# Estimates for constant mean curvature graphs in $M\times\mathbb{R}$

José M. Manzano  José Miguel Manzano, Dpto. Geometría y Topología, Universidad
de Granada. Email address<EMAIL_ADDRESS>
###### Abstract.

We will discuss some sharp estimates for CMC graphs $\Sigma$ in a Riemannian
3-manifold $M\times\mathbb{R}$ whose boundary $\partial\Sigma$ is contained in
a slice $M\times\\{t_{0}\\}$. We will start by giving sharp lower bounds for
the geodesic curvature of the boundary and improve these bounds when assuming
additional restrictions on the maximum height that such a surface reaches in
$M\times\mathbb{R}$. We will also give a bound for the distance from an
interior point to the boundary in terms of the height at that point, and
characterize when these bounds are attained.

###### Key words and phrases:

product manifolds, constant mean curvature, invariant surfaces, boundary
curvature estimates, height estimates

###### 2000 Mathematics Subject Classification:

Primary 53A10, Secondary 49Q05, 53C42

Research partially supported by a Spanish MEC-FEDER Grant no. MTM2007-61775
and a Regional J. Andalucía Grant no. P06-FQM-01642.

##  1\. Introduction

Constant mean curvature surfaces in several $3$-manifolds have been
extensively studied in recent times. One of the most important families of
such $3$-manifolds are product spaces $M\times\mathbb{R}$, $M$ being a
Riemannian surface, which includes some homogeneous spaces as
$\mathbb{R}^{3}$, $\mathbb{H}^{2}\times\mathbb{R}$ and
$\mathbb{S}^{2}\times\mathbb{R}$. It was Rosenberg in [12] who started the
study of minimal surfaces in $M\times\mathbb{R}$ and, since then, many papers
in this setting have appeared.

We will focus on constant mean curvature $H>0$ graphs $\Sigma$ in
$M\times\mathbb{R}$ whose boundary $\partial M$ lies in some slice
$M\times\\{t_{0}\\}$ and, if we denote by $c$ the infimum of the Gaussian
curvature of the domain of $M$ over which $\Sigma$ is a graph, we will assume
the hypothesis $4H^{2}+c>0$. As CMC graphs are stable, it is possible to apply
Theorem 2.8 in [6] to conclude that the distance function
$d(p,\partial\Sigma)$, $p\in\Sigma$, is bounded, so the height function is
also bounded. In the case $4H^{2}+c\leq 0$, this property fails to be true as
invariant examples in $\mathbb{H}^{2}\times\mathbb{R}$ given in [7] and [13]
show.

To fix notation, let $M$ be a Riemannian surface without boundary and consider
$\Sigma\subseteq M\times\mathbb{R}$ an embedded constant mean curvature $H>0$
surface (an $H$-surface in the sequel). Let us also consider the height
function $h\in C^{\infty}(\Sigma)$ given by $h(p,t)=t$ and the angle function
$\nu\in C^{\infty}(\Sigma)$ given by $\nu=\langle N,E_{3}\rangle$, where $N$
is the unit normal vector field to $\Sigma$ for which the mean curvature of
$\Sigma$ is $H$, and $E_{3}=\partial_{t}$ is the vertical Killing vector
field. Throughout this paper, we will denote by $K$ the intrinsic curvature of
$\Sigma$ and $K_{M}$ will stand for the intrinsic curvature of $M$ extended to
$M\times\mathbb{R}$ by making it constant along the vertical geodesics.
Besides, $\sigma$ and $A$ will be the second fundamental form and the shape
operator of $\Sigma$, respectively. In this situation, the Gauss equation
reads $\det(A)=K-K_{M}\nu^{2}$.

Aledo, Espinar and Gálvez proved in [1] that if $\Sigma\subseteq
M\times\mathbb{R}$ is a constant mean curvature $H>0$ graph (or $H$-graph for
short) over a compact open domain that extends to its boundary with $h=0$ and
$\nu=\nu_{0}$ in $\partial\Omega$, and we denote by
$c=\inf\\{K_{M}(p):p\in\Omega\\}>-4H^{2}$, then $\Sigma$ can reach at most
height $\alpha(c,H,\nu_{0})$, where

(1)
$\alpha(c,H,\nu_{0})=\begin{cases}\frac{4H}{\sqrt{-4cH^{2}-c^{2}}}\left(\arctan\left(\frac{\sqrt{-c}}{\sqrt{c+4H^{2}}}\right)+\arctan\left(\frac{\nu_{0}\sqrt{-c}}{\sqrt{c+4H^{2}}}\right)\right)&\text{if
}c<0,\\\ \frac{1+\nu_{0}}{H}&\text{if }c=0,\\\
\frac{4H}{\sqrt{4cH^{2}+c^{2}}}\left(\operatorname{arctanh}\left(\frac{\sqrt{c}}{\sqrt{c+4H^{2}}}\right)+\operatorname{arctanh}\left(\frac{\nu_{0}\sqrt{c}}{\sqrt{c+4H^{2}}}\right)\right)&\text{if
}c>0.\end{cases}$

Indeed, they gave the estimate for the case $\nu_{0}=0$ but their argument can
be directly generalized to this more general case. They also proved that this
bound is the best one in terms of $c$ and $H$ in the sense that in the
homogeneous space $\mathbb{M}^{2}(c)\times\mathbb{R}$ the only such $H$-graphs
for which equality holds are rotationally invariant spheres for $\nu_{0}=0$,
and in the general case they are spherical caps of rotationally invariant
spheres which meet the boundary with constant angle function $\nu_{0}$.

We will restrict ourselves to the capillarity problem, i.e. when the surface
has constant angle function $\nu=\nu_{0}$ for some $-1<\nu_{0}\leq 0$ along
its boundary. This situation includes compact embedded CMC surfaces for
$\nu_{0}=0$ because of the Alexandroff reflection principle, and the more
general case of embedded $H$-bigraphs, that is to say, (not necessarily
compact) connected embedded $H$-surfaces which are made up of two graphs,
symmetric with respect to some slice $M\times\\{t_{0}\\}$. Ritoré [10] and
Große-Brauckmann [2] constructions are examples of this kind of surfaces in
$\mathbb{R}^{3}$. We will prove the following results, where we denote by
$\mathbb{M}^{2}(c)$ the simply-connected constant curvature $c$ surface.

  * •

The geodesic curvature $\kappa_{g}$ of $\partial\Omega$ in $M$, with respect
to the outer conormal vector field, satisfies the lower bound

$\kappa_{g}\geq\frac{-4H^{2}+c(1-\nu_{0})^{2}}{4H\sqrt{1-\nu_{0}^{2}}},$

and, when $M=\mathbb{M}^{2}(c)$, equality holds only for rotationally
invariant spheres.

  * •

If we additionally suppose that $|h|\leq m\cdot\alpha(c.H,\nu_{0})$ for some
constant $0<m\leq\frac{1}{2}$, then the previous bound is improved to the
following one:

$\kappa_{g}\geq\frac{(4-8m)H^{2}+c(1-\nu_{0})^{2}}{4mH\sqrt{1-\nu_{0}^{2}}}.$

In this case, when $M=\mathbb{M}^{2}(c)$, equality holds if, and only if,
$m=\frac{1}{2}$ and $\Sigma$ is the boundary of some neighborhood of a
geodesic of $\mathbb{M}^{2}(c)\times\\{0\\}$, examples which will be fully
described in section 2.

  * •

In the last section, we will give an application of the techniques used in the
above two items to obtain a sharp lower bound for the distance from a point in
$\Sigma$ to $\partial\Sigma$. As in the results above, equality holds when the
surface is rotationally invariant.

The author would like to thank Joaquín Pérez, Magdalena Rodríguez and
Francisco Torralbo for some helpful conversations.

##  2\. Invariant surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ and
$\mathbb{S}^{2}\times\mathbb{R}$

In this section, we will study surfaces that are invariant by $1$-parameter
groups of isometries in $\mathbb{M}^{2}(c)\times\mathbb{R}$ which act
trivially on the vertical lines. In fact, among these, we are interested in
surfaces which are $H$-bigraphs (i.e. embedded $H$-surfaces symmetric with
respect to a horizontal slice), for $H>0$ and $4H^{2}+c>0$. Thus, these groups
of isometries can be identified with $1$-parameter groups of isometries of the
base $\mathbb{M}^{2}(c)$.

In $\mathbb{H}^{2}$, there exist three different types of $1$-parameter groups
of isometries, namely, rotations around a point, parabolic translations (i.e.
rotations about a point at infinity) and hyperbolic translations. The family
of rotationally invariant CMC surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ was
studied by Hsiang and Hsiang [5] and those invariant by the other two families
(including screw motion) were also studied by Sa Earp [13] but it was Onnis
[7] who gave a full classification of all invariant CMC surfaces in
$\mathbb{H}^{2}\times\mathbb{R}$. The case of $\mathbb{S}^{2}$ is quite
different, because the only $1$-parameter groups of isometries of
$\mathbb{S}^{2}$ are the rotations around a certain point and, up to
conjugation, this point can be supposed to be the north pole. Such
rotationally invariant $H$-surfaces were classified by Pedrosa [9].

Finally, the only $1$-parameter groups of isometries of $\mathbb{R}^{2}$ are
rotations around a point and translations; the former give rise in
$\mathbb{R}^{3}=\mathbb{R}^{2}\times\mathbb{R}$ to Euclidean spheres of radius
$\frac{1}{H}$, the latter to horizontal cylinders of radius $\frac{1}{2H}$.

For the sake of completeness, we will now derive the parametrizations and
formulas that we will need in each of these situations. We will begin with
rotations in both $\mathbb{H}^{2}\times\mathbb{R}$ and
$\mathbb{S}^{2}\times\mathbb{R}$ and then proceed to parabolic and hyperbolic
translations in $\mathbb{H}^{2}\times\mathbb{R}$. Let us recall that, up to a
homothety, we can suppose $c\in\\{-1,0,1\\}$ and, in the cases $c=1$ and
$c=0$, the condition $4H^{2}+c>0$ is meaningless (as $H>0$) but, for $c=-1$,
it implies that $H>\frac{1}{2}$.

###  2.1. Rotationally invariant surfaces in $\mathbb{H}^{2}\times\mathbb{R}$
and $\mathbb{S}^{2}\times\mathbb{R}$

To start with, let us consider the model
$\mathbb{H}^{2}\times\mathbb{R}=\\{(x,y,z,t)\in\mathbb{R}^{4}:x^{2}+y^{2}-z^{2}=-1,z>0\\}$
endowed with the metric $dx^{2}+dy^{2}-dz^{2}+dt^{2}$. It was shown by Hsiang
and Hsiang that, for any $H>\frac{1}{2}$, the only rotationally invariant
$H$-bigraphs are the rotationally invariant CMC spheres. If we suppose the
axis of rotation to be $\\{(0,0,1)\\}\times\mathbb{R}$, the upper half of such
a sphere is parametrized by $X(r,u)=\left(\sinh r\cos u,\sinh r\sin u,\cosh
r,h(r)\right)$, where $u\in\mathbb{R}$ and

$h(r)=\frac{4H}{\sqrt{4H^{2}-1}}\arcsin\sqrt{\frac{1-(4H^{2}-1)\sinh^{2}\frac{r}{2}}{4H^{2}}},\quad
r\in\left[0,2\mathop{\rm arcsinh}\nolimits\frac{1}{\sqrt{4H^{2}-1}}\right]$

(see figure 1 where some examples have been depicted).

Figure 1. On the left, rotationally invariant CMC spheres (the horizontal axis
represents the intrinsic length in $\mathbb{H}^{2}$ and the vertical one is
the real line) and, on the right, CMC cylinders invariant under hyperbolic
translations, where we see their intersection with the plane $y=1$ in the
halfspace model. In both cases, the represented values of $H$ are 0.54, 0.6,
0.7, 0.8, 0.9 and 1.

On the other hand, we will consider the standard model of
$\mathbb{S}^{2}\times\mathbb{R}$ as a submanifold of $\mathbb{R}^{4}$, given
by
$\mathbb{S}^{2}\times\mathbb{R}=\\{(x,y,z,t)\in\mathbb{R}^{4}:x^{2}+y^{2}+z^{2}=1\\}$
with the induced Riemannian metric. It is well-known that every $1$-parameter
group of ambient isometries consists only of rotations so, up to an isometry,
they may be supposed to be rotations around the axis
$\\{(0,0,1,t):t\in\mathbb{R}\\}$. Hence, the orbit space can be identified
with the totally geodesic surface
$\\{(x,y,z,t)\in\mathbb{S}^{2}\times\mathbb{R}:x=0\\}\cong\mathbb{S}^{1}\times\mathbb{R}$,
and we will take the generating curve as

$\gamma(t)=\left(0,\sin r(t),\cos r(t),h(t)\right)$

for some functions $r,h$ defined on some interval of the real line. Pedrosa
[9] showed that the generated surface has constant mean curvature
$H\in\mathbb{R}$ if, and only if, certain ODE system is satisfied. In fact, he
proved that in the intervals where $r$ is invertible, we can take it as the
parameter and the corresponding ODE system becomes

(2) $\left\\{\begin{array}[]{l}h^{\prime}(r)=\cot(\sigma(r)),\\\
\sigma^{\prime}(r)=\frac{2H+\cot(r)\cos(\sigma(r))}{\operatorname{sen}(\sigma(r))},\end{array}\right.$

for an auxiliary function $\sigma$. The second equation can be easily solved
as it only depends on $r$ and $\sigma$ and we obtain

$\sigma(r)=\arccos(2H(c_{0}+\cos r)\csc r)$

for some $c_{0}\in\mathbb{R}$, where
$r\in[a(c_{0}),b(c_{0})]\subseteq[-\pi,\pi]$ is the maximal interval in which
$\sigma$ is defined. By plugging this expression into the first equation in
(2), we arrive to

(3) $h(r)=\int_{a(c_{0})}^{r}\frac{2H(c_{0}+\cos s)\csc
s}{\sqrt{1-4H^{2}(c_{0}+\cos s)^{2}\csc^{2}s}}\,\mathrm{d}s.$

The only two cases which lead to $H$-bigraphs are the following:

  * •

For $c_{0}=-1$, rotationally invariant spheres are obtained. More explicitly,

$h(r)=\frac{4H}{\sqrt{1+4H^{2}}}\operatorname{arccosh}\left(\frac{\sqrt{1+4H^{2}}}{4H}\cos\frac{r}{2}\right)$

where $r$ lies in the interval $[-2\arctan\frac{1}{H},2\arctan\frac{1}{H}]$.
Thus, the maximum height is attained for $r=0$ and the sphere is a bigraph
over a domain whose boundary has constant geodesic curvature in
$\mathbb{S}^{2}$ with respect to the outer conormal vector field, equal to
$-H+\frac{1}{4H}$.

  * •

For $c_{0}=0$, we obtain rotationally invariant tori instead. In this case,

$h(r)=\frac{2H}{\sqrt{1+4H^{2}}}\operatorname{arccosh}\left(\frac{\sqrt{1+4H^{2}}}{2H}\sin
r\right)$

where
$r\in[\frac{\pi}{2}-\arctan\frac{1}{2H},\frac{\pi}{2}+\arctan\frac{1}{2H}]$.
The maximum height is attained when $r=\frac{\pi}{2}$ and the boundary of the
domain over which the torus is a bigraph has two connected components which
have constant geodesic curvature $\frac{1}{2H}$ in
$\mathbb{S}^{2}\times\mathbb{R}$ (with respect to the outer conormal vector
field).

These two families are represented in figure 2. We remark that the maximum
height of a CMC torus is exactly a half of that of the corresponding sphere
for the same mean curvature.

Figure 2. On the left, rotationally invariant CMC spheres and, on the right,
rotationally invariant CMC tori, in $\mathbb{S}^{2}\times\mathbb{R}$. In both
cases, the represented values of $H$ are 0.05, 0.12, 0.331372, 0.6, 1 and 2.
The horizontal axis measures the intrinsic distance in $\mathbb{S}^{2}$ while
the vertical one is the real line. The maximum height is attained for
$H\approx 0.331372$, which is drawn as a dashed line.

###  2.2. Invariant surfaces under hyperbolic translations in
$\mathbb{H}^{2}\times\mathbb{R}$

In this section, we will work in the upper halfplane model
$\mathbb{H}^{2}\times\mathbb{R}=\\{(x,y,t)\in\mathbb{R}^{3}:y>0\\}$ endowed
with the metric $(dx^{2}+dy^{2})/y^{2}+dt^{2}$. Up to conjugation by an
ambient isometry, the $1$-parameter group of hyperbolic translations may be
considered to be $\\{\Phi^{h}_{s}\\}_{s\in\mathbb{R}}$, where

$\Phi_{s}^{h}:\mathbb{H}^{2}\times\mathbb{R}\rightarrow\mathbb{H}^{2}\times\mathbb{R},\quad\quad\Phi_{s}^{h}(x,y,t)=(xe^{s},ye^{s},t).$

First of all, we observe that, as the orbit of any point is the horizontal
Euclidean straight line which joins the point to a point in the axis $x=y=0$,
we can consider the plane $y=1$ as the orbit space of this group of
transformations.

Let us take a curve $\gamma(t)=(x(t),1,h(t))$ for some $C^{2}$ functions $x,h$
defined in some interval of the real line. Thus, a surface invariant by
$\\{\Phi^{h}_{s}\\}_{s\in\mathbb{R}}$ can be parametrized as

(4) $X(u,t)=\left(x(t)e^{u},e^{u},h(t)\right).$

It is a straightforward computation to check that the mean curvature of this
parametrization is given by

(5)
$H=\frac{-x^{3}(h^{\prime})^{3}-xh^{\prime}((h^{\prime})^{2}+2(x^{\prime})^{2})+x^{2}(h^{\prime}x^{\prime\prime}-x^{\prime}h^{\prime\prime})-x^{\prime}h^{\prime\prime}+h^{\prime}x^{\prime\prime}}{2((1+x^{2})(h^{\prime})^{2}+(x^{\prime})^{2})^{3/2}}.$

In order to simplify this equation, we will reparametrize the curve $\gamma$
in such a way the denominator simplifies. Observe that we can suppose that
$(h^{\prime})^{2}+(x^{\prime})^{2}/(1+x^{2})=1$ so there exists a $C^{1}$
function $\alpha$ such that $h^{\prime}=\cos\alpha$ and
$x^{\prime}=\sqrt{1+x^{2}}\sin\alpha$. Now, we can obtain expressions for
$x^{\prime\prime}$ and $h^{\prime\prime}$ just by taking derivatives in these
identities. If we substitute the results in equation (5), we get

$H=\frac{\sqrt{1+x^{2}}\alpha^{\prime}-x\cos\alpha}{2\sqrt{1+x^{2}}}.$

The proof of the following lemma is now trivial.

###### Lemma 2.1.

The parametrized surface defined in (4) has constant mean curvature
$H\in\mathbb{R}$ if, and only if, the functions $(x,h,\alpha)$ satisfy the
following ODE system

(6) $\left\\{\begin{array}[]{l}h^{\prime}=\cos\alpha\\\
x^{\prime}=\sqrt{1+x^{2}}\sin\alpha\\\
\alpha^{\prime}=2H+\frac{x\cos\alpha}{\sqrt{1+x^{2}}}\end{array}\right.$

Furthermore, the energy function $E=-2Hx-\sqrt{1+x^{2}}\cos\alpha$ is constant
along any solution.

We will restrict ourselves to the case $H>1/2$. Plugging the expression of the
energy into the second equation in (6), it is not difficult to conclude that
$x$ verifies the equation

(7) $(x^{\prime})^{2}=(1-E^{2})+4HEx+(1-4H^{2})x^{2}.$

As $H>\frac{1}{2}$, the RHS has two different real roots as a polynomial in
$x$ and, if we factor it, the equation can be expressed, up to a sign, as

$\frac{x^{\prime}}{\sqrt{(4H^{2}+E^{2}-1)-\left((4H^{2}-1)x-2HE\right)^{2}}}=\frac{\pm
1}{\sqrt{4H^{2}-1}},$

from where it is easy to deduce that there exists $c_{0}\in\mathbb{R}$ such
that

(8)
$x(t)=\frac{2HE}{4H^{2}-1}+\frac{\sqrt{4H^{2}+E^{2}-1}}{4H^{2}-1}\sin\left(\pm
t\sqrt{4H^{2}-1}+c_{0}\right).$

After a translation and a reflection in the parameter $t$, we will suppose
without loss of generality that $c_{0}=0$ and the $\pm$ sign is positive. Now,
we can integrate $h$ by taking into account the identity
$h^{\prime}=\cos\alpha=(E+2H)/\sqrt{1+x^{2}}$, and we get

(9)
$h(t)=h(0)+\int_{0}^{t}\frac{(8H^{2}-1)E+2H\sqrt{4H^{2}+E^{2}-1}\sin\left(s\sqrt{4H^{2}-1}\right)}{\sqrt{\left(1-4H^{2}\right)^{2}+\left(\sqrt{4H^{2}+E^{2}-1}\sin(s\sqrt{4H^{2}-1})+2HE\right)^{2}}}\,\mathrm{d}s.$

Finally, we are able to characterize the surfaces we were looking for. Some
pictures of them are drawn in Figure 1.

###### Proposition 2.2.

Let $(x,h,\alpha)$ be a solution of (6) with energy $E\in\mathbb{R}$ for some
$H>\frac{1}{2}$. Then, the generated invariant surface can be extended to an
$H$-bigraph if, and only if, $E=0$. In this case, the generating curve can be
reparametrized, up to an ambient isometry, as

$\left.\begin{array}[]{l}x(r)=\displaystyle\frac{1}{\sqrt{4H^{2}-1}}\sin r\\\
h(r)=\displaystyle\frac{2H}{\sqrt{4H^{2}-1}}\arctan\frac{\cos
r}{\sqrt{4H^{2}-1+\sin^{2}r}}\end{array}\right\\},\quad r\in\mathbb{R}.$

###### Proof.

Let us split $h(t)=h_{1}(t)+h_{2}(t)$ in (9) by splitting the integrand in two
additive terms which correspond to the two terms in its numerator. The first
term does not vanish unless $E=0$ so $h_{1}$ is monotonic and the second one
is an odd periodic function in $s$ which vanishes at $s=k\pi/\sqrt{4H^{2}-1}$
for any $k\in\mathbb{Z}$. On the other hand, if the parametrization interval
contains $t=0$, from (8) we deduce that $|t|\leq\pi/(2\sqrt{4H^{2}-1})$ so the
surface is a graph and, furthermore, the points at which the normal vector
field is horizontal must satisfy $x^{\prime}=0$, so the parametrization
interval must be $|t|\leq\pi/(2\sqrt{4H^{2}-1})$. Now, as the integral of
$h_{2}^{\prime}$ over $[-\pi/(2\sqrt{4H^{2}-1}),\pi/(2\sqrt{4H^{2}-1})]$
vanishes, we have

$h\left(\frac{\pi}{2\sqrt{4H^{2}-1}}\right)-h\left(\frac{-\pi}{2\sqrt{4H^{2}-1}}\right)=h_{1}\left(\frac{\pi}{2\sqrt{4H^{2}-1}}\right)-h_{1}\left(\frac{-\pi}{2\sqrt{4H^{2}-1}}\right).$

The RHS term vanishes if, and only if, $h_{1}$ identically vanishes as it is
monotonic and $h_{1}$ vanishes if, and only if, $E=0$. The expressions given
in the statement follow from a direct computation in (8) and (9) for $E=0$ and
from the substitution $r=t\sqrt{4H^{2}-1}$. Observe that there is no
restriction in taking $r\in\mathbb{R}$ because this parametrization generates
the whole bigraph. ∎

In the parametrization given in the statement of Proposition 2.2, observe that
the maximum height is attained for $r=0$ and the surface is a bigraph over a
domain whose boundary consists in two hypercycles which have constant geodesic
curvature in $\mathbb{H}^{2}$ with respect to the outer conormal vector field,
equal to $\frac{-1}{2H}$. Furthermore, the maximum height is exactly a half of
that of the corresponding CMC sphere.

###  2.3. Invariant surfaces under parabolic translations

In this case, we will also consider the upper halfplane model for
$\mathbb{H}^{2}$ so, up to conjugation by an ambient isometry, the
$1$-parameter group of parabolic translations is
$\\{\Phi_{s}^{p}\\}_{s\in\mathbb{R}}$, where

$\Phi_{s}^{p}:\mathbb{H}^{2}\times\mathbb{R}\rightarrow\mathbb{H}^{2}\times\mathbb{R},\quad\quad\Phi_{s}^{p}(x,y,t)=(x+s,y,t).$

Hence, the orbit of any point in $\mathbb{H}^{2}$ is a horizontal Euclidean
line parallel to the plane $y=0$. Thus, the orbit space may be considered to
be the Euclidean plane $x=0$ so the generating curve can be thought as
$\gamma(t)=(0,y(t),h(t))$ and a surface invariant by
$\\{\Phi^{p}_{s}\\}_{s\in\mathbb{R}}$ can be parametrized as

$X(u,t)=\left(s,y(t),h(t)\right).$

It is straightforward to check that the mean curvature of this parametrization
is

(10)
$H=-\frac{y^{2}\left(-h^{\prime\prime}y^{\prime}+h^{\prime}y^{\prime\prime}+y(h^{\prime})^{3}\right)}{2\left(y^{2}(h^{\prime})^{2}+(y^{\prime})^{2}\right)^{3/2}}.$

Furthermore, there is no loss of generality in supposing that the curve
$\gamma$ is parametrized by its arc-length, i.e.
$1=\|\alpha^{\prime}\|^{2}=(y^{\prime})^{2}/y^{2}+(h^{\prime})^{2}$. Hence, we
can take an auxiliary function $\alpha$, determined by
$y^{\prime}=y\sin\alpha$, $h^{\prime}=\cos\alpha$. Substituting these
equalities in (10), it simplifies to the following ODE system

(11) $\left\\{\begin{array}[]{l}y^{\prime}=y\sin\alpha\\\
h^{\prime}=\cos\alpha\\\ \alpha^{\prime}=-2H-\cos\alpha.\end{array}\right.$

Observe that, if we assume an initial condition
$\alpha(0)=\alpha_{0}\in[0,2\pi]$, the third equation has a unique solution.
Let us focus in the case $H>\frac{1}{2}$ which is the most interesting for our
purposes and allows us to integrate the function $\alpha$ as

(12)
$\alpha(t)=2\arctan\left(\frac{(2H+1)}{\sqrt{4H^{2}-1}}\tan\left(\frac{1}{2}\sqrt{4H^{2}-1}(t-c_{0})\right)\right)$

for some $c_{0}\in\mathbb{R}$ depending on $\alpha_{0}$. We emphasize that
this formula defines $\alpha:\mathbb{R}\rightarrow\mathbb{R}$ as a strictly
increasing diffeomorphism by considering all the branches of the function
$\arctan$ and extending it by continuity, so the uniqueness of solution
guarantees that every solution is considered in (12). We will suppose, after a
translation in the parameter $t$, that $c_{0}=0$. By plugging expression (12)
into the first two equations in (11), we can integrate $h$ and $y$ to obtain

(13)
$\begin{array}[]{l}y(t)=c_{1}\left(\cos\left(t\sqrt{4H^{2}-1}\right)+2H\right)\\\
h(t)=\alpha(t)+2Ht+c_{2},\end{array}$

for some constants $c_{1}>0$ and $c_{2}\in\mathbb{R}$ which can be supposed to
be $c_{1}=1$ (after a hyperbolic translation) and $c_{2}=0$ (after a vertical
translation).

###### Proposition 2.3.

There are no invariant embedded bigraphs under parabolic translations with
constant mean curvature $H>\frac{1}{2}$.

###### Proof.

Observe that such a graph must be given by a triple $(y,h,\alpha)$ satisfying
(11) so (12) and (13) are also satisfied. The values of $t\in\mathbb{R}$ for
which $y^{\prime}=0$ are $t_{k}=k\pi/\sqrt{4H^{2}+1}$ for any $k\in\mathbb{Z}$
(these ones correspond to the points in the surface whose tangent plane is
vertical). Now from (12) and (13) it is easy to check that $h(t_{k})\neq
h(t_{k+1})$ for every $k\in\mathbb{Z}$, which makes impossible the surface to
be a bigraph. ∎

##  3\. Boundary curvature estimates

Let us suppose along this section that $\Sigma\subseteq M\times\mathbb{R}$ is
a graph over a domain $\Omega\subseteq M$ with constant mean curvature $H>0$.
Following the ideas given in [1], for any given $c\in\mathbb{R}$ with
$c+4H^{2}>0$, we will consider the function $g:[-1,1]\rightarrow\mathbb{R}$
determined by

(14) $g^{\prime}(t)=\frac{4H}{4H^{2}+c(1-t^{2})},\quad\quad g(0)=0,$

which is strictly increasing and allows us to define the smooth function
$\psi=h+g(\nu)\in C^{\infty}(\Sigma)$, where $h$ and $\nu$ are the height and
angle functions, respectively. We are interested in applying the boundary
maximum principle for the laplacian to $\psi$ so we will need to work out
$\Delta\psi$ (where the laplacian is computed in the surface $\Sigma$) and
$\frac{\partial\psi}{\partial\eta}$, where $\eta$ is some outer conormal
vector field to $\partial\Sigma$. The following formulas will be useful.

###### Lemma 3.1.

In the previous situation, the following equalities hold.

  * i)

$\nabla h=E_{3}^{\top}$,

  * ii)

$\Delta h=2H\nu$,

  * iii)

$\nabla\nu=-AE_{3}^{\top}$,

  * iv)

$\Delta\nu=\left(2K-4H^{2}-K_{M}(1+\nu^{2})\right)\nu$.

###### Proof.

The identities for the gradient and the laplacian of $h$ are easy to check as
$h$ is the restriction to $\Sigma$ of the height function in
$M\times\mathbb{R}$ (see also [12, Lemma 3.1]). On the other hand, the
gradient of $\nu=\langle N,E_{3}\rangle$ satisfies

$\langle\nabla\nu,X\rangle=X(\langle
N,E_{3}\rangle)=\langle\nabla_{X}N,E_{3}\rangle=\langle-
AX,E_{3}\rangle=\langle X,-AE_{3}^{\top}\rangle$

for any vector field $X$ on $\Sigma$, so $\nabla\nu=-AE_{3}^{\top}$. Finally,
since the vertical translations are isometries of $M\times\mathbb{R}$, $\nu$
is a Jacobi function, i.e. $\nu$ lies in the kernel of the linearized mean
curvature operator $L=\Delta-[|\sigma|^{2}+\mathop{\rm Ric}\nolimits(N)]$ on
$\Sigma$ so we can compute its laplacian from $L\nu=0$ and obtain

$\Delta\nu=\left(|\sigma|^{2}+\mathop{\rm
Ric}\nolimits(N)\right)\nu=\left(2K-4H^{2}-K_{M}(1+\nu^{2})\right)\nu,$

where we have used the Gauss equation and the well-known identities
$|\sigma|^{2}=4H^{2}-2\det(A)$ and $\mathop{\rm
Ric}\nolimits(N)=K_{M}(1-\nu^{2})$. ∎

On the other hand, we need to obtain some suitable expression for the modulus
of the Abresch-Rosenberg differential, in the case $M=\mathbb{M}^{2}(c)$. If
we take a conformal parametrization $(U,z)$ in $\Sigma$, this quadratic
differential can be written as

$Q=(2Hp-ch_{z}^{2})\,\mathrm{d}z^{2}$

(see [4]), where
$p\,\mathrm{d}z^{2}=\langle-\nabla_{\partial_{z}}N,\partial_{z}\rangle\,\mathrm{d}z^{2}$
is the Hopf differential and $h_{z}=\frac{\partial h}{\partial z}$. Although
this expression depends on the parametrization, we may consider the function

$\displaystyle q=\frac{4}{\lambda^{2}}|Q|^{2}$
$\displaystyle=\tfrac{4}{\lambda^{2}}\left(4H^{2}|p|^{2}+c^{2}|h_{z}|^{4}-2cH(ph_{\bar{z}}^{2}-\bar{p}h_{z}^{2})\right)$
(15)
$\displaystyle=4H^{2}(H^{2}-\det(A))+\tfrac{c^{2}}{4}(1-\nu^{2})^{2}-c(\|\nabla\nu\|^{2}-(2H^{2}-\det(A))(1-\nu^{2})),$

which is well-defined and smooth on the whole $\Sigma$.

Back to the computation of $\Delta\psi$ and taking into account the formulas
in Lemma 3.1 and identity (15), we get

(16) $\Delta\psi=\Delta
h+g^{\prime}(\nu)\Delta\nu+g^{\prime\prime}(\nu)\|\nabla\nu\|^{2}=\frac{-8q\nu}{\left(4H^{2}+c(1-\nu^{2})\right)^{2}}-\frac{4H\nu\left(1-\nu^{2}\right)(K_{M}-c)}{4H^{2}+c(1-\nu^{2})}.$

Finally, we are interested in working out $\frac{\partial\psi}{\partial\eta}$
along $\partial\Sigma$, where we have considered the outer conormal vector
field to $\partial\Sigma$ in $\Sigma$ given by $\eta=-E_{3}^{\top}$ (it does
not matter which outer conormal vector field is chosen as the only needed
information is the sign of $\frac{\partial\psi}{\partial\eta}$). Hence,

$\displaystyle\frac{\partial h}{\partial\eta}$ $\displaystyle=\langle\nabla
h,\eta\rangle=\langle
E_{3}^{\top},-E_{3}^{\top}\rangle=-\|E_{3}^{\top}\|^{2},$
$\displaystyle\frac{\partial\nu}{\partial\eta}$
$\displaystyle=\langle\nabla\nu,\eta\rangle=\langle-
AE_{3}^{\top},-E_{3}^{\top}\rangle=\langle\overline{\nabla}_{E_{3}^{\top}}E_{3}^{\top},N\rangle.$

However, if we parametrize $\partial\Sigma$ by $\gamma$ with
$\|\gamma^{\prime}\|=1$, then
$\\{E_{3}^{\top}/\|E_{3}^{\top}\|,\gamma^{\prime}\\}$ is an orthonormal basis
of $T\Sigma$, and it is clear that

$2H=\left\langle\frac{1}{\|E_{3}^{\top}\|^{2}}\overline{\nabla}_{E_{3}^{\top}}E_{3}^{\top}+\overline{\nabla}_{\gamma^{\prime}}\gamma^{\prime},N\right\rangle$

so
$\frac{\partial\nu}{\partial\eta}=2H\|E_{3}^{\top}\|^{2}+\|E_{3}^{\top}\|^{3}\kappa_{g}$.
Here, $\kappa_{g}$ denotes the geodesic curvature of
$\partial\Omega=\partial\Sigma$ in the base $M$ with respect to
$\|E_{3}^{\top}\|^{-1}(N-\nu E_{3})$, the outer conormal vector field to
$\partial\Omega$ in $M$. Finally, we obtain

(17) $\frac{\partial\psi}{\partial\eta}=\frac{\partial
h}{\partial\eta}+g^{\prime}(\nu)\frac{\partial\nu}{\partial\eta}=\|E_{3}^{\top}\|^{2}\left(-1+g^{\prime}(\nu)(2H+\|E_{3}^{\top}\|\kappa_{g})\right).$

Note that any outer conormal vector field to $\partial\Omega$ in $M$ is a
linear combination of $N$ and $E_{3}$, which is the key property to relate the
geometries of $\Sigma$ and $M$. Moreover, these computations allow us to give
an optimal bound for the geodesic curvature of the boundary of the domain of a
compact $H$-graph with a capillarity boundary condition.

###### Theorem 3.2.

Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a compact regular domain $\Omega\subseteq M$ with zero values in
$\partial\Omega$. Let us consider $c=\inf\\{K_{M}(p):p\in\Omega\\}$ and
suppose that $c+4H^{2}>0$ and $\nu=\nu_{0}$ in $\partial\Omega$ for some
$-1<\nu_{0}\leq 0$. Then

(18) $\kappa_{g}\geq\frac{-4H^{2}+c(1-\nu_{0}^{2})}{4H\sqrt{1-\nu_{0}^{2}}},$

where $\kappa_{g}$ is the geodesic curvature of $\partial\Omega$ in $M$ with
respect to the outer conormal vector field.

Furthermore, if there exists $p\in\partial\Omega$ such that equality holds in
(18), then $\Omega$ has constant curvature and $\Sigma$ is invariant by a
$1$-parameter group of isometries.

###### Proof.

Let us consider the function $\psi=h+g(\nu)$, defined in terms of (14). Since
$\nu\leq 0$ and $K_{M}\geq c$ in $\Sigma$, equation (16) insures that
$\Delta\psi\geq 0$ in $\Sigma$. Let $p_{0}\in\Sigma$ a point where $\psi$
attains its maximum. If $p_{0}$ is interior to $\Sigma$, then the maximum
principle for the laplacian guarantees that $\psi$ is constant, so from (16),
we get that $q=0$ and $K_{M}=c$ in $\Omega$. These conditions imply that
$\Omega$ has constant curvature $c$ and that $\Sigma$ is invariant by a
1-parameter subgroup of isometries which acts trivially on the vertical lines
(see [3, Lemma 6.1]).

If, on the contrary, $p_{0}\in\partial\Omega$ and $\psi(p_{0})>\psi(p)$ for
every interior point $p\in\Omega$, then such maximum is attained in the whole
boundary as $\psi$ is constant along it, so the maximum principle in the
boundary insures that $\frac{\partial\psi}{\partial\eta}>0$ in
$\partial\Omega$. Then the desired strict inequality for $\kappa_{g}$ can be
deduced from (17). ∎

######  Remark 3.3.

In the situation of the statement of Theorem 3.2, if we suppose that
$-1<\nu\leq\nu_{0}$ in $\partial\Omega$ for some $\nu_{0}\leq 0$ (instead of
$\nu=\nu_{0}$ in $\partial\Omega$) and there exists a point
$p\in\partial\Omega$ such that $\nu(p)=\nu_{0}$ and at which the inequality
for $\kappa_{g}$ becomes and equality, then $\Omega$ has constant curvature
and $\Sigma$ is a spherical cap of a standard rotational sphere. This can be
easily seen as a consequence of the maximum principle in the boundary.

Observe that, in case $\Sigma\subseteq M\times\mathbb{R}$ is a compact
embedded constant mean curvature $H>0$ surface, it is possible to apply the
Alexandrov reflection principle to vertical reflections and conclude that
$\Sigma$ is a symmetric bigraph with respect to some slice
$M\times\\{t_{0}\\}$ which, after a vertical translation, may be supposed to
be $M\times\\{0\\}$. In this setting, it is obvious that $\Sigma$ intersects
orthogonally such a slice.

###### Corollary 3.4.

Let $H>0$ and $\Sigma\subseteq M\times\mathbb{R}$ be a compact embedded
$H$-bigraph over a domain $\Omega\subseteq M$, symmetric with respect to
$M\times\\{0\\}$. If we suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}>-4H^{2}$, then $\partial\Omega$ is a curve
whose geodesic curvature in $M$ with respect to the outer conormal vector
field satisfies

$\kappa_{g}\geq-H+\frac{c}{4H}.$

Furthermore, if equality holds at some point in $\partial\Omega$, then
$\Omega$ has constant curvature and $\Sigma$ is invariant under a
$1$-parameter isometry group.

We now adjust the value of $H$ for which the lower bound is exactly zero.

###### Corollary 3.5.

Let $M$ be a orientable complete Riemannian surface with $K_{M}\geq c>0$ in
$M$. Then, each compact embedded $H$-surface in $M\times\mathbb{R}$ with
$0<H<\frac{1}{2}\sqrt{c}$ is an $H$-bigraph over a connected domain $\Omega$
and $M\smallsetminus\Omega$ is a finite union of disjoint convex disks.
Furthermore, either

  * •

these disks are strictly convex or

  * •

$M=\mathbb{S}^{2}(c)$, $\Omega$ is a closed hemisphere and $\Sigma$ is a
rotationally invariant $H$-sphere for $H=\frac{1}{2}\sqrt{c}$ (in this case
$\kappa_{g}$ identically vanishes).

We recall that, in the case $M=\mathbb{S}^{2}(c)$, each of these disks must
lie in an open hemisphere because they are convex.

##  4\. Further boundary curvature estimates

In this section, we will obtain better estimates for the geodesic curvature of
the boundary by assuming restrictions on the maximum height that the surface
can reach. In order to achieve this, we will use a technique which has its
origins in a paper by Payne and Philippin [8] and which has also been used by
Ros and Rosenberg in [11].

Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
surface which is a graph over a domain $\Omega\subseteq M$ and extends
continuously to the boundary of $\Omega$ with zero values. For any given
$m>0$, let consider the function $g_{m}:[-1,1]\rightarrow\mathbb{R}$
determined by

(19) $g_{m}^{\prime}(t)=\frac{4mH}{4H^{2}+c(1-t^{2})},\quad\quad g(0)=0.$

This function allows us to take

$X=\frac{2H\nu(2m-1)}{m(1-\nu)^{2}}E_{3}^{\top}-\frac{2H\nu
g_{m}^{\prime}(\nu)}{m(1-\nu^{2})}AE_{3}^{\top},$

which is a smooth vector field defined on $\Sigma\smallsetminus V$, where
$V=\\{p\in\Sigma:\nu(p)=-1\\}$ is the subset of $\Sigma$ with vertical Gauss
map. We will consider the second order elliptic operator $L$ on
$C^{\infty}(\Sigma\smallsetminus V)$ given by $Lf=\Delta f+X(f)$, and the
function $\psi_{m}=h+g_{m}(\nu)\in C^{\infty}(\Sigma)$. We are now interested
in working out $L\psi_{m}$. By using Lemma 3.1, we obtain

(20)
$L\psi_{m}=-\frac{(m-1)(m-\frac{1}{2})H\nu}{m}-\frac{4Hm\nu(1-\nu^{2})(K_{M}-c)}{4H^{2}+c(1-\nu^{2})}.$

We observe that the second term in the RHS is positive because $K_{M}\geq c$
is satisfied. Moreover, for $m\geq 1$ or $m\leq\frac{1}{2}$, the first term is
also positive so the function $\psi_{m}$ verifies $L\psi_{m}\geq 0$ in
$\Sigma\smallsetminus V$. Thus, it is possible to apply the maximum principle
for the operator $L$ in $\Sigma\smallsetminus V$, which insures that
$\psi_{m}$ cannot achieve an interior maximum in $\Sigma\smallsetminus V$
unless it is constant.

###### Lemma 4.1.

Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a (not necessarilly compact) domain $\Omega\subseteq M$ which
extends continuously to $\partial\Omega$ with zero values and suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}>-4H^{2}$. Aditionally, if $\psi_{m}$ is
constant in $\Sigma$ for some $m\leq\frac{1}{2}$, then

  * a)

$m=\frac{1}{2}$ and $K_{M}$ is constant in $\Omega$,

  * b)

$\Sigma$ is invariant by a $1$-parameter group of isometries.

In particular, if $c>0$ and $M=\mathbb{S}^{2}(c)$, then $\Sigma$ is a compact
rotationally invariant torus and, if $c\leq 0$ and $M=\mathbb{H}^{2}(c)$, then
$\Sigma$ is an invariant horizontal cylinder, both described in Section 2.

###### Proof.

If $\psi_{m}$ is constant, then $L\psi_{m}=0$ so from (20) we get that
$(m-1)(m-\frac{1}{2})\leq 0$ which is only possible if $m=\frac{1}{2}$. Then,
as equality in (20) holds, $K_{M}$ must be constant in $\Sigma\smallsetminus
V$ so it is constant in $\Sigma$ as $K_{M}$ is continuous and $V$ has empty
interior.

Now, suppose that $m=\frac{1}{2}$ and $\psi_{m}$ is constant. On one hand,
from $\nabla\psi_{m}=0$ we obtain
$AE_{3}^{\top}=\frac{1}{g^{\prime}(\nu)}E_{3}^{\top}$ so $E_{3}^{\top}$ must
be a principal direction and $\frac{1}{g^{\prime}(\nu)}$ its corresponding
principal curvature. We also deduce the following expressions:

(21) $\displaystyle\det(A)$
$\displaystyle=\frac{1}{g^{\prime}(\nu)}\left(2H-\frac{1}{g^{\prime}(\nu)}\right),$
$\displaystyle\|\nabla\nu\|^{2}$ $\displaystyle=\langle
AE_{3}^{\top},AE_{3}^{\top}\rangle=\frac{1-\nu^{2}}{g^{\prime}(\nu)^{2}}.$

If we consider the differentiable function $f:\ ]-1,1[\ \rightarrow\mathbb{R}$
determined by

$f^{\prime}(t)=\frac{1}{\sqrt{(1-t^{2})(4H^{2}+c(1-t^{2}))}},\quad\quad
f(0)=0,$

and take into account (21) and Lemma 3.1, it is easy to check that

$\displaystyle\Delta(f(\nu))$
$\displaystyle=f^{\prime\prime}(\nu)\|\nabla\nu\|^{2}+f^{\prime}(\nu)\Delta\nu$
$\displaystyle=\frac{f^{\prime\prime}(\nu)}{g^{\prime}(\nu)^{2}}(1-\nu^{2})+f^{\prime}(\nu)(2\det(A)-4H^{2}-c(1-\nu^{2}))\nu=0$

where we have also used that $K_{M}$ is constant by item (a). As $f(\nu)$ is a
non-constant harmonic function on $\Sigma\smallsetminus V$, we can (at least
locally) take a conformal parameter $z$ on $\Sigma\smallsetminus V$ such that
$\mathop{\rm Re}\nolimits(z)=f(\nu)$. Now, we can repeat the arguments given
in [3, Lemma 6.1] to conclude that $\Sigma$ is invariant by a $1$-parameter
group of isometries of $\mathbb{M}^{2}(c)\times\mathbb{R}$. As $\Sigma$ is an
embedded $H$-graph over a domain $\Omega\subseteq M$ which continuously
extends to $\partial\Omega$ with zero values, the isometries in this group act
trivially on the vertical lines so, if $c\neq 0$, we are in the situation
studied in Section 2 and the only posibilities for $\Sigma$ are those
mentioned in the statement. If $c=0$, it is well-known that $\Sigma$ must be a
horizontal cylinder. ∎

###### Theorem 4.2.

Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a compact regular domain $\Omega$ with zero values in
$\partial\Omega$ and suppose that $c=\inf\\{K_{M}(p):p\in\Omega\\}$ satisfies
that $4H^{2}+c>0$ and $\nu=\nu_{0}$ in $\partial\Omega$ for some
$-1<\nu_{0}\leq 0$.

If there exists $0<m\leq\frac{1}{2}$ such that $|h|\leq
m\cdot\alpha(c,H,\nu_{0})$, then the following lower bound for the geodesic
curvature of $\partial\Omega$ in $M$ (with respect to the outer conormal
vector field) holds:

$\kappa_{g}\geq\frac{(4-8m)H^{2}+c(1-\nu_{0}^{2})}{4mH\sqrt{1-\nu_{0}^{2}}}.$

###### Proof.

Let us consider the function $\psi_{m}=h+g_{m}(\nu)\in C^{\infty}(\Sigma)$,
which verifies that $L\psi_{m}\geq 0$ in view of (20). As $\Sigma$ is compact,
there exists a point $p_{0}\in\Sigma$ where $\psi_{m}$ attains its maximum. We
distinguish three possibilities:

  * •

If $p_{0}$ is an interior point of $\Sigma\smallsetminus V$, then $\psi_{m}$
is constant in $\Sigma$, which implies that $m=\frac{1}{2}$, $K_{M}$ is
constant in $\Omega$ and $\Sigma$ is invariant by a $1$-parameter isometry
group because of Lemma 4.1.

  * •

If $p_{0}\in\partial\Omega$, then such a maximum is attained in the whole
boundary $\partial\Omega$ since $(\psi_{m})_{|\partial\Omega}$ is constant.
Then the boundary maximum principle for the operator $L$ guarantees that
$\frac{\partial\psi_{m}}{\partial\eta}\geq 0$ along $\partial\Omega$. It is
straightforward to check from (17) that this is equivalent to the inequality
in the statement above.

  * •

If $p_{0}\in V$, then $\nu(p_{0})=-1$. The inequality we are looking for would
be proved if we discarded this case, but it turns out that $h\leq
m\cdot\alpha(c,H,\nu_{0})=-g_{m}(-1)$ so
$\psi_{m}\leq\psi_{m}(p_{0})=h(p_{0})+g_{m}(-1)=h(p_{0})-m\cdot\alpha(c,H,\nu_{0})+g_{m}(\nu_{0})\leq
g_{m}(\nu_{0})$ and, since $\psi_{m}$ is equal to $g_{m}(\nu_{0})$ in
$\partial\Omega$, the maximum is also attained in the boundary, which reduces
this case to the previous one.

∎

We now apply the theorem to the compact embedded case, where we lay in the
same situation of Corollary 3.4.

###### Corollary 4.3.

Let $\Sigma\subseteq M\times\mathbb{R}$ be a compact constant mean curvature
$H>0$ surface, symmetric with respect to $M\times\\{0\\}$, and suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}$ satisfies $4H^{2}+c>0$. If there exists
$0<m\leq\frac{1}{2}$ such that $h\leq m\cdot\alpha(c,H,0)$, then the following
lower bound for the geodesic curvature of $\partial\Omega$ in $M$ (with
respect to the outer conormal vector field) holds:

$\kappa_{g}\geq\frac{(4-8m)H^{2}+c}{4mH}.$

We now adjust the constant $0<m<\frac{1}{2}$ to guarantee the convexity of the
boundary, as we did in Corollary 3.5.

###### Corollary 4.4.

Let $\Sigma\subseteq M\times\mathbb{R}$ be an $H$-graph over a compact regular
domain $\Omega$ with zero boundary values. Suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}$ satisfies that $4H^{2}+c>0$ and
$\nu=\nu_{0}$ in $\partial\Omega$ for some $-1<\nu_{0}\leq 0$. In any of the
following two situations,

  * i)

$c\geq 0$ and $h\leq\frac{1}{2}\alpha(c,H,\nu_{0})$ in $\Sigma$ or

  * ii)

$c<0$ and $h\leq\frac{4H^{2}+c(1-\nu_{0})^{2}}{8H^{2}}\alpha(c,H,\nu_{0})$ in
$\Sigma$,

the boundary $\partial\Omega$ is convex in $M$ with respect to the outer
conormal vector field.

We finally wonder if the compactness hypothesis for the domain of the graph
can be removed. In order to achieve this, we will restrict ourselves to the
ambient space $\mathbb{M}^{2}(c)\times\mathbb{R}$ and $\nu_{0}=0$ (that is,
$\Sigma$ is an $H$-bigraph), where the technique developed by Ros and
Rosenberg in [11] can be easily adapted.

###### Theorem 4.5.

Let $\Sigma\subseteq M(c)\times\mathbb{R}^{+}$ be a properly embedded
$H$-bigraph over a domain $\Omega\subseteq M$ with $4H^{2}+c>0$, symmetric
with respect to $M\times\\{0\\}$, and suppose that there exists
$0<m\leq\frac{1}{2}$ such that $|h|\leq m\cdot\alpha(c,H,\nu_{0})$ in
$\Sigma$. Then, the following lower bound for the geodesic curvature of
$\partial\Omega$ in $M$ (with respect to the outer conormal vector field)
holds:

$\kappa_{g}\geq\frac{(4-8m)H^{2}+c}{4mH}.$

Furthermore, if there exists $p\in\partial\Omega$ for which equality holds,
then

  * i)

$\Sigma$ is a rotationally invariant torus (described in Section 2.1) if
$c>0$,

  * ii)

$\Sigma$ is a invariant cylinder under horizontal translations if $c=0$, and

  * iii)

$\Sigma$ is a invariant cylinder under hyperbolic translations (described in
Section 2.2) if $c<0$.

###### Proof.

Let us consider the same function $\psi_{m}\in C^{\infty}(\Sigma)$ as before.
If $\psi_{m}$ attained its maximum or $\sup_{\Sigma}\psi_{m}\leq 0$, we could
reason in the same way we did for the compact case and the proof would be
finished. Otherwise, let us take a sequence $\\{p_{n}\\}\subseteq\Sigma$ such
that $\\{\psi_{m}(p_{n})\\}$ converges to $\sup\psi_{m}$, and distinguish two
cases.

  * •

If $\lim\\{h(p_{n})\\}=0$, then
$\psi_{m}(p_{n})=h(p_{n})+g_{m}(\nu(p_{n}))\leq h(p_{n})\rightarrow 0$ from
where $\sup_{\Sigma}\psi_{m}\leq 0$ and we are done.

  * •

If $\\{h(p_{n})\\}$ does not converge to zero, we can suppose that
$\\{h(p_{n})\\}\rightarrow a>0$ without loss of generality. Now, ambient
isometries allow us to translate $\Sigma$ horizontally so that $p_{n}$ is over
some fixed point $q_{0}\in M$ and standard convergence arguments make possible
to consider $\Sigma_{\infty}$, the limit $H$-graph of these translated
surfaces (note that $\Sigma$ has uniform curvature estimates around $p_{n}$ by
stability, since the distance from $p_{n}$ to $\partial\Sigma$ is bounded away
from zero). The corresponding function in $\Sigma_{\infty}$, given by
$\psi_{m,\infty}=h_{\infty}+g_{m}(\nu_{\infty})\in
C^{\infty}(\Sigma_{\infty})$, attains its maximum at the interior point
$p_{\infty}=\lim\\{p_{n}\\}$ (which is not in the boundary of
$\Sigma_{\infty}$ because $h_{\infty}(p_{\infty})=a>0$) , so it identically
vanishes since $\psi_{\infty}$ vanishes at $\partial\Sigma_{\infty}$. This
implies that $\sup_{\Sigma}\psi_{m}=\psi_{m,\infty}(p_{\infty})=0$ and we are
also done.

In case that equality holds, the description in the statement follows from
Lemma 4.1. ∎

Observe that, if the maximum heights of a sequence $\\{\Sigma_{n}\\}$ of such
$H$-bigraphs tend to zero, then Theorem 4.5 insures that the geodesic
curvatures of the boundaries diverge uniformly, in the sense that the bound
only depends on that maximum height. Thus, the sequence of domains
$\Omega_{n}\subseteq M$ over which $\Sigma_{n}$ is a graph cannot eventually
omit any set in $M$ with nonempty interior.

##  5\. Intrinsic length estimates

Let $\Sigma\subseteq M\times\mathbb{R}$ be an $H$-graph over a compact domain
$\Omega\subseteq M$ which extends continuously to the boundary with zero
values. Suppose that $K_{M}\geq c>-4H^{2}$ in $\Sigma$ for some $c>0$. In
Section 3 we proved that $\psi=h+g(\nu)$ is subharmonic in $\Sigma$, where $g$
is defined in (14) so, if we suppose that $\nu\leq\nu_{0}$ along
$\partial\Sigma$, then $h+g(\nu)\leq g(\nu_{0})$, as a consequence of that $g$
is strictly increasing and $h$ vanishes on $\partial\Sigma$. Therefore, as $g$
is also an odd function, we derive that $g(-\nu)\geq h-g(\nu_{0})$. Now we can
invert the function $g$ and square both sides to obtain

(22)
$\nu^{2}\geq\zeta(h,\nu_{0}):=\begin{cases}\frac{c+4H^{2}}{c}\tanh^{2}\left(\frac{\sqrt{c^{2}+4H^{2}c}}{4H}(h-g(\nu_{0}))\right)&\text{if
}c<0,\\\ H^{2}(h-g(\nu_{0}))^{2}&\text{if }c=0,\\\
\frac{c+4H^{2}}{-c}\tan^{2}\left(\frac{\sqrt{-c^{2}-4H^{2}c}}{4H}(h-g(\nu_{0}))\right)&\text{if
}c>0.\\\ \end{cases}$

Let $\gamma:[a,b]\rightarrow\Sigma$ be a smooth curve which is parametrized by
arc-length and let $\eta$ be a smooth unit vector field along $\gamma$,
orthogonal to $\gamma^{\prime}$. Then, as $\\{N,\alpha^{\prime},\eta\\}$ is an
orthonormal frame, we have

$E_{3}=\langle N,E_{3}\rangle
E_{3}+\langle\gamma^{\prime},E_{3}\rangle\gamma^{\prime}+\langle\eta,E_{3}\rangle\eta,$

and, since $\langle N,E_{3}\rangle=\nu$ and
$\langle\gamma^{\prime},E_{3}\rangle=h^{\prime}(\gamma)$, we deduce that
$1=\nu^{2}+h^{\prime}(\gamma)^{2}+\langle\eta,E_{3}^{\top}\rangle^{2}$. Taking
into account that $\langle\eta,E_{3}^{\top}\rangle^{2}\geq 0$, we finally get
$|h^{\prime}|\leq 1-\nu^{2}$. Thus, plugging (22) into this inequality, we
have

(23) $\mathop{\rm
Long}\nolimits(\gamma)\geq\int_{0}^{a}\frac{|h^{\prime}|}{\sqrt{1-\nu^{2}}}\,\mathrm{d}t\geq\int_{0}^{a}\frac{-h^{\prime}}{\sqrt{1-\zeta(h,\nu_{0})}}\,\mathrm{d}t=\int_{h(a)}^{h(0)}\frac{ds}{\sqrt{1-\zeta(s,\nu_{0})}}.$

Considering all the curves that join a point $p$ with the boundary (along
which the height vanishes), we obtain the following result:

###### Theorem 5.1.

Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a compact domain $\Omega\subseteq M$ which extends continuously to
the boundary with zero values and suppose that
$c=\inf\\{K_{M}(p):p\in\Sigma\\}>-4H^{2}$. If $\nu\leq\nu_{0}$ in
$\partial\Omega$ for some $-1<\nu_{0}\leq 0$, then

$\mathop{\rm
dist}\nolimits(p,\partial\Sigma)\geq\int_{0}^{h(p)}\frac{ds}{\sqrt{1-\zeta(s,\nu_{0})}}.$

Furthermore, if there exists $p\in\Sigma$ such that equality holds, then
$\Omega$ has constant curvature and $\Sigma$ is a spherical cap of a
rotationally invariant sphere.

In other words, Theorem 5.1 is a comparison result which claims that
rotationally invariant spheres in the corresponding homogeneous space
$\mathbb{M}^{2}(c)\times\mathbb{R}$ minimize the distance from a point to the
boundary in terms of the height of that point.

We remark that the bound given in the statement can be worked out explicitly
in terms of elementary functions, but the result of that computation is a
large formula which does not contribute to a better understanding so we have
preferred to leave it in this way.

## References

  * [1] J. Aledo, J. Espinar, and J. Gálvez.  Height estimates for surfaces with positive constant mean curvature in $\mathbb{M}^{2}\times\mathbb{R}$.  Illinois J. Math., 52(1):203–211, 2008.
  * [2] K. Große Brauckmann.  New surfaces of constant mean curvature.  Math. Z., 214:527–565, 1993.  MR1248112, Zbl 0806.53005.
  * [3] J. M. Espinar and H. Rosenberg.  Complete constant mean curvature surfaces in homogeneous spaces.  To appear in Comment. Math. Helv., 2009.
  * [4] I. Fernandez and P. Mira.  A characterization of constant mean curvature surfaces in homogeneous 3-manifolds.  Diff. Geom. Appl., 25:281–289, 2007.
  * [5] W. T. Hsiang and W. Y. Hsiang.  On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I.  Invent. Math., 98:39–58, 1989.  MR1010154 (90h:53078).
  * [6] W. H. Meeks III, J. Pérez, and A. Ros.  Stable constant mean curvature surfaces.  In Handbook of Geometrical Analysis, volume 1, pages 301–380. International Press, edited by Lizhen Ji, Peter Li, Richard Schoen and Leon Simon, ISBN: 978-1-57146-130-8, 2008.  MR2483369, Zbl 1154.53009.
  * [7] Irene I. Onnis.  Invariant surfaces with constant mean curvature in $\mathbb{H}^{2}\times\mathbb{R}$.  Ann. Mat. Pura Appl., 187(4):667–682, 2008.
  * [8] L. E. Payne and G. A. Philippin.  Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature.  Nonlinear Analysis: Theory, Methods & Applications, 3(2):193–211, 1979.
  * [9] R. Pedrosa.  The isoperimetric problem in spherical cylinders.  Annals of Global Analysis and Geometry, 26(4):333–354, 2004.
  * [10] M. Ritoré.  Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere.  Math. Z., 226:127–146, 1997.
  * [11] A. Ros and H. Rosenberg.  Properly embedded surfaces with constant mean curvature.  Preprint.
  * [12] H. Rosenberg.  Minimal surfaces in ${M}^{2}\times\mathbb{R}$.  Illinois J. of Math., 46:1177–1195, 2002.  MR1988257, Zbl 1036.53008.
  * [13] R. Sa Earp.  Parabolic and hyperbolic screw motion surfaces in $\mathbb{H}^{2}\times\mathbb{R}$.  J. Aust. Math. Soc., 85(1):113–143, 2008.


Normalising Lustre Preserves Security

Indian Institute of Technology Delhi, INDIA
Sanjiva Prasad

R. Madhukar Yerraguntla

The synchronous reactive data flow language  is an expressive language, equipped with a suite of tools for modelling, simulating and model-checking a wide variety of safety-critical systems.
A critical intermediate step in the formally certified compilation of  involves translation to a well-behaved sub-language called “Normalised ” ().
Recently, we proposed a simple Denning-style lattice-based secure information flow type system for , and proved its soundness by establishing that security-typed programs are non-interfering with respect to the co-inductive stream semantics.

In this paper, we propose a similar security type system for unrestricted , and show that Bourke et al.'s semantics-preserving normalisation transformations from  to  are security-preserving as well.
A novelty is the use of refinement security types for node calls.
The main result is the preservation of security types by the normalisation transformations.
The soundness of our security typing rules is shown by establishing that well-security-typed programs are non-interfering, via a reduction to type-preservation (here), semantics-preservation (Bourke et al.) and our previous result of non-interference for .

Synchronous reactive data flow,
Compiler transformation,
Security type system,
Security preservation.

§ CONCLUSIONS

We have presented a novel security type system for  using the notion of constraint-based refinement (sub)types.
Using security-type preservation and earlier results, we have shown its semantic soundness, expressed in terms of non-interference, with respect to the language's stream semantics.

We are developing mechanised proofs of these results, which can be integrated into the Velús verified compiler framework [5].

While 's value type system is quite jejune, this security type system is not.
It is therefore satisfying to see that it satisfies a subject reduction property[
At SYNCHRON 2020, De Simone asked Jeanmaire and Pesin whether the terminology “normalisation” used in their work  [4] was related in any way to notions of normalisation seen in, e.g., the $\lambda$-calculus. It is!].
A difficult aspect encountered during the transcription phase  [4] concerns alignment of clocks in the presence of complex clock dependencies.
We clarify that our type system, being static, only considers security levels of clocks, not actual clock behaviour, and therefore is free from such complications.
Further, the clocks induce no timing side-channels since the typing rules enforce, a fortiori, that the security type of any (clocked) expression is at least as high as that of its clock.

[1]
Auger, C.
Certified compilation of SCADE / LUSTRE.
Theses, Université Paris Sud - Paris XI, Feb. 2013.

[2]
Bourke, T., Brun, L., Dagand, P.-E., Leroy, X., Pouzet, M., and Rieg, L.
A Formally Verified Compiler for Lustre.
In Proceedings of the 38th ACM SIGPLAN Conference on Programming
Language Design and Implementation (New York, NY, USA, 2017), PLDI 2017,
Association for Computing Machinery, p. 586–601.

[3]
Bourke, T., Brun, L., and Pouzet, M.
Mechanized Semantics and Verified Compilation for a Dataflow
Synchronous Language with Reset.
Proc. ACM Program. Lang. 4, POPL (Dec. 2019).

[4]
Bourke, T., Jeanmaire, P., Pesin, B., and Pouzet, M.
Normalisation vérifiée du langage lustre.
In 32ièmes Journées Francophones des
Langages Applicatifs (JFLA 2021) (online, Apr. 2021), Y. Regis-Gianas and
C. Keller, Eds.

[5]
Brun, L., Bourke, T., and Pouzet, M.
Vélus compiler repository.
<https://github.com/INRIA/velus>, 2020.
Accessed: 2020-01-20.

[6]
Caspi, P., Pilaud, D., Halbwachs, N., and Plaice, J. A.
LUSTRE: A Declarative Language for Programming
Synchronous Systems.
In Proc. 14th Symposium on Principles of Programming Languages
(POPL’87). ACM (1987).

[7]
Colaço, J., Pagano, B., and Pouzet, M.
SCADE 6: A Formal Language for Embedded Critical
Software Development (invited paper).
In 2017 International Symposium on Theoretical Aspects of
Software Engineering (TASE) (Sep. 2017), pp. 1–11.

[8]
Denning, D. E.
A Lattice Model of Secure Information Flow.
Commun. ACM 19, 5 (May 1976), 236–243.

[9]
Goguen, J. A., and Meseguer, J.
Security Policies and Security Models.
In 1982 IEEE Symposium on Security and Privacy, Oakland, CA,
USA, April 26-28, 1982 (1982), IEEE Computer Society, pp. 11–20.

[10]
Halbwachs, N., Caspi, P., Raymond, P., and Pilaud, D.
The Synchronous Data Flow Programming Language LUSTRE.
Proceedings of the IEEE 79, 9 (Sep. 1991), 1305–1320.

[11]
Jahier, E.
The Lurette V2 User guide, V2 ed.
Verimag, October 2015.

[12]
Kind 2 group.
Kind 2 User Documentation, version 1.2.0 ed.
Department of Computer Science, The University of Iowa, April 2020.

[13]
Knuth, D. E., and Bendix, P. B.
Simple word problems in universal algebras.
In Computational Problems in Abstract Algebra, J. LEECH, Ed.
Pergamon, 1970, pp. 263–297.

[14]
Leroy, X.
Formal verification of a realistic compiler.
Commun. ACM 52, 7 (2009), 107–115.

[15]
Prasad, S., Yerraguntla, R. M., and Sharma, S.
Security types for synchronous data flow systems.
In 2020 18th ACM-IEEE International Conference on Formal Methods
and Models for System Design (MEMOCODE) (2020), pp. 1–12.

[16]
Raymond, P.
Synchronous Program Verification with Lustre/Lesar.
Wiley, 2010, pp. 171 – 206.

[17]
Viry, P.
Equational rules for rewriting logic.
Theor. Comput. Sci. 285, 2 (2002), 487–517.

[18]
Volpano, D., Irvine, C., and Smith, G.
A Sound Type System for Secure Flow Analysis.
J. Comput. Secur. 4, 2–3 (Jan. 1996), 167–187.