# Cuspidal crosscaps and folded singularities on a maxface and a minface

Rivu Bardhan, Anu Dhochak, Pradip Kumar  Department of Mathematics, Shiv Nadar
University, Dadri 201314, Uttar Pradesh, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For a given zero mean curvature surface $X$ (in the Lorentz Minkowski space)
having folded singularity, we construct a family of maxface and minface,
having increasing cuspidal crosscaps, converging to $X$. We include a general
discussion of this.

###### Key words and phrases:

mix-type zmc surface, zero mean curvature, maxface, minface

###### 2020 Mathematics Subject Classification:

53A35

##  1\. Introduction

Space-like (time-like) zero mean curvature surfaces in the Lorentz-Minkowski
space $\mathbb{E}_{1}^{3}$ are locally area maximizing (minimizing
respectively) surfaces. These are similar to the minimal surface in
$\mathbb{R}^{3}$. But in contrast to the minimal surface in $\mathbb{R}^{3}$,
these surfaces appear with non isolated singularities [shintaro1],
[Estudillo1992], [fujimori2015zero], [KOBAYASHI1983]. The generalized maximal
(time-like minimal) immersion having only non-isolated singularities are
called maxface (minface respectively).

These non-isolated singularities are of various types. As front or frontal, on
these surfaces, we see singularities as cuspidal edge, cuspidal crosscaps,
swallowtails, etc. There are other space-like (time-like) zero mean curvature
surfaces that are neither front nor frontal but appear with other types of
singularities, such as folded singularities, cone-like, etc.

In [Kim2006], the authors gave an example of a family of maxface $X_{n}$ for
each natural number $n$, that has swallowtails in an increasing order.
Further, in [Fujimori2009], the authors talked about Trinoids (a maxface) with
swallowtails whose computer graphics look cone-like singularities. Moreover,
they remarked that the computer graph of many swallowtails together looks
cone-like. In [anupradip], the authors justified this phenomenon by producing
a sequence of maxface with an increasing number of swallowtails, converging to
the maxface with cone-like singularity.

We see similar phenomena (at least graphically) in cuspidal crosscaps. The
graph of cuspidal crosscaps looks like the folded singularity. Similar to
swallowtails and cone-like discussed in [anupradip], we can ask a question:
given a maxface (minface) having folded singularity, does there exist a family
of maxface (minface respectively) having an increasing number of cuspidal
crosscaps and ‘converging’ to the former one?

In this article, in section 4, we give such a family and prove the following:

######  Theorem (Theorem 4.1).

Let X be a maxface (minface) having folded singularity. Then there is a
sequence $X_{n}$ of maxfaces (sequence $Y_{n}$ of minfaces) having an
increasing number of cuspidal crosscaps, and the sequence ‘converges’ (in the
norm $\|.\|_{\Omega}$) to X.

To construct such a family, we will find the condition on the singular Björing
data to have a cuspidal crosscap. For the maxface, such conditions are already
derived in [SaiPradip]. In this article, in section 3, we find similar
conditions on the singular Björling data for the minface, such that they have
a cuspidal edge, swallowtails, and cuspidal crosscaps.

Besides finding the sequence above, such conditions may be useful in various
situations. For example, these are required initial conditions to find
interpolating surfaces containing given two curves with the prescribed nature
of singularities.

##  2\. Preliminaries

The Lorentz Minkowski space is a vector space $\mathbb{R}^{3}$ with the metric
$ds^{2}:=dx^{2}+dy^{2}-dz^{2}$; we denote it by $\mathbb{E}_{1}^{3}$. Space-
like zero mean curvature immersion is called maximal immersion. If we allow
singularities, then we call it a generalized maximal immersion. Moreover, if
singularities are due to those points where the limiting tangent plane
contains a light-like vector, then we call such a generalized maximal
immersion as maxface [UMEHARA2006]. In the following, we will discuss two
methods to construct the generalized maximal immersion [Estudillo1992,
UMEHARA2006, RPR2016, Kim2007].

###  2.1. Weierstrass-Enneper representation for the generalized maximal
immersion

Let $M^{2}$ be a Riemann surface. Let $g$ be a meromorphic function and
$\omega$ a holomorphic 1-form on $M^{2}$ such that
$Re\int_{\gamma}\left(\frac{1}{2}(1+g^{2}),\frac{i}{2}(1-g^{2}),-g\right)\omega=0$
for all loops $\gamma$ on $M^{2}$. Then the map

$f(z):=Re\int_{z_{0}}^{z}\left(\frac{1}{2}(1+g^{2}),\frac{i}{2}(1-g^{2}),-g\right)\omega$

is well-defined and gives a generalized maximal immersion [Estudillo1992,
UMEHARA2006]. Moreover, any generalized maximal immersion can be written in
this manner.

###  2.2. Singular Björling representation for the generalized maximal
immersion

Let $\gamma:I\to\mathbb{E}_{1}^{3}$ be a real analytic null curve and
$L:I\to\mathbb{E}_{1}^{3}$ be a real analytic null vector field such that
$\langle\gamma^{\prime}(u),L(u)\rangle=0$ and $\gamma^{\prime}\not\equiv
0\text{ or }L\not\equiv 0$. Let $g(z)$ be the analytic extension of

$g(t):=\begin{cases}\frac{\gamma_{1}^{\prime}(t)+i\gamma_{2}^{\prime}(t)}{\gamma_{3}^{\prime}(t)}\;\text{
if }\gamma_{3}^{\prime}(t)\not\equiv 0\\\
\frac{L_{1}(t)+iL_{2}(t)}{L_{3}(t)}\text{ if }L_{3}(t)\not\equiv
0.\end{cases}$

defined on $U\subset\mathbb{C}$, $I\subset U$. If $|g(z)|\not\equiv 1$, then
the map

(2.1)
$X_{max,\gamma,L}(u+iv):=\frac{1}{2}(\gamma(u+iv)+\gamma(u-iv))-\frac{i}{2}\int_{u-iv}^{u+iv}L(w)dw$

is the generalized maximal surface with $u-v$ as a conformal parameter and
$X_{max,\gamma,L}(u,0)=\gamma(u),\;\frac{\partial}{\partial
v}\lvert_{(u,0)}X_{max,\gamma,L}=L(u)$ [RPR2016, Kim2007].

Similar to the maximal immersion, we have time-like zero mean curvature
immersion and minface. Below we will recall the Weierstrass Enneper
representation and the Björling formula for a minface [shintaro1,
KYKSEYY2011].

###  2.3. Weierstrass-Enneper representation for minface [shintaro1]

Let $M^{2}$ be a smooth manifold. A smooth map $X:M^{2}\to\mathbb{E}_{1}^{3}$
is a minface if and only if at each point of $M^{2}$, there exists a local
coordinate system $(u,v)$ in a domain $U$, the functions $g_{1}(u)$,
$g_{2}(v)$, and the 1-forms $\omega_{1}=\hat{\omega_{1}}du$,
$\omega_{2}=\hat{\omega_{2}}dv$ such that

  1. (1)

$g_{1}g_{2}\not\equiv 1$ on an open dense subset of $U$

  2. (2)

$\hat{\omega_{1}}\neq 0$, $\hat{\omega_{2}}\neq 0$ at each point on $U$.

Moreover, $X$ can be decomposed into two regular null curves.

(2.2)
$X(u,v)=\frac{1}{2}\left(\int_{u_{0}}^{u}(2g_{1},1-{g_{1}}^{2},-1-{g_{1}}^{2})\omega_{1}+\int_{v_{0}}^{v}(-2g_{2},1-{g_{2}}^{2},1+{g_{2}}^{2})\omega_{2}\right)+X(u_{0},v_{0}).$

The quadruple $(g_{1},g_{2},\omega_{1},\omega_{2})$ is called the real
Weierstrass data. The set of singular points on $U$ of a minface $X$
correspond to the set $\\{(u,v)\in U:g_{1}(u)g_{2}(v)=1\\}.$

###  2.4. Singular Björling representation for the generalized time-like
minimal surface

Let $\gamma:(a,b)\to\mathbb{E}_{1}^{3}$ be a smooth null curve, and
$L:(a,b)\to\mathbb{E}_{1}^{3}$ be a smooth null vector field such that
$\langle\gamma^{\prime}(t),L(t)\rangle\equiv 0$ and $\gamma^{\prime}\not\equiv
0\text{ or }L\not\equiv 0\text{ and }\gamma^{\prime}\not\equiv\pm L$. Then the
map

(2.3)
$X_{min,\gamma,L}(u,v):=\frac{1}{2}(\gamma(u+v)+\gamma(u-v))+\frac{1}{2}\int_{u-v}^{u+v}L(s)ds,$

defined on $\Diamond_{a,b}:=\\{(u,v)\in\mathbb{R}^{2}:a<u+v<b,\text{ and
}a<u-v<b\\}$ is a generalized time-like minimal surface. Furthermore,
$X_{min,\gamma,L}(u,0)=\gamma(u),\;\frac{\partial}{\partial
v}X_{min,\gamma,L}(u,0)=L(u),\;\;\forall u\in(a,b).$

Both maxface and minface admit various singularities. In the table (1), we
mention the criterion given in [SaiPradip] on the singular Björling data for
maxface having various singularities. But for the minface we do not have a
criterion to check the type of singularity in terms of the Björling data till
now.

Nature of singularities at $p$ Function’s value at $p$ | $\gamma^{\prime}$ | $L$ | $\gamma^{\prime\prime}$ | $\gamma^{\prime\prime\prime}$ | $L^{\prime}$ | $L^{\prime\prime}$ | $\gamma_{1}^{\prime}\gamma_{2}^{\prime\prime}-\gamma_{1}^{\prime\prime}\gamma_{2}^{\prime}$ | $L_{1}L_{2}^{\prime}-L_{1}^{\prime}L_{2}$
---|---|---|---|---|---|---|---|---
Cuspidal-edge | $\neq 0$ | $\neq 0$ | – | – | – | – | $\neq 0$ | $\neq 0$
Swallowtails | $=0$ | $\neq 0$ | $\neq 0$ | – | – | – | – | $\neq 0$
Cuspidal-Crosscaps | $\neq 0$ | $=0$ | – | – | $\neq 0$ | – | $\neq 0$ | –
Table 1. Criterion on the Singular Björling data for maxface

For the minface, the criterion to check the type of singularity in terms of
the Weierstrass data is given in Fact 4.1 of [shintaro1]. We recall it here.
The minface with Weierstrass data
$(g_{1},\hat{\omega_{1}},g_{2},\hat{\omega_{2}})$ has

  1. (1)

Cuspidal edge at $p$ if and only if at $p$,

(2.4)
$\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}-\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}\neq
0,\;\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}+\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}\neq
0.$

  2. (2)

Cuspidal crosscap at $p$ if and only if at $p$,

(2.5)
$\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}-\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}=0,\;\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}+\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}\neq
0,\;\text{ and
}\left(\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}\right)^{\prime}\frac{g_{2}^{\prime}}{g_{2}}+\left(\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}\right)^{\prime}\frac{g_{1}^{\prime}}{g_{1}}\neq
0.$

  3. (3)

Swallowtails at $p$ if and only if at $p$,

(2.6)
$\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}-\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}\neq
0,\;\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}+\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}=0,\;\text{
and
}\left(\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}\right)^{\prime}\frac{g_{2}^{\prime}}{g_{2}}+\left(\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}\right)^{\prime}\frac{g_{1}^{\prime}}{g_{1}}\neq
0.$

The above singularities of a maxface and minface appear when we consider these
as front or frontal. Now we discuss another type of singularity called folded
singularity that appears on maxface and minface but not as frontal or front.

######  Definition 2.1 (Folded singularity).

[Kim2007]. The $X_{max,\gamma,L}$ and $X_{min,\gamma,L}$ have folded
singularities along $\gamma$ if $L=0$ and $\gamma$ is non degenerate curve.

There are many other singularities, but we recalled only selected one, as per
the required discussion of this article. For more details on the topic, we
refer [PSV] and [KYKSEYY2011].

##  3\. Conditions on the singular Björling data for minface

We start with $\\{\gamma,L\\}$ the singular Björling data such that

  * •

At those points where $\gamma^{\prime}\neq 0$, we have
$\gamma_{2}^{\prime}\neq\pm\gamma_{3}^{\prime}$, and

  * •

A those points where $L\neq 0$, we have $L_{2}\neq\pm L_{3}$.

Let $X_{min,\gamma,L}$ be the generalized time-like minimal surface as in
(2.3). Below we will find the Weierstrass quadruple for $X_{min,\gamma L}$ in
terms of $\gamma$ and $L$.

###  3.1. Relation between the singular Björling data and Weierstrass
quadruple for the minface

After changing the coordinate $(u,v)$ to $(t,s)$ by $s=u+v$ and $t=u-v$ and
for some $x_{0}\in(a,b)$, $X_{min,\gamma,L}$ as in (2.3), changes to

(3.1)
$\tilde{X}(t,s)=\frac{1}{2}\left(\gamma(t)+\int_{t}^{x_{0}}L(\eta)d\eta\right)+\frac{1}{2}\left(\gamma(s)+\int_{x_{0}}^{s}L(\eta)d\eta\right).$

We take
$\alpha(t):=\frac{1}{2}\left(\gamma(t)+\int_{t}^{x_{0}}L(\eta)d\eta\right)$
and
$\beta(s):=\frac{1}{2}\left(\gamma(s)+\int_{x_{0}}^{s}L(\eta)d\eta\right)$. It
turns out that both $\alpha$ and $\beta$ are null curves.
$\alpha^{\prime}(t)=\frac{1}{2}(\gamma^{\prime}(t)-L(t))$, and
$\beta^{\prime}(s)=\frac{1}{2}(\gamma^{\prime}(s)+L(s))$.
For those point $t$, where $\gamma^{\prime}\neq 0$, we have
$L(t)=c(t)\gamma^{\prime}(t)$. This gives

$\displaystyle\alpha_{2}^{\prime}(t)-\alpha_{3}^{\prime}(t)=\frac{1}{2}(\gamma_{2}^{\prime}(t)-\gamma_{3}^{\prime}(t))(1-c(t))$
$\displaystyle\beta_{2}^{\prime}(s)+\beta_{3}^{\prime}(s)=\frac{1}{2}(\gamma_{2}^{\prime}(s)+\gamma_{3}^{\prime}(s))(1+c(s)).$

Similarly at those points where $L\neq 0$, $\gamma^{\prime}(t)=d(t)L(t)$, we
get

$\displaystyle\alpha_{2}^{\prime}(t)-\alpha_{3}^{\prime}(t)=\frac{1}{2}(L_{2}(t)-L_{3}(t))(-1+d(t))$
$\displaystyle\beta_{2}^{\prime}(s)+\beta_{3}^{\prime}(s)=\frac{1}{2}(L_{2}(s)+L_{3}(s))(1+d(s)).$

The conditions on the singular Björling data as mentioned in the starting of
this section and in the subsection 2.4, imply that for all points
$\alpha_{2}^{\prime}-\alpha_{3}^{\prime}\neq 0$,
$\beta_{2}^{\prime}+\beta_{3}^{\prime}\neq 0$.

Let $\left(g_{1},g_{2},\hat{\omega_{1}},\hat{\omega_{2}}\right)$ be the
Weierstrass quadruple for $\tilde{X}$. $\tilde{X}$ has the representation as
in (2.2). By comparing the two representations (2.2) and (3.1), we get
components of $\alpha^{\prime}$ and $\beta^{\prime}$ as the following:

(3.2)
$\alpha_{1}^{\prime}(u)=2g_{1}(u)\hat{\omega_{1}}(u),\;\;\alpha_{2}^{\prime}(u)=(1-{g_{1}}^{2}(u))\hat{\omega_{1}}(u),\;\;\alpha_{3}^{\prime}(u)=(-1-{g_{1}}^{2}(u))\hat{\omega_{1}}(u)$
(3.3)
$\beta_{1}^{\prime}(v)=-2g_{2}(v)\hat{\omega_{2}}(v),\;\;\beta_{2}^{\prime}(v)=(1-{g_{2}}^{2}(v))\hat{\omega_{2}}(v),\;\;\beta_{3}^{\prime}(v)=(1+{g_{2}}^{2}(v))\hat{\omega_{2}}(v).$

From (3.2) and since $\hat{\omega_{1}}\neq 0$, we get
$g_{1}=\frac{\alpha_{1}^{\prime}}{2\hat{\omega_{1}}}$. Also
$\alpha_{2}^{\prime}=(1-g_{1}^{2})\hat{\omega_{1}}$. Solving this, we get
$\hat{\omega_{1}}=\frac{\alpha_{2}^{\prime}\pm\alpha_{3}^{\prime}}{2}$ and
then
$g_{1}=\frac{\alpha_{1}^{\prime}}{\alpha_{2}^{\prime}\pm\alpha_{3}^{\prime}}$.

Now for the case when
$g_{1}=\frac{\alpha_{1}^{\prime}}{\alpha_{2}^{\prime}-\alpha_{3}^{\prime}}$
and $\hat{\omega_{1}}=\frac{\alpha_{2}^{\prime}-\alpha_{3}^{\prime}}{2}$, we
get

$-(1+g_{1}^{2})\hat{\omega_{1}}=\frac{-1}{2}\left[\frac{\alpha_{2}^{\prime
2}+\alpha_{3}^{\prime
2}-2\alpha_{2}^{\prime}\alpha_{3}^{\prime}+\alpha_{1}^{\prime
2}}{\alpha_{2}^{\prime}-\alpha_{3}^{\prime}}\right]=\alpha_{3}^{\prime}.$

For other choices of $\hat{\omega_{1}},g_{1}$, we see that
$-(1+g_{1}^{2})\hat{\omega_{1}}\neq\alpha_{3}^{\prime}$. Therefore we must
have

$g_{1}=\frac{\alpha_{1}^{\prime}}{\alpha_{2}^{\prime}-\alpha_{3}^{\prime}}\;\text{and}\;\hat{\omega_{1}}=\frac{\alpha_{2}^{\prime}-\alpha_{3}^{\prime}}{2}.$

Similarly we have

$g_{2}(v)=\frac{-\beta_{1}^{\prime}}{\beta_{2}^{\prime}+\beta_{3}^{\prime}}\;\text{and}\;\hat{\omega_{2}}(v)=\frac{\beta_{2}^{\prime}+\beta_{3}^{\prime}}{2}.$

Now to get the Weierstrass quadruple in terms of the singular Björling data,
we replace the values of $\alpha$ and $\beta$ in terms of $\gamma$ and $L$. We
will do this case by case.

If at $u$, $\gamma^{\prime}\neq 0$, then $L(u)=c(u)\gamma^{\prime}(u)$ for the
real function $c=\frac{L_{3}}{\gamma_{3}^{\prime}}$ and we get,

(3.4)
$\hat{\omega_{1}}=\frac{\gamma_{2}^{\prime}-\gamma_{3}^{\prime}}{4}(1-c),\;\;g_{1}=-\frac{\gamma_{2}^{\prime}+\gamma_{3}^{\prime}}{\gamma_{1}^{\prime}},\;\;\hat{\omega_{2}}=\frac{\gamma_{2}^{\prime}+\gamma_{3}^{\prime}}{4}(1+c),\;\;g_{2}=\frac{\gamma_{2}^{\prime}-\gamma_{3}^{\prime}}{\gamma_{1}^{\prime}}.$

We recall here the fact that at points where $\gamma^{\prime}\neq 0$, we have
$\gamma_{3}^{\prime}\neq 0$, and as
$\gamma_{2}^{\prime}\neq\gamma_{3}^{\prime}$, and $\gamma$ a null curve, this
gives $\gamma_{1}^{\prime}\neq 0$.

Similarly when $L\neq 0$, with $d=\frac{\gamma_{3}^{\prime}}{L_{3}}$, we have

(3.5)
$\hat{\omega_{1}}=\frac{L_{2}-L_{3}}{4}(-1+d),\;\;g_{1}=-\frac{L_{2}+L_{3}}{L_{1}},\;\;\hat{\omega_{2}}=\frac{L_{2}+L_{3}}{4}(1+d),\;\;g_{2}=\frac{L_{2}-L_{3}}{L_{1}}.$

Thus, we found the Weierstrass quadruple of $X_{min,\gamma,L}$ in terms of
$\\{\gamma,L\\}$. In the next subsection we will convert conditions in terms
of the singular Björling data so that a time-like minimal surface has various
singularities.

###  3.2. Conditions on the singular Björling data of minface for various
singularity

Let $\\{\gamma\text{, }L\\}$ be the singular Björling data for the generalized
time-like minimal surface $X_{min,\gamma,L}$, as defined in the section 2.4.
We denote

$\delta_{1}:=\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}\text{ ,
}\delta_{2}:=\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}\text{ ,
}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime}):=\gamma_{2}^{\prime}\gamma_{3}^{\prime\prime}-\gamma_{3}^{\prime}\gamma_{2}^{\prime\prime}\text{
, }D(L_{23}L_{23}^{\prime}):=L_{2}L_{3}^{\prime}-L_{3}L_{2}^{\prime}.$

We again start with the case when $\gamma^{\prime}\neq 0$ and
$L=c\gamma^{\prime}$. Using (3.4) we have

$\displaystyle\delta_{1}$
$\displaystyle=\frac{4g_{1}^{\prime}}{g_{1}^{2}(1-c)(\gamma_{2}^{\prime}-\gamma_{3}^{\prime})}$
$\displaystyle=\frac{4[(\gamma_{2}^{\prime\prime}+\gamma_{3}^{\prime\prime})\gamma_{1}^{\prime}-\gamma_{1}^{\prime\prime}(\gamma_{2}^{\prime}+\gamma_{3}^{\prime})]}{(\gamma_{1}^{\prime
2})(1-c)(\gamma_{2}^{\prime}+\gamma_{3}^{\prime})}$
$\displaystyle=\frac{4}{\gamma_{1}^{\prime
3}(1-c)}\left[-\gamma_{1}^{\prime\prime}\gamma_{1}^{\prime}-\gamma_{2}^{\prime\prime}\gamma_{2}^{\prime}+\gamma_{3}^{\prime\prime}\gamma_{3}^{\prime}+\gamma_{2}^{\prime}\gamma_{3}^{\prime\prime}-\gamma_{3}^{\prime}\gamma_{3}^{\prime\prime}\right]$
$\displaystyle=\frac{-4}{\gamma_{1}^{\prime
3}(1-c)}\left[\langle\gamma^{\prime},\gamma^{\prime\prime}\rangle+D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\right].$

Since $\gamma$ is a singular curve,
$\langle\gamma^{\prime},\gamma^{\prime\prime}\rangle=0$. This gives
$\delta_{1}=-4\frac{D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})}{\gamma_{1}^{\prime
3}(1-c)}$.

Moreover at those points where $c(u)=0$, we have

$\delta_{1}^{\prime}=-4\frac{\gamma_{1}^{\prime}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime\prime})-D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})(3\gamma_{1}^{\prime\prime}-\gamma_{1}^{\prime}c^{\prime})}{\gamma_{1}^{\prime
4}}.$

A similar calculation gives
$\delta_{2}=-4\frac{D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})}{\gamma_{1}^{\prime
3}(1+c)}$, and at those points where $c(u)=0$,

$\delta_{2}^{\prime}=-4\frac{\gamma_{1}^{\prime}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime\prime})-D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})(3\gamma_{1}^{\prime\prime}+\gamma_{1}^{\prime}c^{\prime})}{\gamma_{1}^{\prime
4}}.$

As $c(u)=\frac{L_{3}(u)}{\gamma_{3}^{\prime}(u)}$, we get

$\displaystyle\delta_{1}+\delta_{2}$
$\displaystyle=\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}+\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}$
$\displaystyle=\frac{-8}{\gamma_{1}^{\prime
3}(1-c^{2})}\left[\langle\gamma^{\prime},\gamma^{\prime\prime}\rangle
c+D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\right]$
$\displaystyle=\frac{-8\gamma_{3}^{\prime}}{\gamma_{1}^{\prime
3}(\gamma_{3}^{\prime 2}-L_{3}^{\prime
2})}\left[\langle\gamma^{\prime},\gamma^{\prime\prime}\rangle
L_{3}+\gamma_{3}^{\prime}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\right]$
$\displaystyle=\frac{-8\gamma_{3}^{\prime
2}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})}{\gamma_{1}^{\prime
3}(\gamma_{3}^{\prime 2}-L_{3}^{\prime 2})},\text{ and }$
$\displaystyle\delta_{1}-\delta_{2}$
$\displaystyle=\frac{g_{1}^{\prime}}{g_{1}^{2}\hat{\omega_{1}}}-\frac{g_{2}^{\prime}}{g_{2}^{2}\hat{\omega_{2}}}=\frac{-8\gamma_{3}^{\prime}L_{3}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})}{\gamma_{1}^{\prime
3}(\gamma_{3}^{\prime 2}-L_{3}^{\prime 2})}.$

Below we find the conditions for a generalized time-like minimal surface
having singularities like cuspidal edge, cuspidal crosscaps and swallowtails
in term of singular Björling data $\\{\gamma,L\\}$.

####  3.2.1. Cuspidal edge

Using (2.4), the time-like minimal surface $X_{min,\gamma,L}$ for the data
$\\{\gamma,L\\}$ has cuspidal edge at $u$ if and only if $\gamma^{\prime}\neq
0$, $L\neq 0$, $D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\neq 0$.

####  3.2.2. Cuspidal crosscaps

At those point where $\gamma^{\prime}\neq 0$, $\delta_{1}-\delta_{2}=0,\text{
and }\delta_{1}+\delta_{2}\neq 0$ give

(3.6) $\gamma_{3}^{\prime}\neq
0,\;\;L_{3}=0,\;\;D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\neq 0.$

Moreover at those point where (3.6) satisfies, we have

$\delta_{1}^{\prime}\frac{g_{2}^{\prime}}{g_{2}}+\delta_{2}^{\prime}\frac{g_{1}^{\prime}}{g_{1}}=\frac{g_{1}^{\prime}}{g_{1}}(\delta_{2}^{\prime}-\delta_{1}^{\prime})=-\frac{4}{\gamma_{1}^{\prime
6}}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\left[D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})(2c^{\prime}\gamma_{1}^{\prime})\right]$

and
$c^{\prime}=\frac{L_{3}^{\prime}\gamma_{3}^{\prime}-L_{3}\gamma_{3}^{\prime\prime}}{\gamma_{3}^{\prime
2}}.$ Therefore

$\delta_{1}^{\prime}\frac{g_{2}^{\prime}}{g_{2}}+\delta_{2}^{\prime}\frac{g_{1}^{\prime}}{g_{1}}\neq
0\iff c^{\prime}\neq 0\iff L^{\prime}\neq 0.$

Using (2.5), at $u$, where $\gamma^{\prime}\neq 0$, the time-like minimal
surface $X_{min,\gamma,L}$ for the data $\\{\gamma,L\\}$ has cuspidal crosscap
if and only if $\gamma^{\prime}\neq 0,\;\;L=0,\;\;L^{\prime}\neq 0,\text{ and
}D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\neq 0.$

####  3.2.3. Swallowtails

Similar to the cuspidal crossscaps, at $u$, where $\gamma^{\prime}\neq 0$, the
time-like minimal surface $X_{min,\gamma,L}$ for the data $\\{\gamma,L\\}$ has
swallowtails if and only if $\gamma^{\prime}(u)=0,\;\;L(u)\neq
0,\;\;\gamma^{\prime\prime}(u)\neq 0\text{ and }D(L_{23},L_{23}^{\prime})\neq
0.$

For a non-degenerate curve (the curve is non-degenerate if and only if for all
$u\in I$, $D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\neq 0$ or
$D(\gamma_{12}^{\prime},\gamma_{12}^{\prime\prime})\neq 0$), these conditions
turn out to be similar to the one given in the table (1) for the maxface.

##  4\. Family of maxfaces and minfaces

We start this section with an example: Helicoid. This is an example of how the
time-like and space-like Helicoid patches smoothly along the folded
singularity. For details, we refer to [KRSUY2005].

###  4.1. Mix-type Helicoid and a sequence

Let $\gamma:I\to\mathbb{E}_{1}^{3};\;\;\gamma(t):=(\cos{t},\sin{t},t)$. With
$L=0$, $\\{\gamma,L\\}$ turns out to be the singular Björling data for both
generalized time-like minimal surface and generalized space-like maximal
surface.

For this data, we have the following generalized space-like maximal surface

(4.1) $X_{max,\gamma,L}(u,v)=(\cos{u}\cosh{v},\sin{u}\cosh{v},u)$

on $\Omega$ and $\gamma$ is its singular curve corresponding to the points
$v=0$. Also, for the same data, we have a generalized time-like minimal
surface as following

(4.2) $X_{min,\gamma,L}(u,v)=(\cos u\cos v,\sin u\cos v,u).$

The generalized immersions as in (4.1), (4.2) are parametrizations of the
helicoid of type space-like and time-like, respectively. We see that both
these surfaces contain $\gamma$ and have folded singularity on $I$.

Without loss of generality we assume that $I=(0,2\pi)$ and let $r>0$ be such
that $\overline{\Omega},\overline{\Diamond}\subset B(0,r)$. With this $r$ and
$n\geq 1$, we take the singular Björling data as

(4.3)
$\left\\{\gamma,\gamma^{\prime}\frac{1}{n(r+1)^{n}}\prod_{i=1}^{n}(w-\frac{1}{i})\right\\}.$

For each $n\in\mathbb{N}$, let $X_{max,\gamma,L,n}$, be the generalized
maximal immersion as in (2.1) and $X_{min,\gamma,L,n}$ be the generalized
time-like minimal immersion as in (2.3). This defines a pair
$\\{X_{max,\gamma,L,n},X_{min,\gamma,L,n}\\}$ of generalized space-like
maximal surfaces and the generalized time-like minimal surface containing
$\gamma$ as a singular curve at $v=0$.

Using the condition we found in section 3, it is easy to verify that both the
generalized maximal immersions and generalized minimal immersions
$\\{X_{max,\gamma,L,n},X_{min,\gamma,L,n}\\}$ has $n$ cuspidal crosscaps at
$t=\frac{1}{k}$. Moreover corresponding cuspidal crosscaps are
$\\{(\cos{\frac{1}{k}},\sin{\frac{1}{k}},\frac{1}{k})\\}_{k=1}^{n}\subset\gamma(I)$.

Helicoid is an example of a mix-type zero mean curvature surface, where the
casual character changes along the folded singularity (see [KYKSEYY2011]). We
have generated a sequence of immersions
$\\{X_{max,\gamma,L,n},X_{min,\gamma,L,n}\\}_{n\in\mathbb{N}}$ with the
increasing number of cuspidal crosscaps. For each $n$, $X_{max,\gamma,L,n}$
and $X_{min,\gamma,L,n}$ are joined along $\gamma$. Moreover the sequence
$\\{X_{max,\gamma,L,n}\\}$ ‘converges’ towards the mix-type helicoid (that is
smoothly patched along a folded singularity) as $n\rightarrow\infty$.

In the next subsection, we will start with a generalized mixed-type surface
that is smoothly joined along a folded singularity (for example: helicoid).
Next, we will find a sequence of mixed-type surfaces joined along a null curve
with increasing cuspidal crosscaps and this sequence ‘converges’ to the
surface we started with.

###  4.2. Family for a general mix-type zero mean curvature surfaces

Authors in [anupradip, SaiPradip] have defined and discussed the convergence
of maxfaces in a particular norm. We would recall the norm here. Let
$\Omega\subset\mathbb{C}$ be a bounded simply connected domain,
$\overline{\Omega}$ be its closure. Let $X\in
C(\overline{\Omega},\mathbb{R}^{3})$, the space of continuous maps. For each
$z\in\overline{\Omega}$, we define

$\|X(z)\|:={\rm max}\\{\lvert X_{1}(z)\rvert,\lvert X_{2}(z)\rvert,\lvert
X_{3}(z)\rvert\\}\,{\rm
and}\,\|X\|_{\Omega}:=\sup_{z\in{\overline{\Omega}}}\|X(z)\|.$

We start with the singular Björling data $\\{\gamma,L=0\\}$, defined on a
closed and bounded interval $I$, without loss of generality we will assume
$I=[0,1]$. Moreover, $X_{max,\gamma,0}$ and $X_{min,\gamma,0}$ be the
generalized maximal immersion and generalized time-like minimal immersion,
respectively, as in (2.1) and (2.3) defined on some bounded open subsets
$\Omega_{1}$ and $\Omega_{2}$ (of $\mathbb{R}^{2}$) that contains $I$. We also
assume both have folded singularity along $\gamma$. Since $L=0$,
$X_{max,\gamma,0}$ and $X_{min,\gamma,0}$ have folded singularity at $v=0$.
Therefore $\gamma$ is non-degenerate (see definition 2.1 and the curve is non-
degenerate if and only if for all $u\in I$,
$D(\gamma_{23}^{\prime},\gamma_{23}^{\prime\prime})\neq 0$ or
$D(\gamma_{12}^{\prime},\gamma_{12}^{\prime\prime})\neq 0$).

As both $\Omega_{1}$ and $\Omega_{2}$ are bounded and
$I=[0,1]\subset\Omega_{1}\cap\Omega_{2}$, there exists some $r>0$ and
$\Omega$, $\Diamond$ such that
$I\subset\Omega\subset\overline{\Omega_{1}}\subset B(0,r)$, and
$I\subset\Diamond\subset\overline{\Omega_{2}}\subset B(0,r)$. With this $r$
and $n\geq 1$, we take the singular Björling data as in (4.3). For this data,
we have corresponding immersions as follows.

(4.4)
$X_{max,\gamma,L,n}(u,v)=\frac{1}{2}(\gamma(u+iv)+\gamma(u-iv))-\frac{i}{2n(r+1)^{n}}\int_{u-iv}^{u+iv}\gamma^{\prime}(s)\prod_{k=1}^{n}(s-\frac{1}{k})ds$

on $\Omega_{1}$.

(4.5)
$X_{min,\gamma,L,n}(u,v)=\frac{1}{2}(\gamma(u+v)+\gamma(u-v))+\frac{1}{2n(r+1)^{n}}\int_{u-v}^{u+v}\gamma^{\prime}(s)\prod_{k=1}^{n}(s-\frac{1}{k})ds$

on $\Omega_{2}$.

Both the generalized immersions have cuspidal-crosscap singularities at
$\\{\frac{1}{k}:k\in\\{1,2,...,n\\}\\}$ for each $n$. Moreover following the
similar calculations as in [anupradip, SaiPradip], we see that

$\displaystyle\lVert
X_{max,\gamma,L,n}-X_{max,\gamma,0}\rVert_{\Omega_{1}}\leq
M(\lVert\gamma_{n}^{k\prime}-\gamma^{k\prime}\rVert_{\Omega_{1}}+\lVert
L_{n}^{k}-L^{k}\rVert_{\Omega_{1}})\leq M\lVert L_{n}\rVert_{\Omega_{1}}\to
zero.$ $\displaystyle\text{Similarly, we have }\lVert
X_{min,\gamma,L,n}-X_{min,\gamma,0}\rVert_{\Omega_{2}}\leq M^{\prime}\lVert
L_{n}\rVert_{\Omega_{2}}\to zero.$

Thus we summarize all our discussions in this section in the following
theorem.

######  Theorem 4.1.

Let $\\{X_{max},X_{min}\\}$ be a maxface and a minface having a folded
singularity along $\gamma$. Then there exists two sequences
$X_{max,\gamma,L,n}$ and $X_{min,\gamma,L,n}$ of maxfaces and minfaces
respectively such that

  1. (1)

On the trace of $\gamma$ both immersions have an increasing number of cuspidal
crosscaps at the same points.

  2. (2)

$X_{max,\gamma,L,n}$ converges to $X_{max}$ in $\lVert.\rVert_{\Omega_{1}}$
and $X_{min,\gamma,L,n}$ converges to $X_{min}$ in
$\lVert.\rVert_{\Omega_{2}}$.

## References