# Discrete Weierstrass-type representations

Mason Pember and Denis Polly and Masashi Yasumoto

(Date: August 27, 2024)

###### Abstract.

Discrete Weierstrass representations yield a construction method in discrete
differential geometry for certain classes of discrete surfaces. We show that
the known discrete Weierstrass-type representations of certain surface classes
can be viewed as applications of the $\Omega$-dual transform to lightlike
Gauss maps in Laguerre geometry. From this construction, further Weierstrass-
type representations arise.

###### 2020 Mathematics Subject Classification:

53A70 (primary); 51B15 (secondary)

##  1\. Introduction

Weierstrass-type representations play an important role in differential
geometry. The classical representations, like the Weierstrass-Enneper
representation for minimal surfaces [27] or the representation of Bryant for
constant mean curvature (cmc) $1$ surfaces in hyperbolic space [6], [25], are
convenient tools to generate interesting examples of these surface classes.
They became so fundamental, that in a time when there was no notion of mean
curvature for discrete surfaces, authors used the existence of discrete
Weierstrass representations as further evidence to justify their definitions
of discrete constant mean curvature surfaces. Thus, discrete Weierstrass-type
representations played an important role from the very beginnings of the
field. Examples of discrete surface classes with Weierstrass-type
representations111 In this paper, the term “Weierstrass-type representation”
refers to representations that are built from discrete holomorphic functions.
However, this term can also refer to other representations, such as in [20]
where a discrete analogue of the Kenmotsu representation is developed for
discrete nonzero cmc surfaces in $\mathbb{R}^{3}$ using discrete harmonic
maps.  include

  * •

discrete isothermic minimal surfaces in $\mathbb{R}^{3}$ [2, Thm 9],

  * •

discrete maximal surfaces in $\mathbb{R}^{2,1}$ [28, Thm 1.1],

  * •

discrete cmc $1$ surfaces in hyperbolic space $\mathbb{H}^{3}$ [13, Thm 4.4],

  * •

discrete Bryant type surfaces [15], [29].

In the smooth theory, all these surfaces belong to the class of L-isothermic
surfaces. It is therefore natural to consider Laguerre geometry to gain a
unified geometric understanding of Weierstrass representations, which was done
in [21].

To suit our goal of a geometric and unified understanding of the existing
discrete Weierstrass representations, we give in this paper a Laguerre
geometric discussion of discrete L-isothermic surfaces which are defined as
$\Omega$-surfaces that are $\Omega$-dual to their lightlike Gauss map, see
Subsection 3.1. Our main result is that, via the $\Omega$-dual transformation,
discrete surfaces with Weierstrass representations can be constructed from
given discrete lightlike Gauss maps, which arise from holomorphic data.
Applying this to discrete surfaces in hyperplanes of $\mathbb{R}^{3,1}$, we
obtain all known Weierstrass-type representations for discrete zero mean
curvature surfaces and a new case arises, see Subsection 4.1. In Subsection
4.2 we apply this scheme to construct surfaces in subquadrics of
$\mathbb{R}^{3,1}$ to obtain the Weierstrass representations of discrete cmc
surfaces in hyperbolic and de Sitter spaces which have been studied recently
in [18].

To see that even the representations of discrete linear Weingarten surfaces of
Bryant and Bianchi type appear as $\Omega$-dual transformations, we employ the
rich transformation theory of discrete L-isothermic surfaces originally
described in [8] (see also [7]), which we will develop in the realm of
Laguerre geometry. In particular, we will obtain formulas for the Calapso
transformation of discrete L-isothermic surfaces in the Hermitian model of
$\mathbb{R}^{3,1}$.

##  2\. Preliminaries

We give a brief introduction to Laguerre geometry, the setting for our
considerations: Denote by $\mathbb{R}^{3,1}$ the $4$-dimensional space with
basis $(e_{0},e_{1},e_{2},e_{3})$, equipped with the Minkowski product $(,)$,
given by

$\displaystyle(e_{0},e_{0})=-1,(e_{i},e_{i})=1,\leavevmode\nobreak\
i\in\\{1,2,3\\},\leavevmode\nobreak\ \textrm{and}\leavevmode\nobreak\
(e_{i},e_{j})=0,\leavevmode\nobreak\ \textrm{for}\leavevmode\nobreak\ i\neq
j.$ (1)

We call a vector $v\in\mathbb{R}^{3,1}$ _spacelike_ , _timelike_ or
_lightlike_ if $(v,v)$ is positive, negative or equal to zero respectively.

The group of orthogonal transformations of $\mathbb{R}^{3,1}$, denoted
$O(3,1)$, is a Lie group and its Lie algebra $\mathfrak{o}(3,1)$ is the
algebra of skew-symmetric endomorphisms. We identify
$\Lambda^{2}\mathbb{R}^{3,1}$ with $\mathfrak{o}(3,1)$ via

$\displaystyle(v_{1}\wedge v_{2})w=(v_{1},w)v_{2}-(v_{2},w)v_{1}.$

A basis of $\Lambda^{2}\mathbb{R}^{3,1}$ (and thereby of $\mathfrak{o}(3,1)$)
is given by $\\{e_{nm}\\}_{n<m}$, with ${e_{nm}=e_{n}\wedge e_{m}}$.

In the Hermitian model of $\mathbb{R}^{3,1}$, points are identified with
Hermitian $2\times 2$ matrices: Let $\mathbbm{i}$ denote an imaginary unit.
Then, the basis vectors of $\mathbb{R}^{3,1}$ become

$\displaystyle e_{0}=\begin{pmatrix}1&0\cr
0&1\end{pmatrix},\leavevmode\nobreak\ e_{1}=\begin{pmatrix}0&1\cr
1&0\end{pmatrix},\leavevmode\nobreak\
e_{2}=\begin{pmatrix}0&\mathbbm{i}\cr-\mathbbm{i}&0\end{pmatrix},\leavevmode\nobreak\
e_{3}=\begin{pmatrix}1&0\cr 0&-1\end{pmatrix}$

and the inner product (1) is polarization of $|v|^{2}=-\det v$. Each skew-
symmetric endomorphism is identified with a matrix
$A\in\mathfrak{sl}(2,\mathbb{C})$ via

$\displaystyle A\cdot v=Av+vA^{\ast},$

where $A^{\ast}=\overline{A}^{T}$. Thereby, the elements of the basis of
$\mathfrak{o}(3,1)$ are identified as

$\displaystyle e_{0n}=-\frac{1}{2}e_{n},\leavevmode\nobreak\
e_{12}=\frac{\mathbbm{i}}{2}e_{3},\leavevmode\nobreak\
e_{13}=-\frac{\mathbbm{i}}{2}e_{2},\leavevmode\nobreak\
e_{23}=\frac{\mathbbm{i}}{2}e_{1}.$ (2)

Furthermore, every matrix $B\in SL(2,\mathbb{C})$ induces an orthogonal
transformation via

$\displaystyle B\cdot v=BvB^{\ast}.$ (3)

In Laguerre geometry, isotropy projection is used to identify points in
$\mathbb{R}^{3,1}$ with oriented spheres (see [12, Sect 3.4]). Two spheres
$s_{1},s_{2}$ are in oriented contact, if the difference vector $s_{1}-s_{2}$
is lightlike. _Laguerre transformations_ are affine transformations of the
form

$\displaystyle x\mapsto Bx+b,$

where the _linear part_ $B\in O(3,1)$ is orthogonal (thus, Laguerre
transformations preserve oriented contact) and the _translation part_
$b\in\mathbb{R}^{3,1}$ is any vector. A Laguerre transformation fixes
orientation of spheres if its linear part is time-orientation preserving and
therefore fixes

$\displaystyle\mathbb{H}^{3}:=\\{v\in\mathbb{R}^{3,1}:(v,v)=-1,\leavevmode\nobreak\
v_{0}>0\\},$

which is the model space of the Minkowski model of hyperbolic geometry.
Thereby, isometries of $\mathbb{H}^{3}$ are the linear parts of Laguerre
transformations that fix orientation of spheres.

In the light cone model of Möbius geometry, points in the $2$-sphere
$\mathbb{S}^{2}$ are represented by lines in the light cone

$\displaystyle\mathcal{L}:=\\{v\in\mathbb{R}^{3,1}\colon(v,v)=0\\}.$

A Möbius transformation is obtained by the action of an orthogonal
transformation on points in $\mathbb{P}\mathcal{L}$. Thereby we obtain a
$1$-to-$1$ correspondence222For $B\in O(3,1)$, the same Möbius transformation
is induced by $\pm B$, precisely one of which preserves time orientation.
between Möbius transformations of $\mathbb{S}^{2}$ and isometries of
$\mathbb{H}^{3}$.

Let $B=\begin{pmatrix}a&b\cr c&d\end{pmatrix}$ be an $SL(2,\mathbb{C})$ matrix
inducing a complex Möbius transformation

$\displaystyle z\mapsto\frac{az+b}{cz+d}.$

We aim to show that this is the Möbius transformation obtained by the action
of $B$ on $\mathcal{L}$ in the Hermitian model333Note that complex Möbius
transformations are even in the sense that each complex Möbius transformation
is the product of an even number of reflections in a circle. Therefore,
complex Möbius transformations in the light cone model correspond to $SO(3,1)$
transformations and induce isometries on $\mathbb{H}^{3}$.

Via inverse stereographic projection, every $z\in\mathbb{C}$ is identified
with a point in $\mathcal{L}$ by

$\displaystyle\begin{split}z&\mapsto\left\langle(|z|^{2}+1)e_{0}+(z+\overline{z})e_{1}-\mathbbm{i}(z-\overline{z})e_{2}+(|z|^{2}-1)e_{3}\right\rangle\\\
&=\left\langle\begin{pmatrix}|z|^{2}&z\cr\overline{z}&1\end{pmatrix}\right\rangle,\end{split}$
(4)

where $\left\langle\cdot\right\rangle$ denotes the linear span of vectors. The
action of $B$ on $\mathbb{R}^{3,1}$, $X\mapsto B\cdot X$ as defined in (3),
induces a Möbius transformation via

$\displaystyle B\cdot z=\left\langle
B\cdot\begin{pmatrix}|z|^{2}&z\cr\overline{z}&1\end{pmatrix}\right\rangle=\left\langle\begin{pmatrix}\left|\tfrac{az+b}{cz+d}\right|^{2}&\tfrac{az+b}{cz+d}\vspace{5pt}\cr\overline{\left(\tfrac{az+b}{cz+d}\right)}&1\end{pmatrix}\right\rangle,$

which is indeed the action of $B$ on $\mathbb{C}$.

######  Example 2.1.

A hyperbolic rotation $\Gamma^{\left\langle g_{1}\right\rangle}_{\left\langle
g_{2}\right\rangle}(t)$ in $\mathbb{H}^{3}$ is an orthogonal transformations
for which there are two lightlike points $g_{1},g_{2}$, such that

$\displaystyle\Gamma^{\left\langle g_{1}\right\rangle}_{\left\langle
g_{2}\right\rangle}(t)=\left\\{\begin{array}[]{ll}t\operatorname{Id}&\textrm{on}\leavevmode\nobreak\
\left\langle g_{1}\right\rangle\\\
\operatorname{Id}&\textrm{on}\leavevmode\nobreak\ \left\langle
g_{1},g_{2}\right\rangle^{\perp}\\\
t^{-1}\operatorname{Id}&\textrm{on}\leavevmode\nobreak\ \left\langle
g_{2}\right\rangle.\end{array}\right.$

On any point $v\in\mathbb{R}^{3,1}$, we have

$\displaystyle\Gamma^{\left\langle g_{1}\right\rangle}_{\left\langle
g_{2}\right\rangle}(t)v=v+\frac{1-t}{(g_{1},g_{2})}\left(\frac{1}{t}(g_{1},v)g_{2}-(g_{2},v)g_{1}\right).$

Assume, $q=\begin{pmatrix}1&0\cr 0&0\end{pmatrix}\notin\left\langle
g_{1},g_{2}\right\rangle$. Then $g_{1},g_{2}$ and $q$ correspond to three
distinct points $z_{1},z_{2}$ and $\infty$ in $\mathbb{C}\cup\\{\infty\\}$,
respectively444Choosing $\infty$ may sound surprising, but in terms of a
complex Möbius transformation, $\infty\mapsto a/c$.. Since any complex Möbius
transformation is uniquely defined by the images of three points, and

$\displaystyle\\{z_{1},z_{2},\infty\\}\mapsto\left\\{z_{1},z_{2},\frac{z_{2}-tz_{1}}{1-t}\right\\},$

we learn that $\Gamma^{\left\langle g_{1}\right\rangle}_{\left\langle
g_{2}\right\rangle}(t)$ corresponds to the matrix $B\in SL(2,\mathbb{C})$ with
coefficients $a,b,c,d$ given by its action

$\displaystyle B:\leavevmode\nobreak\ z_{1}\mapsto z_{1},\leavevmode\nobreak\
z_{2}\mapsto z_{2},\leavevmode\nobreak\ \infty\mapsto\frac{a}{c}.$

These three conditions amount to a solution matrix

$\displaystyle\tilde{B}=\begin{pmatrix}tz_{1}-z_{2}&z_{1}z_{2}(1-t)\cr
t-1&z_{1}-tz_{2}\end{pmatrix}.$

Computing the determinant of $\tilde{B}$, we learn that

$\displaystyle
B=\frac{1}{(z_{1}-z_{2})\sqrt{t}}\begin{pmatrix}tz_{1}-z_{2}&z_{1}z_{2}(1-t)\cr
t-1&z_{1}-tz_{2}\end{pmatrix},$

up to a choice of sign.

##  3\. Discrete Legendre immersions in Laguerre geometry

A _discrete surface_ is a map $x:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$. Since
points in $\mathbb{R}^{3,1}$ represent spheres in Laguerre geometry, we also
call $x$ a _discrete sphere congruence_. We define

$\displaystyle dx_{ij}=x_{i}-x_{j},\leavevmode\nobreak\
\textrm{and}\leavevmode\nobreak\ x_{ij}=\frac{x_{i}+x_{j}}{2},$

on edges and

$\displaystyle\delta x_{ik}=x_{i}-x_{k},$

for diagonal vertices of an elementary quadrilateral $(ijkl)$ consisting of
neighboring points in $\mathbb{Z}^{2}$. We call a discrete surface $x$ (cf.
[7, Def 4.6 and Rem 4.7])

  * -

_planar_ if the vertices of each elementary quadrilateral are mapped to
coplanar points in $\mathbb{R}^{3,1}$ and

  * -

_circular_ if it is planar and there exists $c\in\mathbb{R}^{3,1}$ for each
elementary quadrilateral, such that

$\displaystyle(x-c,x-c)\equiv const.$

Note that we do not ask that $c$ lies on the corresponding face of $x$.

Let $\mathcal{Z}_{\textrm{aff}}$ denote the set of affine null-lines in
$\mathbb{R}^{3,1}$. Each such line is spanned by two oriented spheres in
contact and represents all oriented spheres in contact with the spanning
spheres. Thus, every $\Lambda\in\mathcal{Z}_{\textrm{aff}}$ is identified with
a contact element. In accordance with [7] we pose the following definition.

######  Definition 3.1.

A map $L:\mathbb{Z}^{2}\to\mathcal{Z}_{\textrm{aff}}$ is called a _discrete
Legendre immersion_ , if for each edge $(ij)$, $L_{i}\cap L_{j}\neq\emptyset$.

A discrete sphere congruence $x:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$ is
_enveloped by $L$_, if there is a map
$G:\mathbb{Z}^{2}\to\mathbb{P}\mathcal{L}$ into the projective light cone such
that $L=x+G$.

The map $L$ may be interpreted as an affine null line bundle. The set of
sections of $L$ is denoted by $\Gamma L$ and consists of maps
$x:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$ such that $x_{i}\in L_{i}$ for all
$i\in\mathbb{Z}^{2}$. Thereby, $\Gamma L$ is easily seen to be the set of
enveloped sphere congruences of $L$. The map
$G:\mathbb{Z}^{2}\to\mathbb{P}\mathcal{L}$ such that $L=x+G$ is called the
_(discrete) lightlike Gauss map of $L$_.

We define the _mixed area_ of two edge-parallel discrete surfaces
$x,x^{\ast}:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$ as a map on faces that takes
values in $\wedge^{2}\mathbb{R}^{3,1}$ via

$\displaystyle A(x,x^{\ast})_{ijkl}=\frac{1}{4}(\delta x_{ik}\wedge\delta
x^{\ast}_{jl})-(\delta x_{jl}\wedge\delta x^{\ast}_{ik}).$

The mixed area is used to define the curvature of discrete surfaces with
respect to given parallel surfaces, see [3]. To condense the curvature notions
of certain subgeometries of Laguerre geometry, we introduce a curvature notion
for discrete Legendre immersions; a discrete version of the light cone mean
curvature given in [16].

######  Definition 3.2.

Let $x:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$ be a discrete sphere congruence
enveloped by $L=x+G$, with lightlike Gauss map $G$ such that there exists a
$g\in\Gamma G$ that is edge parallel to $x$. The _light cone mean curvature
$H_{g}$ of $x$ with respect to $g$_ is then defined on faces via

$\displaystyle A(x,g)+H_{g}A(x,x)=0.$

######  Examples 3.3.

  1. 1)

Let $\mathfrak{p}\in\mathbb{R}^{3,1}$ be a unit timelike vector. A circular
discrete surface $f:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ may be interpreted as a
circular discrete surface $x:\mathbb{Z}^{2}\to\mathfrak{p}^{\perp}$. Since $f$
is circular, it allows for a parallel surface $n:\mathbb{Z}^{2}\to
S^{2}\subset\mathfrak{p}^{\perp}$ and the _discrete mean curvature $H$_ of $f$
with respect to its _discrete Gauss map_ $n$ is defined on faces via

$\displaystyle A(f,n)+HA(f,f)=0.$ (5)

The net $g=\mathfrak{p}+n$ takes values in the light cone $\mathcal{L}$,
hence, $x$ is enveloped by $L=x+\left\langle g\right\rangle$ and $g$ is edge
parallel to $x$. Since $\delta g=\delta n$, we have

$\displaystyle H=H_{g}.$

The same construction of a lightlike Gauss map works for discrete surfaces
with timelike Gauss maps in the Lorentz $3$-space
$\mathbb{R}^{2,1}\cong\mathfrak{p}^{\perp}$ for spacelike $\mathfrak{p}$. We
call a discrete surface in $\mathbb{R}^{3}$ ($\mathbb{R}^{2,1}$) with
vanishing mean curvature _minimal_ (_maximal_).

  2. 2)

For lightlike $\mathfrak{p}$, fix $\mathfrak{o}\in\mathcal{L}$ with
$(\mathfrak{o},\mathfrak{p})=-1$ and $\mathfrak{b}_{1},\mathfrak{b}_{2}$ as an
orthonormal basis of
$\left\langle\mathfrak{o},\mathfrak{p}\right\rangle^{\perp}$. The subgroup of
Laguerre transformations consisting of a vector in
$\left\langle\mathfrak{p}\right\rangle^{\perp}$ and an orthogonal matrix
$A\in\operatorname{O}(3,1)$ with $A\mathfrak{p}=\mathfrak{p}$ fixes

$\displaystyle
I_{\mathfrak{p},\mathfrak{o}}:=\mathfrak{o}+\left\langle\mathfrak{p}\right\rangle^{\perp},$

and acts on a point
$y=\mathfrak{o}+y_{1}\mathfrak{b}_{1}+y_{2}\mathfrak{b}_{2}+y_{3}\mathfrak{p}+\in
I_{\mathfrak{p},\mathfrak{o}}$ as

$\displaystyle y_{1}$ $\displaystyle\mapsto a+y_{1}\cos\phi-y_{2}\sin\phi,$
$\displaystyle y_{2}$ $\displaystyle\mapsto b+y_{1}\sin\phi+y_{2}\cos\phi,$
$\displaystyle y_{3}$ $\displaystyle\mapsto c+c_{1}y_{1}+c_{2}y_{2}+y_{3}.$

Thus, the isotropic geometry of [23], [24], a discrete curvature theory for
which has been developed in [22], is a subgeometry of Laguerre geometry.

Let $f:\mathbb{Z}^{2}\to I_{\mathfrak{p},\mathfrak{o}}$ be a circular discrete
surface, enveloped by a discrete Legendre immersion $L=f+G$ and let
$g\in\Gamma G$ also take values in $I_{\mathfrak{p},\mathfrak{o}}$, that is

$\displaystyle
g=\mathfrak{o}+g_{1}\mathfrak{b}_{1}+g_{2}\mathfrak{b}_{2}+\frac{g_{1}^{2}+g_{2}^{2}}{2}\mathfrak{p}.$

Since $L$ is a discrete Legendre immersion, we have $\\{x_{ij}\in L_{i}\cap
L_{j}\\}_{ij}$ satisfying

$\displaystyle x_{ij}=f_{i}+\lambda_{i}g_{i}=f_{j}+\lambda_{j}g_{j},$

for suitable $\lambda_{i},\lambda_{j}$. Because $f$ and $g$ take values in
$I_{\mathfrak{p},\mathfrak{o}}$ we have $(f,\mathfrak{o})=(g,\mathfrak{o})=-1$
and learn $\lambda_{i}=\lambda_{j}=:\lambda_{ij}$. This implies

$\displaystyle df_{ij}=-\lambda_{ij}dg_{ij},$

hence $f$ and $g$ are edge-parallel and we can define the light cone curvature
$H_{g}$ for $f$. According to [22, Sect 4], in isotropic space
$I_{\mathfrak{p},\mathfrak{o}}$, the mean curvature of $f$ with respect to $g$
is defined by

$\displaystyle A(f,g)+HA(f,f)=0,$

which immediately implies $H=H_{g}$. We call a discrete surface _i-minimal_ ,
if its mean curvature vanishes.

  3. 3)

Let $x$ be a circular discrete surface taking values in a quadric

$\displaystyle Q_{\mu}:=\\{x\in\mathbb{R}^{3,1}:(x,x)=\mu,\leavevmode\nobreak\
\textrm{with}\leavevmode\nobreak\ \mu\neq 0\\},$

which may be considered as

     * •

de Sitter space of curvature $\tfrac{1}{\sqrt{\mu}}$ for $\mu>0$, or

     * •

two copies of a hyperbolic space form with curvature $-\tfrac{1}{\sqrt{-\mu}}$
for $\mu<0$.

For example, $Q_{-1}$ is two copies of hyperbolic space $\mathbb{H}^{3}$ and
$Q_{1}$ is _de Sitter space_ , which we denote by $\mathbb{S}^{2,1}$.

The circular discrete surface $x$ allows for a $2$-parameter choice of
_discrete Gauss map_ ${n:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}}$ with opposite
causal character, so that $(n,x)=0$ and $L=x+\left\langle
x+\sqrt{|\mu|}n\right\rangle$ is a discrete Legendre immersion. We call
$G=\left\langle x+\sqrt{|\mu|}n\right\rangle$ the _discrete hyperbolic Gauss
map of $x$_. The _discrete mean curvature $H_{\mu}$_ of $x$ with respect to
$n$ is then defined by equation (5). For later reference, we also define the
_discrete extrinsinc Gauss curvature $K$_ on faces via

$\displaystyle A(n,n)-KA(f,f)=0.$ (6)

Because $x$ and $n$ are edge-parallel, $g=x+\sqrt{|\mu|}n$ and $x$ are edge-
parallel as well and the light cone mean curvature of $x$ with respect to $g$
satisfies

$\displaystyle H_{g}+1=\sqrt{|\mu|}H_{\mu}.$

Note that, conversely, for a discrete Legendre immersion $L=x+G$ enveloping a
circular discrete surface $x$ in $Q_{\mu}$, a discrete hyperbolic Gauss map is
obtained from any edge-parallel lift $g\in\Gamma G$:

$\displaystyle
n=\tfrac{x}{\sqrt{|\mu|}}-\operatorname{sgn}\mu\tfrac{g}{(x,g)}.$

  4. 4)

The (smooth) extrinsic geometry of spacelike surfaces in the light cone
$\mathcal{L}$ has been developed in [17] (see also [19]), culminating in their
_Theorema Egregium_ that a surface is intrinsically flat if and only of its
light cone mean curvature vanishes, [17, Thm 9.3]. Motivated by this, we call
a discrete surface $x$ in the light cone with vanishing light cone mean
curvature (with respect to some edge-parallel $g\in\Gamma G$) _intrinsically
flat (with respect to $g$)_.

###  3.1. Discrete L-isothermic surfaces

In Laguerre geometry a _discrete isothermic sphere congruence_ is a discrete
surface $x:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$ that possesses a _Christoffel
dual_ $x^{\ast}:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$, which is characterised by
(see [7, Thm 4.11])

  * •

$x$ and $x^{\ast}$ are edge parallel circular nets, and

  * •

$A(x,x^{\ast})=0$.

Clearly, this relation is symmetric in $x$ and $x^{\ast}$, hence, $x^{\ast}$
is itself a discrete isothermic sphere congruence.

Analogous to Demoulin’s definition of $\Omega$-surfaces ([10], [11]), we give
the following definition.

######  Definition 3.4 ([8, Def 3.1]).

A _discrete $\Omega$-surface_ is an envelope $L=x+G$ of a discrete isothermic
sphere congruence.

For a more general treatment of discrete $\Omega$-surfaces, see [7].

######  Definition 3.5.

We say that two discrete $\Omega$-surfaces $L,L^{\ast}$ are _$\Omega$ -dual_
if there are Christoffel dual surfaces $x\in\Gamma L$ and $x^{\ast}\in\Gamma
L^{\ast}$ and the discrete lightlike Gauss maps coincide, i.e., $G^{\ast}=G$.

If $L=x+G$ is a discrete $\Omega$-surface, then we may take a Christoffel dual
$x^{\ast}$ of $x$ and define $L^{\ast}=x^{\ast}+G$. $L^{\ast}$ is
automatically an envelope of $x^{\ast}$ and, thus, an $\Omega$-dual of $L$. A
special class of discrete $\Omega$-surfaces are discrete L-isothermic
surfaces, which are characterised by being $\Omega$-dual to their lightlike
Gauss maps.

######  Definition 3.6.

A _discrete L-isothermic surface_ is a discrete $\Omega$-surface $L$ that is
$\Omega$-dual to its discrete lightlike Gauss map $G$, that is, there exist
$x\in\Gamma L$ and $g\in\Gamma G$ such that $x$ and $g$ are Christoffel dual.

For a discrete L-isothermic surface, $x$ and $g$ are edge parallel with
$A(x,g)=0$. Accordingly, the light cone mean curvature of $x$ with respect to
$g$ vanishes on all quadrilaterals.

######  Remark 3.7.

In [4, Def 16], the authors introduce a notion of L-isothermic discrete
conical surfaces, that is, certain discrete maps into the space of hyperplanes
in $\mathbb{R}^{3}$. These plane congruences allow for a $2$-parameter choice
of an enveloped circular discrete surface $f$ such that the resulting contact
element map is a discrete L-isothermic surface.

Let $g$ be a planar discrete surface in the light cone. Then, $g$ is circular,
as is any edge parallel $x$ (for details, see [7]). Therefore, let
$L=x+\left\langle g\right\rangle$ be a discrete Legendre immersion with $x$
and $g$ edge parallel and planar. Then we can define a $1$-form $\zeta$ by

$\displaystyle\zeta_{ij}:=g_{ij}\wedge dx_{ij}\in G_{i}\wedge G_{j}.$

The closure of this $1$-form is equivalent to $A(x,g)=0$: A simple computation
shows that

$\displaystyle\zeta_{ij}+\zeta_{jk}+\zeta_{kl}+\zeta_{li}=2A(x,g)_{ijkl}.$

We conclude:

######  Proposition 3.8.

A discrete Legendre immersion $L$ with discrete lightlike Gauss map $G$ is
L-isothermic if and only if there exist discrete surfaces $g\in\Gamma G$ and
$x\in\Gamma L$ such that

  * •

$x$ and $g$ are planar edge-parallel and

  * •

the $1$-form $\zeta$, defined on edges via $\zeta_{ij}=g_{ij}\wedge dx_{ij}$,
is closed.

###  3.2. The Calapso transform

The Calapso transformation of discrete $\Omega$-surfaces has been given a Lie
sphere geometric treatment in [8, Section 3], culminating in [8, Thm and Def
3.9]. However, discrete L-isothermic surfaces can be considered in Laguerre
geometry, viewed as a subgeometry of Lie geometry. For self-consistency, we
give a Laguerre geometric treatment of the Calapso transformation of discrete
L-isothermic surfacess which coincides with the projection of the Lie
geometric setting. We will start by collecting some properties of discrete
isothermic surfaces in the projective light cone.

Let $g$ be a discrete isothermic surface in the light cone. Then,
$G=\left\langle g\right\rangle$ is a isothermic in the sense that there is a
cross ratio factorizing edge-labelling (see [5, Sect 4.3]): As $G$ is a planar
discrete surface in the projective space $\mathbb{P}\mathcal{L}$, we can
define the cross ratio of each elementary quadrilateral

$\displaystyle cr_{ijkl}=[G_{i},G_{j},G_{k},G_{l}],$

and because $G$ is isothermic, there is an edge-function $m$ which is constant
across opposite edges of faces, i.e., $m_{ij}=m_{lk}$ and $m_{il}=m_{jk}$,
such that

$\displaystyle cr_{ijkl}=\frac{m_{il}}{m_{ij}}<0.$

We call $m$ the _cross ratio factorizing edge-labelling_ 555Note that many
publications, such as [8], use the cross ratio factorizing function
$a_{ij}=\tfrac{1}{m_{ij}}$..

Also, $G$ can be characterized as isothermic by the existence of a _Moutard
lift_ (see [5, Thms 2.31 and 2.32]): There is a lift $\tilde{g}\in\Gamma G$
such that

$\displaystyle\delta\tilde{g}_{ik}||\delta\tilde{g}_{jl}.$

In our investigation we will use the following relationship to the cross ratio
factorizing edge-labelling

$\displaystyle(\tilde{g}_{i},\tilde{g}_{j})=\frac{1}{m_{ij}},$

a proof for which may be found in [8, Sec 3].

$G$ comes with a family of flat connections: On the edge $(ij)$, define the
map

$\displaystyle\Gamma(t)_{ij}:\\{j\\}\times\mathbb{R}^{3,1}$
$\displaystyle\to\\{i\\}\times\mathbb{R}^{3,1}$ $\displaystyle v$
$\displaystyle\mapsto\Gamma^{G_{i}}_{G_{j}}\left(1-\frac{t}{m_{ij}}\right)v,$

that is, on each edge, $\Gamma(t)$ acts as the hyperbolic rotation (defined by
$G_{i}$ and $G_{j}$) with parameter $1-\tfrac{t}{m_{ij}}$. We may write

$\displaystyle\Gamma(t)_{ij}v=v+\frac{t}{m_{ij}(g_{i},g_{j})}\left(\frac{1}{1-t/m_{ij}}(g_{i},v)g_{j}-(g_{j},v)g_{i}\right).$
(7)

It was shown in [8, Lemma 3.7] that $\Gamma$ is flat, that is,

$\displaystyle\Gamma(t)_{ij}\Gamma(t)_{jk}\Gamma(t)_{kl}\Gamma(t)_{li}=id,$

around each elementary quadrilateral. Via integration, we can obtain a map
${T(t):\mathbb{Z}^{2}\to O(3,1)}$ such that666One can think of $T(t)$ as a
trivialisation of $\Gamma$ in the sense that
$T(t)_{i}\Gamma(t)_{ij}T(t)_{j}^{-1}=id_{ij}$, where $id_{ij}$ denotes the
trivial map $\\{j\\}\times\mathbb{R}^{3,1}\to\\{i\\}\times\mathbb{R}^{3,1}$.

$\displaystyle T(t)_{j}=T(t)_{i}\Gamma(t)_{ij}.$ (8)

Note that $T(t)$ is only unique up to a choice of initial condition.

######  Lemma 3.9.

Let $G:\mathbb{Z}^{2}\to\mathbb{P}\mathcal{L}$ be discrete isothermic with
Moutard lift $\tilde{g}$. Define a map

$G(t)=T(t)G:=\left\langle T(t)\tilde{g}\right\rangle.$

Then, $G(t)$ is also discrete isothermic with Moutard lift
$\tilde{g}(t)=T(t)\tilde{g}$.

###### Proof.

Because $\tilde{g}$ is a Moutard lift of $G$, there is $\lambda$ such that

$\displaystyle\delta\tilde{g}_{ik}=\lambda\delta\tilde{g}_{jl}.$

We compute

$\displaystyle T(t)_{j}\tilde{g}_{j}-T(t)_{l}\tilde{g}_{l}$
$\displaystyle=T(t)_{i}\left[\Gamma(t)_{ij}\tilde{g}_{j}-\Gamma(t)_{il}\tilde{g}_{l}\right]$
$\displaystyle=T(t)_{i}\left[\frac{m_{il}}{m_{il}-t}(\tilde{g}_{j}-\tilde{g}_{l})-\frac{t(m_{ij}-m_{il})}{(m_{ij}-t)(m_{il}-t)}\tilde{g}_{j}\right],$

and, using (7),

$\displaystyle T(t)_{i}\tilde{g}_{i}-T(t)_{k}\tilde{g}_{k}$
$\displaystyle=T(t)_{i}\left[\tilde{g}_{i}-T(t)_{i}^{-1}T(t)_{j}\Gamma_{jk}\tilde{g}_{k}\right]$
$\displaystyle=T(t)_{i}\left[\frac{m_{jk}}{m_{jk}-t}(\tilde{g}_{i}-\tilde{g}_{k})-\frac{tm_{jk}m_{ij}}{(m_{jk}-t)(m_{ij}-t)}(\tilde{g}_{i},\tilde{g}_{k})\tilde{g}_{j}\right].$

Now, because of

$\displaystyle(\tilde{g}_{i},\tilde{g}_{k})=(\tilde{g}_{i},\tilde{g}_{i}-\delta\tilde{g}_{ik})=\lambda\left(\frac{1}{m_{il}}-\frac{1}{m_{ij}}\right)=\lambda\frac{m_{ij}-m_{il}}{m_{ij}m_{il}},$

we see that

$\displaystyle\delta(T(t)\tilde{g})_{ik}=\lambda\delta(T(t)\tilde{g})_{jl},$

proving the lemma. ∎

######  Definition 3.10.

Let $L=x+G$ be a discrete L-isothermic surface, $\Gamma(t)$ the family of flat
connections of its discrete lightlike Gauss map $G$, and $T(t)$ such that
$T(t)_{j}=T(t)_{i}\Gamma(t)_{ij}$. The _Calapso transformation_ of $L$ is
$L(t)=x(t)+G(t)$, where $x(t)$ and $G(t)$ are given by

$\displaystyle dx(t)_{ij}$
$\displaystyle:=T(t)_{i}x_{i}-T(t)_{j}x_{j}+\frac{t}{m_{ij}(g_{i},g_{j})}\left((g_{i},x_{i})T(t)_{j}g_{j}-(g_{j},x_{j})T(t)_{i}g_{i}\right)$
(9) $\displaystyle G(t)$ $\displaystyle:=T(t)G.$ (10)

We call $T(t)$ the _lightlike part of the Calapso transformation_.

######  Proposition 3.11.

The Calapso transformation $L(t)$ of a discrete L-isothermic surface $L$ is
well-defined and L-isothermic.

###### Proof.

We have $dx_{ij}=\lambda dg_{ij}$. Using (9) and (10) one finds
$dx(t)_{ij}=\lambda d(T(t)g)_{ij}$. The Moutard equation satisfied by
$\tilde{g}(t)$ is then seen to be the integrability condition of (9). Further,
$x(t)$ and $g(t):=T(t)g$ are Christoffel dual. Thus $L(t)$ is L-isothermic. ∎

$L(t)$ itself defines a family of flat connections $\Gamma^{t}(s)$ (belonging
to the discrete isothermic surface $G(t)$) and a map $T^{t}(s)$ satisfying

$\displaystyle T^{t}(s)_{j}=T^{t}(s)_{i}\Gamma^{t}(s)_{ij}.$

Since the Moutard lift of $G(t)$ is given by $T(t)\tilde{g}$, we can compute
the cross ratio factorizing edge-labelling $m(t)$ of $G(t)$ to be

$\displaystyle m_{ij}(t)=m_{ij}-t.$

Now, according to (7), we have

$\displaystyle\Gamma^{t}(s)_{ij}v=v+\tfrac{s}{(m_{ij}-t)(T(t)_{i}g_{i},T(t)_{j}g_{j})}\left(\tfrac{m_{ij}-t}{m_{ij}-t-s}(T(t)_{i}g_{i},v)T(t)_{j}g_{j}-(T(t)_{j}g_{j},v)T(t)_{i}g_{i}\right),$

and $(T(t)_{i}g_{i},T(t)_{j}g_{j})=(g_{i},g_{j})\frac{m_{ij}}{m_{ij}-t}$ can
be used to show

$\displaystyle T(t)_{i}^{-1}\Gamma^{t}_{ij}(s)T(t)_{j}=\Gamma(s+t)_{ij}.$

Therefore, we have

$\displaystyle\Gamma^{t}_{ij}(s)=T^{t}(s)_{i}^{-1}T^{t}(s)_{j}=T(t)_{i}T(t+s)_{i}^{-1}T(t+s)_{j}T(t)_{j}^{-1},$

hence, with the right choice of initial condition,

$\displaystyle T^{t}(s)T(t)=T(t+s).$

We proved the following:

######  Proposition 3.12.

The lightlike part of the Calapso transformation is additive in the following
sense: Let $L(t)$ be a Calapso transform of a discrete L-isothermic surface
$L$ with parameter $t$ and let $T^{t}(s)$ denote the lightlike part of the
Calapso transformation of $L(t)$. Then,

$\displaystyle T(s+t)=T^{t}(s)T(t).$

In particular,

$\displaystyle T(t)^{-1}=T^{t}(-t).$

######  Remark 3.13.

The lightlike part of the Calapso transform is the Calapso transformation of
the $\mathbb{P}\mathcal{L}\cong S^{2}$-valued map $G$. Hence Proposition 3.12
is a special case of [13, Thm 3.13] for which we gave a proof within our
setup.

##  4\. Discrete Weierstrass type represenations

Let $\phi:\mathbb{Z}^{2}\to\mathbb{C}$ be a _discrete holomorphic function_ ,
that is, there exists an edge-labelling $m$ such that the cross ratio of
$\phi$ satisfies

$\displaystyle[\phi_{i},\phi_{j},\phi_{k},\phi_{l}]=\frac{m_{il}}{m_{ij}}<0,$

on any face $(ijkl)$. We can lift $\phi$ into $\mathcal{L}$ via inverse
stereographic projection. The resulting map (see (4))

$\displaystyle g=2\begin{pmatrix}|\phi|^{2}&\phi\\\
\overline{\phi}&1\end{pmatrix},$ (11)

gives rise to a discrete isothermic surface $G:=\left\langle
g\right\rangle:\mathbb{Z}^{2}\to\mathbb{P}\mathcal{L}$ with the same cross
ratio factorizing edge-labelling $m$, that is,

$\displaystyle[G_{i},G_{j},G_{k},G_{l}]=\frac{m_{il}}{m_{ij}}<0.$

One then obtains a closed $1$-form $\zeta$ on edges with $\zeta_{ij}\in
G_{i}\wedge G_{j}$ by defining for an arbitrary section $g\in\Gamma G$

$\displaystyle\zeta_{ij}:=\frac{1}{m_{ij}(g_{i},g_{j})}g_{ij}\wedge dg_{ij}.$

The closedness is easily seen using a Moutard lift $\tilde{g}$:

$\displaystyle\zeta_{ij}+\zeta_{jk}+\zeta_{kl}+\zeta_{li}=\delta\tilde{g}_{ik}\wedge\delta\tilde{g}_{lj}=0.$

For the lift in (11), we have

$\displaystyle\begin{split}\zeta_{ij}&=\operatorname{Re}\left\\{\frac{1}{m_{ij}(\phi_{i}-\phi_{j})}\leavevmode\nobreak\
\left((1-\phi_{i}\phi_{j})\leavevmode\nobreak\ e_{0}\wedge
e_{1}+\mathbbm{i}(1+\phi_{i}\phi_{j})\leavevmode\nobreak\ e_{0}\wedge
e_{2}+(\phi_{i}+\phi_{j})\leavevmode\nobreak\ e_{0}\wedge
e_{3}\right.\right.\\\
&\left.\left.\quad+\mathbbm{i}(\phi_{i}+\phi_{j})\leavevmode\nobreak\
e_{1}\wedge e_{2}+(1+\phi_{i}\phi_{j})\leavevmode\nobreak\ e_{1}\wedge
e_{3}+\mathbbm{i}(1-\phi_{i}\phi_{j})\leavevmode\nobreak\ e_{2}\wedge
e_{3}\right)\right\\}\end{split}$ (12)

###  4.1. Discrete surfaces in hyperplanes of $\mathbb{R}^{3,1}$

Chose a non-zero $\mathfrak{p}\in\mathbb{R}^{3,1}$. We then have that
$\zeta\mathfrak{p}$ is a closed $1$-form with values in $\mathbb{R}^{3,1}$ and
we may (locally) integrate to obtain a discrete surface
$x:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$ such that
$dx_{ij}=-\zeta_{ij}\mathfrak{p}$ (see [7, Prop. 2.17]). Then,
$dx\perp\mathfrak{p}$ because $\zeta$ is skew-symmetric. Hence, $x$ takes
values in an affine hyperplane with normal $\mathfrak{p}$. If at a vertex
$i\in\mathbb{Z}^{2}$ $\mathfrak{p}\perp G_{i}$, then
$\zeta_{ij}\mathfrak{p}\in G_{i}$ and $\zeta_{il}\mathfrak{p}\in G_{i}$ are
parallel, hence the quadrilateral $(ijkl)$ degenerates.

Thus, for non-degeneracy, we assume $\mathfrak{p}\not\perp G_{i}$ and we may
choose $g^{n}\in\Gamma G$ such that ${(g^{n},\mathfrak{p})=-1}$. Then

$\displaystyle
dx_{ij}=-\zeta_{ij}\mathfrak{p}=\frac{1}{m_{ij}(g^{n}_{i},g^{n}_{j})}dg^{n}_{ij},$

hence, $x$ is edge-parallel to $g^{n}$. Moreover, we may now write
$\zeta_{ij}=g^{n}_{ij}\wedge dx_{ij}$. By Proposition 3.8 $L=x+G$ is a
discrete L-isothermic surface and $x$ and $g^{n}$ are Christoffel dual. In
particular, the light cone mean curvature of $x$ with respect to $g^{n}$
vanishes, and we learn the following from the examples given in 3.3:

  * •

If $\mathfrak{p}$ is timelike, $x$ is discrete minimal in Euclidean $3$-space.

  * •

If $\mathfrak{p}$ is spacelike, $x$ is discrete maximal in Lorentz $3$-space.

  * •

If $\mathfrak{p}$ is lightlike, $x$ is discrete i-minimal in isotropic
$3$-space.

(a) The discrete power function $z\mapsto z^{2/3}$ used to create the discrete
surfaces in 1(b)

(b) Using the discrete holomorphic power function in 1(a) and (13) we obtain a
discrete minimal surface in Euclidean space ($\mu=1$) on the left, a discrete
maximal surface in Lorentz space ($\mu=-1$) on the right and a discrete
i-minimal surface ($\mu=0$) in the middle

(c) The discrete power function $z\mapsto z^{2/3}$, translated and used to
create the discrete surfaces in 1(d)

(d) Using the translated discrete holomorphic power function in 1(c) and (13)
we obtain a discrete minimal surface in Euclidean space ($\mu=1$) on the left,
a discrete maximal surface in Lorentz space ($\mu=-1$) on the right and a
discrete i-minimal surface ($\mu=0$) in the middle

(e) The translated discrete exponential map and used to create the discrete
surfaces in 1(f)

(f) Using the translated discrete exponential map in 1(e) and (13) we obtain a
discrete minimal surface in Euclidean space ($\mu=1$) on the left, a discrete
maximal surface in Lorentz space ($\mu=-1$) on the right and a discrete
i-minimal surface ($\mu=0$) in the middle

Figure 1. Thre discrete holomorphic functions and the respective discrete zero
mean curvature surfaces in Euclidean, Lorentz and isotropic space.

We obtain explicit formulas for $x$ in terms of the discrete holomorphic
function $\phi$ by using the formula for $\zeta$ in (12). For

$\displaystyle\mathfrak{p}=\tfrac{1}{2}(1-\mu)e_{3}+\tfrac{1}{2}(1+\mu)e_{0},$

we obtain

$\displaystyle
dx_{ij}=\operatorname{Re}\left\\{\left(\tfrac{1-\mu}{2}(\phi_{i}+\phi_{j})e_{0}+(1-\mu\phi_{i}\phi_{j})e_{1}+\mathbbm{i}(1+\mu\phi_{i}\phi_{j})e_{2}+\tfrac{1+\mu}{2}(\phi_{i}+\phi_{j})e_{3}\right)\tfrac{1}{m_{ij}(\phi_{i}-\phi_{j})}\right\\},$
(13)

as representation of a circular discrete surface in
$\mathbb{R}^{3}\cong\left\langle\mathfrak{p}\right\rangle^{\perp}$. For
$\mu=1$, that is $\mathfrak{p}=e_{0}$, we recover the usual discrete
Weierstrass representation for discrete isothermic minimal surfaces in
Euclidean space (see [2, Thm 9]). If we set $\mu=-1$, that is
$\mathfrak{p}=e_{3}$, we get the Weierstrass representation for discrete
maximal surfaces (see [28, Thm 1.1]). For $\mu=0$, that is
$\mathfrak{p}=\frac{e_{0}+e_{3}}{2}$, we obtain a discrete surface in
isotropic space $I_{\mathfrak{p},\mathfrak{o}}$ if we choose the initial point
of $x$ to satisfy $(x,\mathfrak{p})=-1$. According to Example 3.3.(2), $x$ is
a discrete i-minimal surface as in [22, Sect 4] and we derived a Weierstrass
representation formula for this class, namely,

$\displaystyle
dx_{ij}=\text{Re}\\{(e_{1}+\mathbbm{i}e_{2}-(\phi_{i}+\phi_{j})\mathfrak{p})\tfrac{1}{m_{ij}(\phi_{i}-\phi_{j})}\\}.$

In Figure 1 we used the representation formula (13) and different values of
$\mu$ to obtain discrete minimal, i-minimal and maximal surfaces from given
holomorphic data (a discrete power function and the discrete exponential map).

We summarize the results of this subsection in the following theorem:

######  Theorem 4.1.

The Weierstrass-type representations for

  * •

discrete minimal surfaces in Euclidean $3$-space,

  * •

discrete maximal surfaces in Lorentzian $3$-space, and

  * •

discrete i-minimal surfaces in isotropic $3$-space

can be viewed as certain applications of the $\Omega$-dual transformation to a
prescribed discrete lightlike Gauss map.

###  4.2. Discrete surfaces in quadrics in $\mathbb{R}^{3,1}$

For the discrete isothermic surface $G$ given by (11), we defined its family
of flat connections as

$\displaystyle\Gamma(t)_{ij}:=\Gamma^{G_{i}}_{G_{j}}\left(1-\tfrac{t}{m_{ij}}\right).$

For fixed $m\in\mathbb{R}^{\times}$, because $\Gamma(m)$ is flat, there
(locally) exist parallel sections $x:\mathbb{Z}^{2}\to\mathbb{R}^{3,1}$, that
is, discrete surfaces such that on each edge $(ij)$

$\displaystyle x_{j}=\Gamma(m)_{ji}x_{i}.$

For such a surface $(x_{i},x_{i})=(x_{j},x_{j})$ because $\Gamma(m)_{ij}\in
O(3,1)$, so $(x,x)=const.$ and $x$ takes values in a quadric in
$\mathbb{R}^{3,1}$.

If $x_{i}\perp G_{i}$, then one has that $dx_{ij},dx_{il}\in G_{i}$ (see (7))
and the quadrilateral $(ijkl)$ degenerates. Away from such points, we may
choose $g^{n}\in\Gamma G$ so that $(g^{n},x)=-1$. We will now show that $x$
and $g^{n}$ are edge parallel and $A(x,g^{n})=0$:

Firstly,

$\displaystyle(x_{i},g^{n}_{j})=(\Gamma(m)_{ij}x_{j},g^{n}_{j})=(x_{j},\Gamma(m)_{ji}g^{n}_{j})=-\left(1-\tfrac{m}{m_{ij}}\right),$

implying, by (7), that we have that

$\displaystyle
x_{j}=\Gamma(m)_{ji}x_{i}=x_{i}-\tfrac{m}{m_{ij}(g^{n}_{i},g^{n}_{j})}dg^{n}_{ij},$

thus, $dx_{ij}=\tfrac{m}{m_{ij}(g^{n}_{i},g^{n}_{j})}dg^{n}_{ij}$. This tells
us that $x$ is edge-parallel to $g^{n}$ and that we can write
${\zeta_{ij}=\frac{1}{m}g^{n}_{ij}\wedge dx_{ij}}$. This implies
$A(x,g^{n})=0$ and by Proposition 3.8, $L:=x+G$ is a discrete $L$-isothermic
surface.

Assume $x$ takes values in a quadric $Q_{\mu}$ of constant curvature, given by
$\mu:=(\mathfrak{p},\mathfrak{p})\neq 0$ (see example (3) in 3.3). A Gauss map
of $x$ is determined by $g^{n}$ via

$\displaystyle
n=\frac{1}{\sqrt{|\mu|}}\left(x+\operatorname{sgn}(\mu)\sqrt{|\mu|}g^{n}\right),$

and $x$ is a _discrete constant mean curvature (cmc) surface_ in the
respective space form:

  * •

If $\mu<0$, $x$ is cmc $\frac{1}{\sqrt{-\mu}}$ in the hyperbolic space
$Q_{\mu}$.

  * •

If $\mu>0$, $x$ is cmc $\frac{1}{\sqrt{\mu}}$ in de Sitter space $Q_{\mu}$.

If $\mu=0$, $x$ is intrinsically flat in the light cone (see example 3.4). The
representation we obtain for these discrete surfaces is thus a discrete
analogue of the one given in [21, Sect. 4.2].

######  Remark 4.2.

For $\mu<0$, $Q_{\mu}$ consists of two copies of hyperbolic space
$\mathbb{H}^{3}$. A net $x$ takes values in $\mathbb{H}^{3}$ if
$(e_{0},x)<-1$, that is if its Hermitian representative $X$ satisfies
$\operatorname{tr}X>2$. Given an initial point $x_{0}\in\mathbb{H}^{3}$, a
discrete cmc $1$ surface in $Q_{-1}$ is formed by applying the hyperbolic
rotations $\Gamma(m)$. Thus, $x$ takes values in $\mathbb{H}^{3}$ and we have:
_Every discrete cmc 1 surface in $\mathbb{H}^{3}$ is constructed by the
application of hyperbolic rotations to a seed point_.

As we have seen in Example 2.1, $\Gamma(m)$ takes the following form in the
Hermitian model:

$\displaystyle
B(m)_{ij}=\frac{1}{(\phi_{i}-\phi_{j})\sqrt{m_{ij}}\sqrt{m_{ij}-m}}\begin{pmatrix}m_{ij}(\phi_{i}-\phi_{j})-m\phi_{i}&m\phi_{i}\phi_{j}\\\
-m&m_{ij}(\phi_{i}-\phi_{j})+m\phi_{j}\end{pmatrix}.$ (14)

Define a map $F(m):\mathbb{Z}^{2}\to SL(2,\mathbb{C})$ by

$\displaystyle F(m)_{j}=F(m)_{i}B(m)_{ij}$

Thus, $F$ is the Hermitian representative of $T(m)$, as defined via (8).

Since $x$ is parallel with respect to $\Gamma(m)$, there exists a fixed vector
$\mathfrak{p}\in\mathbb{R}^{3,1}$ such that

$\displaystyle x=T^{-1}(m)\mathfrak{p}.$

Now, take

$\displaystyle\mathfrak{p}=\tfrac{1}{2}(1+\mu)e_{3}+\tfrac{1}{2}(1-\mu)e_{0},$

and $x$ is given in the Hermitian model by

$\displaystyle X=F(m)^{-1}\begin{pmatrix}1&0\cr
0&-\mu\end{pmatrix}\left(F(m)^{-1}\right)^{\ast}.$

Denoting $\Phi=F(m)^{-1}$, we obtain the following Weierstrass-type
representation

$\displaystyle X=\Phi\begin{pmatrix}1&0\cr
0&-\mu\end{pmatrix}\Phi^{\ast},\leavevmode\nobreak\
\textrm{for}\leavevmode\nobreak\ \Phi_{j}=B(m)_{ij}^{-1}\Phi_{i},$ (15)

giving

  * •

discrete cmc $1$ surfaces in hyperbolic space $\mathbb{H}^{3}$ for $\mu=-1$,

  * •

discrete cmc $1$ surfaces in de Sitter space $\mathbb{S}^{2,1}$ for $\mu=1$,
and

  * •

discrete intrinsically flat surfaces in the light cone for $\mu=0$.

In Figure 2 there are two examples of discrete cmc 1 surfaces in
$\mathbb{H}^{3}$, obtained via (15), pictured in the Poincaré ball model.

(a) A discrete cmc $1$ surface in hyperbolic space, obtained from the discrete
power function 1(a) via (15)

(b) A discrete cmc $1$ surface of revolution in hyperbolic space, obtained
from the discrete exponential map via (15) with a particular choice of initial
condition

Figure 2. Two examples of discrete cmc $1$ surfaces in $\mathbb{H}^{3}$ shown
in the Poincaré ball model

In the cases of discrete cmc 1 surfaces in $\mathbb{H}^{3}$ our construction
coincides with the Möbius geometric construction of [1, Section 5], where it
was also noted that these nets in $\mathbb{H}^{3}$ are obtained as Darboux
transforms of their hyperbolic Gauss maps. However, this Weierstrass-type
representation does not coincide with the one given in [15]. Using the Lawson
correspondence, we will bridge the gap between the two representations in the
next subsection.

###  4.3. The Lawson correspondence

Let $x=T(m)^{-1}\mathfrak{p}$ be a discrete surface of vanishing light cone
mean curvature, constructed as a parallel section of the flat connection
$\Gamma(m)$ of its discrete lightlike Gauss map as in the last section. Using
(9) the Calapso transform $x(m)$ then satisfies

$\displaystyle dx(m)_{ij}$
$\displaystyle=\frac{m}{(g_{i},g_{j})m_{ij}}\left((g(m)_{i},\mathfrak{p})g(m)_{j}-(g(m)_{j},\mathfrak{p})g(m)_{i})\right)$
$\displaystyle=-m\zeta(m)_{ij}\mathfrak{p}.$

Hence, $x(m)$ is of vanishing mean curvature in a hyperplane according to
Subsection 4.1. We thus have proved the following Laguerre geometric
interpretation of the Lawson correspondence noted in [13, Lem 4.2] and [9,
Sect 5.5] which was generalized in [8, Sec 4.2].

######  Proposition 4.3.

The Calapso transformation of discrete L-isothermic surfaces perturbs

  * •

discrete cmc $1$ surfaces in $\mathbb{H}^{3}$ and discrete minimal surfaces in
$\mathbb{R}^{3}$,

  * •

discrete cmc $1$ surfaces in $\mathbb{S}^{2,1}$ and discrete maximal surfaces
in $\mathbb{R}^{2,1}$ and

  * •

discrete intrinsically flat surfaces in the light cone $\mathcal{L}$ and
discrete i-minimal surfaces in isotropic $3$-space.

We will use this to bridge the gap between the representation (15) for
discrete cmc $1$ surfaces in $\mathbb{H}^{3}$ and the one given in [15]:

For $x:\mathbb{Z}^{2}\to\mathbb{H}^{3}$ with hyperbolic Gauss map $G$, we call
$\widehat{G}=T(m)\cdot G$ the _discrete secondary Gauss map of $x$_. As
$\hat{G}$ is conformally equivalent to $G$, it is isothermic and for a
suitable holomorphic map $\psi:\Sigma\to\mathbb{C}$ (obtained via the inverse
of (4)) we can write

$\displaystyle\widehat{G}=\left\langle\begin{pmatrix}|\psi|^{2}&\psi\\\
\overline{\psi}&1\end{pmatrix}\right\rangle.$

By Proposition 3.12, we have $T(m)^{-1}=T^{m}(-m)$. Hence, the discrete cmc
surface $x$ in $\mathbb{H}^{3}$ satisfies

$\displaystyle x=T^{-1}(m)\mathfrak{p}=T^{m}(-m)\mathfrak{p},$ (16)

with $\mathfrak{p}=e_{0}$. Denoting the Hermitian representative of
$T^{m}(-m)$ by $\Psi$ we obtain the following representation

$\displaystyle X=\Psi\Psi^{\ast},\leavevmode\nobreak\
\textrm{with}\leavevmode\nobreak\ \Psi_{j}=\Psi_{i}B^{m}(-m)_{ij},$ (17)

where $B^{m}(-m)$ denotes the Hermitian representation of $\Gamma^{m}(-m)$ as
given in (14).

This corresponds to the representation given in [15]: There, a discrete cmc
$1$ surface in $\mathbb{H}^{3}$ is given via

$\displaystyle Y=\frac{1}{\det E}EE^{\ast},$

where $E$ satisfies

$\displaystyle E_{j}-E_{i}=E_{i}\begin{pmatrix}\psi_{i}&-\psi_{i}\psi_{j}\\\
1&-\psi_{j}\end{pmatrix}\frac{\lambda}{m_{ij}(\psi_{j}-\psi_{i})},$

that is

$\displaystyle
E_{j}=E_{i}\begin{pmatrix}m_{ij}(\psi_{i}-\psi_{j})-\lambda\psi_{i}&\lambda\psi_{i}\psi_{j}\\\
-\lambda&m_{ij}(\psi_{i}-\psi_{j})+\lambda\psi_{j}\end{pmatrix}\frac{1}{m_{ij}(\psi_{i}-\psi_{j})}$

It is readily seen that

$\displaystyle\det E_{j}=\frac{m_{ij}-\lambda}{m_{ij}}\det E_{i},$

and by setting $\Psi=\frac{1}{\sqrt{\det E}}E$ and $\lambda=-m$, we recover
(17).

######  Remark 4.4.

The hyperbolic Gauss map $G$ of $x$ is given by $G=T^{m}(-m)\cdot\widehat{G}$.
Hence $G$ is the projection of the discrete function

$\displaystyle\phi:=\frac{a\psi+b}{c\psi+d},\leavevmode\nobreak\ \textrm{where
}\Psi=\begin{pmatrix}a&b\cr c&d\end{pmatrix}$

Then, we can directly show that $E_{i}^{-1}$ satisfies

$\displaystyle
E_{j}^{-1}-E_{i}^{-1}=E_{i}^{-1}\begin{pmatrix}\phi_{i}&-\phi_{i}\phi_{j}\cr
1&-\phi_{j}\end{pmatrix}\frac{-\lambda}{(m_{ij}-\lambda)(\phi_{j}-\phi_{i})}$

which is the discrete analogue of the duality formula described in [26, Prop
4]. Then, $E^{-1}$ defines the _discrete dual cmc 1 surface_ of $x$ via its
Hermitian representative

$\displaystyle
X^{\\#}=(\operatorname{det}E)E^{-1}\left(E^{-1}\right)^{\ast}=\Psi^{-1}\left(\Psi^{-1}\right)^{\ast}.$

Since $\Psi^{-1}_{j}=B(m)_{ij}\Psi^{-1}_{i}$, $X^{\\#}$ is obtained via (15).
The hyperbolic Gauss map of $X^{\\#}$ is given by $\widehat{G}$. In Figure 3
we see a pair of dual cmc 1 surfaces in $\mathbb{H}^{3}$.

Note that representations like (17) for other space forms are easily obtained
by applying (16) to
${\mathfrak{p}=\tfrac{1}{2}(1+\mu)e_{3}+\tfrac{1}{2}(1-\mu)e_{0}}$ for
different $\mu$.

(a) The discrete power function $z\mapsto z^{2/3}$ is the hyperbolic Gauss map
of the surface shown in 3(d) and the secondary Gauss map of the surface shown
in 3(b)

(b) The discrete cmc $1$ surface obtained (via (17)) from the discrete power
function 3(a) which is also its secondary Gauss map. Its holomorphic Gauss map
is shown in 3(c).

(c) The hyperboic Gauss map of the surface shown in 3(b) is used as data to
obtain the surface in 3(d)

(d) The dual surface of the one shown in 3(b), obtained from the discrete
holomorphic map in 3(c) via (17).

Figure 3. A discrete cmc $1$ surface in $\mathbb{H}^{3}$ and its dual surface
in the Poincaré ball model. The discrete holomorphic functions can be regarded
of the hyperbolic and secondary Gauss maps of the surfaces.

######  Theorem 4.5.

The Weierstrass-type representations for discrete cmc $1$ surfaces in
hyperbolic space $\mathbb{H}^{3}$ or de Sitter space $\mathbb{S}^{2,1}$ as
well as the Weierstrass representation for discrete intrinsically flat
surfaces in the light cone can be viewed as certain applications of the
$\Omega$-dual transformation to a prescribed discrete lightlike Gauss map.

###  4.4. Discrete linear Weingarten surfaces of Bryant and Bianchi type

In this subsection we employ the Lawson correspondence described in Subsection
4.3 to recover the Weierstrass representation of discrete linear Weingarten
surfaces of Bryant and Bianchi type (abbreviated as BrLW and BiLW,
respectively) given in [29] (see [8] for a geometric interpretation in the
realm of Lie sphere geometry). Thus, let $G=\left\langle g\right\rangle$ be a
discrete isothermic surface in the projectified light cone with flat
connection $\Gamma(m)$ and define $T(m)$ as in (8). Choose a vector
$\mathfrak{p}$ with $(\mathfrak{p},\mathfrak{p})=\mu$. Then
$x^{M}=T^{-1}(m)\mathfrak{p}$ is a discrete cmc $\mu$ surface in a quadric of
constant curvature $\frac{\operatorname{sgn}(\mu)}{\sqrt{|\mu|}}$ (for
$\mu\neq 0$). Now define for an arbitrary lift $g\in\Gamma G$

$\displaystyle x^{\pm}:=x^{M}-\frac{\mu\pm 1}{2(g,x^{M})}g.$

Clearly, $L=x^{\pm}+G=x^{M}+G$ is discrete L-isothermic and since

$\displaystyle(x^{\pm},x^{\pm})=\mp 1,$

$x^{\pm}$ takes values in hyperbolic space $\mathbb{H}^{3}$ or de Sitter space
$\mathbb{S}^{2,1}$. $G$ is then a discrete hyperbolic Gauss map of $x^{\pm}$
and induces the Gauss maps $n^{\pm}$ of $x^{\pm}$ via

$\displaystyle n^{\pm}=\mp\tilde{g}-x^{\pm},$

where $\tilde{g}$ is the lift of $G$ satisfying $(x^{\pm},\tilde{g})=1$.

######  Lemma 4.6.

The discrete surface $x^{\pm}$ is BrLW/BiLW in hyperbolic/de Sitter space with
respect to the Gauss map induced by $G$, that is,

$\displaystyle(\mu\pm 1)K-2\mu H+(\mu\mp 1)=0,$ (18)

where $H$ and $K$ denote the discrete mean and discrete extrinsic Gauss
curvature with respect to $n^{\pm}$, as defined in (5) and (6) respectively.

###### Proof.

It is straightforward to see that

$\displaystyle 0$ $\displaystyle=A(x^{M},\tilde{g})=A\left(-\frac{\mu\pm
1}{2}n^{\pm}-\frac{\mu\mp 1}{2}x^{\pm},-n^{\pm}-x^{\pm}\right)$
$\displaystyle=\frac{A(x^{\pm},x^{\pm})}{2}\left\\{(\mu\pm 1)K-2\mu H+(\mu\mp
1)\right\\},$

proving $x^{\pm}$ to be BrLW/BiLW. ∎

######  Remark 4.7.

The two maps $x^{\pm}$ are orthogonal, hence, form a symmetric pair of
discrete surface and Gauss map. Lemma 4.6 therefore proves that the Gauss map
of a discrete BrLW surfacee is BiLW (and vice versa). Of course, $n$ is only
defined up to its sign, which we chose in alignment with [29]. Note that

$\displaystyle n^{+}=-x^{-},$

that is, for each discrete BrLW surface $x^{+}$ in $\mathbb{H}^{3}$, the Gauss
map $n^{+}$ and the discrete surface $x^{-}$ are antipodal discrete BiLW
surfaces in $\mathbb{S}^{2,1}$.

######  Example 4.8.

In the case that $\mu=0$, we have that $x^{\pm}$ are discrete flat fronts and
$x^{M}$ is a map into the light cone. In [14] it is shown that discrete flat
fronts in $\mathbb{H}^{3}$ are the orthogonal nets of the cyclic congruence
associated to a Darboux pair of maps into the two sphere. We recover this
Darboux pair as $(G,\hat{G})$, where $G$ is the discrete lightlike Gauss map
of $x^{+}$ and $\hat{G}:=\left\langle x^{M}\right\rangle$.

In the Hermitian model, $x^{\pm}$ corresponds to

$\displaystyle X^{\pm}=X^{M}-\frac{\mu\pm
1}{2(g,x^{M})}g=\Psi\cdot\left(\mathfrak{p}-\frac{\mu\pm
1}{2(g,x^{M})}\Psi^{-1}\cdot g\right),$

where $X^{M}$ denotes the Hermitian representative of $x^{M}$. The discrete
secondary Gauss map of $x^{\pm}$, namely ${\widehat{G}=\Psi^{-1}\cdot
G=\left\langle\hat{g}\right\rangle}$, is obtained via inverse stereographic
projection from a discrete holomorphic map
${\psi:\mathbb{Z}^{2}\to\mathbb{C}}$ as

$\displaystyle\widehat{g}=2\begin{pmatrix}|\psi|^{2}&\psi\\\
\overline{\psi}&1\end{pmatrix}.$

If we choose

$\displaystyle\mathfrak{p}=\tfrac{1}{2}(1-\mu)e_{0}+\tfrac{1}{2}(1+\mu)e_{3}=\begin{pmatrix}1&0\cr
0&-\mu\end{pmatrix},$

we have

$\displaystyle(g,x^{M})=(\Psi\cdot\hat{g},\Psi\mathfrak{p})=(\hat{g},\mathfrak{p})=\mu|\psi|^{2}-1,$

and thus

$\displaystyle\begin{split}X^{\pm}&=\Psi\cdot C^{\pm}\\\ \textrm{with}\quad
C^{\pm}&=\begin{pmatrix}1&0\\\ 0&-\mu\end{pmatrix}+\frac{\mu\pm
1}{1-\mu|\psi|^{2}}\begin{pmatrix}|\psi|^{2}&\psi\\\
\overline{\psi}&1\end{pmatrix}=\frac{1}{1-\mu|\psi|^{2}}\begin{pmatrix}1\pm|\psi|^{2}&(\mu\pm
1)\psi\\\ (\mu\pm 1)\overline{\psi}&\pm
1+\mu^{2}|\psi|^{2}\end{pmatrix}.\end{split}$ (19)

We see this representation in Figure 4 where discrete holomorphic functions
are used to obtain pairs of discrete BrLW/BiLW surfaces in hyperbolic and de
Sitter space.

(a) A discrete cmc $1$ surface in $\mathbb{H}^{3}$ (left, see 2(a)) and a
discrete surface of constant harmonic mean curvature $1$ in $\mathbb{S}^{2,1}$
(right)

(b) A discrete cmc $1$ surface in $\mathbb{S}^{2,1}$ (right) and a discrete
surface of constant harmonic mean curvature $1$ in $\mathbb{H}^{3}$ (left)

(c) Two discrete intrinsically flat surfaces in $\mathbb{H}^{3}$ (left) and
$\mathbb{S}^{2,1}$ (right)

Figure 4. Examples of discrete BrLW/BiLW surfaces in $\mathbb{H}^{3}$ or
$\mathbb{S}^{2,1}$ which form a pair of discrete surface and hyperbolic Gauss
map (see Remark 4.7). All examples are created using the representation (19)
with the discrete power function as data. We show surfaces in hyperbolic space
in the Poincaré ball model and surfaces in de Sitter space in the hollow ball
model.

To see that we have recovered the representation of linear Weingarten surfaces
given in [29, Prop 5.1], recall the Weierstrass representation formulas for
discrete BrLW and BiLW surfaces, as given there: Choose $t$ and $\lambda$ so
that

$\displaystyle\mathcal{T}_{i}:=1+t|\psi_{i}|^{2}\neq 0,\quad\textrm{and}\quad
1-\frac{\lambda}{m_{ij}}\neq 0.$

We define $E:\mathbb{Z}^{2}\to SL(2,\mathbb{C})$ by integration of

$\displaystyle
E_{i}^{-1}E_{j}=\frac{1}{\sqrt{1-{\lambda}{m_{ij}}}}\begin{pmatrix}1&\psi_{j}-\psi_{i}\\\
\frac{\lambda}{m_{ij}(\psi_{j}-\psi_{i})}&1\end{pmatrix},$

and another matrix-valued map

$\displaystyle L_{i}=\begin{pmatrix}0&\sqrt{\mathcal{T}_{i}}\\\
-\frac{1}{\sqrt{\mathcal{T}_{i}}}&-\frac{t\overline{\psi}_{i}}{\sqrt{\mathcal{T}_{i}}}\end{pmatrix}.$

Then, we may define two discrete surfaces
$(f^{+},f^{-}):\mathbb{Z}^{2}\to\mathbb{H}^{3}\times\mathbb{S}^{2,1}$ by

$\displaystyle
f_{i}^{\pm}=\operatorname{sgn}(\mathcal{T}_{i})E_{i}L_{i}\begin{pmatrix}1&0\\\
0&\pm 1\end{pmatrix}\left(E_{i}L_{i}\right)^{\ast},$

which are BrLW and BiLW respectively.

The map $\widetilde{\Psi}$, defined by

$\displaystyle\widetilde{\Psi}=E\begin{pmatrix}1&-\psi\\\ 0&1\end{pmatrix},$

satisfies

$\displaystyle\widetilde{\Psi}_{i}^{-1}\widetilde{\Psi}_{j}=\begin{pmatrix}1&\psi_{i}\cr
0&1\end{pmatrix}\left(E_{i}^{-1}E_{j}\right)\begin{pmatrix}1&-\psi_{j}\cr
0&1\end{pmatrix}=B^{m}(\lambda)_{ij}$

which is the system (17) for $\lambda=-m$. Hence, under the right initial
conditions, $\Psi=\widetilde{\Psi}$ and the discrete surfaces $f^{\pm}$ are
given as

$\displaystyle
f^{\pm}=\operatorname{sgn}(\mathcal{T})\Psi\cdot\left[\begin{pmatrix}1&\psi\\\
0&1\end{pmatrix}L\right]\begin{pmatrix}1&0\\\ 0&\pm
1\end{pmatrix}\left[\begin{pmatrix}1&\psi\\\ 0&1\end{pmatrix}L\right]^{\ast}.$

We compute

$\displaystyle\left[\begin{pmatrix}1&\psi\cr
0&1\end{pmatrix}L\right]\begin{pmatrix}1&0\cr 0&\pm
1\end{pmatrix}\left[\begin{pmatrix}1&\psi\cr
0&1\end{pmatrix}L\right]^{\ast}=\frac{1}{|\mathcal{T}|}\begin{pmatrix}|\psi|^{2}\pm
1&\psi(1\mp t)\cr\overline{\psi}(1\mp t)&1\pm t^{2}|\psi|^{2}\end{pmatrix},$

whereby

$\displaystyle
f^{\pm}=\Psi\cdot\frac{1}{1+t|\psi|^{2}}\begin{pmatrix}|\psi|^{2}\pm
1&\psi(1\mp t)\\\ \overline{\psi}(1\mp t)&1\pm t^{2}|\psi|^{2}\end{pmatrix},$

which coincides with (19) under $t=-\mu$ up to the sign ambiguity mentioned in
Remark 4.7.

We summarize this result in the following theorem:

######  Theorem 4.9.

The Weierstrass-type representations for discrete BrLW surfaces in hyperbolic
space $\mathbb{H}^{3}$ and discrete BiLW surfaces in de Sitter space
$\mathbb{S}^{2,1}$ can be viewed as certain applications of the $\Omega$-dual
transformation to a prescribed discrete lightlike Gauss map.

## Acknowledgements

The authors would like to thank Joseph Cho for fruitful discussions during an
impromptu stay in Kobe that sparked many ideas in this paper. Furthermore, we
gratefully acknowledge financial support from the FWF research project
P28427-N35 "Non-rigidity and symmetry breaking" and the JSPS/FWF Joint Project
I3809-N32 "Geometric shape generation". The first author was also supported by
the GNSAGA of INdAM and the MIUR grant "Dipartimenti di Eccellenza" 2018–2022,
CUP: E11G18000350001, DISMA, Politecnico di Torino. The third author was
partly supported by JSPS KAKENHI Grant Numbers JP18H04489, JP19J02034,
JP20K14314, JP20H01801, JP20K03585, and Osaka City University Advanced
Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and
Theoretical Physics JPMXP0619217849).

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Mason Pember
Dipartimento di Matematica

Politecnico di Torino

Corso Duca degli Abruzzi 24

I-10129 Torino, Italy

<EMAIL_ADDRESS>
Denis Polly
Institute of discrete Mathematics and Geometry

TU Vienna

Wiedner Hauptstraße 8-10/104

A-1040 Vienna, Austria

<EMAIL_ADDRESS>
Masashi Yasumoto
Institute of Mathematics for Industry

Kyushu University

744 Motooka, Nishi-ku

Fukuoka 819-0395, Japan

<EMAIL_ADDRESS>