# Gaussian curvature of minimal graphs in $M\times\mathbb{R}$

David Kalaj  University of Montenegro, Faculty of Natural Sciences and
Mathematics, Cetinjski put b.b. 81000 Podgorica, Montenegro<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider minimal graphs in the three-dimensional Riemannian
manifold $M\times\mathbb{R}$. We mainly estimate the Gaussian curvature of
such surfaces. We consider the minimal disks and minimal graphs bounded by two
Jordan curves in parallel planes. The key to the proofs is the Weierstrass
representation of those surfaces via $\wp-$harmonic mappings. We also prove
some Schwarz lemma type results and some Heinz type results for harmonic
mappings between geodesic disks in Riemannian surfaces.

###### Key words and phrases:

Harmonic mappings, minimal surfaces

###### Contents

  1. 1 Introduction
    1. 1.1 Conformal minimal immersion
    2. 1.2 Minimal graphs
  2. 2 Preliminary results from harmonic mappings and related topics
    1. 2.1 Schwarz lemma for harmonic mappings
    2. 2.2 Schwarz lemma for dilatation of harmonic mappings
  3. 3 Minimizers and minimal surfaces in Riemannian manifold
    1. 3.1 Stationary mappings and Noether harmonic maps
  4. 4 Curvature estimates of minimal graphs
    1. 4.1 Gaussian curvature
    2. 4.2 The curvature estimate of minimal graphs between parallel planes
    3. 4.3 The curvature of rotationally symmetric minimal surface

##  1\. Introduction

Assume that $\Omega$ is a domain in the complex plane. Moreover, assume that
$\wp$ is a smooth positive metric defined in $\Omega$ with bounded Gauss
curvature $\mathcal{K}$ where

(1.1) $\mathcal{K}_{\wp}(z)=-\frac{\Delta\log\wp(z)}{\wp^{2}(z)}.$

We assume $\sup_{z\in\Omega}|\mathcal{K}_{\wp}(z)|<\infty$ and $\wp$ has a
finite area defined by

$\mathcal{A}(\wp)=\int_{\Omega}\wp^{2}(w)dudv,\ \ w=u+iv.$

Then we define the Riemannian manifold $M=M_{0}\times\mathbb{R}$, where
$M_{0}=(\Omega,\wp)$, where the metric of $M$ is

(1.2) $d\hbar^{2}(z)=\wp^{2}(z)|dz|^{2}+dt^{2}.$

Let $N_{0}=(D,\mathbf{1})$ and $M_{0}=(\Omega,\wp)$ be Riemann surfaces with
metrics $\mathbf{1}$ and $\wp$, respectively, where $D$ and $\Omega$ are
planar domains. Here $\mathbf{1}$ is the Euclidean metric. If a mapping
$f:N_{0}\to M_{0},$ is $C^{2}$, then $f$ is said to be harmonic (to make
difference with Euclidean harmonic we will sometimes say $\wp$-harmonic) if

(1.3) $f_{z\overline{z}}+{(\log\wp^{2})}_{w}\circ f\cdot
f_{z}\,f_{\bar{z}}=0,$

where $z$ and $w$ are the local parameters on $N_{0}$ and $M_{0}$
respectively. Also $f$ satisfies (1.3) if and only if its Hopf differential

(1.4) $\mathrm{Hopf}(f)=\wp^{2}\circ ff_{z}\overline{f_{\bar{z}}}$

is a holomorphic quadratic differential on $M$. Let

$|\partial f|^{2}:=\wp^{2}(f(z))\left|\frac{\partial f}{\partial
z}\right|^{2}\text{ and
}|\bar{\partial}f|^{2}:={\wp^{2}(f(z))}\left|\frac{\partial
f}{\partial\bar{z}}\right|^{2}$

where $\frac{\partial f}{\partial z}$ and $\frac{\partial f}{\partial\bar{z}}$
are standard complex partial derivatives. The $\wp-$Jacobian is defined by

$J(z,f):=|\partial f(z)|^{2}-|\partial\bar{f}(z)|^{2}.$

If $f$ is sense preserving, then its $\wp-$Jacobian is positive. For
$g:N_{0}\mapsto M_{0}$ the $\wp-$ _Dirichlet energy_ is defined by

(1.5) $\mathcal{E}^{\wp}[g]=\int_{N_{0}}(|\partial
f|^{2}+|\bar{\partial}f|^{2})dx\wedge
dy=\int_{D}\wp^{2}(f(z))(|f_{z}|^{2}+|f_{\bar{z}}|^{2})dxdy.$

Assume that the energy integral of $f$ is bounded. Then a stationary point $f$
of $\mathcal{E}^{\wp}[g_{t}]$, i.e. the solution of the equation
$\partial_{t}\mathcal{E}^{\wp}[g_{t}]|_{t=0}=0$, $g_{t}=g+th$,
$t\in(-\epsilon,\epsilon)$, where $h$ has a compact support in $D$, is a
harmonic mapping i.e. it satisfies equation (1.3). A different variation
produces the so-called Noether harmonic maps in Section 3.1. Recall that
$N_{0}=(D,\mathbf{1})$. For the last definition and some important properties
of harmonic maps see [10, Chapter 8] and [9].

It follows from the definition that, if $a$ is conformal and $f$ is harmonic,
then $f\circ a$ is harmonic. Moreover, if $b$ is conformal, $b\circ f$ is also
harmonic but with respect to (possibly) an another metric $\wp_{1}$.

The Bochner formula [9, p. 113] for a harmonic mapping $h$ are

$\Delta\log|\partial h|^{2}=-\mathcal{K}_{\wp}J(h)$

and

$\Delta\log|\bar{\partial}h|^{2}=\mathcal{K}_{\wp}J(h).$

In this paper, we will prove some curvature estimates of a minimal graph
$\Sigma$ in $M_{0}\times\mathbb{R}$. Firstly we will derive an analogous
divergence formula for its non-parametric parametrization (Propositions 3.2
and (1.8)). Further, we consider disk-type minimal graphs and estimate the
”central” curvature (Theorem 4.4). To prove Theorem 4.4 we previously prove
some Schwarz lemma type estimates of the dilatation of $\wp-$harmonic
mappings, which we believe are interesting on their own (Theorem 2.6). Also,
we prove a Schwarz lemma type estimate for the $\wp-$distance of the
$\wp-$harmonic mapping (Theorem 2.3). We also need some Heinz type estimate
proved in the same theorem. The role of $\wp-$harmonic mappings in a minimal
surface $\Sigma$ in $(\Omega,\wp)\times\mathbb{R}$ concerning Weierstrass
representation is analogous to the role of Euclidean harmonic mappings in
Euclidean minimal surfaces (cf. [20]).

Moreover, minimizers of the $\wp-$energy of the diffeomorphic mappings between
certain doubly connected domains produce minimal surfaces spanning two Jordan
curves lying in parallel planes, provided the original annulus has a modulus
bigger than the image annulus. This is done in Section 3, where is made a
connections with minimizers of $\wp-$energy and minimal surfaces in Riemannian
manifolds.

By using the results in Section 3, we get the following Gaussian curvature at
the ”equator” of minimal annuli $\Sigma$ (see below Theorem 4.5 for its
reformulation ).

######  Theorem 1.1.

Assume that $\Sigma\subset\mathbb{R}^{2}\times\mathbb{R}$ is a double-
connected minimal graph bounded by two Jordan curves $\gamma_{1},\gamma_{2}$,
lying in two horizontal planes $\Pi_{1},\Pi_{2}$ whose distance is
$\mathbf{d}$. Let $\Pi$ be the equidistant plane between $\Pi_{1},\Pi_{2}$.
Then for every point $\zeta\in\Sigma\bigcap\Pi$ we have

$|\mathcal{K}_{\Sigma}(\zeta)|\leq\frac{\pi^{2}}{\mathbf{d}^{2}}.$

Moreover, the tangential plane $T\Sigma_{\zeta}$ at a point
$\zeta\in\Sigma\setminus(\gamma_{1}\cup\gamma_{2})$ is never horizontal nor
vertical.

At the end at Subsection 4.3, some explicit curvature estimates for
rotationally symmetric minimal annuli in Riemannian manifolds
$(\mathbb{A}(R,1),\wp)\times\mathbb{R}$, are obtained. Here $\wp$ is also
rotationally symmetric and $\mathbb{A}(R,1):=\\{z:R<|z|<1\\}$.

###  1.1. Conformal minimal immersion

A mapping $\chi(z)=(f(z),h(z)):\Omega\to(M,\wp)\times\mathbb{R}$ is conformal
minimal if and only if $(h_{w})^{2}=-(\wp\circ h)^{2}f_{w}\bar{f}_{w}$.
Furthermore, the induced metric is given by

$ds^{2}=(\wp\circ h)^{2}(|f_{z}|+|f_{\bar{z}}|)^{2}|dz|^{2}.$

Moreover, the unit normal $N$ at the tangent plane of the surface
$\Sigma=\\{\chi(z):z\in\Omega\\}$ is given by the formula

$N=\frac{1}{1+|F|^{2}}\left(\frac{2}{\wp}\Re F,\frac{2}{\wp}\Im
F,|F|^{2}-1\right)$

where

(1.6) $F^{2}=-\frac{f_{w}}{\bar{f}_{w}}.$

For those facts we refer to [20]. We want to mention that a class of minima
surfaces on the hyperbolic space $\mathbb{H}\times\mathbb{R}$ has been
considered in [16] and [19].

###  1.2. Minimal graphs

For a surface $S=\\{(u,v,\omega(u,v)):(u,v)\in\Omega\\}\subset
M=(\Omega,\wp)\times\mathbb{R}$, the $\wp$-area is defined by

(1.7)
$\|\omega\|:=\mathrm{Area}_{\wp}(S)=\int_{\Omega}\sqrt{\wp^{2}(u,v)+|\nabla\omega(u,v)|^{2}}dudv.$

By setting $\left(\frac{d}{dt}\|\omega_{t}\|\right)_{t=0}=0$, where
$\omega_{t}$ is the outer variation of the form $\omega_{t}=\omega+t\varpi$,
$t\in(-\epsilon,\epsilon)$, where $\varpi$ has a compact support in $\Omega$,
the following non-linear elliptic partial differential equation comes out

(1.8)
$\mathrm{div}\left(\frac{\nabla\omega}{\sqrt{1+\frac{|\nabla\omega(u,v)|^{2}}{\wp^{2}(u,v)}}}\right)=0.$

Therefore, the graph

$\Sigma=\\{(u,v,\omega(u,v)):(u,v)\in\Omega\\}\subset M\times\mathbb{R}$

is called minimal if $\omega$ solves the the equation (1.8). For $\wp\equiv
1$, the equation is the standard minimal surface equation.

We will relate the minimal graphs and conformal minimal immersions in
Proposition 3.2.

##  2\. Preliminary results from harmonic mappings and related topics

###  2.1. Schwarz lemma for harmonic mappings

Let $o\in\Omega$ and let $D_{R}(o)=D(o,R)\subset(\Omega,\wp)$ be a geodesic
disk of radius $R>0$ centered at $o$ and assume that the cut-locus of $o$ is
outside of $D_{R}(o)$ and $2R\kappa<\pi$, where $\kappa^{2}$ is an upper bound
of the Gaussian curvature and define

$\mathcal{H}_{\kappa}(o,R)=\\{f:\mathbb{D}\to\Omega:f(\mathbb{D})\subset
D_{R}(o),f\text{ is }\wp-\text{ harmonic }\\}.$

One of main results in [8] of Hildebrandt, Jost and Widman for harmonic maps
between two-dimensional manifolds reads.

######  Proposition 2.1.

[8, Theorem 3] The class $\mathcal{H}_{\kappa}(o,R)$ is Hölder continuous on
every compact subset of the unit disk, i.e. for every $s<1$ there are
constants $C>0$ and $0<\alpha<1$ depending on $s$, $R$ and $\kappa$ such that

(2.1) $d_{\wp}(f(z),f(w))\leq C|z-w|^{\alpha}$

for $|z|,|w|<s$, $f\in\mathcal{H}_{\kappa}(o,R)$.

Further we use the following theorem of the author

######  Theorem 2.2.

[14] Let

$\mathcal{H}_{\kappa}(o,R)=\\{f:\mathbb{D}\to\Omega:f(\mathbb{D})\subset
D_{R}(o),\,2R\kappa<\pi,f\text{ is }\wp-\text{ harmonic }\\}.$

a) Then there is an absolutely continuous homeomorphism

$\phi_{\wp}=\phi_{o,R}:[0,1]\to[0,R]$

convex in $\log r$, with $\phi^{\prime}(1^{-})>0$ and
$0<\phi^{\prime}(0^{+})<\infty$, such that

$d_{\wp}(f(z),f(0))\leq\phi_{\wp}\left(\left|z\right|\right),$

for $f\in\mathcal{H}_{\kappa}(o,R)$ with $f(0)=o$. Then ${\mathbb{D}}\ni
z\to\phi_{\wp}(|z|)$ is subharmonic.

b) Assume that the cut-locus of $o$ is outside of $D(o,R)$. Then there is an
absolutely continuous homeomorphism

$\varphi_{\wp}=\varphi_{R}:[0,1]\to[0,2R]$

convex in $\log r$, with $\varphi_{\wp}^{\prime}(1^{-})>0$ and
$0<C^{\prime}_{\wp}=\varphi_{\wp}^{\prime}(0^{+})<\infty$, such that

$d_{\wp}(f(z),f(0))\leq\varphi_{\wp}\left(\left|z\right|\right),$

for $f\in\mathcal{H}_{\kappa}(o,R)$. Moreover

$\wp(f(z))|\nabla f(z)|\leq\frac{C^{\prime}_{\wp}}{1-|z|^{2}},\
z\in{\mathbb{D}}.$

Here $|\nabla f(z)|:=|f_{z}(z)|+|f_{\bar{z}}(z)|$.

Let $\mathfrak{M}_{\kappa}$ be the family of all smooth metrics $\wp$ in
$\Omega$ so that $|\mathcal{K}_{\wp}|\leq\kappa^{2}$, and so that the geodesic
disk $D_{1}(o)=D_{\wp}(o,1)\subset\Omega$ is disjoint from its cut-locus and
$2\kappa<\pi$. Let
$\mathcal{H}_{\kappa}(o):=\\{f:\mathbb{D}\to\Omega:f(\mathbb{D})\subset
D_{1}(o),\,2\kappa<\pi,f\text{ is }\wp-\text{ harmonic for some
}\wp\in\mathfrak{M}_{\kappa}\text{ and }f(0)=o\\}.$ Define

(2.2) $\phi_{\kappa}(r)=\sup\\{\phi_{\wp}(r),\wp\in\mathfrak{M}_{\kappa}\\}.$

The continuity of $\phi_{\kappa}$ follows from Theorem 2.2, because the
constant $C$ in (2.1) depends on $\kappa$ only. To do so, observe that

$\phi_{\wp}(r_{j})=\sup_{|z|=r_{j},f\in H_{\wp}}d_{\wp}(f(z),f(0)),\ j=1,2,$

and assume without loss of generality that
$\phi_{\wp}(r_{2})\geqslant\phi_{\wp}(r_{1})$. Then

$\begin{split}|\phi_{\wp}(r_{2})-\phi_{\wp}(r_{1})|&=(\phi_{\wp}(r_{2})-\phi_{\wp}(r_{1}))\\\
&=\sup_{f}d_{\wp}(f(r_{1}),o)-\sup_{g}d_{\wp}(g(r_{2}),o)\\\
&\leq\sup_{f}\left[d_{\wp}(f(r_{1}),o)-d_{\wp}(f(r_{2}),o)\right]\leq\sup_{f}d_{\wp}(f(r_{1}),f(r_{2}))\\\
&\leq C|r_{1}-r_{2}|^{\alpha}.\end{split}$

Similarly we get

$|\phi_{\kappa}(r_{2})-\phi_{\kappa}(r_{1})|\leq C|r_{1}-r_{2}|^{\alpha}.$

Further $\phi_{\kappa}$ is not constant, because $\phi_{\kappa}(0)=0$.

Then by well-known result for subharmonic functions, $\mathbb{D}\ni
z\to\phi_{\kappa}(|z|)$ is a subharmonic absolutely continuous function on
$r=|z|$ and $\phi_{\kappa}(r)=\phi_{o}(r):[0,1]\to[0,1]$ is convex in $\log
r$, with

(2.3) $\phi_{\kappa}^{\prime}(1^{-})>0$

and

(2.4) $0<\phi_{\kappa}^{\prime}(0^{+})<\infty,$

such that

$d_{\wp}(f(z),f(0))\leq\phi_{\kappa}\left(\left|z\right|\right),$

for $f\in\mathcal{H}(o)$ with $o=f(0)$ and $\wp\in\mathfrak{M}_{\kappa}$.

Notice that $\phi_{\kappa}$ is a nonconstant function. Since $\phi_{\kappa}$
is an increasing convex function of $\log r$, it follows that for $r>s$

$r\phi_{\kappa}^{\prime}(r^{+})\geqslant r\phi^{\prime}(r^{-})\geqslant
s\phi_{\kappa}^{\prime}(s^{+})\geqslant
s\phi_{\kappa}^{\prime}(s^{-})\geqslant 0.$

Since it is non-constant, it satisfies in particular that
$\phi^{\prime}(1^{-})>0$ which proves (2.3). The relation (2.4) can be proved
in the same way as the analogous relation for $\phi_{\wp}$ proved in [14].

Moreover, from Schwarz lemma for harmonic Euclidean harmonic functions ([3, p.
76]) we have $\phi_{k}(r)\geqslant\phi_{0}(r)=\frac{4}{\pi}\arctan r$. Since
$\phi_{k}(1)=1$, we get

$\frac{1-\phi_{k}(r)}{1-r}\leq\frac{1-\frac{4}{\pi}\arctan r}{1-r}.$

By letting $r\to 1$ we get

(2.5) $\phi_{\kappa}^{\prime}(1^{-})\leq\frac{2}{\pi}.$

So we have proved the first part of the following Schwarz lemma type estimate
and Heinz type estimate for harmonic mappings.

######  Theorem 2.3.

Let $w=f(z)$ be a harmonic mapping of the unit disk $\mathbb{D}$ into the
geodesic disk $D(o,1)\subset(\mathcal{M},\wp)$ of the Reimannian surface
$(\mathcal{M},\wp)$ with a Gaussian curvature
$-\kappa^{2}\leq\mathcal{K}_{\wp}\leq\kappa^{2}$ and assume that $2\kappa<\pi$
and that the cut-locus of $o$ is outside of $D(o,1)$. Let $\varrho$ be the
distance function from the fixed point $o\in\mathcal{M}$. Define
$\rho(z)=\varrho(f(z))$. Assume also that $o=f(0)$.

a) Then we have the Schwarz lemma type inequality

(2.6) $|\rho(z)|\leq\phi_{\kappa}(|z|),$

where
$\phi_{\kappa}:[0,1]\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}[0,1]$
is a bi-Lipschitz homeomorphism with $\phi_{\kappa}^{\prime}(1^{-})<2/\pi$.
Moreover,

$|\nabla\rho(0)|\leq C_{\kappa},$

where $C_{\kappa}=\phi_{\kappa}^{\prime}(0)\geqslant\frac{4}{\pi}$ and
$\phi_{\kappa}$ depend on $\kappa$ only.

$a^{\prime}$) If the cut-locus of $o$ is outside of $D(o,1)$ we have the
inequality

(2.7) $d_{\wp}(f(z),f(0))|\leq\varphi_{\kappa}(|z|),\ z\in{\mathbb{D}}$

where
$\varphi_{\kappa}:[0,1]\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}[0,2]$
is a homeomorphism locally Lipschitz in $[0,1)$. Moreover,

(2.8) $\wp(f(z))|\nabla f(z)|\leq\frac{C^{\prime}_{\kappa}}{1-|z|^{2}},\
z\in{\mathbb{D}}$

where $C^{\prime}_{\kappa}=\varphi_{\kappa}^{\prime}(0)<\infty$ and
$\varphi_{\kappa}$ depend on $\kappa$ only.

b) If in addition the function
$f:\mathbb{D}\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}D(o,1)$ is a
diffeomorphism with $f(0)=o$, and $\wp$ has a non-negative Gaussian curvature,
then

(2.9) $|\partial f(z)|\geqslant c_{\kappa}>0,\ \ z\in\mathbb{D}.$

The function $\kappa\to c_{\kappa}$ is non-increasing.

######  Remark 2.4.

a) Instead of $D(o,1)\subset(\mathcal{M},\wp)$, we can assume that
$D(o,R)\subset(\mathcal{M},\wp)$, with $2\kappa R<\pi$. In that case we define
$\wp_{R}(z)=R\cdot\wp(z)$. Then $D_{\wp}(o,R)=D_{\wp_{R}}(o,1)$ and

(2.10)
$\mathcal{K}_{\wp_{R}}=\frac{1}{R^{2}}\mathcal{K}_{\wp}<\frac{\pi^{2}}{\kappa^{2}}.$

In this case we get from (2.9) that

(2.11) $|\partial f(z)|\geqslant Rc_{\kappa},\ \ z\in\mathbb{D}.$

A similar remark can be stated for $a^{\prime})$.

If $\phi^{R}_{\kappa}(r)=R\phi_{\kappa}(r)$ is the corresponding homeomorphism
between $[0,1]$ and $[0,R]$, then the inequality
$\rho(z)\leq\phi^{R}_{\kappa}(|z|)$ is sharp, provided that $f$ is a
$\wp-$harmonic mapping of the unit disk onto $D_{\wp}(o,R)$ satisfying
$f(0)=o$ under the constraint (2.10).

b) Inequality (2.9) for $z=0$, with some absolute constant $c$ instead of
$c_{\kappa}$ has been proved by Yau and Schoen in [21] for complete surfaces
with positive curvature. We also want to observe that for $\kappa=0$, i.e. for
flat metrics, we have $\phi_{0}(r)=\frac{4}{\pi}\arctan r$. Moreover,
$C_{0}=4/\pi$. This follows from Heinz classical result [7]. We want also to
point out that the assumption $\kappa<\pi/2$ is essential. Namely it follows
from the example in [14], that $C_{\kappa}\to\infty$ as $\kappa\to\pi/2$.

c) It follows from Wan result [23, Theorem 13] the following Heinz type sharp
inequality $|\partial f(z)|\geqslant 1,z\in{\mathbb{D}}$ (cf. (2.9) above) for
every hyperbolic harmonic diffeomorphism of the unit disk onto itself. Observe
that $\mathcal{K}_{\lambda}=-1$, for the hyperbolic metric $\lambda$.

###### Proof of Theorem 2.3.

The part a) of the theorem is already proved before its formulation. The part
$a^{\prime}$) can be proved by imitating the proof of $a)$ by setting
$\varphi_{\kappa}(r)=\sup\\{d_{\wp}(f(z),f(0)):|z|=r,\wp\in\mathfrak{M}_{\kappa},f:\mathbb{D}\to\Omega:f(\mathbb{D})\subset
D_{1}(o),\,2\kappa<\pi,f\text{ is }\wp-\text{ harmonic }\\}.$ Moreover (2.8)
follows from (2.7) by dividing it by $|z|$ and letting $z\to 0$.

To prove $b)$, assume first that $f$ has a smooth extension up to the
boundary. We have by the triangle inequality that

$\frac{d_{\wp}(f(z),f(1))}{|z-1|}\geqslant\frac{\varrho(f(1))-\varrho(f(z))}{|z-1|}\geqslant\frac{1-\phi_{\kappa}(|z|)}{1-|z|}.$

Then we obtain that

$\wp(f(e^{it}))|\partial_{r}f(re^{it})|_{r=1}\geqslant 2\tilde{c}_{\kappa}>0,$

where

(2.12)
$\tilde{c}_{\kappa}=\frac{1}{2}\inf_{r\in[0,1)}\frac{1-\phi_{\kappa}(r)}{1-r}>0,$

because

$\lim_{r\to
1-0}\frac{1-\phi_{\kappa}(r)}{1-r}=\phi_{\kappa}^{\prime}(1^{-})>0.$

Therefore

$\begin{split}\wp(f(e^{it}))|\partial_{z}f(e^{it})|&\geqslant\frac{1}{2}\wp(f(e^{it}))(|\partial_{z}f(e^{it})|+|\partial_{\bar{z}}f(e^{it})|)\\\
&\geqslant\frac{1}{2}\wp(f(e^{it}))|\partial_{r}f(re^{it})|_{r=1}\geqslant\tilde{c}_{\kappa}.\end{split}$

Further since $\mathcal{K}_{\wp}\geqslant 0$ and $J(z,f)=|\partial
f(z)|^{2}-|\bar{\partial}f(z)|^{2}>0$, it follows that $\partial f(z)\neq 0$.
Then by Bochner’s formula we have

$\Delta\log\frac{1}{|\partial f|}=\mathcal{K}_{\wp}J_{f}\geqslant 0,$

where

$|\partial f(z)|=\wp(f(z))|f_{z}(z)|.$

From the maximum principle for subharmonic functions we have

$\max_{|z|\leq 1}\log\frac{1}{|\partial f|}=\max_{|z|=1}\log\frac{1}{|\partial
f|}.$

So

$\min_{|z|\leq 1}{|\partial f|}=\min_{|z|=1}{|\partial
f|}\geqslant\tilde{c}_{\kappa}.$

If $f$ has not a smooth extension up to the boundary then we use an
approximation argument. We define the sequence
$f_{n}(z)=\Psi^{-1}_{n}(f(\Phi_{n}(z)))$, where
$\Phi_{n}:\mathbb{D}\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}f^{-1}(D(o,n/(n+1)))$,
and
$\Psi^{-1}_{n}:D(o,n/(n+1))\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}D(o,1)$
are conformal mappings so that $\Phi_{n}(0)=0$, $\Psi_{n}(o)=o$,
$\Phi_{n}^{\prime}(0)>0$ and $\Psi_{n}^{\prime}(o)>0$. Then
$f_{n}:\mathbb{D}\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}D(o,1)$ is
$\wp_{n}-$harmonic. Here $\wp_{n}(w)=\wp(\Psi_{n}(w))|\Psi_{n}^{\prime}(w)|$,
and

$\mathcal{K}_{\wp_{n}}(w)=\mathcal{K}_{\wp}(\Psi(w)),$

so $0\leq\mathcal{K}_{\wp_{n}}\leq\kappa^{2}$ if
$0\leq\mathcal{K}_{\wp}\leq\kappa^{2}.$ Thus

$|\partial f_{n}(z)|\geqslant\tilde{c}_{\kappa},\ z\in\mathbb{D}$

and therefore

$|\partial f(z)|\geqslant\tilde{c}_{\kappa},\ z\in\mathbb{D}$

because $\Psi_{n}$ and $\Phi_{n}$ converges to identity in compacts. Now we
define

(2.13) $c_{\kappa}=\inf\\{|\partial
f(z)|,z\in{\mathbb{D}},f\in\mathcal{H}_{\kappa}(o)\\}.$

Then $c_{\kappa}\geqslant\tilde{c}_{\kappa}>0$. Moreover, in view of (2.13),
because $\mathcal{H}_{\kappa_{1}}(o)\subset\mathcal{H}_{\kappa_{2}}(o)$, for
$\kappa_{1}<\kappa_{2}$, we obtain that $\kappa_{1}<\kappa_{2}\Rightarrow
c_{\kappa_{2}}\leq c_{\kappa_{1}}$. ∎

###  2.2. Schwarz lemma for dilatation of harmonic mappings

######  Proposition 2.5.

[21, p. 10-11] Assume that $h$ is harmonic with respect to a non-vanishing
$C^{2}$ metric $\rho$. Then the functions $|\partial h|$ and
$|\bar{\partial}h|$ are identically zero or they have the isolated zeros with
well defined orders. More precisely $|\partial h|=|z-z_{0}|^{n}u(z)$ and
$|\bar{\partial}h|=|z-z_{0}|^{m}v(z)$ for continuous functions $u$ and $v$
such that $u(z_{0})\neq 0$ and $v(z_{0})\neq 0$.

######  Theorem 2.6.

Let $M=(\Omega,\wp)$ be a non-negatively curved Riemannian surface. Assume
that $f:\mathbb{D}\to M$ is a $\wp-$harmonic mapping that preserves the
orientation. Let $m(z)=|{f_{\bar{z}}}/{f_{z}}|$. If $f_{\bar{z}}(0)=0$, then

$|m(z)|\leq|z|^{2},$

or what is the same

$|f_{\bar{z}}(z)|\leq|z|^{2}|f_{z}(z)|.$

Further if $\mu=\sqrt{m}$ then

(2.14) $|\mu(z)|\leq|z|,\text{ and }|\nabla\mu(0)|\leq 1.$

If $\mu(0)\neq 0$, $f:{\mathbb{D}}\to D_{1}(o)$, and if $D_{1}(o)$ is disjoint
from the cut-locus of $o$, and $K_{\wp}\leq\kappa^{2}<\pi^{2}/4$, then we have

(2.15) $|\nabla\mu(0)|\leq C_{\kappa}^{\ast},$

for a constant $C_{\kappa}^{\ast}\geqslant 1$ depending on $\kappa$.

###### Proof.

Prove first (2.15). Let $r\in(0,1)$,
$P_{r}(z)=\frac{r^{2}-|z|^{2}}{|z-r|^{2}}$. By using the well-known formula we
have

(2.16)
$\mu(z)=h_{r}(z)+\frac{1}{r^{2}}\int_{r{\mathbb{D}}}\log\left|\frac{z-w}{1-\frac{1}{r^{2}}z\overline{w}}\right|\Delta\mu(w)dudv,\
\ w=u+iv,$

where $h_{r}:r{\mathbb{D}}\to{\mathbb{D}}$ is the harmonic function defined by

$h_{r}(z)=\int_{0}^{2\pi}P_{r}(ze^{it})\mu[re^{it}]\frac{dt}{2\pi}.$

Further

$\Delta\mu=K_{\wp}\wp^{2}(f(z))(|f_{z}|^{2}-|f_{\bar{z}}|^{2}).$

From Schwarz lemma for Euclidean harmonic functions ([3, p. 76]) we obtain
that $|\nabla h_{r}(0)|\leq\frac{4}{r\pi}$.

On the other hand

$\left|\nabla_{z}\log\left|\frac{z-w}{1-\frac{1}{r^{2}}z\overline{w}}\right|\right|_{z=0}=\frac{\sqrt{r^{2}-|w|^{2}}}{r^{2}|w|}.$

Thus by (2.8) we have

$|\nabla\mu(0)|\leq\frac{4}{r\pi}+\kappa^{2}(C^{\prime}_{\kappa})^{2}\frac{1}{r^{2}}\int_{r{\mathbb{D}}}\frac{\sqrt{r^{2}-|w|^{2}}}{r^{2}|w|}\frac{1}{(1-|w|^{2})^{2}}dudv.$

Then for $r=1/2$ we obtain

(2.17) $|\nabla\mu(0)|\leq
C^{*}_{\kappa}:=\frac{8}{\pi}+\pi\kappa^{2}(C^{\prime}_{\kappa})^{2}(5\coth^{-1}(2)-2).$

Now we improve the constant for the case $\mu(0)=0$. If
$\mathcal{K}_{\wp}\geqslant 0$, and $f:\mathbb{D}\to\Omega$, then by Bochner
formula we have

$\Delta\log\mu(z)=\frac{\mathcal{K}_{\wp}}{2}J\geqslant 0.$

Since

$\mu=\sqrt{\frac{|f_{\bar{z}}|}{|f_{z}|}}=\frac{|\Phi(z)|}{\wp(f(z))|f_{z}|},$

on account of Proposition 2.5, it follows that also $\mu$ has well-defined
order of the zero. So $\log\left(\mu(z)/|z|\right)$ is subharmonic.

Assume now that $\mu$ is continuous on $\overline{\mathbb{D}}$. By the maximum
principle

$\log(\mu(z)/|z|)\leq 0.$

Thus

$\mu(z)/|z|\leq 1.$

Therefore,

$|\mu(z)|\leq|z|.$

Further

(2.18)
$\mu(u,v)=\mu(0,0)+\partial_{x}\mu(0,0)u+\partial_{y}\mu(0,0)+o(\sqrt{u^{2}+v^{2}}).$

If $\partial_{x}\mu(0,0)=\partial_{y}\mu(0,0)=0$ we have nothing to prove. If
the last relation is not true then we choose
$u=t\partial_{x}\mu(0,0)/|\nabla\mu(0,0)|$ and
$v=t\partial_{y}\mu(0,0)/|\nabla\mu(0,0)|.$ By inserting $u$ and $v$ in
(2.18), dividing by $t$ and letting $t\to 0$ we get

$|\nabla\mu(0,0)|\leq 1.$

If $\mu(z)/|z|$ is only upper-semicontinuous in $\overline{\mathbb{D}}$, then
by [6, Theorem 1.4], there exists a decreasing sequence of mappings
$\mu_{k}(z)$ continuous in $\overline{\mathbb{D}}$ so that

$\lim_{k\to\infty}\mu_{k}(z)=\mu(z)/|z|.$

Thus the proof can be reduced to the proof when $\mu$ is continuous up to the
boundary.

∎

##  3\. Minimizers and minimal surfaces in Riemannian manifold

Assume that $\mathbb{X}$ and $\mathbb{Y}$ are double connected domains and
that $\mathscr{W}_{\wp}^{1,2}(\mathbb{X},\mathbb{Y})$ is the class of mappings
that belongs to $\mathscr{W}_{\mathrm{loc}}^{1,2}$ and satisfy the inequality

$\int_{\mathbb{X}}\wp^{2}(f(z))(|f_{z}|^{2}+|f_{\bar{z}}|^{2})dxdy+\int_{\mathbb{X}}\wp^{2}(f(z))|f(z)|^{2}dxdy<\infty.$

Assume that
$\mathcal{H}^{\wp}(\mathbb{X},\mathbb{Y})\subset\mathscr{W}_{\wp}^{1,2}(\mathbb{X},\mathbb{Y})$
is the class of homeomorphic mappings between $\mathbb{X}$ and $\mathbb{Y}$,
that map the inner boundary onto inner boundary and outer boundary onto the
outer boundary.

Let $\overline{\mathcal{H}}^{\wp}(\mathbb{X},\mathbb{Y})$ be the closure of
${\mathcal{H}}^{\wp}(\mathbb{X},\mathbb{Y})$ in the strong topology of
$\mathscr{W}_{\wp}^{1,2}(\mathbb{X},\mathbb{Y})$.

###  3.1. Stationary mappings and Noether harmonic maps

We call a mapping $h\in\overline{\mathcal{H}}^{\wp}(\mathbb{X},\mathbb{Y})$
_stationary_ if

(3.1) $\frac{d}{dt}\bigg{|}_{t=0}{\mathcal{E}^{\wp}}[h\circ\phi_{t}^{-1}]=0$

for every family of diffeomorphisms
$t\to\phi_{t}\colon\mathbb{X}\to\mathbb{X}$ which depend smoothly on the
parameter $t\in\mathbb{R}$ and satisfy $\phi_{0}=\operatorname{id}$. The
latter mean that the mapping
$\mathbb{X}\times[0,\epsilon_{0}]\ni(z,t)\to\phi_{t}(z)\in\mathbb{X}$ is a
smooth mapping for some $\epsilon_{0}>0$. We now have.

######  Lemma 3.1.

[11] Let $\mathbb{X}=\mathbb{A}(r,R)$ be a circular annulus, $0<r<R<\infty$,
and assume that $\mathbb{Y}$ is a doubly connected domain. If
$h\in\overline{\mathcal{H}}^{\wp}(\mathbb{X},\mathbb{Y})$ is a stationary
mapping, then

(3.2)
$\wp^{2}(h(z))h_{z}\overline{h_{\bar{z}}}\equiv\frac{c}{z^{2}}\qquad\text{in
}\mathbb{X}$

where $\mathbf{c}\in\mathbb{R}$ is a constant. Moreover, $\mathbf{c}\geqslant
0$ if $\mathrm{Mod}(\mathbb{X})\leq\mathrm{Mod}(\mathbb{Y})$ and
$\mathbf{c}<0$ if $\mathrm{Mod}(\mathbb{X})>\mathrm{Mod}(\mathbb{Y})$. Further
every stationary mapping that is a diffeomorphism is a minimizer.

Notice first that a change of variables $w=f(z)$ in (1.5) yields

(3.3)
${\mathcal{E}^{\wp}}[f]=2\int_{\mathbb{X}}\wp^{2}(f(z))J_{f}(z)\,dz+4\int_{\mathbb{X}}\wp^{2}(f(z))\lvert
f_{\bar{z}}\rvert^{2}dz\geqslant 2\mathcal{A}(\wp)$

where $J_{f}$ is the Jacobian determinant and $\mathcal{A}(\wp)$ is the area
of $\mathbb{Y}$ and $dz:=dx\wedge dy$ is the area element w.r. to Lebesgue
measure on the complex plane. A conformal mapping of
$f:\mathbb{X}\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}\mathbb{Y}$;
that is, a homeomorphic solution of the Cauchy-Riemann system $f_{\bar{z}}=0$,
would be an obvious choice for the minimizer of (3.3). For arbitrary multiply
connected domains there is no such mapping. In this way the $\wp-$harmonic
mappings comes to the stage.

The following lemma makes a relation with minimizers and minimal surfaces in
manifold $M=M_{0}\times\mathbb{R}$.

######  Proposition 3.2.

Assume that $f=a+\imath b:D\to\Omega$ is a diffeomorphism so that

$\wp^{2}(f(z))f_{z}\bar{f}_{z}=\Phi^{2}(z),$

where $\Phi$ is a holomorphic function in $D$. Assume that for $z\in D$, the
following function $\Phi(z)=\int\Phi(z)dz$ is well-defined. Then for

(3.4) $\omega(z)=2\Re(\Phi(f^{-1}(z)))$

we have

(3.5)
$\mathrm{div}\left(\frac{\nabla\omega}{\sqrt{1+\frac{|\nabla\omega(u,v)|^{2}}{\wp^{2}(u,v)}}}\right)=0.$

In other words, $f$ defines the conformal parameters of the $\wp-$minimal
graph $\Sigma=\\{(u,v,\omega(u,v)),(u,v)\in\Omega\\}$. The conformal
parametrization $\chi:D\to(\Omega,\wp)\times\mathbb{R}$ is given by

$\chi(z)=(\Re f,\Im f,2\Re(\Phi(z))).$

This means that

$|\chi_{x}|^{2}_{\hbar}-|\chi_{y}|^{2}_{\hbar}=\left<\chi_{x},\chi_{y}\right>_{\hbar}=0,$

where the metric $d\hbar^{2}$ is given in (1.2).

###### Proof.

Let $f=a+\imath b$. Then

$2\mathrm{Im}(f_{z}{f_{\bar{z}}})=-(a_{x}b_{x}+a_{y}b_{y}).$

The $\wp-$ harmonic equation (1.3) can be written

(3.6) $f_{z\bar{z}}+2\frac{\partial\wp(w)}{\wp(w)}\circ f\cdot
f_{z}f_{\bar{z}}=0.$

Equation (3.6) is equivalent with the following system of PDE

(3.7) $\Delta
b=\left(\frac{4\wp_{u}(f)}{\wp}\left(\mathrm{Im}(f_{z}f_{\bar{z}})\right)+\frac{\wp_{v}(f)}{\wp}\left(4\mathrm{Re}(f_{z}f_{\bar{z}}\right)\right)$

(3.8) $\Delta
a=\left(\frac{4\wp_{v}(f)}{\wp}\left(\mathrm{Im}(f_{z}f_{\bar{z}})\right)+\frac{\wp_{u}(f)}{\wp}\left(-4\mathrm{Re}(f_{z}f_{\bar{z}})\right)\right).$

From (3.4) we obtain that the function $H(z)=\omega(f(z))$ is a harmonic
function, so its Laplacian vanishes and this can be written in the form

$\begin{split}(b_{y}^{2}&+b_{x}^{2})\omega_{vv}(w)+2(a_{x}b_{x}+a_{y}b_{y})\omega_{uv}(w)+\left(a_{y}^{2}+a_{x}^{2}\right)\omega_{uu}(w)\\\
&+\omega_{u}(w)\left(a_{yy}+a_{xx}\right)+\omega_{v}(w)\left(b_{xx}+b_{yy}\right)=0\end{split}$

which can be written as

(3.9)
$\begin{split}(|f_{z}-&\bar{f}_{z}|^{2})\omega_{vv}(w)-4\mathrm{Im}(f_{z}{f_{\bar{z}}})\omega_{uv}(w)+\left(|f_{z}+\bar{f}_{z}|^{2}\right)\omega_{uu}(w)\\\
&+\omega_{u}(w)\left(4\frac{\wp_{v}(f)}{\wp}\left(\mathrm{Im}(f_{z}f_{\bar{z}})\right)+\frac{\wp_{u}(f)}{\wp}\left(-4\mathrm{Re}(f_{z}f_{\bar{z}})\right)\right)\\\
&+\omega_{v}(w)\left(\frac{4\wp_{u}(f)}{\wp}\left(\mathrm{Im}(f_{z}f_{\bar{z}})\right)+\frac{\wp_{v}(f)}{\wp}\left(4\mathrm{Re}(f_{z}f_{\bar{z}}\right)\right)=0.\end{split}$

From

$\wp^{2}(f(z))f_{z}\bar{f}_{z}=\Psi^{2}(z),$

we have

(3.10) $\bar{f}_{z}=\frac{\Psi^{2}(z)}{f_{z}\wp^{2}(f(z))}.$

Let

$\Phi=\int\Psi(z)dz.$

By differentiating the equation

$\omega(f(z))=\psi(z)=2\Re(\Phi(z))$

we have

(3.11) $\partial\omega
f_{z}+\bar{\partial}\omega\bar{f}_{z}=\Phi^{\prime}(z)=\Psi(z).$

Now recall $f=a+\imath b$ and so $f_{z}+\bar{f}_{z}=a_{x}-\imath a_{y}$ and
$f_{z}-\bar{f}_{z}=-b_{y}+\imath b_{x}$. Further from (3.10) and (3.11) we
obatin

$f_{z}=\frac{\Psi(z)\left(\sqrt{\wp^{2}+|\nabla\omega|^{2}}-\wp\right)}{2\partial\omega\wp}$

and

$\bar{f}_{z}=-\frac{\Psi(z)\left(\sqrt{\wp^{2}+|\nabla\omega|^{2}}+\wp\right)}{2\bar{\partial}\omega\wp}$

and so

$f_{z}f_{\bar{z}}=\frac{-|\Psi(z)|^{2}}{4\wp^{2}(\partial\omega)^{2}},$

and

(3.12)
$4\mathrm{Im}(f_{z}f_{\bar{z}})=\frac{-8|\Psi(z)|^{2}}{\wp^{2}}\frac{\omega_{u}\omega_{v}}{|\nabla\omega|^{4}}$

and

(3.13)
$4\mathrm{Re}(f_{z}f_{\bar{z}})=\frac{-4|\Psi(z)|^{2}}{|z|^{2}\wp^{2}}\frac{\omega_{u}^{2}-\omega_{v}^{2}}{|\nabla\omega|^{4}}.$

If

$\omega_{u}-\imath\omega_{v}=Re^{\imath s},$

then

(3.14)
$\begin{split}|f_{z}+f_{\bar{z}}|^{2}=\frac{|\Psi(z)|^{2}\left(\wp^{2}+\omega_{v}^{2}\right)}{\wp^{2}R^{2}}\end{split}$

and

(3.15)
$\begin{split}|f_{z}-f_{\bar{z}}|^{2}&=\frac{|\Psi(z)|^{2}\left(\wp^{2}+\omega_{u}^{2}\right)}{\wp^{2}R^{2}}.\end{split}$

Inserting (3.14),(3.15), (3.12) and (3.13) to (3.9) we get

$\begin{split}&(\wp^{3}+\wp\omega_{v}(w)^{2})\omega_{uu}(w)+(\wp^{3}+\wp\omega_{u}(w)^{2})\omega_{vv}(w)-2\omega_{u}(w)\wp\omega_{v}(w)\omega_{uv}(w)\\\
&+(\wp_{u}(w)\omega_{u}(w)+\wp_{v}(w)\omega_{v}(w))\left(\omega_{v}(w)^{2}+\omega_{u}(w)^{2}\right)=0\end{split}$

which can be compactly written as (1.8).

∎

######  Example 3.3.

If
$f:\mathbb{A}(1/R,R)\xrightarrow[]{{}_{\\!\\!\textnormal{onto}\\!\\!}}\Omega$
is a minimizer of the $\wp$ energy, then, by a result of Kalaj and Lamel in
[13], it defines a catenoid type minimal surface

$X_{1}=\\{(f_{1}(z),h_{1}(z)):h_{1}(z)=2\sqrt{|\mathbf{c}|}\log{|z|}:z\in\mathbb{A}(1/R,R)\\}$

or a helicoid type minimal surface

$X_{2}=\\{(f_{1}(z),h_{2}(z)):h_{2}(z)=2\sqrt{|\mathbf{c}|}\,\mathrm{arg}z:z\in\mathbb{A}(1/R,R)\\}.$

Those two minimal surfaces are conjugate to each other in the notation of [5].
Namely their Hopf differentials satisfy the relation

$\mathrm{Hopf}(f_{1})=-\mathrm{Hopf}(f_{2})=\frac{|\mathbf{c}|}{z^{2}}.$

Now Proposition 3.2 has this specific form for minimal surfaces from [13].

######  Proposition 3.4.

Assume that $f=a+\imath b:\Omega\to D$ is a diffeomorphism so that

$\wp^{2}(f(z))f_{z}\bar{f}_{z}=\frac{\mathbf{c}}{z^{2}},$

with $\mathbf{c}\in\mathbb{R}$. Assume that for $z\in\Omega$, the following
function is well-defined

(3.16)
$\omega(z)=\left\\{\begin{array}[]{ll}2\sqrt{-\mathbf{c}}\log\frac{1}{|f^{-1}(z)|},&\hbox{if
$\mathbf{c}<0$;}\\\
\hbox{$2\sqrt{\mathbf{c}}\,\mathrm{arg}(f^{-1}(z))$},&\hbox{if
$\mathbf{c}>0$.}\end{array}\right.$

Then $\omega$ satisfies the minimal surface equation (1.8).

######  Example 3.5.

If $\wp=\rho(|w|)$, then from [12], a radial $\wp-$harmonic mapping $f$ of the
annulus $\mathbb{A}(r,1)$ onto $\mathbb{A}(R,1)$, $r,R<1$ is given implicitly
by the formula

$f^{-1}(w)=e^{\imath\tau}\exp\int_{1}^{s}\frac{\rho(y)}{\sqrt{4\mathbf{c}+y^{2}\rho^{2}(y)}}dy,w=se^{\imath\tau},$

where

$\max_{R\leq|y|\leq 1}4\mathbf{c}+y^{2}\rho^{2}(y)\geqslant 0,$

and

$r=\exp\int_{1}^{R}\frac{\rho(y)}{\sqrt{4\mathbf{c}+y^{2}\rho^{2}(y)}}dy.$

Then for $\mathbf{c}<0$

$\omega(z)=\sqrt{-\mathbf{c}}\log\frac{1}{|f^{-1}(z)|}=\int_{|z|}^{1}\frac{\rho(y)}{\sqrt{4\mathbf{c}+y^{2}\rho^{2}(y)}}dy.$

Thus after straightforward computation we get

$\left(\frac{\nabla\omega}{\sqrt{1+\frac{|\nabla\omega(x,y)|^{2}}{\rho^{2}(x,y)}}}\right)=\frac{2\sqrt{-\mathbf{c}}}{\bar{z}}$

and so that

$\mathrm{div}\frac{2\sqrt{-\mathbf{c}}}{\bar{z}}=2\sqrt{-\mathbf{c}}\left(\frac{2}{|z|^{2}}-\frac{2}{|z|^{2}}\right)=0.$

##  4\. Curvature estimates of minimal graphs

In [18] Meeks and White considered minimal surfaces in
$\mathbb{R}^{2}\times\mathbb{R}$ bounded by convex curves in parallel planes
and give some improvement of some earlier results on this topic by Shiffman
[22]. This result has its counterpart to the minimizers [15] (a result of
Ngin-Tee Koh). In this section we consider some curvature estimates of minimal
graphs bounded by Jordan curves in parallel planes in
$\mathbb{R}^{2}\times\mathbb{R}$ and also in $(\Omega,\wp)\times\mathbb{R}$.

In the following proposition, we obtain a connection between minimizers of
energy and minimal graphs bounded by Jordan curves in parallel planes.

######  Proposition 4.1.

Assume that $\Sigma$ is a minimal graph over a double connected domain
$\Omega$ in the complex plane. Assume also that $\Sigma$ spans two planar
curves $\gamma_{1}$ and $\gamma_{2}$ that belongs to two horizontal planes
whose distance is $\mathbf{d}$.

  1. (1)

Then there is a conformal minimal parametrisation
$\chi:\mathbb{A}(1/R,R)\to\Sigma$ where $R>1$. In this case

$\operatorname{Mod}(\Sigma)=\operatorname{Mod}(\mathbb{A}(1/R,R))=\log R^{2}.$

  2. (2)

The mapping $f(z)=\chi_{1}(z)+\imath\chi_{2}(z)$ is a harmonic diffeomorphism
between $A=\mathbb{A}(1/R,R)$ and $\Omega$ with
$\mathrm{Mod}(A)>\mathrm{Mod}(\Omega)$.

  3. (3)

Then there is a constant $\mathbf{c}<0$ so that $\Sigma$ is the graphic of a
real function $u:\Omega\to\mathbb{R}$, of the form
$u(z)=C+2\sqrt{-\mathbf{c}}\log|f^{-1}(z)|$, which satisfies the equation
(3.5) above where $\wp\equiv 1$. Moreover,
$f_{z}\cdot\bar{f}_{z}=\frac{\mathbf{c}}{z^{2}},$ where

(4.1) $\mathbf{c}=-\frac{\mathbf{d}^{2}}{4\mathrm{Mod}(\Sigma)^{2}}.$

######  Remark 4.2.

The question arises for which Jordan curves $\gamma_{1}$ and $\gamma_{2}$
lying to different parallel planes, there is a minimal graph spanning them. A
more general question is whether exist a minimal surface spanning $\gamma_{1}$
and $\gamma_{2}$. By a classical result of Courant, if $m(\gamma_{i})$ is the
area of the domain inside $\gamma_{i}$, $i=1,2$ and $m(\gamma_{1},\gamma_{2})$
is the least area of all surfaces spanning $\gamma_{1}$ and $\gamma_{2}$, then
the condition $m(\gamma_{1},\gamma_{2})<m(\gamma_{1})+m(\gamma_{2})$ implies
that there exist a minimal doubly connected surface spanning $\gamma_{1}$ and
$\gamma_{2}$. If the projection of such surface in the plane $P$ parallel to
both $\gamma_{1},\gamma_{2}$ is a doubly connected domain $\Omega$, then we
conjecture that there is a curve $\gamma_{0}\subset\Sigma$ parallel to
$\gamma_{1},\gamma_{2}$ so that $\Sigma(\gamma_{1},\gamma_{0})$ and
$\Sigma(\gamma_{2},\gamma_{0})$ are minimal graphs, one of which, denote it by
$\Sigma^{\prime}$, has a projection the whole domain $\Omega$. In that case

$\operatorname{Mod}(\Omega)<\operatorname{Mod}(\Sigma^{\prime})<\operatorname{Mod}(\Sigma).$

Moreover, such a setting gives rise to a harmonic diffeomorphism $f$ of the
annulus $\mathbb{A}(1/R,R)$ onto $\Omega$, where

(4.2) $\operatorname{Mod}(\mathbb{A}(1/R,R))>\operatorname{Mod}(\Omega).$

We mention here that when

(4.3) $\operatorname{Mod}(\mathbb{A}(1/R,R))\leq\operatorname{Mod}(\Omega)$

the existence of harmonic diffeomorphisms between $\mathbb{A}(1/R,R)$ and
$\Omega$ has been established in [11]. We refer to [17] where Kovalev and Li
considered the existence of harmonic diffeomorphisms under the condition
(4.2), under some smoothness conditions on the boundaries and some constraint
on their conformal modulus. The general problem of determining the conditions
under which there is a harmonic diffeomorphism with given constraints (4.2) on
the annuli remains open.

###### Proof.

The item a) follows from the fact that $\Sigma$ is smooth double connected
surface and therefore it exist a conformal mapping
$\chi=(\chi_{1},\chi_{2},\chi_{3})$ of an annulus $\mathbb{A}(1/R,R)$ onto
$\Sigma$ by virtue of theorem [9, Theorem 3.1] see also [18, Lemma 2.2].
Moreover, all coordinates of $\chi$ are harmonic. Since $\Sigma$ is a graph of
a real function $u$ defined in a double connected domain $\Omega$, it follows
that $\Sigma=\\{(x,y,u(x,y)):(x,y)\in\Omega\\}$. Therefore the mapping
$f(z)=\chi_{1}(z)+\imath\chi_{2}(z)$ is a diffeomorphism as the composition of
the following two diffeomorphisms $\chi$ and $(x,y,u(x,y))\to(x,y)$. Further
$\chi_{3}(z)=u(f(z))$ and $\chi_{3}(1/Re^{\imath t})=\mathbf{h}$ and
$\chi_{3}(Re^{\imath t})=\mathbf{h}+\mathbf{d}$. Since $\chi_{3}$ is harmonic
in $\mathbb{A}(1/R,R)$ it has the following form

(4.4)
$\chi_{3}(z)=h(z)=\frac{\mathbf{d}}{2}+\mathbf{h}+\frac{\mathbf{d}}{\log\left[R^{2}\right]}\log|z|.$

Thus by conformal condition we have

$(f_{z}(z)+\bar{f}_{z}(z))^{2}+\left(\imath(f_{z}(z)-\bar{f}_{z}(z))\right)^{2}+\left(\frac{\mathbf{d}}{z\log\left[R^{2}\right]}\right)^{2}=0.$

Therefore,

$f_{z}(z)\bar{f}_{z}(z)=-\left(\frac{\mathbf{d}}{2z\log\left[R^{2}\right]}\right)^{2}=\frac{\mathbf{c}}{z^{2}},$

where

$\mathbf{c}=-\frac{\mathbf{d}^{2}}{4\mathrm{Mod}(\Sigma)^{2}}.$

Now Lemma 3.1 implies that if $\mathbf{c}<0$ then
$\operatorname{Mod}A\geqslant\operatorname{Mod}\Omega$. But
$\operatorname{Mod}A=\operatorname{Mod}\Omega$ is not possible, because in
that case $f$ is conformal, and thus $\mathbf{c}=0$ which is also not
possible. So $\operatorname{Mod}A>\operatorname{Mod}\Omega$. The item (3)
follows from Lemma 3.4. ∎

Similarly we can prove the following proposition

######  Proposition 4.3.

Assume that $\Sigma\subset(\Omega,\wp)\times\mathbb{R}$ is a minimal graph
over a double connected domain $\Omega$ in the complex plane. Assume also that
$\Sigma$ spans two planar curves $\gamma_{1}$ and $\gamma_{2}$ that belongs to
two horizontal planes whose distance is $\mathbf{d}$.

  1. (1)

Then there is a conformal minimal parametrisation
$\chi:\mathbb{A}(1/R,R)\to\Sigma$ where $R>1$. In this case

$\operatorname{Mod}(\Sigma)=\operatorname{Mod}(\mathbb{A}(1/R,R))=\log R^{2}.$

  2. (2)

The mapping $f(z)=\chi_{1}(z)+\imath\chi_{2}(z)$ is a $\wp-$harmonic
diffeomorphism between $A=\mathbb{A}(1/R,R)$ and $\Omega$ with
$\mathrm{Mod}(A)>\mathrm{Mod}(\Omega)$.

  3. (3)

Then there is a constant $\mathbf{c}<0$ so that $\Sigma$ is the graphic of a
real function $u:\Omega\to\mathbb{R}$, of the form
$u(z)=C+2\sqrt{-\mathbf{c}}\log|f^{-1}(z)|$ satisfying equation (3.5) above.
Moreover, $\wp^{2}(f(z))f_{z}\cdot\bar{f}_{z}=\frac{\mathbf{c}}{z^{2}},$ where
$\mathbf{c}$ is given by (4.1).

###  4.1. Gaussian curvature

Assume that $\Sigma=\\{(f(z),h(z)):z\in\Omega\\}$ is a minimal surface in the
manifold $M=(\Omega,\wp)\times\mathbb{R}$. From a corresponding formula in [5]
we get

$ds^{2}_{\Sigma}=\wp(f(z))^{2}(|f_{z}|+|f_{\bar{z}}|)^{2}|dz|^{2}=\frac{1}{4}\left(|\mu|+|\mu|^{-1}\right)^{2}|\eta|^{2}|dz|^{2}=\lambda|dz|^{2},$

where

$\eta=\pm 2\rho(f(z))\sqrt{f_{z}\bar{f}_{z}}$

and

$\mu=\sqrt{\left|\frac{\bar{f}_{z}}{f_{z}}\right|}.$

Let $m=\mu^{2}.$ Bochner formula implies that

$\Delta\log m=\mathcal{K}_{\wp}J_{f},$

where

$J(f)=|\partial_{\wp}f|^{2}-|\bar{\partial}_{\wp}f|^{2}=\wp(f(z))^{2}(|f_{z}|^{2}-|f_{\bar{z}}|^{2}).$

Then the Gaussian curvature of $\Sigma$ at $\zeta=\chi(z)=(f(z),h(z))$ can be
expressed as

$\mathcal{K}=\mathcal{K}_{\Sigma}(\zeta)=-\frac{\vartriangle\log\lambda(z)}{2\lambda(z)}.$

Thus

(4.5)
$\mathcal{K}=\frac{-\Delta\log\lambda}{2\lambda}=\frac{-2A+X\mu^{2}\left(1-\mu^{4}\right)}{2\lambda\mu^{2}(1+\mu^{2})^{2}}$

where

$A=|\nabla m|^{2}$

and

$X=2\Delta\log m=2\mathcal{K}_{\wp}J(f).$

So

$\mathcal{K}=\frac{-A}{\lambda\mu^{2}(1+\mu^{2})^{2}}+\frac{\mathcal{K}_{\wp}J(f)\left(1-\mu^{4}\right)}{\lambda(1+\mu^{2})^{2}}.$

Observe that, if $\wp$ has negative Gaussian curvature, then $\Sigma$ is also
negatively curved minimal surface. Furthermore

$0\leq\frac{J(f)\left(1-\mu^{4}\right)}{\lambda(1+\mu^{2})^{2}}=\frac{\left(1-\mu^{2}\right)^{2}}{\left(1+\mu^{2}\right)^{3}}\leq
1.$

Further

$\frac{-A}{\lambda\mu^{2}(1+\mu^{2})^{2}}=\frac{-B}{\wp^{2}(f(z))(|f_{z}|+|f_{\bar{z}}|)^{2}\left(1+\mu^{2}\right)^{4}},$

where

(4.6) $B=4|\nabla\mu|^{2}.$

Thus we get

(4.7)
$|\mathcal{K}|\leq\max\left\\{\left|\frac{-B}{\wp^{2}(f(z))(|f_{z}|+|f_{\bar{z}}|)^{2}}\right|,\left|\frac{-B}{\wp^{2}(f(z))(|f_{z}|+|f_{\bar{z}}|)^{2}}+\mathcal{K}_{\wp}\right|\right\\}.$

By [20, Proposition 4], see (1.1) above, we know that a conformal
parametrization $\chi(z)=(f(z),h(z))$ of a minimal surface $\Sigma\subset
M\times\mathbb{R}$ defines the unit normal at a given point. Moreover, if
$\bar{\partial}f(0)=0$, then the unit normal $N$ at a point $\chi(0)=o$ is
$N=(0,0,1)$ and it defines a horizontal tangential plane. Now we formulate the
following Finn-Osserman type result ([4]):

######  Theorem 4.4.

Let $D_{\wp}(o,R)\subset M_{0}=(\Omega,\wp)$ be a geodesic disk so that
$0\leq\mathcal{K}_{\wp}\leq\kappa^{2}<\frac{\pi^{2}}{4R^{2}}$. Assume further
that $D_{\wp}(o,R)$ is disjoint from the cut-locus of $o$. Assume that
$\Sigma$ is a minimal surface over $D_{\wp}(o,R)$ in the Riemannian manifold
$M_{0}\times\mathbb{R}$. Then at the point $O$ above $o$, we have the
following Gaussian curvature estimate of $\Sigma$

$|\mathcal{K}_{\Sigma}(O)|\leq\frac{4C_{\kappa}^{\ast}}{R^{2}c_{k}^{2}},$

where $c_{\kappa}$ is defined in (2.13) and $C_{\kappa}^{\ast}\geqslant 1$ is
defined in (2.17). If the tangential plane at $O$ is horizontal, then we can
choose $C_{\kappa}^{\ast}=1$.

###### Proof.

Observe that, from Remark 2.4 we have that $c_{\kappa}\leq 2/\pi$. Further,
having in mind Theorem 2.6 we have $|B|\leq 4C_{\kappa}^{\ast}$, where $B$ is
defined in (4.6). Now by (4.7) and Theorem 2.3 (equation (2.11)) we get
$|\mathcal{K}|\leq\frac{4C_{\kappa}^{\ast}}{R^{2}c_{k}^{2}}$, because
$\frac{4}{c_{k}^{2}}>\pi^{2}$ and $\mathcal{K}_{\wp}<\pi^{2}/(4R^{2})$. ∎

###  4.2. The curvature estimate of minimal graphs between parallel planes

In this case, as it is discussed in Propositions 4.1 and 4.3 we have

$\eta=\pm 2\rho(f(z))\sqrt{f_{z}\bar{f}_{z}}=\pm
2\imath\sqrt{\frac{\mathbf{c}}{z^{2}}}.$

For our surface

$\mathcal{K}=\frac{-A|z|^{2}}{|c|\left(1+\mu^{2}\right)^{4}}+\frac{\mathcal{K}_{\wp}J(f)\left(1-\mu^{4}\right)}{2\lambda(1+\mu^{2})^{2}}.$

If $\wp\equiv 1$, then $\mathcal{K}_{\wp}=0$ and

$\frac{-\Delta\log\lambda}{\lambda}=\frac{-A}{2\lambda\mu^{2}(1+\mu^{2})^{2}}.$

In this case $m=|\mathbf{q}(z)|^{2}$, where $\mathbf{q}$ is a holomorphic
function so that $|\mathbf{q}(z)|<1$, and so

$|\nabla m|^{2}=4|\mathbf{q}(z)|^{2}\cdot|\mathbf{q}^{\prime}(z)|^{2}.$

Moreover,

$\lambda=(1+|\mathbf{q}|^{2})^{2}|\mathbf{p}|^{2}.$

In this case we get

$\mathcal{K}=-\frac{4|\mathbf{q}^{\prime}|^{2}}{|\mathbf{p}|^{2}(1+|\mathbf{q}|^{2})^{4}}.$

We also have for $\omega=\mathbf{q}^{2}$, ([3, p. 184])

(4.8)
$\mathcal{K}=-\frac{|\omega^{\prime}|^{2}}{|f_{z}\bar{f}_{z}|(1+|\omega|)^{4}}.$

Now we have the make the following equivalent formulation of Theorem 1.1

######  Theorem 4.5.

Assume that $\Sigma\subset\mathbb{R}^{2}\times\mathbb{R}$ is a double-
connected minimal graph bounded by two Jordan curves in parallel planes
$z=\pm\mathbf{d}/2.$ Then for every point
$\zeta\in\Sigma\bigcap\mathbb{R}^{2}\times\\{0\\}$ we have

$|\mathcal{K}(\zeta)|\leq\frac{\pi^{2}}{\mathbf{d}^{2}}.$

Moreover, the tangential plane $T\Sigma_{\zeta}$ at a point
$\zeta\in\Sigma\setminus\mathbb{R}^{2}\times\\{-\mathbf{d}/2,\mathbf{d/2}\\}$
is never horizontal nor vertical.

###### Proof.

Assume that $\Sigma$ is defined over a double connected domain $\Omega$ and
assume that $\chi(z)=(f(z),h(z)):\mathbb{A}(1/R,R)\to\Sigma$ is its conformal
minimal parameterisation. By (4.4) $h$ is given by

(4.9) $h(z)=\frac{\mathbf{d}}{\log\left[R^{2}\right]}\log|z|.$

Let $\omega(z)=(\mathbf{q}(z))^{2}$. From (4.8) we have

(4.10)
$\mathcal{K}(\chi(z)))=-\frac{|\omega^{\prime}(z)|^{2}}{|f_{z}\bar{f}_{z}|(1+|\omega|)^{4}}=-\frac{|\omega^{\prime}(z)|^{2}|z|^{2}}{|\mathbf{c}|(1+|\omega|)^{4}}.$

Schwarz-Pick inequality for analytic functions of the annulus
$\mathbb{A}(1/R,R)$ into the unit disk, $R>1$ can be formulated as ([2])

(4.11)
$\left|\omega^{\prime}(z)\right|<\frac{\pi\left(1-\left|\omega(z)\right|^{2}\right)\sec\left[\frac{\pi\log|z|}{2\log
R}\right]}{4|z|\log R}.$

(Notice that: For $\omega(z)=z^{2}/R^{2}$ the above inequality is sharp for
$s=|z|\to R$. For $\omega(z)=1/(R^{2}z^{2})$ it is sharp for $s=|z|\to 1/R$.)

So

$|\mathcal{K}|\leq\frac{|\omega^{\prime}(z)|^{2}|z|^{2}}{|\mathbf{c}|(1+|\omega|)^{4}}\leq\left(\pi\frac{(1-|\omega(z)|)}{1+|\omega(z)|}\frac{\sec\left[\frac{\pi\log|z|}{2\log
R}\right]}{4\log R}\right)^{2}\frac{1}{|\mathbf{c}|}.$

So for every $\zeta\in\Sigma$ we have

$|\mathcal{K}(\zeta)|\leq\frac{|\omega^{\prime}(z)|^{2}|z|^{2}}{|\mathbf{c}|(1+|\omega|)^{4}}\leq\left(\pi\frac{\sec\left[\frac{\pi\log|z|}{2\log
R}\right]}{4\log R}\right)^{2}\frac{1}{|\mathbf{c}|}.$

So for $|z|=1$, on account of (4.9), we have that
$\zeta=\chi(z)\in\Sigma\bigcap\mathbb{R}^{2}\times\\{0\\}$. For such $z$ we
have

(4.12) $|\mathcal{K}(\zeta)|\leq\left(\frac{\pi}{4\log
R}\right)^{2}\frac{1}{|\mathbf{c}|}.$

From (4.12) and (4.1) we obtain

$\begin{split}|\mathcal{K}(\zeta)|&\leq\left(\frac{\pi}{4\log
R}\right)^{2}\frac{4\mathrm{Mod}(\Sigma)^{2}}{\mathbf{d}^{2}}\\\
&=\left(\frac{\pi}{2\log
R^{2}}\right)^{2}\frac{4\mathrm{Mod}(\Sigma)^{2}}{\mathbf{d}^{2}}\\\
&=\frac{\pi^{2}}{\mathbf{d}^{2}}.\end{split}$

To prove the last assertion of the theorem, observe that
$\omega^{\prime}(z)=2\mathbf{q}^{\prime}(z)\mathbf{q}^{\prime}(z)$. Then the
unit normal at $\zeta=\chi(z)\in\Sigma$, in view of [3, p. 169] or [1, eq.
2.81] is given by

$\mathbf{n}_{\zeta}=-\frac{1}{1+|\mathbf{q}(z)|^{2}}(2\Im\mathbf{q}(z),2\Re\mathbf{q}(z),-1+|\mathbf{q}(z)|^{2}).$

Further $T\Sigma_{\zeta}$ is horizontal if and only if $\mathbf{q}(z)=0$ at
$\zeta=\chi(z)$. In this case $\omega(z)=0$, but this is impossible because
$f_{z}\bar{f}_{z}=c/z^{2}$ and so $f_{z}\neq 0$ and $f_{\bar{z}}\neq 0$. Now
the tangent plane $T\Sigma_{\zeta}$ is vertical at $\zeta=\chi(z)$, if and
only if $|\mathbf{q}(z)|=1$, but then the maximum principle this implies that
$\mathbf{q}\equiv e^{\imath\tau}$. This is not possible. ∎

###  4.3. The curvature of rotationally symmetric minimal surface

We continue the notation of Example (3.5). Assume that $p:[r,1]\to[R,1]$ is a
function defined by

$s=\exp\int_{1}^{p(s)}\frac{\rho(y)}{\sqrt{4\mathbf{c}+y^{2}\rho^{2}(y)}}dy$

Define for $\mathbf{c}<0$:

$\chi(z)=(p(s)\cos\theta,p(s)\sin\theta,2\sqrt{-\mathbf{c}}\log\frac{1}{s})$

and for $\mathbf{c}>0$:

$\chi(z)=(p(s)\cos\theta,p(s)\sin\theta,2\sqrt{\mathbf{c}}\theta),$

where $z=se^{\imath\theta}$. Then $\chi(z)$ is a conformal parametrisations of
a corresponding minimal surface $\Sigma_{\mathbf{c}}$. Observe that for
$\rho\equiv 1$, i.e. for $\wp-$being an Euclidean metric, the surface
$\Sigma_{\mathbf{c}}$ is a catenoid for $\mathbf{c}<0$ and it is a helicoid
for $\mathbf{c}>0$.

If we set $f(z)=p(s)e^{\imath\theta}$, $z=se^{\imath\theta}$, then $f$ is
$\rho-$harmonic mapping between $\mathbb{A}(r,1)$ and $\mathbb{A}(R,1).$ Since

$p(s)^{2}-(sp^{\prime}(s))^{2}=-\frac{4\mathbf{c}}{\rho(p(s))^{2}},$

we get for $\mathbf{c}<0$ that

$\mu_{f}=\frac{|f_{\bar{z}}|}{|f_{z}|}=\frac{p(s)-sp^{\prime}(s)}{p(s)+p^{\prime}(s)}.$

By having in mind the equation

$|\eta|=\frac{|4\mathbf{c}|}{|z|^{2}},$

and using the formula (4.5), after long but straightforward calculations we
get

$\begin{split}\mathcal{K}=\mathcal{K}_{\mathbf{c}}(\chi(z))&=-\frac{\operatorname{sign}(\mathbf{c})}{t^{4}\rho^{6}}\bigg{(}8\mathbf{c}t^{2}\rho^{\prime
2}+\rho^{2}\left(4\mathbf{c}+t^{4}\rho^{\prime 2}\right)\\\
&+4\mathbf{c}t\rho\left(\rho^{\prime}-t\rho^{\prime\prime}\right)-t^{3}\rho^{3}\left(\rho^{\prime}+t\rho^{\prime\prime}\right)\bigg{)}\end{split}$

where $t=p(s)$ and $\wp=\rho(t)$, where $\mathbf{c}$ is a constant with

$4\mathbf{c}+\min_{t\in[R,1]}t^{2}\rho^{2}(t)\geqslant 0.$

Then for $\mathbf{c}<0$

$\mathcal{K}=\frac{4\mathbf{c}(\rho+t\rho^{\prime})^{2}}{t^{4}\rho^{6}}+\mathcal{K}_{\wp}\left(1+\frac{4\mathbf{c}}{t^{2}\rho^{2}}\right).$

For $\mathbf{c}>0$ we have

$\mathcal{K}=-\frac{4\mathbf{c}\left(\rho+t\rho^{\prime}\right)^{2}}{\left(4\mathbf{c}\rho+t^{2}\rho^{3}\right)^{2}}+\mathcal{K}_{\wp}\frac{t^{2}\rho^{2}}{4\mathbf{c}+t^{2}\rho^{2}}.$

For $\wp$ being an Euclidean metric we have

$\mathcal{K}=\left\\{\begin{array}[]{ll}\frac{4\mathbf{c}}{t^{4}},&\hbox{if
$\mathbf{c}<0$;}\\\
-\frac{4\mathbf{c}}{\left(4\mathbf{c}+t^{2}\right)^{2}},&\hbox{if
$\mathbf{c}>0$.}\end{array}\right.$

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