# Ends, fundamental tones, and capacities of minimal submanifolds via
extrinsic comparison theory

Vicent Gimeno  Departament de Matemàtiques- INIT-IMAC, Universitat Jaume I,
Castelló, Spain<EMAIL_ADDRESS>and  S. Markvorsen  DTU Compute, Mathematics,
Technical University of Denmark<EMAIL_ADDRESS>
###### Abstract.

We study the volume of extrinsic balls and the capacity of extrinsic annuli in
minimal submanifolds which are properly immersed with controlled radial
sectional curvatures into an ambient manifold with a pole. The key results are
concerned with the comparison of those volumes and capacities with the
corresponding entities in a rotationally symmetric model manifold. Using the
asymptotic behavior of the volumes and capacities we then obtain upper bounds
for the number of ends as well as estimates for the fundamental tone of the
submanifolds in question.

###### Key words and phrases:

First Dirichlet eigenvalue, Capacity, Effective resistance, Minimal
submanifolds

###### 2010 Mathematics Subject Classification:

Primary 53A, 53C

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

##  1\. Introduction

Let $M$ be a complete non-compact Riemannian manifold. Let $K\subset M$ be a
compact set with non-empty interior and smooth boundary. We denote by
$\mathcal{E}_{K}(M)$ the number of connected components
$E_{1},\cdots,E_{\mathcal{E}_{K}(M)}$ of $M\setminus K$ with non-compact
closure. Then $M$ has $\mathcal{E}_{K}(M)$ ends
$\\{E_{i}\\}_{i=1}^{\mathcal{E}_{K}(M)}$ with respect to $K$ (see e.g.
[GSC09]), and the _global_ number of ends $\mathcal{E}(M)$ is given by

(1.1) $\mathcal{E}(M)=\sup_{K\subset M}\mathcal{E}_{K}(M)\quad,$

where $K$ ranges on the compact sets of $M$ with non-empty interior and smooth
boundary.

The number of ends of a manifold can be bounded by geometric restrictions. For
example, in the particular setting of an $m-$dimensional minimal submanifold
$P$ which is properly immersed into Euclidean space $\mathbb{R}^{n}$, the
number of ends $\mathcal{E}(P)$ is known to be related to the extrinsic
properties of the immersion. Indeed, V. G. Tkachev proved in [Tka94, Theorem
2] (see also [Che95]) that for any properly immersed $m-$dimensional minimal
submanifold $P$ in $\mathbb{R}^{n}$ with finite volume growth
$V_{w_{0}}(P)<\infty$ the number of ends is bounded from above by

(1.2) $\mathcal{E}(P)\leq C_{m}V_{w_{0}}(P)\quad,$

where $C_{m}=1$ ($C_{m}={2^{m}}$ in the original [Tka94]) and the volume
growth $V_{w_{0}}(P)$ is

(1.3) $V_{w_{0}}(P)=\lim_{R\to\infty}\frac{\operatorname{Vol}\left(P\cap
B_{R}^{\mathbb{R}^{n}}\left(o\right)\right)}{\operatorname{Vol}\left(B_{R}^{\mathbb{R}^{m}}\left(o\right)\right)}\quad.$

Here $\operatorname{Vol}\left(B_{R}^{\mathbb{R}^{m}}\left(o\right)\right)$ is
the volume of a geodesic ball $B_{R}^{\mathbb{R}^{m}}\left(o\right)$ of radius
$R$ centered at $o$ in $\mathbb{R}^{m}$. The inequality (1.2) thus shows a
significant relation between the number of ends (i.e. a topological property)
and the behavior of a quotient of volumes (i.e. a metric property).

Motivated by Tkachev’s application of the _volume quotient_ appearing in
equation (1.3), we will consider the corresponding _flux quotient_ and
_capacity quotient_ of the minimal submanifolds. These quotients are
constructed in the same way as indicated by the _volume quotient_ but here we
generalize the setting as well as Tkachev’s result to minimal submanifolds in
more general ambient spaces as alluded to in the abstract. Specifically we
assume that the minimal immersion goes into an ambient manifold $N$ with a
pole and with sectional curvatures $K_{N}$ bounded from above by the radial
curvatures $K_{w}$ of a rotationally symmetric model space
$M_{w}^{n}=\mathbb{R}^{+}\times\mathbb{S}_{1}^{n-1}$, with warped metric
tensor $g_{M_{w}^{n}}$ constructed using a positive warping function
$w:\mathbb{R}^{+}\to\mathbb{R}^{+}$ in such a way that
$g_{M_{w}^{n}}=dr^{2}+w(r)^{2}g_{\mathbb{S}_{1}^{n-1}}$ is also balanced from
below (see [MP06] and §3 for a precise definitions).

Our generalization of inequality (1.2) is thence the following:

######  Theorem 1.1.

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

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

Suppose moreover, that $w^{\prime}>0$ and there exist $R_{0}$ such that
$K_{M_{w}^{n}}(R)\leq 0$ for any $R>R_{0}$. Then, the number of ends
$\mathcal{E}_{D_{R}}(P)$ with respect to the extrinsic ball $D_{R}=P\cap
B_{R}^{N}(o)$ for $R>R_{0}$ is bounded from above by

(1.5)
$\mathcal{E}_{D_{R}}(P)\leq\left(\frac{2}{1-\frac{R}{t}}\right)^{m}\left(\frac{\int_{0}^{t}w(s)^{m-1}ds}{t^{m}/m}\right)\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B^{w}_{t})}\quad,$

for any $t>R$.

Using the above theorem we can estimate the global number of ends as follows:

######  Corollary 1.2.

Under the assumptions of theorem 1.1, suppose moreover

(1.6)
$\limsup_{t\to\infty}\left(\frac{\int_{0}^{t}w(s)^{m-1}ds}{t^{m}/m}\right)=C_{w}<\infty\quad,$

and suppose also that the submanifold has finite volume growth, namely

(1.7)
$\operatorname{Vol}_{w}(P)=\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B^{w}_{t})}<\infty\quad.$

Then

(1.8) $\mathcal{E}(P)\leq 2^{m}C_{w}\operatorname{Vol}_{w}(P)\quad.$

######  Remark a.

If we choose $w(t)=w_{0}(t)=t$, the model space becomes $\mathbb{R}^{m}$,
which is balanced from below, and the hypothesis of theorem 1.1 are therefore
automatically fulfilled for any complete minimal submanifold properly immersed
in a Cartan-Hadamard ambient manifold. Inequality (1.5) becomes

(1.9)
$\mathcal{E}_{D_{R}}(P)\leq\left(\frac{2}{1-\frac{R}{t}}\right)^{m}\frac{\operatorname{Vol}(D_{t})}{V_{m}t^{m}}\quad,$

For any $R>0$ and any $t>R$, being $V_{m}$ the volume of a geodesic ball of
radius $1$ in $\mathbb{R}^{m}$. From inequality (1.6) we get

(1.10) $C_{w_{0}}=1\quad.$

Thus inequality (1.8) becomes

(1.11) $\mathcal{E}(P)\leq
2^{m}\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{V_{m}t^{m}}\quad,$

which is the original inequality obtained by Tkachev (inequality (1.2)), but
now inequality (1.11) is valid for any minimal submanifold properly immersed
in a Cartan-Hadamard ambient manifold with finite volume growth.

In [GP12, Che95] are also obtained lower bounds for the number of ends, but we
note that those lower bounds seem to need stronger assumptions: Dimension
greater than $2$, or embeddedness of the ends and codimension $1$, decay on
the second fundamental form, and a rotationally symmetric ambient manifold. As
a counterpart, those lower bounds are associated to the so-called gap type
theorems.

Combining the results of [GP12, Theorem 3.5] and corollary 1.2, and taking
into account the role of sectional curvatures of the model space (see [GP12,
Proposition 2.6]) we have

######  Corollary 1.3.

Let $\varphi:P^{m}\to M^{n}_{w}$ be a minimal and proper immersion into a
balanced from below model space $M^{n}_{w}$ with increasing warping function
$w$ satisfying the following conditions:

(1.12)
$\displaystyle\limsup_{t\to\infty}\left(\frac{\int_{0}^{t}w(s)^{m-1}ds}{t^{m}/m}\right)=C_{w}<\infty\quad,$
$\displaystyle\text{ there exist }R_{0}\text{ such that for any }R>R_{0}$
$\displaystyle\begin{cases}-\frac{w^{\prime\prime}(R)}{w(R)}\leq 0\\\
\frac{1-\left(w^{\prime}(R)\right)^{2}}{w(R)^{2}}\leq 0\end{cases}$

Suppose moreover that $m>2$, that the center $o_{w}$ of $M_{w}^{n}$ satisfies
$\varphi^{-1}(o_{w})\neq\emptyset$ and that the norm of the second fundamental
form $\|B^{P}\|$ of the immersion is bounded for large $r$ by

(1.13) $\|B^{P}\|\leq\frac{\epsilon(r)}{w^{\prime}(r)w(r)}\quad,$

where $\epsilon$ is a positive function such that $\epsilon(r)\to 0$ when
$r\to\infty$.

Then the number of ends is bounded from below and from above by

(1.14) $\operatorname{Vol}_{w}(P)\leq\mathcal{E}(P)\leq
2^{m}C_{w}\operatorname{Vol}_{w}(P)\quad.$

Figure 1. Two examples of extrinsic annuli in $\mathbb{R}^{3}$: A catenoid on
the left and the singly periodic Scherk surface on the right. The extrinsic
annuli are constructed by cutting the surfaces with two spheres (with the same
center but of different radii) in the ambient manifold ($\mathbb{R}^{3}$). The
catenoid has two ends and finite total curvature. Hence, by theorem 1.4, the
capacity of the extrinsic annulus of the catenoid is greater than the capacity
of the corresponding annulus of the Euclidean $2$-plane but is smaller than
two times that capacity. The same is true for the extrinsic annulus of the
singly periodic Scherk surface (we refer the reader to the introduction of
[MW07] for the area growth of the singly periodic Scherk surface).

By using our results about the behavior of the comparison quotients we can
also estimate the capacity of an extrinsic annulus
$\text{A}_{\rho,R}=P\cap\left(B_{R}^{N}(o)\setminus B_{\rho}^{N}(o)\right)$
(see figure 1 and, §2 and §3 for a precise definition of _capacity_ and
_extrinsic annulus_):

######  Theorem 1.4.

Let $\varphi:P^{m}\to\mathbb{R}^{n}$ denote a complete and proper minimal
immersion into the Euclidean space $\mathbb{R}^{n}$. Then, for any $R>\rho>0$,
the capacity of the extrinsic annulus $A_{\rho,R}$ is bounded by

(1.15)
$\frac{\operatorname{Vol}(D_{\rho})}{V_{m}\rho^{m}}\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A^{\mathbb{R}^{m}}_{\rho,R})}\leq\frac{\operatorname{Vol}(D_{R})}{V_{m}R^{m}}\quad,$

where $\operatorname{Cap}(A^{\mathbb{R}^{m}}_{\rho,R})$ is the capacity of the
geodesic annulus $A^{\mathbb{R}^{m}}_{\rho,R}$ in $\mathbb{R}^{m}$ of inner
radius $\rho$ and outer radius $R$.

######  Remark b.

Since, from Theorem 2.1, the quotient
$\frac{\operatorname{Vol}(D_{s})}{V_{m}s^{m}}$ is a non-decreasing function on
$s$, we can state Theorem 1.4 in the limit case ( $\rho\to 0$ and
$R\to\infty$) and inequality (1.15) there becomes

(1.16)
$1\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A^{\mathbb{R}^{m}}_{\rho,R})}\leq\lim_{R\to\infty}\frac{\operatorname{Vol}(D_{R})}{V_{m}R^{m}}=V_{w_{0}}(P)\quad.$

When we deal with a minimal surface $\Sigma\subset\mathbb{R}^{3}$ which is
properly _embedded_ into the Euclidean space $\mathbb{R}^{3}$ the limit

(1.17) $V_{w_{0}}(\Sigma)=\lim_{R\to\infty}\frac{\operatorname{Vol}(\Sigma\cap
B_{R}^{\mathbb{R}^{3}}(o))}{\pi R^{2}}$

is well understood. For instance, the above limit corresponds to the number of
ends if the surface $\Sigma$ has finite total curvature.

######  Remark c.

In order to bound the capacity quotient, our theorems do not make use of the
volume quotient as in Theorem 1.4, but instead they make use of the flux
quotient (see Theorems 2.2 and 2.3). In the special case when the ambient
manifold is $\mathbb{R}^{n}$ (such as in Theorem 1.4) the volume quotient
agrees however with the flux quotient (see equation (6.28) and theorem 6.1).

###  1.1. Outline of the paper

In §2 we show our main theorems concerning the flux quotients, the volume
quotients, and the capacity quotients. In §3 we state the preliminary concepts
in order to prove the main theorems of §2 in §4. This allows us then to prove
Theorem 1.1 and Corollary 1.2 in §5. Finally, in §6, we present several
corollaries and examples of applications of the extrinsic theory and results
which have been established in §2.

##  2\. Extrinsic theory: Flux, Capacity and Volume comparison for extrinsic
balls

Let $(M^{n},g)$ be a Riemannian manifold. For any oriented hypersurface
$\Sigma\subset M$ with unit normal vector field $\nu$, we define the flux
$F_{X}(\Sigma)$ of the vector field $X$ through $\Sigma$ by

(2.1) $F_{X}(\Sigma):=\int_{\Sigma}\langle X,\nu\rangle d\mu_{\Sigma}\quad,$

where $d\mu_{\Sigma}$ is the associated Riemannian density determined by the
metric $g_{\Sigma}=i^{*}g$ (being $i:\Sigma\to M$ the inclusion map).

By the divergence theorem (see [Cha93] for instance), if one has an oriented
domain $\Omega$ in $M$ with smooth boundary $\partial\Omega$, and the vector
field $X$ is $C^{1}$ in $\overline{\Omega}$ and with compact support in
$\overline{\Omega}$, the flux of $X$ through $\partial\Omega$ is related to
the divergence of $X$ by

(2.2) $\int_{\Omega}\text{div}Xd\mu=\int_{\partial\Omega}\langle X,\nu\rangle
d\mu_{\partial\Omega}=F_{X}(\partial\Omega)\quad.$

Given a smooth function $u:M\to\mathbb{R}$, we can also define the flux of a
function $u$, but then the flux $J_{u}(t)$ is the flux of the gradient $\nabla
u$ (i.e. the metric dual vector to $du$, $du(X)=\langle\nabla u,X\rangle$)
through the level set $\Sigma^{u}_{t}:=\\{x\in M\,|\,u(x)=t\\}$ so that:

(2.3) $J_{u}(t):=F_{\nabla u}(\Sigma_{t}^{u})\quad.$

Taking into account that the unit normal vector field $\nu$ of
$\Sigma^{u}_{t}$ is $\nu=\frac{\nabla u}{|\nabla u|}$, it is easy to see that

(2.4) $J_{u}(t)=\int_{\Sigma_{t}}|\nabla u|d\mu_{\Sigma_{t}^{u}}\quad.$

Observe moreover, that by the Sard theorem and by the regular set theorem we
need no further restrictions on the smoothness of $\Sigma_{t}^{u}$ and on the
smoothness of the unit normal vector field $\nu$.

The overall goal of this work is to characterize the _isoperimetric
inequalities for extrinsic balls_ , and the _capacity of minimal submanifolds_
in terms of the flux of extrinsic distance functions. Actually we are
interested on the flux of the extrinsic distance function on minimal
submanifolds in an ambient manifold $N$ which possesses a pole and has the
radial curvatures bounded form above by the radial curvatures of rotationally
symmetric model space $K_{N}\leq K_{M_{w}^{n}}=-\frac{w^{\prime\prime}}{w}$,
see [MP06] or section 3 of this paper for precise definitions. It is the
behavior of this particular flux that allows us to study the mean exit time
function, the capacity, the conformal type, the fundamental tone, and in
special cases also the number of ends of the submanifold.

###  2.1. Flux and volume comparison: isoperimetric inequalities and the mean
exit time function

Given an isometric immersion $\varphi:P\to(N,o)$ into a manifold with a pole
$o\in N$, the flux $J_{r}$ of the extrinsic distance function $r_{o}$ (i.e.,
the restriction by the immersion of the ambient distance function to the
submanifold) is given by

$J_{r}(R)=\int_{\partial D_{R}}|\nabla^{P}r_{o}|d\mu\quad,$

where $\partial D_{R}$ is the level set $\partial D_{R}=r_{o}^{-1}(R)$, and
therefore, $D_{R}=r_{o}^{-1}([0,R))$ is the extrinsic ball of radius $R$.

When the immersion is minimal and the ambient manifold has its radial
sectional curvatures $K_{N}$ bounded from above by the radial sectional
curvatures of a rotationally symmetric model space $M_{w}^{n}$ that is
balanced from below (see [MP06] and section 3), $K_{N}\leq K_{M_{w}^{n}}$, we
can compare the volume quotient
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}$ and the flux
quotient $\frac{J_{r}(R)}{J_{r}^{w}(R)}$ . The volume quotient is the quotient
between the volume of a extrinsic ball $D_{R}$ of radius $R$ in $P^{m}$ and
the volume of a geodesic ball $B_{R}^{w}$ of the same radius $R$ in
$M_{w}^{m}$. The flux quotient is the quotient between the flux of the
extrinsic distance in $P^{m}$ and the flux of the geodesic distance in
$M_{w}^{m}$. These two quotients are related by the following theorem

######  Theorem 2.1.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose
that the $o-$radial sectional curvatures of $N$ are bounded from above by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\,,$

and the model space $M_{w}^{m}$ is balanced from bellow. Then

  1. (1)

$J_{r}(R)$ is related with $\operatorname{Vol}(D_{R})$ by

(2.5)
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}\leq\frac{J_{r}(R)}{J_{r}^{w}(R)}.$

  2. (2)

The functions
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}$ and
$\frac{J_{r}(R)}{J_{r}^{w}(R)}$ are non decreasing functions on $R$.

  3. (3)

Denoting by $E_{R}^{P}(x)$ the mean time for the first exit from the extrinsic
ball $D_{R}(o)$ for a Brownnian particle starting at $o\in P^{m}$, and
denoting by $E_{R}^{w}$ the mean exit time function for the $R-$ball
$B_{R}^{w}$ in the model space $M_{w}^{m}$, if equality holds in (2.5) for
some fixed radius $R>0$, then for any $x\in D_{R}$,
$E_{R}^{P}(x)=E_{R}^{w}(r(x))$, where $r(x)$ the extrinsic distance from $o$
to the point $x\in P$.

###  2.2. Capacity and flux comparison: conformal type

Given a compact set $F\subset M$ in a Riemannian manifold $M$ and an open set
$G\subset M$ containing $F$, we call the couple $(F,G)$ a _capacitor_. Each
capacitor then has its capacity defined by

(2.6) $\operatorname{Cap}(F,G):=\inf_{u}\int_{G\setminus F}\|\nabla
u\|d\mu\quad,$

where the $\inf$ is taken over all Lipschitz functions $u$ with compact
support in $G$ such that $u=1$ on $F$.

When $G$ is precompact, the infimum is attained for the function $u=\Psi$
which is the solution of the following Dirichlet problem in $G-F$:

(2.7) $\begin{cases}\Delta\Psi=0\\\ \Psi|_{\partial F}=0\\\ \Psi|_{\partial
G}=1\end{cases}$

From a physical point of view, the capacity of the capacitor $(F,G)$
represents the total electric charge (generated by the electrostatic potential
$\Psi$) flowing into the domain $G-F$ through the interior boundary $\partial
F$. Since the total current stems from a potential difference of $1$ between
$\partial F$ and $\partial G$, we get from Ohm’s Law that the effective
resistance of the domain $G-F$ is

(2.8) $R_{\text{eff}}(G-F)=\frac{1}{\operatorname{Cap}(F,G)}\quad.$

The exact value of the capacity of a set is known only in a few cases, and so
its estimation in geometrical terms is of great interest, not only in
electrostatic, but in many physical descriptions of flows, fluids, heat, or
generally where the Laplace operator plays a key role, see [CFG05, HPR12].

Given a capacitor $(F,G)$, if we have a smooth function $u$ with $u=a$ on
$\partial F$ and $u=b$ on $\partial G$. The capacity and the flux are then
related by (see [Gri99b]):

(2.9)
$\operatorname{Cap}(F,G)\leq\left(\int_{a}^{b}\frac{ds}{J_{u}(s)}\right)^{-1}\quad.$

In this paper we are interested on the $o$-centered _extrinsic annulus_
$A_{\rho,R}(o)\subset P^{m}$ for $0<\rho<R$ given by

(2.10) $A_{\rho,R}(o):=D_{R}(o)-D_{\rho}(o)\quad.$

To be more precise, we are interested on the behavior of the flux and the
capacity of those extrinsic domains. In the following theorems we provide
upper and lower bounds for the capacity quotient in terms of the flux
quotient.

######  Theorem 2.2.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N$ and satisfying
$\varphi^{-1}(o)\neq\emptyset$. Let us suppose that the $o-$radial sectional
curvatures of $N$ are bounded from above by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\quad,$

and the warping function $w$ satisfies

$w^{\prime}\geq 0\quad.$

Then

(2.11)
$\frac{J_{r}(\rho)}{J_{r}^{w}(\rho)}\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\quad,$

where $A^{w}_{\rho,R}$ is the intrinsic annulus in $M_{w}^{m}$.

######  Theorem 2.3.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose
that the $o-$radial sectional curvatures of $N$ are bounded from above by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\,,$

and the model space $M_{w}^{m}$ is balanced from bellow. Then

(2.12)
$\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\leq\frac{J_{r}(R)}{J_{r}^{w}(R)}\quad,$

where $A^{w}_{\rho,R}$ is the intrinsic annulus in $M_{w}^{m}$. Moreover, if
equality holds in (2.12) for some fixed $R>0$, then $D_{R}$ is a minimal cone
in $N^{n}$.

Geometric estimates of the capacity are sufficient to obtain large scale
consequences such as as the parabolic or hyperbolic character of the manifold,
[Ich82b, Ich82a, MP03, MP05]. We note here the following important equivalent
conditions about the conformal type:

######  Theorem A.

Let (M,g) be a given Riemannian manifold, Then the following conditions are
equivalent

  * •

There is a precompact open domain $K$ in $M$, such that the Brownian motion
$X_{t}$ starting from $K$ does not return to $K$ with probability $1$, i.e.

(2.13) $P_{x}\left\\{\omega|X_{t}(\omega)\in K\text{ for some }t>0\right\\}<1$

  * •

$M$ has positive capacity: There exist in $M$ a compact domain $K$, such that

(2.14) $\operatorname{Cap}(K,M)>0$

  * •

$M$ has finite resistance to infinity: There exist in $M$ a compact domain
$K$, such that

(2.15) $R_{\text{eff}}(M-K)<\infty$

A manifold satisfying the conditions of the above theorem will be called a
hyperbolic manifold, otherwise it is called a parabolic manifold.

As a consequence of the above theorem we can state the following corollary for
minimal submanifolds:

######  Corollary 2.4.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose
that the $o-$radial sectional curvatures of $N$ are bounded from above by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\quad,$

and the warping function $w$ satisfies

$w^{\prime}\geq 0\quad.$

Then

  1. (1)

If $M_{w}^{m}$ is a hyperbolic manifold, then $P$ is a hyperbolic manifold.

  2. (2)

In consequence, if $P$ is parabolic, then $M_{w}^{m}$ is also parabolic.

Since $\frac{J_{r}(R)}{J_{r}^{w}(R)}$ and
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}$ are non-
decreasing functions under our hypothesis, we can define two expressions which
are analogous to the projective volume defined by V. G. Tkachev in [Tka94]

######  Definition 2.5.

Given $\varphi:P^{m}\to N^{n}$ an immersion into a manifold $N$ with a pole
$o\in N$. The $w$-_flux_ $\text{Flux}_{w}(P)$ and the $w$-_volume_
$\operatorname{Vol}_{w}(P)$ of the submanifold $P$ are defined by :

(2.16) $\displaystyle\text{Flux}_{w}(P)$
$\displaystyle:=\sup_{R\in\mathbb{R}^{+}}\frac{J_{r}(R)}{J_{r}^{w}(R)}\quad,$
$\displaystyle\operatorname{Vol}_{w}(P)$
$\displaystyle:=\sup_{R\in\mathbb{R}^{+}}\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}\quad.$

We will say that $P$ has _finite_ $w-$ _flux_ ( resp. _finite_ $w-$ _volume_)
if and only if $\text{Flux}_{w}(P)<\infty$ ( or
$\operatorname{Vol}_{w}(P)<\infty$).

We refer to theorem 6.1 for the relation between the $w-$flux and the
$w-$volume of a submanifold.

From theorem A and theorem 2.3 we can now state that for minimal submanifolds
with finite $w-$flux we have:

######  Corollary 2.6.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N^{n}$. Let us suppose
that the $o-$radial sectional curvatures of $N^{n}$ are bounded from above as
follows

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\quad,$

and that the model space $M_{w}^{m}$ is balanced from below. Suppose moreover
that $P$ has finite $w-$flux. Then

  1. (1)

If $M_{w}^{m}$ is a parabolic manifold, then $P$ is a parabolic manifold.

  2. (2)

If $P$ is an hyperbolic manifold, then $M_{w}^{n}$ is an hyperbolic manifold.

Joining the previous two corollaries together we get:

######  Corollary 2.7.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N^{n}$ . Let us suppose
that the $o-$radial sectional curvatures of $N^{n}$ are bounded from above,

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\quad,$

that the warping function $w$ satisfies

$w^{\prime}\geq 0\quad,$

and that the model space $M_{w}^{m}$ is balanced from below, and that $P$ has
finite $w-$flux. Then $P$ is hyperbolic (parabolic) if and only if $M_{w}^{m}$
is hyperbolic (parabolic).

##  3\. Preliminaires

We assume throughout 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 $o\in N^{n}$ . 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
$r(x)=r_{o}(x)=\operatorname{dist}_{N}(o,x)$, since $o$ is a pole 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 $P^{m}$.

With the extrinsic distance we can construct the extrinsic ball $D_{R}(o)$ of
radius $R$ centered at $o$ as

$D_{R}(o):=\\{x\in P:r(\varphi(x))<R\\}\quad.$

Since $\partial D_{t}(o)=\Sigma_{t}^{r}$, the flux of the extrinsic distance
function $r$ on $P$ is

$J_{r}(t)=\int_{\partial D_{t}}|\nabla^{P}r|d\rho\quad,$

where the gradients of $r$ in $N$ and $r|_{P}$ in $P$ are denoted by
$\nabla^{N}r$ and $\nabla^{P}r$, respectively. These two gradients 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 $x\in P$.

We now present the curvature restrictions which constitute the geometric
framework of the present study.

######  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 $K_{o,N}(\sigma_{x})$.

###  3.1. Model spaces

Throughout this paper we shall assume that the ambient manifold $N^{n}$ has
its $o$-radial sectional curvatures $K_{o,N}(x)$ bounded from above by the
expression $K_{w}(r(x))=-w^{\prime\prime}(r(x))/w(r(x))$, which are precisely
the radial sectional curvatures of the $w$-model space $\,M^{m}_{w}\,$ we are
going to define.

######  Definition 3.2 (See [O’N83, Gri99a, GW79]).

A $w-$model $M_{w}^{m}$ is a smooth warped product with base
$B^{1}=[0,\Lambda[\,\subset\mathbb{R}$ (where $0<\Lambda\leq\infty$), fiber
$F^{m-1}=\mathbb{S}^{m-1}_{1}$ (i.e. the unit $(m-1)$-sphere with standard
metric), and warping function
$w\colon[0,\Lambda[\to\mathbb{R}_{+}\cup\\{0\\}$, with $w(0)=0$,
$w^{\prime}(0)=1$, and $w(r)>0$ for all $r>0$. The point $o_{w}=\pi^{-1}(0)$,
where $\pi$ denotes the projection onto $B^{1}$, is called the center point of
the model space. If $\Lambda=\infty$, then $o_{w}$ is a pole of $M_{w}^{m}$.

######  Proposition 3.3.

The simply connected space forms $\mathbb{K}^{m}(b)$ of constant curvature $b$
are $w-$models with warping functions

$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 $b<0$}.\end{cases}$

Note that for $b>0$ the function $w_{b}(r)$ admits a smooth extension to
$r=\pi/\sqrt{b}$.

######  Proposition 3.4 (See [O’N83, GW79, Gri99a]).

Let $M_{w}^{m}$ be a $w-$model space with warping function $w(r)$ and center
$o_{w}$. The distance sphere of radius $r$ and center $o_{w}$ in $M_{w}^{m}$
is the fiber $\pi^{-1}(r)$. This distance sphere has the constant mean
curvature $\eta_{w}(r)=\frac{w^{\prime}(r)}{w(r)}$ On the other hand, 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

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

######  Remark d.

The $w-$model spaces are completely determined via $w$ by the mean curvatures
of the spherical fibers $S^{w}_{r}$:

$\,\eta_{w}(r)=w^{\prime}(r)/w(r)\,\quad,$

by the volume of the fiber

$\,\operatorname{Vol}(S^{w}_{r})\,=V_{0}\,w^{m-1}(r)\,\quad,$

and by the volume of the corresponding ball, for which the fiber is the
boundary

$\,\operatorname{Vol}(B^{w}_{r})\,=\,V_{0}\,\int_{0}^{r}\,w^{m-1}(t)\,dt\,\quad.$

Here $V_{0}$ denotes the volume of the unit sphere $S^{0,m-1}_{1}$, (we denote
in general as $S^{b,m-1}_{r}$ the sphere of radius $r$ in the real space form
$\mathbb{K}^{m}(b)$) . The latter two functions define the isoperimetric
quotient function as follows

$\,q_{w}(r)\,=\,\operatorname{Vol}(B^{w}_{r})/\operatorname{Vol}(S^{w}_{r})\quad.$

We observe moreover that the flux of the geodesic distance function $r_{o}$
from the center to the model space is

$J^{w}_{r}(R)=\int_{S^{w}_{R}}|\nabla
r|d\sigma=\operatorname{Vol}(S^{w}_{R})\quad.$

Besides the already defined comparison controllers for the radial sectional
curvatures of $N^{n}$, we shall need two further purely intrinsic conditions
on the model spaces:

######  Definition 3.5.

A given $w-$model space $\,M^{m}_{w}\,$ is called balanced from below and
balanced from above, respectively, if the following weighted isoperimetric
conditions are satisfied:

$\displaystyle\text{Balance from below:}\quad q_{w}(r)\,\eta_{w}(r)$
$\displaystyle\geq 1/m\quad\text{for all}\quad r\geq 0\quad;$
$\displaystyle\text{Balance from above:}\quad q_{w}(r)\,\eta_{w}(r)$
$\displaystyle\leq 1/(m-1)\quad\text{for all}\quad r\geq 0\quad.$

A model space is called totally balanced if it is balanced both from below and
from above.

###  3.2. Laplacian comparison for radial functions

Let us recall the expression of the Laplacian on model spaces for radial
functions

######  Proposition 3.6 (See [O’N83], [GW79] and [Gri99a]).

Let $M_{w}^{n}$ be a model space, denote by
$r:M_{w}^{n}-\\{o_{w}\\}\to\mathbb{R}^{+}$ the geodesic distance from the
center $o_{w}$, let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function, then

(3.2) $\Delta^{M_{w}^{n}}\left(f\circ r\right)=f^{\prime\prime}\circ
r+\left(n-1\right)\left(f^{\prime}\cdot\eta_{w}\right)\circ r\quad.$

Applying the Hessian comparison theorems given in [GW79] we can obtain (see
[MP06] for instance)

######  Proposition 3.7.

Let $\varphi:P^{m}\to N^{n}$ be an immersion into a manifold $N$ with a pole.
Suppose the the radial sectional curvatures $K_{N}$ of $N$ are bounded from
above by the radial sectional curvatures of a model space $M_{w}^{m}$ as
follows:

(3.3) $K_{N}\leq-\frac{w^{\prime\prime}}{w}\circ r\quad.$

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function with $f^{\prime}\geq 0$,
and dennote by $r:P\to\mathbb{R}^{+}$ the extrinsic distance function. Then

(3.4) $\displaystyle\Delta^{P}\left(f\circ r\right)\geq$
$\displaystyle|\nabla^{P}r|\left(f^{\prime\prime}-f^{\prime}\cdot\eta_{w}\right)\circ
r$ $\displaystyle+m\left(f^{\prime}\cdot\eta_{w}\right)\circ
r+m\langle\nabla^{N}r,H_{P}\rangle f^{\prime}\circ r\quad,$

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

####  3.2.1. Capacity and the Mean Exit Time function on Model spaces

One key purpose of this paper is to compare the capacity of extrinsic annuli
of an immersed minimal submanifold with the capacity in an adequate model
space. In the model space we can obtain the value of the capacity directly:

######  Proposition 3.8 (See [Gri99b]).

Let $M_{w}^{n}$ be a model space. Then

(3.5)
$\operatorname{Cap}(A^{w}_{\rho,R})=\left(\int_{\rho}^{R}\frac{ds}{\operatorname{Vol}(S_{s}^{w})}\right)^{-1}=V_{n}\left(\int_{\rho}^{R}\frac{ds}{w^{n-1}}\right)^{-1}\quad.$

We note that the radial function $\Psi:M_{w}^{m}\to\mathbb{R}$ given by

(3.6) $\Psi(p):=\Psi^{w}_{\rho,R}(r(p))\quad,$

being

(3.7)
$\Psi^{w}_{\rho,R}(t)=\int_{\rho}^{t}\frac{\operatorname{Cap}(A^{w}_{\rho,R})}{\operatorname{Vol}(S^{w}_{s})}ds\quad,$

is the solution to the Dirichlet problem given in 2.7 for the annular region
$A^{w}_{\rho,R}$, namely

(3.8) $\begin{cases}\Delta^{M_{w}^{m}}\Psi=0\\\ \Psi|_{S^{w}_{\rho}}=0\\\
\Psi|_{S^{w}_{R}}=1\end{cases}$

Another important tool in this paper is the comparison result for the mean
exit time. Let now $E_{R}^{w}$ denote the mean time of the first exit from
$B_{R}^{w}$ for a Brownnian particle starting at $o_{w}$. A remark due to
Dynkin in [Dyn65] claims that $E_{R}^{w}$ is the continuous solution to the
following Poisson equation with Dirichlet boundary data,

(3.9) $\displaystyle\Delta^{M_{w}^{n}}E_{R}^{w}=$ $\displaystyle-1$
$\displaystyle E_{R}^{w}|_{S_{R}^{w}}=$ $\displaystyle 0.$

Since the ball $B_{R}^{w}$ has maximal isotropy at the center $o_{w}$, so we
have that $E_{R}^{w}$ only depends on the extrinsic distance $r$. Therefore,
we will write $E_{R}^{w}=E_{R}^{w}(r)$ and

######  Proposition 3.9 (See [MP06]).

Let $M_{w}^{n}$ be a model space of dimension $n$ then

(3.10) $E^{w}_{R}(r)=\int_{r}^{R}q_{w}(t)dt$

##  4\. Proof of the main theorems of §2

###  4.1. Proof of theorem 2.1

Since the mean time function $E_{R}^{w}$ is a radial function, we can
transplant it to $P$ using the extrinsic distance, hence, we also denote as
$E_{R}^{w}:P\to\mathbb{R}$ the function given by
$E_{R}^{w}(x)=E_{R}^{w}(r(x))$. To compare the mean exit time function, we
need the following comparison for the mean exit time

######  Proposition 4.1.

([MP06]) Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and
minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$
into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$. Let us
suppose that the $o-$radial sectional curvatures of $N$ are bounded from above
by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\,,$

and the model space $M_{w}^{m}$ is balanced from below, then

(4.1) $\Delta^{P}E^{w}_{R}\leq-1=\Delta^{P}E_{R}.$

Applying now the divergence theorem to inequality (4.1) we obtain

(4.2) $\displaystyle-\operatorname{Vol}(D_{R})=$
$\displaystyle\int_{D_{R}}\Delta^{P}E^{P}_{R}(r)d\mu\geq\int_{D_{R}}\Delta^{P}E^{w}_{R}(r)d\mu$
$\displaystyle=$ $\displaystyle\int_{\partial
D_{R}}E^{w}_{R}(r)^{\prime}\langle\nabla^{P}r,\nu\rangle
d\sigma=-q_{w}(R)\int_{\partial D_{R}}\|\nabla^{P}r\|d\sigma$

Therefore,

(4.3)
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}\leq\frac{J_{r}(R)}{\operatorname{Vol}(S^{w}_{R})}=\frac{J_{r}(R)}{J_{r}^{w}(R)}\quad.$

Observe that equality in inequality (4.3) implies equality in inequality (4.2)
and therefore, in inequality (4.1). Taking, thus, into account that
$E_{R}^{P}=E_{R}^{w}$ in $x\in\partial D_{R}$, $\Delta E_{R}^{P}=\Delta
E_{R}^{w}$ in $x\in D_{R}$, and the maximum principle, we obtain that equality
in (4.3) implies

(4.4) $E_{R}^{P}=E_{R}^{w}\quad,$

for all $x\in D_{R}$.

In order to obtain the monotonicity of the quotient
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}$, we note
that by the co-area formula we get:

(4.5)
$\displaystyle\left(\ln\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}\right)^{\prime}=$
$\displaystyle\frac{\int_{\partial
D_{R}}\frac{1}{\|\nabla^{P}r\|}d\sigma}{\operatorname{Vol}(D_{R})}-\frac{\operatorname{Vol}(S_{R}^{w})}{\operatorname{Vol}(B_{R}^{w})}$
$\displaystyle\geq$ $\displaystyle\frac{\int_{\partial
D_{R}}\|\nabla^{P}r\|d\sigma}{\operatorname{Vol}(D_{R})}-\frac{\operatorname{Vol}(S_{R}^{w})}{\operatorname{Vol}(B_{R}^{w})}$
$\displaystyle=$
$\displaystyle\frac{\operatorname{Vol}(S_{R}^{w})}{\operatorname{Vol}(D_{R})}\left(\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}-\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}\right)$
$\displaystyle\geq$ $\displaystyle 0\quad.$

Hence $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}$ is a
monotone non-decreasing function. To prove that also
$\frac{J_{r}(R)}{J_{r}^{w}(R)}$ is a monotone nondecreasing function we need
the following lemma

######  Lemma 4.2.

(4.6) $\text{div
}\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)\leq
0\quad.$

###### Proof.

Taking into account the product rule for the divergence and the mean exit time
comparison result

(4.7) $\displaystyle\text{div
}\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)=$
$\displaystyle\frac{\Delta^{P}E_{R}^{w}(r)}{\operatorname{Vol}(B_{r}^{w})}-\frac{\operatorname{Vol}(B_{r}^{w})^{\prime}}{\operatorname{Vol}(B_{r}^{w})^{2}}\langle\nabla^{P}r,\nabla^{P}E_{R}^{w}(r)\rangle$
$\displaystyle=$
$\displaystyle\frac{\Delta^{P}E_{R}^{w}(r)}{\operatorname{Vol}(B_{r}^{w})}-\frac{\operatorname{Vol}(S_{r}^{w})}{\operatorname{Vol}(B_{r}^{w})^{2}}E_{R}^{w}(r)^{\prime}\|\nabla^{P}r\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{-1}{\operatorname{Vol}(B_{r}^{w})}+\frac{\|\nabla^{P}r\|^{2}}{\operatorname{Vol}(B_{r}^{w})}\leq
0\quad.$

∎

Using now this lemma and the divergence theorem in the extrinsic annulus
$A_{\rho,R}$ for $\rho<R$

(4.8) $\displaystyle 0\geq$ $\displaystyle\int_{A_{\rho,R}}\text{div
}\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)d\mu$
$\displaystyle=$ $\displaystyle\int_{\partial
D_{R}}\frac{E^{w}_{R}(r)^{\prime}\|\nabla^{P}r\|}{\operatorname{Vol}(B_{r}^{w})}d\sigma-\int_{\partial
D_{\rho}}\frac{E^{w}_{R}(r)^{\prime}\|\nabla^{P}r\|}{\operatorname{Vol}(B_{r}^{w})}d\sigma$
$\displaystyle=$
$\displaystyle-\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}+\frac{J_{r}(\rho)}{\operatorname{Vol}(S_{\rho}^{w})}\quad.$

Therefore,

(4.9)
$\frac{J_{r}(R)}{J^{w}_{r}(R)}\geq\frac{J_{r}(\rho)}{J_{r}^{w}(\rho)}\quad,$

for any $R>\rho$, and the theorem is proven.

###  4.2. Proof of theorem 2.2

The corresponding Dirichlet problem for the capacity of the extrinsic annulus
$A_{\rho,R}$ is

(4.10) $\begin{cases}\Delta^{P}\Psi=0\\\ \Psi|_{\partial D_{\rho}}=0\\\
\Psi|_{\partial D_{R}}=1\end{cases}$

Let us transplant the function $\Psi^{w}_{\rho,R}$ with the extrinsic distance
function $r$:

(4.11) $\Psi^{w}(p):A_{\rho,R}\to\mathbb{R},\quad
p\to\Psi^{w}(p):=\Psi^{w}_{\rho,R}(r(p))\quad.$

Then, applying proposition 3.7

(4.12) $\Delta^{P}\Psi^{w}\geq
m\left(1-|\nabla^{P}r|\right)\left((\Psi^{w}_{\rho,R})^{\prime}\cdot\eta_{w}\right)\circ
r\quad.$

Taking into account that $\eta_{w}\geq 0$

(4.13) $\Delta^{P}\Psi^{w}\geq 0=\Delta^{P}\Psi\quad.$

Since $\Delta^{P}\left(\Psi^{w}-\Psi\right)\geq 0$ and since $\Psi_{\partial
A_{\rho,R}}=\Psi^{w}_{\partial A_{\rho,R}}$, we have by the Maximum Principle
that $\Psi^{w}\leq\Psi$ on $A_{\rho,R}$, and, since $\Psi_{\partial
D_{\rho}}=\Psi^{w}_{\partial D_{\rho}}=0$, we obtain

(4.14) $|\nabla^{P}\Psi^{w}|\leq|\nabla^{P}\Psi|\quad\text{ on }\partial
D_{\rho}\quad.$

Finally, we can estimate the capacity

(4.15) $\displaystyle\operatorname{Cap}(A_{\rho,R})$
$\displaystyle=\int_{\partial D_{\rho}}|\nabla^{P}\Psi|d\sigma$
$\displaystyle\geq\int_{\partial D_{\rho}}|\nabla^{P}\Psi^{w}|d\sigma$
$\displaystyle=(\Psi^{w}(\rho))^{\prime}\int_{\partial
D_{\rho}}|\nabla^{P}r|d\sigma$
$\displaystyle=\operatorname{Cap}(A_{\rho,R}^{w})\frac{J_{r}(\rho)}{J^{w}_{r}(\rho)}\quad,$

and the theorem follows.

###  4.3. Proof of theorem 2.3

With the flux we can provide an upper bound for the capacity (see inequality
(2.9) ). Using theorem 2.1 we obtain that

(4.16) $\displaystyle\operatorname{Cap}(A_{\rho,R})\leq$
$\displaystyle\frac{1}{\int_{\rho}^{R}\frac{ds}{\int_{\partial
D_{s}}\|\nabla^{P}r\|d\sigma}}=\frac{1}{\int_{\rho}^{R}\frac{ds}{\frac{J_{r}(s)}{\operatorname{Vol}(S_{s}^{w})}\operatorname{Vol}(S_{s}^{w})}}$
$\displaystyle\leq$
$\displaystyle\frac{\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}}{\int_{\rho}^{R}\frac{ds}{\operatorname{Vol}(S_{s}^{w})}}=\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}\operatorname{Cap}(A^{w}_{\rho,R}).$

For the bounds from below, see [MP03]. Observe moreover that equality in the
above inequality implies that

(4.17)
$\int_{t}^{R}\left(\frac{\frac{J_{r}(R)}{\operatorname{Vol}(S^{w}_{R})}}{\frac{J_{r}(s)}{\operatorname{Vol}(S^{w}_{s})}}-1\right)\frac{1}{\operatorname{Vol}(S_{s}^{w})}ds=0.$

Therefore

(4.18)
$\frac{J_{r}(R)}{\operatorname{Vol}(S^{w}_{R})}=\frac{J_{r}(s)}{\operatorname{Vol}(S^{w}_{s})},$

for any $s\in[\rho,R]$. Then, by inequality (4.8)

(4.19) $\text{div
}\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)=0,$

for any $p\in A_{\rho,R}$. From inequality (4.7)

(4.20) $\|\nabla^{P}r\|=1,$

for any $p\in A_{\rho,R}$, and hence, $D_{R}$ is a minimal cone.

##  5\. Proof of Theorem 1.1 and Corollary 1.2

This proof mimics the argument given in [Tka94, Theorem 2], so we merely give
a sketch emphasizing the points where the line of reasoning from [Tka94] is
modified to hold in the present more general setting.

First of all, note that we can construct the following order-preserving
bijection

$F:\mathbb{R}^{+}\to\mathbb{R}^{+},\quad F(t)=\int_{0}^{t}w(s)ds\quad.$

Since $\varphi:P^{m}\to N^{n}$ is a complete proper and minimal immersion
within a manifold with a pole $N^{n}$, applying proposition 3.7 we have

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

Hence, by using the assumption $w^{\prime}>0$, the extrinsic distance has no
local maximum. Therefore for any $R$, $P^{m}\setminus D_{R}$ has no bounded
components, being each component of $P^{m}\setminus D_{R}$ non compact, and
the number of ends $\mathcal{E}_{D_{R}}(P)$ with respect to $D_{R}$ is the
number of connected components of $P^{m}\setminus D_{R}$.

Let us denote by
$\displaystyle\left\\{\Omega^{i}\right\\}_{i=1}^{\mathcal{E}_{D_{R}}(P)}$ the
set of $\mathcal{E}_{D_{R}}(P)$ connected components of $P^{m}\setminus D_{R}$
(every one of them is a minimal submanifold with boundary). Now we need the
following lemma

######  Lemma 5.1.

For any connected component $\Omega_{i}$ of $P^{m}\setminus D_{R}$ the volume
of the set

$D_{t}^{\Omega_{i}}=D_{t}\cap\Omega_{i}\quad.$

is bounded from below by

(5.2) $\operatorname{Vol}(D_{t}^{\Omega_{i}})\geq
V_{m}\left(\frac{t-R}{2}\right)^{m}\quad.$

###### Proof.

Now pick a point $o^{\prime}\in D_{t}^{\Omega_{i}}$ such that its extrinsic
distance is $r_{o}(o^{\prime})=\frac{R+t}{2}$, then the extrinsic ball
$D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime})$ in $\Omega_{i}$ centered at
$o^{\prime}$ with radius $\frac{t-R}{2}$ satisfies

(5.3) $D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime})\subset
D_{t}^{\Omega_{i}}\quad.$

Hence,

(5.4)
$\operatorname{Vol}(D_{t}^{\Omega_{i}})\geq\operatorname{Vol}(D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime}))\quad.$

Since $r(o^{\prime})>R$ and the sectional curvatures of any tangent $2-$plane
of the tangent space at every point in the geodesic ball
$B^{N}_{\frac{t-R_{0}}{2}}(\varphi(o^{\prime}))$ of the ambient manifold are
non-positive, we can make use of the behavior of the volume quotient (claim
(2) in theorem 2.1) for extrinsic balls to the immersion
$\varphi:D^{\Omega_{i}}_{\frac{t-R_{0}}{2}}(o^{\prime})\to
B^{N}_{\frac{t-R_{0}}{2}}(\varphi(o^{\prime}))$ with the new model comparison
$w(r)=w_{0}(r)=r$ (namely $M_{w}^{m}=\mathbb{R}^{m}$)

(5.5)
$\frac{\operatorname{Vol}(D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime}))}{V_{m}\left(\frac{t-R}{2}\right)^{m}}\geq\lim_{s\to
0}\frac{\operatorname{Vol}(D^{\Omega_{i}}_{s}(o^{\prime}))}{V_{m}s^{m}}\geq
1\quad.$

And the lemma is proved. ∎

Summing now in inequality (5.2) we obtain

(5.6)
$\operatorname{Vol}(A_{R,t})=\sum_{i=1}^{\mathcal{E}_{D_{R}}(P)}\operatorname{Vol}(D_{t}^{\Omega_{i}})\geq\mathcal{E}_{D_{R}}(P)V_{m}\left(\frac{t-R}{2}\right)^{m}\quad.$

Taking into account that
$\operatorname{Vol}(A_{R,t})\leq\operatorname{Vol}(D_{t})$ and dividing by
$\operatorname{Vol}(B_{t}^{w})$ we obtain

(5.7)
$\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}\geq\mathcal{E}_{D_{R}}(P)V_{m}\frac{\left(\frac{t-R}{2}\right)^{m}}{\operatorname{Vol}(B_{t}^{w})}\quad.$

We can split the last quotient by division and multiplication by $t^{m}$

(5.8)
$\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}\geq\mathcal{E}_{D_{R}}(P)V_{m}\left(\frac{1-R/t}{2}\right)^{m}\frac{t^{m}}{\operatorname{Vol}(B_{t}^{w})}\quad.$

Hence, finally, using the explicit expression for
$\operatorname{Vol}(B_{t}^{w})$ the theorem follows.

In order to prove corollary 1.2, note that by the maximum principle
$\mathcal{E}_{D_{R}}^{P}$ is a non-decreasing function with respect to $R$. By
inequality (1.5) and the assumptions of the corollary we can conclude that
$\mathcal{E}_{D_{R}}^{P}$ is stabilized, i.e. $\mathcal{E}_{D_{R}}^{P}=\text{
constant for sufficient large }R$.

Now let $F\subset P$ be an arbitrary compact subset. Using again the maximum
principle of the immersion, we conclude that $\mathcal{E}_{F}(P)$ is a non-
decreasing function of the compact set $F$ (namely, if $F_{1}\subset F_{2}$
then $\mathcal{E}_{F_{1}}(P)\leq\mathcal{E}_{F_{2}}(P)$). Taking into account
that for any compact set $K$ there exist $R_{K}$ such that $K\subset
D_{R_{K}}$, we finally obtain

(5.9) $\mathcal{E}(P)=\lim_{R\to\infty}\mathcal{E}_{D_{R}}(P)\quad,$

and the corollary follows.

##  6\. Corollaries and application of the extrinsic comparison theory

###  6.1. Relation between $w-$volume and $w-$flux of submanifolds

Under the hypotesis of theorem 2.1, if the submanifold has finite $w$-flux,
the submanifold has finite $w$-volume. But in particular settings we can also
state a reverse:

######  Theorem 6.1.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose
that the $o-$radial sectional curvatures of $N$ are bounded from above by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\,.$

Suppose that the model space $M_{w}^{m}$ is balanced from bellow with warping
function satisfying

$w^{\prime}(r)\geq 0\quad\forall r\in\mathbb{R}_{+}.$

Then, if the submanifold has finite $w$-volume, we have:

  1. (1)

The submanifold has finite $w$-flux.

  2. (2)

$\text{Flux}_{w}(P)=\operatorname{Vol}_{w}(P)$.

###### Proof.

To prove the theorem let us state the following metric property for geodesic
balls and geodesic spheres in a rotationally symmetric model space

######  Lemma 6.2.

Let $M^{m}_{w}$ be a model space with

$w^{\prime}(r)\geq 0\quad\forall r\in\mathbb{R}_{+}.$

Then

$q_{w}(s)=\frac{\operatorname{Vol}(B^{w}_{S})}{\operatorname{Vol}(S^{w}_{s})}\leq
s.$

###### Proof.

Observe that

(6.1) $q_{w}(0)=0,$

and, since $w^{\prime}\geq 0$ ,

(6.2) $q_{w}^{\prime}(t)\leq 1,\,\forall t\geq 0.$

Hence, by integrating the above inequality, the lemma follows. ∎

Now, since $P$ has finite $w$-volume, then there exists $S\in\mathbb{R}^{+}$
such that

(6.3)
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}\leq\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}=S<\infty\quad.$

By inequality (4.5)

(6.4)
$\displaystyle\left(\ln\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B^{w}_{s})}\right)^{\prime}\geq$
$\displaystyle\frac{\operatorname{Vol}(S_{s}^{w})}{S\operatorname{Vol}(B_{s}^{w})}\left(\frac{J_{r}(s)}{J_{r}^{w}(s)}-\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B_{s}^{w})}\right)\geq
0\quad.$

Therefore, taking lemma 6.2 into acount we get:

(6.5) $\displaystyle
0\leq\left(\frac{J_{r}(s)}{J_{r}^{w}(s)}-\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B_{s}^{w})}\right)\leq
S\left(\ln\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B^{w}_{s})}\right)^{\prime}s\quad.$

But since

(6.6)
$\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}=\frac{\operatorname{Vol}(D_{R_{1}})}{\operatorname{Vol}(B_{R_{1}}^{w_{b}})}e^{\int_{R_{1}}^{\infty}\left(\ln\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B^{w}_{s})}\right)^{\prime}ds}=S<\infty\quad,$

then, for any $\epsilon>0$ there exist a sequence $\\{t_{i}\\}_{i=1}^{\infty}$
with $t_{i}\to\infty$ when $i\to\infty$, and $R_{\epsilon}$, such that

(6.7)
$\left(\ln\frac{\operatorname{Vol}(D_{t_{i}})}{\operatorname{Vol}(B^{w}_{t_{i}})}\right)^{\prime}t_{i}<\epsilon\quad.$

This holds for any $t_{i}>R_{\epsilon}$. Applying inequality (6.5) taking into
account the monotonicity of the flux and volume comparison quotients

(6.8)
$0\leq\text{Flux}_{w_{b}}(P)-\operatorname{Vol}_{w_{b}}(P)\leq\epsilon\quad,$

for any $\epsilon>0$ . Letting $\epsilon$ tend to $0$, the theorem is proven.
∎

###  6.2. Intrinsic versions

In this subsection we consider the intrinsic versions of Theorems 2.1 and 2.3
assuming that $P^{m}=N^{n}$. In this case, the extrinsic distance to the pole
$p$ becomes the intrinsic distance in $N^{n}$, hence, for all $R$ the
extrinsic domains $D_{R}$ become the geodesic balls $B^{N}_{R}$ of the ambient
manifold $N^{n}$. Then, for all $x\in P$

$\displaystyle\nabla^{P}r(x)$ $\displaystyle=$ $\displaystyle\nabla^{N}r(x).$

As a consequence, $\|\nabla^{P}r\|=1$.

From this intrinsic viewpoint, we have the following isoperimetric and volume
comparison inequalities.

######  Theorem 6.3.

Let $N^{n}$ denote a complete Riemannian manifold with a pole $p$. Suppose
that the $p$-radial sectional curvatures of $N^{n}$ are bounded from above by
the $p_{w}$-radial sectional curvatures of a $w$-model space $M^{n}_{w}$.
Assume that

(6.9) $w^{\prime}\geq 0\quad.$

Then the capacity of the intrinsic annulus $A_{\rho,R}$ is bounded from below
by

$\frac{\operatorname{Vol}(\partial
B^{N}_{\rho})}{\operatorname{Vol}(S_{\rho}^{w})}\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}$

And, furthermore, if $M_{w}^{n}$ is hyperbolic, then $N^{n}$ is also
hyperbolic.

######  Theorem 6.4.

Let $N^{n}$ denote a complete Riemannian manifold with a pole $p$. Suppose
that the $p$-radial sectional curvatures of $N^{n}$ are bounded from above by
the $p_{w}$-radial sectional curvatures of a $w$-model space $M^{n}_{w}$.
Assume that $M_{w}^{n}$ is balanced from below. Then,

  1. (1)

for all $R>0$

(6.10)
$\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B^{w}_{R})}\leq\frac{\operatorname{Vol}(\partial
B^{N}_{R})}{\operatorname{Vol}(S^{w}_{R})}\quad.$

  2. (2)

The functions
$\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B^{w}_{R})}$ and
$\frac{\operatorname{Vol}(\partial B^{N}_{R})}{\operatorname{Vol}(S^{w}_{R})}$
are non decreasing on $R$.

  3. (3)

Denoting by $E_{R}^{N}(x)$ the mean exit time function for the geodesic ball
$B^{N}_{R}$ in $N$ and denoting by $E_{R}^{w}$ the mean exit time function in
the $R-$ball $B_{R}^{w}$ in the model space $M_{w}^{n}$. If equality holds in
(6.10) for some fixed $R>0$ then for any $x\in B^{N}_{R}$,
$E_{R}^{N}(x)=E_{R}^{w}(r(x))$.

  4. (4)

The capacity of the intrinsic annulus $A_{\rho,R}$ is bounded from above by

$\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\leq\frac{\operatorname{Vol}(\partial
B^{N}_{R})}{\operatorname{Vol}(S_{R}^{w})}$

Furthermore, if we suppose that there exist a finite real constant $C<\infty$
such that
$\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B_{R}^{w})}<C$ (or
$\frac{\operatorname{Vol}(\partial
B^{N}_{R})}{\operatorname{Vol}(S_{R}^{w})}<C$) then if $M_{w}^{n}$ is
parabolic, $N$ is parabolic, and

$\lim_{R\to\infty}\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B_{R}^{w})}=\lim_{R\to\infty}\frac{\operatorname{Vol}(\partial
B^{N}_{R})}{\operatorname{Vol}(S_{R}^{w})}.$

###  6.3. Upper bounds for the fundamental tone

S.T Yau suggested in [Yau00] the “very interesting” question to find an upper
estimate to the first Dirichlet eigenvalue of minimal surfaces.

Recall that for any precompact region $\Omega\subset M$ in a Riemannian
manifold $M$, the first eigenvalue $\lambda_{1}(\Omega)$ of the Dirichlet
problem in $\Omega$ for the Laplace operator is defined by the variational
property

(6.11) $\lambda_{1}(\Omega)=\inf_{u}\frac{\int\|\nabla
u\|^{2}d\mu}{\int\|u\|^{2}}$

where the $\inf$ is taken over all Lipschitz functions $u\neq 0$ compactly
supported in $\Omega$.

The fundamental tone $\lambda^{*}(M)$ of a complete Riemannian manifold can be
obtained as the limit of the first Dirichlet eigenvalues of the precompact
open sets in any exhaution sequence $\\{\Omega_{n}\\}_{n\in\mathbb{N}}$ for
$M$, see [Gri99a]

(6.12) $\lambda^{*}(M)=\lim_{n\to\infty}\lambda_{1}(\Omega_{n})\quad.$

In this section, we shall impose flux and volume restrictions not on the
submanifold $P$ but on one end $V$ of the submanifold with respect to the
extrinsic ball $D_{R_{0}}$. Let us denote $D_{R}^{V}$ the intersection of the
extrinsic ball $D_{R}$ with the end $V$ with respect to $D_{R_{0}}$

(6.13) $D_{R}^{V}=D_{R}\cap V\quad.$

Let us denote $J^{V}_{r}(R)$ the flux of the extrinsic distance in the end
$V$, namely

(6.14) $J^{V}_{r}(R)=\int_{\partial D_{R}\cap V}|\nabla^{P}r|d\sigma\quad.$

With this setting we then have:

######  Theorem 6.5.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N$. Let us suppose that
the $o-$radial sectional curvatures of $N$ are bounded from above by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\,,$

and the model space $M_{w}^{m}$ is balanced from below. Suppose moreover that
there exists an end $V$ with respect to an extrinsic ball $D_{R_{0}}$ with
finite $w$-flux. Then

(6.15)
$\lambda^{*}(P)\leq\frac{\text{Flux}_{w}(V)}{\operatorname{Vol}_{w}(V)}\limsup_{t\to\infty}\left(\frac{1}{\operatorname{Vol}(B_{t}^{w})\int_{t}^{\infty}\frac{ds}{\operatorname{Vol}(S^{w}_{s})}}\right)\quad.$

###### Proof.

Due to the relation between the first Dirichlet eigenvalue and the capacity
given in [Gri99b] we can conclude for the extrinsic ball $D_{R}^{V}$ that

(6.16)
$\lambda_{1}(D_{R}^{V})\leq\frac{\operatorname{Cap}(A^{V}_{t,R})}{\operatorname{Vol}(D_{t}^{V})}\quad.$

Being $t<R$ and $A^{V}_{t,R}$ the extrinsic annulus in $V$. Hence, by the
theorem 2.3

(6.17)
$\lambda_{1}(D_{R}^{V})\leq\frac{\frac{J_{r}^{V}(R)}{J_{r}^{w}(R)}}{\frac{\operatorname{Vol}(D_{t}^{V})}{\operatorname{Vol}(B_{t}^{w})}}\frac{\operatorname{Cap}(A^{w}_{t,R})}{\operatorname{Vol}(B^{w}_{t})}\quad.$

For any $t<R$. Finally, taking into account that
$\lambda(D_{R})\leq\lambda_{1}(D_{R}^{V})$ (by the monotonicity of the first
eigenvalue), and letting $R$ tend to infinity we have

(6.18)
$\lambda^{*}(P)\leq\frac{\text{Flux}_{w}(V)}{\frac{\operatorname{Vol}(D_{t}^{V})}{\operatorname{Vol}(B_{t}^{w})}}\frac{1}{\operatorname{Vol}(B^{w}_{t})\int_{t}^{\infty}\frac{ds}{\operatorname{Vol}(S_{s}^{w})}}\quad.$

Taking limits, the theorem follows. ∎

Obviously, by using theorem 6.1 we also have the following:

######  Corollary 6.6.

Under the assumptions of theorem 6.5 suppose moreover

$w^{\prime}\geq 0.$

Then,

(6.19)
$\lambda^{*}(P)\leq\limsup_{t\to\infty}\left(\frac{1}{\operatorname{Vol}(B_{t}^{w})\int_{t}^{\infty}\frac{ds}{\operatorname{Vol}(S^{w}_{s})}}\right)\quad.$

Using the Cheeger isoperimetric constant we can deduce the following lower
bounds

######  Theorem 6.7.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
complete Riemannian manifold $N^{n}$ with a pole $o\in N^{n}$ . Let us suppose
that the $o-$radial sectional curvatures of $N^{n}$ are bounded from above by

$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall
x\in P\,,$

and the model space $M_{w}^{m}$ is balanced from below. Suppose moreover that

$L:=\sup_{t\in\mathbb{R}^{+}}q_{w}(t)<\infty.$

Then

(6.20) $\frac{1}{4L^{2}}\leq\lambda^{*}(P)\quad.$

###### Proof.

Consider $\Omega\subset P^{m}$ a smooth domain with smooth boundary
$\partial\Omega$. Using the transplanted mean exit function in a similar way
as in the proof of theorem 2.1 we obtain:

(6.21) $\displaystyle-\operatorname{Vol}(\Omega)=$
$\displaystyle\int_{\Omega}\Delta^{P}E_{R}d\mu\geq\int_{\Omega}\Delta^{P}E_{R}^{w}d\mu=\int_{\partial\Omega}E_{R}^{w}(r)^{\prime}\langle\nabla^{P}r,\nu\rangle
d\sigma$ $\displaystyle\geq$
$\displaystyle-\int_{\partial\Omega}q_{w}(r)\langle\nabla^{P}r,\nu\rangle
d\sigma\geq-\int_{\partial\Omega}q_{w}(r)d\sigma$ $\displaystyle\geq$
$\displaystyle-L\operatorname{Vol}(\partial\Omega)\quad.$

Hence, for any $\Omega\subset P$,

(6.22)
$\frac{\operatorname{Vol}(\partial\Omega)}{\operatorname{Vol}(\Omega)}\geq\frac{1}{L}\quad.$

Thence the Cheeger constant $h(P)$ (see [Cha84]) satisfies

(6.23) $h(P)\geq\frac{1}{L}\quad.$

Taking into account that

(6.24) $\lambda^{*}(P)\geq\frac{1}{4}\left(h(P)\right)^{2}\quad,$

the theorem follows. ∎

As an immediate consequence of the previous theorems and corollaries in the
particular setting of a minimal submanifold within a Cartan-Hadamard ambient
manifold is the following:

######  Corollary 6.8.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be a complete minimal immersion into
a simply connected Cartan-Hadamard manifold $N^{n}$ with sectional curvatures
$K_{N}\leq b\leq 0$. Suppose moreover that there exists an end $V$ with
respect to an extrinsic ball $D_{R_{0}}$ with finite $w_{b}$-volume. Then

(6.25) $\frac{-(m-1)^{2}b}{4}\leq\lambda^{*}(P)\leq-(m-1)^{2}b\quad.$

######  Remark e.

Note that if $b=0$ in the above theorem, $\lambda^{*}(P)=0$. See also [Gim13].

###  6.4. Applications to minimal submanifolds in $\mathbb{R}^{n}$

If $P^{m}$ is a minimal submanifold in $\mathbb{R}^{n}$, it is well known that
the extrinsic distance $r$ satisfies

(6.26) $\Delta^{P}r^{2}=2m$

Applying the divergence theorem

(6.27) $\displaystyle 2m\operatorname{Vol}(D_{R})=$
$\displaystyle\int_{D_{R}}\Delta^{p}r^{2}d\mu=\int_{\partial
D_{R}}2r\langle\nabla r,\nu\rangle d\sigma$ $\displaystyle=$ $\displaystyle
2R\int_{\partial D_{R}}\langle\nabla r,\frac{\nabla r}{|\nabla r|}\rangle
d\sigma=2R\int_{\partial D_{R}}|\nabla r|d\sigma$ $\displaystyle=$
$\displaystyle
2m\frac{\operatorname{Vol}(B_{R}^{w_{0}})}{\operatorname{Vol}(S_{R}^{w_{0}})}\int_{\partial
D_{R}}|\nabla r|d\sigma$

hence, the volume comparison quotient
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w_{0}})}$ is just

(6.28)
$\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w_{0}})}=\frac{J_{r}(R)}{J^{w_{0}}_{r}(R)}\quad.$

And therefore, we can state that

######  Corollary 6.9.

Let $P^{m}$ be a minimal submanifold properly immersed in the Euclidean space
$\mathbb{R}^{n}$. Then

$E^{P}_{R}(x)=E^{\mathbb{R}^{m}}_{R}(r(x))\quad,$

where $E^{P}_{R}(x)$ denotes the first exit from $D_{R}$ for a Brownian
particle starting at $x\in D_{R}$, and $E_{R}^{w}(r)$ denotes the
(rotationally symmetric) mean exit time function for the $R-$ball $B_{R}^{w}$
in the model space $M_{w}^{m}$

If we have finite $w_{0}$-volume
($\sup_{R\in\mathbb{R}^{+}}\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w_{0}})}<\infty$)
we also get:

######  Corollary 6.10.

Let $P^{m}$ be a minimal submanifold immersed in $\mathbb{R}^{n}$, suppose
moreover that $P$ has finite $w_{0}$-volume then:

  1. (1)

$P$ is parabolic if $m=2$ and if $m\geq 3$, $P$ is hyperbolic.

  2. (2)

$\lambda^{*}(P)=0$.

On the other hand, in special geometric settings the finiteness of the
$w_{0}$-volume is related to the number of ends

######  Theorem B.

(See [And84] and [Che95]) Let $P^{m}$ be a minimal submanifold properly
immersed in $\mathbb{R}^{n}$ with finite total scalar curvature i.e.
$\int_{P}\|B^{P}\|^{m}d\mu<\infty$ where $\|B^{P}\|$ denotes the norm of the
second fundamental form in $P$, then

(6.29) $\frac{J_{r}(R)}{J_{r}^{w_{0}}(R)}\leq\mathcal{E}(P),$

provided either of the following two conditions hold

  1. (1)

$m=2$, $n=3$ and each end of $P$ is embedded.

  2. (2)

$m\geq 3$.

Where $\mathcal{E}(P)$ denotes the finite number of ends of $P$.

This relation between the number of ends and the flux quotient allow us to
state

######  Corollary 6.11.

Let $P^{m}$ be a minimal submanifold properly immersed in $\mathbb{R}^{n}$
with finite total scalar curvature and either $m\geq 3$, or $m=2$ $n=3$ and
each end of $P$ is embedded, then for any $\rho>0$ and any $R>\rho$

(6.30)
$1\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\leq\mathcal{E}(P)\quad.$

And for the fundamental tone

(6.31) $\lambda^{*}(P)=0\quad.$

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