PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(117) (2018), 175–180 DOI: https://doi.org/10.2298/PIM1817175P

ISOTHERMIC SURFACES OBTAINED FROM HARMONIC MAPS IN S^{6}

Rui Pacheco

Abstract. The harmonicity of a smooth map from a Riemann surface into the 6-dimensional sphere S^{6} amounts to the closeness of a certain 1-form that can be written in terms of the nearly Kähler structure of S^{6}. We will prove that the immersions F in ℝ^{7} obtained from superconformal harmonic maps in S^{3} ⊂ S^{6} by integration of the corresponding closed 1-forms are isothermic. The isothermic surfaces so obtained include a certain class of constant mean curvature surfaces in ℝ^{3} that can be deformed isometrically through isothermic surfaces into non-spherical pseudo-umbilical surfaces in ℝ^{7}.

1. Introduction

It is a well-known fact that any non-conformal harmonic map φ from a simplyconnected Riemann surface Σ into the round 2-sphere S^{2} is the Gauss map of a constant Gauss curvature surface, F : Σ → ℝ^{3}, and of two parallel constant mean curvature surfaces, F^{±} = F ± φ : Σ → ℝ^{3}; the surface F integrates the closed 1-form ω = φ × ∗dφ, where × denotes the standard cross product of ℝ^{3}.

Again, the harmonicity of a smooth map φ : Σ → S^{6} amounts to the closeness of the differential 1-form ω = φ × ∗dφ, where × stands now for the 7-dimensional cross product. This means that we can integrate on simply-connected domains in order to obtain a map F : Σ → ℝ^{7}. If φ is a conformal harmonic immersion, then F is a conformal immersion; and, in contrast with the 3-dimensional case, where F is necessarily a totally umbilical surface, F can exhibit a wide variety of geometrical behaviors in the 7-dimensional case [3].

Recall [4] that a surface in ℝ^{n} is isothermic if, away from umbilic points, it admits conformal curvature line (CCL) coordinates, that is, conformal coordinates with respect to which each second fundamental form is diagonal. In this short note, we prove that the immersions F in ℝ^{7} obtained from superconformal [1, 3] harmonic maps in S^{3} = S^{6} ∩ W by integration of the corresponding 1-forms ω, with W a 4-dimensional subspace of ℝ^{7}, are isothermic. We will also see that the isothermic surfaces so obtained include a certain class of constant mean curvature surfaces in ℝ^{3} that can be deformed isometrically through isothermic surfaces into non-spherical pseudo-umbilical surfaces in ℝ^{7}.

2010 Mathematics Subject Classification: 53C43,53C42, 53A10,53A07.

Key words and phrases: Harmonic maps, isothermic surfaces, parallel mean curvature, pseudo-umbilical surfaces, seven dimensional cross product.

This work was supported by CMA-UBI through the project UID/MAT/00212/2013.

2. Harmonic maps from Riemann surfaces into S^{6}

Let · be the standard inner product on ℝ^{7} and e_{1}, …, e_{7} be the canonical basis of ℝ^{7}. Fix the 7-dimensional cross product × defined by the multiplication table

(2.1)

× e_{1} e_{2} e_{3} e_{4} e_{5} e_{6} e_{7}
e_{1} 0  e_{3} −e_{2} e_{5} −e_{4} −e_{7} e_{6}
e_{2} −e_{3}  0  e_{1} e_{6} e_{7} −e_{4} −e_{5}
e_{3} e_{2} −e_{1}  0  e_{7} −e_{6} e_{5} −e_{4}
e_{4} −e_{5} −e_{6} −e_{7}  0  e_{1} e_{2} e_{3}
e_{5} e_{4} −e_{7} e_{6} −e_{1}  0 −e_{3} e_{2}
e_{6} e_{7} e_{4} −e_{5} −e_{2} e_{3}  0 −e_{1}
e_{7} −e_{6} e_{5} e_{4} −e_{3} −e_{2} e_{1}  0

The cross product × satisfies the following identities, for all x, y ∈ ℝ^{7}:

(P1) x · (x × y) = (x × y) · y = 0;

(P2) (x × y) · (x × y) = (x · x)(y · y) − (x · y) ^{2};

(P3) x × y = −y × x ;

(P4) x · (y × z) = y · (z × x) = z · (x × y) ;

(P5) (x × y) × (x × z) = ((x × y) × z) × x + ((y × z) × x) × x + ((z × x) × x) × y; (P6) x × (x × y) = −(x · x)y + (x · y)x ; (P7) x × (y × z) + (x × y) × z = 2(x · z)y − (x · y)z − (y · z)x .

Extend the inner product · and the cross product × by complex bilinearity to ℂ^{7} = ℝ^{7} ⊗ ℂ. We also denote these complex bilinear extensions by · and ×, respectively.

The standard nearly Kähler structure J on the 6-dimensional unit sphere S^{6} can be written in terms of the cross product × as follows: for each x ∈ S^{6}, Ju = x × u, for all u ∈ T_{x}S^{6}. Let Σ be a Riemann surface with local conformal coordinate z = x + iy, and let φ : Σ → S^{n−1} be a harmonic map, that is, △φ ⊥ T_{φ} S^{n−1} = {u ∈ ℝ^{n}| φ · u = 0}. For n = 7, this means that φ × △φ = 0, which is equivalent to the closeness of the one form ω = φ × ∗dφ = J ∗ dφ.

3. Isothermic surfaces from harmonic maps into S^{6}

If φ : Σ → S^{6} is a harmonic immersion and Σ is simply-connected, we can integrate to obtain an immersion F : Σ → ℝ^{7} such that dF = φ × ∗dφ. In local conformal coordinates z = x + iy of Σ, this can be written in the form

F_{z} = iφ × φ_{z}, Fz̄ = −iφ × φz̄.

Making use of the properties for the cross product, we obtain the following formulae for the first and second fundamental forms of the immersion F in terms of φ and its derivatives.

ISOTHERMIC SURFACES FROM HARMONIC MAPS IN S^{6} 177

Proposition 3.1. [3] Let I_{F} and II_{F} be the first and the second fundamental forms of F : Σ → ℝ^{7}, respectively. Let N be a vector field of the normal bundle T F^{⊥} . Then, with respect to the local conformal coordinates z = x + iy of Σ, we have

(3.1)  I_{F} = ( \begin{matrix}|φ_{y}|^{2} & −φ_{x} · φ_{y} \\ −φ_{x} · φ_{y} & |φ_{x}|^{2} \end{matrix} )

(3.2)

II _{F}^{N} := II_{F} · N = ( \begin{matrix}(φ_{x} × φ_{y}) · N + (φ × φ_{xy}) · N & (φ × φ_{yy}) · N \\ (φ × φ_{yy}) · N & (φ_{x} × φ_{y}) · N − (φ × φ_{xy}) · N \end{matrix} ) .

If φ is conformal, F is also conformal. Let e^{2α} be the common conformal factor of φ and F. One can check [3] that the mean curvature vector of F is given by

(3.3) h_{F} = \frac{1}{2}trI_{F}^{−1} II_{F} = \frac{e^{−2α}}{2} \{II_{F} (F_{x}, F_{x}) + II_{F} (F_{y}, F_{y})\} = e^{−2α} φ_{x} × φ_{y}.

Next we establish our main result. Recall that a harmonic map in S^{3} is superconformal if it has finite isotropy r = 3 [1, 2, 3].

Theorem 3.1. If φ : Σ → S^{6} ∩ W is a superconformal harmonic immersion, where W is a 4-dimensional subspace of ℝ^{7}, then F is isothermic.

Proof. Let z = x + iy be local conformal coordinates on Σ and consider the harmonic sequence \{φ_{j} \}_{j∈ℤ} associated to φ (see [1, 2, 3]). For each j, φ_{−j} = \overline{φ}_{j} and there exists a local meromorphic section f_{j} of φ_{j} (with respect to the Koszul-Malgrange holomorphic structure) such that [2]:

\frac{∂f_{j}}{∂z} = f_{j+1} + \frac{∂}{∂z} log|f_{j} |^{2}f_{j}; \frac{∂f_{j+1}}{∂z̄} = −\frac{|f_{j+1} |^{2}}{|f_{j} |^{2}} f_{j}; |f_{j} ||f_{−j} | = 1 (if f_{j} ≠ 0 ).

Since φ is superconformal in S^{3} = S^{6} ∩ W, we have: the harmonic line bundles φ_{−1} , φ, φ_{1}, φ_{2} are mutually orthogonal; φ_{2} is real, that is, φ_{−2} = φ_{2}; and

f_{−2} = \frac{P_{φ_{2}} (φ_{z̄z̄})}{|P_{φ_{2}} (φ_{zz})|^{2}} , f_{2} = P_{φ_{2}} (φ_{zz}),

where P_{φ_{2}} denotes the orthogonal projection onto φ_{2}. Both f_{−2} and f_{2} are local meromorphic sections of φ_{2}. Hence there exists a meromorphic function g on Σ such that f_{−2} = gf_{2}. Equivalently,

(3.4)  \frac{P_{φ_{2}} (φ_{z̄z̄})}{|P_{φ_{2}} (φ_{zz})|^{2}} = gP_{φ_{2}} (φ_{zz}), \frac{P_{φ_{2}} (φ_{zz})}{|P_{φ_{2}} (φ_{zz})|^{2}} = ḡP_{φ_{2}} (φ_{z̄z̄}).

Locally, away from the isolated zeros of P_{φ_{2}} (φ_{zz}), we can consider the holomorphic non-vanishing function q = \sqrt{g} and rewrite (3.4) as

\overline{q}P_{φ_{2}} (φ_{z̄z̄}) = qP_{φ_{2}} (φ_{zz}).

Define new local conformal coordinates w = u + iv by \frac{dw}{dz} = \frac{e^{πi/4}}{\sqrt{q}} . With respect to these coordinates, we have P_{φ_{2}} (φ_{w̄ w̄}) = −P_{φ_{2}} (φ_{ww}). Then,

F_{w̄ w̄}^{⊥} = −iφ × P_{φ_{2}} (φ_{w̄ w̄}) = iφ × P_{φ_{2}} (φ_{ww}) = F_{ww}^{⊥} ,

which implies F_{uv}^{⊥} = 0, where ⊥ denotes the component in the normal bundle. Then \frac{∂}{∂u} and \frac{∂}{∂v} diagonalize II _{F}^{N} for any normal vector field N to F.

In view of (3.2), the umbilic points of II _{F}^{N}, for each normal section N, are precisely the points where P_{φ_{2}} (φ_{zz}) = 0. □

4. Isometric deformations of CMC into pseudo-umbilical surfaces

A 4-dimensional subspace W of ℝ^{7} is coassociative if V = W^{⊥} is closed with respect to ×. It can be shown [3] that, if W is a coassociative 4-space, then V × W = W and W × W = V.

If φ is superconformal in some 3-dimensional sphere S^{3} = S^{6} ∩ W, where W is a coassociative 4-space, then, up to translation, F is a constant mean curvature surface in the 3-space V = W^{⊥} . This is a consequence of the following result.

Theorem 4.1. [3] Let φ : Σ → S^{6} be a conformal harmonic immersion. Then F : Σ → ℝ^{7} has a parallel mean curvature vector field and it is not pseudo-umbilical if, and only if, φ is superconformal in some 3-dimensional sphere S^{3} = S^{6} ∩ W, where W is a coassociative 4-space.

Example 4.1. Let W = span{e_{4}, e_{5}, e_{6}, e_{7}} and φ : ℂ → S^{3} = S^{6} ∩ W

be defined by φ(x, y) = \frac{1}{\sqrt{2}}( cosx e_{4} + sinx e_{5} + cosy e_{6} + siny e_{7}), which is a superconformal harmonic map and parameterizes a Clifford torus. Taking into account multiplication table (2.1), one can check that the associated surface F : ℂ → V ⊂ ℝ^{7} is the cylinder given by

(4.1)  F (x, y) = \frac{1}{2}( − (x + y) e_{1} − cos(x − y) e_{2} + sin(x − y) e_{3}).

Remark 4.1. Since the Gauss map of a CMC surface without umbilical points is a non-conformal harmonic map, we see that theorem 4.1 also gives a procedure to obtain non-conformal harmonic maps into S^{2} from superconformal harmonic maps into S^{3}: starting with a superconformal harmonic map φ : Σ → S^{3} = S^{6}∩W, where W is a coassociative 4-space, the Gauss map of F is precisely the mean curvature vector field h_{F} : Σ → S^{2} given by (3.3); this is a non-conformal harmonic map.

Given a 4-dimensional subspace W of ℝ^{7}, an orthogonal direct sum decomposition W = W_{1} ⊕ W_{2}, with dimW_{1} = dimW_{2} = 2, is said to be ×-compatible if W_{1} × W_{1} ⊥ W_{2} × W_{2}. For example, W = span{e_{1}, e_{2}, e_{3}, e_{4}} admits ×-compatible decompositions.

Recall that an immersion F is said to be pseudo-umbilical if II_{F} · h_{F} = λI_{F} for some smooth function λ on Σ.

Theorem 4.2. [3] If φ is a superconformal harmonic map in S^{3} = S^{6} ∩ W, for some 4-space W admitting a ×-compatible decomposition, then F is pseudoumbilical with non-parallel mean curvature vector field.

For a general dimension, a pseudo-umbilical submanifold M^{n} of ℝ^{m} has mean curvature vector field parallel in the normal bundle if, and only if, M^{n} is either a minimal submanifold of ℝ^{m} or a minimal submanifold of a hypersphere of ℝ^{m} [6]. Hence, the pseudo-umbilical surfaces of theorem 4.2 are neither minimal in ℝ^{7} nor minimal in hyperspheres of ℝ^{7}. By exploiting the notion of III -deformation, Vlachos [5] established a method that gives examples of full pseudo-umbilical surfaces in ℝ^{4} in the same conditions. Next we apply theorem 4.2 in order to obtain an example of a full pseudo-umbilical surface in ℝ^{5}.

Example 4.2. Let W = span{e_{1}, e_{2}, e_{3}, e_{4}} and φ : ℂ → S^{3} = S^{6} ∩ W

be the Clifford torus φ(x, y) = \frac{1}{\sqrt{2}}( cosx e_{1} + sinx e_{2} + cosy e_{3} + siny e_{4}). The corresponding immersion F : ℂ → ℝ^{7} is given by

(4.2)  F (x, y) = \frac{1}{2}(cosx siny e_{1} + sinx siny e_{2} − y e_{3}
+ sinx cosy e_{5} − cosx cosy e_{6} + x e_{7}).

By theorem 4.2, since W admits ×-compatible decompositions and φ is superconformal in S^{3} = S^{6} ∩ W, the immersion F is pseudo-umbilical with a non-parallel mean curvature vector field. It can be shown (by straightforward computation of the derivatives) that F is full in some 5-dimensional subspace of ℝ^{7}.

Taking theorem 4.1 and theorem 4.2 into account, we also can obtain examples of isometric deformations of CMC surfaces into pseudo-umbilical surfaces as follows. Let φ be a superconformal harmonic map from Σ into S^{3} = S^{6} ∩ W, where W is a coassociative 4-space. Let g : [0, a] → SO(7) be a smooth map with g(0) the identity of SO(7) and g(a)W = W̃, where W is a coassociative 4-space and W̃ is a 4-space admitting ×-compatible decompositions. Then we have a one-parameter family of congruent superconformal harmonic maps: for each λ ∈ [0, a], φ_{λ} = g(λ)φ : Σ → g(λ)W. We can integrate in order to obtain a smooth one-parameter family of conformal immersions F_{λ} (with I_{F_{0}} = I_{F_{λ}} for all λ, because {φ_{λ}} is a one-parameter family of congruent harmonic maps), where F_{0} is a CMC surface and F_{a} is a pseudo-umbilical surface with non-parallel mean curvature vector field.

Example 4.3. Let g : [0, \frac{π}{2} ] → SO(7) be the smooth map defined as follows: for each λ ∈ [0, \frac{π}{2} ], g(λ) = [v_{1}(λ), …, v_{7}(λ)] is the matrix (written with respect to the canonical basis of ℝ^{7}) whose columns are the vectors

v_{1}(λ) = cosλ e_{1} + sinλ e_{7},  v_{2}(λ) = cosλ e_{2} − sinλ e_{5},
v_{3}(λ) = cosλ e_{3} − sinλ e_{6},  v_{4}(λ) = sin^{2} λ e_{1} + cosλ e_{4} − sinλ cosλ e_{7},
v_{5}(λ) = cosλ e_{5} + sinλ e_{2},  v_{6}(λ) = cosλ e_{6} + sinλ e_{3},
v_{7}(λ) = − sinλ cosλ e_{1} + sinλ e_{4} + cos^{2} λ e_{7}.

It is clear that g(0) is the identity of SO(7) and g(\frac{π}{2} )W = W̃, with

W = span{e_{4}, e_{5}, e_{6}, e_{7}}, W̃ = span{e_{1}, e_{2}, e_{3}, e_{4}}.

Hence, after integration, we obtain the following isometric deformation of the cylinder (4.1) into the pseudo-umbilical surface (4.2):

F_{λ}(x, y) = \frac{1}{2}( − cosx sinyv_{4}(λ) × v_{6}(λ) + cosx cosyv_{4}(λ) × v_{7}(λ)
− sinx sinyv_{5}(λ) × v_{6}(λ) + sinx cosyv_{5}(λ) × v_{7}(λ)
+ xv_{6}(λ) × v_{7}(λ) − yv_{4}(λ) × v_{5}(λ) ).

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Centro de Matemática e Aplicações (CMA-UBI)

Universidade da Beira Interior

Covilhã Portugal

    rpacheco@ubi.pt