Hypersurfaces of constant higher-order mean curvature in $M\times {\mathbb{R}}$

Abstract

We consider hypersurfaces of products $M\times {\mathbb{R}}$ with constant rth mean curvature ${H_{r}} \ge 0$ (to be called ${{H}_{r}}$-hypersurfaces), where M is an arbitrary Riemannian n-manifold. We develop a general method for constructing them and employ it to produce many examples for a variety of manifolds M,  including all simply connected space forms and the hyperbolic spaces ${\mathbb{H}}_{{\mathbb{F}}}^{m}$ (rank one symmetric spaces of noncompact type). We construct and classify complete rotational ${{H}_{r}}(\ge 0)$-hypersurfaces in ${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$ and in ${\mathbb{S}}^{n}\times {\mathbb{R}}$ as well. They include spheres, Delaunay-type annuli and, in the case of ${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}},$ entire graphs. We also construct and classify complete ${{H}_{r}}(\ge 0)$-hypersurfaces of ${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and strictly convex ${{H}_{r}}$-hypersurface of ${\mathbb{H}}^{n}\times {\mathbb{R}}$ or ${\mathbb{S}}^{n}\times {\mathbb{R}}$ $(n\ge 3)$ is a rotational embedded sphere. Other uniqueness results for complete ${{H}_{r}}$-hypersurfaces of these ambient spaces are obtained.

1 Introduction

In his pioneering work [29], H. Rosenberg initiated the study of minimal and constant mean curvature hypersurfaces of product spaces $M\times {\mathbb{R}} ,$ where M is an arbitrary Riemannian n-manifold. Since then, many results on this subject have been obtained by many authors, mostly in the particular case M is a simply connected space form.

Following this path, we approach here hypersurfaces of $M\times {\mathbb{R}}$ with constant rth mean curvature ${{H}_{r}}\ge 0$ (for some $r\in \{1,\dots , n\}$), which we call ${{H}_{r}}$-hypersurfaces. Let us recall that the (nonnormalized) rth mean curvature ${{H}_{r}}$ of a hypersurface is the rth elementary symmetric polynomial of its principal curvatures, so that it constitutes a natural extension of the mean curvature ($r=1$) and the Gauss–Kronecker curvature ($r=n$).

We focus on constructing and classifying ${{H}_{r}}$-hypersurfaces of products $M\times {\mathbb{R}} .$ With this purpose, we use a special type of graph built on families of parallel hypersurfaces of M. In fact, for any given constant ${{H}_{r}}\ge 0,$ we obtain ${{H}_{r}}$-graphs in $M\times {\mathbb{R}}$ for those Riemannian manifolds M which admit a local family of parallel hypersurfaces, each of them having constant principal curvatures. Following [4], such hypersurfaces are called isoparametric. We point out that many Riemannian manifolds M admit isoparametric hypersurfaces, such as space forms, hyperbolic spaces, warped products and ${\mathbb{E}}(\kappa ,\uptau )$ spaces.

By suitably “gluing” pieces of ${{H}_{r}}$-graphs, we construct properly embedded ${{H}_{r}}$-hypersurfaces in $M\times {\mathbb{R}}$ when M is either the standard n-sphere ${\mathbb{S}}^{n}$ or one of the hyperbolic spaces ${\mathbb{H}}_{{\mathbb{F}}}^{m}$ (rank one symmetric spaces of noncompact type). In this setting, we show that there exists a rotational ${{H}_{r}}$-sphere in ${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$ if and only if ${{H}_{r}}>C_{{\mathbb{F}}}(r),$ where the constant $C_{{\mathbb{F}}}(r)$ is defined as the limit of the rth mean curvature of a geodesic sphere of ${\mathbb{H}}_{{\mathbb{F}}}^{m}$ as its radius goes to infinity. (In particular, $C_{{\mathbb{F}}}(r)$ is positive for $1\le r<n=\dim {\mathbb{H}}_{{\mathbb{F}}}^{m}$ and vanishes for $r=n$.) On the other hand, as we also show, for any $r\in \{1,\dots ,n\}$ and any constant ${{H}_{r}}>0,$ there exists a rotational ${{H}_{r}}$-sphere in ${\mathbb{S}}^{n}\times {\mathbb{R}} .$

We remark that rotational hypersurfaces of a general product $M\times {\mathbb{R}}$ are defined here as those which are foliated by horizontal geodesic spheres centered at an axis $\{o\}\times {\mathbb{R}} ,$ $o\in M.$

We provide other examples of properly embedded rotational ${H_{r}} (>0)$-hypersurfaces in ${\mathbb{H}}_{{\mathbb{F}}}^{m} \times {\mathbb{R}}$ and in $\mathbb{S}^{n} \times {\mathbb{R}}$ as well, including Delaunay-type annuli and, in the case of ${\mathbb{H}}_{{\mathbb{F}}}^{m} \times {\mathbb{R}}$, entire graphs over $\mathbb{H}_{{\mathbb{F}}}^{m}.$ Then, we classify those complete connected rotational ${{H}_{r}}(>0)$-hypersurfaces of these product spaces whose height functions are Morse type (i.e., have isolated critical points), which include all the properly embedded rotational ${{H}_{r}}(>0)$-hypersurfaces we obtain here.

We also construct and classify complete connected ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ with no horizontal points (critical points of the height function) which are invariant by either parabolic or hyperbolic isometries of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$

(in the latter case, only the real hyperbolic space $\mathbb{H}_{\mathbb{R}}^{m}:=\mathbb{H}^{n}$ is considered).

Our methods work equally well for ${{H}_{r}}$-hypersurfaces with ${{H}_{r}}=0,$ the so called r-minimal hypersurfaces. By applying them, we obtain a one-parameter family of rotational, properly embedded catenoid-type r-minimal n-annuli in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$ Similarly, we obtain a one-parameter family of rotational, properly embedded Delaunay-type r-minimal n-annuli in $\mathbb{S}^{n} \times {\mathbb{R}} .$ Then, we show that these annuli are the only complete connected r-minimal rotational hypersurfaces of these product spaces (besides horizontal hyperplanes and, in the case $r=n,$ cylinders over geodesic spheres).

Analogously to the case of $H_{r} (>0)$-hypersurfaces, we construct and classify the complete connected r-minimal hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which are invariant by either parabolic or hyperbolic isometries of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$

The study of ${{H}_{r}}$-hypersurfaces of a Riemannian manifold leads naturally to considerations on their uniqueness properties. On this matter, Montiel and Ros [23] (see also [21]) showed the following Alexandrov-type theorem:

The only compact, connected and embedded ${{H}_{r}}$-hypersurfaces in $\mathbb{R} ^{n},$ $\mathbb{H}^{n},$ or an open hemisphere of  $\mathbb{S}^{n}$ are geodesic spheres.

In [15], this result was extended to the context of ${{H}_{r}}$-hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R} ,$ where the geodesic spheres in the statement are replaced by rotational spheres.

Here, we establish uniqueness results for rotational ${{H}_{r}}$-spheres of $\mathbb{H}^{n}\times \mathbb{R}$ and $\mathbb{S}^{n}\times \mathbb{R}$, $n\ge 3.$ The case $n=2$ was settled in [1] (for $r=1$) and in [17] (for $r=2$). More precisely, we show that, for $n\ge 3,$ any compact connected strictly convex ${{H}_{r}}$-hypersurface $\varSigma$ of either $\mathbb{H}^{n}\times \mathbb{R}$ or $\mathbb{S}^{n}\times \mathbb{R}$ is necessarily an embedded rotational sphere. Assuming $\varSigma$ complete, instead of compact, the same conclusion holds if, in addition, the height function of $\varSigma$ has a critical point and, in the case $\varSigma \subset \mathbb{H}^{n}\times \mathbb{R}$, the least principal curvature of $\varSigma$ is bounded away from zero. Finally, we show that, for $n\ge 3,$ any connected, properly immersed and strictly convex ${{H}_{r}}(>0)$-hypersurface of  $\mathbb{S}^{n}\times \mathbb{R}$ is necessarily an embedded rotational ${{H}_{r}}$-sphere.

It is worth mentioning that these uniqueness results constitute applications of the main theorems in [8], which concern convexity properties of hypersurfaces in $M\times \mathbb{R} ,$ M being either a Hadamard manifold or the sphere $\mathbb{S}^{n}.$ Besides, the noncompact cases are based on height estimates we establish here for strictly convex vertical graphs in arbitrary products $M\times \mathbb{R} .$

The paper is organized as follows. In Sect. 2, we set notation and some basic concepts. In Sect. 3, we introduce graphs on parallel hypersurfaces and establish two key lemmas. In Sect. 4 (resp. Sect. 5), we construct and classify complete rotational ${{H}_{r}}(>0)$-hypersurfaces (resp. r-minimal hypersurfaces) in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ and $\mathbb{S}^{n}\times \mathbb{R} ,$ whereas in Sect. 6 (resp. Sect. 7) we do the same for complete translational ones, i.e, invariant by either parabolic or hyperbolic isometries. In Sect. 8, we prove the aforementioned uniqueness results for rotational ${{H}_{r}}$-spheres of $\mathbb{Q} _{\epsilon}^{n} \times \mathbb{R}$ .

2 Preliminaries

Let $\varSigma$ be an oriented hypersurface of a Riemannian manifold $\overline{M}^{n+1},$ $n\ge 2.$ Set $\overline{\nabla }$ for the Levi-Civita connection of $\overline{M},$ N for the unit normal field of $\varSigma$ and A for its shape operator with respect to N,  so that

$$\begin{aligned} AX=-\overline{\nabla }_XN, \,\, X\in T\varSigma , \end{aligned}$$

where $T\varSigma$ stand for the tangent bundle of $\varSigma$. The principal curvatures of $\varSigma ,$ that is, the eigenvalues of the shape operator A,  will be denoted by $k_1\,, \dots ,k_n$.

Given an integer $r\ge 0,$ we define the (nonnormalized) rth mean curvature ${{H}_{r}}$ of the hypersurface $\varSigma \subset \overline{ M}$ as:

$$\begin{aligned} {{H}_{r}}:=\left\{ \begin{array}{cl} 1 &{} \mathrm{if} \,\, r=0. \\ \displaystyle \sum _{i_1<\cdots <i_r}k_{i_1}\dots k_{i_r} &{} \mathrm{if} \,\, 1\le r\le n. \\ 0 &{} \mathrm{if} \,\, r>n. \end{array} \right. \end{aligned}$$

(1)

Notice that $H_1$ and $H_n$ are the nonnormalized mean curvature and Gauss–Kronecker curvature functions of $\varSigma ,$ respectively, i.e.,

$$\begin{aligned} H_1=\mathrm{trace}\,A\quad \text {and}\quad H_n=\det A. \end{aligned}$$

Definition 1

With the above notation, given a constant ${{H}_{r}}\in \mathbb{R} ,$ we say that $\varSigma \subset \overline{ M}$ is an ${{H}_{r}}$-hypersurface of $\overline{ M}$ if its rth mean curvature is constant and equal to ${{H}_{r}}.$ In the case ${{H}_{r}}=0$, we say that $\varSigma$ is an r-minimal hypersurface of $\overline{ M}.$

Definition 2

A hypersurface $\varSigma \subset \overline{ M}$ is said to be convex at $x\in \varSigma$ if, at this point, all the nonzero principal curvatures have the same sign. If, in addition, none of these principal curvatures is zero, then $\varSigma$ is said to be strictly convex at x. We call $\varSigma$ convex (resp. strictly convex) if it is convex (resp. strictly convex) at all of its points.

In some of our proofs, we shall consider the tangency principle obtained in [18]. Roughly speaking, this principle asserts that two ${{H}_{r}}$-hypersurfaces $\varSigma _1$ and $\varSigma _2$ which are tangent at a point—with one “above” the other in a neighborhood V of this point—must coincide on V,  provided that one of them is strictly convex at one of its points (see [15, 18] for details). If, in addition, $\varSigma _1$ and $\varSigma _2$ are complete, then the continuation principle applies and gives that $\varSigma _1=\varSigma _2.$ Essentially, these tangency and continuation principles are due to the fact that the equation for prescribed rth mean curvature is elliptic (cf. [7, 27]).

2.1 Hypersurfaces of $M\times {\mathbb{R}}$

The ambient spaces we shall consider are the products $\overline{ M}^{n+1}=M^{n}\times \mathbb{R}$—where $M^{n}$ is some Riemannian manifold—endowed with the standard product metric:

$$\begin{aligned}\langle \,,\,\rangle =\langle \,,\,\rangle _{M}+dt^2.\end{aligned}$$

In this setting, we denote by $\partial _t$ the gradient of the projection $\pi _{\mathbb{R} }$ of $M\times \mathbb{R}$ on its second factor. Clearly, $\partial _t$ is parallel on $M\times \mathbb{R} ,$ that is,

$$\begin{aligned} \overline{\nabla }_X\partial _t=0 \,\,\, \forall X\in T(M\times \mathbb{R} ). \end{aligned}$$

(2)

Let $\varSigma$ be a hypersurface of $M\times \mathbb{R}$. Its height function $\xi$ and its angle function $\varTheta$ are defined by the following identities:

$$\begin{aligned} \xi :=\pi _{\scriptscriptstyle \mathbb{R} }|_\varSigma \quad \text {and}\quad \varTheta (x):=\langle N(x),\partial _t\rangle , \,\, x\in \varSigma . \end{aligned}$$

A critical point of $\xi$ is called horizontal, whereas a point on which $\varTheta$ vanishes is called vertical. Notice that $x\in \varSigma$ is horizontal if and only if $\varTheta (x)=\pm 1.$

We shall denote the gradient field and the Hessian of a function $\zeta$ on $\varSigma$ by $\nabla \zeta$ and $\mathrm{Hess}\,\zeta ,$ respectively, that is,

$$\begin{aligned} \mathrm{Hess}\,\zeta (X,Y):=\langle \overline{\nabla }_X\nabla \zeta ,Y\rangle \,\,\, \forall X, Y\in T(\varSigma ). \end{aligned}$$

It is easily checked that

$$\begin{aligned} \nabla \xi =\partial _t-\varTheta N \quad \text {and}\quad \nabla \varTheta =-A\nabla \xi . \end{aligned}$$

(3)

Also, from (2) and (3), one has $\overline{\nabla }_{X}\,\nabla \xi =-(\varTheta \overline{\nabla }_XN+X(\varTheta )N).$ Hence,

$$\begin{aligned} \mathrm{Hess}\,\xi (X,Y)=\varTheta \langle AX,Y\rangle \,\,\, \forall X, Y\in T\varSigma . \end{aligned}$$

(4)

Given $t\in \mathbb{R} ,$ the set $P_t:=M\times \{t\}$ is called a horizontal hyperplane of $M\times \mathbb{R} .$ Horizontal hyperplanes are all isometric to M and totally geodesic in $M\times \mathbb{R} .$ In this context, we call a transversal intersection a horizontal section of $\varSigma .$ Any horizontal section $\varSigma _t$ is a hypersurface of $P_t$ . So, at any point $x\in \varSigma _t\subset \varSigma ,$ the tangent space $T_x\varSigma$ of $\varSigma$ at x splits as the orthogonal sum

$$\begin{aligned} T_x\varSigma =T_x\varSigma _t\oplus \mathrm{Span}\{\nabla \xi \}. \end{aligned}$$

(5)

We will denote by $\mathbb{Q} _{\epsilon}^{n}$ the simply connected space form of constant sectional curvature $\epsilon \in \{0,1,-1\}:$ the Euclidean space $\mathbb{R} ^{n}$ ($\epsilon =0$), the unit sphere $\mathbb{S}^{n}$ ($\epsilon =1$), and the hyperbolic space $\mathbb{H}^{n}$ ($\epsilon =-1$).

2.2 The hyperbolic spaces $\mathbb{H}_{{\mathbb{F}}}^{m}$

Mostly, the first factor manifold M of the Riemannian products $M\times \mathbb{R}$ considered here is either the unit n-sphere $\mathbb{S}^{n}$ or one of the Riemannian manifolds known as hyperbolic spaces (which include the canonical n-dimensional hyperbolic space $\mathbb{H}^{n}$ of constant sectional curvature $-1$) .

The hyperbolic spaces are precisely the rank one symmetric spaces of noncompact type, which can be described through the four normed division algebras: $\mathbb{R}$ (real numbers), ${\mathbb{C}}$ (complex numbers), ${\mathbb{K}}$ (quaternions) and ${\mathbb{O}}$ (octonions). They are denoted by $\mathbb{H}_\mathbb{R} ^{m}, \,\,\mathbb{H}_{\mathbb{C}}^{m}, \,\, \mathbb{H}_{{\mathbb{K}}}^{m}$ and $\mathbb{H}_{{\mathbb{O}}}^2$ and called real hyperbolic space, complex hyperbolic space, quaternionic hyperbolic space and Cayley hyperbolic plane, respectively.

We will adopt the unified notation $\mathbb{H}_{{\mathbb{F}}}^{m}$ for the hyperbolic spaces, where $m=2$ for ${\mathbb{F}}={\mathbb{O}}.$ The real dimension of $\mathbb{H}_{{\mathbb{F}}}^{m}$ is $n=m\dim {\mathbb{F}}.$ In particular, $\mathbb{H}_{{\mathbb{O}}}^2$ has dimension $n=16.$ We will keep the standard notation $\mathbb{H}^{n}$ for the real hyperbolic space $\mathbb{H}_\mathbb{R} ^{n}.$

We remark that, being symmetric, the hyperbolic spaces $\mathbb{H}_{{\mathbb{F}}}^{m}$ are homogeneous. In addition, they are included in a distinguished class of Lie groups known as Damek–Ricci spaces (see Example 3 in the next section). In this context, it can be shown that any hyperbolic space $\mathbb{H}_{{\mathbb{F}}}^{m}$ is a Hadamard–Einstein manifold with nonconstant (except for $\mathbb{H}^{n}$) sectional curvatures pinched between $-1$ and $-1/4$ (cf. [4, Sects. 4.1.9 and 4.2]).

3 ${{H}_{r}}$-graphs on parallel hypersurfaces

In this section, we give a detailed description of graphs in $M\times \mathbb{R}$ which are built on families of parallel hypersurfaces of M. As we mentioned before, they will constitute our main tool for constructing ${{H}_{r}}$-hypersurfaces in product spaces $M\times \mathbb{R} .$

With this purpose, consider an isometric immersion

$$\begin{aligned} f:M_0^{n-1}\rightarrow M^{n} \end{aligned}$$

between two Riemannian manifolds $M_0^{n-1}$ and $M^{n},$ and suppose that there exists a neighborhood $\mathscr {U}$ of $M_0$ in $TM_0^\perp$ without focal points of f,  that is, the restriction of the normal exponential map $\exp ^\perp _{M_0}:TM_0^\perp \rightarrow M$ to $\mathscr {U}$ is a diffeomorphism onto its image. In this case, denoting by $\eta$ the unit normal field of f,  there is an open interval $I\ni 0$ such that, for all $p\in M_0,$ the curve

$$\begin{aligned} \gamma _p(s)=\exp _{\scriptscriptstyle M}(f(p),s\eta (p)), \, s\in I, \end{aligned}$$

(6)

is a well-defined geodesic of M without conjugate points. Thus, for all $s\in I,$

$$\begin{aligned} \begin{array}{cccc} f_s: &{} M_0 &{} \rightarrow &{} M\\ &{} p &{} \mapsto &{} \gamma _p(s) \end{array} \end{aligned}$$

is an immersion of $M_0$ into M,  which is said to be parallel to f. Observe that, given $p\in M_0$, the tangent space $f_{s_*}(T_p M_0)$ of $f_s$ at p is the parallel transport of $f_{*}(T_p M_0)$ along $\gamma _p$ from 0 to s. We also remark that, with the induced metric, the unit normal $\eta _s$ of $f_s$ at p is given by

$$\begin{aligned}\eta _s(p)=\gamma _p'(s).\end{aligned}$$

Definition 3

Let $\phi :I\rightarrow \phi (I)\subset \mathbb{R}$ be an increasing diffeomorphism, i.e., $\phi '>0.$ With the above notation, we call the set

$$\begin{aligned} \varSigma :=\{(f_s(p),\phi (s))\in M\times \mathbb{R} \,;\, p\in M_0, \, s\in I\}, \end{aligned}$$

(7)

the graph determined by $\{f_s\,;\, s\in I\}$ and $\phi ,$ or $(f_s,\phi )$-graph, for short.

We remark that, for a given $(f_s,\phi )$-graph $\varSigma$, and for any $s\in I$ , $f_s(M_0)$ is the projection on M of the horizontal section $\varSigma _{\phi (s)}\subset \varSigma ,$ that is, these sets are the level hypersurfaces of $\varSigma .$

For an arbitrary point $x=(f_s(p),\phi (s))$ of such a graph $\varSigma ,$ one has

$$\begin{aligned}T_x\varSigma =f_{s_*}(T_p M_0)\oplus \mathrm{Span}\,\{\partial _s\}, \,\,\, \partial _s=\eta _s+\phi '(s)\partial _t.\end{aligned}$$

So, a unit normal to $\varSigma$ is

$$\begin{aligned} N=\frac{-\phi '}{\sqrt{1+(\phi ')^2}}\eta _s+\frac{1}{\sqrt{1+(\phi ')^2}}\partial _t\,. \end{aligned}$$

(8)

In particular, its angle function is

$$\begin{aligned} \varTheta =\frac{1}{\sqrt{1+(\phi ')^2}}\,\cdot \end{aligned}$$

(9)

A key property of $(f_s,\phi )$-graphs is that the trajectories of $\nabla \xi$ on them are lines of curvature, that is, $\nabla \xi$ is one of its principal directions. (Notice that, by (9), $0<\varTheta <1,$ so $\nabla \xi$ never vanishes on an $(f_s,\phi )$-graph.) More precisely (cf. [9, 31]),

$$\begin{aligned} A\nabla \xi =\frac{\phi ''}{(\sqrt{1+(\phi ')^2})^3}\nabla \xi . \end{aligned}$$

(10)

We point out that, besides being lines of curvature, the trajectories of $\nabla \xi$ on an $(f_s,\phi )$-graph $\varSigma$, when properly reparametrized, are also geodesics. This follows from the fact that $\varTheta$, and consequently $\Vert \nabla \xi \Vert ,$ is constant along the horizontal sections of $\varSigma$ (see [31, Lemma 5]). It should also be noticed that these trajectories project on the geodesics $\gamma _p=\gamma _p(s)$ given by (6) (Fig. 1).

Fig. 1

Trajectory of $\nabla \xi$ on an $(f_s,\phi )$-graph

Let us compute now the principal curvatures of an $(f_s,\phi )$-graph $\varSigma .$ For that, let $\{X_1\,,\dots ,X_n\}$ be the orthonormal frame of principal directions of $\varSigma$ in which $X_n=\nabla \xi /\Vert \nabla \xi \Vert .$ In this case, for $1\le i\le n-1,$ the fields $X_i$ are all horizontal, that is, tangent to M (cf. (5)). Therefore, setting

$$\begin{aligned} \varrho :=\frac{\phi '}{\sqrt{1+(\phi ')^2}} \end{aligned}$$

(11)

and considering (8), we have, for all $i=1,\dots ,n-1,$ that

$$\begin{aligned} k_i=\langle AX_i,X_i\rangle =-\langle \overline{\nabla }_{X_i}N,X_i\rangle =\varrho \langle \overline{\nabla }_{X_i}\eta _s,X_i\rangle =-\varrho k_i^s, \end{aligned}$$

where $k_i^s$ is the ith principal curvature of $f_s\,.$ Also, it follows from (10) that $k_n=\varrho '.$ Thus, the array of principal curvatures of the $(f_s,\phi )$-graph $\varSigma$ is

$$\begin{aligned} k_i=-\varrho k_i^s \,\, (1\le i\le n-1) \quad \text {and}\quad k_n=\varrho '. \end{aligned}$$

(12)

Now, considering the above identities and writing, for $1\le r\le n,$

$$\begin{aligned} {H_{r}} = \sum _{i_1<\cdots<i_r\ne n}k_{i_1}\dots k_{i_r}+\sum _{i_1<\cdots <i_{r-1}}k_{i_1}\dots k_{i_{r-1}}k_{n}\,, \end{aligned}$$

we have that the rth mean curvature of the $(f_s,\phi )$-graph $\varSigma$ is

$$\begin{aligned} {{H}_{r}}= (-1)^r{{H}_{r}}^s\varrho ^r+(-1)^{r-1}H_{r-1}^s\varrho ^{r-1}\varrho ', \end{aligned}$$

(13)

where ${{H}_{r}}^s$ denotes the rth mean curvature of $f_s.$

Due to equality (13), the function defined in (11)—to be called the $\varrho$-function of the $(f_s,\phi )$-graph $\varSigma$—will play a major role in the sequel. We remark that, up to a constant, the $\varrho$-function of $\varSigma$ determines its $\phi$-function. Indeed, it follows from equality (11) that

$$\begin{aligned} \phi (s)=\int _{s_0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du+\phi (s_0), \,\,\, s_0\in I. \end{aligned}$$

(14)

We introduce now a special type of family of parallel hypersurfaces which will be used for constructing ${{H}_{r}}$-hypersurfaces in $M\times \mathbb{R} .$

Definition 4

Following [4], we call a family of parallel hypersurfaces

$$\begin{aligned}{\mathscr {F}}:=\{f_s:M_0\rightarrow M\,;\, s\in I\}\end{aligned}$$

isoparametric if, for each $s\in I,$ any principal curvature $k_i$ of $f_s$ is constant (possibly depending on i and s). If so, each hypersurface $f_s$ is also called isoparametric.

We should mention that there is some mismatch regarding the nomenclature for isoparametric hypersurfaces. In some contexts, isoparametric hypersurfaces are defined as those which, together with its parallel nearby hypersurfaces, have constant mean curvature. It is shown that some manifolds M admit hypersurfaces which are isoparametric in this sense and nonisoparametric as we defined.

Let $\varSigma$ be an $(f_s,\phi )$-graph such that the family ${\mathscr {F}}=\{f_s\,;\, s\in I\}$ is isoparametric. Then, for any $r=1,\dots ,n-1,$ the rth mean curvature of $f_s$ is a function of s alone, which we assume to be no vanishing. In this setting, writing $\uptau :=\varrho ^r$, and considering (13) with ${{H}_{r}}$ constant, we obtain the following result, which turns out to be our main lemma.

Lemma 1

Let ${\mathscr {F}}=\{f_s:M_0\rightarrow M\,;\, s\in I\}$ be an isoparametric family of hypersurfaces whose $r(<n)$-mean curvatures ${{H}_{r}}^s$ never vanish. Given $r\in \{1,\dots ,n\}$ and ${{H}_{r}}\in \mathbb{R} ,$ let $\uptau$ be a solution of the first-order differential equation

$$\begin{aligned} y'=a(s)y+b(s), \,\,\, s\in I, \end{aligned}$$

(15)

where the coefficients a and b are the functions

$$\begin{aligned} a(s):=\frac{r{{H}_{r}}^s}{H_{r-1}^s}\quad \mathrm{and}\quad b(s):=\frac{(-1)^{r-1}r{{H}_{r}}}{H_{r-1}^s}\,\cdot \end{aligned}$$

(16)

Then, if  $0<\uptau <1,$ the $(f_s,\phi )$-graph $\varSigma$ with $\varrho$-function $\uptau ^{1/r}$ is an ${{H}_{r}}$-hypersurface of the product $M\times \mathbb{R} .$ Conversely, if an $(f_s,\phi )$-graph $\varSigma$ has constant rth mean curvature ${{H}_{r}}$ , then $\uptau :=\varrho ^r$ is a solution of (15).

Regarding Eq. (15), recall that its general solution is

$$\begin{aligned} \uptau (s)=\frac{1}{\mu (s)}\left( \uptau _0+\int _{s_0}^{s}{b(u)}{\mu (u)}du\right) , \,\, s_0, s\in I, \, \uptau _0\in \mathbb{R} , \end{aligned}$$

(17)

where $\mu$ is the exponential function

$$\begin{aligned} \mu (s)=\exp \left( -\int _{s_0}^sa(u)du\right) , \,\,\, s\in I. \end{aligned}$$

It follows from Lemma 1 that, as long as M admits isoparametric hypersurfaces with nonvanishing rth mean curvature, for any ${{H}_{r}}\in \mathbb{R} ,$ there exist ${{H}_{r}}$-graphs in $M\times \mathbb{R}$. (Notice that the interval I and the constant $\uptau _0$ in (17) can be chosen in such a way that the corresponding solution $\uptau$ of (15) satisfies $0<\uptau <1$.) This includes, as a trivial case, the Euclidean space $\mathbb{R} ^{n}.$ In the next examples, we shall consider other manifolds M to which Lemma 1 applies.

Example 1

(sphere $\mathbb{S}^{n}$) It is a well-known fact that isoparametric hypersurfaces in $\mathbb{S}^{n}$ are abundant and include all its geodesic spheres (see, e.g., [12]).

Example 2

(warped products) Let $M=I\times _\omega F^{n-1}$ be a warped product, where the basis $I\subset \mathbb{R}$ is an open interval and the fiber F is an arbitrary $(n-1)$-dimensional manifold. For each $s\in I,$ define $f_s$ as the standard immersion $F\hookrightarrow \{s\}\times _\omega F\subset M.$ It is well known that, with the induced metric, ${\mathscr {F}}=\{f_s\,;\, s\in I\}$ is a family of parallel totally umbilical hypersurfaces of M with constant principal curvatures $\omega '/\omega$ (see, e.g., [5]). In particular, ${\mathscr {F}}$ is isoparametric. Hence, if $\omega '$ never vanishes, Lemma 1 applies to M.

Example 3

(Damek–Ricci spaces) Let us consider the Riemannian manifolds known as Damek–Ricci spaces. These are Lie groups endowed with a left invariant metric with especial properties (see [4, 12]). For instance, all Damek–Ricci spaces are both Hadamard and Einstein manifolds. As we have mentioned, the hyperbolic spaces $\mathbb{H}_{{\mathbb{F}}}^{m}$ are Damek–Ricci spaces. In fact, they are the only ones which are symmetric. Their isoparametric hypersurfaces include their geodesic spheres, as well as their horospheres. Also, as shown in [10], there exist families of isoparametric hypersurfaces with nonvanishing rth curvatures in Damek–Ricci harmonic spaces.

Example 4

($E(\kappa ,\uptau )$-spaces) In [13], it was proved that there exist isoparametric families of parallel surfaces with nonzero constant principal curvatures in ${\mathbb{E}}(k,\uptau )$ spaces satisfying $k-4\uptau ^2\ne 0.$ (Those include the products $\mathbb{H}^2\times \mathbb{R}$ and $\mathbb{S}^2\times \mathbb{R}$, the Heisenberg space $\mathrm{Nil}_3$, the Berger spheres and the universal cover of the special linear group $\mathrm{SL}_2(\mathbb{R} )$.)

In the next two sections, we construct properly embedded ${{H}_{r}}$-hypersurfaces in products $M\times \mathbb{R}$ by suitably “gluing” ${{H}_{r}}$-graphs. To this task, the following result will be considerably helpful.

Lemma 2

Let $\varrho :[a_1,a_2]\rightarrow \mathbb{R}$ be a differentiable function which satisfies:

$$\begin{aligned} 0<\varrho {|_{(a_1,a_2)}}<1, \end{aligned}$$

and consider the following conditions:

• (i) $\varrho (a_2)=1$ and $\varrho '(a_2)>0.$

• (ii) $\varrho (a_1)=1$ and $\varrho '(a_1)<0.$

Then, if  (i) occurs, there exists $\delta >0$ such that the improper integral

$$\begin{aligned} \int _{a_2-\delta }^{a_2}\frac{\varrho (s)ds}{\sqrt{1-\varrho ^2(s)}} \end{aligned}$$

(18)

is convergent. Analogously, the improper integral

$$\begin{aligned} \int _{a_1}^{a_1+\delta }\frac{\varrho (s)ds}{\sqrt{1-\varrho ^2(s)}} \end{aligned}$$

is convergent if (ii) occurs.

Proof

Assume that (i) occurs. In this case, there exist positive constants, $\delta$ and C,  such that $\varrho '(s)\ge C>0\,\forall s\in (a_2-\delta ,a_2).$ Therefore, since $0<\varrho {|_{(a_1,a_2)}}<1,$

$$\begin{aligned} \int _{a_2-\delta }^{a_2}\frac{\varrho (s)ds}{\sqrt{1-\varrho ^2(s)}}\le & {} \int _{a_2-\delta }^{a_2}\frac{\varrho '(s)ds}{\varrho '(s)\sqrt{1-\varrho ^2(s)}} \le \frac{1}{C} \int _{\varrho (a_2-\delta )}^{1}\frac{d\varrho }{\sqrt{1-\varrho ^2}} \\= & {} \frac{1}{C}\left( \frac{\pi }{2}-\arcsin (\varrho (a_2-\delta ))\right) \le \frac{\pi }{2C}\,, \end{aligned}$$

which proves the convergence of the integral (18). The case (ii) is analogous. $\square$

4 Rotational ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ and $\mathbb{S}^{n}\times \mathbb{R} .$

Rotational hypersurfaces in simply connected space forms $\mathbb{Q} ^{n}_\epsilon$ or products $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ are among the most classical hypersurfaces of these spaces. In the case of $\mathbb{Q} ^{n}_\epsilon \times \mathbb{R} ,$ they are obtained by rotating (with the aid of the group of isometries of $\mathbb{Q} _{\epsilon}^{n}$) a plane curve about an axis $\{o\}\times \mathbb{R} ,$ $o\in \mathbb{Q} _{\epsilon}^{n}$. Consequently, any connected component of any horizontal section $\varSigma _t$ of a rotational hypersurface $\varSigma$ in $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ lies in a geodesic sphere of $\mathbb{Q} _{\epsilon}^{n}\times \{t\}$ with center at the axis. This fact suggests the following definition.

Definition 5

A hypersurface $\varSigma \subset M\times \mathbb{R}$ is called rotational, if there exists a fixed point $o\in M$ such that any connected component of any horizontal section $\varSigma _t$ of $\varSigma$ is contained in a geodesic sphere of $M\times \{t\}$ with center at $o\times \{t\}.$ If so, the set $\{o\}\times \mathbb{R}$ is called the axis of $\varSigma .$ In particular, any horizontal hyperplane $P_t:=M\times \{t\}$ is a rotational hypersurface of $M\times \mathbb{R}$ with axis at any point $o\in M.$

In what follows, we construct and classify complete rotational ${{H}_{r}}(>0)$-hypersurfaces in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ and $\mathbb{S}^{n}\times \mathbb{R} .$

4.1 Rotational ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$

Let us consider a family

$$\begin{aligned} {\mathscr {F}}:=\{f_s:\mathbb{S}^{n-1}\rightarrow \mathbb{H}_{{\mathbb{F}}}^{m}\,; \,\,\, s\in (0,+\infty )\} \end{aligned}$$

(19)

of isoparametric concentric geodesic spheres of $\mathbb{H}_{{\mathbb{F}}}^{m}$ indexed by their radiuses, that is, for a fixed point $o\in \mathbb{H}_{{\mathbb{F}}}^{m}$, and for each $s\in (0,+\infty ),$ $f_s(\mathbb{S}^{n-1})$ is the geodesic sphere $S_s(o)$ of $\mathbb{H}_{{\mathbb{F}}}^{m}$ with center at o and radius s.

We remark that any sphere $f_s\in {\mathscr {F}}$ is strictly convex. Also, in accordance to the notation of Sect. 3, for each $s\in (0,+\infty ),$ we choose the outward orientation of $f_s$ , so that any principal curvature $k_i^s$ of  $f_s$ is negative. In this setting, the function $s\in (0,+\infty ) \mapsto {{H}_{r}}^s$ is positive for r even and negative for r odd. Hence, for any constant ${{H}_{r}}>0,$ the coefficients a and b in (16) are given by

$$\begin{aligned} a(s)=-\frac{r|{{H}_{r}}^s|}{|H_{r-1}^s|} \quad \text {and}\quad b(s)=\frac{r {{H}_{r}}}{|H_{r-1}^s|} \,\cdot \end{aligned}$$

(20)

The principal curvatures $k_i^s$ of the geodesic spheres $f_s\in {\mathscr {F}}$ are ($n=\dim \mathbb{H}_{{\mathbb{F}}}^{m}$):

$$\begin{aligned} \begin{aligned} k_1^s&= -\frac{1}{2}\coth (s/2) \,\,\,\text {with multiplicity}\,\,\, n-p-1 \\ k_2^s&=-\coth (s) \,\,\, \text {with multiplicity} \,\,\, p \end{aligned}\,\,, \end{aligned}$$

(21)

where $p=n-1$ for $\mathbb{H}^{n}$, $p=1$ for $\mathbb{H}_{\mathbb{C}}^{m}$, $p=3$ for $\mathbb{H}_{{\mathbb{K}}}^{m}$, and $p=7$ for $\mathbb{H}_{{\mathbb{O}}}^2$ (see, e.g., [6, pgs. 353, 543] and [20]).

From equalities (21), we obtain the r-mean curvatures ${{H}_{r}}^s$ of the geodesic spheres $f_s$ of $\mathbb{H}_{{\mathbb{F}}}^{m}.$ For instance, in $\mathbb{H}^{n},$ $n\ge 2,$ we have

$$\begin{aligned} {{H}_{r}}^s=(-1)^r{{n-1}\atopwithdelims (){r}}\coth ^r(s) \quad (1\le r\le n-1), \end{aligned}$$

(22)

whereas for $\mathbb{H}_{\mathbb{C}}^{m},\, m>1,$ one has

$$\begin{aligned} {{H}_{r}}^s= & {} \left( -\frac{1}{2}\right) ^r{{n-2}\atopwithdelims (){r}}\coth ^r(s/2)\nonumber \\+ & {} (-1)^r\left( \frac{1}{2}\right) ^{r-1}{{n-2}\atopwithdelims (){r-1}}\coth ^{r-1}(s/2)\coth (s) \end{aligned}$$

(23)

if $1\le r\le n-2$, and

$$\begin{aligned} H_{n-1}^s=(-1)^{n-1}\coth (s)\coth ^{n-2}(s/2)/2^{n-2}. \end{aligned}$$

(24)

Analogously, one obtains the rth mean curvature functions ${{H}_{r}}^s$ for the other hyperbolic spaces. A direct computation from these data yields the following

Lemma 3

The functions a e b defined in (20) have the following properties:

• (i) a is negative and increasing for $1\le r\le n-1$, and vanishes for $r=n.$

• (ii) b is positive and increasing for $1<r\le n$, and $b=H_1$ for $r=1.$

In particular, we have the inequalities

$$\begin{aligned} a'(s)\ge 0, \,\,\,\, b'(s)\ge 0, \,\, \,\,\, \text {and}\ \,\,\,\, a'(s)+b'(s)>0 \,\, \, \forall s\in (0,+\infty ). \end{aligned}$$

(25)

We point out that, in the above setting, one has (cf. [22])

$$\begin{aligned} |{{H}_{r}}^s|={{{n-1}\atopwithdelims (){r}}}s^{-r}+{\mathcal {O}}(s^{2-r}) \end{aligned}$$

(26)

in a neighborhood of $s=0.$ In particular,

$$\begin{aligned} \lim _{s\rightarrow 0}|{{H}_{r}}^s|=+\infty . \end{aligned}$$

(27)

In what follows, by means of the family ${\mathscr {F}},$ we will construct complete rotational ${{H}_{r}}$-hypersurfaces in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which are made of pieces of $(f_s,\phi )$-graphs. To this end, we will look for solutions $\uptau (s)$ of the equation $y'=ay+b$ (with a and b as in (20)) satisfying suitable initial conditions. Let us recall that, in this context, the general solution of $y'=ay+b$ is

$$\begin{aligned} \begin{aligned} \uptau (s)&=\frac{1}{\mu (s)}\left( \uptau _0+\int _{s_0}^{s}{b(u)}{\mu (u)}du\right) \\ \mu (s)&=\exp \left( -\int _{s_0}^sa(u)du\right) \end{aligned}\,\,, \quad s_0\,, s\in (0,+\infty ), \,\,\uptau _0\in \mathbb{R} . \end{aligned}$$

(28)

Concerning the solutions $\uptau (s)$, we will be also interested in those which can be extended to $s=0.$ Notice that, in principle, neither a nor b are defined at $s=0$, which makes this point a singularity. However, the function b is easily extendable to $s=0.$ Indeed, we can set $b(0)={{H}_{r}}$ if $r=1$ and $b(0)=0$ if $r>1$ (from (27)). As for a,  it follows from (26) that, for $1\le r<n,$

$$\begin{aligned} \lim _{s\rightarrow 0}|a(s)|=+\infty \quad \text {and}\quad \lim _{s\rightarrow 0}s|a(s)|<+\infty , \end{aligned}$$

(29)

which characterizes $s=0$ as a regular singular point of $y'=ay+b.$ This means that, despite the fact that a is not defined at $s=0,$ this equation has a nonnegative solution $\uptau$ defined at $s=0$ that satisfies $\uptau (0)=0$ (cf. [28, Theorem 3.1], [30, Lemma 4.4]). More precisely, this solution is

$$\begin{aligned} \uptau (s):=\left\{ \begin{array}{lcl} \frac{1}{\mu (s)}\int _{0}^{s}{b(u)}{\mu (u)}du &{} \text {if} &{} s\in (0,+\infty ) \\ 0 &{} \text {if} &{} s=0, \end{array} \right. \end{aligned}$$

(30)

where $\mu$ is a solution of $y'+ay=0.$ Notice that the function $\uptau$ defined in (30) is also the solution of $y'=ay+b$ in the case $r=n,$ i.e., for $a=0.$ (Just set $\mu (s)=1.$)

As we shall see, the geometry of the ${{H}_{r}}$-hypersurfaces we construct from $(f_s,\phi )$-graphs is closely related to the growth of $\uptau$ as $s\rightarrow +\infty .$ Taking that into account, for a given family of parallel geodesic spheres ${\mathscr {F}}$ in $\mathbb{H}_{{\mathbb{F}}}^{m},$ we define

$$\begin{aligned} C_{{\mathbb{F}}}(r):=\lim _{s\rightarrow +\infty }|{{H}_{r}}^s|, \,\,\, r=1,\dots ,n. \end{aligned}$$

(31)

In particular, $C_{{\mathbb{F}}}(n)=0.$ Notice that, since $\mathbb{H}_{{\mathbb{F}}}^{m}$ is homogeneous, the constant $C_{{\mathbb{F}}}(r)$ is well defined, that is, it does not depend on the family ${\mathscr {F}}$ of geodesic spheres.

It follows from equalities (22)–(24) that

• (i) $C_\mathbb{R} (r)={{n-1}\atopwithdelims (){r}} \quad (1\le r\le n-1).$

• (ii) $C_{{\mathbb{C}}}(r)=\left( \frac{1}{2}\right) ^r{{n-2}\atopwithdelims (){r}}+\left( \frac{1}{2}\right) ^{r-1}{{n-2}\atopwithdelims (){r-1}} \quad (1\le r\le n-2).$

• (iii) $C_{{\mathbb{C}}}(n-1)=\frac{1}{2^{n-2}}\,\cdot$

Similarly, one can compute the other constants $C_{{\mathbb{F}}}(r)$ and easily conclude that

$$\begin{aligned} C_{\mathbb{F}}(r)>0 \,\,\, \forall r\in \{1,\dots ,n-1\}. \end{aligned}$$

The next proposition shows the relation between the solution $\uptau$ of $y'=ay+b$ and the constant $C_{{\mathbb{F}}}(r).$ Notice that, for $1\le r\le n-1,$ the identities (20) yield:

$$\begin{aligned} \lim _{s\rightarrow +\infty }\frac{-b(s)}{a(s)}=\frac{{{H}_{r}}}{C_{{\mathbb{F}}}(r)}\,\cdot \end{aligned}$$

(32)

Proposition 1

The following assertions hold:

• (i) The solution $\uptau$ defined in (30) is increasing, i.e., $\uptau '>0$ in $(0,+\infty ).$

• (ii) Both the solutions $\uptau$ in (28) and (30) satisfy the following equality:

  $$\begin{aligned} \lim _{s\rightarrow +\infty }\uptau (s)= \left\{ \begin{array}{lcl} +\infty &{} \text {if} &{} r=n.\\ {{H}_{r}}/C_{{\mathbb{F}}}(r) &{} \text {if} &{} 1\le r\le n-1. \end{array} \right. \end{aligned}$$

  (33)

Proof

To prove (i), let us first observe that, since the solution $\uptau$ in (30) is positive in $(0,+\infty )$ and $\uptau (0)=0,$ we have that $\uptau$ is increasing near 0. Assume that $\uptau$ is not increasing in $(0,+\infty ).$ In this case, $\uptau$ has a first critical point $s_0$ in $(0,+\infty )$ which is necessarily a local maximum. However, considering (25) and the equality $\uptau '=a\uptau +b,$ we have

$$\begin{aligned} \uptau ''(s_0)=a'(s_0)\uptau (s_0)+a(s_0)\uptau '(s_0)+b'(s_0)=a'(s_0)\uptau (s_0)+b'(s_0)>0, \end{aligned}$$

which implies that $s_0$ is a local minimum—a contradiction. Therefore, $\uptau$ is increasing in $(0,+\infty ),$ which proves (i).

Suppose that $\uptau$ is as in (28). Since b is increasing, if $r=n$ one has

$$\begin{aligned} \uptau (s)=\uptau _0+\int _{s_0}^sb(u)du\ge \uptau _0+b(s_0)(s-s_0), \end{aligned}$$

which implies that $\uptau (s)\rightarrow +\infty$ as $s\rightarrow +\infty .$

Now, assume $1\le r\le n-1.$ We claim that, in this case, $\mu (s)\rightarrow +\infty$ as $s\rightarrow +\infty$. Indeed, for any fixed $s_0>0,$ and $s>s_0$ ,

$$\begin{aligned} |a(s)|=\frac{r|{{H}_{r}}^s|}{|H_{r-1}^s|}>\frac{rC_{{\mathbb{F}}}(r)}{|H_{r-1}^{s_0}|}>0. \end{aligned}$$

Since |a| is decreasing, this inequality gives that $\inf |a|>0$ in $(0,+\infty ).$ Hence,

$$\begin{aligned} \mu (s)=e^{-\int _{s_0}^{s}a(u)du}=e^{\int _{s_0}^{s}|a(u)|du}>e^{(\inf |a|) (s-s_0)} \,\,\, \forall s>s_0, \end{aligned}$$

which clearly implies the claim on $\mu .$

From the expression of $\uptau ,$ we have (apply l’Hôpital to the second summand):

$$\begin{aligned} \lim _{s\rightarrow +\infty }\uptau (s)=\lim _{s\rightarrow +\infty }\frac{\uptau _0}{\mu (s)}+\lim _{s\rightarrow +\infty }\frac{b(s)\mu (s)}{\mu '(s)}= \lim _{s\rightarrow +\infty }\frac{b(s)\mu (s)}{-a(s)\mu (s)}=\frac{{{H}_{r}}}{C_{{\mathbb{F}}}(r)}\,, \end{aligned}$$

where the last equality comes from (32). This finishes the proof of (33) when $\uptau$ is defined as in (28). The proof for the solution $\uptau$ in (30) is completely analogous. $\square$

Now, it is clear from Proposition 1 that a solution $\uptau$ as in (30) reaches the value 1 if and only if ${{H}_{r}}>C_{{\mathbb{F}}}(r).$ More specifically, we have the following

Corollary 1

Let $\uptau$ be as in (30). Then, there exists $s_0\in (0, +\infty )$ satisfying

$$\begin{aligned} 0<\uptau |_{(0,s_0)}<1 \,\,\,\, \text {and} \,\,\,\,\, \uptau (s_0)=1 \end{aligned}$$

if and only if ${{H}_{r}}>C_{{\mathbb{F}}}(r)$ (Fig. 2a). Consequently, for $1\le r\le n-1$ and any constant ${{H}_{r}}\in (0,C_{{\mathbb{F}}}(r)),$ the following inequality holds (Fig. 2b):

$$\begin{aligned} 0<\uptau (s)<1 \,\,\, \forall s\in (0,+\infty ). \end{aligned}$$

Fig. 2

Graphs of $\uptau$ (as in (30)) according to the sign of ${{H}_{r}}-C_{{\mathbb{F}}}(r)$

Now, we are in position to establish our first existence result.

Theorem 1

Given $r\in \{1,\dots , n\}$ and a constant ${{H}_{r}}>0,$ the following hold:

• (i) If  ${{H}_{r}}>C_{{\mathbb{F}}}(r),$ there exists an embedded strictly convex rotational ${{H}_{r}}$-sphere in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which is symmetric with respect to a horizontal hyperplane.

• (ii) If  $0<{{H}_{r}}\le C_{{\mathbb{F}}}(r),$ there exists an entire strictly convex rotational ${{H}_{r}}$-graph in $\mathbb{H}_{{\mathbb{F}}}^{m}\times [0,+\infty )$ which is tangent to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\}$ at a single point, and whose height function is unbounded above. Consequently, there are no compact ${{H}_{r}}$-hypersurfaces of  $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ for such values of ${{H}_{r}}$ .

Proof

Let ${\mathscr {F}}$ be an arbitrary family of parallel geodesic spheres of $\mathbb{H}_{{\mathbb{F}}}^{m}$ as in (19). Consider the functions a and b defined in (20) and let $\uptau$ be the solution (30) of the ODE $y'=ay+b.$

If ${{H}_{r}}>C_{{\mathbb{F}}}(r),$ we have from Corollary 1 that there exists $s_0\in (0,+\infty )$ satisfying

$$\begin{aligned} \uptau (0)=0<\uptau |_{(0,s_0)}<1=\uptau (s_0). \end{aligned}$$

Hence, by Lemma 1, the $(f_s,\phi )$-graph $\varSigma '$ with $\varrho$-function $\varrho :=\root r \of {\uptau |_{[0,s_0)}}$ is a rotational ${{H}_{r}}$-graph of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ over the open ball $B_{s_0}(o)\subseteq \mathbb{H}_{{\mathbb{F}}}^{m}$ such that

$$\begin{aligned} \phi (s)=\int _{0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du, \,\,\,\, s\in [0,s_0). \end{aligned}$$

(34)

From Proposition 1-(i), one has $\uptau '(s_0)>0,$ which implies that $\varrho '(s_0)>0.$ Hence, by Lemma 2, $\phi$ extends to $s_0$ , i.e.,

$$\begin{aligned} \phi (s_0):=\lim _{s\rightarrow s_0}\phi (s) \end{aligned}$$

is well defined. In particular, $\partial \varSigma '=S_{s_0}\times \{\phi (s_0)\}.$

Fig. 3

Trajectories of $\nabla \xi$ on $\varSigma '$ emanate from o and meet $\partial \varSigma '$ orthogonally

Notice that $o\in \varSigma '$ is an isolated minimum of the height function $\xi$ of $\varSigma '.$ Thus, $\varSigma '$ is strictly convex at o. In addition, by the identities (12), at any point of $\varSigma '-\{o\},$ all the principal curvatures are positive. Therefore, $\varSigma '$ is strictly convex.

As we know, the angle function $\varTheta$ of $\varSigma '$ is given by

$$\begin{aligned} \varTheta =\frac{1}{\sqrt{1+(\phi ')^2}}\,\cdot \end{aligned}$$

(35)

Since $\varrho (s_0)=1$, we have from (34) that $\phi '(s)\rightarrow +\infty$ as $s\rightarrow s_0$ . This, together with (35), implies that the tangent spaces of $\varSigma '$ along $\partial \varSigma '$ are vertical. Hence, the trajectories of $\nabla \xi$ all emanate from o and meet $\partial \varSigma '$ orthogonally (Fig. 3). Recall that these trajectories are geodesics which foliate $\varSigma '.$

Now, set $\varSigma ''$ for the reflection of $\varSigma '$ with respect to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{\phi (s_0)\}$ and define

$$\begin{aligned} \varSigma :=\mathrm{closure}\,(\varSigma ')\cup \mathrm{closure}\,(\varSigma ''), \end{aligned}$$

that is, $\varSigma$ is the “gluing” of $\varSigma '$ and $\varSigma ''$ along the $(n-1)$-sphere $S_{s_0}(o)\times \{\phi (s_0)\}.$ Since the tangent spaces along $S_{s_0}(o)\times \{\phi (s_0)\}$ are all vertical, we have that $\varSigma$ is a well-defined rotational strictly convex ${{H}_{r}}$-hypersurface of  $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which is homeomorphic to the n-sphere $\mathbb{S}^{n}$ and is symmetric with respect to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{\phi (s_0)\}.$ This proves (i).

Under the hypotheses in (ii), it follows from Corollary 1 that $\uptau$ satisfies:

$$\begin{aligned} \uptau (0)=0<\uptau |_{(0,+\infty )}<1, \end{aligned}$$

so that the $(f_s,\phi )$-graph $\varSigma$ with $\varrho$-function $\varrho :=\root r \of {\uptau |_{[0,+\infty )}}$ is an entire rotational ${{H}_{r}}$-graph of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ over $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\}.$ Since $\phi (0)=0$ and $\phi (s)>0$ for any $s>0,$ $\varSigma$ is contained in the closed half-space $\mathbb{H}_{{\mathbb{F}}}^{m}\times [0,+\infty )$ and is tangent to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\}$ at o. Also, the height function of $\varSigma$ is unbounded above. Indeed, from Proposition 1-(i), $\uptau$, and so $\varrho$, is increasing. Thus, for a fixed $\delta >0,$ and any $s>\delta ,$ one has

$$\begin{aligned} \phi (s)=\int _{0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du\ge \int _{\delta }^{s}\varrho (u)du\ge \varrho (\delta )(s-\delta ), \end{aligned}$$

which implies that $\phi$ is unbounded above. Also, arguing as for the graph $\varSigma '$ in the preceding paragraphs, we conclude that $\varSigma$ is strictly convex.

Observe that the mean curvature vector of $\varSigma$ “points upwards,” that is, its mean convex side $\Lambda$ is the connected component of $(\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} )-\varSigma$ which contains the axis $\{o\}\times \mathbb{R} .$ In particular, $\Lambda$ is foliated by the balls $B_s(o)\times \{\phi (s)\}, \,\, s\in (0,+\infty ).$

Fig. 4

After a downward translation of ${\tilde{\varSigma }},$ it has a contact with $\varSigma$

Let us suppose that there exists a compact ${{H}_{r}}$-hypersurface ${\widetilde{\varSigma }}$ such that $0<{{H}_{r}}\le C_{{\mathbb{F}}}(r).$ Considering the fact that $\Lambda$ is “horizontally and vertically unbounded,” it is easily seen that, after a suitable vertical translation, we can assume ${\widetilde{\varSigma }}\subset \Lambda$ (Fig. 4). Now, translate ${\widetilde{\varSigma }}$ downward until it has a first contact with $\varSigma .$ Since $\varSigma$ is strictly convex, the tangency principle applies (cf. Sect. 2) and gives that $\varSigma$ coincides with ${\widetilde{\varSigma }},$ which is clearly impossible. This shows that such a ${\widetilde{\varSigma }}$ cannot exist and finishes the proof of (ii). $\square$

Remark 1

Let $\varSigma \subset \mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ be an entire ${{H}_{r}}$-graph as in Theorem 1-(ii). Since $C_{{\mathbb{F}}}(n)=0$, we must have $r<n.$ Also, the associated function $\uptau :[0,+\infty )\rightarrow \mathbb{R}$ is positive, bounded and increasing, so that

$$\begin{aligned} \lim _{s\rightarrow +\infty }\uptau '(s)=0, \end{aligned}$$

which implies that $\varrho '(s)\rightarrow 0$ as $s\rightarrow +\infty .$ This, together with (12), gives that the principal curvature $k_n=\varrho '(s)$ goes to zero as $s\rightarrow +\infty .$ In particular, the least principal curvature function of $\varSigma$ is not bounded away from zero.

Next, we apply the method of $(f_s,\phi )$-graphs to produce one-parameter families of ${{H}_{r}}(>0)$-annuli in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} ,$ $1\le r<n.$ For that, fix $s_0=\lambda >0$ and consider the solution $\uptau (s)$ of $y'=ay+b$ given in (28), which satisfies the initial condition $\uptau _0=\uptau (\lambda )=1.$ Since we must have $0<\uptau <1,$ we look at those values of $\lambda$ for which $\uptau '(\lambda )<0.$ From (20), we have

$$\begin{aligned} \uptau '(\lambda )=a(\lambda )+b(\lambda )=\frac{r({{H}_{r}}-|{{H}_{r}}^\lambda |)}{|H_{r-1}^\lambda |}\,, \end{aligned}$$

(36)

so that $\uptau '(\lambda )<0$ if and only if $|{{H}_{r}}^\lambda |>{{H}_{r}}.$ This (and equality 27) suggest us to define, for $1\le r<n$ and a given ${{H}_{r}}>0,$ the following constant:

$$\begin{aligned} \delta _{{{H}_{r}}}:=\sup \{\lambda>0\,;\, |{{H}_{r}}^\lambda |>{{H}_{r}}\}. \end{aligned}$$

(37)

In this setting, we have the following result.

Fig. 5

Two types of solutions $\uptau$, as in (28), satisfying $\uptau (\lambda )=1$

Theorem 2

Given $r\in \{1,\dots ,n-1\}$ and ${{H}_{r}}>0,$ let $\delta _{{{H}_{r}}}>0$ be as in (37). Then, there exists a one-parameter family

$$\begin{aligned} {\mathscr {S}}\,=\{\varSigma (\lambda )\,;\,\lambda \in (0,\delta _{{{H}_{r}}})\} \end{aligned}$$

of properly embedded rotational ${{H}_{r}}$-hypersurfaces in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which are all homeomorphic to the n-annulus $\mathbb{S}^{n-1}\times \mathbb{R} .$ In addition, the following assertions hold:

• (i) If  ${{H}_{r}}>C_{{\mathbb{F}}}(r),$ each $\varSigma (\lambda )\in {\mathscr {S}}$ is Delaunay type, i.e., it is periodic in the vertical direction, and has unduloids as the trajectories of the gradient of its height function.

• (ii) If $0<{{H}_{r}}\le C_{{\mathbb{F}}}(r)$, each hypersurface $\varSigma (\lambda )\in {\mathscr {S}}$ is symmetric with respect to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\}$ and has unbounded height function.

Proof

Given $\lambda \in (0,\delta _{{{H}_{r}}}),$ let $\uptau$ be the solution (28) such that $s_0=\lambda$ and $\uptau (\lambda )=1.$ By (36) and the definition of $\delta _{{{H}_{r}}},$ one has $\uptau '(\lambda )<0,$ so that $\uptau$ is decreasing near $\lambda .$

It is clear from (28) that $\uptau$ is positive in $(\lambda ,+\infty ).$ Thus, if ${{H}_{r}}>C_{{\mathbb{F}}}(r),$ it follows from (33) that there exists ${\bar{\lambda }}\in (\lambda ,+\infty )$ such that (Fig. 5a)

$$\begin{aligned} 0<\uptau |_{(\lambda ,{\bar{\lambda }})}< 1=\uptau ({\bar{\lambda }})=\uptau (\lambda ). \end{aligned}$$

Let us observe that a critical point $s_1$ of $\uptau$ is necessarily a strict minimum, since $\uptau ''(s_1)=a'(s_1)\uptau (s_1)+b'(s_1)>0.$ Therefore, $\uptau$ must have a unique local minimum at a point between $\lambda$ and $\overline{\lambda }.$ In particular, $\uptau '({\bar{\lambda }})>0.$

Fig. 6

“Block” of a Delaunay-type ${{H}_{r}}$-hypersurface in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$

Setting $\uptau _\lambda :=\uptau |_{(\lambda ,{\bar{\lambda }})}$, it follows from the above considerations and Lemmas 1 and  2 that the $(f_s,\phi )$-graph $\varSigma '(\lambda )$ with $\varrho$-function $\varrho (s)=\root r \of {\uptau _\lambda }$ is a bounded ${{H}_{r}}$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$ Moreover, $\varSigma '(\lambda )$ is homeomorphic to $\mathbb{S}^{n-1}\times (\lambda ,{\bar{\lambda }})$ and has boundary (see Fig. 6):

$$\begin{aligned} \partial \varSigma '(\lambda )=(S_\lambda (o)\times \{0\})\cup (S_{{\bar{\lambda }}}(o)\times \{\phi ({\bar{\lambda }})\}). \end{aligned}$$

We also have that the tangent spaces of $\varSigma '(\lambda )$ are vertical along its boundary $\partial \varSigma '(\lambda )$, for $\varrho (\lambda )=\varrho ({\bar{\lambda }})=1.$ Therefore, we obtain a properly embedded rotational ${{H}_{r}}$-hypersurface $\varSigma (\lambda )$ from $\varSigma '(\lambda )$ by continuously reflecting it with respect to the horizontal hyperplanes $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{k\phi ({\bar{\lambda }})\}, \, k\in \mathbb{Z} .$ This proves (i).

Now, let us assume $0<{{H}_{r}}\le C_{{\mathbb{F}}}(r).$ In this case, (33) gives that (Fig. 5b)

$$\begin{aligned} 0<\uptau |_{(\lambda ,+\infty )}<1, \end{aligned}$$

so that the $(f_s,\phi )$-graph $\varSigma '(\lambda )$ determined by $\varrho =\uptau ^{1/r}|_{(\lambda ,+\infty )}$ is a rotational ${{H}_{r}}$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ with boundary $\partial \varSigma '(\lambda )=S_\lambda (o)\times \{0\}$ (Fig. 7). By reflecting $\varSigma '(\lambda )$ with respect to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\},$ as we did before, we obtain the embedded ${{H}_{r}}$-hypersurface $\varSigma (\lambda )$ as stated.

It remains to show that the height function of $\varSigma (\lambda )$ is unbounded. For that, we have just to observe that the infimum of $\uptau$ in $[\lambda ,+\infty )$ is positive, since $\uptau$ itself is positive in this interval, and its limit as $s\rightarrow +\infty$ is ${{H}_{r}}/C_{{\mathbb{F}}}(r)>0.$ So, the same is true for $\varrho =\uptau ^{1/r}.$ Therefore,

$$\begin{aligned} \phi (s)=\int _{\lambda }^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du> \int _{\lambda }^{s}\varrho (u)du > \inf \varrho |_{[\lambda ,+\infty )}(s-\lambda ), \end{aligned}$$

from which we conclude that $\phi$ is unbounded. $\square$

Remark 2

The case ${\mathbb{F}}=\mathbb{R}$ of Theorem 1 was previously established in [15], whereas the case ${\mathbb{F}}=\mathbb{R}$ and $r=1$ of Theorem 2 was considered in [2]. Nevertheless, the methods employed in these works are different from ours and are not applicable to the products $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} ,$ ${\mathbb{F}}\ne \mathbb{R} .$

Fig. 7

$(f_s,\phi )$-graph $\varSigma '(\lambda ),$ on which all the trajectories of $\nabla \xi$ emanate from $\partial \varSigma '(\lambda )$ orthogonally

We proceed now to the classification of the complete rotational ${{H}_{r}}$-hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ whose height functions are Morse type, i.e., have isolated critical points, if any. As we shall see, besides cylinders over geodesic spheres, these hypersurfaces are precisely the ones we obtained in Theorems 1 and 2. In particular, any of them is embedded. We point out that, in [2], it was shown that, for any $H_1>0,$ there exist complete rotational $H_1$-hypersurfaces in $\mathbb{H}^{n}\times \mathbb{R}$ which are not embedded. In accordance with our results, the height function of none of these $H_1$-hypersurfaces is Morse type.

Firstly, let us recall that the ${{H}_{r}}$-hypersurfaces in Theorems 1 and 2 were constructed from a single $(f_s,\phi )$-graph whose associated $\uptau$-function is a solution of the ODE $y'=ay+b,$ where a and b are as in (20). For such a $\uptau$, there is a maximal interval $(s_0,s_1)$, $0\le s_0<s_1\le +\infty ,$ such that $0<\uptau |_{(s_0,s_1)}<1.$

Notice that each choice of ${{H}_{r}}$ determines the function b and, so, the equation $y'=ay+b.$ The corresponding graph, then, is determined by the ordering of the constants ${{H}_{r}}$ and $C_{{\mathbb{F}}}(r),$ as well as by the values of $s_0$ and $\uptau (s_0).$

Below, we list all the occurrences of $s_0$ and $\uptau (s_0)$ in Theorems 1 and 2 with respect to the ordering of ${{H}_{r}}$ and $C_{{\mathscr {F}}}(r)$:

• (C1) $s_0=0$, $\uptau (s_0)=0$, ${{H}_{r}}>C_{{\mathbb{F}}}(r)$.

• (C2) $s_0=0$, $\uptau (s_0)=0$, ${{H}_{r}}\le C_{{\mathbb{F}}}(r)$.

• (C3) $s_0>0$, $\uptau (s_0)=1$, ${{H}_{r}}>C_{{\mathbb{F}}}(r)$.

• (C4) $s_0>0$, $\uptau (s_0)=1$, ${{H}_{r}}\le C_{{\mathbb{F}}}(r)$.

The cases C1 and C2 correspond to Theorem 1-(i) and Theorem 1-(ii), respectively, whereas C3 and C4 correspond to Theorem 2-(i) and Theorem 2-(ii). We also remark that $s_1<+\infty$ in the cases C1 and C3, with $\uptau (s_1)=1,$ and that $s_1=+\infty$ in the cases C2 and C4.

Let $M_0$ be a hypersurface of a Riemannian manifold M. It is easily seen that $\varSigma :=M_0\times \mathbb{R}$ is a hypersurface of $M\times \mathbb{R}$ whose tangent spaces are all vertical, so that $\partial _t$ is a principal direction of $\varSigma$ with vanishing principal curvature. In particular, $H_n=0$ on $\varSigma .$ Also, for all $r\in \{1,\dots ,n-1\},$ the rth mean curvatures of $M_0$ and $\varSigma$ at $x\in M_0$ and $(x,t)\in \varSigma$ coincide. In particular, $M_0$ is an $H_{r(<n)}$-hypersurface of M if and only if $\varSigma$ is an $H_{r(<n)}$-hypersurface of $M\times \mathbb{R} .$ We call $\varSigma :=M_0\times \mathbb{R}$ the cylinder over $M_0.$

Theorem 3

Let $\varSigma$ be a connected complete rotational ${{H}_{r}}(>0)$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ whose height function is Morse type. Then, $\varSigma$ is either a cylinder over a geodesic sphere of  $\mathbb{H}_{{\mathbb{F}}}^{m}$ or one of the embedded ${{H}_{r}}$-hypersurfaces of Theorems 1–2.

Proof

Suppose that $\varSigma$ is not a cylinder. In this case, we have that the open set $\varSigma _0\subset \varSigma$ on which $\varTheta \nabla \xi$ never vanishes is nonempty. Since $\varSigma _0$ contains no vertical points, for a given $x_0\in \varSigma _0$ , there is an open neighborhood $\varSigma '$ of $x_0$ in $\varSigma _0$ which is a graph over an open set $\Omega$ of $\mathbb{H}_{{\mathbb{F}}}^{m}$. Thus, since $\varSigma$ is rotational and $\varSigma _0$ contains no horizontal points, after possibly a reflection with respect to a horizontal hyperplane, we can assume that $\varSigma '$ is an $(f_s,\phi )$-graph over $\Omega .$ (Recall that, in our setting, the $\phi$-function of an $(f_s,\phi )$-graph is required to be radially increasing.)

By Lemma 1, the function $\uptau =\varrho ^r$ associated with $\varSigma '$ is a solution of $y'=ay+b,$ with a and b as in (20). In addition, since $\varSigma$ is complete, there exists a maximal interval $(s_0,s_1)$, $0\le s_0<s_1\le +\infty ,$ such that $0<\uptau |_{(s_0,s_1)}<1.$ In particular, we have the following two possibilities:

$$\begin{aligned} \uptau (s_0)=0 \quad \text {or}\quad \uptau (s_0)=1. \end{aligned}$$

Suppose that $\uptau (s_0)=0.$ After a vertical translation, we can assume that

$$\begin{aligned} \phi (s)=\int _{s_0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du, \end{aligned}$$

which yields $\phi (s_0)=\phi '(s_0)=0.$ If $s_0>0,$ these equalities imply that the sphere $S_{s_0}(o)\times \{0\}$ of $\mathbb{H}_{{\mathbb{F}}}^{m}$ is contained in $\partial \varSigma ',$ and that $\nabla \xi$ vanishes at all of its points. This, however, contradicts that the height function of $\varSigma$ is Morse type. Hence, $s_0=0,$ so that the $\uptau$-function of $\varSigma '$ satisfies the initial condition $\uptau (0)=0.$

If ${{H}_{r}}>C_{{\mathbb{F}}}(r),$ by the uniqueness of solutions of linear ODE’s satisfying an initial condition, the function $\uptau$ such that $\uptau (0)=0$ coincides with the one in the case C1 above. Thus, the corresponding $\phi$-functions also coincide, which clearly implies that $\varSigma '$ is an open set of the (strictly convex) ${{H}_{r}}$-sphere obtained in Theorem 1-(i). Therefore, by the tangency principle, $\varSigma$ coincides with this ${{H}_{r}}$-sphere. If ${{H}_{r}}\le C_{{\mathbb{F}}}(r),$ then $\uptau$ coincides with the solution of case C2. Analogously, we conclude that $\varSigma$ is an entire graph as in Theorem 1-(ii).

Let us suppose now that $\uptau (s_0)=1.$ Since $0<\uptau <1$ in $(s_0,s_1),$ $\uptau$ is decreasing near $s_0$ , which implies that $r<n.$ (Indeed, for $r=n,$ we have $\uptau '=b>0.$) In this case, as we have discussed, $|a(s)|\rightarrow +\infty$ as $s\rightarrow 0$ (cf. (29)), and b(0) is 0 or $H_1$. In particular, any solution $\uptau$ of $y'=ay+b$ at $s=0$ must satisfy $\uptau (0)=0,$ so that $s_0\ne 0.$ Hence, $s_0\in (0,\delta _{{{H}_{r}}}),$ where $\delta _{{{H}_{r}}}$ is the positive constant defined in (37). Otherwise, $\uptau$ would not be decreasing near $s_0.$

Setting $s_0=\lambda$ and observing that any of the hypersurfaces obtained in Theorem 2 is strictly convex at some of its points, we can argue as in the second from the last paragraph and conclude that $\varSigma$ is the ${{H}_{r}}$-hypersurface $\varSigma (\lambda )$ of either Theorem  2-(i) or Theorem 2-(ii), according to whether ${{H}_{r}}>C_{{\mathbb{F}}}(r)$ or ${{H}_{r}}\le C_{{\mathbb{F}}}(r).$ This finishes the proof. $\square$

4.2 Rotational ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{S}^{n}\times \mathbb{R}$

In this section, we apply the method of $(f_s,\phi )$-graphs to construct and classify rotational ${{H}_{r}}(>0)$-hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R} .$

As we did before, let us fix a point $o\in \mathbb{S}^{n}$ and consider a family

$$\begin{aligned} {\mathscr {F}}:=\{f_s:\mathbb{S}^{n-1}\rightarrow \mathbb{S}^{n}\,;\, s\in (0,\pi )\} \end{aligned}$$

(38)

of parallel geodesic spheres $f_s$ of $\mathbb{S}^{n}$ with radius s and center at o. As is well known, each $f_s$ is totally umbilical, having principal curvatures all equal to $-\cot s$ with respect to the outward orientation. In particular, ${\mathscr {F}}$ is isoparametric.

From a direct computation, we get that the coefficients a and b of the ODE $y'=ay+b$ determined by ${\mathscr {F}}$ and any given ${{H}_{r}}>0$ are

$$\begin{aligned} a(s)=-(n-r)\cot s \quad \text {and}\quad b(s)=b_r\tan ^{r-1}(s)\,, \,\,\, b_r=r{{H}_{r}}{{{n-1}\atopwithdelims (){r-1}}}^{-1}, \end{aligned}$$

(39)

and that the corresponding general solution is:

$$\begin{aligned} \uptau (s):= \left( \frac{\sin s_0}{\sin s}\right) ^{n-r}\left( \uptau _0+ \frac{b_r}{\sin ^{n-r}(s_0)}\int _{s_0}^{s}\frac{\sin ^{n-1}(u)}{\cos ^{r-1}(u)}du\right) , \,\,s_0\,, s\in (0,{\mathcal {R}}), \end{aligned}$$

(40)

where $\uptau _0=\uptau (s_0)\in \mathbb{R}$ and

$$\begin{aligned} {\mathcal {R}}:=\left\{ \begin{array}{ccc} \pi /2 &{} \text {if} &{} r>1.\\ \pi &{} \text {if} &{} r=1. \end{array} \right. \end{aligned}$$

Also, it is easily checked that

$$\begin{aligned} \uptau (s):=\left\{ \begin{array}{lcl} \frac{b_r}{\sin ^{n-r}(s)}\int _{0}^{s}\frac{\sin ^{n-1}(u)}{\cos ^{r-1}(u)}du&{} \text {if} &{} s\in (0,{\mathcal {R}}) \\ 0 &{} \text {if} &{} s=0 \end{array} \right. \end{aligned}$$

(41)

is a well-defined solution of $y'=ay+b$ satisfying $y(0)=0.$

Given an integer $n\ge 2,$ it will be convenient to introduce the following constant:

$$\begin{aligned} S(n)=\int _{0}^{\pi /2}\sin ^{n-1}(s)ds, \,\, n\ge 2. \end{aligned}$$

(42)

Proposition 2

Let $\uptau$ be the solution (41). Then, the following hold:

• (i) $\uptau '>0$ in $(0,{\mathcal {R}}).$

• (ii) $\displaystyle \lim _{s\rightarrow {\mathcal {R}}}\uptau (s)=+\infty .$

• (iii) $\displaystyle \lim _{s\rightarrow \frac{\pi }{2}}\uptau (s)=H_1S(n)$ if $r=1.$

Proof

Since the functions a and b in (39) are both increasing (when nonconstant), the proof of (i) is entirely analogous to the one given in Proposition 1-(i).

To prove (ii), let us first assume $r=1.$ In this case, since $\sin \pi =0$ and the integral $\int _{0}^{\pi }\sin ^{n-1}(u)du$ is positive, we have that $\uptau$ satisfies (ii) for ${\mathcal {R}}=\pi .$

If $r>1$, for a fixed $\delta \in (0,\pi /2)$ and any $s\in (\delta ,\pi /2),$ one has

$$\begin{aligned} \int _{0}^{s}\frac{\sin ^{n-1}(u)}{\cos ^{r-1}(u)}du\ge \int _{\delta }^{s}\tan ^{r-1}(u){\sin ^{n-r}(u)}du\ge \sin ^{n-r}(\delta )\int _{\delta }^{s}\tan ^{r-1}(u)du, \end{aligned}$$

which implies that the first integral on the left goes to infinity as $s\rightarrow \pi /2,$ since the same is true for the integral $\int _{\delta }^{s}\tan ^{r-1}(u)du$. It follows from this fact that $\uptau (s)\rightarrow +\infty$ as $s\rightarrow \pi /2$ if $r>1,$ which proves (ii).

The identity in (iii) follows directly from the definitions of $\uptau$ (for $r=1$) and S(n) (as in (42)). $\square$

From the above proposition, we get the following existence result for ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{S}^{n}\times \mathbb{R} .$

Theorem 4

Given $r\in \{1,\dots , n\}$ and a constant ${{H}_{r}}>0,$ there exists a rotational ${{H}_{r}}$-sphere $\varSigma$ in $\mathbb{S}^{n}\times \mathbb{R}$ which is symmetric with respect to a horizontal hyperplane. Furthermore:

• (i) $\varSigma$ is strictly convex if either $r=1$ and $H_1 > 1/S(n)$ or $r>1.$

• (ii) $\varSigma$ is convex if $r=1$ and $H_1=1/S(n).$

• (iii) $\varSigma$ is nonconvex if $r=1$ and $0<H_1 < 1/S(n).$

Proof

Let ${\mathscr {F}}$ be an arbitrary family of parallel geodesic spheres of $\mathbb{S}^{n}$ as given in (38). Consider the functions a and b defined in (39) and let $\uptau$ be the solution (41) of the ODE $y'=ay+b.$

From Proposition 2-(ii), there exists a positive $s_0<{\mathcal {R}}$ such that

$$\begin{aligned} 0=\uptau (0)<\uptau |_{(0,s_0)}<1=\uptau (s_0), \end{aligned}$$

so that $\uptau |_{[0,s_0)}$ determines an $(f_s,\phi )$-graph $\varSigma '$ over $B_{s_0}(0)\subset \mathbb{S}^{n}.$ Since $\uptau (s_0)=1$ and $\uptau '(s_0)>0$ (by Proposition 2-(i)), we can proceed just as in the proof of Theorem 1-(i) to obtain from $\varSigma '$ the embedded ${{H}_{r}}$-sphere $\varSigma$ of $\mathbb{S}^{n}\times \mathbb{R}$ which is symmetric with respect to $P_{\phi (s_0)}:=\mathbb{S}^{n}\times \{\phi (s_0)\}.$

If either $r=1$ and $H_1 > 1/S(n)$ or $r>1,$ we have from Proposition 2, items (ii) and (iii), that $0<s_0<\pi /2.$ Hence, for $s\in (0, s_0),$ all spheres $f_s$ have negative principal curvatures, which, together with equalities (12), gives that $\varSigma$ is strictly convex. This proves (i).

If $r=1$ and $H_1=1/S(n),$ Proposition 2-(iii) yields $s_0=\pi /2.$ However, $f_{\pi /2}$ is totally geodesic in $\mathbb{S}^{n},$ which implies that, except for $k_n=H_1>0,$ the principal curvatures of $\varSigma$ vanish at all points of the horizontal section $\varSigma _{\phi (\pi /2)}=\varSigma \cap P_{\phi (\pi /2)}.$ Therefore, $\varSigma$ is convex on $\varSigma _{\phi (\pi /2)}$ and strictly convex on $\varSigma -\varSigma _{\phi (\pi /2)}$ .

Finally, assuming $r=1$ and $0<H_1<1/S(n),$ we have from Proposition 2-(iii) that $s_0>\pi /2.$ Observing that, for $\pi /2<s<\pi ,$ $f_s$ has positive principal curvatures, we conclude, as in the last paragraph, that $\varSigma$ is strictly convex (resp. convex, nonconvex) on $\varSigma _{\phi (s)}$ if $s<\pi /2$ (resp. $s=\pi /2,$ $s>\pi /2$). In particular, $\varSigma$ is nonconvex. This shows (iii) and concludes our proof. $\square$

Remark 3

Except for the assumptions on the convexity of $\varSigma ,$ the case $r=1$ of Theorem 4 was proved in [25]. The case $r=n=2$ was considered in [7]. It should also be mentioned that, for $n=2$ and $r=1,$ the nonconvexity of $\varSigma$ as stated in (iii) was pointed out in [1, Remark 2.8].

In our next theorem, we show the existence of one-parameter families of rotational Delaunay-type ${{H}_{r}}(>0)$-annuli in $\mathbb{S}^{n}\times \mathbb{R} .$ This result, then, generalizes the analogous one obtained in [26] for $r=1.$

First, let us introduce the constant

$$\begin{aligned} C_r:=\frac{n-r}{n}{{n}\atopwithdelims (){r}} \end{aligned}$$

and observe that, for $1\le r<n,$ ${{H}_{r}}$ , $b_r$ (as in (39)), and $C_r$ satisfy:

$$\begin{aligned} \frac{n-r}{b_r}=\frac{C_r}{{{H}_{r}}}\,\cdot \end{aligned}$$

(43)

In this setting, if we define

$$\begin{aligned} \delta _{{{H}_{r}}}:=\mathrm{arctan}\,(C_r/{{H}_{r}})^{1/r}\,\in (0,\pi /2), \end{aligned}$$

(44)

then a solution $\uptau$ of $y'=ay+b$ such that $\uptau (s_0)=1$, $s_0\in (0,\pi /2),$ satisfies:

$$\begin{aligned} \uptau '(s_0)<0 \quad \Leftrightarrow \quad 0<s_0<\delta _{{{H}_{r}}}. \end{aligned}$$

(45)

Theorem 5

Given $n\ge 2,$ $r\in \{1,\dots ,n-1\}$, and ${{H}_{r}} > 0,$ there exists a one-parameter family  ${\mathscr {S}}=\{\varSigma (\lambda )\,;\,0<\lambda <\delta _{{{H}_{r}}}\}$ of properly embedded Delaunay-type rotational ${{H}_{r}}$-hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}$.

Proof

Given $\lambda \in (0,\delta _{{{H}_{r}}}),$ consider the solution $\uptau$ as in (40) such that $s_0=\lambda$ and $\uptau _0=\uptau (\lambda )=1.$ From (45), we have that $\uptau$ is decreasing in a neighborhood of $\lambda .$

Observe that $\uptau$ is positive in $(0,{\mathcal {R}}).$ Also, setting

$$\begin{aligned} \mu (s)=\left( \frac{\sin s}{\sin \lambda }\right) ^{n-r}\,, \end{aligned}$$

we have that $\mu >1$ on $(\lambda ,\pi /2).$ So, for $r>1$ and $s>\lambda ,$

$$\begin{aligned} \uptau (s)>\frac{1}{\mu (s)}\left( 1+\int _{\lambda }^{s}b(u)du\right) =\frac{1}{\mu (s)}\left( 1+b_r\int _{\lambda }^{s}\tan ^{r-1}(u)du\right) , \end{aligned}$$

which implies that $\uptau (s)\rightarrow +\infty$ as $s\rightarrow \pi /2.$

If $r=1,$ since $\sin \pi =0$ and $\int _{\lambda }^{\pi }\mu (s)ds$ is positive, we have that $\uptau (s)\rightarrow +\infty$ as $s\rightarrow \pi .$

It follows from the above considerations that there exists $\overline{\lambda }\in (0,{\mathcal {R}})$ such that

$$\begin{aligned} \uptau (\lambda )=\uptau ({\bar{\lambda }})=1 \,\,\, \text {and} \,\,\, \uptau '(\lambda )<0<\uptau '({\bar{\lambda }}). \end{aligned}$$

(46)

From this point on, the proof is entirely analogous to that of Theorem 2-(i). $\square$

A classification result for rotational ${{H}_{r}}$-hypersurfaces of $\mathbb{S}^{n}\times \mathbb{R}$ can be achieved as it was for their congeners in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$ To see this, assume that $\varSigma$ is a complete connected rotational ${{H}_{r}}(>0)$-hypersurface of $\mathbb{S}^{n}\times \mathbb{R}$ whose height function is Morse type. Assuming that $\varSigma$ is noncylindrical, we have, as before, that there exists an open set $\varSigma '\subset \varSigma$ which is an $(f_s,\phi )$-graph, $f_s\in {\mathscr {F}}.$ The corresponding $\uptau$-function, restricted to a maximal interval $(s_0,s_1),$ satisfies:

$$\begin{aligned} 0<\uptau |_{(s_0\,,s_1)}<1, \,\,\, 0\le s_0<s_1\le {\mathcal {R}}, \end{aligned}$$

which yields $\uptau (s_0)=0$ or $\uptau (s_0)=1.$

If $\uptau (s_0)=0,$ then $s_0=0.$ (Otherwise, the height function of $\varSigma$ would not be Morse type.) In this case, $\uptau$ coincides with the $\uptau$-function of the ${{H}_{r}}$-sphere of Theorem 4, and then, $\varSigma$ itself coincides with this sphere. (Notice that any of the spheres obtained in Theorem 4 is strictly convex on an open set.)

If $\uptau (s_0)=1,$ then $\uptau$ is decreasing in a neighborhood of $s_0$ . Thus, $r<n.$ In particular, $|a(s)|\rightarrow +\infty$ as $s\rightarrow 0,$ so that $s_0\in (0,\delta _{{{H}_{r}}}).$ Analogously, this gives that $\varSigma$ coincides with the ${{H}_{r}}$-annulus $\varSigma (\lambda )$ of Theorem 5, $\lambda =s_0.$

Summarizing, we have the following result.

Theorem 6

Let $\varSigma$ be a connected complete rotational ${{H}_{r}}(>0)$-hypersurface of $\mathbb{S}^{n}\times \mathbb{R}$ whose height function is Morse type. Then, $\varSigma$ is either a cylinder over a strictly convex geodesic sphere of  $\mathbb{S}^{n}$ or one of the embedded ${{H}_{r}}$-hypersurfaces of Theorems 4–5.

Remark 4

Regarding the hypothesis on the height function of $\varSigma$ in Theorem 6, we point out that a rotational embedded $H_1(>0)$-torus in $\mathbb{S}^{n}\times \mathbb{R}$ whose height function is non-Morse type was obtained in [25].

5 Rotational r-minimal hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ and $\mathbb{S}^{n}\times \mathbb{R} .$

In this section, we shall see that the method of $(f_s,\phi )$-graphs can be used for constructing and classifying rotational r-minimal hypersurfaces of  $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ and $\mathbb{S}^{n}\times \mathbb{R} .$ A major distinction from the case of ${{H}_{r}}(>0)$-hypersurfaces is that the tangency principle is no longer available.

Theorem 7

Given $r\in \{1,\dots ,n\},$ there exists a one-parameter family

$$\begin{aligned} {\mathscr {S}}=\{\varSigma (\lambda )\,;\, \lambda >0\} \end{aligned}$$

of complete rotational r-minimal n-annuli in  $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ with the following properties:

• (i) If $r=n,$ $\varSigma (\lambda )$ is a cylinder over a geodesic sphere of  $\mathbb{H}_{{\mathbb{F}}}^{m}$ of radius $\lambda >0.$

• (ii) If $r<n$, $\varSigma (\lambda )$ is catenoid type. More precisely, it is symmetric with respect to $P_0=\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\},$ and $P_0\cap \varSigma (\lambda )$ is the geodesic sphere of  $\mathbb{H}_{{\mathbb{F}}}^{m}$ of radius $\lambda$ centered at the point $o\in \mathbb{H}_{{\mathbb{F}}}^{m}$ of the axis. In addition, each of the parts of $\varSigma (\lambda )$ above and below $P_0$ is a rotational graph over $\mathbb{H}_{{\mathbb{F}}}^{m}-B_\lambda (o)$ (Fig. 8).

Furthermore, up to ambient isometries, any complete connected rotational r-minimal hypersurface of  $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ is either an element of ${\mathscr {S}}$ or a horizontal hyperplane.

Proof

Given $\lambda >0,$ is immediate that a cylinder over a geodesic sphere of $\mathbb{H}_{{\mathbb{F}}}^{m}$ of radius $\lambda$ is an n-minimal rotational annulus of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} ,$ which yields (i).

Assume that $1\le r<n$ and let ${\mathscr {F}}=\{f_s\,;\, s\in (0,+\infty )\}$ be the parallel family of geodesic spheres of $\mathbb{H}_{{\mathbb{F}}}^{m}$ centered at the axis point $o\in \mathbb{H}_{{\mathbb{F}}}^{m}.$ The ODE determined by ${\mathscr {F}}$ and ${{H}_{r}}=0$ is

$$\begin{aligned} y'=ay, \quad a(s)=-\frac{r|{{H}_{r}}^s|}{|{{H}_{r}}^{s-1}|}\,, \,\,\, s\in (0,+\infty ). \end{aligned}$$

(47)

Since $a<0,$ given $\lambda >0,$ the function

$$\begin{aligned} \uptau _\lambda (s)=\exp \left( \int _{\lambda }^{s}a(u)du\right) , \,\,\, s\in [\lambda ,+\infty ), \end{aligned}$$

is clearly a solution of (47) which satisfies

$$\begin{aligned} 0<\uptau _\lambda (s)\le \uptau _\lambda (\lambda )=1 \,\,\, \forall s\in [\lambda ,+\infty ). \end{aligned}$$

In addition, $\uptau _\lambda '(\lambda )=a(\lambda )<0.$ So, setting $\varrho _\lambda =\uptau _\lambda ^{1/r},$ it follows from Lemma 2 that

$$\begin{aligned} \phi _\lambda (s):=\int _{\lambda }^{s}\frac{\varrho _\lambda (u)}{\sqrt{1-\varrho _\lambda ^2(u)}}du, \,\,\, s\in [\lambda ,+\infty ), \end{aligned}$$

is well defined. Therefore, by Lemma 1, the $(f_s,\phi _\lambda )$-graph $\varSigma '(\lambda )$ is an r-minimal hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$ Notice that $\varSigma '(\lambda )$ is a graph over $\mathbb{H}_{{\mathbb{F}}}^{m}-{\overline{B}_\lambda (o)}$ with boundary $\partial \varSigma '(\lambda )=S_\lambda (o)$ (Fig. 8).

Fig. 8

Half r-minimal catenoid $\varSigma '(\lambda ),$ on which all the trajectories of $\nabla \xi$ emanate from $\partial \varSigma '(\lambda )$ orthogonally

Also, since $\varrho _\lambda (\lambda )=1,$ the tangent spaces of $\varSigma '(\lambda )$ along $\partial \varSigma '(\lambda )$ are all vertical. Thus, considering the reflection $\varSigma ''(\lambda )$ of $\varSigma '(\lambda )$ with respect to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\},$ as before, we have that $\varSigma (\lambda ):=\mathrm{closure}\,(\varSigma '(\lambda ))\cup \mathrm{closure}\,(\varSigma ''(\lambda ))$ is the desired r-minimal hypersurface.

Suppose now that $\varSigma$ is a complete connected rotational r-minimal hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$, $r\in \{1,\dots , n\},$ and set

$$\begin{aligned} \varSigma _0:=\{x\in \varSigma \,;\, \varTheta (x)\nabla \xi (x)\ne 0\}. \end{aligned}$$

Notice that $\varSigma$ is either a horizontal hyperplane or a cylinder if and only if $\varSigma _0=\emptyset .$ So, we can assume $\varSigma _0\ne \emptyset .$ We can also assume, without loss of generality, that $\varSigma$ and all elements of ${\mathscr {S}}$ share the same axis $\{o\}\times \mathbb{R} .$

As we argued in previous proofs, under the above hypotheses, there exists an $(f_s,\phi )$-graph $\varSigma '\subset \varSigma _0$ and a maximal open interval $(s_0,s_1),$ $0\le s_0<s_1\le +\infty ,$ such that the $\uptau$-function of $\varSigma '$ satisfies $0<\uptau |_{(s_0,s_1)}<1.$

For $r=n,$ we have that $\uptau ,$ and so $\varrho$, is constant. Hence, up to a vertical translation, one has $\phi (s)=cs, \,\, s>0,$ for some constant $c>0.$ However, $\phi '(0)=c>0,$ which implies that the closure of $\varSigma '$ in $\varSigma$ meets the rotation axis nonorthogonally, i.e., $\varSigma$ is not smooth at $\partial \varSigma '$—a contradiction. So, $\varSigma _0=\emptyset$ if $r=n.$

For $r<n,$ we have that $\uptau$ is a solution of (47). In particular, $\uptau$ is decreasing, which implies that $\uptau (s_0)=1.$ As before, this yields $s_0>0.$ Thus, setting $s_0=\lambda ,$ we have $\uptau =\uptau _\lambda$, which implies that, up to a vertical translation, $\phi =\phi _\lambda$ and, then, $\varSigma '$ coincides with the half-catenoid $\varSigma '(\lambda ).$

We conclude from the above that $\varSigma _0$ is the union of open half-catenoids $\varSigma '(\lambda )$, where $\varSigma (\lambda )\in {\mathscr {S}}.$ In particular, the boundary $\partial \varSigma _0$ of $\varSigma _0$ in $\varSigma$ has no horizontal points.

Now we prove that $\varSigma _1:=\varSigma -\varSigma _0$ has empty interior in $\varSigma .$ Indeed, assuming otherwise, consider a nonempty maximum open set $U\subset \varSigma _1$ of $\varSigma$ whose boundary intersects $\partial \varSigma _0.$ Then, U is either horizontal or vertical. Since $\partial \varSigma _0$ has no horizontal points, U should be vertical and, so, part of a vertical rotational cylinder. However, rotational cylinders in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ are r-minimal if and only if $r=n.$

It follows from the above that $\varSigma _0$ is open and dense in $\varSigma .$ Clearly, the intersection of two distinct elements of ${\mathscr {S}}$ is always transversal. This, together with the connectedness of $\varSigma$ and the density of $\varSigma _0$  in $\varSigma ,$ gives that $\varSigma$ coincides with some $\varSigma (\lambda )\in {\mathscr {S}}$. This finishes the proof. $\square$

Theorem 8

Given $r\in \{1,\dots ,n\},$ there exist ${\mathcal {R}}_0>0$ and a one-parameter family

$$\begin{aligned} {\mathscr {S}}=\{\varSigma (\lambda )\,;\,0<\lambda <{\mathcal {R}}_0\} \end{aligned}$$

of complete rotational r-minimal n-annuli in  $\mathbb{S}^{n}\times \mathbb{R}$ with the following properties:

• (i) If $r=n,$ then ${\mathcal {R}}_0=\pi$ and $\varSigma (\lambda )$ is a cylinder over a geodesic sphere of  $\mathbb{S}^{n}$ of radius $\lambda .$

• (ii) If $r<n$, then ${\mathcal {R}}_0=\pi /2$ and $\varSigma (\lambda )$ is Delaunay type.

Furthermore, up to ambient isometries, any complete connected rotational r-minimal hypersurface of  $\mathbb{S}^{n}\times \mathbb{R}$ is either an element of ${\mathscr {S}}$ or a horizontal hyperplane.

Proof

Statement (i) is trivial. So, assume $r<n$ and let ${\mathscr {F}}=\{f_s\,;\, s\in (0,\pi )\}$ be the family of parallel geodesic spheres of $\mathbb{S}^{n}$ centered at some point $o\in \mathbb{S}^{n}.$ In this setting, the ODE determined by ${\mathscr {F}}$ and ${{H}_{r}}=0$ is

$$\begin{aligned} y'=ay, \quad a(s)=-(n-r)\cot s, \quad s\in (0,\pi ). \end{aligned}$$

(48)

Given $\lambda \in (0,\pi /2),$ the function

$$\begin{aligned} \uptau _\lambda (s)=\left( \frac{\sin \lambda }{\sin s}\right) ^{n-r}\,, \,\, s\in (0,\pi ), \end{aligned}$$

is easily seen to be the solution of (48) satisfying

$$\begin{aligned} 0<\uptau _\lambda |_{(\lambda ,\pi -\lambda )}<1=\uptau (\lambda )=\uptau (\pi -\lambda ). \end{aligned}$$

Henceforth, the reasoning in the proof of Theorem 5 applies and leads to the construction of the Delaunay-type r-minimal hypersurface $\varSigma (\lambda )$ as stated in (ii).

Now, suppose that $\varSigma$ is a complete connected rotational r-minimal hypersurface of $\mathbb{S}^{n}\times \mathbb{R} .$ Under this assumption, define

$$\begin{aligned} \varSigma _0:=\{x\in \varSigma \,;\, \varTheta (x)\nabla \xi (x)\ne 0\}. \end{aligned}$$

As in the preceding proof, $\varSigma _0=\emptyset$ if $r=n.$ Thus, in this case, $\varSigma$ is either a horizontal hyperplane or a cylinder over a geodesic sphere of $\mathbb{S}^{n}.$

Suppose that $r<n$ and that the axis of $\varSigma$ is $\{o\}\times \mathbb{R} .$ If $\varSigma _0\ne \emptyset ,$ then $\varSigma$ is neither a horizontal hyperplane nor a cylinder. In addition, there exists an $(f_s,\phi )$-graph $\varSigma '\subset \varSigma _0$ and a maximal interval $(s_0,s_1),$ $0\le s_0<s_1\le \pi$, such that the $\uptau$-function of $\varSigma '$ satisfies $0<\uptau |_{(s_0,s_1)}<1$. So, $\uptau (s_0)=0$ or $\uptau (s_0)=1.$

The formula of the general solution of the ODE (48) gives that $\uptau$ is positive, not defined at $s=0,\pi$, and bounded away from zero. In particular, $s_0\ne 0, s_1\ne \pi$ and $\uptau (s_0)=\uptau (s_1)=1,$ so that $\uptau$ is given by

$$\begin{aligned} \uptau (s)=\left( \frac{\sin s_0}{\sin s}\right) ^{n-r}\,, \,\, s_0\,, s\in (0,\pi ). \end{aligned}$$

It is clear from this last equality and the considerations preceding it that $s_0<\pi /2<s_1<\pi ,$ which implies that $\uptau$ coincides with $\uptau _\lambda$ , $\lambda =s_0$ . Therefore, $\varSigma '$ coincides with the “block” $\varSigma '(\lambda )$ that generates $\varSigma (\lambda ),$ so that $\varSigma _0$ is a union of open sets of elements of ${\mathscr {S}}.$ In particular, the boundary $\partial \varSigma _0$ of $\varSigma _0$ in $\varSigma$ has no horizontal points, and any of its vertical points lies on a geodesic sphere of a hyperplane of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ of radius different from $\pi /2$ (see Fig. 6).

It follows from the considerations in the last paragraph that $\varSigma _1:=\varSigma -\varSigma _0$ has empty interior in $\varSigma .$ Indeed, assume by contradiction that $U\subset \varSigma _1$ is a nonempty maximal open set of $\varSigma$ intersecting $\partial \varSigma _0.$ Then, U must be vertical, since $\partial \varSigma _0$ has no horizontal points. This gives that U is part of the totally geodesic cylinder $S_{\pi /2}\times \mathbb{R} .$ In this case, a boundary point of U in $\partial \varSigma _0$ is vertical and lies on a geodesic sphere of radius $\pi /2$ in a hyperplane of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} ,$ which is impossible, as we have shown.

We conclude from the above that $\varSigma _0$ is open and dense in $\varSigma .$ Since $\varSigma$ is connected and two distinct elements of ${\mathscr {S}}$ are never tangent, it follows that, for some $\lambda \in (0,\pi /2),$ $\varSigma$ coincides with $\varSigma (\lambda )\in {\mathscr {S}}.$ $\square$

Remark 5

The case ${\mathbb{F}}=\mathbb{R}$ of Theorem 7 was considered in [16], whereas Theorem 8 was proved in [26] for $r=1.$ Again, the methods employed in these works is different from ours and cannot be applied to general products $M\times \mathbb{R}$, since they all rely on the Euclidean and Lorentzian geometries of the underlying spaces of $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times \mathbb{R} .$

6 Translational ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$

Given a Hadamard manifold M,  recall that the Busemann function ${\mathfrak {b}}_\gamma$ of M corresponding to an arclength geodesic $\gamma :(-\infty ,+\infty )\rightarrow M$ is defined as

$$\begin{aligned} {\mathfrak {b}}_\gamma (p):=\lim _{s\rightarrow +\infty }(\mathrm{dist}_M(p,\gamma (s))-s), \,\,\, p\in M. \end{aligned}$$

The level sets $\mathscr {H}_s:={\mathfrak {b}}_\gamma ^{-1}(s)$ of a Busemann function ${\mathfrak {b}}_\gamma$ are called horospheres of M. In this setting, as is well known, $\{{\mathscr {H}}_s\,;\, s\in (-\infty , +\infty )\}$ is a family of parallel hypersurfaces which foliates M. Furthermore, any horosphere $\mathscr {H}_s$ is homeomorphic to $\mathbb{R} ^{n-1}$, and any geodesic of M which is asymptotic to $\gamma$—i.e., with the same point $p_\infty$ on the asymptotic boundary $M(\infty )$ of M—is orthogonal to each horosphere ${\mathscr {H}}_s$ . In this case, we say that the horospheres ${\mathscr {H}}_s$ are centered at $p_\infty$ .

Fig. 9

A “ball model” for a Hadamard manifold

Therefore, in what concerns its horospheres, a Hadamard manifold can be pictured just as the Poincaré ball model of hyperbolic space $\mathbb{H}^{n},$ where the horospheres centered at a point $p_\infty \in \mathbb{H}^{n}(\infty )$ are the Euclidean $(n-1)$-spheres in $\mathbb{H}^{n}$ which are tangent to $\mathbb{H}^{n}(\infty )$ at $p_\infty$ (Fig. 9).

In the real hyperbolic space $\mathbb{H}^{n},$ any horosphere is totally umbilical with constant principal curvatures equal to 1. As shown in [4, Proposition-(vi), pg. 88], any horosphere of $\mathbb{H}_\mathbb{F}^{m},$ ${\mathbb{F}}\ne \mathbb{R} ,$ has principal curvatures 1 and 1/2 with multiplicities 1 and $n-2,$ respectively. Therefore, any family ${\mathscr {F}}$ of parallel horospheres of  $\mathbb{H}_\mathbb{F}^{m}$ is isoparametric and its elements are pairwise congruent. In addition, for any integer $r\in \{1,\dots , n-1\},$ all horospheres of  $\mathbb{H}_\mathbb{F}^{m}$ have the same (positive) r-th mean curvature, which we denote by ${{H}_{r}}^0.$

Theorem 9

Let ${\mathscr {F}}:=\{{\mathscr {H}}_s\,;\, s\in (-\infty , +\infty )\}$ be a family of parallel horospheres in hyperbolic space $\mathbb{H}_{{\mathbb{F}}}^{m}.$ Then, for any even integer  $r\in \{2,\dots ,n-1\},$ and any constant ${{H}_{r}}\in (0,{{H}_{r}}^0),$ there exists two properly embedded, everywhere nonconvex ${{H}_{r}}$-hypersurfaces $\varSigma _1$ and $\varSigma _2$ in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} ,$ which are both homeomorphic to  $\mathbb{R} ^{n}.$ In addition, the following hold:

• (i) $\varSigma _1$ is a convex and nowhere strictly convex entire graph over $\mathbb{H}_{{\mathbb{F}}}^{m}$ with constant angle function and unbounded height function.

• (ii) $\varSigma _2$ is foliated by horospheres, is symmetric with respect to the horizontal hyperplane $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\},$ and has unbounded height function.

Furthermore, as ${{H}_{r}}\rightarrow {{H}_{r}}^0,$ both $\varSigma _1$ and $\varSigma _2$ converge to a ${{H}_{r}}^0$-cylinder over a horosphere of the family ${\mathscr {F}}.$

Proof

For each $s\in (-\infty ,\infty )$, consider the isometric immersion $f_s:\mathbb{R} ^{n-1}\rightarrow \mathbb{H}_{{\mathbb{F}}}^{m}$ such that $f_s(\mathbb{R} ^{n-1})={\mathscr {H}}_s$ . Since all the principal curvatures of $f_s$ are constant and independent of s,  the coefficients a e b of the ODE $y'=ay+b$ associated with this family are constants. Also, since r is even and $0<{{H}_{r}}<{{H}_{r}}^0$ , we have

$$\begin{aligned} b<0<a \quad \text {and}\quad 0<-\frac{b}{a}=\frac{{{H}_{r}}}{{{H}_{r}}^0}<1. \end{aligned}$$

(49)

Therefore, the constant function

$$\begin{aligned} \uptau (s)=-\frac{b}{a}\,, \,\,\, s\in (-\infty ,+\infty ), \end{aligned}$$

is a trivial solution of $y'=ay+b$ satisfying $0<\uptau <1.$ This, together with Lemma 1 and equalities (9) and (14), gives that the $(f_s,\phi )$-graph $\varSigma _1$ determined by $\varrho =\uptau ^{1/r}$ is entire, unbounded and constitutes an ${{H}_{r}}$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ with constant angle function. Also, from identities (12), all principal curvatures of $\varSigma _1$ are nonpositive, with $k_n=\varrho '=0,$ so that $\varSigma _1$ is convex and nowhere strictly convex. Finally, it follows from (49) that $\uptau =-b/a\rightarrow 1$ as ${{H}_{r}}\rightarrow {{H}_{r}}^0,$ which implies that the angle function of $\varSigma _1$ goes to 0 as ${{H}_{r}}\rightarrow {{H}_{r}}^0.$ Consequently, $\varSigma _1$ converges to a cylinder over a horosphere of ${\mathscr {F}}$ as ${{H}_{r}}\rightarrow {{H}_{r}}^0.$

Now, let us consider the following solution $\uptau :(-\infty , 0]\rightarrow \mathbb{R}$ of $y'=ay+b$:

$$\begin{aligned} \uptau (s)=e^{a(s-s_0)}-\frac{b}{a}\,, \,\, \, s_0=\log (1+b/a)^{-1/a}. \end{aligned}$$

(50)

It is easily seen that

$$\begin{aligned} 0<-\frac{b}{a}<\uptau (s)\le 1=\uptau (0) \,\,\, \forall s\in (-\infty , 0]. \end{aligned}$$

(51)

Fig. 10

A piece of the graph $\varSigma '$, on which all the trajectories of $\nabla \xi$ meet ${\mathscr {H}}_{0}\times \{0\}$ orthogonally

So, by Lemma 1, the $(f_s,\phi )$-graph $\varSigma '$ with $\varrho =\root r \of {\uptau }$ is an ${{H}_{r}}$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$ The function $\phi ,$ in this case, is given by

$$\begin{aligned} \phi (s):=-\int _{s}^{0}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du, \,\,\, s\in (-\infty , 0). \end{aligned}$$

Notice that, by (51), one has

$$\begin{aligned} \uptau '(s)=a\uptau (s)+b>a\frac{-b}{a}+b=0, \end{aligned}$$

so that $\uptau '(0)>0.$ Thus, by Lemma 2, $\phi$ is well defined. Also, $\phi$ is negative on $(-\infty , 0)$ and is unbounded. Indeed, for all $s\in (-\infty , 0),$

$$\begin{aligned} -\phi (s) = \int _{s}^{0}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du \ge \int _{s}^{0}\varrho (u)du \ge -\inf \varrho |_{[s,0]}s=-s\varrho (s)\,, \end{aligned}$$

which implies that $\phi$ is unbounded, since $\varrho (s)\rightarrow (-b/a)^{1/r}>0$ as $s\rightarrow -\infty .$

Denoting by $B_0$ the horoball of $\mathbb{H}_{{\mathbb{F}}}^{m}$ with boundary ${\mathscr {H}}_0$ , it follows from the above considerations that $\varSigma '$ is an ${{H}_{r}}$-graph over $\mathbb{H}_{{\mathbb{F}}}^{m}-B_0$ which is unbounded and has boundary $\partial \varSigma '={\mathscr {H}}_0\times \{0\}$ (Fig. 10). In particular, $\varSigma '$ is homeomorphic to $\mathbb{R} ^{n}$ and, from the identities (12), $\varSigma '$ is everywhere nonconvex.

Since $\varrho (0)=1,$ as in the previous theorems, we have that any trajectory of $\nabla \xi$ on $\varSigma '$ meets $\partial \varSigma '$ orthogonally. Consequently, setting $\varSigma ''$ for the reflection of $\varSigma '$ with respect to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\}$ and defining

$$\begin{aligned} \varSigma _2:=\mathrm{closure}\,(\varSigma ')\cup \mathrm{closure}\,(\varSigma ''), \end{aligned}$$

we have that $\varSigma$ is a properly embedded ${{H}_{r}}$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which is foliated by horospheres and is homeomorphic to $\mathbb{R} ^{n}$ (Fig. 11).

To finish the proof, let us just observe that, from (49) and (50), one has

$$\begin{aligned} \lim _{{{H}_{r}}\rightarrow {{H}_{r}}^0}\uptau (s)=1 \,\,\, \forall s\in (-\infty ,0]. \end{aligned}$$

Therefore, just as the graph $\varSigma _1,$ $\varSigma _2$ converges to a cylinder over a horosphere of ${\mathscr {F}}$ as ${{H}_{r}}\rightarrow {{H}_{r}}^0.$ $\square$

Fig. 11

A piece of a properly embedded everywhere nonconvex ${{H}_{r}}(> 0)$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which is foliated by horospheres

Our next result establishes that the conditions on the parity of r and on the sign of ${{H}_{r}}-{{H}_{r}}^0$ in Theorem 9 are, in fact, necessary.

Theorem 10

Let ${\mathscr {F}}$ be a family of parallel horospheres in $\mathbb{H}_{{\mathbb{F}}}^{m}.$ Assume that, for some $r\in \{1,\dots , n\},$ $\varSigma$ is a complete connected ${{H}_{r}}(>0)$-hypersurface of  $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ with no horizontal points and that each connected component of any horizontal section $\varSigma _t\subset \varSigma$ is a (vertically translated) horosphere of ${\mathscr {F}}.$ Under these conditions, one has $r<n.$ Assume, in addition, that either of the following assertions holds:

• (i) r is even and ${{H}_{r}}\ge {{H}_{r}}^0.$

• (ii) r is odd.

Then, $\varSigma ={\mathscr {H}}_s\times \mathbb{R}$ for some ${\mathscr {H}}_s\in {\mathscr {F}}.$ In particular, ${{H}_{r}}={{H}_{r}}^0.$

Proof

Let $\varSigma _0\subset \varSigma$ be the open set of points $x\in \varSigma$ satisfying $\varTheta (x)\ne 0.$ Our aim is to prove that $\varSigma _0$ is empty. Assuming otherwise, choose $x_0\in \varSigma _0$ . Since $\varSigma$ has no horizontal points, we can suppose (after possibly a reflection about a horizontal hyperplane) that there is an open neighborhood $\varSigma '\subset \varSigma _0$ of $x_0$ which is an $(f_s,\phi )$-graph, $f_s\in {\mathscr {F}}.$

The $\uptau$-function associated with $\varSigma '$ satisfies $\uptau '=a\uptau +b,$ where a and $b\ne 0$ are the (constant) functions (16) determined by ${\mathscr {F}}$ and ${{H}_{r}}$ . Also, there is a maximal interval $I=(s_1, s_2)\subset \mathbb{R} ,$ $-\infty \le s_1<s_2\le +\infty$, such that $\uptau (I)\subset (0,1).$

Let us suppose that $r=n.$ In this case, we have $a=0,$ which gives $\uptau '(s)=b\ne 0,$ that is, $\uptau (s)=bs+c,$ $c\in \mathbb{R} .$ In particular, $s_1>-\infty$ and $s_2<+\infty$, and $\uptau$ is increasing (if $b>0$), or decreasing (if $b<0$) in $(s_1,s_2).$ So, $\uptau$ vanishes at $s_1$ or at $s_2$ . Assuming the former, we have that $\phi$ is defined at $s_1$ and $\phi '(s_1)=0.$ Thus, for any $p\in \mathbb{R} ^{n-1},$ the point $x=(f_{s_1}(p),\phi (s_1))\in \varSigma '$ is horizontal, contrary to our assumption. Therefore, if $r=n,$ then $\varSigma _0=\emptyset ,$ which implies that $\varSigma ={\mathscr {H}}_s\times \mathbb{R}$ for some $s\in \mathbb{R} .$ But this contradicts the assumed positiveness of $H_n$ . Hence, we must have $r<n.$

Let us assume now that (i) holds. Then, we have $b<0<a.$ Also, on $(s_1,s_2),$

$$\begin{aligned} \uptau '=a\uptau +b<a+b=\frac{r({{H}_{r}}^0-{{H}_{r}})}{H_{r-1}^0}\le 0 \quad \text {and}\quad \uptau ''=a\uptau '+b<0, \end{aligned}$$

that is, $\uptau$ is decreasing and concave in $(s_1,s_2),$ which clearly implies that $s_2<+\infty$ and $\uptau (s_2)=0.$ As in the preceding paragraph, this leads to the existence of a horizontal point of $\varSigma .$ Therefore, $\varSigma _0=\emptyset$ if (i) holds, which implies that $\varSigma ={\mathscr {H}}_s\times \mathbb{R}$ for some $s\in \mathbb{R} .$

Finally, let us assume that (ii) holds. In this case, one has $a, b>0,$ which gives that $\uptau$ is increasing and convex. From this point, we get easily to the conclusion by reasoning just as in the last paragraph. $\square$

An isometry $\varphi$ of $\mathbb{H}_{{\mathbb{F}}}^{m}$ which fixes only one point $p_\infty \in \mathbb{H}_{{\mathbb{F}}}^{m}(\infty )$ is called parabolic. Such isometries have the following fundamental property: The horospheres of $\mathbb{H}_{{\mathbb{F}}}^{m}$ centered at $p_\infty \in \mathbb{H}_{{\mathbb{F}}}^{m}(\infty )$ are invariant by parabolic isometries of $\mathbb{H}_{{\mathbb{F}}}^{m}$ that fix $p_\infty$ (cf. [14, Proposition 7.8]).

We point out that any isometry $\varphi$ of $\mathbb{H}_{{\mathbb{F}}}^{m}$ has a natural extension to an isometry $\Phi$ of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$ Namely,

$$\begin{aligned} \Phi (p,t)=(\varphi (p),t), \,\,\, (p,t)\in \mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} . \end{aligned}$$

We call $\Phi$ parabolic if $\varphi$ is parabolic. More specifically, if

$$\begin{aligned} {\mathscr {F}}=\{f_s:\mathbb{R} ^{n-1}\rightarrow \mathbb{H}_{{\mathbb{F}}}^{m}\,;\, s\in (-\infty ,+\infty )\}, \,\,\, f_s(\mathbb{R} ^{n-1})={\mathscr {H}}_s\,, \end{aligned}$$

is the family of parallel horospheres which are invariant by $\varphi ,$ we say that $\varphi$ and $\Phi$ are ${\mathscr {F}}$-parabolic isometries.

In the upper half-space model of $\mathbb{H}^{n}=\mathbb{H}_\mathbb{R} ^{n}$, Euclidean horizontal translations in a fixed direction are parabolic. As for the other hyperbolic spaces, the parabolic isometries are more involved (see, e.g., [19]). Nevertheless, inspired by the real case, we say that parabolic isometries are translational.

Finally, let us remark that, given a family ${\mathscr {F}}$ of parallel horospheres in $\mathbb{H}_{{\mathbb{F}}}^{m},$ if a hypersurface $\varSigma$ of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ is invariant by ${\mathscr {F}}$-parabolic isometries of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$, then any connected component of any horizontal section $\varSigma _t\subset \varSigma$ is contained in a (vertically translated) horosphere of ${\mathscr {F}}.$

Now, we are in position to classify all complete connected ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ with no horizontal points which are invariant by parabolic isometries.

Theorem 11

Let ${\mathscr {F}}$ be a family of parallel horospheres of $\mathbb{H}_{{\mathbb{F}}}^{m}.$ Assume that $\varSigma$ is a complete connected ${{H}_{r}}(>0)$-hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$, $r\in \{1,\dots , n\},$ with no horizontal points, which is invariant by ${\mathscr {F}}$-parabolic isometries. Then, up to ambient isometries, $\varSigma$ is either a cylinder over a horosphere of  $\mathbb{H}_{{\mathbb{F}}}^{m}$ or one of the embedded ${{H}_{r}}$-hypersurfaces obtained in Theorem 9.

Proof

Assume that $\varSigma$ is not a cylinder over a horosphere of $\mathbb{H}_{{\mathbb{F}}}^{m}.$ By Theorem 10, we have that $r(<n)$ is even and $0<{{H}_{r}}<{{H}_{r}}^0.$

In this case, the open set

$$\begin{aligned} \varSigma _0:=\{x\in \varSigma \,;\, \varTheta (x)\ne 0\} \end{aligned}$$

is dense in $\varSigma .$ Otherwise, there would be a vertical open set $U\subset \varSigma$ which would be necessarily contained in a cylinder over a horosphere of ${\mathscr {F}}.$ But such a cylinder has constant $r(<n)$th mean curvature ${{H}_{r}}^0>{{H}_{r}}.$ Hence, the open set U cannot exist.

Given $x_0\in \varSigma _0\,,$ there exists an $(f_s,\phi )$-graph $\varSigma '\ni x_0$ in $\varSigma _0$ with $f_s\in {\mathscr {F}}.$ As we know, its associated $\uptau$-function is a solution of the ODE $y'=ay+b$ determined by ${\mathscr {F}}$ and ${{H}_{r}},$ which satisfies $0<\uptau <1$ when restricted to a maximal interval $(s_0,s_1),$ $-\infty \le s_0<s_1\le +\infty .$

It is easily seen that $\varSigma _0,$ and so $\varSigma ,$ coincides with the entire graph $\varSigma _1$ of Theorem 9 if $\uptau$ is the constant solution. Hence, we can assume that $\uptau$ is nonconstant. Then, the conditions on the parity of r and the sign of ${{H}_{r}}-{{H}_{r}}^0$, as in the proof of Theorem 9, give that $\uptau$ is increasing and convex, which implies that $s_1<+\infty$ and that $\uptau (s_1)=1.$

Consider now the solution (50) of $y'=ay+b$ and denote it by ${\tilde{\uptau }}.$ Since ${\tilde{\uptau }}(0)=\uptau (s_1)=1,$ and the coefficients a and b are constants, by the uniqueness of solutions satisfying initial conditions, we have that $\uptau (s)={\tilde{\uptau }}(s-s_1).$ This, together with the homogeneity of the horospheres of $\mathbb{H}_{{\mathbb{F}}}^{m},$ implies that $\varSigma '$ coincides with the $(f_s,\phi )$-graph determined by ${\tilde{\uptau }}.$ From this fact and the density of $\varSigma _0$ in $\varSigma ,$ we conclude that $\varSigma$ coincides with the embedded ${{H}_{r}}$-hypersurface $\varSigma _2$ obtained in Theorem 9, as we wished to prove. $\square$

Given a totally geodesic hyperplane ${\mathscr {E}}_0$ of $\mathbb{H}^{n},$ let us recall that there exists a family ${\mathscr {F}}:=\{{\mathscr {E}}_s\,;\, s\in (-\infty , +\infty )\}$ of parallel hypersurfaces of $\mathbb{H}^{n}$ such that the distance of any point of ${\mathscr {E}}_s$ to ${\mathscr {E}}_0$ is |s|. The family ${\mathscr {F}}$ foliates $\mathbb{H}^{n},$ and each element ${\mathscr {E}}_s$ of ${\mathscr {F}}$, which is called an equidistant hypersurface, is properly embedded and homeomorphic to $\mathbb{R} ^{n-1}$ (Fig. 12).

Fig. 12

Equidistant hypersurfaces in the Poincaré ball model of $\mathbb{H}^{n}$

We shall also write ${\mathscr {F}}$ as a family of immersions:

$$\begin{aligned} {\mathscr {F}}=\{f_s:\mathbb{R} ^{n-1}\rightarrow \mathbb{H}^{n}\,;\, s\in (-\infty ,+\infty )\}, \end{aligned}$$

(52)

that is, for each $s\in (-\infty ,+\infty ),$ $f_s(\mathbb{R} ^{n-1})$ is the equidistant ${\mathscr {E}}_s$ to ${\mathscr {E}}_0=f_0(\mathbb{R} ^{n-1})$ .

Given a geodesic $\gamma _p$ orthogonal to the elements of ${\mathscr {F}},$ $p\in {\mathscr {E}}_0,$ any equidistant hypersurface ${\mathscr {E}}_s$ is totally umbilical with constant principal curvatures all equal to

$$\begin{aligned} k^s=-\tanh (s) \end{aligned}$$

with respect to the unit normal $\eta _s=\gamma _p'$ (see Sect. 3). In particular, ${\mathscr {F}}$ is isoparametric. Also, given a constant ${{H}_{r}}$ , the coefficients a and b of the differential equation $y'=ay+b$ associated with ${\mathscr {F}}$ and ${{H}_{r}}$ are:

$$\begin{aligned} a(s)=-(n-r)\tanh (s)\quad \text {and}\quad b(s)=b_r\tanh ^{1-r}(s), \,\,\, b_r=r{{H}_{r}}{{n-1}\atopwithdelims (){r-1}}^{-1}. \end{aligned}$$

(53)

It will be convenient to reconsider the constant

$$\begin{aligned} C_r:=\frac{n-r}{n}{{n}\atopwithdelims (){r}} \end{aligned}$$

and recall that, for $1\le r<n,$ the following identity holds:

$$\begin{aligned} \frac{b_r}{n-r}=\frac{{{H}_{r}}}{C_r}\,\cdot \end{aligned}$$

(54)

Our next result establishes that, for ${{H}_{r}}\in (0, C_r)$, $1\le r<n,$ there exists a one-parameter family of properly embedded ${{H}_{r}}$-hypersurfaces in $\mathbb{H}^{n}\times \mathbb{R}$ which are foliated by (vertical translations of) parallel equidistant hypersurfaces of $\mathbb{H}^{n}.$ In this setting, we have $0<{{H}_{r}}/C_r<1,$ so we can define:

$$\begin{aligned} s_r:=\mathrm{arctanh}\,({{H}_{r}}/C_r)^{1/r}. \end{aligned}$$

(55)

Notice that, for $1\le r<n,$ one has $C_r={{n-1}\atopwithdelims (){r}}.$ This, together with (55), yields

$$\begin{aligned} |{{H}_{r}}^{s_r}|={{H}_{r}} \end{aligned}$$

(56)

Theorem 12

For any given integer $r\in \{1,\dots ,n-1\}$ and any constant ${{H}_{r}}\in (0,C_r),$ there exists a one-parameter family

$$\begin{aligned} {\mathscr {S}}:=\{\varSigma (\lambda )\,;\, \lambda \in (s_r,+\infty )\} \end{aligned}$$

of properly embedded and everywhere nonconvex ${{H}_{r}}$-hypersurfaces of  $\mathbb{H}^{n}\times \mathbb{R} .$ Each element $\varSigma (\lambda )$ of ${\mathscr {S}}$ is homeomorphic to $\mathbb{R} ^{n}$ and is foliated by equidistant hypersurfaces. Moreover, $\varSigma (\lambda )$ is symmetric with respect to $\mathbb{H}^{n}\times \{0\}$, and its height function is unbounded.

Proof

Let ${\mathscr {F}}$ be the family of parallel equidistant hypersurfaces of $\mathbb{H}^{n}$ as in (52). Given $\lambda \in (s_r,+\infty ),$ let $\uptau _\lambda$ be the solution of $y'=ay+b$ satisfying $y(\lambda )=1,$ where a and b are the functions in (53). From (54) and the definition of $s_r,$ one has

$$\begin{aligned} \tanh ^r(\lambda )>\frac{b_r}{n-r}\,, \end{aligned}$$

(57)

which implies that

$$\begin{aligned} \uptau _\lambda '(\lambda )=-(n-r)\tanh (\lambda )+b_r\tanh ^{1-r}(\lambda )<0, \end{aligned}$$

i.e., $\uptau _\lambda$ is decreasing near $\lambda .$

We claim that $\uptau _\lambda$ is decreasing on the whole interval $[\lambda ,+\infty )$. To show that, it suffices to prove that $\uptau _\lambda$ has no critical points in $(\lambda ,+\infty ).$ Assuming otherwise, consider $s_1>\lambda$ satisfying $\uptau _\lambda '(s_1)=0.$ Since $\uptau _\lambda '(s_1)=a(s_1)\uptau _\lambda (s_1)+b(s_1),$ we have that $\uptau _\lambda (s_1)=-b(s_1)/a(s_1)>0.$ We also have $a'<0$ and $b'\le 0.$ Thus,

$$\begin{aligned} \uptau _\lambda ''(s_1)=a'(s_1)\uptau _\lambda (s_1)+b'(s_1)<0, \end{aligned}$$

which implies that $s_1$ is necessarily a local maximum for $\uptau _\lambda .$ This proves the claim, for $\uptau _\lambda$ is decreasing near $\lambda ,$ so that a local maximum $s_1>\lambda$ for $\uptau _\lambda$ should be preceded by a local minimum.

We also have that $\uptau _\lambda$ is positive in $[\lambda ,+\infty ).$ Indeed, if we had $\uptau _\lambda (s)\le 0$ for some $s>\lambda ,$ it would give $\uptau _\lambda '(s)=a(s)\uptau (s)+b(s)>0,$ and then $\uptau _\lambda$ would be increasing near s.

It follows from the above considerations that

$$\begin{aligned} 0<\uptau _\lambda (s)\le 1=\uptau _\lambda (\lambda ) \,\,\, \forall s\in [\lambda ,+\infty ). \end{aligned}$$

Furthermore, since $\uptau _\lambda$ is decreasing and positive, one has $\uptau _\lambda '(s)\rightarrow 0$ as $s\rightarrow +\infty .$ This, together with the equalities $\uptau _\lambda '=a\uptau _\lambda +b$ and $\varrho _\lambda ^r=\uptau _\lambda$, gives

$$\begin{aligned} \lim _{s\rightarrow +\infty }\varrho _\lambda (s)=\left( {{{H}_{r}}}/{C_r}\right) ^{1/r}>0. \end{aligned}$$

(58)

Therefore, the $(f_s,\phi )$ graph $\varSigma '(\lambda )$ associated with $\varrho _\lambda$ (see Fig. 13) is an ${{H}_{r}}$-hypersurface of $\mathbb{H}^{n}\times \mathbb{R}$ whose $\phi$-function is

$$\begin{aligned} \phi _\lambda (s)=\int _{\lambda }^{s}\frac{\varrho _\lambda (u)}{\sqrt{1-\varrho _\lambda ^2(u)}}du, \,\,\, s\in [\lambda ,+\infty ). \end{aligned}$$

Fig. 13

A piece of the graph $\varSigma '$, on which all trajectories of $\nabla \xi$ emanate from ${\mathscr {E}}_{\lambda }\times \{0\}$ orthogonally

As in the preceding proofs, we obtain a properly embedded ${{H}_{r}}$-hypersurface $\varSigma (\lambda )$ of $\mathbb{H}^{n}\times \mathbb{R}$ by reflecting $\varSigma '(\lambda )$ with respect to $\mathbb{H}^{n}\times \{0\},$ since $\varrho _\lambda (\lambda )=1$ and $\varrho _\lambda '(\lambda )<0$. It is also clear from equalities (12) that, except for $k_n=\varrho '$, its principal curvatures $k_i$ are all positive, so that $\varSigma (\lambda )$ is nowhere convex.

Finally, considering (58) and the fact that $\varrho _\lambda$ is decreasing, we have

$$\begin{aligned} \phi _\lambda (s)=\int _{\lambda }^{s}\frac{\varrho _\lambda (u)}{\sqrt{1-\varrho _\lambda ^2(u)}}du\ge \int _{\lambda }^{s}\varrho _\lambda (u)du\ge \left( {{{H}_{r}}}/{C_r}\right) ^{1/r}(s-\lambda ), \end{aligned}$$

which clearly implies that the height function of $\varSigma (\lambda )$ is unbounded. $\square$

Let us see now that, for any ${{H}_{r}}\in (0, C_{r}),$ there exists an ${{H}_{r}}$-hypersurface $\varSigma$ in $\mathbb{H}^{n}\times \mathbb{R}$ which, as the ones in the above theorem, is foliated by equidistant hypersurfaces. However, $\varSigma$ is not symmetric with respect to any horizontal hyperplane. Instead, it is asymptotic to a half-cylinder over an equidistant hypersurface of $\mathbb{H}^{n}.$ The precise statement is as follows.

Theorem 13

Let ${\mathscr {F}}$ be the family of parallel equidistant hypersurfaces of  $\mathbb{H}^{n}$ as in (52). Given $r\in \{1,\dots ,n-1\}$ and ${{H}_{r}}\in (0, C_{r}),$ let $s_r>0$ be the constant defined in (55). Then, there exists a complete everywhere nonconvex ${{H}_{r}}$-hypersurface $\varSigma$ in $\mathbb{H}^{n}\times \mathbb{R}$ which is an $(f_s,\phi )$-graph, $s\in (s_r,+\infty ).$ Furthermore, the height function of $\varSigma$ is unbounded above and below, and $\varSigma$ is asymptotic to ${\mathscr {E}}_{s_r}\times (-\infty , 0).$

Proof

Let $\uptau$ be the solution of the differential equation $y'=ay+b$ associated with ${{H}_{r}}$ and ${\mathscr {F}}$ (i.e., with a and b as in (53)) which satisfies the initial condition $\uptau (s_r)=1.$

From its definition, we have that $s_r$ satisfies $\uptau '(s_r)=0.$ In addition,

$$\begin{aligned} \uptau ''(s_r)=a'(s_r)+b'(s_r)<0, \end{aligned}$$

so that $s_r$ is a local maximum of $\uptau .$ Reasoning as in the preceding proof, we get that $\uptau$, and so $\varrho =\uptau ^{1/r}$, is positive and decreasing in $(s_r,+\infty ).$ From this, we conclude analogously that $\varrho (s)\rightarrow ({{H}_{r}}/C_r)^{1/r}$ as $s\rightarrow +\infty .$

Now, for a fixed $s_0>s_r$ , define

$$\begin{aligned} \phi (s):=\int _{s_0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du, \,\,\, s\in (s_r,+\infty ), \end{aligned}$$

and let $\varSigma$ be the corresponding $(f_s,\phi )$-graph. As before, we have that $\varSigma$ is nowhere convex. Denoting by $\Omega _{s_r}$ the convex connected component of $\mathbb{H}^{n} -{\mathscr {E}}_{s_r},$ we also have that $\varSigma$ is a graph over $\mathbb{H}^{n}-\Omega _{s_r}$ (Fig. 14).

For $s>s_0$ , we have

$$\begin{aligned} \phi (s)=\int _{s_0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du\ge \int _{s_0}^{s}\varrho (u)du\ge ({{H}_{r}}/C_r)^{1/r}(s-s_0), \end{aligned}$$

which gives that $\phi$, and so the height function of $\varSigma ,$ is unbounded above.

Finally, given a constant $C>0,$ there exists ${\bar{s}}\in (s_r, s_0)$ such that

$$\begin{aligned} \frac{1}{\varrho '(s)}<-C \,\,\,\forall s\in (s_r,{\bar{s}}), \end{aligned}$$

for $\varrho '(s_r)=0.$ Also, since $\varrho (s_r)=1,$ we can choose ${\bar{s}}$ sufficiently close to $s_r$ in such a way that, for a fixed small $\delta >0,$ the inequalities

$$\begin{aligned} \arcsin \varrho ({\bar{s}})-\arcsin \varrho (s_0)>(\pi /2-\arcsin \varrho (s_0))-\delta>0 \quad \text {and}\quad \varrho ({\bar{s}})>1-\delta >0 \end{aligned}$$

hold. In this manner, for all $s\in (s_r,{\bar{s}})$, one has

$$\begin{aligned} \phi (s)= & {} \int _{s_0}^{s}\frac{\varrho '(u)\varrho (u)}{\varrho '(u)\sqrt{1-\varrho ^2(u)}}du \le -C\varrho ({\bar{s}})\int _{s_0}^{s}\frac{\varrho '(u)}{\sqrt{1-\varrho ^2(u)}}du \\= & {} -C\varrho ({\bar{s}})\int _{\varrho (s_0)}^{\varrho (s)}\frac{d\varrho }{\sqrt{1-\varrho ^2}}= -C\varrho ({\bar{s}})(\arcsin \varrho (s)-\arcsin \varrho (s_0))\\\le & {} -C\varrho ({\bar{s}})(\arcsin \varrho ({\bar{s}})-\arcsin \varrho (s_0)) \\\le & {} -C(1-\delta )((\pi /2-\arcsin \varrho (s_0))-\delta ), \end{aligned}$$

which implies that $\phi (s)\rightarrow -\infty$ as $s\rightarrow s_r$ , since $\delta$ is fixed and the positive constant C is arbitrary. Therefore, the height function of $\varSigma$ is unbounded below, and $\varSigma$ is asymptotic to ${\mathscr {E}}_{s_r}\times (-\infty , 0)$ in $\mathbb{H}^{n}\times (-\infty , 0),$ as we wished to prove. $\square$

Fig. 14

A piece of the $(f_s,\phi )$-graph $\varSigma$ which is above $\mathbb{H}^{n}\times \{0\}.$ As $s\rightarrow -\infty$, the trajectories of $\nabla \xi$ converge asymptotically to ${\mathscr {E}}_{s_r}\times (0,-\infty )$

A family ${\mathscr {F}}=\{{\mathscr {E}}_s\,;\, s\in (-\infty ,+\infty )$ of equidistant hypersurfaces in $\mathbb{H}^{n}$ determines a group of translational isometries which we shall call ${\mathscr {F}}$-hyperbolic. In the upper half-space model of $\mathbb{H}^{n},$ taking ${\mathscr {E}}_0$ as a Euclidean half vertical hyperplane orthogonal to $\partial _\infty \mathbb{H}^{n}$ through the “origin” o,  we have that the ${\mathscr {F}}$-hyperbolic isometries are the Euclidean homotheties from o. It should be noticed that the equidistant hypersurfaces of ${\mathscr {F}}$ are all invariant by ${\mathscr {F}}$-hyperbolic isometries.

The natural extension of an ${\mathscr {F}}$-hyperbolic isometry of $\mathbb{H}^{n}$ to $\mathbb{H}^{n}\times \mathbb{R}$ will also be called ${\mathscr {F}}$-hyperbolic. If $\varSigma$ is a hypersurface of $\mathbb{H}^{n}\times \mathbb{R}$ which is invariant by ${\mathscr {F}}$-hyperbolic isometries, it is clear that any connected component of any horizontal section $\varSigma _t$ of $\varSigma$ is contained in ${\mathscr {E}}_s\times \{t\}$ for some $s\in (-\infty ,+\infty ).$

Next, we classify ${{H}_{r}}(>0)$-hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R}$ (without horizontal points or totally geodesic horizontal sections) which are invariant by hyperbolic translations.

Theorem 14

Let ${\mathscr {F}}$ be the family of parallel equidistant hypersurfaces of $\mathbb{H}^{n}$ as in (52). Assume that, for some $r\in \{1,\dots , n\},$ $\varSigma$ is a complete connected ${{H}_{r}}(>0)$-hypersurface of  $\mathbb{H}^{n}\times \mathbb{R}$ which is invariant by ${\mathscr {F}}$-hyperbolic translations. Assume further that $\varSigma$ has no horizontal points and that no horizontal section $\varSigma _t$ of $\varSigma$ is totally geodesic in $\mathbb{H}^{n}\times \{t\}$ (i.e., $\varSigma _t\not \subset {\mathscr {E}}_0\times \{t\}).$ Under these conditions, the following assertions hold:

• (i) $r<n$.

• (ii) $0<{{H}_{r}}<C_r$ .

• (iii) $\varSigma$ is either the cylinder over the equidistant  ${\mathscr {E}}_{s_r}$ or, up to an ambient isometry, one of the embedded hypersurfaces obtained in Theorems 12–13.

Proof

Set $\varSigma _0:=\{x\in \varSigma \,;\, \varTheta (x)\ne 0\}$ and assume $\varSigma _0\ne \emptyset .$ Given $x_0\in \varSigma _0$ , as in previous proofs, we can assume there is an open set $\varSigma '\subset \varSigma _0$ which is an $(f_s,\phi )$-graph containing $x_0$ . Its $\uptau$ function satisfies $\uptau '=a\uptau +b$, where a and b are the functions given in (53). Also, $\uptau$ is defined in a maximal interval $(s_0\,, s_1)\subset \mathbb{R}$ such that $0<\uptau |_{(s_0,s_1)}<1.$ Since no horizontal section of $\varSigma$ is totally geodesic, we can assume $0<s_0<s_1\le +\infty .$

The maximality of $(s_0,s_1)$ gives that $\uptau (s_0)=0$ or $\uptau (s_0)=1.$ In the former case, we have $\uptau '(s_0)=b(s_0)>0.$ Then, $\phi (s_0)$ is well defined (by Lemma 2) and $\phi '(s_0)=0,$ so that $x=(f_{s_0}(p),\phi (s_0)),$ $p\in \mathbb{R} ^{n-1},$ is a horizontal point of $\varSigma$, contrary to our hypothesis. Then, we must have $\uptau (s_0)=1.$ In particular, near $s_0$ , $\uptau$ is decreasing in $(s_0,s_1),$ which implies that $r<n.$ Indeed, for $r=n,$ $\uptau '=b>0.$

Assume now that ${{H}_{r}}\ge C_r$, $r<n.$ Then, we have

$$\begin{aligned} \uptau '(s_0)= & {} a(s_0)+b(s_0)=-(n-r)\tanh (s_0)+b_r\tanh ^{1-r}(s_0)\\= & {} (n-r)(({{H}_{r}}/C_r)\tanh ^{1-r}(s_0)-\tanh (s_0))\\\ge & {} (n-r)(\tanh ^{1-r}(s_0)-\tanh (s_0))>0, \end{aligned}$$

which contradicts that $\uptau$ is decreasing near $s_0$ .

It follows from the above considerations that, if $\varSigma _0\ne \emptyset ,$ then $r<n$ and ${{H}_{r}}<C_r$ . Furthermore, a direct computation gives that $\uptau '(s_0)\le 0$ if and only if $s_0\ge s_r$ . If $s_0=\lambda >s_r$ , then $\uptau$ coincides with the function $\uptau _\lambda$ of the $(f_s,\phi )$-graph associated with the hypersurface $\varSigma (\lambda )$ of Theorem 12. From this, arguing as in preceding proofs, we conclude that $\varSigma =\varSigma (\lambda ).$ By the same token, if $s_0=s_r,$ then $\varSigma =\varSigma '$ is the complete graph obtained in Theorem 13.

Let us suppose now that $\varSigma _0=\emptyset .$ In this case, we must have $\varSigma ={\mathscr {E}}_s\times \mathbb{R} ,$ where ${\mathscr {E}}_s=f_s(\mathbb{R} ^{n-1})$ is an equidistant hypersurface with rth mean curvature ${{H}_{r}}^s={{H}_{r}}$ , so that $s=s_r$ (see (56)). Therefore, we have $r<n$ (since we are assuming ${{H}_{r}}>0$) and

$$\begin{aligned} {{H}_{r}}=|{{H}_{r}}^{s_r}|={{n-1}\atopwithdelims (){r}}\tanh ^r(s_r)=C_r \tanh ^r(s_r)<C_r\,, \end{aligned}$$

which concludes our proof. $\square$

7 Translational r-minimal Hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R} .$

In this section, we construct and classify r-minimal hypersurfaces in $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which are invariant by translational isometries. It will be convenient to consider first the case of hyperbolic isometries of $\mathbb{H}^{n}.$

Theorem 15

Let ${\mathscr {F}}=\{f_s\,;\, s\in (-\infty ,+\infty )\}$ be a family of parallel equidistant hypersurfaces to a totally geodesic hyperplane ${\mathscr {E}}_0=f_0(\mathbb{R} ^{n-1})$ of  $\mathbb{H}^{n}.$ Then, for each $r\in \{1, \dots ,n\},$ there exists a one-parameter family ${\mathscr {S}}=\{\varSigma (\lambda )\,;\, \lambda >0\}$ of properly embedded r-minimal hypersurfaces of  $\mathbb{H}^{n}\times \mathbb{R}$ which are all homeomorphic to $\mathbb{R} ^{n}$ and invariant by ${\mathscr {F}}$-hyperbolic translations. Each element $\varSigma (\lambda )\in {\mathscr {S}}$ has the following additional properties:

• (i) For $r=n,$ $\varSigma (\lambda )$ is a constant angle entire r-minimal graph over $\mathbb{H}^{n}$ whose height function is unbounded above and below.

For $r<n,$ we distinguish the following cases:

• (ii) $\lambda >1:$ $\varSigma (\lambda )$ is symmetric with respect to the horizontal hyperplane $P_0=\mathbb{H}^{n}\times \{0\}$ and is contained in a slab $\mathbb{H}^{n}\times (-\alpha ,\alpha ),$ $\alpha >0.$

• (iii) $\lambda =1:$ $\varSigma (\lambda )$ is an $(f_s,\phi )$-graph ($s>0$) which is bounded above, unbounded below, and asymptotic to ${\mathscr {E}}_0\times (0,-\infty ).$

• (iv) $\lambda <1:$ $\varSigma (\lambda )$ is an entire graph over $\mathbb{H}^{n}$ which is symmetric with respect to ${\mathscr {E}}_0$, and is contained is a slab $\mathbb{H}^{n}\times (-\alpha ,\alpha ),$ $\alpha >0.$

Furthermore, except for the cylinders ${\mathscr {E}}_s\times \mathbb{R} , \,s\ne 0$ (in the case $r=n$), and up to ambient isometries, the elements of ${\mathscr {S}}$ are the only complete nontotally geodesic r-minimal hypersurfaces of  $\mathbb{H}^{n}\times \mathbb{R}$ which are invariant by hyperbolic translations.

Proof

The Eq. (15) determined by ${\mathscr {F}}$ and ${{H}_{r}}=0$ is:

$$\begin{aligned} y'=a(s)y, \,\,\, a(s)=-(n-r)\tanh (s). \end{aligned}$$

(59)

For $r=n,$ its solution $\uptau$ is constant. So, given $\lambda >0,$ defining

$$\begin{aligned} \phi (s)= \lambda s, \,\, s\in (-\infty ,+\infty ), \end{aligned}$$

we have that the corresponding $(f_s,\phi )$-graph $\varSigma (\lambda )$ is an entire n-minimal graph whose level hypersurfaces are the leaves of ${\mathscr {F}}.$ Clearly, the height function of $\varSigma (\lambda )$ is unbounded above and below. Moreover, it follows from (9) that $\varSigma (\lambda )$ is a constant angle hypersurface. This proves (i).

Let us suppose now that $1\le r<n.$ Given $\lambda >0,$ set

$$\begin{aligned} \uptau _\lambda (s):=\lambda {\cosh ^{r-n}(s)}, \,\,\, s\in (-\infty , +\infty ). \end{aligned}$$

It is easily checked that $\uptau _\lambda$ is the solution of (59) satisfying $\uptau _\lambda (0)=\lambda .$

Assume that $\lambda >1.$ Then, defining $s_\lambda :=\mathrm{arccosh}\,(\lambda ^{1/(n-r)}),$ one has

$$\begin{aligned} 0<\uptau _\lambda (s)\le 1=\uptau _\lambda (s_\lambda ) \,\,\, \forall s\in [s_\lambda ,+\infty ). \end{aligned}$$

Hence, setting

$$\begin{aligned} \phi _\lambda (s):=\int _{s_\lambda }^{s}\frac{\varrho _\lambda (u)}{\sqrt{1-\varrho _\lambda ^2(u)}}du, \quad \varrho _\lambda =\uptau _\lambda ^{1/r}, \quad s\in (s_\lambda ,+\infty ), \end{aligned}$$

(60)

we have that the $(f_s,\phi _\lambda )$-graph $\varSigma '(\lambda )$ is a well-defined r-minimal hypersurface, for $\uptau '(s_\lambda )<0.$ Also, since $\uptau _\lambda (s_\lambda )=1,$ the closure of $\varSigma '(\lambda )$ intersects $P_0$ orthogonally. Thus, we obtain an r-minimal hypersurface $\varSigma (\lambda )$ by reflecting $\varSigma '(\lambda )$ about $P_0.$

As for the boundedness of $\phi _\lambda$, we first observe that, from the equalities $\uptau _\lambda =\varrho _\lambda ^r$ and $\uptau _\lambda '=a\uptau _\lambda ,$ we have $\varrho _\lambda =(r/a)\varrho _\lambda '.$ In addition, the function 1/a is bounded above by $-1/(n-r)$ in $(0,+\infty ).$ Hence,

$$\begin{aligned} \phi _\lambda (s)= & {} \int _{s_\lambda }^{s}\frac{r\varrho _\lambda '(u)}{a(u)\sqrt{1-\varrho _\lambda ^2(u)}}du \le -\frac{r}{n-r}\int _{\varrho _\lambda (s_\lambda )}^{\varrho _\lambda (s)}\frac{d\varrho _\lambda }{\sqrt{1-\varrho _\lambda ^2}}\nonumber \\= & {} \frac{r}{n-r}(\arcsin \varrho _\lambda (s_\lambda )-\arcsin \varrho _\lambda (s))\le \frac{\pi r}{2(n-r)}\,, \end{aligned}$$

(61)

which finishes the proof of (ii).

Assuming now $\lambda =1,$ let us fix $s_0>0$ and define

$$\begin{aligned} \phi (s)=\int _{s_0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du, \quad \varrho =\uptau _1^{1/r}, \quad s\in (0,+\infty ). \end{aligned}$$

Since $\uptau '(0)=0,$ we can mimic the final part of the proof of Theorem 13 and conclude that $\phi$ is unbounded below and that the corresponding $(f_s,\phi )$-graph $\varSigma$ is asymptotic to ${\mathscr {E}}_0\times (-\infty , 0).$ Also, proceeding as in (61), we can show that $\phi$ is bounded above. This proves (iii).

Given $0<\lambda <1,$ we have that $0<\uptau _\lambda <1$ in $(0,+\infty ).$ So, we can define $\phi _\lambda$ as in (60), replacing $s_\lambda$ by 0. Analogously, we have that $\phi _\lambda$ is bounded above and that the boundary of the $(f_s,\phi _\lambda )$-graph $\varSigma '(\lambda )$ is ${\mathscr {E}}_0\times \{0\}\subset P_0$ .

Notice that $\phi _\lambda '(0)=\varrho _\lambda (0)/\sqrt{1-\varrho _\lambda ^2(0)}$ is well defined and positive, since $\varrho _\lambda (0)$ is neither 0 nor 1. Thus, we obtain a complete properly embedded r-minimal hypersurface $\varSigma (\lambda )$ from $\varSigma '(\lambda )$ by reflecting it with respect to $P_0$ , and then with respect to the totally geodesic vertical hyperplane ${\mathscr {E}}_0\times \mathbb{R}$ (Fig. 15). This shows (iv).

Assume now that $\varSigma$ is a complete nontotally geodesic r-minimal hypersurface of $\mathbb{H}^{n}\times \mathbb{R}$ which is invariant by ${\mathscr {F}}$-hyperbolic translations. Set

$$\begin{aligned} \varSigma _0:=\{x\in \varSigma \,;\, \varTheta (x)\nabla \xi (x)\ne 0\} \end{aligned}$$

and suppose that $1\le r<n.$ Then, $\varSigma _0\ne \emptyset .$ Otherwise, $\varSigma$ would be either a horizontal hyperplane or a cylinder over the hyperplane ${\mathscr {E}}_0$ of $\mathbb{H}^{n}.$ In both cases, $\varSigma$ would be totally geodesic, which is contrary to our assumption.

Therefore, if $1\le r<n,$ for each $x_0\in \varSigma _0$ , there is an $(f_s,\phi )$-graph $\varSigma '\subset \varSigma _0$ which contains $x_0$ , and whose $\uptau$-function is a solution of (59). More precisely, for some $c>0,$ this solution is given by

$$\begin{aligned} \uptau (s)=c\left( \frac{\cosh s_0}{\cosh s} \right) ^{n-r}, \,\, s_0\,, s\in (-\infty , +\infty ). \end{aligned}$$

Now, recall that in the cases (ii)–(iv) above, the corresponding function $\uptau _\lambda$ satisfies $\uptau _\lambda (0)=\lambda .$ Thus, for $\lambda :=\uptau (0)=c\cosh ^{n-r}(s_0),$ the function $\uptau$ coincides with $\uptau _\lambda ,$ which implies that $\varSigma '\subset \varSigma (\lambda ).$ In addition, no $\varSigma (\lambda )\in {\mathscr {S}}$ has horizontal or totally geodesic points, which gives that $\varSigma _0$ is open and dense in $\varSigma .$ Therefore, $\varSigma$ coincides with $\varSigma (\lambda ).$

Finally, let us suppose that $r=n.$ If $\varSigma _0=\emptyset ,$ then $\varSigma ={\mathscr {E}}_s$ for some $s\ne 0.$ If $\varSigma _0\ne \emptyset ,$ then there exists an $(f_s,\phi )$-graph $\varSigma '\subset \varSigma _0$ whose $\uptau$-function is constant. In particular, up to a vertical translation, we have $\phi (s)=\lambda s$ for some $\lambda >0,$ so that $\varSigma '=\varSigma$ is the entire graph $\varSigma (\lambda )$ given in (i). $\square$

Fig. 15

Half of the $(f_s,\phi )$-graph $\varSigma '(\lambda )$ (above $\mathbb{H}^{n}\times \{0\}$) and half of its reflection with respect to ${\mathscr {E}}_0$ (below $\mathbb{H}^{n}\times \{0\}$)

Remark 6

It should be mentioned that, with an approach different from ours, the particular case $r=1$ of Theorem 15 was considered in [3].

Next, we obtain all complete nontotally geodesic r-minimal hypersurfaces of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which are invariant by parabolic isometries.

Theorem 16

Let ${\mathscr {F}}=\{f_s\,;\, s\in (-\infty ,+\infty )\}$ be a family of parallel horospheres in $\mathbb{H}_{{\mathbb{F}}}^{m}.$ Then, for any $r\in \{1,\dots ,n\},$ there exists a properly embedded r-minimal hypersurface $\varSigma$ of  $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ which is invariant by ${\mathscr {F}}$-parabolic isometries. In addition, $\varSigma$ is homeomorphic to $\mathbb{R} ^{n}$ and has the following properties:

• (i) For $r=n,$ $\varSigma$ is a constant angle entire graph over $\mathbb{H}_{{\mathbb{F}}}^{m}$ whose height function is unbounded above and below.

• (ii) For $r<n,$ $\varSigma$ is symmetric with respect to $\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\}$ and is contained in a slab $\mathbb{H}_{{\mathbb{F}}}^{m}\times (-\alpha ,\alpha ).$

Furthermore, except for the cylinders ${\mathscr {H}}_s\times \mathbb{R}$ (in the case $r=n$), and up to ambient isometries, $\varSigma$ is the only complete nontotally geodesic r-minimal hypersurface of  $\mathbb{H}^{n}\times \mathbb{R}$ which is invariant by parabolic isometries.

Proof

The proof of the existence of $\varSigma$ as in (i) is analogous to the one given in the preceding theorem. So, let us assume $r<n.$ In this case, the Eq. (15) determined by ${\mathscr {F}}$ and ${{H}_{r}}=0$ takes the form

$$\begin{aligned} y'=ay, \,\,\, a=\frac{r{{H}_{r}}^0}{H_{r-1}^0}>0\,, \end{aligned}$$

(62)

and its positive solutions are

$$\begin{aligned} \uptau _\lambda (s)=\lambda e^{as}, \, \,\,\, \lambda >0, \,\, s\in (-\infty ,+\infty ). \end{aligned}$$

(63)

It is easily checked that, since the horospheres of $\mathbb{H}_{{\mathbb{F}}}^{m}$ are pairwise congruent, an $(f_s,\phi )$ graph with $\uptau$-function $\uptau _\lambda$ does not depend on $\lambda .$ More precisely, two such graphs obtained from functions $\uptau _{\lambda _1}$ and $\uptau _{\lambda _2}$ , $\lambda _1\ne \lambda _2,$ are isometric. Therefore, we can assume $\lambda =1$ and set $\uptau :=\uptau _1$ . Then, we have

$$\begin{aligned} 0<\uptau (s)< 1=\uptau (0) \,\,\, \forall s\in (-\infty , 0). \end{aligned}$$

Since $\uptau '(0)=a\uptau (0)=a>0,$ writing $\varrho ^r=\uptau |_{(-\infty , 0)}$, we have

$$\begin{aligned} \phi (s)=\int _{0}^{s}\frac{\varrho (u)}{\sqrt{1-\varrho ^2(u)}}du, \,\,\, s\in (-\infty ,0), \end{aligned}$$

is well defined, and so is the corresponding $(f_s,\phi )$-graph $\varSigma '.$ Also, $\uptau (0)=1,$ so that the tangent spaces of $\varSigma '$ along its boundary are all vertical. Therefore, we obtain the stated r-minimal hypersurface $\varSigma$ by reflecting $\varSigma '$ about $P_0=\mathbb{H}_{{\mathbb{F}}}^{m}\times \{0\}.$

Observe that, for all $s\in (-\infty , 0),$ one has

$$\begin{aligned} -\phi (s)=\int _{s}^{0}\frac{e^{au/r}}{\sqrt{1-e^{2au/r}}}du=\frac{r}{a}(\pi /2-\arcsin (e^{as/r})). \end{aligned}$$

Hence, setting $\alpha :={\pi r}/{2a}>0,$ we have that $\phi (s)\rightarrow -\alpha$ as $s\rightarrow -\infty ,$ which proves that $\varSigma$ is contained in the slab $\mathbb{H}_{{\mathbb{F}}}^{m}\times (-\alpha ,\alpha ).$

As for the uniqueness of $\varSigma$, notice that the following hold:

• The $\uptau$-function of any r-minimal $(f_s,\phi )$-graph, $f_s\in {\mathscr {F}},$ is a positive solution of (62) (if $r<n$) or is a positive constant (if $r=n$).

• $\varSigma$ has no horizontal points.

• A vertical ${\mathscr {F}}$-invariant hypersurface of $\mathbb{H}_{{\mathbb{F}}}^{m}\times \mathbb{R}$ is r-minimal if and only if $r=n.$

• The graph in (i) has no vertical points.

These facts allow us to argue just as in preceding proofs, and then show the uniqueness of $\varSigma$ as asserted. $\square$

8 Uniqueness of Rotational ${{H}_{r}}$-spheres of $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$

In this concluding section, we concern the uniqueness of the rotational ${{H}_{r}}$-spheres we constructed in Sect. 4. We restrict ourselves to $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$, with $\epsilon \in \{-1,1\}$ and $n\ge 3.$ As we mentioned before, the case $n=2$ was considered in [1, 17].

We obtain a Jellett–Liebmann-type theorem by showing that a compact, connected and strictly convex ${{H}_{r}}$-hypersurface of $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ is a rotational embedded sphere (cf. Theorem 17). We also show the uniqueness of these spheres under completeness or properness assumptions, instead of compactness (cf. Theorem 18 and Corollary 2).

For the proof of Theorem 18, we make use of a height estimate for convex graphs in $M\times \mathbb{R}$ which we establish in the next proposition. First, we compute the Laplacian of both the height function $\xi$ and the angle function $\varTheta$ of an arbitrary hypersurface $\varSigma$ of a general product $M\times \mathbb{R} .$

Given a smooth function $\zeta$ on $\varSigma ,$ let us denote its Laplacian by $\Delta \zeta ,$ i.e.,

$$\begin{aligned} \Delta \zeta :=\mathrm{trace}(\mathrm{Hess}\,\zeta ). \end{aligned}$$

In particular, from Eq. (4), the Laplacian of $\xi$ is given by

$$\begin{aligned} \Delta \xi = \varTheta H, \,\,\, H=H_1\,. \end{aligned}$$

(64)

Recall that, for $X, Y\in T\varSigma ,$ the Codazzi Eq. reads as

$$\begin{aligned} (\overline{R}(X,Y)N)^\top = \left( \nabla _YA\right) X - \left( \nabla _XA\right) Y, \end{aligned}$$

where $\overline{R}$ is the curvature tensor of $M\times \mathbb{R} ,$ $\top$ denotes the tangent component of the tangent bundle $T\varSigma$ of $\varSigma ,$ and, by definition,

$$\begin{aligned} \left( \nabla _YA\right) X:=\nabla _YAX-A\nabla _YX. \end{aligned}$$

Observing that

$$\begin{aligned} \nabla _X\nabla \xi =\left( \overline{\nabla }_X\nabla \xi \right) ^\top = -\left( \overline{\nabla }_X\varTheta N\right) ^\top =\varTheta AX, \end{aligned}$$

we have from (3) that

$$\begin{aligned}-\nabla _X\nabla \varTheta = \nabla _X A\nabla \xi =\left( \nabla _XA\right) \nabla \xi +A\nabla _X\nabla \xi =\left( \nabla _XA\right) \nabla \xi +\varTheta A^2X,\end{aligned}$$

which yields

$$\begin{aligned} \nabla _X\nabla \varTheta =-(\overline{R}(\nabla \xi ,X)N)^\top - \left( \nabla _{\nabla \xi }A\right) X-\varTheta A^2X. \end{aligned}$$

(65)

Now, let us fix $x\in \varSigma$ and an orthonormal frame $\{X_1\,, \dots ,X_n\}$ in a neighborhood of x in $\varSigma ,$ which is geodesic at x,  that is

$$\begin{aligned}\nabla _{X_i}X_j\,(x)=0 \,\,\,\forall i, j=1,\dots ,n.\end{aligned}$$

Writing $\xi _j=X_j(\xi ),$ we have $\nabla \xi =\sum _j \xi _jX_j.$ Therefore,

$$\begin{aligned} \sum _{i=1}^{n}\langle \left( \nabla _{\nabla \xi }A\right) X_i\,,\,X_i\rangle= & {} \sum _{i=1}^{n}\left( \langle \nabla _{\nabla \xi }AX_i\,,\,X_i\rangle - \langle A\nabla _{\nabla \xi }X_i\,,\,X_i\rangle \right) \\= & {} \sum _{i,j=1}^{n}\xi _j(\langle \nabla _{X_j}AX_i\,,\,X_i\rangle -\langle A\nabla _{X_j}X_i\,,\,X_i\rangle ) \\= & {} \sum _{i,j=1}^{n}\xi _j(X_j\langle AX_i,X_i\rangle -\langle AX_i,\nabla _{X_j}X_i\rangle -\langle A\nabla _{X_j}X_i\,,\,X_i\rangle ) \\= & {} \langle \nabla \xi ,\nabla H \rangle -\sum _{i,j=1}^{n}\xi _j(\langle AX_i,\nabla _{X_j}X_i\rangle -\langle A\nabla _{X_j}X_i\,,\,X_i\rangle ), \end{aligned}$$

which implies that, at the chosen point $x\in \varSigma ,$

$$\begin{aligned} \sum _{i=1}^{n}\langle \left( \nabla _{\nabla \xi }A\right) X_i\,,\,X_i\rangle = \langle \nabla \xi ,\nabla H\rangle . \end{aligned}$$

Since x is arbitrary, we get from this last equality and (65) that, on $\varSigma ,$

$$\begin{aligned} \Delta \varTheta =\overline{\mathrm{Ric}}(\nabla \xi ,N)-\langle \nabla \xi ,\nabla H\rangle -\varTheta \Vert A\Vert ^2, \end{aligned}$$

(66)

where $\overline{\mathrm{Ric}}$ denotes the Ricci curvature tensor of $M\times \mathbb{R}$ and $\Vert A\Vert ^2:=\mathrm{trace}\,A^2.$

Remark 7

For the next results, except for Theorem 19, we order the principal curvatures of a hypersurface $\varSigma$ of $M\times \mathbb{R}$ as

$$\begin{aligned}k_1\le k_2\le \cdots \le k_{n-1}\le k_n\,.\end{aligned}$$

Proposition 3

Consider an arbitrary Riemannian manifold  M,  and let $\varSigma \subset M\times \mathbb{R}$ be a compact vertical graph of a nonnegative function defined on a domain $\Omega \subset M\times \{0\}.$ Assume $\varSigma$ strictly convex up to $\partial \varSigma \subset M\times \{0\}.$ Under these conditions, the following height estimate holds:

$$\begin{aligned} \xi (x)\le \frac{1}{\inf _\varSigma k_1} \,\,\, \forall x\in \varSigma . \end{aligned}$$

(67)

Proof

Consider in $\varSigma$ the “inward” orientation, so that its angle function $\varTheta$ is nonpositive. Choose $\delta >0$ satisfying $1/\delta <\inf _\varSigma k_1$ and define on $\varSigma$ the function

$$\begin{aligned} \varphi =\xi +\delta \varTheta . \end{aligned}$$

We claim that $\varphi$ has no interior maximum. Indeed, assuming otherwise, let $x\in \varSigma -\partial \varSigma$ be a maximum point of $\varphi$. In this case, from (3), we have

$$\begin{aligned} 0=\nabla \varphi (x)=\nabla \xi (x)+\delta \nabla \varTheta (x)=\nabla \xi (x)-\delta A\nabla \xi (x). \end{aligned}$$

Hence, if we had $\nabla \xi (x)\ne 0,$ then $1/\delta$ would be an eigenvalue of A at x,  which is impossible, by our choice of $\delta .$ Thus, x is a critical point of $\xi .$ Since $\varSigma$ is strictly convex, x is necessarily its highest point. In particular, $\varTheta (x)=-1.$ This, together with identities (64) and (66), gives that, at x,

$$\begin{aligned} 0\ge \Delta \varphi =-H+\delta \Vert A\Vert ^2. \end{aligned}$$

(68)

However, from our choice of $\delta$ and the strict convexity of $\varSigma$, we have

$$\begin{aligned} \frac{H}{\delta }<k_1H=k_1(k_1+\cdots +k_n)\le k_1^2+\cdots +k_n^2=\Vert A\Vert ^2, \end{aligned}$$

which contradicts (68). Therefore, $\varphi$ attains its maximum on $\partial \varSigma ,$ which implies that $\varphi \le 0$ on $\varSigma ,$ for $\varphi |_{\partial \varSigma }=\delta \varTheta \le 0.$ Hence,

$$\begin{aligned} \xi (x)\le -\delta \varTheta (x)\le \delta \,\,\, \forall x\in \varSigma . \end{aligned}$$

The result, then, follows from this last inequality, since it holds for any positive $\delta >1/ \inf _\varSigma k_1.$ $\square$

Remark 8

Proposition 3 has its own importance, since it establishes height estimates for vertical graphs in $M\times \mathbb{R}$ making no assumptions on M. In addition, no curvature of such a graph is assumed to be constant.

In the next two theorems, we apply the Alexandrov reflection technique. Since the arguments are standard, the proofs will be somewhat sketchy on this matter (see, e.g., [7, Theorems 4.2 and 5.1] and [24, Theorem 1.1]). We add that the proof of Theorem 17 is, essentially, the one for [8, Corollary 1], in which the case $r=1$ was considered.

Theorem 17

(Jellett–Liebmann-type theorem) Let $\varSigma$ be a compact connected strictly convex ${{H}_{r}}(>0)$-hypersurface of  $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ ($n\ge 3$). Then, $\varSigma$ is an embedded rotational ${{H}_{r}}$-sphere.

Proof

Since $\varSigma$ is compact, its height function $\xi$ has a maximal point x. This, together with the strict convexity of $\varSigma$, allows us to apply [8, Theorems 1 and 2] and conclude that $\varSigma$ is embedded and homeomorphic to $\mathbb{S}^{n}$. Thus, for $\epsilon =-1,$ the result follows from [15, Theorem 7.6], the Alexandrov-type theorem we mentioned in the introduction.

For $\epsilon =1,$ we can perform Alexandrov reflections on $\varSigma$ with respect to horizontal hyperplanes $P_t:=\mathbb{S}^{n}\times \{t\}$ coming down from above $\varSigma .$ For some $t_0<\xi (x),$ the reflection of the part of $\varSigma$ above $P_{t_0}$ will have a first contact with $\varSigma .$ Then, by the tangency principle $\varSigma$ is symmetric with respect to $P_{t_0}$. Therefore, assuming $t_0=0$ and identifying $\mathbb{S}^{n}\times \{0\}$ with $\mathbb{S}^{n},$ we conclude that $\varSigma$ is a bigraph over its projection $\pi (\varSigma )$ to $\mathbb{S}^{n}$. As a consequence, $\varSigma _0:=\varSigma \cap \mathbb{S}^{n}$ is the boundary of $\pi (\varSigma )$ in $\mathbb{S}^{n}$.

By [8, Lemma 1], the second fundamental form of $\varSigma _{0}$, as a hypersurface of  $\mathbb{S}^{n}$, is positive definite. In particular, $\varSigma _{0}$ is nontotally geodesic in  $\mathbb{S}^{n}$. Thus, by [11, Theorem 1], $\varSigma _{0}$ is contained in an open hemisphere $\mathbb{S}_+^{n}$ of $\mathbb{S}^{n}$, which implies that the same is true for $\pi (\varSigma ),$ that is, $\varSigma \subset \mathbb{S}_+^{n}\times \mathbb{R} .$ In this setting, we can apply Alexandrov reflections on “vertical hyperplanes” $(\mathbb{S}^{n-1}\cap \mathbb{S}_+^{n})\times \mathbb{R} ,$ where $\mathbb{S}^{n-1}\subset \mathbb{S}^{n}$ is a totally geodesic $(n-1)$-sphere of $\mathbb{S}^{n}$ , and conclude that $\varSigma$ is rotational. $\square$

Let us show now that, regarding Theorem 17, the compactness hypothesis can be replaced by completeness if we add a one point condition on the height function of $\varSigma .$ In the case $\epsilon =-1$, we also have to impose a condition on the second fundamental form of $\varSigma ,$ which turns out to be a necessary hypothesis (see Remark 10, below).

Theorem 18

Let $\varSigma$ be a complete connected strictly convex ${{H}_{r}}(>0)$-hypersurface of $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ ($n\ge 3$) whose height function $\xi$ has a local extreme point. For $\epsilon =-1$, assume further that the least principal curvature $k_1$ of $\varSigma$ is bounded away from zero. Then, $\varSigma$ is an embedded rotational sphere.

Proof

As in the previous theorem, $\varSigma$ fulfills the hypotheses of [8, Theorems 1 and 2], which implies that $\varSigma$ is properly embedded and homeomorphic to either $\mathbb{S}^{n}$ or $\mathbb{R} ^{n}.$ Furthermore, in the latter case, the height function of $\varSigma$ is unbounded and has a single extreme point x,  which we assume to be a maximum.

For $\epsilon =1,$ the height estimates obtained in [7, Theorem 4.1-(i)] forbid $\xi$ to be unbounded. Thus, in this case, $\varSigma$ is homeomorphic to $\mathbb{S}^{n}$ and the result follows from Theorem 17.

Let us consider now the case $\epsilon =-1.$ Assume, by contradiction, that $\varSigma$ is homeomorphic to $\mathbb{R} ^{n},$ so that $\xi$ is unbounded below. Hence, given a horizontal hyperplane $P_t=M\times \{t\}$ with $t<\xi (x),$ the part $\varSigma _t^+$ of $\varSigma$ which lies above $P_t$ must be a vertical graph with boundary in $P_t$. If not, for some $t'$ between t and $\xi (x),$ $P_{t'}$ would be orthogonal to $\varSigma$ at one of its points. Then, the Alexandrov reflection method would give that $\varSigma$ is symmetric with respect to $P_{t'},$ which is impossible, since we are assuming $\xi$ unbounded, and the closure of $\varSigma _{t'}^+$ in $\varSigma$ is compact.

It follows from the above that, for |t| sufficiently large, one has

$$\begin{aligned}\xi (x)-t>\frac{1}{\inf _\varSigma k_1}\ge \frac{1}{\inf _{\varSigma _t^+} k_1}\,,\end{aligned}$$

which clearly contradicts Proposition 3. Therefore, $\varSigma$ is homeomorphic to $\mathbb{S}^{n}$ and, again, the result follows from Theorem 17. $\square$

Remark 9

In Theorems 17 and 18, the hypothesis of strict convexity of $\varSigma$ is automatically satisfied for $r=n,$ so it can be dropped in this case. Indeed, in both theorems, the height function $\xi$ has a critical point $x\in \varSigma$, which can be assumed to be a maximum. Then, taking the inward orientation on $\varSigma ,$ we have that $\varTheta (x)=-1,$ which, together with equality (4), yields

$$\begin{aligned}\langle AX,X\rangle =-\mathrm{Hess}\,\xi (X,X)\ge 0 \,\,\, \forall X\in T_{x}\varSigma .\end{aligned}$$

However, $H_n=\det A>0$ on $\varSigma$. Thus, at x,  and then on all of $\varSigma$, the second fundamental form is positive definite, that is, $\varSigma$ is strictly convex.

Remark 10

It follows from the considerations of Remark 1 in Sect. 4 that, for $r<n,$ the hypothesis on the least principal curvature of $\varSigma$ in Theorem 18 is necessary. As shown by Theorem 5, the same is true for the hypothesis on the height function $\xi$ in the case $\epsilon =1$ and $r<n.$

Now, we consider the dual case of Theorem 18, assuming that the height function of the hypersurface $\varSigma$ has no critical points. First, we recall that if M is an arbitrary Riemannian manifold, a hypersurface $\varSigma \subset M\times \mathbb{R}$ is said to be cylindrically bounded if there exists a closed geodesic ball $B\subset M$ such that $\varSigma \subset B\times \mathbb{R} .$ In particular, if M is compact, any hypersurface $\varSigma \subset M\times \mathbb{R}$ is cylindrically bounded.

Theorem 19

Assume $n\ge 3,$ and let $\varSigma$ be a proper, convex, connected ${{H}_{r}}(> 0)$-hypersurface of  $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ with no horizontal points. Then, if $\varSigma$ is cylindrically bounded, it is a cylinder over a geodesic sphere of  ${\mathbb{Q}}_{\epsilon}^{n}$. In particular, $r<n.$

Proof

From the hypothesis and [8, Theorem 3], $\varSigma =\varSigma _0\times \mathbb{R} ,$ where $\varSigma _0$ is an embedded convex topological sphere of $\mathbb{Q} _{\epsilon}^{n}$ . Moreover, in the case $\epsilon =1,$ $\varSigma _0$ is contained in an open hemisphere of $\mathbb{S}^{n}.$

At a given point $x\in \varSigma ,$ the principal curvatures are $k_1, \,\dots ,k_{n-1}, 0,$ where $k_1\,, \dots ,k_{n-1}$ are the principal curvatures of $\varSigma _0\subset \mathbb{Q} _{\epsilon}^{n}$ at $\pi _{\mathbb{Q} ^{n}_\epsilon }(x)\in \varSigma _0$ . In particular, $\varSigma _0$ has constant rth mean curvature ${{H}_{r}}$ if $r<n,$ which implies that, in this case, $\varSigma _0$ is a geodesic sphere of $\mathbb{Q} _{\epsilon}^{n}$ (see [21, 23]). Also, $H_n=0$ on $\varSigma$, so we must have $r<n,$ since we are assuming ${{H}_{r}}>0.$ $\square$

Since a cylinder $\varSigma _0\times \mathbb{R} \subset \mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ is nowhere strictly convex, it follows from the above theorem that, for $n\ge 3,$ a connected, proper, cylindrically bounded and strictly convex ${{H}_{r}}$-hypersurface of $\mathbb{Q} _{\epsilon}^{n}\times \mathbb{R}$ must have a horizontal point. This fact, together with Theorem 18, gives our last result:

Corollary 2

For $n\ge 3,$ any connected, properly immersed and strictly convex ${{H}_{r}}(>0)$-hypersurface $\varSigma \subset \mathbb{S}^{n}\times \mathbb{R}$ is necessarily an embedded rotational ${{H}_{r}}$-sphere. The same is true for $\varSigma \subset \mathbb{H}^{n}\times \mathbb{R}$ if we assume further that $\varSigma$ is cylindrically bounded and has least principal curvature bounded away from zero.