# Volume growth, number of ends and the topology of a complete submanifold

Vicent Gimeno  Departament de Matemàtiques- Institut of New Imaging
Technologies, Universitat Jaume I, Castellon, Spain<EMAIL_ADDRESS>and
Vicente Palmer  Departament de Matemàtiques- Institut of New Imaging
Technologies, Universitat Jaume I, Castellon, Spain<EMAIL_ADDRESS>
###### Abstract.

Given a complete isometric immersion $\varphi:P^{m}\longrightarrow N^{n}$ in
an ambient Riemannian manifold $N^{n}$ with a pole and with radial sectional
curvatures bounded from above by the corresponding radial sectional curvatures
of a radially symmetric space $M^{n}_{w}$, we determine a set of conditions on
the extrinsic curvatures of $P$ that guarantees that the immersion is proper
and that $P$ has finite topology in the line of the results in [24] and [25].
When the ambient manifold is a radially symmetric space, it is shown an
inequality between the (extrinsic) volume growth of a complete and minimal
submanifold and its number of ends which generalizes the classical inequality
stated in [1] for complete and minimal submanifolds in $\mathbb{R}^{n}$. We
obtain as a corollary the corresponding inequality between the (extrinsic)
volume growth and the number of ends of a complete and minimal submanifold in
the Hyperbolic space together with Bernstein type results for such
submanifolds in Euclidean and Hyperbolic spaces, in the vein of the work [12].

###### Key words and phrases:

volume growth, minimal submanifold, end, Hessian-Index comparison theory,
extrinsic distance, total extrinsic curvature, second fundamental form, gap
theorem, Bernstein-type theorem.

###### 2000 Mathematics Subject Classification:

Primary 53A20 53C40; Secondary 53C42

* Work partially supported by the Caixa Castelló Foundation, and DGI grant MTM2010-21206-C02-02.

##  1\. Introduction

A natural question in Riemannian geometry is to explore the influence of the
curvature conduct of a complete Riemannan manifold on its geometric and
topological properties. Classical results concernig this are the gap theorems
showed by Greene and Wu in [7], (see too [8]), and, when it is considered a
minimal submanifold (properly) immersed in the Euclidean space
$\mathbb{R}^{n}$, the Berstein-type theorems showed by Anderson in [1] and by
Schoen in [32]. Greene and Wu’s results states, roughly speaking, that a
Riemannian manifold with a pole and with faster than quadratic decay of its
sectional curvatures is isometric to the Euclidean space. On the other hand,
Anderson proved, as a corollary of a generalization of the Chern-Osserman
theorem on complete and minimal submanifolds of $\mathbb{R}^{n}$ with finite
total (extrinsic) curvature, that any of such submanifolds having one end is
an affine $n$-plane. More examples concerning submanifolds immersed in an
ambient Riemannian manifold and the analysis of its (intrinsic and extrinsic)
curvature behavior are the gap results, (of Bernstein-type), given by Kasue
and Sugahara in [12] (see Theorems A and B), where an accurate (extrinsic)
curvature decay forces to minimal, (or not) submanifolds with one end of the
Euclidean and Hyperbolic spaces to be totally geodesic, and the gap results
for minimal submanifolds in the Euclidean space with controlled scalar
curvature given by Kasue in [13].

The estimation of the number of ends of these submanifolds plays a fundamental
rôle in all the Bernstein-type results above mentioned. In this way, it is
proved in [1] (see Theorems 4.1 and 5.1 in that paper) that given a complete
and minimal submanifold $\varphi:P^{m}\longrightarrow\mathbb{R}^{n}$, ($m>2$)
having finite total curvature $\int_{P}\|B^{P}\|^{m}d\sigma<\infty$, its
(extrinsic) volume growth, defined as the quotient
$\frac{\operatorname{Vol}(\varphi(P)\cap B^{0,n}_{t})}{\omega_{n}t^{n}}$ is
bounded from above by the number of ends of $P$, $\mathcal{E}(P)$, namely

(1.1) $\lim_{t\to\infty}\frac{\operatorname{Vol}(\varphi(P)\cap
B^{0,n}_{t})}{\omega_{n}t^{n}}\leq\mathcal{E}(P)$

where $B^{b,n}_{t}$ denotes the metric $t-$ ball in the real space form of
constant curvature $b$, $I\\!\\!K^{n}(b)$, and $\|B^{P}\|$ denotes the
Hilbert-Schmidt norm of the second fundamental form of $P$ in
$\mathbb{R}^{n}$. If moreover $\mathcal{E}(P)=1$, it is concluded (using
inequality (1.1)) the Bernstein-type result above alluded, namely, that
$P^{m}$ is an affine plane, i.e. totally geodesic in $\mathbb{R}^{n}$, (see
Theorem 5.2 in [1]).

In the paper [3] it was proved that inequality (1.1) is in fact an equality
when the minimal submanifold in $\mathbb{R}^{n}$ exhibits an accurate decay of
its extrinsic curvature $\|B^{P}\|$ and in the paper [12] it was proved that,
if the submanifold $P$ has only one end and the decay of its extrinsic
curvature $\|B^{P}\|$ is faster than linear, (when the ambient space is
$\mathbb{R}^{n}$) or than exponential, (when the ambient space is
$\mathbb{H}^{n}(b)$), then it is is totally geodesic.

Within this study of the behavior at infinity of complete and minimal
submanifolds with finite total curvature immersed in the Euclidean space, it
was proved also in [1] and in [22] that the immersion of a complete and
minimal submanifold $P$ in $\mathbb{R}^{n}$ or $\mathbb{H}^{n}(b)$ satisfying
$\int_{P}\|B^{P}\|^{m}d\sigma<\infty$ is proper and that $P$ is of finite
topological type.

We should mention here the results in [24] and in [25], where has been stated
new conditions on the decay of the extrinsic curvature for a completely
immersed submanifold $P$ in the Euclidean space ([24]) and in a Cartan-
Hadamard manifold ([25]) which guarantees the properness of the submanifold
and the finiteness of its topology.

In view of these results, it seems natural to consider the following three
issues:

  1. (1)

Can the properness/finiteness results in [24] and [25] be extended to
submanifolds immersed in spaces which have not necessarily non-positive
curvature?,

  2. (2)

Do we have an analogous to inequality (1.1) between the extrinsic volume
growth and the number of ends when we consider a minimal submanifold
(properly) immersed in Hyperbolic space which exhibit an accurate extrinsic
curvature decay?.

  3. (3)

Moreover, is it possible to deduce from this inequality a Bernstein-type
result in the line of [1] and [12]?.

We provide in this paper a (partial) answer to these questions, besides other
lower bounds for the number of ends for (non-minimal) submanifolds in the
Euclidean and Hyperbolic spaces and other gap results related with these
estimates. As a preliminary view of our results, we have the following
theorems, Theorem 1.1 and Theorem 1.2, which follows directly from our Theorem
3.5. In Theorem 1.1 we have the answer to the two last questions, namely,
setting equation (1.1), but in the Hyperbolic case, and a Bernstein-type
result for minimal submanifolds in the Hyperbolic space, in the line studied
by Kasue and Sugahara in [12], (see assertion (A-iv) of Theorem A). On the
other hand, Theorem 1.2 encompasses a slightly less general version of
assertion (A-i) of Theorem A in [12].

######  Theorem 1.1.

Let $\varphi:P^{m}\longrightarrow\mathbb{H}^{n}(b)$ be a complete, proper and
minimal immersion with $m>2$. Let us suppose that for sufficiently large
$R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e. outside a
compact),

$\|B^{P}_{x}\|\leq\frac{\delta(r(x))}{e^{2\sqrt{-b}\,r(x)}}$

where $r(x)=d_{\mathbb{H}^{n}(b)}(o,\varphi(x))$ is the (extrinsic) distance
in $\mathbb{H}^{n}(b)$ of the points in $\varphi(P)$ to a fixed pole
$o\in\mathbb{H}^{n}(b)$ such that $\varphi^{-1}(o)\neq\emptyset$ and
$\delta(r)$ is a smooth function such that $\delta(r)\to 0$ when $r\to\infty$.
Then:

  1. (1)

The finite number of ends $\mathcal{E}(P)$ is related with the volume growth
by

$\operatorname{Sup}_{t>0}\frac{D_{t}(o)}{\operatorname{Vol}(B_{t}^{b,m})}\leq\mathcal{E}(P)$

where $D_{t}(o)=\\{x\in P:r(x)<t\\}=\\{x\in P:\varphi(x)\in B^{b,n}_{t}(o)\\}$
is the extrinsic ball of radius $t$ in $P$, (see Definition 2.1).

  2. (2)

If $P$ has only one end, $P$ is totally geodesic in $\mathbb{H}^{n}(b)$

When the ambient manifold is $\mathbb{R}^{n}$, we have the following
Bernstein-type result as in [12]:

######  Theorem 1.2.

Let $\varphi:P^{m}\longrightarrow\mathbb{R}^{n}$ be a complete non-compact,
minimal and proper immersion with $m>2$. Let us suppose that for sufficiently
large $R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e.
outside the compact extrinsic ball $D_{R_{0}}(o)$ with
$\varphi^{-1}(o)\neq\emptyset$),

$\|B^{P}_{x}\|\leq\frac{\epsilon(r(x))}{r(x)}$

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

  1. (1)

The finite number of ends $\mathcal{E}(P)$ is related with the volume growth
by

$\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,m})}\leq\mathcal{E}(P)$

  2. (2)

If $P$ has only one end, $P$ is totally geodesic in $\mathbb{R}^{n}$.

These results, that we shall prove in Section 8, (together the corollaries of
Section 4), follows from two main theorems, stablished in Section 3. In the
first (Theorem 3.1) we show that a complete isometric immersion
$\varphi:P^{m}\longrightarrow N^{n}$, ($m>2$), with controlled second
fundamental form in a complete Riemannian manifold which possess a pole and
has controlled radial sectional curvatures is proper and has finite topology.
In the second (Theorem 3.4) it is proved that a complete and proper isometric
immersion $\varphi:P^{m}\longrightarrow M^{n}_{w}$, ($m>2$), with controlled
second fundamental form in a radially symmetric space $M^{n}_{w}$ with
sectional curvatures bouded from below by a radial function has its volume
growth bounded from above by a quantity which involve its (finite) number of
ends.

The proof of both theorems follows basically the argumental lines of the
proofs given in [24] and [25] and some ideas in [3]. An important difference
to these results is that, on our side, we allow to the ambient manifold to
have positive sectional curvatures, bounding from above only the sectional
curvatures of the planes containing radial directions. However, to show the
properness of the immersion in [25], the ambient manifold must have non-
positive sectional curvatures, and to assure the finiteness of the topology of
the immersion $P$, this ambient manifold must be, in addition, simply
connected, (i.e. a Cartan-Hadamard manifold). This difference is based in
following considerations.

To obtain the finiteness of the topology in Theorem 3.1, we show that the
restricted, (to the submanifold) extrinsic distance to a fixed pole (in the
ambient manifold) has no critical points outside a compact and then, we apply
classical Morse theory. To show that the extrinsic distance function has no
critical points we compute its Hessian as we can find it in [16] and [27].
These results are, in its turn, based in the Jacobi-Index analysis for the
Hessian of the distance function given in [6], in particular, its Theorem A,
(see Subsection 2.3). This comparison theorem is different of the Hessian
comparison Theorem 1.2 used in [25]: while in this last theorem, the space
used as a model to compare is the real space form with constant sectional
curvature equal to the bound on the sectional curvatures of the given
Riemannian manifold, in our adaptation of Theorem A in [6], (see Theorem
2.10), only the sectional curvatures of the planes containing radial
directions from the pole are bounded by the corresponding radial sectional
curvatures in a radially symmetric space used as a model.

We also note at this point that although we use the definition of pole given
by Greene and Wu in [6], (namely, the exponential must be a diffeomorphism at
a pole), in fact, the comparison of the Hessians in Theorem A holds along
radial geodesics from the poles defined as those points which have not
conjugate points, as in [25].

###  1.1. Outline

The outline of the paper is the following. In Section §.2 we present the
definiton of extrinsic ball, together the basic facts about the Hessian
comparison theory of restricted distance function we are going to use and an
isoperimetric inequality for the extrinsic balls which plays an important rôle
in the proof of Theorem 3.4 . Section §.3 is devoted to the statement of the
main results (Theorem 3.1, Theorem 3.4 and Theorem 3.5). We shall present in
Section 4 two lists of results based in Theorems 3.1, 3.4 and 3.5: the first
set of consequences is devoted to bound from above the volume growth of a
submanifold by the number of its ends, in several contexts, obtaining moreover
some Bernstein-type results. In the second set of corollaries are stated some
compactification theorems for submanifolds in $\mathbb{R}^{n}$, in
$\mathbb{H}^{n}$ and in $\mathbb{H}^{n}\times\mathbb{R}^{l}$. Sections §.5,
§.6, §.7 are devoted to the proof of Theorems 3.1, 3.4, and 3.5, respectively.
Theorem 1.1, Theorem 1.2 and the corollaries stated in Section §.4 are proved
in Section §.8.

##  2\. Preliminaires

###  2.1. The extrinsic distance

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 $o\in N$, (this is the
precise meaning we shall give to the word submanifold along the text) . 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 $\varphi_{*_{x}}(X)\in T_{\varphi(x)}N$

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

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

######  Definition 2.1.

Given $\varphi:P^{m}\longrightarrow N^{n}$ an isometric immersion of a
complete and connected Riemannian $m$-manifold $P^{m}$ into a complete
Riemannian manifold $N^{n}$ with a pole $o\in N$, we denote the extrinsic
metric balls of radius $t>0$ and center $o\in N$ by $D_{t}(o)$. They are
defined as the subset of $P$:

$D_{t}(o)=\\{x\in P:r(\varphi(x))<t\\}=\\{x\in P:\varphi(x)\in
B^{N}_{t}(o)\\}$

where $B^{N}_{t}(o)$ denotes the open geodesic ball of radius $t$ centered at
the pole $o$ in $N^{n}$. Note that the set $\varphi^{-1}(o)$ can be the empty
set.

######  Remark 2.2.

When the imersion $\varphi$ is proper, the extrinsic domains $D_{t}(o)$ are
precompact sets, with smooth boundary $\partial D_{t}(o)$. The assumption on
the smoothness of $\partial D_{t}(o)$ makes no restriction. Indeed, the
distance function $r$ is smooth in $N-\\{o\\}$ since $N$ is assumed to possess
a pole $o\in N$. Hence the composition $r|_{P}$ is smooth in $P$ and
consequently the radii $t$ that produce smooth boundaries $\partial D_{t}(o)$
are dense in $\mathbb{R}$ by Sard’s theorem and the Regular Level Set Theorem.

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

######  Definition 2.3.

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

###  2.2. 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 2.4 (See [23], [10] and [6]).

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 2.5.

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 $Q_{b}(r)$ admits a smooth extension to
$r=\pi/\sqrt{b}$.

######  Proposition 2.6 (See Proposition 42 in Chapter 7 of [23]. See also [6]
and [10]).

Let $M_{w}^{m}$ be a $w-$model with warping function $w(r)$ and center
$o_{w}$. The distance sphere $S^{w}_{r}$ 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

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

and the sectional curvatures of $M_{w}^{m}$ at every $x\in\pi^{-1}(r)$ (for
$r>0$) of the tangent planes to the fiber $S^{w}_{r}$ are also all identical
and determined by

$K(r)=K_{M_{w}}(\Pi_{S^{w}_{r}})=\frac{1-(w^{\prime}(r))^{2}}{w^{2}(r)}.$

######  Remark 2.7.

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
$I\\!\\!K^{m}(b)$) . The latter two functions define the isoperimetric
quotient function as follows

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

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

######  Definition 2.8.

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.

######  Remark 2.9.

If $\,K_{w}(r)\geq-\eta_{w}^{2}(r)\,$ then $\,M_{w}^{m}\,$ is balanced from
above. If $\,K_{w}(r)\leq 0\,$ then $\,M_{w}^{m}\,$ is balanced from below,
see the paper [16] for a detailed list of examples.

###  2.3. Hessian comparison analysis

The 2.nd order analysis of the restricted distance function $r_{|_{P}}$
defined on manifolds with a pole is governed by the Hessian comparison Theorem
A in [6].

This comparison theorem can be stated as follows, when one of the spaces is a
model space $M^{m}_{w}$, (see [27]):

######  Theorem 2.10 (See [6], Theorem A).

Let $N=N^{n}$ be a manifold with a pole $o$, let $M=M_{w}^{m}$ denote a
$w-$model with center $o_{w}$. Suppose that every $o$-radial sectional
curvature at $x\in N\setminus\\{o\\}$ is bounded from above by the
$o_{w}$-radial sectional curvatures in $M_{w}^{m}$ as follows:

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

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

(2.2) $\displaystyle{{\rm Hess}\,}^{N}(r(x))(X,X)$ $\displaystyle\,\geq\,{{\rm
Hess}\,}^{M}(r(y))(Y,Y)$
$\displaystyle=\eta_{w}(r)\left(\|X\|^{2}-\langle\nabla^{M}r(y),Y\rangle_{M}^{2}\right)$
$\displaystyle=\eta_{w}(r)\left(\|X\|^{2}-\langle\nabla^{N}r(x),X\rangle_{N}^{2}\right)$

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

######  Remark 2.11.

As we mentioned in the Introduction, inequality (2.2) is true along the
geodesics emanating from $o$ and $o_{w}$ which are free of conjugate points of
$o$ and $o_{w}$, (see Remark 2.3 in [6]). Other relevant observation is that
the bound given in inequality (2.2) does not depend on the dimension of the
model space, (see Remark 3.7 in [27]).

We present now a technical result concerning the Hessian of a radial function,
namely, a function which only depends on the distance function $r$. For the
proof of this result, and the rest of the results in this subsection, we refer
to the paper [27].

######  Proposition 2.12.

Let $N=N^{n}$ be a manifold with a pole $o$. Let
$r=r(x)=\operatorname{dist}_{N}(o,x)$ be the distance from $o$ to $x$ in $N$.
Let $F:\mathbb{R}\longrightarrow\mathbb{R}$ a smooth function. Then, given
$q\in N$ and $X,Y\in T_{q}N$,

(2.3) $\begin{split}{{\rm Hess}\,}^{N}F\circ
r|_{q}(X,Y)&=F^{\prime\prime}(r)(\nabla^{N}r\otimes\nabla^{N}r)(X,Y)\\\
&+F^{\prime}(r){{\rm Hess}\,}^{N}r|_{q}(X,Y)\end{split}$

Now, let us consider a complete isometric immersion
$\varphi:P^{m}\longrightarrow N$ in a Riemannian ambient manifold $N^{n}$ with
pole $o$, and with distance function to the pole $r$. We are going to see how
the Hessians (in $P$ and in $N$), of a radial function defined in the
submanifold are related via the second fundamental form $B^{P}$ of the
submanifold $P$ in $N$. As before, we identify, given any $q\in P$, the
tangent vectors $X\in T_{q}P$ with $\varphi_{*_{q}}X\in T\varphi(q)N$ along
the next results.

######  Proposition 2.13.

Let $N^{n}$ be a manifold with a pole $o$, and let us consider an isometric
immersion $\varphi:P^{m}\longrightarrow N$. If $r|_{P}$ is the extrinsic
distance function, then, given $q\in P$ and $X,Y\in T_{q}P$,

(2.4) ${{\rm Hess}\,}^{P}r|_{q}(X,Y)={{\rm
Hess}\,}^{N}r|_{\varphi(q)}(X,Y)+\langle
B^{P}_{q}(X,Y),\nabla^{N}r|_{q}\rangle$

where $B^{P}_{q}$ is the second fundamental form of $P$ in $N$ at the point
$q\in P$.

Now, we apply Proposition 2.12 to $F\circ r|_{P}=F\circ r\circ\varphi$,
(considering $P$ as the Riemannian manifold where the function is defined), to
obtain an expression for ${{\rm Hess}\,}^{P}F\circ r|_{P}(X,Y)$ . Then, let us
apply Proposition above to ${{\rm Hess}\,}^{P}r|_{P}(X,Y)$, and we finally
get:

######  Proposition 2.14.

Let $N=N^{n}$ be a manifold with a pole $o$, and let $P^{m}$ denote an
immersed submanifold in $N$. Let $r|_{P}$ be the extrinsic distance function.
Let $F:\mathbb{R}\longrightarrow\mathbb{R}$ be a smooth function. Then, given
$q\in P$ and $X,Y\in T_{q}P$,

(2.5) $\begin{split}{{\rm Hess}\,}^{P}F\circ
r|_{q}(X,Y)&=F^{\prime\prime}(r(q))\langle\,\nabla^{N}r|_{q},X\,\rangle\langle\,\nabla^{N}r|_{q},Y\,\rangle\\\
&+F^{\prime}(r(q))\\{{{\rm Hess}\,}^{N}r|_{q}(X,Y)\\\
&+\langle\nabla^{N}r|_{q},B^{P}_{q}(X,Y)\,\rangle\,\\}\end{split}$

###  2.4. Comparison constellations and Isoperimetric inequalities

The isoperimetric inequalities satisfied by the extrinsic balls in minimal
submanifolds are on the basis of the monotonicity of the volume growth
function $f(r)=\frac{Vol(D_{r})}{Vol(B_{r}^{w})}$, a key result to prove
Theorem 1.1. We have the following theorem.

######  Theorem 2.15 (See [16], [17], [18], [19] and [26]).

Let $\varphi:P^{m}\longrightarrow N^{n}$ be a complete, proper and minimal
immersion in an ambient Riemannian manifold $N^{n}$ which possess at least one
pole $o\in N$. Let us suppose that the $o-$radial sectional curvatures of $N$
are bounded from above by the $o_{w}-$radial sectional curvatures of the
$w-$model space $M_{w}^{m}$:

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

and assume that $M^{m}_{w}$ is balanced from below. Let $D_{r}$ be an
extrinsic $r$-ball in $P^{m}$, with center at a pole $o\in N$ in the ambient
space $N$. Then:

(2.6) $\frac{\operatorname{Vol}(\partial
D_{r})}{\operatorname{Vol}(D_{r})}\geq\frac{\operatorname{Vol}(S^{w}_{r})}{\operatorname{Vol}(B^{w}_{r})}\,\,\,\,\,\textrm{for
all}\,\,\,r>0\quad.$

Furthermore, if $\varphi^{-1}(o)\neq\emptyset$,

(2.7)
$\operatorname{Vol}(D_{r})\geq\operatorname{Vol}(B^{w}_{r})\,\,\,\,\textrm{for
all}\,\,\,r>0\quad.$

Moreover, if equality in inequalities (2.6) or (2.7) holds for some fixed
radius $R$ and if the balance of $M^{m}_{w}$ from below is sharp
$q_{w}(r)\,\eta_{w}(r)\,>\,1/m\,$ for all $r$, then $D_{R}$ is a minimal cone
in the ambient space $N^{n}$, so if $N^{n}$ is the hyperbolic space
$\,\mathbb{H}^{n}(b)\,$, $\,b<0\,$, then $P^{m}\,$ is totally geodesic in
$\mathbb{H}^{n}(b)$.

If, on the other hand, the ambient space is $\mathbb{R}^{n}$ and equality in
inequalities (2.6) or (2.7) holds for all radius $r>0$ then $P^{m}$ is totally
geodesic in $\mathbb{R}^{n}$.

On the other hand, and also as a consequence of inequality (2.6), the volume
growth function $f(r)=\frac{Vol(D_{r})}{Vol(B_{r}^{w})}$ is a non-decreasing
function of $r$.

##  3\. Main Results

We prove in this section our main results, stablishing a set of conditions
that assures that our submanifolds are properly immersed and have finite
topology and bounding from below, under certain conditions, the number of its
ends.

######  Theorem 3.1.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric immersion of a
complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete
Riemannian manifold $N^{n}$ with a pole $o\in N$ and satisfying
$\varphi^{-1}(o)\neq\emptyset$. Let us suppose that:

  1. (1)

The $o-$radial sectional curvatures of $N$ are bounded from above by the
$o_{w}-$radial sectional curvatures of the $w-$model space $M_{w}^{m}$:

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

  2. (2)

The second fundamental form $B^{P}_{x}$ in $x\in P$ satisfies that, for
sufficiently large radius $R_{0}$, and for some constant $c\in]0,1[$:

$\|B^{P}_{x}\|\leq c\,\eta_{w}(\rho^{P}(x))\,\,\,\forall x\in
P-B^{P}_{R_{0}}(x_{o})$

where $\rho^{P}(x)$ denotes the intrinsic distance in $P$ from some fixed
$x_{o}\in\varphi^{-1}(o)$ to $x$.

  3. (3)

For any $r>0$, $w^{\prime}(r)\geq d>0$ and $(\eta_{w}(r))^{\prime}\leq 0$.

Then $P$ is properly immersed in $N$ and it is $C^{\infty}$\- diffeomorphic to
the interior of a compact smooth manifold $\overline{P}$ with boundary.

######  Remark 3.2.

To show that $\varphi$ is proper, we shall use Theorem 2.10. Hence, it is
enough to assume that $o$ is a pole in the sense that there are not conjugate
points along any geodesic emanating from $o$, (see [5] and [30]). Therefore
our statement about the properness of the immersion includes ambient manifolds
$N$ that admit non-negative sectional curvatures, unlike the ambient manifold
in Theorem 1.2 in [25]. On the other hand, to prove the finiteness of the
topology of $P$ we need to assume that the ambient manifold $N$ posses a pole
as it is defined in [6], namely, a point $p\in N$ where $exp_{p}$ is a
$C^{\infty}$ diffeomorphism. However, although our ambient manifold must be
diffeomorphic to $\mathbb{R}^{n}$ in this case, (as in Theorem 1.2 in [25],
where the ambient space must be a Cartan-Hadamard manifold), also admits non-
negative sectional curvatures.

To complete the benchmarking with the hypotheses in [24] and [25], we are
going to compare the assumptions (2) and (3) in Theorem 3.1 with the notion of
“submanifold with tamed second fundamental form” introduced in [24]. It is
straightforward to check that if $\varphi:P^{m}\longrightarrow N^{n}$ is an
immersion of a complete Riemannian $m$\- manifold $P^{m}$ into a complete
Riemannian manifold $N^{n}$ with sectional curvatures $K_{N}\leq b\leq 0$, and
$P$ has tamed second fundamental form, in the sense of Definition 1.1 in [25],
then there exists $R_{0}>0$ such that for all $r\geq R_{0}$, the quantity

$a_{r}:=\operatorname{Sup}\\{\frac{w_{b}}{w_{b}^{\prime}}(\rho^{P}(x))\|B^{P}_{x}\|:x\in
P-B^{P}_{r}\\}$

satisfies $a_{r}<1$.

Hence, taking $r=R_{0}$, we have that for all $x\in P-B^{P}_{R_{0}}$, and some
$c\in(0,1)$,

$\|B^{P}_{x}\|\leq c\eta_{w_{b}}(\rho^{P}(x))\,.$

On the other hand, when $b\leq 0$, then $w_{b}^{\prime}(r)\geq 1>0\,\,\forall
r>0$ and $(\eta_{w_{b}}(r))^{\prime}\leq 0\,\,\forall r>0$.

All these observations make us consider our Theorem 3.1 as a natural and
slight generalization of assertions (b) and (c) of Theorem 1.2 in [25].

Observe that if we assume the properness of the immersion we obtain the
following version of Theorem 3.1, where we can remove the hypothesis about the
decrease of the function $\eta_{w}(r)$ because the norm of the second
fundamental form $\|B^{P}_{x}\|$ is bounded by the value of $\eta_{w}$ at
$r(x)$ instead of $\rho^{P}(x)$ :

######  Theorem 3.3.

Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric and proper 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, as in Theorem 3.1, the
$o-$radial sectional curvatures of $N$ are bounded from above as

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

and for any $r>0$, $w^{\prime}(r)\geq d>0$. Let us assume moreover that the
second fundamental form $B^{P}_{x}$ in $x\in P$ satisfies that, for
sufficiently large radius $R_{0}$:

$\|B^{P}_{x}\|\leq c\,\eta_{w}(r(x))\,\,\,\forall x\in P-D_{R_{0}}(o)$

where $c$ a positive constant such that $c<1$ .

Then $P$ is $C^{\infty}$\- diffeomorphic to the interior of a compact smooth
manifold $\overline{P}$ with boundary.

We are going to see how to estimate the area growth function of $P$, defined
as $g(r)=\frac{Vol(\partial D_{r})}{Vol(S_{r}^{w})}$ by the number of ends of
the immersion $P$, $\mathcal{E}(P)$, when the ambient space $N$ is a radially
symmetric space.

######  Theorem 3.4.

Let $\varphi:P^{m}\longrightarrow M^{n}_{w}$ be an isometric and proper
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
model space $M^{n}_{w}$ with pole $o_{w}$. Suppose that
$\varphi^{-1}(o_{w})\neq\emptyset$, $m>2$ and moreover:

  1. (1)

The norm of second fundamental form $B^{P}_{x}$ in $x\in P$ is bounded from
above outside a (compact) extrinsic ball $D_{R_{0}}(o)\subseteq P$ with
sufficiently large radius $R_{0}$ by:

$\|B^{P}_{x}\|\,\leq\,\frac{\epsilon(r(x))}{(w^{\prime}(r(x)))^{2}}\eta_{w}(r(x))\,\,\,\forall
x\in P-D_{R_{0}}$

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

  2. (2)

For $r$ sufficiently large, $w^{\prime}(r)\geq d>0$.

Then, for sufficiently large $r$, we have:

(3.1) $\frac{Vol(\partial
D_{r})}{Vol(S_{r}^{w})}\leq\frac{\mathcal{E}(P)}{\left(1-4\epsilon(r)\right)^{\frac{(m-1)}{2}}}$

where $\mathcal{E}(P)$ is the (finite) number of ends of $P$.

When we consider minimal immersions in the model spaces, we have the following
result, which is an inmediate corollary from the above theorem, and Theorem
2.15 in Section 2.

######  Theorem 3.5.

Let $\varphi:P^{m}\longrightarrow M^{n}_{w}$ be a complete non-compact, proper
and minimal immersion into a ballanced from below model space $M^{n}_{w}$ with
pole $o_{w}$. Suppose that $\varphi^{-1}(o_{w})\neq\emptyset$ and $m>2$. Let
us assume moreover the hypotheses (1) and (2) in Theorem 3.4.

Then

  1. (1)

The (finite) number of ends $\mathcal{E}(P)$ is related with the (finite)
volume growth by

(3.2)
$1\leq\lim_{r\to\infty}\frac{Vol(D_{r})}{Vol(B_{r}^{w})}\leq\mathcal{E}(P)$

  2. (2)

If $P$ has only one end, P is a minimal cone in $M_{w}^{n}$.

##  4\. Corollaries

As we have said in the Introduction, we have divided the list of results based
in Theorem 3.1 and in Theorem 3.4 in two series of corollaries. The first set
of consequences follows the line of Theorem 1.1 and Theorem 1.2, (which are in
fact the main representatives of these results) presenting upper bounds for
the volume and area growth of a complete and proper immersion in the real
space form $I\\!\\!K^{n}(b)$, ($b\leq 0$), in terms of the number of its ends.
In the second set of corollaries, are stated compactification theorems for
complete and proper immersions in $\mathbb{R}^{n}$, $\mathbb{H}^{n}(b)$ and
$\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$.

The first of these corollaries constitutes a non-minimal version of Theorem
1.1:

######  Corollary 4.1.

Let $\varphi:P^{m}\longrightarrow\mathbb{H}^{n}(b)$ be a complete non-compact
and proper immersion with $m>2$. Let us suppose that for sufficiently large
$R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e. outside the
compact extrinsic ball $D_{R_{0}}(o)$ with $\varphi^{-1}(o)\neq\emptyset$),

$\|B^{P}_{x}\|\leq\frac{\delta(r(x))}{e^{2\sqrt{-b}\,r(x)}}$

where $r(x)=d_{\mathbb{H}^{n}(b)}(o,\varphi(x))$ is the (extrinsic) distance
in $\mathbb{H}^{n}(b)$ of the points in $\varphi(P)$ to a fixed pole
$o\in\mathbb{H}^{n}(b)$ and $\delta(r)$ is a smooth function such that
$\delta(r)\to 0$ when $r\to\infty$.Let $\\{t_{i}\\}_{i=1}^{\infty}$ be any
non-decreasing sequence such that $t_{i}\to\infty$ when $i\to\infty$. Then the
finite number of ends $\mathcal{E}(P)$ is related with the area growth of $P$
by:

$\liminf_{i\to\infty}\frac{\operatorname{Vol}(\partial
D_{t_{i}})}{\operatorname{Vol}(S_{t_{i}}^{b,m-1})}\leq\mathcal{E}(P)$

The corresponding non-minimal statement of Theorem 1.2 is:

######  Corollary 4.2.

Let $\varphi:P^{m}\longrightarrow\mathbb{R}^{n}$ be a complete non-compact and
proper immersion with $m>2$. Let us suppose that for sufficiently large
$R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e. outside the
compact extrinsic ball $D_{R_{0}}(o)$ with $\varphi^{-1}(o)\neq\emptyset$),

$\|B^{P}_{x}\|\leq\frac{\epsilon(r(x))}{r(x)}$

where $r(x)=d_{\mathbb{R}^{n}}(o,\varphi(x))$ is the (extrinsic) distance in
$\mathbb{R}^{n}$ of the points in $\varphi(P)$ to a fixed pole
$o\in\mathbb{R}^{n}$ and $\epsilon(r)$ is a smooth function such that
$\epsilon(r)\to 0$ when $r\to\infty$. Let $\\{t_{i}\\}_{i=1}^{\infty}$ be any
non-decreasing sequence such that $t_{i}\to\infty$ when $i\to\infty$. Then the
finite number of ends $\mathcal{E}(P)$ is related with the area growth by:

$\liminf_{i\to\infty}\frac{\operatorname{Vol}(\partial
D_{t_{i}})}{\operatorname{Vol}(S_{t_{i}}^{0,m-1})}\leq\mathcal{E}(P)$

Concerning the compactification results we have the following result given by
Bessa, Jorge and Montenegro in [24] and by Bessa and Costa in [25]:

######  Corollary 4.3.

Let $\varphi:P^{m}\longrightarrow I\\!\\!K^{n}(b)$ be a complete non-compact
immersion in the real space form $I\\!\\!K^{n}(b)$, ($b\leq 0$). Let us
suppose that for all points $x\in P\setminus B^{P}_{R_{0}}(o)$ (for
sufficientlty large $R_{0}$, where $o$ is a pole in $I\\!\\!K^{n}(b)$ such
that $\varphi^{-1}(o)\neq\emptyset$) :

$\|B^{P}_{x}\|\leq c\,h_{b}(\rho^{P}(x))$

where $\rho^{P}(x)$ is the (intrinsic) distance to a fixed
$x_{o}\in\varphi^{-1}(o)$ and $c$ is a positive constant such that $c<1$ and

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

is the mean curvature of the geodesic spheres in $I\\!\\!K^{n}(b)$. Then $P$
is properly immersed in $I\\!\\!K^{n}(b)$ and it is diffeomorphic to the
interior of a compact smooth manifold $\overline{P}$ with boundary.

Our last result concerns isometric immersions in
$\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$:

######  Corollary 4.4.

Let $\varphi:P^{m}\longrightarrow\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ be a
complete non-compact immersion. Let us consider a pole
$o\in\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ such that
$\varphi^{-1}(o)\neq\emptyset$. Let us suppose that for all points $x\in
P\setminus B^{P}_{R_{0}}(x_{o})$, where $x_{o}\in\varphi^{-1}(o)$ and for
$R_{0}$ sufficiently large:

$\|B_{x}\|\leq\frac{c}{\rho^{P}(x)}\,\,.$

Here $\rho^{P}(x)$ denotes the intrinsic distance in $P$ from the fixed
$x_{o}\in\varphi^{-1}(o)$ to $x$ and $c$ is a positive constant such that
$c<1$. Then $P$ is properly immersed in
$\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ and it is diffeomorphic to the
interior of a compact smooth manifold $\overline{P}$ with boundary.

##  5\. Proof of Theorem 3.1

###  5.1. $P$ is properly immersed

Let us define the following function:

(5.1) $F(r):=\int_{0}^{r}w(t)dt$

Observe that $F$ is injective, because $F^{\prime}(r)=w(r)>0\,\,\,\forall
r>0$, and $F(r)\to\infty$ when $r\to\infty$. Applying Theorem 2.10 and
Proposition 2.14, we obtain, for all $x\in P$, and given $X\in T_{x}P$,

(5.2) $\displaystyle{{\rm Hess}\,}^{P}_{x}F(r)(X,X)$ $\displaystyle\geq
w^{\prime}(r(x))\|X\|^{2}+w(r(x))\langle B^{P}_{x}(X,X),\nabla^{N}r\rangle$
$\displaystyle\geq
w^{\prime}(r(x))\|X\|^{2}-w(r(x))\|B^{P}_{x}\|\,\,\|X\|^{2}$

By hypotesis there exist a geodesic ball $B^{P}_{r_{1}}(x_{0})$ in $P$, with
$r_{1}\geq R_{0}$, such that for any $x\in P\setminus B^{P}_{r_{1}}(x_{0})$,
$\|B^{P}_{x}\|\ \leq c\eta_{w}(\rho^{P}(x))$. On the other hand, as
$\eta_{w}(r)$ is non-increasing and $r(x)\leq\rho^{P}(x)$ because $\varphi$ is
isometric, we have $c\eta_{w}(\rho^{P}(x))\leq c\eta_{w}(r(x))$, so if $x\in
P\setminus B^{P}_{r_{1}}$ :

(5.3) $\displaystyle{{\rm Hess}\,}^{P}_{x}F(r)(X,X)$ $\displaystyle\geq
w^{\prime}(r(x))\|X\|^{2}-w(r)c\eta_{w}(\rho^{P}(x))\,\|X\|^{2}$
$\displaystyle\geq w^{\prime}(r(x))\|X\|^{2}\left(1-c\right)\geq
d\left(1-c\right)>0$

The above result implies that there exists $r_{1}\geq R_{0}$ such that $F\circ
r$ is a strictly convex function outside the geodesic ball in $P$ centered at
$x_{0}$, $B^{P}_{r_{1}}(x_{0})$. And hence, as $r(x)\leq\rho^{P}(x)$ for all
$x\in P$, (and therefore $B^{P}_{r_{1}}(x_{0})\subseteq D_{r_{1}}$), $F\circ
r$ is a strictly convex function outside the extrinsic disc $D_{r_{1}}$.

Let $\sigma:[0,\rho^{P}(x)]\to P^{m}$ be a minimizying geodesic from $x_{0}$
to $x$.

If we denote as $f=F\circ r$, let us define $h:\mathbb{R}\to\mathbb{R}$ as

$h(s)=F(r(\sigma(s)))=f(\sigma(s))$

Then,

(5.4)
$(f\circ\sigma)^{\prime}(s)=h^{\prime}(s)=\sigma^{\prime}(s)(f)=\langle\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle$

and hence,

(5.5) $\displaystyle(f\circ\sigma)^{\prime\prime}(s)$
$\displaystyle=h^{\prime\prime}(s)=\sigma^{\prime}(s)(\langle\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle)=\langle\nabla^{P}_{\sigma^{\prime}(s)}\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle$
$\displaystyle+\langle\nabla^{P}f(\sigma(s)),\nabla^{P}_{\sigma^{\prime}(s)}\sigma^{\prime}(s)\rangle=Hess^{P}_{\sigma(s)}f(\sigma(s))(\sigma^{\prime}(s),\sigma^{\prime}(s))$

We have from (5.3) that $(f\circ\sigma)^{\prime\prime}(\tau)={{\rm
Hess}\,}^{P}f(\sigma(\tau))(\sigma^{\prime},\sigma^{\prime})\geq d(1-c)$ for
all $\tau\geq r_{1}$ . And for $\tau<r_{1}$,
$(f\circ\sigma)^{\prime\prime}(\tau))\geq a=\inf_{x\in B^{P}_{r_{1}}}\\{{{\rm
Hess}\,}^{P}f(x)(\nu,\nu),\,|\nu|=1\\}$. Then

$\displaystyle(f\circ\sigma)^{\prime}(s)$ $\displaystyle=$
$\displaystyle(f\circ\sigma)^{\prime}(0)+\int_{0}^{s}(f\circ\sigma)^{\prime\prime}(\tau)d\tau$
$\displaystyle\geq$
$\displaystyle(f\circ\sigma)^{\prime}(0)+\int_{0}^{r_{1}}a\,d\tau+d\,\int_{r_{1}}^{s}(1-c)d\tau$
$\displaystyle\geq$
$\displaystyle(f\circ\sigma)^{\prime}(0)+a\,r_{1}+d\,(1-c)(s-r_{1})$

On the other hand, as

(5.7)
$\nabla^{P}f(\sigma(s))=\nabla^{P}F(r(\sigma(s)))=F^{\prime}(r(\sigma(s)))\nabla^{P}r|_{\sigma(s)}=w(r(\sigma(s)))\nabla^{P}r|_{\sigma(s)}$

then

$\nabla^{P}f(\sigma(0))=w(r(\sigma(0)))\nabla^{P}r|_{\sigma(0)}=w(0)\nabla^{P}r|_{\sigma(0)}=0$

so we have that

(5.8)
$(f\circ\sigma)^{\prime}(0)=\langle\nabla^{P}f(\sigma(0)),\sigma^{\prime}(0)\rangle=0$

We also have that $(f\circ\sigma)(0)=F(r(\sigma(0)))=F(0)=0$. Hence, applying
inequality (5.1),

(5.9)
$f(\sigma(s))=(f\circ\sigma)(0)+\int_{0}^{s}(f\circ\sigma)^{\prime}(\tau)d\tau\geq
ar_{1}s+d(1-c)\\{\frac{1}{2}s^{2}-r_{1}s\\}$

Therefore,

$\displaystyle F(r(x))$ $\displaystyle=$ $\displaystyle
f(x)=f(\sigma(\rho^{P}(x)))=\int_{0}^{\rho^{P}(x)}(f\circ\sigma)^{\prime}(s)\,ds$
$\displaystyle\geq$
$\displaystyle\int_{0}^{\rho^{P}(x)}a\,r_{1}+d\,(1-c)(s-r_{1})\,ds$
$\displaystyle=$ $\displaystyle a\,r_{1}\rho^{P}(x)\
+d\,(1-c)\left(\frac{\rho^{P}(x)^{2}}{2}-r_{1}\,\rho^{P}(x)\right)$

Hence, if $\rho^{P}\to\infty$ then $F(r(x))\to\infty$ and then, as $F$ is
strictly increasing, $r\to\infty$ so the immersion is proper.

###  5.2. $P$ has finite topology

We are going to see that $\nabla^{P}r$ never vanishes on $P\setminus
D_{r_{1}}$. To show this, we consider, as in the previous subsection, any
geodesic in $P$ emanating from the pole $o$, $\sigma(s)$. We have, using
inequality (5.1), that

(5.11)
$\langle\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle=(f\circ\sigma)^{\prime}(s)\geq
a\,r_{1}+d\,(1-c)(s-r_{1})>0\,\,\forall s>r_{1}$

Hence, as $\|\sigma^{\prime}(s)\|=1\,\,\forall s$, then
$\|\nabla^{P}f(\sigma(s))\|>0$ for all $s>r_{1}$. But we have computed
$\nabla^{P}f(\sigma(s))=w(r(\sigma(s)))\nabla^{P}r|_{\sigma(s)}$, so, as
$w(r)>0\,\,\forall r>0$, then $\|\nabla^{P}r|_{\sigma(s)}\|>0\,\,\forall
s>r_{1}$ and hence, $\nabla^{P}r|_{\sigma(s)}\neq 0\,\,\forall s>r_{1}$. We
have proved that $\nabla^{P}r$ never vanishes on $P\setminus B^{P}_{r_{1}}$,
so we have too that $\nabla^{P}r$ never vanishes on $P\setminus D_{r_{1}}$.
Let

$\phi:\partial D_{r_{1}}\times[r_{1},+\infty)\to P\setminus D_{r_{1}}$

be the integral flow of a vector field
$\frac{\nabla^{P}r}{\|\nabla^{P}r\|^{2}}$ with

$\phi(p,r_{1})=p\in\partial D_{r_{1}}$

It is obvious that $r(\phi(p,t))=t$ and

$\phi(\cdot,t):\partial D_{r_{1}}\to\partial D_{t}$

is a diffeomorphism. So $P$ has finitely many ends, and each of its ends is of
finite topological type.

In fact, applying Theorem 3.1 in [20], we conclude that, as the extrinsic
annuli $A_{r_{1},R}(o)=D_{R}(o)\setminus D_{r_{1}}(o)$ contain no critical
points of the extrinsic distance function $r:P\longrightarrow\mathbb{R}^{+}$,
then $D_{R}(o)$ is diffeomorphic to $D_{r_{1}}(o)$ for all $R\geq r_{1}$ and
hence the annuli $A_{r_{1},R}(o)$ are diffeomorphic to $\partial
D_{r_{1}}\times[r_{1},R]$.

######  Remark 5.1.

To show Theorem 3.3, we argue as in the beginning of the proof of Theorem 3.1:
with the same function $F(r)$ we obtain inequality (5.2). But now we have as
hypothesis that $\|B^{P}_{x}\|\leq c\,\eta_{w}(r(x))$, so we don’t need that
$\eta_{w}^{\prime}(r)\leq 0$ to get inequality (5.3).

##  6\. Proof of Theorem 3.4

We are going to see first that $P$ has finite topology. As $P$ is properly
immersed, we shall apply Theorem 3.3 and for that, it must be checked that
hypotheses in that theorem are acomplished. First, we have hypothesis (1) in
Theorem 3.3 because $N=M^{n}_{w}$. On the other hand, as $w^{\prime}(r)\geq
d>0\forall r>0$ and, for some $R_{0}$, we have that
$\|B^{P}_{x}\|\leq\frac{\epsilon(r(x))}{(w^{\prime}(r(x)))^{2}}\eta_{w}(r(x))\,\,\,\forall
x\in P-D_{R_{0}}$ where $\epsilon$ is a positive function such that
$\epsilon(r)\to 0$ when $r\to\infty$, hence
$0\leq\lim_{r\to\infty}\frac{\epsilon(r)}{(w^{\prime}(r))^{2}}\leq\lim_{r\to\infty}\frac{\epsilon(r)}{d^{2}}=0$.
Therefore, for some constant $c<1$, there exist $R_{0}$ such that
$\|B^{P}_{x}\|\leq c\eta_{w}(r(x))\,\,\,\forall x\in P-D_{R_{0}}$. Therefore,
as $\varphi:P\longrightarrow M^{n}_{w}$ is a proper immersion, we have by
Theorem 3.3 that $P$ has finite topological type and thus $P$ has finitely
many ends, each of finite topological type. Hence we have, in an analogous way
than in [1], and for $r_{1}\geq R_{0}$ as in Section 5:

(6.1) $P-D_{r_{1}}=\cup_{k=1}^{\mathcal{E}(P)}V_{k}$

where $V_{k}$ are disjoint, smooth domains in $P$. Along the rest of the
proof, we will work on each end $V_{k}$ separately. Let $V$ denote one element
of the family $\\{V_{k}\\}_{k=1}^{\mathcal{E}(P)}$, and, given a fixed radius
$t>r_{1}$, let $\partial V(t)$ denote the set $\partial V(t)=V\cap\partial
D_{t}=V\cap S^{w}_{t}$, where $S^{w}_{t}$ is the geodesic $t$-sphere in
$M^{n}_{w}$. This set is a hypersurface in $P^{m}$, with normal vector
$\frac{\nabla^{P}r}{\|\nabla^{P}r\|}$, and we are going to estimate its
sectional curvatures when $t\to\infty$.

Suppose that $e_{i},e_{j}$ are two orthonormal vectors of $T_{p}\partial V(t)$
on the point $p\in\partial V(t)$. Then the sectional curvature of the plane
expanded by $e_{i},e_{j}$ is, using Gauss formula:

(6.2) $\displaystyle K_{\partial V(t)}$
$\displaystyle(e_{i},e_{j})=K_{P}(e_{i},e_{j})+\langle B^{\partial
V-P}(e_{i},e_{i}),B^{\partial V-P}(e_{j},e_{j})\rangle$
$\displaystyle-\|B^{\partial
V-P}(e_{i},e_{j})\|^{2}=K_{N}(e_{i},e_{j})+\langle B^{\partial
V-P}(e_{i},e_{i}),B^{\partial V-P}(e_{j},e_{j})\rangle$
$\displaystyle-\|B^{\partial V-P}(e_{i},e_{j})\|^{2}+\langle
B^{P}(e_{i},e_{i}),B^{P}(e_{j},e_{j})\rangle-\|B^{P}(e_{i},e_{j})\|^{2}$
$\displaystyle\geq K_{N}(e_{i},e_{j})+\langle B^{\partial
V-P}(e_{i},e_{i}),B^{\partial V-P}(e_{j},e_{j})\rangle$
$\displaystyle-\|B^{\partial V-P}(e_{i},e_{j})\|^{2}-2\|B^{P}\|^{2}$

where $B^{\partial V-P}$ is the second fundamental form of $\partial V(t)$ in
$P$. But this second fundamental form is for two vector fields $X,Y$ in
$T\partial V(t)$:

(6.3) $\displaystyle B^{\partial V-P}(X,Y)$
$\displaystyle=\langle\nabla_{X}^{P}Y,\frac{\nabla^{P}r}{||\nabla^{P}r||}\rangle\frac{\nabla^{P}r}{||\nabla^{P}r||}=\langle\nabla_{X}^{P}Y,\nabla^{P}r\rangle\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}$
$\displaystyle=X(\langle
Y,\nabla^{P}r\rangle)\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}-\langle
Y,\nabla^{P}_{X}\nabla^{P}r\rangle\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}$
$\displaystyle=-{{\rm
Hess}\,}^{P}r(X,Y)\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}$

Then, since, for all $X,Y\in T_{p}M^{n}_{w}$

(6.4) ${{\rm Hess}\,}^{M^{n}_{w}}r(X,Y)=\eta_{w}(r)\langle X,Y\rangle-\langle
X,\nabla^{M^{n}_{w}}r\rangle\langle Y,\nabla^{M^{n}_{w}}r\rangle$

we have, (using the fact that $e_{i}$ are tangent to the fiber $S^{w}_{t}$,
and Proposition 2.6), that

(6.5) $K_{M^{n}_{w}}(e_{i},e_{j})=K(t)=\frac{1}{w^{2}(t)}-\eta^{2}_{w}(t)$

so for any $p\in\partial V(t)$ such that $t=r(p)$ is sufficiently large:

(6.6) $\displaystyle K_{\partial V(t)}(e_{i},e_{j})\geq$ $\displaystyle
K_{M^{n}_{w}}(e_{i},e_{j})+\frac{{{\rm Hess}\,}^{P}_{p}r(e_{i},e_{i}){{\rm
Hess}\,}^{P}_{p}r(e_{j},e_{j})}{||\nabla^{P}r||^{2}}$
$\displaystyle-\frac{{{\rm
Hess}\,}^{P}_{p}r(e_{i},e_{j})^{2}}{||\nabla^{P}r||^{2}}-2\|B^{P}\|^{2}$
$\displaystyle\geq
K(t)+\frac{\left(\eta_{w}(t)-\|B^{P}\|\right)^{2}-\|B^{P}\|^{2}}{||\nabla^{P}r||^{2}}-2\|B^{P}\|^{2}$
$\displaystyle\geq\eta^{2}_{w}(t)\left(1-2\frac{\|B^{P}\|}{\eta_{w}(t)}-2\left(\frac{\|B^{P}\|}{\eta_{w}(t)}\right)^{2}+\frac{K(t)}{\eta^{2}_{w}(t)}\right)$
$\displaystyle\geq\eta^{2}_{w}(t)\left(1-4\frac{\|B^{P}\|}{\eta_{w}(t)}+\frac{K(t)}{\eta^{2}_{w}(t)}\right)$
$\displaystyle=\eta^{2}_{w}(t)\left(1+\frac{K(t)}{\eta^{2}_{w}(t)}\right)\left(1-4\frac{\frac{\|B^{P}\|}{\eta_{w}(t)}}{1+\frac{K(t)}{\eta^{2}_{w}(t)}}\right)$
$\displaystyle\geq\frac{1}{w^{2}(t)}\left(1-4\|B^{P}\|w^{\prime}(t)w(t)\right)\geq\frac{1}{w^{2}(t)}\left(1-4\epsilon(t)\right)$

where we recall that, by hypothesis,
$\|B^{P}\|\leq\frac{\epsilon(t)}{(w^{\prime}(t))^{2}}\eta_{w}(t)$ for all
$t=r(x)>R_{0}$, and $\epsilon$ is a positive function such that
$\epsilon(r)\to 0$ when $r\to\infty$.

If we denote as $\delta(t)=\frac{1}{w^{2}(t)}\left(1-4\epsilon(t)\right)$ we
have for each $t$ sufficiently large that $K_{\partial
V(t)}(e_{i},e_{j})\geq\delta(t)$ holds everywhere on $\partial V(t)$ and
$\delta(t)$ is a positive constant. Then, the Ricci curvature of $\partial
V(t)$ is bounded from below, for these sufficiently large radius $t$ as

$Ricc_{\partial V(t)}(\xi,\xi)\geq\delta(t)(m-1)\|\xi\|^{2}>0\,\,\forall\xi\in
T\partial V(t)$

so, applying Myers’ Theorem $\partial V(t)$ is compact and has diameter
$d(\partial V(t))\leq\frac{\pi}{\sqrt{\delta(t)}}$ (see [30]). Applying on the
other hand Bishop’s Theorem, (see Theorem 6 in [2]), we obtain:

(6.7) $\displaystyle\operatorname{Vol}(\partial
V(t))\leq\frac{\operatorname{Vol}(S^{0,m-1}(1))}{\sqrt{\delta(t)^{m-1}}}$

and hence

(6.8) $\displaystyle\frac{\operatorname{Vol}(\partial
V(t))}{\operatorname{Vol}(S_{t}^{w})}\leq$
$\displaystyle\frac{1}{w(t)^{m-1}\sqrt{\delta(t)^{m-1}}}$
$\displaystyle=\frac{1}{\left(1-4\epsilon(t)\right)^{(m-1)/2}}$

Therefore, since for $t$ large enough $Vol(\partial
D_{t}(o))\leq\sum_{i=1}^{\mathcal{E}(P)}Vol(\partial V_{i}(t))$ where $V_{i}$
denotes each end of $P$ then:

(6.9) $\displaystyle\frac{\operatorname{Vol}(\partial
D_{t}(o))}{\operatorname{Vol}(S_{t}^{w})}\leq\frac{\mathcal{E}(P)}{\left(1-4\epsilon(t)\right)^{(m-1)/2}}$

##  7\. Proof of Theorem 3.5

To show assertion (1) we apply Theorem 2.15 and inequality (3.1) in Theorem
3.4 to obtain, for $r$ sufficiently large, (we suppose that
$\varphi^{-1}(o_{w})\neq\emptyset$, and take $o\in\varphi^{-1}(o_{w})$ in
order to have that
$\operatorname{Vol}(D_{r}(o))\geq\operatorname{Vol}(B^{w}_{r})$ for all $r>0$)
:

(7.1) $\displaystyle 1\leq$
$\displaystyle\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}\leq\frac{\operatorname{Vol}(\partial
D_{r}(o))}{\operatorname{Vol}(S_{r}^{w})}$
$\displaystyle\leq\frac{\mathcal{E}(P)}{\left(1-4\epsilon(r)\right)^{(m-1)/2}}$

Moreover, we know (again using Theorem 2.15) that the volume growth function
is non-decreasing.

Therefore, taking limits in (7.1) when $r$ goes to $\infty$, we obtain:

(7.2)
$1\leq\lim_{r\to\infty}\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}=\operatorname{Sup}_{r>0}\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}\leq\mathcal{E}(P)$

Now, to prove assertion (2), we have, if $P$ has one end, that

(7.3)
$1\leq\operatorname{Sup}_{r>0}\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}\leq
1$

Hence, as
$f(r)=\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}$ is
non-decreasing, then $f(r)=1\,\,\forall r>0$, so we have equality in
inequality (2.6) for all $r>0$, and $P$ is a minimal cone, (see [17] for
details).

##  8\. Proof of Theorems 1.1 and 1.2 and the Corollaries

###  8.1. Proof of Theorem 1.1

We are going to apply Theorem 3.5. To do that, we must to check hypotheses (1)
and (2) in Theorem 3.4.

We have, in this case, that the ambient manifold is the hyperbolic space
$\mathbb{H}^{n}(b)$. Therefore all of its points are poles, so there exist at
least $o\in\mathbb{H}^{n}(b)$ such that $\varphi^{-1}(o)\neq\emptyset$. As it
is known, Hyperbolic space $\mathbb{H}^{n}(b)$ is a model space with
$w(r)=w_{b}(r)=\frac{1}{\sqrt{-b}}\sinh\sqrt{-b}r$ so
$w_{b}^{\prime}(r)=\cosh\sqrt{-b}r\geq 1\,\,\forall r>0$.

Therefore, hypothesis (2) in Theorem 3.4 is fulfilled in this context.
Concerning hypothesis (1), it is straightforward that

(8.1) $\displaystyle\|B^{P}_{x}\|$
$\displaystyle\leq\frac{\delta(r(x))}{e^{2\sqrt{-b}\,r(x)}}\leq\frac{\epsilon(r)\sqrt{-b}}{\sinh\sqrt{-b}r\cosh\sqrt{-b}r}$
$\displaystyle=\frac{\epsilon(r)}{\cosh^{2}\sqrt{-b}r}\sqrt{-b}\coth\sqrt{-b}r=\frac{\epsilon(r)}{(w_{b}^{\prime}(r))^{2}}\eta_{w_{b}}(r)$

where $\epsilon(r)=\frac{\delta(r(x))}{4\sqrt{-b}}$ goes to $0$ when $r$ goes
to $\infty$.

Hence, also hypothesis (1) in Theorem 3.4 is fulfilled so, applying inequality
(3.2) in Theorem 3.5, (because $P$ is minimal)

(8.2)
$1\leq\lim_{r\to\infty}\frac{Vol(D_{r})}{Vol(B_{r}^{w_{b}})}\leq\mathcal{E}(P)$

Finally, when $P$ has one end, then
$\lim_{r\to\infty}\frac{Vol(D_{r})}{Vol(B_{r}^{w_{b}})}=1$. Since $P$ is
minimal, by Theorem 2.15, $f(r)=\frac{Vol(D_{r})}{Vol(B_{r}^{w_{b}})}$ is a
monotone non-decreasing function, and, on the other hand, $f(r)\geq
1\,\,\forall r>0$ because inequality (2.7). Hence $f(r)=1\,\,\forall r>0$, so
$f^{\prime}(r)=0\,\,\forall r>0$. This last equality implies the equality in
inequality (2.6) for all $r>0$, (see [17] or [18] for details), and we apply
equality assertion in Theorem 2.15 to conclude that $P$ is totally geodesic in
$\mathbb{H}^{n}(b)$.

###  8.2. Proof of Theorem 1.2

In this case, we apply Theorem 3.5, being $M^{n}_{w}=\mathbb{R}^{n}$, i.e.,
being $w(r)=w_{0}(r)=r$, ($b=0$). Hence, $w_{0}^{\prime}(r)=1>0\,\,\forall
r>0$ and $\eta_{0}(r)=\frac{1}{r}$ and hypotheses (1) and (2) in this theorem
are trivially satisfied.

When $P$ has only one end we conclude as before that the volume growth
function is constant so we conclude equality in (2.6) for all radius $r>0$.
Hence $P$ is totally geodesic in $\mathbb{R}^{n}$ applying the corresponding
equality assertion in Theorem 2.15.

###  8.3. Proof of Corollary 4.1

We are considering now a complete and proper immersion in $\mathbb{H}^{n}(b)$,
as in Theorem 1.1, but $P$ is not necessarily minimal. In this setting
hypotheses (1) and (2) in Theorem 3.4 are fulfilled (as we have checked in the
proof above, without using minimality). Hence taking limits in (3.1) when we
consider an increasing sequence $\\{t_{i}\\}_{i=1}^{\infty}$ such that
$t_{i}\to\infty$ when $i\to\infty$, we have:

$\liminf_{i\to\infty}\frac{\operatorname{Vol}(\partial
D_{t_{i}})}{\operatorname{Vol}(S_{t_{i}}^{b,m-1})}\leq\mathcal{E}(P)$

###  8.4. Proof of Corollary 4.2

Hypotheses (1) and (2) in Theorem 3.4 are trivially satisfied and we argue as
in the proof of Corollary 4.1 to obtain the result.

###  8.5. Proof of Corollary 4.3

We apply Theorem 3.1. Our ambient manifold is $I\\!\\!K^{n}(b)$, ($b\leq 0$),
so hypothesis (1) about the bounds for the radial sectional curvature holds,
and as $w(r)=w_{b}(r)$ hence $w_{b}^{\prime}(r)\geq 1>0\,\,\forall r>0$ and
$\eta_{w_{b}}^{\prime}(r)\leq 0\,\,\forall r>0$. This means that hypothesis
(3) is fulfilled. Hypothesis (2) in Theorem 3.1 holds because

$\|B^{P}_{x}\|\leq c\,h_{b}(\rho^{P}(x))$

where $\rho^{P}(x)$ is the (intrinsic) distance to a fixed
$x_{o}\in\varphi^{-1}(o)$ and $c$ is a positive constant such that $c<1$.

###  8.6. Proof of Corollary 4.4

We apply again Theorem 3.1, having into account that the ambient space is the
Cartan-Hadamard manifold $\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ and the model
space used to compare is $\mathbb{R}^{m}$, with $w(r)=w_{0}(r)=r$.

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