# Entropy bounds for self-shrinkers with symmetries

John Man Shun Ma, Ali Muhammad

###### Abstract.

In this work we derive explicit entropy bounds for two classes of closed self-
shrinkers: the class of embedded closed self-shrinkers recently constructed in
[28] using isoparametric foliations of spheres, and the class of compact non-
spherical immersed rotationally symmetric self-shrinkers. These bounds
generalize the entropy bounds found in [19] on the space of complete embedded
rotationally symmetric self-shrinkers.

The authors were supported by the Danish National Research Foundation CPH-
GEOTOP-DNRF151.

##  1\. Introduction

A smooth hypersurface $\Sigma$ in $\mathbb{R}^{n+1}$ is called a self-shrinker
if

(1.1) $H_{\Sigma}(x)=\frac{1}{2}\langle x,\nu\rangle,\ \ \text{ for all
}x\in\Sigma.$

Here $\nu$ is the unit normal vector field on $\Sigma$,
$H_{\Sigma}=\operatorname{div}\nu$ is the mean curvature of $\Sigma$ and $x$
is the position vector. Self-shrinkers are central objects of study in the
mean curvature flow (MCF) since they serve as singularity models [14], [15].
Besides the generalized cylinders, different techniques have been used to
construct explicit examples of self-shrinkers [1], [2], [16], [6], [25], [12],
[17], [28].

In the seminal paper [8] (see also [20]), Colding and Minicozzi defined the
entropy $\lambda(\Sigma)$ of any smooth hypersurface $\Sigma$. The entropy is
a scaling and translation invariant quantity. More importantly, if
$\\{\Sigma_{t}\\}$ is a solution to the MCF, then
$t\mapsto\lambda(\Sigma_{t})$ is non-increasing and is constant if and only if
(up to rescaling and space-time translation) $\Sigma_{t}=\sqrt{-t}\Sigma$ for
$t\in(-\infty,0)$, where $\Sigma$ is a self-shrinker.

The entropy of the generalized cylinders were computed in [29], and the
entropy of the Angenent doughnuts were numerically computed in [5], [13]. In
general, upper bounds for the entropy are of great interest, since they
normally imply surprising topological and geometrical consequences [22], [4],
[7], [19].

In [19], the authors and Møller proved an entropy upper bound for complete
embedded rotationally symmetric self-shrinkers in $\mathbb{R}^{n+1}$. Using
that, several compactness and finiteness results for the space of such self-
shrinkers were obtained. The goal of this paper is to generalize the entropy
upper bounds obtained in [19, Theorem 1.1] in two directions.

Using the theory of isoparametric foliations of spheres, Riedler constructed
in [28] new examples of closed embedded self-shrinkers in $\mathbb{R}^{n+1}$
with topology $\mathbb{S}^{1}\times M$, where $M\subset\mathbb{S}^{n}$ is
diffeomorphic to an isoparametric hypersurface of $\mathbb{S}^{n}$ for which
the multiplicities of the principal curvatures agree. This is done by applying
a reduction theorem from [27] to reduce the self-shrinker equation to a
geodesic equation in an open subset of $\mathbb{R}^{2}$. Indeed, the
construction generalizes the classical work by Angenent [2] for rotationally
symmetric self-shrinkers. Our first result is the following theorem which
provides entropy bounds for this class of self-shrinkers.

######  Theorem 1.1.

Let $\mathscr{M}=\\{M_{\varphi}\\}_{\varphi\in(0,\pi/g)}$ be an isoparametric
foliation of $\mathbb{S}^{n}$ of type $(g,m,m)$, and let $N$ be a
$f$-invariant embedded closed self-shrinker diffeomorphic to
$\mathbb{S}^{1}\times M$, where $M$ is diffeomorphic to a regular fiber of the
foliation $\mathscr{M}$. Then there is a positive number $E_{g,m}$ such that
$\lambda(N)\leq E_{g,m}$.

We remark that the constants $E_{g,m}$ are explicit and depend only on the
pair $(g,m)$. We refer the reader to section 3 for the terminology on
isoparametric foliations appearing in Theorem 1.1.

Our second result is Theorem 1.2 below which gives entropy bounds for immersed
rotationally symmetric self-shrinkers diffeomorphic to
$\mathbb{S}^{1}\times\mathbb{S}^{n-1}$. Drugan and Kleene constructed in [11]
infinitely many complete immersed rotationally symmetric self-shrinkers of
various topological types. In the following theorem, we obtain an entropy
upper bound for compact immersed non-spherical rotationally symmetric self-
shrinkers in terms of the number of self-intersections of the corresponding
profile curve. For each $n\geq 2$, let $E_{n}$ denote the entropy upper bound
for complete embedded rotationally symmetric self-shrinkers in
$\mathbb{R}^{n+1}$ obtained in [19, equation (3.1)].

######  Theorem 1.2.

Let $\Sigma=\Sigma_{\sigma}$ be a compact immersed non-spherical rotationally
symmetric self-shrinker in $\mathbb{R}^{n+1}$, where the profile curve
$\sigma$ has $k$ self-intersection points counted with multiplicity. Then

(1.2) $\lambda(\Sigma)\leq(k+1)E_{n}.$

We refer the reader to section 4 and 5 for the terminology on rotationally
symmetric self-shrinkers appearing in Theorem 1.2. Both Theorem 1.1 and
Theorem 1.2 are proven using comparison theorems ([19, Theorem 3.1] and
Theorem 4.8 respectively), which provide estimates for the length of closed
geodesics $\sigma$ in a simply connected Riemannian surface with strictly
positive Gaussian curvature. Theorem 4.8 also implies an entropy bound for a
class of closed immersed $f$-invariant self-shrinkers (see remark 5.1).

The organization of this paper is as follows. In section 2, we recall the
definitions of self-shrinkers and the entropy of a hypersurface. In section 3,
we recall the construction in [28] and prove Theorem 1.1. In section 4, we
prove a general comparison theorem (Theorem 4.8) and in the last section, we
prove Theorem 1.2.

The first author would like to thank Hojoo Lee for their discussion on
immersed rotationally symmetric self-shrinkers. The authors would also like to
thank Alexander Mramor for fruitful discussions.

##  2\. Self-shrinkers and the entropy

Let $\Sigma$ be a properly immersed smooth hypersurface in $\mathbb{R}^{n+1}$.
For any $x_{0}\in\mathbb{R}^{n+1}$ and $t_{0}>0$, the functional
$F_{x_{0},t_{0}}$ is defined in [8] as

(2.1) $F_{x_{0},t_{0}}(\Sigma)=\frac{1}{(4\pi
t_{0})^{n/2}}\int_{\Sigma}e^{-\frac{\|x-x_{0}\|^{2}}{4t_{0}}}d\mu,$

Where $d\mu$ is the volume form of $\Sigma$. It is well known (e.g. [9,
section 1]) that the following statements are equivalent:

  * •

$\Sigma$ is a self-shrinker;

  * •

$\Sigma$ is a minimal hypersurface in $\mathbb{R}^{n+1}$ with respect to the
conformal (Gaußian) metric $g_{\text{\ss}}$, where

(2.2) $(g_{\text{\ss}})_{ij}=e^{-\frac{\|x\|^{2}}{2n}}\delta_{ij}$

and $\delta$ denotes the Euclidean metric on $\mathbb{R}^{n+1}$;

  * •

$\Sigma$ is a critical point of the $F$-functional $F=F_{0,1}$.

The entropy of a hypersurface $\Sigma$ is defined by ([8], [20])

(2.3)
$\lambda(\Sigma)=\sup_{(x_{0},t_{0})\in\mathbb{R}^{n+1}\times\mathbb{R}_{>0}}F_{x_{0},t_{0}}(\Sigma).$

We also need the following fact about the entropy of self-shrinkers [8,
section 7.2].

######  Lemma 2.1.

Let $\Sigma$ be a properly immersed self-shrinker in $\mathbb{R}^{n+1}$. Then
$\lambda(\Sigma)=F_{0,1}(\Sigma)$.

##  3\. Closed embedded self-shrinkers constructed by isoparametric foliations

In this section we prove Theorem 1.1. First, we recall the construction of the
closed embedded self-shrinkers in [28] using the theory of isoparametric
foliations of spheres, which generalizes the classical construction of
rotationally symmetric self-shrinkers by Angenent [2]. The definition of
isoparametric foliations and the properties of such foliations of spheres, in
particular, the structural theorems proved by Münzner in [23], [24], are
neatly summarized in [28, section 2] and we closely follow their notation. For
a general introduction, see for example [10, Chapter 3]. We start by briefly
recalling the necessary background on the theory. We then proceed to describe
how the problem of finding self-shrinkers in the setting of [28] is reduced to
finding geodesics in a certain open subset of $\mathbb{R}^{2}$ equipped with a
family of (incomplete) metrics $h_{g,m}=h$ with Gaussian curvature bounded
from below by strictly positive numbers.

###  3.1. Background on Isoparametric Foliations of $\mathbb{S}^{n}$

A smooth function $F:\mathbb{S}^{n}\to\mathbb{R}$ is called isoparametric if
there exist smooth functions
$\phi_{1},\phi_{2}:F(\mathbb{S}^{n})\to\mathbb{R}$ such that

$\displaystyle\|\text{grad}F\|^{2}=\phi_{1}\circ F,\quad\Delta F=\phi_{2}\circ
F,$

where $\text{grad}F$ and $\Delta F$ denote the gradient and the Laplacian of
$F$, respectively. Foliations of $\mathbb{S}^{n}$ that arise from a family of
level sets of an isoparametric function are called isoparametric foliations of
$\mathbb{S}^{n}$. Such level sets are then the fibers of a given foliation. By
the work of Münzner [23], the isoparametric function of a given foliation is a
restriction to $\mathbb{S}^{n}$ of a homogeneous polynomial
$F:\mathbb{R}^{n+1}\to\mathbb{R}$ of degree $g$ which satisfies certain
differential equations, where $g\in\\{1,2,3,4,6\\}$. Such a polynomial $F$ is
called the Cartan-Münzner polynomial. The foliation has exactly two singular
fibers $V_{\pm}=(F|_{\mathbb{S}^{n}})^{-1}(\pm 1)$, and each regular fiber is
given by

(3.1)
$M_{\varphi}=\\{x\in\mathbb{S}^{n}|\operatorname{dist}_{0}(x,V_{-})=\varphi\\},$

where $\varphi\in(0,\pi/g)$ and $\operatorname{dist}_{0}$ is the distance
function on the round $\mathbb{S}^{n}$. On each $M_{\varphi}$, there are $g$
(the degree of the Cartan-Münzner polynomial $F$) distinct principal
curvatures, and they each assume a constant value on a given regular fiber.
Furthermore, there are at most two distinct multiplicities for the principal
curvatures which we denote by $m_{1}$ and $m_{2}$, and we have the following
relation $n-1=\frac{g}{2}(m_{1}+m_{2})$. Also, for each $\varphi\in(0,\pi/g)$
we have:

(3.2)
$\operatorname{Vol}_{0}(M_{\varphi})=c\sin\left(\frac{g}{2}\varphi\right)^{m_{1}}\cos\left(\frac{g}{2}\varphi\right)^{m_{2}},$

where $c$ is some constant independent of $\varphi$ and
$\operatorname{Vol}_{0}(M_{\varphi})$ is the volume of $M_{\varphi}$ with
respect to the round metric on $\mathbb{S}^{n}$.

We shall call an isoparametric foliation $\mathscr{M}=\\{M_{\varphi}\\}$ of
type $(g,m_{1},m_{2})$ if the fibers have $g$ distinct principal curvatures
with corresponding multiplicities $m_{1}$ and $m_{2}$. In the following we
shall restrict to the case where the multiplicities agree, i.e. $m_{1}=m_{2}$
and hence consider foliations of type $(g,m,m)$. In this case we have $n=mg+1$
and we shall use $n$ and $m$ interchangeably through this relation.

In the next lemma we determine the number $c$ in (3.2) in terms of $g$ and
$m$. We start by recalling the following useful result: let
$\pi:(E,g_{E})\to(B,g_{B})$ be a surjective Riemannian submersion between two
Riemannian manifolds with compact fibers. Let $v:B\to\mathbb{R}$ be
$v(x)=\operatorname{Vol}_{g_{E}}(\pi^{-1}(x))$. By the co-area formula [26,
Theorem 2.1], for all submanifolds $M$ of $B$ we have

(3.3) $\operatorname{Vol}_{g_{E}}(\widetilde{M})=\int_{M}v(x)d\mu_{M}(x),$

where $\widetilde{M}=\pi^{-1}(M)$ and $d\mu_{M}$ is the volume form of $M$ in
$(B,g_{B})$.

######  Lemma 3.1.

Given an isoparametric foliation $\\{M_{\varphi}\\}$ of $\mathbb{S}^{n}$ of
type $(g,m,m)$, we have

(3.4)
$\operatorname{Vol}_{0}(M_{\varphi})=\frac{g\omega_{n}}{s(m)}\sin^{m}(g\varphi)$

for every regular fiber $M_{\varphi}$, $\varphi\in(0,\pi/g)$. Here $n=mg+1$,
$\omega_{n}$ is the volume of $\mathbb{S}^{n}$ and

(3.5) $s(m)=\int_{0}^{\pi}\sin^{m}t\;\mathrm{d}t.$

###### Proof.

Since $m_{1}=m_{2}=m$ we have from (3.2)

(3.6) $\operatorname{Vol}_{0}(M_{\varphi})=\frac{c}{2^{m}}\sin^{m}(g\varphi),$

hence it suffices to find $c$. By the relation (3.1) we know that the mapping

$\displaystyle\mathbb{S}^{n}\setminus(V_{-}\cup V_{+})$
$\displaystyle\to\left(0,\frac{\pi}{g}\right),$ $\displaystyle x\in
M_{\varphi}$ $\displaystyle\mapsto\varphi,$

is a Riemannian submersion, with respect to the standard metric on
$\mathbb{S}^{n}$ and the Euclidean metric on the interval $(0,\pi/g)$. Using
(3.3) and (3.6),

$\displaystyle\omega_{n}:=\operatorname{Vol}_{0}(\mathbb{S}^{n})=\int_{0}^{\pi/g}\operatorname{Vol}_{0}(M_{\varphi})d\varphi=\frac{c}{g2^{m}}\int_{0}^{\pi}\sin^{m}t\;\mathrm{d}t.$

Hence $c=g2^{m}\omega_{n}s(m)^{-1}$ and Lemma 3.1 is proved. ∎

For simplicity, from now on we write

(3.7) $c_{g,m}=\frac{g\omega_{n}}{s(m)}.$

###  3.2. The Reduction Theory; Proof of Theorem 1.1

Equip $\mathbb{R}^{n+1}\backslash\mathbb{R}_{\geq 0}\cdot(V_{+}\cup V_{-})$
with the metric $g_{\text{\ss}}$ defined in (2.2), and let

(3.8) $\displaystyle
g_{\text{S}}=e^{-\frac{r^{2}}{2n}}(dr^{2}+r^{2}d\varphi^{2})$

be a metric on $(0,\infty)\times(0,\pi/g)$. Define the map

(3.9) $\displaystyle f:\mathbb{R}^{n+1}\backslash\mathbb{R}_{\geq
0}\cdot(V_{+}\cup V_{-})$
$\displaystyle\to(0,\infty)\times\left(0,\frac{\pi}{g}\right),$ $\displaystyle
x$ $\displaystyle\mapsto\left(\|x\|,\frac{\arccos(F(x/\|x\|))}{g}\right).$

Here $F$ is the corresponding Cartan-Münzner polynomial and
$V_{\pm}=(F|_{\mathbb{S}^{n}})^{-1}(\pm 1)$ are the singular fibers of the
foliation. It can be shown that $f$ is a Riemannian submersion. A set
$N\subset\mathbb{R}^{n+1}$ is called $f$-invariant if there exists a set
$C\subset(0,\infty)\times(0,\pi/g)$ such that $N=f^{-1}(C)$.

Using [27, Theorem 4], it is proved in [28, Proposition 2.4] that an
$f$-invariant hypersurface $N\subset\mathbb{R}^{n+1}$ is a closed self-
shrinker if and only if $C:=f(N)$ is a closed geodesic in
$(0,\infty)\times(0,\pi/g)$ with respect to the metric $h$ defined by

(3.10)
$h(r,\phi):=\operatorname{Vol}_{g_{\text{\ss}}}(f^{-1}(r,\varphi))^{2}g_{\text{S}}(r,\phi).$

Note that for any $(r,\varphi)\in(0,\infty)\times(0,\pi/g)$, we have by Lemma
3.1

$\displaystyle\operatorname{Vol}_{g_{\text{\ss}}}(f^{-1}(r,\varphi))$
$\displaystyle=\operatorname{Vol}_{g_{\text{\ss}}}(rM_{\varphi})$
$\displaystyle=e^{-\frac{(n-1)r^{2}}{4n}}\operatorname{Vol}_{0}(rM_{\varphi})$
$\displaystyle=r^{n-1}e^{-\frac{(n-1)r^{2}}{4n}}\operatorname{Vol}_{0}(M_{\varphi})$
$\displaystyle=c_{g,m}r^{n-1}e^{-\frac{(n-1)r^{2}}{4n}}\sin^{m}(g\varphi).$

Using (3.8) we hence obtain the following expression for the metric $h$111In
equation (4) of [28], the metric has the term $e^{-r^{2}}$ instead of
$e^{-r^{2}/2}$ since they use a different scaling convention for self-
shrinkers.

(3.11)
$h(r,\phi)=c^{2}_{g,m}r^{2mg}e^{-r^{2}/2}\sin\left(g\varphi\right)^{2m}(dr^{2}+r^{2}d\varphi^{2}).$

By a direct calculation, the Gaussian curvature $K=K_{g,m}$ of $h$ is given by

$K=\frac{e^{r^{2}/2}}{c^{2}_{g,m}r^{2n}\sin\left(g\varphi\right)^{2m}}\left(r^{2}+\frac{(n-1)g}{\sin^{2}(g\varphi)}\right).$

It is clear that $K$ is strictly positive and that $K\to+\infty$ as
$(r,\varphi)$ tend to the boundary of $(0,\infty)\times(0,\pi/g)$. By simple
calculus, the Gaussian curvature $K$ is bounded from below by
$\kappa=\kappa_{g,m}>0$, where

(3.12)
$\kappa_{g,m}=\frac{e^{\frac{y_{g,m}}{2}}}{c_{g,m}^{2}y_{g,m}^{n}}\left(y_{g,m}+(n-1)g\right)$

and

(3.13) $y_{g,m}=\frac{(n-1)(2-g)+\sqrt{(g-2)^{2}(n-1)^{2}+8ng(n-1)}}{2}.$

Moreover, by (3.10), (3.3) and (2.2), the length $L_{h}(C)$ of the curve $C$
with respect to the metric $h$ is given by

$\displaystyle L_{h}(C)$
$\displaystyle=\int_{C}\operatorname{Vol}_{g_{\text{\ss}}}(f^{-1}(r,\varphi))d\mu_{C}(r,\phi)$
$\displaystyle=\operatorname{Vol}_{g_{\text{\ss}}}(f^{-1}(C))$
$\displaystyle=\operatorname{Vol}_{g_{\text{\ss}}}(N)$ (3.14)
$\displaystyle=\int_{N}e^{-\frac{\|x\|^{2}}{4}}d\mu_{N}.$

Where $d\mu_{C}$ is the volume form of $C$ in
$((0,\infty)\times(0,\pi/g),g_{\text{S}})$.

###### Proof of Theorem 1.1.

For each pair $(g,m)$ let

(3.15) $E_{g,m}=\frac{2\pi}{(4\pi)^{n/2}\sqrt{\kappa_{g,m}}}.$

Let $N$ be a $f$-invariant embedded self-shrinker in $\mathbb{R}^{n+1}$. Then
$C:=f(N)$ is the image of an embedded geodesic in $(0,\infty)\times(0,\pi/g)$
with metric $h$ given in (3.10). Note that $N$ is diffeomorphic to
$\mathbb{S}^{1}\times M$ if and only if $f(N)$ is an embedded closed geodesic.
Hence we can apply [19, Theorem 3.1] to conclude that

$L_{h}(C)\leq\frac{2\pi}{\sqrt{\kappa_{g,m}}}.$

Together with (3.2) and Lemma 2.1, one obtains the entropy bound
$\lambda(N)\leq E_{g,m}$. ∎

######  Remark 3.1.

When $g=1$, the isoparametric foliation of $\mathbb{S}^{n}$ is given (up to
congruence) by a family of $(n-1)$-spheres

(3.16) $M_{\varphi}=\\{\cos\varphi\\}\times\sin\varphi\ \mathbb{S}^{n-1},\ \
\varphi\in(0,\pi)$

and $f$-invariant hypersurfaces are percisely the rotationally symmetric
hypersurfaces. In [19], the authors and Møller derived entropy bounds for
complete embedded rotationally symmetric self-shrinkers. We remark that the
entropy bounds found in [19] are a special case of the ones obtained here:
when $g=1$, then $m=n-1$ and we obtain from (3.15) that

$E_{1,n-1}=\frac{2\pi}{(4\pi)^{n/2}\sqrt{\kappa_{1,n-1}}}.$

From (3.12) and $\kappa_{n}$ found in [19, section 3], we have

$\sqrt{\kappa_{1,n-1}}=\frac{1}{c_{1,n-1}}\sqrt{\kappa_{n}}.$

Together with the identity $\omega_{n}=s(n-1)\omega_{n-1}$, (3.7) and the
expression of $E_{n}$ in [19, equation (3.1)], we have $E_{1,n-1}=E_{n}$ for
all $n\geq 2$.

##  4\. Length upper bound on immersed closed geodesics in Riemannian surfaces
with positive curvature

In this section we prove Theorem 4.8, which gives an upper bound on the length
of closed immersed geodesics in a simply connected Riemannian surface with
Gaussian curvature bounded below by a strictly positive constant $\kappa$. We
start with a short outline of the section.

An immersed closed geodesic $\sigma$ encloses a number of domains. In
Proposition 4.4 we determine the number of these domains in terms of the
number of self-intersection points of $\sigma$. The proof of Theorem 4.8 then
reduces to proving an estimate of the length of the boundary of each domain.
This is done by showing that the domains are complete length spaces of
curvature $\geq\kappa$, for which one can prove an analogous result as in [19,
Theorem 3.1]. The necessary technical details are provided in Lemma 4.5 and
Proposition 4.6.

Let $\mathbb{V}$ be a Riemannian $2$-manifold homeomorphic to
$\mathbb{R}^{2}$.

######  Definition 4.1.

Let $\gamma:\mathbb{S}^{1}\to\mathbb{V}$ be an immersed closed curve. Let
$\operatorname{Image}(\gamma)$ denote the image of $\gamma$. A point $p$ in
$\operatorname{Image}(\gamma)$ is called a self-intersection point if
$\gamma^{-1}(p)$ has more than one element. A self-intersection point $p$ is
transverse if for all $t,s\in\gamma^{-1}(p)$, the pair
$\gamma^{\prime}(s),\gamma^{\prime}(t)$ are not parallel to each other. Let
$S$ be the set of all self-intersection points of $\gamma$ and let
$k\in\mathbb{N}$. We say that $\gamma$ has $k$ self-intersection points
counted with multiplicity if

(4.1) $\sum_{p\in S}(|\gamma^{-1}(p)|-1)=k.$

######  Definition 4.2.

Let $\gamma$ be an immersed closed curve in $\mathbb{V}$ with only transverse
self-intersections. A pre-compact connected component of
$\mathbb{V}\setminus\operatorname{Image}(\gamma)$ is called a domain enclosed
by $\gamma$.

######  Proposition 4.3.

Let $\gamma:\mathbb{S}^{1}\to\mathbb{V}$ be an immersed closed curve in
$\mathbb{V}$ with only transverse self-intersections. Then the boundary
$\partial U$ of each domain $U$ enclosed by $\gamma$ is piece-wise smooth and
at each corner of $\partial U$, the interior angle is less than $\pi$.

###### Proof.

It is clear that the boundary of $U$ is piece-wise smooth. At each corner, the
interior angle $\theta$ is not $\pi$ since $\gamma$ has transverse self-
intersections. If $\theta>\pi$, then there is a sub-arc $\beta$ of $\gamma$
which lies in the interior of $U$ and $\beta$ is not part of $\partial U$. Let
$\beta$ be the boundary of a domain $\tilde{U}$ enclosed by $\gamma$. But
since $\beta$ lies in the interior of $U$, we must have $U=\tilde{U}$, which
is a contradiction that $\beta$ is not part of $\partial U$. Hence we have
$\theta<\pi$. ∎

######  Proposition 4.4.

Let $\gamma$ be an immersed closed curve in $\mathbb{V}$ with $k$ transverse
self-intersections counted with multiplicity. Then $\gamma$ encloses exactly
$k+1$ domains.

###### Proof.

Let $U_{1},\cdots,U_{f}$ be the domains enclosed by $\gamma$. For every
$j=1,\cdots,f$ the closure of the domain $U_{j}$ has a piece-wise smooth
boundary consisting of subarcs of $\gamma$. Let $\alpha_{1},\cdots,\alpha_{e}$
be the collection of those sub-arcs. Let $S=\\{p_{1},\cdots,p_{\ell}\\}$ be
the set of self-intersection points of $\gamma$ and let
$W=\overline{U_{1}\cup\cdots\cup U_{f}}$. Since $W$ is simply connected, the
Euler formula implies

(4.2) $\chi(W)=1=\ell-e+f,$

where $\chi(W)$ is the Euler characteristic of $W$. Let
$s_{i}=|\gamma^{-1}(p_{i})|$. By definition there are $2s_{i}$ sub-arcs with
vertices $p_{i}$ (in the case where there is a loop, i.e., a sub-arc which
starts and ends at $p_{i}$, that sub-arc would be counted twice). Since each
sub-arc has exactly two vertices (also counted with multiplicity), we have

$e=\sum_{i=1}^{\ell}s_{i}$

and hence

(4.3) $f=1+e-\ell=1+\sum_{i=1}^{\ell}(s_{i}-1)=1+k$

by (4.1). This finishes the proof of the proposition. ∎

Let $h$ be a Riemannian metric on $\mathbb{V}$ such that the Gaussian
curvature $K$ satisfies $K\geq\kappa>0$ for some positive constant $\kappa$,
and let $\sigma$ be an immersed closed geodesic in $(\mathbb{V},h)$. By the
uniqueness theorem of solutions to ODE, any self-intersection point of
$\sigma$ must be transverse. We shall now describe the type of domains
$U\subset\mathbb{V}$ that can be realized as the domains enclosed by $\sigma$.

Let $U$ be a precompact, connected open subset in $(\mathbb{V},h)$ and let
$\Omega:=\overline{U}\subset\mathbb{V}$. Assume further that the boundary
$\partial\Omega$ is parameterized by a curve
$\beta:\mathbb{S}^{1}\to(\mathbb{V},h)$ consisting of piece-wise smooth
geodesic arcs, and which furthermore satisfies the following properties:
identifying $\mathbb{S}^{1}$ with $[0,d]$, with $0$ and $d$ identified, then
there is a partition

$\mathscr{P}=\\{0=t_{0}<t_{1}<\cdots<t_{a}=d(=0)\\}$

so that

(4.4) $\begin{cases}\beta|_{\mathbb{S}^{1}\setminus\mathscr{P}}\text{ is
injective,}\\\ \text{for each }j=0,\cdots,a-1,\ \beta|_{[t_{j},t_{j+1}]}\text{
is a geodesic in }(\mathbb{V},h),\text{ and}\\\ \text{for each
}j=0,\cdots,a-1\text{ the interior angle at }q_{j}:=\beta(t_{j})\text{ is less
than }\pi.\end{cases}$

We remark that if $\beta$ is injective, then each vertex $q_{j}$ is joined by
only two geodesic arcs $\beta|_{[t_{j-1},t_{j}]}$, $\beta|_{[t_{j},t_{j+1}]}$.

Assume first that $\partial\Omega$ is parametrized by an injective curve
$\beta$ which satisfies (4.4). Using Lemma 4.5 below, we will define as in
[19, Section 3] a metric structure $d^{\Omega}$ on $\Omega$ using the
Riemannian metric $h$ and show that $(\Omega,d^{\Omega})$ is a complete length
space with curvature $\geq\kappa$, where $\kappa$ is the strictly positive
lower bound of the Gaussian curvature of $(\mathbb{V},h)$. See [3] or [19,
Section 3] for the terminology and notation needed from metric geometry.

For each $p,q\in\Omega$, let

$d^{\Omega}(p,q)=\inf_{\gamma}L_{h}(\gamma),$

where the infimum is taken among piece-wise $C^{1}$ curves
$\gamma:[0,1]\to\Omega$ with $\gamma(0)=p$, $\gamma(1)=q$ and

(4.5)
$L_{h}(\gamma)=\int_{0}^{1}\sqrt{h(\gamma^{\prime}(t),\gamma^{\prime}(t))}dt$

is the length of $\gamma$ in $(\mathbb{V},h)$. The following lemma is proved
in [19, Lemma 3.3] when $\partial\Omega$ has no corners.

######  Lemma 4.5.

Let $\Omega$ be as above, i.e. a compact connected set in $(\mathbb{V},h)$ so
that the boundary $\partial\Omega$ is parametrized by an injective curve
$\beta$ which satisfies (4.4). Let $p,q\in\Omega$. Then there is a simple
geodesic $\gamma$ in $\Omega$ joining $p$ and $q$ which is shortest among all
piece-wise $C^{1}$ curves in $\Omega$ joining $p$ and $q$. Moreover,

  * (i)

$\gamma(t)\notin\\{q_{1},\cdots,q_{a}\\}$ for all $t\in(0,1)$.

  * (ii)

if one of $p,q$ is in the interior of $\Omega$, then $\gamma$ also lies in the
interior of $\Omega$ (except possibly at the other end point),

  * (iii)

if both $p,q$ are in $\operatorname{Image}(\beta)=\partial\Omega$, then either
$\gamma$ lies completely in the image of $\beta|_{[t_{j},t_{j+1}]}$ for some
$j=0,\cdots,a-1$, or the interior of $\gamma$ lies inside the interior of
$\Omega$.

###### Proof.

The Lemma follows from the same arguments used in the proof of [19, Lemma
3.3]. We only sketch the minor modifications here.

Let $p,q\in\Omega$, $p\neq q$. As in the proof of [19, Lemma 3.3], using a
minimizing sequence we construct a Lipschitz length minimizing curve
$\gamma:[0,1]\to\Omega$ with $\gamma(0)=p$, $\gamma(1)=q$, and we may assume
that $\gamma^{-1}(q)=\\{1\\}$, $\gamma^{-1}(p)=\\{0\\}$.

First we show (i) by contradiction: if not, let $s_{0}\in(0,1)$ be the first
time such that $\gamma(s_{0})=q_{j}$ for some $j$. Let $B$ be the closed
geodesic ball of radius $\epsilon$ in $(\mathbb{V},h)$ centered at $q_{j}$.
For $\epsilon$ small enough, $p,q$ are not in $B$. Hence there are
$s_{1},s_{2}\in(0,1)$ so that $s_{1}<s_{0}<s_{2}$ and
$\gamma(s_{i})\in\partial B$ for $i=1,2$. Note that since
$\beta|_{[t_{j-1},t_{j}]}$ and $\beta_{[t_{j},t_{j+1}]}$ are both geodesics,
under geodesic polar coordinates centered at $q_{j}$, the set $B\cap\Omega$ is
of the form

(4.6) $\\{(r,\theta):0\leq
r\leq\epsilon,\theta_{1}\leq\theta\leq\theta_{2}\\}$

with $\theta_{2}-\theta_{1}<\pi$ by the third assumption in (4.4). Assuming
$\epsilon$ is small enough so that $g$ is nearly Euclidean in $U$. Together
with $\theta_{2}-\theta_{1}<\pi$, the (Euclidean) straight line $\ell$ joining
$\gamma(s_{1})$, $\gamma(s_{2})$ has length

$\displaystyle
L_{h}(\ell)<d_{h}(\gamma(s_{1}),q_{j})+d_{h}(q_{j},\gamma(s_{2}))\leq
L_{h}(\gamma|_{[s_{1},s_{2}]}).$

Thus $\gamma$ cannot be length minimizing in $\Omega$ and hence (i) is proved.
Part (ii) and (iii) of the Lemma can be proven similarly as in the proof of
[19, Lemma 3.3] and are skipped. ∎

From Lemma 4.5 it follows that $(\Omega,d^{\Omega})$ is a complete length
space.

######  Proposition 4.6.

Let $\Omega$ be as in Lemma 4.5. Then the complete length space
$(\Omega,d^{\Omega})$ has curvature $\geq\kappa$, where $\kappa$ is the lower
bound of the Gaussian curvature of $(\mathcal{S},h)$.

###### Proof.

Let $x\in\Omega$ and let $V_{x}$ be a geodesically convex neighborhood in
$(\mathbb{V},h)$ centered at $x$. When $x$ is in the interior of $\Omega$, we
choose $V_{x}\subset\Omega$. Let $U_{x}=\Omega\cap V_{x}$. For any $p,q\in
U_{x}$, let $\gamma$ be the unique shortest geodesic in $V_{x}$ joining $p$
and $q$. Arguing as in the proof of [19, Proposition 3.6], it suffices to show
that $\gamma$ lies in $U_{x}$. The argument there works in our case when
$x\notin\\{q_{1},\cdots,q_{a}\\}$. For the case
$x\in\\{q_{1},\cdots,q_{a}\\}$, one may assume that $V_{x}$ is of the form
(4.6) for some $r_{0}>0$ and $\theta_{2}-\theta_{1}<\pi$. Arguing as in the
proof of [19, Proposition 3.6], one shows that $\gamma$ lies in
$H_{\theta_{0}}$ for all $\theta_{0}\in[\theta_{2},\theta_{1}+\pi]$, where in
geodesic polar coordinates

$H_{\theta_{0}}=\\{(r,\theta):0\leq r\leq
r_{0},\theta_{0}-\pi\leq\theta\leq\theta_{0}\\}.$

Since

$U_{x}=\bigcap_{\theta_{0}\in(\theta_{2},\theta_{1}+\pi)}H_{\theta_{0}},$

one concludes that $\gamma$ lies in $U_{x}$. ∎

Using exactly the same proof, by approximating the boundary of $\Omega$ by
geodesic polygons, one has the following generalization of [19, Theorem 3.1].

######  Theorem 4.7.

Let $\Omega$ be as in Lemma 4.5. Then the boundary $\partial\Omega$ satisfies

(4.7) $|\partial\Omega|:=L_{h}(\beta)\leq\frac{2\pi}{\sqrt{\kappa}}.$

Now we are ready to prove the main theorem in this section.

######  Theorem 4.8.

Let $\sigma$ be an immersed geodesic loop in $(\mathbb{V},h)$ with $k$ self-
intersection points counted with multiplicity. Then the length $L_{h}(\sigma)$
of $\sigma$ satisfies

(4.8) $L_{h}(\sigma)\leq(k+1)\frac{2\pi}{\sqrt{\kappa}}.$

###### Proof.

By Proposition 4.4, the immersed geodesic loop $\sigma$ encloses $k+1$
domains, denote the closure of these domains by
$\Omega_{1},\cdots,\Omega_{k+1}$. Fix each $i=1,\cdots,k+1$ and write
$\Omega=\Omega_{i}$ for simplicity. The boundary $\partial\Omega$ of $\Omega$
is parametrized by a closed curve $\beta:\mathbb{S}^{1}\to(\mathbb{V},h)$
consisting of piece-wise smooth geodesic arcs which satisfies (4.4), but which
may not be injective. Let $q$ be any vertex of $\partial\Omega$ (i.e.
$q=\beta(t_{j})$ for some $j=0,\cdots,a$). Let $s=|\beta^{-1}(q)|$. For any
small $\epsilon>0$, let $B=B_{\epsilon}(q)$ be the closed geodesic ball in
$(\mathbb{V},h)$ centered at $q$ with radius $\epsilon$. Then $\beta$
intersects $\partial B$ at $2s$ points, given by $\exp_{q}(\epsilon
e^{i\theta_{0}}),\cdots,\exp_{q}(\epsilon e^{i\theta_{2s-1}})$ in geodesic
polar coordinates $(r,\theta)$ centered at $q$. Moreover, we have

$\Omega\cap B=\\{\exp_{q}(re^{i\theta}):0\leq r\leq\epsilon,\ \
\theta_{2m}\leq\theta\leq\theta_{2m+1},m=0,\cdots,s-1\\}.$

For each $m$, let $\beta_{m}$ be the shortest geodesic in $B$ joining
$\exp_{q}(\epsilon e^{i\theta_{2m}})$ and $\exp_{q}(\epsilon
e^{i\theta_{2m+1}})$. Arguing as in the proof of Proposition 4.6, $\beta_{m}$
also lies inside $\Omega$. Let $\Delta_{m}$ be the geodesic triangle in
$\Omega$ with vertices $q$, $\exp_{q}(\epsilon e^{i\theta_{2m}})$ and
$\exp_{q}(\epsilon e^{i\theta_{2m+1}})$. Let $D_{\epsilon}$ be the collection
of all such triangles constructed for all vertices of $\partial\Omega$. Let
$\Omega_{\epsilon}=\Omega\setminus D_{\epsilon}$. Note that the interior of
$\Omega$ is homeomorphic to the interior of $\Omega_{\epsilon}$. Hence
$\Omega_{\epsilon}$ is connected. Moreover, the boundary
$\partial\Omega_{\epsilon}$ is parametrized by an injective curve consisting
of piece-wise smooth geodesic arcs which satisfies (4.4). Thus we can apply
Theorem 4.7 to conclude

$|\partial\Omega_{\epsilon}|\leq\frac{2\pi}{\sqrt{\kappa}}.$

Taking $\epsilon\to 0$, we obtain

$|\partial\Omega|\leq\frac{2\pi}{\sqrt{\kappa}}$

and hence

$L_{h}(\sigma)\leq\sum_{i=1}^{k+1}|\partial\Omega_{i}|\leq(k+1)\frac{2\pi}{\sqrt{\kappa}}.$

∎

##  5\. Entropy bounds for immersed non-spherical closed self-shrinkers

We recall that for any immersed curve $\sigma:I\to\mathbb{H}$, where
$\mathbb{H}$ is the upper half space, the immersed rotationally symmetric
hypersurface $\Sigma_{\sigma}$ in $\mathbb{R}^{n+1}$ with profile curve
$\sigma(s)=(x(s),r(s))$ is given by

$\Sigma_{\sigma}:=\\{(x(s),r(s)\omega):s\in I,\ \omega\in\mathbb{S}^{n}\\}.$

Now we prove Theorem 1.2.

###### Proof of Theorem 1.2.

By [19, Proposition 2.3], $\Sigma_{\sigma}$ is a self-shrinker if and only if
$\sigma$ is a geodesic in $(\mathbb{H},g_{A})$, where $g_{A}$ is given by

(5.1) $\displaystyle
g_{A}=r^{2(n-1)}e^{-\frac{x^{2}+r^{2}}{2}}(dx^{2}+dr^{2}).$

Since $\Sigma_{\sigma}$ is compact, either $I=\mathbb{S}^{1}$ and
$\Sigma_{\sigma}$ is an immersion from $\mathbb{S}^{1}\times\mathbb{S}^{n-1}$.
Or $I=(a,b)$, $\sigma(a^{+}),\sigma(b^{-})$ lie in $\partial\mathbb{H}$ and
$\Sigma_{\sigma}$ is an immersion from $\mathbb{S}^{n}$. Since
$\Sigma_{\sigma}$ is non-spherical, the latter case is excluded and thus
$\sigma$ is a closed immersed geodesic. By Theorem 4.8, the length of $\sigma$
satisfies

$L(\sigma)\leq(k+1)\frac{2\pi}{\sqrt{\kappa_{n}}},$

where $\kappa_{n}$ is the lower bound of the Gaussian curvature of $g_{A}$
computed in [19, section 3]. Together with

(5.2)
$\displaystyle\lambda(\Sigma_{\sigma})=(4\pi)^{-n/2}\omega_{n-1}L_{A}(\sigma)$

and the numbers $E_{n}$ from [19, equation (3.1)], given by

$\displaystyle E_{n}=\frac{2\pi\omega_{n-1}}{(4\pi)^{n/2}\sqrt{\kappa_{n}}}$

we obtain (1.2). ∎

######  Remark 5.1.

As in the proof of Theorem 1.2, one can use Theorem 4.8 to prove an entropy
upper bound for the class of $f$-invariant closed self-shrinkers $N$ such that
$f(N)$ is a closed immersed geodesic with $k$ self-intersections in
$(0,\infty)\times(0,\pi/g)$ with respect to the metric $h$ given in (3.11).
However, as of now no such examples are known to the authors’ knowledge.

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John Man Shun Ma
Department of Mathematics, Southern University of Science and Technology
No 1088, xueyuan Rd., Xili, Nanshan District, Shenzhen, Guangdong, China
518055
email<EMAIL_ADDRESS>

Ali Muhammad
Department of Mathematical Sciences, University of Copenhagen
Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
email<EMAIL_ADDRESS>