Minimal translation graphs in semi-Euclidean space

1.   Introduction

As is wellknown the first non-trivial examples of minimal surfaces in 3-dimensional Euclidean space ${{E}^{3}}$ are the catenoids, the helicoids and the minimal translation surfaces. A surface is called a translation surface if it is given by an immersion

where $z = f(x)+g(y).$ Scherk proved in 1835 that the only minimal translation surfaces (besides the planes) are the surfaces given by

where $a$ is a non-zero constant ^[1].

In ^[2], it has been shown that the minimal translation surfaces are generalized to minimal translation hypersurfaces as follows:

"Let ${{M}^{n}}\, \, (n\ge 2)$ be a translation hypersurface in ${{E}^{n+1}}$ i.e. ${{M}^{n}}$ is the graph of a function

where ${{f}_{i}}$ is a smooth function of one real variable for $i = 1, \ldots, n.$ Then ${{M}^{n}}$ is minimal if and only if either ${{M}^{n}}$ is a hyperplane or a product submanifold ${{M}^{n}} = {{M}^{2}}\times {{E}^{n-2}}$, where ${{M}^{2}}$ is a minimal translation surface of Scherk in ${{E}^{3}}$."

In ^[3], Woestyne parameterized minimal translation surfaces in the 3-dimensional Minkowski space $\mathbb{R}_{1}^{3}$ with metric $g = dx_{1}^{2}+dx_{2}^{2}-dx_{3}^{2}$, in the following theorems:

Theorem 1. Every minimal, spacelike surface of translation in $\mathbb{R}_{1}^{3}$, is congruent to a part of one of the following surfaces:

1. A spacelike plane,

2. The surface of Scherk of the first kind, a parametrization of the surface is $F(x, y) = (x, y, {{a}^{-1}}\log (\cosh (ay)/\cosh (ax))),$ with $\tanh^{2}(ax)+\tanh^{2}(ay) < 1$ ^[3].

Theorem 2. Every minimal, timelike surface of translation in $\mathbb{R}_{1}^{3}$, is congruent to a part of one of the following surfaces:

1. A timelike plane,

2. The surface of Scherk of the first kind, a parametrization of the surface is $F(x, y) = (x, y, {{a}^{-1}}\log (\cosh (ay)/\cosh (-ax)))\,$with $\tan {{h}^{2}}(-ax)+\tan {{h}^{2}}(ay) > 1$.

3. The surface of Scherk of the second kind, a parametrization of the surface is $F(x, y) = (x, y, {{a}^{-1}}\log (\cosh (ay)/\sinh (-ax)))\,$.

4. The surface of Scherk of the third kind, a parametrization of the surface is $F(x, y) = (x, y, {{a}^{-1}}\log (\sinh (ay)/\sinh (-ax)))\,$.

5. A flat B-scroll over a null curve, a parametrization of the surface is $F(x, y) = (x, y, \pm x+g(y))\,$with $g(y)$ an arbitrary function ^[3].

Seo, gave a classification of the translation hypersurfaces with constant mean curvature or constant Gauss–Kronecker curvature in Euclidean space or Lorentz– Minkowski space in ^[4]. Also they characterized the minimal translation hypersurfaces in the upper half-space model of hyperbolic space. In particular, they proved the following theorem:

Theorem 3. Let $M$ be a translation hypersurface with constant mean curvature $H$ in ${{\mathbb{R}}^{n+1}}$. Then $M$ is congruent to a cylinder $\Sigma \times {{\mathbb{R}}^{n-2}}$, where $\Sigma$ is a constant mean curvature surface in ${{\mathbb{R}}^{3}}$. In particular, if $H = 0$, then $M$ is either a hyperplane or $M = \Sigma \times {{\mathbb{R}}^{n-2}}$, where $\Sigma$ is a Scherk's minimal translation surface in ${{\mathbb{R}}^{3}}$^[4].

And they can obtained a similar result in the Lorentz–Minkowski space as follows:

Theorem 4. Let $M$ be a spacelike translation hypersurface with constant mean curvature $H$ in ${{L}^{n+1}}.$ Then $M$ is congruent to a cylinder $\Sigma \times {{\mathbb{R}}^{n-2}}$, where $\Sigma$ is a constant mean curvature surface in ${{L}^{3}}$. In particular, if $H = 0$, then $M$ is either a hyperplane or $M = \Sigma \times {{\mathbb{R}}^{n-2}}$, where $\Sigma$ is a Scherk's maximal spacelike translation surface in ${{L}^{3}}$ ^[4].

In ^[5], Hasanis and Lopez classified and described the construction of all minimal translation surfaces in ${{\mathbb{R}}^{3}}$. In 2019, Aydın and Ogrenmis investigated translation hypersurfaces generated by translating planar curves and classified these translation hypersurfaces with constant Gauss-Kronecker and mean curvature ^[6]. Recently, many authors have studied the geometry of minimal translational hypersurface ^{[7,8,9,10,11,12]}.

In ^[12], Yang, Zhang and Fu gave a characterization of a class of minimal translation graphs which are generalization of minimal translation hypersurfaces in Euclidean space. In this paper we study a characterization of minimal translation graphs in semi-Euclidean space.

2.   Characterization of minimal translation graphs in semi-Euclidean space

By the ($n+1$)-dimensional semi-Euclidean space with index $\nu$, denoted by $\mathbb{R}_{\nu }^{n+1}$, mean ${{\mathbb{R}}^{n+1}}$ equipped with the semi-Euclidean metric

for which ${{\varepsilon }_{i}}, \, (i = 1, 2, \ldots, n+1)$ is either $-1$ or $1.$ We assume $n\ge 2.$ The number of minus signs is equal to the index $\nu$ and is given by

Let ${{M}^{n}}$ be a hypersurface of $\mathbb{R}_{\nu }^{n+1}$ for which the induced metric is non-degenerate. Then ${{M}^{n}}$ can locally always be seen as the graph of a function $F:{{\mathbb{R}}^{n}}\to \mathbb{R}$. In what follows, we will assume that $f$ is a function of the coordinates ${{x}_{1}}, \ldots, {{x}_{n}}$. This can easily be achieved possibly by rearranging the coordinates of $\mathbb{R}_{\nu }^{n+1}$. So ${{M}^{n}}$ is locally given by

Assume that ${{M}^{n}}$ is minimal. This means that the mean curvature vector vanishes at every point. The graph ${{M}^{n}}$ in the semi-Euclidean space $\mathbb{R}_{\nu }^{n}$ is minimal if and only if

One easily calculates that minimality condition above ^[13].

A hypersurface ${{M}^{n}}$ in the semi-Euclidean space $\mathbb{R}_{\nu }^{n+1}$ is called translation graph if it is the graph of the function given by

where $u = \sum\limits_{i = 1}^{n}{{{c}_{i}}{{x}_{i}}, }$ ${{c}_{i}}$ are constants, ${{c}_{n}}\ne 0$ and each ${{f}_{i}}$ is a smooth function of one real variable for $i = 1, 2, \ldots, n$. Additionally in this paper, we assume that the following condition are provided:

${{f}_{i}}$ vanishes nowhere for $i = 1, 2, \ldots, n$, otherwise ${{M}^{n}}$ is a non-degenerate hyperplane.

The minimality condition (2.3) can be rewritten as

Then we calculate the partial derivatives in the Eq (2.5) for the translation graph,

for $1\le i\le n-1.$ Since ${{M}^{n}}$ is minimal, we substitute (2.6)–(2.8) into (2.5) and we obtain

Since ${{c}_{n}}\ne 0,$ we take the derivative of the Eq (2.9) with respect to ${{x}_{n}}$, we have

According to the Eq (2.10), we get following cases:

Case 1. $f_{n}^{\prime\prime\prime} = 0.$

With proper translation, ${{f}_{n}} = m{{u}^{2}}$ for a constant $m\ne 0$ such that ${{f}_{n}} = m{{u}^{2}}.$ If $m = 0$, then ${{M}^{n}}$ would not be a translation graph. According to this, we rewrite (2.10)

Since $m\ne 0$, we get

In the first equation of (2.12), each $f_{i}^{\prime\prime}$ depends on a different variable, then $f_{i}^{\prime\prime}$ has to be a constant for $i = 1, \ldots, n-1.$ Also, let be ${{f}_{i}}({{x}_{i}}) = {{a}_{i}}x_{i}^{2},$ where ${{a}_{i}}$ is constant. Then from (2.12) we obtain following equations

Now we substitute ${{f}_{n}} = m{{u}^{2}}$ and ${{f}_{i}}({{x}_{i}}) = {{a}_{i}}x_{i}^{2}$ for $i = 1, \ldots, n-1$ in the Eq (2.9), we find

According to (2.13) and (2.14), we obtain

Since the above equation is a quadratic polynomial with ${{x}_{1}}, \ldots, {{x}_{n-1}}$, by the arbitrariness of ${{x}_{i}}$, we get

and

According to (2.16) and (2.17), we find

From (2.18), we can see that at most one ${{a}_{k}}{{c}_{k}}\ne 0$. Without loss of generality, we assume ${{a}_{{{k}_{0}}}}{{c}_{{{k}_{0}}}}\ne 0$ and every ${{a}_{k}}{{c}_{k}} = 0$ for $k\ne {{k}_{0}}$. From (2.19), we get ${{a}_{{{k}_{0}}}}\ne 0$ and ${{a}_{k}} = 0$ for $k\ne {{k}_{0}}$. According to this, with the first equation of (2.13) and the assumption (2.4), we have a contradiction. Therefore, we obtain that every ${{a}_{k}} = 0$ for $k = 1, \ldots, n-1$ and $f_{i}^{\prime\prime}({{x}_{i}}) = 0$. By substituting this equalities in (2.9) we obtain $m\sum\limits_{i = 1}^{n}{{{\varepsilon }_{i}}c_{i}^{2}} = 0$, which is a contradiction with the assumption (2.4).

Case 2. $f_{n}^{\prime\prime\prime}\ne 0$.

If we divide by $f_{n}^{\prime\prime\prime}$ on both sides of the Eq (2.10), we obtain

Differentiating (2.20) with respect to $u,$ we get

We have 3 possibilities.

Case 2a. ${{\left(\frac{f_{n}^{\prime\prime}}{f_{n}^{\prime\prime\prime}} \right)}_{u}}\ne 0$.

In this case,

where $a, b$ are constants. Since $f_{n}^{\prime\prime\prime}\ne 0$, then $a\ne 0$. By solving this equation we obtain

where $k$ is a nonzero constant. According to this equation, we get

Thus, according to (2.21), we obtain

Since each $f_{i}^{\prime\prime}$ depends on a different variable, then $f_{i}^{\prime\prime}$ has to be a constant for $i = 1, \ldots, n-1.$ Let be ${{f}_{i}}({{x}_{i}}) = {{a}_{i}}x_{i}^{2},$ where ${{a}_{i}}$ is constant. From (2.23) we obtain

Hence, from (2.20) we have

This is a contradiction.

Case 2b. $\sum\limits_{i = 1}^{n-1}{{{\varepsilon }_{i}}(\sum\limits_{\begin{smallmatrix} j = 1 \\ j\ne i \end{smallmatrix}}^{n}{{{\varepsilon }_{j}}c_{j}^{2}})f_{i}^{\prime\prime}}\ne 0$.

Let be $\sum\limits_{i = 1}^{n-1}{{{\varepsilon }_{i}}(\sum\limits_{\begin{smallmatrix} j = 1 \\ j\ne i \end{smallmatrix}}^{n}{{{\varepsilon }_{j}}c_{j}^{2}})f_{i}^{\prime\prime}} = 0$. Then each $f_{i}^{\prime\prime}$ is constant for $i = 1, \ldots, n-1.$ Also we can write ${{f}_{i}}({{x}_{i}}) = {{a}_{i}}x_{i}^{2},$ where ${{a}_{i}}$ is constant. From the assumption, we have

According to (2.21), we get

From (2.20), we obtain

which is a contradiction. Also it must be $\sum\limits_{i = 1}^{n-1}{{{\varepsilon }_{i}}(\sum\limits_{\begin{smallmatrix} j = 1 \\ j\ne i \end{smallmatrix}}^{n}{{{\varepsilon }_{j}}c_{j}^{2}})f_{i}^{\prime\prime}}\ne 0$. According to Cases 2a and 2b, we can rewrite (2.21)

where $m$ is constant. Thus we have

By integration of this equation, we get

where $c$ is a constant. Thus we have $f_{n}^{\prime}f_{n}^{\prime\prime} = -mf_{n}^{\prime\prime}+cf_{n}^{\prime\prime\prime}.$ By integration of this equation, we obtain

where ${{c}_{0}}$ is a constant. By solving this ODE, after a translation, we find

Moreover, from (2.24), we get

Since

all $f_{i}^{\prime\prime}$ functions don't vanish for $i = 1, \ldots, n-1.$ Let be $f_{{{i}_{0}}}^{\prime\prime}\ne 0.$ By differentiating the Eq (2.27) with respect to ${{x}_{{{i}_{0}}}}$, we obtain

Thus we get the following case.

Case 2c. $m\sum\limits_{\begin{smallmatrix} i = 1 \\ i\ne {{i}_{0}} \end{smallmatrix}}^{n}{{{\varepsilon }_{i}}c_{i}^{2}}-\sum\limits_{\begin{smallmatrix} i = 1 \\ i\ne {{i}_{0}} \end{smallmatrix}}^{n-1}{{{\varepsilon }_{i}}{{c}_{i}}f_{i}^{\prime}} = 0$.

Let be $m\sum\limits_{\begin{smallmatrix} i = 1 \\ i\ne {{i}_{0}} \end{smallmatrix}}^{n}{{{\varepsilon }_{i}}c_{i}^{2}}-\sum\limits_{\begin{smallmatrix} i = 1 \\ i\ne {{i}_{0}} \end{smallmatrix}}^{n-1}{{{\varepsilon }_{i}}{{c}_{i}}f_{i}^{\prime}}\ne 0$. From (2.28), we get

where $a$ is a constant. From (2.29), we obtain

where ${{b}_{{{i}_{0}}}}$ and ${{d}_{{{i}_{0}}}}$ are constants. According to (2.29), we get

for each $i = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1.$ By solving the equation, we obtain

where ${{b}_{i}}$ are constants for $i = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1.$ By differentiating (2.27) with respect to ${{x}_{k}}$ for $k = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1,$ we get

We substitute (2.30) and (2.31) in (2.32) and we obtain

where

for $k = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1.$ Since ${{x}_{k}}$ is arbitrary and from (2.31), we get $c_{k}^{3}{{b}_{k}}c_{i}^{2}{{b}_{i}} = 0$. Hence ${{c}_{k}}{{b}_{k}}{{c}_{i}}{{b}_{i}} = 0$ for $i, \, k = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1$ and $i\ne k$. Thus, there is at most one ${{c}_{k}}{{b}_{k}}\ne 0$. Let be all ${{c}_{k}}{{b}_{k}} = 0$ for $k = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1$ and $k\ne {{j}_{0}}.$ It follows

and

for $k = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1$ and $k\ne {{j}_{0}}.$ By substituting (2.30), (2.34) and (2.35) in (2.27), we obtain

where

We rewrite the Eq (2.20) for $n = 3$

According to (2.25) and (2.27), (2.36) becomes

By differentiating the Eq (2.37) with respect to ${{x}_{1}}$, we obtain

If we arrange the equation above, then we get

From this equation, $f_{2}^{\prime\prime}$ is constant and $f_{2}^{\prime\prime} = 0$. This is a contradiction. Also it must be

We showed that $m\sum\limits_{\begin{smallmatrix} i = 1 \\ i\ne {{i}_{0}} \end{smallmatrix}}^{n}{{{\varepsilon }_{i}}c_{i}^{2}}-\sum\limits_{\begin{smallmatrix} i = 1 \\ i\ne {{i}_{0}} \end{smallmatrix}}^{n-1}{{{\varepsilon }_{i}}{{c}_{i}}f_{i}^{\prime}} = 0$ and $f_{i}^{\prime\prime}$ are constants for $1\le i\le n-1, \, \, i\ne {{i}_{0}}$. Then $f_{i}^{\prime}({{x}_{i}}) = {{a}_{i}}$, where ${{a}_{i}}$ are constants for $i = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1$ and

We rewrite (2.20),

According to (2.25) and (2.39), we rewrite (2.40)

We arrange this equation

with

From (2.39), we rewrite (2.42)

By solving the equation, we find

with

Moreover $f_{i}^{\prime}({{x}_{i}}) = {{a}_{i}}$, where ${{a}_{i}}$ are constants for $i = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1$ and we rewrite (2.9) again

According to (2.41) and (2.44), we get

Since $f_{{{i}_{0}}}^{\prime\prime}\ne 0$, by substituting (2.39) into (2.45) and we obtain

When considered this equation with (2.26), then

According to (2.39), we get

Depending on the epsilones (-1 or +1) in the above equation, ${{m}^{2}}+{{c}_{0}}$ can be positive, negative and zero. With suitable translation, we get ${{f}_{i}} = 0$ for $i = 1, \ldots, {{i}_{0}}-1, {{i}_{0}}+1, \ldots, n-1$ and following the equations

and

where

Therefore we complete the proof of the following main theorem.

Main theorem. $M^n$ is a non-degenerate minimal translation graph in semi-Euclidean space $\mathbb{R}_{\nu }^{n+1}$, if it is congruent to a part of one of the following surfaces:

1. A non-degenerate hyperplane,

2. A hypersurface parameterized by

where $u = \sum\limits_{i = 1}^{n}{{{c}_{i}}{{x}_{i}}, }$ ${{c}_{i}}$ are constants, ${{c}_{n}}\ne 0$, with the conditions in the Eq (2.4), for a unique ${{i}_{0}}$, $1\le {{i}_{0}}\le n-1$, such that ${{f}_{{{i}_{0}}}}$ and ${{f}_{n}}$ one of the previous forms in (2.46) and (2.47), respectively. In additionally, ${{f}_{k}}({{x}_{k}}) = 0$ for $k\ne {{i}_{0}}$ and $1\le k\le n-1$.

3.   Conclusions

Semi-Euclidean spaces are important in applications of general relativity which is the explanation of gravity in modern physics. In this study, we have a characterization of minimal translation graphs which are generalization of minimal translation hypersurfaces in semi-Euclidean space. Also, we obtain the main theorem by which we classify all non-degenerate minimal translation graphs.

Acknowledgments

The author would like to thank the referees for their valuable suggestions.

Conflict of interest

The author declares no conflict of interest.