Minimal Lorentz Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

TL;DR

This paper classifies a special class of minimal Lorentz surfaces in pseudo-Euclidean 4-space with neutral metric, introducing canonical parameters and invariant functions to characterize them through PDEs.

Contribution

It introduces a new class of minimal Lorentz surfaces of general type and provides a method to classify and construct them using invariant functions and PDEs.

Findings

Canonical parameters are introduced for these surfaces.

Surfaces are characterized by two invariant functions satisfying PDEs.

An explicit example of such a surface is constructed.

Abstract

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that any minimal Lorentz surface of general type is determined (up to a rigid motion) by two invariant functions satisfying a system of two natural partial differential equations. Using a concrete solution to this system we construct an example of a minimal Lorentz surface of general type.

Equations

$$K+|\varkappa |\le ||H|{|}^2.$$

$$\begin{array}[]{l}\displaystyle{(K^{2}-\varkappa^{2})^{\frac{1}{4}}\,\Delta\ln(K^{2}-\varkappa^{2})}=8K\\[5.69054pt] \displaystyle{(K^{2}-\varkappa^{2})^{\frac{1}{4}}\,\Delta\ln\frac{K-\varkappa}{K+\varkappa}=-4\varkappa}\end{array}$$

$$\begin{array}[]{l}\displaystyle{(K^{2}+\varkappa^{2})^{\frac{1}{4}}\,\Delta\ln(K^{2}+\varkappa^{2})}=8K\\[5.69054pt] \displaystyle{(K^{2}+\varkappa^{2})^{\frac{1}{4}}\,\Delta\arctan\frac{\varkappa}{K}=2\varkappa}\end{array}$$

$$\begin{array}[]{l}\displaystyle{(K^{2}+\varkappa^{2})^{\frac{1}{4}}\,\Delta^{h}\ln(K^{2}+\varkappa^{2})}=8K\\[5.69054pt] \displaystyle{(K^{2}+\varkappa^{2})^{\frac{1}{4}}\,\Delta^{h}\arctan\frac{\varkappa}{K}=2\varkappa}\end{array}$$

$$\begin{array}[]{l}\vspace{2mm}\displaystyle{\sqrt{\left|\mu^{2}-\nu^{2}\right|}}\,\Delta^{h}\ln\left|\mu^{2}-\nu^{2}\right|=-4\varepsilon(\mu^{2}+\nu^{2})\\ \vspace{2mm}\displaystyle{\sqrt{\left|\mu^{2}-\nu^{2}\right|}}\,\Delta^{h}\ln\left|\frac{\mu+\nu}{\mu-\nu}\right|=-4\,\varepsilon\mu\,\nu\end{array}\qquad\quad(\varepsilon=\pm 1),$$

$$\begin{array}[]{l}\vspace{2mm}\displaystyle{\sqrt[4]{K^{2}-\varkappa^{2}}}\,\Delta^{h}\ln\left(K^{2}-\varkappa^{2}\right)=8K\\ \vspace{2mm}\displaystyle{\sqrt[4]{K^{2}-\varkappa^{2}}}\,\Delta^{h}\ln\frac{K+\varepsilon\,\varkappa}{K-\varepsilon\,\varkappa}=4\varepsilon\varkappa\end{array}$$

$$g_0=d{x}_1^2+d{x}_2^2-d{x}_3^2-d{x}_4^2,$$

$${\mathbb{E}}_2^4=T_p{M}_1^2\oplus N_p{M}_1^2$$

$$\displaystyle\begin{array}[]{l}\vspace{2mm}\nabla^{\prime}_{x}y=\nabla_{x}y+\sigma(x,y);\\ \vspace{2mm}\nabla^{\prime}_{x}\xi=-A_{\xi}x+D_{x}\xi,\end{array}$$

$$H=\frac{1}{2}\mathrm{tr}\sigma .$$

$$g=f^2(u,v)(du\otimes du-dv\otimes dv)$$

$$E=⟨z_u,z_u⟩=f^2(u,v),F=⟨z_u,z_v⟩=0,G=⟨z_v,z_v⟩=-f^2(u,v).$$

$$\begin{array}[]{l}\vspace{2mm}\nabla_{z_{u}}z_{u}=\frac{f_{u}}{f}\,z_{u}+\frac{f_{v}}{f}\,z_{v}=f_{u}\,x+f_{v}\,y;\\ \vspace{2mm}\nabla_{z_{u}}z_{v}=\frac{f_{v}}{f}\,z_{u}+\frac{f_{u}}{f}\,z_{v}=f_{v}\,x+f_{u}\,y;\\ \vspace{2mm}\nabla_{z_{v}}z_{v}=\frac{f_{u}}{f}\,z_{u}+\frac{f_{v}}{f}\,z_{v}=f_{u}\,x+f_{v}\,y.\\ \end{array}$$

$$\begin{array}[]{l}\vspace{2mm}\nabla^{\prime}_{x}x=\quad\quad\;\frac{f_{v}}{f^{2}}\,y+\sigma(x,x);\\ \vspace{2mm}\nabla^{\prime}_{x}y=\frac{f_{v}}{f^{2}}\,x\qquad\;+\sigma(x,y);\\ \vspace{2mm}\nabla^{\prime}_{y}x=\quad\quad\;\frac{f_{u}}{f^{2}}\,y+\sigma(x,y);\\ \vspace{2mm}\nabla^{\prime}_{y}y=\frac{f_{u}}{f^{2}}\,x\qquad\;+\sigma(y,y).\end{array}$$

$$z(u,v)=\alpha (u)+\beta (v),$$

$$\mathrm{Im}{\sigma }_p=\mathrm{span}\{\sigma (X,Y):X,Y\in T_p{M}_1^2\}.$$

$$\begin{array}[]{l}\vspace{2mm}\sigma(x,x)=\sigma(y,y)=\nu\,n;\\ \vspace{2mm}\sigma(x,y)=\mu\,n.\end{array}$$

$$\begin{array}[]{ll}\vspace{2mm}\nabla^{\prime}_{x}x=\quad\quad\;-\gamma_{1}\,y+\mu a\,e_{1}+\mu b\,e_{2};&\qquad\quad\nabla^{\prime}_{x}e_{1}=-\mu a\,x+\nu a\,y\quad\,\quad\quad+\beta_{1}\,e_{2};\\ \vspace{2mm}\nabla^{\prime}_{x}y=-\gamma_{1}\,x\quad\quad\;+\nu a\,e_{1}+\nu b\,e_{2};&\qquad\quad\nabla^{\prime}_{y}e_{1}=-\nu a\,x+\mu a\,y\quad\,\quad\quad+\beta_{2}\,e_{2};\\ \vspace{2mm}\nabla^{\prime}_{y}x=\quad\quad\;-\gamma_{2}\,y+\nu a\,e_{1}+\nu b\,e_{2};&\qquad\quad\nabla^{\prime}_{x}e_{2}=\mu b\,x-\nu b\,y+\beta_{1}\,e_{1};\\ \vspace{2mm}\nabla^{\prime}_{y}y=-\gamma_{2}\,x\quad\quad\;+\mu a\,e_{1}+\mu b\,e_{2};&\qquad\quad\nabla^{\prime}_{y}e_{2}=\nu b\,x-\mu b\,y+\beta_{2}\,e_{1},\end{array}$$

$$\begin{array}[]{l}\vspace{2mm}x(\nu a)-y(\mu a)=2\nu a\gamma_{2}-2\mu a\gamma_{1}-\nu b\beta_{1}+\mu b\beta_{2};\\ \vspace{2mm}x(\nu b)-y(\mu b)=2\nu b\gamma_{2}-2\mu b\gamma_{1}-\nu a\beta_{1}+\mu a\beta_{2};\\ \vspace{2mm}x(\mu a)-y(\nu a)=2\mu a\gamma_{2}-2\nu a\gamma_{1}-\mu b\beta_{1}+\nu b\beta_{2};\\ \vspace{2mm}x(\mu b)-y(\nu b)=2\mu b\gamma_{2}-2\nu b\gamma_{1}-\mu a\beta_{1}+\nu a\beta_{2};\\ \vspace{2mm}x(\beta_{2})-y(\beta_{1})+\gamma_{1}\beta_{1}-\gamma_{2}\beta_{2}=0.\end{array}$$

$$\begin{array}[]{l}\vspace{6mm}K=\displaystyle{\frac{\langle\sigma(x,x),\sigma(y,y)\rangle-\langle\sigma(x,y),\sigma(x,y)\rangle}{\langle x,x\rangle\,\langle y,y\rangle-\langle x,y\rangle^{2}}};\\ \vspace{2mm}\varkappa=\displaystyle{\frac{\langle R^{\bot}(x,y)\,n_{1},n_{2}\rangle}{\langle x,x\rangle\,\langle y,y\rangle-\langle x,y\rangle^{2}}},\end{array}$$

$$K=-({\mu }^2-{\nu }^2)(a^2-b^2);$$

$$\varkappa =x({\beta }_2)-y({\beta }_1)+{\gamma }_1{\beta }_1-{\gamma }_2{\beta }_2.$$

$$\begin{array}[]{l}\vspace{2mm}(\mu^{2}-\nu^{2})(bx(a)-ax(b))=\beta_{1}(\mu^{2}-\nu^{2})(a^{2}-b^{2});\\ \vspace{2mm}(\mu^{2}-\nu^{2})(by(a)-ay(b))=\beta_{2}(\mu^{2}-\nu^{2})(a^{2}-b^{2}).\end{array}$$

$${\beta }_1=\frac{bx(a)-ax(b)}{a^2-b^2};{\beta }_2=\frac{by(a)-ay(b)}{a^2-b^2}.$$

$$\bar{n}=\frac{b}{\sqrt{|a^2-b^2|}}{e}_1+\frac{a}{\sqrt{|a^2-b^2|}}{e}_2.$$

$$\begin{array}[]{l}\vspace{2mm}\sigma(x,x)=a\,e_{1}+b\,e_{2};\\ \vspace{2mm}\sigma(x,y)=c\,e_{1}+d\,e_{2};\\ \vspace{2mm}\sigma(y,y)=a\,e_{1}+b\,e_{2}\end{array}$$

$$\begin{array}[]{ll}\vspace{2mm}\nabla^{\prime}_{x}x=\quad\quad\;-\gamma_{1}\,y+a\,e_{1}+b\,e_{2};&\qquad\quad\nabla^{\prime}_{x}e_{1}=-a\,x+c\,y\quad\quad\quad+\beta_{1}\,e_{2};\\ \vspace{2mm}\nabla^{\prime}_{x}y=-\gamma_{1}\,x\quad\quad\;+c\,e_{1}+d\,e_{2};&\qquad\quad\nabla^{\prime}_{y}e_{1}=-c\,x+a\,y\quad\quad\quad+\beta_{2}\,e_{2};\\ \vspace{2mm}\nabla^{\prime}_{y}x=\quad\quad\;-\gamma_{2}\,y+c\,e_{1}+d\,e_{2};&\qquad\quad\nabla^{\prime}_{x}e_{2}=b\,x-d\,y\quad+\beta_{1}\,e_{1};\\ \vspace{2mm}\nabla^{\prime}_{y}y=-\gamma_{2}\,x\quad\quad\;+a\,e_{1}+b\,e_{2};&\qquad\quad\nabla^{\prime}_{y}e_{2}=d\,x-b\,y\quad+\beta_{2}\,e_{1}.\end{array}$$

$$K=b^2-a^2+c^2-d^2.$$

$$\varkappa =2(bc-ad).$$

$$K^2-{\varkappa }^2=(a^2-b^2{)}^2+(c^2-d^2{)}^2-2(bc-ad{)}^2-2(ac-bd{)}^2.$$

Full text

1. Introduction

The study of minimal surfaces is one of the main topics in classical differential geometry which goes back to the latter part of the 18th century. Lagrange was the first who initiated in 1760 the study of minimal surfaces in Euclidean 3-space and found the minimal surface equation when he looked for a necessary condition for minimizing a certain integral. Actually, the notion of mean curvature was first formally defined by Meusnier in 1776. Throughout the 19th century grate mathematicians such as Gauss and Weierstrass devoted much of their studies to these surfaces. The theory of minimal surfaces in real space forms have been attracting the attention of many mathematicians for more than two centuries (see [4], [11], and the references therein).

In the last years, great attention is also paid to Lorentz surfaces in pseudo-Euclidean spaces, since pseudo-Riemannian geometry has many important applications in Physics. Minimal Lorentz surfaces in $\displaystyle\mathbb{C}^{2}_{1}$ have been classified recently by B.-Y. Chen [5]. Classification results for minimal Lorentz surfaces in pseudo-Euclidean space $\displaystyle\mathbb{E}^{m}_{s}$ with arbitrary dimension $\displaystyle m$ and arbitrary index $\displaystyle s$ are obtained in [6].

Metrics of neutral signature $\displaystyle(+,+,-,-)$ in dimension four appear in many geometric and physics problems as well as in string theory. In the present paper we study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric $\displaystyle\mathbb{E}^{4}_{2}$. Our aim is to characterize the minimal Lorentz surfaces in terms of a pair of smooth functions satisfying a system of two natural partial differential equations. This aim is motivated by similar results concerning minimal surfaces in the four-dimensional Euclidean space $\displaystyle\mathbb{E}^{4}$ (see [14]) and spacelike or timelike surfaces with zero mean curvature in the Minkowski 4-space $\displaystyle\mathbb{E}^{4}_{1}$ (see [1] and [8]). Our approach to the study of minimal Lorentz surfaces in $\displaystyle\mathbb{E}^{4}_{2}$ is based on the introducing of special geometric parameters which we call canonical parameters.

It is known that the Gauss curvature $\displaystyle K$, the curvature of the normal connection $\displaystyle\varkappa$, and the mean curvature vector field $\displaystyle H$ of an arbitrary surface in the Euclidean space $\displaystyle\mathbb{E}^{4}$ satisfy the Wintgen inequality [15]

[TABLE]

Following [12], a surface in $\displaystyle\mathbb{E}^{4}$ is called Wintgen ideal surface, if it satisfies the equality case of the Wintgen’s inequality identically. A surface $\displaystyle M$ is called minimal, if its mean curvature vector vanishes identically. Therefore, the invariants $\displaystyle K$ and $\displaystyle\varkappa$ of any minimal surface in $\displaystyle\mathbb{E}^{4}$ satisfy the inequality $\displaystyle K^{2}-\varkappa^{2}\geq 0$, which divides the minimal surfaces into two basic classes:

• •

the class of minimal Wintgen ideal surfaces characterized by $\displaystyle K^{2}-\varkappa^{2}=0$;

• •

the class of minimal surfaces of general type characterized by $\displaystyle K^{2}-\varkappa^{2}>0$.

The Wintgen ideal surfaces are characterized by circular ellipse of normal curvature [9]. A surface $\displaystyle M$ in $\displaystyle\mathbb{E}^{4}$ is called super-conformal [3] if at any point of $\displaystyle M$ the ellipse of curvature is a circle. Hence, the class of minimal Wintgen ideal surfaces coincides with the class of minimal super-conformal surfaces. According to a result of Eisenhart [7], the class of minimal super-conformal surfaces in $\displaystyle\mathbb{E}^{4}$ is locally equivalent to the class of holomorphic curves in $\displaystyle\mathbb{C}^{2}\equiv\mathbb{E}^{4}$.

In [14], de Azevedo Tribuzy and Guadalupe proved that the Gauss curvature $\displaystyle K$ and the curvature of the normal connection $\displaystyle\varkappa$ of any minimal non-super-conformal surface parametrized by special isothermal parameters in the Euclidean space $\displaystyle\mathbb{E}^{4}$ satisfy the following system of partial differential equations

[TABLE]

and conversely, any solution ($\displaystyle K$, $\displaystyle\varkappa$) to this system determines a unique (up to a rigid motion in $\displaystyle\mathbb{E}^{4}$) minimal non-super-conformal surface with Gauss curvature $\displaystyle K$ and normal curvature $\displaystyle\varkappa$.

Similar results hold for surfaces with zero mean curvature in the 4-dimensional Minkowski space $\displaystyle\mathbb{E}^{4}_{1}$. In [1], Alías and Palmer studied spacelike surfaces with zero mean curvature in $\displaystyle\mathbb{E}^{4}_{1}$ and proved that these surfaces are described by the following system of partial differential equations

[TABLE]

where $\displaystyle K$ and $\displaystyle\varkappa$ are the Gauss curvature and the normal curvature, respectively.

In [8], Ganchev and the second author developed the local theory of timelike surfaces with zero mean curvature in the Minkowski 4-space $\displaystyle\mathbb{E}^{4}_{1}$ and proved that the Gauss curvature $\displaystyle K$ and the normal curvature $\displaystyle\varkappa$ of any timelike surface (parametrized by special, so called canonical parameters) with zero mean curvature satisfy the following system of natural partial differential equations

[TABLE]

where $\displaystyle\Delta^{h}$ denotes the hyperbolic Laplace operator. Conversely, any solution ($\displaystyle K$, $\displaystyle\varkappa$) to the above system, determines a unique (up to a rigid motion in $\displaystyle\mathbb{E}^{4}_{1}$) timelike surface with zero mean curvature such that $\displaystyle K$ is the Gauss curvature and $\displaystyle\varkappa$ is the normal curvature of the surface.

Minimal Lorentz surfaces in the pseudo-Euclidean space $\displaystyle\mathbb{E}^{4}_{2}$ can be divided into three basic classes:

• •

surfaces characterized by $\displaystyle K^{2}-\varkappa^{2}>0$;

• •

surfaces characterized by $\displaystyle K^{2}-\varkappa^{2}=0$;

• •

surfaces characterized by $\displaystyle K^{2}-\varkappa^{2}<0$.

In the present paper we study minimal Lorentz surfaces from the first class, i.e. satisfying the inequality $\displaystyle K^{2}-\varkappa^{2}>0$. In the special case when the first normal space is one-dimensional at each point, the minimal surface is either a degenerate hyperplane, a flat umbilic surface, a quasi-umbilic surface, or lies in a non-degenerate hyperplane. We focus our attention on minimal Lorentz surfaces whose first normal space is two-dimensional and whose Gauss curvature $\displaystyle K$ and normal curvature $\displaystyle\varkappa$ satisfy $\displaystyle K^{2}-\varkappa^{2}>0$ on a dense open subset. We call these surfaces minimal surfaces of general type, since their local theory can be developed analogously to the local theory of minimal surfaces in $\displaystyle\mathbb{E}^{4}$ and spacelike or timelike surfaces with zero mean curvature vector in $\displaystyle\mathbb{E}^{4}_{1}$. We introduce special geometric (canonical) parameters $\displaystyle(u,v)$ on any minimal Lorentz surface of general type. With respect to these parameters all coefficients of the first and the second fundamental form are expressed by two invariant functions $\displaystyle\mu(u,v)$ and $\displaystyle\nu(u,v)$. Using the canonical parameters we introduce a geometrically determined moving frame field $\displaystyle\{x,y,n_{1},n_{2}\}$ at each point of the surface, where $\displaystyle x$ and $\displaystyle y$ are unit tangent vector fields determining the so-called canonical directions of the surface; $\displaystyle n_{1}$ and $\displaystyle n_{2}$ are unit normal vector fields which are uniquely determined by the canonical tangents. We prove a fundamental existence and uniqueness theorem (Theorem 4.2) stating that any minimal Lorentz surface of general type is determined up to a rigid motion in $\displaystyle\mathbb{E}^{4}_{2}$ by the invariants $\displaystyle\mu(u,v)$ and $\displaystyle\nu(u,v)$ satisfying the following system of natural partial differential equations

[TABLE]

where $\displaystyle\Delta^{h}=\frac{\partial^{2}}{\partial u^{2}}-\frac{\partial^{2}}{\partial v^{2}}$ is the hyperbolic Laplace operator, $\displaystyle\varepsilon=1$ corresponds to the case the geometric normal vector field $\displaystyle n_{1}$ is spacelike, $\displaystyle\varepsilon=-1$ corresponds to the case $\displaystyle n_{1}$ is timelike.

Further, expressing the Gauss curvature $\displaystyle K$ and the normal curvature $\displaystyle\varkappa$ by the invariants $\displaystyle\mu$ and $\displaystyle\nu$, we prove that $\displaystyle K$ and $\displaystyle\varkappa$ satisfy the following system of natural partial differential equations

[TABLE]

Conversely, any solution $\displaystyle(K,\varkappa)$ to this system determines a unique (up to a rigid motion in $\displaystyle\mathbb{E}^{4}_{2}$) minimal Lorentz surface of general type with Gauss curvature $\displaystyle K$ and normal curvature $\displaystyle\varkappa$ and such that the given parameters are canonical.

The above system is the background system of partial differential equations describing the class of minimal Lorentz surfaces of general type. Equalities (2) follow also from a result of M. Sakaki [13] for Lorentz stationary surfaces in 4-dimensional space forms of index 2. We obtain system (2) for the invariants $\displaystyle K$ and $\displaystyle\varkappa$ as a consequence of system (1) for the invariants $\displaystyle\mu$ and $\displaystyle\nu$. We prefer to describe the class of minimal Lorentz surfaces of general type by the system for the invariants $\displaystyle\mu$ and $\displaystyle\nu$ since the proof of Theorem 4.2 gives a procedure for constructing a minimal surface given a solution $\displaystyle(\mu,\nu)$ to system (1). As an application of this procedure, in the last section we obtain an example of a minimal Lorentz surface of general type using a concrete solution $\displaystyle(\mu,\nu)$ to system (1).

The class of minimal Lorentz surfaces satisfying the equality $\displaystyle K^{2}-\varkappa^{2}=0$ is the analogue of the class of minimal super-conformal surfaces in the Euclidean space $\displaystyle\mathbb{E}^{4}$. Minimal surfaces from the third class, i.e. satisfying the inequality $\displaystyle K^{2}-\varkappa^{2}<0$ need a different approach to be studied with. Our idea for introducing a geometric frame field based on the canonical directions cannot be applied for this class of minimal surfaces, since in the case $\displaystyle K^{2}-\varkappa^{2}<0$ there do not exist canonical (in our sense) directions. Note that such class of minimal surfaces does not exist neither in the Euclidean space $\displaystyle\mathbb{E}^{4}$ nor in the Minkowski space $\displaystyle\mathbb{E}^{4}_{1}$.

2. Preliminaries

Let $\displaystyle\mathbb{E}^{4}_{2}$ be the pseudo-Euclidean 4-space endowed with the canonical pseudo-Euclidean metric of index 2 given in local coordinates by

[TABLE]

where $\displaystyle(x_{1},x_{2},x_{3},x_{4})$ is a rectangular coordinate system of $\displaystyle\mathbb{E}^{4}_{2}$. We denote by $\displaystyle\langle.\,,.\rangle$ the indefinite inner scalar product associated with $\displaystyle g_{0}$. Since $\displaystyle g_{0}$ is an indefinite metric, a vector $\displaystyle v\in\mathbb{E}^{4}_{2}$ can have one of the three casual characters: spacelike, if $\displaystyle\langle v,v\rangle>0$ or $\displaystyle v=0$, timelike if $\displaystyle\langle v,v\rangle<0$, and lightlike if $\displaystyle\langle v,v\rangle=0$ and $\displaystyle v\neq 0$. This terminology is inspired by General Relativity and the Minkowski 4-space $\displaystyle\mathbb{E}^{4}_{1}$.

A surface $\displaystyle M^{2}_{1}$ in $\displaystyle\mathbb{E}^{4}_{2}$ is called Lorentz if the induced metric $\displaystyle g$ on $\displaystyle M^{2}_{1}$ is Lorentzian, i.e. at each point $\displaystyle p\in M^{2}_{1}$ we have the following decomposition

[TABLE]

with the property that the restriction of the metric onto the tangent space $\displaystyle T_{p}M^{2}_{1}$ is of signature $\displaystyle(1,1)$, and the restriction of the metric onto the normal space $\displaystyle N_{p}M^{2}_{1}$ is of signature $\displaystyle(1,1)$.

Denote by $\displaystyle\nabla$ and $\displaystyle\nabla^{\prime}$ the Levi Civita connections of $\displaystyle M^{2}_{1}$ and $\displaystyle\mathbb{E}^{4}_{2}$, respectively. Let $\displaystyle x$ and $\displaystyle y$ be vector fields tangent to $\displaystyle M^{2}_{1}$ and $\displaystyle\xi$ be a normal vector field. The formulas of Gauss and Weingarten are given respectively by

[TABLE]

where $\displaystyle\sigma$ is the second fundamental form of $\displaystyle M^{2}_{1}$, $\displaystyle D$ is the normal connection on the normal bundle, and $\displaystyle A_{\xi}$ is the shape operator with respect to $\displaystyle\xi$. In general, $\displaystyle A_{\xi}$ is not diagonalizable. The mean curvature vector field $\displaystyle H$ of $\displaystyle M^{2}_{1}$ in $\displaystyle\mathbb{E}^{4}_{2}$ is defined as

[TABLE]

A Lorentz surface in an indefinite space form is called totally geodesic if its second fundamental form vanishes identically. It is called minimal if its mean curvature vector vanishes identically, i.e. $\displaystyle H=0$.

3. Minimal Lorentz surfaces whose first normal space is one-dimensional

Let $\displaystyle M^{2}_{1}$ be a Lorentz surface in $\displaystyle\mathbb{E}^{4}_{2}$. According to a result of Larsen [10], locally there exist isothermal coordinates $\displaystyle(u,v)$ such that the metric tensor $\displaystyle g$ of $\displaystyle M^{2}_{1}$ takes the form

[TABLE]

for some positive function $\displaystyle f(u,v)$. Let $\displaystyle z=z(u,v),(u,v)\in\mathcal{D}\,(\mathcal{D}\subset\mathbb{R}^{2})$ be a local parametrization on $\displaystyle M^{2}_{1}$ with respect to such isothermal parameters. Then, the coefficients of the first fundamental form are

[TABLE]

We consider the orthonormal tangent frame field given by $\displaystyle x=\displaystyle{\frac{z_{u}}{f}}$, $\displaystyle y=\displaystyle{\frac{z_{v}}{f}}$. Obviously, $\displaystyle\langle x,x\rangle=1$, $\displaystyle\langle x,y\rangle=0$, $\displaystyle\langle y,y\rangle=-1$. It can easily be checked that the Levi-Civita connection of the metric tensor (3) satisfies

[TABLE]

Then equalities (4) imply the following derivative formulas

[TABLE]

The mean curvature vector field of $\displaystyle M^{2}_{1}$ is given by $\displaystyle H=\displaystyle{\frac{\sigma(x,x)-\sigma(y,y)}{2}}$, since $\displaystyle\langle x,x\rangle=1$, $\displaystyle\langle y,y\rangle=-1$. Hence, $\displaystyle M^{2}_{1}$ is a minimal surface in $\displaystyle\mathbb{E}^{4}_{2}$ if and only if $\displaystyle\sigma(x,x)=\sigma(y,y)$.

The classification of minimal surfaces in an arbitrary indefinite pseudo-Euclidean space $\displaystyle\mathbb{E}^{m}_{s}$ is given by B.-Y. Chen in the next theorem.

Theorem 3.1**.**

[6]** A Lorentz surface in a pseudo-Euclidean m-space $\displaystyle\mathbb{E}^{m}_{s}$ is minimal if and only if, locally the surface is parametrized by

[TABLE]

where $\displaystyle\alpha$ and $\displaystyle\beta$ are null curves satisfying $\displaystyle\left\langle\alpha^{\prime}(u),\beta^{\prime}(v)\right\rangle\neq 0$.

We study minimal Lorentz surfaces in $\displaystyle\mathbb{E}^{4}_{2}$ in terms of geometric (canonical) parameters and geometric frame field which allow us to determine each minimal surface of general type by two smooth functions satisfying a system of two natural partial differential equations.

Let $\displaystyle M^{2}_{1}$ be a minimal surface parametrized by isothermal parameters such that the metric tensor $\displaystyle g$ is given by (3) and hence formulas (5) hold true. Since $\displaystyle M^{2}_{1}$ is minimal, we have $\displaystyle\sigma(x,x)=\sigma(y,y)$.

At a given point $\displaystyle p\in M^{2}_{1}$, the first normal space of $\displaystyle M^{2}_{1}$ in $\displaystyle\mathbb{E}^{4}_{2}$, denoted by $\displaystyle\rm{Im}\,\sigma_{p}$, is the subspace given by

[TABLE]

In this section we consider minimal Lorentz surfaces for which the first normal space at each point is one-dimensional. A point $\displaystyle p$ of such surface is called degenerate if the Gauss curvature $\displaystyle K$ and the curvature of the normal connection $\displaystyle\varkappa$ (the normal curvature) are both zero at $\displaystyle p$. According to a result in [2], a minimal Lorentz surface consisting of degenerate points either belongs to a degenerate hyperplane or is a flat umbilic or quasi-umbilic surface.

In the next theorem we describe the minimal Lorentz surfaces whose first normal space is one-dimensional.

Theorem 3.2**.**

Let $\displaystyle M^{2}_{1}$ be a minimal Lorentz surface in $\displaystyle\mathbb{E}^{4}_{2}$ for which the first normal space at each point is one-dimensional. Then in a neighbourhood of a non-degenerate point $\displaystyle M^{2}_{1}$ is a non-flat surface lying in a non-degenerate hyperplane of $\displaystyle\mathbb{E}^{4}_{2}$.

Proof.

Since the first normal space is one-dimensional, at least locally there exist a normal vector field $\displaystyle n$ and smooth functions $\displaystyle\nu$, $\displaystyle\mu$ such that

[TABLE]

Note that if $\displaystyle\nu=\mu=0$ then the surface $\displaystyle M^{2}_{1}$ is totally geodesic, and hence $\displaystyle M^{2}_{1}$ is contained in a two-dimensional plane $\displaystyle\mathbb{E}^{2}_{1}$. So, we assume that at least one of the functions $\displaystyle\nu$, $\displaystyle\mu$ is non-zero.

Let $\displaystyle\{e_{1},e_{2}\}$ be a local normal frame field of $\displaystyle M^{2}_{1}$, defined in $\displaystyle\mathcal{D}$, such that $\displaystyle\langle e_{1},e_{1}\rangle=1$, $\displaystyle\langle e_{2},e_{2}\rangle=-1$, $\displaystyle\langle e_{1},e_{2}\rangle=0$ and $\displaystyle n=a\,e_{1}+b\,e_{2}$ for some smooth functions $\displaystyle a$ and $\displaystyle b$, where $\displaystyle a^{2}+b^{2}\neq 0$. Then with respect to the frame field $\displaystyle\{x,y,e_{1},e_{2}\}$ we have the following Frenet-type derivative formulas

[TABLE]

where $\displaystyle\gamma_{1}=-\frac{f_{v}}{f^{2}}$, $\displaystyle\gamma_{2}=-\frac{f_{u}}{f^{2}}$, $\displaystyle\beta_{1}$ and $\displaystyle\beta_{2}$ are smooth functions determining the components of the normal connection.

Since the Levi-Civita connection $\displaystyle\nabla^{\prime}$ is flat, from $\displaystyle R^{\prime}(x,y,x)=0,R^{\prime}(x,y,y)=0,R^{\prime}(x,y,e_{1})=0$, using (6) we obtain that the functions $\displaystyle\gamma_{1}$, $\displaystyle\gamma_{2}$, $\displaystyle\nu$, $\displaystyle\mu$, $\displaystyle\beta_{1}$, $\displaystyle\beta_{2}$, $\displaystyle a$, $\displaystyle b$ satisfy the following conditions

[TABLE]

The Gauss curvature $\displaystyle K$ and the normal curvature $\displaystyle\varkappa$ are given by the following formulas

[TABLE]

where $\displaystyle R^{\bot}(x,y)\,n_{1}=D_{x}D_{y}\,n_{1}-D_{y}D_{x}\,n_{1}-D_{[x,y]}\,n_{1}$. So, using (6) we obtain the following expressions for the Gauss curvature and the normal curvature of the surface

[TABLE]

[TABLE]

The expression for the Gauss curvature can be established also from the choice of the parameters and $\displaystyle e_{1},e_{2}$. Note that (8) and the last equality of (7) imply $\displaystyle\varkappa=0$, i.e. the surface has flat normal connection. Hence, points of the surface at which $\displaystyle\mu^{2}-\nu^{2}=0$ or $\displaystyle a^{2}-b^{2}=0$ are degenerate.

In a neighbourhood $\displaystyle\mathcal{D}_{0}\subset\mathcal{D}$ of a non-degenerate point we have $\displaystyle\mu^{2}-\nu^{2}\neq 0$ and $\displaystyle a^{2}-b^{2}\neq 0$. From the first four equalities of (7) we obtain

[TABLE]

The last two equalities imply

[TABLE]

Let us consider the normal vector field $\displaystyle\bar{n}$ defined by

[TABLE]

Since $\displaystyle a^{2}-b^{2}\neq 0$, the vector fields $\displaystyle\bar{n}$ and $\displaystyle n$ are non-lightlike in $\displaystyle\mathcal{D}_{0}$. Obviously, $\displaystyle\bar{n}$ is orthogonal to $\displaystyle n$. Using (6) and (9) we calculate that $\displaystyle\nabla^{\prime}_{x}\bar{n}=0$, $\displaystyle\nabla^{\prime}_{y}\bar{n}=0$, i.e. the normal vector field $\displaystyle\bar{n}$ is constant. Hence, the surface $\displaystyle M^{2}_{1}{/_{\mathcal{D}_{0}}}$ lies in a constant 3-dimensional space parallel to $\displaystyle{\rm span}\{x,y,n\}$. Moreover, since $\displaystyle K\neq 0$, the surface is non-flat. Consequently, $\displaystyle M^{2}_{1}{/_{\mathcal{D}_{0}}}$ is a non-flat surface lying in a hyperplane $\displaystyle\mathbb{E}^{3}_{1}$ or $\displaystyle\mathbb{E}^{3}_{2}$ of $\displaystyle\mathbb{E}^{4}_{2}$.

∎

4. Minimal Lorentz surfaces of general type

In this section we study minimal Lorentz surfaces for which the first normal space is two-dimensional at each point.

Let $\displaystyle M^{2}_{1}$ be a minimal surface parametrized by isothermal parameters such that the metric tensor $\displaystyle g$ is given by (3). We choose a local normal frame field $\displaystyle\{e_{1},e_{2}\}$, defined in $\displaystyle\mathcal{D}$, such that $\displaystyle\langle e_{1},e_{1}\rangle=1$, $\displaystyle\langle e_{1},e_{2}\rangle=0$, $\displaystyle\langle e_{2},e_{2}\rangle=-1$. Then the components of the second fundamental form are expressed as follows

[TABLE]

for some smooth functions $\displaystyle a(u,v)$, $\displaystyle b(u,v)$, $\displaystyle c(u,v)$, and $\displaystyle d(u,v)$.

Let us note that $\displaystyle a\,d-b\,c\neq 0$, $\displaystyle a^{2}+b^{2}\neq 0$, $\displaystyle c^{2}+d^{2}\neq 0$, since the first normal space is two-dimensional.

With respect to the frame field $\displaystyle\{x,y,e_{1},e_{2}\}$ we have the following derivative formulas

[TABLE]

It follows from equalities (10) that the Gauss curvature $\displaystyle K$ is expressed as

[TABLE]

Using the equation of Ricci and formulas (10) we get that the normal curvature is

[TABLE]

Consequently,

[TABLE]

We shall consider minimal surfaces whose first normal space is two-dimensional and whose Gauss curvature $\displaystyle K$ and normal curvature $\displaystyle\varkappa$ satisfy the following inequality

[TABLE]

on a dense open subset of $\displaystyle M^{2}_{1}$. Such minimal surfaces we call minimal surfaces of general type. For this class of surfaces we shall introduce a local orthonormal frame field $\displaystyle\{x,y,n_{1},n_{2}\}$ such that the vector fields $\displaystyle\sigma(x,x)$ and $\displaystyle\sigma(x,y)$ are collinear to $\displaystyle n_{1}$ and $\displaystyle n_{2}$, respectively, i.e.

[TABLE]

for some smooth functions $\displaystyle\nu(u,v)\neq 0$ and $\displaystyle\mu(u,v)\neq 0$.

Using that $\displaystyle R^{\prime}(x,y,x)=0$, $\displaystyle R^{\prime}(x,y,y)=0$, $\displaystyle R^{\prime}(x,y,e_{1})=0$, from (10) we obtain that the functions $\displaystyle\gamma_{1}$, $\displaystyle\gamma_{2}$, $\displaystyle\beta_{1}$, $\displaystyle\beta_{2}$, $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, $\displaystyle d$ satisfy the following conditions

[TABLE]

If we suppose that $\displaystyle a=b$ and $\displaystyle c=d$ (or $\displaystyle a=-b$ and $\displaystyle c=-d$), we get a contradiction with the assumption $\displaystyle ad-bc\neq 0$. If we suppose that $\displaystyle a=b$ and $\displaystyle c=-d$ (similarly for $\displaystyle a=-b$ and $\displaystyle c=d$), then from the first four equalities of (12) we obtain

[TABLE]

On the other hand, $\displaystyle\gamma_{1}=-y(\ln f),\gamma_{2}=-x(\ln f)$. A direct calculation shows that $\displaystyle x(\beta_{2})-y(\beta_{1})+\gamma_{1}\beta_{1}-\gamma_{2}\beta_{2}=0$. So, the last equality of (12) implies that $\displaystyle ac=0$, which contradicts the assumption that the first normal space is two-dimensional.

Now, suppose that $\displaystyle a^{2}-b^{2}\neq 0$ and $\displaystyle c=d$ (or $\displaystyle c=-d$). Then the first four equalities of (12) imply

[TABLE]

or

[TABLE]

The last expressions of $\displaystyle\beta_{1}$ and $\displaystyle\beta_{2}$ together with $\displaystyle\gamma_{1}=-y(\ln f),\gamma_{2}=-x(\ln f)$ imply that $\displaystyle x(\beta_{2})-y(\beta_{1})+\gamma_{1}\beta_{1}-\gamma_{2}\beta_{2}=0$. Then from the last equality of (12) we obtain $\displaystyle c(a-b)=0$ (or $\displaystyle c(a+b)=0$), which contradicts the assumption that the first normal space is two-dimensional. Similarly, in the case $\displaystyle a^{2}=b^{2}$ and $\displaystyle c^{2}-d^{2}\neq 0$ we also get a contradiction.

Hence, $\displaystyle a^{2}-b^{2}\neq 0$ and $\displaystyle c^{2}-d^{2}\neq 0$ for each $\displaystyle(u,v)\in\mathcal{D}$, i.e. $\displaystyle\sigma(x,x)$ and $\displaystyle\sigma(x,y)$ are both non-lightlike vector fields. If $\displaystyle ac-bd=0$ for each $\displaystyle(u,v)\in\mathcal{D}$, then $\displaystyle\langle\sigma(x,x),\sigma(x,y)\rangle=0$ and hence $\displaystyle\sigma(x,x)$ and $\displaystyle\sigma(x,y)$ are orthogonal. If $\displaystyle ac-bd\neq 0$ at a point $\displaystyle p=z(u_{0},v_{0})$ of $\displaystyle M^{2}_{1}$, then there exists a subdomain $\displaystyle\mathcal{D}_{0}\subset\mathcal{D}$, $\displaystyle(u_{0},v_{0})\in\mathcal{D}_{0}$, such that $\displaystyle ac-bd\neq 0$ in $\displaystyle\mathcal{D}_{0}$. In this case, we consider an orthonormal tangent frame field $\displaystyle\bar{x},\bar{y}$, defined in $\displaystyle\mathcal{D}_{0}$, such that

[TABLE]

for some smooth function $\displaystyle\varphi$. Then

[TABLE]

where the functions $\displaystyle\bar{a},\bar{b},\bar{c},\bar{d}$ are expressed as follows

[TABLE]

Straightforward computations show that

[TABLE]

If we suppose that $\displaystyle b^{2}-a^{2}+d^{2}-c^{2}=0$, then $\displaystyle a^{2}+c^{2}=b^{2}+d^{2}$ and hence there exist functions $\displaystyle p$, $\displaystyle q$, $\displaystyle r$, such that $\displaystyle a=r\cos p$, $\displaystyle c=r\sin p$, $\displaystyle b=r\cos q$, $\displaystyle d=r\sin q$. Then, using (11) we get

[TABLE]

which contradicts the condition $\displaystyle K^{2}-\varkappa^{2}>0$. Hence, we can assume that $\displaystyle b^{2}-a^{2}+d^{2}-c^{2}\neq 0$. Then, $\displaystyle\langle\sigma(\bar{x},\bar{x}),\sigma(\bar{x},\bar{y})\rangle=0$ if and only if there exists a function $\displaystyle\varphi$ such that

[TABLE]

Denote $\displaystyle A=\displaystyle{\frac{2(ac-bd)}{b^{2}-a^{2}+d^{2}-c^{2}}}$. Then

[TABLE]

and after some computations we obtain

[TABLE]

Having in mind (11), we get

[TABLE]

Since we study surfaces satisfying $\displaystyle K^{2}-\varkappa^{2}>0$, we get that $\displaystyle-1<A<1$, $\displaystyle A\neq 0$ for $\displaystyle(u,v)\in{\mathcal{D}}_{0}$. So, there exists a function $\displaystyle\varphi(u,v)$ defined in $\displaystyle{\mathcal{D}}_{0}$ such that $\displaystyle A=\tanh 4\varphi$. Hence, $\displaystyle\langle\sigma(\bar{x},\bar{x}),\sigma(\bar{x},\bar{y})\rangle=0$, i.e. $\displaystyle\sigma(\bar{x},\bar{x})$ and $\displaystyle\sigma(\bar{x},\bar{y})$ are orthogonal. Consequently, there exists an orthonormal normal frame field $\displaystyle\{n_{1},n_{2}\}$ such that

[TABLE]

for some functions $\displaystyle\nu\neq 0$ and $\displaystyle\mu\neq 0$.

Finally, we obtain that in a neighbourhood of any point of a minimal Lorentz surface of general type we can introduce a special orthonormal frame field $\displaystyle\{x,y,n_{1},n_{2}\}$ such that

[TABLE]

where $\displaystyle\nu\mu\neq 0$ and $\displaystyle\langle x,x\rangle=1$, $\displaystyle\langle y,y\rangle=-1$, $\displaystyle\langle n_{1},n_{1}\rangle=\varepsilon$, $\displaystyle\langle n_{2},n_{2}\rangle=-\varepsilon$ ($\displaystyle\varepsilon=\pm 1$). We call this orthonormal frame field a geometric frame field of the surface and the tangent directions determined by the tangent vector fields $\displaystyle x$ and $\displaystyle y$ we call canonical directions of the surface. Obviously, the canonical directions are uniquely determined at any point of a minimal Lorentz surface of general type. The normal frame field $\displaystyle\{n_{1};n_{2}\}$ is also uniquely determined by the canonical tangents, and the functions $\displaystyle\mu$ and $\displaystyle\nu$ are geometric functions of the surface.

With respect to the geometric frame field $\displaystyle\{x,y,n_{1},n_{2}\}$ the following Frenet-type derivative formulas hold true

[TABLE]

Using that the Levi-Civita connection $\displaystyle\nabla^{\prime}$ is flat, from (13) we obtain that the functions $\displaystyle\gamma_{1}$, $\displaystyle\gamma_{2}$, $\displaystyle\nu$, $\displaystyle\mu$, $\displaystyle\beta_{1}$, and $\displaystyle\beta_{2}$ satisfy the following conditions

[TABLE]

The Gauss curvature $\displaystyle K$ and the normal curvature $\displaystyle\varkappa$ are expressed in terms of the invariants $\displaystyle\mu$ and $\displaystyle\nu$ as follows

[TABLE]

Since $\displaystyle\nu\mu\neq 0$, we can formulate the following statement.

Proposition 4.1**.**

Let $\displaystyle M^{2}_{1}$ be a Lorentz minimal surface of general type. Then at each point of the surface the Gauss curvature $\displaystyle K$ and the normal curvature $\displaystyle\varkappa$ are non-zero.

It follows from (15) that

[TABLE]

Remark. Minimal Lorentz surfaces satisfying $\displaystyle K^{2}-\varkappa^{2}=0$ (or equivalently $\displaystyle\mu^{2}=\nu^{2}$) are the analogue of minimal super-conformal surfaces in the Euclidean space $\displaystyle\mathbb{E}^{4}$. Instead of ellipse of normal curvature defined for surfaces in $\displaystyle\mathbb{E}^{4}$, in the pseudo-Euclidean space $\displaystyle\mathbb{E}^{4}_{2}$ we can use the notion of curvature hyperbola associated to the second fundamental form of a Lorentz surface (see [2]). Using the terminology from the Euclidean space, we call a minimal Lorentz surfaces super-conformal if at each point the Gauss curvature and the normal curvature satisfy the equality $\displaystyle K^{2}-\varkappa^{2}=0$.

In this paper we study surfaces for which $\displaystyle K^{2}-\varkappa^{2}>0$, so we assume that $\displaystyle\mu^{2}-\nu^{2}\neq 0$. Then, from (14) we obtain that

[TABLE]

Taking into account that $\displaystyle\gamma_{1}=-\displaystyle{\frac{f_{v}}{f^{2}}}$, $\displaystyle\gamma_{2}=-\displaystyle{\frac{f_{u}}{f^{2}}}$, we get

[TABLE]

which imply that the function $\displaystyle\left|\mu^{2}-\nu^{2}\right|f^{4}$ is constant. Hence,

[TABLE]

where $\displaystyle c$ is a non-zero constant. After the change of the parameters

[TABLE]

we may assume that

[TABLE]

Following the terminology in [8], we call the parameters $\displaystyle(u,v)$ canonical if

[TABLE]

Note that any minimal Lorentz surface of general type locally admits canonical parameters.

Now, we assume that the surface is parametrized by canonical parameters. Then by use of (14) and (16) we obtain that the functions $\displaystyle\gamma_{1}$, $\displaystyle\gamma_{2}$, $\displaystyle\beta_{1}$, and $\displaystyle\beta_{2}$ are expressed in terms of $\displaystyle\mu$ and $\displaystyle\nu$ as follows

[TABLE]

Using (17) and the first two equalities of (14) we obtain the following partial differential equations for the functions $\displaystyle\mu$ and $\displaystyle\nu$

[TABLE]

where $\displaystyle\Delta^{h}=\frac{\partial^{2}}{\partial u^{2}}-\frac{\partial^{2}}{\partial v^{2}}$ is the hyperbolic Laplace operator.

Now we can prove the following fundamental Bonnet-type theorem for the class of minimal Lorentz surfaces of general type.

Theorem 4.2**.**

Let $\displaystyle\mu(u,v)$ and $\displaystyle\nu(u,v)$ be two smooth functions, defined in a domain $\displaystyle{\mathcal{D}},\,\,{\mathcal{D}}\subset{\mathbb{R}}^{2}$, and satisfying the conditions:

[TABLE]

where $\displaystyle\varepsilon=\pm 1$. Let $\displaystyle\{x_{0},\,y_{0},\,(n_{1})_{0},\,(n_{2})_{0}\}$ be an orthonormal frame at a point $\displaystyle p_{0}\in\mathbb{E}^{4}_{2}$ such that $\displaystyle\langle x_{0},x_{0}\rangle=1$, $\displaystyle\langle y_{0},y_{0}\rangle=-1$, $\displaystyle\langle(n_{1})_{0},(n_{1})_{0}\rangle=\varepsilon$, $\displaystyle\langle(n_{2})_{0},(n_{2})_{0}\rangle=-\varepsilon$. Then there exist a subdomain $\displaystyle{\mathcal{D}}_{0}\subset{\mathcal{D}}$ and a unique minimal Lorentz surface of general type $\displaystyle M^{2}_{1}:z=z(u,v),\,\,(u,v)\in{\mathcal{D}}_{0}$, such that $\displaystyle M^{2}_{1}$ passes through $\displaystyle p_{0}$, the functions $\displaystyle\mu(u,v)$, $\displaystyle\nu(u,v)$ are the geometric functions of $\displaystyle M^{2}_{1}$, $\displaystyle\{x_{0},\,y_{0},\,(n_{1})_{0},\,(n_{2})_{0}\}$ is the geometric frame of $\displaystyle M^{2}_{1}$ at the point $\displaystyle p_{0}$, $\displaystyle\varepsilon=1$ (resp. $\displaystyle\varepsilon=-1$) in the case the geometric normal vector field $\displaystyle n_{1}$ is spacelike (resp. timelike). Furthermore, $\displaystyle(u,v)$ are canonical parameters of the surface.

Proof.

Let us define the following functions

[TABLE]

We consider the following system of partial differential equations for the unknown vector functions $\displaystyle x=x(u,v),\,y=y(u,v),\,n_{1}=n_{1}(u,v),\,n_{2}=n_{2}(u,v)$ in $\displaystyle\mathbb{E}_{2}^{4}$

[TABLE]

Denote

[TABLE]

[TABLE]

Using matrices $\displaystyle A$ and $\displaystyle B$ we can rewrite system (20) in the form

[TABLE]

The integrability conditions of system (21) are

[TABLE]

or equivalently,

[TABLE]

where $\displaystyle a_{i}^{j}$ and $\displaystyle b_{i}^{j}$ are the elements of the matrices $\displaystyle A$ and $\displaystyle B$. By use of (19) and equations (18) it can be checked that equalities (22) are fulfilled. Hence, there exist a subset $\displaystyle\mathcal{D}_{1}\subset\mathcal{D}$ and unique vector functions $\displaystyle x=x(u,v),\,y=y(u,v),\,n_{1}=n_{1}(u,v),\,n_{2}=n_{2}(u,v),\,\,(u,v)\in\mathcal{D}_{1}$, which satisfy system (20) and the initial conditions

[TABLE]

Now, we shall prove that the vectors $\displaystyle x(u,v),\,y(u,v),\,n_{1}(u,v),\,n_{2}(u,v)$ form an orthonormal frame in $\displaystyle\mathbb{E}_{2}^{4}$ for each $\displaystyle(u,v)\in\mathcal{D}_{1}$. Let us consider the following functions

[TABLE]

defined for $\displaystyle(u,v)\in\mathcal{D}_{1}$. Since $\displaystyle x(u,v),\,y(u,v),\,n_{1}(u,v),\,n_{2}(u,v)$ satisfy (20), for the functions $\displaystyle\varphi_{i}(u,v),\,\,i=1,\dots,10$ we obtain the following system

[TABLE]

where $\displaystyle\alpha_{i}^{j},\beta_{i}^{j},\,\,i,j=1,\dots,10$ are functions of $\displaystyle(u,v)\in\mathcal{D}_{1}$. This is a linear system of partial differential equations satisfying the initial conditions $\displaystyle\varphi_{i}(u_{0},v_{0})=0,\,\,i=1,\dots,10$. Hence, $\displaystyle\varphi_{i}(u,v)=0,\,\,i=1,\dots,10$ for each $\displaystyle(u,v)\in\mathcal{D}_{1}$. Consequently, the vector functions $\displaystyle x(u,v),\,y(u,v),\,n_{1}(u,v),\,n_{2}(u,v)$ form an orthonormal frame in $\displaystyle\mathbb{E}_{2}^{4}$ for each $\displaystyle(u,v)\in\mathcal{D}_{1}$.

Finally, we consider the following system of partial differential equations for the vector function $\displaystyle z=z(u,v)$

[TABLE]

Using (20) we obtain that the integrability conditions $\displaystyle z_{uv}=z_{vu}$ of system (23) are fulfilled. Hence, there exist a subset $\displaystyle\mathcal{D}_{0}\subset\mathcal{D}_{1}$ and a unique vector function $\displaystyle z=z(u,v)$, defined for $\displaystyle(u,v)\in\mathcal{D}_{0}$ and satisfying $\displaystyle z(u_{0},v_{0})=p_{0}$.

Consequently, the surface $\displaystyle M^{2}_{1}:z=z(u,v),\,\,(u,v)\in\mathcal{D}_{0}$ satisfies the assertion of the theorem.

∎

The meaning of Theorem 4.2 is that any minimal Lorentz surface of general type is determined up to a rigid motion in $\displaystyle\mathbb{E}^{4}_{2}$ by two invariant functions $\displaystyle\mu$ and $\displaystyle\nu$ satisfying the system of natural partial differential equations (18).

Using equalities (15) we can express the functions $\displaystyle\mu$ and $\displaystyle\nu$ in terms of the Gauss curvature $\displaystyle K$ and the normal curvature $\displaystyle\varkappa$. More precisely, the following relations hold true

[TABLE]

Hence, equations (18) can be rewritten in terms of $\displaystyle K$ and $\displaystyle\varkappa$ as follows

[TABLE]

Then the fundamental theorem for minimal Lorentz surfaces of general type can be stated in terms of the curvatures $\displaystyle K$ and $\displaystyle\varkappa$ as follows.

Theorem 4.3**.**

Let $\displaystyle K(u,v)\neq 0$ and $\displaystyle\varkappa(u,v)\neq 0$ be two smooth functions, defined in a domain $\displaystyle{\mathcal{D}},\,\,{\mathcal{D}}\subset{\mathbb{R}}^{2}$, such that $\displaystyle K^{2}-\varkappa^{2}>0$, and satisfying the equations

[TABLE]

where $\displaystyle\varepsilon=\pm 1$. Then there exists a unique (up to a rigid motion) minimal Lorentz surface of general type such that $\displaystyle K(u,v)$ and $\displaystyle\varkappa(u,v)$ are the Gauss curvature and the normal curvature, respectively; $\displaystyle\varepsilon=1$ (resp. $\displaystyle\varepsilon=-1$) in the case the geometric normal vector field $\displaystyle n_{1}$ is spacelike (resp. timelike). Furthermore, $\displaystyle(u,v)$ are the canonical parameters of the surface.

Finally, the background system of partial differential equations for minimal Lorentz surfaces of general type in $\displaystyle\mathbb{E}^{4}_{2}$ is system (24) or equivalently, system (18).

5. Example

In this section we shall construct an example of a minimal Lorentz surface of general type applying the procedure given in the proof of Theorem 4.2 to a concrete solution to system (18).

Let us consider the following functions

[TABLE]

where $\displaystyle u\in\left(\sqrt[4]{3}\,\frac{\pi}{3};2\sqrt[4]{3}\,\frac{\pi}{3}\right)$. For simplicity we denote $\displaystyle t=\frac{u}{\sqrt[4]{3}}$, $\displaystyle t\in\left(\frac{\pi}{3};2\frac{\pi}{3}\right)$. Using the functions $\displaystyle\mu$ and $\displaystyle\nu$ defined above, we obtain

[TABLE]

Direct computations show that $\displaystyle\mu(u)$ and $\displaystyle\nu(u)$ satisfy system (18) in the case $\displaystyle\varepsilon=-1$. Let us consider the following functions

[TABLE]

Now, system (20) for the vector functions $\displaystyle x=x(u,v),\,y=y(u,v),\,n_{1}=n_{1}(u,v),\,n_{2}=n_{2}(u,v)$ takes the form

[TABLE]

Denoting

[TABLE]

we rewrite the above system as follows

[TABLE]

Let us consider the vector functions $\displaystyle X=x-n_{1}$ and $\displaystyle Y=y-n_{2}$. It follows from (26) that

[TABLE]

Hence, for each coordinate function $\displaystyle X^{k}$ ($\displaystyle k=1,\dots,4$) of $\displaystyle X$ we have $\displaystyle X^{k}_{u}=-\varphi X^{k}$. The last equality implies that

[TABLE]

where $\displaystyle C^{k}(v)$, $\displaystyle k=1,\dots,4$ is a function of $\displaystyle v$. Calculating the integral in (27) we obtain

[TABLE]

for some functions $\displaystyle\xi^{k}(v)$. We denote $\displaystyle a(t)=\frac{\cos 2t-\cos t}{\sin t\sqrt{1-4\cos^{2}t}}$. Then the vector function $\displaystyle X(u,v)$ is expressed as

[TABLE]

for some vector function $\displaystyle\xi(v)$.

Similarly, for each coordinate function $\displaystyle Y^{k}$ ($\displaystyle k=1,\dots,4$) of $\displaystyle Y$ we have $\displaystyle Y^{k}_{u}=-\psi Y^{k}$, which implies that

[TABLE]

where $\displaystyle D^{k}(v)$, $\displaystyle k=1,\dots,4$ is a function of $\displaystyle v$. Calculating the integral in (29) we get

[TABLE]

$\displaystyle\eta^{k}(v)$ being a function of $\displaystyle v$. Denoting $\displaystyle b(t)=\frac{1+2\cos t}{\sqrt{1-4\cos^{2}t}}$, we obtain that the vector function $\displaystyle Y(u,v)$ is expressed as

[TABLE]

for some vector function $\displaystyle\eta(v)$.

Having in mind that $\displaystyle x$, $\displaystyle y$, $\displaystyle n_{1}$, $\displaystyle n_{2}$ should satisfy $\displaystyle\langle x,x\rangle=1$, $\displaystyle\langle y,y\rangle=1$, $\displaystyle\langle n_{1},n_{1}\rangle=-1$, $\displaystyle\langle n_{2},n_{2}\rangle=1$, we get $\displaystyle\langle X,X\rangle=0$, $\displaystyle\langle Y,Y\rangle=0$, $\displaystyle\langle X,Y\rangle=0$, and hence $\displaystyle\langle\xi,\xi\rangle=0$, $\displaystyle\langle\eta,\eta\rangle=0$, $\displaystyle\langle\xi,\eta\rangle=0$. Let us consider the following vector functions

[TABLE]

Since $\displaystyle X=x-n_{1}$, $\displaystyle Y=y-n_{2}$, from (28) and (30) we get

[TABLE]

which imply $\displaystyle x_{v}=a\,\xi_{v}+(n_{1})_{v}$, $\displaystyle y_{v}=b\,\eta_{v}+(n_{2})_{v}$. Now, using (26) and (32), we obtain

[TABLE]

Equalities (33) together with (25) and (31) allow us to find the vector functions $\displaystyle n_{1}$ and $\displaystyle n_{2}$. They are expressed as follows

[TABLE]

where $\displaystyle s=\frac{v}{\sqrt[4]{3}}$.

Using (32) and (34) we find the vector functions $\displaystyle x$ and $\displaystyle y$:

[TABLE]

Now we consider system (23). In our example it takes the form

[TABLE]

Using that $\displaystyle t=\frac{u}{\sqrt[4]{3}}$, $\displaystyle s=\frac{v}{\sqrt[4]{3}}$, we rewrite the above system as

[TABLE]

Now, system (36) together with (35) imply

[TABLE]

where $\displaystyle C$ is a constant vector.

The vector function $\displaystyle z(t,s)$ given by (37) determines a minimal Lorentz surface of general type in $\displaystyle\mathbb{E}^{4}_{2}$. Note that $\displaystyle(t,s)$ are not the canonical parameters of the surface. With respect to canonical parameters the surface is parametrized as follows

[TABLE]

Finally, given a concrete solution $\displaystyle(\mu,\nu)$ to system (18) we obtained an example of a minimal Lorentz surface of general type parametrized by canonical parameters.

Acknowledgments: The authors are partially supported by the National Science Fund, Ministry of Education and Science of Bulgaria under contract DN 12/2.