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\begin{document}




\title{\LARGE An extension of a theorem of Serrin
to\\ graphs in warped products
\footnotetext{{\it Math. Subject Classification.}
53A10, 35J60, 53C42}
\footnotetext{{\it Key words and phrases.} Constant mean curvature graph, warped product.}
}\author{By {\it Marcos Dajczer and Jaime Ripoll}}
\date{}
\maketitle



\begin{abstract}
In this article we extend a well known theorem of  J.
Serrin about existence and uniqueness of  graphs of constant mean
curvature in  Euclidean space to a broad class of  Riemannian manifolds.
Our result also generalizes several others proved recently
and includes the new case of  Euclidean ``rotational" graphs with
constant mean curvature.
\end{abstract}

\medskip

A well known theorem on elliptic PDE's due to J. Serrin \cite {se}
establishes  existence and uniqueness of a solution $u\in
C^{2,\alpha}(\bar \Omega)$ to the Dirichlet problem
\begin{equation}\label{serrin1}
\left\{ \begin{array}{ll}
div\,\displaystyle{\frac{grad\, u}
{\sqrt{1+|grad\, u|^{2}}}+2H=0} \\
u|_{\partial \Omega}=\varphi
\end{array}\right.
\end{equation}
for any given $\varphi \in C^{2,\alpha}(\partial \Omega)$. Here $H\ge 0$
is a real number and $\Omega$ a bounded $C^{2,\alpha }$ domain in the $xy$
plane satisfying the geometric condition $k_{\Gamma }\geq 2H$, being
$k_{\Gamma }$ the curvature of the boundary curve $\Gamma =\partial
\Omega$. Moreover, $div$ and $grad$ are the usual divergence and gradient
operators on the Euclidean plane. It follows that the graph of a solution
$u$ of (\ref{serrin1}) is a surface in $\R^3$ with constant mean curvature
$H$ ({\it CMC $H$-graph}) when oriented with a unit normal vector $\eta$
satisfying $\left\langle\eta,(0,0,1)\right\rangle \leq 0$.

 Serrin's result has been recently extended from  Euclidean
to other ambient spaces. In \cite{gs} and \cite{ni} extensions to
hyperbolic space $\Hy^{\,3}$ are obtained for two different types of
graphs. Theorem $6$ in \cite{gs} considers graphs along horocycles ({\it
parabolic graphs\/}) whereas  Theorem $4$ in \cite{ni} along hypercycles
({\it hyperbolic graphs\/}), both,  horocycles and hypercycles being
orthogonal to a bounded domain contained in a totally geodesic surface. If
the ``cylinder" generated by  horocycles or hypercycles over the boundary
of the domain  has mean curvature larger than or equal to a given $H\geq
0$ ({\em strictly\/} larger in Theorem~$4$) then it is proved that the
associated Dirichlet problem has a unique solution for any smooth boundary
data. Existence and uniqueness for $H$-graphs in $\Sf_{+}^{2}\times\R$
over a bounded domain $\Omega\in\Sf_{+}^{2}$  was established in
\cite{hlr} when $\partial\Omega\times\R$ has mean curvature {\em
strictly\/} larger  than $H\ge 0$.


In this article we obtain a unifying result that applies
to a broad class of ambient manifolds. It  extends Serrin's theorem, the one in \cite{gs}
and  generalizes the aforementioned theorems in \cite{hlr} and \cite{ni}.


 It is a standard fact that $\Sf^3,$ $\R^3\,$ and $\Hy^{\,3}$ admit  \mbox{{\it
 warped product representations\/}} as $S\times_\rho L$
where $L$ is any complete circle (i.e., a totally umbilical curve) and $S$
an open subset of the unique totally geodesic surface orthogonal to $L$ at
a given point $p\in L$. In the spherical case, considering the standard
inclusion $\Sf^3\subset\R^4$ then the  warping function $\rho$ is the
restriction to $S\subset\Sf^3$ of the height function $y\mapsto
-\<y,\delta(p)\>$ in $\R^{\,4}$, where $\delta(p)$ is the curvature vector
of $L$ at $p$ in $\R^{\,4}$. Hence, $\rho(x)=-\<x,\delta(p)\>.$ The
hyperbolic case is similar but now $L$ is a circle, a hypercycle or a
horocycle, and $\Hy^3\subset\Les^4$ where $\Les^4$ is the standard flat
Lorentzian space. Finally, in the flat case, $\rho(x)=
1-\<x-p,\delta(p)\>$ where $\delta(p) $ is the curvature vector of $L$ in
$\R^3$ at a given point $p\in L$. See \cite{no} for more details.


Warped product representations allow a natural notion of {\it graph\/} by
taking as \mbox{$L$-coordinate} a function on $S$. When $L$ is compact
($L=\Sf^1$) and since we are working with graphs, we may replace $L$ by
$\R$ and work in the covering space with the induced metric. Since the
circles $L$  are the trajectories of a Killing field  orthogonal to $S$,
such a graph has an equivalent description (see \cite{bh}) as a {\it
section transversal to the~orbits~of a one-parameter subgroup of
isometries orthogonal to a domain contained in a totally geodesic
surface}. In a special case, a graph of this type was properly called a
{\it Killing graph\/} in \cite{nse}. This discussion naturally suggests to
work in the following general setting that includes the ones in \cite{gs},
\cite{hlr}, \cite{ni} and \cite{se}.


Let $S$ be a  two-dimensional simply connected
Riemannian manifold and let $N$ be the Riemannian warped product $N=S\times_\rho\R$ with warping function
$\rho\in C^{\infty }(S)$.
 We define the {\it graph\/} $\mbox{Gr}(u)\subset
{\cal C}=\bar{\Omega}\times _{\rho |_{\bar{\Omega}}}\R$
 of $u\in C^{0}(\bar{\Omega})$ on a domain $\Omega\subset S$ as
$$
\mbox{Gr}(u)=\{(x,u(x))\in N:x\in \bar{\Omega}\}.
$$
Let $T$ be the Killing field $T=d/dt$, where $t$
is the parameter for $\R$. We denote
$$
R_0=\min_{\cal C}\mbox{Ric}_{N}\;\;\;\;\mbox{and}\;\;\;\;
K_0=\max_{\cal C}\mbox{K}_{N}
$$
where $\mbox{Ric}_{N}$ stands for the normalized Ricci curvature tensor
and $\mbox{K}_{N}$ the sectional curvature of $N$.

\newpage

\begin{theorem}\po\label{main} Let  $\Omega\subset S$ be a $C^{2,\alpha }$
domain contained in a normal geodesic disk $D\subset S$ with radius
$r_0$. Let $\varphi\in C^{2,\alpha}(\partial\Omega)$ and $H\geq 0$ be given.
We require  that $H^{\prime}\geq H$, where
$H^{\prime }$ is the mean curvature of
$\partial {\cal C}$ with respect to the inner orientation. If the Ricci and
sectional curvature of $N$ satisfy $R_0<\min\,\{0, 2K_0\}$ and
$H^{2}\leq-R_0/2$ we assume, in addition, that
\begin{equation}\label{comp}
r_0\leq\left\{
\begin{array}
[c]{ll}
\frac{1}{\sqrt{K_0}}\cot^{-1}
\,(\frac{H}{\sqrt{K_0}}) & \;\;\;
\mbox{if }K_0>0\vspace*{1ex}\\
\frac{1}{H} & \;\;\;\mbox{if }K_0=0\vspace*{1ex}\\
\frac{1}{\sqrt{-K_0}}\coth^{-1}\,(\frac{H}{\sqrt{-K_0}}) & \;\;\;\mbox{if }K_0<0.
\end{array}\right.
\end{equation}
Then the following holds:
\begin{itemize}
\item[(i)]  There exists a unique function $u\in C^{2,\alpha}(\bar\Omega)$
satisfying $u|_{\partial\Omega}=\varphi$ such that the graph $M=\mbox{Gr}(u)$
has constant mean curvature $H$ with respect to the unit normal vector field $\eta$ satisfying $\<\eta,T\>\leq 0$.
\item[(ii)]  If $\tilde{M}\subset {\cal C}$ is a compact connected cmc $H$-surface with
$\partial\tilde M=\partial M$ that induces on $\partial M$ the same orientation as $M$, then~$\tilde{M}=M$.
\end{itemize}
\end{theorem}



Observe that there is no restriction on the domain besides lying in a
normal disk in the following situations: $(a)$ $R_0\geq 0$, $(b)$
$2K_0\leq R_0<0$ and $(c)$ $H^2>-K_0>0$. In particular, there is no
restriction on the domain when the ambient space has constant sectional
curvature. Moreover, in certain cases it is sufficient to assume
(\ref{comp}) only for $H^{2}<-R_0/2$ (see Remark \ref{re}) .

If the assumption
$H\leq H^{\prime }$ fails at some point then the result in part $(i)$ of
Theorem \ref{main} is no longer true:
there is a boundary data $\varphi $ for which no solution exists. This is
well known in the Euclidean space for standard graphs and has  been proved
for two kinds of graphs in the hyperbolic space (see \cite{gs} and \cite{ni}).
Notice that the condition $H\leq H^{\prime }$
is equivalent to $k_{\Gamma }+\alpha _{\Gamma}\geq 2H$, where
$\alpha _{\Gamma }$ is the normal curvature of the factor $\R$
with respect to the inner normal vector of~$\Omega$.

It is well known that part $(ii)$ of Theorem \ref{main} holds in Euclidean
space for standard CMC graphs and is no longer true if $\tilde{M}$ is not
assumed to be contained in ${\cal C}$. The former claim follows from
Rado's Theorem. For the second statement consider a usual right circular
cylinder in $\R^{3}$, and take two spherical caps, a small  and a large
one, with a common circle as boundary.

Notice that Theorem \ref{main} is interesting even  for $\R^{3}$ since we
may consider (rotational) CMC $H$-graphs contained inside a torus obtained
by rotating $\bar{\Omega}\subset\R^{2}$ around a line in $\R^{2}$ disjoint
from  $\bar{\Omega}$ and apply the theorem to the universal cover with the
induced metric. Observe that the assumptions of the theorem do {\it not\/}
imply the convexity of $\Omega$.

Any CMC $H$-surface given by Theorem \ref{main} is stable since the
argument given in Remark 2.1 of \cite{nr} works in our case.

In the particular case of Euclidean graphs with boundary $\partial \Omega$
we show that any $C^1$ (up to $\partial \Omega$) rotational CMC $H$-graph
must be a standard graph. This result combined with Theorem~\ref{main}
provides a sufficient condition for the existence of standard CMC
\mbox{$H$-graphs} with planar but not necessarily convex boundary; see
Theorem \ref{planar}. Results of this nature but under different
assumptions were given in \cite{ri}.

Other extensions of Serrin's theorem to hyperbolic space were obtained in
\cite{lm} and \cite{ns}. These articles  solve the Dirichlet problem for a
CMC surface equation for graphs that are  sections transversal to the
geodesics orthogonal to a domain contained in a horosphere. Since the
geodesics are not the trajectories of a Killing field the graphs are not
of the type considered in this paper. However, making use of the
uniqueness part of Theorem \ref{main}, we show that those with strictly
convex boundary and $|H|\le 1$ are also CMC \mbox{$H$-graphs} in our
sense. More precisely, as an immediate consequence of Theorem \ref{last}
below we prove that they are also hyperbolic~graphs.  \vspace{3ex}





\noindent {\bf\Large \S 1. The proofs}
\vspace{3ex}


The proof of the existence part of Theorem \ref{main} is
obtained by formulating it in terms of a Dirichlet problem
of a quasilinear elliptic PDE and applying standard techniques.


\begin{proposition}\po\label{three}
Given $H\geq 0$, the graph $\Gr(u)$ of $u\in C^{2}(\Omega)$ in $N=S\times_\rho\R$ has constant
mean curvature $H$ with respect to a unit normal vector
$\eta$ satisfying $\<\eta,T\> \leq 0$ if and only
if $u$ satisfies the PDE

\be\label{expr}
Q_{H}(u):=div\,\frac{\rho\,grad\,  u}
{\sqrt{1+\rho^{2}\,|grad\,  u|^{2}}}  +
\frac{\<grad\,  u,grad\, \rho\> }
{\sqrt{1+\rho^{2}\,|grad\,  u|^{2}}}+2H=0
\end{equation}
where $div\,$ and $grad $ are the divergence and gradient in $S$.
\end{proposition}

\proof Consider the smooth functions
$u^*, z\colon\,\Omega\times\R\to\R$ defined as $u^*(x,t)=u(x)$ and $z(x,t)=t$,
and set $F(x,t)=z(x,t)-u^*(x,t)$.
Then $\Gr(u)=F^{-1}(0)$, and hence
$$
2H=div\, \frac{grad\, F}{|grad\, F|}.
$$
A straightforward computation now yields (\ref{expr}).
\qed\vspace{2ex}

We proceed by first obtaining a priori $C^{1}$ estimates for the
solutions $u\in C^{2,\alpha}(\bar\Omega)$ of the Dirichlet problem
\be\label{dirichlet}
\left\{
\begin{array}
[c]{ll}
Q_{H}(u)=0
\vspace{1ex}\\
u|_{\partial \Omega}=\varphi &
\end{array}
\right.
\end{equation}
for a given $\varphi\in C^{2,\alpha}(\partial \Omega)$.
We begin with $C^{0}$ estimates.
\vspace{3ex}


\noindent{\bf\large 1.1\, Height estimate}\vspace{1ex}



\begin{lemma}\po\label{height} Under the assumptions of Theorem
\ref{main} there is a constant $C=C(\Omega,H)$ such that
$$
|u|_{0}\leq C+|\varphi|_0
$$
if $u\in C^{2}(\Omega)\cap C^0(\bar \Omega)$ satisfies $Q_{H}(u)=0$ and
$u|_{\partial \Omega}=\varphi$.
\end{lemma}

\proof  We first assume that $2H^2>-R_0$.
With the notation in the proof of Proposition \ref{three}
we show that the function $f\in C^\infty(M)$ given by
$$
f=cu^*|_M+\<\eta,T\>
$$
is subharmonic for a choice of a constant $c>0$ that depends only on $H$ and $\Omega$.

\noindent Since  $M$ is a graph, one has
that $u^{\ast}|_{M}=z|_{M}$, and thus
$$
\Delta(u^{\ast}|_{M})=\Delta(z|_{M}).
$$
Choose $q\in M$, and let $\{E_{1},E_{2}\}$ be an orthonormal
frame of $M$ in a neighborhood of $q$ that is geodesic at $q$.
For simplicity, in the sequel, we frequently use the notation
$a=1/\rho$. Since the gradient of $z$ in $N$ is
$grad\, z=a^{2}T$,
we obtain
$$
grad\, (z|_{M})=\sum_{j=1,2}\<a^2T,E_{j}\>E_{j}.
$$
It follows that
\bea
\Delta(z|_{M})(q)\!\!&=&\!\! \sum_{j=1,2}E_{j}
\left(\< a^{2}T,E_{j}\> \right)
=\sum_{j=1,2}\< \nabla_{E_{j}}a^{2}T,E_{j}\>
+a^{2}\sum_{j=1,2}\< T,\nabla_{E_{j}}E_{j}\> \\
\!\!&=&\!\!\sum_{j=1,2}\< \nabla_{E_{j}}a^{2}T,E_{j}\>
+2Ha^{2}\<\eta,T\>= \<grad\, a^2,T\> + 2Ha^2\<\eta,T\>\\
\!\!&=&\!\!
(-\<\overline{grad}\, a^2,\eta\> +  2Ha^2)\<\eta,T\>
\eea
since $grad\, a^2 =
\overline{grad}\,a^2-\<\overline{grad\,} a^2,\eta\>\eta$ and $\<\overline{grad}\, a^2,T\>=0$, where $\overline{grad\,}$ denotes the gradient in $N$. On the other hand,
it follows from Proposition $1$ of \cite{for} that
\be\label{sus}
\Delta\<\eta,T\>=-(\Ric_N(\eta)+|B|^{2})\<\eta,T\>
\end{equation}
where $|B|$ denotes the norm of the second fundamental form of $M$. Thus,
$$
\Delta f=(c\,(-\<\overline{grad\,} a^2,\eta\>+2Ha^2)
-\Ric_N(\eta)-|B|^{2})\<\eta,T\>.
$$
Since $\<\eta,T\>\le 0$, then $f$ will be subharmonic if
\be\label{condi}
c\,(|\overline{grad\,} a^2|+2Ha^2)
-\Ric_N(\eta)-|B|^{2}\le 0.
\ee
Using that $|B|^2\geq 2H^2$, we have
$$
\Ric_N(\eta)+|B|^{2}\ge \Ric_N(\eta)+ 2H^2\ge -R_0+2H^2>0.
$$
Thus, we may choose $c>0$ depending only on $H$ and $\Omega$ such that~(\ref{condi}) holds.

We have from the maximum principle for subharmonic
functions that
$$
\sup_M f=\sup_{\partial M}f\leq c\,\sup\varphi
+ \sup_{\cal C}\rho.
$$
Since $\sup_M f\geq c\,\sup_\Omega u -\sup_{\cal C}\rho$,
we conclude that
$$
\sup_\Omega u\leq 2c^{-1}\sup_{\cal C}\rho + \sup\varphi.
$$
Since $z=z_0=\inf\varphi$ is a subsolution of $Q_H$ we have that $u\ge z_0$, and therefore
$\inf_\Omega u=\inf \varphi$.



In the sequel me may assume that $H^{2}\leq-R_0/2$.
Let $B\subset N$ be the normal geodesic ball of radius
$r_0$ such that  $D=\bar B\cap S$.
From the formula for the second variation of energy we have
that the second fundamental form $A$ of $\partial B$ at
$p\in \partial B$ with respect to the inner normal vector satisfies
$$
\<Av,v\>=I_{\gamma(r)}(v,v)\;\;\;\mbox{for all}\;\;\; v\in T_p\partial B
$$
where $I_\gamma$ is the index form along the geodesic
$\gamma$ joining the center of $B$ to $p$.
We now compare the index form of $N$ with the index form of
a simply connected  space of constant sectional curvature $K_0$
as in \cite{mo} or p.\ 223 of \cite{ca}.
It follows that $H_r$, the minimum  of the mean curvature
of $\partial B$  with respect to the inner normal vector,
satisfies
\be\label{comp2}
H_r\geq\left\{
\begin{array}
[c]{ll}
\sqrt{K_0}\cot\,(r_0\sqrt{K_0}) & \;\;\;\mbox{if }K_0>0\vspace*{1ex}\\
1/r_0 & \;\;\;\mbox{if }K_0=0\vspace*{1ex}\\
\sqrt{-K_0}\coth\,(r_0\sqrt{-K_0}) & \;\;\;\mbox{if }K_0<0.
\end{array}\right.
\ee
We claim that $H_r\geq H$. From $H^{2}\leq-R_0/2$ we have $R_0\leq 0$.
Moreover, $R_0=0$ implies $H=0$, and the claim holds in this case.
Thus, we may assume $R_0<0$. If
$R_0\geq 2K_0$ then $K_0<0$, and we have from (\ref{comp2})
that
$$
H_r\geq\sqrt{-K_0}\coth\,(r_0\sqrt{-K_0})
\geq\sqrt{-R_0/2}\coth\,(r_0\sqrt{-R_0/2})
\geq\sqrt{-R_0/2}\geq H.
$$
Thus, we may assume that $R_0<\min\,\{0, 2K_0\}$. Since $H^{2}\leq-R_0/2$
is also satisfied we have that (\ref{comp}) holds and the claim follows
from (\ref{comp2}). We obtain from the claim and the tangency principle
that any CMC $H$-graph with boundary $\partial \Omega$ lies inside $B$. We
have that
$$
\inf \varphi\le u\le \sup_B z + \sup\varphi,
$$
and this concludes the proof.\qed

\begin{remark}\po\label{re}{\em
If $N=S\times R$ where $S$ has negative Gaussian curvature (thus
(\ref{comp}) is required) then the argument at the beginning of the last
proof holds for $H^2=-R_0/2$. To see this it suffices to observe that
$\Ric_N(\eta)>-R_0$ outside  points where $\<\eta, T\>=0$ and that at such
points we have  $\Delta f=0$. }\end{remark} \vspace{1ex}




\noindent{\bf\large 1.2\, Gradient estimates}
\vspace{3ex}


 Our next goal is to obtain global gradient estimates
of a solution of~(\ref{dirichlet}). We begin with a boundary estimate.\vspace{1ex}

Let $\gamma\colon\,[0,\ell]\rightarrow\Gamma$ be a parametrization of $\Gamma=\partial \Omega$ by arc length. If $\nu$ stands for the unit normal vector to $\Gamma$ pointing to $\Omega$, then the curvature of $\Gamma$ is
$$
k_{\Gamma}=\<\nabla_{\gamma^{\prime}}
\gamma^{\prime},\nu\>
$$
and the normal curvature of the factor $\R$ is
$$
\alpha_{\Gamma}=\<\nabla_{T/\rho}T/\rho,\nu\> =
-\<grad\, \ln \rho,\nu\>=\<grad\, \ln a,\nu\>.
$$

\begin{lemma}\po\label{gradbound}
Let $H\geq 0$ and $\varphi\in C^{2}(\Gamma)$ be given. Assume that
$k_{\Gamma}+\alpha_{\Gamma}\geq 2H$
everywhere. Then there is a constant $K=K(H,\Omega)$ such that if $u\in C^{2}(\Omega)\cap C^{1}(\bar \Omega)$ satisfies
$Q_{H}(u)=0$ in $\Omega$ and $u|_{\Gamma}=\varphi$, then
$\sup_{\,\Gamma}|grad\, u|\leq K$.

\end{lemma}

\proof
By means of the normal exponential map along $\Gamma$ we have coordinates on $S$ \mbox{$P=P(s,t)$} for $(s,t)\in [0,\ell]\times\lbrack0,\epsilon]$, $\epsilon>0$, such that
$$
|P_{t}|=1,\;\;\< P_{s},P_{t}\> =0\;\;\mbox{and}
\;\;|P_{s}|=\phi(s,t),
$$
where $\phi$ is a smooth function.
Given  $\psi\in C^\infty(\lbrack 0,\epsilon])$, we define
$w\in C^\infty([0,\ell]\times\lbrack0,\epsilon])$ by
$$
w(s,t)=w(P(s,t))=\psi(t)+\varphi(s).
$$
Hence,
$$
grad\, w=\psi_{t}P_{t}
+\frac{\varphi_{s}}{\phi^{2}}
P_{s}\;\;\;\mbox{and}\;\;\;
|\,grad\, \,w|^{2}=\psi_{t}
^{2}+\frac{\varphi_{s}^{2}}{\phi^{2}}.
$$
We need to compute
$$
Q_{H}(w)=div\,  \frac{grad\, w}{\sqrt
{\beta+\psi_{t}^{2}}} -\frac{\< grad\,
w,grad\, \ln a\> }{\sqrt{\beta+\psi_{t}^{2}}}+2H,
$$
where $\beta=a^{2}+(\varphi_{s}/\phi)^{2}$. We have,
$$
div\, P_{t}=\phi^{-2}\< \nabla_{P_{s}}P_{t}
,P_{s}\> =\phi_{t}/\phi\;\;\;\mbox{and}\;\;\;
div\, P_{s}=\phi_{s}/\phi.
$$
Thus,
$$
div\,  \frac{grad\, w}{\sqrt{\beta+\psi_{t}^{2}}}
=\left(\frac{\psi_{t}}{\sqrt{A}}\right)_{t}
+\frac{\psi_{t}\phi_{t}}{\phi\sqrt{A}}
+\left(  \frac{\varphi_{s}}{\phi^{2}\sqrt{A}}\right)_{s}
+\frac{\varphi_{s}\phi_{s}}{\phi^{3}\sqrt{A}}
$$
where $A=\beta+\psi_{t}^{2}$. Moreover,
$$
\left(  \frac{\psi_{t}}{\sqrt{A}}\right)  _{t}=\frac{2\beta\psi_{tt}-\beta_{t}\psi_{t}}{2A^{3/2}}
$$
and
$$
\left( \frac{\varphi_{s}}{\phi^{2}\sqrt{A}}\right)_{s}
=\frac{\phi\varphi_{ss}-2\varphi_{s}\phi_{s}}
{\phi^{3}\sqrt{A}}
-\frac{\varphi_{s}\beta_{s}}{2\phi^{2}A^{3/2}}.
$$
We also have
$$
\< grad\, w,
grad\, \ln a\>
=\frac{a_t\psi_{t}}{a} +\frac{a_s\varphi_{s}}{a\phi^{2}}.
$$
Setting $$
\alpha^{T}=a^{2}\<\nabla_TT,P_t\>=
a^{-1}a_t,
$$
we obtain that
$$
A^{3/2}Q_{H}(w)=\beta\psi_{tt}-(k_{t}
+\alpha^{T})A\psi_{t}-\frac{1}{2}\beta_{t}\psi_{t}
-\frac{\varphi_{s}\beta_{s}}{2\phi^{2}}+BA+2HA^{3/2},
$$
where
$$
k_{t}=-\frac{\phi_{t}}{\phi}=\frac{1}{\phi^{2}}\< \nabla_{P_{s}}P_{s},P_{t}\>
$$
and
$B=B(s,t)=\phi^{-3}(\varphi_{ss}\phi-\varphi_{s}\phi_{s}
-\varphi_{s}\phi a^{-1}a_s)$.
It follows that
$$
A^{3/2}Q_{H}(w)   =\beta\psi_{tt}-(k_{t}+\alpha^{T})\psi_{t}^{3}
+B\psi_{t}^{2}+C\psi_{t}+D
+2HA^{3/2},
$$
where the functions $B,C,D$ of $(s,t)$ depend on the functions $a,\varphi,\phi$ and its derivatives but
{\it not} on $\psi$ or any of its derivatives.

For positive constants $L$ and $K$ we define $\psi\in C^\infty(\lbrack0,\infty))$ by
$$
\psi(t)=L\ln(1+K^{2}t).
$$
Then $w$ satisfies the condition
$$
w(s,0)=\varphi(s).
$$
Moreover, $\psi_{tt}=-\psi_{t}^{2}/L$. Therefore,
\be\label{eq}
A^{3/2}Q_{H}(w)=-(k_{t}+\alpha^{T})\psi_{t}^{3}
+2H(\beta+\psi_{t}^{2})^{3/2}
-(\beta L^{-1}-B)\psi_{t}^{2}
+C\psi_{t}+D.
\end{equation}
For a given $M>0$ choose $L=(c+M)/\ln(1+K)$ where $c=|\varphi|_{0}$.
Thus,
$$
\psi(t)=\frac{c+M}{\ln(1+K)}\ln(1+K^{2}t).
$$
Hence,
$$
\psi(1/K)=c+M
$$
and
$$
\psi_{t}(0)=\frac{(c+M)K^{2}}{\ln(1+K)}.
$$

We claim that we may choose $K>1/\epsilon$ such that
$Q_{H}(w)<0$ for all
$(s,t)\in [0,\ell]\times\lbrack0,1/K]$. Observe that the functions $B,C,D$ are
bounded on $[0,\ell]\times [0,\epsilon]$ and  that
$\beta$ is bounded away form
zero since, by assumption, $\beta\geq a^{2}$ and $T$ does not vanish at any point.
Therefore, since $L\rightarrow 0$ as $K\rightarrow\infty$, we can choose $K$
large enough so that $\beta L^{-1}-B$ in (\ref{eq}) is positive in $[0,1/K]$.
At $t=0$ we have that $k_{t}=k_{\Gamma}$ and  $\alpha^{T}=\alpha_{\Gamma}$.
Hence,  for $t=0$
and any $s\in [0,\ell]$ we have by assumption that
$$
k_{t}+\alpha^{T}\geq 2H\geq 0,
$$
and the claim  now follows easily.  Moreover, we also have that $w(s,1/K)\geq M$.
These facts allow us to use $w$ as
a barrier from above to $Q_{H}$. Replacing $L$ by $-L$ we get a barrier from below. In particular,
we have the boundary gradient estimate
$$
\sup_{\partial\Omega}|\,grad\, u|
\leq\sup_{\partial\Omega}|\,grad\, w|
$$
if $u\in C^{2}(\Omega)\cap C^{1}(\bar \Omega)$ satisfies
$Q_{H}(u)=0$ in $\Omega$, $u|_{\Gamma}=\varphi$ and
$|u(s,1/K)|\leq M$ for $s\in[0,\ell]$.
By Lemma \ref{height},
the last condition is satisfied by choosing $M=C+|\varphi|_{0}$.
\qed\vspace{2ex}



The next result guarantees that global estimates of the gradient reduces to boundary estimates. Its proof uses techniques developed in \cite{cns}.

\begin{lemma}\po\label{gradglob}
Let $u\in C^{3}(\Omega)\cap C^{1}(\bar \Omega)$ be a solution of (\ref{expr}). Assume
that $u$ is bounded in $\Omega$ and that $|\,grad\, u|$ is
bounded in $\Gamma=\partial\Omega$. Then $|\,grad\, u|$ is bounded in $\Omega$ by a constant that depends
only on $|u|_0$ and $\sup_{\,\Gamma}|\,grad\, u|$.
\end{lemma}

\proof
We use a global isothermal coordinate system $\{x_{1},x_{2}\}$ and set
$\lambda=|\partial/\partial x_{j}|.$
Then (\ref{expr}) takes the form
\be\label{uno}
w^{2}(u_{11}+u_{22})-ww_{1}u_{1}-ww_{2}u_{2}
+\lambda^{2}w^{2}\psi=0,
\end{equation}
where $w=\sqrt{a^{2}+|\,grad\, u|^{2}}$ and $\psi=2Hw
-\<grad\, u,grad\,\ln a\> $.

Let $\nabla_{e}u=\lambda^2grad\, u=u_{1}\partial/\partial x_{1}+u_{2}\partial/\partial x_{2}$ be the
Euclidean gradient of $u$. Thus,
$$
\sup_{\Omega}|\,grad\, u|= \sup_{\Omega}\{\lambda^{-2}|\nabla_{e}u|\}.
$$
Suppose that $\sup_{\Gamma}|\,grad\, u|\leq C_{1}$. Then
$$
\sup_{\Gamma}|\nabla_{e}u|
\leq\sup_{\Gamma}\{\lambda^2\}C_{1}.
$$
Therefore, it is enough to prove that if \mbox{$\sup_{\Omega}|u|\leq C_{0}$} and $\sup_{\Gamma}|\nabla_{e}u|\leq K$, then
$$
\sup_{\Omega}|\nabla_{e}u|\leq C(C_{0},K).
$$

To estimate $|\nabla_{e}u|$ in the interior of $\Omega$ it suffices
to obtain
an estimate for \mbox{$z=|\nabla_{e}u|e^{Au}$} where $A$ is a positive constant to be
chosen later. If $z$ achieves its maximum on $\Gamma$ then we have the desired bound. Otherwise, $z$ must reach its maximum at an interior point $y$ in $\Omega$. At $y$ we may
assume that $\lambda=1$,
$|\,grad\, u|=u_{1}>0$ and $u_{2}=0$.
The function $\ln z=\ln|\nabla_{e}u|+Au$ also takes a maximum
at $y\in \Omega$. That
\mbox{$(\ln z)_{k}(y)=0$} for $k=1,2$, yields
\be\label{es}
u_{11}=-Au_{1}^{2}\;\;\;\;\;\mbox{and}\;\;\;\;\;u_{12}=0.
\ee
Moreover, $(\ln z)_{kk}(y)\leq 0$ gives
\be\label{dos*}
u_{111}\leq2A^{2}u_{1}^{3}
\end{equation}
and
\be\label{tres}
u_{122}\leq -Au_{1}u_{22}-\frac{u_{22}^{2}}{u_{1}}.
\end{equation}
The derivative of (\ref{uno}) with respect to $x_1$
at $y$  yields
\be\label{seis}
w^{2}(u_{111}+u_{122})+ww_{1}(u_{11}+2u_{22})
-(ww_{1})_{1}u_{1}+(\lambda^{2}w^{2}\psi)_{1}=0.
\end{equation}

We give next the result of several straightforward computations  at $y\in \Omega$. Notice that
$w^{2}=a^{2}+u_{1}^{2}$.  For $j\geq 0$ let $G(j)$ denote a
function of the form
$$
G(j)=\sum_{k+\ell\leq j}B_{k,\ell}w^ku_1^\ell,
$$
where $\ell\geq 0$ and the $B_{k,\ell}$'s depend on the
functions $a$,  $\lambda$ and its derivatives at $y$
but not on $A$. Hence,
$$
\lim_{u_{1}\rightarrow\infty}\frac{G(j)}{u_{1}^{j}}<\infty.
$$

We have that
\be\label{cuatro}
ww_{1}=-Au_{1}^{3}-\lambda_{1}u_{1}^{2}+G(0).
\end{equation}
From (\ref{uno}), (\ref{es}) and (\ref{cuatro}) we obtain
\be\label{cinco}
u_{22}=Aa^{2}\frac{u_{1}^{2}}{w^{2}} -2Hw+du_{1}-\lambda_{1}\frac{u_{1}^{3}}{w^{2}}+G(0),
\end{equation}
where $d=a_{1}/a$. Now (\ref{tres}) and (\ref{cinco}) give
\be\label{tres*}
w^{2}u_{122}\leq-\frac{A^{2}}{w^{2}}(2a^{4}u_{1}^{3}
+a^{2}u_{1}^{5}
)+A(\lambda_{1}u_{1}^{4}+2Hw^{3}u_{1}-dw^{2}u_{1}^{2})+G(1). \end{equation}
Further computations show that
\be\label{siete}
(ww_{1})_{1}=u_{1}u_{111}+A^{2}u_{1}^{4}
+4A\lambda_{1}u_{1}^{3}+G(2)
\end{equation}
and
\be\label{ocho}
(\lambda^{2}w^{2}\psi)_{1}=3A(du_{1}^4-2Hwu_{1}^{3})+G(3).
\end{equation}
We introduce (\ref{siete}) into (\ref{seis}) and compute
the coefficients of $u_{111}$ and $u_{122}$.
$$
a^{2}u_{111}+w^{2}u_{122}-A^{2}u_{1}^{5}
-4A\lambda_{1}u_{1}^{4}+ww_{1}
(u_{11}+2u_{22})+(\lambda^{2}w^{2}\psi)_{1}+G(3)=0.
$$
Since the coefficients are positive, we can use inequalities (\ref{dos*}) and (\ref{tres*}), and then use (\ref{cuatro}), (\ref{cinco}) and (\ref{ocho}) to obtain
an inequality that turns out to be of the form
$$
\frac{u_{1}^{5}}{w^{2}}\leq\frac{1}{A^{2}}(AG_{1}(3)
+G_{2}(3)).
$$
Since $\lim_{u_{1}\rightarrow\infty}G(3)/u_{1}^{3}<\infty$,
we can choose $A$ such that
$$
\frac{u_{1}^2}{w^2}\leq\frac{1}{2}.
$$
This gives an upper bound for $u_{1}$ and hence for $|\nabla_{e}u|e^{Au}$.
\qed\vspace{3ex}



\noindent{\bf\large 1.3\, A flux formula}
\vspace{3ex}

For the proof of the uniqueness part of Theorem \ref{main} we use the following result that is basically known.

\begin{lemma}\po Let $M_{1}$ and $M_{2}$ be compact
oriented  surfaces with boundary immersed with the same constant
mean curvature $H$ (with respect to the given
orientations) in an oriented  three-dimensional Riemannian manifold $N$. Assume $\Gamma=\partial M_{1}=\partial M_{2}$ and that $M_{1}$ and $M_{2}$ induce the same orientation on $\Gamma$. Let $V$ be a Killing vector field of $N$ and assume that $M_{1}\cup M_{2}$ is a homologically trivial two cycle in $N$. Then,
\be\label{flux}
\int_{\Gamma}\< V,\nu_{1}\> =\int_{\Gamma}\<V,\nu_{2}\>
\end{equation}
where $\nu_{i}$ for $i=1,2$, is the outer pointing conormal to $M_{i}$ along $\Gamma$.
\end{lemma}

\proof
Since $M_{1}\cup M_{2}$ is a homologically trivial two cycle in $N$ then it is the
boundary of an immersed domain $W$ in $N$. Denote by $\eta_i$ the unit
normal vector field to $M_{i}$ in the given orientation. Since the orientations
induced by $M_{i}$ on $\Gamma$, $i=1,2$, are the same, we may assume that $\eta_1$
and $-\eta_2$ point to the exterior of $W$ along $M_{1}$ and $M_{2}$,
respectively. By Stokes theorem and  $div\, V=0$ it follows that
$$
0=\int_{W}div\, V=
\int_{M_{1}}\< V,\eta_1\> -
\int_{M_{2}}\<V,\eta_2\> .
$$
Denoting by $\delta_{V}(M_{i})$ the first variation of the area of $M_{i}$
having $V$ as the variation vector field, we have
$$
\delta_{V}(M_{i})=\int_{M_{i}}div\, V=0
$$
since $V$ is a Killing field. We obtain,
\bea
0\!\!\!&=&\!\!\!\int_{M_{i}}div\, V^{T}+\int_{M_{i}}div\, V^{\perp }
=\int_{M_{i}}div_{M_{i}}V^{T}
+\int_{M_{i}}div\, V^{\perp}\\
\!\!\!&=&\!\!\!-\int_{\Gamma}\< V,\nu_{i}\> -2H\int_{M_{i}}\< V,\eta_{M_{i}}\>,
\eea
and this concludes the proof.\qed\vspace{3ex}



\noindent{\bf\large 1.4\, Final arguments}
\vspace{3ex}

We now give the proof of Theorem \ref{main}.
 We apply the continuity method, namely, we
prove that
$$
Z:=\{t\in\lbrack0,1]:\exists\, u_{t}\in C^{2,\alpha}(\bar \Omega)
\mbox{ such that }
Q_{tH}(u_{t})=0 \mbox{ and } u_{t}|_{\Gamma}=t\varphi\}
$$
is nonempty, open and closed in $[0,1]$. We have that  $Z$ is not empty
since \mbox{$0\in Z$}.  The openness of $Z$ is a direct consequence of the
implicit function theorem since the derivative of $Q_{H}$ computed in
local coordinates is a linear homeomorphism. Standard regularity PDE
results guarantee that any solution of $Q_{H}(u)=0$ is smooth in $\Omega$.
The closeness of $Z$ follows from Lemmas \ref{height} to \ref{gradglob}
and linear elliptic  PDE theory. This proves item $(i)$ in Theorem
\ref{main}. For proving item $(ii)$, let $\tilde{M}$ be a compact CMC
$H$-surface contained in ${\cal C}$ with $\partial\tilde{M}=\Gamma$ and
inducing on $\Gamma$ the same orientation as the one induced by $M$. By
the flux formula (\ref{flux}), we have
$$
\int_{\Gamma}\< T,\nu_{1}\> =
\int_{\Gamma}\<T,\nu_{2}\>
$$
where $\nu_{1}$ (respec.\ $\nu_{2}$) is the outer pointing conormal to $M$ (respec.\ $\tilde{M}$) along $\Gamma$.
It follows  that $\< T,\nu_{1}\> =\<T,\nu_{2}\>$ at some point of $p\in\Gamma$.
Since both $M$ and $\tilde{M}$ are contained in ${\cal C}$
it is easy to see that $T_pM=T_p\tilde M$.
From the boundary tangency principle (see Theorem $3.6$ of \cite{gt})
we conclude that $M=\tilde{M}$.\qed\vspace{3ex}



\noindent{\bf\large 1.5\, A comment}
\vspace{3ex}

The proof of Theorem \ref{main} can be greatly simplified assuming the
strict inequality $H<H^{\prime}$ instead of $H\leq H^{\prime}$ as we did.
In fact, under this stronger assumption we may  obtain height and boundary
gradient estimates  by the use  of geometric barriers, and thus avoiding
the use of Lemmas \ref{height} and \ref{gradbound}. We give the reasons
only in the Euclidean space for standard graphs since the proof in this
particular case is enough to indicate the proof for the general situation
of warped products. Let $\Omega$ be a open bounded $C^{2}$ domain in the
plane $z=0$ in $\mathbb{R}^{3}$ and let $\varphi\in C^{2}(\partial
\Omega)$ be given. Choose $(x_{0},y_{0},0)\in \Omega.$ Given
$z\in\mathbb{R},$ let $C_{z}$ be the cone with vertex
$v_{z}:=(x_{0},y_{0},z)$ constituted of straight line segments from
$v_{z}$ to points of the graph of $\varphi.$ If the right cylinder over
the boundary $\partial \Omega$ of $\Omega$ has mean curvature
$H^{\prime}>H$ then, choosing $z$ large enough, say $z=z_{1}$ and $z$
small enough, say $z=z_{2},$ then the cones
$C_{z_{i}}\backslash\{v_{z_{i}}\},$ $i=1,2,$ have mean curvature strictly
larger than $H,$ being therefore upper and lower barriers for the CMC $H$
graph equation on $\Omega$. These barriers provide a priori height and
gradient boundary estimates of any solution of the CMC $H$-equation on
$\Omega$ assuming $\varphi$ at $\partial \Omega$.

\vspace{3ex}



\noindent{\bf\Large \S 2. Some special cases}
\vspace{4ex}

We first apply Theorem \ref{main} to get an interesting condition for the existence of a  standard Euclidean $H$-graph with planar but not necessarily convex boundary.  We begin with the following fact.



\begin{lemma}\po\label{idem} Any  rotational CMC
$H$-graph in $\R^3$ with planar boundary that is of class $C^1$ up to the boundary is also a standard graph.
\end{lemma}

\proof Let $M$ be a $C^1$ rotational $H$-graph over $\bar \Omega$ with
$\partial M=\partial \Omega$. Clearly, $M$ is a standard $C^1$ $H$-graph
over a neighborhood in $\bar \Omega$ of $\partial \Omega$. Now using
Alexandrov`s reflexion technique with respect to planes parallel to the
boundary plane we easily see that $M$ must be a standard CMC $H$-graph.
\qed\vspace{2ex}

Let $\gamma\colon\,I\rightarrow\R^2$ be a parametrization
of $\partial \Omega$ in a canonical system of coordinates in $\R^2$ with origin $O\in \Omega$.
Define the functions
$$
h(s,\theta)=\left\langle\gamma(s),
e(\theta)\right\rangle \>\;\;\;
\mbox{and}\;\;\;g(s,\theta)
=1/\left\langle n(s),e(\theta )\right\rangle
$$
where $e(\theta )=(\sin\theta ,\cos\theta )$ for
$\theta\in [0,2\pi )$ and $n(s)$ denotes the unit vector field normal to $\partial \Omega$ pointing inward. We consider the subset
$$
B=\{(\theta ,d)\in \lbrack 0,2\pi )\times
\R\colon \,d>\max_{s\in I}h(s,\theta)\},
$$
and define
$$
A(\theta ,d)=\inf_{s\in I}\,\{k(s)
+((d-h(s,\theta ))g(s,\theta))^{-1}:\,(\theta ,d)\in B\}
$$
where $k(s)n(s)$ is the curvature vector of $\gamma(s)$.\vspace{1ex}


\begin{theorem}\po\label{planar}
Assume that $H>0$ satisfies the condition \be\label{cond} 2H\leq \sup
\,\{A(\theta ,d)\colon \,(\theta ,d)\in \Omega\}. \label{ordinarygraph}
\ee Then there is a standard CMC $H$-graph over $\Omega$ with boundary
$\partial \Omega$.
\end{theorem}

\proof It is easy to see that  Theorem \ref{main} applies under
our assumption (\ref{cond}) and
the result follows from Lemma \ref{idem}.
\qed\vspace{2ex}

To conclude we have the following characterization of CMC $H$-graphs with
mean curvature $|H|\le 1$ and with boundary contained in an umbilical
surface in hyperbolic $\Hy^{\,3}$ (with normalized constant sectional
curvature $-1$).




\begin{theorem}\label{last}\po
Let $M$ be a compact embedded smooth surface with boundary in $\Hy^{\,3}$
such that $\partial M$ lies in a complete noncompact umbilical surface $U$. Assume that
\begin{itemize}
\item[\em (i)] $\partial M$ is strictly convex in $U,$ that is, given $p\in\partial M$
there is a geodesic circle in $U$ passing through $p$ and containing $\partial M$
on the closure of its interior.
\item[\em (ii)] $M$ has constant mean curvature $H$ satisfying $|H|\leq 1$.
\end{itemize}
Then $M$ is a CMC hyperbolic $H$-graph.
\end{theorem}


\proof  Recall that a Killing field  in hyperbolic
space is of hyperbolic type if it leaves invariant a geodesic $\gamma$
and the trajectories are hypercycles equidistant to $\gamma$.
The two points of the asymptotic boundary of $\gamma$ are referred as the
points at infinity left fixed by the Killing field.
Let $p\in U$ be a point contained in the interior of the
bounded connected component of $U\backslash\partial M$, and let $\gamma$
be the geodesic through $p$ orthogonal to $U$.
Let $V$ be a hyperbolic Killing vector field translating $\gamma$
and denote by $\phi$ the one parameter subgroup
of isometries determined by $V$.
We denote by $C(\partial M)$ the hyperbolic cylinder over
$\partial M$ determined by $\phi$, i.e.,
$$
C(\partial M)=\{\phi_{t}(q) : q\in\partial M\;
\mbox{and}\; t\in\R\}.
$$

We claim that the mean curvature $H_{C(\partial M)}$ of
$C(\partial M)$ with respect to the inner unit normal vector is strictly
larger than $1$. In fact, consider an arbitrary point $q$ of $\partial M$
and compare the mean curvature of
$C(\partial M)$ with the mean curvature
of a hyperbolic cylinder over a geodesic circle in $U$ tangent
to $\partial M$ at $q$ and
containing $\partial M$. Since, as it is not difficult to see, any hyperbolic cylinder over a geodesic circle has constant mean curvature bigger than 1, the claim follows.
A straightforward application of the tangency principle
using $H_{C(\partial M)}>1\geq|H|$ proves that $M$ lies inside $C(\partial M)$.

Let $a_{1},a_{2}$ be the two points at infinity left fixed by $V$ and let $S_{i},\;i=1,2$,  be two totally geodesic spheres everywhere orthogonal to $V$
such that $M\subset X_i$ where $X_i$ is the connected component of $\Hy^{\,3}\backslash S_{i}$ whose asymptotic boundary does not contain $a_i$.  Thus $M\subset X_1\cap X_2$.


Since we may represent the ambient space as a warp product
$\bar X_i=S_{i}\times_{\rho_{i}}\R$ for a convenient
choice of $\rho_{i}$ with $d/dt_1=V$ and $d/dt_2
=-V,$ we may apply Theorem~\ref{main} to
assert the existence of hyperbolic graphs $\tilde{M}_{1}$ and $\tilde{M}_2$ over  the appropriate domains in $S_1$ and $S_2$ (i.e., warped graphs in $S_i\times_{\rho_{i}}\R$ with constant mean curvature $|H|$ such that $\partial\tilde{M}_{i}=\partial M$) inducing opposite orientation on $\partial M$. We conclude from part $(ii)$ of Theorem \ref{main} that either $M=\tilde{M}_1$ or $M=\tilde{M}_2$.\qed



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\vspace*{-1ex}

{\renewcommand{\baselinestretch}{1}
\hspace*{-20ex}\begin{tabbing}
\indent \= Marcos Dajczer\hspace{25,5ex} Jaime Ripoll\\
\> IMPA \hspace{35ex}Instituto de Matematica\\
\> Estrada Dona Castorina, 110\hspace{12ex}
Univ. Federal do Rio Grande do Sul\\
\> 22460-320 --- Rio de Janeiro ---RJ
\hspace{6ex} Av. Bento Gon\c calves 9500\\
\> Brazil\hspace{35ex} 91501-970 --- Porto Alegre --- RS\\
\> marcos@impa.br\hspace{24,6ex} Brazil\\
\>\hspace{41ex} ripoll@mat.ufrgs.br
\end{tabbing}}

\end{document}
