0.057 | 0.593 | 0.688
$\text{MOON}_{2,t}$ |  |  | -0.257 | 0.068
$\text{YOLO}_{1,t}$ |  |  |  | 0.574
Table 4: AMC Correlation NOK | $\text{MOON}_{1,t}$ | $\text{MOON}_{2,t}$ | $\text{YOLO}_{1,t}$ | $\text{YOLO}_{2,t}$
---|---|---|---|---
$V_{t}$ | 0.608 | -0.061 | 0.509 | 0.391
$\text{MOON}_{1,t}$ |  | -0.118 | 0.464 | 0.779
$\text{MOON}_{2,t}$ |  |  | -0.102 | 0.137
$\text{YOLO}_{1,t}$ |  |  |  | 0.401
Table 5: NOK Correlation BB | $\text{MOON}_{1,t}$ | $\text{MOON}_{2,t}$ | $\text{YOLO}_{1,t}$ | $\text{YOLO}_{2,t}$
---|---|---|---|---
$V_{t}$ | 0.740 | -0.044 | 0.658 | 0.517
$\text{MOON}_{1,t}$ |  | 0.101 | 0.691 | 0.778
$\text{MOON}_{2,t}$ |  |  | -0.109 | 0.187
$\text{YOLO}_{1,t}$ |  |  |  | 0.439
Table 6: BB Correlation

## References

  * (1)
  * Abendschein & Watfe (2022) Abendschein, M. & Watfe, G. (2022), ‘From cashtag to hashcrash-predicting financial market volatility with twitter’, Available at SSRN 4141474 .
  * Alothali et al. (2018) Alothali, E., Zaki, N., Mohamed, E. A. & Alashwal, H. (2018), Detecting social bots on twitter: a literature review, in ‘2018 International conference on innovations in information technology (IIT)’, IEEE, pp. 175–180.
  * Audrino et al. (2020) Audrino, F., Sigrist, F. & Ballinari, D. (2020), ‘The impact of sentiment and attention measures on stock market volatility’, International Journal of Forecasting 36(2), 334–357.
  * Barber et al. (2022) Barber, B. M., Huang, X., Odean, T. & Schwarz, C. (2022), ‘Attention-induced trading and returns: Evidence from robinhood users’, The Journal of Finance 77(6), 3141–3190.
  * Behrendt & Schmidt (2018) Behrendt, S. & Schmidt, A. (2018), ‘The twitter myth revisited: Intraday investor sentiment, twitter activity and individual-level stock return volatility’, Journal of Banking & Finance 96, 355–367.
  * Bollen et al. (2011) Bollen, J., Mao, H. & Zeng, X. (2011), ‘Twitter mood predicts the stock market’, Journal of computational science 2(1), 1–8.
  * Bonta & Janardhan (2019) Bonta, V. & Janardhan, N. K. N. (2019), ‘A comprehensive study on lexicon based approaches for sentiment analysis’, Asian Journal of Computer Science and Technology 8(S2), 1–6.
  * Bujari et al. (2017) Bujari, A., Furini, M. & Laina, N. (2017), On using cashtags to predict companies stock trends, in ‘2017 14th IEEE Annual Consumer Communications & Networking Conference (CCNC)’, IEEE, pp. 25–28.
  * Campbell (2016) Campbell, J. Y. (2016), ‘Restoring rational choice: The challenge of consumer financial regulation’, American Economic Review 106(5), 1–30.
  * Chavoshi et al. (2016) Chavoshi, N., Hamooni, H. & Mueen, A. (2016), Debot: Twitter bot detection via warped correlation., in ‘Icdm’, Vol. 18, pp. 28–65.
  * Chen et al. (2014) Chen, H., De, P., Hu, Y. J. & Hwang, B.-H. (2014), ‘Wisdom of crowds: The value of stock opinions transmitted through social media’, The Review of Financial Studies 27(5), 1367–1403.
  * Chohan (2021) Chohan, U. W. (2021), ‘Yolo capitalism’, Available at SSRN 3775127 .
  * Christoffersen & Diebold (2000) Christoffersen, P. F. & Diebold, F. X. (2000), ‘How relevant is volatility forecasting for financial risk management?’, Review of Economics and Statistics 82(1), 12–22.
  * Chu et al. (2010) Chu, Z., Gianvecchio, S., Wang, H. & Jajodia, S. (2010), Who is tweeting on twitter: human, bot, or cyborg?, in ‘Proceedings of the 26th annual computer security applications conference’, pp. 21–30.
  * Cresci et al. (2019) Cresci, S., Lillo, F., Regoli, D., Tardelli, S. & Tesconi, M. (2019), ‘Cashtag piggybacking: Uncovering spam and bot activity in stock microblogs on twitter’, ACM Transactions on the Web (TWEB) 13(2), 1–27.
  * Dumas et al. (2009) Dumas, B., Kurshev, A. & Uppal, R. (2009), ‘Equilibrium portfolio strategies in the presence of sentiment risk and excess volatility’, The Journal of Finance 64(2), 579–629.
  * Elbagir & Yang (2019) Elbagir, S. & Yang, J. (2019), Twitter sentiment analysis using natural language toolkit and vader sentiment, in ‘Proceedings of the international multiconference of engineers and computer scientists’, Vol. 122, p. 16.
  * Fan et al. (2020) Fan, R., Talavera, O. & Tran, V. (2020), ‘Social media bots and stock markets’, European Financial Management 26(3), 753–777.
  * French (2017) French, J. H. (2017), Image-based memes as sentiment predictors, in ‘2017 International Conference on Information Society (i-Society)’, IEEE, pp. 80–85.
  * Garman & Klass (1980) Garman, M. B. & Klass, M. J. (1980), ‘On the estimation of security price volatilities from historical data’, Journal of business pp. 67–78.
  * Hentschel & Alonso (2014) Hentschel, M. & Alonso, O. (2014), ‘Follow the money: A study of cashtags on twitter’, First Monday 19(8).
  * Ho et al. (2019) Ho, S. Y., Choi, K. W. S. & Yang, F. F. (2019), ‘Harnessing aspect-based sentiment analysis: how are tweets associated with forecast accuracy?’, Journal of the Association for Information Systems 20(8), 2.
  * Hutto & Gilbert (2014) Hutto, C. & Gilbert, E. (2014), Vader: A parsimonious rule-based model for sentiment analysis of social media text, in ‘Proceedings of the international AAAI conference on web and social media’, Vol. 8, pp. 216–225.
  * Jones et al. (2021) Jones, C. M., Reed, A. V. & Waller, W. (2021), ‘When brokerages restrict retail investors, does the game stop?’, Columbia Business School Research Paper Forthcoming .
  * Kelleher et al. (2022) Kelleher, D. M., Grimes, J. & Chovil, A. (2022), ‘Securities—democratizing equity markets with and without exploitation: Robinhood, gamestop, hedge funds, gamification, high frequency trading, and more’, Western New England Law Review 44(1), 51.
  * Long et al. (2023) Long, S., Lucey, B., Xie, Y. & Yarovaya, L. (2023), ‘“i just like the stock”: The role of reddit sentiment in the gamestop share rally’, Financial Review 58(1), 19–37.
  * Lyócsa et al. (2022) Lyócsa, Š., Baumöhl, E. & Vỳrost, T. (2022), ‘Yolo trading: Riding with the herd during the gamestop episode’, Finance Research Letters 46, 102359.
  * Pano & Kashef (2020) Pano, T. & Kashef, R. (2020), ‘A complete vader-based sentiment analysis of bitcoin (btc) tweets during the era of covid-19’, Big Data and Cognitive Computing 4(4), 33.
  * Parkinson (1980) Parkinson, M. (1980), ‘The extreme value method for estimating the variance of the rate of return’, Journal of business pp. 61–65.
  * Pasztor (2021) Pasztor, J. (2021), ‘Robinhood–hero or villain?’, Journal of financial service professionals 75(5).
  * Roesslein (n.d.) Roesslein, J. (n.d.), ‘Tweepy documentation’, https://docs.tweepy.org/en/stable/.  Accessed: 2023-02-15.
  * Rogers & Satchell (1991) Rogers, L. C. G. & Satchell, S. E. (1991), ‘Estimating variance from high, low and closing prices’, The Annals of Applied Probability pp. 504–512.
  * Souza et al. (2015) Souza, T. T. P., Kolchyna, O., Treleaven, P. C. & Aste, T. (2015), ‘Twitter sentiment analysis applied to finance: A case study in the retail industry’, arXiv preprint arXiv:1507.00784 .
  * Subramanian et al. (2022) Subramanian, H., Angle, P. & Nagaraj, N. (2022), Do sentiment indices outperform quantitative indicators as predictors for cryptocurrency prices?, in ‘Proceedings of PACIS 2022’.
  * Sul et al. (2014) Sul, H., Dennis, A. R. & Yuan, L. I. (2014), Trading on twitter: The financial information content of emotion in social media, in ‘2014 47th Hawaii International Conference on System Sciences’, IEEE, pp. 806–815.
  * Tafti et al. (2016) Tafti, A., Zotti, R. & Jank, W. (2016), ‘Real-time diffusion of information on twitter and the financial markets’, PloS one 11(8), e0159226.
  * Tan (2021) Tan, G. K. S. (2021), ‘Democratizing finance with robinhood: Financial infrastructure, interface design and platform capitalism’, Environment and Planning A: Economy and Space 53(8), 1862–1878.
  * Yahoo Finance - Stock Market Live, quotes, Business & Finance News (n.d.) Yahoo Finance - Stock Market Live, quotes, Business & Finance News (n.d.), https://finance.yahoo.com/.  Accessed: 2023-02-16.
  * Zheludev et al. (2014) Zheludev, I., Smith, R. & Aste, T. (2014), ‘When can social media lead financial markets?’, Scientific reports 4(1), 4213.