text:
\section{Introduction}
In the last years, the study of surfaces in product spaces $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a Riemannian surface, has undergone an important development, especially since in the work \cite{AR} Abresch and
Rosenberg generalized Hopf's theorem to homogeneous product spaces. Regarding minimal surfaces in such ambient spaces, they have been of considerable interest since the former works by Rosenberg \cite{Ro} and Meeks and Rosenberg
\cite{MR}. For instance, entire minimal graphs and, more generally, constant mean curvature graphs in $\mathbb{M}^2\times\mathbb{R}$ have been recently studied by several authors, establishing certain Bernstein type results in this
context \cite{Ro,ADR,ER}. Also constant Gaussian curvature surfaces \cite{AEG1} and constant extrinsic curvature surfaces \cite{EGR} in the homogeneous product spaces $\mathbb{H}^2 \times \mathbb{R}$ and $\mathbb{S}^2 \times
\mathbb{R}$ have been recently studied. In fact, in \cite{AEG1} the authors showed that the only complete surfaces of constant Gaussian curvature $K>0$ in $\mathbb{H}^2 \times \mathbb{R}$ (resp. $K>1$ in $\mathbb{S}^2 \times
\mathbb{R}$) are rotational surfaces. In addition, they proved the non-existence of complete surfaces with constant Gaussian curvature $K<-1$ in $\mathbb{H}^2 \times \mathbb{R}$ and $\mathbb{S}^2 \times \mathbb{R}$. On the other hand,
in \cite{EGR} the authors established that the only complete surfaces of constant extrinsic curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$ are rotational spheres.

Both $\mathbb{H}^2 \times \mathbb{R}$ and $\mathbb{S}^2 \times\mathbb{R}$, and more generally any product manifold $\mbox{$\mathbb{M}^n$} \times\mathbb{R}$ endowed with the product metric, are manifolds with a globally defined Killing vector field
$\partial_t$. In this work we deal with compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space \mbox{$\mathbb{N}^{n+1}$}\, ($n\geq 2$), endowed with a conformal Killing field, which in the case where \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian will be
assumed to be timelike. In fact, we obtain an integral formula for such hypersurfaces (Theorem \ref{thintegral}) which, especially in the simpler case when $\mbox{$\mathbb{N}^{n+1}$}=\mbox{$\mathbb{M}^n$} \times\mathbb{R}$ is a product space, allows us to develop a nice study in
terms of $S_\mathbb{M}$ and $S$, the scalar curvatures of $\mathbb{M}$ and the hypersurface, respectively. More specifically, we are able to characterize the slices of such product spaces from assumptions on $S_\mathbb{M}$ and $S$
(Subsections \ref{SS41} and \ref{SS42}), which become quite powerful either when $n=2$ or when $n\geq 3$ and $\mbox{$\mathbb{M}^n$}$ is Einstein with non-zero scalar curvature.

For instance, when $n=2$ and \mbox{$\mathbb{M}^2$}\ is either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$, we prove that the slices are the only compact two-sided surfaces in the Riemannian product $\mbox{$\mathbb{M}^2$}\times\mathbb{R}$
whose angle function $\Theta$ does not change sign and have constant Gaussian curvature (Corollary\rl{corriem}). It is worth pointing out that the condition that the angle function does not change sign is the natural generalization to
our ambient spaces of the condition that the image of the Gauss map of a surface in $\mathbb{R}^3$ is contained in a hemisphere, and it is a natural condition to consider if one wants to conclude that the surface is a slice
\cite{AD,ADR,ER}. When \mbox{$\mathbb{M}^2$}\ is either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$, we also prove that the slices are the only compact spacelike surfaces in the Lorentzian product $\mbox{$\mathbb{M}^2$}\times\mathbb{R}_1$ with constant Gaussian curvature (Corollary\rl{corlor}), without any additional hypothesis. In
general, when $n\geq 3$ we derive analogous results under the assumption that $\mathbb{M}^n$ is Einstein.

Finally, in Section\rl{S5} we apply our results to the case of entire graphs over $\mathbb{M}^2$ with prescribed Gaussian curvature, obtaining several uniqueness results (Corollaries\rl{coro10} and\rl{coro11}). We also give examples
of non-trivial entire graphs over $\mathbb{H}^2$ with constant Gaussian curvature which show the sharpness of our results (Example\rl{exam}).

\section{Preliminaries}
Let \mbox{$\mathbb{N}^{n+1}$}\ be an $(n+1)$-dimensional either Riemannian or Lorentzian space ($n\geq 2$). In order to simplify our notation, we will denote by $\g{}{}$, without distinction, the corresponding metric tensor, and we will assume that \mbox{$\mathbb{N}^{n+1}$}\ is
endowed with a conformal Killing field $T \in \mathfrak{X}(\mbox{$\mathbb{N}^{n+1}$})$, which in the case where \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian will be assumed to be timelike. Recall that a vector field $T$ on \mbox{$\mathbb{N}^{n+1}$}\ is called a conformal Killing field if the Lie derivative
of the metric tensor $\g{}{}$ with respect to $T$ satisfies $\mathcal{L}_T\g{}{}=2\phi\g{}{}$ for some smooth function $\phi \in \mathcal{C}^\infty(\mbox{$\mathbb{N}^{n+1}$})$. Equivalently, \begin{equation} \label{conformal}
\g{\mbox{$\overline{\nabla}$}_VT}{W}+\g{V}{\mbox{$\overline{\nabla}$}_WT}=2\phi\g{V}{W} \end{equation} for all vector fields $V,W \in \mathfrak{X}(\mbox{$\mathbb{N}^{n+1}$})$, where $\mbox{$\overline{\nabla}$}$ denotes the Levi-Civita connection in \mbox{$\mathbb{N}^{n+1}$}. In particular, $T$ is called a homothetic Killing field if the
function $\phi$ is constant, and just a Killing field whenever that constant vanishes.

Let us consider $\Sigma^n$ a connected hypersurface immersed into \mbox{$\mathbb{N}^{n+1}$}. In the case where \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian, we will assume that $\star$ is a spacelike hypersurface, that is, the metric induced on $\star$ via the immersion is a Riemannian
metric. Since $T$ is a globally defined timelike vector field on \mbox{$\mathbb{N}^{n+1}$}, it follows that there exists a unique unitary timelike normal field $N$ globally defined on \star\ which is in the same time-orientation as $T$, so that
\[
\g{N}{T}\leq -|T|=-\sqrt{-\g{T}{T}}<0 \quad \mathrm{on} \quad \star.
\]
In that case we will refer to $N$ as the future-pointing Gauss map of \star. On the other hand, when \mbox{$\mathbb{N}^{n+1}$}\ is Riemannian \star\ is assumed to be a \textit{two-sided} hypersurface in \mbox{$\mathbb{N}^{n+1}$}. This condition means that there is a globally defined
unit normal vector field $N$. In both cases, we will denote by $\Theta:\star \mbox{$\rightarrow$} \mathbb{R}$ the smooth function on \star\ given by $\Theta=\g{N}{T}$, which in the Lorentzian case will be always negative.

Let $\nabla$ denotes the Levi-Civita connection on \star. Then the Gauss and Weingarten formulae for the hypersurface in \mbox{$\mathbb{N}^{n+1}$}\ are given, respectively, by \begin{equation} \label{fgauss} \mbox{$\overline{\nabla}$}_XY=\nabla_XY+\varepsilon\g{AX}{Y}N, \end{equation} and \begin{equation}
\label{fwein} A(X)=-\mbox{$\overline{\nabla}$}_XN, \end{equation} for all tangent vector fields $X,Y \in \mathfrak{X}(\star)$. Here $A:\mathfrak{X}(\star) \mbox{$\rightarrow$} \mathfrak{X}(\star)$ denotes the shape operator (or second fundamental form) of \star\ with respect to $N$, and
$\varepsilon=\g{N}{N}$. As is well known, the curvature tensor $R$ of the hypersurface \star\ is described in terms of the shape operator and the curvature tensor $\overline{R}$ of the ambient space \mbox{$\mathbb{N}^{n+1}$}\ by the so-called Gauss equation,
which can be written as \begin{equation} \label{eqgauss} R(X,Y)Z=(\overline{R}(X,Y)Z)^\top+\varepsilon(\g{AX}{Z}AY - \g{AY}{Z}AX) \end{equation} for all tangent vector fields $X,Y,Z \in \mathfrak{X}(\star)$, where $(\;)^\top$ denotes the tangential component
of a vector field in $\mathfrak{X}(\mbox{$\mathbb{N}^{n+1}$})$ along $\star^n$. Observe that our criterion here for the definition of the curvature tensor is the one in \cite{ONe}. Consider a local orthonormal frame $\{E_1,...E_n\}$ of $\mathfrak{X}(\star)$. We
can also assume that $\{E_1,...,E_n\}$ diagonalizes the shape operator, that is $AE_i=\kappa_iE_i, \, 1\leq i\leq n$ where $\kappa_i,\,1\leq i \leq n$, stand for the principal curvatures of $\star^n$. Then, it follows from the Gauss
equation \rf{eqgauss} that
\begin{eqnarray*}
\mathop\mathrm{Ric}\nolimits(E_j,E_j)&=&\sum_{i \neq j}\g{\overline{R}(E_i,E_j)E_i}{E_j}+\varepsilon \sum_{i \neq j}\kappa_i\kappa_j\\
&=&\overline{\mathop\mathrm{Ric}\nolimits}(E_j,E_j)-\varepsilon\g{\overline{R}(E_j,N)E_j}{N}+\varepsilon \sum_{i \neq j}\kappa_i\kappa_j
\end{eqnarray*}
for $1 \leq j \leq n$, where $\mathop\mathrm{Ric}\nolimits$ and $\overline{\mathop\mathrm{Ric}\nolimits}$ stand for the Ricci operators of $\star^n$ and \mbox{$\mathbb{N}^{n+1}$}\, respectively. Taking traces, we can also obtain the analogous relation between the scalar curvatures $S$ and $\overline{S}$ of
$\star^n$ and \mbox{$\mathbb{N}^{n+1}$}, which is given by
\begin{eqnarray}
\label{eqgaussaux}
\nonumber S & = & \sum_{j=1}^n \overline{\mathop\mathrm{Ric}\nolimits}(E_j,E_j)-\varepsilon\overline{\mathop\mathrm{Ric}\nolimits}(N,N)+2\varepsilon\sum_{i<j}\kappa_i\kappa_j\\
{} & = & \overline{S}-2\varepsilon\overline{\mathop\mathrm{Ric}\nolimits}(N,N)+2\varepsilon\sum_{i<j}\kappa_i\kappa_j.
\end{eqnarray}
Recall that the mean curvature function of the hypersurface $\Sigma$ is defined as
\[
H=\frac{\varepsilon}{n}\mathrm{tr}(A)=\frac{\varepsilon}{n}(\kappa_1+...+\kappa_n).
\]
Therefore, the following identity is verified, \begin{equation} \label{eq0} n^2H^2-\|A\|^2=2\sum_{i<j}\kappa_i\kappa_j \end{equation} being $\|A\|^2=\mathop\mathrm{trace}\nolimits{A^2}$ and so \rf{eqgaussaux} becomes
\[
S=\overline{S}-2\varepsilon\overline{\mathop\mathrm{Ric}\nolimits}(N,N)+\varepsilon(n^2H^2-\|A\|^2).
\]
In other words, \begin{equation} \label{eqgausssc} \|A\|^2+\overline{\mathop\mathrm{Ric}\nolimits}(N,N)=\varepsilon(\overline{S}-S)-\overline{\mathop\mathrm{Ric}\nolimits}(N,N)+n^2H^2. \end{equation}

On the other hand, the Codazzi equation of the hypersurface describes the normal component of $\overline{R}(X,Y)Z$ in terms of the derivative of the shape operator, and it is given by \begin{equation} \label{eqcodazzi}
\g{\overline{R}(X,Y)Z}{N}=\g{(\nabla_YA)X-(\nabla_XA)Y}{Z}, \end{equation} where $\nabla_XA$ denotes the covariant derivative of $A$, that is
\[
(\nabla_XA)Y=\nabla_X(AY)-A(\nabla_XY).
\]

\section{An integral formula}
In all what follows, \mbox{$\mathbb{N}^{n+1}$}\ denotes an $(n+1)$-dimensional either Riemannian or Lorentzian space endowed with a conformal Killing field $T$ (which is assumed to be timelike when \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian), and $\Sigma^n$ is an immersed
hypersurface in \mbox{$\mathbb{N}^{n+1}$}, which is assumed to be spacelike when \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian and two-sided when \mbox{$\mathbb{N}^{n+1}$}\ is Riemannian. Before giving our main result of this section, we need to give an expression for the Laplacian of the function
$\Theta=\g{N}{T}$. In the case where \mbox{$\mathbb{N}^{n+1}$}\ is a Riemannian manifold, Al\'{\i}as, Dajczer and Ripoll gave in \cite{ADR} an expression for the Laplacian of $\Theta$ in terms of the Ricci tensor of the ambient space and of the norm of the
second fundamental form. Barros, Brasil and Caminha proved in \cite{BBC} the analogous expression when \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian. For our purpose, we need this expression in a slightly different way. For that reason, and for the sake of
completeness, we will give here an alternative proof for the expression we will need.

\begin{proposition}\label{prlaplaT}
Under all the assumptions above, the Laplacian of the smooth function $\Theta\in \mbox{$\mathcal{C}^\infty(\s)$}$, $\Theta=\g{N}{T}$, is given by \begin{equation} \label{laplatheta} \Delta\Theta=-\varepsilon n \g{\nabla H}{T}+\Theta (S-\overline{S}+\varepsilon
(\overline{\mathop\mathrm{Ric}\nolimits}(N,N)-n^2H^2))-n(\varepsilon H\phi+\frac{\partial \phi}{\partial N}), \end{equation} where $\nabla H$ denotes the gradient of the mean curvature function on \star.
\end{proposition}

\begin{proof}
As $T$ is a conformal Killing vector field, from \rf{conformal} and Weingarten formula \rf{fwein} we have
\[
X(\Theta)=\g{\mbox{$\overline{\nabla}$}_XN}{T}+\g{N}{\mbox{$\overline{\nabla}$}_XT}=-\g{AX}{T}-\g{X}{\mbox{$\overline{\nabla}$}_NT}
\]
for every tangent vector field $X \in \mathfrak{X}(\star)$. Therefore, the gradient of $\Theta$ on \star\ is given by \begin{equation} \label{gradtheta} \nabla \Theta=-A T^\top-(\mbox{$\overline{\nabla}$}_NT)^\top, \end{equation} and its Laplacian is given by
\begin{equation}
\label{laplaT}
\Delta \Theta=-\mathop\mathrm{div}\nolimits(AT^\top)-\mathop\mathrm{div}\nolimits((\mbox{$\overline{\nabla}$}_NT)^\top).
\end{equation}
We will start by computing $\mathop\mathrm{div}\nolimits(AT^\top)$. By Codazzi equation, it holds that for any tangent vector field $X \in \mathfrak{X}(\star)$
\begin{eqnarray*}
\g{\nabla_X(AT^\top)}{X}&=&\g{(\nabla_XA)T^\top}{X}+\g{A(\nabla_XT^\top)}{X}\\
&=&\g{(\nabla_{T^\top}A)X}{X}-\g{\overline{R}(T^\top,X)N}{X}+\g{\nabla_XT^\top}{AX}.
\end{eqnarray*}
Consequently, \begin{equation} \label{eq1} \mathop\mathrm{div}\nolimits(AT^\top)=\mathop\mathrm{trace}\nolimits(\nabla_{T^\top}A)-\overline{\mathop\mathrm{Ric}\nolimits}(T^\top,N)+\sum_{i=1}^n\g{\nabla_{E_i}T^\top}{AE_i} \end{equation} where $\{E_1,..,E_n\}$ is again a local orthonormal frame of $\mathfrak{X}(\star)$ which diagonalizes $A$.
From the decomposition $T=T^\top+\varepsilon \Theta N$, we also obtain
\[
\mbox{$\overline{\nabla}$}_XT=\mbox{$\overline{\nabla}$}_XT^\top+\varepsilon X(\Theta)N+\varepsilon \Theta\mbox{$\overline{\nabla}$}_XN,
\]
for every tangent vector field $X \in \mathfrak{X}(\star)$. Then, applying the Weingarten formula \rf{fwein} we get \begin{equation} \label{eq2} \nabla_XT^\top=(\mbox{$\overline{\nabla}$}_XT)^\top+\varepsilon\Theta AX. \end{equation} Therefore,
\[
\sum_{i=1}^n\g{\nabla_{E_i}T^\top}{AE_i}=\sum_{i=1}^n\g{\mbox{$\overline{\nabla}$}_{E_i}T}{AE_i}+\varepsilon\Theta \|A\|^2.
\]
Since $\{E_1,...,E_n\}$ diagonalizes $A$, it holds that $\g{\mbox{$\overline{\nabla}$}_{E_i}T}{AE_i}=\g{\mbox{$\overline{\nabla}$}_{AE_i}T}{E_i}$, and so from \rf{conformal}
\[
\g{\mbox{$\overline{\nabla}$}_{E_i}T}{AE_i}=\phi\g{AE_i}{E_i}.
\]

Then, \rf{eq1} becomes
\begin{eqnarray} \label{div1}
\mathop\mathrm{div}\nolimits(AT^\top)&=&\mathop\mathrm{trace}\nolimits(\nabla_{T^\top}A)+\varepsilon nH\phi+\varepsilon\Theta \|A\|^2-\overline{\mathop\mathrm{Ric}\nolimits}(T^\top,N)\\
\nonumber &=&\varepsilon n \g{\nabla H}{T}+\varepsilon n H \phi+\varepsilon \Theta \|A\|^2-\overline{\mathop\mathrm{Ric}\nolimits}(T^\top,N),
\end{eqnarray}
where we have used the fact that the trace commutes with the covariant derivative.

It remains to obtain an expression for $\mathop\mathrm{div}\nolimits((\mbox{$\overline{\nabla}$}_NT)^\top)$. Observe that from \rf{conformal} we have
\[
\mbox{$\overline{\nabla}$}_NT=(\mbox{$\overline{\nabla}$}_NT)^\top+\varepsilon\g{\mbox{$\overline{\nabla}$}_NT}{N}N=(\mbox{$\overline{\nabla}$}_NT)^\top+\phi N,
\]
which by the Gauss and Weingarten formulae yields that
\[
\mbox{$\overline{\nabla}$}_X\mbox{$\overline{\nabla}$}_NT=\nabla_X(\mbox{$\overline{\nabla}$}_NT)^\top+\varepsilon\g{AX}{\nabla_NT}N+X(\phi)N-\phi AX.
\]
Therefore, \begin{equation} \label{eq2.5} \mathop\mathrm{div}\nolimits((\mbox{$\overline{\nabla}$}_NT)^\top)=\sum_{i=1}^n \g{\mbox{$\overline{\nabla}$}_{E_i}(\mbox{$\overline{\nabla}$}_NT)}{E_i}+\varepsilon nH\phi. \end{equation} Observe that
\[
\overline{R}(E_i,N)T=
-\mbox{$\overline{\nabla}$}_{AE_i}T-\mbox{$\overline{\nabla}$}_{\mbox{$\overline{\nabla}$}_NE_i}T-\mbox{$\overline{\nabla}$}_{E_i}\mbox{$\overline{\nabla}$}_NT+\mbox{$\overline{\nabla}$}_N\mbox{$\overline{\nabla}$}_{E_i}T.
\]
Taking traces in the last expression we easily obtain from \rf{eq2.5}
\begin{eqnarray}\label{eq3}
\hspace*{0.7cm}\mathop\mathrm{div}\nolimits((\mbox{$\overline{\nabla}$}_NT)^\top)&= &\varepsilon n \phi H+\overline{\mathop\mathrm{Ric}\nolimits}(T,N)\\
\nonumber&+&\sum_{i=1}^n\g{\mbox{$\overline{\nabla}$}_N\mbox{$\overline{\nabla}$}_{E_i}T}{E_i}-\sum_{i=1}^n\g{\mbox{$\overline{\nabla}$}_{AE_i}T}{E_i}-\sum_{i=1}^n\g{\mbox{$\overline{\nabla}$}_{\mbox{$\overline{\nabla}$}_NE_i}T}{E_i}.
\end{eqnarray}
To simplify the different terms in \rf{eq3}, applying \rf{conformal} we get the relation
\[
\g{\mbox{$\overline{\nabla}$}_N\mbox{$\overline{\nabla}$}_{E_i}T}{E_i}=N(\g{\mbox{$\overline{\nabla}$}_{E_i}T}{E_i})-\g{\mbox{$\overline{\nabla}$}_{E_i}T}{\mbox{$\overline{\nabla}$}_NE_i}=\frac{\partial \phi}{\partial N}+\g{\mbox{$\overline{\nabla}$}_{\mbox{$\overline{\nabla}$}_NE_i}T}{E_i}.
\]
On the other hand, we have
\[
\sum_{i=1}^n\g{\mbox{$\overline{\nabla}$}_{AE_i}T}{E_i}=\phi \mathop\mathrm{trace}\nolimits(A)=\varepsilon n \phi H.
\]
Turning back to \rf{eq3}, it becomes \begin{equation} \label{div2} \mathop\mathrm{div}\nolimits((\mbox{$\overline{\nabla}$}_NT)^\top)=\overline{\mathop\mathrm{Ric}\nolimits}(T,N)+n \frac{\partial \phi}{\partial N}. \end{equation} Finally, from \rf{eqgausssc}, \rf{laplaT}, \rf{div1} and \rf{div2} and since
$T=T^\top+\varepsilon \Theta N$,
\begin{eqnarray*}
\Delta \Theta&=&-\varepsilon n \g{\nabla H}{T}-\varepsilon \Theta \|A\|^2+\overline{\mathop\mathrm{Ric}\nolimits}(T^\top,N)-\overline{\mathop\mathrm{Ric}\nolimits}(T,N)-n(\varepsilon H \phi+\frac{\partial \phi}{\partial N})\\
{} & = &-\varepsilon n \g{\nabla H}{T}-\varepsilon \Theta (\|A\|^2+\overline{\mathop\mathrm{Ric}\nolimits}(N,N))-n(\varepsilon H\phi+\frac{\partial \phi}{\partial N})\\
&=& -\varepsilon n \g{\nabla H}{T}+\Theta (S-\overline{S}+\varepsilon (\overline{\mathop\mathrm{Ric}\nolimits}(N,N)-n^2H^2))-n(\varepsilon H\phi+\frac{\partial \phi}{\partial N}).
\end{eqnarray*}
\end{proof}

We can give now the following integral formula for compact hypersurfaces in \mbox{$\mathbb{N}^{n+1}$}.
\begin{theorem}
\label{thintegral} Let $\Sigma^n$ be a compact hypersurface immersed into \mbox{$\mathbb{N}^{n+1}$}\ with the general assumptions stated at the beginning of this section. Then \begin{equation} \label{integralaux} \int_{\Sigma}
\Theta(S-\overline{S}+\varepsilon\overline{\mathop\mathrm{Ric}\nolimits}(N,N))d\star=n\int_{\Sigma}\frac{\partial \phi}{\partial N}d\Sigma-n(n-1)\varepsilon\int_{\Sigma}H\phi d\Sigma. \end{equation}
\end{theorem}
\begin{proof}
From \rf{eq2}, and from the fact that $T \in \mathfrak{X}(\mbox{$\mathbb{N}^{n+1}$})$ is a conformal Killing field it is immediate to see that
\[
\mathop\mathrm{div}\nolimits(T^\top)=n\phi+nH\Theta.
\]
Therefore, \begin{equation} \label{div} \mathop\mathrm{div}\nolimits(HT^\top)=H\mathop\mathrm{div}\nolimits(T^\top)+\g{\nabla H}{T^\top}=nH\phi+nH^2\Theta+\g{\nabla H}{T^\top}. \end{equation} Now, from \rf{laplatheta} and \rf{div} we get
\[
\Delta \Theta+\varepsilon n \mathop\mathrm{div}\nolimits(HT^\top)=\Theta(S-\overline{S}+\varepsilon\overline{\mathop\mathrm{Ric}\nolimits}(N,N))+n(n-1)\varepsilon H\phi-n\frac{\partial \phi}{\partial N}.
\]
Finally, integrating the last expression over the compact hypersurface $\star^n$ and applying the divergence theorem the result follows.
\end{proof}

Theorem\rl{thintegral} becomes especially simple when the field $T$ is a Killing vector field. In that case, \rf{integralaux} is simplified to \begin{equation} \label{integral} \int_{\Sigma}
\Theta(S-\overline{S}+\varepsilon\overline{\mathop\mathrm{Ric}\nolimits}(N,N))d\star=0. \end{equation} On the other hand, when $T$ is a homothetic Killing vector field with $\phi\neq 0$, we may assume without loss of generality that $\phi=1$ and \rf{integralaux} becomes
\[
\int_{\Sigma} \Theta(S-\overline{S}+\varepsilon\overline{\mathop\mathrm{Ric}\nolimits}(N,N))d\star=-n(n-1)\varepsilon \int_{\Sigma}H d\Sigma.
\]

It is especially interesting the case where $T$ is a Killing field, because it will allow us to give some nice consequences of \rf{integral}. Therefore, from now on $T$ will be assumed to be Killing. As a first application, we can
state the following consequence in the case where \mbox{$\mathbb{N}^{n+1}$}\ is Einstein.
\begin{proposition}
\label{thkilling} Let \mbox{$\mathbb{N}^{n+1}$}\ be an Einstein space with non-zero scalar curvature $\overline{S}\neq 0$, and endowed with a Killing field (which is assumed to be timelike when \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian).
\begin{itemize}
\item[(i)] If \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian and $\overline{S}<0$ (respectively, $\overline{S}>0$) there does not exist any compact spacelike hypersurface satisfying $S\leq \overline{S}$ (respectively, $S\geq \overline{S}$).
\item[(ii)] If \mbox{$\mathbb{N}^{n+1}$}\ is Riemannian and $\overline{S}<0$ (respectively, $\overline{S}>0$) there does not exist any compact hypersurface satisfying $S\leq \overline{S}$ (respectively, $S\geq \overline{S}$) and having $\Theta\neq 0$.
\end{itemize}
\end{proposition}
\begin{proof}
Let us suppose that $\overline{S}<0$ and that there exists a compact hypersurface $\star^n$ such that $S \leq \overline{S}$ in any of the cases (i) or (ii). \mbox{$\mathbb{N}^{n+1}$}\ being Einstein yields that
$\overline{\mathop\mathrm{Ric}\nolimits}(N,N)=\varepsilon\frac{\overline{S}}{n+1}$, so \rf{integral} becomes \begin{equation} \label{intaux} \int_{\Sigma} \Theta\left(S-\overline{S}+\frac{\overline{S}}{n+1}\right)d\star=0. \end{equation} Recall that when \mbox{$\mathbb{N}^{n+1}$}\ is Lorentzian the
function $\Theta$ satisfies $\Theta\leq -1<0$. Then, since in the Riemannian case $\Theta$ is assumed to be non-vanishing, we can assume without loss of generalization that $\Theta$ is a negative function over \star. Therefore,
\[
\Theta\left(S-\overline{S}+\frac{\overline{S}}{n+1}\right) \geq 0 \quad \mathrm{on} \quad \star,
\]
and by \rf{intaux} it must vanish. In particular, it must be $S=\overline{S}$ and $\overline{S}=0$, which contradicts the assumption of the theorem. The proof for the case when $\overline{S}>0$ is analogous.
\end{proof}

\section{Hypersurfaces in a product space}
A particular family of spaces with a Killing vector field is that of product spaces. Let $\mbox{$\mathbb{M}^n$}$ be a connected Riemannian surface and consider the product manifold $\mbox{$\mathbb{M}^n$} \times \mathbb{R}$ endowed with the metric
\[
\g{}{}=\g{}{}_\mathbb{M}+\varepsilon dt^2
\]
being $\varepsilon=\pm 1$. When necessary, we will denote the Lorentzian product by $\mbox{$\mathbb{M}^n$} \times \mathbb{R}_1$ in order to distinguish it to the Riemannian one. Observe that $\partial_t$ is a globally defined Killing field over any product
space $\mbox{$\mathbb{M}^n$} \times \mathbb{R}$, with $\g{\partial_t}{\partial_t}=\varepsilon$. Moreover, $\partial_t$ is a parallel vector field, and $\mbox{$\overline{\nabla}$}\pi_{\mathbb{R}}=\varepsilon\partial_t$, where $\pi_\mathbb{R}$ denotes the projection of $\mbox{$\mathbb{M}^n$}
\times \mathbb{R}$ onto the factor \mathbb{R}. The height function $h$ of an immersed hypersurface $\psi:\star^n \mbox{$\rightarrow$} \mbox{$\mathbb{M}^n$} \times\mathbb{R}$ is the smooth function $h \in \mbox{$\mathcal{C}^\infty(\s)$}$ defined as the projection of the immersion over the factor \mathbb{R}, that is,
$h=\pi_\mathbb{R} \circ\psi$. Therefore, the gradient of $h$ on \star\ is given by
\[
\nabla h=(\mbox{$\overline{\nabla}$}\pi_\mathbb{R})^\top=\varepsilon \partial_t^\top,
\]
and from the decomposition $\partial_t=\partial_t^\top+\varepsilon \Theta N$ we immediately get \begin{equation} \label{normgradh} \|\nabla h\|^2=\varepsilon(1-\Theta^2). \end{equation} On the other hand, since $\partial_t$ is parallel in $\mbox{$\mathbb{M}^n$} \times \mathbb{R}$, we
obtain from \rf{eq2} that \begin{equation} \label{hessh} \nabla_X \nabla h=\Theta AX \end{equation} and therefore $\Delta h=\varepsilon n H \Theta$.

Here and in what follows \mbox{$\mathbb{M}^n$}\ is assumed to be Einstein whenever $n\geq 3$. For simplicity, $\kappa$ will denote either the (non-necessarily constant) Gaussian curvature of \mbox{$\mathbb{M}^2$}\ along \star, when $n=2$, or the constant
$S_\mathbb{M}/n$ when $n\geq 3$, being $S_\mathbb{M}$ the scalar curvature of \mbox{$\mathbb{M}^n$}. Therefore,
\[
\overline{\mathop\mathrm{Ric}\nolimits}(U,U)=\kappa |U^\ast|^2
\]
for any vector field $U$ in $\mathfrak{X}(\mbox{$\mathbb{M}^n$} \times \mathbb{R})$, where $U^\ast$ stands for the projection of the vector field $U$ onto the factor $\mathbb{M}$, that is, $U=U^\ast+\varepsilon \g{U}{\partial_t}\partial_t$, and $|U^\ast|$
denotes its norm with respect to the original metric $\g{}{}_\mathbb{M}$. Then,
\[
\overline{\mathop\mathrm{Ric}\nolimits}(N,N)=\kappa |N^\ast|^2=\kappa \varepsilon (1-\Theta^2),
\]
and
\[
\overline{S}=n \kappa.
\]
Consequently, when \mbox{$\mathbb{N}^{n+1}$}\ is a product space $\mbox{$\mathbb{M}^n$} \times \mathbb{R}$, \rf{eqgaussaux} becomes \begin{equation} \label{eqgaussprod} S=(n-2)\kappa+2\kappa\Theta^2+2\varepsilon\sum_{i<j}\kappa_i\kappa_j. \end{equation} On the other hand, the integral \rf{integral} is
written as \begin{equation} \label{integralproduct} \int_\Sigma\Theta((S-n\kappa)+\kappa(1-\Theta^2))=0. \end{equation}

\subsection{Hypersurfaces in a Riemannian product space}\label{SS41}
Let us first consider the case where $\star^n$ is a hypersurface immersed into a Riemannian product space $\mbox{$\mathbb{M}^n$}\times\mathbb{R}$. Recall that we are assuming that \star\ is two-sided and the function $\Theta=\g{N}{\partial_t}$ is globally defined.
We will refer to $\Theta$ as the angle function. If \star\ is locally a graph over \mbox{$\mathbb{M}^n$}\ (that it, transversal to $\partial_t$) then  either $\Theta<0$ or $\Theta>0$ over \star. Thus, the assumption \textit{the angle function does not
change sign is on} \star\ is weaker than \star\ \textit{is a local graph}. As already observed by other authors (see for instance \cite{AD,ADR} and \cite{ER}), the angle function is a natural function to consider if one wants to conclude that a hypersurface is necessarily
a slice since, in that case, by \rf{normgradh} we must have that $\Theta^2=1$.

As a first application of our integral formula \rf{integralproduct}, we get the following result for surfaces. Observe that we are not assuming that neither the Gaussian curvature of \mbox{$\mathbb{M}^2$}\ nor the Gaussian curvature of $\star^2$ are
constant.
\begin{theorem}
\label{th4} Let \mbox{$\mathbb{M}^2$}\ be a compact Riemannian surface with non-negative Gaussian curvature, $K_{\mathbb{M}}\geq 0$ (respectively, non-positive Gaussian curvature $K_{\mathbb{M}}\leq 0$), and assume that $K_{\mathbb{M}}>0$
(respectively, $K_{\mathbb{M}}<0$) on a dense subset of \mbox{$\mathbb{M}^2$}. The only compact two-sided surfaces \star\ in the Riemannian product $\mbox{$\mathbb{M}^2$}\times\mathbb{R}$ such that its angle function $\Theta$ does not change sign and satisfying $K\geq
K_{\mathbb{M}}$ along $\Sigma$ (respectively, $K\leq K_{\mathbb{M}}$) are the slices.
\end{theorem}
\begin{proof}
We can assume without loss of generality that the angle function satisfies $\Theta\leq 0$. Let us consider the case where $K_{\mathbb{M}}\geq 0$ (the case $K_{\mathbb{M}}\leq 0$ is similar). From our hypothesis we have that
$K\geq\kappa\geq 0$ on $\star$. Hence, since $1-\Theta^2\geq 0$, we get that
\[
\Theta(2(K-\kappa)+\kappa(1-\Theta^2))\leq 0,
\]
and it vanishes at a point $p\in\Sigma$ if and only if
\begin{itemize}
\item[(i)] either $\Theta(p)=0$,
\item[(ii)] or $\Theta(p)<0$, $K(p)=\kappa(p)$ and $\kappa(p)(1-\Theta^2(p))=0$.
\end{itemize}
On the other hand, by \rf{integralproduct} we have
\[
\int_\Sigma\Theta(2(K-\kappa)+\kappa(1-\Theta^2))=0,
\]
and then $\Theta(2(K-\kappa)+\kappa(1-\Theta^2))=0$ at every point. This implies that at every $p\in\Sigma$, either $\Theta(p)=0$ or $\Theta(p)=-1$ (see below). Therefore, by continuity the function $\Theta$ must be constant on
$\Sigma$, with either $\Theta=0$ or $\Theta=-1$. But observe that the case $\Theta=0$ cannot happen, since, from \rf{normgradh} we know that there always exist points at which $\Theta=-1$, the critical points of $h$. It follows that
$\Theta=-1$ on $\Sigma$ which means that the surface is a slice.

It remains to prove that $\Theta(2(K-\kappa)+\kappa(1-\Theta^2))=0$ on $\Sigma$ implies that either $\Theta(p)=0$ or $\Theta(p)=-1$  at every $p\in\Sigma$. When $\Theta(p)=0$ there is nothing to prove. Then, let us assume that
$\Theta(p)<0$. If $\kappa(p)\neq 0$, by (ii) above we have that $\Theta(p)=-1$. On the other hand, if $\kappa(p)=0$ we reason as follows. Since $\Theta(p)<0$, we know that $\Sigma$ is locally a graph around  $p$, that is, there exists
an open subset $\Omega\subset\mathbb{M}^2$ and a smooth function $u:\Omega\mbox{$\rightarrow$}\mathbb{R}$ such that $W=\{ (x,u(x)) : x\in\Omega \}\subset\Sigma$ is an open neighborhood of $p$, with $p=(x_0,u(x_0))$ for a certain $x_0\in\Omega$. In
particular, $\Theta<0$ on $W$. Recall now that we are assuming that $K_{\mathbb{M}}>0$ on a dense subset of \mbox{$\mathbb{M}^2$}. Therefore, we may find a sequence $\{x_k\}_{k=1}^{\infty}\subset\Omega$ converging to $x_0$ such that
$K_\mathbb{M}(x_k)>0$ at every $k\geq 1$. Let $p_k=(x_k,u(x_k))\in W$, and observe that $\kappa(p_k)=K_\mathbb{M}(x_k)>0$ and $\Theta(p_k)<0$ for every $k\geq 1$. Then, by (ii) above we get that $\Theta(p_k)=-1$ for every $k\geq 1$,
and by continuity $\Theta(p)=-1$. This finishes the proof.
\end{proof}

For the general $n$-dimensional case, we have the following.
\begin{theorem}
\label{th5} Let \mbox{$\mathbb{M}^n$}\ be a compact Einstein Riemannian manifold, $n\geq 3$, with positive scalar curvature $S_{\mathbb{M}}>0$ (respectively, negative scalar curvature $S_{\mathbb{M}}<0$). The only compact two-sided hypersurfaces \star\
in the Riemannian product $\mbox{$\mathbb{M}^n$}\times\mathbb{R}$ such that its angle function $\Theta$ does not change sign and satisfying $\inf_\Sigma S\geq S_{\mathbb{M}}$ (respectively, $\sup_\Sigma S\leq S_{\mathbb{M}}$) are the slices.
\end{theorem}
\begin{proof}
We use the same ideas of the previous proof, taking into account that the scalar curvature $S_\mathbb{M}$ is necessarily constant. We may assume again that $\Theta\leq 0$ and consider the case where $S_{\mathbb{M}}> 0$ is a
positive constant. From our hypothesis we have that $S\geq n\kappa>0$ on \star, so that
\[
\Theta((S-n\kappa)+\kappa(1-\Theta^2))\leq 0,
\]
and by \rf{integralproduct} it vanishes at every point $p\in\Sigma$. In this case, since $\kappa$ is a positive constant we easily have that $\Theta(p)((S(p)-n\kappa)+\kappa(1-\Theta^2(p)))=0$ at a point $p$ if and only if either
$\Theta(p)=0$ or $\Theta(p)=-1$. The same argument as in the case $n=2$ implies then that $\Theta=-1$ and $\Sigma$ is a slice.
\end{proof}

As is well known, the only complete Riemannian surfaces with positive constant Gaussian curvature are, up to a homothety, the sphere $\mathbb{S}^2$ and the real projective plane $\mathbb{RP}^2$. Therefore, we have the following
consequence of Theorem\rl{th4}
\begin{corollary}
\label{corriem} Let \mbox{$\mathbb{M}^2$}\ be either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$.
\begin{itemize}
\item[(i)]  The only compact two-sided surfaces \star\ in the Riemannian product $\mbox{$\mathbb{M}^2$}\times\mathbb{R}$ such that its angle function $\Theta$ does not change sign and having constant Gaussian curvature are the slices.
\item[(ii)]  The only complete two-sided surfaces \star\ in the Riemannian product $\mbox{$\mathbb{M}^2$}\times\mathbb{R}$ such that its angle function $\Theta$ is bounded away from 0 and having constant Gaussian curvature are the slices.
\end{itemize}
\end{corollary}

Observe that a cylinder $\gamma\times\mathbb{R}$ over a complete curve $\gamma$ in \mbox{$\mathbb{M}^2$}\ is an example of a complete and non-compact two-sided surface having $\Theta=0$ and constant Gaussian curvature $K=0$. This
shows that our hypothesis on $\Theta$ in (ii) is necessary. In \cite{AEG1} Aledo, Espinar and G\'{a}lvez posed the following question: are the slices the only complete surfaces in $\mathbb{S}^2 \times \mathbb{R}$ with constant Gaussian
curvature $K$ such that $0<K\leq 1$? Later, in \cite{AEG2}, the same authors proved that there do not exist such surfaces when $0<K<1$, but the question remains open for the case $K=1$. Corollary\rl{corriem} answers partially that
question in the following sense: the only complete surfaces in $\mathbb{S}^2 \times \mathbb{R}$ with constant Gaussian curvature and such that its angle function does not change sign are the slices. In particular, this is also true for entire
complete graphs over $\mathbb{S}^2 \times \mathbb{R}$.

\begin{proof}[Proof of Corollary\rl{corriem}]
We can always assume that $\Theta\leq 0$. Our first objective is to see that in case (ii) the surface \star\ is necessarily compact, and then case (ii) follows directly from case (i). Our hypothesis on $\Theta$ in (ii) means that
$\Theta(p)\leq -\delta<0$ at every point of \star\ for certain $\delta>0$. We claim that this implies that \star\ is necessarily compact. Actually, consider the projection $\Pi:\star \mbox{$\rightarrow$} \mbox{$\mathbb{M}^2$}$ of \star\ onto the factor \mbox{$\mathbb{M}^2$}. Given $p\in
\Sigma$ and a tangent vector $v\in T_p\star$, it is not difficult to see that \begin{equation} \label{des} \g{d\Pi_p(v)}{d\Pi_p(v)}_\mathbb{M} \geq c \g{v}{v} \end{equation} where $c=\frac{\delta^2}{1+\delta^2}>0$. Since $\g{}{}$ is a complete Riemannian
metric on \star\, the same holds for the homothetic metric $\widetilde{\g{}{}}=c\g{}{}$. Then, by \rf{des} the map
\[
\Pi:(\Sigma,\widetilde{\g{}{}})\mbox{$\rightarrow$}(\mbox{$\mathbb{M}^2$},\g{}{}_\mathbb{M})
\]
is a local diffeomorphism which increases the distance. Hence, by \cite[Chapter VIII, Lemma 8.1]{KN} $\Pi$ is a covering map. Therefore, since the universal covering of \mbox{$\mathbb{M}^2$}\ is $\mathbb{S}^2$, we conclude that \star\ must also be
compact.

Let us see now how to prove the case (i). By compactness of \star, there exists a point $p_0\in\star$ such that $h(p_0)=\min_\Sigma h$. Then, by \rf{normgradh} $\Theta(p_0)=-1$ and $\nabla^2 h_{p_0}(v,v) \geq 0$ for any $v\in T_{p_0}\star$, where
$\nabla^2 h$ stands for the Hessian operator on \star. In particular, if we consider the orthonormal frame $\{e_1,e_2\}$ of the principal directions, from \rf{hessh} we get that
\[
\nabla^2h_{p_0} (e_i,e_i)=-\kappa_i(p_0)\geq 0.
\]
Therefore, $\mathop\mathrm{det}\nolimits(A)(p_0)\geq 0$ and from the Gauss equation \rf{eqgaussprod} we conclude that
\[
K=K(p_0)=K_\mathbb{M}\Theta^2(p_0)+\mathop\mathrm{det}\nolimits(A)(p_0)\geq K_\mathbb{M}.
\]
The result follows now as a direct consequence of Theorem\rl{th4}.
\end{proof}

For general $n$ we get.
\begin{corollary}
Let \mbox{$\mathbb{M}^n$}\ be a compact Einstein Riemannian manifold, $n\geq 3$, with positive scalar curvature. The only compact two-sided hypersurfaces \star\ in the Riemannian product $\mbox{$\mathbb{M}^n$}\times\mathbb{R}$  whose angle function $\Theta$ does not
change sign and having constant scalar curvature are the slices.
\end{corollary}
\begin{proof}
The proof is similar to the proof of Corollary\rl{corriem}. In fact, taking again $p_0\in\star$ a point where $p_0=\min_\Sigma h$ we have that $\Theta(p_0)=-1$ and $\kappa_i(p_0)\leq 0$ for every $i=1,\ldots,n$. Therefore, from the Gauss
equation \rf{eqgaussprod} we obtain that
\[
S=S(p_0)\geq
n\kappa=S_\mathbb{M}.
\]
The result is now a direct consequence of Theorem\rl{th5}.
\end{proof}

\subsection{Hypersurfaces in a Lorentzian product space} \label{SS42}
Let us consider now the case where $\star^n$ is a spacelike hypersurface immersed into a Lorentzian product space $\mbox{$\mathbb{M}^n$}\times\mathbb{R}_1$. Recall that in this case, $N$ denotes the future-directed Gauss map of \star\ and the function $\Theta$
satisfies $\Theta=\g{N}{\partial_t}\leq -1<0$. In particular, we always have $\Theta<0$, and $\Theta(p)=-1$ if and only if $p$ is a critical point of $h$. Reasoning now as in the Riemannian case, we can state the following results.
\pagebreak
\begin{theorem}
\label{th8} \
\begin{itemize}
\item[(i)] Let \mbox{$\mathbb{M}^2$}\ be a compact Riemannian surface with non-negative Gaussian curvature, $K_{\mathbb{M}}\geq 0$ (respectively, non-positive Gaussian curvature $K_{\mathbb{M}}\leq 0$), and assume that $K_{\mathbb{M}}>0$
(respectively, $K_{\mathbb{M}}<0$) on a dense subset of \mbox{$\mathbb{M}^2$}. The only compact spacelike surfaces \star\ in the Lorentzian product $\mbox{$\mathbb{M}^2$}\times\mathbb{R}_1$ satisfying $K\leq K_{\mathbb{M}}$ (respectively, $K\geq K_{\mathbb{M}}$) are
the slices.
\item[(ii)] Let \mbox{$\mathbb{M}^n$}\ be a compact Einstein Riemannian manifold, $n\geq 3$, with positive scalar curvature $S_{\mathbb{M}}>0$ (respectively, negative scalar curvature $S_{\mathbb{M}}<0$). The only compact spacelike hypersurfaces \star\ in the
Lorentzian product $\mbox{$\mathbb{M}^n$}\times\mathbb{R}_1$ satisfying $\sup_\Sigma S\leq S_{\mathbb{M}}$ (respectively, $\inf_\Sigma S\geq S_{\mathbb{M}}$) are the slices.
\end{itemize}
\end{theorem}

\begin{corollary}
\label{corlor} \
\begin{itemize}
\item[(i)] Let \mbox{$\mathbb{M}^2$}\ be either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$. The only complete spacelike surfaces \star\ in the Lorentzian product $\mbox{$\mathbb{M}^2$}\times\mathbb{R}_1$ with constant
Gaussian curvature are the slices.
\item[(ii)] Let \mbox{$\mathbb{M}^n$}\ be a compact Einstein Riemannian manifold, $n\geq 3$, with positive scalar curvature. The only compact spacelike hypersurfaces \star\ in the
Lorentzian product $\mbox{$\mathbb{M}^n$}\times\mathbb{R}_1$  with constant scalar curvature are the slices.
\end{itemize}
\end{corollary}

\section{Entire graphs in a product space}
\label{S5} As a nice consequence of the results stated in the previous section, in the $n=2$ dimensional case is the possibility of stating them in terms of entire graphs in a product space $\mbox{$\mathbb{M}^2$} \times \mathbb{R}$. Let $\Omega\subseteq\mbox{$\mathbb{M}^2$}$ be a
connected domain. Every smooth function $u\in\mathcal{C}^\infty(\Omega)$ determines a graph over $\Omega$ given by $\Sigma(u)=\{ (x,u(x)) : x\in\Omega \}\subset\mbox{$\mathbb{M}^2$}\times\mathbb{R}$. The metric induced on $\Omega$ from the metric on the
ambient space via $\Sigma(u)$ is given by \begin{equation} \label{gu} \g{}{}=\g{}{}_\mathbb{M}+\varepsilon du^2. \end{equation} In particular, in the case when \mbox{$\mathbb{M}^2$}\ is Lorentzian $\Sigma(u)$ is a spacelike surface in $\mbox{$\mathbb{M}^2$} \times \mathbb{R}_1$ if and only if
$|Du|^2<1$ everywhere on $\Omega$, where $Du$ denotes the gradient of $u$ with respect to the metric $\g{}{}_\mathbb{M}$ on $\Omega$ and $|Du|^2=\g{Du}{Du}_\mathbb{M}$. A graph is said to be entire if $\Omega=\mbox{$\mathbb{M}^2$}$.

Let $\star(u)$ be a graph over a domain $\Omega$, which is assumed to be spacelike when the ambient space is Lorentzian, and let us orient it by the normal field given by
\[
N=\frac{1}{\sqrt{1+\varepsilon|Du|^2}}(-\varepsilon\partial_t+Du),
\]
so that
\[
\Theta=\frac{-1}{\sqrt{1+\varepsilon|Du|^2}}<0.
\]
From the Gauss equation \rf{eqgaussprod}, we know that
\[
K=K_\mathbb{M}\Theta^2+\varepsilon\mathop\mathrm{det}\nolimits(A)=\frac{K_\mathbb{M}}{1+\varepsilon|Du|^2}+\varepsilon\mathop\mathrm{det}\nolimits(A).
\]
Take a local orthonormal frame $\{E_1, E_2\}$ on $\Omega$ (with respect to the metric $\g{}{}_\mathbb{M}$) and observe that
\[
\mathop\mathrm{det}\nolimits(A)=\frac{\mathop\mathrm{det}\nolimits(h_{ij})}{\mathop\mathrm{det}\nolimits(g_{ij})}
\]
where, by \rf{gu}, $g_{ij}=\g{E_i}{E_j}=\delta_{ij}+\varepsilon E_i(u)E_j(u)$, and, by a straightforward computation,
\[
h_{ij}=\g{AE_i}{E_j}=-\g{\mbox{$\overline{\nabla}$}_{E_i}N}{E_j}=\frac{-D^2u(E_i,E_j)}{\sqrt{1+\varepsilon|Du|^2}}.
\]
Therefore, the Gaussian curvature of $\Sigma(u)$ is given by
\[
K=\frac{K_\mathbb{M}}{1+\varepsilon|Du|^2}+\varepsilon\frac{\mathop\mathrm{det}\nolimits(D^2u)}{(1+\varepsilon|Du|^2)^2},
\]
with the restriction $|Du|^2<1$, when $\varepsilon=-1$.

Taking this into account, Corollaries\rl{corriem} and\rl{corlor} can be formulated in terms of graphs in the following way.
\begin{corollary}
\label{coro10} Let \mbox{$\mathbb{M}^2$}\ be either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$, and let $K$ be a real constant. The only entire solutions on \mbox{$\mathbb{M}^2$}\ to either
\[
(1+|Du|^2)^2K=1+|Du|^2+\mathop\mathrm{det}\nolimits(D^2u)
\]
or
\[
(1-|Du|^2)^2K=1-|Du|^2-\mathop\mathrm{det}\nolimits(D^2u), \quad |Du|^2<1,
\]
where $Du$ and $D^2u$ stand for the gradient and the Hessian of a function $u$ on \mbox{$\mathbb{M}^2$}, are the constant functions, and $K=1$ necessarily. In particular, if $K\neq 1$, there exist no entire solutions on \mbox{$\mathbb{M}^2$}\ to the two equations
above.
\end{corollary}
More generally, our Theorems\rl{th4} and\rl{th8} allow us to state the following result, where we do not assume that neither the Gaussian curvature of \mbox{$\mathbb{M}^2$}\ nor the Gaussian curvature of the graph are constant.
\begin{corollary}
\label{coro11} Let \mbox{$\mathbb{M}^2$}\ be a compact Riemannian surface with non-negative Gaussian curvature, $K_{\mathbb{M}}\geq 0$ (respectively, non-positive Gaussian curvature $K_{\mathbb{M}}\leq 0$), and assume that $K_{\mathbb{M}}>0$
(respectively, $K_{\mathbb{M}}<0$) on a dense subset of \mbox{$\mathbb{M}^2$}.
\begin{itemize}
\item[(i)] The only entire solutions on \mbox{$\mathbb{M}^2$}\ to
\[
(1+|Du|^2)^2K=(1+|Du|^2)K_\mathbb{M}+\mathop\mathrm{det}\nolimits(D^2u),
\]
where $K$ is a smooth function on \mbox{$\mathbb{M}^2$}\ satisfying $K\geq K_\mathbb{M}$ (respectively, $K\leq K_{\mathbb{M}}$) are the constant functions, and $K=K_\mathbb{M}$ necessarily.
\item[(ii)] The only entire solutions on \mbox{$\mathbb{M}^2$}\ to
\[
(1-|Du|^2)^2K=(1-|Du|^2)K_\mathbb{M}-\mathop\mathrm{det}\nolimits(D^2u), \quad |Du|^2<1,
\]
where $K$ is a smooth function on \mbox{$\mathbb{M}^2$}\ satisfying $K\leq K_\mathbb{M}$ (respectively, $K\geq K_{\mathbb{M}}$) are the constant functions, and $K=K_\mathbb{M}$ necessarily.
\end{itemize}
\end{corollary}

In contrast to Corollary\rl{coro10}, when $\mbox{$\mathbb{M}^2$}=\mathbb{H}^2$ is the hyperbolic plane, there exist entire non-trivial solutions to the corresponding constant Gaussian curvature equation, \begin{equation} \label{grafo1}
K=\frac{-1}{1+\varepsilon|Du|^2}+\varepsilon\frac{\mathop\mathrm{det}\nolimits(D^2u)}{(1+\varepsilon|Du|^2)^2}, \end{equation} with the restriction $|Du|^2<1$, when $\varepsilon=-1$. To see that, it will be appropriate to use the Minkowskian model of the hyperbolic
plane. Write $\mathbb{R}^{3}_1$ for $\mathbb{R}^{3}$, with canonical coordinates $x=(x_0,x_1,x_2)$, endowed with the Lorentzian metric \begin{equation} \label{lor} \g{}{}=-dx_0^2+dx_1^2+dx_2^2. \end{equation} The hyperbolic plane $\mathbb{H}^2$ is the
complete simply connected Riemannian surface with sectional curvature $-1$, which is realized as the hyperboloid
\[
\mathbb{H}^2=\{ x\in{\mathbb R}^{3}_1 : \g{x}{x}=-1, x_0\geq 1 \}\subset\mathbb{R}^3_1
\]
endowed with the Riemannian metric induced from ${\mathbb R}^{3}_1$.

\begin{example}
\label{exam}
\begin{rm}
Let us look for non-trivial solutions on $\mathbb{H}^2$ of \rf{grafo1} of the type $u(x)=f(x_0)$ for a certain smooth function $f(x_0)$ with $x_0\geq 1$. For such a function $u$, its gradient is $Du(x)=-f'(x_0)e_0^\top$,
where $e_0^\top$ is the tangent part of $e_0=(1,0,0)$ along $\mathbb{H}^2$, that is, \begin{equation} \label{grafo2} e_0=e_0^\top+x_0x. \end{equation} In particular, $|Du(x)|^2=f'(x_0)^2(x_0^2-1)$. On the other hand, taking covariant derivatives in
\rf{grafo2} we get
\[
D_X(Du)=f''(x_0)\g{X}{e_0^\top}e_0^\top+f'(x_0)x_0X
\]
for every vector field $X$ tangent to $\mathbb{H}^2$. It follows that
\[
\mathop\mathrm{det}\nolimits(D^2u)=x_0f'(x_0)f''(x_0)(x_0^2-1)+x_0^2f'(x_0)^2.
\]
Hence, equation \rf{grafo1} becomes \begin{equation} \label{grafo3} (1+\varepsilon f'(x_0)^2(x_0^2-1))^2K=-1-\varepsilon f'(x_0)^2(x_0^2-1)+\varepsilon(x_0f'(x_0)f''(x_0)(x_0^2-1)+x_0^2f'(x_0)^2). \end{equation} It can be easily checked that the solution
to the differential equation \rf{grafo3} is given, up to a constant, by
\[
f(x_0)=\sqrt{\frac{\varepsilon(1+K)}{-K}}\, \mathrm{log}\left(\sqrt{1-K(x_0^2-1)}+\sqrt{-K}x_0\right)
\]
where $K<0$ and $\varepsilon(1+K)>0$. Specifically,
\begin{itemize}
\item[$\bullet$] when $\varepsilon=1$, $-1<K<0$, and
\item[$\bullet$] when $\varepsilon=-1$, $K<-1$.
\end{itemize}
\end{rm}
\end{example}

Let us observe that when $\varepsilon=1$, any entire graph in $\mathbb{H}^2\times\mathbb{R}$ is complete. In particular, the entire graphs in our Example\rl{exam} are complete. However, when $\varepsilon=-1$ an entire spacelike
graph in $\mathbb{H}^2\times\mathbb{R}_1$ is not necessarily complete (see, for instance, \cite[Example 3.3]{Al}). Even more, there exist entire spacelike graphs with constant Gaussian curvature which are not complete; for instance
\cite[Example 3.1]{Al} is an example of a maximal entire graph with constant Gaussian curvature $-1$ in $\mathbb{H}^2\times\mathbb{R}$ which is not complete.

In our case, the completeness of the entire spacelike graphs given in Example\rl{exam} follows from the more general technical result.
\begin{lemma}
Let $\Sigma(u)$ be an entire spacelike graph in $\mathbb{M}^2\times\mathbb{R}_1$, where \mbox{$\mathbb{M}^2$}\ is a complete Riemannian surface. If
\[
\sup_{x\in\mathbb{M}}|Du(x)|^2<1,
\]
where $|Du|^2=\g{Du}{Du}_\mathbb{M}$, then $\Sigma(u)$ is complete.
\end{lemma}
\begin{proof}
Observe that the induced metric on $\mathbb{M}^2$ via the graph $\Sigma(u)$ is given by
\[
g(v,v)=\g{v}{v}_{\mathbb{M}}-\g{v}{Du}^2_{\mathbb{M}},
\]
and by Cauchy-Schwarz inequality we have
\[
g(v,v)\geq \g{v}{v}_{\mathbb{M}}(1-|Du|^2)\geq \g{v}{v}_{\mathbb{M}}(1-\sup_{x\in\mathbb{M}}|Du(x)|^2).
\]
Finally, since $(1-\sup_{x\in\mathbb{M}}|Du(x)|^2)>0$ and $\g{}{}_\mathbb{M}$ is complete, the metric $g$ is also complete on $\mathbb{M}^2$.
\end{proof}
Observe that in our case, for every $x\in\mathbb{H}^2$
\[
|Du(x)|^2=f'(x_0)^2(x^2_0-1)=\frac{-(1+K)(x^2_0-1)}{1-K(x^2_0-1)}, \quad \mathrm{with} \quad K<-1,
\]
and then
\[
\sup_{x\in\mathbb{H}^2}|Du(x)|^2=\sup_{x_0\geq 1}f'(x_0)^2(x_0^2-1)=1+\frac{1}{K}<1.
\]

\section*{Acknowledgements}
The authors would like to thank to the referee for valuable suggestions which improved the paper.


text:
\section{Introduction}
 Next-generation wireless communication technologies (IEEE $802.11$ ay, $5$G and beyond) can enable extremely high throughput applications due to their operation in the mm-Wave spectrum. They promise low latency and ultra-reliable packet delivery.
To overcome high path loss,
mm-Wave devices use directional radio beams for communication.
The device modems typically use small-sized arrays as they offer very high gains and a multitude of radiation patterns.
Deploying a large number of Base Stations (BS) and Access Points can therefore provide Line of Sight (LoS) communication with the mobiles.

However, the mm-Wave spectrum not only suffers from high path loss, but also high penetration loss \cite{penloss, _2017_study}.
Common media such as drywall, wood, glass, concrete, and the human body, cause severe signal degradation \cite{block1}.
Penetration losses increase
with the frequency of operation and can completely disrupt LoS communication.
During a blockage event, the Received Signal Strength (RSS) of the mobile reduces drastically and can result in link outage.
If the BS has to reacquire the link, the re-connection to a 5G New Radio (NR) Base Station can take up to a second \cite{_2017_nr}.\footnote{During initial access the mobile performs a spatial scan; it searches for the BS beam with the best RSS. The BS also periodically sweeps through all its beams. After acquisition, both BS and the mobile communicate in scheduled epochs using the beam acquired during the initial access. Sudden onset of blockage results in the mobile losing communication with BS over these epochs.
}
Such high connection latency severely impacts applications \cite{enable}.
In addition, the network reconnection process is also power hungry. Therefore, it is important to minimize the
need for such re-acquisitions.

In this paper we distinguish between ``transient" blockages and ``permanent" blockages. Transient blockages are defined as blockages which last
on the order of a hundred milliseconds, and which end after the blockage event is over.
Transient blockages can occur as the user walks past obstacles or other users walk across the directional mm-Wave link.
Permanent blockage occurs when the LoS beam between the mobile and the BS is permanently (i.e., for a very long time) blocked, and the only solution for
maintaining a connection is to switch to a LoS beam to a different BS that is not blocked, i.e., handoff.

The focus of this paper is on sustaining a communication link for control messages during a transient blockage.
Poor RSS during blockage events results in the mobile losing time synchronization with the BS. This results in the mobile missing future scheduled epochs for communication with the BS, during which the BS would have been able to align
its directional beam towards the mobile. The mobile relies on the signals transmitted during such epochs to combat high phase noise in the mm-wave spectrum.
It is therefore critical to sustain the control plane communication link during such blockage events so as to maintain time-synchronization with the BS.

 The ability to electronically steer the direction of beams can help the device to sustain a communication link by using alternate or Non-Line of Sight (NLoS) paths, when the direct LoS beam is blocked. NLoS paths are reflections from scatterers that exist indoors and urban outdoors. In our extensive experiments in office-like environments, we have found that there typically are available usable NLoS paths between the BS and the mobile. By maintaining the time synchronization and control plane communication with the base station using NLoS paths, mobile devices can sustain a control link through a transient blockage. This then allows recovery to a LoS link as soon as the blockage disappears.
 {Our experiments} have shown that the RSS on the NLoS paths is about  $10$ dB less than the LoS path in an indoor environment.  This is sufficient for sustaining a NLoS link over which control packets are transmitted.

To employ a NLoS link during blockage events, both the BS and the mobile need a mechanism to identify {the viable} NLoS paths. As the scattering environment changes with user mobility, a one-time environment scan to extract NLoS paths is not sufficient for mobiles. Moreover, such an approach increases the memory usage of the modem. Thus it is necessary to dynamically determine NLoS paths.

The UNBLOCK protocol is a low complex blockage recovery protocol which enables both BS and mobile to dynamically identify NLoS paths, and to then utilize them to maintain time-synchronization so as to preserve the link without need for re-acquisition. UNBLOCK's design is based on extensive mobility experiments using mm-wave software defined radios in indoor {office} environments. It uses an appropriate beam scanning interval that helps in conserving the mobile battery. Unblock employs the right NLoS beam discovery interval for both BS and mobile by observing the duration of transient blockage events, and beam coherence time of NLoS paths from pedestrian mobility experiments. The protocol fits into the framework of $5$G NR standards on which we elaborate in Section \ref{UNBLOCK application in 5G Newradio}.

The rest of the paper is organised as follows. Section \ref{challenges} presents the challenges to overcome to avoid outages to mobiles caused by blockages. Testbed and Experimentation is elaborated in Section \ref{testbed}. Our protocol is described in Section \ref{protocol}. Section \ref{results} provides implementation details and performance of protocol. Section \ref{5gap} maps the protocol  to 5G cellular standards. Section \ref{related}  presents existing work in the domain and Section \ref{conclude} concludes our work.

\section{Challenges}\label{challenges}

Communication devices operating in the mm-Wave spectrum, whether BS or mobile, utilize narrow radio beams to overcome high path loss, and need a beam alignment protocol to handle user mobility \cite{BM}.
Several in-band and sensor-assisted beam alignment protocols requiring the angle of arrival/departure predict the next best-aligned beam \cite{BA1, BA2,BA3,BA4,BA5}. Although these works address beam misalignment for static links, such predictive mechanisms do not recover link RSS   during LoS blockage events while the user is moving. Maintaining the communication link's RSS during an LoS blockage event perforce requires an NLoS/reflected path.
As the scattering environment changes with user mobility, a one-time environment scan is not sufficient to identify usable reflected paths.
Such reflected paths can therefore
be discovered only by active probing.

Let $ N_{BS} $ and $ N_{MS}$ be the number of  beams\footnote{In the description of
the following protocol, the BS is using its beams in transmit mode, while the mobile is using its beams in receive mode. By reciprocity, their roles can be revered to obtain the reverse link.}  available at the BS and the mobile respectively. For a given BS transmit beam, a mobile might discover that one or more of these $N_{MS}$ beams (used in receive mode) are NLoS paths. Similarly, for a given receive beam of the mobile, the BS might find one or more of its transmit beams as NLoS paths to the mobile. Both the BS and the mobile must periodically search for and \emph{store} the discovered NLoS paths,  while  communicating on the LoS beams. Storing the discovered paths for later use is necessary since the blockage events are unpredictable. When blockage happens the discovered NLoS beams in the memory are employed to revive link RSS during blockage. Taking the example of a static user where the scattering environment remains the same throughout, for a given LoS transmit beam of the BS, the mobile sweeps through all its $N_{MS}$ receive beams to identify available NLoS beams. Due to the presence of a rich scattering environment indoors, especially due to the walls, the mobile potentially discovers multiple NLoS paths. From our extensive experiments, we found that the RSS of these NLoS beams is typically about $10$ dB less than that of the LoS beams. By listening on the discovered NLoS beam, the mobile can help the BS discover its available NLoS transmit beams {corresponding to the receive beam discovered by the mobile}. To accomplish this, the BS transmits over all the $N_{BS}$ beams while the mobile is making measurements on the NLoS beam. At the end of this process, an NLoS BS-mobile beam pair is found. The BS and the mobile thereby store in their memory at least $N_{BS}$ NLoS beam pairs, one for each BS transmit beam, and use the appropriate stored pair, corresponding to the particular BS transmit beam at the onset of blockage, in case of blockage. Due to mobility,
the scattering environment changes over time, and so
the above protocol is periodically employed to discover NLoS beams at appropriate $100$ ms intervals.
\begin{comment}

{\color{red} Santosh I believe that this entire paragraph can be deleted.
Basically it is a table with $N_{BS}$ entries, that's all. In any case at the end of the day we anyway have no recommendation on how to optimize on-chip storage} Equipped with phased arrays  of a large number of antenna elements that provide high gain and more bits to control element weights, both BS and mobile can steer beams with a greater spatial resolution.  Typically, each phased array sweeps a single base station sector. Increasing antenna elements reduces the beamwidth of the array, often resulting in large numbers of non-overlapping beams within a sector. Storing a backup NLoS beam pair for each of the LoS beam pairs takes a good amount of on-chip memory. To avoid that, it benefits to find the least number of NLoS beams required to store that can reliably avoid link outage during the transient blockage.
\end{comment}

An important additional aspect to consider while designing a protocol for battery-driven mobiles is the reduction of modem power consumption. Frequent measurements using the beams to discover NLoS paths not only consume wireless resources, but persistent usage of the radio front end also consumes a significant amount of device power. To discover an NLoS beam pair, the mobile must first perform a spatial scan, measure RSS, and identify a good NLoS receive beam.
Then it initiates a BS transmit beam sweep to discover a good NLoS beam for the BS. With a large number of beams both at the BS and the mobile, this process requires the mobile to make a significant number of measurements. For example, with a beamwidth of $5^\circ$ to cover a $120^\circ$ sector, at least $24$ beams are needed; therefore the NLoS beam discovery process may require up to $48$ measurements. As the scattering environment changes with mobility, this process must be repeated periodically. The periodicity of this process depends on the Beam Coherence Time (BCT) \cite{BCT}, defined as the duration after which a beam change is necessary to restore the link RSS to the previous maximum. The NLoS beam discovery protocol must run at least once within each BCT on the mobile. The BCT is dependent on motion -- the faster a user is moving, the lesser is the BCT.

To summarize,  the complete protocol to recover the link RSS during blockage events has the following features:
\begin{itemize}
    \item A dynamic mechanism to discover NLoS beams both at BS and mobile.
    \item Storage of the most appropriate backup NLoS beam pairs.
    \item Optimization of the device power consumption by choosing the optimal NLoS beam pair re-scanning interval.

    \item Most importantly, employing the discovered NLoS beam pair when blockage happens, to avoid the need for network re-connection after transient blockage events.
\end{itemize}

The UNBLOCK protocol's parameters have been tuned based on extensive measurement data and it has been validated in various indoor office environments. We first ran extensive measurements in  Wisenbaker  Engineering Building, a three-storied, 30,000 sq. ft. department building \cite{_aggie}. Experiments have been performed in several classrooms, corridors, and graduate offices to study the RSS of reflected paths from various surfaces. We found that the RSS of the reflected beams is around $10$ to $14$ dB less than that of LoS Beams in these environments. Along with measurement studies, we did experiments to study the BCT using NLoS beams for several pedestrian motion patterns. Based on these observations, we have designed the UNBLOCK protocol to be a low complexity, power-efficient, and dynamic blockage recovery protocol.
\captionsetup[subfigure]{labelformat=empty}
  \begin{figure*}[t]
	\centering
	\begin{subfigure}[t]{.3\linewidth}
		\centering
		\includegraphics[width=2.2in, height=1.5in]{pic1.jpg}

		\label{fig:b1}
	\end{subfigure}
	\quad
  \begin{subfigure}[t]{.3\linewidth}
		\centering
		\includegraphics[width=2.2in, height=1.5in]{pic2.jpg}

	\end{subfigure}
	\quad
	\begin{subfigure}[t]{.3\linewidth}
		\centering
		\includegraphics[width=2.2in, height=1.5in]{pic3.jpg}

				\label{blockage}

	\end{subfigure}
		\caption{Indoor Measurement Environments ({From Left to Right: A large lab, a common area, and a corridor)}}
		\label{blockagemeasurement}
\end{figure*}
\section{Testbed and Experimentation}\label{testbed}
The fundamental goal of our experiments is to understand both the signal and temporal characteristics of  multipath/NLoS paths in mm-wave spectrum.
Our experiments have been focused on NLoS path behaviour in indoor environments where  reflected paths are typically present, as our experiments have shown. Accurate understanding of multipath characteristics is important to design a protocol to recover link RSS during blockage events. In crowded indoor environments like conference halls and classrooms, humans act as primary blockers, whereas in household and workplace environments, mobility of the user can result in transient blockage. We conducted experiments in various locations
of our large department building as a representative of office building environments, using $60$ GHz software defined radio testbed \cite{instruments_2020_introduction}.

\subsection{Testbed}\label{testbedsection}
The testbed is built around a National Instruments' software-defined mm-wave transceiver system with two nodes \cite{instruments_2020_introduction}.
The node has a chassis with high speed backplane interconnections between several slots. The FPGA cards inserted in these slots communicate with each other through backplace interconnections and are programmed to form transmit and receive chains of mm-Wave radios.  A Sibeam phased array is interfaced with each node. These phased arrays have $24$ antenna elements, $12$ each for transmit
and receive beamforming. With independent transmit and receive RF chains, this provides analog beamforming. Although it is possible to create a large number of beams with arrays, we used $25$ equally spaced narrow beams, roughly covering a sector of $120$ degrees, obtained by programming  $2$ bit weights for each element. The element phase weights are predetermined to obtain the desired beam patterns of approximately $20$ degrees beamwidth. The $25$ predetermined beams form the beamforming codebook. For ease of understanding, we refer to the two bidirectional nodes as ``base station" and ``user".

  In the measurement experiments, we transmit single carrier symbols of $2$ GHz bandwidth at $60$ GHz carrier frequency in a slot that has a duration of $100$ microseconds. A frame of $10$ ms duration is used for time division duplexed transmission. A frame is divided into $100$ slots, with the first $50$ slots for downlink and the rest for the uplink operation. $4$ slots in every $50$ with reference signals are used to time synchronize the base station and user nodes.

  During our measurement studies, the user node measures the RSS and signal-to-noise ratio (SNR) of transmission from the base station node. A beam index from the chosen codebook can be changed every slot; therefore, our codebook which has $25$ different indices can be swept in $25$ slots, i.e., $2.5$ ms. We performed experiments with two different codebooks, covering azimuth and full space. Beams in each codebook approximately cover $120^{\circ}$ sector. Our narrow beam codebook has an approximate beamwidth of $20^{\circ}$.

\subsection{RSS of Reflected paths}

Upon encountering a medium,
electromagnetic radiation may be transmitted, reflected, or absorbed. The reflectivity of radiation from a surface usually increases as the frequency increases. Based on the penetration depth, radiation may transmit through or get completely absorbed by the medium. Because of its shorter wavelength, mm-wave radiation is reflected by common building materials. Penetration and reflection losses \cite{a1_analysis} from our experiments for some of the common obstacles are tabulated in the Table \ref{losses}.

\begin{table}[H]
\small
    \caption{Penetration and Reflection Losses}\label{losses}
	\centering

		\begin{tabular}{ |P{2 cm}|P{2cm}|P{2cm}|}
\hline
\hline
    {Material } & {Penetration Loss (dB)} & {Reflection Loss (dB)} \\
 \hline\hline
  Drywall - single layer & 8  &  10 \\
 \hline
Drywall - double layer & 16 & 10 \\
 \hline

Wooden door & NF & 11 \\
 \hline

 Concrete wall & NF & 13 \\
  \hline
Human Body & NF & 14-20 \\
 \hline
\hline
\end{tabular}
\end{table}

\begin{comment}

{\color{red} It would be good to write the next three sentences more directly. Say that the reflected radiation from several surfaces exceeds the noise floor of SPECIFY dBm , and so these reflected paths can be used to maintain the link, perhaps at a reduced rate, during the temporary blockage}
\end{comment}
The reflected radiation from typical building surfaces is measurable and usually above the noise threshold. The total noise power from all the sources in our system at the room temperature for a $2$ GHz bandwidth system is
$-73$ dBm, in the scenarios where RSS is below this noise threshold, we indicated Noise Floor (NF){\footnote{Wireless communication system cannot detect any signal below its NF.}}.
Therefore we can rely on these reflected paths to maintain the link, perhaps at a reduced rate, during the temporary blockage.

To quantify the RSS of reflected signals, measurements were taken at more than $50$ locations inside the building. In particular, several measurements were taken in office-like environments, corridors, and spaces with concrete walls and dry walls. Fig. \ref{blockagemeasurement} shows some of the test environments. We observed that, on average, the RSS of reflected paths from wooden walls is $10$ dB lesser than that of the LoS path, whereas it is $13$ dB lesser from concrete walls. To make accurate measurements of RSS on the reflected beams, it was ensured that the additional path loss between the reflected path and LoS path was negligible. Through observations, it was found that narrow NLoS beams have better RSS than wider beams, but system designer must also take into account the trade-off between NLoS beam discovery latency and the gains offered for a given beamwidth. As the spatial scan area is lesser for narrow beams, more measurements are required to discover NLoS beams. From the experiments, it was observed that to recover from blockage during mobility, it suffices to keep just one beam,  the best reflected/NLoS beam, in the memory.

Another important observation from the experiments is that during the blockage events, NLoS beam operation at the mobile alone
can recover RSS and maintain the link in the scenarios where the RSS is above the noise floor of the mobile's receiver. Optimizing both NLoS beams, both at the BS and the mobile, however, can improve the link RSS up to $10$ dB more than that of using NLoS beam alone at the mobile.

\subsection{Beam Coherence Time}
It is equally important to study how the RSS of a particular reflected beam changes with user mobility. As the user moves from one location to another, the receive beam of the mobile has to be changed to better align with that of the BS beam. As the alignment is disrupted over time, the RSS of the beam is degraded. How long a particular mobile beam remains aligned with that of the BS depends on the mobility of the user. As we rely on scatterers for blockage recovery, which are more common in indoor rather than outdoor environments, the focus was on pedestrian mobility. The temporal changes in RSS for both translational and rotational motion patterns of the user were studied. For any beam pattern, the high gain region is within  $3$ dB  from the maximum or between the  half power points of the main lobe. Beyond the the $3$ dB beamwidth, the gain in the angular direction falls off significantly from the peak and is often unpredictable. Fig. \ref{bmp}
shows the radiation pattern of the beams used in the experiments. Therefore, it is necessary to change the beam after misalignment results in  more than $3$ dB loss. We formally define this duration as the ``Beam Coherence Time" \cite{bct1, bct2}. It is the duration after which the RSS on a beam drops by $3$ dB from the maximum.
\begin{figure}[h]
  \centering
 \includegraphics[width=1\linewidth, height=1.5in]{bm1.PNG}
  \caption{Radiation Pattern }
  \label{bmp}

\end{figure}

Similar to the measurement experiments, mobility experiments were performed at several locations in the building. Different mobility patterns like human walk, hand movements, and array rotation at multiple angular speeds, were studied. Knowledge of BCT helps in determining the NLoS beam discovery interval. Choosing the right interval for beam discovery not only helps in reliably preserving communication during the blockage but also reduces the device power consumption. Table \ref{BCT}a presents the BCT for users walking at distances of $5$ m and $10$ m from the BS, with the mobile using a fixed NLoS beam to communicate with the BS. Table \ref{BCT}b shows the BCT for multiple angular speeds using a NLoS beam,  when the distance between BS and mobile is $5$ m.
\begin{table}[h]
	\centering
	\small
	\caption{Mean BCT}\label{BCT}
	\begin{subtable}[t]{1.5in}
		\centering
		\begin{tabular}{ |p{.75cm}|p{1.1cm}|p{.75cm}| }
\hline
 \hline
 Speed (m/s)  & Distance (m) & BCT (ms)\\
 \hline\hline
  .8 & 5 & 600  \\
 .8 & 10 & 800  \\
  1.4 & 5 & 470  \\
 1.4 & 10 & 750  \\

 \hline
\hline
\end{tabular}
		\caption{Translational Motion}\label{table:1a}
	\end{subtable}
	\quad
	\begin{subtable}[t]{1.5in}
		\centering
		\begin{tabular}{ |p{.75cm}|p{1.1cm}|p{.75cm}| }
\hline
 \hline
 Speed (rad/s)  & Distance (m) & BCT (ms)\\
 \hline\hline
  2$\pi$/9 & 5 & 284  \\
 $\pi$/3 & 5 & 200  \\
 2$\pi$/3 & 5 & 141  \\
 4$\pi$/3 & 5 & 101  \\
 \hline
\hline
\end{tabular}
		\caption{Rotational Motion}\label{table:1b}
	\end{subtable}
\label{table2}
\end{table}
The fundamental idea behind using BCT to determine the NLoS the beam re-scanning interval is that there is likely to be a LoS beam update after this duration, either at BS or user, to overcome the misalignment caused by the user mobility. If the LoS beam either at the BS or user is changed, then the previously discovered NLoS beam is no longer  valid to recover from blockage, and must be updated to account for  mobility of the user. From the experiments, it was observed that rotational motion of the user quickly degrades the RSS, and hence the BCT is lesser compared to lateral user motion patterns like walking. Also, use of narrow beams demand frequent updates of NLoS beams, just as it does for LoS beams. It is expected that rapid hand movements while users are interacting  with the mobile result in faster angular speeds, and so have shorter BCT duration. This can especially happen while users are playing mobile games or running Virtual Reality (VR) applications. Other movements, such as swinging hands, head roll, and body rotation, although slower, can cause intermittent  blockage.

Unlike RSS improvement with NLoS beam operation at the BS in conjunction with NLoS beam at the user, the BCT remains unchanged, as it is purely dependent on the motion pattern of the user. Additionally, the BCT duration becomes larger when the user moves farther from the BS because of the smaller subtended angle.

\section{UNBLOCK}\label{protocol}
Before describing the protocol for blockage recovery, we elaborate on the NLoS beam discovery procedure both at the BS and the mobile. First, the mobile, while listening to a  BS beam using its LoS beam, performs a spatial scan during the available communication epochs with the BS. The mobile measures the RSS on all its beams for a particular BS LoS transmit beam, and discovers NLoS beams with RSS  within   $10$ dB of its LoS Beam. At the end of this procedure, the mobile has obtained an NLoS path to the BS. Next the BS needs to optimize its NLoS beam; this can be achieved by the mobile measuring its RSS using the discovered NLoS beam, while the BS is sweeping over its transmit beams. The mobile communicates back to the BS the identity of the transmit beam that has the highest RSS. Thereby, both BS and mobile discover their respective NLoS beams that can be used in case of blockage.

Based on the BCT data in Table \ref{BCT}, the {smallest} BCT is around $100$ ms. Therefore, the NLoS beam discovery procedure is repeated every $100$ milliseconds. Fig. \ref{statemachine} shows the State machine of the UNBLOCK protocol.
\begin{figure}[h]
  \centering
 \includegraphics[width=.8\linewidth]{statemachine.PNG}
  \caption{UNBLOCK State Machine }
  \label{statemachine}

\end{figure}
\begin{figure*}[t]
  \centering
 \includegraphics[width=\textwidth]{frame.PNG}
  \caption{5G NR Frame}
  \label{5gframe}

\end{figure*}
To limit the discussion to the scope of this work, we assume that both BS and mobile run a beam alignment protocol to align their LoS beams to account for user mobility. This happens in the State of ``Beam Adaptation" (BA). The ``NO" state denotes Normal Operation, in which both the mobile and BS continue to use an aligned LoS beam pair. These LoS beams are acquired during the initial acquisition. By the end of the initial acquisition process, both BS and mobile determine their respective LoS beams. Over time, this alignment gets disturbed because of user mobility. Whenever the mobile finds that its current RSS has dropped by $3$ dB from the previous measurement occasion, it moves to the ``Beam Adaptation" (BA) state. In the BA state, the mobile first tries to recover the RSS by aligning its LoS beam, and, later, the BS adapts the LoS beam on an as-needed basis. Upon successful alignment, they continue in the ``NO" state.

Periodically, in our case, every $100$ ms, the mobile moves to the ``MS NBD" state, i.e, ``Mobile Side NLoS Beam Discovery. In this state, the mobile performs a spatial scan to find an NLoS receive beam for the current LoS beam of the BS. The mobile measures the RSS on each of its beams and identifies beams with RSS within 10 dB of the current LoS beam. Among all the NLoS beams discovered, it stores the beam index with the highest RSS. While the mobile is listening to the BS on the NLoS beam, ``Base Station NLoS Beam Discovery" (BS NBD) is triggered. In ``BS NBD", the BS sweeps over all beams, one at a time, and the mobile measures the RSS of the discovered NLoS beam. Upon the mobile finding the BS beam with the highest RSS, it communicates the beam information to BS. The BS stores the NLoS beam information. This process is repeated every $100$ ms to account for changing scatterers caused by user mobility.

If at anytime during ``NO" operation, the current RSS of the mobile drops by $10$ dB from that of the previous measurement, the mobile first moves to the ``NBO" state. In this ``NLoS Beam Operation" state, the mobile first reverts to the stored NLoS beam and informs the BS via the control packet communication to switch the BS beam to the NLoS beam. Upon detecting the blockage event from the control packets, the BS switches to the NLoS beam, retrieving the stored information from its memory. There are ample measurement opportunities in the 3GPP NR standards to discover NLoS beams. Section \ref{UNBLOCK application in 5G Newradio} presents how UNBLOCK can run in adherence to the existing standards.

\section{Implementation and Evaluation} \label{results}
    We implemented the UNBLOCK protocol on a $60$ GHz software defined radio tested. In the testbed, a beam can be changed every $100$ $\mu$s. With $25$ beams in the beamforming codebook, it takes $2.5$ ms each at mobile and BS to search over all the beams. Every $100$ ms, the UNBLOCK protocol on the testbed needs $5$ ms to identify an NLoS backup beam pair. The Beam Alignment protocol proposed in \cite{ganji_10_beamsurfer} for LoS Beam Adaptation was implemented to evaluate UNBLOCK under user mobility. The UNBLOCK protocol was programmed in labVIEW software both at the BS and the mobile nodes. UNBLOCK was evaluated in two different scenarios.
    \begin{itemize}
        \item BS and mobile are stationary, while a human walks around and blocks the LoS beam pair.
        \item BS and blockers are stationary, while a user with a phased array walks in such a way that it obstructs the LoS path to the BS.
    \end{itemize}
     The experiments were repeated in several locations in the building. It is observed that in both scenarios, UNBLOCK is successful in recovering the link during blockage, and thereby preserving the time synchronization.

Fig. \ref{result1} shows a blockage event in an experiment in which a human walks across the path between a static BS and a mobile and briefly blocks the link. It can be observed that the blockage occurs quickly; therefore, it is not possible to predict such events. In Fig. \ref{result1}, the RSS of the LoS link RSS, which is $-58$ dBm, falls to -$72$ dBm in $35$ milliseconds. The noise floor, which is the total noise power from all the sources in our transceiver is $-74$ dBm
  system with $2$ GHz bandwidth is . In the communication system with $2$ GHz operating bandwidth, the thermal noise at room temperature alone is $-80.8$ dBm. When the RSS hits NF, the mobile loses time synchronization and connectivity with the BS. Without the UNBLOCK protocol, one can observe that the mobile would lose synchronization with the BS and will need to re-initiate the initial search process. However, when the UNBLOCK protocol is running, it detects the RSS drop at around $50$ ms,  and the backup beam pair in the memory maintains link RSS to preserve time synchronization. The experiments were repeated at $50$ different locations with the BS remaining static, while the user with the phased array in hand either turns away from the BS to obstruct the LoS links with his body, or walks past a metal board that blocks LoS link. It was observed that the UNBLOCK protocol preserves time synchronization $96\%$ of the experiments.
    \begin{figure}[h]
  \centering
 \includegraphics[width=.8\linewidth]{result1.eps}
  \caption{Blockage Recovery}
  \label{result1}

\end{figure}
\section{UNBLOCK for 5G NR } \label{5gap}

 \label{UNBLOCK application in 5G Newradio} \label{UNBLOCK application in 5G Newradio}
Beam Management at the base station, called gNodeB, and user equipment (UE) in $5$G Newradio standards can be broadly divided into two phases:

\begin{itemize}
    \item Beam Selection phase. During this phase, a mobile performs Acquisition/Re-Acquisition.
    \item Beam Refinement phase. This allows both the BS and mobile to perform Receive/Transmit Beam Adaptation periodically, determine a good reflected beam, and perform blockage recovery upon onset of blockage.
\end{itemize}

The 3GPP standards \cite{_2017_nr} make provisions to complete these two phases at gNodeB and UE to identify an aligned beam pair.
We describe how UNBLOCK can be implemented using the mechanisms available in the standards for these phases.

The gNodeB transmits  synchronization signal blocks (SSB), control information, and broadcast information necessary for UEs to discover and connect to it. This is done with up to $64$ different beams within a sector, every two frames.   Communication between gNodeB and several UEs happens within a frame, which is of $10$ ms duration.  Each frame is further divided into slots. Based on the configuration of deployment, the number of slots in a frame can be chosen from the set $\{10, 20, 40, 80, 160\}$.
mm-Wave 5G standards are time-duplexed, meaning the gNodeB and the UE communicate in a synchronized order, but not simultaneously. In a slot, either downlink/uplink data or control information
 are transmitted. Within each slot, each SSB is transmitted on a particular beam, and there can be up to $4$ SSB blocks within a slot. Fig. \ref{5gframe} shows a $5G$ NR frame.

 When UE is turned on, it first searches for these SSB beams and time synchronizes with  gNodeB. As there is no prior synchronization with gNodeB, the schedule of these SSBs is unknown and UE has to wait at least $20$ ms on a receive beam. As there are $64$ SSBs,
 it takes $1.28$ seconds to test all the gNodeB beams.

 However, upon successfully connecting to gNodeB, UE is time-synchronized with gNodeB, and is also aware of the SSB beams' temporal locations. In addition, the gNodeB changes their periodicity to $5$ ms from $20$ ms for the particular UE after initial access procedure. Hence, UE has ample opportunity to harvest the reflected beams of the transmitting/gNodeB beam every $100$ ms.

\subsection{Beam Selection in 5G Newradio} \label{Beam Selection in 5G Newradio}
As the first step in the network connection procedure, a user device operating in the mm-wave spectrum
initiates gNodeB discovery by searching for synchronization signals. It is a directional search using several beams to identify the direction of the best possible RSS. The gNodeB sweeps all $64$ SSBs consecutively, and repeats the sweep every $20$ ms.  UE discovers either one or multiple SSB beams and uses one of those beams to communicate back to gNodeB to complete
the connection. gNodeB changes the  periodicity of these SSB bursts to $5$ ms for UEs after acquistion. Therefore, using the UNBLOCK protocol, it has enough opportunities not only to make measurements on the current receive beam but also to identify the reflected NLoS beams of a transmitting beam of the gNodeB.
Upon listening to UE's connection request on one of its SSB beams,  gNodeB designates that SSB beam as a communicating beam to that UE. By the end of this procedure, both  gNodeB and UE identify the transmit-receive beam pair for communication.  UE receives all subsequent SSB, channel state information reference signals (CSI-RS) and scheduling information once it connects to the gNodeB.

\subsection{Beam Refinement in 5G Newradio} \label{Beam Refinement in 5G Newradio}
After Beam Selection, the gNodeB assigns each user a set of beams with either SSB or CSI-RS resources for channel quality reporting.
This helps the gNodeB manage its beam  as and when necessary,
since user motion or hand movements quickly disrupt beam alignment.  In the beam refinement phase, the gNode and UE adapt the transmit-receive configuration identified during the initial connection to handle mobility and to improve the quality of service.
UE also reports channel state information to gNodeB, which gNodeB uses to adapt its transmit beams.
As the user moves, the previously identified beams might become stale, either through misalignment or blockage, and new beams are necessary to sustain connectivity. When beam misalignment is detected either at gNodeB through reports
from UE, or at UE from measurements, any beam adaptation protocol can be employed to manage the Transmit and Receive beams.

UNBLOCK at the mobile side can identify NLoS beams for a given gNode B transmit beam either using SSB resources or CSI-RS signals scheduled on that beam. To find the NLoS transmit beam of gNode, UNBLOCK can use the resources available either for Beam Selection or Beam Refinement. In the case of the Beam Selection phase, gNode B sweeps over all its $64$ transmit beams in $5$ ms. UNBLOCK can also identify the NLoS gNodeB transmit beam from an exclusive set of beams allocated by the gNodeB. It is also possible to use beams in both Beam Selection phase and beam refinement phase to quickly narrow down the NLoS beams. Unlike our implementation, beams can be changed on a per symbol basis. Symbols within SSB are of either $4.46$ or $8.92$ microseconds duration, and CSI RS duration can be $8.92$ microseconds. Therefore, the mobile can use a large number of narrow beams to harvest reflected paths.  Although faster search methods like hierarchical search can be employed, to present the fundamental idea of UNBLOCK we limit the discussion to an exhaustive search.

\section{Related Work} \label{related}
The authors of \cite{blockage_study} have proposed a measurement system and conducted human blockage experiments using a similar testbed \cite{instruments_2020_introduction}. Our experiments and the NYU measurement campaign \cite{blockage_study} confirm the existence of rich multipaths in indoor environments at mm-wave frequencies. The work \cite{adapoon} modelled the common causes of packet decoding errors in mm-wave communications as a linear dynamical system and proposed tests to identify the cause of an error. It identifies blockage after its occurrence, and is a reactive solution unlike the pro-active UNBLOCK protocol which always maintains a backup beam pair. Using multihop mm-Wave network to avoid blockage by forwarding packets to the destination via unblocked links is studied in the work \cite{8580773}.
In the paper \cite{antenna_diversity}, hybrid beamforming is explored to instantaneously recover from blockage. The work proposes antenna diversity to design and maintain several beams in different directions at the transmitter and receiver. In case an antenna beam suffers from blockage, the proposed method invokes one of the beams from the available. Despite requiring simultaneously active beams, the method still needs a search to identify the unblocked beams.  Another reactive approach to overcome blockage is presented in \cite{8589087}. Once blockage occurs, several beams are used simultaneously in this approach \cite{8580773}. Most of the works in the literature propose reactive mechanisms to overcome blockage; in contrast, the UNBLOCK protocol is a pro-active dynamic approach to avoid outage during user mobility and always keep a backup beam pair to preserve time synchronization for immediate use when blockage happens.
\section{Conclusion} \label{conclude}
mm-Wave signals are highly susceptible to blockage. Avoiding link outage and preserving time synchronization during blockage events is important to conserving mobile device power and circumventing high reconnection latency. The UNBLOCK protocol harvests viable
NLoS paths in a timely fashion and immediately employs them upon the onset of blockage. Thereby, it ensures that time synchronization with the BS for mobile devices is maintained during transient blockage events. When the transient blockage disappears, the LoS link is immediately restored by communicating
at those epochs dedicated by the BS for the particular mobile. Measurement experiments and evaluation on software defined radios are reported to assess the efficacy of the UNBLOCK protocol.

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