surfaces with moebius syndrrome

8/18/2019 Surfaces With Moebius Syndrrome

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672 Li H. Z. and Wang C. P.

(ii) The standard flat minimal surfaces in an odd dimensional unit sphere S n with n ≥ 5;(iii) The pre-image of the stereographic projection of the circular cylinder in R3;

(iv) The standard flat torus in S 3;

(v) The pre-image of the stereographic projection of the circular cone in R3.

We organize the paper as follows. In §1, we give Moebius invariant system and structureequations for surfaces in S n. In §2, we prove the main theorem.

1 Moebius Invariants for Surfaces in S n

In this section, we give Moebius invariants and structure equations for surfaces in S n. For

details refer to [2] and [1].

Let Rn+21 be the Lorentz space Rn+2 with the inner product , given by

X, Y = −x0y0 + x1y1 + · · · + xn+1yn+1, (1.1)

where X = (x0, x1, . . . , xn+1), Y = (y0, y1, . . . , yn+1) ∈ Rn+2. We denote by C n+1+ the half cone in Rn+21 :

C n+1+ = {X ∈ Rn+2|X, X = 0, x0 = p(X ) > 0}, (1.2)where p : Rn+21 → R is the projection of X to its first coordinate x0. We denote by O+(n + 1, 1)the Lorentz group of Rn+21 preserving the inner product , and C n+1+ invariant.

Let x : M → S n be a umbilic-free surface with I and II as the first and the secondfundamental form. We define the positive function ρ : M → R by

ρ =√

2II − HI .Then the map Y = ρ(1, x) = (ρ,ρx) : M

→ C n+1+ is called the canonical lift of x. The following

result is a key theorem in the study of surface theory:

Theorem 1.1 Two surfaces x, x̃ : M → S n are Moebius equivalent if and only if there exists T ∈ O+(n + 1, 1) such that their canonical lifts Y and Ỹ satisfy Y = Ỹ T : M → C n+1+ .

Let Y be the canonical lift of x. The Moebius invariant g = dY,dY = ρ2dx · dx is calledthe Moebuis metric of x. We denote by ∆ the Laplacian of g and define N : M → Rn+21 by

N = −12

∆Y − 18∆Y, ∆Y Y ; (1.3)

then we have

Y, Y = N, N = 0, Y, N = 1, N,dY = dN,Y = 0. (1.4)Now let {e1, e2} be a local orthonormal basis of Moebius metric. We denote by V the Lorentzianorthogonal complement to the subspace

span{Y, N } ⊕ span{e1(Y ), e2(Y )} ⊂ Rn+21 .Then we have

Rn+21 = span{Y, N } ⊕ span{e1(Y ), e2(Y )} ⊕ V. (1.5)We call V the Moebius normal bundle of M . Let {E α} be a local orthonormal basis of V with


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Surfaces with Vanishing Moebius Form in S n 673

respect to the Lorentz metric in Rn+21 . Then we have a Moebius moving frame {Y , N , ei(Y ), E α}in Rn+21 for x : M → S n. Using this Moebius frame we can give structure equations and Moebiusinvariants for surfaces in S n.

Let z = u + iv be a local complex coordinate on M with respect to g . We can write

g = 1

2e2ω(dz ⊗ dz̄ + dz̄ ⊗ dz) = e2ω|dz|2 (1.6)

for some locally defined smooth function ω . From (1.6) and the fact g = dY,dY we get

Y z, Y z = Y ̄z, Y ̄z = 0, Y z, Y ̄z = 12

e2ω. (1.7)

We denote by K the Gaussian curvature of g . Then by (1.6), we have

∆Y = 4e−2ωY zz̄, K = −4e−2ωωzz̄. (1.8)It follows from Theorem 1.2 and (2.1) in [1] that

∆Y, ∆Y = 1 + 4K. (1.9)Since {Y , N , Y z, Y ̄z, E α} is a moving frame in Rn+21 , for any vector W ∈ Rn+21 we have theformula

W = W, N Y + W, Y N + 2e−2ωW, Y ̄zY z

+ 2e−2ωW, Y zY ̄z +n−2α=1

W, E αE α. (1.10)

We define

ψ = 2N z, Y z, φα = N z, E α, (1.11)Ωα = 2Y zz , E α, Aαβ = (E α)z, E β = −Aβα . (1.12)

It is clear that Ψ = ψdz2, Φc =

α φαdz ⊗ E α and Ω = α Ωαdz2 ⊗ E α are globally definedMoebius invariants. Using the formula (1.9) and Equations (1.3)–(1.12) we can write the

structure equations as follows (see [2]):

N z = 1

8(1 + 4K )Y z + e

−2ωψY ̄z +α

φαE α,

Y zz = −12

ψY + 2ωzY z + 1

2

α

ΩαE α,

Y zz̄ = − 116

e2ω(1 + 4K )Y − 12

e2ωN

(E α)z = −φαY − e−2ωΩαY ̄z +β

AαβE β . (1.13)

Using the identities N zz̄ = N ̄zz , Y zzz̄ = Y zz̄z and (E α)zz̄ = (E α)z̄z , we get the following

integrability conditions for the linear PDE system (1.13):

ψz̄ = 1

2e2ωK z −

α

Ωαφ̄α, (1.14)

(φα)z̄ − 12

e−2ω ψ̄Ωα +β

φβ Āβα = (φ̄α)z̄ − 12

e−2ωψΩ̄α +β

φ̄βAβα , (1.15)


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e−4ωα

|Ωα|2 = 14

, (1.16)

(Ωα

)z̄

= −β Ωβ

Āβα −

e2ωφα

, (1.17)

(Aαβ)z̄ − ( Āαβ)z = 12

e−2ω(ΩαΩ̄β − Ω̄αΩβ) +γ

( Āαγ Aγβ − Aαγ Āγβ). (1.18)

According to [1], we define Moebius form Φ =

iα C αi ωi⊗E α, where C αi = −ρ−2{H αi +

j(h

αij−

H αδ ij)ej(log ρ)}. It can be reformulated as Φ =

αdN,E αE α. Since dN = N zdz + N ̄zdz̄,we get in the surface case that

Φ =α

(φαdz + φαdz̄) ⊗ E α. (1.19)

Thus x : M → S n has vanishing Moebius form if and only if φα = N z, E α = 0 for all α.

Remark 1.1 Let x : M → S n be a surface. The Willmore functional for x is defined byW (x) =

M

(||II ||2 − 2||H ||2)dM, (1.20)

where dM is the volume form for x. It is known that W (x) is a Moebius invariant (see Willmore

[3] and references there). The Euler–Lagrange equation for the Willmore functional W (x) can

be written as (cf. also Weiner [4], Burstall–Pedit–Pinkall [5])

(φα)z̄ − 12

e−2ω ψ̄Ωα +β

φβ Āβα = 0, ∀ α. (1.21)

x : M → S n is called a Willmore surface if it satisfies (1.21). About the results of Willmoresurfaces, readers can see Bryant [6], Castro–Urbano [7], Li [8], Montiel [9] and Pinkall [10], etc.

Remark 1.2 Some related results about Moebius hypersurfaces in S n+1 can be found in

Hu–Li [11], Li–Liu–Wang–Zhao [12] and Liu–Wang–Zhao [13].

2 The Classification of Surfaces in S n with Φ = 0

Let x : M → S n be a surface in an n-dimensional unit sphere with vanishing Moebius form,i.e.,

φα = 0, 1 ≤ α ≤ n − 2. (2.1)

Then from (1.14)–(1.17) we obtain

ψz̄ = 12

e2ωK z, (2.2)

ψ̄Ωα = ψΩ̄α, (2.3)

e−4ωα

|Ωα|2 = 14

, (2.4)

(Ωα)z̄ = −β

Ωβ Āβα . (2.5)


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unit sphere S n (also see Calabi [16] and Wallach [17]); (ii) the flat minimal surfaces in an odd

dimensional unit sphere S n as an orbit of a 2-dimensional abelian subgroup of SO(n + 1) (also

see Kenmotsu [18]).

If η is light-like, we have

η, η

= −

1

4

(1 + 4K ) = 0. Thus we can find a transformation

T ∈ O+(n + 1, 1) such that ηT = (−1, 1, 0). By making a Moebius transformation, if necessary,we may assume that η = (−1, 1, 0). It follows from (2.8), (1.3) and (1.9) that

η = (−1, 1, 0) = N = −12

∆Y. (2.11)

We write x = (x0, x1) ∈ R ×Rn. Then Y = ρ(1, x) = (ρ,ρx0, ρx1). From the second equation of (2.9) we get ρ(1+x0) = 1. Thus x0 = −1 and (−1, 0) /∈ x(M ) ⊂ S n. Now let σ : Rn∪{∞} → S nbe the stereographic projection

x = σ(u) =

1 − |u|21 + |u|2 ,

2u

1 + |u|2

. (2.12)

We define u = σ−1

◦x : M

→ Rn. Since ρ = 1/(1 + x0) = (1 +

|u|2)/2, by a direct calculation,

we get

g = ρ2dx · dx = du · du. (2.13)

Thus the induced metric of u : M → Rn is exactly the Moebius metric g of x : M → S n. Inparticular, it has constant curvature. Comparing with the third coordinate in (2.11), we get

∆u = ∆(ρx1) = 0,

which implies that u : M → Rn is a minimal surface with constant curvature. By Proposition4.1 of Bryant [15] we know that it is a part of a 2-plane in Rn, thus x : M → S n is a part of around 2-sphere, which contradicts our assumption that x is umbilic-free. Therefore, this case

cannot occur.If η is space-like, we write η, η = − 14 (1 + 4K ) = r2 with r = 12

−(1 + 4K ) > 0. Bymaking a Moebius transformation, if necessary, we may assume that η = (0, r, 0). We write

x = (x0, x1), then Y = (ρ,ρx0, ρx1). From the second equation of (2.9) we get ρrx0 = 1, which

implies that x0 > 0, thus x(M ) contains in the half sphere S n+ of S

n. Now let

H n = {(y0, y1) ∈ R+ × Rn | −y20 + y1 · y1 = −1}be the n-dimensional hyperbolic space. We define a conformal map τ : H n → S n+ by

x = (x0, x1) = τ (y0, y1) =

1

y0, y1y0

. (2.14)

Then y = τ −1

◦ x : M → H n

is a surface. Since ρ = r−1

x−10 , we have

Y = (ρ,ρx0, ρx1) =

y0r

, 1

r, y1

r

, (2.15)

which implies that

g = dY,dY = r−2(−dy0 · dy0 + dy1 · dy1) := r−2gM .Since g is of constant curvature K , then the induced metric gM of y : M → H n has constantcurvature. We denote by ∆M the Laplacian of gM , then we have ∆M = r

−2∆. It follows from


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Surfaces with Vanishing Moebius Form in S n 677

(2.8) with η = (0, r, 0) and (1.3) that ∆M y − 2y = 0. Thus y : M → H n is a minimal surfaceof constant curvature. By Theorem 4.2 of Bryant [15] we know that y(M ) is totally geodesic

in H n, which implies that x(M ) is totally umbilic in S n and thus contradicts our assumption.

This case cannot occur. Thus we have completed the discussion for the case (a).

Now we come to discuss the case (b): the zero points of Ψ are isolated. In this case we can

cut M by some disjoint curves C i to get a simply connected domain U = M \

i C i such that

x : U → S n is a surface with Ψ = 0 on U .Since Ψ = ψdz2 = 0 is a holomorphic 2-form on the simply connected domain U , by

changing complex coordinate, if necessary, we may assume that ψ ≡ 1 on U . It follows from(2.6) and (2.7) that K = −4e−2ωωzz̄ = 0. By (2.3), we know that {Ωα} are real functions. Wedefine a global real vector field E ∈ V by

E = 2e−2ωα

ΩαE α, (2.16)

then by (2.4), we have

E, E

= 1. By choosing a local orthonormal basis {

E 1

, E 2

, . . . , E n−2}in the Moebius normal bundle V with E 1 ≡ E we get

Ω1 = 1

2e2ω, Ω2 = · · · = Ωn−2 = 0. (2.17)

Using (1.17) and (2.17), we get

(Ω1)z̄ = −n−2β=1

Ωβ Āβ1 = −Ω1 Ā11 = 0, (2.18)

0 = (Ωα)z̄ = −β=α

Ωβ Āβα = −Ω1 Ā1α, α ≥ 2. (2.19)

Thus ω is a constant and

A1α = 0, α = 2, . . . , n − 2. (2.20)Now the structure equations with respect to the local frame {Y , N , Y z, Y ̄z, E , E 2, . . . , E n−2} read

Y z = Y z, Y ̄z = Y ̄z, (2.21)

N z = 1

8Y z + e

−2ωY ̄z, N ̄z = e−2ωY z +

1

8Y ̄z, (2.22)

Y zz = −12

Y + Y z + 1

2e2ωE 1, (2.23)

Y zz̄ = − 116

e2ωY − 12

e2ωN, (2.24)

Y ̄zz̄ =

−1

2

Y + Y ̄z + 1

2

e2ωE 1, (2.25)

(E 1)z = −12

Y ̄z, (E 1)z̄ = −12

Y z, (2.26)

(E α)z =β=α

AαβE β , (E α)z̄ =β=α

ĀαβE β, α = 2, . . . , n − 2. (2.27)

A surface is said to be full in S n if x(M ) does not lie in any totally umbilic S n−1 of S n.

Now we claim that for a full surface in S n with the assumption of the case (b) we must have

n = 3.


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We assume that n ≥ 4. By fixing a point p ∈ U , we can find a constant vector ξ ∈ Rn+21with ξ, ξ = 1 such that

Y ( p), ξ = N ( p), ξ = Y u( p), ξ = Y v( p), ξ = E 1( p), ξ = 0, (2.28)

where z = u + iv is the complex coordinate on U . We define real functions

f 1 = Y, ξ , f 2 = N, ξ , f 3 = Y u, ξ , f 4 = Y v, ξ , f 5 = E 1, ξ .Then by (2.21)–(2.26) we can find constants {aλµ} and {bλµ} such that

(f λ)u =µ

aλµf µ, (f λ)v =µ

bλµf µ, 1 ≤ λ, µ ≤ 5. (2.29)

By (2.28) and the uniqueness of the linear PDE (2.29) we get f λ ≡ 0. In particular, f 1 =Y, ξ = 0 on U , which implies that Y, ξ = 0 on M . Thus x : M → S n is not full, whichcontradicts our assumption. Therefore n = 3 and x : M → S 3 is a surface with vanishingMoebius form. By Theorem 5.1 of Wang [1], we know that x is Moebius equivalent to one

of the following surfaces in S 3: (iii) the pre-image of the stereographic projection of a circularcylinder in R3; (iv) a standard flat torus in S 3; (v) the pre-image of the stereographic projection

of a circular cone in R3.

Thus we complete the proof of the Main Theorem.

References

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[2] Li, H., Wang, C. P., Wu, F.: A Moebius characterization of Veronese surfaces in S n. Math. Ann., 319,

707–714 (2001)

[3] Willmore, T. J.: Surfaces in conformal geometry. Ann. Global Anal. Geom., 18, 255–264 (2000)

[4] Weiner, J.: On a problem of Chen, Willmore, et al.. Indiana Univ. Math. J., 27, 19–35 (1978)

[5] Burstall, F., Pedit, F., Pinkall, U.: Schwarzian derivatives and flows of surfaces. arXiv. org/math. DG/0111169

[6] Bryant, R.: A duality theorem for Willmore surfaces. J. Differential Geom., 20, 23–53 (1984)

[7] Castro, I., Urbano F.: Willmore surfaces of R4 and the Whitney sphere. Ann. Global Anal. Geom., 19,

153–175 (2001)

[8] Li, H.: Willmore surfaces in S n. Ann. Global Anal. Geom., 21, 203–213 (2002)

[9] Montiel, S.: Willmore two-spheres in the four-sphere. Trans. AMS , 352, 4469–4486 (2000)

[10] Pinkall, U.: Hopf tori in S 3. Invent. Math., 8, 379–386 (1985)

[11] Hu, Z. J., Li, H.: Submanifolds with constant Moebius scalar curvature in S n. Manuscripta Math., 111,

287–302 (2003)

[12] Li, H., Liu, H. L., Wang, C. P., Zhao, G. S.: Moebius isoparametric hypersurfaces in S n+1 with two distinct

principal curvatures. Acta Math. Sinica, English Series, 18, 437–446 (2002)

[13] Liu, H. L., Wang, C. P., Zhao, G. S.: Moebius isotropic submanifolds in S n. Tohoku Math. J., 53, 553–569

(2001)

[14] Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan , 18, 380–385 (1966)

[15] Bryant, R.: Minimal surfaces of constant curvature in S n. Trans. AMS , 290, 259–271 (1985)

[16] Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. Diff. Geom., 1, 111–125 (1967)

[17] Wallach, N.: Extension of locally defined minimal immersions of spheres into spheres. Arch. Math., 21,

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