Minimal Surfaces in ${\Mathbb {R}}^{4} $ Foliated by Conic Sections

$Minimal Surfaces in ${\Mathbb {R}}^{4} $ Foliated by Conic Sections$

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n 3 We ﬁrst review the classical deformations of minimal surfaces in R ≥ induced by deformations of holomorphic null curves in Cn. We use the complex parabolic rotations (Lemma 4.1) of holomorphic null curves in C4 to show that the deformation of classical catenoids in R3 to the Hoffman-Osserman catenoids in R4 can be generalized to deformations of arbitrary minimal surfaces in R3 to a family of degenerate minimal surfaces in R4 (Theorem 5.1). As an application of our deformation, generalizing helicoids in R3 foliated by lines, we construct new minimal surfaces in R4 foliated by conic sections with eccentricity grater than 1: hyperbolas or lines (Example 6.1). We establish the existence of minimal surfaces in R4 foliated by ellipses, which converges to circles at inﬁnity (Theorem 6.3). Finally, we also provide examples of minimal surfaces in R4 foliated by conic sections with eccentricity 1: parabolas (Example 6.5).

2. DEFORMATIONS OF MINIMAL SURFACES IN Rn AND HOLOMORPHIC CURVES IN Cn Since two dimensional minimal surfaces in real Euclidean space is parametrized by conformal harmonic mappings, they can be constructed from holomorphic null curves in complex Euclidean space. Let’s review classical deformations of simply connected minimal surfaces in Rn induced by deformations of the holomorphic null curves in Cn. Example 2.1 (Associate family of minimal surfaces in R3). Let X :Σ R3 denote a conformal → harmonic immersion of a simply connected minimal surfaces Σ induced by the holomorphic null curve (φ1, φ2, φ3): ζ ζ ζ X(ζ)= X(ζ0)+ Re φ1(ζ)dζ, Re φ2(ζ)dζ, Re φ3(ζ)dζ , Zζ0 Zζ0 Zζ0 ! or concisely,

X(ζ)= Re φ1(ζ)dζ, Re φ2(ζ)dζ, Re φ3(ζ)dζ . Z Z Z Given a real angle constant θ, we rotate the initial holomorphic null curve (φ1, φ2, φ3) to associate by the new holomorphic null curve φ1, φ2, φ3 :

iθ φ1 φ1 ee−e e 0 0 φ1 iθ φ2 φ2 = 0 e− 0 φ2 .   7→    iθ   φ3 φe3 0 0 e− φ3   e        θ The holomorphic null curve φ1, φ2,eφ3 induces the minimal surface Σ is given by the conformal harmonic immersion e e e θ X (ζ)= Re φ1(ζ)dζ, Re φ2(ζ)dζ, Re φ3(ζ)dζ . Z Z Z π 0 The minimal surface Σ 2 is called thee conjugate surfacee of Σ = Σe. Under the conjugate trans- π π formation Σ Σ 2 , the lines of curvature on Σ map to the asymptotic lines on Σ 2 and the 7→ π asymptotic lines on Σ map to the lines of curvature on Σ 2 . Example 2.2 (Lawson’s isometric deformation [11] of minimal surfaces in R3 to minimal surfaces in R6). Given a simply connected minimal surface Σ in R3, we construct Lawson’s two- parameter family of minimal surfaces in R6, which are isometric to Σ. Suppose that the minimal 3 surface Σ is obtained by integrating the holomorphic null curve (φ1, φ2, φ3) in C . First, given a 6 real angle constant β, we obtain the minimal surface Σβ in R induced by the holomorphic null curve in C6: (ϕ , , ϕ ) = cos β (φ , 0, φ , 0, φ , 0) + sin β (0, iφ , 0, iφ , 0, iφ ) . 1 ··· 6 1 2 3 − 1 − 2 − 3 MINIMAL SURFACES IN R4 AND HOLOMORPHIC NULL CURVES IN C4 3

π R6 We notice that the minimal surface Σ 4 in can be identiﬁed with the holomorphic curve 1 (φ , φ , φ ) C3 α √2 1 2 3 in . Second, given another real angle constant , we rotate the holomorphic null curve (ϕ , , ϕ ) to get the holomorphic null curve 1 ··· 6 iα (ϕ , , ϕ )= e− (ϕ , , ϕ ) . 1 ··· 6 1 ··· 6 6 3 Integrating this, we obtain the minimal surface Σ(α,β) in R , which is isometric to Σ in R . Lawson [11, Theorem 1] provedf that thisf is the only way to deform a minimal surface Σ in R3 n 3 isometrically to minimal surfaces in Euclidean space R ≥ . Example 2.3 (Goursat’s transformation of minimal surfaces in R3 [4, 5, 7, 10]). The Goursat deformation transforms minimal surfaces in R3 to minimal surfaces in R3. Let X(ζ):Σ R3 → be the immersion of a simply connected minimal surfaces Σ induced by the holomorphic null curve (φ1, φ2, φ3). For each t R, we associate the holomorphic null curve φ , φ , φ by the linear map: ∈ 1 2 3 φ φ1 cosh t i sinh t e0 e φe 1 − 1 φ2 φ = i sinh t cosh t 0 φ2 .   7→  2     φ3 φe3 0 0 1 φ3     e      This induces the conformal harmonice map of the minimal surface

t X (ζ)= Re φ1(ζ)dζ, Re φ2(ζ)dζ, Re φ3(ζ)dζ . Z Z Z The nullity of the holomorphic curvee φ , φ , φ comese from thee identity, for t R, 1 2 3 ∈ z cosh t i sinh t z cos(it) sin (it) z z 2 + z 2 = z 2 + z 2, 1 = e e e 1 = 1 . 1 2 1 2 z i sinh t −cosh t z sin (it)− cos(it) z 2 2 2 e See also [27, Sectione 2] bye P. Romon. e Example 2.4 (L´opez-Ros deformation [13] of minimal surfaces in R3). In terms of Weierstrass datum of holomorphic null curves, we rewrite Goursat’s deformation more geometrically. (For instance, see [25, Section 2.1.1].) Let Σ be a simply connected minimal surface in R3, up to translations, parametrized by the conformal harmonic map

(2.1) (x1(ζ), x2(ζ), x3(ζ)) = Re φ1(ζ)dζ, Re φ2(ζ)dζ, Re φ3(ζ)dζ , Z Z Z where the curve (φ1, φ2, φ3) is determined by the Weierstrass data (G(ζ), Ψ(ζ)dζ): 1 i (φ , φ , φ )= 1 G2 Ψ, 1+ G2 Ψ, GΨ . 1 2 3 2 − 2 Geometrically, the meromorphic function G is the complexiﬁed Gauss map under the stereo- graphic projection of the induced unit normal on the minimal surface. Given a constant λ> 0, taking t = ln λ in Goursat transfromation (in Example 2.3), we have the linear deformation of − holomorphic null curves:

φ 1 1 i 1 φ1 1 2 λ + λ 2 λ λ 0 φ1 i 1 −1 1 − φ2 φ = λ + λ 0 φ2 .   7→  2  2 − λ 2 λ    φ3 φe3 0 01 φ3     e      e 4 HOJOO LEE

This induces the conformal harmonic map of the minimal surface Σλ :

λ λ λ λ λ λ x1 (ζ), x2 (ζ), x3 (ζ) = Re φ1 (ζ)dζ, Re φ2 (ζ)dζ, Re φ3 (ζ)dζ . Z Z Z The Goursat deformation of holomorphic null curves yields the deformation of the Weierstrass datum of the so called López-Ros deformation: 1 (G(ζ), Ψ(ζ)dζ) Gλ(ζ), Ψλ(ζ)dζ = λG(ζ), Ψ(ζ)dζ . 7→ λ Under the López-Ros deformation Σ Σλ, the lines of curvature maps into the lines of curva- 7→ ture and the asymptotic lines maps into the asymptotic lines. Since this deformation preserves the height differential 1 φ dζ = G(ζ) Ψ(ζ)dζ = (λG(ζ)) Ψ(ζ)dζ = φλ dζ, 3 λ 3 we immediately find that the third component of the conformal harmonic map is also preserved (up to vertical translations): λ x3(ζ)= x3 (ζ). Another important and useful property of the López-Ros deformation is that if a component of a horizontal level set Σ x = constant is convex, then the same property holds for the related ∩{ 3 } component at the corresponding height on Σλ. The López-Ros deformation admits number of interesting applications [1, 13, 19, 24, 25, 28].

3. GAUSS MAP OF DEGENERATE MINIMAL SURFACES IN R4 Given a conformal harmonic immersion X = (x , x , x , x ):Σ R4 with the local coordi- 0 1 2 3 → nate ζ, we associate the complex curve φ (ζ):Σ C4 deﬁned by → dX (3.1) φ = (φ , φ , φ , φ ) := 2 . 0 1 2 3 dζ We ﬁnd that the harmonicity of X guarantees that the curve φ is holomorphic and that the conformality of the immersion X implies that φ lies on the complex null cone (3.2) Q = (z ,z ,z ,z ) C4 z 2 + z 2 + z 2 + z 2 =0 . 2 0 1 2 3 ∈ | 0 1 2 3 Such immersion X(ζ) can be recovered, up to translations, by the integration

(3.3) X = Re ω0, Re ω1, Re ω2, Re ω3 , Z Z Z Z where we introduce holomorphic one forms (ω0,ω1,ω2,ω3) = (φ0 dζ, φ1 dζ, φ2 dζ, φ3dζ). The induced metric on the surface Σ reads g = 1 φ 2 + φ 2 + φ 2 + φ 2 dζ 2. Σ 2 | 0| | 1| | 2| | 3| | | We introduce the Gauss map of minimal surfaces in R4. Inside the complex projective space CP3, we take the complex null cone Q := z = [z : z : z : z ] CP3 z 2 + z 2 + z 2 + z 2 =0 . 2 { 0 1 2 3 ∈ | 0 1 2 3 } Deﬁnition 3.1 (Gauss map of minimal surfaces in R4 [21, Chapter 3, p.35]). Let Σ be a minimal surface in R4. Consider a conformal harmonic immersion X :Σ R4 with the local coordinate → ζ. The Gauss map of Σ is the map G :Σ Q CP3 deﬁned by → 2 ⊂ ∂X G(ζ) := = [ φ : φ : φ : φ ] CP3. ∂ζ 0 1 2 3 ∈ MINIMAL SURFACES IN R4 AND HOLOMORPHIC NULL CURVES IN C4 5

Following [21, Chapter 4], we shall review the notion of degeneracy of Gauss map of minimal surfaces in R4. Let Σ be a minimal surface, which lies fully in R4, in the sense that it does not lie in any proper affine subpace of R4. The surface Σ is degenerate when the image of its Gauss map lies in a hyperplane of CP3. Otherwise, it is non-degenerate. A degenerate minimal surface is k- degenerate, if k 1, 2, 3 is the largest integer such that the Gauss map image lies in a projective ∈{ } subspace of codimension k in CP3. We recall the fundamental theorem [21, Proposition 4.6] that the surface Σ is 2-degenerate if and only if there exists an orthogonal complex structure on R4 such that it becomes a complex analytic curve lying fully in C2. Bernstein’s beautiful theorem says that the only entire minimal graphs in R3 are planes. More generally, Osserman solved the codimension two generalization of Bernstein type problem, and showed that examples of 1-degenrate and 2-generate minimal surfaces in R4 naturally appear in the classification of entire minimal minimal graphs. Example 3.2 (Osserman’s entire, non-planar, minimal graph in R4 [22, Chapter 5]). We prepare a complex constant µ = a ib with a R and b > 0. For any entire holomorphic function − ∈ F : C C, we define the minimal surface Σ with the patch → X (ζ)= Re φ 0 (ζ) dζ, Re φ 1 (ζ) dζ, Re φ 2 (ζ) dζ, Re φ 3 (ζ) dζ , R R R R where the holomorphic curveb φ 0, φ 1 =bµ φ 0, φ 2, φ 3 breads b 1 F(ζ) 2 F(ζ) i F(ζ) 2 F(ζ) 1, µ, e 1+ µ e− , e + 1+ µ e− . 2 −b b b b 2 b One can check that Σ becomes the entire graph (x1, x2, A(x1, x2), B(x1, x2)) defined on the 4 whole x1x2-plane. Osserman proved that any entire, non-planar, minimal graphs in R should admit the above representation with the entire holomorphic function F. When µ i, i , the ∈ { − } minimal surface Σ becomes the complex analytic curve. Remark 3.3. As known in [21, Theorem 4.7], degenerate minimal surfaces in R4 admit a general representation formula, which is analogous to the classical Enneper-Weierstrass representation formula for minimal surfaces in R3.

4. COMPLEX PARABOLIC ROTATIONS OF HOLOMORPHIC NULL CURVES IN C4 Given a holomorphic null curve φ in C4, for any linear mapping M O (4, C), we can as- ∈ sociate new holomorphic null curve φ = Mφ. The purpose of this section is to construct the so-called parabolic rotations of holomorphic null curves in C4 to construct explicit deformations of minimal surfaces in R3 to degeneratee minimal surfaces in R4. Lemma 4.1 (Complex parabolic rotations of holomorphic null curves in C4). Given a non- constant holomorphic curve φ (ζ):Σ C4 and a constant c C, we associate the holomorphic curve → ∈ φ :Σ C4 by the linear transformation → φ0 φ0 1 c ci 0 φ0 b − 2 − 2 φ φ c c c i φ 1  1 1 2 2 0 1 (4.1)   =  −c2 − c2    φ2 b φ2 7→ φ2 ci 2 i 1+ 2 0 φ   − φ  3 φb3  0 0 01  3           b      If the curve φ lies on the null cone Q = (z ,z ,z ,z ) C4 z 2 + z 2 + z 2 + z 2 =0 , then b 2 0 1 2 3 ∈ | 0 1 2 3 φ Q the curve also lies on 2. Proof. It is straightforward to check the algebraic identity b 2 2 2 2 2 2 (4.2) φ0 + φ1 + φ2 = φ0 + φ1 + φ2 .

b b b 6 HOJOO LEE

Since φ3 = φ3, this implies 2 2 2 2 2 2 2 2 b φ0 + φ1 + φ2 + φ3 = φ0 + φ1 + φ2 + φ3 . Since the curve φ lies on the null cone Q2, the curve φ also lies on Q2. b b b b The identity (4.2) in the proof of Lemma 4.1 is the key idea of our deformation (4.1) in Lemma b 4.1. We shall illustrate ideas behind the identity (4.2) and Lemma 4.1. Remark 4.2 (Real light cone x2 + y2 z2 = 0 in L3, complex cone z 2 + z 2 + z 2 = 0 in C3, − 0 1 2 parabolic rotations, and Wick rotation). We explain that the algebraic identity (4.2) in the proof of Lemma 4.1 can be obtained by the complexiﬁcation of parabolic rotational isometries in L3 via Wick rotation. Let L3 denote the Lorentz-Minkowski (2 + 1)-space, which is the real vector space R3 endowed with the Lorentzian metric dx2 +dy2 dz2. The light cone sitting in L3 given − by the quadratic variety (x,y,z) R3 x2 + y2 z2 =0 ∈ | − is invariant under parabolic rotations Lt R with respect to the null line spanned by light-like ∈ vector (1, 0, 1). The isometry Lt is given by (for instance, see [18, Section 2]) 2 2 x x x 1 t t t x − 2 2 (4.3) y y = Lt y = t 1 t y , z 7→ z z  −t2 t2  z b b 2 t 1+ 2        −    for each t R. We have the realb quadraticb form identity ∈ b b x x 2 2 2 2 2 2 (4.4) x + y z = x + y z , where y = Lt y . − −     bz zb So far, we viewed x, y, z, x, y, z as realb variables.b b However,b clearly, theb algebraic identity (4.4) also holds when we treat them as complex variables. We perforb m theb so-called Wick rotation. Replace z by iz, and z byb bizbin (4.2) to have the transformation − − 2 2 x x 1 t t t x − 2 2 b y b y = t 1 t y ,   7→    − 2 2    iz iz t t 1+ t iz − −b 2 2 −      −    which can be rewritten as b b 2 2 x x x 1 t t t i x − 2 − 2 y y = Rt y = t 1 ti y , z 7→ z z  −t2 − t2  z b 2 i ti 1+ 2        −    Now, the real variables identityb (4.4) induces the complex quadratic form identity b x x 2 2 2 2 2 2 (4.5) x + y + z = x + y + z , where y = Rt y .     bz bz     Finally, taking (x,y,z) = (z1,z0,z2), b(x, y,bz) = (bz1, z0, z2), andb t = c, web have b b z0 1 c ci z0 2 2 2 2 2b b b2 b b b − c2 −c2 (4.6) z0 + z1 + z2 = z0 + z1 + z2 , where z1 = c 1 2 2 i z1 ,    −c2 − c2    z2 z2 b ci 2 i 1+ 2    −    which gives the algebraic identityb b (4.2).b We see that (zb,z ,z ) (z , z , z ) becomes the well- 1 0 2 7→ 1 0 2 deﬁned linear transformation from the complex null coneb to itself. See also [14, Section 3. Com- plex rotations and minimal surfaces]. b b b MINIMAL SURFACES IN R4 AND HOLOMORPHIC NULL CURVES IN C4 7

Remark 4.3 (Parabolic rotations of null curves in C4 in terms of Segre coordinates). We begin with the algebraic identity (4.7) z 2 + z 2 + z 2 + z 2 = (z + iz ) (z iz ) (z + iz ) ( z + iz ) . 0 1 2 3 1 2 1 − 2 − 3 0 − 3 0 Using the Segre transformation (t1,t2,t3,t0) = (z1 + iz2,z1 iz2,z3 + iz0, z3 + iz0), we are 2 2 2 2 − − able to identify the null cone z0 + z1 + z2 + z3 =0 as the determinant variety

t1 t0 t1 t0 (4.8) S = M2 (C) det =0 . t3 t2 ∈ | t3 t2 Given a pair (L, R) of complex constants, we ﬁnd the implication t t t t 1 0 t t 1 R (4.9) 1 0 S 1 0 := 1 0 S. t3 t2 ∈ ⇒ t t L 1 t3 t2 0 1 ∈ 3 2 b b Writing t , t , t , t = (z + iz , z iz , z + iz , z + iz ), we see that this implication in- 1 2 3 0 1 2 1 −b 2b 3 0 − 3 0 duces the linear map from the quadratic cone z 2 + z 2 + z 2 + z 2 =0 to itself: 0 1 2 3 b b b b b b b b b i b b 1b z0 z0 1 2 (L + R) 2 (L + R) 0 z0 i − LR i 1 z1 z1 2 (L + R) 1+ 2 2 LR 2 (L R) z1 (4.10)     =  1 i LR − i −    z2 7→ zb2 2 (L + R) 2 LR 1 2 2 (L R) z2 − 1 i − − − z3 z3  0 (L R) (L R) 1  z3   b   2 − 2 −            In particular, taking (L,b R) = ( ci, ci), we obtain the linear transformation b − − z0 z0 1 c ci 0 z0 − c2 −c2 z1 z1 c 1 2 2 i 0 z1 (4.11)     =  −c2 − c2    , z2 7→ zb2 ci i 1+ 0 z2 − 2 2 z3 z3   z3   b   0 0 01           which recovers the deformation (4.1)b in Lemma 4.1. b

5. FROM MINIMAL SURFACES IN R3 TO DEGENERATE MINIMAL SURFACES IN R4 Theorem 5.1 (Deformations of non-planar minimal surfaces in R3 to degenerate minimal surfaces in R4). Let Σ be a simply connected non-planar minimal surface in R3, up to translations, parametrized by the conformal harmonic immersion

(5.1) X (ζ)= Re φ1 (ζ) dζ, Re φ2 (ζ) dζ, Re φ3 (ζ) dζ , Z Z Z where the holomorphic null curve φ = (φ1, φ2, φ3) admits the Weierstrass data (G(ζ), Ψ(ζ)dζ): 1 i (5.2) φ = 1 G2 Ψ, 1+ G2 Ψ, GΨ , 2 − 2 Given a constant c C, there exists a minimal surface Σc in R4, up to translations, parametrized by the ∈ conformal harmonic immersion

c (5.3) X (ζ)= Re φ 0 (ζ) dζ, Re φ 1 (ζ) dζ, Re φ 2 (ζ) dζ, Re φ 3 (ζ) dζ , Z Z Z Z b b b b where the holomorphic curve φ = φ 0, φ 1, φ 2, φ 3 is determined by 1 i (5.4) φ = c G2Ψb, 1+b cb2 1b G2b Ψ, 1+ c2 +1 G2 Ψ, GΨ 2 − 2 b 8 HOJOO LEE

The induced metric on Σc by the patch Xc reads 2 2 1 2 2 2 2 G G 2 gΣc = Ψ 1+ c G 1+ | | 2 1+ | | 2 dζ . 4 | | | | 1+ icG ! 1 icG !! | | | | | − | c R4 Due to the identity φ 0 + c φ 1 + ic φ 2 =0, the minimal surface Σ in is degenerate. Proof. To prove that Σc is a minimal surface in R4, we show that the holomorphic curve b b b 1 i c G2Ψ, 1+ c2 1 G2 Ψ, 1+ c2 +1 G2 Ψ, GΨ 2 − 2 3 3 is null. Take φ0 =0. Regard Σ in R induced by the holomorphic null curve (φ1, φ2, φ3) in C as a minimal surface in R4 induced by the holomorphic null curve in C4 : 1 i (φ , φ , φ , φ )= 0, 1 G2 Ψ, 1+ G2 Ψ, GΨ . 0 1 2 3 2 − 2 Then, by Lemma 4.1, we ﬁnd that the holomorphic curve in C4 :

2 1 c ci 0 φ0 c G Ψ φ0 − 2 − 2 c c c i φ 1 c2 G2 φ 1 2 2 0 1 2 1+ 1 Ψ  1  −c2 − c2    =  i 2 − 2  = ci i 1+ 0 φ2 1+ c +1 G Ψ φb2 − 2 2 2  0 0 01 φ3  GΨ  b        φ3       b  should be also null. The conformal factor of the conformal metric on Σc is equal  to b 3 2 2 1 2 1 2 G G φ = Ψ 2 1+ c2G2 1+ | | 1+ | | . 2 | k| 4 | | | | 2 2 k ! 1+ icG ! 1 icG !! X=0 | | | − | The deﬁnition of φb gives the following equality, which implies the degeneracy of Σc: φ + c φ + ic φ = φ =0. b 0 1 2 0 b b b Remark 5.2. We used the parabolic rotations of holomorphic null curves in C4 in Lemma 4.1 to construct deformations (Theorem 5.1) of minimal surfaces in R3 to a family of minimal surfaces in R4. These deformations of simply connected minimal surfaces in R3 to minimal surfaces in R3 or R4 are non-isometric deformations, in general. Indeed, H. Lawson [11, Theorem 1] used n 3 E. Calabi’s Theorem [2] to determine when minimal surfaces in R ≥ are isometric to minimal surfaces in R3. See also [17, Theorem 1.2]. Remark 5.3. We point out that the holomorphic null curves in C4 also naturally appears in the theory of superconformal surfaces in R4. For instance, see [6, 16]. Taking the constant c = tan θ R in Theorem 5.1 and rotating coordinate system in the ∈ ambient space R4, we have the following deformation: Corollary 5.4 (Degenerate minimal surfaces in R4). Let Σ be a simply connected minimal surface in R3, up to translations, parametrized by the conformal harmonic immersion

(5.5) X (ζ)= Re φ1 (ζ) dζ, Re φ2 (ζ) dζ, Re φ3 (ζ) dζ , Z Z Z where the holomorphic null curve φ = (φ1, φ2, φ3) admits the Weierstrass data (G(ζ), Ψ(ζ)dζ): 1 i (5.6) φ = 1 G2 Ψ, 1+ G2 Ψ, GΨ , 2 − 2 MINIMAL SURFACES IN R4 AND HOLOMORPHIC NULL CURVES IN C4 9

For each angle constant θ π , π , there exists a degenerate minimal surface Σtan θ in R4, up to ∈ − 2 2 translations, parametrized by the conformal harmonic immersion tan θ (5.7) X (ζ)= Re φ 0 (ζ) dζ, Re φ 1 (ζ) dζ, Re φ 2 (ζ) dζ, Re φ 3 (ζ) dζ , Z Z Z Z where the holomorphic curve φe = i (sin θ) φ e, φ , φ , φ eis determined by e 0 − 2 1 2 3 sin θ G2 cos θ G2 i G2 (5.8) φ = 1+ e Ψ, b1 e e Ψb , 1+ Ψ, GΨ . 2 cos2θ 2 − cos2θ 2 cos2θ Proof. Wee take c = tan θ in Theorem 5.1. With respect to the standard frame e0 = (1, 0, 0, 0), tan θ e1 = (0, 1, 0, 0), e2 = (0, 0, 1, 0), e3 = (0, 0, 0, 1), the surface Σ admits the patch

Re φ0 (ζ) dζ e0 + Re φ1 (ζ) dζ e1 + Re φ2 (ζ) dζ e2 + Re φ3 (ζ) dζ e3, Z Z Z Z b b b b where the holomorphic curve φ = φ 0, φ 1, φ 2, φ 3 reads 1 i φ = tan θ G2Ψ, 1+b 1b + tanb 2θ bG2 bΨ, 1+ 1 + tan2θ G2 Ψ, GΨ . 2 − 2 With theb new frame E0 = cos θ e0 + sin θ e1, E1 = sin θ e0 + cos θ e1, E2 = e2, E3 = e3, the − minimal surface Σtan θ admits the patch

Re φ0 (ζ) dζ E0 + Re φ1 (ζ) dζ E1 + Re φ2 (ζ) dζ E2 + Re φ3 (ζ) dζ E3. Z Z Z Z e e e e

6. MINIMAL SURFACES IN R4 FOLIATED BY CONIC SECTIONS We present applications of our deformations of minimal surfaces in R3 to produce new minimal surfaces in R4. Applying our deformation to holomorphic null curves in C3 C4 induced ⊂ by helicoids in R3, we discover minimal surfaces in R4 foliated by conic sections with eccentricity grater than 1: hyperbolas or straight lines. Applying our deformation to holomorphic null curves in C3 induced by catenoids in R3, we can rediscover the Hoffman-Osserman minimal surfaces in R4 foliated by conic sections with eccentricity smaller than 1: ellipses or circles. Example 6.1 (From helicoids in R3 to minimal surfaces in R4 foliated by conic sections with eccentricity grater than 1: hyperbolas or straight lines). Let’s begin with the helicoid Σ = ( sinh u sin v, sinh u cos v, v) R3 u, v R foliated by horizontal lines. The global confor- { − ∈ | ∈ } mal coordinate ζ = u + iv C on Σ induces the Weierstrass data ∈ ζ ζ (G(ζ), Ψ(ζ)dζ)= e , ie− dζ . − The helicoid Σ is, up to translations, given by the conformal harmonic immersion 1 i X (ζ)= Re 1 G2 Ψdζ, 1+ G2 Ψdζ, GΨdζ . 2 − 2 Z Z Z Let c = α + iβ R + iR be the deformation constant. Applying Theorem 5.1 to the Weierstrass ∈ ζ ζ c 4 data (G(ζ), Ψ(ζ)) = e , ie− , we obtain the minimal surface Σ in R , up to translations, − parametrized by the conformal harmonic map Xc (ζ): 1 i Xc = Re c G2Ψdζ, 1+ c2 1 G2 Ψdζ, 1+ c2 +1 G2 Ψdζ, GΨdζ . 2 − 2 Z Z Z Z 10 HOJOO LEE

With the frame e0 = (1, 0, 0, 0), e1 = (0, 1, 0, 0), e2 = (0, 0, 1, 0), e3 = (0, 0, 0, 1), we write

c X = X0e0 + X1e1 + X2e2 + X3e3 and obtain

u X0(u, v) = e (α sin v + β cos v) , 2 2 u α β 1 u sin v X (u, v) = e − − sin v + αβ cos v + e− , 1 2 2 2 2 u αβ sin v + α β +1 u cos v X (u, v) = e − − cos v + e− − , 2 2 2 X3(u, v) = v. We show that the minimal surface Σc is foliated by hyperbolas or lines. We introduce new orthonormal frame 1 E0 = (e0 + αe1 βe2) , α2 + β2 +1 − 1 E1 = p ( αe0 + e1) , √α2 +1 − 1 E2 = ( βe0 + e2) , β2 +1 E e , 3 = p3 and prepare two auxiliary functions

1 1 Ch(u) = α2 + β2 +1 eu + = cosh u + ln α2 + β2 +1 , 2 α2 + β2 +1 eu ! p p 1 p 1 Sh(u) = α2 + β2 +1 eu = sinh u + ln α2 + β2 +1 . 2 − α2 + β2 +1 eu ! p p p c Each components of the patch X (u, v) = Ξ0E0 + Ξ1E1 + Ξ2E2 + Ξ3E3 are given by

Ξ0(u, v) = (α sin v + β cos v) Ch(u), α2 + β2 +1 Ξ1(u, v) = sin v Sh(u), 2 −p √α +1 α2 + β2 +1 Ξ2(u, v) = cos v Sh(u), 2 −p β +1 Ξ3(u, v) = v. p

When the deformation constant c = α + iβ R + iR is equal to zero, the surface Σc given by ∈ =0 the patch Xc=0 recovers the helicoid. Now now on, consider the case (α, β) = (0, 0). Fixing the c 6 last coordinate v = v on Σ, we examine the level set Cv := Σ Ξ = v given by 0 0 ∩{ 3 0} Ξ0 (u, v0) E0 + Ξ1 (u, v0) E1 + Ξ2 (u, v0) E2 + v0E3.

Translating the level set Cv0 in the E0 direction yields the curve given by

c(u) = Ξ0 (u, v0) E0 + Ξ1 (u, v0) E1 + Ξ2 (u, v0) E2. MINIMAL SURFACES IN R4 AND HOLOMORPHIC NULL CURVES IN C4 11

We introduce the new orthonormal frame

ǫ0 = E0,

1 sin v0 cos v0 ǫ1 = ǫ1 (v0)= E1 + E2 , 2 2 2 2 sin v0 cos v0 √α +1 β +1 ! α2+1 + β2+1 q p 1 cos v0 sin v0 ǫ2 = ǫ2 (v0)= E1 E2 . 2 2 2 2 sin v0 cos v0 β +1 − √α +1 ! α2+1 + β2+1 q p to rewrite c(u)= x(u)ǫ0 + y(u)ǫ1 + z(u)ǫ2 with components x = c(u) ǫ = (α sin v + β cos v ) Ch(u), · 0 0 0 sin2 v cos2 v y = c(u) ǫ = (α2 + β2 + 1) 0 + 0 Sh(u), · 1 − α2 +1 β2 +1 s z = c(u) ǫ =0, · 2 We distinguish two cases:

(1) When α sin v + β cos v =0, the level set Cv is congruent to the hyperbola 0 0 6 0 2 x 2 y   =1, α sin v + β cos v − 2 2 0 0 2 2 sin v0 cos v0 (α + β + 1) 2 + 2  α +1 β +1    r  which has two orthogonal asymptotic lines x = α sin v0+β cos v0 y. sin2 v cos2 v ± 0 + 0 r α2+1 β2+1 (2) When α sin v + β cos v = 0, we have x 0 and z 0. This means that the level set 0 0 0 ≡ 0 ≡ curve Cv0 becomes a line. Example 6.2 (From catenoids in R3 to the Hoffman-Osserman minimal surfaces in R4 foliated by conic sections with eccentricity smaller than 1: ellipsesor circles). The catenoid of the neck radius 1 is, up to translations, given by 1 i X (ζ)= Re 1 G2 Ψdζ, 1+ G2 Ψdζ, GΨdζ , 2 − 2 Z Z Z 1 with the Weierstrass data (G(ζ), Ψ(ζ)dζ) = ζ, 2 dζ , where ζ = u + iv C 0 . Let c C ζ ∈ −{ } ∈ be the deformation constant. Applying Theorem 5.1 to the Weierstrass data (G(ζ), Ψ(ζ)dζ), we obtain the minimal surface Σc in R4 with the patch

c X (ζ)= Re φ0 dζ, φ1 dζ, φ2 dζ, φ3 dζ . Z Z Z Z Here, the holomorphic null curve is givenb by b b b

2 φ0 c G Ψ c φ 1 c2 G2 1 1 c2  1 2 1+ 1 Ψ 2 z2 + 1 (6.1) =  i 2 − 2  =  i 1 2 −  , φb 1+ c +1 G Ψ 2 + c +1 2 2 2 z b   GΨ   1  φ3    z  b      which recovers the holomorphic  data [21, Proposition 6.6] of the Hoffman-Osserman minimal b surfaces foliated by ellipses or circles [21, Remark 1]. The conformal harmonic immersion of the 12 HOJOO LEE

Hoffman-Osserman minimal surfaces in R5 should read C C X = Re d ζ , d ζ i , α log ζ, d z, d z . 1 − ζ 2 − ζ 4 5 Here, d1, d2, C, d4, d5 are complex constants and α is a positive real constant satisfying C α2 C α2 (d , d )= d 2 + d 2 ,i d 2 + d 2 + , 1 2 α2 4 5 − 4C α2 4 5 4C which guarantees that the induced holomorphic null curve in C5 : C C α (6.2) d + , d + i , , d , d . 1 ζ2 2 ζ2 ζ 4 5 Taking the normalization (d , d ,C,d , d , α) = 1 (c2 1), i (c2 + 1), 1 , c, 0, 1 in the Hoffman- 1 2 4 5 2 − 2 2 Osserman curve (6.2), we recover the holomorphic curve, which is equivalent to (6.1). Theorem 6.3 (Minimal surfaces in R4 spanned by circles at infinity). Given a constant θ π tan θ ∈ 0, 2 , we define the minimal surface Σ defined by the conformal harmonic mapping

θ sin θ 1 Xtan (U, V )= sinh U cos V, cosh U cos V, cosh U sin V,U , (U, V ) R2. cos θ cos θ ∈ Then, we have the following properties. tan θ (1) For each constant height x = U , the level set CU = Σ x = U is an ellipse. In 4 0 0 ∩{ 4 0} particular, the neck C =Σtan θ x =0 is congruent to x2 + cos2θ y2 =1. 0 ∩{ 4 } (2) When U approaches to (or ), the ellipse CU converges to a circle. ∞ −∞ Proof. We recall the classical Weierstrass data of the catenoid in R3 with neck size 1:

1 ζ ζ z, dz = e ,e− dζ =: (G(ζ), Ψ(ζ)dζ) . z2 Applying Corollary 5.4, we have the minimal surface in R4, up to translations, given by

(x0, x1, x2, x3)= Re φ 0 (ζ) dζ, Re φ 1 (ζ) dζ, Re φ 2 (ζ) dζ, Re φ 3 (ζ) dζ , Z Z Z Z e e e e where the holomorphic curve φ = φ 0, φ 1, φ 2, φ 3 is determined by

ζ ζ ζ sin θ ζ e cos θ ζ e i ζ e φ = e− + e , e e e−e b Ψ, e− + Ψ, 1 . 2 cos2θ 2 − cos2θ 2 cos2θ We writeeζ = u + iv and introduce the coordinates (U, V ) := (u ln (cos θ) , v). Applying − reﬂection (x , x , x , x ) (x , x , x , x ) and translation x x ln (cos θ), the above 0 1 2 3 → 0 − 1 − 2 3 3 → 3 − patch recovers the desired conformal harmonic mapping Xtan θ(U, V ). One can easily check tan θ that the level set CU =Σ x = U is congruent to the ellipse 0 ∩{ 4 0} x 2 y 2 sin θ 2 1 + =1, (r , r )= sinh U + cosh2U, cosh U . r r 1 2  cos θ cosh θ  1 2 s   When U (or tanh U 1), the ellipse CU should converge to a circle: | | → ∞ | | → 2 1 1 r 2 2 2 cos θ cos θ . lim 2 = lim 2 = 2 =1 U r1 U sin θ tan θ +1 | |→∞ | |→∞ cos θ tanh U +1 MINIMAL SURFACES IN R4 AND HOLOMORPHIC NULL CURVES IN C4 13

Example 6.4 (Lagrangian catenoid in R4). Rotating and dilating the Lagrangian catenoid

1 2 w w 2 ζ, C ζ C 0 = e ,e− C w C ζ ∈ | ∈ −{ } ∈ | ∈ we have the holomorphic curve in C2:

1 1 w 1 w 1 w 1 w 2 e e− , e + e− = (sinh w, cosh w) C w C , √2 √2 − √2 √2 √2 ∈ | ∈ which can be identiﬁed as a minimal surface in R4, with coordinates (u, v) = (Re w, Im w), Σ= X(u, v) = (sinh u cos v, cosh u sin v, cosh u cos v, sinh u sin v) R4 (u, v) R2 . ∈ | ∈ We obtain two different families of planar curves on Σ: (1) Fixing v = v , the curve u X(u, v ) can be viewed as 0 7→ 0 u X(u, v ) = cosh u (0, sin v , cos v , 0) + sinh u (cos v , 0, 0, sin v ) , 7→ 0 0 0 0 0 which is congruent to the hyperbola x2 y2 =1, as unit vectors (0, sin v , cos v , 0) and − 0 0 (cos v0, 0, 0, sin v0) are orthogonal to each other. (2) Fixing u = u , the curve v X(u , v) can be viewed as 0 7→ 0 v X(u, v ) = cos v (sinh u , 0, cosh u , 0) + sinh u (0, cosh u , 0, sinh u ) , 7→ 0 0 0 0 0 2 2 2 2 which is congruent to the circle x + y = cosh u0 + sinh u0, as two orthogonal vectors 2 2 (sinh u0, 0, cosh u0, 0) and (0, cosh u0, 0, sinh u0) have the same length cosh u0 + sinh u0. R More generally, one can easily check that, given constants λ1, λ2, λ3, λ4 p, the holomorphic w w w w 2 ∈ curve (λ e + λ e− , λ e + λ e− ) C w C admit similar properties. 1 2 3 4 ∈ | ∈ Example 6.5 (Minimal surfaces in R4 foliated by conic sections with eccentricity 1: parabolas). Let µ C 0 be a constant. We shall see that the holomorphic curve z = µw2 in C2 becomes ∈ −{ } a minimal surface Σµ in R4 foliated by a family of parabolas y = λx2 with λ (0, µ ). With ∈ | | the frame E1 = (1, 0, 0, 0), E2 = (0, 1, 0, 0), E3 = (0, 0, 1, 0), E4 = (0, 0, 0, 1), we prepare the immersion Xµ of the complex parabola Σµ, with the conformal coordinate ζ = u + iv C : ∈ µ 2 2 X (u, v)= x1E1 + x2E2 + x3E3x3 + x4E4 = Re ζ E1 + Im ζ E2 + Re µζ E3 + Im µζ E4. µ µ Fix u = u . Let’s look at the slice Cu = Σ x = u given by c(v) = X (u , v). Writing 0 0 ∩{ 1 0} 0 (a,b) = (Re µ, Im µ) = (0, 0) and introducing the new orthonormal frame 6 e1 = E1, 1 e2 = e2(u0)= (E2 2bE3 +2aE4) , 2 2 2 1+4(a + b ) u0 − 1 e3 = e3(u0)= p ( aE3 bE4) , √a2 + b2 − − 1 e4 = e4(u0)= (E2 2bE3 +2aE4) , 2 2 2 (a + b )(1+4u0 ) − p we rewrite the patch of the level set curve Cu0 : c(v)= u e + v 1+4(a2 + b2) u 2 e + v2 u 2 a2 + b2 e . 0 1 0 2 − 0 3 C This indicates that the slice u0pis congruent to the planar curve p (x(v), y(v)) = (v 1+4(a2 + b2) u 2 , v2 u 2 a2 + b2 ). 0 − 0 √a2+b2 2 2 2 2 This curve represents the parabolapy = 2 2 2 x u0 √a +pb . 1+4(a +b )u0 − 14 HOJOO LEE

Remark 6.6. D. Joyce [9] constructed special Lagrangian submanifolds in R2n evolving quadrics. In the case when n =2, those examples become holomorphic curves in C2. Remark 6.7. Motivated by M. Shiffman’s theorems for minimal surfaces in R3 bounded by two convex curves [30], F. L´opez, R. L´opez, and R. Souam [12] generalized Riemann’s minimal surfaces [20, 26] in R3 foliated by circles and lines to maximal surfaces in Lorentz-Minkowski space L3 = R3, dx 2 + dx 2 dx 2 foliated by conic sections. Unlike Euclidean space, since 1 2 − 3 Lorentz-Minkowski space L3 admits three different rotational isometries (elliptic, hyperbolic, parabolic rotations), there exist fruitful examples of maximal surfaces foliated by conic sections.

ACKNOWLEDGEMENTS The author would like to thank Marc Soret and Marina Ville for enlightening discussion. Part of this work was carried out while he was visiting Universit´e Francois Rabelais in December, 2016. He would like to warmly thank M. S. and M. V. for their hospitality and support.

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