Annals of Global Analysis and Geometry 10: 219-226, 1992. 219© 1992 Kluwer Academic Publishers. Printed in the Netherlands.

There exist no 2-type surfaces in E3

which are images under stereographic projectionof minimal surfaces in S3

MANUEL BARROS*

Abstract: In this paper we deal with the following particular case of a weaker conjecture byB.Y. Chen: Are there 2-type Willmore surfaces in E3 ? In particular we prove that the abovequestion has a negative answer when the surface is the image under stereographic projectionof a minimal surface in S3 .Key words: Willmore surface, 2-type surface, mznimal surfaceMSC 1991: 53C40, 53A05

1. Introduction

The notion of a finite type submanifold (or a finite type isometric immersion) in theEuclidean space was introduced by B.Y. Chen (see [Ch.2] for details) in 1983. Thiskind of submanifolds are essentially constructed in Em by making use of a finitenumber of Em-valued eigenfunctions of their Laplacians (k-type if one uses exactlyk different eigenspaces of the Laplacian). In terms of this terminology, 1-type sub-manifolds in E"' correspond to either minimal submanifolds in Em (constructed bymeans of harmonic functions) or minimal submanifolds in some hypersphere of Em

(see [Ta]). Certainly, finite type submanifolds closest in simplicity to minimal sub-manifolds are the 2-type submanifolds. Many people obtained important results on 2-type submanifolds in the sphere and gave nice applications to: the inverse problem inspectral geometry; the variational problem of Chen-Willmore; the study of minimalsurfaces in the sphere etc. (See for instance: [BC.1,2],[BU],[G.2],[BG],[Mi],[Ro.1,2]etc.). However, few results are kown about 2-type submanifolds which are not spher-ical, namely 2-type surfaces in E3. Perhaps this is so because the partial differentialequations involved are very difficult to deal with. Let me recall some results in thisdirection: In [Ch.3], B.Y. Chen proved that circular cylinders are the only tubesof finite type one can find in E3 (they are exactly of 2-type and involve the eigen-value A = 0, in this sense they are called null-two-type surfaces). Also Chen, [Ch.1],showed that circular cylinders are the only null-two-type surfaces in E3 . In [Ga.1],O.J.Garay classified the surfaces of revolution in E3 whose coordinate functions areeigenfunctions of the Laplacian. In [Ch.4], Chen stated the following conjecture: Theonly compact finite type surface in E3 is the sphere. A weaker conjecture is to trythe 2-type case and even so the solution becomes very difficult. In this short paper

* Partially supported by a DGICYT Grant No.PS87-0115

M. BARROS

I will deal with the following particular case of the above weaker conjecture: Arethere 2-type Willmore surfaces in E3 ?

It is well known, (see [La]), that there exist infinitely many compact minimalsurfaces in the 3-sphere S3 . In fact, Lawson proved that one can find compact(embedded) minimal surfaces in S3 of arbitrary genus. The images of these surfacesunder stereographic projection of S3 onto E3 are examples of stationary surfaces inE3 (also called Willmore surfaces) (see Section 5 and [We] for more details). In thispaper we will prove that None of the surfaces obtained in this way are of 2-type inE3 . Moreover, we will prove: There exist no 2-type Willmore surfaces in E3 withnon-negative Gaussian curvature. This result also holds for non-compact surfaces.

2. The Laplacian of the Mean Curvature Vector Fieldfor Surfaces in E3

Let z : M ---+ E3 be an isometric immersion of a surface M into the 3-dimensionalEuclidean space. Denote by H, a, and A, the mean curvature vector field, themean curvature and the shape operator of z, respectively. Therefore, H = aN fora unit normal vector field N. If V denotes the gradient operator of M (of coursewith respect to the induced metric), the Laplacian of M is given by A = -div.V.We choose p E M and an orthonormal basis in the tangent bundle of M such thatVE,E,(p) = 0, where V also denotes the Levi-Civita connection of M. Now we willcompute AH in terms of these data.

AH = 2A(Va) + a(VA) + (Aa + a A 12 )N, (1)

where VA = i21(VE,A)Ei and I A I denotes the length of A.In order to obtain a nice formula for VA, we choose {El, E2} to be principal

directions with AE, = ,Ei (, being the principal curvatures). We put

VE, E = w(E,)Ek (2)

and then it is easy to see that

VA = (E1(J1i + ( 1 - L2 )W2(E 2))E1 + (E2( 2) + (1 - Pu2)W12(El))E 2. (3)

On the other hand, Codazzi's equation yields

El(2) = ( 1 - ~P2 )w2(E2 ), (4)

E2(Pl) = (mU - 2)W(El). (5)

Therefore, we get

VA = 2Va. (6)

Now we combine (1) and (2) to obtain the following

Lemma 1. Let x : M - E3 be an isometric immersion of a surface in the3-dimensional Euclidean space, then

AH = 2A(Va) + Va 2 + (Aao + a I A 12 )N.

220

(7)

THERE EXIST NO 2-TYPE SURFACES IN E3

...

3. Two Type Surfaces in E3

Let x: M - E3 be a 2-type surface in E3 , that means x = xp+Xq with Axp = ApXpand Axq = Aqxq. It is known that Ax = -2H and so

1H = -1(Apxp + qq),

1

AH = -- (AP2 + Aqq)

Consequently, we have

AH = bH + cx, (8)

where b = Ap + Aq and 2c = ApAq.Now we use (7) and (8) to obtain the following

Lemma 2. Let x: M - E3 be a 2-type surface in the 3-dimensional Euclideanspace. Then

cx = 2A(Va) + Va 2 + (Aa + a I A 12 -ba)N. (9)

Remark 1. In the whole paper we will assume that c $ 0. Otherwise M must bea circular cylinder, cp. [Ch.1]. Surfaces are assumed to be connected, too.

Let V = {p e Mla(p) 0}. Then, as a consequence of (9), we have

Lemma 3. Let x : M - E3 be a 2-type surface. Then V is dense in M.

Proof. It is clear that V 4 0, otherwise M is minimal in E3 and, consequently, of1-type. Consider the open subset M \ V (V being the closure of V in M). If M \ Vis not empty, then (9) implies cx = 0 on M \ V and coincides with the origin of E3 ,which is impossible.

Finally, by direct computation, we prove the following formula

A(x, N) = 2a + -(A(Va), Va) + 4 I Va 12 + I A 12 (x, N). (10)C c

Furthermore, it is easy to check that

a(x, H) = 22 + b(x, H)+ c z 12. (11)

4. Conformal Changes of Metrics and Willmore Surfaces

Let (N,g) be a Riemannian manifold and p a positive function on N. We define anew metric

g* = p29 (12)

and we say that g* is obtained by a conformal change of the metric on N.Let M be a submanifold of N with dimension n; we will denote by g and g* the

metrics on M induced from and g*, respectively. Also h and h* will denote thesecond fundamental forms associated with both isometric immersions. It is knownthat

h*(x, y) = h(x, y)- g(x, y)UN

221

(13)

for x, y tangent to M, where U = Vp, and UN denotes the normal component of Urestricted to M. An easy computation yields the relation between H and H* (themean curvature vector fields associated with the isometric immersions):

p2 H* = H - UN. (14)

Let f M - N be an isometric immersion of a surface M into a Riemannianmanifold N. We denote by a and R the mean curvature of f and the sectionalcurvature of N with respect to the tangent space of M. Define r(f) by

r(f) = M(a2 + R)dv. (15)

It was proved in [Ch.2] that r(f) is an invariant under conformal changes of the met-ric of N. In [BC.1], r(f) was called the conformal total mean curvature of f. Thevariation of r(f) was calculated in [We]. The critical points of r are called stationarysurfaces. In particular, if N is the 3-dimensional Euclidean space, stationary sur-faces are called Willmore surfaces and they are characterized by the following Eulerequation

Aa = a(I A 12 -2a2). (16)

It is well-known that minimal surfaces in a real space-form are stationary surfaces.Further, the sphere is the only stationary surface in E3 with non-zero constant meancurvature.

5. Surfaces in E3 that are Images of Minimal Surfaces in S3 underStereographic Projection

Let f: M , S3 be an isometric immersion of a surface into the 3-dimensionalunit sphere. Assume that f is minimal and, hence, a stationary surface. Now, wechoose q E S3 \ f(M) and consider E: S3 \ q} E3 to be the stereographicprojection, then x = E f : M - E3 is a Willmore surface. We denote by4): E3 ' ,3 \ {q} the inverse of E and by g = (,) and go the canonical metrics onE3 and S3 , respectively. It is very easy to check that

g* = *go - (1+I 12) 2g (17)

Next we use (12) and (14) with p2Next we use (12) and (14) with p2 =(1+1412 )2 to compute the mean curvature vectorfield H of x: M -- E3 as:

H 4 N, (18)

(1+ I X12) 2

where xN denotes the normal component of the position vector z of M in E3 .

Lemma 4. Let f: M -+ S3 be a minimal surface. If x = E. f : M E3 is of2-type, then

4b = c(l+ I 12)2 + 8(3a2 - 2G), (19)

where G denotes the Gaussian curvature of M.

Proof. First we use (9), (16) and the Gauss equation to obtain

222 M. BARROS

THERE EXIST NO 2-TYPE SURFACES IN E3

...

c(x, N) = a(6a2 - 4G - b). (20)

On the other hand, (18) yields

a(l+ I x 12)2 = -4(x, N). (21)

Now one combines (20) and (21) to obtain

a{c(l+ I x 12)2 + 4(6a2 - 4G - b)} = 0. (22)

This proves that (19) holds on V (the subset of M where a does not vanish). SinceV is dense in M, (19) holds everywhere on M.

Theorem 1. There exist no 2-type surfaces in E3 that are the images of minimalsurfaces in S3 under stereographic projection.

Proof. First, we prove that the zeros of a are isolated in M. Without loss ofgenerality, one can assume that the origin of E3 does not belong to x(M). Supposethat C - (s) is a level curve of a in a - (0). Then the shape operator of M alongC is

(t (s) ( -(s) )We apply A to (21) and obtain

-4(x, N) = Aa(l+ I 12)2 + aA(1+ I x 12)2 - 2(Va, V(1+ I 12)2). (23)

Now we use (16) to see that Aa(s) = 0 along C and so

A(x, N) = (l+[ I2 )(Va, V I 12) (24)

along C. Moreover, one can use (9) to find

VI 12= 4 A(Va) (25)

along C. Therefore

A(x,N) = -(1+ ] X 12 )(A(Va),Va) (26)

along C. We also use (20) to see that (x, N) = 0 along C and so (10) yields

A(x, N) = (A(Va), Va) (27)

along C. As a consequence of (25) and (26) we obtain

(A(Va), Va) = 0 (28)

along C. The last formula allows us to write

/(El(a) 2 - E2(a0)2) = 0 (29)

along C. On the other hand, using (9) and (14), the position vector of C is

cy = 2A(Va). (30)

This proves that /u(s) 0 Vs because c 0, and, moreover, that El(a)(s) $ 0 Vs.So we can assume El(a) = E2 (a) according to formula (28). Consequently, one has

223

M. BARROS

Va = El(a)(Ei + E2 ) (31)

and

A(Va) = pVa' (32)

along C, where Va ± = El(a)(E1 - E2 ). Now, from (29) and (31), we find

2(5) = /(s)Val(s) + /(s)V(s)(Val) (33)

and so

( - (s))Va(s) = i(s)Vj(8)(Va'). (34)

From (30) and (19) we obtain

c2 1 12= 42 Va 2 (35)

and

4b = c(l+ I 12)2 + 16/t 2 (36)

along C. On the other hand, we can apply A to a 2(1+ Z 12)2 = -4 < x, H > andthen use (11) to get

-2 1 Va 12 (1+ I x 12)2 = -4c I x 12 (37)

along C. At this point, one can use (35),(36) and (37) to obtain that p(s), I x 12 (s)and I Va(s) 12 are actually constant along C. This proves that I Val(s) 12 is aconstant on C and so (34) implies

-Val(s) = 0, (38)

which is impossible unless C reduces to a point. Consequently, a has isolated zeros.Let p E M be a point such that a(p) = 0, then one can find a neighbourhood, say

U, of p such that a does not change the sign on U and so Va(p) = 0. Now we use(9) and (16) to get a contradiction. Therefore V = M and then (16) implies that Mmust be a sphere, which is impossible.

6. 2-Type Willmore Surfaces

In this section I will deal with 2-type Willmore surfaces with non-negative Gaussiancurvature. Given an isometric immersion x: M E3 of a surface into the Eu-clidean space, we define a function h : M -- R by h(x) =1 x 12 (the square of thedistance to the origin of E3 ). It is known that the Laplacian and the Hessian of hsatisfy

Ah = -4 - 4(x, H) (39)

and

det(Hess(h)) = 4G(x, N)2 - Ah + 4(x, H). (40)

Let me assume that x: M - E3 is of 2-type, then an easy computation showsthe following:

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THERE EXIST NO 2-TYPE SURFACES IN E3

...

Lemma 5. If z: M , E3 is of 2-type, then

2Vh = 2A(Va) + Va2 (41)

and

4Ah = aAa - 3 Va 12 -Trace(Hess(a) A). (42)

Next we assume that the Gaussian curvature G of M is non-negative to prove:

Lemma 6. Let x: M --- E3 be a 2-type surface with G > O. Then either M is acircular cylinder or h and a2 have the same critical points.

Proof. As usual in this paper, consider c $ 0, that is, M is not a circular cylinder.It is not difficult to see that

(A(Va), Va 2 ) =1 A(Va) 12 +G I Va 12> 0. (43)

If p is a critical point of a 2 , then (42) implies A(Va)(p) = 0 and therefore (41)yields Vh(p) = 0. Conversely, assume Vh(p) = 0 and use (41) to get A(Va)(p) =-a(p)Va(p), which combined with (43), implies -1 I Va 2 12> 0 and so Va 2 (p) = 0.

Theorem 2. There exist no complete 2-type Willmore surfaces in E3 with non-negative Gaussian curvature.

Proof. First we improve Lemma 3. Let x be a 2-type isometric immersion of a surfaceM into E3 with G > 0. Let p E M \ V, then a(p) = 0, G(p) = 0, A(p) = 0 and from(39) we see Ah(p) = -4. On the other hand, from (40) one has det(Hess(h))(p) = 4.Therefore, if M is not a circular cylinder, then Lemma 6 implies that p is a criticalpoint of h and actually a non-degenerate critical point of h. Consequently, the zerosof a (if they exist) are isolated. Next we prove that V = M. In fact, if p E M \ V,then M is not a circular cylinder, and an easy argument proves that Va(p) = 0.Then one uses (42) to get cAh(p) = 0, which is impossible. Hence V = M and onecan suppose (without loss of generality) that a > 0. Finally, if M is a Willmoresurface, we use (16) to obtain Aa > 0. Now, if M is compact, then a is certainlyconstant, otherwise and because M is complete, the theorem of Blanc-Fiala-Huber,(see [Hu]), states that M is parabolic and consequently a is constant. In both casesone obtains that a must be constant. Since circular cylinders are not Willmoresurfaces, we conclude that M is minimal in E3 or it is some sphere in E3 , hence itis of 1-type, and this finishes the proof.

References

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[BC.2] BARROS, M.; CHEN, B.Y.: Spherical submanifolds which are of 2-type via thesecond standard immersion of the sphere. Nagoya Math. J. 108 (1987), 77-91.

[BG] BARROS, M.; GARAY, O.J.: 2-type surfaces in S3. Geom. Dedicata 24 (1987), 329-336.

[BU] BARROS, M.; URBANO, F.: Spectral geometry of minimal surfaces in the sphere.Tbhoku Math. J. 39 (1987), 575-588.

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226 M. BARROS

[Ch.1] CHEN, B.Y.: Null 2-type surfaces in E3 are circular cylinders. Kodai Math. J. 11(1988), 295-299.

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[KO] KLOTZ, T.; OSSERMANN, R.: Complete surfaces in E3 with constant mean curva-ture. Comment. Math. Helv. 41 (1966-67), 313-318.

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[Ro.1] Ros, A.: On spectral geometry of Kaehler submanifolds. J. Math. Soc. Japan 36(1984), 433-448.

[Ro.2] Ros, A.: Eigenvalue inequalities for minimal submanifolds and P-manifolds. Math.Z. 187 (1984), 393-404.

[Ta] TAKAHASHI, T.. Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan18 (1966), 380-385.

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MANUEL BARROSDepartamento de Geometria y TopologiaUniversidad de Granada18071 Granada, Spain.

(Received April 10, 1991)