Polar varieties and bipolar surfaces of minimal surfaces in the n-sphere

For a given minimal surface in the n-sphere, two ways to construct a minimal surface in the m-sphere are given. One way constructs a minimal immersion. The other way constructs a minimal immersion which may have branch points. The branch points occur exactly at each point where the original minimal surface is geodesic. If a minimal surface in the 3-sphere is given, then these ways construct Lawson’s polar variety and bipolar surface.


Introduction
Minimal surfaces in the unit (r − 1)-sphere S r−1 (r ≥ 4) are a classical research subject common to different research fields. In the theory of surfaces, they are surfaces with vanishing mean curvature vector. In the theory of harmonic maps, parametrized minimal surfaces in S r−1 are conformal harmonic maps. In the theory of spectral geometry, parametrized minimal spheres in S r−1 embedded in the Euclidean r-space r are eigenmaps of the Laplace-Beltrami operator.
Giving parametrizations f ∶ Σ → S r−1 of minimal surfaces in S r−1 explicitly is one simple but important problem. We consider this problem by restricting ourselves in the case where Σ is oriented and compact without boundary in this paper.
Parametrizations have been given for each minimal sphere. A minimal sphere is a minimal surface of constant positive Gaussian curvature with genus zero. Minimal spheres in S 3 are totally geodesic spheres. Borůvka [4][5][6] constructed linearly full immersions from S 2 into S 2r−2 for each r ≥ 2 . Calabi [7] showed that Borůvka's spheres were the only minimal spheres in S r−1 for each r ≥ 3.
For minimal surfaces which are not of constant Gaussian curvature or not genus zero, parametrizations were given under some conditions. Bobenko [2] gave a parametrization for each minimal torus in S 3 . Sharipov [14] gave a parametrization for each complex normal minimal torus in S 5 .
Since it is difficult to completely solve the problem at present, it is meaningful to define transforms of minimal surfaces. Lawson's polar varieties and bipolar surfaces [11] are transforms of minimal surfaces in S 3 . Polar varieties are minimal surfaces in S 3 and bipolar surfaces are minimal surfaces in S 5 . Bolton and Vrancken [3] generalized polar varieties and defined two transforms between minimal surfaces in S 5 with noncircular ellipse of curvature. Antić and Vrancken [1] generalized Bolton and Vrancken's transforms and defined transforms between minimal surfaces in S 2r−1 for each r ≥ 2.
In this paper, we will define two transforms for a given conformal immersion f ∶ Σ → S r−1 (r ≥ 4) which is not necessarily minimal. If a given conformal immersion is minimal into S 3 , then our polar varieties and bipolar surfaces are Lawson's polar varieties and bipolar surfaces, respectively. Our polar varieties are neither Bolton and Vrancken's transforms nor Antić and Vrancken's transforms.
Lawson used the exterior algebra in order to define a polar variety and a bipolar surface. For a minimal immersion f ∶ Σ → S 3 ⊂ 4 with local holomorphic coordinate z and the metric 2F|dz| 2 on Σ induced by f, the polar variety f * ∶ Σ → S 3 is the Hodge dual of the map and the bipolar surface is the map We use the Clifford algebra C ( r ) instead to define our polar variety and bipolar surface (Definition 1). Let m n be the binomial coefficient. For a conformal immersion f ∶ Σ → S r−1 ⊂ r ⊂ C ( r ) which is not necessarily minimal, our polar variety and bipolar surface are maps respectively. The map df (TΣ) is the oriented volume form of df (TΣ) . It is a counterpart of ( f ∧̄f )∕iF and considered as the generalized Gauss map of f. We show that the differential map of a polar variety vanishes at the points where the metric induced by a given conformal immersion is equal to that of its bipolar surface (Theorem 1).
Lawson showed that polar varieties and bipolar surfaces of minimal surfaces in S 3 are minimal. We show that if a given conformal immersion from Σ to S r−1 is minimal, then our polar variety and bipolar surface are minimal (Theorems 2, 3). For the proof, we write the equation for a map being a conformal map and that for a map being a harmonic map in terms of the Clifford algebra. These are variants of equations written in terms of quaternions [9,13].
Lawson showed that the singularities of a polar variety of a minimal surface in S 3 occur precisely at the points where the Gaussian curvature of a given minimal surface in S 3 is equal to one. These points are geodesic points of the given minimal surface. We show that a geodesic point of a given minimal immersion into S r−1 corresponds to a branch point of our polar variety (Theorem 4).

The Clifford algebra
We review the Clifford algebra [12] which plays an important role in this paper. For a real vector space W, we denote the dimension of W by dim W. Let V r be an oriented real vector space with dim V r = r . We fix a positively-oriented basis e 1 , … , e r of V r . Let Q be a positive definite quadratic form on V r such that e 1 , … , e r is an orthonormal basis. A positive definite inner product B of V r is associated with Q by The vector space V r with inner product B is the r-dimensional Euclidean space r . We denote the unit hypersphere in r centered at the origin by S r−1 : The Clifford algebra C (V r ) is the algebra generated by e 1 , … , e r subject to the relations Then, C (V r ) is a 2 r -dimensional real vector space. A different choice of a positively-oriented basis ẽ 1 , … ,ẽ r generates an algebra being isomorphic to C (V r ) . Hence, Let C (V r ) × be the group of all invertible elements of C (V r ) and F r be the finite subgroup of C (V r ) × generated by e 1 , … , e r . The set F r forms a basis of C (V r ) . We denote by C (V r ) ⟨j⟩ the linear subspace of C (V r ) of dimension r j spanned by Then, C (V r ) ⟨0⟩ = ℝ and C (V r ) ⟨1⟩ = V r . The Clifford algebra C (V r ) has a direct decomposition For an element ∈ C (V r ) , we denote by ⟨i⟩ the C (V r ) ⟨i⟩ -part of . We see that is defined by extending the map Let (V r ) be a subgroup of C × (V r ) that consists of all products The spin group Spin(V r ) of V r is the subgroup of (V r ) of dimension r(r − 1)∕2 defined by e i e j + e j e i = −2 ij (i, j = 1, … , r).

3
The set {Ad̂v ∶v ∈ Spin(V r )} forms the special orthogonal group of V r . The element is called the oriented volume form of V r . The oriented volume form of V r is independent of the choice of orthonormal positively-oriented basis of V r .

Orthogonal complex structures
We use a complex structure of a vector space in order to interpret conformality of maps in terms of the Clifford algebra. In this section, we extend the quadratic form Q on V r to C (V r ) and explain an orthogonal complex structure of C (V r ) by the Clifford algebra.
We use Q to denote the positive definite quadratic form on C (V r ) . The restriction of Q to V r is the positive definite quadratic form in the definition of C (V r ) . We use B for the symmetric bilinear form associated with Q. Since the F r forms an orthonormal basis of C (V r ). The set is the unit hypersphere with center at the origin in C (V r ) . Then, S r ⊂S 2 r −1 .
We consider linear subspaces of V r . If 1 ≤ n ≤ r , then we may regard V n as a subspace of V r spanned by e 1 , … , e n . For v ∈ Spin(V r ) , we have The element − V r is the volume form of V r with the opposite orientation. We may consider G n (V r ) as the oriented Grassmannian of n-dimensional linear subspaces in V r . We have Let GL(C (V r )) be the general linear group of the real vector space C (V r ) and O(C (V r )) be the orthogonal group of C (V r ) . Let L ∶ C (V r ) × → GL(C (V r )) and , respectively.
Let W 2 be a two-dimensional oriented linear subspace V r . If r = 2 , then we assume that the orientation of W 2 is the same as that of V 2 . If ∈ ℭ and W 2 = W 2 , then L ( ) is an orthogonal complex structure of W 2 . Similarly if ∈ ℭ and W 2 = W 2 , then R ( ) is an orthogonal complex structure of W 2 . The volume form W 2 of W 2 is associated with W 2 .

Conformal immersions
We explain a conformal immersion from a two-dimensional oriented manifold to Euclidean space by the Clifford algebra. We assume that r ≥ 3 and that a linear map is not the zero map. A linear map We have the following lemma by Lemma 2 immediately.

Lemma 5 A linear map ∶ V 2 → V r is conformal if and only if
Proof By Lemma 3, the endomorphisms L ( (V 2 ) ) and R ( (V 2 ) ) are complex structures of (V 2 ) and L ( ( . The converse is trivial. ◻ The wedge product of C (V r ) explains a relations between two conformal linear maps.

Lemma 6
Let ∶ V 2 → V r be a conformal linear map such that (V 2 ) is a two-dimensional subspace. Assume that ∶ V 2 → C (V r ) be a linear map.
. We use Lemma 4, Lemma 5 and Lemma 6 in order to explain conformal immersions from a two-dimensional Riemannian oriented manifold to the Clifford algebra.
In the following, we assume that all manifolds and all maps are smooth. Let M be a manifold. For a vector bundle E over M, we denote a fiber of E at p by E p . We denote the tangent bundle of M by TM and the cotangent bundle by T * M.
Let Σ be a two-dimensional Riemannian oriented manifold. The metric on Σ and the orientation of Σ determines an orthogonal complex structure J Σ on Σ such that the orientation of the ordered frame field X, J Σ X is positive for any local nowhere-vanishing section X of TΣ . Define an operator * on one-forms on Σ by * ∶= •J Σ .
An immersion f ∶ Σ → C (V r ) is conformal if the differential map is conformal at each point p ∈ Σ . Applying Lemmas 4, 5 and 6 for df, we have the following Lemmas 7, 8 and 9 immediately.
Lemma 7 Let f ∶ Σ → C (V r ) be an immersion. If there exists a map ∶ Σ → ℭ with * df = df or * df = df , then f is conformal.

Lemma 8 Let f ∶ Σ → V r be an immersion. The differential map df satisfies equation
if and only if f is conformal.
The immersion g is conformal with * dg = dg df (TΣ) if and only if dg ∧ df = 0.

The immersion g is conformal with
We omit the proofs of these lemmas. Lawson [11] defined a polar variety and a bipolar surface of a minimal immersion into S 3 . We define a polar variety and a bipolar surface of a conformal immersion which is not necessarily minimal into S r−1 (r ≥ 4).

Polar varieties and bipolar surfaces
We review Lawson's polar varieties and bipolar surfaces in terms of the Clifford algebra. Let f ∶ Σ → S 3 ⊂ V 4 be a minimal immersion. Since f is conformal, we have * df = df (TΣ) df = −df df (TΣ) by Lemma 8. The five-dimensional sphere C (V 4 ) ⟨2⟩ ∩S 15 is a codomain of df (TΣ) . The map f is orthogonal to df (TΣ) . Let 1 , 2 be an orthonormal positively-oriented basis of (TΣ) p and U f p be a three-dimensional linear subspace of V 4 spanned by df p ( 1 ) , df p ( 2 ) , f(p). We call the orientation of the ordered basis Lawson defined a polar variety and a bipolar surface without using the minimality of f. We define analogues of a polar variety and a bipolar surface for a conformal immersion as follows.
We see that C (V r ) ⟨3⟩ ∩S 2 r −1 is a sphere with dimension r (1).
A conformal immersion f, the polar variety U f of f, the bipolar surface df (TΣ) of f and their differential maps are related as follows. (3) Proof Let 1 , 2 be a local orthonormal positively-oriented basis of (TΣ) p . Since r ≥ 4 , there exists v ∈ Spin(V r ) such that Then, Hence, Eq. (4) holds.
Lawson's bipolar surface is an immersion and his polar variety is an immersion admitting singularities. The same is true for our bipolar surface and polar variety.
Theorem 1 Let f ∶ Σ → S r−1 be a conformal immersion. Then, the bipolar surface of f is an immersion. The differential map of the polar variety of f vanishes at p ∈ Σ if and only if the metric on Σ induced by f is equal to the metric on Σ induced by the bipolar surface of f at p.

Minimal surfaces in a sphere
Lawson [11] showed that if a conformal immersion f ∶ Σ → S 3 is minimal, then the polar variety of f and the bipolar surface of f are minimal. We show that if a conformal immersion f ∶ Σ → S r−1 (r ≥ 4) is minimal, then the polar variety of f and the bipolar surface of f are minimal.
A conformal harmonic map f ∶ Σ → S r−1 (r ≥ 4) is a minimal immersion or a minimal branched immersion [8]. In the beginning, we write the harmonic map equation in terms of the Clifford algebra.
Let Δ be the Laplace-Beltrami operator with respect to the metric of Σ . Then, an immersion f ∶ Σ →S 2 r −1 ⊂ C (V r ) is a harmonic map [8] if and only if Let dA be the area element of the metric induced by f on Σ . The Hodge dual of Eq. (7) is

Theorem 2
If a conformal immersion f ∶ Σ → S r−1 ⊂ V r with is minimal, then the polar variety U f ∶ Σ → C (V r ) ⟨3⟩ ∩S 2 r −1 is a minimal immersion with or a minimal branched immersion with (10).
Since by Lemma 12 and (4), the polar variety U f is conformal with by Lemma 9, except at the points where d U f vanishes. We have (3) and (4). Hence, the polar variety U f is a harmonic map by Lemma 12. Since U f is a conformal harmonic map, it is a minimal immersion or a minimal branched immersion. ◻ Next, we show that a bipolar surface of a minimal immersion is minimal.
The bipolar surface df (TΣ) is an immersion by Theorem 1.
Since U f is conformal by Theorem 2, there exists a non-negative funcntion such that ( U f ) * B = (f * B) . Then, by Lemma 11. Then, df (TΣ) is conformal.
We have by (5), Lemma 12 and Theorem 2. Then, the map df (TΣ) is a harmonic map by Lemma 12.
Since df (TΣ) is a conformal harmonic map, it is minimal. ◻ We may think of showing that df (TΣ) is conformal by the existence of a map such that * d df (TΣ) = d df (TΣ) = −d df (TΣ) like Lemma 8. However, we see that there does not exist such a map from the proof of Theorem 4 later.
We obtain more information about the metrics induced by f, U f and df (TΣ) on Σ.
For a vector space U, we denote by U the trivial vector bundle over Σ with fiber U. The vector bundle C (V r ) is a real trivial vector bundle of rank 2 r over Σ . The real vector bundle V r is a subbundle of C (V r ) . A map f ∶ Σ → C (V r ) is regarded as a section of C (V r ).
Let 1 , … , r be a local orthonormal frame field of V r . Assume that 1 2 = df (TΣ) and r = f . Then, there exist one-forms 1 and 2 on Σ such that The metric on Σ induced by f is Then, the metric induced by df (TΣ) is (11). By Lemma 11, the metric induced by U f is (12