Classification of Complete Degenerate Stationary Surfaces in ${\mathbb{R}}^{3,1}$ Whose Gauss Maps Are Injective

Abstract

In this paper, we classify the complete degenerate stationary surfaces in ${\mathbb{R}}^{3,1}$ with injective Gauss maps.

1. Introduction

Let $\mathbf{x}:M\to {\mathbb{R}}^{3,1}$ be an oriented surface in the Minkowski space. M is space-like if the induced metric $d{s}^2:=\langle d\mathbf{x},d\mathbf{x}\rangle$ is positive definite everywhere. Moreover, if the mean curvature vector field H vanishes everywhere, M is called a stationary surface. Such surfaces are neither a local minimizer nor maximizer of the functional area. Asperti–Vilhena [1] obtained some explicit examples of maximal surfaces in ${\mathbb{R}}^{3,1}$. Then, Kaya–López [2] provided a new explicit parametrizations of maximal surfaces in ${\mathbb{R}}^{3,1}$, which are the solutions of the Björling problem. The global geometry properties of stationary surfaces in 4-dimensional Minkowski space ${\mathbb{R}}^{3,1}$ have been researched by Ma–Wang–Wang in [3]. They generalized the classical theory of minimal surfaces in ${\mathbb{R}}^3$ to stationary surfaces in ${\mathbb{R}}^{3,1}$. For a stationary surface $M\subset {\mathbb{R}}^{3,1}$, the Weierstrass representation formula can be stated as: [3]:

$$\mathbf{x}=2Re\int ({\psi }_1+{\psi }_2,-i({\psi }_1-{\psi }_2),1-{\psi }_1{\psi }_2,1+{\psi }_1{\psi }_2)dh,$$

where ${\psi }_1,{\psi }_2$ are meromorphic functions on M, and $dh$ is a holomorphic 1-form on M.

A surface in ${\mathbb{R}}^n$ or ${\mathbb{R}}^{n,1}$ is degenerate if the Gauss image lies in a hyperplane of ${\mathbb{CP}}^{n-1}$. Hoffman–Osserman [4] investigated the degenerate minimal surfaces in ${\mathbb{R}}^4$, which are either a complex analytic curve lying fully in ${\mathbb{C}}^2$ or can be derived by a minimal surface in ${\mathbb{R}}^3$. In ${\mathbb{R}}^{3,1}$, Asperti–Vilhena [5] classified the space-like degenerate surfaces. In a previous paper [6], we studied the complete degenerate stationary surfaces in ${\mathbb{R}}^{3,1}$, which can be classified to the totally real type, hyperbolic type, elliptic type and parabolic type (see Definition 1). We solved the value distribution problems for the Gauss maps of complete degenerate stationary surfaces and obtained many properties about such surfaces in [6].

In this paper, to further investigate the properties of complete degenerate stationary surfaces in ${\mathbb{R}}^{3,1}$, we consider the complete degenerate stationary surfaces with injective Gauss maps on the basis of previous research in [6]. For totally real type, there are three cases: minimal surfaces in ${\mathbb{R}}^3$, maximal surfaces in ${\mathbb{R}}^{2,1}$ and zero mean curvature surfaces in ${\mathbb{R}}^{2,0}:=\{\mathbf{x}\in {\mathbb{R}}^{3,1}:x_3=x_4\}$ endowed with the induced degenerate inner product. A minimal surface M in ${\mathbb{R}}^3$ with injective Gauss map has finite total curvature $-4\pi$, and Ossserman [7] proved that the complete minimal surfaces in ${\mathbb{R}}^3$ with $\int KdA=-4\pi$ must be the Enneper surfaces or the Catenoid surfaces. The only complete maximal surface in ${\mathbb{R}}^{2,1}$ is the affine plane [8,9]. The third case: M is a zero mean curvature surface in ${\mathbb{R}}^{2,0}$, which is a graph of a harmonic function over the entire space-like plane and can be expressed as:

$$\mathbf{x}=(x_1,\pm x_2,x_1^2-x_2^2,x_1^2-x_2^2).$$

For the hyperbolic case, we proved [6] that each complete degenerate stationary surface in ${\mathbb{R}}^{3,1}$ of the hyperbolic type must be congruent to the entire stationary graph of $F:{\mathbb{R}}^2\to {\mathbb{R}}^{1,1}$. Furthermore, if M is non-flat, then ${\psi }_1=C{e}^{-\beta }$ with a constant C and an entire function $\beta$, takes each point in ${\mathbb{S}}^2$ for infinite times, with the exception of exactly 2 points. Therefore, there exists no complete degenerate stationary surface of hyperbolic type whose Gauss map is injective.

For the elliptic case, the degenerate stationary surfaces of the elliptic type with injective Gauss maps must be algebraic, and the total Gauss curvature is $-4\pi$. Ma–Wang classified the complete stationary surfaces with total curvature $\int KdM=-4\pi$ in [10]. Such surfaces must be oriented and be congruent to either the generalized Catenoids or the generalized Enneper surfaces [3], which generalized Osserman’s Theorem in ${\mathbb{R}}^3$. In Section 3.3, we consider such surfaces from the degenerate point of view and obtain the Theorem 6, which can be congruent to the Ma–Wang result in [10].

Finally, we discuss the parabolic case. We first proved that the degenerate stationary surfaces of parabolic type with injective Gauss map must be algebraic. As shown in Theorem 4.14 of [6], each complete, algebraic, degenerate stationary surface in ${\mathbb{R}}^{3,1}$ of parabolic type has to be an entire graph over a light-like plane. Thus, up to a constant factor, such surface can be represented as:

$$\mathbf{x}=\mathrm{Re}\left(z^2+z,iz,z-\frac{z^2}{2}-\frac{z^3}{3},z+\frac{z^2}{2}+\frac{z^3}{3}\right).$$

2. Preliminaries

2.1. The Conformal and Metric Structure of $G_{1,1}^2$

Let ${\mathbb{R}}^{3,1}$ be the 4-dimensional Minkowski space. The Minkowski inner product for $\mathbf{u}=(u_1,u_2,u_3,u_4)$ and $\mathbf{v}=(v_1,v_2,v_3,v_4)$ is given by:

$$\langle \mathbf{u},\mathbf{v}\rangle =u_1{v}_1+u_2{v}_2+u_3{v}_3-u_4{v}_4.$$

$\mathbf{u}\in {\mathbb{R}}^{3,1}$ is space-like if $\langle \mathbf{u},\mathbf{u}\rangle >0$; $\mathbf{u}$ is time-like if $\langle \mathbf{u},\mathbf{u}\rangle <0$; $\mathbf{u}$ is called a null vector if $\langle \mathbf{u},\mathbf{u}\rangle =0$. ${\mathbb{C}}^{3,1}$ is the complexification of ${\mathbb{R}}^{3,1}$ equipped with a complex bilinear form:

$$\langle \mathbf{z},\mathbf{w}\rangle =z_1{w}_1+z_2{w}_2+z_3{w}_3-z_4{w}_4,$$

where $\mathbf{z}=(z_1,z_2,z_3,z_4),\mathbf{w}=(w_1,w_2,w_3,w_4)\in {\mathbb{C}}^{3,1}$. ${\mathbb{CP}}^{2,1}$ is the complex projective space equipped with an analog of the Fubini–Study metric:

$$h_{FS}=\frac{2\langle \mathbf{z}\wedge d\mathbf{z},\overline{\mathbf{z}}\wedge d\overline{\mathbf{z}}\rangle }{{\langle \mathbf{z},\overline{\mathbf{z}}\rangle }^2}.$$

Denote ${\mathbf{G}}_{1,1}^2$ as the Lorentz–Grassmann manifold consisting of all oriented space-like 2-plane in ${\mathbb{R}}^{3,1}$. Given $\Pi \in {\mathbf{G}}_{1,1}^2$, $\mathbf{u}\wedge \mathbf{v}$ is the Plücker coordinate of $\Pi$ with $\mathbf{u},\mathbf{v}$, an oriented orthonormal basis of $\Pi$. The canonical pseudo-Riemannian metric on ${\mathbf{G}}_{1,1}^2$ is

$$\langle {\mathbf{u}}_1\wedge {\mathbf{v}}_1,{\mathbf{u}}_2\wedge {\mathbf{v}}_2\rangle =\left|\begin{array}{cc}\langle {\mathbf{u}}_1,{\mathbf{v}}_1\rangle & \langle {\mathbf{u}}_1,{\mathbf{v}}_2\rangle \\ \langle {\mathbf{u}}_2,{\mathbf{v}}_1\rangle & \langle {\mathbf{u}}_2,{\mathbf{v}}_2\rangle \end{array}\right|.$$

On the other hand, for $\Pi =\mathrm{span}\{\mathbf{u},\mathbf{v}\}$, there is a complex vector $\mathbf{z}=\mathbf{u}-i\mathbf{v}\in {\mathbb{C}}^{3,1}$. Noting that another oriented orthonormal basis $cos\theta \mathbf{u}+sin\theta \mathbf{v},-sin\theta \mathbf{u}+cos\theta \mathbf{v}$ of $\Pi$ corresponds to $e^{i\theta }\mathbf{z}$, then

$$i:\Pi =\mathrm{span}\{\mathbf{u},\mathbf{v}\}\in {\mathbf{G}}_{1,1}^2\mapsto \left[\mathbf{z}\right]=[\mathbf{u}-i\mathbf{v}]\in Q_{1,1}^{+}\subset {\mathbb{CP}}^{2,1}$$

is well-defined and injective. Here,

$$Q_{1,1}=\{\left[\mathbf{z}\right]\in {\mathbb{CP}}^3:z_1^2+z_2^2+z_3^2-z_4^2=0\},$$

$$Q_{1,1}^{+}=\{\left[\mathbf{z}\right]\in Q_{1,1}:|z_1{|}^2+|z_2{|}^2+|z_3{|}^2-|z_4{|}^2>0\}.$$

Then i gives a one-to-one correspondence between ${\mathbf{G}}_{1,1}^2$ and $Q_{1,1}^{+}$.

Proposition 1

([6]). For $Q_{1,1}$ and $Q_{1,1}^{+}$, we have:

(1) 

$Q_{1,1}$ is biholomorphic to ${\mathbb{C}}^{*}\times {\mathbb{C}}^{*}={\mathbb{S}}^2\times {\mathbb{S}}^2$, with ${\mathbb{C}}^{*}:=\mathbb{C}\cup \{\infty \}$ the extended complex plane.

(2) 

$Q_{1,1}^{+}$ is biholomorphic to $\{(w_1,w_2)\in {\mathbb{C}}^{*}\times {\mathbb{C}}^{*}:w_2\ne {\overline{w}}_1\}$, where $\overline{\infty }=\infty$.

By the direct calculation, we obtain the metric of $Q_{1,1}^{+}$ in terms of $w_1$ and $w_2$:

$$g=\mathrm{Re}\left[\frac{4d{\overline{w}}_{1}d{w}_2}{{({\overline{w}}_1-w_2)}^2}\right].$$

(1)

2.2. The Geometry of Hyperplanes in $Q_{1,1}$

For $\left[A\right]\in {\mathbb{CP}}^{2,1}$,

$$H_A:=\{\left[\mathbf{z}\right]\in {\mathbb{CP}}^{2,1}:\langle A,\mathbf{z}\rangle =0\}$$

is the corresponding hyperplane in ${\mathbb{CP}}^{2,1}$. Furthermore, $H_A\cap Q_{1,1}$ (or $H_A\cap Q_{1,1}^{+}$) is a hyperplane in $Q_{1,1}$ (or $Q_{1,1}^{+}$).

Proposition 2

([6]). Let $\Psi :Q_{1,1}\to {\mathbb{S}}^2\times {\mathbb{S}}^2$ be the biholomorphic map. For each hyperplane $H=H_A\cap Q_{1,1}$, we have

(1) 

$\left[A\right]\notin Q_{1,1}$ if and only if $\Psi \left(H\right)\subset {\mathbb{C}}^{*}\times {\mathbb{C}}^{*}$ is the graph of a Möbius transformation.

(2) 

$\left[A\right]\in Q_{1,1}$ if and only if $\Psi \left(H\right)=\{(w_1,w_2)\in {\mathbb{C}}^{*}\times {\mathbb{C}}^{*}:w_1=c_1\mathrm{or}{w}_2=c_2\}$. Especially, $\left[A\right]\in Q_{1,1}^{+}$ if and only if $c_2\ne {\overline{c}}_1$.

Let

$$O(3,1):=\{\sigma \in GL\left({\mathbb{R}}^{3,1}\right):\langle \sigma \left(\mathbf{u}\right),\sigma \left(\mathbf{v}\right)\rangle =\langle \mathbf{u},\mathbf{v}\rangle \};$$

$$SO(3,1):=\{\sigma \in O(3,1):det\sigma =1\};$$

$$S{O}^{+}(3,1):=\{\sigma \in SO(3,1):{\left(\sigma \left({ϵ}_4\right)\right)}_4>0\}.$$

Theorem 1

([6]). For $\left[A\right]=[X+iY]\in {\mathbb{CP}}^{2,1}$, the orbits of $\left[A\right]$ under the action of $O(3,1)$, $SO(3,1)$, $S{O}^{+}(3,1)$ are just the same one. For $\left[A\right],\left[B\right]\in {\mathbb{CP}}^{2,1}$, denote [A]∼[B] if $\exists \sigma$ in $S{O}^{+}(3,1)$ such that $\left[\sigma \right(A\left)\right]=\left[B\right]$. Under this relation, all elements in ${\mathbb{CP}}^{2,1}$ can be classified as follows:

1. 

Totally real type: $\left[A\right]=\left[{\mathbf{u}}_0\right]$, with a real vector ${\mathbf{u}}_0$ ⇔ X and Y are linearly dependent.

(a) 

$\left[A\right]$∼$\left[{ϵ}_1\right]$, where ${ϵ}_1=(1,0,0,0)\iff {\mathbf{u}}_0$ is a space-like vector.

(b) 

$\left[A\right]$∼$\left[{ϵ}_4\right]$, where ${ϵ}_4=(0,0,0,1)\iff {\mathbf{u}}_0$ is a time-like vector.

(c) 

$\left[A\right]$∼$[{ϵ}_3+{ϵ}_4]=\left[(0,0,1,1)\right]\iff {\mathbf{u}}_0$ is a null vector.

2. 

Non-totally real type: X and Y are linearly independent, denoted as $P_A=span\{X,Y\}$.

(a) 

Hyperbolic type: $P_A$ is a space-like 2-plane, $\left[A\right]$∼$\left[\right(tanhu,i,0,0\left)\right],u\in (0,+\infty ]$. (Here, $tanh(+\infty )=1$.)

(b) 

Elliptic type: $P_A$ is a time-like two-plane, $\left[A\right]$∼$\left[(0,0,1,itan\alpha )\right],\alpha \in (0,\frac{\pi }{2})$.

(c) 

Parabolic type: $P_A$ is a light-like two-plane, $\left[A\right]$∼$\left[\right(1,0,i,i\left)\right]$.

The following diagram commutes [6]:

$$\begin{array}{c}{Q}_{1,1}\stackrel{\hspace{1em}\hspace{1em}\hspace{1em}\sigma \hspace{1em}\hspace{1em}\hspace{1em}}{\to }{Q}_{1,1}\\ \downarrow \mathsf{\Psi }\downarrow \mathsf{\Psi }\\ {\mathbb{C}}^{*}\times {\mathbb{C}}^{*}\stackrel{\hspace{1em}\psi \hspace{1em}}{\to }{\mathbb{C}}^{*}\times {\mathbb{C}}^{*}.\end{array}$$

(2)

Here, $\sigma \in S{O}^{+}(3,1)$ and

$$\psi :(w_1,w_2)\mapsto (\frac{a{w}_1+b}{c{w}_1+d},\frac{\overline{a}{w}_1+\overline{b}}{\overline{c}{w}_1+\overline{d}})$$

(3)

with $T:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in SL(2,\mathbb{C})$.

Proposition 3

([6]). Let $H_1,H_2$ be hyperplanes in $Q_{1,1}$, such that $\Psi \left(H_1\right)$ and $\Psi \left(H_2\right)$ are the graphs of the Möbius transformations $M_{S_1}$ and $M_{S_2}$, respectively, with $S_1,S_2\in SL(2,\mathbb{C})$. Then, $H_1$∼$H_2$ if and only if there exists $T\in SL(2,\mathbb{C})$, such that $S_2=\pm \overline{T}{S}_1{T}^{-1}$. In this case, $S_1$ and $S_2$ are said to be conjugate similar to each other, and we denote $S_1\stackrel{conj}{\sim }{S}_2$.

Theorem 2

([6]). Each hyperplane H in $Q_{1,1}$ can be classified into the following types:

• $H=H_A$ with $\left[A\right]\in Q_{1,1}$, i.e., $\Psi \left(H\right)=\{(w_1,w_2)\in {\mathbb{C}}^{*}\times {\mathbb{C}}^{*}:w_1=c_1\mathrm{or}{w}_2=c_2\}$:

  (a) 

  $\left[A\right]\in Q_{1,1}^{+}$ if and only if $c_2\ne {\overline{c}}_1$. In this case, $\left[A\right]$∼$\left[\right(1,i,0,0\left)\right]$, and $\Psi \left(H^{\prime }\right)$$=\{(c_1,w_2):w_2\ne {\overline{c}}_1\}\cup \{(w_1,c_2):w_1\ne {\overline{c}}_2\}$.

  (b) 

  $\left[A\right]\notin Q_{1,1}^{+}$ if and only if $c_2={\overline{c}}_1$. In this case, $\left[A\right]$∼$[{ϵ}_3+{ϵ}_4]$, and $\Psi \left(H^{\prime }\right)$$=\{(c_1,w_2):w_2\ne {\overline{c}}_1\}\cup \{(w_1,{\overline{c}}_1):w_1\ne c_1\}$.

• $H=H_A$ with $\left[A\right]\notin Q_{1,1}$, i.e., $\Psi \left(H\right)=\{(w,M_S\left(w\right)):w\in {\mathbb{C}}^{*}\}$:

  (a) 

  $\left[A\right]$∼$\left[{ϵ}_1\right]$ if and only if $S\stackrel{conj}{\sim }\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ and H is conformally equivalent to $\mathbb{C}\backslash \mathbb{R}$.

  (b) 

  $\left[A\right]$∼$\left[{ϵ}_4\right]$ if and only if $S\stackrel{conj}{\sim }\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ and H is conformally equivalent to ${\mathbb{C}}^{*}$.

  (c) 

  $\left[A\right]$∼$\left[\right(tanhu,i,0,0\left)\right]$ if and only if $S\stackrel{conj}{\sim }\left(\begin{array}{cc}{e}^u& 0\\ 0& e^{-u}\end{array}\right)$, where $u\in (0,+\infty )$ and H is conformally equivalent to $\mathbb{C}\backslash \left\{0\right\}$.

  (d) 

  $\left[A\right]$∼$\left[\right(0,0,1,itan\alpha \left)\right]$ if and only if $S\stackrel{conj}{\sim }\left(\begin{array}{cc}0& i{e}^{-i\alpha }\\ i{e}^{i\alpha }& 0\end{array}\right)$, where $\alpha \in (0,\frac{\pi }{2})$ and H is conformally equivalent to ${\mathbb{C}}^{*}$.

  (e) 

  $\left[A\right]$∼$\left[\right(1,0,i,i\left)\right]$ if and only if $S\stackrel{conj}{\sim }\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ and H is conformally equivalent to $\mathbb{C}$.

2.3. Metrics of Hyperplanes in $Q_{1,1}^{+}$

Let $H=H_A\cap Q_{1,1}^{+}$ be a hyperplane in $Q_{1,1}^{+}$ and $\Psi \left(H\right)$ be a graph of the Möbius transformation $M_S(S\in SL(2,\mathbb{C}))$. Diagram (2) implies

$$\Psi \left(\sigma \left(H\right)\right)=\{(w,M_{\overline{T}S{T}^{-1}\left(w\right)}):w\in {\mathbb{C}}^{*}\},$$

where $\sigma \in S{O}^{+}(3,1)$, and $M_T(T\in SL(2,\mathbb{C}))$ is the corresponding Möbius transformation. Then, the isometric transformation group of H is

$$G_S=\{T\in SL(2,\mathbb{C}):\overline{T}S{T}^{-1}=\pm S\},$$

which is a Lie group, and the $G_S$-action on H is

$$T\times {\Psi }^{-1}(w,M_S\left(w\right))={\Psi }^{-1}(M_T\left(w\right),M_{\overline{T}S}\left(w\right)),\forall \overline{w}\ne M_S\left(w\right).$$

Let ${𝖌}_S$ be the Lie algebra of $G_S$, then

$${𝖌}_S=\{X\in sl(2,\mathbb{C}):\overline{X}S-SX=0\}.$$

Any hyperplanes $H_1$∼$H_2$ are isometric to each other, so it suffices to consider the metrics of representative hyperplanes as follows:

• Case I. $S=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$. Then ${𝖌}_S=sl(2,\mathbb{R})$ and $G_S=SL(2,\mathbb{R})\cup \mathrm{diag}(i,-i)SL(2,\mathbb{R})$. Substituting $w_2=w_1=w$ into (1) gives

  $$g=-\frac{{\left|dw\right|}^2}{{\left(\mathrm{Im}w\right)}^2}w\in \mathbb{C}\backslash \mathbb{R}.$$
  Therefore, $(H,-g)$ has two connected components, and each component is isometric to the complete hyperbolic plane with the constant Gauss curvature $-1$.
• Case II. $S=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Then, ${𝖌}_S=su(2,\mathbb{C})$ and $G_S=SU(2,\mathbb{C})$. Substituting $w_1=w$ and $w_2=-\frac{1}{w}$ into (1) gives:

  $$g=\frac{{4\left|dw\right|}^2}{{(1+|w|}^2{)}^2}.$$
  Therefore, $(H,g)$ is isometric to the unit sphere equipped with the canonical metric.
• Case III. $S=\left(\begin{array}{cc}{e}^u& 0\\ 0& e^{-u}\end{array}\right)$ with $u\in (0,+\infty )$. Then, ${𝖌}_S=\mathbb{R}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, $G_S=\left\{\left(\begin{array}{cc}\lambda & 0\\ 0& {\lambda }^{-1}\end{array}\right):\lambda \in \mathbb{R}\mathrm{or}i\mathbb{R},\lambda \ne 0\right\}$. Substituting $w_1=w$ and $w_2=e^{2u}w$ into (1) and then letting $w=e^{t+i\theta }$ with $t\in \mathbb{R},\theta \in \mathbb{R}/\left(2\pi \mathbb{Z}\right)$ implies:

  $$g=\mathrm{Re}\left[\frac{4}{{(e^{-u}{e}^{-i\theta }-e^u{e}^{i\theta })}^2}\right](d{t}^2+d{\theta }^2).$$
  Therefore, H is diffeomorphic to $\mathbb{C}\backslash \left\{0\right\}$ and the metric g is invariant under the scaling $z\mapsto kz$ for each $k\in \mathbb{R}\backslash \left\{0\right\}$. The associated area form $d{A}_g$ is also invariant under the scalings, and hence, ${\int }_{H}d{A}_g$ is divergent.
• Case IV. $S=\left(\begin{array}{cc}0& i{e}^{-i\alpha }\\ i{e}^{i\alpha }& 0\end{array}\right)$ with $\alpha \in (0,\frac{\pi }{2})$. Then, ${𝖌}_S=\mathbb{R}\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)$, $G_S=\left\{\left(\begin{array}{cc}{e}^{it}& 0\\ 0& e^{-it}\end{array}\right):t\in \mathbb{R}/\left(2\pi \mathbb{Z}\right)\right\}$. Substituting $w_1=w$ and $w_2=e^{-2i\alpha }{w}^{-1}$ into (1) and then letting $w=e^{t+i\theta }$ with $t\in \mathbb{R}$, $\theta \in \mathbb{R}/\left(2\pi \mathbb{Z}\right)$ implies:

  $$g=-\mathrm{Re}\left[\frac{4}{{(e^{-t}{e}^{-i\alpha }-e^t{e}^{i\alpha })}^2}\right](d{t}^2+d{\theta }^2).$$
  Therefore, H is diffeomorphic to ${\mathbb{C}}^{*}$, and the metric g is invariant under the rotation $z\mapsto e^{i\beta }z$ for each $\beta \in \mathbb{R}/\left(2\pi \mathbb{Z}\right)$. Moreover, we have:

  $${\int }_{H}d{A}_g=4\pi .$$

  (4)
• Case V. $S=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$. Then, ${𝖌}_S=\mathbb{R}\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, $G_S=\left\{\pm \left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right):t\in \mathbb{R}\right\}$. Substituting $w_1=w$ and $w_2=w+1$ into (1) and then letting $w=x+iy$ with $x,y\in \mathbb{R}$ implies

  $$g=\mathrm{Re}\left[\frac{4}{{(1+2iy)}^2}\right](d{x}^2+d{y}^2).$$
  Therefore, H is diffeomorphic to $\mathbb{C}$ and the metric g is invariant under the parallel translation $z\mapsto z+t$ for each $t\in \mathbb{R}$. Since $d{A}_g$ is also invariant under the parallel translations, ${\int }_{H}d{A}_g$ should be divergent.

For $H=H_A\cap Q_{1,1}^{+}$ with $\left[A\right]\in Q_{1,1}$, $\Psi \left(H\right)$ lies in $(\left\{c_1\right\}\times {\mathbb{C}}^{*})\cup ({\mathbb{C}}^{*}\times \left\{c_2\right\})$ with $c_1,c_2\in {\mathbb{C}}^{*}$. Then, a direct calculation shows that the pull-back metric on H vanishes everywhere.

2.4. Stationary Surfaces in ${\mathbb{R}}^{3,1}$

Let $\mathbf{x}:M\to {\mathbb{R}}^{3,1}$ be an oriented space-like stationary surface in the Minkowski space. Namely, the mean curvature vector field H of M vanishes everywhere. M is stationary if and only if the restriction of each coordinate function on M is harmonic.

The Gauss map of M is defined by

$$p\in M\mapsto T_{p}M\in {\mathbf{G}}_{1,1}^2.$$

Let $(u,v)$ be local isothermal parameters in a neighborhood of p, then $T_{p}M=span\{{\mathbf{x}}_u,{\mathbf{x}}_v\}$ and $\left[{\mathbf{x}}_z\right]=\left[\frac{1}{2}({\mathbf{x}}_u-i{\mathbf{x}}_v)\right]\in Q_{1,1}^{+}$, where $z=u+iv$.

Denote

$$\varphi =({\varphi }_1,{\varphi }_2,{\varphi }_3,{\varphi }_4):=\left(\frac{\partial x_1}{\partial z},\frac{\partial x_2}{\partial z},\frac{\partial x_3}{\partial z},\frac{\partial x_4}{\partial z}\right)dz;$$

then, the harmonicity of $x_k$ forces ${\varphi }_k$ to be a holomorphic 1-form that can be globally defined on M. $\left[{\mathbf{x}}_z\right]\in Q_{1,1}^{+}$ is equivalent to saying that

$$\begin{array}{ccc}\hfill {\varphi }_1^2+{\varphi }_2^2+{\varphi }_3^2-{\varphi }_4^2& =& 0,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill |{\varphi }_1{|}^2+|{\varphi }_2{|}^2+|{\varphi }_3{|}^2-{\left|{\varphi }_4\right|}^2& >& 0.\hfill \end{array}$$

(6)

Let $({\psi }_1,{\psi }_2)=\Psi \left(\varphi \right),dh=\frac{1}{2}({\varphi }_3+{\varphi }_4)$; then, ${\psi }_1,{\psi }_2$ are meromorphic functions on M, and $dh$ is a holomorphic 1-form on M. If $dh\equiv 0$, then (5) implies ${\varphi }_1-i{\varphi }_2\equiv 0$ or ${\varphi }_1+i{\varphi }_2\equiv 0$. Hence, $\varphi =({\varphi }_1,\pm i{\varphi }_1,{\varphi }_3,-{\varphi }_3)$ and the zeros of ${\varphi }_1$ and ${\varphi }_3$ do not coincide. Otherwise, the zeros of $dh$ are discrete.

Thus, the Weierstrass representation of space-like stationary surfaces in ${\mathbb{R}}^{3,1}$ can be stated as follows:

Theorem 3

([3]). Let ${\psi }_1,{\psi }_2$ be meromorphic functions on M and $dh$ be a holomorphic 1-form on M. If ${\psi }_1,{\psi }_2,dh$ satisfy the regularity conditions $\left(1\right),\left(2\right)$ and the period condition $\left(3\right)$:

(1) 

${\psi }_1\ne \overline{{\psi }_2}$ on M, and their poles do not coincide;

(2) 

The zeros of $dh$ coincide with the poles of ${\psi }_1$ or ${\psi }_2$ with the same order;

(3) 

For any closed curve γ on M:

$${\int }_{\gamma }{\psi }_{1}dh=-\overline{{\int }_{\gamma }{\psi }_{2}dh},Re{\int }_{\gamma }dh=0=Re{\int }_{\gamma }{\psi }_1{\psi }_{2}dh.$$

Then, the following equation defines a stationary surface $\mathbf{x}:M\to {\mathbb{R}}^{3,1}$:

$$x=2Re\int \left({\psi }_1+{\psi }_2,-i({\psi }_1-{\psi }_2),1-{\psi }_1{\psi }_2,1+{\psi }_1{\psi }_2\right)dh.$$

(7)

Conversely, every stationary surface $\mathbf{x}:M\to {\mathbb{R}}^{3,1}$ can be represented as (7), where $dh$, ${\psi }_1$, ${\psi }_2$ satisfy the conditions $\left(1\right),\left(2\right)$, and $\left(3\right)$.

The induced metric is:

$${\left|ds\right|}^2=\langle {\mathbf{x}}_z,{\mathbf{x}}_{\overline{z}}\rangle =\langle \varphi ,\overline{\varphi }\rangle =2|{\psi }_1-\overline{{\psi }_2}{|}^2{\left|dh\right|}^2.$$

(8)

Actually, as shown in [6], ${\psi }_1$ and ${\psi }_2$ correspond to the two null vectors $\mathbf{y},{\mathbf{y}}^{*}$, respectively, which span the normal plane of M at the considered point. Moreover, ${\psi }_2\ne {\overline{\psi }}_1$ is equivalent to saying that $\mathbf{y}\ne {\mathbf{y}}^{*}$ everywhere.

Remark 1.

If we let ${\psi }_1\equiv -\frac{1}{{\psi }_2}$, (7) yields the representation for a minimal surface in ${\mathbb{R}}^3$. If ${\psi }_1\equiv \frac{1}{{\psi }_2}$, then we can obtain the Weierstrass representation for a maximal surface in ${\mathbb{R}}^{2,1}$.

Remark 2.

The induced action on ${\mathbb{S}}^2$ of the Lorentz transformation is just the Möbius transformation on ${\mathbb{S}}^2$ (i.e., a fractional linear transformation on ${\mathbb{C}}^{*}$). Since $({\psi }_1,{\psi }_2)=\Psi \left(\left[x_{\mathbf{z}}\right]\right)$, then the Gauss maps ${\psi }_1,{\psi }_2$ and the height differential $dh$ can be transformed as below:

$${\psi }_1\Rightarrow \frac{a{\psi }_1+b}{c{\psi }_1+d},{\psi }_2\Rightarrow \frac{\overline{a}{\psi }_2+\overline{b}}{\overline{c}{\psi }_2+\overline{d}},dh\Rightarrow (c{\psi }_1+d)(\overline{c}{\psi }_2+\overline{d})dh,$$

where $S=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in SL(2,\mathbb{C})$.

Definition 1

([6]). Let M be a space-like stationary surface in ${\mathbb{R}}^{3,1}$. If the Gauss image of M lies in a hyperplane $H=H_A\cap Q_{1,1}^{+}$ with $\left[A\right]\in {\mathbb{CP}}^{2,1}$, then M is called degenerate. Moreover, M is said to be a degenerate stationary surface of totally real (or hyperbolic, elliptic, parabolic) type whenever $\left[A\right]$ is totally real (or hyperbolic, elliptic, parabolic). If the Gauss image of M lies in the intersection of k linear independent hyperplanes in $Q_{1,1}^{+}$, then we say M is$\mathbf{k}$-degenerate.

3. Complete Degenerate Stationary Surfaces Whose Gauss Maps Are Injective

3.1. Degenerate Surfaces of Totally Real Type

Proposition 4

([6]). For each degenerate stationary surface M of totally real type whose Gauss image lies in $H=H_A\cap Q_{1,1}^{+}$, we have:

(1) 

$\left[A\right]$∼$\left[{ϵ}_1\right]$ if and only if M is a maximal surface in ${\mathbb{R}}^{2,1}$.

(2) 

$\left[A\right]$∼$\left[{ϵ}_4\right]$ if and only if M is a minimal surface in ${\mathbb{R}}^3$.

(3) 

$\left[A\right]$∼$[{ϵ}_3+{ϵ}_4]$ if and only if M is a zero mean curvature surface in ${\mathbb{R}}^{2,0}:=\{\mathbf{x}\in {\mathbb{R}}^{3,1}:$$x_3=x_4\}$ endowed with the induced degenerate inner product.

For the maximal surfaces, Calabi [8] showed that the only complete maximal surface in ${\mathbb{R}}^{2,1}$ is the affine plane. Let M be a minimal surface in ${\mathbb{R}}^3$ with injective Gauss map, then M has finite total curvature $-4\pi$. Ossserman [7] proved that the complete minimal surfaces in ${\mathbb{R}}^3$ with $\int KdA=-4\pi$ must be the Enneper surfaces or the Catenoid surfaces. Thus, we only have to consider the third case: M is a zero mean curvature surface in ${\mathbb{R}}^{2,0}$.

Theorem 4

([6]). For each space-like stationary surface M in ${\mathbb{R}}^{3,1}$, the following statements are equivalent:

• The Gauss image of M lies in $H_A\cap Q_{1,1}^{+}$ with $\left[A\right]$∼$\left[\right(0,0,1,1\left)\right]$.
• The Gauss image of M lies in $H_A\cap Q_{1,1}^{+}$ with $\left[A\right]\in Q_{1,1}$.
• Either ${\psi }_1$ or ${\psi }_2$ is a constant function on M.
• The Gauss curvature K of M is 0 everywhere.
• M is 2-degenerate.

Moreover, a complete surface satisfying the above conditions has to be the entire graph of $F:(x_1,x_2)\in {\mathbb{R}}^2\mapsto h(x_1,x_2){\mathbf{y}}_0\in {\mathbb{R}}^{1,1}$, where h is a harmonic function and ${\mathbf{y}}_0$ is a null vector.

As showed in the proof of Theorem $4.3$ in [6], the Gauss map of such surface is:

$$\varphi =\frac{1}{2}(1,\pm i,f,f)dz,$$

with an entire function f, so ${\psi }_1=\frac{1}{f}$. If ${\psi }_1$ is injective, then we can assume $f\left(z\right)$$=z=x_1+i{x}_2$, without loss of generality. Thus, up to a constant, the surface can be represented as:

$$\begin{array}{cc}\hfill \mathbf{x}& =Re\int (1,\pm i,z,z)dz\hfill \\ \hfill & =Re\left(z,\pm z,\frac{z^2}{2},\frac{z^2}{2}\right)\hfill \\ \hfill & =(x_1,\pm x_2,x_1^2-x_2^2,x_1^2-x_2^2).\hfill \end{array}$$

3.2. Complete Degenerate Stationary Surfaces of Hyperbolic Type Whose Gauss Maps Are Injective

As shown in [6], the complete degenerate space-like stationary surface in ${\mathbb{R}}^{3,1}$ of hyperbolic type must be congruent to the entire stationary graph of $F:{\mathbb{R}}^2\to {\mathbb{R}}^{1,1}$. Moreover, if M is non-flat, then ${\psi }_1=C{e}^{-\beta }$ with a constant C and an entire function $\beta$ takes each point in ${\mathbb{S}}^2$ for infinite times, with the exception of exactly 2 points. Thus, there exists no complete degenerate stationary surface of hyperbolic type whose Gauss map is injective.

3.3. Complete Degenerate Stationary Surfaces of Elliptic Type Whose Gauss Maps Are Injective

Let M be a degenerate stationary surface of elliptic type, i.e., the Gauss image of M lies in $H=H_A\cap Q_{1,1}^{+}$ of elliptic type. In this case, $\left[A\right]\sim \left[\right(0,0,1,itan\alpha \left)\right]$, where $\alpha \in (0,\frac{\pi }{2})$ is called the elliptic argument of M. The two Gauss maps ${\psi }_1,{\psi }_2$ satisfy ${\psi }_2=e^{-2i\alpha }{\psi }_1^{-1}$. Let $\psi :={\psi }_1$; then, $\psi$ is a meromorphic function, and the Weierstrass representation for degenerate stationary surfaces of elliptic type is:

Theorem 5

([6]). Given a holomorphic 1-form ω and a meromorphic function ψ globally defined on a Riemann surface M:

• p is a zero of ω if and only if p is a zero or a pole of ψ, with the same order;
• ${\oint }_{\gamma }\psi \omega =-\overline{{\oint }_{\gamma }{\psi }^{-1}\omega }$ and ${\oint }_{\gamma }\omega =0$ for each closed path in M.

Then,

$$\mathbf{x}:=\mathrm{Re}\int (e^{i\alpha }\psi +e^{-i\alpha }{\psi }^{-1},-i(e^{i\alpha }\psi -e^{-i\alpha }{\psi }^{-1}),2isin\alpha ,2cos\alpha )\omega$$

(9)

defines a degenerate stationary surface with the elliptic argument $\alpha \in (0,\frac{\pi }{2})$. Conversely, all degenerate stationary surfaces of elliptic type can be expressed in this form.

The induced metric is:

$$d{s}^2=2|\overline{\psi }-e^{-2i\alpha }{\psi }^{-1}{|}^2{\left|dh\right|}^2=2\left[{\left|\psi \right|}^2+{\left|\psi \right|}^{-2}-2cos\left(2\alpha \right)\right]{\left|\omega \right|}^2.$$

(10)

Proposition 5

([6]).  Let M be a complete degenerate stationary surface of elliptic type, then the total Gauss curvature of M is either ∞ or $-4\pi m$ with m a non-negative number, i.e., ${\int }_{M}Kd{A}_M$$=-4\pi m$.

Let M be a complete degenerate stationary surface of elliptic type with an injective Gauss map. Then, M has finite total Gauss curvature $-4\pi$ due to Proposition 5, and M is conformally equivalent to ${\mathbb{S}}^2\backslash \{p_1,\dots ,p_m\}$. The injectivity of the Gauss map implies $\psi$ can be extended to a biholomorphism between ${\mathbb{S}}^2$ and ${\mathbb{C}}^{*}$. By using Lemma 9.6 of [7], we know $\omega$ can also be extended to a meromorphic differential on ${\mathbb{S}}^2$. If $p_i$ is a removable singularity of $\omega$, we can consider the completeness of M or the period condition ${\oint }_{\gamma }\psi \omega =-\overline{{\oint }_{\gamma }{\psi }^{-1}\omega }$, which causes a contradiction. Therefore, $p_i$ has to be a pole of $\omega$ of order $v_i\ge 2$, due to the period condition ${\int }_{\gamma }\omega =0$. On the other hand, by Theorem 5, each zero of $\omega$ coincides with the zero (if it exists) or the pole (if it exists) of $\psi$ in M, whose order must be 1. The Riemann relation says the number of poles minus the number of zeros of $\omega$ equals 2. That is,

$$2=\sum _{i=1}^m{v}_i-\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{zeros}\ge 2m-2.$$

Hence, we obtain $m=1$ or 2.

Since any $\sigma \in S{O}^{+}(3,1)$, preserving H invariant yields a rotation $z\mapsto e^{i\beta }z$ on ${\pi }_1\circ \Psi \left(H\right)\cong {\mathbb{C}}^{*}$. Hence, we can assume the pole of $\psi$ is $0,\infty ,t\ne 0,\infty$, and $t\in {\mathbb{R}}^{+}$. Now we consider $\psi \left(M\right)$ case by case:

• Case I. $\psi \left(M\right)=\mathbb{C}$. Then, we can take

  $$M=\mathbb{C},\psi \left(z\right)=z.$$

  and the only zero of $\omega =f\left(z\right)dz$ is 0 of order 1. This means $\omega =\lambda zdz$ with $\lambda \ne 0$. The period conditions are automatically satisfied since M is simply connected.
• Case II. $\psi \left(M\right)={\mathbb{C}}^{*}\backslash \left\{0\right\}$. Similarly to above, the Weierstrass data can be taken as:

  $$M=\mathbb{C},\psi \left(z\right)=\frac{1}{z},\omega =\lambda zdz.$$
• Case III. $\psi \left(M\right)={\mathbb{C}}^{*}\backslash \left\{t\right\}$ with $t\in {\mathbb{R}}^{+}$ without loss of generality. Now we take:

  $$M=\mathbb{C},\psi \left(z\right)=\frac{tz+1}{z};$$

  then, the zeros of $\omega =f\left(z\right)dz$ are 0 and $-\frac{1}{t}$, each of which has order 1. This implies $\omega =\lambda z(tz+1)dz$ with $\lambda \ne 0$.
• Case IV. The complement of $\psi \left(M\right)$ consists of exact 2 points. Let $v_i,i=1,2$ be the order of the two poles of $\omega$, respectively. By the Riemann relation, we have:

  $$\begin{array}{cc}\hfill 2& =v_1+v_2-\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{zeros}\hfill \\ \hfill & \ge 4-\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{zeros}.\hfill \end{array}$$
  Thus, $\omega$ has at most two zeros. Since $\psi$ is injective, $\psi$ must have a zero and a pole in M, and the extended meromorphic differential $\omega$ has exactly two poles of order 2. Then, $\mathbb{C}\backslash \psi \left(M\right)=\{t,s\}$ with $t,s\in \mathbb{C}\backslash \left\{0\right\}$, and we can assume $t\in {\mathbb{R}}^{+}$ without loss of generality. Now, we take

  $$M=\mathbb{C}\backslash \left\{0\right\},\psi \left(z\right)=\frac{sz+t}{z+1},\omega =\frac{\lambda (sz+t)(z+1)}{z^2}dz,$$

  with $\lambda$ be a nonzero constant to be chosen. A straightforward calculation shows:

  $${\oint }_M\omega =2\lambda (t+s)\pi i,{\oint }_M\psi \omega =4\lambda ts\pi i,{\oint }_M{\psi }^{-1}\omega =4\lambda \pi i.$$
  Then, the period conditions force $t=1,s=-1$ and $\lambda \in \mathbb{R}i,\lambda \ne 0$.

Thereby, we obtain the following result:

Theorem 6.

Let M be a complete degenerate stationary surface with elliptic argument α, such that the Gauss map is injective; then, M is congruent to one of the representative surfaces given by (9) with respect to the following W-data:

(1) 

$M=\mathbb{C},\psi \left(z\right)=z$, $\omega =\lambda zdz$, $\lambda \in \mathbb{C}\backslash \left\{0\right\}$;

(2) 

$M=\mathbb{C},\psi \left(z\right)=\frac{tz+1}{z}$, $\omega =\lambda z(tz+1)dz$, where $t\in {\mathbb{R}}^{+}\cup \left\{0\right\}$ and $\lambda \in \mathbb{C}\backslash \left\{0\right\}$;

(3) 

$M=\mathbb{C}\backslash \left\{0\right\},\psi \left(z\right)=\frac{1-z}{1+z},\omega =\frac{\lambda (1-z)(1+z)}{z^2}dz,\lambda \in \mathbb{R}i\backslash \left\{0\right\}$.

Remark 3.

Actually, item 1 and item 2 in the above Theorem are the generalized Enneper surfaces, and item 3 is the generalized Catenoid, which are given by Ma–Wang–Wang [3]. The generalized Enneper surface is:

$${\psi }_1=z,{\psi }_2=\frac{c}{z},dh=\lambda zdz$$

(11)

or

$${\psi }_1=z+1,{\psi }_2=\frac{c}{z},dh=\lambda zdz$$

(12)

over $\mathbb{C}$ with complex parameters $c,\lambda \in \mathbb{C}\backslash \left\{0\right\}$. $\mathbf{x}$ has no singular points if and only if the parameter $c=c_1+i{c}_2$ is not zero or positive real numbers in $\left(11\right)$ or

$$c_1-c_2^2+\frac{1}{4}<0$$

in $\left(12\right)$. In addition, the generalized catenoid is defined over $\mathbb{C}\backslash \left\{0\right\}$ with

$${\psi }_1=z+t,{\psi }_2=\frac{-1}{z-t},dh=s\frac{z-t}{z^2}dz.(-1<t<1,s\in \mathbb{R}\backslash \left\{0\right\}).$$

(13)

Thus, the item 1 in Theorem 6 can be identified with the generalized Enneper surface $\left(11\right)$. The generalized Enneper surface $\left(12\right)$ satisfies ${\psi }_2=\frac{c}{{\psi }_1-1}=M_{S_1}\left({\psi }_1\right)$ with

$$S_1=\left(\begin{array}{cc}0& \frac{c}{\sqrt{-c}}\\ \frac{1}{\sqrt{-c}}& \frac{-1}{\sqrt{-c}}\end{array}\right)\in SL(2,\mathbb{C}).$$

Then, $S_1\stackrel{conj}{\sim }{S}_2=\left(\begin{array}{cc}0& i{e}^{-i\alpha }\\ i{e}^{i\alpha }& 0\end{array}\right)$ with $e^{2i\alpha }=\frac{1+2c_1+i\sqrt{4c_2^2-4c_1-1}}{2\left|c\right|}$. Let $T\in SL(2,\mathbb{C})$ satisfy $S_2=\overline{T}{S}_1{T}^{-1}$ and $\widetilde{{\psi }_1}=M_T\left({\psi }_1\right)$. By a direct calculation, we have $\widetilde{{\psi }_1}\left(M\right)={\mathbb{C}}^{*}\backslash \left\{a\right\},a\in \mathbb{C}$. Thus, the surface $\left(12\right)$ is congruent with the item 2 in Theorem 6.

In $\left(13\right)$, ${\psi }_2=\frac{-1}{{\psi }_1-2t}=M_{S_1}\left({\psi }_1\right)$ with $S_1=\left(\begin{array}{cc}0& -1\\ 1& -2t\end{array}\right)\in SL(2,\mathbb{C})$. Then, $S_1\stackrel{conj}{\sim }{S}_2=\left(\begin{array}{cc}0& i{e}^{-i\alpha }\\ i{e}^{i\alpha }& 0\end{array}\right)$ with $e^{i\alpha }=t+i\sqrt{1-t^2}$. Let $T\in SL(2,\mathbb{C})$ satisfy $S_2=\overline{T}{S}_1{T}^{-1}$ and $\widetilde{{\psi }_1}=M_T\left({\psi }_1\right)$. By a direct calculation, we have $\widetilde{{\psi }_1}\left(M\right)={\mathbb{C}}^{*}\backslash \{a,b\},a,b\in \mathbb{C}\backslash \left\{0\right\}$. Thus, the surface $\left(13\right)$ is congruent with the item 3 in Theorem 6.

3.4. Complete Degenerate Stationary Surfaces of Parabolic Type Whose Gauss Maps Are Injective

Let M be a degenerate stationary surface of parabolic type, i.e., its Gauss image lies in $H=H_A\cap Q_{1,1}^{+}$ of parabolic type. As shown in Theorem 2 and Proposition 3, there exists $S\stackrel{conj}{\sim }\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, such that $\Psi \left(H\right)$ with $H:=H_A\cap Q_{1,1}$ is a graph of $M_S$. Thus, we can assume the two Gauss maps ${\psi }_1,{\psi }_2$ satisfy ${\psi }_2={\psi }_1+1$. Let $\psi :={\psi }_1$; then, $\psi$ is a holomorphic function. Thereby, we obtain the Weierstrass representation for degenerate stationary surfaces of parabolic type as follows:

Theorem 7

([6]). Given a holomorphic 1-form $dh$ and a holomorphic function ψ globally defined on a Riemann surface M, if

• $dh$ has no zeros;
• ${\oint }_{\gamma }dh=0$ and $\mathrm{Re}{\oint }_{\gamma }\psi dh=\mathrm{Re}{\oint }_{\gamma }{\psi }^{2}dh=0$ for each closed path in M,

then

$$\mathbf{x}:=\mathrm{Re}\int (2\psi +1,i,1-\psi -{\psi }^2,1+\psi +{\psi }^2)dh$$

(14)

defines a degenerate stationary surface of parabolic type. Conversely, all degenerate stationary surfaces of hyperbolic type can be expressed in this form.

The induced metric is

$$d{s}^2=2(1+4{\left(\mathrm{Im}\psi \right)}^2){\left|dh\right|}^2.$$

Let M be a complete degenerate stationary surface of parabolic type with injective Gauss maps. Then, M is conformally equivalent to ${\mathbb{S}}^2\backslash \{p_1,\dots ,p_m\}$. Similarly, the injectivity of the Gauss map implies that $\psi$ can be extended to a biholomorphism between ${\mathbb{S}}^2$ and ${\mathbb{C}}^{*}$. Since

$$d{s}^2=2(1+4{\left(\mathrm{Im}\psi \right)}^2){\left|dh\right|}^2\le {C(1+|\psi |}^2{\left)\right|dh|}^2,$$

and $dh$ has no zeros, we know $dh$ can be extended to a meromorphic differential on ${\mathbb{S}}^2$ by considering the completeness of M and using Lemma 9.6 of [7]. Then, such surfaces are algebraic. As shown in Theorem 4.14 of [6], each complete, algebraic, degenerate stationary surface in ${\mathbb{R}}^{3,1}$ of parabolic type has to be an entire graph over a light-like plane. We can assume

$$M=\mathbb{C},\psi \left(z\right)=z,dh=\lambda dz,\lambda \in \mathbb{C}\backslash \left\{0\right\}.$$

Thus, up to a constant factor, such surfaces (14) can be represented as:

$$\begin{array}{cc}\hfill \mathbf{x}:& =\mathrm{Re}{\int }_M(2\psi +1,i,1-\psi -{\psi }^2,1+\psi +{\psi }^2)dh\hfill \\ \hfill & =\mathrm{Re}{\int }_{\mathbb{C}}(2z+1,i,1-z-z^2,1+z+z^2)dz\hfill \\ \hfill & =\mathrm{Re}\left(z^2+z,iz,z-\frac{z^2}{2}-\frac{z^3}{3},z+\frac{z^2}{2}+\frac{z^3}{3}\right).\hfill \end{array}$$

(15)

Therefore, we obtain the following conclusion:

Theorem 8.

Each complete degenerate stationary surface of parabolic type with an injective Gauss map has to be an entire graph over a light-like plane and can be represented as (15).