# Density and spectrum of minimal submanifolds in space forms

B. Pessoa Lima  Dep. Matemática, UFPI, Campus Ministro Petrônio Portela,
64049-550 Teresina - PI<EMAIL_ADDRESS>,  L. Mari  Dep. Matemática, UFC,
Campus do Pici - Bloco 914, 60.455-760, Fortaleza - CE<EMAIL_ADDRESS>,  J.
Fabio Montenegro  Dep. Matemática, UFC, Campus do Pici - Bloco 914,
60.455-760, Fortaleza - CE<EMAIL_ADDRESS>and  F. de Brito Vieira  Dep.
Matemática, UFC, Campus do Pici - Bloco 914, 60.455-760, Fortaleza - CE
<EMAIL_ADDRESS>
###### Abstract.

Let $\varphi:M^{m}\rightarrow N^{n}$ be a minimal, proper immersion in an
ambient space suitably close to a space form $\mathbb{N}^{n}_{k}$ of curvature
$-k\leq 0$. In this paper, we are interested in the relation between the
density function $\Theta(r)$ of $M$ and the spectrum of its Laplace-Beltrami
operator. In particular, we prove that if $\Theta(r)$ has subexponential
growth (when $k<0$) or sub-polynomial growth ($k=0$) along a sequence, then
the spectrum of $M^{m}$ is the same as that of the space form
$\mathbb{N}^{m}_{k}$. Notably, the result applies to Anderson’s (smooth)
solutions of Plateau’s problem at infinity on the hyperbolic space,
independently of their boundary regularity. We also give a simple condition on
the second fundamental form that ensures $M$ to have finite density. In
particular, we show that minimal submanifolds with finite total curvature in
the hyperbolic space also have finite density.

###### Key words and phrases:

density spectrum Laplace-Beltrami minimal submanifolds monotonicity

###### 2010 Mathematics Subject Classification:

Primary 58J50; Secondary 58C21, 35P15

The second author is supported by the grant PRONEX - Núcleo de Análise
Geométrica e Aplicacões Processo nº PR2-0054-00009.01.00/11

The third author is partially supported by CNPq

##  1\. Introduction

Let $M^{m}$ be a minimal, properly immersed submanifold in a complete ambient
space $N^{n}$. In the present paper, we are interested in the case when $N$ is
close, in a sense made precise below, to a space form $\mathbb{N}_{k}^{n}$ of
curvature $-k\leq 0$. In particular, our focus is the study of the spectrum of
the Laplace Beltrami operator $-\Delta$ on $M$ and its relationship with the
density at infinity of $M$, that is, the limit as $r\rightarrow+\infty$ of the
(monotone) quantity

(1.1) $\Theta(r)\doteq\frac{\mathrm{vol}(M\cap B_{r})}{V_{k}(r)},$

where $B_{r}$ indicates a geodesic ball of radius $r$ in $N^{n}$ and
$V_{k}(r)$ is the volume of a geodesic ball of radius $r$ in
$\mathbb{N}^{m}_{k}$. Hereafter, we will say that $M$ has finite density if

$\Theta(+\infty)\doteq\lim_{r\rightarrow+\infty}\Theta(r)<+\infty.$

To properly put our results into perspective, we briefly recall few facts
about the spectrum of the Laplacian on a geodesically complete manifold. It is
known by works of P. Chernoff [15] and R.S. Strichartz [49] that $-\Delta$ on
a complete manifold is essentially self-adjoint on the domain
$C^{\infty}_{c}(M)$, and thus it admits a unique self-adjoint extension, which
we still call $-\Delta$. Since $-\Delta$ is positive and self-adjoint, its
spectrum is the set of $\lambda\geq 0$ such that $\Delta+\lambda I$ does not
have bounded inverse. Sometimes we say spectrum of $M$ rather than spectrum of
$-\Delta$ and we denote it by $\sigma(M)$. The well-known Weyl’s
characterization for the spectrum of a self-adjoint operator in a Hilbert
space implies the following

######  Lemma 1.

[19, Lemma 4.1.2] A number $\lambda\in\mathbb{R}$ lies in $\sigma(M)$ if and
only if there exists a sequence of nonzero functions
$u_{j}\in\mathrm{Dom}(-\Delta)$ such that

(1.2) $\|\Delta u_{j}+\lambda
u_{j}\|_{2}=o\big{(}\|u_{j}\|_{2}\big{)}\qquad\text{as
}\,j\rightarrow+\infty.$

In the literature, characterizations of the whole $\sigma(M)$ are known only
in few special cases. Among them, the Euclidean space, for which
$\sigma(\mathbb{R}^{m})=[0,\infty)$, and the hyperbolic space
$\mathbb{H}^{m}_{k}$, for which

(1.3)
$\sigma(\mathbb{H}^{m}_{k})=\left[\frac{(m-1)^{2}k}{4},+\infty\right)\\!.$

The approach to guarantee that $\sigma(M)=[c,+\infty)$, for some $c\geq 0$,
usually splits into two parts. The first one is to show that
$\inf\sigma(M)\geq c$ via, for instance, the Laplacian comparison theorem from
below ([42], [5]), and the second one is to produce a sequence like in lemma 1
for each $\lambda>c$. This step is accomplished by considering radial
functions of compact support, and, at least in the first results on the topic
like the one in [21], uses the comparison theorems on both sides for
$\Delta\rho$, $\rho$ being the distance from a fixed origin $o\in M$.
Therefore, the method needs both a pinching on the sectional curvature and the
smoothness of $\rho$, that is, that $o$ is a pole of $M$ (see [21], [25],[37]
and Corollary 2.17 in [8]), which is a severe topological restriction. Since
then, various efforts were made to weaken both the curvature and the
topological assumptions. We briefly overview some of the main achievements.

In [35], Kumura observed that to perform the second step (and just for it) it
is enough that there exists a relatively compact, mean convex, smooth open set
$\Omega$ with the property that the normal exponential map realizes a global
diffeomorphism $\partial\Omega\times\mathbb{R}_{0}^{+}\rightarrow
M\backslash\Omega$. Conditions of this kind seem, however, unavoidable for his
techniques to work. On the other hand, in [36] the author drastically weakened
the curvature requirements needed to establish Step 2, by replacing the two-
sided pinching on the sectional curvature with a combination of a lower bound
on a suitably weighted volume and an $L^{p}$-bound on the Ricci curvature.

Regarding the need for a pole, major recent improvements have been made in a
series of papers ([50], [55], [41], [11]): their guiding idea was to replace
the $L^{2}$-norm in relation (1.2) with the $L^{1}$-norm, which via a trick in
[55], [41] enables to use smoothed distance functions to construct sequences
as in Lemma 1. Building on deep function-theoretic results due to Sturm [50]
and Charalambous-Lu [11], in [55], [41] the authors proved that
$\sigma(M)=[0,\infty)$ when

(1.4) $\liminf_{\rho(x)\rightarrow+\infty}\mathrm{Ricc}_{x}=0$

in the sense of quadratic forms, without any topological assumption. This
remarkable result improves on [37] and [25] (see also Corollary 2.17 in [8]),
where $M$ was assumed to have a pole. Further refinements of (1.4) have been
given in [11]. However, when (1.4) does not hold, the situation is more
delicate and is still the subject of an active area of research. In this
respect, we also quote the general function-theoretic criteria developed by H.
Donnelly [22], and K.D. Elworthy and F-Y. Wang [24] to ensure that a half-line
belongs to the spectrum of $M$.

The main concern in this paper is to achieve, in the above-mentioned setting
of minimal submanifolds $\varphi:M\rightarrow N$, a characterization of the
whole $\sigma(M)$ free from curvature or topological conditions on $M$ (in
this respect, observe that the completeness of $M$ follows from that of $N$
and the properness of $\varphi$). It is known by [18] and [5] that for a
minimal immersion $\varphi:M^{m}\rightarrow\mathbb{N}^{n}_{k}$ the fundamental
tone of $M$, $\inf\sigma(M)$, is at least that of $\mathbb{N}^{m}_{k}$, i.e.,

(1.5) $\inf\sigma(M)\geq\frac{(m-1)^{2}k}{4}.$

Moreover, as a corollary of [35] and [4], [6], if the second fundamental form
$\mathrm{II}$ satisfies the decay estimate

(1.6)
$\begin{array}[]{ll}\displaystyle\lim_{\rho(x)\rightarrow+\infty}\rho(x)|\mathrm{II}(x)|=0&\quad\text{if
}\,k=0\\\\[8.5359pt]
\displaystyle\lim_{\rho(x)\rightarrow+\infty}|\mathrm{II}(x)|=0&\quad\text{if
}\,k>0\\\\[5.69046pt] \end{array}$

($\rho(x)$ being the intrinsic distance with respect to some fixed origin
$o\in M$), then $M$ has the same spectrum that a totally geodesic submanifold
$\mathbb{N}^{m}_{k}\subset\mathbb{N}^{n}_{k}$, that is,

(1.7) $\sigma(M)=\left[\frac{(m-1)^{2}k}{4},+\infty\right)\\!.$

According to [1], [20], (1.6) is ensured when $M$ has finite total curvature,
that is, when

(1.8) $\int_{M}|\mathrm{II}|^{m}<+\infty.$

######  Remark 1.

_A characterization of the essential spectrum, similar to ( 1.7), also holds
for submanifolds of the hyperbolic space $\mathbb{H}^{n}_{k}$ with constant
(normalized) mean curvature $H<\sqrt{k}$. There, condition (1.8) is replaced
by the finiteness of the $L^{m}$-norm of the traceless second fundamental
form. For deepening, see [10]. _

Condition (1.6) is a quite binding requirement for (1.7) to hold, since it
needs a pointwise control of the second fundamental form, and the search for
more manageable conditions has been at the heart of the present paper. Here,
we identify a suitable growth on the density function $\Theta(r)$ _along a
sequence_ as a natural candidate to replace it, see (1.10). As a very special
case, (1.7) holds when $M$ has finite density. It might be interesting that
just a volume growth condition along a sequence could control the whole
spectrum of $M$; for this to happen, the minimality condition enters in a
crucial and subtle way.

Regarding the relation between (1.8) and the finiteness of $\Theta(+\infty)$,
we remark that their interplay has been investigated in depth for minimal
submanifolds of $\mathbb{R}^{n}$, but the case of $\mathbb{H}^{n}_{k}$ seems
to be partly unexplored. In the next section, we will briefly discuss the
state of the art, to the best of our knowledge. As a corollary of Theorem 2
below, we will show the following

######  Corollary 1.

Let $M^{m}$ be a minimal properly immersed submanifold in
$\mathbb{H}^{n}_{k}$. If $M$ has finite total curvature, then
$\Theta(+\infty)<+\infty$.

As far as we know, this result was previously known just in dimension $m=2$
via a Chern-Osserman type inequality, see the next section for further
details.

We now come to our results, beginning with defining the ambient spaces which
we are interested in: these are manifolds with a pole, whose radial sectional
curvature is suitably pinched to that of the model $\mathbb{N}^{n}_{k}$.

######  Definition 1.

Let $N^{n}$ possess a pole $\bar{o}$ and denote with $\bar{\rho}$ the distance
function from $\bar{o}$. Assume that the radial sectional curvature
$\bar{K}_{\mathrm{rad}}$ of $N$, that is, the sectional curvature restricted
to planes $\pi$ containing $\bar{\nabla}\bar{\rho}$, satisfies

(1.9)
$-G\big{(}\bar{\rho}(x)\big{)}\leq\bar{K}_{\mathrm{rad}}(\pi_{x})\leq-k\leq
0\qquad\forall\,x\in N\backslash\\{\bar{o}\\},$

for some $G\in C^{0}(\mathbb{R}^{+}_{0})$. We say that

  * $(i)$

_$N$ has a pointwise (respectively, integral) pinching to $\mathbb{R}^{n}$ if
$k=0$ and_

$sG(s)\rightarrow 0\ \text{ as
}\,s\rightarrow+\infty\qquad\big{(}\textrm{respectively, $\,sG(s)\in
L^{1}(+\infty)$}\big{)};$

  * $(ii)$

_$N$ has a pointwise (respectively, integral) pinching to $\mathbb{H}^{n}_{k}$
if $k>0$ and_

$G(s)-k\rightarrow 0\ \text{ as
}\,s\rightarrow+\infty\qquad\big{(}\textrm{respectively, $\,G(s)-k\in
L^{1}(+\infty)$}\big{)}.$

Hereafter, given an ambient manifold $N$ with a pole $\bar{o}$, the density
function $\Theta(r)$ will always be computed by taking extrinsic balls
centered at $\bar{o}$.

Our main achievements are the following two theorems. The first one
characterizes $\sigma(M)$ when the density of $M$ grows subexponentially
(respectively, sub-polynomially) along a sequence. Condition (1.10) below is
very much in the spirit of a classical growth requirement due to R. Brooks [9]
and Y. Higuchi [31] to bound from above the infimum of the essential spectrum
of $-\Delta$. However, we stress that our Theorem 1 seems to be the first
result in the literature characterizing the whole spectrum of $M$ under just a
mild volume assumption.

######  Theorem 1.

Let $\varphi:M^{m}\rightarrow N^{n}$ be a minimal properly immersed
submanifold, and suppose that $N$ has a pointwise or an integral pinching to a
space form. If either

(1.10) $\begin{array}[]{ll}\text{$N$ is pinched to $\mathbb{H}^{n}_{k}$,
and}&\qquad\displaystyle\liminf_{s\rightarrow+\infty}\frac{\log\Theta(s)}{s}=0,\quad\text{or
}\\\\[11.38092pt] \text{$N$ is pinched to $\mathbb{R}^{n}$,
and}&\qquad\displaystyle\liminf_{s\rightarrow+\infty}\frac{\log\Theta(s)}{\log
s}=0.\end{array}$

then

(1.11) $\sigma(M)=\left[\frac{(m-1)^{2}k}{4},+\infty\right)\\!.$

The above theorem is well suited for minimal submanifolds constructed via
Geometric Measure Theory since, typically, their existence is guaranteed by
controlling the density function $\Theta(r)$. As an important example, Theorem
1 applies to all solutions of Plateau’s problem at infinity
$M^{m}\rightarrow\mathbb{H}^{n}_{k}$ constructed in [2], provided that they
are smooth. Indeed, because of their construction, $\Theta(+\infty)<+\infty$
(see [2], part [A] at p. 485) and they are proper (it can also be deduced as a
consequence of $\Theta(+\infty)<+\infty$, see Remark 5). By standard
regularity theory, smoothness of $M^{m}$ is automatic if $m\leq 6$.

######  Corollary 2.

Let $\Sigma\subset\partial_{\infty}\mathbb{H}^{n}_{k}$ be a closed, integral
$(m-1)$ current in the boundary at infinity of $\mathbb{H}^{n}_{k}$ such that,
for some neighbourhood $U\subset\mathbb{H}^{n}_{k}$ of
$\operatorname{supp}(\Sigma)$, $\Sigma$ does not bound in $U$, and let
$M^{m}\hookrightarrow\mathbb{H}^{n}_{k}$ be the solution of Plateau’s problem
at infinity constructed in [2] for $\Sigma$. If $M$ is smooth, then (1.11)
holds.

An interesting fact of Corollary 2 is that $M$ is _not_ required to be regular
up to $\partial_{\infty}\mathbb{H}^{n}_{k}$, in particular it might have
infinite total curvature. In this respect, we observe that if $M$ be $C^{2}$
up to $\partial_{\infty}\mathbb{H}^{n}_{k}$, then $M$ would have finite total
curvature (Lemma 5 in Appendix 1). By deep regularity results, this is the
case if, for instance, $M^{m}\rightarrow\mathbb{H}^{m+1}_{k}$ is a smooth
hypersurface that solves Plateau’s problem for $\Sigma$, and $\Sigma$ is a
$C^{2,\alpha}$ (for $\alpha>0$), embedded compact hypersurface of
$\partial_{\infty}\mathbb{H}^{n}_{k}$. See Appendix 1 for details.

The spectrum of solutions of Plateau’s problems has also been considered in
[3] for minimal surfaces in $\mathbb{R}^{3}$. In this respect, it is
interesting to compare Corollary 2 with $(3)$ of Corollary 2.6 therein.

######  Remark 2.

_The solution $M$ of Plateau’s problem in [2] is constructed as a weak limit
of a sequence $M_{j}$ of minimizing currents for suitable boundaries
$\Sigma_{j}$ converging to $\Sigma$. and property $\Theta(+\infty)<+\infty$ is
a consequence of a uniform upper bound for the mass of a sequence $M_{j}$
(part [A], p. 485 in [2]). Such a bound is achieved because of the way the
boundaries $\Sigma_{j}$ are constructed, in particular, since they are all
sections of the same cone. One might wonder whether $\Theta(+\infty)<+\infty$,
or at least the subexponential growth in (1.10), is satisfied by all solutions
of Plateau’s problem. In this respect, we just make this simple observation:
in the hypersurface case $n=m+1$, if $M\cap B^{m+1}_{r}$ is volume-minimizing
then clearly_

$\Theta(r)=\frac{\mathrm{vol}(M\cap
B^{m+1}_{r})}{V_{k}(r)}\leq\frac{\mathrm{vol}(\partial
B_{r}^{m+1}\subset\mathbb{H}_{k}^{m+1})}{V_{k}(r)}=c_{k}\frac{\sinh^{m}(\sqrt{k}r)}{V_{k}(r)},$

_but this last expression diverges exponentially fast as $r\rightarrow+\infty$
(differently from its Euclidean analogous, which is finite). This might
suggest that a general solution of Plateau’s problem does not automatically
satisfies $\Theta(+\infty)<+\infty$, and maybe not even (1.10). _

In our second result we focus on the particular case when
$\Theta(+\infty)<+\infty$, and we give a sufficient condition for its validity
in terms of the decay of the second fundamental form. Towards this aim, we
shall restrict to ambient spaces with an integral pinching.

######  Theorem 2.

Let $\varphi:M^{m}\rightarrow N^{n}$ be a minimal immersion, and suppose that
$N$ has an integral pinching to a space form. Denote with $\rho(x)$ the
intrinsic distance from some reference origin $o\in M$. Assume that there
exist $c>0$ and $\alpha>1$ such that the second fundamental form satisfies,
for $\rho(x)>>1$,

(1.12)
$\begin{array}[]{ll}\displaystyle|\mathrm{II}(x)|^{2}\leq\frac{c}{\rho(x)\log^{\alpha}\rho(x)}&\qquad\text{if
$N$ is pinched to $\mathbb{H}^{n}_{k}$;}\\\\[11.38092pt]
\displaystyle|\mathrm{II}(x)|^{2}\leq\frac{c}{\rho(x)^{2}\log^{\alpha}\rho(x)}&\qquad\text{if
$N$ is pinched to $\mathbb{R}^{n}$.}\end{array}$

Then, $\varphi$ is proper, $M$ is diffeomorphic to the interior of a compact
manifold with boundary, and $\Theta(+\infty)<+\infty$.

The assertions that $\varphi$ be proper and $M$ have finite topology is well-
known under assumptions even weaker than (1.12) and not necessarily requiring
the minimality, see for instance [4], [6]. Former results are due to [1]
($N=\mathbb{R}^{n}$) and [20], [10] ($N=\mathbb{H}^{n}_{k}$). Here, our
original contribution is to show that $M$ has finite density. Because of a
result in [20], [46], if $\varphi:M\rightarrow\mathbb{H}^{n}_{k}$ has finite
total curvature then $|II(x)|=o(\rho(x)^{-1})$ as $\rho(x)\rightarrow+\infty$.
Hence, (1.12) is met and Corollary 1 follows at once.

We briefly describe the strategy of the proof of Theorem 1. In view of (1.5),
it is enough to show that each $\lambda>(m-1)^{2}k/4$ lies in $\sigma(M)$. To
this end, we follow an approach inspired by a general result due to K.D.
Elworthy and F-Y. Wang [24]. However, Elworthy-Wang’s theorem is not
sufficient to conclude, and we need to considerably refine the criterion in
order to fit in the present setting. To construct the sequence as in Lemma 1,
a key step is to couple the volume growth requirement (1.10) with a sharpened
form of the monotonicity formula for minimal submanifolds, which improves on
the classical ones in [48], [2]. Indeed, in Proposition 3 we describe three
monotone quantities other than $\Theta(s)$, that might be useful beyond the
purpose of the present paper. For example, in the very recent [27] the authors
discovered and used some of the relations in Proposition 3 to show interesting
comparison results for the capacity and the first eigenvalue of minimal
submanifolds.

###  1.1. Finite density and finite total curvature in $\mathbb{R}^{n}$ and
$\mathbb{H}^{n}$

The first attempt to extend the classical theory of finite total curvature
surfaces in $\mathbb{R}^{n}$ (see [44], [32], [16], [17]) to the higher-
dimensional case is due to M.T. Anderson. In [1], the author drew from (1.8) a
number of topological and geometric consequences, and here we focus on those
useful to highlight the relationship between total curvature and density.
First, he showed that (1.8) implies the decay

(1.13) $\lim_{\rho(x)\rightarrow+\infty}\rho(x)|\mathrm{II}(x)|=0,$

where $\rho(x)$ is the intrinsic distance from a fixed origin, and as a
consequence $M$ is proper, the extrinsic distance function $r$ has no critical
points outside some compact set and $|\nabla r|\rightarrow 1$ as $r$ diverges,
so by Morse theory $M$ is diffeomorphic to the interior of a compact manifold
with boundary. Moreover, he proved that $M$ has finite density via a higher-
dimensional extension of the Chern-Osserman identity [16], [17], namely the
following relation linking the Euler characteristic $\chi(M)$ and the Pfaffian
form $\Omega$ ([1], Theorem 4.1):

(1.14)
$\chi(M)=\int_{M}\Omega+\lim_{r\rightarrow+\infty}\frac{\mathrm{vol}(M\cap\partial
B_{r})}{V_{0}^{\prime}(r)}.$

Observe that, since $|\nabla r|\rightarrow 1$, by coarea’s formula the limit
in the right hand-side coincides with $\Theta(+\infty)$. We underline that
property $\Theta(+\infty)<+\infty$ plays a fundamental role to apply the
machinery of manifold convergence to get information on the limit structure of
the ends of $M$ ([1], [47], [54]). For instance, $\Theta(+\infty)$ is related
to the number $\mathcal{E}(M)$ of ends of $M$: if we denote with
$V_{1},\ldots,V_{\mathcal{E}(M)}$ the (finitely many) ends of $M$, (1.8)
implies for $m\geq 3$ the identities

(1.15)
$\Theta(+\infty)=\sum_{i=1}^{\mathcal{E}(M)}\lim_{r\rightarrow+\infty}\frac{\mathrm{vol}(V_{i}\cap\partial
B_{r})}{V_{0}^{\prime}(r)}\equiv\mathcal{E}(M),$

and thus $M$ is totally geodesic provided that it has only one end and finite
total curvature ([1], Thm 5.1 and its proof). Further information on the
mutual relationship between the finiteness of the total curvature and
$\Theta(+\infty)<+\infty$ can be deduced under the additional requirement that
$M$ is stable or it has finite stability index. For example, by work of J.
Tysk [54], if $M^{m}$ has finite index and $m\leq 6$, then

(1.16) $\Theta(+\infty)<+\infty\qquad\text{if and only
if}\qquad\int_{M}|\mathrm{II}|^{m}<+\infty.$

######  Remark 3.

_Indeed, the main result in[54] states that, when $\Theta(+\infty)<+\infty$
and $m\leq 6$, $M$ has finite index if and only if it has finite total
curvature. However, since the finite total curvature condition alone implies
both that $M$ has finite index and $\Theta(+\infty)<+\infty$ (in any
dimension***As said, finite total curvature implies $\Theta(+\infty)<+\infty$
by (1.14), while the finiteness of the index can be seen as an application of
the generalized Cwikel-Lieb-Rozembljum inequality (see [38]) to the stability
operator $L=-\Delta-|\mathrm{II}|^{2}$, recalling that a minimal submanifold
$M^{m}\rightarrow\mathbb{R}^{n}$ satisfies a Sobolev inequality. We refer to
[45] for deepening.), the characterization in (1.16) is equivalent to Tysk’s
theorem. We underline that it is still a deep open problem whether or not, for
$m\geq 3$, stability or finite index alone implies the finiteness of the
density at infinity. _

Since then, efforts were made to investigate analogous properties for minimal
submanifolds of finite total curvature immersed in $\mathbb{H}^{n}_{k}$.
There, some aspects show strong analogy with the $\mathbb{R}^{n}$ case, while
others are strikingly different. For instance, minimal immersions
$\varphi:M^{m}\rightarrow\mathbb{H}^{n}_{k}$ with finite total curvature enjoy
the same decay property (1.13) with respect to the intrinsic distance
$\rho(x)$ ([20], see also [46]), which is enough to deduce that they are
properly immersed and diffeomorphic to the interior of a compact manifold with
boundary. Moreover, Anderson [2] proved the monotonicity of $\Theta(r)$ in
(1.1). In order to show (among other things) that complete, finite total
curvature surfaces $M^{2}\hookrightarrow\mathbb{H}^{n}$ have finite density,
in [13], [14] the authors obtained the following Chern-Osserman type
inequality:

(1.17) $\chi(M)\geq-\frac{1}{4\pi}\int_{M}|\mathrm{II}|^{2}+\Theta(+\infty),$

see also [28]. However, in the higher dimensional case we found no analogous
of (1.14), (1.17) in the literature, and adapting the proof of (1.14) to the
hyperbolic ambient space seems to be subtler than what we expected. In fact,
an _equality_ like (1.14) is not even possible to obtain, since there exist
minimal submanifolds of $\mathbb{H}^{n}_{k}$ with finite density but whose
density at infinity depends on the chosen reference origin [26]. We point out
that, on the contrary, inequality (1.17) holds for each choice of the
reference origin in $\mathbb{R}^{n}$. This motivated the different route that
we follow to prove Theorem 2 and Corollary 1. Among the results in [1] that
could not admit a corresponding one in $\mathbb{H}_{k}^{n}$, in view of the
solvability of Plateau’s problem at infinity on $\mathbb{H}_{k}^{n}$ we stress
that a relation like (1.15) cannot hold for each minimal submanifold of
$\mathbb{H}^{n}_{k}$ with finite total curvature. Indeed, there exist a wealth
of properly immersed minimal submanifolds in $\mathbb{H}^{n}_{k}$ with finite
total curvature and one end: for example, referring to the upper half-space
model, the graphical solution of Plateau’s problem for
$\Sigma^{m-1}\subset\partial_{\infty}\mathbb{H}^{n}_{k}$ being the boundary of
a convex set (constructed at the end of [2]) has finite total curvature, as
follows from Lemma 5 and the regularity results recalled in Appendix 1. It
shall be observed, however, that when $\mathrm{II}$ decays sufficiently fast
at infinity with respect to the extrinsic distance function $r(x)$:

(1.18) $\lim_{r(x)\rightarrow+\infty}e^{2\sqrt{k}r(x)}|\mathrm{II}(x)|=0,$

then the inequality $\Theta(+\infty)\leq\mathcal{E}(M)$ still holds for
minimal hypersurfaces in $\mathbb{H}^{n}_{k}$ as shown in [29], and in
particular $M$ is totally geodesic provided that it has only one end, as first
observed in [33], [34]. We remark that there exists an infinite family of
complete minimal cylinders
$\varphi_{\lambda}:\mathbb{S}^{1}\times\mathbb{R}\rightarrow\mathbb{H}^{3}$
whose second fundamental form $\mathrm{II}_{\lambda}$ decays exactly of order
$\exp\\{-2r(x)\\}$, see [43].

##  2\. Preliminaries

Let $\varphi:(M^{m},\langle\,,\,\rangle)\rightarrow(N^{n},(\,,\,))$ be an
isometric immersion of a complete $m$-dimensional Riemannian manifold $M$ into
an ambient manifold $N$ of dimension $n$ and possessing a pole $\bar{o}$. We
denote with $\nabla,\mathrm{Hess}\,,\Delta$ the connection, the Riemannian
Hessian and the Laplace-Beltrami operator on $M$, while quantities related to
$N$ will be marked with a bar. For instance,
$\bar{\nabla},\overline{\mathrm{dist}},\overline{\mathrm{Hess}\,}$ will
identify the connection, the distance function and the Hessian in $N$. Let
$\bar{\rho}(x)=\overline{\mathrm{dist}}(x,\bar{o})$ be the distance function
from $\bar{o}$. Geodesic balls in $N$ of radius $R$ and center $y$ will be
denoted with $B_{R}^{N}(y)$. Moreover, set

(2.1) $r\ :\ M\rightarrow\mathbb{R},\qquad
r(x)=\bar{\rho}\big{(}\varphi(x)\big{)},$

for the extrinsic distance from $\bar{o}$. We will indicate with
$\Gamma_{\\!s}$ the extrinsic geodesic spheres restricted to $M$:
$\Gamma_{\\!s}\doteq\\{x\in M;\;r(x)=s\\}$. Fix a base point $o\in M$. In what
follows, we shall also consider the intrinsic distance function
$\rho(x)=\mathrm{dist}(x,o)$ from a reference origin $o\in M$.

###  2.1. Target spaces

Hereafter, we consider an ambient space $N$ possessing a pole $\bar{o}$ and,
setting $\bar{\rho}(x)\doteq\mathrm{dist}(x,\bar{o})$, we assume that (1.9) is
met for some $k\geq 0$ and some $G\in C^{0}(\mathbb{R}^{+}_{0})$. Let
$\mathrm{sn}_{k}(t)$ be the solution of

(2.2)
$\left\\{\begin{array}[]{l}\mathrm{sn}_{k}^{\prime\prime}-k\,\mathrm{sn}_{k}=0\quad\text{on
}\,\mathbb{R}^{+},\\\\[2.84544pt]
\mathrm{sn}_{k}(0)=0,\quad\mathrm{sn}_{k}^{\prime}(0)=1,\end{array}\right.$

that is

(2.3) $\mathrm{sn}_{k}(t)=\left\\{\begin{array}[]{ll}t&\quad\text{if
}\,k=0,\\\\[2.84544pt] \sinh(\sqrt{k}t)/\sqrt{k}&\quad\text{if
}\,k>0.\end{array}\right.$

Observe that $\mathbb{R}^{n}$ and $\mathbb{H}^{n}_{k}$ can be written as the
differentiable manifold $\mathbb{R}^{n}$ equipped with the metric given, in
polar geodesic coordinates
$(\rho,\theta)\in\mathbb{R}^{+}\times\mathbb{S}^{n-1}$ centered at some
origin, by

$\mathrm{d}s^{2}_{k}=\mathrm{d}\rho^{2}+\mathrm{sn}^{2}_{k}(\rho)\,\mathrm{d}\theta^{2},$

$\mathrm{d}\theta^{2}$ being the metric on the unit sphere $\mathbb{S}^{n-1}$.
We also consider the model $M^{n}_{g}$ associated with the lower bound $-G$
for $\bar{K}_{\mathrm{rad}}$, that is, we let $g\in C^{2}(\mathbb{R}^{+}_{0})$
be the solution of

(2.4) $\left\\{\begin{array}[]{l}g^{\prime\prime}-Gg=0\quad\text{on
}\,\mathbb{R}^{+},\\\\[2.84544pt] g(0)=0,\quad
g^{\prime}(0)=1,\end{array}\right.$

and we define $M^{n}_{g}$ as being $(\mathbb{R}^{n},\mathrm{d}s^{2}_{g})$ with
the $C^{2}$-metric
$\mathrm{d}s_{g}^{2}=\mathrm{d}\rho^{2}+g^{2}(\rho)\mathrm{d}\theta^{2}$ in
polar coordinates. Condition (1.9) and the Hessian comparison theorem (Theorem
2.3 in [45], or Theorem 1.15 in [8]) imply

(2.5)
$\frac{\mathrm{sn}_{k}^{\prime}(\bar{\rho})}{\mathrm{sn}_{k}(\bar{\rho})}\Big{(}(\,,\,)-\mathrm{d}\bar{\rho}\otimes\mathrm{d}\bar{\rho}\Big{)}\leq\overline{\mathrm{Hess}\,}(\bar{\rho})\leq\frac{g^{\prime}(\bar{\rho})}{g(\bar{\rho})}\Big{(}(\,,\,)-\mathrm{d}\bar{\rho}\otimes\mathrm{d}\bar{\rho}\Big{)}.$

The next proposition investigates the ODE properties that follow from the
assumptions of pointwise or integral pinching.

######  Proposition 1.

Let $N^{n}$ satisfy (1.9), and let $\mathrm{sn}_{k},g$ be solutions of (2.3),
(2.4). Define

(2.6)
$\zeta(s)\doteq\frac{g^{\prime}(s)}{g(s)}-\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}.$

Then, $\zeta(0^{+})=0$, $\zeta\geq 0$ on $\mathbb{R}^{+}$. Moreover,

  * $(i)$

If $N$ has a pointwise pinching to $\mathbb{H}^{n}_{k}$ or $\mathbb{R}^{n}$,
then $\zeta(s)\rightarrow 0$ as $s\rightarrow+\infty$.

  * $(ii)$

If $N$ has an integral pinching to $\mathbb{H}^{n}_{k}$ or $\mathbb{R}^{n}$,
then $g/\mathrm{sn}_{k}\rightarrow C$ as $s\rightarrow+\infty$ for some
$C\in\mathbb{R}^{+}$, and

(2.7) $\zeta(s)\in
L^{1}(\mathbb{R}^{+}),\qquad\zeta(s)\frac{\mathrm{sn}_{k}(s)}{\mathrm{sn}_{k}^{\prime}(s)}\rightarrow
0\ \text{ as }\,s\rightarrow+\infty.$

###### Proof.

The non-negativity of $\zeta$, which in particular implies that
$g/\mathrm{sn}_{k}$ is non-decreasing, follows from $G\geq k$ via Sturm
comparison, and $\zeta(0^{+})=0$ depends on the asymptotic relations
$\mathrm{sn}_{k}^{\prime}/\mathrm{sn}_{k}=s^{-1}+o(1)$ and
$g^{\prime}/g=s^{-1}+o(1)$ as $s\rightarrow 0^{+}$, which directly follow from
the ODEs satisfied by $\mathrm{sn}_{k}$ and $g$. To show $(i)$,
differentiating $\zeta$ we get

(2.8) $\zeta^{\prime}(s)=R(s)-\zeta(s)B(s),$

where $R(s)\doteq G(s)-k$ and $\displaystyle
B(s)\doteq\frac{g^{\prime}(s)}{g(s)}+\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}$.
Thus, integrating on $[1,s]$, we can rewrite $\zeta$ as follows:

(2.9)
$\zeta(s)=\zeta(1)e^{-\int_{1}^{s}B}+e^{-\int_{1}^{s}B}\int_{1}^{s}R(\sigma)e^{\int_{1}^{\sigma}B}\mathrm{d}\sigma$

Using that $B\not\in L^{1}([1,+\infty))$, and applying de l’Hopital’s theorem,
we infer

$\lim_{s\rightarrow+\infty}\zeta(s)=\lim_{s\rightarrow+\infty}\frac{R(s)}{B(s)}\leq\lim_{s\rightarrow+\infty}\frac{\mathrm{sn}_{k}(s)[G(s)-k]}{\mathrm{sn}_{k}^{\prime}(s)}.$

In our pointwise pinching assumptions on $G(s)$, for both $k=0$ and $k>0$ the
last limit is zero, hence $\zeta(s)\rightarrow 0$ as $s$ diverges. To show
$(ii)$, suppose that $N$ has an integral pinching to $\mathbb{H}^{n}_{k}$ or
to $\mathbb{R}^{n}$. We first observe that the boundedness of
$g/\mathrm{sn}_{k}$ on $\mathbb{R}^{+}$ equivalent to the property $\zeta\in
L^{1}(+\infty)$, as it follows from

(2.10)
$\log\frac{g(s)}{\mathrm{sn}_{k}(s)}=\int_{0}^{s}\frac{\mathrm{d}}{\mathrm{d}\sigma}\log\left(\frac{g(\sigma)}{\mathrm{sn}_{k}(\sigma)}\right)\mathrm{d}s=\int_{0}^{s}\zeta$

(we used that $(g/\mathrm{sn}_{k})(0^{+})=1$). The boundedness of
$g/\mathrm{sn}_{k}$ is the content of Corollary 4 and Remark 16 in [7], but we
prefer here to present a direct proof. Integrating (2.9) on $[1,s]$ and using
Fubini’s theorem, the monotonicity of $g/\mathrm{sn}_{k}$ and the expression
of $B$ we obtain

(2.11)
$\begin{array}[]{lcl}\displaystyle\int_{1}^{s}\zeta&=&\displaystyle\zeta(1)\int_{1}^{s}\frac{g(1)\mathrm{sn}_{k}(1)}{g(\sigma)\mathrm{sn}_{k}(\sigma)}\,\mathrm{d}\sigma+\int_{1}^{s}e^{-\int_{1}^{\sigma}B}\int_{1}^{\sigma}R(\tau)e^{\int_{1}^{\tau}B}\mathrm{d}\tau\,\mathrm{d}\sigma\\\\[14.22636pt]
&\leq&\displaystyle\zeta(1)\mathrm{sn}_{k}(1)^{2}\int_{1}^{s}\frac{\mathrm{d}\sigma}{\mathrm{sn}_{k}^{2}(\sigma)}+\int_{1}^{s}\left[\int_{\tau}^{s}e^{-\int_{1}^{\sigma}B}R(\tau)e^{\int_{1}^{\tau}B}\mathrm{d}\sigma\right]\mathrm{d}\tau\\\\[14.22636pt]
&\leq&\displaystyle
C+\int_{1}^{s}R(\tau)g(\tau)\mathrm{sn}_{k}(\tau)\left[\int_{\tau}^{s}\frac{\mathrm{d}\sigma}{g(\sigma)\mathrm{sn}_{k}(\sigma)}\right]\mathrm{d}\tau\\\\[14.22636pt]
&\leq&\displaystyle
C+\int_{1}^{s}R(\tau)g(\tau)\mathrm{sn}_{k}(\tau)\left[\int_{\tau}^{+\infty}\frac{\mathrm{d}\sigma}{g(\sigma)\mathrm{sn}_{k}(\sigma)}\right]\mathrm{d}\tau\\\\[14.22636pt]
\end{array}$

for some $C>0$, where we have used that
$\mathrm{sn}_{k}^{-2},g^{-1}\mathrm{sn}_{k}^{-1}\in L^{1}(+\infty)$. Next,
since $g\,\mathrm{sn}_{k}/\mathrm{sn}_{k}^{2}$ is non-decreasing, Proposition
3.12 in [8] ensures the validity of the following inequality:

$g(\tau)\mathrm{sn}_{k}(\tau)\left[\int_{\tau}^{+\infty}\frac{\mathrm{d}\sigma}{g(\sigma)\mathrm{sn}_{k}(\sigma)}\right]\leq\mathrm{sn}_{k}^{2}(\tau)\left[\int_{\tau}^{+\infty}\frac{\mathrm{d}\sigma}{\mathrm{sn}_{k}^{2}(\sigma)}\right].$

It is easy to show that the last expression is bounded if $k>0$, and diverges
at the order of $\tau$ if $k=0$. In other words, it can be bounded by
$C_{1}\mathrm{sn}_{k}/\mathrm{sn}_{k}^{\prime}$ on $[1,+\infty)$, for some
large $C_{1}>0$. Therefore, by (2.11)

$\displaystyle\int_{1}^{s}\zeta\leq\displaystyle
C+C_{1}\int_{1}^{s}R(\tau)\frac{\mathrm{sn}_{k}(\tau)}{\mathrm{sn}_{k}^{\prime}(\tau)}\mathrm{d}\tau=\displaystyle
C+C_{1}\int_{1}^{s}\big{[}G(\tau)-k\big{]}\frac{\mathrm{sn}_{k}(\tau)}{\mathrm{sn}_{k}^{\prime}(\tau)}\mathrm{d}\tau.$

In our integral pinching assumptions, both for $k=0$ and for $k>0$ it holds
$(G-k)\mathrm{sn}_{k}/\mathrm{sn}_{k}^{\prime}\in L^{1}(+\infty)$, and thus
$\zeta\in L^{1}(+\infty)$. Next, we use (2.8) and the non-negativity of
$\zeta,B$ to obtain

$\begin{array}[]{lcl}\displaystyle\left(\frac{\zeta(s)\mathrm{sn}_{k}(s)}{\mathrm{sn}_{k}^{\prime}(s)}\right)^{\prime}&=&\displaystyle\big{[}G(s)-k-\zeta(s)B(s)\big{]}\frac{\mathrm{sn}_{k}(s)}{\mathrm{sn}_{k}^{\prime}(s)}+\zeta(s)\left[1-k\left(\frac{\mathrm{sn}_{k}(s)}{\mathrm{sn}_{k}^{\prime}(s)}\right)^{2}\right]\\\\[14.22636pt]
&\leq&\displaystyle\frac{\big{[}G(s)-k\big{]}\mathrm{sn}_{k}(s)}{\mathrm{sn}_{k}^{\prime}(s)}+\zeta(s)\in
L^{1}(+\infty),\end{array}$

hence $\zeta\mathrm{sn}_{k}/\mathrm{sn}_{k}^{\prime}\in
L^{\infty}(\mathbb{R}^{+})$ by integrating. This implies that the function $B$
in (2.8) satisfies $B\leq C\mathrm{sn}_{k}^{\prime}/\mathrm{sn}_{k}$ for some
constant $C>0$. Therefore, from (2.8) we get $\zeta^{\prime}\geq-\zeta
B\geq-C\zeta\mathrm{sn}_{k}^{\prime}/\mathrm{sn}_{k}$. Integrating on $[s,t]$
and using the monotonicity of $\mathrm{sn}_{k}^{\prime}/\mathrm{sn}_{k}$ we
obtain

$-C\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\int_{s}^{t}\zeta\leq\zeta(t)-\zeta(s).$

Since $\zeta\in L^{1}(\mathbb{R}^{+})$, we can choose a divergent sequence
$\\{t_{j}\\}$ such that $\zeta(t_{j})\rightarrow 0$ as $j\rightarrow+\infty$.
Setting $t=t_{j}$ into the above inequality and taking limits we deduce

$\zeta(s)\leq
C\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\int_{s}^{+\infty}\zeta,$

thus letting $s\rightarrow+\infty$ we get the second relation in (2.7). ∎

###  2.2. A transversality lemma

This subsection is devoted to an estimate of the measure of the critical set

$S_{t,s}=\Big{\\{}x\in M\ :\ t\leq r(x)\leq s,\ |\nabla r(x)|=0\Big{\\}},$

with the purpose of justifying some coarea’s formulas for integrals over
extrinsic annuli. We begin with the next

######  Lemma 2.

Let $\varphi:M^{m}\rightarrow N^{n}$ be an isometric immersion, and let
$r(x)=\overline{\mathrm{dist}}(\varphi(x),\bar{o})$ be the extrinsic distance
function from $\bar{o}\in N$. Denote with $\Gamma_{\\!\sigma}\doteq\\{x\in
M;\;r(x)=\sigma\\}$. Then, for each $f\in L^{1}(\\{t\leq r\leq s\\})$,

(2.12) $\int_{\\{t\leq r\leq
s\\}}f\,\mathrm{d}x=\int_{S_{t,s}}\\!f\,\mathrm{d}x+\int_{t}^{s}\left[\int_{\Gamma_{\\!\sigma}}\frac{f}{|\nabla
r|}\right]\mathrm{d}\sigma.$

In particular, if

(2.13) $\mathrm{vol}(S_{t,s})=0,$

then

(2.14) $\int_{\\{t\leq r\leq
s\\}}f\,\mathrm{d}x=\int_{t}^{s}\left[\int_{\Gamma_{\\!\sigma}}\frac{f}{|\nabla
r|}\right]\mathrm{d}\sigma.$

###### Proof.

We prove (2.12) for $f\geq 0$, and the general case follows by considering the
positive and negative part of $f$. By the coarea’s formula, we know that for
each $g\in L^{1}(\\{t\leq r\leq s\\})$,

(2.15) $\int_{\\{t\leq r\leq s\\}}g|\nabla
r|\,\mathrm{d}x=\int_{t}^{s}\left[\int_{\Gamma_{\\!\sigma}}g\right]\mathrm{d}\sigma.$

Fix $j$ and consider $A_{j}=\\{|\nabla r|>1/j\\}$ and the function

$g=f1_{A_{j}}/|\nabla r|\in L^{1}(\\{t\leq r\leq s\\}).$

Applying (2.15), letting $j\rightarrow+\infty$ and using the monotone
convergence theorem we deduce

(2.16) $\int_{\\{t\leq r\leq s\\}\backslash
S_{t,s}}f\,\mathrm{d}x=\int_{t}^{s}\left[\int_{\Gamma_{\\!\sigma}\backslash
S_{t,s}}\frac{f}{|\nabla
r|}\right]\mathrm{d}\sigma=\int_{t}^{s}\left[\int_{\Gamma_{\\!\sigma}}\frac{f}{|\nabla
r|}\right]\mathrm{d}\sigma,$

where the last equality follows since $\Gamma_{\sigma}\cap S_{t,s}=\emptyset$
for a.e. $\sigma\in[t,s]$, in view of Sard’s theorem. Formula (2.12) follows
at once.

∎

Let now $N$ possess a pole $\bar{o}$ and satisfy (1.9), and consider a minimal
immersion $\varphi:M\rightarrow N$. Since, by the Hessian comparison theorem,
geodesic spheres in $N$ centered at $\bar{o}$ are positively curved, it is
reasonable to expect that the “transversality” condition (2.13) holds. This is
the content of the next

######  Proposition 2.

Let $\varphi:M^{m}\rightarrow N^{n}$ be a minimal immersion, where $N$
possesses a pole $\bar{o}$ and satisfies (1.9). Then,

(2.17) $\mathrm{vol}(S_{0,+\infty})=0.$

###### Proof.

Suppose by contradiction that $\mathrm{vol}(S_{0,+\infty})>0$. By Stampacchia
and Rademacher’s theorems,

(2.18) $\nabla|\nabla r|(x)=0\qquad\text{for a.e. }\,x\in S_{0,+\infty}.$

Pick one such $x$ and a local Darboux frame $\\{e_{i}\\},\\{e_{\alpha}\\}$,
$1\leq i\leq m$, $m+1\leq\alpha\leq n$ around $x$, that is, $\\{e_{i}\\}$ is a
local orthonormal frame for $TM$ and $\\{e_{\alpha}\\}$ is a local orthonormal
frame for the normal bundle $TM^{\perp}$. Since $\nabla r(x)=0$, then
$\bar{\nabla}\bar{\rho}(x)\in T_{x}M^{\perp}$. Up to rotating
$\\{e_{\alpha}\\}$, we can suppose that $\bar{\nabla}\bar{\rho}(x)=e_{n}(x)$.
Fix $i$ and consider a unit speed geodesics
$\gamma:(-\varepsilon,\varepsilon)\rightarrow M$ such that $\gamma(0)=x$,
$\dot{\gamma}(0)=e_{i}$. Identify $\gamma$ with its image $\varphi\circ\gamma$
in $N$. By Taylor’s formula and (2.18),

$|\nabla r|(\gamma(t))=o(t)\qquad\text{as }\,t\rightarrow 0^{+}.$

Using that $|\nabla
r|=\sqrt{1-\sum_{\alpha}(\bar{\nabla}\bar{\rho},e_{\alpha})^{2}}$, we deduce

(2.19)
$1-\sum_{\alpha}(\bar{\nabla}\bar{\rho},e_{\alpha})_{\gamma(t)}^{2}=o(t^{2}).$

Since $\bar{\nabla}\bar{\rho}(x)=e_{n}(x)$, we deduce from (2.20) that also

(2.20) $u(t)\doteq 1-(\bar{\nabla}\bar{\rho},e_{n})_{\gamma(t)}^{2}=o(t^{2}),$

thus $\dot{u}(0)=\ddot{u}(0)=0$. Computing,

$\begin{array}[]{lcl}\dot{u}(t)&=&2(\bar{\nabla}\bar{\rho},e_{n})\left[(\bar{\nabla}_{\dot{\gamma}}\bar{\nabla}\bar{\rho},e_{n})+(\bar{\nabla}\bar{\rho},\bar{\nabla}_{\dot{\gamma}}e_{n})\right]\\\\[5.69046pt]
\ddot{u}(t)&=&2\left[(\bar{\nabla}_{\dot{\gamma}}\bar{\nabla}\bar{\rho},e_{n})+(\bar{\nabla}\bar{\rho},\bar{\nabla}_{\dot{\gamma}}e_{n})\right]^{2}\\\\[5.69046pt]
&&+2(\bar{\nabla}\bar{\rho},e_{n})\left[\displaystyle(\bar{\nabla}_{\dot{\gamma}}\bar{\nabla}_{\dot{\gamma}}\bar{\nabla}\bar{\rho},e_{n})+2(\bar{\nabla}_{\dot{\gamma}}\bar{\nabla}\bar{\rho},\bar{\nabla}_{\dot{\gamma}}e_{n})+(\bar{\nabla}\bar{\rho},\bar{\nabla}_{\dot{\gamma}}\bar{\nabla}_{\dot{\gamma}}e_{n})\right].\end{array}$

Evaluating at $t=0$ we deduce

$0=\ddot{u}(0)/2=\displaystyle(\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho},\bar{\nabla}\bar{\rho})+2(\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho},\bar{\nabla}_{e_{i}}e_{n})+(e_{n},\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{i}}e_{n}).$

Differentiating twice $1=|e_{n}|^{2}=|\bar{\nabla}\bar{\rho}|^{2}$ along
$e_{i}$ we deduce the identities
$(e_{n},\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{i}}e_{n})=-|\bar{\nabla}_{e_{i}}e_{n}|^{2}$
and
$(\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho},\bar{\nabla}\bar{\rho})=-|\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho}|^{2}$,
hence

$0=\ddot{u}(0)/2=\displaystyle-|\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho}|^{2}+2(\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho},\bar{\nabla}_{e_{i}}e_{n})-|\bar{\nabla}_{e_{i}}e_{n}|^{2}=-\big{|}\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho}-\bar{\nabla}_{e_{i}}e_{n}\big{|}^{2},$

which implies
$\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho}=\bar{\nabla}_{e_{i}}e_{n}$.
Therefore, at $x$,

$(\mathrm{II}(e_{i},e_{i}),e_{n})=-(\bar{\nabla}_{e_{i}}e_{n},e_{i})=-(\bar{\nabla}_{e_{i}}\bar{\nabla}\bar{\rho},e_{i})=\overline{\mathrm{Hess}\,}(\bar{\rho})(e_{i},e_{i}).$

Tracing with respect to $i$, using that $M$ is minimal and (2.5) we conclude
that

$0\geq\frac{\mathrm{sn}_{k}^{\prime}(r(x))}{\mathrm{sn}_{k}(r(x))}(m-|\nabla
r(x)|^{2})=m\frac{\mathrm{sn}_{k}^{\prime}(r(x))}{\mathrm{sn}_{k}(r(x))}>0,$

a contradiction. ∎

##  3\. Monotonicity formulae and conditions equivalent to
$\Theta(+\infty)<+\infty$

Our first step is to improve the classical monotonicity formula for
$\Theta(r)$, that can be found in [48] (for $N=\mathbb{R}^{n}$) and [2] (for
$N=\mathbb{H}^{n}_{k}$). For $k\geq 0$, let $v_{k},V_{k}$ denote the volume
function, respectively, of geodesic spheres and balls in the space form of
sectional curvature $-k$ and dimension $m$, i.e.,

(3.1) $v_{k}(s)=\omega_{m-1}\mathrm{sn}_{k}(s)^{m-1},\qquad
V_{k}(s)=\int_{0}^{s}v_{k}(\sigma)\mathrm{d}\sigma,$

where $\omega_{m-1}$ is the volume of the unit sphere $\mathbb{S}^{m-1}$.
Although we shall not use all the four monotone quantities in (3.3) below,
nevertheless they have independent interest, and for this reason we state the
result in its full strength. We define the _flux_ $J(s)$ of $\nabla r$ over
the extrinsic sphere $\Gamma_{s}$:

(3.2) $J(s)\doteq\frac{1}{v_{k}(s)}\int_{\Gamma_{s}}|\nabla r|.$

######  Proposition 3 (The monotonicity formulae).

Suppose that $N$ has a pole $\bar{o}$ and satisfies (1.9), and let
$\varphi:M^{m}\rightarrow N^{n}$ be a proper minimal immersion. Then, the
functions

(3.3) $\Theta(s),\qquad\frac{1}{V_{k}(s)}\int_{\\{0\leq r\leq s\\}}|\nabla
r|^{2}$

are absolutely continuous and monotone non-decreasing. Moreover, $J(s)$
coincides, on an open set of full measure, with the absolutely continuous
function

$\bar{J}(s)\doteq\frac{1}{v_{k}(s)}\int_{\\{r\leq s\\}}\Delta r$

and $\bar{J}(s)$, $V_{k}(s)\big{[}\bar{J}(s)-\Theta(s)\big{]}$ are non-
decreasing. In particular, $J(s)\geq\Theta(s)$ a.e. on $\mathbb{R}^{+}$.

######  Remark 4.

_To the best of our knowledge, the monotonicity of $J(s)$ (aside from its
differentiability properties) has first been shown, in the Euclidean setting,
in a paper by V. Tkachev [51]. _

###### Proof.

We first observe that, in view of Lemma 2 and Proposition 2 applied with
$f=\Delta r$,

(3.4) $v_{k}(s)\bar{J}(s)\doteq\int_{\\{r\leq s\\}}\Delta
r\equiv\int_{0}^{s}\left[\int_{\Gamma_{\sigma}}\frac{\Delta r}{|\nabla
r|}\right]\mathrm{d}\sigma$

is absolutely continuous, and by the divergence theorem it coincides with
$v_{k}(s)J(s)$ whenever $s$ is a regular value of $r$. Consider

(3.5)
$f(s)=\int_{0}^{s}\frac{V_{k}(\sigma)}{v_{k}(\sigma)}\mathrm{d}\sigma=\int_{0}^{s}\frac{1}{v_{k}(\sigma)}\left[\int_{0}^{\sigma}v_{k}(\tau)\mathrm{d}\tau\right]\mathrm{d}\sigma$

which is a $C^{2}$ solution of

$f^{\prime\prime}+(m-1)\frac{\mathrm{sn}_{k}^{\prime}}{\mathrm{sn}_{k}}f^{\prime}=1\quad\text{on
}\,\mathbb{R}^{+},\quad f(0)=0,\quad f^{\prime}(0)=0,$

and define $\psi(x)=f(r(x))\in C^{2}(M)$. Let $\\{e_{i}\\}$ be a local
orthonormal frame on $M$. Since $\varphi$ is minimal, by the chain rule and
the lower bound in the Hessian comparison theorem 2.5

(3.6) $\Delta
r=\sum_{j=1}^{m}\overline{\mathrm{Hess}\,}(\bar{\rho})\big{(}\mathrm{d}\varphi(e_{j}),\mathrm{d}\varphi(e_{j})\big{)}\geq\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}\big{(}m-|\nabla
r|^{2}\big{)}.$

We then compute

(3.7) $\begin{array}[]{lcl}\displaystyle\Delta\psi&=&\displaystyle
f^{\prime\prime}|\nabla r|^{2}+f^{\prime}\Delta r\geq\displaystyle
f^{\prime\prime}|\nabla
r|^{2}+f^{\prime}\frac{\mathrm{sn}_{k}^{\prime}}{\mathrm{sn}_{k}}(m-|\nabla
r|^{2})\\\\[8.5359pt] &=&\displaystyle\displaystyle 1+\left(1-|\nabla
r|^{2}\right)\left(f^{\prime}(r)\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}-f^{\prime\prime}(r)\right).\end{array}$

It is not hard to show that the function

$z(s)\doteq
f^{\prime}(s)\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}-f^{\prime\prime}(s)=\frac{m}{m-1}\frac{V_{k}(s)v_{k}^{\prime}(s)}{v_{k}^{2}(s)}-1.$

is non-negative and non-decreasing on $\mathbb{R}^{+}$. Indeed, from

(3.8) $z(0)=0,\qquad
z^{\prime}(s)=\frac{m}{v_{k}(s)}\left[kV_{k}(s)-\frac{1}{m-1}v_{k}^{\prime}(s)z(s)\right]$

we deduce that $z^{\prime}>0$ when $z<0$, which proves that $z\geq 0$ on
$\mathbb{R}^{+}$. Fix $0<t<s$ regular values for $r$. Integrating (3.7) on the
smooth compact set $\\{t\leq r\leq s\\}$ and using the divergence theorem we
deduce

(3.9) $\frac{V_{k}(s)}{v_{k}(s)}\int_{\Gamma_{\\!s}}|\nabla
r|-\frac{V_{k}(t)}{v_{k}(t)}\int_{\Gamma_{t}}|\nabla
r|\geq\mathrm{vol}\big{(}\\{t\leq r\leq s\\}\big{)}.$

By the definition of $J(s)$ and $\Theta(s)$, and since $J(s)\equiv\bar{J}(s)$
for regular values, the above inequality rewrites as follows:

$V_{k}(s)\bar{J}(s)-V_{k}(t)\bar{J}(t)\geq
V_{k}(s)\Theta(s)-V_{k}(t)\Theta(t),$

or in other words,

$V_{k}(s)\big{[}\bar{J}(s)-\Theta(s)\big{]}\geq
V_{k}(t)\big{[}\bar{J}(t)-\Theta(t)\big{]}.$

Since all the quantities involved are continuous, the above relation extends
to all $t,s\in\mathbb{R}^{+}$, which proves the monotonicity of
$V_{k}[\bar{J}-\Theta]$. Letting $t\rightarrow 0$ we then deduce that
$\bar{J}(s)\geq\Theta(s)$ on $\mathbb{R}^{+}$. Next, by using $f\equiv 1$ and
$f\equiv|\nabla r|^{2}$ in Lemma 2 and exploiting again Proposition 2 we get

(3.10) $\mathrm{vol}\big{(}\\{t\leq r\leq
s\\}\big{)}=\int_{t}^{s}\left[\int_{\Gamma_{\\!\sigma}}\frac{1}{|\nabla
r|}\right]\mathrm{d}\sigma,\qquad\int_{\\{0\leq r\leq s\\}}|\nabla
r|^{2}=\int_{0}^{s}\left[\int_{\Gamma_{\sigma}}|\nabla
r|\right]\mathrm{d}\sigma,$

showing that the two quantities in (3.3) are absolutely continuous. Plugging
into (3.9), letting $t\rightarrow 0$ and using that $z\geq 0$ we deduce

(3.11) $\frac{V_{k}(s)}{v_{k}(s)}\int_{\Gamma_{\\!s}}|\nabla
r|\geq\int_{0}^{s}\left[\int_{\Gamma_{\\!\sigma}}\frac{1}{|\nabla
r|}\right]\mathrm{d}\sigma,$

for regular $s$, which together with the trivial inequality $|\nabla
r|^{-1}\geq|\nabla r|$ and with (3.10) gives

(3.12) $\begin{array}[]{l}\quad\displaystyle
V_{k}(s)\int_{\Gamma_{\\!s}}|\nabla r|\geq
v_{k}(s)\int_{0}^{s}\left[\int_{\Gamma_{\\!\sigma}}|\nabla
r|\right]\mathrm{d}\sigma,\\\\[11.38092pt] \quad\displaystyle
V_{k}(s)\left[\frac{\mathrm{d}}{\mathrm{d}s}\mathrm{vol}\big{(}\\{r\leq
s\\}\big{)}\right]\geq v_{k}(s)\mathrm{vol}\big{(}\\{r\leq
s\\}\big{)}.\end{array}$

Integrating the second inequality we obtain the monotonicity of $\Theta(s)$,
while integrating the first one and using (3.10) we obtain the monotonicity of
the second quantity in (3.3). To show the monotonicity of $\bar{J}(s)$, by
(3.6) and using the full information coming from (2.5) we obtain

(3.13) $\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}\big{(}m-|\nabla
r|^{2}\big{)}\leq\Delta r\leq\frac{g^{\prime}(r)}{g(r)}\big{(}m-|\nabla
r|^{2}\big{)}.$

In view of the identity (3.4), we consider regular $s>0$, we divide (3.13) by
$|\nabla r|$ and integrate on $\Gamma_{s}$ to get

(3.14)
$\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\int_{\Gamma_{\\!s}}\frac{m-|\nabla
r|^{2}}{|\nabla
r|}\leq\big{(}v_{k}(s)\bar{J}(s)\big{)}^{\prime}\leq\frac{g^{\prime}(s)}{g(s)}\int_{\Gamma_{\\!s}}\frac{m-|\nabla
r|^{2}}{|\nabla r|}.$

Writing $m-|\nabla r|^{2}=m(1-|\nabla r|^{2})+(m-1)|\nabla r|^{2}$, setting
for convenience

(3.15) $v_{g}(s)=\omega_{m-1}g(s)^{m-1},\qquad
T(s)\doteq\frac{\int_{\Gamma_{\\!s}}|\nabla
r|^{-1}}{\int_{\Gamma_{\\!s}}|\nabla r|}-1,$

rearranging we deduce the two inequalities

(3.16)
$\begin{array}[]{rcl}\big{(}v_{k}(s)\bar{J}(s)\big{)}^{\prime}&\geq&\displaystyle
v_{k}^{\prime}(s)\bar{J}(s)+m\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}T(s)v_{k}(s)\bar{J}(s)\\\\[11.38092pt]
\big{(}v_{k}(s)\bar{J}(s)\big{)}^{\prime}&\leq&\displaystyle\frac{v_{g}^{\prime}(s)}{v_{g}(s)}v_{k}(s)\bar{J}(s)+m\frac{g^{\prime}(s)}{g(s)}T(s)v_{k}(s)\bar{J}(s).\end{array}$

Expanding the derivative on the left-hand side, we deduce

(3.17) $\begin{array}[]{rcl}\bar{J}^{\prime}(s)&\geq&\displaystyle
m\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}T(s)\bar{J}(s),\\\\[11.38092pt]
\displaystyle\left(\frac{v_{k}(s)}{v_{g}(s)}\bar{J}(s)\right)^{\prime}&\leq&\displaystyle
m\frac{g^{\prime}(s)}{g(s)}T(s)\left(\frac{v_{k}(s)}{v_{g}(s)}\bar{J}(s)\right).\end{array}$

The first inequality together with the non-negativity of $T$ implies the
desired $\bar{J}^{\prime}\geq 0$, concluding the proof. The second inequality
in (3.17), on the other hand, will be useful in awhile. ∎

######  Remark 5.

_The properness of $\varphi$ is essential in the above proof to justify
integrations by parts. However, if $\varphi$ is non-proper, at least when $N$
is Cartan-Hadamard with sectional curvature $\bar{K}\leq-k$ the function
$\Theta$ is still monotone in an extended sense. In fact, as it has been
observed in [54] for $N=\mathbb{R}^{m+1}$, $\Theta(s)=+\infty$ for each $s$
such that $\\{r<s\\}$ contains a limit point of $\varphi$. Briefly, if
$\bar{x}\in N$ is a limit point with $\bar{\rho}(\bar{x})<s$, choose
$\varepsilon>0$ such that $2\varepsilon<s-\bar{\rho}(\bar{x})$, and a
diverging sequence $\\{x_{j}\\}\subset M$ such that
$\varphi(x_{j})\rightarrow\bar{x}$. We can assume that the balls
$B_{\varepsilon}(x_{j})\subset M$ are pairwise disjoint. Since
$\overline{\mathrm{dist}}(\varphi(x),\varphi(x_{j}))\leq\mathrm{dist}(x,x_{j})$,
we deduce that $\varphi(B_{\varepsilon}(x_{j}))\subset\\{r<s\\}$ for $j$ large
enough, and thus_

$\mathrm{vol}\big{(}\\{r\leq
s\\}\big{)}\geq\sum_{j}\mathrm{vol}(B_{\varepsilon}(x_{j})).$

_However, using that $\bar{K}\leq-k$ and since $N$ is Cartan-Hadamard, we can
apply the intrinsic monotonicity formula (see Proposition 7 in Appendix 2
below) with chosen origin $\varphi(x_{j})$ to deduce that
$\mathrm{vol}(B_{\varepsilon}(x_{j}))\geq V_{k}(\varepsilon)$ for each $j$,
whence $\mathrm{vol}(\\{r\leq s\\})=+\infty$. _

We next investigate conditions equivalent to the finiteness of the density.

######  Proposition 4.

Suppose that $N$ has a pole and satisfies (1.9). Let $\varphi:M^{m}\rightarrow
N^{n}$ be a proper minimal immersion. Then, the following properties are
equivalent:

  * (1)

$\Theta(+\infty)<+\infty$;

  * (2)

$\bar{J}(+\infty)<+\infty$.

Moreover, both $(1)$ and $(2)$ imply that

($3$)
$\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\left[\frac{\int_{\Gamma_{\\!s}}|\nabla
r|^{-1}}{\int_{\Gamma_{\\!s}}|\nabla r|}-1\right]\in L^{1}(\mathbb{R}^{+}).$

If further $N$ has an integral pinching to $\mathbb{R}^{n}$ or
$\mathbb{H}^{n}_{k}$, then $(1)\Leftrightarrow(2)\Leftrightarrow(3)$.

###### Proof.

We refer to the proof of the previous proposition for notation and formulas.
$(2)\Rightarrow(1)$ is obvious since, by the previous proposition,
$\bar{J}(s)\geq\Theta(s)$.
$(1)\Rightarrow(2)$. Note that the limit in $(2)$ exists since $\bar{J}$ is
monotone. Suppose by contradiction that $\bar{J}(+\infty)=+\infty$, let $c>0$
and fix $s_{c}$ large enough that $\bar{J}(s)\geq c$ for $s\geq s_{c}$. From
(3.10) and (3.2), and since $\bar{J}\equiv J$ a.e.,

$\begin{array}[]{lcl}\Theta(s)&=&\displaystyle\frac{1}{V_{k}(s)}\int_{0}^{s}\left[\int_{\Gamma_{\\!\sigma}}\frac{1}{|\nabla
r|}\right]\mathrm{d}\sigma\geq\frac{1}{V_{k}(s)}\int_{0}^{s}v_{k}(\sigma)J(\sigma)\mathrm{d}\sigma\\\\[14.22636pt]
&\geq&\displaystyle\frac{1}{V_{k}(s)}\int^{s}_{s_{c}}v_{k}(\sigma)J(\sigma)\mathrm{d}\sigma\geq
c\frac{V_{k}(s)-V_{k}(s_{c})}{V_{k}(s)}.\end{array}$

Letting $s\rightarrow+\infty$ we get $\Theta(+\infty)\geq c$, hence
$\Theta(+\infty)=+\infty$ by the arbitrariness of $c$, contradicting $(1)$.
$(2)\Rightarrow(3)$. Integrating (3.17) on $[1,s]$ we obtain

(3.18)
$c_{1}\exp\left\\{m\int_{1}^{s}\frac{\mathrm{sn}_{k}^{\prime}(\sigma)}{\mathrm{sn}_{k}(\sigma)}T(\sigma)\mathrm{d}\sigma\right\\}\leq\bar{J}(s)\leq
c_{2}\frac{v_{g}(s)}{v_{k}(s)}\exp\left\\{m\int_{1}^{s}\left[\frac{g^{\prime}(\sigma)}{g(\sigma)}\right]T(\sigma)\mathrm{d}\sigma\right\\},$

for some constants $c_{1},c_{2}>0$, where $v_{g}(s),T(s)$ is as in (3.15). The
validity of $(2)$ and the first inequality show that
$\mathrm{sn}_{k}^{\prime}T/\mathrm{sn}_{k}\in L^{1}(+\infty)$, that is, $(3)$
is satisfied.
$(3)\Rightarrow(2)$. In our pinching assumptions on $N$, $(ii)$ in Proposition
1 gives

$\frac{g^{\prime}}{g}=\frac{\mathrm{sn}_{k}^{\prime}}{\mathrm{sn}_{k}}+\zeta,\quad\text{with}\quad\zeta\leq
C\frac{\mathrm{sn}_{k}^{\prime}}{\mathrm{sn}_{k}}\ \text{ on
}\,\mathbb{R}^{+},\quad\text{and}\quad g\leq C\mathrm{sn}_{k}\ \text{ on
}\,\mathbb{R}^{+},$

for some $C>0$. Plugging into (3.18) and recalling the definition of $v_{g}$
we obtain

$\displaystyle\bar{J}(s)\leq
c_{3}\exp\left\\{c_{4}\int_{1}^{s}\left[\frac{\mathrm{sn}_{k}^{\prime}(\sigma)}{\mathrm{sn}_{k}(\sigma)}\right]T(\sigma)\mathrm{d}\sigma\right\\},$

for some $c_{3},c_{4}>0$, and $(3)\Rightarrow(2)$ follows by letting
$s\rightarrow+\infty$. ∎

######  Remark 6.

_A version of Propositions 3 and 4 that covers most of the material presented
above has also been independently proved in the very recent [27], see Theorems
2.1 and 6.1 therein. We mention that their results are stated for more general
ambient spaces subjected to specific function-theoretic requirements, and
that, in Proposition 4, it holds in fact
$\bar{J}(+\infty)\equiv\Theta(+\infty)$. For an interesting characterization,
when $N=\mathbb{R}^{n}$, of the limit $\bar{J}(+\infty)$ in terms of an
invariant called the projective volume of $M$ we refer to [51]. _

##  4\. Proof of Theorem 1

Let $M^{m}$ be a minimal properly immersed submanifold in $N^{n}$, and suppose
that $N$ has a pointwise or integral pinching to a space form. Because of the
upper bound in (1.9), by [18] and [5] the bottom of $\sigma(M)$ satisfies

(4.1) $\inf\sigma(M)\geq\frac{(m-1)^{2}k}{4}.$

Briefly, the lower bound in (3.13) implies

$\Delta
r\geq(m-1)\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}\geq(m-1)\sqrt{k}\qquad\text{on
}\,M.$

Integrating on a relatively compact, smooth open set $\Omega$ and using the
divergence theorem and $|\nabla r|\leq 1$, we deduce
$\mathcal{H}^{m-1}(\partial\Omega)\geq(m-1)\sqrt{k}\mathrm{vol}(\Omega)$. The
desired (4.1) then follows from Cheeger’s inequality:

$\inf\sigma(M)\geq\frac{1}{4}\left(\inf_{\Omega\Subset
M}\frac{\mathcal{H}^{m-1}(\partial\Omega)}{\mathrm{vol}(\Omega)}\right)^{2}\geq\frac{(m-1)^{2}k}{4}.$

To complete the proof of the theorem, since $\sigma(M)$ is closed it is
sufficient to show that each $\lambda>(m-1)^{2}k/4$ lies in $\sigma(M)$.

Set for convenience $\beta\doteq\sqrt{\lambda-(m-1)^{2}k/4}$ and, for $0\leq
t<s$, let $A_{t,s}$ denote the extrinsic annulus

$A_{t,s}\doteq\big{\\{}x\in M\ :\ r(x)\in[t,s]\big{\\}}.$

Define the weighted measure $\mathrm{d}\mu_{k}\doteq v_{k}(r)^{-1}\mathrm{d}x$
on $\\{r\geq 1\\}$. Hereafter, we will always restrict to this set. Consider

(4.2) $\psi(s)\doteq\frac{e^{i\beta s}}{\sqrt{v_{k}(s)}},\qquad\text{which
solves}\qquad\psi^{\prime\prime}+\psi^{\prime}\frac{v_{k}^{\prime}}{v_{k}}+\lambda\psi=a(s)\psi,$

where

(4.3)
$a(s)\doteq\frac{(m-1)^{2}k}{4}+\frac{1}{4}\left(\frac{v_{k}^{\prime}(s)}{v_{k}(s)}\right)^{2}-\frac{1}{2}\frac{v_{k}^{\prime\prime}(s)}{v_{k}(s)}\rightarrow
0$

as $s\rightarrow+\infty$. For technical reasons, fix $R>1$ large such that
$\Theta(R)>0$. Fix $t,s,S$ such that

$R+1<t<s<S-1,$

and let $\eta\in C^{\infty}_{c}(\mathbb{R})$ be a cut-off function satisfying

$\begin{array}[]{l}\displaystyle 0\leq\eta\leq 1,\quad\eta\equiv 0\ \text{
outside of }\,(t-1,S),\quad\eta\equiv 1\ \text{ on }\,(t,s),\\\\[5.69046pt]
|\eta^{\prime}|+|\eta^{\prime\prime}|\leq C_{0}\ \ \text{ on
}\,[t-1,s],\qquad|\eta^{\prime}|+|\eta^{\prime\prime}|\leq\frac{C_{0}}{S-s}\ \
\text{ on }\,[s,S]\end{array}$

for some absolute constant $C_{0}$ (the last relation is possible since
$S-s\geq 1$). The value $S$ will be chosen later in dependence of $s$. Set
$u_{t,s}\doteq\eta(r)\psi(r)\in C^{\infty}_{c}(M)$. Then, by (4.2),

$\begin{array}[]{lcl}\Delta u_{t,s}+\lambda
u_{t,s}&=&\displaystyle(\eta^{\prime\prime}\psi+2\eta^{\prime}\psi^{\prime}+\eta\psi^{\prime\prime})|\nabla
r|^{2}+(\eta^{\prime}\psi+\eta\psi^{\prime})\Delta
r+\lambda\eta\psi\\\\[5.69046pt]
&=&\displaystyle\left(\eta^{\prime\prime}\psi+2\eta^{\prime}\psi^{\prime}-\frac{v_{k}^{\prime}}{v_{k}}\eta\psi^{\prime}-\lambda\eta\psi+a\eta\psi\right)(|\nabla
r|^{2}-1)+a\eta\psi\\\\[8.5359pt]
&&\displaystyle+(\eta^{\prime}\psi+\eta\psi^{\prime})\left(\Delta
r-\frac{v_{k}^{\prime}}{v_{k}}\right)+\displaystyle\left(\eta^{\prime\prime}\psi+2\eta^{\prime}\psi^{\prime}+\eta^{\prime}\psi\frac{v_{k}^{\prime}}{v_{k}}\right).\\\\[8.5359pt]
\end{array}$

Using that there exists an absolute constant $c$ for which
$|\psi|+|\psi^{\prime}|\leq c/\sqrt{v_{k}}$, the following inequality holds:

$\begin{array}[]{lcl}\|\Delta u_{t,s}+\lambda
u_{t,s}\|^{2}_{2}&\leq&\displaystyle C\left(\int_{A_{t-1,S}}\left[(1-|\nabla
r|^{2})^{2}+\left(\Delta
r-\frac{v_{k}^{\prime}}{v_{k}}\right)^{2}+a(r)^{2}\right]\mathrm{d}\mu_{k}\right.\\\\[14.22636pt]
&&\displaystyle\left.+\frac{\mu_{k}(A_{s,S})}{(S-s)^{2}}+\mu_{k}(A_{t-1,t})\right),\end{array}$

for some suitable $C$ depending on $c,C_{0}$. Since
$\|u_{t,s}\|^{2}_{2}\geq\mu_{k}(A_{t,s})$ and $(1-|\nabla r|^{2})^{2}\leq
1-|\nabla r|^{2}$, we obtain

(4.4) $\begin{array}[]{lcl}\displaystyle\frac{\|\Delta u_{t,s}+\lambda
u_{t,s}\|^{2}_{2}}{\|u_{t,s}\|_{2}^{2}}&\leq&\displaystyle
C\left(\frac{1}{\mu_{k}(A_{t,s})}\int_{A_{t-1,S}}\left[1-|\nabla
r|^{2}+\left(\Delta
r-\frac{v_{k}^{\prime}}{v_{k}}\right)^{2}+a(r)^{2}\right]\mathrm{d}\mu_{k}\right.\\\\[14.22636pt]
&&\displaystyle\left.+\frac{1}{(S-s)^{2}}\frac{\mu_{k}(A_{s,S})}{\mu_{k}(A_{t,s})}+\frac{\mu_{k}(A_{t-1,t})}{\mu_{k}(A_{t,s})}\right)\end{array}$

Next, using (2.5),

$\begin{array}[]{lcl}\Delta
r&=&\displaystyle\sum_{j=1}^{m}\overline{\mathrm{Hess}\,}(\bar{\rho})(e_{i},e_{i})=\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}(m-|\nabla
r|^{2})+\mathcal{P}(x)\\\\[8.5359pt]
&=&\displaystyle\frac{v_{k}^{\prime}(r)}{v_{k}(r)}+\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}(1-|\nabla
r|^{2})+\mathcal{P}(x),\end{array}$

where, by Proposition 1,

(4.5)
$\begin{array}[]{lcl}0\leq\mathcal{P}(x)&\doteq&\displaystyle\sum_{j=1}^{m}\overline{\mathrm{Hess}\,}(\bar{\rho})(e_{i},e_{i})-\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}(m-|\nabla
r|^{2})\\\\[8.5359pt]
&\leq&\displaystyle\left(\frac{g^{\prime}(r)}{g(r)}-\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}\right)(m-|\nabla
r|^{2})=\zeta(r)(m-|\nabla r|^{2})\leq m\zeta(r).\end{array}$

We thus obtain, on the set $\\{r\geq 1\\}$,

(4.6) $\begin{array}[]{lcl}\displaystyle\left(\Delta
r-\frac{v_{k}^{\prime}}{v_{k}}\right)^{2}+1-|\nabla
r|^{2}+a(r)^{2}&\leq&\displaystyle\left[\frac{\mathrm{sn}_{k}^{\prime}(r)}{\mathrm{sn}_{k}(r)}(1-|\nabla
r|^{2})+m\zeta(r)\right]^{2}\\\\[11.38092pt] &&\displaystyle+1-|\nabla
r|^{2}+a(r)^{2}\\\\[8.5359pt] &\leq&\displaystyle
C\Big{(}\zeta(r)^{2}+1-|\nabla r|^{2}+a(r)^{2}\Big{)}\end{array}$

for some absolute constant $C$. Note that, in both our pointwise or integral
pinching assumptions on $N$, by Proposition 1 it holds $\zeta(s)\rightarrow 0$
as $s\rightarrow+\infty$. Set

$F(t)\doteq\sup_{\sigma\in[t-1,+\infty)}[a(\sigma)^{2}+\zeta(\sigma)^{2}],$

and note that $F(t)\rightarrow 0$ monotonically as $t\rightarrow+\infty$.
Integrating (4.6) we get the existence of $C>0$ independent of $s,t$ such that

(4.7) $\begin{array}[]{l}\displaystyle\int_{A_{t-1,S}}\left[\left(\Delta
r-\frac{v_{k}^{\prime}}{v_{k}}\right)^{2}+1-|\nabla
r|^{2}+a(r)^{2}\right]\mathrm{d}\mu_{k}\\\\[14.22636pt]
\qquad\qquad\leq\displaystyle
C\left(F(t)\int_{A_{t-1,S}}\frac{1}{v_{k}(r)}+\int_{A_{t-1,S}}\frac{1-|\nabla
r|^{2}}{v_{k}(r)}\right).\end{array}$

Using the coarea’s formula and the transversality lemma, for each $0\leq a<b$

(4.8)
$\mu_{k}(A_{a,b})=\int_{A_{a,b}}\frac{1}{v_{k}(r)}=\int_{a}^{b}J\big{[}1+T\big{]},\qquad\int_{A_{a,b}}\frac{1-|\nabla
r|^{2}}{v_{k}(r)}=\int_{a}^{b}JT,$

where $J$ and $T$ are defined, respectively, in (3.2) and (3.15). Summarizing,
in view of (4.7) and (4.8) we deduce from (4.4) the following inequalities:

(4.9) $\begin{array}[]{lcl}\displaystyle\frac{\|\Delta u_{t,s}+\lambda
u_{t,s}\|^{2}_{2}}{\|u_{t,s}\|_{2}^{2}}&\leq&\displaystyle
C\left(\frac{1}{\int_{t}^{s}J\big{[}1+T\big{]}}\left[F(t)\int_{t-1}^{S}J\big{[}1+T\big{]}+\int_{t-1}^{S}JT\right]\right.\\\\[14.22636pt]
&&\displaystyle\left.+\frac{\int_{s}^{S}J\big{[}1+T\big{]}}{(S-s)^{2}\int_{t}^{s}J\big{[}1+T\big{]}}+\frac{\int_{t-1}^{t}J\big{[}1+T\big{]}}{\int_{t}^{s}J\big{[}1+T\big{]}}\right)\doteq\mathcal{Q}(t,s).\end{array}$

If we can guarantee that

(4.10)
$\liminf_{t\rightarrow+\infty}\liminf_{s\rightarrow+\infty}\frac{\|\Delta
u_{t,s}+\lambda u_{t,s}\|^{2}_{2}}{\|u_{t,s}\|_{2}^{2}}=0,$

then we are able to construct a sequence of approximating eigenfunctions for
$\lambda$ as follows: fix $\varepsilon>0$. By (4.10) there exists a divergent
sequence $\\{t_{i}\\}$ such that, for $i\geq i_{\varepsilon}$,

$\liminf_{s\rightarrow+\infty}\frac{\|\Delta u_{t_{i},s}+\lambda
u_{t_{i},s}\|^{2}_{2}}{\|u_{t_{i},s}\|_{2}^{2}}<\varepsilon/2.$

For $i=i_{\varepsilon}$, pick then a sequence $\\{s_{j}\\}$ realizing the
liminf. For $j\geq j_{\varepsilon}(i_{\varepsilon},\varepsilon)$

(4.11) $\|\Delta u_{t_{i},s_{j}}+\lambda
u_{t_{i},s_{j}}\|^{2}_{2}<\varepsilon\|u_{t_{i},s_{j}}\|_{2}^{2},$

Writing $u_{\varepsilon}\doteq u_{t_{i_{\varepsilon}},s_{j_{\varepsilon}}}$,
by (4.11) from the set $\\{u_{\varepsilon}\\}$ we can extract a sequence of
approximating eigenfunctions for $\lambda$, concluding the proof that
$\lambda\in\sigma(M)$. To show (4.10), by (4.9) it is enough to prove that

(4.12)
$\liminf_{t\rightarrow+\infty}\liminf_{s\rightarrow+\infty}\mathcal{Q}(t,s)=0.$

Suppose, by contradiction, that (4.12) were not true. Then, there exists a
constant $\delta>0$ such that, for each $t\geq t_{\delta}$,
$\liminf_{s\rightarrow+\infty}\mathcal{Q}(t,s)\geq 2\delta$, and thus for
$t\geq t_{\delta}$ and $s\geq s_{\delta}(t)$

(4.13)
$F(t)\int_{t-1}^{S}J\big{[}1+T\big{]}+\int_{t-1}^{S}JT+\int_{s}^{S}\frac{J\big{[}1+T\big{]}}{(S-s)^{2}}+\int_{t-1}^{t}J\big{[}1+T\big{]}\geq\delta\int_{t}^{s}J\big{[}1+T\big{]},$

and rearranging

(4.14)
$(F(t)+1)\int_{t-1}^{S}J\big{[}1+T\big{]}-\int_{t-1}^{S}J+\int_{s}^{S}\frac{J\big{[}1+T\big{]}}{(S-s)^{2}}+\int_{t-1}^{t}J\big{[}1+T\big{]}\geq\delta\int_{t}^{s}J\big{[}1+T\big{]}.$

We rewrite the above integrals in order to make $\Theta(s)$ appear.
Integrating by parts and using again the coarea’s formula and the
transversality lemma,

(4.15)
$\begin{array}[]{lcl}\displaystyle\int_{a}^{b}J\big{[}1+T\big{]}&=&\displaystyle\int_{A_{a,b}}\frac{1}{v_{k}(r)}=\displaystyle\int_{a}^{b}\frac{1}{v_{k}(\sigma)}\left[\int_{\Gamma_{\sigma}}\frac{1}{|\nabla
r|}\right]\mathrm{d}\sigma=\int_{a}^{b}\frac{\big{(}V_{k}(\sigma)\Theta(\sigma)\big{)}^{\prime}}{v_{k}(\sigma)}\mathrm{d}\sigma\\\\[14.22636pt]
&=&\displaystyle\frac{V_{k}(b)}{v_{k}(b)}\Theta(b)-\frac{V_{k}(a)}{v_{k}(a)}\Theta(a)+\int_{a}^{b}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta.\end{array}$

To deal with the term containing the integral of $J$ alone in (4.14), we use
the inequality $J(s)\geq\Theta(s)$ coming from the monotonicity formulae in
Proposition 3. This passage is crucial for us to conclude. Inserting (4.15)
and $J\geq\Theta$ into (4.14) we get

(4.16)
$\begin{array}[]{l}\displaystyle(F(t)+1)\displaystyle\frac{V_{k}(S)}{v_{k}(S)}\Theta(S)-(F(t)+1)\frac{V_{k}(t-1)}{v_{k}(t-1)}\Theta(t-1)+(F(t)+1)\int_{t-1}^{S}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta\\\\[14.22636pt]
\displaystyle-\int_{t-1}^{S}\Theta+\displaystyle\frac{1}{(S-s)^{2}}\left[\frac{V_{k}(S)}{v_{k}(S)}\Theta(S)-\frac{V_{k}(s)}{v_{k}(s)}\Theta(s)+\int_{s}^{S}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta\right]+\frac{V_{k}(t)}{v_{k}(t)}\Theta(t)\\\\[14.22636pt]
\displaystyle-\frac{V_{k}(t-1)}{v_{k}(t-1)}\Theta(t-1)+\int_{t-1}^{t}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta\\\\[14.22636pt]
\qquad\qquad\qquad\geq\quad\delta\displaystyle\frac{V_{k}(s)}{v_{k}(s)}\Theta(s)-\delta\frac{V_{k}(t)}{v_{k}(t)}\Theta(t)+\delta\int_{t}^{s}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta.\end{array}$

The idea to reach the desired contradiction is to prove that, as a consequence
of (4.16),

(4.17) $\int_{t-1}^{S}\Theta$

(hence, $\Theta(S)$) must grow faster as $S\rightarrow+\infty$ than the bound
in (1.10). To do so, we need to simplify (4.16) in order to find a suitable
differential inequality for (4.17).
We first observe that, both for $k>0$ and for $k=0$, there exists an absolute
constant $\hat{c}$ such that $\hat{c}^{-1}\leq
V_{k}v_{k}^{\prime}/v_{k}^{2}\leq\hat{c}$ on $[1,+\infty)$. Furthermore, by
the monotonicity of $\Theta$,

(4.18)
$\int_{s}^{S}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta\leq\hat{c}(S-s)\Theta(S).$

Next, we deal with the two terms in the left-hand side of (4.16) that involve
(4.17):

$\begin{array}[]{lcl}\displaystyle(F(t)+1)\int_{t-1}^{S}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta-\int_{t-1}^{S}\Theta&=&\displaystyle
F(t)\int_{t-1}^{S}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta+\int_{t-1}^{S}\frac{V_{k}v_{k}^{\prime}-v_{k}^{2}}{v_{k}^{2}}\Theta\\\\[14.22636pt]
&\leq&\displaystyle\hat{c}F(t)\int_{t-1}^{S}\Theta+\int_{t-1}^{S}\frac{V_{k}v_{k}^{\prime}-v_{k}^{2}}{v_{k}^{2}}\Theta.\end{array}$

The key point is the following relation:

(4.19)
$\frac{V_{k}(s)v_{k}^{\prime}(s)-v_{k}(s)^{2}}{v_{k}(s)^{2}}\left\\{\begin{array}[]{ll}=-1/m&\quad\text{if
}\,k=0;\\\\[5.69046pt] \rightarrow 0\ \text{as
}\,s\rightarrow+\infty,&\quad\text{if }\,k>0.\end{array}\right.$

Define

$\omega(t)\doteq\sup_{[t-1,+\infty)}\frac{V_{k}v_{k}^{\prime}-v_{k}^{2}}{v_{k}^{2}},\qquad\chi(t)\doteq\hat{c}F(t)+\omega(t).$

Again by the monotonicity of $\Theta$,

(4.20)
$\begin{array}[]{lcl}\displaystyle(F(t)+1)\int_{t-1}^{S}\frac{V_{k}v_{k}^{\prime}}{v_{k}^{2}}\Theta-\int_{t-1}^{S}\Theta&\leq&\displaystyle\big{[}\hat{c}F(t)+\omega(t)\big{]}\int_{t-1}^{S}\Theta=\chi(t)\int_{t-1}^{S}\Theta\\\\[8.5359pt]
&\leq&\displaystyle\chi(t)\Theta(t)+\chi(t)\int_{t}^{S}\Theta.\end{array}$

For simplicity, hereafter we collect all the terms independent of $s$ in a
function that we call $h(t)$, which may vary from line to line. Inserting
(4.18) and (4.20) into (4.16) we infer

(4.21)
$\begin{array}[]{l}\displaystyle\left[\left(F(t)+1+\frac{1}{(S-s)^{2}}\right)\frac{V_{k}(S)}{v_{k}(S)}+\frac{\hat{c}}{S-s}\right]\displaystyle\Theta(S)+\chi(t)\int_{t}^{S}\Theta\\\\[14.22636pt]
\displaystyle\geq
h(t)+\left(\delta+\frac{1}{(S-s)^{2}}\right)\frac{V_{k}(s)}{v_{k}(s)}\Theta(s)+\delta\hat{c}^{-1}\int_{t}^{s}\Theta.\end{array}$

Summing $\delta\hat{c}^{-1}(S-s)\Theta(S)$ to the two sides of the above
inequality, using the monotonicity of $\Theta$ and getting rid of the term
containing $\Theta(s)$ we obtain

(4.22)
$\begin{array}[]{l}\displaystyle\left[\left(F(t)+1+\frac{1}{(S-s)^{2}}\right)\frac{V_{k}(S)}{v_{k}(S)}+\frac{\hat{c}}{S-s}+\delta\hat{c}^{-1}(S-s)\right]\displaystyle\Theta(S)+\chi(t)\int_{t}^{S}\Theta\\\\[11.38092pt]
\displaystyle\geq h(t)+\delta\hat{c}^{-1}\int_{t}^{S}\Theta.\end{array}$

Using (4.19), the definition of $\chi(t)$ and the properties of
$\omega(t),F(t)$, we can choose $t_{\delta}$ sufficiently large to guarantee
that

(4.23) $\delta\hat{c}^{-1}-\chi(t)\geq
c_{k}\doteq\left\\{\begin{array}[]{ll}\frac{1}{m}+\frac{\delta\hat{c}^{-1}}{2}&\quad\text{if
}\,k=0,\\\\[5.69046pt] \frac{\delta\hat{c}^{-1}}{2}&\quad\text{if
}\,k>0,\end{array}\right.$

hence

(4.24)
$\displaystyle\left[\left(F(t)+1+\frac{1}{(S-s)^{2}}\right)\frac{V_{k}(S)}{v_{k}(S)}+\frac{\hat{c}}{S-s}+\delta\hat{c}^{-1}(S-s)\right]\displaystyle\Theta(S)\geq
h(t)+c_{k}\int_{t}^{S}\Theta.$

We now specify $S(s)$ depending on whether $k>0$ or $k=0$.
_The case $k>0$._
We choose $S\doteq s+1$. In view of the fact that $V_{k}/v_{k}$ is bounded
above on $\mathbb{R}^{+}$, (4.24) becomes

(4.25) $\bar{c}\Theta(s+1)\geq
h(t)+c_{k}\int_{t}^{s+1}\Theta\geq\frac{c_{k}}{2}\int_{t}^{s+1}\Theta,$

for some $\bar{c}$ independent of $t,s$. Note that the last inequality is
satisfied provided $s\geq s_{\delta}(t)$ is chosen to be sufficiently large,
since the monotonicity of $\Theta$ implies that $\Theta\not\in
L^{1}(\mathbb{R}^{+})$. Integrating and using again the monotonicity of
$\Theta$, we get

$(s+1-t)\Theta(s+1)\geq\int_{t}^{s+1}\Theta\geq\left[\int_{t}^{s_{0}+1}\Theta\right]\exp\left\\{\frac{c_{k}}{2\bar{c}}(s-s_{0})\right\\},$

hence $\Theta(s)$ grows exponentially. Ultimately, this contradicts our
assumption (1.10).
_The case $k=0$._
We choose $S\doteq s+\sqrt{s}$. Since $V_{k}(S)/v_{k}(S)=S/m$, from (4.24) we
infer

(4.26)
$\displaystyle\left[\left(F(t)+1+\frac{1}{s}\right)\frac{S}{m}+\frac{\hat{c}}{\sqrt{s}}+\delta\hat{c}^{-1}\sqrt{s}\right]\displaystyle\Theta(S)\geq
h(t)+c_{k}\int_{t}^{S}\Theta.$

Using the expression of $c_{k}$ and the fact that $F(t)\rightarrow 0$, up to
choosing $t_{\delta}$ and then $s_{\delta}(t)$ large enough we can ensure the
validity of the following inequality:

$\left[\left(F(t)+1+\frac{1}{s}\right)\frac{S}{m}+\frac{\hat{c}}{\sqrt{s}}+\delta\hat{c}^{-1}\sqrt{s}\right]<\left[\frac{1}{m}+\frac{\delta\hat{c}^{-1}}{4}\right]S=\left[c_{k}-\frac{\delta\hat{c}^{-1}}{4}\right]S$

for $t\geq t_{\delta}$ and $s\geq s_{\delta}(t)$. Plugging into (4.24), and
using that $\Theta\not\in L^{1}(\mathbb{R}^{+})$,

$S\Theta(S)\geq
h(t)+\frac{c_{k}}{c_{k}-\delta\hat{c}^{-1}/4}\int_{t}^{S}\Theta\geq(1+\varepsilon)\int_{t}^{S}\Theta,$

for a suitable $\varepsilon>0$ independent of $t,S$, and provided that $S\geq
s_{\delta}(t)$ is large enough. Integrating and using again the monotonicity
of $\Theta$,

$S\Theta(S)\geq(S-t)\Theta(S)\geq\int_{t}^{S}\Theta\geq\left[\int_{t}^{S_{0}}\Theta\right]\left(\frac{S}{S_{0}}\right)^{1+\varepsilon},$

hence $\Theta(S)$ grows polynomially at least with power $\varepsilon$,
contradicting (1.10).
Concluding, both for $k>0$ and for $k=0$ assuming (4.13) leads to a
contradiction with our assumption (1.10), hence (4.10) holds, as required.

##  5\. Proof of Theorem 2

We first show that $\varphi$ is proper and that $M$ is diffeomorphic to the
interior of a compact manifold with boundary. Both the properties are
consequence of the following lemma due to [6], which improves on [1], [20],
[10], [4].

######  Lemma 3.

Let $\varphi:M^{m}\rightarrow N^{n}$ be an immersed submanifold into an
ambient manifold $N$ with a pole and suppose that $N$ satisfies (1.9) for some
$k\geq 0$. Denote by $B_{s}=\\{x\in M;\;\rho(x)\leq s\\}$ the intrinsic ball
on $M$. Assume that

(5.1)
$\begin{array}[]{rll}(i)&\quad\displaystyle\limsup_{s\rightarrow+\infty}s\|\mathrm{II}\|_{L^{\infty}(\partial
B_{s})}<1&\quad\text{if }\,k=0\text{ in \eqref{pinchsectio},
or}\\\\[11.38092pt]
(ii)&\quad\displaystyle\limsup_{s\rightarrow+\infty}\|\mathrm{II}\|_{L^{\infty}(\partial
B_{s})}<\sqrt{k}&\quad\text{if }\,k>0\text{ in
\eqref{pinchsectio}}.\end{array}$

Then, $\varphi$ is proper and there exists $R>0$ such that $|\nabla r|>0$ on
$\\{r\geq R\\}$, where $r$ is the extrinsic distance function. Consequently,
the flow

(5.2) $\Phi:\mathbb{R}^{+}\times\\{r=R\\}\rightarrow\\{r\geq
R\\},\qquad\frac{\mathrm{d}}{\mathrm{d}s}\Phi_{s}(x)=\frac{\nabla r}{|\nabla
r|^{2}}\big{(}\Phi_{s}(x)\big{)}$

is well defined, and $M$ is diffeomorphic to the interior of a compact
manifold with boundary.

The properness of $\varphi$ enables us to apply Proposition 4. Therefore, to
show that $\Theta(+\infty)<+\infty$ it is enough to check that

(5.3)
$\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\frac{\int_{\Gamma_{s}}\big{[}|\nabla
r|^{-1}-|\nabla r|\big{]}}{\int_{\Gamma_{s}}|\nabla r|}\in L^{1}(+\infty).$

To achieve (5.3), we need to bound from above the rate of approaching of
$|\nabla r|$ to $1$ along the flow $\Phi$ in Lemma 3. We begin with the
following

######  Lemma 4.

Suppose that $N$ has a pole and radial sectional curvature satisfying (1.9),
and that $\varphi:M^{m}\rightarrow N^{n}$ is a proper minimal immersion such
that $|\nabla r|>0$ outside of some compact set $\\{r\leq R\\}$. Let $\Phi$
denote the flow of $\nabla r/|\nabla r|^{2}$ as in (5.2) and let
$\gamma:[R,+\infty)\rightarrow M$ be a flow line starting from some
$x_{0}\in\\{r=R\\}$. Then, along $\gamma$,

(5.4)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\big{(}\mathrm{sn}_{k}(r)\sqrt{1-|\nabla
r|^{2}}\big{)}\leq\displaystyle\mathrm{sn}_{k}(r)|\mathrm{II}(\gamma(s))|$

###### Proof.

Observe that $r(\gamma(s))=s-R$. By the chain rule and the Hessian comparison
theorem 2.5,

$\begin{array}[]{lcl}\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}|\nabla
r|^{2}&=&\displaystyle 2\mathrm{Hess}\,r(\nabla
r,\dot{\gamma})=\frac{2}{|\nabla r|^{2}}\mathrm{Hess}\,r(\nabla r,\nabla
r)\\\\[11.38092pt] &=&\displaystyle\frac{2}{|\nabla
r|^{2}}\overline{\mathrm{Hess}\,}(\bar{\rho})\big{(}\mathrm{d}\varphi(\nabla
r),\mathrm{d}\varphi(\nabla r)\big{)}+\frac{2}{|\nabla
r|^{2}}\big{(}\bar{\nabla}\bar{\rho},\mathrm{II}(\nabla r,\nabla
r)\big{)}\\\\[11.38092pt] &\geq&\displaystyle
2\frac{\mathrm{sn}^{\prime}_{k}(r)}{\mathrm{sn}_{k}(r)}(1-|\nabla
r|^{2})-2|\bar{\nabla}^{\perp}\bar{\rho}||\mathrm{II}|,\end{array}$

where $\bar{\nabla}^{\perp}\bar{\rho}$ is the component of $\bar{\rho}$
perpendicular to $\mathrm{d}\varphi(TM)$ and
$|\bar{\nabla}^{\perp}\rho|=\sqrt{1-|\nabla r|^{2}}$. Then,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}|\nabla r|^{2}\geq\displaystyle
2\frac{\mathrm{sn}^{\prime}_{k}(r)}{\mathrm{sn}_{k}(r)}(1-|\nabla
r|^{2})-2|\mathrm{II}|\sqrt{1-|\nabla r|^{2}}.$

Multiplying by $\mathrm{sn}_{k}^{2}(r)$ gives

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\big{(}\mathrm{sn}^{2}_{k}(r)(1-|\nabla
r|^{2})\big{)}\leq 2\mathrm{sn}_{k}^{2}(r)|\mathrm{II}|\sqrt{1-|\nabla
r|^{2}},$

which implies (5.4). ∎

The above lemma relates the behaviour of $|\nabla r|$ to that of the second
fundamental form. The next result makes this relation explicit in the two
cases considered in Theorem 2.

######  Proposition 5.

In the assumptions of the above proposition, suppose further that either

(5.5)
$\begin{array}[]{rll}(i)&\quad\displaystyle\|\mathrm{II}\|_{L^{\infty}(\partial
B_{s})}\leq\frac{C}{s\log^{\alpha/2}s}&\quad\text{if }\,k=0\text{ in
\eqref{pinchsectio}, or}\\\\[11.38092pt]
(ii)&\quad\displaystyle\|\mathrm{II}\|_{L^{\infty}(\partial
B_{s})}\leq\frac{C}{\sqrt{s}\log^{\alpha/2}s}&\quad\text{if }\,k>0\text{ in
\eqref{pinchsectio}.}\end{array}$

for $s\geq 1$ and some constants $C>0$ and $\alpha>0$. Here, $\partial B_{s}$
is the boundary of the intrinsic ball $B_{s}(o)$. Then, $|\nabla
r|(\gamma(s))\rightarrow 1$ as $s$ diverges, and if $s>2R$ and $R$ is
sufficiently large,

(5.6) $\begin{array}[]{ll}\displaystyle\text{in the case
$(i)$,}&\displaystyle\qquad 1-|\nabla
r(\gamma(s))|^{2}\leq\frac{\hat{C}}{\log^{\alpha}s}\\\\[11.38092pt]
\displaystyle\text{in the case $(ii)$,}&\displaystyle\qquad 1-|\nabla
r(\gamma(s))|^{2}\leq\frac{\hat{C}}{s\log^{\alpha}s}\end{array}$

for some constant $\hat{C}$ depending on $R$.

###### Proof.

We begin by observing that, in (5.5), $\partial B_{s}$ can be replaced by
$\Gamma_{s}$. Indeed, since $r(x)\leq r(o)+\rho(x)$, we can choose $R$ large
enough depending on $r(o),\alpha$ in such a way that, for instance in $(i)$,

$|\mathrm{II}(x)|\leq\frac{C}{\rho(x)\log^{\alpha/2}\rho(x)}\leq\frac{C_{1}}{r(x)\log^{\alpha/2}r(x)}$

for some absolute $C_{1}$ and for each $r\geq R$. Thus, from $(i)$ and $(ii)$
we infer the bounds

(5.7)
$\|\mathrm{II}\|_{L^{\infty}(\Gamma_{s})}\leq\frac{C_{1}}{s\log^{\alpha/2}s}\quad\text{for
}\,(i),\qquad\|\mathrm{II}\|_{L^{\infty}(\Gamma_{s})}\leq\frac{C_{1}}{\sqrt{s}\log^{\alpha/2}s}\quad\text{for
}\,(ii).$

Because of (5.7), up to enlarging $R$ further there exists a uniform constant
$C_{2}>0$ such that, on $[R,+\infty)$,

(5.8)
$\mathrm{sn}_{k}(s)|\mathrm{II}(\gamma(s))|\leq\left\\{\begin{array}[]{ll}\displaystyle\frac{C_{1}}{\log^{\alpha/2}s}\leq
C_{2}\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{s}{\log^{\alpha/2}s}\right)&\quad\text{if
}\,k=0;\\\\[14.22636pt]
\displaystyle\frac{C_{1}\mathrm{sn}_{k}(s)}{\sqrt{s}\log^{\alpha/2}s}\leq
C_{2}\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{\mathrm{sn}_{k}(s)}{\sqrt{s}\log^{\alpha/2}s}\right)&\quad\text{if
}\,k>0.\end{array}\right.$

Integrating on $[R,s]$ and using (5.4) we get

$\sqrt{1-|\nabla
r(\gamma(s))|^{2}}\leq\left\\{\begin{array}[]{ll}\displaystyle\frac{C_{3}(R)}{s}+\frac{C_{4}}{\log^{\alpha/2}s}\leq\frac{C_{5}}{\log^{\alpha/2}s}&\quad\text{if
}\,k=0,\\\\[14.22636pt]
\displaystyle\frac{C_{3}(R)}{\mathrm{sn}_{k}(s)}+\frac{C_{4}}{\sqrt{s}\log^{\alpha/2}s}\leq\frac{C_{5}}{\sqrt{s}\log^{\alpha/2}s}&\quad\text{if
}\,k>0,\end{array}\right.$

for some absolute constants $C_{4},C_{5}>0$ and if $s>2R$ and $R$ is large
enough. The desired (5.6) follows by taking squares.

∎

We are now ready to conclude the proof of Theorem 2 by showing that $M$ has
finite density or, equivalently, that (5.3) holds.

Let $\eta(s)$ be either

(5.9) $\frac{1}{\log^{\alpha}s}\ \,\text{ when }k=0,\text{ or
}\,\frac{1}{s\log^{\alpha}s}\ \,\text{ when }k>0,$

where $\alpha>1$ and $C$ is a large constant. In our assumptions, we can apply
Lemma 4 and Proposition 5 to deduce, according to (5.6), that, for large
enough $R$,

$1-|\nabla r(\gamma(s))|^{2}\leq C\eta(s)\qquad\text{on }\,(R,+\infty),$

where $\gamma(s)$ is a flow curve of $\Phi$ in (5.2) and $C=C(R)$ is a large
constant. In particular, $|\nabla r(\gamma(s))|\rightarrow 1$ as
$s\rightarrow+\infty$. We therefore deduce the existence of a constant
$C_{2}(R)>0$ such that

$\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\frac{\int_{\Gamma_{s}}\big{[}|\nabla
r|^{-1}-|\nabla r|\big{]}}{\int_{\Gamma_{s}}|\nabla r|}\leq
C\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\eta(s)\frac{\int_{\Gamma_{s}}|\nabla
r|^{-1}}{\int_{\Gamma_{s}}|\nabla r|}\leq
C_{2}\frac{\mathrm{sn}_{k}^{\prime}(s)}{\mathrm{sn}_{k}(s)}\eta(s).$

In both our cases $k=0$ and $k>0$, since $\alpha>1$ it is immediate to check
that $\mathrm{sn}_{k}^{\prime}\eta/\mathrm{sn}_{k}\in L^{1}(+\infty)$, proving
(5.3).

## Appendix 1: finite total curvature solutions of Plateau’s problem

In this appendix, we show that (smooth) solutions of Plateau’s problem at
infinity $M^{m}\rightarrow\mathbb{H}^{n}$ have finite total curvature whenever
$M$ is a hypersurface and the boundary datum
$\Sigma\subset\partial_{\infty}\mathbb{H}^{n}$ is sufficiently regular.
Consider the Poincaré model of $\mathbb{H}^{n}$, and let
$M\rightarrow\mathbb{H}^{n}$ be a proper minimal submanifold. We say that $M$
is $C^{k,\alpha}$ up to $\partial_{\infty}\mathbb{H}^{n}$ if its closure
$\overline{M}$ in the topology of the closed unit ball
$\overline{\mathbb{H}^{n}}=\mathbb{H}^{n}\cup\partial_{\infty}\mathbb{H}^{n}$
is a $C^{k,\alpha}$-manifold with boundary. We begin with a lemma, whose proof
have been suggested to the second author by L. Mazet.

######  Lemma 5.

Let $\varphi:M^{m}\rightarrow\mathbb{H}^{n}$ be a proper minimal submanifold.
If $M$ is of class $C^{2}$ up to $\partial_{\infty}\mathbb{H}^{n}$, then $M$
has finite total curvature.

###### Proof.

The Euclidean metric $\overline{\langle\,,\,\rangle}$ is related to the
Poincaré metric $\langle\,,\,\rangle$ by the formula

$\overline{\langle\,,\,\rangle}=\lambda^{2}\langle\,,\,\rangle,\qquad\text{with}\quad\lambda=\frac{1-|x|^{2}}{2}.$

Given a proper, minimal submanifold
$\varphi:(M^{m},g)\rightarrow(\mathbb{H}^{n},\langle\,,\,\rangle)$, we
associate the isometric immersion
$\bar{\varphi}:(M,(\lambda^{2}\circ\varphi)g)\rightarrow(\mathbb{H}^{n},\overline{\langle\,,\,\rangle})$,
$\bar{\varphi}(x)\doteq\varphi(x)$. Fix a local Darboux frame
$\\{e_{i},e_{\alpha}\\}$ on $(M,g)$ for $\varphi$, with $\\{e_{i}\\}$ tangent
to $M$ and $\\{e_{\alpha}\\}$ in the normal bundle, and let
$\bar{e}_{i}=e_{i}/\lambda$, $\bar{e}_{\alpha}=e_{\alpha}/\lambda$ be the
corresponding Darboux frame on $(M,\lambda^{2}g)$ for $\bar{\varphi}$. Let
$\mathrm{d}V$ and $\mathrm{d}\bar{V}=\lambda^{m}\mathrm{d}V$ be the volume
forms of $(M,g)$ and $(M,\lambda^{2}g)$, and denote with $h^{\alpha}_{ij}$ and
$\bar{h}^{\alpha}_{ij}$ the coefficients of the second fundamental forms of
$\varphi$ and $\bar{\varphi}$, respectively. A standard computation shows that

$\bar{h}^{\alpha}_{ij}=\frac{1}{\lambda}h^{\alpha}_{ij}-\frac{\lambda_{\alpha}}{\lambda}\delta_{ij},$

where $\lambda_{\alpha}=e_{\alpha}(\lambda)$. Evaluating the norms of
$\mathrm{II}$ and $\bar{\mathrm{II}}$, since $h^{\alpha}_{ij}$ is trace-free
by minimality we obtain

$|\bar{\mathrm{II}}|^{2}=\lambda^{-2}|\mathrm{II}|^{2}+m|\nabla^{\perp}\log\lambda|^{2}\geq\lambda^{-2}|\mathrm{II}|^{2},$

and thus
$|\bar{\mathrm{II}}|^{m}\mathrm{d}\bar{V}\geq|\mathrm{II}|^{m}\mathrm{d}V$.
Integrating on $M$ it holds

$\int_{M}|\mathrm{II}|^{m}\mathrm{d}V\leq\int_{M}|\bar{\mathrm{II}}|^{m}\mathrm{d}\bar{V}.$

However, the last integral is finite since $M$ is $C^{2}$ up to
$\partial_{\infty}\mathbb{H}^{n}$, and thus $\varphi$ has finite total
curvature. ∎

In view of Lemma 5, we briefly survey on some boundary regularity results for
solutions of Plateau’s problem. To the best of our knowledge, we just found
regularity results for hypersurfaces. Let $M^{m}\rightarrow\mathbb{H}^{m+1}$
be a solution of Plateau’s problem for a compact, $(m-1)$-dimensional
submanifold $\Sigma^{m-1}\subset\partial_{\infty}\mathbb{H}^{m+1}$. Then, a
classical result of Hardt and Lin [30] states that if
$\Sigma^{m-1}\hookrightarrow\partial_{\infty}\mathbb{H}^{m+1}$ is properly
embedded and $C^{1,\alpha}$, with $0\leq\alpha\leq 1$, near $\Sigma$ each
solution $M^{m}\rightarrow\mathbb{H}^{m+1}$ of Plateau’s problem is a finite
collection of $C^{1,\alpha}$-manifolds with boundary, which are disjoint
except at the boundary. Therefore, near $\Sigma$, $M$ can locally be described
as a graph, and the higher regularity theory in [39] [40], [53], [52] applies
to give the following: if $\Sigma$ is $C^{j,\alpha}$, then $M$ is
$C^{j,\alpha}$ up to $\partial_{\infty}\mathbb{H}^{m+1}$ whenever

  * -

$1\leq j\leq m-1$ and $0\leq\alpha\leq 1$, or

  * -

$j=m$ and $0<\alpha<1$, or

  * -

$j\geq m+1$ and $0<\alpha<1$ (if $m$ is odd, under a further condition on
$\Sigma$).

The reader can consult the statement and references in [40]. In particular,
because of Lemma 5, if $\Sigma$ is $C^{2,\alpha}$ for some $0<\alpha<1$ then
$M$ has finite total curvature (provided that it is smooth).

## Appendix 2: the intrinsic monotonicity formula

We conclude by recalling an intrinsic version of the monotonicity formula. To
state it, we premit the following observation due to H. Donnelly and N.
Garofalo, Proposition 3.6 in [23].

######  Proposition 6.

For $k\geq 0$, the function

(5.10) $\frac{V_{k}(s)}{v_{k}(s)}\qquad\text{is non-decreasing on
}\,\mathbb{R}^{+}.$

###### Proof.

The ratio $v_{k}^{\prime}/v_{k}$ is monotone decreasing by the very definition
of $v_{k}$. Then, since $v_{k}^{\prime}>0$, the desired monotonicity follows
from a lemma at p. 42 of [12]. ∎

######  Proposition 7 (The intrinsic monotonicity formula).

Suppose that $N$ has a pole $\bar{o}$ and satisfies (1.9), and let
$\varphi:M^{m}\rightarrow N^{n}$ be a complete, minimal immersion. Suppose
that $\bar{o}\in\varphi(M)$, and choose $o\in M$ be such that
$\varphi(o)=\bar{o}$. Then, denoting with $\rho$ the intrinsic distance
function from $o$ and with $B_{s}=\\{\rho\leq s\\}$,

(5.11) $\frac{\mathrm{vol}(B_{s})}{V_{k}(s)}$

is monotone non-decreasing on $\mathbb{R}^{+}$.

###### Proof.

We refer to Proposition 3 for definitions and computations. We know that the
function $\psi=f\circ r$, with $f$ as in (3.5), solves $\Delta\psi\geq 1$ on
$M$. Integrating on $B_{s}$ and using the definition of $\psi$ we obtain

$\mathrm{vol}(B_{s})\leq\int_{B_{s}}\Delta\psi=\int_{\partial
B_{s}}\langle\nabla\psi,\nabla\rho\rangle\leq\int_{\partial
B_{s}}\frac{V_{k}(r)}{v_{k}(r)}.$

Next, since $\bar{o}=\varphi(o)$, it holds $r(x)\leq\rho(x)$ on $M$. Using
then Proposition 6, we deduce

$\mathrm{vol}(B_{s})\leq\frac{V_{k}(s)}{v_{k}(s)}\mathrm{vol}(\partial
B_{s}).$

Integrating we obtain the monotonicity of the desired (5.11). ∎

Acknowledgements.
The second author is supported by the grant PRONEX - Núcleo de Análise
Geométrica e Aplicacões Processo nº PR2-0054-00009.01.00/11. The third author
is partially supported by CNPq. The second author would like to thank L.
Hauswirth and L. Mazet for an interesting discussion on finite total curvature
submanifolds of $\mathbb{H}^{n}$, A. Figalli and G.P. Bessa for a hint, P.
Castillon for a bibliographical suggestion and V. Gimeno for pleasant
conversations and various comments that lead to several improvements after we
posted a first version of the paper on arXiv.

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