Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF
M. Carmen Domingo-Juan and Vicente Miquel \footnote{Mathematics Subject Classification(2010) 53C42
52C21
Let $(\oM,\<,\>,e^\psi)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold of $\oM$ with the induced metric and density. We give an upper bound on the first eigenvalue $\lambda_1$ of the closed eigenvalue problem for $\Delta_\psi$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector $\vHp+ \ona\psi$ with respect to the volume form induced by the density, where $\vHp$ is the mean curvature of $M$ associated to the density $e^\psi$. \\ When $\oM=\re^{n+k$ or $\oM=S^{n+k-1$, the equality between $\lambda_1$ and its bound implies that $e^\psi$ is a Gaussian density ($\psi(x) = \frac{C{2 |x|^2$, $C<0$), and $M$ is a shrinker for the mean curvature flow (MCF) on $\re^{n+k$. \\ We prove also that $\lambda_1 =-C$ on the standard shrinker torus of revolution. \\ Based on this and on the Yau's conjecture on the first eigenvalue of minimal submanifolds of $S^n$, we conjecture that the equality $\lambda_1=-C$ is true for all the shrinkers of MCF in $R^{n+k$.
1503.01332.tex
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\begin{document}


\title{Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF}
 

\author{ M. Carmen Domingo-Juan and Vicente Miquel 
\footnote{Mathematics Subject
Classification(2010) 53C42, 52C21}}

\date{}



\maketitle




\vspace{-1cm}
\begin{abstract}  
Let $(\oM,\<,\>,e^\psi)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold  of $\oM$ with the induced metric and density. We give an upper bound on the first eigenvalue $\lambda_1$ of the closed eigenvalue problem for $\Delta_\psi$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector $\vHp+ \ona\psi$ with respect to the volume form induced by the density, where $\vHp$ is the mean curvature of $M$ associated to the density $e^\psi$. \\
When $\oM=\re^{n+k}$ or $\oM=S^{n+k-1}$, the equality between $\lambda_1$ and its bound implies that $e^\psi$ is a Gaussian density ($\psi(x) = \frac{C}{2} |x|^2$, $C<0$), and $M$ is a shrinker for the mean curvature flow (MCF) on $\re^{n+k}$. \\
We prove also that  $\lambda_1 =-C$ on the standard shrinker torus of revolution. \\
Based on this and on the Yau's conjecture on the first eigenvalue of minimal submanifolds of $S^n$, we conjecture that the equality $\lambda_1=-C$ is true for all the shrinkers of MCF in $R^{n+k}$.
\end{abstract}



\section{Introduction }\lb{In}

After the seminal works of Bleecker, Weiner (\cite{BW}) and  Reilly (\cite{Re}), the obtention of extrinsic upper bounds for the eigenvalues of the Laplacian $\Delta$  on a submanifold in the Euclidean Space or other more general ambient spaces has become an interesting and fruitful problem. As examples of these works, one can look  at the references in \cite{GMO} and \cite{CESG}. 

Specially relevant for us are the quoted work of Reilly and  the paper \cite{he} by Heintze, where they got sharp bounds for the first eigenvalue of the Laplacian on a closed manifold embedded in the euclidean space, in the sphere or in spaces with sectional curvature bounded from above. They also prove that equality is attained only at minimal submanifolds of some sphere, in particular, at the sphere when the codimension of the submanifold is $1$ and the ambient space is the euclidean space.

In the last years  the study of the spectrum of the Laplacian associated to a density $\Delta_\psi$ has received an increasing interest. A sample is the works of Ma, Liu, Du (\cite{ma14,maliu,madu}), Cheng, Mejia, Zhu (\cite{chmezh12b}), Pigola, Rimoldi (\cite{piri}), Ding and Xin 
(\cite{dixi}). Recently, Batista, Cavalcante and Pyo (\cite{BCP}) have studied Reilly's type inequalities for $\Delta_\psi$, generalizing to this operator the Heintze's results on submanifolds of Riemannian manifolds with curvature bounded from above.
 The Laplacian associated to a density has also been used   in the study of Kähler-Einstein manifolds by Li (\cite{lilo}) and it appears in a natural way as a part of the operator used in \cite{comi} by Colding and Minicozzi in the study of the $F$-stability of shrinkers in mean curvature flow.

Our contribution to this line of research is: i)  a deeper understanding of the Reilly's type inequalities for the Laplacian associated to a density obtained in \cite{BCP}, with the observation that, in the euclidean space, equality holds only on shrinkers of the mean curvature flow (theorems \ref{TF} and \ref{CorF}), ii) the determination of the first eigenvalue of $\Delta_\psi$ on Angenent's type torus, proving that it attains the bound given by the previous inequality (Theorem \ref{l1A}), and iii) the observation of a relation  between the problem of the characterization of the submanifolds that attain the bound with a conjecture of Yau on the first eigenvalue of the Laplacian in a closed submanifold of the sphere. 

Before stating our results, we introduce the basic concepts.  


A $m$-dimensional Riemannian manifold $(\overline{M} , \overline{g})$ with a density is a Riemannian manifold where volumes are measured with a weighted (smooth) function 
$
 e^{\psi} : \overline{M} \flecha \Bbb R 
$
in the following way. If $\Omega$ is a domain in $\overline{M}$, $M$ is a closed submanifold of dimension $n$, and  $\mu_\og$ and $\mu_g$ are their respective riemannian volume elements, the volume elements associated to the density $e^\psi$ are $\overline\mu_\psi = e^{\psi} \ \mu_{\overline{g}}$ and $\mu_\psi = e^{\psi} \ \mu_{g}$ respectively. This gives, for the corresponding $m$-volume $V_{\psi} (\Omega)$ and $n$-volume $V_{\psi} (M)$,
\begin{equation}
V_{\psi} (\Omega) = \int_{\Omega} e^\psi \ \mu_{\overline{g}}, \qquad \quad 
V_{\psi} (M) = \int_{M} e^\psi \ \mu_{g} \label{VA}.
\end{equation}

The generalization of mean curvature vector for  manifolds with densities is denoted by $\vec{H_{\psi}}$ and given by 
\begin{equation}
\vec{H_{\psi}} = \vec H -  \(\overline{\nabla} \psi\)^\bot , \label{curvden}
\end{equation}
where $\(\overline{\nabla} \psi\)^\bot$ denotes the component of $\overline{\nabla} \psi$ orthogonal to $M$. With this definition $\vec{H_{\psi}}$ is the $L^2$-gradient (with respect to the measure $\mu_\psi$) of the functional $V_\psi$ defined on the $n$-dimensional submanifolds of $\oM$. 

When $\psi(x)=a + \frac12\ C\ |x|^2$ (the Gaussian density), it is well known (for instance see \cite{sm12}) that the submanifolds $F:M\flecha \re^m$ satisfying $\vec{H_{\psi}}=0$, that is
$\overrightarrow{H} = C \ F^{\perp}$ (where $^\perp$ denotes the component orthogonal to $M$), are the self-similar solutions of the mean curvature flow. They can be: {\it shrinkers}, when $C<0$, which contract to a point under the flow, {\it minimal submanifolds}, when $C=0$, and {\it expanders}, when $C>0$, which enlarge under the flow.

Along this paper we shall use  $\odv$,  $\oDelta$ and $\ona$ to denote the divergence, Laplacian and gradient or covariant derivative respectively in the ambient manifold $\oM$, and $\dv$,  $\Delta$, $\nabla$ to denote the corresponding operators on the submanifold $M$.

In a manifold with density, the divergence and Laplacian associated to the volume form $\mu_{\psi} $  are defined as
\begin{equation}
\mathrm{div}_{\psi} X \mu_{\psi} = \mathcal{L}_X \mu_{\psi}, \qquad \quad 
\Delta _{\psi} f = \mathrm{div} _{\psi} \mathrm{grad} \ f ,
\label{divlap}
\end{equation}
and called $\psi$-divergence and $\psi$-Laplacian (drifted or drifting Laplacian in some references). Our first result is  a remark on the   Reilly's type theorem obtained in \cite{BCP}. 

\begin{teor}[cf. Th. 1.5 in \cite{BCP}]
\lb{TF} Let $M$ be an $n$-dimensional compact submanifold of the euclidean space $\re^{n+k}$ endowed with a density $e^\psi$. The first eigenvalue $\lambda_1$ of the $\psi$-Laplacian $\Delta_\psi$ on $M$ associated to the metric and density induced on $M$ by those of $\re^{n+k}$ satisfies the inequality
\begin{equation}\lb{ineqTF}
\lambda_1 \le  \fracc{\int_M |\vHp + \ona\psi|^2 \ \mu_\psi}{n\  V_\psi(M)}.
\end{equation}
Moreover, if the equality holds then $M$ is a shrinker for the mean curvature flow in $\re^{n+k}$, and there is a point $p\in \re^{n+k}$ such that $\psi$ restricted to $M$ has the form $\psi|_M = a - \frac12 \lambda_1 r_p^2$, where $r_p$ denotes the euclidean distance to $p$,  and $\vec{H}(F(x)) = - \lambda _1 (F(x) - p)^{\perp}$, where $F$ is the embedding $F:M\flecha \re^{n+k}$. 

As a consequence, if $k=1$ and $H>0$, or if $n=2$, $k=1$ and $M$ has genus $0$, the equality  holds if and only if $M$ is a round sphere.
\end{teor}

The inequality \eqref{ineqTF} was proved, in a more general form, in Theorem 1.5 in \cite{BCP}. What is new in the statement of the Theorem \ref{TF} is the observation that the equality in \eqref{ineqTF} implies that $M$ must be an shrinker of the mean curvature flow in $\re^{n+k}$. This essentially appears when we consider as a condition on the function $\psi|_M$ what in  \cite{BCP} is written as: \lq\lq $M$ is contained in the hypersurface $\lambda \psi+\int^r s_\delta(t) dt =c$" and, moreover, restricts the ambient space to be $\re^{n+k}$.

\begin{nota}  When $\oM=\re^{n+k}$, the inequality of Lemma 3.2 in \cite{BCP} becomes
$\ds
0   = n \ V_{\psi} (M) 
+ \int_M \< \vHp + \overline{\nabla} \psi, F_p \>  \ \mu_{\psi}
$, where $F_p(x) := F(x)-p$.
If $M$ is a shrinker, this formula gives  \ $\ds \frac{\int_M |\vHp + \ona\psi|^2 \ \mu_\psi}{n\  V_\psi(M)} = \fracc{ C^2 \int_M r_p^2 \ \mu_\psi}{n\ V_\psi(M)}$ $=-C$. This $-C$  is an eigenvalue of $\Delta_\psi$ on $M$  and the coordinates $x_p^{i} = (x-p)^i$ of $M$  are the corresponding eigenfunctions (see \eqref{xieig}). 

It remains unproved if $-C$ is precisely the first eigenvalue. We shall go back to this question after we state our next theorem.
\end{nota}

\begin{teor}\lb{CorF}
Let $(\oM;\<,\>;\psi)$ be a Riemannian manifold with metric $\<,\>$ and density $e^\psi$. Let $M$ be an $n$-dimensional compact submanifold of $\oM$ of codimension $k$. The first eigenvalue $\lambda_1$ of the $\psi$-Laplacian $\Delta_\psi$ on $M$ associated to the induced metric and density on $M$ satisfies the inequality
\begin{equation}\lb{ineq}
\lambda_1 \le b^2 + \fracc{\int_M |\vHp + \ona\psi|^2 \ \mu_\psi}{n V_\psi(M)} \text{ for some constant $b$ which depends on } \oM
\end{equation}
Moreover, if $\oM$ is a sphere $S^{n+k}$ of radius $R$ in $\re^{n+k+1}$, then $b^2 = n/R^2$ and the equality implies: 
\begin{description}
\item[ i)] If the  center of $\psi$-mass, $p$, is the origin, then $\psi|_M$ is constant and $M$ is a minimal submanifold of $S^{n + k}$. 

\item[ ii)] If $p$ is not the origin and $k = 1$, then  $\psi|_M$ is constant and $M$ is a parallel of $S^{n + 1}$ with center at $p$.

\item[ iii)] If $p$ is not the origin and $k > 1$, then $\psi|_M$ is constant and $M$ is a  minimal submanifold of $S^{n + k -1}$, which is a parallel of $S^{n + k}$ with center at $p$. 
\end{description}
In case (ii) the converse is also true. In cases i) and iii) the converse is equivalent to the Yau's conjecture on the first eigenvalue of a minimal surface of a sphere (see, for instance, \cite{br13} section 5 for the statement of the conjecture and some history). 
\end{teor}

The concept of center of $\psi$-mass, its existence and uniqueness is explained in section \ref{pre}. By the moment the reader can look at it as a generalization of the center of mass when the volume elements associated to a density are used instead of the standard volume elements.

 Theorem \ref{CorF} has some overlapping with Theorem 1.5 in \cite{BCP}. But they are different results: our inequality \eqref{ineq} is weaker than the corresponding inequality in \cite{BCP}, but valid for any ambient space and any submanifold, without restrictions on the bound of the sectional curvature on the ambient space nor on the size of the submanifold. When the ambient space is the sphere, the inequality is the same in both works, but in \cite{BCP} there is restrictions on the size of the submanifold, whereas Theorem \ref{CorF} is valid for any compact submanifold $M$ of $S^{n+k}$. More concisely, the theorems 1.4 and 1.5 in \cite{BCP} generalize the theorems 2.3 and 2.1 in \cite{he}, whereas our theorem \ref{CorF} generalizes the theorem 1.1 in \cite{he}.

In view of the equivalence of the converse in cases i) and iii) of the above theorem with the Yau's conjecture on the first eigenvalue of a minimal surface of a sphere, we conjecture that this is also true for all compact shrinkers in $\re^{n+k}$, that is, we conjecture that if $M$ is a compact shrinker in $\re^{n+k}$ ($\vec{H} =  C \ (F-p)^{\perp}$, $C<0$), then $\lambda_1 = -C$. In \cite{dixi} it is proved that $\lambda_1\in[-C/2,-C]$, and other related bounds for $\lambda_1$ are obtained in \cite{madu} and \cite{piri}. Those estimates are in agreement of our conjecture, but, as a better
support of it, we give the following theorem, which states it in the best known example of shrinker which is not contained in a sphere.

\begin{teor}\lb{l1A} If $\psi$ is of Gaussian type (that is, $\psi(x)=a+\frac12 C \ r_p(x)^2$ with $a\in \re$ and $C<0$), then
the first eigenvalue of $\Delta_{\varphi}$ in an embedded shrinker of revolution in $\re^{3}$ symmetric respect to plane orthogonal to the axis of revolution  is $\lambda_1 = -C=\ds\fracc{\int_M |\vHp + \ona\psi|^2 \ \mu_\psi}{n\  V_\psi(M)}$. 
\end{teor}

We did the statement of Theorem \ref{l1A} in the more general way, but it is possible that, apart from the sphere, it refers only to a unique (up to homothety) surface, the Angenent's torus. In fact, Moller \cite{mo11} proves the existence of a revolution torus satisfying the hypothesis of Theorem \ref{l1A}, and it conjectures that this torus is unique (up to a homothety). If it is so, these will be Angenent's torus.

What is striking in these results is the fact that equality requires a Gaussian density and a shrinker. In previous theorems on bounds of the $\lambda_1$ for the $\psi$-Laplacian (\cite{huru}) the equality holds only when $\psi$ is constant. 
We found specially interesting the fact that, when $\psi$ is a Gaussian density, for the $\psi$-Laplacian, both, the sphere and the  Angenent's torus  have the same first eigenvalue (Theorem \ref{l1A}).

{\bf Acknowledgments:} 
Research partially supported by  the
DGI (Spain) and FEDER  project MTM2013-46961-P. and the Generalitat Valenciana Project  PROMETEOII/2014/064



\section{Preliminaries}\lb{pre}

\begin{defi} Let $f: \Bbb R \flecha \Bbb R$ be a ${C}^2$-function.
Let $M$ be a compact submanifold of $\re^m$ endowed with a density $e^\psi$, and let us denote by $\mathcal{F}: \re^m \flecha \Bbb R$ the function defined by $p \mapsto \mathcal{F}(p) = \int_M f(r_p(x)) \ \mu_{\psi}(x)$. The point  $x_0 \in \Bbb \re^m$ where $\mathcal F$  attains its minimum is called the $f$-center of $\psi$-mass of $M$.
\end{defi}

The following theorem of existence and unicity of the $f$-center of $\psi$-mass can be proved following the steps for the usual center of mass (see \cite{BeGo},\cite{Ka}, \cite{BK} and \cite{GMO})

\begin{teor}\lb{com}
Let $f: \Bbb R \flecha \Bbb R$ be a ${C}^2$-function satisfying $f'(t) > 0$ and $f''(t) > 0$ for every $t > 0$. If $M$ is a compact submanifold of $\re^m$, then there is a unique $f$-center of $\psi$-mass of $M$. 
\end{teor}

Let $p$ be the $f$-center of $\psi$-mass of $M$. Because $p$ is a minimum of $\mathcal F$ , the gradient of $\mathcal{F}$ must be $0$ at $p$, that is, for every $\zeta \in \re^m$, $0 = \<\grad\mathcal{F}(p),\zeta\> = \int_M f'(r_p(x)) \< \overline{\nabla} r_p, \zeta  \> \ \mu_{\psi}(x)$. In particular, if we take  $f(t) = \ds\frac{1}{2} t^2$,  $x_p^i$ are  the standard coordinates of $\re^m$ with origin at $p$, and $\zeta = \partial _{i}$ in $\re^m$, we have the following equality 
\begin{equation}\lb{xip0}
0 = \int_M r_p \< \overline{\nabla} r_p, \partial _{i}  \> \ \mu_{\psi} 
= \int_M x^{i}_p  \ \mu_{\psi} ,
\end{equation}
and we can use the $x_p^i$ as test functions to apply the Rayleigh priciple to the $\psi$-Laplacian. From now one we shall refer to the $\ds\frac{1}{2} t^2$-center of $\psi$-mass as the center of $\psi$-mass.

The usual divergence Theorem and the formula of integration by parts  take the following form for $\Delta_\psi$. 
\begin{equation}
\int_{M} {\mathrm{div}}_{\psi} X \  \mu_\psi = 0
\label{intdiv}
\end{equation}

\begin{align}
\int_{M} f {\Delta}_{\psi} h \  \mu_\psi &= 
\int_{M}  h {\Delta}_{\psi} f \  \mu_\psi = - \int_{M}  \< \nabla h, \nabla f \>  \mu_\psi 
\label{intlap}
\end{align}

 Thanks to formula \eqref{intlap}, the basic properties of the eigenvalues and eigenfunctions of $\Delta_\psi$ are the same that for the usual Laplacian (see, for instance, \cite{Ch}, pages 8 and 9), just changing the volume form $\mu$ by $\mu_\psi$. 
 
 On $C^\infty(M)$ we have the scalar product $(f,h)_{\psi 1} = \int_M f h \mu_\psi + \int_M \<\nabla f, \nabla h\> \mu_\psi$. 
 Instead of $L^2(M)$, when there is a density $e^\psi$ we use the space 
$L^2_\psi(M) = \{ f:M\rightarrow \re \ ;\  \int_M f^2 \mu_\psi< \infty\}$.
  One defines also $\mc{H}_\psi(M)$ as the $|| \cdot ||_{\psi 1}$-completion of $\{f\in C^\infty(M);\ ||f||_{\psi 1}<\infty\}$ in $L^2_\psi(M)$.

The standard Rayleigh's principle is still true for $\psi$-Laplacians just changing the volume elements as above.


\section{Proof of Theorem \ref{TF}}\lb{sec12}
 Let be $p \in M$ the center of $\psi$-mass of $M$. Using the Rayleigh's principle and the coordinates functions $x^i_p$ as test functions, it is proved in subsection 3.2 of \cite{BCP} that the inequality \eqref{ineqTF} holds. We advertise the reader that Batista, Cavalcante and Pyo call $-f$ what we have denoted by $\psi$. Morevoer, they observe that the equality in \eqref{ineqTF} implies
  that there is a constant $C$ such that
\begin{equation}\lb{56}
\vHp + \overline{\nabla} \psi = C \ F_p
\end{equation}
The tangent part of the equality \eqref{56} is
\begin{equation}\lb{57}
\nabla \psi = C \ r_p \nabla r_p,
\end{equation}
and its normal part is
\begin{equation}\lb{57-1}
\vHp + \left( \overline{\nabla} \psi \right) ^{\perp}
= C \ (F_p)^{\perp} \qquad \text{ that is } \qquad 
\overrightarrow{H} = C \ (F_p)^{\perp},
\end{equation}
which is the equation of a self-similar solution of the mean curvature flow.

If $C = 0$,  $\overrightarrow{H} = 0$, the submanifold $M$ is minimal and compact in $\re^{n+k}$, which is not possible. 

If $C > 0$,  $M$ is a compact expander. In this case  $M$ will expand  with time under mean curvature flow, then  the volume of $M$ will grow, which is impossible because the mean curvature flow decreases volume. 

If $C < 0$, $M$ is  a shrinker, as claimed in the statement of Theorem \ref{TF}. 
Moreover, from (\ref{57}) 
\begin{align}\lb{57-2}
\nabla \psi = C \ r_p \nabla r_p = \frac{1}{2}\ C \ \nabla r_p^2
\end{align}
then \ $\nabla (\psi - \frac{1}{2}\ C \ r_p^2) = 0$, \ therefore  
\ $\psi|_M = a + \frac{1}{2}\ C \ r_p^2$ for some constant $a$. That is, the equality in \eqref{55} also implies that the density $e^\psi$ restricted to the submanifold $M$ is of Gaussian type.

Now, we check that, under conditions \eqref{57-1} and \eqref{57-2}, the $x_p^i$ are eigenfunctions of $\Delta_\psi$ with eigenvalue  $-C$. In fact
\begin{align}
\Delta _{\psi} x_p^{i} &= \Delta x_p^{i} + \< \nabla \psi, \nabla x_p^{i} \> = \vec{H}^{i} + \< \nabla \psi, \nabla x_p^{i} \>  \nn \\ 
& 
= C \ ((F-p)^\bot)^i    + C \ ((F-p)^{\top})^{i} 
= C \ (F-p)^{i} = C \ x_p^{i}, \lb{xieig}
\end{align}
Then, if we have equality in \eqref{ineqTF}, $M$ is a shrinker soliton ($\vec{H}= C F_p^\bot$, with $C<0$), $\psi$ restricted to $M$ is of Gaussian type, and 
$\lambda _1 = - C$. All these are necessary conditions to have equality in \eqref{ineqTF}, but we have not proved that they are sufficient. We know that in a shrinker $-C$ is an eigenvalue of $\Delta_\psi$ with $\psi$ Gaussian, but we do not know if $-C$ is precisely its first eigenvalue. We shall prove that this is true for the Angenent's torus type in Theorem \ref{l1A} and we conjecture that it is true for all the shrinkers.

When $k = 1$ and $H > 0$, it is known  (see \cite{hui}) that the unique compact shrinking soliton is the sphere, where we know that the equality holds in \eqref{ineqTF}. Also, for $k=1$, $n=2$ and $M$ of genus $0$, it has been proved recently in \cite{br15} that $M$ must be a sphere. Then, when $k=1$ and $H>0$ or $n=2$ and genus of $M$ is zero, the theorem can be stated with a \lq\lq if and only if"\ for the case of equality in \eqref{ineqTF}. 

\bigskip

\section{Proof of Theorem \ref{CorF}}

In this setting $M$ is a $n$-dimensional submanifold of a Riemannian $(n+k)$-dimensional riemannian manifold $\oM$. By Nash embedding theorem, we can consider $\oM$ as a riemannian submanifold of $\re^m$. We shall denote by $D$ the covariant derivative and the gradient in $\re^m$. If $\oM$ has a density $e^\psi$, we can consider on $\re^m$ an extension of $\psi$ that we shall denote by the same symbol. We choose this extension of $\psi$ satisfying 
\begin{equation}\lb{61}
D \psi|_\oM = \overline{\nabla} \psi.
\end{equation}
This can be done, for instance, taking $\psi(p+ t \zeta) = \psi(p)$ for $p+t\zeta$ in a small tubular neighborhood of $\oM$, where $p\in \oM$ and $\zeta$ is a unit vector normal to $\oM$ at $p$. 

We have then the following chain of extrinsic curvatures: 

$\alpha $ \ is the second fundamental form of $M$ in $\overline{M}$ with $\psi$-mean curvature $\vHp$.

$\alpha_{\overline{M}} $ \ is the second fundamental form of  $\overline{M}$ in 
$\re^m$.

$\widetilde{\alpha} $ \ is the second fundamental form of  $M$ in 
$\re^m$, with $\psi$-mean curvature $\wHp$.



Since  $M$ is also a riemannian submanifold of $\re^m$,  we can apply Theorem \ref{TF} to obtain
\begin{equation}\lb{62}
\lambda_1 (M) \le  \fracc{\int_M |\wHp + D\psi|^2 \ \mu_\psi}{n\  V_\psi(M)}
\end{equation}

Let $\{e_i\}_{i = 1}^n$ be an orthonormal local frame of $M$. From the definitions of the $\alpha$'s and $\vec{H}$'s it follows
\begin{align} \lb{60}
& \wvH = \ds\sum_{i = 1}^{n} \wal(e_i, e_i) 
= \ds\sum_{i = 1}^{n} \alpha_{\overline{M}} (e_i, e_i)   + \vec{H}
\end{align}

From (\ref{61}), (\ref{62}) and \eqref{60}, taking into account that  $\vHp$ and 
$\ona\psi$ are tangent to $\overline{M}$ and 
$\ds\sum_{i = 1}^{n} \alpha_{\overline{M}} (e_i, e_i)$ is orthogonal to 
$\overline{M}$, we obtain  
\begin{align} \lb{61-1}
|\wHp + D \psi|^2  
= |\vHp + \ona\psi|^2 + \left|\ds\sum_{i = 1}^{n} \alpha_{\overline{M}} (e_i, e_i) \right| ^2
\end{align}

Denoting by $c = \ds\max_{v \in TM, |v|=1} \{ \alpha_{\overline{M}}  (v, v) \} $, we have the estimate
\begin{align} \lb{61-2}
\ds\int_M  \left|\ds\sum_{i = 1}^{n} \alpha_{\overline{M}} (e_i, e_i) 
\right| ^2 \ \mu_\psi \leq  n^2 \ c^2 \ V_\psi(M)
\end{align}

By substitution of (\ref{61-1}) in (\ref{62}), having in mind (\ref{61-2}), we obtain 
\begin{align}\lb{63}
\lambda_1 (M) & \le   n \ c^2 
+  \fracc{\int_M |\vHp + \ona\psi|^2 \ \mu_\psi}{n\  V_\psi(M)}  
\end{align}
which is \eqref{ineq}, with $b^2 = n c^2$. 

Now, let us consider that  $\oM$ is a round sphere $S^{n+k}$ of radius $R$. Then we can take $m=n+k+1$ and the sphere centered at the origin of $\re^m$. In this case the inequality \eqref{61-2} is a equality and $c=1/R$.  Let us suppose that, in this setting, we have the equality in \eqref{63}, then we have equality in \eqref{62}, which, by Theorem \ref{TF}, implies that  $M$ is a shrinker in $\re^m$ and $\psi|_M = a - \frac12 \lambda_1 r_p^2$ for some point $p\in\re^{n+k+1}$, which is the center of $\psi$-mass of $M$. 

Now we consider two possibilities:

i) $p$ is the center of $S^{n+k}$, then $\psi|_M$ is constant and $M$ is a shrinker of $\re^{n+k+1}$ contained in $S^{n+k}$. But, as Smoczyk noticed (\cite{sm05}), the shrinkers contained in a sphere are the minimal submanifolds of the sphere. Then part i) of Theorem \ref{CorF} is proved.

ii) $p$ is not the center of $S^{n+k}$.

Since $M$ is a shrinker with center of $\psi$-mass in $p$, there is a constant $C  < 0$ such that  
\begin{equation} \lb{10*}
\wvH = C (F_p)^{\perp} = C (F - p)^{\perp} = C \< F - p, N \> N +C  \< F - p, \zeta \> \zeta 
\end{equation}
where $\perp$ means the component orthogonal to $M$ in $\Bbb R^{ n + k +1}$, $N$ is a unit vector orthogonal to $S^{n+k}$ in $\re^{n+k+1}$ and $\zeta$ is a unit vector in the direction of the component of $F-p$ orthogonal to $M$ in $S^{n+k}$.

In the sphere, \eqref{60} becomes 
\begin{align}\lb{11*}
\wvH = n \ \frac{1}{R} \ N + \vec{H}, 
\end{align}
From (\ref{10*}) and (\ref{11*}) we obtain 
\begin{align}\lb{12*}
&\frac{n}{R} = C \< F - p, N \>
\end{align}
Now we show that 
(\ref{12*}) implies that $M$ is contained in a sphere of center $p$ of dimension $n+k-1$. In fact, let $\beta(s)$ be a curve in $M$, by (\ref{12*}) one has $\ds\frac{n}{ C R} = \< \beta(s) - p, N \>$.
Taking derivative respect to $s$, and using that $\beta(s)$ is also contained in $S^{n+k}$,
\begin{equation}
0 = \< \beta', N \> + \< \beta - p, (N \circ \beta)' \> = \< \beta - p, \frac{1}{R} \beta' \>. 
\end{equation}
Then  $0=\< \beta - p, \beta' \> =  \ds\frac{1}{2} \< \beta- p, \beta - p \> '= \fracc12 \ds\frac{d}{ds}\left( |\beta - p|^2 \right)$, so $|\beta - p|$ is constant, and $\beta(s)$, therefore $M$,  is contained in the intersection of a sphere centered at  $p$ with $S^{n+k}$, which proves our claim. 

If $k=1$, $M$ is just the sphere where it is contained. If $k>1$, it is a shrinker contained in this sphere, then a minimal submanifold of the sphere. This finishes the proof of parts ii) and iii) of Theorem \ref{CorF}.



\bigskip

\section{Proof of Theorem \ref{l1A}}
In this section we shall work in $\re^3$. Then, for simplicity, we shall use $x$, $y$, $z$ to denote the coordinates $x^1$, $x^2$, $x^3$.

It was proved by Brendle in \cite{br15} that the unique embedded shrinker of genus $0$ in $\re^3$ is the standard round sphere. Since an embedded surface of revolution only can be of genus $0$ and $1$, and Theorem \ref{l1A} is well known for the round sphere, we shall suppose that the embedded shrinkers in the hypothesis of the theorem are tori of  revolution $T$ around the axis $Z$. They are warped products  $T = E \times_{\rho} S^1$ with $E$ a simple closed curve in the plane $XZ$ invariant respect to the symmetry   $z \longrightarrow - z$, where $\rho: E \flecha \Bbb R$ denotes the distance, in the plane, to the axis $Z$, and also the distance $\rho_3$ in $\re^3$ to the axis $Z$ restricted to $T$. From now on we shall use the same letter for $\rho$ and $\rho_3$ restricted to $E$ and on $T$ respectively, because $\rho_3(s,\theta) = \rho(s)$ for every $(s,\theta)\in E\times_\rho S^1$. 

Using the expression of the standard Laplacian on a warped product (see, for instance, \cite{Ej}), the $\psi$-Laplacian on the torus $T$ can be written as 
\begin{align}\label{lap}
\Delta_{\psi} ^T &= \Delta ^{E} +  \frac{1}{\rho} \ \nabla{\rho}
+  \frac{1}{\rho^2} \ \Delta ^{S^1} +   \nabla \psi \\
& = \Delta _{\ln \rho + \psi}^{E} + \frac{1}{\rho ^2} \Delta ^{S^1}, \label{lap2}
\end{align}
where $\Delta^{E}$ and $\Delta^{S^1}$ denote the usual Laplacians in $E$ and $S^1$ respectively. 

To study the spectrum of $\Delta_\psi^T$  we shall follow the procedure of Ejiri \cite{Ej} on warped products.
 Let $\{ \psi_0, \psi_1, \psi_2,  \dots\  \}$ be a complete orthonormal  basis of $L^2(S^1)$ formed by eigenfunctions of $\Delta^{S^1}$ with corresponding eigenvalues  $\lambda_0=0<\lambda_1 =\lambda_2 =1 <\lambda_3 =\lambda_4 =4 < \cdots< \lambda_{2k-1}=\lambda_{2k}=k^2 < \cdots $.  
For every $\lambda_i$, we define the operator  $L_{\lambda_i}^{\psi}$ acting on smooth real functions defined on $E$
 \begin{align}\lb{llp}
 L_{\lambda_i}^{\psi} = \Delta ^{E} +  \ds\frac{1}{\rho} \ \nabla \rho  - \ds\frac{\lambda_i}{\rho^2} +  \nabla \psi .
 \end{align}
The same arguments given in Lemma 2.3 of \cite{Ej}  show that $L_{\lambda_i}^{\psi}$ is a strongly elliptic self-adjoint operator on the space $L_{\ln\rho+\psi}^2 (E)$  of the  smooth functions on $E$ with bounded square norm under the measure $\rho e^\psi \mu = dv_{\ln\rho+\psi}$. Then, there exists a complete orthonormal basis of $L_{\ln\rho+\psi}^2 (E)$ formed by eigenfunctions $\phi_j^{\lambda_i}$ of $L_{\lambda_i}^{\psi}$ with eigenvalues  $\mu_0^{\lambda_i} \le \mu_1^{\lambda_i}\le  \mu_2^{\lambda_i}  \le \cdots $.

Now, we use \eqref{lap} to compute
\begin{align}\lb{dphit}
\Delta_{\psi} ^T \left( \psi_i \phi_j^{\lambda_i} \right)   \nonumber 
& = \left( \Delta ^{E} +  \frac{1}{\rho} \ \nabla \rho  - \frac{\lambda_i}{\rho^2} + \frac{\lambda_i}{\rho^2} 
+  \frac{1}{\rho^2} \ \Delta ^{S^1} +  \< \nabla \psi, \nabla \cdot \>\right) \left( \psi_i \phi_j^{\lambda_i} \right)  \\ 
& = L_{\lambda_i}^{\psi} \left( \psi_i \phi_j^{\lambda_i} \right) 
+ \left( \frac{\lambda_i}{\rho^2} +  \frac{1}{\rho^2} \ \Delta ^{S^1} \right) 
\left( \psi_i \phi_j^{\lambda_i} \right) = L_{\lambda_i}^{\psi} \left( \psi_i \phi_j^{\lambda_i} \right) 
\end{align}
because $\phi_j^{\lambda_i}$ depends only on  $E$, $\psi_i$ only depends $S^1$ and
$\Delta ^{S^1} \psi_i = - \lambda _i \psi_i$. 
Moreover, 
\begin{align}\label{pg3 (22-9)}
L_{\lambda_i}^{\psi} \left( \psi_i \phi_j^{\lambda_i} \right) 
= \Delta ^{E} \left( \phi_j^{\lambda_i}  \right) \ \psi_i + \frac{1}{\rho} \nabla \rho \left( \phi_j^{\lambda_i}  \right) \ \psi_i - \frac{\lambda_i}{\rho^2} \ \psi_i \phi_j^{\lambda_i} + \< \nabla \psi, \nabla \left(\psi_i \phi_j^{\lambda_i}\right) \>
\end{align}

But $\nabla \psi$ is in the direction of the position vector, this is orthogonal to the curves $S^1$ in the warped product, and $\nabla \psi_i$ is tangent to these $S^1$, then $\<\nabla \psi, \nabla\psi_i\>=0$ and 
$\< \nabla \psi, \nabla \left(\psi_i \phi_j^{\lambda_i}\right) \> = \< \nabla \psi, \nabla (\phi_j^{\lambda_i}) \> \psi_i $.

Plugging this equality into \eqref{dphit} and \eqref{pg3 (22-9)} and taking into account that  $\mu_j ^{\lambda_i}$ is an eigenvalue of $L_{\lambda_i}^{\psi}$ with eigenfunction $\phi_j^{\lambda_i}$ 
\begin{align}\label{eigT}
\Delta_{\psi} ^T \left( \psi_i \phi_j^{\lambda_i} \right)= L_{\lambda_i}^{\psi} \left( \psi_i \phi_j^{\lambda_i} \right) 
& = - \mu_j ^{\lambda_i} \phi_j^{\lambda_i} \psi_i
\end{align}
that is, $\psi_i \phi_j^{\lambda_i} $ are eigenfunctions of $\Delta _{\psi} ^T$ with eigenvalues $\mu_j ^{\lambda_i}$. 
From this, using the same arguments that in the proof of Theorem 2.1 in \cite{Ej},  we conclude that $\{\psi_i \phi_j^{\lambda_i} ; \ i,j=0,1,2,\dots\}$  is a complete orthonormal basis of $L^2_\psi(T)$ and the $\{\mu_j^{\lambda_i}\}$ are all the eigenvalues of $\Delta_\psi^T$.

To finish the proof of Theorem \ref{l1A} it remains only to check  that $\inf\{\mu_j^{\lambda_i}\ne 0; \ i,j=0,1,2,\dots\} = - C$. We shall prove this in a series of lemmas.

\begin{lema}\lb{e0Llp} Constant functions are eigenfunctions of $L_{\lambda_i}^\psi$ with eigenvalue $0$ if and only if $i=0$
\end{lema}
\begin{demo}
If we write $L_{\lambda_i}^\psi$ in the form
\bec\lb{LlnD}
L_{\lambda_i}^\psi =  \Delta _{\ln \rho + \psi} ^{E} - \frac{\lambda_i}{\rho ^2},
\eec
by \eqref{LlnD} and \eqref{intlap}, we have 
\begin{align}\label{pg.8 (29-9)}
 \int_E \(L_{\lambda_i}^\psi u\) \ u \ dv_{\ln \rho + \psi} 
&= \int_E \left( \Delta _{\ln \rho + \psi} ^{E} u \right)  \ u \ dv_{\ln \rho + \psi} 
- \int_E  \frac{\lambda_i}{\rho ^2} \ u^2 \ dv_{\ln \rho + \psi} \nn\\ &= - \int_E |du|^2 \ dv_{\ln \rho + \psi} 
- \int_E  \frac{\lambda_i}{\rho ^2} \ u^2 \ dv_{\ln \rho + \psi} \le 0
\end{align}
which vanishes if and only if $u$ is constant and $i=0$. 
\end{demo}

The next two lemmas  try to check, adapting the exposition given in \cite{Ch},  that the well known Nodal Domains Theorem of Courant is still valid for a Laplacian with density. First we recall the concept of nodal domain and the classical Sturm theorem.

The {\it nodal domains of a function} $f:M\flecha \re$ are the connected components of the set $M-\{ x \in M \ / \ f(x) = 0 \}$.

\textbf{Sturm's Theorem} (\cite{CL}, page 214) {\it The eigenvalue problem $(p y')'+ \mu \ p\ y =0$ for periodic functions $y$ on $\re$ of period $L$  has solution for a sequence of values of $\mu$
$$ 0 = \mu_0 < \mu_1 \le \mu_2 < \mu_3 \le \mu_4 < ... < \mu_{2j-1} \le \mu_{2j} < ...$$
where each eigenvalue $\mu_k$ has multiplicity $1$ and the eigenfunctions corresponding to $\mu_{2j-1}$ and $\mu_{2j}$ have $2 j$ zeros.} 

In Lemma \ref{lema1} we shall write the above eigenvalue problem as the eigenvalue problem of the operator $y\mapsto \fracc{(p\ y')'}{p}$. We shall call it the {\it Sturm operator}.

\begin{lema}\lb{Cour}
 Let $\{\phi_0, \phi_1, \phi_2, ... \}$ be a complete orthonormal basis of $L^2_\varphi(M)$ with each $\phi_j$ a eigenfunction of $\Delta_\p$ associated to an eigenvalue $\lambda_j$, $j=0,1,2, ... $. Then the number of nodal domains of $\phi_k$ is less or equal to $k+1$, for every $k=0,1,2,...$.  Moreover, the number of nodal domains of $\phi_0$ and  $\phi_1$ are $1$ and  $2$ respectively.
\end{lema}
\begin{demo}
It follows with exactly the same arguments used for the ordinary Laplacian (see \cite{Ch} pages 19-20), just changing the corresponding spaces associated to the riemannian measure $\mu_g$ by those spaced associated to the measure with density $\mu_\psi$ indicated in the Preliminaries.
\end{demo}

These lemmas will be used in the proof of the next two ones, which give the final steps in the proof of the theorem.

\begin{lema}\lb{lema1} $\inf\{\mu_j^{\lambda_i}\ne 0; \ i,j=0,1,2,\dots\} = \min\{\mu_1^{\lambda_0},\  \mu_2^{\lambda_0},\ \mu_0^{\lambda_1}\}$. If this minimum is $ \mu_0^{\lambda_1}$, then $\mu_0^{\lambda_1}=-C$.
\end{lema}
\begin{demo}
By Lemma \ref{Cour}, the eigenfunctions associated to the first nonzero eigenvalue of $\Delta_\psi^T$ have  exactly two nodal domains. Then we shall look for  products $\psi_i\phi_j^{\lambda_i}$ with two nodal domains.

Let us denote by $N(f)$ the number of nodal domains of a function. Obviously we have $N(\psi_i\phi_j^{\lambda_i})= N(\psi_i) N(\phi_j^{\lambda_i})$. We discuss the possibilities for the different values of $i$ and $j$.


\begin{enumerate}
\item For $i=0$,  $\psi_0$ is constant, then $N(\psi_0 \phi_j^{\lambda_0})= N(\phi_j^{\lambda_0})$. Moreover, from \eqref{llp}, $L_{\lambda_0}^{\psi} (y) = (\Delta ^{E} +  \ds\frac{1}{\rho} \ \nabla \rho   +  \nabla \psi) y= y'' + \(\frac{\rho'}{\rho}+\psi'\)y'$ has the form of  the Sturm operator, with $p = \rho e^\psi$. Then, by Sturm's Theorem, the eigenfunctions  $\phi_{2j-1}^{\lambda_0}$ and $\phi_{2j}^{\lambda_0}$ have $2j$ nodal domains for $j\ge 1$, whereas, by Lemma \ref{e0Llp},  $\phi_{0}^{\lambda_0}$ is constant and has one nodal domain.  By Lemma \ref{Cour} the first nonzero eigenvalue of $\Delta^T_\psi$ has an associated eigenfunction with two nodal domains. Therefore the unique candidates to be this eigenvalue when $i=0$ are $\mu^{\lambda_0}_1$ and $\mu^{\lambda_0}_2$.
\item For $i\in\{1,2\}$, $N(\psi_1)=2$, then $\mu^{\lambda_1}_j=\mu^{\lambda_2}_j$ is a candidate to be the first nonzero eigenvalue of $\Delta_\psi^T$ only if  $N(\phi_{j}^{\lambda_1})=1$. 
Since $T$ is of revolution around $Z$ and symmetric respect to the plane through the origin which is orthogonal to $Z$,  $T$  has central symmetry and, because $\psi$ is radial, the origin is the center of $\psi$-mass of $T$. It follows from  \eqref{xieig} that the coordinates functions $x,y,z$ restricted to $T$ are eigenfunctions of $\Delta_\psi^T$ with eigenvalue $-C$. Considering the relation $x(s,\theta) = \rho(s) \cos \theta , \quad 
y(s,\theta) = \rho(s) \sin \theta,$  $s\in E$  and $\theta\in S^1$   in the surface of revolution $T$, using  the expression \eqref{lap2} and the fact  that $\cos \theta$ is an eigenfunction of $\Delta^{S^1}$ with eigenvalue $\lambda_1=1$, we obtain 
\begin{align}
- C \rho(s) \cos \theta &= \Delta_{\psi}^T (\rho \ \cos \theta) = \left( \Delta _{\ln \rho + \psi}^{E} + \frac{1}{\rho ^2} \Delta ^{S^1} \right) (\rho \ \cos \theta)\nn \\
& = \cos \theta \ \Delta _{\ln \rho + \psi}^{E} \rho  + \frac{\rho}{\rho ^2} \Delta ^{S^1} \cos \theta = \(\Delta _{\ln \rho + \psi}^{E} \rho - \frac1\rho\) \cos \theta . \lb{Cc}
\end{align}

It follows from \eqref{Cc} that 
\begin{align}
L_{\lambda_1}^{\psi} \rho = \left( \Delta _{\ln \rho + \psi}^{E} \rho - \frac{1}{\rho}\right) \rho  =   - C \ \rho ,
\end{align}
that is $\rho$ is an eigenfunction of $L_{\lambda_1}^{\psi}$. Moreover $\rho>0$ because $E$ does not touch the axis $Z$, then $N(\rho)=1$, and any other eigenfunction of $L_{\lambda_1}^{\psi}$ orthogonal to $\rho$ must have at least $2$ nodal domains. We conclude that the unique $\mu_j^{\lambda_1}$ candidate to be the first nonzero eigenvalue of $\Delta_\psi^T$ is $\mu_0^{\lambda_1}=-C$ (with $j=0$ because these eigenvalues are ordered). As a consequence, $\psi_0^{\lambda_1}$ is a multiple of $\rho$.
\item For $i\ge 3$, $N(\psi_i)\ge 4$, then none of the $\mu_j^{\lambda_i}$ is a candidate to be the first nonzero eigenvalue of $\Delta_\psi^T$.
\end{enumerate}
 \end{demo}
 

\begin{lema} $\mu_1^{\lambda_0} = -C$ or $\mu_1^{\lambda_0} =\mu_2^{\lambda_0} = -C$.
\end{lema}
\begin{demo}
As we observed in the previous lemma, $z$ is an eigenfunction of $\Delta^T_\psi$ with eigenvalue $-C$.  This, together with  the expressions \eqref{lap} and \eqref{llp} and the facts that $\lambda_0=0$ and $z$ restricted to $S^1$ is constant give that $z$ is an eigenfunction of $L_{\lambda_0}^\psi$ with eigenvalue $-C$. Since the generatrix curve $E$ of $T$ is symmetric respect to the axis $X$, the function $z$ restricted to $E$ has two nodal domains. Then, by Sturm's Theorem (see case 1 in the proof of Lemma \ref{lema1}), its associated eigenvalue can be $\mu_1^{\lambda_0}$ or $\mu_2^{\lambda_0}$. Since $\mu_1^{\lambda_0} \le\mu_2^{\lambda_0}$, the Lemma is proved if $\mu_1^{\lambda_0}=-C$. Let us suppose that $\mu_2^{\lambda_0}=-C$.

We shall prove that, if $\mu_2^{\lambda_0}$ is the eigenvalue associated to $z$, then there is another eigenfunction of  $L_{\lambda_0}^\psi$, linearly independent of $z$ with the same eigenvalue $\mu_2^{\lambda_0}$. According to Sturm's Theorem, this will imply that $\mu_1^{\lambda_0} =\mu_2^{\lambda_0}$ and the lemma will be proved.

As we pointed out in the proof of Lemma \ref{lema1}, the eigenvalue problem $L_{\lambda_0}^\psi z = - \mu_2^{\lambda_0}z $ can be written as $(p z')'+ \mu_2^{\lambda_0}\ p\ z =0$, with $p= \rho\ e^\psi$. Let $L$ denote the length of $E$. Since $E$ is symmetric respect to the axis $X$ and the functions $p$ and $z$ restricted to $E$ are  well defined smooth functions, we have that $p(s)= p(L-s)$, $z(s)=-z(L-s)$ and $p$ and $z$ can be extended to periodic smooth functions on $\re$ with period $L$. Moreover, $z(0)=0=z(L/2)=z(L)$, $z>0$ on $[0,L/2]$ and $z<0$ on $[L/2,L]$. Then $z'(0)=-z'(L/2)=\delta>0$. By multiplication of $z$ by a constant if necessary, we can suppose that $z'(0)=-z'(L/2)=1/p(0)>0$.   

Let $\zeta(s)$ be the solution of the ordinary differential equation on $\re$ 
\begin{align}\lb{SLl}
(p \zeta')'+ \mu_2^{\lambda_0} \ p\ \zeta &=0\\
\zeta(0)&=1 \nn\\
\zeta'(0)&=0. \nn
\end{align}
  As a consequence of the initial conditions, it is clear that $\zeta$ and $z$ are linearly independent. If we can prove that $\zeta$ is periodic with period $L$, then $\zeta$ will be also a solution of the same Sturm eigenvalue problem that $z$, with the same associated eigenvalue, and our claim will be proved. Let us prove that $\zeta$ is periodic of period $L$.

Since $\zeta$ is the solution of a differential equation with periodic coefficients, it will be enough to prove that $\zeta(L)=\zeta(0)=1$ and $\zeta'(L)=\zeta'(0)=0$. 

Using the fact that $z$ and $\zeta$ are solutions of \eqref{SLl}, we compute
\begin{align}
((\zeta z'-\zeta' z)p)' &= (\zeta z''-\zeta ''z)p + (\zeta z'-\zeta 'z)p' \nn \\
&= \zeta  (z'' p + z' p') - z (\zeta ''p + \zeta ' p') = - \zeta  \mu_2^{\lambda_0} p z + z \mu_2^{\lambda_0} p \zeta  =0. \lb{fs}
\end{align}
The initial conditions of $z$ and $\zeta $ give
 \begin{align}
 \((\zeta z'-\zeta 'z)\ p\)(0) = (1 \frac1{p(0)} - 0 )p(0) = 1. \lb{fe}
 \end{align}
From \eqref{fs} and \eqref{fe}
\begin{equation}\lb{zeta z}
(\zeta z'-\zeta 'z)p=1.
\end{equation}

 
 Then, $\zeta $ satisfies the linear differential equation $\zeta ' - \fracc{z'}{z} \zeta  = -\frac1{pz}$, whose general solution is 
 $$ \zeta (s) = \frac{1}{e^{-\int\frac{z'}{z}ds}} \int e^{-\int\frac{z'}{z}ds} \(-\frac1{pz}\) ds =  - z(s) \ \eta(s) \quad  \text{ where }\quad  \eta(s) = \int \frac1{pz^2} ds$$
 and
 $$\zeta '(s) = -z'(s) \eta(s) - z(s) \eta'(s) = -z'(s) \eta(s) - \frac1{p(s) z(s)}.$$
 The initial conditions for $\zeta $ give
 $1 = \zeta (0) = -\ds \lim_{s\to 0} (z(s) \eta(s))$, that is
 \begin{align}
 \lim_{s\to 0} \frac{\eta(s)}{-1/z(s)} = 1, \lb{lim}
 \end{align}
and $0= \zeta '(0) = \ds \lim_{s\to 0}\( -z'(s) \eta(s) - \fracc1{p(s) z(s)}\) = \ds\lim_{s\to 0} \(z'(s)  - \fracc1{p(s)}\)\fracc1{z(s)}$, an equality which always occurs because $\ds \lim_{s\to 0} \fracc{pz'-1}{pz} = \lim_{s\to 0} \fracc{(pz')'}{p'z+pz'} =\lim_{s\to 0} \fracc{-\mu_2^{\lambda_0} p z}{p'z+pz'}=0$. 

On the other hand, $\fracc1{pz^2}(s) = \fracc1{pz^2}(s+L) = \fracc1{pz^2}(L-s)$, then $\eta'(s) = \eta'(s+L) = \eta'(L-s)$, and, integrating, $\eta(s+L) = k_++ \eta(s)$ and $\eta(L-s)= k_--\eta(s)$ for some constants $k_+$ and $k_-$, which, using \eqref{lim},  gives:
\begin{align}
\lim_{s\to0}(\eta(L-s)\ z(s)) &= \lim_{s\to 0}((k_-  \eta(s))z(s)) = 1, \nn \\ 
\lim_{s\to0}(\eta(s+L)\ z(s))&= \lim_{s\to 0}((k_+ + \eta(s))z(s)) = - 1.\nn
\end{align}
Therefore
\begin{align*}
\lim_{s\to 0} \zeta (s+L) &= -  \lim_{s\to 0} z(s+L) \eta(s+L) =  -\lim_{s\to 0} z(s) \eta(s) = 1 =\zeta (0), \text{ and } \\
\lim_{s\to 0} \zeta (L-s) &= -  \lim_{s\to 0} z(L-s) \eta(L-s) =  \lim_{s\to 0} z(s) (-\eta(s)) = 1 =\zeta (0)
\end{align*}
For the first derivative we have
\begin{align*}
\lim_{s\to 0} \zeta '(s+L) &=   \lim_{s\to 0} \(-z'(s+L) \eta(s+L)-\frac1{pz}(s+L)\) \nn \\ &=  -\lim_{s\to 0} \(-z'(s) \eta(s)-\frac1{pz}(s)\) = \lim_{s\to 0} \(z'(s) -\frac1{p}(s)\)\frac1{z(s)}=0, \text{ and } \\
\lim_{s\to 0} \zeta '(L-s) &=  \lim_{s\to 0} \(-z'(L-s) \eta(L-s)-\frac1{pz}(L-s)\) \nn \\ &=  -\lim_{s\to 0} \(z'(s) \eta(s)+\frac1{pz}(s)\) = \lim_{s\to 0} \(-z'(s) +\frac1{p}(s)\)\frac1{z(s)}=0
\end{align*}
Then $\zeta (0)=1=\zeta (L)$ and $\zeta '(0) = \zeta '(L)=0$, which finishes the proof.
\end{demo} 


\bibliographystyle{alpha}
{\small
\begin{thebibliography}{99}

\bibitem{BCP} Batista, M.; Cavalcante, M. P.; Pyo, J.; Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds. {\it J. Math. Anal. Appl.} 419 (2014), no. 1, 617–626.

\bibitem{BeGo} Berger, Marcel; Gostiaux, Bernard; {\it Differential geometry: manifolds, curves, and surfaces}. GTM, 115. Springer-Verlag, New York, 1988.

\bibitem{BW}Bleecker, D., Weiner, J.: Extrinsic bounds on $\lambda_1$ of $\Delta$ on a compact manifold. {\it Comment. Math. Helv.} 51  (1976), 601-609.

\bibitem{br13} Brendle, Simon; Minimal surfaces in $S^3$: a survey on recent results. {\it Bull. Math. Sci.} 3 (2013), no.1, 133–171.

\bibitem{br15} Brendle, Simon;  
Embedded self-similar shrinkers of genus 0; Preprint,
arXiv:1411.4640

\bibitem {BK} Buser, P. and Karcher, H.; {\em
Gromov's almost flat manifolds},
 société mathematique de france, astérisque num. 81, Paris
 (1981).

\bibitem{CESG} Colbois, Bruno; El Soufi, Ahmad; Girouard, Alexandre; Isoperimetric control of the spectrum of a compact hypersurface. {\it J. Reine Angew. Math.} 683 (2013), 49–65.

\bibitem{Ch} Chavel, Isaac; {\it Eigenvalues in Riemannian geometry}. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp.

\bibitem{chmezh12b} Cheng, Xu;  Mejia, Tito and  Zhou, Detang; {Eigenvalue estimate and compactness for closed $f$-minimal surfaces},  {\it Pacific J. Math.} 271 (2014), no. 2, 347–367 
 
 \bibitem{CL} Coddington, Earl A.; Levinson, Norman; {\it Theory of ordinary differential equations}. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. xii+429 pp.
 
\bibitem{comi} Colding, Tobias H.; Minicozzi, William P., II;  Generic mean curvature flow I: generic singularities. {\it Ann. of Math.} (2) 175 (2012), no. 2, 755–833.

\bibitem{dixi} Ding, Qi; Xin, Y. L. Volume growth, eigenvalue and compactness for self-shrinkers. {\it Asian J. Math.} 17 (2013), no. 3, 443–456
 
\bibitem{Ej} Ejiri, Norio;  A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric. {\it Math. Z.} 168 (1979), no. 3, 207–212.

\bibitem{GMO} Giménez, Fernando; Miquel, Vicente; Orengo, J. Javier;  Upper bounds of the first eigenvalue of closed hypersurfaces by the quotient area/volume. {\it Arch. Math}. 83 (2004), no. 3, 279--288.

\bibitem{he} Heintze, Ernst; Extrinsic upper bounds for $\lambda_1$. {\it Math. Ann.} 280 (1988), no. 3, 389–402.

\bibitem{huru} Huang, Qin; Ruan, Qihua;  Applications of some elliptic equations in Riemannian manifolds. {\it J. Math. Anal. Appl.} 409 (2014), no. 1, 189–196.

\bibitem{hui} Huisken, G.; Asymptotic behavior for singularities of the mean curvature flow, {\it J. Differential Geom. }31 (1990), 285–299. 

\bibitem{Ka} Karcher, H.;  Riemannian Center of Mass and Mollifier Smoothing {\em Comm. Pure and Applied Math.} {\bf XXX}, (1977) 509--541

\bibitem{lilo} Long, Li; On the spectrum of weighted Laplacian operator and its applications to uniqueness of Kähler Einstein metrics, {\it Math. Ann.} Published online 2014, DOI: 10.1007/s00208-014-1152-1

\bibitem{ma14} Ma, Li; Eigenvalue estimates and L1 energy on closed manifolds. {\it Acta Math. Sin.} (Engl. Ser.) 30 (2014), no. 10, 1729–1734.

\bibitem{madu} Ma, Li; Du, Sheng-Hua; Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. {\it C. R. Math. Acad. Sci. Paris} 348 (2010), no. 21-22, 1203–1206.

\bibitem{maliu} Ma, Li; Liu, Baiyu; Convexity of the first eigenfunction of the drifting Laplacian operator and its applications. {\it New York J. Math.} 14 (2008), 393–401.

\bibitem{mo11} Møller, Niels Martin ; Closed self-shrinking surfaces in $\mathbb{R}^3$ via the torus, Preprint, arXiv:1111.7318

\bibitem{piri} Pigola, Stefano; Rimoldi, Michele; Complete self-shrinkers confined into some regions of the space. {\it Ann. Global Anal. Geom.} 45 (2014), no. 1, 47–65

\bibitem{Re} Reilly, R.: On the first eigenvalue of the Laplacian for compact submanifold of
Euclidean space. {\it Comment. Math. Helv.} 52, 525-533 (1977)



\bibitem {sm05} Smoczyk, Knut;  Self-shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 2005, no. 48, 2983–3004.

\bibitem{sm12} Smoczyk, Knut; {\it Mean curvature flow in higher codimension: Introduction and survey}. in Global Differential Geometry, Springer Proceedings in Mathematics, vol. 17, part 2,(2012) 231-274. 


\end{thebibliography}

}

\bigskip


{\small 

\begin{tabular}{ l l }
&Universidad de Valencia \\
&Departamento de Matem\'aticas para la Econom\'{\i}a y la Empresa \\
&Avda Tarongers s/n\\
& 46022-Valencia (Spain)\\
      \       & email: carmen.domingo@uv.es\\
      
& and \\

&Universidad de Valencia.\\
&Departamento de Geometr\'{\i}a y Topolog\'{\i}a\\
& Avda. Andr\'es Estell\'es, 1, 46100-Burjassot (Valencia) Spain \\
      \       & email: miquel@uv.es 
\end{tabular}

}





\end{document}






1503.01332.extracted_bib
{BCP Batista, M.; Cavalcante, M. P.; Pyo, J.; Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds. {J. Math. Anal. Appl. 419 (2014), no. 1, 617–626.
1503.01332.extracted_bib
{BeGo Berger, Marcel; Gostiaux, Bernard; {Differential geometry: manifolds, curves, and surfaces. GTM, 115. Springer-Verlag, New York, 1988.
1503.01332.extracted_bib
{BWBleecker, D., Weiner, J.: Extrinsic bounds on $\lambda_1$ of $\Delta$ on a compact manifold. {Comment. Math. Helv. 51 (1976), 601-609.
1503.01332.extracted_bib
{br13 Brendle, Simon; Minimal surfaces in $S^3$: a survey on recent results. {Bull. Math. Sci. 3 (2013), no.1, 133–171.
1503.01332.extracted_bib
{br15 Brendle, Simon; Embedded self-similar shrinkers of genus 0; Preprint, arXiv:1411.4640
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{BK Buser, P. and Karcher, H.; {\em Gromov's almost flat manifolds, société mathematique de france, astérisque num. 81, Paris (1981).
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{CESG Colbois, Bruno; El Soufi, Ahmad; Girouard, Alexandre; Isoperimetric control of the spectrum of a compact hypersurface. {J. Reine Angew. Math. 683 (2013), 49–65.
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{Ch Chavel, Isaac; {Eigenvalues in Riemannian geometry. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp.
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{chmezh12b Cheng, Xu; Mejia, Tito and Zhou, Detang; {Eigenvalue estimate and compactness for closed $f$-minimal surfaces, {Pacific J. Math. 271 (2014), no. 2, 347–367
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{CL Coddington, Earl A.; Levinson, Norman; {Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. xii+429 pp.
1503.01332.extracted_bib
{comi Colding, Tobias H.; Minicozzi, William P., II; Generic mean curvature flow I: generic singularities. {Ann. of Math. (2) 175 (2012), no. 2, 755–833.
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{dixi Ding, Qi; Xin, Y. L. Volume growth, eigenvalue and compactness for self-shrinkers. {Asian J. Math. 17 (2013), no. 3, 443–456
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{Ej Ejiri, Norio; A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric. {Math. Z. 168 (1979), no. 3, 207–212.
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{GMO Giménez, Fernando; Miquel, Vicente; Orengo, J. Javier; Upper bounds of the first eigenvalue of closed hypersurfaces by the quotient area/volume. {Arch. Math. 83 (2004), no. 3, 279--288.
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{he Heintze, Ernst; Extrinsic upper bounds for $\lambda_1$. {Math. Ann. 280 (1988), no. 3, 389–402.
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{huru Huang, Qin; Ruan, Qihua; Applications of some elliptic equations in Riemannian manifolds. {J. Math. Anal. Appl. 409 (2014), no. 1, 189–196.
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{hui Huisken, G.; Asymptotic behavior for singularities of the mean curvature flow, {J. Differential Geom. 31 (1990), 285–299.
1503.01332.extracted_bib
{Ka Karcher, H.; Riemannian Center of Mass and Mollifier Smoothing {\em Comm. Pure and Applied Math. {XXX, (1977) 509--541
1503.01332.extracted_bib
{lilo Long, Li; On the spectrum of weighted Laplacian operator and its applications to uniqueness of Kähler Einstein metrics, {Math. Ann. Published online 2014, DOI: 10.1007/s00208-014-1152-1
1503.01332.extracted_bib
{ma14 Ma, Li; Eigenvalue estimates and L1 energy on closed manifolds. {Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 10, 1729–1734.
1503.01332.extracted_bib
{madu Ma, Li; Du, Sheng-Hua; Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. {C. R. Math. Acad. Sci. Paris 348 (2010), no. 21-22, 1203–1206.
1503.01332.extracted_bib
{maliu Ma, Li; Liu, Baiyu; Convexity of the first eigenfunction of the drifting Laplacian operator and its applications. {New York J. Math. 14 (2008), 393–401.
1503.01332.extracted_bib
{mo11 Møller, Niels Martin ; Closed self-shrinking surfaces in $\mathbb{R^3$ via the torus, Preprint, arXiv:1111.7318
1503.01332.extracted_bib
{piri Pigola, Stefano; Rimoldi, Michele; Complete self-shrinkers confined into some regions of the space. {Ann. Global Anal. Geom. 45 (2014), no. 1, 47–65
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{Re Reilly, R.: On the first eigenvalue of the Laplacian for compact submanifold of Euclidean space. {Comment. Math. Helv. 52, 525-533 (1977)
1503.01332.extracted_bib
{sm05 Smoczyk, Knut; Self-shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 2005, no. 48, 2983–3004.
1503.01332.extracted_bib
{sm12 Smoczyk, Knut; {Mean curvature flow in higher codimension: Introduction and survey. in Global Differential Geometry, Springer Proceedings in Mathematics, vol. 17, part 2,(2012) 231-274.