CLASSIFICATION OF WINTGEN IDEAL SURFACES

CLASSIFICATION OF WINTGEN IDEAL SURFACES IN EUCLIDEAN 4-SPACE WITH EQUAL GAUSS AND NORMAL CURVATURES BANG-YEN CHEN

Abstract. Wintgen proved in [24] that the Gauss curvature K and the normal curvature K D of a surface in Euclidean 4-space E4 satisfy K + |K D | ≤ H 2 , where H 2 is the squared mean curvature. A surface in E4 is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in E4 form an important family of surfaces; namely, surfaces with circular ellipse of curvature. In this paper, we completely classify Wintgen ideal surfaces in E4 satisfying |K| = |K D | identically.

1. Introduction. For surfaces M in a Euclidean 3-space E3 , the Euler inequality K ≤ H 2 , whereby K is the intrinsic Gauss curvature of M and H 2 is the extrinsic squared mean curvature of M in E3 , at once follows from the fact that K = k1 k2 and H = 1 3 2 (k1 + k2 ) whereby k1 and k2 denote the principal curvatures of M in E . And, 2 obviously, K = H everywhere on M if and only if the surface M is totally umbilical in E3 , i.e. k1 = k2 at all points of M , or still, by a theorem of Meusnier, if and only if M is a part of a plane E 2 or of a round sphere S 2 in E3 . ˜ 4 of a surface M into a Riemannian Consider an isometric immersion ψ : M → M 4 ˜ , the ellipse of curvature at a point p of M is defined as 4-manifold M Ep = {h(X, X) | X ∈ Tp M, kXk = 1}, ˜ 4 . The ellipse of curvature is where h is the second fundamental form of M in M the analogue of the Dupin indicatrix of an ordinary surface in E3 . ˜ 4 is superminimal if and only if, at each point p ∈ M , A surface ψ : M → M the ellipse of curvature Ep is a circle with center 0 (see [16]). Simple examples of superminimal surfaces in the Euclidean 4-space E4 are R-surfaces, i.e., graphs of holomorphic functions: {(z, f (z)) : z ∈ U }, where U ⊂ C ≈ R2 is an open subset of the complex plane and f is a holomorphic function. Th. Friedrich proved in 1984 that superminimal surfaces are characterized by the property that the lift into the twistor space is holomorphic and horizontal (see 2000 Mathematics Subject Classification. Primary: 53A05; Secondary 53C40, 53C42. Key words and phrases. Gauss curvature, normal curvature, squared mean curvature, Wintgen ideal surface, superminimal surface, Whitney sphere. 1

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B.-Y. CHEN

˜ 4 is a space of constant curvature, [16, 17] for details). When the ambient space M O. Bor˚ uvka [2] proved in 1928 that the family of superminimal immersions ψ : M → 4 ˜ M depends (locally) on two holomorphic functions. In [3], R. Bryant proved that every compact Riemann surface admits a conformal superminimal immersion into the ordinary 4-sphere S 4 . For an oriented plane E in E4 , let E ⊥ denote the orthogonal complement with the orientation given by the condition E ⊕ E ⊥ = E4 . Two oriented planes E, F are called oriented-isoclinic if either E = F ⊥ (as oriented planes) or the projection prF : E → F is a non-trivial, conformal map preserving the orientations. Consider ˜ 4 . If γ is a curve in M ˜ 4 , denote by τγ the parallel an oriented surface ψ : M → M ˜ 4 . The surface M 2 is called a displacement along γ in the tangent bundle T M negatively oriented-isoclinic surface if, for every curve γ in M from x to y, the planes ˜ 4 . It τψ◦γ (Tψ(x) M ) and Tψ(y) M are negatively oriented isoclinic planes in Tψ(y) M ˜ 4 is superminimal if and only if was proved in [16, 21] that an immersion ψ : M → M it is negatively oriented-isoclinic. Moreover, M. Dajczer and R. Tojeiro established very recently a representation formula for superminimal surfaces in E4 in terms of pairs (g, h) of conjugate minimal surfaces in E4 (see [14] for details). In 1979, P. Wintgen [24] proved a basic relationship between the intrinsic Gauss curvature K, the extrinsic normal curvature K D , and squared mean curvature H 2 of any surface M in a Euclidean 4-space E4 ; namely, (1.1)

K + |K D | ≤ H 2 ,

with the equality holding if and only if the curvature ellipse is a circle. Following L. Verstraelen et al. [15, 23], a surface M in E4 is called Wintgen ideal if it satisfies the equality case of Wintgen’s inequality identically. Obviously, Wintgen ideal surfaces in E4 are exactly superminimal surfaces. In this paper, we completely classify Wintgen ideal surfaces in E4 with equal Gauss and normal curvatures, i.e., Wintgen ideal surfaces which satisfy |K| = |K D | identically. Our main result is stated as Theorem 5.1. 2. Preliminaries. 2.1. Basic formulas and definitions. Let ψ : M → E4 be an isometric immer˜ the Levi-Civita connections on sion of a surface M into E4 . Denote by ∇ and ∇ 4 M and E , respectively. For vector fields X, Y tangent to M and ξ normal to M , the formulas of Gauss and Weingarten are given respectively by (cf. [6, 7]): (2.1)

˜ X Y = ∇X Y + h(X, Y ), ∇

(2.2)

˜ X ξ = −Aξ X + DX ξ, ∇

where h, A and D are the second fundamental form h, the shape operator A, and the normal connection D of M in E4 . The shape operator and the second fundamental form are related by (2.3)

hh(X, Y ), ξi = hAξ X, Y i .

WINTGEN IDEAL SURFACES

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→ − → − The mean curvature vector H is defined by H = 12 trace h. The squared mean → − → − curvature H 2 is defined to be h H , H i. The equations of Gauss, Codazzi and Ricci are given respectively by (2.4) (2.5) (2.6)

R(X, Y )Z = hX, Zi Y − hY, Zi X + Ah(Y,Z) X − Ah(X,Z) Y, ¯ X h)(Y, Z) = (∇ ¯ Y h)(X, Z), (∇

D R (X, Y )ξ, η = h[Aξ , Aη ]X, Y i

¯ is defined by for vector fields X, Y, Z tangent to M and ξ normal to M , where ∇h (2.7)

¯ X h)(Y, Z) = DX h(Y, Z) − h(∇X Y, Z) − h(Y, ∇X Z), (∇

and RD is the curvature tensor associated with the normal connection D, i.e., (2.8)

RD (X, Y )ξ = DX DY ξ − DY DX ξ − D[X,Y ] ξ.

The normal curvature K D is given by

(2.9) K D = RD (e1 , e2 )e4 , e3 . 2.2. Connection forms. For a surface M in E4 , let {e1 , e2 } be an orthonormal frame of the tangent bundle T M of M . Then we have (2.10)

he1 , e1 i = he2 , e2 i = 1, he1 , e2 i = 0.

We may choose an orthonormal normal frame {e3 , e4 } of M in E4 such that (2.11)

he3 , e3 i = he4 , e4 i = 1, he3 , e4 i = 0.

For the orthonormal frame {e1 , e2 , e3 , e4 }, we put (2.12) ∇X e1 = ω12 (X)e2 , ∇X e2 = ω21 (X)e1 , DX e3 = ω34 (X)e4 , DX e4 = ω43 (X)e3 , where ωij and ωαβ (i, j = 1, 2; α, β = 3, 4) are the connection forms of the tangent and normal bundles. 2.3. Gauss map and Gauss image. The Gauss map G of a surface ψ : M → E4 is the map which assigns to each point p ∈ M the oriented tangent space ψ∗ (Tp M ) ⊂ E4 . The Gauss map G can be considered as a map from M into the Grassmannian manifold G(4, 2) = SO(4)/SO(2) × SO(2) of oriented 2-planes in E4 , which in turn can be identified with the complex quadric Q2 : ( ) Q2 (C) =

(z0 , z1 , z2 , z3 ) ∈ CP 3 :

3 X 2

zj = 0

j=0

in the complex projective space CP 3 in a natural way. The complex projective space admits a unique K¨ ahler metric with constant holomorphic sectional curvature 2. The induced metric on Q2 defines a metric gˆ on G(4, 2) satisfying G∗ (ˆ g ) = −Kg for any minimal surface M in E4 , where g is the metric and K the Gauss curvature of M . Thus, the Gauss map G is conformal for a minimal surface in E4 .

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B.-Y. CHEN

ˆ = G(M ) denote the Gauss image of M For a minimal surface M in E4 let M ˆ , we have a well-defined Gauss under its Gauss map. At all nonsingular points of M ˆ ˆ ≤ 2. curvature K. It follows from the normalization of the metric on CP 3 that K D. Hoffman and R. Osserman proved in [19] that Theorem 2.1. Let M be a minimal surface in E4 whose Gauss image has constant ˆ = c. Then c = 1 or c = 2, and curvature K ˆ = 1 if and only if M lies fully in some affine hyperplane of E4 , (i) K ˆ = 2 if and only if M is a complex curve lying fulling in C2 , where C2 (ii) K denotes E4 endowed with some orthogonal complex structure. In 1983, B.-Y. Chen and S. Yamaguchi [11] classified surfaces with totally geodesic Gauss image in a Euclidean space with arbitrary codimension (see also [12]). In particular, for surfaces in E4 they obtained the following. Theorem 2.2. Let M be a surface in E4 with regular Gauss map. Then the Gauss image of M is totally geodesic if and only if M is one of the following surfaces: (a) a surface in an affine hyperplane of E4 . (b) a surface in E4 with parallel second fundamental form. (c) a surface which is locally the Riemannian product of two plane curves of nonzero curvature. (d) a complex curve lying fully in C2 , where C2 denotes an affine E4 endowed with some orthogonal almost complex structure. 2.4. Jacobi’s elliptic functions. We review briefly some known facts on Jacobi’s elliptic functions for later use (for details, see, for instance, [4, 22]). Put Z x Z 1 dt dt p p u= (2.13) , K= , 2 2 2 2 (1 − t )(1 − k t ) (1 − t )(1 − k 2 t2 ) 0 0 where first we suppose that x and k satisfy 0 < k < 1, −1 ≤ x ≤ 1. Equation (2.13) defines u as an odd function of x which is positive, increasing from 0 to K as x increases from 0 to 1. Inversely, the same equation defines x as an odd function of u which increases from 0 to 1 as u increase from 0 to K; this function is a Jacobi’s elliptic function, denoted by sn(u, k) (or simply by sn(u)), so that we can put (2.14)

u = sn−1 (x),

x = sn(u).

The other two main Jacobi’s functions sn(u, k) and dn(u, k) (simply denoted respectively by sn(u) and dn(u)) are defined by p p (2.15) cn(u) = 1 − sn2 (u), dn(u) = 1 − k 2 sn2 (u), the square roots are positive when u is confined to the interval −K < u < K, so that cn(u) and dn(u) are even functions of u. The Jacobi’s elliptic functions depend on the variable u as well as on the parameter k, which is called the modulus.


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Using cn(u), dn(u), sn(u), one can define 1 1 1 nc(u) = , nd(u) = , ns(u) = , cn(u) dn(u) sn(u) (2.16) cn(u) cn(u) dn(u) cs(u) = , cd(u) = , dc(u) = , · · · , etc. sn(u) dn(u) cn(u) Jacobi also defined a function am(x) by means of the equation: Z x (2.17) am(x) = dn(u)du, 0

which is known as Jacobi’s amplitude. 3. Wintgen inequality We recall the following result of P. Wintgen [24]. For the proof of our main theorem, we restate this result in slightly different way as in [24]. Theorem 3.1. Let M be a surface in Euclidean 4-space E4 . Then we have (3.1)

H 2 ≥ K + |K D |

at every point in M . Moreover, we have (i) If K D ≥ 0 holds at a point p ∈ M , then the equality sign of (3.1) holds at p if and only if, with respect to some suitable orthonormal frame {e1 , e2 , e3 , e4 } at p, the shape operator at p satisfies ! ! µ + 2γ 0 0 γ (3.2) Ae 3 = , A e4 = . 0 µ γ 0 (ii) If K D < 0 holds at p ∈ M , then the equality only if, with respect to some suitable orthonormal shape operator at p satisfies ! µ − 2γ 0 (3.3) Ae 3 = , A e4 = 0 µ

sign of (3.1) holds at p if and frame {e1 , e2 , e3 , e4 } at p, the 0 γ

! γ . 0

Proof. Let M be a surface in E4 . At a point p ∈ M , we may choose an orthonormal frame {e1 , e2 , e3 , e4 } at p such that the shape operator satisfies ! ! α 0 δ γ (3.4) Ae 3 = , Ae4 = 0 µ γ −δ for some functions α, γ, δ, µ, with respect to {e1 , e2 , e3 , e4 }. From (3.4) we find → − α+µ (3.5) e3 . K(p) = αµ − γ 2 − δ 2 , K D (p) = γ(α − µ), H (p) = 2 If K D (p) ≥ 0, we have 1 (3.6) H 2 = K + K D + (2γ + µ − α)2 + δ 2 ≥ K + K D 4 2 at p, which implies that H = K + K D at p if and only if δ = 0 and α = 2γ + µ hold at p. So, we obtain (3.2). In this case, we get K D = 2γ 2 at p.

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B.-Y. CHEN

If K D (p) < 0, we have (3.7)

1 H 2 = K + |K D | + (2γ + α − µ)2 + δ 2 ≥ K + |K D | 4

at p, which implies that H 2 = K + |K D | holds at p if and only if δ = 0 and α = µ − 2γ hold at p. So, we obtain (3.3). In this case, we have K D = −2γ 2 at p. The converse is easy to verify. 4. Wintgen ideal surfaces with constant K and H 2 . We need the following result for later use. Proposition 4.1. Let M be a Wintgen ideal surface in E4 . Then M has constant mean curvature and constant Gauss curvature if and only if M is totally umbilical. Proof. Let ψ : M → E4 be a Wintgen ideal surface. Then, according to Theorem 3.1, there is an orthonormal frame {e1 , e2 , e3 , e4 } such that the second fundamental form h satisfies either (4.1)

h(e1 , e1 ) = (µ + 2γ)e3 , h(e1 , e2 ) = γe4 , h(e2 , e2 ) = µe3 , or

(4.2)

h(e1 , e1 ) = (µ − 2γ)e3 , h(e1 , e2 ) = γe4 , h(e2 , e2 ) = µe3

for some functions γ and µ, according to K D ≥ 0 or K D < 0. First, assume that K D ≥ 0 and M has constant Gauss and mean curvatures, then γ and µ are constant. Thus, after using (4.1) and Codazzi’s equation, we find (4.3)

2γω12 (e1 ) = −µω34 (e1 ) = γω34 (e1 ),

(4.4)

2γω12 (e2 ) = γω34 (e) = (2γ + µ)ω34 (e2 ).

From (4.3) and (4.4) we obtain (γ + µ)ω34 = 0. So, we have either (a) γ + µ = 0, or (b) γ + µ 6= 0 and ω34 = 0. If γ + µ = 0 holds, then M is minimal in E4 . Because the Gauss curvature is constant, the surface must be totally geodesic in E4 . If γ + µ 6= 0 and ω34 = 0 hold, we get K D = 2γ 2 = 0. Thus, (4.1) implies that M is totally umbilical in E4 . Similarly, if K D < 0 and M has constant Gauss and mean curvatures, we find from (4.2) that (µ − γ)ω34 = 0, which implies that M is totally umbilical. The converse is trivial. 5. Wintgen ideal surfaces in E4 with |K| = |K D |. Lemma 5.1. Let M be a Wintgen ideal surface in E4 . If M satisfies |K| = |K D | at a point p ∈ M , then there exists an orthonormal frame {e1 , e2 , e3 , e4 } at p such


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that the shape operator at p satisfies either (5.1) or (5.2) given by ! ! −γ 0 0 γ Ae 3 = (5.1) , A e4 = ; 0 γ γ 0 ! ! 3γ 0 0 γ (5.2) Ae 3 = , A e4 = . 0 γ γ 0 Proof. Let M be a Wintgen ideal surface in E4 . (i) If K = K D > 0 holds at a point p ∈ M , then (3.2) yields µ2 +2γµ−γ 2 = 2γ 2 . So, we have µ = −3γ or µ = γ. Combining these with (3.2) gives (5.1) and (5.2), respectively. (ii) If K = −K D < 0 holds at p ∈ M , (3.3) yields µ = −γ. This gives (5.1). (iii) If K = −K D > 0 holds at p, we find from (3.2) that µ = −γ or µ = 3γ. In the first case, after replacing e3 by −e3 , we have (5.2). In the second case, we get (5.2) after interchanging e1 and e2 . (iv) If K = K D < 0 holds at p, we obtain from (3.3) that µ = γ. This implies K = K D = 0, which is a contradiction. The main result of this article is the following classification theorem. Theorem 5.1. Let ψ : M → E4 be a Wintgen ideal surface in E4 . Then |K| = |K D | holds identically if and only if one of the following four cases occurs: (1) M is an open portion of a totally geodesic plane in E4 . (2) M is a complex curve lying fully in C2 , where C2 is the Euclidean 4-space E4 endowed with some orthogonal almost complex structure. (3) Up to dilations and rigid motions on E4 , M is an open portion of the Whitney sphere defined by sin u ψ(u, v) = sin v, cos v, cos u sin v, cos u cos v . 1 + cos2 u (4) Up to dilations and rigid motions of E4 , M is a surface with K = K D = 21 H 2 defined by √ x 2 y√ 1 x ψ(x, y) = cos x cos cos(ln y) cos tanh−1 tan 5 2 2 2 1 x x × tan tanh−1 tan (2 − tan(ln y)) + tan (1 + 2 tan(ln y)), 2 2 2 x 1 x −1 tan tanh tan (1 + 2 tan(ln y)) − tan (2 − tan(ln y)), 2 2 2 x 1 x tan tan tanh−1 tan (1 + 2 tan(ln y)) + tan(ln y) − 2, 2 2 2 ! x 1 x −1 tan tan tanh tan (tan(ln y) − 2) − 2 tan(ln y) − 1 . 2 2 2

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B.-Y. CHEN

Proof. It is direct to verify that surfaces given in (1)-(4) of the theorem are Wintgen ideal surfaces satisfying |K| = |K D | identically. Conversely, assume that ψ : M → E4 is a Wintgen ideal surface. Then, Lemma 5.1 implies that there exists an orthonormal frame {e1 , e2 , e3 , e4 } such that the shape operator A satisfies either (5.1) or (5.2) for some nonzero function γ with respect to {e1 , e2 , e3 , e4 }. If M is totally geodesic in E4 , we obtain (1). So, from now on, we may assume that M is non-totally geodesic. Thus, we have γ 6= 0, If (5.1) is satisfied, then M is a minimal surface. In the case, γ vanishes only at isolated points. Thus, the Gauss map of M is regular except at some isolated points. From (4.30) and (4.31) of [11, page 153], it follows that the Gauss curvature ˆ of the Gauss image of ψ satisfy of M and Gauss curvature K (5.3)

ˆ 412 = 2(h311 )2 h412 , K(1 − K)h ˆ 311 = 2(h412 )2 h311 K(1 − K)h

with hrij = h(ei , ej ), i, j = 1, 2; r = 3, 4. On the other hand, we find from (5.1) that K = −2γ 2 , h412 = −h311 = γ. Hence, ˆ = 2. Therefore, after (5.3) shows that the Gauss image has Gauss curvature K applying Theorem 2.1, we conclude that M is a complex curve lying fully in C2 . So, we get (2). Next, let us assume that the shape operator A satisfies (5.2). Then we have (5.4)

h(e1 , e1 ) = 3γe3 , h(e1 , e2 ) = γe4 , h(e2 , e2 ) = γe3 .

After applying the equation of Codazzi, we find from (5.4) that (5.5)

e1 γ = γω12 (e2 ),

(5.6)

ω34 (e1 ) = −ω12 (e1 ),

e2 γ = γω12 (e1 ), ω34 (e2 ) = ω12 (e2 ).

Case (a): e1 γ = e2 γ = 0. In this case, γ is constant. So, it follows from (5.4) that M has constant mean curvature and constant Gauss curvature. Thus, M is totally umbilical in E4 according to Proposition 4.1. Therefore, (5.4) implies that γ = 0, which is a contradiction. Case (b): e1 γ 6= 0 and e2 γ = 0. We find from (5.5) that [e1 , γe2 ] = 0. Therefore, ∂ ∂ there exists a coordinate system {x, y} such that ∂x = e1 , ∂y = γe2 . Hence, the metric tensor of M is given by g = dx2 + γ 2 (x)dy 2 .

(5.7)

Thus, the Levi-Civita connection satisfies (5.8)

∇

∂ ∂x

∂ ∂ γ0 ∂ ∂ ∂ = 0, ∇ ∂ = , ∇∂ = −γγ 0 . ∂x ∂y ∂x ∂y γ ∂y ∂y ∂x

By applying the equation of Gauss, we obtain (5.9)

γ 00 (x) + 2γ 3 (x) = 0.


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After solving (5.9) and applying a suitable translation in x, we obtain √ sn( 2x/a) √ γ= , 0 6= a ∈ R, a(1 + cn2 ( 2x/a))1/2

(5.10)

√ where the Jacobi elliptic functions have module k = 1/ 2. After applying a suitable dilations of x and of E4 , we may choose a = 1. Thus, we have √ sn( 2x) √ γ= (1 + cn2 ( 2x))1/2

(5.11)

It follows from (5.4), (5.7) and (5.11) that √ 3 sn( 2x)e3 √ = , h (1 + cn2 ( 2x))1/2 √ ∂ ∂ sn2 ( 2x)e4 √ h , , = ∂x ∂y 1 + cn2 ( 2x) √ ∂ ∂ sn3 ( 2x)e3 h , = √ 3/2 . ∂y ∂y 1 + cn2 ( 2x)

(5.12)

∂ ∂ , ∂x ∂x

Also, from (5.6), (5.7), (5.8) and (5.11), we find

(5.13)

ω34

∂ ∂x

= 0,

ω34

∂ ∂y

√ √ √ 2 2 cn( 2x)dn( 2x) = √ 3/2 . 1 + cn2 ( 2x)

Hence, we obtain from (5.4) and (5.7)-(5.11) that

(5.14)

√ √ √ ∂ ∂ 2 2 cn( 2x)dn( 2x) ∂ √ √ = 0, ∇ ∂ = , ∇∂ ∂x ∂x ∂x ∂y 1 + cn2 ( 2x) sn( 2x) ∂y √ √ √ √ 2 2 cn( 2x)dn( 2x)sn( 2x) ∂ ∂ ∇∂ =− . √ 2 ∂y ∂y ∂x 1 + cn2 ( 2x)

From (2.1), (5.12) and (5.14), we know that the immersion ψ : M → E4 satisfies

(5.15)

√ 3 sn( 2x)e3 √ ψxx = , (1 + cn2 ( 2x))1/2 √ √ √ √ 2 2 cn( 2x)dn( 2x) sn2 ( 2x)e4 √ √ √ ψxy = ψy + , 1 + cn2 ( 2x) sn( 2x) 1 + cn2 ( 2x) √ √ √ √ √ 2 2 cn( 2x)dn( 2x)sn( 2x) sn3 ( 2x)e3 ψyy = − ψx + √ 2 √ 3/2 . 1 + cn2 ( 2x) 1 + cn2 ( 2x)

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Also, it follows from (5.7), (5.12) and (5.13) that √ 3 sn( 2x) √ ψx , ∂ e3 = − ∂x (1 + cn2 ( 2x))1/2 √ √ √ √ sn( 2x) 2 2 cn( 2x)dn( 2x) ˜ ∂ e3 = − √ ∇ ψy − √ 3/2 e4 , ∂y (1 + cn2 ( 2x))1/2 1 + cn2 ( 2x) ˜ ∂ e4 = −ψy , ∇ ∂x √ √ √ √ 2 2 2 cn( 2x)dn( 2x) ˜ ∂ e4 = − sn ( 2x) √ ∇ ψx + √ 3/2 e3 . ∂y 1 + cn2 ( 2x) 1 + cn2 ( 2x) ˜ ∇

(5.16)

From (5.15) and (5.16) we obtain (5.17)

ψyyy + ψy = 0.

Solving (5.17) gives (5.18)

ψ(x, y) = A(x) + B(x) cos y + C(x) sin y

for some E4 -valued functions A(x), B(x), C(x). By substituting (5.18) into the last equation in (5.15), we find

(5.19)

√ √ √ 2 2 cn( 2x)dn( 2x) (A0 (x) + B 0 (x) cos y + C 0 (x) sin y) √ √ e3 = (1 + cn2 ( 2x))1/2 sn2 ( 2x) √ (1 + cn2 ( 2x))3/2 (B(x) cos y + C(x) sin y) √ − sn3 ( 2x)

Substituting (5.18) and (5.19) into the first equation in (5.15) gives (5.20) (5.21) (5.22)

√ √ √ 6 2 cn( 2x)ds( 2x) 0 √ A − A = 0, 1 + cn2 ( 2x) √ √ √ √ 6 2 cn( 2x)ds( 2x) 0 3(1 + cn2 ( 2x)) 00 √ √ B − B + B = 0, 1 + cn2 ( 2x) a2 sn2 ( 2x) √ √ √ √ 6 2 cn( 2x)ds( 2x) 0 3(1 + cn2 ( 2x)) √ √ C 00 − C + C=0 1 + cn2 ( 2x) sn2 ( 2x) 00

After solving these second order differential equations we obtain √ c1 cd( 2x) √ (5.23) A(x) = c0 + , (1 + cn2 ( 2x))1/2 √ √ sn( 2x)(c2 + c3 cn( 2x)) √ (5.24) B(x) = , 1 + cn2 ( 2x) √ √ sn( 2x)(c4 + c5 cn( 2x)) √ (5.25) C(x) = . 1 + cn2 ( 2x)


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for some vectors c0 , c1 , . . . , c5 ∈ E4 . Without loss of generality, we may choose c0 = 0. Thus, after combining (5.18) and (5.23)-(5.25), we obtain √ √ sn( 2x) n √ ψ(x, y) = c2 + c3 cn( 2x) cos y 2 1 + cn ( 2x) (5.26) √ o √ c1 cd( 2x) √ . + c4 + c5 cn( 2x) sin y + (1 + cn2 ( 2x))1/2 After using (5.7), (5.11), (5.26), and choosing suitable initial conditions, we have √ √ √ sn( 2x) √ ψ(x, y) = sin y, cos y, cn( 2x) sin y, cn( 2x) cos y . 1 + cn2 ( 2x) √ If we put u = am( 2x), we have √ √ √ cn( 2x)sn( 2x) sin u sin u cos u sn ( 2x) √ √ , . = = 2 2 2 1 + cos u 1 + cos2 u 1 + cn ( 2x) 1 + cn ( 2x) Therefore, the surface can be reparametrization as sin u ψ(u, v) = (5.27) sin v, cos v, cos u sin v, cos u cos v 1 + cos2 u with y = v. Thus, up to dilations and rigid motions, the surface is an open portion of a Whitney sphere (see [8, page 283] or [1]). This gives (3). Case (c): e1 γ = 0 and e2 γ = 0. In this case, (5.5) and (5.6) reduce to (5.28)

ω12 (e2 ) = ω34 (e2 ) = 0, e2 (ln γ) = ω12 (e1 ) = −ω34 (e1 ),

which imply that [γ −1 e1 , e2 ] = 0. Thus, there exists a coordinate system {x, y} ∂ ∂ such that ∂x = γ −1 e1 , ∂y = e2 . Hence, the metric tensor of M is given by (5.29)

g=

dx2 + dy 2 , γ = γ(y). γ2

Therefore, the Gauss curvature is given by (5.30)

K=

γγ 00 − 2γ 02 . γ2

Thus, after applying (5.30) and the equation of Gauss, we get (5.31)

γγ 00 − 2γ 02 − 2γ 4 = 0.

On the other hand, by applying equation (2.6) of Ricci, (5.4) and (5.28), we also find (5.32)

γγ 00 − 2γ 02 + 2γ 4 = 0.

Combining (5.31) and (5.32) gives γ = 0, which is a contradiction. Case (d): e1 γ, e2 γ 6= 0. We find [γ −1 e1 , γe2 ] = 0 by virtue of (5.5). Thus, there exists a coordinate system {u, v} such that (5.33)

∂ e1 = , ∂u γ

∂ = γe2 . ∂v

12

B.-Y. CHEN

Hence, the metric tensor of M is given by (5.34)

g=

du2 + γ 2 dv 2 . γ2

Therefore, the Levi-Civita connection satisfies γu ∂ γv ∂ ∂ =− + , ∂u γ ∂u γ 5 ∂v ∂ γv ∂ γu ∂ ∇∂ =− + , ∂u ∂v γ ∂u γ ∂v ∂ ∂ γv ∂ = −γ 3 γu + . ∇∂ ∂v ∂v ∂u γ ∂v ∇ (5.35)

∂ ∂u

and the Gauss curvature is given by (5.36)

K=

γγvv − 3γv2 − γγuu − γu2 . γ4

After applying equation (2.4) of Gauss, (5.4) and (5.36), we find (5.37)

γγvv − 3γv2 − γγuu − γu2 = 2γ 2 . γ4

On the other hand, it follows from (5.33) and (5.35) that γv ∂ ∂ (5.38) = 3 , ω12 = γγu . ω12 ∂u γ ∂v By combining (5.6) and (5.38), we find ∂ γv 4 (5.39) ω3 = − 3, ∂u γ

ω34

∂ ∂v

= γγu .

It follows from equation (2.6) of Codazzi, (2.12), (5.4) and (5.39) that (5.40)

3γv2 − γγvv − γγuu − γu2 = 2γ 2 . γ4

Now, by combining (5.37) and (5.40), we derive that (5.41)

γγuu + γu2 + 2γ 2 = 0, γγvv = 3γv2 .

Solving the second equation in (5.41) gives (5.42)

γ=p

f (u) v + k(u)

for some functions f, k. After substituting (5.42) into the first equation in (5.41), we find from the coefficients of y 2 that f (u)f 00 (u) + f 0 (u)2 + 2f 2 (u) = 0. p By solving this equation, we get f (u) = a cos(2u + c) for some real numbers a, c √ with a 6= 0. After applying a suitable translation in x, we have f = a cos 2u. Combining this with (5.42) yields √ a cos 2u (5.43) γ=p v + k(u)


13

Now, by substituting (5.43) into (5.41), we get (5.44)

k 00 (u) = 4(tan u)k 0 (u),

(5.45)

k(u)k 00 (u) = 2k 0 (u){k 0 (u) + 2k(u) tan 2u}.

Combining these two equations gives k 0 (u) = 0. Hence, k(u) is constant. Therefore, after applying a suitable translation in y and a suitable dilation of E4 , we obtain √ cos x γ = √ , x = 2u, y = v (5.46) y for some nonzero real number a. Consequently, (5.34) becomes y cos x 2 g= dx + (5.47) dy 2 . 4 cos x y From (5.47), we obtain ∂ tan x ∂ y sec2 x ∂ = − , ∂x 2 ∂x 8 ∂y 1 ∂ tan x ∂ ∂ = − , ∇∂ ∂x ∂y 2y ∂x 2 ∂y ∂ sin 2x ∂ 1 ∂ ∇∂ = − . ∂y ∂y y 2 ∂x 2y ∂y ∇

(5.48)

∂ ∂x

Moreover, from (5.39) and (5.43), we have ∂ sec x ∂ sin x (5.49) ω34 , ω34 . = =− ∂x 4 ∂y y It follows from (5.4), (5.33) and (5.46)-(5.49) that √ 3 y tan x y sec2 x ψx − ψy + √ e3 , 2 8 4 cos x √ ψx tan x cos x = − ψy + √ e4 , 2y 2 2 y

(5.50)

ψxx =

(5.51)

ψxy

(5.52)

(5.53)

sin 2x ψy cos3/2 x ψyy = ψ − + e3 , x y2 2y y 3/2 √  cos x sec x  ˜ ∂ e3 = − 3 √  ψx + e4 , ∇   ∂x y 4   √   cos x sin x   ˜ ∂ e3 = − √ ψy − e4 , ∇ ∂y y √y   ˜ ∂ e4 = − √ y ψy − sec x e3 ,  ∇  ∂x  4 2 cos x   3/2   2 cos x sin x  ˜ ∂ e4 = − ∇ ψx + e3 . 3/2 ∂y y y

From (5.52) and (5.53) we derive that (5.54)

4y 2 ψyyy + 8yψyy + 5ψy = 0.

After solving (5.54) we obtain √ √ ψ(x, y) = y sin(ln y)A(x) + y cos(ln y)B(x) + C(x). (5.55)

14

B.-Y. CHEN

for some E4 -valued functions A.B, C. Also, it follows from (5.50) and (5.52) that (5.56)

4ψxx − 3(sec2 x)y 2 ψyy + 4(tan x)ψx − (sec2 x)yψy = 0.

By substituting (5.55) into (5.56), we find (5.57) (5.58) (5.59)

13 sec2 x (sec2 x)A(x) = B(x), 16 4 13 sec2 x B 00 (x) + (tan x)B 0 (x) + (sec2 x)B(x) = − A(x), 16 4 C 00 (x) + (tan x)C 0 (x) = 0. A00 (x) + (tan x)A0 (x) +

Solving (5.59) yields C(x) = a0 + a1 sin x or some vectors a0 , a1 ∈ E4 . Without loss of generality, we may choose a0 = 0. Thus, we have (5.60)

C(x) = a1 sin x

Moreover, it follows from (5.57) and (5.58) that both A(x) and B(x) satisfy the same fourth order differential equation given by 5 p(4) (x) − 2(tan x)p000 (x) + 1 + sec2 x p00 (x) 8 (5.61) 5 185 + sec2 x − 2 (tan x)p0 (x) + (sec4 x)p(x) = 0. 8 256 After solving this fourth order differential equation, we obtain x x x √ 1 b1 cos A(x) = cos x tanh−1 tan + b2 sin cos 2 2 2 2 x x 1 x tanh−1 tan + b3 cos + b4 sin sin , 2 2 2 2 (5.62) x x x √ 1 c1 cos B(x) = cos x + c2 sin cos tanh−1 tan 2 2 2 2 x x 1 x + c4 sin sin tanh−1 tan + c3 cos 2 2 2 2 for some vectors bi , ci , i = 1, 2, 3, 4. From (5.55), (5.60) and (5.62), we have ( x h x 1 −1 ψ(x, y) = cos tanh tan cos (c1 cos(ln y) + b1 sin(ln y)) 2 2 2 x i + sin (c2 cos(ln y) + b2 sin(ln y)) 2 (5.63) x h x 1 tanh−1 tan cos (c3 cos(ln y) + b3 sin(ln y)) + sin 2 2 2 ) x i √ + sin (c4 cos(ln y) + b4 sin(ln y)) y cos x + a1 sin x. 2


15

From (5.63), we get √ n h 3x y ψx = √ cos u 2 cos ((b2 sin(ln y)+c2 cos(ln y)) 4 cos x 2 x 3x (b3 sin(ln y)+c3 cos(ln y)) − 2 sin (b1 sin(ln y)+c1 cos(ln y))+cos 2 2 x i + sin ((b4 sin(ln y) + c4 cos(ln y)) 2 h 3x + sin u 2 cos ((b4 sin(ln y) + c4 cos(ln y)) 2 x 3x (b1 sin(ln y)+c1 cos(ln y)) (b sin(ln y) +c cos(ln y))−cos − 2 sin 3 3 (5.64) 2 2 x io − sin ((b2 sin(ln y) + c2 cos(ln y)) + a1 cos x, 2 √ n h x cos x ψy = √ [(2b1 + c1 ) cos(ln y) + (b1 − 2c1 ) sin(ln(y))] cos u cos 2 y 2 x i + sin [(2b2 + c2 ) cos(ln y) + (b2 − 2c2 ) sin(ln(y))] h 2 x + sin u cos [(2b3 + c3 ) cos(ln y) + (b3 − 2c3 ) sin(ln(y))] x 2 io + sin [(2b4 + c4 ) cos(ln y) + (b4 − 2c4 ) sin(ln(y))] 2 with u = 21 tanh−1 (tan( x2 )). Now, by applying (5.47), (5.64), and a long computation, we derive that 4 5

|a1 |2 = 0, |bj |2 = |cj |2 = , j = 1, 2, 3, 4, (5.65)

4 5

4 5

hb1 , c4 i = hb2 , c3 i = , hb3 , c2 i = hb4 , c1 i = − , hbi , cj i = 0, otherwise.

Hence, up to rigid motions of E4 , we may choose

(5.66)

a1 = (0, 0, 0, 0), 2 4 4 2 b1 = c4 = 0, 0, , − , b2 = c3 = , , 0, 0 , 5 5 5 5 2 4 4 2 b3 = −c2 = − , , 0, 0 , b4 = −c1 = 0, 0, , . 5 5

5 5

Consequently, we obtain (4) of the theorem from (5.63) and (5.66).

Remark 5.1. As pointed out in [14], the Whitney sphere is the composition I ◦ f of the holomorphic curve f : C∗ → C2 given by f (z) = (z, 1/z) with the inversion I with respect to the sphere of unit radius centered at the origin. The pair (g, h) of conjugate minimal surfaces associated to I ◦ f for Dajczer-Tojeiro’s representation formula, is given by 1 4

i 4

g = (1/¯ z , z¯) and h(z) = (1/¯ z , z¯). According to I. Castro [5], up to rigid motions and dilations the Whitney sphere is the only compact orientable Lagrangian superminimal surface in C2 .

16

B.-Y. CHEN

Remark 5.2. It seems to the author that it is quite difficult to prove our main result (Theorem 5.1) by applying Dajczer-Tojeiro’s representation formula, since there is no simple way to express Gauss curvature K and normal curvature K D of a superminimal surface M ⊂ E4 in terms of curvatures of the pair (g, h) of conjugate minimal surfaces associated with M via the representation formula.

References [1] Borrelli, V., Chen, B.-Y. and Morvan, J.-M.: Une caract´ erisation g´ eom´ etrique de la sph` ere de Whitney. C. R. Acad. Sci. Paris S´ er. I Math. 321, 1485–1490 (1995). [2] Bor˚ uvka, O.: Sur une classe de surfaces minma plon´ ees dans un espace a ´ quatre dimensions ` a courbure constante. C. R. Acad. Sci. 187, 334–336 (1928). [3] Rryant, R. L.: Conformal and minimal immersions of compact surfaces into the 4-sphere. J. Diff. Geom. 17, 455–473 (1982). [4] Byrd, P. F. and Friedman, M. D.: Handbook of elliptic integrals for engineers and scientists. Second edition, Springer-Verlag, New York-Heidelberg (1971). [5] Castro, I.: Lagrangian surfaces with circular ellipse of curvature in complex space forms. Math. Proc. Camb. Phil. Soc. 136, 239–245 (2004). [6] Chen, B.-Y.: Geometry of Submanifolds. Mercer Dekker, New York (1973). [7] Chen, B.-Y.: Total Mean Curvature and Submanifolds of Finite Type. World Scientific, New Jersey (1984). [8] Chen, B.-Y.: Complex extensors and Lagrangian submanifolds in complex Euclidean spaces. Tohoku Math. J. 49, 277–297 (1997). [9] Chen, B.-Y.: Riemannian submanifolds, Handbook of Differential Geometry. Vol. I, 187–418, North-Holland, Amsterdam, (eds. F. Dillen and L. Verstraelen) (2000). [10] Chen, B.-Y.: Exact solutions of a class of differential equations of Lam´ e’s type and its applications to contact geometry. Rocky Mountain J. Math. 30, 497–506 (2000). [11] Chen, B.-Y. and Yamaguchi, S.: Classification of surfaces with totally geodesic Gauss image. Indiana Univ. Math. J. 32, 143–154 (1983). [12] Chen, B.-Y. and Yamaguchi, S.: Submanifolds with totally geodesic Gauss image. Geom. Dedicata 15, 313–322 (1984). [13] De Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, L.: A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35, 115–128 (1999). [14] Dajczer, M. and Tojeiro, R.: All superconformal surfaces in R4 in terms of minimal surfaces. Math. Z. 261, 869-890 (2009). [15] Decu, S., Petrovi´ c–Torgaˇsev, M. and Verstraelen, L.: On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds. Tamkang J. Math. 41(2) (2010) (in print). [16] Friedrich, Th.: On surfaces in four-spaces. Ann. Glob. Anal. Geom. 2, 257-287 (1984). [17] Friedrich, Th.: On superminimal surfaces. Arch. Math. (Brno) 33, 41-56 (1997). [18] Guadalupe, I. V. and Rodriguez, L.: Normal curvature of surfaces in space forms. Pacific J. Math. 106, 95–103 (1983). [19] Hoffman, D. and Osserman, R.: The geometry of the generalized Gauss map. Mem. Amer. Math. Soc. 236 (1980). [20] Kommerell, K.: Rimannsche Fl¨ achen im ebenen Raum von vier Dimensionen. Math. Ann. 60, 548–596 (1905). ¨ [21] Kwietniewski, S.: Uber Fl¨ achen der 4-dimensionalen Raumes deren Tangentialebenen paarweise isoklin sind, Dissertation Z¨ urich, 1902. [22] Lawden, D. F.: Elliptic Functions and Applications. Springer-Verlag, Berlin (1989). [23] Petrovi´ c-Torgaˇsev, M. and Verstraelen, L.: On Deszcz symmetries of Wintgen ideal submanifolds. Arch. Math. (Brno) 44, 57–67 (2008).


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[24] Wintgen, P.: Sur l’in´ egalit´ e de Chen-Willmore. C. R. Acad. Sci. Paris, 288, 993–995 (1979). Department of Mathematics, Michigan State University, East Lansing, Michigan 48824–1027, USA E-mail address: [email protected]