PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 241–246 DOI: https://doi.org/10.2298/PIM1716241S

TRANSFORMS FOR MINIMAL SURFACES IN 5-DIMENSIONAL SPACE FORMS

Makoto Sakaki

Abstract. For a minimal surface in a 5-dimensional space form, we give transforms to get another minimal surface in another 5-or 4-dimensional space form.

1. Introduction

For a minimal surface in the 3-sphere S^{3}, the unit normal vector field, that is, the Gauss map gives another minimal surface in S^{3} possibly with singularities (cf. [5]). It is generalized by Bolton, Pedit and Woodward [2] for superconformal minimal surfaces in odd-dimensional spheres. On the other hand, Bolton and Vrancken [3] discovered new transforms from a minimal surface with non-circular ellipse of curvature in the 5-sphere S^{5}, to another minimal surface in S^{5}, which are called (±)transforms (see also [1, 4]).

In this paper, generalizing them, we give transforms from a minimal surface in a 5-dimensional space form, to another minimal surface in another 5-or 4-dimensional space form.

Let N^{n}(c) be the n-dimensional Riemannian space form of constant curvature c, where c is either 1, 0 or −1. In particular, let N^{n}(1) = S^{n}, N^{n}(0) = R^{n} and N^{n}(−1) = H^{n}. Let R_{1}^{n+1} be the (n + 1)-dimensional Minkowski space with standard coordinate system (x_{1}, ⋯, x_{n}, x_{n+1} ) of signature (+, ⋯, +, −). Then

H^{n} = {(x_{1}, ⋯, x_{n}, x_{n+1} ) ∈ R_{1}^{n+1} | x_{1}^{2} + ⋯+ x_{n}^{2} − x_{n+1}^{2} = −1},

and

S_{1}^{n} = {(x_{1}, ⋯, x_{n}, x_{n+1} ) ∈ R_{1}^{n+1} | x_{1}^{2} + ⋯+ x_{n}^{2} − x_{n+1}^{2} = 1},

where S_{1}^{n} is the n-dimensional de Sitter space.

Let f: M → N^{5}(c) be an immersion of a 2-dimensional manifold M into N^{5}(c). We denote by h the second fundamental form of f. The first normal space T_{1}^{⊥} (x) at x ∈ M is defined by

T_{1}^{⊥} (x) = {h(X, Y) | X, Y ∈ T_{x}M}.

2010 Mathematics Subject Classification: Primary 53A10; Secondary 53B25.

Key words and phrases: minimal surface, space form, transform, ellipse of curvature.

Communicated by Stevan Pilipović.

The ellipse of curvature E(x) at x ∈ M is defined by

E(x) = {h(X, X) | X ∈ T_{x}M, |X| = 1}.

We assume that f: M → N^{5}(c) is a minimal immersion. Suppose that the ellipse of curvature is non-degenerate at any point. Then the dimension of the first normal space is 2 at any point. Let e_{5} be the unit normal vector to f(M) which is orthogonal to the first normal space. Then we can regard G = e_{5} as a map to either S^{5}, S^{4} or S_{1}^{5}, according to when c = 1, 0 or −1. It is the Gauss-like map.

Theorem 1.1. Let f: M → N^{5}(c) be a minimal surface. Suppose that the ellipse of curvature is a non-degenerate circle at any point. If the Gauss-like map G is non-degenerate, then it gives a minimal surface in either S^{5}, S^{4} or S_{1}^{5}.

Remark 1.1. The case c = 1 can be seen in [2].

Next we consider the case where the ellipse of curvature is not a circle. For a minimal surface f: M → N^{5}(c), suppose that the ellipse of curvature is nondegenerate and non-circular at any point. Let a and b be the semi-minor and semimajor axes of the ellipse of curvature, respectively. We choose the local normal orthonormal frame field {e_{α} }_{3⩽α⩽5} so that e_{3} is in the direction of the semi-minor axis and e_{4} is in the direction of the semi-major axis. Now, for ε = +1 or −1, let

f^{ε} = ε\sqrt{1 − ( \frac{a}{b})^{2}} e_{4} + \frac{a}{b}e_{5}.

Then f^{ε} is a map to either S^{5}, S^{4} or S_{1}^{5}, according to when c = 1, 0 or −1.

Theorem 1.2. Let f: M → N^{5}(c) be a minimal surface. Suppose that the ellipse of curvature is non-degenerate and non-circular at any point. Then f^{ε} gives a minimal surface in either S^{5}, S^{4} or S_{1}^{5}.

Remark 1.2. It is a generalization of [3] for S^{5}.

2. Preliminaries

In this section, we recall the method of moving frames for surfaces in 5dimensional space forms. We shall use the following convention on the ranges of indices:

1 ⩽ A, B, … ⩽ 5, 1 ⩽ i, j, … ⩽ 2, 3 ⩽ α, β, … ⩽ 5.

Let {e_{A}} be a local orthonormal frame field in N^{5}(c), and {ω^{A}} be the dual coframe field. Let ω_{B}^{A} denote the connection forms which satisfy ω_{B}^{A} = −ω_{A}^{B}. The structure equations are given by

(2.1)  dω^{A} = − ∑_{B} ω_{B}^{A} ∧ ω^{B},

dω_{B}^{A} = − ∑_{C} ω_{C}^{A} ∧ ω_{B}^{C} + \frac{1}{2} ∑_{C,D} R_{BCD}^{A} ω^{C} ∧ ω^{D}, R_{BCD}^{A} = c(δ_{C}^{A}δ_{BD} − δ_{D}^{A}δ_{BC} ).

Let f: M → N^{5}(c) be a surface in N^{5}(c). When c = 1, f is an R^{6}-valued map with ⟨f, f⟩ = 1. When c = −1, f is an R_{1}^{6}-valued map with ⟨f, f⟩ = −1.

We choose the frame {e_{A}} so that {e_{i}} are tangent to f(M). In the following, the argument will be restricted to f(M). Then ω^{α} = 0 along f(M), and by (2.1), we have

0 = − ∑_{i} ω_{i}^{α} ∧ ω^{i}.

So there exists a symmetric tensor {h_{ij}^{α} } so that

ω_{i}^{α} = ∑_{j} h_{ij}^{α} ω^{j} ,

where h_{ij}^{α} are the components of the second fundamental form h of f.

In the ambient R^{6}(⊃ S^{5}), R^{5} or R_{1}^{6}(⊃ H^{5}), according to when c = 1, 0 or −1, we have

de_{j} = ∑_{i} e_{i}ω_{j}^{i} + ∑_{α} e_{α} ω_{j}^{α} − cfω^{j} ,

and

de_{β} = ∑_{i} e_{i}ω_{β}^{i} + ∑_{α} e_{α} ω_{β}^{α} .

The mean curvature vector H of f is given by

H = \frac{1}{2} ∑_{α} (h_{11}^{α} + h_{22}^{α})e_{α} .

We say that f is minimal if H = 0 identically.

3. Proof of Theorem 1.1

Proof. Since the ellipse of curvature is a non-degenerate circle at any point, we can choose the local orthonormal frame field {e_{A}} so that

(h_{ij}^{3} ) = ( \begin{matrix}a & 0 \\ 0 & −a \end{matrix} ) , (h_{ij}^{4} ) = ( \begin{matrix}0 & a \\ a & 0 \end{matrix}) , (h_{ij}^{5} ) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) ,

where a > 0. Then

ω_{1}^{3} = aω^{1}, ω_{2}^{3} = −aω^{2}, ω_{1}^{4} = aω^{2}, ω_{2}^{4} = aω^{1}, ω_{1}^{5} = ω_{2}^{5} = 0.

We compute that

0 = dω_{1}^{5} = −ω_{3}^{5} ∧ ω_{1}^{3} − ω_{4}^{5} ∧ ω_{1}^{4} = a(ω^{1} ∧ ω_{3}^{5} − ω_{4}^{5} ∧ ω^{2})

and

0 = dω_{2}^{5} = −ω_{3}^{5} ∧ ω_{2}^{3} − ω_{4}^{5} ∧ ω_{2}^{4} = a(ω_{3}^{5} ∧ ω^{2} + ω^{1} ∧ ω_{4}^{5}).

Then, using the notation like

ω_{3}^{5} = (ω_{3}^{5})_{1}ω^{1} + (ω_{3}^{5})_{2}ω^{2}, ω_{4}^{5} = (ω_{4}^{5})_{1}ω^{1} + (ω_{4}^{5})_{2}ω^{2},

we have

(ω_{3}^{5})_{2} − (ω_{4}^{5})_{1} = 0, (ω_{3}^{5})_{1} + (ω_{4}^{5})_{2} = 0.

So we can write

ω_{3}^{5} = pω^{1} + qω^{2}, ω_{4}^{5} = qω^{1} − pω^{2}

for some functions p and q.

For the Gauss-like map G = e_{5}, we have

dG(e_{1}) = de_{5}(e_{1}) = (ω_{5}^{3})_{1} e_{3} + (ω_{5}^{4})_{1} e_{4} = −pe_{3} − qe_{4},
dG(e_{2}) = de_{5}(e_{2}) = (ω_{5}^{3})_{2} e_{3} + (ω_{5}^{4})_{2} e_{4} = −qe_{3} + pe_{4},

and

⟨dG(e_{1}), dG(e_{1})⟩ = ⟨dG(e_{2}), dG(e_{2})⟩ = p^{2} + q^{2}, ⟨dG(e_{1}), dG(e_{2})⟩ = 0.

Assume that G is non-degenerate in the following. Then p^{2} + q^{2} > 0, and G is conformal to f.

Now we have

dG = −e_{3}(pω^{1} + qω^{2}) − e_{4}(qω^{1} − pω^{2}).

Let ∗ denote the Hodge star operator so that ∗ω^{1} = ω^{2} and ∗ω^{2} = −ω^{1}. Then

∗dG = e_{3}(qω^{1} − pω^{2}) − e_{4}(pω^{1} + qω^{2}) = e_{3}ω_{4}^{5} − e_{4}ω_{3}^{5}.

We can compute that

d(∗dG) = −2(p^{2} + q^{2})e_{5}ω^{1} ∧ ω^{2}.

Denoting the Laplacian by Δ, we get ΔG = −2(p^{2} + q^{2})G. So the Gauss-like map G is a conformal harmonic map to either S^{5}, S^{4} or S_{1}^{5}, according to when c = 1, 0 or −1. Thus G gives a minimal surface in either S^{5}, S^{4} or S_{1}^{5}. □

4. Proof of Theorem 1.2

Proof. Since the ellipse of curvature is non-degenerate and non-circular at any point, we can choose the local orthonormal frame field {e_{A}} so that

(h_{ij}^{3} ) = ( \begin{matrix}a & 0 \\ 0 & −a \end{matrix} ) , (h_{ij}^{4} ) = ( \begin{matrix}0 & b \\ b & 0 \end{matrix}) , (h_{ij}^{5} ) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) ,

where 0 < a < b. We note that a and b are the semi-minor and semi-major axes of the ellipse of curvature, respectively. Then we have

ω_{1}^{3} = aω^{1}, ω_{2}^{3} = −aω^{2}, ω_{1}^{4} = bω^{2}, ω_{2}^{4} = bω^{1}, ω_{1}^{5} = ω_{2}^{5} = 0.

We compute that

dω_{1}^{3} = da ∧ ω^{1} − aω_{2}^{1} ∧ ω^{2} = −ω_{2}^{3} ∧ ω_{1}^{2} − ω_{4}^{3} ∧ ω_{1}^{4} = aω_{2}^{1} ∧ ω^{2} − bω_{4}^{3} ∧ ω^{2}.

Using the notation like

ω_{2}^{1} = (ω_{2}^{1})_{1}ω^{1} + (ω_{2}^{1})_{2}ω^{2}, ω_{4}^{3} = (ω_{4}^{3})_{1}ω^{1} + (ω_{4}^{3})_{2}ω^{2},
da = a_{1}ω^{1} + a_{2}ω^{2}, db = b_{1}ω^{1} + b_{2}ω^{2},

we have

2a(ω_{2}^{1})_{1} − b(ω_{4}^{3})_{1} = −a_{2}.

Similarly, from dω_{2}^{3}, dω_{1}^{4} and dω_{2}^{4},

2a(ω_{2}^{1})_{2} − b(ω_{4}^{3})_{2} = a_{1}, 2b(ω_{2}^{1})_{2} − a(ω_{4}^{3})_{2} = b_{1}, 2b(ω_{2}^{1})_{1} − a(ω_{4}^{3})_{1} = −b_{2}.

Thus we get

2aω_{2}^{1} − bω_{4}^{3} = ∗da, 2bω_{2}^{1} − aω_{4}^{3} = ∗db,

TRANSFORMS FOR MINIMAL SURFACES 245

and

ω_{2}^{1} = \frac{1}{4}(∗d log(b^{2} − a^{2})), ω_{4}^{3} = \frac{a(∗db) − b(∗da)}{b^{2} − a^{2}} = −\frac{∗d(a/b)}{1 − (a/b)^{2}} .

Next we compute that

0 = dω_{1}^{5} = −ω_{3}^{5} ∧ ω_{1}^{3} − ω_{4}^{5} ∧ ω_{1}^{4} = aω^{1} ∧ ω_{3}^{5} − bω_{4}^{5} ∧ ω^{2}

and

0 = dω_{2}^{5} = −ω_{3}^{5} ∧ ω_{2}^{3} − ω_{4}^{5} ∧ ω_{2}^{4} = aω_{3}^{5} ∧ ω^{2} + bω^{1} ∧ ω_{4}^{5}.

Then we can write

ω_{3}^{5} = b(pω^{1} + qω^{2}), ω_{4}^{5} = a(qω^{1} − pω^{2})

for some functions p and q.

From dω_{4}^{3} = −ω_{1}^{3} ∧ ω_{4}^{1} − ω_{2}^{3} ∧ ω_{4}^{2} − ω_{5}^{3} ∧ ω_{4}^{5}, we obtain

(4.1)  − \frac{Δ(a/b)}{1 − (a/b)^{2}} − \frac{2(a/b)|d(a/b)|^{2}}{(1 − (a/b)^{2})^{2}} = ab(2 − p^{2} − q^{2}).
Set r = a/b. Then f^{ε} = ε\sqrt{1 − r^{2}} e_{4} + re_{5}. We can compute that
df^{ε}(e_{1}) = − εb\sqrt{1 − r^{2}} e_{2} + ( ε\frac{r_{2}}{\sqrt{1 − r^{2}}} − ap) e_{3}

− ( ε\frac{r_{1}}{\sqrt{1 − r^{2}}} + aq) (re_{4} − ε\sqrt{1 − r^{2}} e_{5}),

and

df^{ε}(e_{2}) = − εb\sqrt{1 − r^{2}} e_{1} − ( ε\frac{r_{1}}{\sqrt{1 − r^{2}}} + aq) e_{3}

− ( ε\frac{r_{2}}{\sqrt{1 − r^{2}}} − ap) (re_{4} − ε\sqrt{1 − r^{2}} e_{5}).

Set

A = ε\frac{r_{1}}{\sqrt{1 − r^{2}}} + aq, B = ε\frac{r_{2}}{\sqrt{1 − r^{2}}} − ap.

Then we have

⟨df^{ε}(e_{1}), df^{ε}(e_{1})⟩ = ⟨df^{ε}(e_{2}), df^{ε}(e_{2})⟩ = b^{2} − a^{2} + A^{2} + B^{2}(> 0)

= b^{2} − a^{2} + \frac{|dr|^{2}}{1 − r^{2}} + \frac{2εa(qr_{1} − pr_{2})}{\sqrt{1 − r^{2}}} + a^{2}(p^{2} + q^{2}),

and ⟨df^{ε}(e_{1}), df^{ε}(e_{2})⟩ = 0. So f^{ε} is conformal to f.

Now we have

df^{ε} = − εb\sqrt{1 − r^{2}} (e_{2}ω^{1} + e_{1}ω^{2}) − εe_{3}(∗d(sin^{−1} r)) − ae_{3}(pω^{1} + qω^{2})
+ εe_{4}d(\sqrt{1 − r^{2}} ) + e_{5}dr − are_{4}(qω^{1} − pω^{2}) + εa\sqrt{1 − r^{2}} e_{5}(qω^{1} − pω^{2}),

and

∗df^{ε} = ε\sqrt{1 − r^{2}} (e_{1}ω_{2}^{4} − e_{2}ω_{1}^{4}) + εe_{3}d(sin^{−1} r) + e_{3}ω_{4}^{5}
+ εe_{4}(∗d(\sqrt{1 − r^{2}} )) + e_{5}(∗dr) − r^{2}e_{4}ω_{3}^{5} + εr\sqrt{1 − r^{2}} e_{5}ω_{3}^{5}.

We need to compute d(∗df^{ε}) to get Δf^{ε}. We note that

Δ(\sqrt{1 − r^{2}} ) = −\frac{rΔr}{\sqrt{1 − r^{2}}} − \frac{|dr|^{2}}{(1 − r^{2})^{3/2}} ,

and by (4.1),

Δr = ab(p^{2} + q^{2} − 2)(1 − r^{2}) − \frac{2r|dr|^{2}}{1 − r^{2}} .

By a little long but straight computation, we can show that

Δf^{ε} = −2( b^{2} − a^{2} + \frac{|dr|^{2}}{1 − r^{2}} + \frac{2εa(qr_{1} − pr_{2})}{\sqrt{1 − r^{2}}} + a^{2}(p^{2} + q^{2})) f^{ε}.

Hence, the map f^{ε} is a conformal harmonic map to either S^{5}, S^{4} or S_{1}^{5}, according to when c = 1, 0 or −1. Thus f^{ε} gives a minimal surface in either S^{5}, S^{4} or S_{1}^{5}. □

References

1. M. Antić, L. Vrancken, Sequences of minimal surfaces in S^{2n+1} , Isr. J. Math. 179 (2010), 493–508.

2. J. Bolton, F. Pedit, L. M. Woodward, Minimal surfaces and the affine Toda field model, J. Reine Angew. Math. 459 (1995), 119–150.

3. J. Bolton, L. Vrancken, Transforms for minimal surfaces in the 5-sphere, Differ. Geom. Appl. 27 (2009), 34–46.

4. B. Dioos, J. Van der Veken, L. Vrancken, Sequences of harmonic maps in the 3-sphere, Math. Nachr. 288 (2015), 2001–2015.

5. H. B. Lawson, Complete minimal surfaces in S^{3}, Ann. Math. (2) 92 (1970), 335–374.

Graduate School of Science and Technology (Received 21 09 2016) Hirosaki University Hirosaki Japan

    sakaki@hirosaki-u.ac.jp