1 Introduction
The study of geometric aspects concerning the horospheres of hyperbolic space
$\mathbb {H}^{n+1}$
constitutes an important and fruitful topic into the theory of differential geometry and geometric analysis. In a famous work, by using the well-known Alexandrov reflexion method, do Carmo and Lawson [Reference do Carmo and Lawson10] showed that any complete hypersurface properly embedded with constant mean curvature in hyperbolic space
$\mathbb {H}^{n+1}$
with a single point at the asymptotic boundary
$\partial _{\infty } \mathbb {H}^{n+1}$
is a horosphere, emphasizing that the same is not true if we replace embedded by immersed. Meanwhile, Alías and Dajczer [Reference Alías and Dajczer3] contributed to the immersed case proving that the horospheres are the only surfaces properly immersed in
$\mathbb {H}^{3}$
with constant mean curvature
$|H| \leq 1$
contained in a slab (that means, contained in the region between two horospheres that share the same point in the asymptotic boundary
$\partial _{\infty } \mathbb {H}^{3}$
).
In 2019, the second author jointly with Aquino et al. [Reference Aquino, Baltazar and de Lima7] studied complete two-sided hypersurfaces of
$\mathbb {H}^{n+1}$
using a maximum principle about integrability for complete noncompact Riemannian manifolds due to Caminha [Reference Caminha8]. In this regard, with appropriate assumptions on the higher order mean curvature, they established several rigidity results for the horospheres of
$\mathbb {H}^{n+1}$
. More recently, in a similar configuration, the second and third authors jointly with Colares et al. [Reference Colares, de Lima and Velásquez9] obtained some new characterization results for the horospheres of
$\mathbb {H}^{n+1}$
via an extension of the generalized maximum principle of Omori and Yau [Reference Omori16, Reference Yau18] under suitable controls on the higher order mean curvatures of a complete two-sided hypersurface of
$\mathbb {H}^{n+1}$
.
To continue this investigation, in this manuscript, we study the behavior of complete two-sided hypersurfaces immersed in hyperbolic space under the light of the concepts of the linearized curvature functions
$\mathcal {F}_{r,s}$
of a two-sided hypersurface, its associated modified Newton transformation
$\mathcal {P}_{r,s}$
, and its natural differential operator
$\mathcal {L}_{r,s}$
(see Definitions 2.1, (2-10) and (2-11) in Section 2). In this setting, in Section 3, we obtain two formulas for differential operators
$\mathcal {L}_{r,s}$
acting on the height function of a two-sided hypersurface (see Proposition 3.2) and for the case where their support functions are related by a negative constant, we establish two new formulas for the Newton transformation
$P_{r}$
and the modified Newton transformation
$\mathcal {P}_{r,s}$
acting on the gradient of the height function (see Proposition 3.3).
Our main results are presented in Section 4. More specifically, in Section 4.1, we use a maximum principle concerning an integrability property to get a new characterization result (see Theorem 4.2); in Section 4.2, we obtain four characterization results using a maximum principle that deal with the notion of convergence to zero at infinity (see Theorems 4.5 and 4.7, and Corollaries 4.6 and 4.8); in Section 4.3, via a maximum principle that deals with the notion of polynomial volume growth, we produce four nonexistence results (see Theorems 4.10 and 4.12, and Corollaries 4.11 and 4.13); and in Section 4.4, we use a suitable extension of the generalized maximum principles of Omori and Yau to get our last four characterization results (see Theorems 4.15, 4.17, and 4.19 and Corollary 4.16).
2 The linearized curvature function
$\mathcal {F}_{r,s}$
of a two-sided hypersurface
Herein, we consider the
$(n+1)$
-dimensional hyperbolic space
$\mathbb H^{n+1}$
as a hyperquadric of the
$(n+2)$
-dimensional Lorentz–Minkowski space
$\mathbb {L}^{n+2}$
. So, we represent by
$\mathbb {L}^{n+2}$
the Euclidean space
$\mathbb {R}^{n+2}$
endowed with the Lorentz metric
$$ \begin{align*} \langle v,w\rangle=\displaystyle\sum_{i=1}^{n+1}v_iw_i-v_{n+2}w_{n+2}, \end{align*} $$
and
$\mathbb H^{n+1}$
is identified with
$$ \begin{align*} \mathbb{H}^{n+1}=\{p\in \mathbb{L}^{n+2}\,; \langle p,p\rangle=-1,\, p_{n+2}\geq 1\} \end{align*} $$
equipped with the induced Riemannian metric from
$\mathbb {L}^{n+2}$
.
In this setting, we deal with a (connected) two-sided hypersurface
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
immersed in
$\mathbb H^{n+1}$
, which means that there exists a globally defined unit normal vector field N on
$\Sigma^n$
. Let us denote by
$A:\mathfrak X(\Sigma^n)\rightarrow \mathfrak X(\Sigma^n)$
the shape operator of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
with respect to the vector field N. At each
$p\in \Sigma^n$
, the operator A restricts to a self-adjoint linear map
$A_p:T_p\Sigma \rightarrow T_p\Sigma $
. For
$j\in \{1,\ldots ,n\}$
, let
$S_j(p)$
denote the j th elementary symmetric function on the eigenvalues of
$A_p$
. Thus, one gets n smooth functions
$S_j:\Sigma^n\rightarrow \mathbb R$
, such that
$$ \begin{align*} \det(tI-A)=\sum_{k=0}^n(-1)^kS_kt^{n-k}, \end{align*} $$
where I is the identity operator on
$\mathfrak {X}(\Sigma^n)$
and, by convention,
$S_0=1$
. If
$p\in \Sigma^n$
and
$\{e_1,\ldots ,e_n\}$
is a basis of
$T_p\Sigma $
formed by eigenvectors of
$A_p$
, with corresponding eigenvalues
$\{\lambda _1,\ldots ,\lambda _n\}$
, one immediately sees that
$$ \begin{align*} S_j=\sigma_j(\lambda_1,\ldots,\lambda_n), \end{align*} $$
where
$\sigma _j\in \mathbb R[X_1,\ldots ,X_n]$
is the j th elementary symmetric polynomial on the indeterminates
$X_1,\ldots ,X_n$
. This allows us to define the higher order mean curvature
$H_j:\Sigma^n\rightarrow \mathbb {R}$
(also known as the j th mean curvature) of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
by
$$ \begin{align*} {n\choose\ j}H_j=S_j, \quad j\in\{0,\ldots,n\}. \end{align*} $$
We observe that
$H_0=1$
, while
$H_1=(1/n)S_1$
is the usual mean curvature H of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
.
For
$j\in \{0,\ldots ,n\}$
, one defines the j th Newton transformation
$$ \begin{align*} P_j:\mathfrak X(\Sigma^n)\rightarrow\mathfrak X(\Sigma^n) \end{align*} $$
associated to
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
by setting
$P_0=I$
(the identity operator) and, for
$j\in \{1,\ldots ,n\}$
, via the recurrence relation
$$ \begin{align} P_j={n\choose\ j}H_jI-A\circ P_{j-1}. \end{align} $$
With a trivial inductive argument, from (2-1), we verify that
$$ \begin{align} P_j={n\choose\ j}H_jI-{n\choose\ j-1}H_{j-1}A+{n\choose\ j-2}H_{j-2}A^2-\cdots+(-1)^{{\kern1pt}j}A^j, \end{align} $$
so that the Cayley–Hamilton theorem gives
$P_n=0$
. Moreover, since
$P_j$
is a polynomial in A for every j, it is also self-adjoint and commutes with A. Therefore, all bases of
$T_p\Sigma $
diagonalizing A at
$p\in \Sigma^n$
also diagonalize all of the
$P_r$
at p. So, let
$\{e_1,\ldots ,e_n\}$
be an orthonormal frame on
$T_p\Sigma $
that diagonalizes
$A_p$
,
$A_p(e_i)=\lambda _i(p)e_i$
, then from (2-2),
$$ \begin{align} (P_j)_p(e_i)=\sum_{i_1<\cdots<i_j, i_k\neq i}\lambda_{i_1}(p)\cdots\lambda_{i_j}(p)e_i. \end{align} $$
Moreover,
$$ \begin{align} \mathrm{tr}(P_j)=c_jH_j, \end{align} $$
where
$c_j=(j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})=(n-j)(\begin {smallmatrix}n\\ j\end {smallmatrix})$
.
However, for each
$j\in \{0,\ldots ,n\}$
, the second-order differential operator
$L_j$
associated to
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is defined as follows:
$$ \begin{align} \begin{array}{rccl} L_j:&\!\!C^{\infty}(\Sigma^n)&\rightarrow&C^{\infty}(\Sigma^n)\\ &f&\mapsto&L_j(f)=\mathrm{tr}(P_j\circ\nabla^{\,2}f), \end{array} \end{align} $$
where
$\nabla ^{\,2}f:\mathfrak {X}(\Sigma^n)\rightarrow \mathfrak {X}(\Sigma^n)$
denotes the self-adjoint linear operator metrically equivalent to the Hessian of f, which is given by
$$ \begin{align*} \langle\nabla^{\,2}f(X),Y\rangle=\langle\nabla_X\nabla f,Y\rangle \end{align*} $$
for all
$X,Y\in \mathfrak {X}(\Sigma^n)$
. In particular, for
$r=0$
, we get the well-known Laplacian operator
$L_0=\Delta $
.
According to [Reference Rosenberg17], we observe that
$$ \begin{align} \mathrm{div}(P_j(\nabla f))&=\sum^{n}_{i=1}\langle(\nabla_{e_i}P_j)(\nabla f),e_i\rangle+\sum^{n}_{i=1}\langle P_j(\nabla_{e_i}\nabla f),e_i\rangle \nonumber\\ &=\langle\mathrm{div}P_j,\nabla f\rangle+L_j(f) \end{align} $$
for any
$f\in C^{\infty }(\Sigma^n)$
, where
$\{e_1,\ldots ,e_n\}$
is a local orthonormal frame on
$\Sigma^n$
and the divergence of
$P_j$
on
$\Sigma^n$
is given by
$$ \begin{align*} \mathrm{div}P_j=\mathrm{tr}(\nabla P_j)=\sum_{i=1}^n(\nabla_{e_i}P_j)(e_i). \end{align*} $$
Consequently, since [Reference Alías, de Lira and Malacarne4, Lemma
$3.1$
] guarantees that
$P_j$
is divergence free when the ambient space has constant sectional curvature, from (2-6),
$$ \begin{align} L_j(f)=\mathrm{div}(P_j(\nabla f)) \end{align} $$
for all
$f\in C^{\infty }(\Sigma^n)$
.
At this point, we are in a position to define our main geometric quantity of study.
Definition 2.1. Let r and s be any integers satisfying the inequalities
$0\leq r\le s\leq n-1$
. Throughout a two-sided hypersurface
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
, we define the linearized curvature function
$\mathcal {F}_{r,s}:\Sigma^n\rightarrow \mathbb {R}$
of its higher order mean curvatures
$H_{r+1}$
and
$H_{s+1}$
by
$$ \begin{align*} \mathcal{F}_{r,s} = A_{r}H_{r+1}+\cdot \cdot \cdot + A_{s}H_{s+1}, \end{align*} $$
where
$A_{r},\ldots ,A_{s}$
are real numbers (at least one of them nonzero) and
$H_{j+1}$
is the
$(j+1)$
th mean curvature of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
for
$j \in \{r,\ldots ,s\}$
. Without loss of generality, we can write
$A_{j} = (j+1)a_{j}c_{j}$
, where
$c_{j} = (j+1)(\begin {smallmatrix}n \\ j+1\end {smallmatrix})$
and
$a_{j} = {A_{j}}/({(j+1)c_{j}})$
for
$j \in \{r,\ldots ,s\}$
. In this setting, we have that the linearized curvature function
$\mathcal {F}_{r,s}$
becomes
$$ \begin{align} \mathcal{F}_{r,s}=(r+1)a_rc_rH_{r+1}+\cdots+(s+1)a_sc_sH_{s+1}, \end{align} $$
where
$a_r,\ldots ,a_s$
are real numbers (at least one of them nonzero) and
$H_{j+1}$
is the
$(j+1)$
th mean curvature of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
for
$j\in \{r,\ldots ,s\}$
. In particular, we call
$$ \begin{align} \mathcal{F}_{0,1}=na_0H+2n(n-1)a_1H_2 \end{align} $$
the first linearized curvature function of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
.
Remark 2.2. According to Definition 2.1, if
$r=s$
in (2-8), then
$\mathcal {F}_{r,r}$
becomes (up to a nonzero real constant) the
$(r+1)$
th mean curvature
$H_{r+1}$
of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
, with
$r\in \{0,\ldots ,n-1\}$
. In turn, if there is a constant
$b\in \mathbb {R}$
such that
$\mathcal {F}_{r,s}=b$
on
$\Sigma^n$
, then we get that
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is an
$(r,s)$
-linear Weingarten two-sided hypersurface. In particular, if
$\mathcal {F}_{0,1}=b$
on
$\Sigma^n$
for some
$b\in \mathbb {R}$
,
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is just a linear Weingarten two-sided hypersurface.
Associated with the linearized curvature function
$\mathcal {F}_{r,s}$
, we define the modified Newton transformation
$$ \begin{align} \begin{array}{rccl} \mathcal{P}_{r,s}:\!\!&\mathfrak{X}(\Sigma^n)&\rightarrow&\mathfrak{X}(\Sigma^n)\\ &X&\mapsto&\mathcal{P}_{r,s}(X)=\displaystyle\sum_{j=r}^{s}(j+1)a_jP_j(X) \end{array} \end{align} $$
and the differential operator
$$ \begin{align} \begin{array}{rccl} \mathcal{L}_{r,s}:\!\!&C^{\infty}(\Sigma^n)&\rightarrow&C^{\infty}(\Sigma^n)\\ &f&\mapsto&\mathcal{L}_{r,s}(f)=\displaystyle\sum_{j=r}^{s}(j+1)a_jL_j(f), \end{array} \end{align} $$
where
$P_j$
and
$L_j$
are given by (2-1) and (2-5), respectively.
For a smooth function
$\varphi :\mathbb {R}\rightarrow \mathbb {R}$
and
$f\in C^{\infty }(\Sigma^n)$
, it follows from the properties of the Hessian of functions that
$$ \begin{align} \mathcal{L}_{r,s}(\varphi\circ f)=\varphi'(f)\mathcal{L}_{r,s}(f)+\varphi"(f)\langle \mathcal{P}_{r,s}(\nabla f),\nabla f\rangle. \end{align} $$
Furthermore, from (2-7), we obtain that
$\mathcal {L}_{r,s}$
is a differential operator of the divergent form, namely,
$$ \begin{align} \mathcal{L}_{r,s}(f)=\mathrm{div}(\mathcal{P}_{r,s}(\nabla f)) \end{align} $$
for all
$f\in C^{\infty }(\Sigma^n)$
.
3 The operators
$\mathcal {L}_{r,s}$
and
$\mathcal {P}_{r,s}$
on the height function
As was observed by López and Montiel in [Reference López and Montiel14], any horosphere of
$\mathbb H^{n+1}$
is given by
$$ \begin{align} L_{\tau}^n= \{p\in\mathbb H^{n+1}; \langle p,a\rangle=\tau\}, \end{align} $$
where
$a\in \mathbb L^{n+2}$
is a nonzero null vector, that is,
$\langle a,a\rangle =0$
, and
$\tau $
is a positive number. When one fixes such a vector
$a\in \mathbb L^{n+2}$
and moves
$\tau \in \mathbb R^+$
, one obtains a complete foliation of
$\mathbb H^{n+1}$
by means of horospheres sharing the same fixed point in the asymptotic boundary
$\partial _{\infty }\mathbb {H}^{n+1}$
. It is not difficult to see that
$$ \begin{align} \xi(p)=-p-\frac{1}{\tau}\,a \end{align} $$
is a unit normal vector field on
$L_{\tau }^n$
with respect to which it has constant j th mean curvature equal to
$1$
for any
$j\in \{0,\ldots ,n\}$
. Thus, for any integers r and s satisfying the inequalities
$0\leq r\le s\leq n$
, we obtain that the linearized curvature function
$\mathcal {F}_{r,s}$
of
$L_{\tau }^n$
is the constant
$$ \begin{align} \mathcal{F}_{r,s}=(r+1)a_rc_r+\cdots+(s+1)a_sc_s\in\mathbb{R}, \end{align} $$
where
$c_j=(j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})$
,
$j\in \{r,\ldots ,s\}$
, and
$a_r,\ldots ,a_s$
are real numbers (at least one of them nonzero).
With a fixed nonzero null vector
$a\in \mathbb L^{n+2}$
, in what follows, we consider two particular functions naturally attached to a two-sided hypersurface
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
, namely, the height and angle functions with respect to a, which are defined by
$$ \begin{align} \begin{array}{rccl} \ell_a:\!\!&\Sigma^n&\rightarrow&\mathbb{R}\\ &p&\mapsto&\ell_a(p)=\langle\psi(p),a\rangle \end{array} \end{align} $$
and
$$ \begin{align} \begin{array}{rccl} f_a:\!\!&\Sigma^n&\rightarrow&\mathbb{R}\\ &p&\mapsto&f_a(p)=\langle N(p),a\rangle, \end{array} \end{align} $$
respectively, where N is the globally defined unit normal vector field on
$\Sigma^n$
. In particular, we have from (3-1) that
$\ell _a$
is always a positive function on
$\Sigma^n$
. Moreover, a direct computation allows us to conclude that the gradients of such functions are given by
$$ \begin{align} \nabla \ell_a=a^\top \end{align} $$
and
$$ \begin{align} \nabla f_a=-A(a^\top), \end{align} $$
where A is the shape operator of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
with respect to the vector field N and
$a^\top $
is the orthogonal projection of
$a\in \mathbb {L}^{n+2}$
onto the tangent bundle of
$\Sigma^n$
, namely,
$$ \begin{align} a^\top=a-f_aN+\ell_a\psi. \end{align} $$
Thus, since
$a\in \mathbb {L}^{n+2}$
is a null vector, from (3-8),
$$ \begin{align} |\nabla \ell_a|^2+f_a^2-\ell_a^{\,2}=0. \end{align} $$
Remark 3.1. In the setup described above, from (3-9), it follows that the functions
$\ell _a$
and
$f_a$
attached to
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
satisfy
$$ \begin{align} \ell_a\geq-f_a, \end{align} $$
on
$\Sigma^n$
. Furthermore, if the equality
$\ell _a=-f_a$
occurs on
$\Sigma^n$
, then
$$ \begin{align*} |\nabla\ell_a|^2=\ell_a^2-f_a^2=0 \end{align*} $$
on
$\Sigma^n$
and, in this case,
$\ell _a$
is constant on
$\Sigma^n$
, which means that
$\psi (\Sigma^n)$
is a horosphere
$L_{\tau }^n$
of
$\mathbb {H}^{n+1}$
orthogonal to
$a\in \mathbb {L}^{n+2}$
for some
$\tau>0$
.
Below, we present appropriate formulas for the differential operator
$\mathcal {L}_{r,s}$
acting on the smooth functions
$\ell _a$
and
$\ell _a^2$
, which are of relevant importance for obtaining our main results.
Proposition 3.2. Let
$a\in \mathbb L^{n+2}$
be a nonzero null vector and let
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
be a two-sided hypersurface immersed into the hyperbolic space
$\mathbb {H}^{n+1}$
. Then,
$$ \begin{align} \mathcal{L}_{r,s} (\ell_a) = \sum_{j=r}^{s}(j+1)a_jc_j (H_{j+1}f_a+H_j\,\ell_a), \end{align} $$
where
$\mathcal {L}_{r,s}$
is the differential operator on
$\Sigma^n$
given in (2-11);
$\ell _a$
and
$f_a$
are respectively the height and angle functions of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
with respect to a defined in (3-4) and (3-5);
$H_j$
and
$H_{j+1}$
are two consecutive higher order mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
and
$c_j=(j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})$
, with
$j\in \{r,\ldots ,s\}$
; and
$a_r,\ldots ,a_s$
are real numbers (at least one of them nonzero). Furthermore, if
$$ \begin{align} \sum_{j=r}^{s}(j+1)a_jc_jH_j>0 \end{align} $$
on
$\Sigma^n$
, then
$$ \begin{align} \mathcal{L}_{r,s} (\ell_a^{\,2}) &= \left(\sqrt{\sum_{j=r}^{s} (j+1)a_jc_jH_j}\,f_a+\frac{\mathcal{F}_{r,s}}{\sqrt{\sum_{j=r}^{s}(j+1)a_jc_jH_j}}\, \ell_a\right)^2\nonumber\\ &\quad +\bigg(\sum_{j=r}^{s}(j+1)a_jc_jH_j-\frac{\mathcal{F}_{r,s}^{\,2}}{\sum_{j=r}^{s}(j+1)a_jc_jH_j}\bigg)\ell_a^{\,2} \nonumber \\ &\quad +|\nabla \ell_a|^2\sum_{j=r}^{s}(j+1)a_jc_jH_j\nonumber \\ &\quad +2\langle\mathcal{P}_{r,s}(\nabla \ell_a),\nabla \ell_a\rangle, \end{align} $$
where
$\mathcal {F}_{r,s}$
and
$\mathcal {P}_{r,s}$
are the linearized curvature function and the modified Newton transformation of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
given in (2-8) and (2-10), respectively.
Proof. For any
$j\in \{0,\ldots ,n-1\}$
, from [Reference Rosenberg17],
$$ \begin{align} L_j(\ell_a)=c_j (H_{j+1}f_a+H_j\,\ell_a). \end{align} $$
Hence, (3-11) is obtained immediately from (2-11) and (3-14).
When (3-12) is valid on
$\Sigma^n$
, from (2-12) and (3-11), we deduce that
$$ \begin{align} \mathcal{L}_{r,s}(\ell_a^{\,2})&= 2\,\ell_a\,\mathcal{L}_{r,s}(\ell_a)+2\langle\mathcal{P}_{r,s}(\nabla \ell_a),\nabla \ell_a\rangle \nonumber \\ &=2\sum_{j=r}^{s}(j+1)a_jc_jH_{j+1}f_a\ell_a+2\sum_{j=r}^{s}(j+1)a_jc_jH_j\,\ell_a^{\,2}\nonumber\\ &\quad +2\langle\mathcal{P}_{r,s}(\nabla \ell_a),\nabla \ell_a\rangle \nonumber\\ &=\left(\sqrt{\sum_{j=r}^{s}(j+1)a_jc_jH_j}\,f_a+\frac{\mathcal{F}_{r,s}}{\sqrt{\sum_{j=r}^{s}(j+1)a_jc_jH_j}}\,\ell_a\right)^2 \nonumber\\ &\quad -\sum_{j=r}^{s}(j+1)a_jc_jH_jf_a^2-\frac{\mathcal{F}_{r,s}^{\,2}}{\sum_{j=r}^{s}(j+1)a_jc_jH_j}\,\ell_a^{\,2}\nonumber\\ &\quad +2\sum_{j=r}^{s}(j+1)a_jc_jH_j\,\ell_a^{\,2}+2\langle\mathcal{P}_{r,s}(\nabla \ell_a),\nabla \ell_a\rangle \nonumber\\ &=\left(\sqrt{\sum_{j=r}^{s}(j+1)a_jc_jH_j}\,f_a+\frac{\mathcal{F}_{r,s}}{\sqrt{\sum_{j=r}^{s}(j+1)a_jc_jH_j}}\,\ell_a\right)^2 \nonumber\\ &\quad +\left(\sum_{j=r}^{s}(j+1)a_jc_jH_j-\frac{\mathcal{F}_{r,s}^{\,2}}{\sum_{j=r}^{s}(j+1)a_jc_jH_j}\right)\ell_a^{\,2} \nonumber\\ &\quad +(\ell_a^{\,2}-f_a^2)\sum_{j=r}^{s}(j+1)a_jc_jH_j+2\langle\mathcal{P}_{r,s}(\nabla \ell_a),\nabla \ell_a\rangle.\\[-30pt]\nonumber \end{align} $$
From (3-2), we observe that the support functions of each horosphere
$L_{\tau }$
of the hyperbolic space
$\mathbb {H}^{n+1}$
satisfy
$\ell _{a} = - f_{a}$
, where a is the nonzero null vector, which defines such a horosphere. For this reason, in what follows, we study two-sided hypersurfaces
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
such that their support functions satisfy
$\ell _a=\lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
. In this setting, since
$\ell _a=\lambda f_a$
, we have that
$\nabla \ell _{a} = \lambda \nabla f_{a}$
. Consequently, from (3-6) and (3-7), we observe that
$$ \begin{align} \lambda^{-1}\nabla \ell_{a} = -A(\nabla \ell_{a}). \end{align} $$
Next, taking into account (2-1) and (3-16), we present our second proposition of this section.
Proposition 3.3. Let
$a\in \mathbb L^{n+2}$
be a nonzero null vector and let
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
be a two-sided hypersurface immersed into the hyperbolic space
$\mathbb {H}^{n+1}$
such that the support functions satisfy
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
. Then,
$$ \begin{align} P_{j}(\nabla \ell_a)=\bigg\{\sum_{i=0}^{j}{n \choose\ i}H_i\lambda^{i-j}\bigg\}\nabla \ell_{a}, \end{align} $$
where
$P_{j}$
is the j th Newton transformation and
$H_i$
stands for the i th mean curvature of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
. Furthermore,
$$ \begin{align} \mathcal{P}_{r,s}(\nabla \ell_{a}) = \bigg\{\sum_{j=r}^{s}\sum_{i=0}^{j}(j+1)a_{j}{n \choose\ i}H_{i}\lambda^{i-j}\bigg\}\nabla \ell_{a}, \end{align} $$
where
$\mathcal {P}_{r,s}$
is the modified Newton transformation defined in (2-10) on
$\Sigma^n$
;
$a_{r},\ldots ,a_{s}$
are real numbers (at least one of them nonzero); and r and s are any integers satisfying the inequalities
$0\leq r\le s\leq n-1$
.
Proof. We use induction on the index j to prove the equality (3-17). Taking into account the definition of
$P_{j}$
in (2-1), we have that
$P_{0}(\nabla \ell _{a}) = \nabla \ell _{a}$
. However, for
$j=0$
, the right-hand side of (3-17) becomes
$$ \begin{align*} {n \choose\ 0}H_{0}\lambda^{0}\nabla \ell_{a} = \nabla \ell_{a}. \end{align*} $$
Let us suppose that (3-17) is true for
$j=k-1$
. Thus,
$$ \begin{align*} P_{k-1}(\nabla \ell_{a}) = \bigg\{\displaystyle\sum_{i=0}^{k-1}{n \choose\ i}H_{i}\lambda^{i-(k-1)}\bigg\} \nabla \ell_{a}. \end{align*} $$
$$ \begin{align*} P_{k}(\nabla \ell_{a}) &= {n \choose\ k}H_{k}\nabla \ell_{a} - A(P_{k-1}(\nabla \ell_{a})) \\ &= {n \choose\ k}H_{k}\nabla \ell_{a} - A\bigg(\bigg\{\sum_{i=0}^{k-1}{n \choose\ i}H_{i}\lambda^{i-(k-1)}\bigg\} \nabla \ell_{a}\bigg) \\ &={n \choose\ k}H_{k}\nabla \ell_{a} + \bigg\{\sum_{i=0}^{k-1}{n \choose\ i}H_{i}\lambda^{i-(k-1)}\bigg\} \lambda^{-1} \nabla \ell_{a} \\ &= \bigg\{\sum_{i=0}^{k}{n \choose\ i}H_{i}\lambda^{i-k} \bigg\}\nabla \ell_{a}. \end{align*} $$
Finally, from (2-10) and (3-17),
$$ \begin{align*} \mathcal{P}_{r,s}(\nabla \ell_{a}) &= \sum_{j=r}^{s}(j+1)a_{j}P_{j}(\nabla \ell_{a}) \\ &= \sum_{j=r}^{s}(j+1)a_{j}\sum_{i=0}^{j}{n \choose\ i}H_{i}\lambda^{i-j}\nabla \ell_{a} \\ &= \bigg\{\sum_{j=r}^{s}\sum_{i=0}^{j}(j+1)a_{j}{n \choose\ i}H_{i}\lambda^{i-j} \bigg\} \nabla \ell_{a} .\\[-3.9pc] \end{align*} $$
4 Main results
This section is devoted to establishing our main results which, essentially, guarantee that a complete two-sided hypersurface of the hyperbolic space
$\mathbb {H}^{n+1}$
must be isometric to a horosphere under appropriate hypotheses on its linearized curvature function (see Definition 2.1). For this purpose, we apply suitable maximum principles as analytical tools.
4.1 Via an integrability criterion
In [Reference Yau19], Yau generalized a previous result due to Gaffney [Reference Gaffney12], establishing the following version of Stokes’ theorem on an n-dimensional, complete noncompact Riemannian manifold
$\Sigma^n$
: if
$\omega \in \Omega ^{n-1}(\Sigma^n)$
is an integrable
$(n-1)$
-differential form on
$\Sigma^n$
, then there exists a sequence
$B_i$
of domains on
$\Sigma^n$
such that
$$ \begin{align*} B_i\subset B_{i+1}, \quad \Sigma^n=\bigcup_{i\geq1}B_i \quad \mathrm{and} \quad \lim_{i\rightarrow+\infty}\displaystyle\int_{B_i}d\omega=0. \end{align*} $$
Suppose
$\Sigma^n$
is oriented by the volume element
$d\Sigma $
, and let
$\mathcal L^1(\Sigma^n)$
be the space of Lebesgue integrable functions on
$\Sigma^n$
. If
$\omega =\iota _Xd\Sigma $
is the contraction of
$d\Sigma $
in the direction of a smooth vector field X on
$\Sigma^n$
, then Caminha obtained the following consequence of Yau’s result (see [Reference Caminha8, Proposition 2.1]).
Lemma 4.1. Let X be a smooth vector field on the n-dimensional complete noncompact oriented Riemannian manifold
$\Sigma^n$
such that
$\mathrm {div}X$
does not change sign on
$\Sigma^n$
. If
$|X|\in \mathcal L^1(\Sigma^n)$
, then
$\mathrm {div}X=0$
.
Next, we use Lemma 4.1 to prove our first main result.
Theorem 4.2. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete noncompact two-sided hypersurface of
$\mathbb {H}^{n+1}$
, with bounded shape operator, and let r and s be any integers satisfying
$0\leq r\le s\leq n-1$
. Suppose that the linearized curvature function
$\mathcal {F}_{r,s}=(r+1)a_rc_rH_{r+1}+\cdots +(s+1)a_sc_sH_{s+1}$
of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is such that
$$ \begin{align} 0\leq\mathcal{F}_{r,s}\leq\sum_{j=r}^{s}(j+1)a_jc_jH_j \end{align} $$
on
$\Sigma^n$
, where
$a_r,\ldots ,a_s$
are real numbers (at least one of them nonzero),
$H_j$
and
$H_{j+1}$
are two consecutive higher order mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
, and
$c_j=(j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})$
,
$j\in \{r,\ldots ,s\}$
. If
$|a^\top |\in \mathcal {L}^1(\Sigma^n)$
for some nonzero null vector
$a\in \mathbb {L}^{n+2}$
, then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tau }^n$
of
$\mathbb {H}^{n+1}$
for some
$\tau>0$
.
Proof. Let
$\ell _a$
be the height function of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
with respect to
$a\in \mathbb {L}^{n+2}$
, given in (3-4). Let us consider the vector field
$X=\mathcal {P}_{r,s}(\nabla \ell _a)\in \mathfrak {X}(\Sigma^n)$
, where
$\mathcal {P}_{r,s}$
is the modified Newton transformation of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
given in (2-10). Taking into account that
$\ell _a$
is a positive function on
$\Sigma^n$
, from (2-13), (3-10), (3-11), and (4-1),
$$ \begin{align} \mathrm{div}X=\mathcal{F}_{r,s}\,f_a+\bigg(\sum_{j=r}^{s}(j+1)a_jc_jH_j\bigg)\ell_a\geq\bigg(\sum_{j=r}^{s}(j+1)a_jc_jH_j-\mathcal{F}_{r,s}\bigg)\ell_a\geq0 \end{align} $$
on
$\Sigma^n$
. Furthermore, since the shape operator of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
is bounded, from (2-2), we get that each j th Newton transformation
$P_j$
is also bounded,
$j\in \{0,\ldots ,n\}$
. Next, taking into account that
$|a^\top |\in \mathcal {L}^1(\Sigma^n)$
, from (2-10) and (3-6),
$$ \begin{align*} |X|\leq\sum_{j=r}^{s}(j+1)|a_j||P_j||a^\top|\in\mathcal{L}^1(\Sigma^n). \end{align*} $$
Next, we can apply Lemma 4.1 to get that
$\mathrm {div}X=0$
on
$\Sigma^n$
. Hence, by returning to (4-2), on
$\Sigma^n$
,
$$ \begin{align*} \ell_a=-f_a \quad\mathrm{and}\quad \mathcal{F}_{r,s}=\sum_{j=r}^{s}(j+1)a_jc_jH_j. \end{align*} $$
Therefore, from Remark 3.1,
$\ell _a$
is constant on
$\Sigma^n$
, which means that
$\psi (\Sigma^n)$
is a horosphere of
$\mathbb H^{n+1}$
orthogonal to
$a\in \mathbb {L}^{n+2}$
.
Remark 4.3. It is worth noting that for the particular case where
$r=s$
, with
$1 \leq r \leq n-1$
, and
$a_{r} = ((r+1)^2(\begin {smallmatrix}n \\ r+1\end {smallmatrix}))^{-1}$
, we get [Reference Aquino, Baltazar and de Lima7, Theorem
$2$
] as a corollary of Theorem 4.2.
4.2 Via a criterion of convergence to zero at infinity
We start this subsection by establishing the concept of convergence to zero at infinity. Given a connected, complete noncompact Riemannian manifold
$\Sigma^n$
and denoting by
$d(\cdot ,o): \Sigma^n \rightarrow [0,+\infty )$
the Riemannian distance of
$\Sigma^n$
measured from a fixed point
$o \in \Sigma^n$
, a function
${f \in C^{\infty }(M)}$
converges to zero at infinity when
$$ \begin{align*}\lim_{d(x,o) \rightarrow \infty} f(x) = 0.\end{align*} $$
In the following, to demonstrate our next results, we can enunciate a lemma due to Alías et al. [Reference Alías, Caminha and do Nascimento2].
Lemma 4.4. Let
$\Sigma^n$
be a complete noncompact Riemannian manifold and let
$X\in \mathfrak {X}(\Sigma^n)$
be a vector field on
$\Sigma^n$
. Assume that there exists a nonnegative, nonidentically vanishing function
$f\in C^{\infty }(\Sigma^n)$
that converges to zero at infinity and such that
$\langle \nabla f,X\rangle \geq 0$
. If
$\mathrm {div}X\geq 0$
on
$\Sigma^n$
, then
$\langle \nabla f,X\rangle \equiv 0$
on
$\Sigma^n$
.
Let
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
be a two-sided hypersurface of
$\mathbb {H}^{n+1}$
and
$\ell _{a}$
the height function of
$\psi (\Sigma^n)$
with respect to a nonzero null vector
$a \in \mathbb {L}^{n+2}$
and
$\tau \in \mathbb {R}$
. We say that
$\ell _{a}$
converges from above to
$\tau $
at infinity when
$\ell _{a} \geq \tau $
and
$\ell _{a} - \tau $
converges to zero at infinity. Now, we are in a position to present four more results.
Theorem 4.5. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete noncompact two-sided hypersurface of
$\mathbb {H}^{n+1}$
, and let r and s be any integers satisfying
$0\leq r\le s\leq n-1$
. Suppose that the modified Newton transformation
$\mathcal {P}_{r,s}$
is positive definite in the direction of
$\nabla \ell _{a}$
and the linearized curvature function
$\mathcal {F}_{r,s}=(r+1)a_rc_rH_{r+1}+\cdots +(s+1)a_sc_sH_{s+1}$
of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is such that
$$ \begin{align} 0\leq\mathcal{F}_{r,s}\leq\sum_{j=r}^{s}(j+1)a_jc_jH_j \end{align} $$
on
$\Sigma^n$
, where
$a_r,\ldots ,a_s$
are real numbers (at least one of them nonzero),
$H_j$
is a higher order mean curvature of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
, and
$c_j=(j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})$
,
$j\in \{r,\ldots ,s\}$
. If the height function
$\ell _{a}$
of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
converges from above to
$\tau $
at infinity, where
$\tau $
is a positive real number, then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tau }^n$
of
$\mathbb {H}^{n+1}$
.
Proof. Suppose for contradiction that
$\psi (\Sigma ^{n})$
is not a horosphere
$L^{n}_{\tau }$
of
$\mathbb {H}^{n+1}$
. We have that
$\ell _{a} - \tau $
is a nonnegative and nonidentically vanishing function of
$C^{\infty }(\Sigma ^{n})$
that converges to zero at infinity.
Let us take the smooth vector field
$X = \mathcal {P}_{r,s}(\nabla \ell _{a}) \in \mathfrak {X}(\Sigma ^{n})$
. From (4-3) and (4-2), we obtain
$\mathrm {div}X \geq 0$
and
$\langle \mathcal {P}_{r,s}(\nabla \ell _{a}),\nabla \ell _{a}\rangle \geq 0$
by hypothesis. Since
$\nabla (\ell _{a} - \tau ) = \nabla \ell _{a}$
, applying Lemma 4.4, we obtain
$\langle \mathcal {P}_{r,s}(\nabla \ell _{a}),\nabla \ell _{a}\rangle = 0$
on
$\Sigma ^{n}$
and therefore
$\nabla \ell _{a} =0$
.
In this way,
$\ell _{a}$
is constant and converges to
$\tau $
at infinity. We conclude that
$\ell _{a} = \tau $
and
$\psi (\Sigma ^{n})$
is a horosphere
$L^{n}_{\tau }$
of
$\mathbb {H}^{n+1}$
, which cannot occur.
Taking again
$r=s$
and choosing
$a_{r} = ((r+1)^2(\begin {smallmatrix}n\\ r+1\end {smallmatrix}))^{-1}$
, returning to Theorem 4.5, we derive the following consequence.
Corollary 4.6. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete noncompact two-sided hypersurface of
$\mathbb {H}^{n+1}$
such that the Newton transformation
$P_{r}$
is positive definite in the direction of
$\nabla \ell _{a}$
and
$$ \begin{align*} 0 \leq H_{r+1} \leq H_{r} \end{align*} $$
on
$\Sigma^n$
, with
$1 \leq r \leq n-1$
, where
$H_{r}$
and
$H_{r+1}$
are two consecutive higher order mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
. If the height function
$\ell _{a}$
of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
converges from above to
$\tau $
at infinity, where
$\tau $
is a positive real number, then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tau }^n$
of
$\mathbb {H}^{n+1}$
.
For our next results, we consider
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
a two-sided hypersurface of
$\mathbb {H}^{n+1}$
such that the support functions satisfy
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
, which results in the following inequality:
$$ \begin{align} H_{j}> -\displaystyle{n \choose\ j}^{-1}\displaystyle\sum_{i=0}^{j-1}{n \choose\ i}H_{i}\lambda^{i-j}, \end{align} $$
with
$0 < r \leq j \leq s$
(or
$r+1 \leq j \leq s$
if
$r=0$
), where
$r \text { and } s$
are integers such that
$0 \leq r \leq s \leq n-1$
.
Our reason to consider this configuration is motivated by the fact that the inequality (4-4) is satisfied on the horosphere
$L^{n}_{\tau }$
, with
$0 < \tau \in \mathbb {R}$
for
$1 \leq j \leq n-1$
. More precisely, on
$L^{n}_{\tau }$
, we have that
$\lambda = -1$
and
$H_{m} = 1$
,
$0 \leq m \leq n$
. So, we want to verify that
$$ \begin{align*} 1>-\displaystyle{n\choose\ j}^{-1}\displaystyle\sum_{i=0}^{j-1}{n\choose\ i}(-1)^{i-j}. \end{align*} $$
For this, we claim that
$$ \begin{align*} -\displaystyle{n \choose\ j}^{-1}\displaystyle\sum_{i=0}^{j-1}{n\choose\ i}(-1)^{i-j}=\frac{j}{n}. \end{align*} $$
Indeed, for
$j=1$
,
$$ \begin{align*} -\displaystyle{n \choose\ 1}^{-1}\displaystyle\sum_{i=0}^{0}{n \choose\ i}(-1)^{i-1} = -\displaystyle{n \choose\ 1}^{-1}{n \choose\ 0}(-1)^{0-1} = \frac{1}{n}. \end{align*} $$
Suppose that
$$ \begin{align*} -\displaystyle{n \choose\ k}^{-1}\displaystyle\sum_{i=0}^{k-1}{n\choose\ i}(-1)^{i-k}=\frac{k}{n}. \end{align*} $$
So, for
$j=k+1$
,
$$ \begin{align*} &-\displaystyle{n \choose\ k+1}^{-1}\!\displaystyle\sum_{i=0}^{k}{n \choose\ i}(-1)^{i-(k+1)}\\&\quad =-\displaystyle{n \choose\ k}^{-1}\bigg(\frac{n-k}{k+1}\bigg)^{-1} \bigg\{\sum_{i=0}^{k-1}{n \choose\ i}(-1)^{i-(k+1)} +{n \choose\ k}(-1)^{-1} \bigg\} \\&\quad = \bigg(\frac{k+1}{n-k}\bigg)\displaystyle{n \choose\ k}^{-1}\sum_{i=0}^{k-1}{n \choose\ i}(-1)^{i-k} + \bigg(\frac{k+1}{n-k}\bigg)\displaystyle{n \choose\ k}^{-1}\displaystyle{n \choose\ k}\\&\quad =\bigg(\frac{k+1}{n-k}\bigg)\bigg(1+\frac{-k}{n}\bigg)\\&\quad =\frac{k+1}{n}. \end{align*} $$
Therefore, we conclude that
$$ \begin{align*} 1> \frac{j}{n} = -\displaystyle{n \choose\ j}^{-1}\displaystyle\sum_{i=0}^{j-1}{n \choose\ i}(-1)^{i-j}. \end{align*} $$
Motivated by the previous digression, we present our next characterization result concerning the horospheres of the hyperbolic space.
Theorem 4.7. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete noncompact two-sided hypersurface of
$\mathbb {H}^{n+1}$
such that the support functions satisfy
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
, and let r and s be any integers satisfying
$0\leq r\le s\leq n-1$
. Suppose that the linearized curvature function
$\mathcal {F}_{r,s}=(r+1)a_rc_rH_{r+1}+\cdots +(s+1)a_sc_sH_{s+1}$
of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is such that
$$ \begin{align} - \lambda^{-1}\mathcal{F}_{r,s}\leq\sum_{j=r}^{s}(j+1)a_jc_jH_j \end{align} $$
on
$\Sigma^n$
, where
$a_r,\ldots ,a_s$
are nonnegative real numbers (at least one of them nonzero),
$H_j$
is a higher order mean curvature of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
, and
$c_j=(j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})$
,
$j\in \{r,\ldots ,s\}$
. If the inequality (4-4) is satisfied and the height function
$\ell _{a}$
of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
converges from above to
$\tau $
at infinity, where
$\tau $
is a positive real number, then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tau }^n$
of
$\mathbb {H}^{n+1}$
.
Proof. Suppose for contradiction that
$\psi (\Sigma ^{n})$
is not a horosphere
$L^{n}_{\tau }$
of
$\mathbb {H}^{n+1}$
. We have that
$\ell _{a} - \tau $
is a nonnegative and nonidentically vanishing function of
$C^{\infty }(\Sigma ^{n})$
that converges to zero at infinity.
Let us take
$X =\mathcal {P}_{r,s}(\nabla \ell _a)\in \mathfrak {X}(\Sigma^n)$
. From (4-5), we verify that
$$ \begin{align*} \mathrm{div}X=\mathcal{F}_{r,s}f_a+\sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j}\ell_a=\bigg(\sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j}+\lambda^{-1}\mathcal{F}_{r,s} \bigg)\ell_{a} \geq 0. \end{align*} $$
Moreover, from (3-18) of Proposition 3.3,
$$ \begin{align} \langle \mathcal{P}_{r,s}(\nabla \ell_{a}),\nabla \ell_{a}\rangle = \bigg\{\sum_{j=r}^{s}\sum_{i=0}^{j}(j+1)a_{j}{n \choose\ i}H_{i}\lambda^{i-j}\bigg\}|\nabla \ell_{a}|^2. \end{align} $$
Since, from (4-4),
$$ \begin{align*} H_j>-\displaystyle{n\choose\ j}^{-1}\displaystyle\sum_{i=0}^{j-1}{n\choose\ i}H_i\lambda^{i-j}, \quad 0<r\leq j\leq s \quad (\mathrm{or}\,\,r+1\leq j\leq s\,\,\mathrm{if}\,\,r=0), \end{align*} $$
we note that
$$ \begin{align*} \sum_{j=r}^{s}\sum_{i=0}^{j}(j+1)a_{j}{n \choose\ i}H_{i}\lambda^{i-j} = \sum_{j=r}^{s}(j+1)a_{j}\bigg\{{n \choose\ j}H_{j} + \sum_{i=0}^{j-1}{n \choose\ i}H_{i}\lambda^{i-j} \bigg\}> 0. \end{align*} $$
In this way, returning to (4-6), we verify
$$ \begin{align*} \langle \mathcal{P}_{r,s}(\nabla \ell_{a}),\nabla \ell_{a}\rangle = \bigg\{\sum_{j=r}^{s}\sum_{i=0}^{j}(j+1)a_{j}{n \choose\ i}H_{i}\lambda^{i-j}\bigg\}|\nabla \ell_{a}|^2 \geq 0. \end{align*} $$
Applying Lemma 4.4, we obtain that
$\langle \mathcal {P}_{r,s}(\nabla \ell _{a}),\nabla \ell _{a}\rangle = 0$
on
$\Sigma ^{n}$
and, therefore,
$\nabla \ell _{a} =0$
. Thus, we verify that
$\ell _{a}$
is constant and converges to
$\tau $
at infinity. We conclude that
$\ell _{a} = \tau $
and
$\psi (\Sigma ^{n})$
is a horosphere
$L^{n}_{\tau }$
of
$\mathbb {H}^{n+1}$
, which leads us to a contradiction.
For the case where
$r=s$
and
$a_{r} = ((r+1)^2(\begin {smallmatrix}n \\ r+1\end {smallmatrix}))^{-1}$
, the inequality (4-4) becomes
$$ \begin{align} H_{r}> -\displaystyle{n \choose\ r}^{-1}\displaystyle\sum_{i=0}^{r-1}{n \choose\ i}H_{i}\lambda^{i-r}, \end{align} $$
with
$1 \leq i \leq r$
and
$1 \leq r \leq n-1$
. In this setting, returning to Theorem 4.7, we obtain the following corollary.
Corollary 4.8. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete two-sided hypersurface of
$\mathbb {H}^{n+1}$
such that the support functions satisfy
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
. Suppose that
$$ \begin{align*} -\lambda^{-1} H_{r+1} \leq H_{r} \end{align*} $$
on
$\Sigma^n$
, with
$1 \leq r \leq n-1$
, where
$H_{r}$
and
$H_{r+1}$
are two consecutive higher order mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
. If the inequality (4-7) is satisfied and the height function
$\ell _{a}$
of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
converges from above to
$\tau $
at infinity, where
$\tau $
is a positive real number, then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tau }^n$
of
$\mathbb {H}^{n+1}$
.
4.3 Via a volume growth criterion
Let
$\Sigma^n$
be a connected, oriented, complete noncompact Riemannian manifold. Given a polynomial function
$\sigma :(0,+\infty )\rightarrow (0,+\infty )$
, we say that
$\Sigma^n$
has polynomial volume growth like
$\sigma (r)$
if there exists
$x_0\in \Sigma^n$
such that
$$ \begin{align*}\mathrm{vol}(B_r(x_0))=\mathcal{O}(\sigma(r)),\end{align*} $$
as
$r\rightarrow +\infty $
, where
$\mathrm {vol}$
stands for the volume, that is,
$$ \begin{align*} \mathrm{vol}(B_{r}(x_{0})) = \int_{B_{r}(x_{0})} d\Sigma. \end{align*} $$
As was observed in the beginning of [Reference Alías, Caminha and do Nascimento1, Section
$2$
], if
$x_0,x_1\in \Sigma^n$
are at distance d from each other, we can verify that
$$ \begin{align*}\frac{\mathrm{vol}(B_r(x_0))}{\sigma(r)}\geq\frac{\mathrm{vol}(B_{r-d}(x_1))}{\sigma(r-d)}\cdot\frac{\sigma(r-d)}{\sigma(r)}.\end{align*} $$
So, the choice of
$x_0$
in the notion of polynomial volume growth is immaterial. For this reason, we just say that
$\Sigma^n$
has polynomial volume growth.
Considering the previous digression, we quote a key lemma which corresponds to a particular case of a more general maximum principle due to Alías, Caminha, and do Nascimento (see [Reference Alías, Caminha and do Nascimento1, Theorem 2.1]).
Lemma 4.9. Let
$\Sigma^n$
be a connected, oriented, complete noncompact Riemannian manifold and let
$X\in \mathfrak {X}(\Sigma )$
be a bounded smooth vector field on
$\Sigma^n$
. Let
$u\in \mathcal {C}^{\infty }(\Sigma )$
be a smooth function such that
$\langle \nabla u,X \rangle \geq 0$
and
$\mathrm {div}X \geq au$
on
$\Sigma^n$
for some positive constant
$a\in \mathbb R$
. If
$\Sigma^n$
has polynomial volume growth, then
$u\leq 0$
on
$\Sigma^n$
.
Inserted in the above context, we use Lemma 4.9 to establish nonexistence results concerning the concept of polynomial volume growth. So, we state and prove our first one.
Theorem 4.10. There is no complete two-sided hypersurface
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
of
$\mathbb {H}^{n+1}$
, with polynomial volume growth and bounded shape operator, such that
$a^{\top }$
is bounded, the operator
$\mathcal {P}_{r,s}$
is positive semidefinite in the direction of
$\nabla \ell _{a}$
, with r and s integers satisfying
$0 \leq r \leq s \leq n-1$
, and
$$ \begin{align} \sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} - \mathcal{F}_{r,s} \geq \beta> 0 \end{align} $$
for some positive constant
$\beta $
, where
$a_{r}, \ldots , a_{s}$
are real numbers (at least one of them nonzero), and
$H_{j}$
and
$\mathcal {F}_{r,s}$
are respectively the j th mean curvature and the linearized curvature function of
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
,
$c_{j} = (j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})$
,
$j \in \{r,\ldots , s\}$
.
Proof. Suppose for contradiction that there exists a complete two-sided hypersurface
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
of
$\mathbb {H}^{n+1}$
that satisfies all hypotheses of Theorem 4.10.
Taking a smooth vector field
$X=\mathcal {P}_{r,s}(\nabla \ell _{a}) \in \mathfrak {X}(\Sigma^n)$
, we verify that
$$ \begin{align*} |X| \leq \displaystyle\sum_{j=r}^{s}(j+1)|a_{j}||P_{j}||a^{\top}|. \end{align*} $$
Since the shape operator is bounded, we obtain that
$P_{j}$
is also bounded for
$j \in \{0,\ldots ,n\}$
, and since
$a^{\top }$
is bounded by hypothesis, we conclude that
$X \in \mathfrak {X}(\Sigma^n)$
is a bounded smooth vector field on
$\Sigma^n$
.
Moreover, by the hypotheses, we have
$\langle \mathcal {P}_{r,s}(\nabla \ell _{a}), \nabla \ell _{a} \rangle \geq 0$
and from (4-8) and (4-2),
$$ \begin{align} \mathrm{div}X \geq \bigg( \sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} - \mathcal{F}_{r,s} \bigg)\ell_{a} \geq \beta \ell_{a}. \end{align} $$
Taking into account that
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
has polynomial volume growth, if
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
is noncompact, applying Lemma 4.9, we conclude that
$\ell _{a} \leq 0$
, which is absurd. If
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
is compact (without boundary), from (4-9),
$$ \begin{align*} 0 = \int_{\Sigma} \mathrm{div}X\, d\Sigma \geq \int_{\Sigma^n}\beta \ell_{a}\,d\Sigma, \end{align*} $$
and hence,
$\ell _{a}$
should be zero, which constitutes a contradiction.
Let us take
$r=s$
and
$a_{r} = ((r+1)^2(\begin {smallmatrix}n \\ r+1\end {smallmatrix}))^{-1}$
. Returning to Theorem 4.10, we obtain the following corollary.
Corollary 4.11. There is no complete two-sided hypersurface
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
of
$\mathbb {H}^{n+1}$
, with polynomial volume growth and bounded shape operator, such that
$a^{\top }$
is bounded, the operator
$P_{r}$
is positive semidefinite in the direction of
$\nabla \ell _{a}$
, and
$$ \begin{align*} 0 < \beta \leq H_{r} - H_{r+1} \end{align*} $$
for some positive constant
$\beta $
, where
$H_{r}$
and
$H_{r+1}$
are two consecutive higher order mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
.
To establish the last results of this subsection, we consider once again a two-sided hypersurface
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
such that the support functions satisfy the relation
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
and the inequality (4-4) holds.
Theorem 4.12. There is no complete two-sided hypersurface
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
of
$\mathbb {H}^{n+1}$
, with bounded shape operator and polynomial volume growth, such that the support functions satisfy the relation
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
,
$\nabla \ell _{a}$
is bounded, the inequality (4-4) is satisfied, and
$$ \begin{align} \sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} + \lambda^{-1} \mathcal{F}_{r,s} \geq \beta> 0 \end{align} $$
for some positive constant
$\beta $
, where
$a_{r}, \ldots , a_{s}$
are nonnegative real numbers (at least one of them nonzero), and
$H_{j}$
and
$\mathcal {F}_{r,s}$
are respectively the j th mean curvature and the linearized curvature function of
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
, with r and s integers satisfying
$0 \leq r \leq s \leq n-1$
and
$c_{j} = (j+1)(\begin {smallmatrix}n\\ j+1\end {smallmatrix})$
for
$j \in \{r,\ldots , s\}$
.
Proof. Suppose for contradiction that there exists a complete two-sided hypersurface
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
of
$\mathbb {H}^{n+1}$
satisfying all hypotheses of Theorem 4.12.
In a similar way to the proof of Theorem 4.7,
$$ \begin{align*} \langle \mathcal{P}_{r,s}(\nabla \ell_{a}),\nabla \ell_{a}\rangle = \bigg\{\sum_{j=r}^{s}\sum_{i=0}^{j}(j+1)a_{j}{n \choose\ i}H_{i}\lambda^{i-j}\bigg\} |\nabla \ell_{a}|^2 \geq 0. \end{align*} $$
However, taking a smooth vector field
$X = \mathcal {P}_{r,s}(\nabla \ell _{a}) \in \mathfrak {X}(\Sigma^n)$
,
$$ \begin{align*} |X| \leq \displaystyle\sum_{j=r}^{s}(j+1)|a_{j}||P_{j}||\nabla \ell_{a}|. \end{align*} $$
Since the shape operator and
$\nabla \ell _{a}$
are bounded on
$\Sigma^n$
, we can reason as in the proof of Theorem 4.10 to conclude that X is a bounded smooth vector field on
$\Sigma^n$
.
Moreover, from (4-10) and (4-2),
$$ \begin{align} \mathrm{div}X = \bigg( \sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} + \lambda^{-1} \mathcal{F}_{r,s} \bigg)\ell_{a} \geq \beta \ell_{a}. \end{align} $$
Taking into account that
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
has polynomial volume growth, if
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
is noncompact, using Lemma 4.9, we conclude that
$\ell _{a} \leq 0$
, which is absurd. If
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
is compact (without boundary), from (4-11),
$$ \begin{align*} 0 = \int_{\Sigma} \mathrm{div}X\, d\Sigma \geq \int_{\Sigma^n}\beta \ell_{a}\,d\Sigma; \end{align*} $$
therefore,
$\ell _{a}$
should be zero, which cannot happen.
Taking again
$r=s$
and choosing
$a_{r} = ((r+1)^2(\begin {smallmatrix}n \\ r+1\end {smallmatrix}))^{-1}$
, returning to Theorem 4.12, we obtain the following consequence.
Corollary 4.13. There is no complete noncompact two-sided hypersurface
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
of
$\mathbb {H}^{n+1}$
, with bounded shape operator and polynomial volume growth, such that the support functions satisfy
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
,
$\nabla \ell _{a}$
is bounded, the inequality (4-7) is satisfied, and
$$ \begin{align*} 0 < \beta \leq H_{r} + \lambda^{-1} H_{r+1} \end{align*} $$
for some positive constant
$\beta $
, where
$H_{r}$
and
$H_{r+1}$
are two consecutive higher order mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
, with
$1 \leq r \leq n-1$
.
4.4 Via an extension of the generalized maximum principle of Omori and Yau
Let
$\Sigma^n$
be a complete Riemannian manifold and let
$\mathcal {T}:\mathfrak {X}(\Sigma^n)\rightarrow \mathfrak {X}(\Sigma^n)$
be a self-adjoint operator. Extending the notion of the differential operators
$L_j$
and
$\mathcal {L}_{r,s}$
defined in (2-5) and (2-11), respectively, we consider the following trace-type second-order linear differential operator:
$$ \begin{align} \begin{array}{rccl} \mathfrak{L}:&\!\!C^{\infty}(\Sigma^n)&\rightarrow&C^{\infty}(\Sigma^n)\\ &f&\mapsto&\mathfrak{L}(f)=\mathrm{tr}(\mathcal{T}\circ\nabla^{\,2}f) \end{array} \end{align} $$
naturally associated to
$\mathcal {T}$
.
From [Reference Alías, Impera and Rigoli5, Lemma
$4.2$
], we have the following generalized maximum principle related to such an operator
$\mathfrak {L}$
, which is an extension of the generalized maximum principle of Omori and Yau [Reference Omori16, Reference Yau18] (see also [Reference Alías, Mastrolia and Rigoli6] for a modern and comprehensive reference to this topic.
Lemma 4.14. Let
$\Sigma^n$
be a complete Riemannian manifold with sectional curvature bounded from below, and
$f\in C^{\infty }(\Sigma^n)$
be a function that is bounded from above on
$\Sigma^n$
. If
$\mathcal {T}:\mathfrak {X}(\Sigma^n)\rightarrow \mathfrak {X}(\Sigma^n)$
is a positive semidefinite self-adjoint operator such that
$\mathrm {tr}(\mathcal {T})$
is bounded from above on
$\Sigma^n$
, then there exists a sequence
$(p_k)_{k\geq 1}$
in
$\Sigma^n$
satisfying
$$ \begin{align*} \lim_kf(p_k)=\sup_{\Sigma^n}f, \quad \lim_k|\nabla f(p_k)|=0 \quad \mathrm{and} \quad \limsup_k\mathfrak{L}f(p_k)\leq0, \end{align*} $$
where
$\mathfrak {L}$
is the trace-type second-order linear differential operator defined in (4-12).
Before the next results, we recall that the closed horoball determined by a horosphere
$L_{\tau }^n$
of
$\mathbb {H}^{n+1}$
is just the closure of the interior domain enclosed by
$L_{\tau }$
.
Theorem 4.15. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete two-sided hypersurface contained in the closed horoball determined by a horosphere
$L_{\tau }$
orthogonal to a nonzero null vector
$a\in \mathbb {L}^{n+2}$
. Suppose that the first linearized curvature function
$\mathcal {F}_{0,1}=na_0H+2n(n-1)a_1H_2$
of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is such that
$$ \begin{align} 0<\mathcal{F}_{0,1}\leq na_0+2n(n-1)a_1H \end{align} $$
on
$\Sigma^n$
, where H and
$H_2$
are the first two mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
,
$a_0$
is any nonnegative real number, and
$a_1$
is a positive real number. Suppose also that H is bounded and
$H_2>0$
on
$\Sigma^n$
. If
$$ \begin{align} |a^\top|\leq C\inf_{\Sigma^n} \{na_0+2n(n-1)a_1H-\mathcal{F}_{0,1}\} \end{align} $$
for some positive constant C, then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tilde {\tau }}^n$
of
$\mathbb {H}^{n+1}$
for some
$\tilde {\tau }\leq \tau $
.
Proof. Since we are assuming that H is bounded, Hypothesis (4-13) guarantees that
$H_2$
is also bounded. Thus, taking into account the algebraic relation
$$ \begin{align} \sum_{i=1}^n\lambda_i^2=|A|^2=n^2H^2-n(n-1)H_2, \end{align} $$
we have that all the principal curvatures
$\lambda _1,\ldots ,\lambda _n$
of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
, are bounded. Consequently, from the Gauss equation,
$$ \begin{align} K_{\Sigma}(e_i,e_j)=-1+\lambda_{i}\lambda_{j}, \end{align} $$
we conclude that the sectional curvature
$K_{\Sigma }$
of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
is bounded from below.
However, since
$a_{0}\geq 0$
,
$a_1>0$
and the functions
$H_2$
and
$na_0+ 2n(n-1)a_1H$
only assume positive values on
$\Sigma^n$
, [Reference Elbert11, Lemma
$3.10$
] gives that
$\mathcal {P}_{0,1}$
is positive definite. Furthermore, from (2-4),
$$ \begin{align*} \mathrm{tr}(\mathcal{P}_{0,1})=(na_0+2n(n-1)a_1H), \end{align*} $$
which is bounded from above, because H is bounded.
Since
$\mathcal {P}_{0,1}$
is positive definite and taking into account that
${\ell _{a}^{2} \geq - f_{a} \ell _{a}}$
, from (3-11) and from (4-13),
$$ \begin{align} \mathcal{L}_{0,1} (\ell_a^{\,2}) &= 2 \ell_{a} \mathcal{L}_{0,1}(\ell_{a}) + 2 \langle \mathcal{P}_{0,1}(\nabla \ell_{a}), \nabla \ell_{a} \rangle \nonumber\\ & \geq (na_0+2n(n-1)a_1H-\mathcal{F}_{0,1} )\ell_a^{\,2}\geq0 \end{align} $$
on
$\Sigma^n$
, where
$\ell _a$
is the height function of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
with respect to
$a\in \mathbb {L}^{n+2}$
defined in (3-4). However, our assumption that
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
is contained in the closed horoball determined by a horosphere of
$\mathbb {H}^{n+1}$
guarantees that
$\ell _a$
is bounded. Thus, we can apply Lemma 4.14 to obtain a sequence
$(p_k)_{k\geq 1}$
in
$\Sigma^n$
such that
$$ \begin{align} \lim_k\ell_a^{\,2}(p_k)=\sup_{\Sigma}\ell_a^{\,2}, \quad \lim_k|\nabla\ell_a^{\,2}(p_k)|=0 \quad \mathrm{and} \quad \limsup_k\mathcal{L}_{0,1}(\ell_a^{\,2})(p_k)\leq0. \end{align} $$
Consequently, from (4-17) and (4-18),
$$ \begin{align} 0\geq\limsup_k\mathcal{L}_{0,1}(\ell_a^{\,2})(p_k)\geq 2 (\sup_{\Sigma^n}\ell_a^{\,2})\limsup_k (na_0+2n(n-1)a_1H-\mathcal{F}_{0,1} )(p_k)\geq0. \end{align} $$
Thus, since
$$ \begin{align*} \sup_{\Sigma^n}\ell_a^{\,2}>0, \end{align*} $$
from (4-19),
$$ \begin{align*} \limsup_k (na_0+2n(n-1)a_1H-\mathcal{F}_{0,1})(p_k)=0 \end{align*} $$
and, in particular, by considering again (4-13),
$$ \begin{align} \inf_{\Sigma^n} \{na_0+2n(n-1)a_1H-\mathcal{F}_{0,1}\}=0. \end{align} $$
Therefore, Hypothesis (4-14) jointly with (4-20) guarantees that
$a^\top $
vanishes identically on
$\Sigma^n$
, which means that
$\psi (\Sigma^n)$
is a horosphere orthogonal to
$a\in \mathbb {L}^{n+2}$
.
Knowing that
$H_{2} \leq H^2$
, from Theorem 4.15, we obtain the following corollary.
Corollary 4.16. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete two-sided hypersurface contained in the closed horoball determined by a horosphere
$L_{\tau }$
orthogonal to a nonzero null vector
$a\in \mathbb {L}^{n+2}$
. Suppose that
$0 < H \leq 1$
,
$H_{2}> 0$
, and
$$ \begin{align*} |a^\top|\leq C\inf_{\Sigma^n}\{na_0+2n(n-1)a_1H-\mathcal{F}_{0,1}\} \end{align*} $$
for some positive constant C. Then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tilde {\tau }}^n$
of
$\mathbb {H}^{n+1}$
for some
$\tilde {\tau }\leq \tau $
.
According to [Reference Aquino, Baltazar and de Lima7], we say that a two-sided hypersurface
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
is locally tangent from below to a horosphere
$L_{\tau }$
of
$\mathbb H^{n+1}$
when there are a point
$p\in \Sigma^n$
and a neighborhood
$\mathcal U\subset \Sigma^n$
of p such that
$\ell _a(p)=\tau $
and
$\ell _a(q)\leq \tau $
for all
$q\in \mathcal U$
. In our next result, we assume that
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
is a complete two-sided hypersuface locally tangent from below to a horosphere. We point out that this hypothesis is not necessary in Theorem 4.15. For this reason, Theorem 4.15 cannot be regarded as a special case of the following result.
Theorem 4.17. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete two-sided hypersurface contained in the closed horoball determined by a horosphere
$L_{\tau }$
orthogonal to a nonzero null vector
$a\in \mathbb {L}^{n+2}$
and locally tangent from below to a horosphere
$L_{\tilde {\tau }}$
, for some
$\tilde {\tau }\leq \tau $
, and let r and s be any integers satisfying
$0\leq r\le s\leq n-1$
. Suppose that the linearized curvature function
$\mathcal {F}_{r,s}=(r+1)a_rc_rH_{r+1}+\cdots +(s+1)a_sc_sH_{s+1}$
of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is such that
$$ \begin{align} 0 < \mathcal{F}_{r,s}\leq\sum_{j=r}^{s}(j+1)a_jc_jH_j \end{align} $$
on
$\Sigma^n$
, where
$a_r,\ldots ,a_s$
are nonnegative real numbers (at least one of them nonzero),
$H_j$
and
$H_{j+1}$
are two consecutive higher order mean curvatures of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
, and
$c_j=(j+1){n\choose\ j+1}$
,
$j\in \{r,\ldots ,s\}$
. If the mean curvature H of
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
is bounded,
$H_{s+1}>0$
on
$\Sigma^n$
, and
$$ \begin{align} |a^\top|\leq C\inf_{\Sigma^n}\bigg\{\sum_{j=r}^{s}(j+1)a_jc_jH_j-\mathcal{F}_{r,s}\bigg\} \end{align} $$
for some positive constant C, then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tilde {\tau }}^n$
for some
$\tilde {\tau }\leq \tau $
.
Proof. Taking the vector field V defined on
$\mathbb {H}^{n+1}$
by
$V(p) = -\langle p, a\rangle \xi (p)$
, from ([Reference Montiel15], Example 4.3), we obtain that V is a complete closed conformal vector field such that
$\mathrm {div}(V) = (n+1)\langle p,a\rangle $
and
$|V|_{\Sigma } = \ell _{a}$
.
Since
$\Sigma^n$
is locally tangent from below to a horosphere
$L_{\tilde {\tau }}$
, we get that
$|V|_{\Sigma }$
attains a local maximum on
$\Sigma^n$
. Applying [Reference Aquino, Baltazar and de Lima7, Lemma 3], we have that there exists an elliptic point in
$\Sigma^n$
. Since by hypothesis
$H_{s+1}> 0$
, it follows from [Reference Aquino, Baltazar and de Lima7, Lemma 2] that
$P_{j}$
is positive definite and
$H_{j}$
is positive for all
$1 \leq j \leq s$
. In this setting, we conclude that
$\mathcal {P}_{r,s}$
defined in (2-10) is positive definite.
In addition, from (4-15) and knowing that
$H_{2}> 0$
,
$$ \begin{align} \sum_{i} \lambda_{i}^{2} \leq n^2H^2, \end{align} $$
and therefore, the boundedness of H implies the boundedness of all principal curvatures of
$\Sigma^n$
. Thus, we verify that
$\mathrm {tr}(\mathcal {P}_{r,s})$
is bounded and from (4-16), we have that
$K_{\Sigma^n}$
is bounded from below.
Considering the operator
$\mathcal {L}_{\,r,s}$
acting on function
$\ell _{a}^2$
, we can use Lemma 4.14 to get a sequence
$(p_{k})_{k \geq 1}$
in
$\Sigma^n$
such that
$$ \begin{align*} \lim_k\ell_{a}^2(p_k)=\sup_{\Sigma^n}\ell_{a}^2, \quad \lim_k|\nabla \ell_{a}^2(p_k)|=0 \quad\mathrm{and}\quad \limsup_k\mathcal{L}_{\,r,s}(\ell_{a}^2)(p_k)\leq0. \end{align*} $$
However, since
$\mathcal {P}_{r,s}$
is positive definite and taking into account that
$\ell ^{2}_{a} \geq - f_{a} \ell _{a}$
, from (3-11) and from (4-21),
$$ \begin{align*} \mathcal{L}_{\,r,s}(\ell_{a}^2) &= 2 \ell_{a} \mathcal{L}_{\,r,s}(\ell_{a}) + 2 \langle \mathcal{P}_{r,s}(\nabla \ell_{a}), \nabla \ell_{a} \rangle \\ &\geq 2 \bigg\{\sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} - \mathcal{F}_{r,s}\bigg\}\ell_{a}^2 \geq 0. \end{align*} $$
Thus, we have that
$$ \begin{align} 0 \geq \limsup_k\mathcal{L}_{\,r,s}(\ell_{a}^2)(p_k) \geq 2 (\sup_{\Sigma^n}\ell_{a}^2)\limsup_k\bigg\{\sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} - \mathcal{F}_{r,s}\bigg\}(p_k)\geq 0. \end{align} $$
Hence, since
$\sup _{\Sigma^n}\ell _{a}^2> 0$
, from (4-24),
$$ \begin{align*} \limsup_k\bigg\{\sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} - \mathcal{F}_{r,s} \bigg\}(p_k)=0; \end{align*} $$
in particular,
$$ \begin{align*} \inf_{\Sigma^n}\bigg\{\sum_{j=r}^{s}(j+1)a_{j}c_{j}H_{j} - \mathcal{F}_{r,s}\bigg\} = 0. \end{align*} $$
Taking into account the hypothesis (4-22), we conclude that
$a^{\top }$
vanishes identically on
$\Sigma^n$
and, therefore,
$\psi (\Sigma^n)$
is a horosphere
$L^{n}_{\tilde {\tau }}$
.
Remark 4.18. It is not difficult to verify that for the particular case
$r=s$
, with
$1 \leq r \leq n-1$
, and
$a_{r} = ((r+1)^2(\begin {smallmatrix}n \\ r+1\end {smallmatrix}))^{-1}$
, we get [Reference Colares, de Lima and Velásquez9, Theorem
$3.6$
] as a corollary of Theorem 4.17.
We close our article returning to the configuration in which
$\psi : \Sigma^n \looparrowright \mathbb {H}^{n+1}$
is a complete two-sided hypersurface such that its support functions satisfy
$\ell _{a} = \lambda f_{a}$
for some a negative constant
$\lambda \in \mathbb {H}^{n+1}$
. In this setting, we present our last characterization result.
Theorem 4.19. Let
$\psi :\Sigma^n\looparrowright \mathbb {H}^{n+1}$
be a complete two-sided hypersurface contained in the closed horoball determined by a horosphere
$L_{\tau }$
orthogonal to a nonzero null vector
$a\in \mathbb {L}^{n+2}$
and locally tangent from below to a horosphere
$L_{\tilde {\tau }}$
for some
$\tilde {\tau }\leq \tau $
, and let r and s be any integers satisfying
$0\leq r\le s\leq n-1$
. Suppose that H is bounded,
${H_{s+1}> 0}$
and is bounded away from zero, and
$\ell _{a} = \lambda f_{a}$
for some negative constant
$\lambda \in \mathbb {R}$
. If the linearized curvature function
$\mathcal {F}_{r,s}$
of
$\psi :\Sigma^n\looparrowright \mathbb H^{n+1}$
is such that
$$ \begin{align} 0 < \mathcal{F}_{r,s}\leq\sum_{j=r}^{s}(j+1)a_jc_jH_j \end{align} $$
on
$\Sigma^n$
, where
$a_{r},\ldots ,a_{s}$
are nonnegative numbers (at least one of them nonzero), then
$\psi (\Sigma^n)$
is a horosphere
$L_{\tilde {\tau }}^n$
for some
$\tilde {\tau }\leq \tau $
.
Proof. Since
$\ell _{a} = \lambda f_{a}$
, from (3-6),
$$ \begin{align} |\nabla \ell_{a}|^2 = (1- \lambda^{-2})\ell_{a}^2 \geq 0. \end{align} $$
Taking into account the operator
$\mathcal {L}_{r,s}$
defined in (2-11) and the hypothesis (4-25), from (3-13) of Proposition 3.2 and from (4-26),
$$ \begin{align} \mathcal{L}_{r,s} (\ell_a^{\,2}) \geq (1-\lambda^{-2})\bigg(\sum_{j=r}^{s}(j+1)a_jc_jH_j \bigg)\ell_{a}^2 + 2\langle\mathcal{P}_{r,s}(\nabla \ell_a),\nabla \ell_a\rangle. \end{align} $$
In a similar way as the proof of Theorem 4.17, we conclude that
$H_{j}$
is positive for all
$1 \leq j \leq s+1$
,
$\mathcal {P}_{r,s}$
is positive definite and
$\mathrm {tr}(\mathcal {P}_{r,s})$
is bounded. Therefore, the inequality (4-27) becomes
$$ \begin{align} \mathcal{L}_{r,s} (\ell_a^{\,2}) \geq (1-\lambda^{-2})\bigg(\sum_{j=r}^{s}(j+1)a_jc_jH_j \bigg)\ell_{a}^2 \geq 0. \end{align} $$
In this setting, it is not difficult to infer that we can apply Lemma 4.14 to
$\mathcal {L}_{r,s}$
acting on
$\ell _{a}^2$
. Since
$H_{j}$
is positive for all
$1 \leq j \leq s+1$
, we can use the Gårding inequalities [Reference Gårding13] to obtain that
$$ \begin{align*} H \geq H_{2}^{\tfrac{1}{2}} \geq \cdot \cdot \cdot \geq H_{r}^{{1}/{r}} \geq \cdot \cdot \cdot \geq H_{s}^{{1}/{s}} \geq H_{s+1}^{{1}/({s+1})}. \end{align*} $$
Thus,
$H_{j}$
is bounded away from zero for all
$1 \leq j \leq s+1$
because
$H_{s+1}$
is bounded away from zero by hypothesis.
Then,
$\sum _{j=r}^{s}(j+1)a_jc_jH_j$
is bounded away from zero and from (4-28), we conclude that there exist a sequence
$(p_{k})_{k \geq 1}$
in
$\Sigma ^{n}$
and a positive constant
$\kappa \in \mathbb {R}$
such that
$$ \begin{align} 0 \geq \limsup_{k}{\mathcal{L}_{r,s}(\ell_{a}^2)(p_{k})} \geq (1 - \lambda^{-2})\kappa\sup_{\Sigma^n}{\ell_{a}^2} \geq 0. \end{align} $$
Hence, since
$\kappa> 0$
and
$\sup _{\Sigma^n}{\ell _{a}^2}> 0$
, returning to (4-29), we conclude that
${\lambda = -1}$
. Therefore, from (4-26), we conclude that
$\ell _{a}$
is constant on
$\Sigma^n$
and, consequently,
$\psi (\Sigma^n)$
must be the horosphere
$L_{\tilde \tau }^n$
.
Acknowledgement
The authors would like to thank the anonymous referee for their valuable suggestions and useful comments which improved the presentation of the manuscript.
