Ramification and unicity theorems for Gauss maps of complete space-like stationary surfaces in four-dimensional Lorentz-Minkowski space

\fnmLi \surOu [email protected] \orgdivSchool of Mathematics, \orgnameSichuan university, \orgaddress\streetNo.24, Renmin South Road, \cityChengdu, \postcode610064, \stateSichuan Province, \countryChina

Abstract

In this paper, we investigate the value distribution properties for Gauss maps of space-like stationary surfaces in four-dimensional Lorentz-Minkowski space $\mathbb{R}^{3,1}$ , focusing on aspects such as the number of totally ramified points and unicity properties. We not only obtain general conclusions similar to situations in four-dimensional Euclidean space, but also consider the space-like stationary surfaces with rational graphic Gauss image, which is an extension of degenerate space-like stationary surfaces.

keywords:

Gauss map, stationary surface, ramification, unicity theorem

pacs:

[

MSC Classification]53A10, 53C42, 32A22, 51B20, 30C15

1 Introduction

It is well known that value distribution properties for Gauss maps of regular minimal surfaces in $\mathbb{R}^{n}$ play a crucial role in the theory of minimal surfaces. For a minimal surface $M$ in $\mathbb{R}^{3}$ , the Gauss map is defined by the unit normal vector $G(p)\in\mathbb{S}^{2}$ , for $p\in M$ . The surface $M$ is canonically considered as an open Riemann surface with some conformally metric $ds^{2}$ and by the minimality of $M$ , the Gauss map is a meromorphic function under the sphere stereographic projection. S. S. Chern [1] introduced the generalized Gauss map $G$ for an oriented regular surface in $\mathbb{R}^{n}$ , which map $p\in M$ to the point in $Q_{n-2}=\{[(z_{1},\cdots,z_{n})]\in\mathbb{CP}^{n-1}:z_{1}^{2}+\cdots+z_{n}^{2}=0\}$ corresponding to the oriented tangent plane of $M$ at $p$ . Then $G$ is a holomorphic map from the Riemann surface $M$ to $\mathbb{CP}^{n-1}$ . So there are many analogous results between Gauss maps of minimal surfaces and meromorphic mappings.

One of them is the Picard theorem. Through the efforts of R. Osserman [2, 3, 4], F. Xavier [5], Mo-Osserman [6], H. Fujimoto [7] finally proved that the Gauss map of a non-flat complete minimal surface in $\mathbb{R}^{3}$ can omit at most $4$ points in $\mathbb{S}^{2}$ . H. Fujimoto [7] also gave the estimate for the number of exceptional values for Gauss map of minimal surfaces in $\mathbb{R}^{4}$ . The number ”four” in $\mathbb{R}^{3}$ is the best possible upper bound since a lot of examples of complete minimal surfaces whose Gauss maps miss $4$ points exist [3, 8]. And the geometry interpretation of the maximal number exceptional value is given by Y. Kawakami [9] (for $\mathbb{R}^{3}$ ) and R. Aiyama, K. Akutagawa, S. Imagawa and Y. Kawakami [10] (for $\mathbb{R}^{4}$ ). The number of exceptional values of Gauss map of minimal surfaces in $\mathbb{R}^{n}$ were investigated by Fujimoto [11] and M. Ru [12].

The second one is the ramification problem. In 1992, H. Fujimoto [13] studied the ramification of Gauss maps of complete regular minimal surfaces in $\mathbb{R}^{3}$ . One says that $g:M\to\bar{\mathbb{C}}=C\cup\{\mathbb{\infty}\}$ is ramified over a point $a_{j}\in\bar{\mathbb{C}}(1\leq j\leq q)$ with multiplicity at least $e_{j}(1\leq j\leq q)$ if all zeros of the function $g(z)-a_{j}$ have orders at least $e_{j}$ . If the image of $g$ omits $a_{j}$ , one will say that $g$ is ramified over $a_{j}$ with multiplicity $e_{j}=\infty$ . Specifically, H. Fujimoto obtained the following result:

Theorem 1.1.

[13] Let M be a non-flat complete minimal surface in $\mathbb{R}^{3}$ . If there are $q(q>4)$ distinct points $a_{1},\cdots,a_{q}\in\bar{\mathbb{C}}$ such that the Gauss map of M is ramified over $a_{j}$ with multiplicity at least $e_{j}$ for each $j$ , then

\sum_{j=1}^{q}(1-1/e_{j})\leq 4.

In particular, if the Gauss map omits five distinct points, then $M$ must be flat. In 1993, M. Ru [14] extended the ramification result to Gauss maps of minimal surfaces in $\mathbb{R}^{n}$ . Afterwards, J. Lu [15], S. J. Kao [16], P. H. Ha, L. B. Phuong, P. D. Thoan, G. Dethloff [17, 18] studied the number of exceptional values and ramification of the Gauss map of complete minimal surfaces in $\mathbb{R}^{n}$ on annular end.

The third one is the unicity problem. H. Fujimoto [19] also investigated the uniqueness theorem for Gauss maps of minimal surfaces in $\mathbb{R}^{n}$ , which is analogue to the Nevanlinna unicity theorem [20] for meromorphic functions on the complex plane $\mathbb{C}$ : Two meromorphic functions on $\mathbb{C}$ sharing $5$ distinct values must be identically equal to each other. Here, two functions $g_{1},g_{2}$ share value $a$ means $g_{1}^{-1}(a)=g_{2}^{-1}(a)$ . For the Gauss map of minimal surface, H. Fujimoto proved the following theorem:

Theorem 1.2.

[19] Let $M$ and $\hat{M}$ be two non-flat minimal surfaces in $\mathbb{R}^{3}$ with their Gauss maps $g$ and $\hat{g}$ respectively. Suppose that there is a conformal diffeomorphism $\Phi$ from $M$ onto $\hat{M}$ and $g,\hat{g}$ share $q$ distinct points $a_{1},a_{2},\cdots,a_{q}$ , then the following statements hold:

•

If $q\geq 7$ and either $M$ or $\hat{M}$ is complete, then $g\equiv\hat{g}\circ\Phi$ .
•

If $q\geq 6$ and both $M$ and $\hat{M}$ are complete and have finite total curvature, then $g\equiv\hat{g}\circ\Phi$ .

The number $7$ is the best possible since H. Fujimoto construct two mutually distinct isometric complete minimal surfaces whose Gauss maps are distinct and have the same inverse images for six points. Later, H. Fujimoto [21], J. Park and M. Ru [22] gave generalizations of the unicity theorem to minimal surfaces in $\mathbb{R}^{n}(n>3)$ . After that, many mathematicians studied unicity theorem for Gauss maps of minimal surfaces (for more details, see [23, 24, 25, 26]).

It is quite natural to study the value distribution problem for Gauss maps of complete space-like stationary surfaces (i.e. surfaces with zero mean curvature and positive-definite metric) in $n$ -dimensional Lorentz space $\mathbb{R}^{n-1,1}$ . E. Calabi [27] showed that any complete maximal space-like surface in $\mathbb{R}^{2,1}$ has to be affine linear, that is the Gauss map must be constant. Furthermore, the ramification problem and the unicity problem of weakly complete maxfaces in $\mathbb{R}^{2,1}$ are investigated by Y. Kawakami in [28]. Here, maxfaces are maximal (or stationary) space-like surfaces with some admissible singularities, as introduced by Umehara and Yamada [29]. In [30], the author, C, Cheng and L. Yang [30] proved the exceptional value theorem for Gauss maps of complete space-like stationary surfaces in $\mathbb{R}^{3,1}$ :

Theorem 1.3.

Let M be a non-flat complete space-like stationary surface in $\mathbb{R}^{3,1}$ , $(\psi_{1},\psi_{2})$ be the Gauss map of $M$ , and $q_{i}$ be the number of exceptional values of $\psi_{i}(i=1,2)$ . If neither $\psi_{1}$ nor $\psi_{2}$ are constant, then $\min\{q_{1},q_{2}\}\leq 3$ or $q_{1}=q_{2}=4$ .

This result cannot be further improved without additional assumptions, since every minimal surface in $\mathbb{R}^{3}$ is a space-like stationary surface in $\mathbb{R}^{3,1}$ . They also considered the situation that Gauss image lies in a graph of a rational function of degree $m$ , which contain the degenerate cases $(m\leq 1)$ . They obtained the following result:

Theorem 1.4.

[30] Let M be a non-flat complete space-like stationary surface in $\mathbb{R}^{3,1}$ , whose Gauss map $(\psi_{1},\psi_{2})$ satisfies $\psi_{2}=f(\psi_{1})$ with $f$ a rational function of degree $m$ , then the number $q_{1}$ of the exceptional values of $\psi_{1}$ should satisfy $|E_{f}|\leq q_{1}\leq m-|E_{f}|+3$ , where $E_{f}=\{z\in\bar{\mathbb{C}},f(z)=\bar{z}\}$ .

In this paper, we study the ramification and the unicity problem for Gauss maps of complete space-like stationary surfaces in $\mathbb{R}^{3,1}$ . Since every complete space-like stationary surface in $\mathbb{R}^{3,1}$ corresponds to a complete regular minimal surface in $\mathbb{R}^{4}$ which has same Gauss map, thus we can easily get Theorem 3.1 and Theorem 4.1 for ramification problem and unicity problem respectively.

Moreover, we extend the ramification result of H. Fujimoto [13] and the unicity theorem of H. Fujimoto [19] to the complete space-like stationary surfaces in $\mathbb{R}^{3,1}$ , whose Gauss map satisfies $\psi_{2}=f(\psi_{1})$ where $f$ is a rational function with degree $m$ . In the case $m\leq 1$ , $M$ is called degenerate, which contains the minimal surfaces in $\mathbb{R}^{3}$ , maximal surfaces in $\mathbb{R}^{2,1}$ , 2-degenerate space-like stationary surface, and three types of space-like stationary graphs in $\mathbb{R}^{3,1}$ (see more details in [30]).

We first give some basic preliminaries in section 2. And in section 3, we study the ramification problem of Gauss map for space-like stationary surface in $\mathbb{R}^{3,1}$ and get the following theorem:

Main Theorem A.

Let $M\subset\mathbb{R}^{3,1}$ is a non-flat complete space-like stationary surface with Gauss map $G=(\psi_{1},\psi_{2})$ satisfying $\psi_{2}=f(\psi_{1})$ , where $f$ is a rational function of degree $m$ . Set $E_{f}=\{z\in\bar{\mathbb{C}},f(z)=\bar{z}\}$ and assume that there are $q(q>m-2|E_{f}|+3)$ distinct points $a_{1},\cdots,a_{q}\in\bar{\mathbb{C}}\setminus E_{f}$ , such that the Gauss map $\psi=\psi_{1}$ of $M$ is ramified over $a_{j}$ with multiplicity at least $e_{j}$ for each $j$ , then

\gamma:=|E_{f}|+\sum_{j=1}^{q}(1-\frac{1}{e_{j}})\leq m-|E_{f}|+3.

(1)

In equation $(\ref{mr})$ , if all $e_{j}(1\leq j\leq q)$ are $\infty$ , then we can easily get the Theorem 1.4.

In section 4, we study the unicity problem for Gauss maps of complete stationary space-like surfaces in $\mathbb{R}^{3,1}$ and get the following theorem:

Main Theorem B.

Let $M$ and $\hat{M}$ be two complete non-flat space-like stationary surfaces in $\mathbb{R}^{3,1}$ with Gauss maps $G=(\psi_{1},\psi_{2})$ and $\hat{G}=(\hat{\psi}_{1},\hat{\psi}_{2})$ respectively. Suppose $\psi_{2}=f(\psi_{1})$ and $\hat{\psi}_{2}=f(\hat{\psi}_{1})$ with $f$ a rational function of degree $m$ and $\Phi:M\to\hat{M}$ is a conformal diffeomorphism. Assume there are $q$ points $a_{1},a_{2},\cdots,a_{q}\in\bar{\mathbb{C}}$ such that

\psi_{1}^{-1}(a_{i})=(\hat{\psi}_{1}\circ\Phi)^{-1}(a_{i}),i=1,\cdots,q.

Then, we have $\psi_{1}\equiv\hat{\psi}_{1}\circ\Phi$ if $q\geq m-|E_{f}|+6$ , where $E_{f}=\{z\in\bar{\mathbb{C}},f(z)=\bar{z}\}$ .

2 Preliminaries

2.1 The Gauss map of space-like stationary surfaces in $\mathbb{R}^{3,1}$

Denote $\mathbb{R}^{3,1}$ be the $4$ -dimensional Minkowski space with the Minkowski inner product:

\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}-u_{4}v_{4},

where $\mathbf{u}=(u_{1},u_{2},u_{3},u_{4}),\mathbf{v}=(v_{1},v_{2},v_{3},v_{4})\in\mathbb{R}^{3,1}$ . $\mathbf{u}\in\mathbb{R}^{3,1}$ 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 or a light-like vector if $\langle\mathbf{u},\mathbf{u}\rangle=0$ .

Let $\mathbf{x}:M\to\mathbb{R}^{3,1}$ be an oriented space-like stationary surface in the Minkowski space. That is the mean curvature vector field $\mathbf{H}$ of $M$ vanishes everywhere, and the pull-back metric $ds^{2}=\langle d\mathbf{x},d\mathbf{x}\rangle$ is positive-definite everywhere. $M$ is stationary if and only if the restriction of each coordinate function on $M$ is harmonic.

Let $(u,v)$ be local isothermal parameters in a neighborhood of $p\in M$ , then the tangent space at $p$ is $T_{p}M=\text{span}\{\mathbf{x}_{u},\mathbf{x}_{v}\}$ . The Gauss map of $M$ is defined by

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

where $\mathbf{G}_{1,1}^{2}$ is the Lorentz–Grassmann manifold consisting of all oriented space-like 2-plane in $\mathbb{R}^{3,1}$ . Denote

Q_{1,1}=\{[\mathbf{z}]\in\mathbb{CP}^{3}:z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\},

Q_{1,1}^{+}=\{[\mathbf{z}]\in Q_{1,1}:|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}-|z_{4}|^{2}>0\}.

Since $\langle\mathbf{x}_{u},\mathbf{x}_{u}\rangle=\langle\mathbf{x}_{v},\mathbf{x}_{v}\rangle>0$ , and $\langle\mathbf{x}_{u},\mathbf{x}_{v}\rangle=0$ , then $[\mathbf{x}_{z}]=[\frac{1}{2}(\mathbf{x}_{u}-i\mathbf{x}_{v})]\in Q_{1,1}^{+}$ , where $z=u+iv$ . We may identify $\mathbf{G}_{1,1}^{2}$ with $Q_{1,1}^{+}$ via the map

i:\Pi=\text{span}\{\mathbf{u},\mathbf{v}\}\in\mathbf{G}_{1,1}^{2}\mapsto[\mathbf{z}]=[\mathbf{u}-i\mathbf{v}]\in Q_{1,1}^{+}.

Proposition 2.1.

[30] For $Q_{1,1}$ and $Q_{1,1}^{+}$ , we have:

(1)

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

$Q_{1,1}^{+}$ is biholomorphic to $\{(w_{1},w_{2})\in\bar{\mathbb{C}}\times\bar{\mathbb{C}}:w_{2}\neq\bar{w}_{1}\}$ , where $\overline{\infty}=\infty$ .

The holomorphic map $\Psi:Q_{1,1}\to\bar{\mathbb{C}}\times\bar{\mathbb{C}}$ is defined by

•

$\Psi([\mathbf{z}])=(\frac{z_{1}+iz_{2}}{z_{3}+z_{4}},\frac{z_{1}-iz_{2}}{z_{3}+z_{4}})$ whenever $z_{3}+z_{4}\neq 0$ ;
•

$\Psi([\mathbf{z}])=(\frac{z_{4}-z_{3}}{z_{1}-iz_{2}},\infty)$ whenever $z_{3}+z_{4}=z_{1}+iz_{2}\equiv 0\text{ and }z_{1}-iz_{2}\neq 0$ ;
•

$\Psi([\mathbf{z}])=(\infty,\frac{z_{4}-z_{3}}{z_{1}+iz_{2}})$ whenever $z_{3}+z_{4}=z_{1}-iz_{2}\equiv 0\text{ and }z_{1}+iz_{2}\neq 0$ ;
•

$\Psi([\mathbf{z}])=(\infty,\infty)$ whenever $z_{3}+z_{4}=z_{1}-iz_{2}=z_{1}+iz_{2}\equiv 0$ .

The metric of $Q_{1,1}^{+}$ in terms of $w_{1}$ and $w_{2}$ is :

g=\text{Re}\left[\frac{4d\bar{w}_{1}dw_{2}}{(\bar{w}_{1}-w_{2})^{2}}\right].

(2)

2.2 Weierstrass representation of space-like stationary Surfaces in $\mathbb{R}^{3,1}$

Denote

\varphi=(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4}):=\Big{(}\frac{\partial x_{1}}{\partial z},\frac{\partial x_{2}}{\partial z},\frac{\partial x_{3}}{\partial z},\frac{\partial x_{4}}{\partial z}\Big{)}dz.

Then, the harmonicity of $x_{k}$ forces $\varphi_{k}$ to be a holomorphic 1-form that can be globally defined on $M$ . $[\mathbf{x}_{z}]\in Q_{1,1}^{+}$ is equivalent to saying that

	$\displaystyle\varphi_{1}^{2}+\varphi_{2}^{2}+\varphi_{3}^{2}-\varphi_{4}^{2}$	$\displaystyle=$	$\displaystyle 0,$		(3)
	$\displaystyle\|\varphi_{1}\|^{2}+\|\varphi_{2}\|^{2}+\|\varphi_{3}\|^{2}-\|\varphi_{4}\|^{2}$	$\displaystyle>$	$\displaystyle 0.$		(4)

Let

(\psi_{1},\psi_{2})=\Psi(\varphi),\quad 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 (3) implies

\varphi_{1}-i\varphi_{2}\equiv 0\text{ 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 2.2.

[31] Let $\psi_{1},\psi_{2}$ be meromorphic functions and $dh$ be a holomorphic 1-form on a Riemann surface $M$ . If $\psi_{1},\psi_{2},dh$ satisfy the regularity conditions $(1),(2)$ and the period condition $(3)$ :

(1)

$\psi_{1}\neq\bar{\psi_{2}}$ 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 $C$ on $M$ :

$\int_{C}\psi_{1}dh=-\overline{\int_{C}\psi_{2}dh}\quad,\quad Re\int_{C}dh=0=Re\int_{C}\psi_{1}\psi_{2}dh\ .$

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

\mathbf{x}=2Re\int\big{(}\psi_{1}+\psi_{2},-i(\psi_{1}-\psi_{2}),1-\psi_{1}\psi_{2},1+\psi_{1}\psi_{2}\big{)}dh\ .

(5)

Conversely, every space-like stationary surface $\mathbf{x}:M\rightarrow\mathbb{R}^{3,1}$ can be represented as equation (5), where $dh$ , $\psi_{1}$ , $\psi_{2}$ satisfy the conditions $(1),(2)$ , and $(3)$ .

The induced metric is:

ds^{2}=\langle\mathbf{x}_{z},\mathbf{x}_{\bar{z}}\rangle=\langle\varphi,\bar{\varphi}\rangle=2|\psi_{1}-\bar{\psi_{2}}|^{2}|dh|^{2}.

(6)

Actually, as shown in [30], $\psi_{1}$ and $\psi_{2}$ correspond to two null vectors $\mathbf{y},\mathbf{y}^{*}$ respectively, which span the normal plane of $M$ at the considered point. Moreover, $\psi_{2}\neq\bar{\psi}_{1}$ is equivalent to saying that $\mathbf{y}\neq\mathbf{y}^{*}$ everywhere.

Remark 2.3.

If we let $\psi_{1}\equiv-\frac{1}{\psi_{2}}$ , (5) 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 $\mathbb{R}^{2,1}$ . Actually, all minimal surfaces in $\mathbb{R}^{3}$ and maximal surfaces in $\mathbb{R}^{2,1}$ are space-like stationary surfaces in $\mathbb{R}^{3,1}$ .

Remark 2.4.

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 $\bar{\mathbb{C}}$ ). Since $(\psi_{1},\psi_{2})=\Psi([\mathbf{x}_{z}])$ , 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},\quad\psi_{2}\Rightarrow\frac{\bar{a}\psi_{2}+\bar{b}}{\bar{c}\psi_{2}+\bar{d}},\quad dh\Rightarrow(c\psi_{1}+d)(\bar{c}\psi_{2}+\bar{d})dh,

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

Let

\varphi^{*}_{k}=\varphi_{k},k=1,2,3,\quad\varphi^{*}_{4}=i\varphi_{4},

we have

\sum_{k=1}^{4}(\varphi^{*}_{k})^{2}=\sum_{k=1}^{4}\varphi^{2}_{k}-\varphi_{4}^{2}=0,

and

\sum_{k=1}^{4}|\varphi^{*}_{k}|^{2}=\sum_{k=1}^{4}|\varphi_{k}|^{2}\geq\sum_{k=1}^{3}|\varphi_{k}|^{2}-|\varphi_{4}|^{2}>0.

(7)

Then

\mathbf{x}^{*}=Re\int(\varphi^{*}_{1},\varphi^{*}_{2},\varphi^{*}_{3},\varphi^{*}_{4})dz

defines a simply-connected regular minimal surface $M^{*}$ in $\mathbb{R}^{4}$ . Conversely, given a simply-connected regular minimal surface $M^{*}$ in $\mathbb{R}^{4}$ , the corresponding stationary surface $M$ may not be space-like. Thus $\mathbf{x}\leftrightarrow\mathbf{x}^{*}$ gives a one-to-one correspondence between all simply-connected generalized stationary surface $M$ in $\mathbb{R}^{3,1}$ and all simply-connected generalized minimal surface $M^{*}$ in $\mathbb{R}^{4}$ .

If $M$ is complete, $M^{*}$ is also complete by the equation (7). Thus, every simply-connected complete space-like stationary surface in $\mathbb{R}^{3,1}$ can be viewed as a simply-connected complete regular minimal surface in $\mathbb{R}^{4}$ .

2.3 Space-like stationary surfaces with rational graphical Gauss image

Let $M$ be a space-like stationary surface in $\mathbb{R}^{3,1}$ . If the Gauss image $G(M)$ of $M$ lies in a hyperplane $H_{A}\subset Q_{1,1}$ with $[A]\in\mathbb{CP}^{2,1}$ , then $M$ is called degenerate. If $M$ is 1-degenerate, then the Gauss map $(\psi_{1},\psi_{2})$ satisfies $\psi_{2}=M_{S}(\psi_{1}):=\frac{a\psi_{1}+b}{c\psi_{1}+d}$ , where $S:=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in SL(2,\mathbb{C})$ . $\psi_{1}\neq\bar{\psi_{2}}$ implies that $\psi_{1}$ must omit all the points in $E_{S}$ , where

E_{S}=\{z\in\overline{\mathbb{C}},M_{S}(z)=\bar{z}\}.

Under the conjugate similarity equivalence relation ( $S_{1}\stackrel{{\scriptstyle conj}}{{\sim}}S_{2}\text{ if and only if }S_{2}=\pm\bar{T}S_{1}T^{-1}$ , where $T=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in SL(2,\mathbb{C})$ . See section 2.4 in [30] for more details ), $M$ is one of the following types:

1.

$M$ is a maximal surface in $\mathbb{R}^{2,1}$ . In this case, $S\stackrel{{\scriptstyle conj}}{{\sim}}\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right)$ and $|E_{S}|=\infty$ ;
2.

$M$ is a degenerate space-like stationary surface of hyperbolic type, which is congruent to entire space-like stationary graph of $F:\mathbb{R}^{2}\to\mathbb{R}^{1,1}$ . In this case, $S\stackrel{{\scriptstyle conj}}{{\sim}}\left(\begin{array}[]{cc}e^{u}&0\\ 0&e^{-u}\end{array}\right)$ with $u\in(0,\infty)$ and $|E_{S}|=2$ ;
3.

$M$ is a minimal surface in $\mathbb{R}^{3}$ or a degenerate space-like stationary surface of elliptic type, which can be deformed by a minimal surface in $\mathbb{R}^{3}$ . In this case, $S\stackrel{{\scriptstyle conj}}{{\sim}}\left(\begin{array}[]{cc}0&ie^{-i\alpha}\\ ie^{i\alpha}&0\end{array}\right)$ with $\alpha\in(0,\frac{\pi}{2}]$ and $|E_{S}|=0$ ;
4.

$M$ is a degenerate space-like stationary surface of parabolic type. In this case, $S\stackrel{{\scriptstyle conj}}{{\sim}}\left(\begin{array}[]{cc}1&1\\ 0&1\end{array}\right)$ and $|E_{S}|=1$ .

If $M$ is 2-degenerate, that is the Gauss image $G(M)$ lies in the intersection of two linear independent hyperplanes in $Q_{1,1}$ . Then $\psi_{1}$ or $\psi_{2}$ is a constant function. By Theorem 4.3 of [30], $M$ has to be an entire graph $F:(x_{1},x_{2})\in\mathbb{R}^{2}\to h(x_{1},x_{2})\mathbf{y}_{0}\in\mathbb{R}^{1,1}$ , where $h(x_{1},x_{2})$ is a harmonic map and $\mathbf{y}_{0}$ is a null vector. Since the Gauss curvature $K\equiv 0$ , the universal covering space $\tilde{M}$ of $M$ is conformally equivalent to the whole complex plane $\mathbb{C}$ .

More generally, if the Gauss map $(\psi_{1},\psi_{2})$ satisfies $\psi_{2}=f(\psi_{1})$ with $f$ a rational function of degree $m$ , then $\psi_{1}\neq\bar{\psi_{2}}$ implies that $\psi_{1}$ must omit all the points in $E_{f}$ , where

E_{f}=\{z\in\overline{\mathbb{C}},f(z)=\bar{z}\}.

Then we have the following results:

Proposition 2.5.

[30] Let $M$ be a non-flat complete space-like stationary surface in $\mathbb{R}^{3,1}$ . If the Gauss map $(\psi_{1},\psi_{2})$ satisfies $\psi_{2}=f(\psi_{1})$ with $f$ a rational function of degree $m$ , then

•

$m\leq 5;$
•

If $m\geq 2$ , then $m-1\leq|E_{f}|\leq\frac{m+3}{2}$ .
•

The number of exceptional values of $\psi_{1}$ in $\bar{\mathbb{C}}$ cannot exceed $m-|E_{f}|+3$ .

2.4 Metrics with negative curvature

Let $\Omega$ be a domain in $\mathbb{C}$ of hyperbolic type which can be endowed with a Poincar $\acute{e}$ metric denoted by

ds^{2}=\lambda_{\Omega}(z)^{2}|dz|^{2},

where $\lambda_{\Omega}(z)$ is a positive $C^{2}$ -function on $\Omega$ satisfying the condition $\Delta log\lambda_{\Omega}=\lambda_{\Omega}^{2}.$ In particular, for a disc $\mathbb{D}(R)=\{z\in\mathbb{C},|z|<R\}$ , we have

\lambda_{\mathbb{D}(R)}(z)=\frac{2R}{R^{2}-|z|^{2}}.

Theorem 2.6.

[32] Let $\Omega$ be a domain in $\mathbb{C}$ and $\lambda$ be a positive $C^{2}$ -function on $\Omega$ satisfying the condition $\Delta log\lambda\geq\lambda^{2}$ . Then, for every holomorphic map $f:\mathbb{D}(R)\to\Omega,$

|f^{\prime}(z)|\lambda(f(z))\leq\frac{2R}{R^{2}-|z|^{2}}.

(8)

Especially, if $f(z)=z$ , then we get

\lambda(z)\leq\frac{2R}{R^{2}-|z|^{2}}.

3 Ramification problem

Let $M$ be a space-like stationary surfaces in $\mathbb{R}^{3,1}$ with Gauss map $G=(\psi_{1},\psi_{2})$ , and the induced metric $(\ref{phi5})$ satisfies

ds^{2}=2|\psi_{1}-\bar{\psi_{2}}|^{2}|dh|^{2}\leq 2(1+|\psi_{1}|^{2})(1+|\psi_{2}|^{2})|dh|^{2}=d\tilde{s}^{2}.

Then $(M,d\tilde{s}^{2})$ is a complete Riemann surface if $ds^{2}$ is complete. Thus the following ramification result for Gauss map of stationary surfaces in $\mathbb{R}^{3,1}$ can be derived from Theorem III in [33] :

Theorem 3.1.

Let $M\subset\mathbb{R}^{3,1}$ is a non-flat complete space-like stationary surface with Gauss map $G=(\psi_{1},\psi_{2})$ . Then if $\psi_{1},\psi_{2}$ are not constant, and $\psi_{i}(i=1,2)$ is ramified over $a^{ij}(1\leq j\leq q_{i})$ with multiplicity at least $m_{ij}$ , then $\min\{\gamma_{1},\gamma_{2}\}\leq 3$ , or $\gamma_{1}=\gamma_{2}=4$ , where

\gamma_{1}:=\Sigma_{j=1}^{q_{1}}(1-\frac{1}{m_{1j}}),\quad\gamma_{2}:=\Sigma_{j=1}^{q_{2}}(1-\frac{1}{m_{2j}}).

Remark 3.2.

If all $m_{ij}=\infty(i=1,2,j=1,\cdots,q_{i})$ , that is $a^{ij}$ are exceptional values of $\psi_{i}$ . Then $\gamma_{i}=q_{i}(i=1,2)$ are the number of exceptional values, and the above theorem is a generalization of Theorem $1.2$ in [30].

Next, we will discuss the ramification problem with condition $\psi_{2}=f(\psi_{1})$ where $f(z)=\frac{P(z)}{Q(z)}$ is a rational function of degree $m$ . Throughout the proof process, we may assume $M$ is simply connected, otherwise we consider its universal covering space. By Koebe’s uniformization theorem, $M$ is conformally equivalent to the whole complex plane $\mathbb{C}$ or to the unit disc $\mathbb{D}$ . For the case $M=\mathbb{C}$ , $\gamma$ defined in (1) satisfies $\gamma\leq 2$ by the following theorem:

Theorem 3.3.

[34] Let $f$ be a non-constant meromorphic function over $\mathbb{C}$ , if there are $q$ distinct points $a_{1},a_{2},\cdots,a_{q}\in\overline{\mathbb{C}}$ such that all the roots of the equation $f(z)=a_{i}$ have multiplicity at least $e_{i}$ ( $e_{i}=\infty$ if $f(z)=a_{i}$ has no root). Then

\sum_{i=1}^{q}(1-\frac{1}{e_{i}})\leq 2.

(9)

Proposition 3.4.

•

If $M$ is 2-degenerate $(m=0)$ or 1-degenerate of hyperbolic type( $m=1,|E_{f}|=2$ ), $M$ is an entire graph over $\mathbb{R}^{2}$ and the universal covering space $\tilde{M}$ of $M$ is conformally equivalent to the whole complex plane. Then $\gamma\leq 2$ by Theorem 3.3.
•

If $M$ is 1-degenerate of elliptic type( $m=1,|E_{f}|=0$ ), $M$ can be deformed by a minimal surface in $\mathbb{R}^{3}$ , thus $\gamma\leq 4$ by Theorem 1.1.

Thus, we just need to consider the 1-degenerate of parabolic type ( $m=1,|E_{f}|=1$ ) and the case $2\leq m\leq 5$ . Since $E_{f}\neq\emptyset$ , without loss of generality, we can assume $\infty\in E_{f}$ under a suitable Möbius transformation. This means $f(\infty)=\infty$ , hence

m=degreeP>degreeQ=n.

Set

E_{f}=\{c_{1},c_{2},\cdots,c_{l},\infty\}

and let $b_{1},\cdots,b_{s}$ be the zeros of $Q(z)$ with orders $r_{1},\cdots,r_{s}$ respectively. Then

|f(z)-\bar{z}|\leq C\left(\Pi_{i=1}^{l}|z-c_{i}|\right)\left(\Pi_{j=1}^{s}|z-b_{j}|^{-r_{j}}\right)(1+|z|^{2})^{\frac{m-l}{2}}.

Let $\psi:=\psi_{1}$ , the induced metric on $M$ is

\begin{split}ds^{2}&=|f(\psi)-\bar{\psi}|^{2}|dh|^{2}\\ &\leq\left(\Pi_{i=1}^{l}|\psi-c_{i}|^{2}\right)\left(\Pi_{j=1}^{s}|\psi-b_{j}|^{-2r_{j}}\right)(1+|\psi|^{2})^{m-l}|dh|^{2}\\ &=\left(\Pi_{i=1}^{l}|\psi-c_{i}|^{2}\right)(1+|\psi|^{2})^{m-l}|\omega|^{2}=d\tilde{s}^{2},\end{split}

(10)

then $d\tilde{s}^{2}$ is also a complete metric on $M$ , where

\omega=\frac{dh}{\Pi_{j=1}^{s}|\psi-b_{j}|^{r_{j}}}

is a holomorphic 1-form on $M$ with no zero. Since

m-1\leq|E_{f}|=l+1\leq\frac{m+3}{2},

we have $0<m-l\leq 2$ .

For a non-zero meromorphic function $f$ , the divisor $\nu_{f}$ of $f$ is defined as follows:

\nu_{f}(a)=\begin{cases}k&\text{ if $f$ has a zero of order $k$ at $a$, }\\ -p&\text{ if $f$ has a pole of order $p$ at $a$, }\\ 0&\text{ otherwise.}\end{cases}

Lemma 3.5.

Assume that there are $q$ distinct points $a_{1},\cdots,a_{q}\in\bar{\mathbb{C}}\setminus E_{f}$ , such that the meromorphic function $\psi$ on $\mathbb{D}(R)$ is ramified over $a_{j}$ with multiplicity at least $e_{j}$ for each j. If $l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-1>\epsilon>(l+q)\epsilon^{\prime}>0$ , where $0<\epsilon^{\prime}<1$ . Set

v=\frac{(1+\psi^{2})^{\frac{(m-l)p}{2}}\psi^{\prime}}{\Pi_{i=1}^{l}(\psi-c_{i})^{1-\epsilon^{\prime}}\Pi_{j=1}^{q}(\psi-a_{j})^{1-\frac{1}{e}_{j}-\epsilon^{\prime}}}

(11)

on $\mathbb{D}(R)\setminus\{z,\psi(z)=a_{j}\text{ for some j}\}$ and $v=0$ on $\mathbb{D}(R)\cap\{z,\psi(z)=a_{j}\text{ for some j}\}$ . Set

p=\frac{l+q-\sum_{j=1}^{q}\frac{1}{e}_{j}-1-\epsilon}{m-l}.

Then $v$ is a continuous function in $\mathbb{D}(R)$ , and there exists a positive constant $B$ such that

|v|\leq B\frac{2R}{R^{2}-|z|^{2}}.

(12)

Proof.

First, we prove the continuity of $v$ . Obviously, $v$ is continuous on

\mathbb{D}(R)\setminus\{z,\psi(z)=a_{j}\text{ for some }0\leq j\leq q\}.

Take a point $z_{0}$ with $\psi(z_{0})=a_{j}$ for some $j$ . Now, we write

\psi(z)-a_{j}=(z-z_{0})^{r_{0}}h(z),

where $r_{0}=\nu_{\psi}(z_{0})\geq e_{j}$ and $h(z)$ is a meromorphic function with $h(z_{0})\neq 0$ . Then

\psi^{\prime}=(z-z_{0})^{r_{0}-1}(r_{0}h(z)+(z-z_{0})h^{\prime}(z)),

(13)

that is $\nu_{\psi^{\prime}}(z_{0})=\nu_{\psi}(z_{0})-1=r_{0}-1$ . Since

	$\displaystyle\nu_{v}(z_{0})$	$\displaystyle=\nu_{\psi^{\prime}}(z_{0})-(1-\frac{1}{e}_{j}-\epsilon^{\prime})\nu_{\psi}(z_{0})$
		$\displaystyle=r_{0}-1-r_{0}(1-\frac{1}{e}_{i}-\epsilon^{\prime})$
		$\displaystyle=\frac{r_{0}}{e_{i}}-1+r_{0}\epsilon^{\prime}>0,$

then $\lim_{z\to z_{0}}v=0$ . This implies that $v$ is continuous on $\mathbb{D}(R)$ .

Obviously, (12) holds if $z\in\mathbb{D}(R)\cap\{z,\psi(z)=a_{j}\text{ for some j}\}$ . If $z\in\mathbb{D}(R)\setminus\{z,\psi(z)=a_{j}\text{ for some j}\}$ , let $\Omega=\mathbb{C}\setminus\{a_{1},\dots,a_{q},c_{1},\dots,c_{l}\}$ and $\lambda_{\Omega}$ be the Poincar $\acute{e}$ metric of $\Omega$ , that is $\lambda_{\Omega}$ satisfies $\Delta log\lambda_{\Omega}=\lambda_{\Omega}^{2},$ then (8) implies that

|\psi^{\prime}(z)|\lambda_{\Omega}(\psi(z))\leq\frac{2R}{R^{2}-|z|^{2}}.

(14)

Moreover, $\lambda_{\Omega}$ satisfies(see e.g. p.250 of [35])

\lambda_{\Omega}(\psi)\sim\frac{1}{|\psi-a_{i}|\log|\psi-a_{i}|^{-1}}\quad\text{ near }a_{i},

\lambda_{\Omega}(\psi)\sim\frac{1}{|\psi-c_{j}|\log|\psi-c_{j}|^{-1}}\quad\text{ near }c_{j},

\lambda_{\Omega}(\psi)\sim\frac{1}{|\psi|\log|\psi|}\quad\text{ near }\infty.

Denote

u=\frac{(1+|\psi|^{2})^{\frac{(m-l)p}{2}}}{\Pi_{i=1}^{l}|\psi-c_{i}|^{1-\epsilon^{\prime}}\Pi_{j=1}^{q}|\psi-a_{j}|^{1-\frac{1}{e}_{j}-\epsilon^{\prime}}\lambda_{\Omega}(\psi)},

then by a direct calculation, we get $u$ is bounded by a constant $B$ .

Thus by Theorem 2.6, we get

v=u|\psi^{\prime}|\lambda_{\Omega}(\psi)\leq B\frac{2R}{R^{2}-|z|^{2}}.

∎

Now, we begin to proof the Main Theorem A.

Proof.

If the equation (1) is wrong, then

l+1+\sum_{j=1}^{q}(1-\frac{1}{e_{j}})>m-l+2,

that is

2l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-m-1>0.

(15)

Take $\epsilon^{\prime}$ with

\frac{2l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-m-1}{l+q}>\epsilon^{\prime}>\frac{2l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-m-1}{q+m},

and set

\tilde{p}=\frac{1}{p}=\frac{m-l}{l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-1-\epsilon}.

Then

0<\tilde{p}=\frac{m-l}{l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-1-\epsilon}<\frac{m-l}{m-l+2-\epsilon}<1,

and

	$\displaystyle\frac{\tilde{p}}{1-\tilde{p}}>\frac{\epsilon^{\prime}\tilde{p}}{1-\tilde{p}}$	$\displaystyle=\frac{(m-l)\epsilon^{\prime}}{l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-1-\epsilon-m+l}$
		$\displaystyle=\frac{(m-l)\epsilon^{\prime}}{2l+q-\sum_{j=1}^{q}\frac{1}{e_{j}}-1-m-\epsilon}$
		$\displaystyle>\frac{(m-l)\epsilon^{\prime}}{(m+q)\epsilon^{\prime}-\epsilon}$
		$\displaystyle=\frac{(m-l)\epsilon^{\prime}}{(m-l)\epsilon^{\prime}+(l+q)\epsilon^{\prime}-\epsilon}>1.$

Now, we consider $M=\mathbb{D}$ and

E=\{z\in\mathbb{D},\psi^{\prime}(z)=0\}.

Non-flatness of $M$ implies $E\neq\mathbb{D}$ .

Let $\omega=gdz$ , where $g$ is a holomorphic function on $\mathbb{D}$ with no zero, and consider the following many-valued function:

\eta=\frac{g^{\frac{1}{1-\tilde{p}}}\Pi_{i=1}^{l}(\psi-c_{i})^{\frac{\tilde{p}(1+p-\epsilon^{\prime})}{1-\tilde{p}}}\Pi_{j=1}^{q}\left(\psi-a_{j}\right)^{\frac{\tilde{p}(1-\frac{1}{e_{j}}-\epsilon^{\prime})}{1-\tilde{p}}}}{\psi^{\prime\frac{\tilde{p}}{1-\tilde{p}}}}

on $\mathbb{D}^{\prime}=\mathbb{D}\setminus E$ .

Take an arbitrary single-value branch of $\eta$ , still denoted by $\eta$ for the matter of convenience. Let

w=F(z)=\int\eta dz

be a holomorphic mapping from $\mathbb{D}^{\prime}$ to $\mathbb{C}$ , satisfying $F(0)=0,F^{\prime}(z)=\eta(z)\neq 0$ . Hence there exists a holomorphic inverse mapping $z=H(w)$ on a neighborhood of $0$ . Let $\mathbb{D}(R)=\{w:|w|\leq R\}$ is the largest ball that $H$ can be defined, then $R<+\infty$ (otherwise, $H$ is a non-constant bounded entire function, which contracts to the Liouville theorem) and there exists a point $a$ on the boundary of $\mathbb{D}(R)$ , such that $H$ cannot be extended beyond a neighborhood of $a$ . Let $l_{a}:=\{ta:0\leq t<1\}$ be the straight line segment starting from $0$ , then $H(l_{a})$ must be a divergent curve in $\mathbb{D}^{\prime}$ as $t$ tends to $1$ .

Lemma 3.6.

There exists a point with $|a|=R$ such that $H(l_{a})$ is a divergent curve in $\mathbb{D}$ .

Proof.

If $H(l_{a})$ tends to a point $z_{0}$ such that $\psi^{\prime}(z_{0})=a_{j}$ for some $0\leq j\leq q$ , then we have two cases:

Case 1: $\psi(z_{0})\neq a_{j}$ for all $1\leq j\leq q$ , then $\nu_{\eta}(z_{0})=-\frac{k\tilde{p}}{1-\tilde{p}}$ . Since

\frac{k\tilde{p}}{1-\tilde{p}}>\frac{\tilde{p}}{1-\tilde{p}}>1,

then

	$\displaystyle R$	$\displaystyle=\int_{l_{a}}\|dw\|=\int_{H(l_{a})}\left\|\frac{dw}{dz}\right\|\|dz\|$
		$\displaystyle=\int_{H(l_{a})}\|\eta\|\|dz\|=\infty,$

which contradicts with $R<\infty$ .

Case 2: $\psi(z_{0})=a_{j}$ for some $j\in\{1,2,\cdots,q\}$ . Then $\nu_{\psi}(z_{0})\geq e_{j}$ , and $\nu_{\psi^{\prime}}(z_{0})=\nu_{\psi}(z_{0})-1$ .

	$\displaystyle\nu_{\eta}(z_{0})$	$\displaystyle=\frac{\tilde{p}}{1-\tilde{p}}(\nu_{\psi}(z_{0})(1-\frac{1}{e_{j}}-\epsilon^{\prime})-\nu_{\psi}(z_{0})+1)$
		$\displaystyle=\frac{\tilde{p}}{1-\tilde{p}}(1-\nu_{\psi}(z_{0})(\frac{1}{e_{j}}+\epsilon^{\prime}))$
		$\displaystyle\leq\frac{\tilde{p}}{1-\tilde{p}}(1-e_{j}(\frac{1}{e_{j}}+\epsilon^{\prime}))$
		$\displaystyle=-\frac{e_{j}\epsilon^{\prime}\tilde{p}}{1-\tilde{p}}<-1$

then $\int_{H(l_{a})}|\eta||dz|=+\infty$ and forces a contradiction.

∎

Observing that

\frac{dw}{dz}=\frac{g\Pi_{i=1}^{l}(\psi-c_{i})^{\tilde{p}(1+p-\epsilon^{\prime})}\Pi_{j=1}^{q}\left({\psi-a_{j}}\right)^{\tilde{p}(1-\frac{1}{e_{j}}-\epsilon^{\prime})}}{(\psi^{\prime})^{\tilde{p}}}\left(\frac{dw}{dz}\right)^{\tilde{p}},

then

\left|\frac{dz}{dw}\right|^{2}=\frac{|(\psi\circ H)^{\prime}|^{2\tilde{p}}\left|\frac{dz}{dw}\right|^{2\tilde{p}}}{|g\circ H|^{2}\Pi_{i=1}^{l}|\psi\circ H-c_{i}|^{2\tilde{p}(1+p-\epsilon^{\prime})}\Pi_{j=1}^{q}|\psi\circ H-a_{j}|^{2\tilde{p}(1-\frac{1}{e_{j}}-\epsilon^{\prime})}},

and the pull back metric on $\mathbb{D}(R)$ is

	$\displaystyle H^{*}d\tilde{s}^{2}$	$\displaystyle=\bigg{(}\left(\Pi_{i=1}^{l}\|\psi-c_{i}\|^{2}\right)(1+\|\psi\|^{2})^{m-l}\|g\|^{2}\bigg{)}\circ H\left\|\frac{dz}{dw}\right\|^{2}\|dw\|^{2}$
		$\displaystyle=\frac{\left(\Pi_{i=1}^{l}\|\psi(w)-c_{i}\|^{2}\right)(1+\|\psi(w)\|^{2})^{m-l}\|g(w)\|^{2}\|\psi^{\prime}(w)\|^{2\tilde{p}}}{\|g(w)\|^{2}\Pi_{i=1}^{l}\|\psi(w)-c_{i}\|^{2\tilde{p}(1+p-\epsilon^{\prime})}\Pi_{j=1}^{q}\|\psi(w)-a_{j}\|^{2\tilde{p}(1-\frac{1}{e_{j}}-\epsilon^{\prime})}}\|dw\|^{2}$
		$\displaystyle=\left(\frac{(1+\|\psi\|^{2})^{\frac{(m-l)p}{2}}\psi^{\prime}}{\Pi_{i=1}^{l}(\psi-c_{i})^{1-\epsilon^{\prime}}\Pi_{j=1}^{q}(\psi-a_{j})^{1-\frac{1}{e}_{j}-\epsilon^{\prime}}}\right)^{2\tilde{p}}\|dw\|^{2}$
		$\displaystyle\leq\left(B\frac{2R}{R^{2}-\|w\|^{2}}\right)^{2\tilde{p}}.$

Thus

\int_{H(l_{a})}d\tilde{s}=\int_{l_{a}}H^{*}d\tilde{s}\leq\int_{0}^{R}\left(\frac{2R}{R^{2}-|r|^{2}}\right)^{\tilde{p}}dr.

(16)

The integral (16) is convergent since $0<\tilde{p}<1$ , which contradicts to the completeness of $(M,d\tilde{s}^{2})$ . ∎

Remark 3.7.

In equation $(\ref{mr})$ ,

•

If $m=0,|E_{f}|=1$ ( $M$ is 2-degenerate), then $\gamma\leq 2$ ;
•

If $m=1,|E_{f}|=2$ ( $M$ is 1-degenerate of hyperbolic type), then $\gamma\leq 2$ ;
•

If $m=1,|E_{f}|=0$ ( $M$ is 1-degenerate of elliptic type), then $\gamma\leq 4$ .

These conclusions are agree with the Proposition 3.4.

Thus, by combining the above proof process and Proposition 3.4, we get Main Theorem A.

4 Unicity problem

Similarly as in Section 3, we first consider Riemann surface $M$ and metric

d\tilde{s}^{2}=2(1+|\psi_{1}|^{2})(1+|\psi_{2}|^{2})|dh|^{2}.

Thus we can easily get the following theorem by Theorem $1.2$ in [23].

Theorem 4.1.

Let $M$ and $\hat{M}$ be two complete non-flat space-like stationary surfaces in $\mathbb{R}^{3,1}$ with Gauss maps $G=(\psi_{1},\psi_{2})$ and $\hat{G}=(\hat{\psi}_{1},\hat{\psi}_{2})$ respectively. Assume $\Phi:M\to\hat{M}$ is a conformal diffeomorphism and $\psi_{1},\psi_{2},\hat{\psi}_{1},\hat{\psi}_{2}$ are not constant. If for each $i(i=1,2)$ , $\psi_{i}$ and $\hat{\psi}_{i}$ share $p_{i}>4$ distinct values and $\psi_{i}\neq\hat{\psi}_{i}\circ\Phi$ , then we have $\min\{p_{1},p_{2}\}\leq 6.$ In particular, if $p_{1}\geq 7$ and $p_{2}\geq 7$ , then either $\psi_{1}\equiv\hat{\psi}_{1}\circ\Phi$ or $\psi_{2}\equiv\hat{\psi}_{2}\circ\Phi$ , or both hold.

Remark 4.2.

Since every minimal surface in $\mathbb{R}^{3}$ is also a space-like stationary surface in $\mathbb{R}^{3,1}$ with Gauss map $G=(\psi_{1},\psi_{2})$ and $\psi_{2}=-\frac{1}{\psi_{1}}$ , then theorem 1.2 and the optimality in $\mathbb{R}^{3}$ tell us the above result is also optimal.

Next, we will consider the space-like stationary surface with Gauss map $G=(\psi_{1},\psi_{2})$ satisfying $\psi_{2}=f(\psi_{1})$ , where $f$ is a rational function with degree $m$ . Now, assume $\Phi:M\to\hat{M}$ be the conformal diffeomorphism, and $(\psi_{1},\psi_{2})$ and $(\hat{\psi_{1}},\hat{\psi_{2}})$ be the Gauss map of $M$ and $\hat{M}$ respectively, where $\psi_{2}=f(\psi_{1})$ , $\hat{\psi_{2}}=f(\hat{\psi_{1}})$ . For brevity, we denote $\psi_{1},\hat{\psi}_{1}\circ\Phi$ by $\psi,\hat{\psi}$ .

If $M,\hat{M}$ are 2-degenerate $(m=0)$ or 1-degenerate of hyperbolic type( $m=1,|E_{f}|=2$ ), then they are entire graphs over $\mathbb{R}^{2}$ and their universal covering spaces are conformally equivalent to the whole complex plane. Then the Main Theorem B is followed by the Nevanlinna unicity theorem [20] for meromorphic functions on the complex plane $\mathbb{C}$ .

If $M,\hat{M}$ are 1-degenerate of elliptic type( $m=1,|E_{f}|=0$ ), then they can be deformed by minimal surfaces in $\mathbb{R}^{3}$ , and the Main Theorem B is followed by Theorem 1.2.

Thus, we only need to consider the proof when $m\geq 2$ and $M,\hat{M}$ are 1-degenerate of parabolic type( $m=1,|E_{f}|=1$ ). For each $\alpha,\beta\in\bar{\mathbb{C}}$ , the chordal distance[13] between $\alpha,\beta$ is

|\alpha,\beta|=\frac{|\alpha-\beta|}{\sqrt{1+|\alpha|^{2}}\sqrt{1+|\beta|^{2}}},

if $\alpha\neq\infty,\beta\neq\infty$ , and $|\alpha,\beta|=|\beta,\alpha|=\frac{1}{\sqrt{1+|\alpha|^{2}}}$ , if $\beta=\infty$ .

Lemma 4.3.

[19] Let $f$ and $\hat{f}$ be two mutually distinct non-constant meromorphic functions on a Riemann surface $M$ and $q$ distinct points $a_{1},a_{2},\cdots,a_{q}(q>4)$ . Assume that $f^{-1}(a_{j})=\hat{f}^{-1}(a_{j})(1\leq j\leq q).$ For $a_{0}>0$ and $\epsilon$ with $q-4>q\epsilon>0$ , set

\lambda:=\left(\Pi_{i=1}^{q}|f,a_{i}|\log\left(\frac{a_{0}}{|f,a_{i}|^{2}}\right)\right)^{-1+\epsilon},

\hat{\lambda}:=\left(\Pi_{i=1}^{q}|\hat{f},a_{i}|\log\left(\frac{a_{0}}{|\hat{f},a_{i}|^{2}}\right)\right)^{-1+\epsilon},

d\tau^{2}:=|f,\hat{f}|^{2}\lambda\hat{\lambda}\frac{f^{\prime}}{1+|f|^{2}}\frac{\hat{f}^{\prime}}{1+|\hat{f}|^{2}}

(17)

outside the $E:=\cup_{i=1}^{q}f^{-1}(a_{i})$ and $d\tau^{2}=0$ on $E$ . Then, for a suitably chosen $a_{0}$ , $d\tau^{2}$ is continuous on $M$ , and has strictly negative curvature on the set $\{d\tau^{2}\neq 0\}$ .

Lemma 4.4.

[19] Let $f$ and $\hat{f}$ be two mutually distinct non-constant meromorphic functions on a Riemann surface $M$ satisfies the same assumption as in Lemma 4.3, then for the metric $d\tau^{2}$ defined by (17), there is a constant $C>0$ such that

d\tau^{2}\leq C\frac{4R^{2}}{(R^{2}-z^{2})^{2}}|dz|^{2}.

Combine the completeness of $M$ and inequality (10), the conformal metric $ds^{2}$ can be stated as follows:

ds^{2}=\Pi_{i=1}^{l}|\psi-c_{i}|(1+|\psi|^{2})^{m-l}|h|^{2}|dz|^{2},

where $h$ is a holomorphic function with no zeros on a simply connected open set $U$ .

Since there exists a local non-zero holomorphic function $\zeta$ on $U$ such that $ds^{2}=|\zeta|^{2}\Phi^{*}(d\hat{s}^{2})$ , thus

\Pi_{i=1}^{l}|\psi-c_{i}|^{2}(1+|\psi|^{2})^{m-l}|h|^{2}|dz|^{2}=|\zeta|^{2}\Pi_{i=1}^{l}|\hat{\psi}-c_{i}|^{2}(1+|\hat{\psi}|^{2})^{m-l}|\hat{h}|^{2}|dz|^{2},

\Rightarrow\Pi_{i=1}^{l}|\psi-c_{i}|(1+|\psi|^{2})^{\frac{m-l}{2}}|h|=|\zeta|\Pi_{i=1}^{l}|\hat{\psi}-c_{i}|(1+|\hat{\psi}|^{2})^{\frac{m-l}{2}}|\hat{h}|.

Let $k$ be a non-zero holomorphic function on $U$ satisfies $k^{2}=h\hat{h}\zeta$ , then

ds^{2}=|k|^{2}\Pi_{i=1}^{l}(|\psi-c_{i}||\hat{\psi}-c_{i}|)(1+|\psi|^{2})^{\frac{m-l}{2}}(1+|\hat{\psi}|^{2})^{\frac{m-l}{2}}|dz|^{2}.

Assume there are $q$ distinct points $a_{1},\cdots,a_{q}\in\bar{\mathbb{C}}$ such that $\psi^{-1}(a_{i})=\hat{\psi}^{-1}(a_{i})$ for all $1\leq i\leq q$ . Since $\psi^{-1}(E_{f})=\hat{\psi}^{-1}(E_{f})=\emptyset$ , then $E_{f}\subset\{a_{1},\cdots,a_{q}\}$ . We may assume $a_{q}=\infty\in E_{f}$ .

Since $q\geq m-|E_{f}|+6=m-l+5$ , then $q-m+l-4>0$ . Take a positive real number $\delta$ with

\frac{q-m+l-4}{q}>\delta>\min\left\{\frac{q-m+l-4}{m-l+q},\frac{q-4-2m+2l}{q}\right\}

and set

p=\frac{m-l}{q-4-q\delta}<1.

Then

\frac{p}{1-p}>1,\quad\frac{\delta p}{1-p}>1.

If $\psi\not\equiv\hat{\psi}$ , consider the function

\tilde{\eta}=k^{\frac{1}{1-p}}\frac{\Pi_{i=1}^{l}\left((\psi-c_{i})(\hat{\psi}-c_{i})\right)^{\frac{1}{2(1-p)}}\Pi_{j=1}^{q-1}\left((\psi-a_{j})(\hat{\psi}-a_{j})\right)^{\frac{p(1-\delta)}{2(1-p)}}}{\left((\psi-\hat{\psi})^{2}\psi^{\prime}\hat{\psi}^{\prime}\Pi_{j=1}^{q-1}(1+|a_{j}|^{2})^{1-\delta}\right)^{\frac{p}{2(1-p)}}}

(18)

on $M^{\prime}=M\setminus E$ , where $E=\{z\in M,\psi^{\prime}(z)\hat{\psi}^{\prime}(z)=0,\text{ or }\psi(z)=\hat{\psi}(z)\}.$

Denote

w=\tilde{F}(z)=\int\tilde{\eta}dz,

(19)

by similar method in the proof of Main Theorem A, we can find a neighborhood $U$ of point $0$ and a positive number R, such that the inverse map $\tilde{H}=(\tilde{F}|_{U})^{-1}:\mathbb{D}(R)\to U\subset M^{\prime}$ is a holomorphic map. Since $d\sigma^{2}=|\tilde{\eta}|^{2}|dz|^{2}$ has strictly negative curvature on $M^{\prime}$ by Lemma 4.3, then $R<\infty$ . And there exists a point $a$ with $|a|=R$ , such that for the line segment

l_{a}=\{ta,0\leq t<1\},

the image $\tilde{\gamma}=\tilde{H}(l_{a})$ is a divergent curve in $M^{\prime}$ as $t$ tends to $1$ .

Lemma 4.5.

There exists a point with $|a|=R$ such that $\tilde{H}(l_{a})$ is a divergent curve in $M$ .

Proof.

It is suffices to consider that $\tilde{H}(l_{a})$ tends to a point $z_{0}\in E$ .

Case 1: If $\psi(z_{0})=\hat{\psi}(z_{0})$ and there exists some $a_{j},$ such that $\psi(z_{0})=\hat{\psi(z_{0})}=a_{j}$ , then

\begin{split}\nu_{\tilde{\eta}}(z_{0})&=\frac{p}{1-p}\left((1-\delta)(\nu_{\psi}(z_{0})+\nu_{\hat{\psi}}(z_{0}))-2\min\{\nu_{\psi}(z_{0}),\nu_{\hat{\psi}}(z_{0})\}+\nu_{\psi}(z_{0})+\nu_{\hat{\psi}}(z_{0})-2\right)\\ &\leq-\frac{\delta p}{1-p}(\nu_{\psi}(z_{0})+\nu_{\hat{\psi}}(z_{0}))\\ &<-\frac{\delta p}{1-p}.\end{split}

Otherwise,

\nu_{\tilde{\eta}}(z_{0})=-\frac{p}{1-p}(2\min\{\nu_{\psi}(z_{0}),\nu_{\hat{\psi}}(z_{0})\})\leq-\frac{2\delta p}{1-p}.

Since $\frac{\delta p}{1-p}>1$ , then

	$\displaystyle R$	$\displaystyle=\int_{l_{a}}\|dw\|=\int_{\tilde{\gamma}}\left\|\frac{dw}{dz}\right\|\|dz\|$
		$\displaystyle=\int_{\tilde{\gamma}}\|\tilde{\eta}\|\|dz\|=\infty,$

which contradicts with $R<\infty$ .

Case 2: $\tilde{H}(l_{a})$ tends to a point $z_{0}$ such that $\psi^{\prime}(z_{0})\hat{\psi^{\prime}}(z_{0})=0$ , we can easily get $\nu_{\tilde{\eta}}(z_{0})<-\frac{p}{1-p}<-1$ . This also contradicts that $R$ is finite. ∎

Combine with (18) and (19), we obtain

\left|\frac{dw}{dz}\right|=\frac{|k|\Pi_{i=1}^{l}\left(|\psi-c_{i}||\hat{\psi}-c_{i}|\right)^{\frac{1}{2}}\Pi_{j=1}^{q-1}\left(|\psi-a_{j}||\hat{\psi}-a_{j}|\right)^{\frac{p(1-\delta)}{2}}}{\left(|\psi-\hat{\psi}|^{2}|\psi^{\prime}||\hat{\psi}^{\prime}|\Pi_{j=1}^{q-1}(1+|a_{j}|^{2})^{1-\delta}\right)^{\frac{p}{2}}}\left|\frac{dw}{dz}\right|^{p}.

Set $g(w)=\psi(\tilde{H}(w)),\hat{g}(w)=\hat{\psi}(\tilde{H}(w))$ , since $g^{\prime}=\psi^{\prime}\frac{dz}{dw},\hat{g}^{\prime}=\hat{\psi}^{\prime}\frac{dz}{dw}$ , then

\left|\frac{dz}{dw}\right|^{2}=\frac{\left(|g-\hat{g}|^{2}|g^{\prime}||\hat{g}^{\prime}|\Pi_{j=1}^{q-1}(1+|a_{j}|^{2})^{1-\delta}\right)^{p}}{|k\circ\tilde{H}|^{2}\Pi_{i=1}^{l}\left(|g-c_{i}||\hat{g}-c_{i}|\right)\Pi_{j=1}^{q-1}\left(|g-a_{j}||\hat{g}-a_{j}|\right)^{p(1-\delta)}}

Therefore, the induced metric of $\mathbb{D}(R)$ from $M$ by $\tilde{H}$ is

	$\displaystyle\tilde{H}^{*}ds^{2}$	$\displaystyle=\|k\circ\tilde{H}\|^{2}\Pi_{i=1}^{l}(\|g-c_{i}\|\|\hat{g}-c_{i}\|)(1+\|g\|^{2})^{\frac{m-l}{2}}(1+\|\hat{g}\|^{2})^{\frac{m-l}{2}}\left\|\frac{dz}{dw}\right\|^{2}\|dw\|^{2}$
		$\displaystyle=\left(\frac{(1+\|g\|^{2})^{\frac{m-l}{2p}}(1+\|\hat{g}\|^{2})^{\frac{m-l}{2p}}\|g-\hat{g}\|^{2}\|g^{\prime}\|\|\hat{g}^{\prime}\|\Pi_{j=1}^{q-1}(1+\|a_{j}\|^{2})^{1-\delta}}{\Pi_{j=1}^{q-1}\|g-a_{j}\|^{1-\delta}\|\hat{g}-a_{j}\|^{1-\delta}}\right)^{p}\|dw\|^{2}$
		$\displaystyle=\left(\mu^{2}\Pi_{i=1}^{q}(\|g,a_{i}\|\|\hat{g},a_{i}\|)^{\epsilon}\left(\Pi_{i=1}^{q}\log\frac{a_{0}}{\|g,a_{i}\|^{2}}\log\frac{a_{0}}{\|\hat{g},a_{i}\|^{2}}\right)^{1-\epsilon}\right)^{p}\|dw\|^{2},$

where $\mu$ is the function with $d\tau^{2}=\mu^{2}|dw|^{2}$ . On the other hand, for a given $\epsilon$ , it holds that

\lim_{x\to 0}x^{\epsilon}\log^{1-\epsilon}\frac{a_{0}}{x^{2}}<\infty.

Thus there exists a constant $C_{1}$ such that

\tilde{H}^{*}ds^{2}\leq C_{1}\mu^{2p}|dw|^{2}.

By Lemma 4.4,

\int_{\tilde{\gamma}}ds=\int_{{l_{a}}}\tilde{H}^{*}ds<C\int_{l_{a}}\left(\frac{R}{R^{2}-|w|^{2}}\right)^{p}|dw|<\infty,

which contradicts with the completeness of $M$ . We have necessarily $\psi=\hat{\psi}$ , and the proof of Main Theorem B is completed.

Remark 4.6.

In this context, we cannot uniformly discuss whether the results are optimal or not, because values of $m$ and $|E_{f}|$ vary. Although for each fixed $m$ , the value of $|E_{f}|$ may also have several situations.

References

\bibcommenthead
Chern [1965] Chern Ss. Minimal surfaces in an Euclidean space of $N$ dimensions. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse). Princeton Univ. Press, Princeton, NJ; 1965. p. 187–198.
Osserman [1959] Osserman R. Proof of a conjecture of Nirenberg. Comm Pure Appl Math. 1959;12:229–232. 10.1002/cpa.3160120203.
Osserman [1961] Osserman R. Minimal surfaces in the large. Comment Math Helv. 1961;35:65–76. 10.1007/BF02567006.
Osserman [1964] Osserman R. Global properties of minimal surfaces in $E^{3}$ and $E^{n}$ . Ann of Math (2). 1964;80:340–364. 10.2307/1970396.
Xavier [1981] Xavier F. The Gauss map of a complete nonflat minimal surface cannot omit $7$ points of the sphere. Ann of Math (2). 1981;113(1):211–214. 10.2307/1971139.
Mo and Osserman [1990] Mo X, Osserman R. On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto’s theorem. J Differential Geom. 1990;31(2):343–355.
Fujimoto [1988] Fujimoto H. On the number of exceptional values of the Gauss maps of minimal surfaces. J Math Soc Japan. 1988;40(2):235–247. 10.2969/jmsj/04020235.
Osserman [1986] Osserman R. A survey of minimal surfaces. 2nd ed. Dover Publications, Inc., New York; 1986.
Kawakami [2013] Kawakami Y. On the maximal number of exceptional values of Gauss maps for various classes of surfaces. Math Z. 2013;274(3-4):1249–1260. 10.1007/s00209-012-1115-8.
Aiyama et al. [2017] Aiyama R, Akutagawa K, Imagawa S, Kawakami Y. Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space. Ann Mat Pura Appl (4). 2017;196(5):1863–1875. 10.1007/s10231-017-0643-6.
Fujimoto [1990] Fujimoto H. Modified defect relations for the Gauss map of minimal surfaces. II. J Differential Geom. 1990;31(2):365–385.
Ru [1991] Ru M. On the Gauss map of minimal surfaces immersed in ${\bf R}^{n}$ . J Differential Geom. 1991;34(2):411–423.
Fujimoto [1992] Fujimoto H. On the Gauss curvature of minimal surfaces. J Math Soc Japan. 1992;44(3):427–439. 10.2969/jmsj/04430427.
Ru [1993] Ru M. Gauss map of minimal surfaces with ramification. Trans Amer Math Soc. 1993;339(2):751–764. 10.2307/2154296.
Jin and Ru [2007] Jin L, Ru M. Values of Gauss maps of complete minimal surfaces in $\mathbb{R}^{m}$ on annular ends. Trans Amer Math Soc. 2007;359(4):1547–1553. 10.1090/S0002-9947-06-04126-2.
Kao [1991] Kao SJ. On values of Gauss maps of complete minimal surfaces on annular ends. Math Ann. 1991;291(2):315–318. 10.1007/BF01445210.
Dethloff and Ha [2014] Dethloff G, Ha PH. Ramification of the Gauss map of complete minimal surfaces in $\mathbb{R}^{3}$ and $\mathbb{R}^{4}$ on annular ends. Ann Fac Sci Toulouse Math (6). 2014;23(4):829–846. 10.5802/afst.1426.
Ha et al. [2016] Ha PH, Phuong LB, Thoan PD. Ramification of the Gauss map and the total curvature of a complete minimal surface. Topology Appl. 2016;199:32–48. 10.1016/j.topol.2015.11.010.
Fujimoto [1993] Fujimoto H. Unicity theorems for the Gauss maps of complete minimal surfaces. J Math Soc Japan. 1993;45(3):481–487. 10.2969/jmsj/04530481.
Nevanlinna [1926] Nevanlinna R. Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen. Acta Math. 1926;48(3-4):367–391. 10.1007/BF02565342.
Fujimoto [1993] Fujimoto H. Unicity theorems for the Gauss maps of complete minimal surfaces. II. Kodai Math J. 1993;16(3):335–354. 10.2996/kmj/1138039844.
Park and Ru [2017] Park J, Ru M. Unicity results for Gauss maps of minimal surfaces immersed in $\mathbb{R}^{m}$ . J Geom. 2017;108(2):481–499. 10.1007/s00022-016-0353-z.
Ha and Kawakami [2018] Ha PH, Kawakami Y. A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space. Canad Math Bull. 2018;61(2):292–300. 10.4153/CMB-2017-015-0.
Ru and Ugur [2017] Ru M, Ugur G. Uniqueness results for algebraic and holomorphic curves into $\mathbb{P}^{n}(\mathbb{C})$ . Internat J Math. 2017;28(9):1740003, 27. 10.1142/S0129167X17400031.
Jin and Ru [2007] Jin L, Ru M. Algebraic curves and the Gauss map of algebraic minimal surfaces. Differential Geom Appl. 2007;25(6):701–712. 10.1016/j.difgeo.2007.06.014.
Ha [2020] Ha PH. Gaussian curvature and unicity problem of Gauss maps of various classes of surfaces. Nagoya Math J. 2020;240:275–297. 10.1017/nmj.2019.5.
Calabi [1970] Calabi E. Examples of Bernstein problems for some nonlinear equations. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968). Amer. Math. Soc., Providence, R.I.; 1970. p. 223–230.
Kawakami [2015] Kawakami Y. Function-theoretic properties for the Gauss maps of various classes of surfaces. Canad J Math. 2015;67(6):1411–1434. 10.4153/CJM-2015-008-5.
Umehara and Yamada [2006] Umehara M, Yamada K. Maximal surfaces with singularities in Minkowski space. Hokkaido Math J. 2006;35(1):13–40. 10.14492/hokmj/1285766302.
Ou et al. [2024] Ou L, Cheng C, Yang L. On complete space-like stationary surfaces in 4-dimensional Minkowski space with graphical Gauss image. Math Z. 2024;306(1):Paper No. 16, 40. 10.1007/s00209-023-03408-1.
Ma et al. [2013] Ma X, Wang C, Wang P. Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space. Adv Math. 2013;249:311–347. 10.1016/j.aim.2013.09.013.
Ahlfors [1973] Ahlfors LV. Conformal invariants: topics in geometric function theory. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg; 1973.
Fujimoto [1989] Fujimoto H. Modified defect relations for the Gauss map of minimal surfaces. J Differential Geom. 1989;29(2):245–262.
Robinson [1939] Robinson RM. A generalization of Picard’s and related theorems. Duke Math J. 1939;5(1):118–132. 10.1215/S0012-7094-39-00513-2.
Nevanlinna [1970] Nevanlinna R. Analytic functions. Die Grundlehren der mathematischen Wissenschaften, Band 162. Springer-Verlag, New York-Berlin; 1970. Translated from the second German edition by Phillip Emig.

	$\displaystyle H^{*}d\tilde{s}^{2}$	$\displaystyle=\bigg{(}\left(\Pi_{i=1}^{l}\|\psi-c_{i}\|^{2}\right)(1+\|\psi\|^{2})^{m-l}\|g\|^{2}\bigg{)}\circ H\left\|\frac{dz}{dw}\right\|^{2}\|dw\|^{2}$
		$\displaystyle=\frac{\left(\Pi_{i=1}^{l}\|\psi(w)-c_{i}\|^{2}\right)(1+\|\psi(w)\|^{2})^{m-l}\|g(w)\|^{2}\|\psi^{\prime}(w)\|^{2\tilde{p}}}{\|g(w)\|^{2}\Pi_{i=1}^{l}\|\psi(w)-c_{i}\|^{2\tilde{p}(1+p-\epsilon^{\prime})}\Pi_{j=1}^{q}\|\psi(w)-a_{j}\|^{2\tilde{p}(1-\frac{1}{e_{j}}-\epsilon^{\prime})}}\|dw\|^{2}$
		$\displaystyle=\left(\frac{(1+\|\psi\|^{2})^{\frac{(m-l)p}{2}}\psi^{\prime}}{\Pi_{i=1}^{l}(\psi-c_{i})^{1-\epsilon^{\prime}}\Pi_{j=1}^{q}(\psi-a_{j})^{1-\frac{1}{e}_{j}-\epsilon^{\prime}}}\right)^{2\tilde{p}}\|dw\|^{2}$
		$\displaystyle\leq\left(B\frac{2R}{R^{2}-\|w\|^{2}}\right)^{2\tilde{p}}.$

	$\displaystyle\tilde{H}^{*}ds^{2}$	$\displaystyle=\|k\circ\tilde{H}\|^{2}\Pi_{i=1}^{l}(\|g-c_{i}\|\|\hat{g}-c_{i}\|)(1+\|g\|^{2})^{\frac{m-l}{2}}(1+\|\hat{g}\|^{2})^{\frac{m-l}{2}}\left\|\frac{dz}{dw}\right\|^{2}\|dw\|^{2}$
		$\displaystyle=\left(\frac{(1+\|g\|^{2})^{\frac{m-l}{2p}}(1+\|\hat{g}\|^{2})^{\frac{m-l}{2p}}\|g-\hat{g}\|^{2}\|g^{\prime}\|\|\hat{g}^{\prime}\|\Pi_{j=1}^{q-1}(1+\|a_{j}\|^{2})^{1-\delta}}{\Pi_{j=1}^{q-1}\|g-a_{j}\|^{1-\delta}\|\hat{g}-a_{j}\|^{1-\delta}}\right)^{p}\|dw\|^{2}$
		$\displaystyle=\left(\mu^{2}\Pi_{i=1}^{q}(\|g,a_{i}\|\|\hat{g},a_{i}\|)^{\epsilon}\left(\Pi_{i=1}^{q}\log\frac{a_{0}}{\|g,a_{i}\|^{2}}\log\frac{a_{0}}{\|\hat{g},a_{i}\|^{2}}\right)^{1-\epsilon}\right)^{p}\|dw\|^{2},$

Ramification and unicity theorems for Gauss maps of complete space-like stationary surfaces in four-dimensional Lorentz-Minkowski space

Abstract

keywords:

pacs:

1 Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

Main Theorem A.

Main Theorem B.

2 Preliminaries

2.1 The Gauss map of space-like stationary surfaces in ℝ3,1\mathbb{R}^{3,1}

Proposition 2.1.

2.2 Weierstrass representation of space-like stationary Surfaces in ℝ3,1\mathbb{R}^{3,1}

Theorem 2.2.

Remark 2.3.

Remark 2.4.

2.3 Space-like stationary surfaces with rational graphical Gauss image

Proposition 2.5.

2.4 Metrics with negative curvature

Theorem 2.6.

3 Ramification problem

Theorem 3.1.

Remark 3.2.

Theorem 3.3.

Proposition 3.4.

Lemma 3.5.

Proof.

Proof.

Lemma 3.6.

Proof.

Remark 3.7.

4 Unicity problem

Theorem 4.1.

Remark 4.2.

Lemma 4.3.

Lemma 4.4.

Lemma 4.5.

Proof.

Remark 4.6.

References

2.1 The Gauss map of space-like stationary surfaces in $\mathbb{R}^{3,1}$

2.2 Weierstrass representation of space-like stationary Surfaces in $\mathbb{R}^{3,1}$