# Jellett-Minkowski’s formula revisited.
Isoperimetric inequalities for submanifolds in an ambient manifold with
bounded curvature

Vicent Gimeno  Department of Mathematics-INIT, Universitat Jaume I, Castelló
de la Plana, Spain<EMAIL_ADDRESS>
###### Abstract.

In this paper we provide an extension to the Jellett-Minkowski’s formula for
immersed submanifolds into ambient manifolds which possesses a pole and radial
curvatures bounded from above or below by the radial sectional curvatures of a
rotationally symmetric model space. Using this Jellett-Minkowski’s generalized
formula we can focus on several isoperimetric problems. More precisely, on
lower bounds for isoperimetric quotients of any precompact domain with smooth
boundary, or on the isoperimetric profile of the submanifold and its modified
volume. In the particular case of a model space with strictly decreasing
radial curvatures, an Aleksandrov type theorem is provided.

###### Key words and phrases:

Jellett-Minkowski-formula and Aleksandrov type theorem and isoperimetric
problem and Mean curvature

###### 1991 Mathematics Subject Classification:

35P15

Work partially supported by DGI grant MTM2010-21206-C02-02.

##  1\. Introduction

Given a precompact domain $\Omega\subset P$ with smooth boundary
$\partial\Omega$ in a $m-$dimensional submanifold $P^{m}$ of the Euclidean
space $\mathbb{R}^{n}$, by the Jellett formula one obtains (see [Jel53, Cha93,
Cha78])

(1.1) $m\text{V}(\Omega)+\int_{\Omega}\langle\tau,H\rangle
dV=\int_{\partial\Omega}\langle\tau,\nu\rangle dA\quad,$

where $\tau$ is the vector position in $\mathbb{R}^{n}$, $V(\Omega)$ is the
volume of $\Omega$, $\nu$ is the unit normal vector pointed outward to
$\partial\Omega$ and, $dV$ and $dA$ are the induced Riemannian volume and area
densities on $\Omega$ and $\partial\Omega$ respectively.

The so-called Minkowski formula (see for instance [MR09, Formula A in Theorem
6.11]) follows from the above formula for the particular case of a closed
$m$-dimensional submanifold $S$ immersed in $\mathbb{R}^{n}$ :

(1.2) $mV(S)+\int_{S}\langle\tau,H\rangle dV=0\quad.$

Making use of this formula, Jellett [Jel53] proved in the middle of the
nineteenth century that a star-shaped constant mean curvature surface
$S\subset\mathbb{R}^{3}$ is a round sphere (see [BS09] for an extension of
results in this direction). In 1956 A. D Aleksandrov [Ale56] improved that
result asserting that any closed, embedded hypersurface in $\mathbb{R}^{n}$
with constant mean curvature is a round sphere.

In this paper we provide an extension of the Jellett-Minkowski’s formula (1.1)
into a more general setting. Our setting will be a $m$-dimensional submanifold
$P^{m}$ immersed in a $n$-dimensional ambient manifold $(N,o)$ with a pole $o$
and radial curvatures $K_{N}$ (see §3) bounded from above or below by the
radial sectional curvatures of a rotationally symmetric model space
$M_{w}^{n}=\mathbb{R}^{+}\times\mathbb{S}_{1}^{n-1}\cup\\{o_{w}\\}$, with
center point $o_{w}$ and warped metric tensor $g_{M_{w}^{n}}$ constructed
using the positive and increasing warping function
$w:\mathbb{R}^{+}\to\mathbb{R}^{+}$ in such a way that
$g_{M_{w}^{n}}=dr^{2}+w(r)^{2}g_{\mathbb{S}_{1}^{n-1}}$ (see §2.1 for precise
definition and for the conditions that $w$ should attain in order to
$M_{w}^{n}$ have smooth metric tensor $g_{M^{n}_{w}}$ around $r=0$). By the
Jellet-Minkowski generalized formula we can obtain -among other results- an
Aleksandrov type theorem for rotationally symmetric model manifolds.

The first step to generalize the Jellett-Minkowski’s formula is to define a
_generalized $w$-vector position_ $\tau_{w}$ using the gradient $\nabla^{N}r$
of the distance function $r$ to the pole $o$ in the ambient manifold $N$

(1.3) $\tau_{w}:=\frac{w}{w^{\prime}}\nabla^{N}r\quad.$

Our second steep is to define a _weighted densities_ $d\mu_{w}$ and
$d\sigma_{w}$ on the submanifold using the warping function $w$ and the
induced Riemannian densities $dV$ and $dA$

(1.4) $\displaystyle d\mu_{w}$ $\displaystyle:=w^{\prime}(r)dV\quad,$
$\displaystyle d\sigma_{w}$ $\displaystyle:=w^{\prime}(r)dA\quad.$

Using the weighted densities $d\mu_{w}$ and $d\sigma_{w}$ for any domain
$\Omega$ with smooth boundary $\partial\Omega$ we can define the $w$-_weighted
volume_ $\mu_{w}(\Omega)$ and the $w$-_weighted area_
${\sigma_{w}}(\partial\Omega)$

(1.5) $\displaystyle\mu_{w}(\Omega)$
$\displaystyle:=\int_{\Omega}d\mu_{w}\quad,$
$\displaystyle\sigma_{w}(\partial\Omega)$
$\displaystyle:=\int_{\partial\Omega}d\sigma_{w}\quad.$

With these previous definitions we can state following Jellett-Minkowski’s
generalized formula

######  Main Theorem (Jellett-Minkowski’s generalized formula).

Let $\varphi:P^{m}\to N^{n}$ be an immersion into a $n-$dimensional ambient
manifold $N$ which possesses a pole and its radial sectional curvatures
$K_{N}$ at any point $p\in N$ are bounded by above (or below ) by the radial
curvatures $K_{w}$ of a model space $M_{w}^{n}$

$K_{N}\left(p\right)\leq
K_{M_{w}^{n}}\left(r\left(p\right)\right)=-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\quad\left(\text{respectively
}K_{N}\left(p\right)\geq-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\right)\quad.$

Suppose moreover, that $w^{\prime}>0$ . Then for any precompact domain
$\Omega$ with smooth boundary $\partial\Omega$

(1.6) $\displaystyle m\mu_{w}(\Omega)+\int_{\Omega}\langle\tau_{w},H\rangle
d\mu_{w}\leq(\geq)\int_{\partial\Omega}\langle\tau_{w},\nu\rangle
d\sigma_{w}\quad,$

where $\tau_{w}$ is the generalized $w$-vector position, $\mu_{w}(\Omega)$ is
the $w$-weighted volume of $\Omega$, $d\mu_{w}$ and $d\sigma_{w}$ are the
$w$-weighted densities, and $\nu$ is the unit normal vector to
$\partial\Omega$.

######  Remark a.

Observe since $w$ is a positive increasing function expressions (1.3), (1.4)
(1.5) are well defined. In the particular case when $w(r)=r$ one obtains

(1.7) $\displaystyle d\mu_{w}$
$\displaystyle=dV\quad\mu_{w}(\Omega)=V(\Omega)$ $\displaystyle d\sigma_{w}$
$\displaystyle=dA\quad\sigma_{w}(\partial\Omega)=A(\partial\Omega)\quad,$

and (1.6) in the above theorem becomes the same expression than in (1.1) but
instead of an equality, one obtains an inequality.

######  Remark b.

Recall that given an isometric immersion $\varphi:(P,g_{P})\to(N,g_{N})$ and
the Levi-Civita connections $\nabla^{N}$ , $\nabla^{P}$ over $N$ and $P$
respectively, the second fundamental form $B^{P}(X,Y)$ for any two vector
fields $X$ and $Y$ of $P$ is

$B^{P}(X,Y)=\nabla^{N}_{X}Y-\nabla^{P}_{X}Y\quad.$

In this paper we use the following convention to define the mean curvature
vector

$H:=\text{tr}_{g}(B^{P})\quad.$

Observe that we do not divide by the dimension of the submanifold.

######  Remark c.

In the particular case when the ambient manifold is a model space inequality
(1.6) becomes an equality (see [Bre13]) and we can rededuce the formula
[Bre13, equation (6)]

The structure of the paper is as follows.

In §2 we explore the application of the above Jellett-Minkowski’s generalized
formula obtaining:

  1. (1)

In the particular case in which the ambient manifold is a model space, we
obtain an equality in the generalized Jellett-Minkowski formula. That allow us
to provide a slight new Aleksandrov type theorem (theorem 2.5) for
rotationally symmetric model spaces in the line of [Bre13] and to study its
application to the isoperimetric problem.

  2. (2)

The relation of the Jellett-Minkowski formula to the $k$-isoperimetric
quotients.

  3. (3)

Bounds for the isoperimetric profile of isometric immersions into a geodesic
ball in a Cartan-Hadamard in terms of the total mean curvature of the
submanifolds and its conformal type.

  4. (4)

A gap result for the modified volume of complete non-compact manifolds into a
Cartan-Hadamard ambient manifold.

  5. (5)

Finally, for minimal immersions, in corollary 2.18 \- that is an extension of
[CG92, theorem 4] \- , we provide $w-$modified isoperimetric inequalities.

In §3 we recall previous results due to the Hessian comparison of the
extrinsic distance function in order to in §4,§5,§6, §7, §8, §9, §10, §11, §12
prove the main theorem, the Aleksandrov type theorem (theorem 2.5) and
corollaries 2.2, 2.3, 2.7, 2.8, 2.10, 2.11, 2.15, 2.17 and 2.18.

##  2\. Applications of the Jellett-Minkowski’s generalized formula

###  2.1. Aleksandrov type theorem on $w-$model spaces

The Jellet-Minkowski generalized formula becomes an equality in the case of an
immersion into a $w-$model space. Model spaces or also called $w-$Model spaces
or rotationally symmetric model spaces, are generalized manifolds of
revolution using warped products. Let us recall here the following definition
of a model space.

######  Definition 2.1 (See [GW79, Gri99, Gri09]).

A $w-$model space $M_{w}^{n}$ is a simply connected $n$-dimensional smooth
manifold $M_{w}^{n}$ with a point $o_{w}\in M_{w}^{n}$ called the _center
point of the model space_ such that $M_{w}^{n}-\\{o_{w}\\}$ is isometric to a
smooth warped product with base $B^{1}=(\,0,\,\Lambda)\,\,\subset\,\mathbb{R}$
(where $\,0<\Lambda\leq\infty$ ), fiber $F^{n-1}=S^{n-1}_{1}$ (i.e. the unit
$(n-1)-$sphere with standard metric), and positive warping function
$w:\,[\,0,\,\Lambda\,)\to\mathbb{R}_{+}$. Namely:

(2.1)
$g_{M_{w}^{n}}=\pi^{*}\left(g_{(\,0,\,\Lambda)}\right)+(w\circ\pi)^{2}\sigma^{*}\left(g_{S^{n-1}_{1}}\right)\quad,$

being $\pi:M_{w}^{n}\to(\,0,\,\Lambda)$ and $\sigma:M_{w}^{n}\to S^{n-1}_{1}$
the projections onto the factors of the warped product.

Despite of the freedom in the choose of the $w$ function in the above
definition there exist certain restrictions around $r\to 0$. In order to
attain $M_{w}^{n}$ a smooth metric tensor around $o_{w}$ the positive warping
function $w$ should hold the following equalities (see [GW79, Pet98]) :

(2.2) $\displaystyle w(0)=0\quad,$ $\displaystyle w^{\prime}(0)=1\quad,$
$\displaystyle w^{(2k)}(0)=0\quad,$

where $w^{(2k)}(r)$ are the even derivatives of $w$.

The parameter $\Lambda$ in the above definition is called the _radius of the
model space_. If $\Lambda=\infty$, then $o_{w}$ is a pole of $M_{w}^{n}$.

The expression of the metric tensor (2.1) is often written as
$g_{M_{w}^{n}}=dr^{2}+\left(w(r)\right)^{2}d\Theta^{2}$, where $d\Theta^{2}$
denotes the standard metric on $S^{n-1}_{1}$ ( $d\Theta^{2}=g_{S^{n-1}_{1}}$).
The usual examples of model spaces are the real space forms

######  Remark d.

The simply connected space forms $\mathbb{K}^{m}(b)$ of constant curvature $b$
can be constructed as $w-$models with any given point as center point using
the warping functions

(2.3) $w(r)=w_{b}(r)=\begin{cases}\frac{1}{\sqrt{b}}\sin(\sqrt{b}\,r)&\text{if
$b>0$}\\\ \phantom{\frac{1}{\sqrt{b}}}r&\text{if $b=0$}\\\
\frac{1}{\sqrt{-b}}\sinh(\sqrt{-b}\,r)&\text{if $b<0$}\quad.\end{cases}$

Note that for $b>0$ the function $w_{b}(r)$ admits a smooth extension to
$r=\pi/\sqrt{b}$. For $\,b\leq 0\,$ any center point is a pole.

Applying now the generalized Jellett-Minkowski formula on embedded
hypersurfaces $\Sigma$ in a model space bounding a domain $\Omega$ we can
state

######  Corollary 2.2.

Let $M_{w}^{n}$ be a $w-$model space with positive warping function $w$ and
positive derivative $w^{\prime}$ on $(0,\Lambda)$ being $\Lambda$ the radius
of the model space. Then for any closed, embedded, orientable hypersurface
$\Sigma$ bounding a domain $\Omega$

(2.4)
$\frac{\sigma_{w}(\Sigma)}{\mu_{w}(\Omega)}\geq\frac{1}{n\sup_{\Sigma}(|\tau_{w}|)}\quad.$

Recall that a constant mean curvature hypersurface is a hypersurface $\Sigma$
with constant pointed inward mean curvature $h$ :

$h=-\langle H,\nu\rangle=\text{ constant on }\Sigma\quad,$

being $\nu$ the unit normal outward vector to $\Sigma$.

For constant mean curvatures hypersurfaces using the generalized Jellett-
Minkowski formula we obtain

######  Corollary 2.3.

Let $M_{w}^{n}$ be a $w-$model space with positive warping function $w$ and
positive derivative $w^{\prime}$ on $(0,\Lambda)$ being $\Lambda$ the radius
of the model space. Then for any closed, embedded, orientable hypersurface
$\Sigma$ with constant mean curvature bounding a domain $\Omega$

(2.5) $h=\frac{(n-1)}{n}\frac{\sigma_{w}(\Sigma)}{\mu_{w}(\Omega)}\quad.$

######  Remark e.

Observe that the above stated constant mean curvature hypersurfaces are mean
convex hypersurfaces ($h>0$).

Given a point $p\in M_{w}^{n}-\\{o_{w}\\}$ in a model space $M_{w}^{n}$ the
distance $r$ to the center point $o_{w}$ is given by $r(p)=\pi(p)$. A
$2$-plane $\Pi_{\text{radial}}$ in $T_{p}M_{w}^{n}$ is a _radial plane_ if it
contains the radial direction ($\nabla r\in\Pi_{\text{radial}}$) and a
$2$-plane $\Pi_{\text{fiber}}$ is a _tangent to the fiber plane_ if it is
perpendicular to the radial direction ($\Pi_{\text{fiber}}\perp\nabla r$). The
sectional curvatures of the radial or tangent to the fibers planes depends
only on the distance to the center of the model space (see proposition 3.2).

For model spaces with an accurate control of the sectional curvatures of the
radial and tangent to the fiber planes. We obtain the following Aleksandrov
type theorem

######  Theorem 2.4 (From Theorem 1.4 of [Bre13]).

Let $(M_{w}^{n},o_{w})$ be a $w-$model space with center $o_{w}$, positive
warping function $w$ and positive derivative $w^{\prime}$ on $(0,\Lambda)$
being $\Lambda$ the radius of the model space. Suppose moreover that the
sectional curvatures $K_{\Pi_{\text{fiber}}}$ of planes tangents to the fiber
and the radial sectional currvatures $K_{\Pi_{\text{rad}}}$ satisfies the
following relation

(2.6) $K_{\Pi_{\text{fiber}}}\geq K_{\Pi_{\text{rad}}}.$

Then, if the radial sectional curvature is a monotone function non increasing
on the distance to the center of the model
($\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$), every closed, embedded, orientable
hypersurface $\Sigma$ with constant mean curvature bounding a domain $\Omega$
is an umbilic submanifold. In the particular case when
$K_{\Pi_{\text{fiber}}}>K_{\Pi_{\text{rad}}}$ and
$\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$, every closed, embedded, orientable
hypersurface $\Sigma$ with constant mean curvature bounding a domain $\Omega$
is a sphere centered at $o_{w}$.

###### Proof.

By proposition 3.2

(2.7)
$\frac{d}{dr}K_{\Pi_{\text{rad}}}=\left(-\frac{w^{\prime\prime}}{w}\right)^{\prime}=\frac{w^{\prime}}{w}\left(-\frac{w^{\prime\prime\prime}}{w^{\prime}}+\frac{w^{\prime\prime}}{w}\right).$

Assuming the positivity of $w^{\prime}$, we obtain that inequality (2.6) and
the condition $\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$ are attained if and
only if the warping function satisfies

(2.8)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}\geq-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad.$

And that implies the condition H3’ of [Bre13, Theorem 1.4]. Finally, the
condition

(2.9) $K_{\Pi_{\text{fiber}}}>K_{\Pi_{\text{rad}}}$

is equivalent to condition H4’. ∎

Following the proof of the Theorem 1.4 of [Bre13] we can state the following
sligth modifiqued theorem

######  Theorem 2.5 (Aleksandrov type theorem).

Let $(M_{w}^{n},o_{w})$ be a $w-$model space with center $o_{w}$, positive
warping function $w$ and positive derivative $w^{\prime}$ on $(0,\Lambda)$
being $\Lambda$ the radius of the model space. Suppose moreover that the
sectional curvatures $K_{\Pi_{\text{fiber}}}$ of planes tangents to the fiber
and the radial sectional currvatures $K_{\Pi_{\text{rad}}}$ satisfies the
following relation

(2.10) $K_{\Pi_{\text{fiber}}}\geq K_{\Pi_{\text{rad}}}.$

Then:

  1. (1)

If the radial sectional curvature is a monotone function non increasing on the
distance to the center of the model ($\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq
0$), every closed, embedded, orientable hypersurface $\Sigma$ with constant
mean curvature bounding a domain $\Omega$ is an umbilic submanifold. In the
particular case when $K_{\Pi_{\text{fiber}}}>K_{\Pi_{\text{rad}}}$ and
$\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$, every closed, embedded, orientable
hypersurface $\Sigma$ with constant mean curvature bounding a domain $\Omega$
is a sphere centered at $o_{w}$.

  2. (2)

If the radial sectional curvature is a strictly decreasing function on the
distance to the center of the model ($\frac{d}{dr}K_{\Pi_{\text{rad}}}<0$),
every closed, embedded, orientable hypersurface $\Sigma$ with constant mean
curvature bounding a domain $\Omega$ is a sphere centered at $o_{w}$.

######  Remark f.

Observe that in the above theorem the new item (2)
($\frac{d}{dr}K_{\Pi_{\text{rad}}}<0$) implies (by proposition 3.2)

(2.11)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}>-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad.$

An example of a model space satisfying the Hypotheses of the above theorem is
a generalized paraboloid

######  Example 2.6 (Generalized paraboloids).

The usual parabolid $P$ in $\mathbb{R}^{3}$

$P=\\{(x,y,z)\in\mathbb{R}^{3}\,|\,z=x^{2}+y^{2}\\}$

can be obtained as a surface of revolution with metric tensor

$g_{P}=dr^{2}+\left(\frac{1}{2}\text{arcsinh}(2r)\right)^{2}d\theta^{2}.$

In this way, we define the generalized paraboloid as a $n-$dimensional model
space $M_{w}^{n}$ with warping function

$w(r)=\frac{1}{2}\text{arcsinh}(2r).$

After a few calculation one can check that

(2.12)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}>-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad.$

Hence, the closed, embedded, orientable and constant mean curvature
hypersurfaces in the generalized paraboloid are spheres centered at the center
of the model.

The above theorem allow us to focus on the isoperimetric problem in such model
spaces. The isoperimetric problem in a Riemannian manifold $M$ consists in
studying, among the compact hypersurfaces $\Sigma\subset M$ enclosing a region
$\Omega$ of volume $\text{V}(\Omega)=v$, those which minimize the area
$\text{A}(\Sigma)$. For a given volume $v$, an _isoperimetric region_ is a
region of volume $v$ and with minimum area on the boundary. Using [Ros05,
Theorem 1.1] we know that if the $M^{n}$ is compact the isoperimetric problem
has solution and moreover for $n\leq 7$ the isoperimetric hypersurface (the
boundary of the isoperimetric region) is smooth and with constant mean
curvature. Therefore,

######  Corollary 2.7.

Let $(M_{w}^{n},o_{w})$ be a $w-$model space with center $o_{w}$, dimension
$n\leq 7$, positive warping function $w$ and positive derivative $w^{\prime}$
on $(0,\Lambda)$ being $\Lambda$ the radius of the model space. Suppose
moreover that the sectional curvatures $K_{\Pi_{\text{fiber}}}$ of planes
tangents to the fiber and the radial sectional currvatures
$K_{\Pi_{\text{rad}}}$ satisfies the following relation

(2.13) $K_{\Pi_{\text{fiber}}}\geq K_{\Pi_{\text{rad}}}.$

Suppose moreover that the radial sectional curvature is a strictly decreasing
function on the distance to the center of the model
($\frac{d}{dr}K_{\Pi_{\text{rad}}}<0$) and $\Lambda<\infty$. Then the
isoperimetric regions are geodesic balls centered at $o_{w}$.

######  Remark g.

Let us emphasize here, that by [HHM99, theorem 9.1] the unique length-
minimizing simple closed curve enclosing a given area is a circle centered at
the origin for any plane with smooth, rotationally symmetric, complete metric
such that the Gauss curvature is a strictly decreasing function from the
origin. Hence, the above corollary is a (partial) generalization on the case
of dimension greater than $2$. See also [BM02, corollary 2.4] and [MM09,
corollary 2.3].

Given a domain $\Omega$ of volume $\text{V}(\Omega)$, denoting by
$\omega_{n-1}$ the volume of the standard sphere $S^{n-1}_{1}$ and using the
following function $\text{Rad}:\mathbb{R}^{+}\to\mathbb{R}^{+}$

(2.14) $v\to\text{Rad}(v)\quad\text{such that}\quad
v=\omega_{n-1}\int_{0}^{\text{Rad}(v)}w^{n-1}(t)dt,$

we obtain by the expression of the volume of a geodesic ball $B^{w}_{R}$
centered at $o_{w}$ (see equation (3.8))

(2.15) $\text{V}(\Omega)=\text{V}(B_{\text{Rad}(\text{V}(\Omega))}^{w}).$

Therefore, taking into account and the area of a geodesic sphere $S^{w}_{R}$
of radius $R$ centered at $o_{w}$ in a model space (see equation (3.7) ),
under the hypotheses of corollary 2.7 for any hypersurface $\Sigma$ bounding a
domain $\Omega$ of volume $V(\Omega)=v$, we have the following isoperimetric
inequality:

(2.16)
$A(\Sigma)\geq\omega_{n-1}w(\text{Rad}(v))^{n-1}=\text{A}(S^{w}_{\text{Rad}(v)}).$

Isoperimetric inequalities have several applications, among them, the above
inequality allow us to obtain an isoperimetric inequality for the first
eigenvalue of the Laplacian for the Dirichlet problem (see [Cha84] for
details).

######  Corollary 2.8.

Under the hypotheses of corollary 2.7 suppose moreover that the function

(2.17) $\mathcal{I}:\mathbb{R}^{+}\to\mathbb{R}^{+},\quad
t\to\mathcal{I}(t)=\frac{\omega_{n-1}w(\text{Rad}(t))^{n-1}}{t}$

is an non-increasing function. Then, for any precompact domain $\Omega\subset
M^{n}_{w}$ with smooth boundary $\partial\Omega$ and volume
$\text{V}(\Omega)=v$, the first eigenvalue of the Laplacian for the Dirichlet
problem $\lambda_{1}(\Omega)$ is bounded from below by

(2.18)
$\lambda_{1}(\Omega)\geq\frac{1}{4}\left[\mathcal{I}\left(v\right)\right]^{2}.$

###  2.2. Isoperimetric inequalities and the Jellett-Minkowski formula

In the setting of a controlled radial curvature ambient manifold with a pole,
and certain restrictions on the mean curvature of the submanifold, the
generalized Jellett-Minkowski formula is enough to get isoperimetric
inequalities (lower bounds to the isoperimetric profile) or to characterize
the $k$-isoperimetric quotient of a domain in the submanifold. The
$k-$isoperimetric quotient is defined as follows

######  Definition 2.9 (See also [CF91, Cha93]).

Given a domain $\Omega$ with smooth boundary $\partial\Omega$, the _$w$
-weighted $k$-isoperimetric quotient_ $\mathcal{I}^{w}_{k}(\Omega)$ is given
by

(2.19)
$\mathcal{I}^{w}_{k}(\Omega):=\frac{\sigma_{w}(\partial\Omega)}{\mu_{w}(\Omega)^{\frac{k-1}{k}}}\quad,$

where $\mu_{w}(\Omega)$ and $\sigma_{w}(\partial\Omega)$ are the $w$-weighted
volume and $w$-weighted area respectively.

To recall the definition of the isoperimetric profile in a Riemannian manifold
$M$, let us denote by $\mathcal{O}_{M}$ the set of relatively compact open
subsets of $M$ with smooth boundary. The isoperimetric profile $I_{M}$ is a
function $I_{M}:[0,\operatorname{Vol}(M)]\to\mathbb{R}^{+}$ such that

(2.20) $I_{M}(v):=\begin{cases}0\text{ if }v=0\\\
\inf_{\Omega\in\mathcal{O}_{M}}\\{\text{A}(\partial\Omega)\,:\,\text{V}(\Omega)=v\\}\end{cases}$

In this subsection we will examine under appropriate settings the relation
between the Jellett-Minkowski’s generalized formula and the $k-$isoperimetric
quotients, the isoperimetric profile, and the modified volume of a
submanifold.

####  2.2.1. Lower bounds to the $k$-isoperimetric quotient of a domain

The next application of the generalized Jellett-Minkowski formula will be to
obtain lower bounds to the $k$-isoperimetric profile of a precompact domain
with smooth boundary. Hence, as a direct consequence of theorem Main Theorem
and the Hölder’s inequality we can state

######  Corollary 2.10.

Let $\varphi:P^{m}\to N^{n}$ be an immersion into a $n-$dimensional ambient
manifold $N$ which possesses a pole and its radial sectional curvatures
$K_{N}$ at any point $p\in N$ are bounded by above by the radial curvatures
$K_{w}$ of a model space $M_{w}^{n}$

$K_{N}\left(p\right)\leq
K_{w}\left(r\left(p\right)\right)=-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\quad.$

Suppose moreover, that $w^{\prime}>0$ . Then for any precompact domain
$\Omega$ with smooth boundary $\partial\Omega$

(2.21)
$\mathcal{I}^{w}_{k}(\Omega)\geq\frac{m}{\sup_{\Omega}|\tau_{w}|}\mu_{w}(\Omega)^{1/k}-\left(\int_{\Omega}|H|^{k}d\mu_{w}\right)^{1/k}\quad.$

####  2.2.2. Immersions into a geodesic ball. Isoperimetric profile, total
mean curvature and Parabolicity

Applying corollary 2.10 to the special setting of an immersion into a geodesic
ball in a Cartan-Hadamard ambient manifold we obtain the following corollary
for the isoperimetric profile.

######  Corollary 2.11.

Let $\varphi:P^{m}\to B_{R}^{N}(o)$ be an complete immersion into a geodesic
ball $B_{R}^{N}(o)$ of radius $R$ centered in $o\in N$ in a Cartan-Hadamart
ambient manifold $N$ with sectional curvatures $K_{N}\leq 0$ bounded from
above. Suppose that the submanifold has finite $k$-norm of the mean curvature,
namely, $\int_{P}|H|^{k}dV<\infty$, for some $k>1$. Then the isoperimetric
profile is bounded from below by

(2.22)
$I_{P}(v)\geq\left(\frac{mv^{1/k}}{R}-\left(\int_{P}|H|^{k}dV\right)^{1/k}\right)v^{\frac{k-1}{k}}\quad,$

where $dV$ is the Riemannian density in $P$.

######  Remark h.

The existence of isometric and complete immersions into a ball in in a Cartan-
Hadamard is out of any doubt. In fact, by the celebrated Nash’s imbedding
theorem [Nas56], any Riemannian $n$-manifold with a $C^{\infty}$ positive
metric has a $C^{\infty}$ isometric imbedding in
$\frac{1}{2}(n+1)(3n+1)$-dimensional Euclidean space, and in fact, in any
small portion of this space (as for instance a geodesic ball of this space).

The existence of immersions with finite integral of the norm of the mean
curvature is also well known (at least in the limit case $H=0$). In [MM05,
MM06] F. Martín and S. Morales in -according to S.T Yau- highly non-trivial
refinement of a Nadirashvili’s method [Nad96] constructed a complete minimal
immersion of a $2$-dimensional disc into every open convex set of
$\mathbb{R}^{3}$.

######  Remark i.

From corollary 2.10 applying as a warping function $w(r)=r$ we can deduce that
if we have a submanifold $P$ immersed in a ball $B_{R}^{N}(o)$ of radius $R$
centered at $o\in N$ in a Cartan-Hadamard ambient manifold $N$ with norm of
the mean curvature vector $|H|$ bounded from above by

(2.23) $|H|\leq\frac{m\,\epsilon}{R}\quad,$

for $0<\epsilon<1$, the isoperimetric quotients are bounded from below by

(2.24) $I_{k}(\Omega)\geq\frac{m(1-\epsilon)}{R}\text{V}(\Omega)^{1/k}\quad,$

for any $k>1$. But that implies

(2.25)
$\frac{\text{A}(\partial\Omega)}{\text{V}(\Omega)}\geq\frac{m(1-\epsilon)}{R}\quad,$

and therefore $P$ has positive Cheeger constant, positive fundamental tone and
$P$ is Hyperbolic (see [Gri99] for the relation of positive fundamental tone
and non-parabolicity).

Hence, the mean curvature of an isometric immersion into a geodesic ball of a
Cartan-Hadamard manifold is closely related to the conformal type problem.

The geodesic balls are the isoperimetric domains in the Euclidean space.
Therefore, in the $n$-dimensional Euclidean space, the isoperimetric profile
is given by

(2.26) $I_{\mathbb{R}^{n}}(v)=C_{n}v^{\frac{n-1}{n}}\quad.$

Applying corollary 2.11, proving by contradiction, we can easily state that
for any immersion of $\mathbb{R}^{n}$ into a ball $B_{R}^{\mathbb{R}^{n+k}}$
of $\mathbb{R}^{n+k}$, $\int_{\mathbb{R}^{n}}|H|^{n}dV=\infty$. In general

######  Corollary 2.12.

Let $\varphi:P^{m}\to B_{R}^{N}(o)$ be an complete immersion into a geodesic
ball $B_{R}^{N}(o)$ of radius $R$ centered in $o\in N$ in a Cartan-Hadamart
ambient manifold $N$ with sectional curvatures $K_{N}\leq 0$ bounded from
above.

Suppose that for some $k>1$ there exist a constant $C_{k}$ such that the
isoperimetric profile is bounded from above by

(2.27) $I_{P}(v)\leq C_{k}v^{\frac{k-1}{k}}\quad.$

Then, either

(2.28) $\int_{P}|H|^{k}dV=\infty\quad.$

or

(2.29)
$\text{V}(P)\leq\left(\frac{C_{k}+\left(\int_{P}|H|^{k}d\text{V}\right)^{\frac{1}{k}}}{m}\right)^{k}R^{k}\quad.$

If moreover the submanifold has the non-shrinking property (see definition
below) we can state further geometric and analytic properties

######  Definition 2.13 (Non-shrinking property).

A manifold $M$ of infinite volume has the _non-shrinking property_ if for any
$v>0$ there exist $R_{v}$ such that

(2.30) $\inf_{x\in M}\text{V}(B_{R_{v}}^{M}(x))\geq v\quad.$

Let us emphasize here, that there exist well known manifolds with that
property

######  Example 2.14 (Non-shrinking manifolds).

The most elemental example of a manifold with non-shrinking property is
$\mathbb{R}^{n}$. Since given a point $x\in\mathbb{R}^{n}$ the volume
$V(B_{R}(x))$ of a geodesic ball of radius $R$ centered at $x$ is an
increasing function of $R$, for any $v>0$ one can easily found $R_{v}$ such
that $V(B_{R_{v}}(x))>v$, and by the homogeneity of $\mathbb{R}^{n}$ we obtain
the non-shrinking property. The same is true for the hyperbolic space
$\mathbb{H}^{n}$, and in general for any Cartan-Hadamard manifold.

Using that non-shrinking property we can state that

######  Corollary 2.15.

Under the assumptions of corollary 2.11, suppose moreover that $P$ has non-
shrinking property. Then

  1. (1)

For any point $x\in P$ the geodesic balls of $P$, $B^{P}_{R}(x)$, of radius
$R$ centered at $x$ satisfies

(2.31) $\liminf_{R\to\infty}R^{-k}\text{V}(B^{P}_{R}(x))>0,$

  2. (2)

If moreover $k>2$, $P$ possesses a positive Green’s function (or equivalently,
it has transient Brownian motion).

As a reverse of the above corollary we can state

######  Corollary 2.16.

Let $\varphi:P^{m}\to B^{N}_{R}(0)$ be an isometric immersion of a parabolic
submanifold $P$ into the ball $B_{R}^{N}(o)$ of radius $R$ centered at $o\in
N$ in a Cartan-Hadamard manifold $N$. Then:

  1. (1)

or, for any $k>2$

$\int_{P}|H|^{k}dV=\infty\quad.$

  2. (2)

Or, $P$ has not the non-shrinking property.

By the above corollary any immersion of a parabolic manifold with non-positive
sectional curvature into a geodesic ball of a Cartan-Hadamard manifold has
infinite norm of its mean curvature vector.

####  2.2.3. The volume of complete non-compact submanifolds into a Cartan-
Hadamard ambient space

In [CMV12] unifying results from [CL98, DCWX10, Fre96, FX10] is proved that
given an isometric immersion $\varphi:P\to N$ of a complete non-compact
manifold $P$ in a manifold $N$ with bounded geometry (i.e., $N$ has sectional
curvature bounded from above and injectivity radius bounded from below by a
positive constant), if any end $E$ of $P$ has finite $L^{p}$-norm of the mean
curvature vector of $\varphi$, $\|H\|_{L^{p}(E)}<\infty$, for some $m\leq
p\leq\infty$ then $E$ must have infinite volume.

Note that $p\geq m$ is not a removable condition. Indeed in Example 4.3 of
[CMV12] is shown a complete non-compact hypersurface $P^{m}$ in
$\mathbb{R}^{m+1}$ , with $m\geq 3$, of finite volume and mean curvature
vector with finite $L^{p}$-norm, for any $0\leq p<m-1$.

In the particular case of a $m$-dimensional submanifold in a Cartan-Hadamard
ambient manifold with sectional curvatures bounded from above by a negative
constant $b$, for any $p\geq 2$, and any dimension $m$ of the submanifold , we
can obtain lower bounds for the $w_{b}$-weighted volume of the submanifold in
terms of the $w_{b}$-weighted $L^{p}$-norm of the norm of the mean curvature
vector, being $w_{b}$ the warping function given in remark d, i.e.,

$w_{b}=\frac{1}{\sqrt{-b}}\sinh(\sqrt{-b}\,r)\quad.$

In such a setting we can state

######  Corollary 2.17.

Let $\varphi:P^{m}\to N$ be an immersion into a Cartan-Hadamard manifold $N$
with sectional curvatures $K_{N}$ bounded from above by a negative constant
$K_{N}\leq b<0$. Then for any $p\geq 2$,

  1. (1)

either

$\mu_{w_{b}}(P)^{\frac{1}{p}}\leq\frac{\|H\|_{L^{p}_{w_{b}}(P)}}{m\sqrt{-b}}\quad,$

  2. (2)

or

$\mu_{w_{b}}(P)=\infty\quad.$

where $\mu_{w_{b}}(P)$ is the $w_{b}$-weighted volume of $P$, and
$\|H\|_{L^{p}_{w_{b}}(P)}$ is the $w_{b}$-weighted $L^{p}$-norm of the norm of
the mean curvature vector, namely

(2.32)
$\|H\|_{L^{p}_{w_{b}}(P)}=\left(\int_{P}|H|^{p}d\mu_{w_{b}}\right)^{\frac{1}{p}}\quad.$

####  2.2.4. Isoperimetric inequalities on minimal submanifolds

Given a simple close curve $C$ in the flat plane, bounding a domain $D$.
Denoting by $L$ and $A$ the length of $C$ and the area of $D$ respectively,
the classical isoperimetric inequality states that

$4\pi A\leq L^{2}\quad.$

For minimal submanifolds $P$ with smooth boundary $\partial P$ lying on a
geodesic ball we can state a similar inequality using the $w$-weighted volume

######  Corollary 2.18.

Let $P^{m}$ be a $m-$dimensional compact manifold with smooth boundary
$\partial P$. Let $\varphi:P^{m}\to N^{n}$ a minimal immersion into a
$n-$dimensional ambient manifold $N$ which possesses a pole $o\in N$ and its
radial sectional curvatures $K_{N}$ at any point $p\in N$ are bounded by above
by the radial curvatures $K_{w}$ of a model space $M_{w}^{n}$

$K_{N}\left(p\right)\leq
K_{M_{w}^{n}}\left(r\left(p\right)\right)=-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\quad.$

Suppose moreover, that $w^{\prime}\geq c>0$. $\varphi^{-1}(o)\in P$ and
$\varphi(\partial P)$ lies in a geodesic sphere centered at the pole $o$. Then

(2.33) $\displaystyle cm^{m}V_{m}\,\mu_{w}(P)^{m-1}\leq A(\partial
P)^{m}\quad,$

where $\mu_{w}(P)$ is the $w$-weighted volume of $P$, $A(\partial P)$ is the
Riemannian area of $\partial P$ and $V_{m}$ is the volume of the unit ball in
$\mathbb{R}^{m}$.

######  Remark j.

Applying the above corollary but using $w=w_{b}$ given in remark d we get, as
a particular case, the isoperimetric inequality of theorem 4 of [CG92]. See
also [Pal99] for an other sort of isoperimetric inequalities on minimal
submanifolds properly immersed into a Cartan-Hadamard ambient manifold.

##  3\. Preliminaries

###  3.1. Manifold with a pole and extrinsic distance function

We assume throughout the most part of the paper that
$\varphi:P^{m}\longrightarrow N^{n}$ is an isometric immersion of a complete
non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian
manifold $N^{n}$ with a pole $o\in N$. Recall that a pole is a point $o$ such
that the exponential map

$\exp_{o}\colon T_{o}N^{n}\to N^{n}$

is a diffeomorphism. For every $x\in N^{n}-\\{o\\}$ we define
$r(x)=r_{o}(x)=\operatorname{dist}_{N}(o,x)$, and this distance is realized by
the length of a unique geodesic from $o$ to $x$, which is the radial geodesic
from $o$. We also denote by $r|_{P}$ or by $r$ the composition
$r\circ\varphi:P\to\mathbb{R}_{+}\cup\\{0\\}$. This composition is called the
extrinsic distance function from $o$ in $P^{m}$. The gradients of $r$ in $N$
and $r|_{P}$ in $P$ are denoted by $\nabla^{N}r$ and $\nabla^{P}r$,
respectively. Then we have the following basic relation, by virtue of the
identification, given any point $x\in P$, between the tangent vector fields
$X\in T_{x}P$ and $\varphi_{*_{x}}(X)\in T_{\varphi(x)}N$

(3.1) $\nabla^{N}r=\nabla^{P}r+(\nabla^{N}r)^{\bot},$

where $(\nabla^{N}r)^{\bot}(\varphi(x))=\nabla^{\bot}r(\varphi(x))$ is
perpendicular to $T_{x}P$ for all $x\in P$.

Since the manifold with a pole has a well defined radial vector field, we cab
define the radial sectional curvatures.

######  Definition 3.1.

Let $o$ be a point in a Riemannian manifold $N$ and let $x\in N-\\{o\\}$. The
sectional curvature $K_{N}(\sigma_{x})$ of the two-plane $\sigma_{x}\in
T_{x}N$ is then called a $o$-radial sectional curvature of $N$ at $x$ if
$\sigma_{x}$ contains the tangent vector to a minimal geodesic from $o$ to
$x$. We denote these curvatures by $K_{o,N}(\sigma_{x})$.

###  3.2. $w-$model spaces

The model spaces has two different roles in this paper, the first of them is
the role as an ambient manifold and the second one is as a controller of the
curvature restrictions. The sectional curvatures of a model space can be
explicitly obtained using the warped function $w$.

######  Proposition 3.2 (See [GW79, Gri99, O’N83]).

Let $M_{w}^{m}$ be a $w-$model with center point $o_{w}$. Then the
$o_{w}$-radial sectional curvatures of $M_{w}^{m}$ at every $x\in\pi^{-1}(r)$
(for $\,r\,>\,0\,$) are all identical and determined by the radial function
$\,K_{w}(r)\,$ defined as follows:

(3.2)
$K_{p_{w},M_{w}}(\sigma_{x})\,=\,K_{w}(r)\,=\,-\frac{w^{\prime\prime}(r)}{w(r)}\quad.$

And the sectional curvatures $K(\Pi_{S^{w}_{r}})$ of the $2-$planes
$\Pi_{S^{w}_{r}}$ tangents to $S_{r}^{w}=\pi^{-1}(r)$ are equal to

(3.3)
$K(\Pi_{S^{w}_{r}})=\frac{1-\left(w^{\prime}\left(r\right)\right)^{2}}{w(r)}\quad.$

We can also explicitly calculate the mean curvature of the geodesic spheres.

######  Proposition 3.3 (See [O’N83] p. 206).

Let $M_{w}^{n}$ be a $w-$model with warping function $w(r)$ and center
$o_{w}$. The distance sphere of radius $r$ and center $o_{w}$ in $M_{w}^{n}$,
denoted as $S^{w}_{r}$, is the fiber $\pi^{-1}(r)$. This distance sphere has
the following constant mean curvature vector in $M_{w}^{n}$

(3.4)
$H_{{\pi^{-1}}(r)}=-n\,\eta_{w}(r)\,\nabla^{M}\pi=-n\,\eta_{w}(r)\,\nabla^{M}r\quad,$

where the mean curvature function $\eta_{w}(r)$ is defined by

(3.5) $\eta_{w}(r)=\frac{w^{\prime}(r)}{w(r)}=\frac{d}{dr}\ln(w(r))\quad.$

In particular we have for the constant curvature space forms
$\mathbb{K}^{m}(b)$:

(3.6) $\eta_{w_{b}}(r)=\begin{cases}\sqrt{b}\cot(\sqrt{b}\,r)&\text{if
$b>0$}\\\ \phantom{\sqrt{b}}1/r&\text{if $b=0$}\\\
\sqrt{-b}\coth(\sqrt{-b}\,r)&\text{if $b<0$}\quad.\end{cases}$

The area of the geodesic sphere $S^{w}_{R}(o_{w})$ of radius $R$ centered at
$o_{w}$ is completely determined via $w$ by the volume of the fiber

(3.7) $A(S^{w}_{R}(o_{w}))=\omega_{n-1}w^{n-1}(R),$

And the volume of the corresponding ball $B_{R}^{w}(o_{w})$, for which the
fiber is the boundary

(3.8) $V(B^{w}_{R}(o_{w}))=\omega_{n-1}\int_{0}^{R}w^{n-1}(t)dt,$

being $\omega_{n-1}$ in (3.7) and (3.8) the volume of the standard sphere
$S^{n-1}_{1}$.

###  3.3. Hessian and Laplacian comparison

The 2.nd order analysis of the restricted distance function $r_{|_{P}}$
defined on manifolds with a pole is governed by the Hessian comparison (see
[GW79, Theorem A]).

The Hessian of a restricted function in a submanifold and the Hessian of the
function in the ambient space are related by the following proposition

######  Proposition 3.4.

Given an isometric immersion $\varphi:P^{m}\to N^{n}$, and given a smooth
function $f:N\to\mathbb{R}$, then:

(3.9)
$\operatorname{Hess}^{P}(f\circ\varphi)(X,Y)=\operatorname{Hess}^{N}f(X,Y)+\langle
B^{P}(X,Y),\nabla^{N}f\rangle.$

In the case of radial functions of a model space

######  Proposition 3.5.

let $M_{w}^{m}$ denote a $w-$model with center $o_{w}$. Let
$r:M_{w}^{n}\to\mathbb{R}^{+}$ denote the distance function to the center
$o_{w}$. Then for any smooth function $F:\mathbb{R}\to\mathbb{R}$,

(3.10) $\displaystyle\operatorname{Hess}^{M_{w}^{n}}F\circ r(X,Y)=$
$\displaystyle\left(F^{\prime\prime}\circ r-\left(F^{\prime}\circ
r\right)\left(\eta_{w}\circ r\right)\right)\langle X,\nabla r\rangle\langle
Y,\nabla r\rangle$ $\displaystyle+\left(F^{\prime}\circ
r\right)\left(\eta_{w}\circ r\right)\left(\langle X,Y\rangle\right).$

Now, we can state a comparison theorem when one of the spaces is a model space
$M^{m}_{w}$ using [GW79]:

######  Theorem 3.6 (See [GW79], Theorem A).

Let $N^{n}$ be a manifold with a pole $p$, let $M_{w}^{m}$ denote a $w-$model
with center $p_{w}$. Suppose that $m\leq n$ and that every $p$-radial
sectional curvature at $x\in N-\\{p\\}$ is bounded from above (or below) by
the $p_{w}$-radial sectional curvatures in $M_{w}^{m}$ as follows:

(3.11)
$K_{p,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}\quad\left(\text{respectively
}K_{p,N}(\sigma_{x})\geq-\frac{w^{\prime\prime}(r)}{w(r)}\right)$

for every radial two-plane $\sigma_{x}\in T_{x}N$ at distance
$r=r(x)=\operatorname{dist}_{N}(p,x)$ from $p$ in $N$. Then the Hessian of the
distance function in $N$ satisfies

(3.12) $\displaystyle\operatorname{Hess}^{N}(r(x))(X,X)$
$\displaystyle\geq(\leq)\operatorname{Hess}^{M_{w}^{m}}(r(y))(Y,Y)$
$\displaystyle=\eta_{w}(r)\left(1-\langle\nabla^{M}r(y),Y\rangle_{M}^{2}\right)$
$\displaystyle=\eta_{w}(r)\left(1-\langle\nabla^{N}r(x),X\rangle_{N}^{2}\right)$

for every unit vector $X$ in $T_{x}N$ and for every unit vector $Y$ in
$T_{y}M$ with $\,r(y)=r(x)=r\,$ and
$\,\langle\nabla^{M}r(y),Y\rangle_{M}=\langle\nabla^{N}r(x),X\rangle_{N}\,$.

Hence, from proposition 3.4, proposition 3.5, and theorem 3.6 after few
calculations one obtains

######  Corollary 3.7.

Given an isometric immersion $\varphi:P^{m}\to N^{n}$. Suppose again that the
assumptions of Theorem 3.6 are satisfied. Then, for every smooth function
$f(r)$ with $f^{\prime}(r)\geq 0\,\,\textrm{for all}\,\,\,r$ :

(3.13) $\displaystyle\Delta^{P}(f\circ r)\,\geq(\leq)$
$\displaystyle\left(\,f^{\prime\prime}(r)-f^{\prime}(r)\eta_{w}(r)\,\right)\|\nabla^{P}r\|^{2}$
$\displaystyle+mf^{\prime}(r)\left(\,\eta_{w}(r)+\frac{1}{m}\langle\,\nabla^{N}r,\,H_{P}\,\rangle\,\right)\quad,$

where $H_{P}$ denotes the mean curvature vector of $P$ in $N$.

##  4\. Proof of the Jellett-Minkowski’s generalized formula (main theorem)

In order to prove the Jellett-Minkowski’s generalized formula we only have to
apply the divergence theorem to an appropriate function. Let us define the
following function $F:\mathbb{R}^{+}\to\mathbb{R}^{+}$ given by

(4.1) $F(t):=\int_{0}^{t}w(s)ds\quad.$

Using corollary 3.7

(4.2) $\displaystyle\Delta^{P}F$
$\displaystyle\geq(\leq)mw(r)\left(\eta_{w}(r)+\frac{1}{m}\langle\nabla^{N}r,H\rangle\right)$
$\displaystyle=$
$\displaystyle\left(m+\langle\tau_{w},H\rangle\right)w^{\prime}(r)\quad.$

Applying the divergence theorem to the domain $\Omega\subset P$

(4.3) $\displaystyle\int_{\partial\Omega}\langle\nabla^{P}F,\nu\rangle
dA\geq(\leq)m\mu_{w}(\Omega)+\int_{\Omega}\langle\tau_{w},H\rangle
d\mu_{w}\quad.$

Therefore

(4.4) $\displaystyle\int_{\partial\Omega}\langle\tau_{w},\nu\rangle
d\sigma_{w}\geq(\leq)m\mu_{w}(\Omega)+\int_{\Omega}\langle\tau_{w},H\rangle
d\mu_{w}\quad.$

And the theorem follows.

##  5\. Proof of corollary 2.2 and corollary 2.3

Since $\Sigma=\partial\Omega$ ($\partial\Sigma=\emptyset$), and $\Omega$ is
totally geodesic submanifold by the generalized Jellett-Minkowsi formula

(5.1) $\displaystyle n\mu_{w}(\Omega)$
$\displaystyle=\int_{\Sigma}\langle\tau_{w},\nu\rangle d\sigma_{w}\quad,$
$\displaystyle(n-1)\sigma_{w}(\Sigma)$
$\displaystyle=-\int_{\Sigma}\langle\tau_{w},H\rangle d\sigma_{w}\quad.$

From the first line of the above equations

(5.2)
$n\mu_{w}(\Omega)\leq\max_{\Sigma}|\tau_{w}|\int_{\Sigma}d\sigma_{w}=\max_{\Sigma}|\tau_{w}|\sigma_{w}(\Sigma)\quad,$

And the corollary 2.2 follows. On the other hand, by the second equality of
(5.1) if $\Sigma$ is a constant mean curvature hypersurface

(5.3) $\displaystyle(n-1)\sigma_{w}(\Sigma)$
$\displaystyle=h\int_{\Sigma}\langle\tau_{w},\nu\rangle
d\sigma_{w}=h\,n\,\mu_{w}(\Omega)\quad.$

And corollary 2.3 is proven.

##  6\. Proof of the theorem 2.5

This proof follows the proof done in [Bre13]. First of all, we need the
following Heintze-Karcher [HK78] inequality

######  Theorem 6.1 (Heintze-Karcher inequality).

Let $M_{w}^{n}$ be a $w-$model space with center $o_{w}$, positive warping
function $w$ and positive derivative $w^{\prime}$ on $(0,\Lambda)$ being
$\Lambda$ the radius of the model space. Suppose moreover that the warping
function satisfies the following inequalities:

(6.1)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}\geq-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad,$

then in every closed, embedded, orientable and convex mean curvature
hypersurface $\Sigma$ bounding a domain $\Omega$ the following inequality
holds

(6.2) $\left(n-1\right)\int_{\Sigma}\frac{1}{h}d\sigma_{w}\geq
n\mu_{w}(\Omega)\quad.$

With equality in 6.2 if $\Sigma$ is umbilic, and in the particular case when
$\frac{w^{\prime\prime}}{w}\neq\frac{w^{\prime\prime\prime}}{w^{\prime}}$,
equality in 6.2 implies that $\Sigma$ is a sphere centered at $o_{w}$.

From corollary 2.3 it is clear that if the hypersurface is a constant mean
hypersurface bounding a domain, the surface is mean convex and attains
equality in inequality (6.2). Hence, the only thing to do in order to prove
theorem 2.5 is to prove the above theorem.

######  Remark k.

The proof of the above theorem follows from the proof of Theorem 3.5 of
[Bre13]. The only new piece is the condition
$\frac{w^{\prime\prime}}{w}\neq\frac{w^{\prime\prime\prime}}{w^{\prime}}$ and
its rigidity consequences arising from remark l. For completeness in order to
attain remark l we have to repeat part of the proof done in section 3 of
[Bre13].

###### Proof.

Since $w^{\prime}>0$ we can use the following conformally modified metric
$g_{c}=\frac{1}{w^{\prime}}g_{M_{w}^{n}}$ being $g_{M_{w}^{n}}$ the metric
tensor in $M_{w}^{n}$. For each point $p\in\overline{\Omega}$, we denote by
$u(p)=d_{g_{c}}(p,\Sigma)$ the distance to $p$ from $\Sigma$ with respect to
the metric $g_{c}$, and we denote by
$\Phi:\Sigma\times[0,\infty)\to\overline{\Omega}$ the normal exponential map
with respect to $g_{c}$. Namely, for each point $x\in\Sigma$, the curve
$t\to\Phi(x,t)$ is a geodesic with respect to $g_{c}$, and we have

(6.3) $\Phi(x,0)=x,\quad\left.\frac{\partial}{\partial
t}\Phi(x,t)\right|_{t=0}=-w^{\prime}(r(x))\nu(x).$

Let us define

(6.4) $\displaystyle A$
$\displaystyle:=\left\\{(x,t\in\Sigma\times[0,\infty)|\,u(\Phi(x,t))=t\right\\}$
$\displaystyle A^{*}$
$\displaystyle:=\left\\{(x,t\in\Sigma\times[0,\infty)|\,(x,t+\delta)\in
A\text{ for some }\delta>0\right\\}$ $\displaystyle\Sigma_{t}^{*}$
$\displaystyle=\Phi(A^{*}\cap(\Sigma\times\\{t\\}).$

We also denote by $h$ and $B^{\Sigma_{t}^{*}}$ the mean curvature and the
second fundamental form of $\Sigma^{*}_{t}$ with respect to the metric
$g_{M_{w}^{n}}$. Hence,

######  Proposition 6.2 (See proposition 3.2 of [Bre13]).

The mean curvature of $\Sigma_{t}^{*}$ is positive and satisfies the
differential inequality

$\frac{\partial}{\partial
t}\left(\frac{w^{\prime}}{h}\right)\leq-\frac{1}{n-1}\left(w^{\prime}\right)^{2}.$

###### Proof.

Since $\nu=-\frac{\nabla u}{|\nabla u|}$ is the outward-pointing unit normal
vector to $\Sigma_{t}^{*}$ with respect to the metric $g_{M_{w}^{n}}$, by the
variation formulas, the mean curvature of $\Sigma_{t}^{*}$ satisfies the
following equation

(6.5) $\frac{\partial}{\partial
t}h=\Delta_{\Sigma^{*}_{t}}w^{\prime}+\left(\text{Ricc}^{M_{w}^{n}}(\nu,\nu)+\|B^{\Sigma^{*}_{t}}\|^{2}\right)w^{\prime}.$

Using proposition 3.4 for any orthonormal basis $\\{e_{i}\\}_{i=1}^{n-1}$ of
$\Sigma_{t}^{*}$

(6.6) $\displaystyle\Delta_{\Sigma_{t}^{*}}w^{\prime}$
$\displaystyle=\sum_{i=1}^{n-1}\operatorname{Hess}^{\Sigma_{t}^{*}}w^{\prime}(e_{i},e_{i})$
$\displaystyle=\sum_{i=1}^{n-1}\operatorname{Hess}^{M_{w}^{n}}w^{\prime}(e_{i},e_{i})+\sum_{i=1}^{n-1}\langle
B^{\Sigma_{t}^{*}}(e_{i},e_{i}),\nabla^{M_{w}^{n}}w^{\prime}\rangle$
$\displaystyle=\Delta^{M_{w}^{n}}w^{\prime}-\operatorname{Hess}^{M_{w}^{n}}w^{\prime}(\nu,\nu)+\langle
H,\nabla^{M_{w}^{n}}w^{\prime}\rangle,$

applying proposition 3.5

(6.7) $\displaystyle\Delta_{\Sigma_{t}^{*}}w^{\prime}=$ $\displaystyle
w^{\prime\prime\prime}+(n-1)w^{\prime\prime}\eta_{w}-\operatorname{Hess}^{M_{w}^{n}}w^{\prime}(\nu,\nu)+\langle
H,\nabla^{M_{w}^{n}}w^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
w^{\prime}\left[\left(\frac{w^{\prime\prime\prime}}{w^{\prime}}-\frac{w^{\prime\prime}}{w}\right)\left(1-\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}\right)+\frac{w^{\prime\prime}}{w}\left(n-1\right)\right]$
$\displaystyle+\langle H,\nabla^{M_{w}^{n}}w^{\prime}\rangle.$

Taking into account that $1-\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}\geq 0$
and $\frac{w^{\prime\prime\prime}}{w^{\prime}}\geq\frac{w^{\prime\prime}}{w}$

(6.8) $\displaystyle\Delta_{\Sigma_{t}^{*}}w^{\prime}\geq$ $\displaystyle
w^{\prime}\frac{w^{\prime\prime}}{w}(n-1)+\langle
H,\nabla^{M_{w}^{n}}w^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
w^{\prime}\frac{w^{\prime\prime}}{w}(n-1)-h\langle\nu,\nabla^{M_{w}^{n}}w^{\prime}\rangle.$

######  Remark l.

If we have equality in inequality (6.8) then

(6.9)
$\left(\frac{w^{\prime\prime\prime}}{w^{\prime}}-\frac{w^{\prime\prime}}{w}\right)\left(1-\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}\right)=0.$

In the particular case when
$\frac{w^{\prime\prime\prime}}{w^{\prime}}\neq\frac{w^{\prime\prime}}{w}$ we
would get

(6.10) $\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}=1$

In order to make use of equality (6.5) we need estimate
$\text{Ricc}^{M_{w}^{n}}(\nu,\nu)$ and $\|B^{\Sigma_{t}^{+}}\|^{2}$. Those
estimates are taken care of in the next two lemmas

######  Lemma 6.3 (From Proposition 3.2).

Let $M_{w}^{n}$ be a $w-$model space. Suppose that

(6.11)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w},$

then for any unit vector $\nu$,

(6.12) $\text{Ricc}^{M_{w}^{n}}(\nu,\nu)\geq-(n-1)\frac{w^{\prime\prime}}{w}.$

######  Lemma 6.4.

For any hypersurface $S$,

(6.13) $h^{2}\leq(n-1)\|B^{S}\|^{2}$

with equality if and only if $S$ is umbilic.

Applying inequalities (6.8),(6.12) and (6.13) to equality (6.5) we get

(6.14) $\frac{\partial h}{\partial
t}\geq-h\langle\nu,\nabla^{M_{w}^{n}}w^{\prime}\rangle+w^{\prime}\frac{h^{2}}{n-1}$

Therefore

(6.15) $\displaystyle\frac{\partial}{\partial
t}\left(\frac{h}{w^{\prime}}\right)=$
$\displaystyle\frac{1}{w^{\prime}}\frac{\partial h}{\partial
t}-\frac{h}{\left(w^{\prime}\right)^{2}}\frac{\partial w^{\prime}}{\partial
t}$ $\displaystyle\geq$
$\displaystyle\frac{-h}{w^{\prime}}\langle\nabla^{M_{w}^{n}}w^{\prime},\nu\rangle+\frac{h^{2}}{n-1}-\frac{h}{\left(w^{\prime}\right)^{2}}\frac{\partial
w^{\prime}}{\partial t}.$

By the evolution equation (6.3)

(6.16) $\frac{\partial w^{\prime}}{\partial
t}=-w^{\prime}\nu(w^{\prime})=-w^{\prime}\langle\nabla^{M_{w}^{n}}w^{\prime},\nu\rangle.$

Hence,

(6.17) $\displaystyle\frac{\partial}{\partial
t}\left(\frac{h}{w^{\prime}}\right)\geq$ $\displaystyle\frac{h^{2}}{n-1}.$

And the proposition follows. ∎

Following the proof of [Bre13, Theorem 3.5] consider the quantity

(6.18) $Q(t)=(n-1)\int_{\Sigma^{*}_{t}}\frac{w^{\prime}}{h}dA.$

Therefore we can use a similar proposition to proposition 3.4 in [Bre13]

######  Proposition 6.5.

(6.19) $Q(0)-Q(\tau)\geq n\int_{u\leq\tau}w^{\prime}dV.$

Letting $\tau\to\infty$ in the above proposition,

(6.20) $\displaystyle(n-1)\int_{\Sigma}\frac{1}{h}d\sigma_{w}$
$\displaystyle=(n-1)\int_{\Sigma^{*}_{t}}\frac{w^{\prime}}{h}dA$
$\displaystyle=Q(0)\geq
n\int_{u<\infty}w^{\prime}dV=n\int_{\Omega}w^{\prime}dV$
$\displaystyle=n\mu_{w}(\Omega).$

And the inequality (6.2) follows.

\begin{picture}(2640.0,2638.0)(9931.0,-1830.0)\end{picture}

Figure 1. By equality (6.10) $\Sigma$ is a sphere or a finite union of
spheres, hence $\Omega$ is either a geodesic ball centered at $o_{w}$ ( A ),
or a geodesic annulus centered at $o_{w}$ ( B ) or a finite union of annuli
(B+C ) or A geodesic ball centered at $o_{w}$ with a finite union of geodesic
annuli centered at $o_{w}$ (A + B + C ).

$\nu$$\nu$

Figure 2. The product $h=-\langle\nu,H\rangle$ is not constant in the boundary
of an annulus centered at the center of the model.

Observe that equality in (6.2) implies equality in (6.13) and hence umbilic
submanifold. If moreover we assume
$\frac{w^{\prime\prime\prime}}{w^{\prime}}\neq\frac{w^{\prime\prime}}{w}$
equality in (6.2) implies equality (6.10). If $\Sigma$ has $k>0$ connected
components $\Sigma_{1},\cdots,\Sigma_{k}$, each one is a sphere centered at
$o_{w}$. Hence, $\Omega$ is of one of the following types (see figure 1):

  1. (1)

A geodesic ball centered at $o_{w}$.

  2. (2)

A finite union of geodesic annuli centered at $o_{w}$.

  3. (3)

A geodesic ball centered at $o_{w}$ with a finite union of geodesic annuli
centered at $o_{w}$.

But we can prove that $\Omega$ does not contain annuli. Because, if $\Omega$
contains an annulus since $\eta_{w}>0$, $H$ is always pointing to $o_{w}$ (see
proposition 3.3) and the unit normal $\nu$ is always pointing outward to
$\Omega$, therefore the product $h=-\langle\nu,H\rangle$ has to be not
constant (see figure 2) in contradiction to the assumption of constant $h$.

Thus finally, $\Omega$ is a ball and $\Sigma$ is a sphere centered at $o_{w}$.

∎

##  7\. Proof of corollary 2.8

By the Cheeger inequality (see [Cha84, theorem 3, chapter IV])

(7.1)
$\lambda_{1}(\Omega)\geq\frac{1}{4}\left(\inf_{\mathcal{O}\subset\Omega}\frac{\text{A}(\partial\mathcal{O})}{\text{V}(\mathcal{O})}\right)^{2},$

where $\mathcal{O}$ ranges on the open subdomains of $\Omega$. By the
isoperimetric inequality (2.16)

(7.2) $\displaystyle\lambda_{1}(\Omega)$
$\displaystyle\geq\frac{1}{4}\left(\inf_{\mathcal{O}\subset\Omega}\frac{\omega_{n-1}w(\text{Rad}(\text{V}(\mathcal{O}))^{n-1}}{\text{V}(\mathcal{O})}\right)^{2}$
$\displaystyle=\frac{1}{4}\left(\inf_{\mathcal{O}\subset\Omega}\mathcal{I}(\text{V}(\mathcal{O}))\right)^{2}.$

Taking into account that $\mathcal{I}$ is non-increasing and
$\text{V}(\mathcal{O})\leq\text{V}(\Omega)$, the corollary follows.

##  8\. Proof of corollary 2.10

For any $p>0$, we use the density $d\mu_{w}$ to define the
$L^{p}_{w}(\Omega)$-space, by declaring a measurable function $f$ to be an
element of $L^{p}_{w}(\Omega)$ if the integral

(8.1) $\left(\int_{\Omega}|f|^{p}d\mu_{w}\right)^{1/p}$

is finite. The $L^{p}_{w}(\Omega)$-norm of $f$, $\|f\|_{L^{p}_{w}(\Omega)}$,
is given by the above expression. Hölder’s inequality states that for $p,q>1$
satisfying

(8.2) $1/p+1/q=1\quad,$

one has for $\phi\in\text{L}^{p}_{w}(\Omega)$,
$\psi\in\text{L}^{q}_{w}(\Omega)$ ,

(8.3)
$\int_{\Omega}|\phi\psi|d\mu_{w}\leq\|\phi\|_{L^{p}_{w}(\Omega)}\|\psi\|_{L^{q}_{w}(\Omega)}\quad.$

Now, from the Jellett-Minkowski’s generalized formula we get

(8.4) $\sup_{\Omega}|\tau_{w}|\sigma_{w}(\partial\Omega)\geq
m\mu_{w}(\Omega)-\sup_{\Omega}|\tau_{w}|\int_{\Omega}|H|d\mu_{w}.$

Applying the Hölder inequality to the last integral in the above inequality

(8.5) $\sup_{\Omega}|\tau_{w}|\sigma_{w}(\partial\Omega)\geq
m\mu_{w}(\Omega)-\sup_{\Omega}|\tau_{w}|\left(\int_{\Omega}|H|^{k}d\mu_{w}\right)^{1/k}\mu_{w}(\Omega)^{\frac{k-1}{k}}\quad,$

and the corollary follows.

##  9\. Proof of corollary 2.11

Applying corollary 2.10 to the setting of corollary 2.11 (namely $w(t)=t$,
$d\mu=dV$) we obtain

(9.1)
$\mathcal{I}_{k}(\Omega)\geq\frac{m\text{V}(\Omega)^{1/k}}{R}-\left(\int_{P}|H|^{k}dV\right)^{1/k}$

Taking into account the definition of $\mathcal{I}_{k}$ and the isoperimetric
profile, the corollary follows.

##  10\. Proof of corollary 2.15

Since $P$ has non-shrinking property, there exist $\rho$ such that for any
$o\in P$

(10.1)
$V(B_{\rho}(o))>\left(\frac{R}{m}\right)^{k}\int_{P}|H|^{k}dV+\epsilon\quad,$

for some $\epsilon>0$.

Therefore, by inequality (9.1)

(10.2) $\mathcal{I}_{k,\rho}(P)>0\quad.$

Applying now [CF91, Theorem 5 and inequality (15)] the corollary is proven.

##  11\. Proof of corollary 2.18

The first thing to do in order to prove the corollary 2.18 is to study the
behavior of the volume of the extrinsic balls. Recall that an extrinsic ball
$D_{R}(o)$ centered to the pole $o\in N$ and with radius $R$ is the sublevel
set of the extrinsic distance function. Namely,

(11.1) $D_{R}(o)=\varphi^{-1}\left(B_{R}^{N}\left(o\right)\right)\quad,$

being $B_{R}^{N}$ the geodesic ball of $N$ of radius $R$ centered at the pole
$o\in N$. Note that we can construct the order-preserving bijection

$F:\mathbb{R}^{+}\to\mathbb{R}^{+}\quad t\to F(t)$

given by equation (4.1).

Since $\varphi:P\to N$ is a minimal immersion into a manifold with a pole $N$,
applying equation (4.2) we have

(11.2) $\Delta^{P}F\circ r\geq mw^{\prime}\circ r\quad.$

Taking into account that $w^{\prime}>0$, by the maximum principle there exist
$R_{T}$ such that

(11.3) $P=D_{R_{T}}(o)\quad.$

Now we need the following monotonicity formula

######  Proposition 11.1.

Under the assumptions of corollary 2.18, the function
$f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ given by

(11.4) $f(R):=\frac{\mu_{w}(D_{R})}{w(R)^{m}}\quad,$

is a nondecreasing function of $R$, and

(11.5) $f(R)\geq cV_{m}\quad.$

###### Proof.

Using theorem Main Theorem taking into account that $P$ is minimal and
$\partial D_{R}$ lies in a geodesic sphere of $N$ of radius $R$, we obtain

(11.6) $m\mu_{w}(D_{R})\leq\frac{1}{\eta_{w}(R)}\sigma_{w}(\partial
D_{R})\quad.$

By the coarea formula we get

(11.7) $\sigma_{w}(\partial D_{R})\leq\frac{d}{dR}\mu_{w}(D_{R})\quad.$

Therefore, using inequalities (11.6) and (11.7) together

(11.8)
$\frac{d}{dR}\ln\left(\mu_{w}\left(D_{R}\right)\right)\geq\frac{d}{dR}\ln\left(w\left(R\right)^{m}\right)\quad.$

Hence, we obtain the desired monotonicity formula. Observe also that

(11.9) $\lim_{R\to 0}\frac{\mu_{w}(D_{R})}{w(R)^{m}}\geq\lim_{R\to
0}c\frac{V(D_{R})}{w(R)^{m}}\geq cV_{m}\quad.$

And the proposition follows.∎

On the other hand, from inequality (11.6)

(11.10) $\sigma_{w}(\partial D_{R})\geq m\mu_{w}(D_{R})\eta_{w}(R)\quad.$

But taking into account the definition of the extrinsic ball and $w$-weighted
area, and using the above proposition

(11.11) $\displaystyle A(\partial D_{R})\geq$
$\displaystyle\frac{m}{w(R)}\mu_{w}(D_{R})=m\left(\frac{\mu_{w}(D_{R})}{w(R)^{m}}\right)^{\frac{1}{m}}\mu_{w}(D_{R})^{1-1/m}$
$\displaystyle\geq$ $\displaystyle
m\left(cV_{m}\right)^{\frac{1}{m}}\mu_{w}(D_{R})^{1-1/m}\quad.$

Hence, finally the corollary follows changing $R$ by $R_{T}$ in the above
inequality.

##  12\. Proof of corollary 2.17

Applying inequality (8.5) to the extrinsic ball $D_{R}$ taking into account
that $\sup_{D_{R}}|\tau_{w}|\leq\frac{1}{\sqrt{-b}}$ we get

(12.1) $m\mu_{w}(D_{R})\leq\frac{\sigma_{w}(\partial
D_{R})}{\sqrt{-b}}+\frac{1}{\sqrt{-b}}\left(\int_{D_{R}}|H|^{p}d\mu_{w}\right)^{1/p}\mu_{w}(D_{R})^{\frac{p-1}{p}}\quad,$

by inequality (11.7),

(12.2)
$m\sqrt{-b}-\|H\|_{L_{w}^{p}(P)}\mu_{w}(D_{R})^{\frac{-1}{p}}\leq\frac{d}{dR}\ln\mu_{w}(D_{R})\quad,$

Given $R_{0}>0$, for any $R\geq R_{0}$

(12.3)
$m\sqrt{-b}-\|H\|_{L_{w}^{p}(P)}\mu_{w}(D_{R_{0}})^{\frac{-1}{p}}\leq\frac{d}{dR}\ln\mu_{w}(D_{R})\quad,$

If the submanifold has finite volume, there exist a divergent sequence
$\\{R_{i}\\}_{i=1}^{\infty}$ such that

(12.4)
$m\sqrt{-b}-\|H\|_{L_{w}^{p}(P)}\mu_{w}(D_{R_{0}})^{\frac{-1}{p}}\leq\limsup\frac{d}{dR}\ln\mu_{w}(D_{R_{i}})=0.$

And therefore the corollary follows letting $R_{0}$ tend to infinity.

## Acknowledgments

The author is very grateful to Simon Brendle for his useful discussion about
his paper [Bre13] and to Antonio Cañete for show me the paper [MM09].

## References

  * [Ale56] A. D. Aleksandrov, _Uniqueness theorems for surfaces in the large. I_ , Vestnik Leningrad. Univ. 11 (1956), no. 19, 5–17. MR 0086338 (19,167c)
  * [BM02] Hubert Bray and Frank Morgan, _An isoperimetric comparison theorem for Schwarzschild space and other manifolds_ , Proc. Amer. Math. Soc. 130 (2002), no. 5, 1467–1472 (electronic). MR 1879971 (2002i:53073)
  * [Bre13] Simon Brendle, _Constant mean curvature surfaces in warped product manifolds_ , Publications mathématiques de l’IHÉS 117 (2013), no. 1, 247–269 (English).
  * [BS09] A. Barros and P. Sousa, _An extension of Jellett’s theorem_ , Bull. Sci. Math. 133 (2009), no. 2, 190–197. MR 2494466 (2010g:53105)
  * [CF91] Isaac Chavel and Edgar A. Feldman, _Isoperimetric constants, the geometry of ends, and large time heat diffusion in Riemannian manifolds_ , Proc. London Math. Soc. (3) 62 (1991), no. 2, 427–448. MR 1085648 (93c:58209)
  * [CG92] Jaigyoung Choe and Robert Gulliver, _Isoperimetric inequalities on minimal submanifolds of space forms_ , manuscripta mathematica 77 (1992), no. 1, 169–189.
  * [Cha78] Isaac Chavel, _On A. Hurwitz’ method in isoperimetric inequalities_ , Proc. Amer. Math. Soc. 71 (1978), no. 2, 275–279. MR 0493885 (58 #12845)
  * [Cha84] by same author, _Eigenvalues in Riemannian geometry_ , Pure and Applied Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984, Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk. MR 768584 (86g:58140)
  * [Cha93] by same author, _Riemannian geometry—a modern introduction_ , Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141 (95j:53001)
  * [CL98] Leung-Fu Cheung and Pui-Fai Leung, _The mean curvature and volume growth of complete noncompact submanifolds_ , Differential Geometry and its Applications 8 (1998), no. 3, 251–256.
  * [CMV12] Marcos P Cavalcante, Heudson Mirandola, and Feliciano Vitorio, _The non-parabolicity of infinite volume ends_ , arXiv preprint arXiv:1201.6391 (2012).
  * [DCWX10] Manfredo P Do Carmo, Qiaoling Wang, and Changyu Xia, _Complete submanifolds with bounded mean curvature in a hadamard manifold_ , Journal of Geometry and Physics 60 (2010), no. 1, 142–154.
  * [Fre96] Katia Rosenvald Frensel, _Stable complete surfaces with constant mean curvature_ , Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society 27 (1996), no. 2, 129–144.
  * [FX10] Hai-Ping Fu and Hong-Wei Xu, _Total curvature and l 2 harmonic 1-forms on complete submanifolds in space forms_ , Geometriae Dedicata 144 (2010), no. 1, 129–140.
  * [Gri99] Alexander Grigor′yan, _Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds_ , Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135–249. MR 1659871 (99k:58195)
  * [Gri09] Alexander Grigor’yan, _Heat kernel and analysis on manifolds_ , AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI, 2009. MR 2569498 (2011e:58041)
  * [GW79] R. E. Greene and H. Wu, _Function theory on manifolds which possess a pole_ , Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983 (81a:53002)
  * [HHM99] Hugh Howards, Michael Hutchings, and Frank Morgan, _The isoperimetric problem on surfaces_ , Amer. Math. Monthly 106 (1999), no. 5, 430–439. MR 1699261 (2000i:52027)
  * [HK78] Ernst Heintze and Hermann Karcher, _A general comparison theorem with applications to volume estimates for submanifolds_ , Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451–470. MR 533065 (80i:53026)
  * [Jel53] J-H Jellett, _Sur la surface dont la courbure moyenne est constante._ , Journal de Mathématiques Pures et Appliquées (1853), 163–167.
  * [MM05] Francisco Martín and Santiago Morales, _Complete proper minimal surfaces in convex bodies of $\mathbb{R}^{3}$_, Duke Math. J. 128 (2005), no. 3, 559–593. MR 2145744 (2006c:53007)
  * [MM06] by same author, _Complete proper minimal surfaces in convex bodies of $\mathbb{R}^{3}$. II. The behavior of the limit set_, Comment. Math. Helv. 81 (2006), no. 3, 699–725. MR 2250860 (2007e:53005)
  * [MM09] Quinn Maurmann and Frank Morgan, _Isoperimetric comparison theorems for manifolds with density_ , Calc. Var. Partial Differential Equations 36 (2009), no. 1, 1–5. MR 2507612 (2010d:53066)
  * [MR09] Sebastián Montiel and Antonio Ros, _Curves and surfaces_ , second ed., Graduate Studies in Mathematics, vol. 69, American Mathematical Society, Providence, RI, 2009, Translated from the 1998 Spanish original by Montiel and edited by Donald Babbitt. MR 2522595 (2010f:53007)
  * [Nad96] Nikolai Nadirashvili, _Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces_ , Invent. Math. 126 (1996), no. 3, 457–465. MR 1419004 (98d:53014)
  * [Nas56] John Nash, _The imbedding problem for Riemannian manifolds_ , Ann. of Math. (2) 63 (1956), 20–63. MR 0075639 (17,782b)
  * [O’N83] Barrett O’Neill, _Semi-Riemannian geometry_ , Pure and Applied Mathematics, vol. 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983, With applications to relativity. MR 719023 (85f:53002)
  * [Pal99] Vicente Palmer, _Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications_ , J. London Math. Soc. (2) 60 (1999), no. 2, 607–616. MR 1724821 (2000j:53050)
  * [Pet98] Peter Petersen, _Riemannian geometry_ , Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 1998. MR 1480173 (98m:53001)
  * [Ros05] Antonio Ros, _The isoperimetric problem_ , Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175–209. MR 2167260 (2006e:53023)