M A N U E L B A R R O S AND OSCAR J. GARAY

2 - T Y P E S U R F A C E S I N S 3

ABSTRACT. It is proved that the Riemannian product of two plane circles of different radii are the only compact surfaces of the 3-spheres in R 4 which can be constructed in R '~ by using eigenfunctions associated with two different eigenvalues of their corresponding Laplacians.

0. I N T R O D U C T I O N

Let us consider the flat torus Tab = RZ/A, where A is the lattice generated

by (2za,0) and (0,27zb). Then Tab is isometric to the Riemannian product of

two plane circles: Tab=Sl (a )× Sl(b). One can define, as usual, the isometric imbedding xab of Tab in R 4 by:

s . s t ~ t , a s l n , b c o s bsin xab(s , t )= aCOSa a b ' "

It is clear that Xab(Tab) is contained in S 3 ( r ) , the 3-sphere in R 4 with its center in the origin of R 4 and radius r = (a 2 + b2) 1/2. Furthermore, it is

contructed in R 4 by using, at most, two different eigenvalues of the Laplacian of Tab. Namely: if a = b, then xa, is constructed in R 4 from only

one eigenspace which corresponds with the minimality of Taa (the Clifford

torus) in S 3 (r) (cf. [9]), otherwise Xab is defined by two different eigenspaces

and, in this sense, we shall say it is of 2-type.

A well-known result of Lawson [63 says that Taa is the only compact non-totally geodesic minimal surface in a 3-sphere of R 4 with constant

Gaussian curvature (and thus the only one that can be constructed in R 4 by

using exactly one eigenspace). Our first approach proves that Tab (a # b) is

the only compact surface in S 3 with constant Gaussian curvature which can be constructed in R 4 by using two different eigenspaces of its Laplacian.

However, the study of 2-type surfaces is S 3 c R 4 is quite different to that for

minimal surfaces. In fact, while one can find minimal surfaces in S 3 for any

genus, this does not occur when we consider two different eigenspaces. In a

second approach we shall see that only one genus is allowed.

A well-known hypothesis due to Lawson (see [I0]) says that the only. torus minimally imbedded into S 3 is the Clifford torus. Therefore, our last

approach (Theorem 2) can be considered to be the answer to the corres- ponding question for surfaces constructed with two different eigenspaces.

Surfaces in this paper are assumed to be compact, connected and imbedded unless otherwise stated.

Geometriae Dedicata 24 (1987), 329-336. © 1987 by D. Reidel Publishing Company.

330 MANUEL BARROS AND OSCAR J. GARAY

1. S O M E BASIC P R E L I M I N A R I E S

Let M be a compact Riemannian surface and A the Laplacian of M acting on differentiable functions in C ~(M). The strongly elliptic operator A has an

infinite sequence of eigenvalues: 0 = 2 o < 2 ~ <22 < ' " < 2 k < ' " 1 " ~ , the spectrum of M. For each /~k, its associated eigenspace, Vk, is finite dimensional. On Coo(M), one defines, as usual, the inner product

( f ,g) = SMf9 dV, dV being the canonical volume element on M. Then the decomposition Z,>~0V~, is orthogonal and dense in Coo(M) (in LZ-sense). Therefore, for each f ~ C OO (M), one can consider its spectral decomposition:

f = Z,>~0f~; Aft = 2t f,, which is convergent in an U-sense. This construction can be extended to R"-valued differentiable functions

on M. In particular, if x : M ~ R " is an isometric immersion of M into a Euclidean m-space, one can talk about its spectral decomposition: x = Xo + E~>~l x,; Ax, = 2tx, (in an LZ-sense), where x o is nothing but the center of mass of M in R" (see [4], for instance, for further details). If the last decomposition consists of only a finite number of non-zero terms, M is

said to be finite. It is of k-type if there are exactly k non-zero xt's in the decomposition (except Xo; cf. [4]).

In terms of finite-type submanifolds, a well-known theorem of Takahashi [9] says that a 1-type submanifold is nothing but a minimal submanifold in some hypersphere of R m. In particular, a 1-type surface is a minimal surface of a hypersphere of R". Furthermore, it is always mass-symmetric; i.e. its center of mass in R m coincides with the center of the hypersphere. So

if the center of the hypersphere is chosen to be the origin of R", then the position vector of such a surface in R" takes the form: x = Xp; Axp = 2pXp.

Many important submanifolds are known to be of 2-type (cf. [1], [2], [4], [8]), however spherical submanifolds of 2-type are not, in general, mass-symmetric (cf. [2]).

Let M be a 2-type surface in R 4, then its position vector takes the form:

X = X 0 "{- Xp -]- Xq; Axp = 2pXp, Axq ~--- ;LqXq, ~p <,~q,

and Xo is a constant vector in R ¢, the center of mass of M in R 4. If H denotes its mean curvature vector, it is well-known that kx = - 2 H , and so one automatically obtains:

(1) A H = bH + c(x - Xo),

where b = )~p + 2q and c = ½2p).q.

Now, let M be a surface in a 3-sphere S 3 ~ R 4. We shall assume, without loss of generality, it is the unit 3-sphere with center the origin of R 4. Its mean curvature vector is H' = H + x. Denote by A ' ,D ' , and a', the

2 - T Y P E S U R F A C E S IN S 3 331

Weingarten map, the normal connection, and the second fundamental form of M in S 3, respectively, and by A, D,.a, the same geometric elements of M in R 4. It is clear that A¢ = A~, for ~ normal to M in S 3.

One can compute H in terms of elements associated with the immersion of M in S 3. So, by using an orthonormal basis {El, E2 } tangent to M and such that VEE~ = 0 at the corresponding point of M, V being the Levi- Civita connection of M, one has:

2 2

A H = A H ' + 2H = - ~ E ,E ,H ' + 2H = Z (VE, An')E, i = l i=1

2 2

+ Z AD'~,n'EI + Z a'(Ei, AH, El) i=1 i=1

2 2

-- ~, < E i , A 1 r E i > x - ~ D' D ' rt' i=1 i = l

and so we get

(2) AH = (AH)r+ AD'H' + {IA[ 2 + 2}H' - 2{1 + (~')Z}x,

where (AH) r is the tangential component of AH; A D' denotes the Laplacian associated with D'; and a' is the mean curvature of M in S 3.

2 . T H E SET OF R E G U L A R P O I N T S OF T H E M E A N C U R V A T U R E

We shall start this section by computing a nice formula for (AH) T In order to do this, we consider an orthonormal basis {El, E2 } tangent to M, such that A E i = piEi(A = A~ and ~ the unit normal vector to M in $3), and so A n,Ei = o~t[Ai E i .

We put V e E j = E~=lco~(Ei)Ek and so

(3) (Ve, AH,)E s = Ei(~')I~jE j + o( Ei(,uj)E j 2

+ % - (e , )Ek. k = l

Furthermore, it is clear that

(4) ~'(VE, A)Ee = (Ve, AH,)E2 - E,(7')p2Ez

~'(VE2A)E, = (V E2 A H,)E1 - E2 (o( )p l E l .

Therefore, by combining (3) and (4) with Codazzi's equation, we have

(5) ~ ' E 2 ( # l ) = 0((/21 - / 1 2 ) ( D 1 2 ( E 1 )

c(EI (P2) = 0~ ' (~1 - ] J2 ) (D2 (E2)'

332 M A N U E L B A R R O S A N D O S C A R J. G A R A Y

R e m e m b e r that

2 2

(AH) ~= 2 (VE, A.,)e, + y. A~,~..,e,; i=1 i=1

and so by using (3) and (5), we finally obtain

(6) (AH) r = 2 Tr AD,u, + V(~') 2

where Tr AD, z, = E2=, AD,~,Ei , and V(~') 2 is the gradient of (~,)2.

We define a function f: M ~ R by f (x ) = (x , x0 ), where ( , ) denotes the Euclidean inner p roduc t on R 4. Then we use (1) and (2) to get

( A l l , x ) = - 2 { 1 + (e,)2 } = c - b - c f

and so

(7) c f = 2(~X') 2 + 2 + c - b .

This formula implies that the following condit ions are equivalent: (i) M has a cons tant mean curvature (in either S 3 or R 4);

(ii) Xo = 0 and so M is mass-symmetr ic in S 3.

Now, for any vector field X tangent to M, we have f rom (1) and (2):

(AH, X ) = <(AH) r , X ) = - C ( X o , X )

therefore, we use (6) and (7) to obtain

(8) (AH) r = - c V f = - 2 V ( ~ ' ) 2

and 3 , z

(9) T r A D H , - ~V(e ) .

Fo rmu la (a) gives

(10) (/q + 3 ~ ' ) E l ( c t ' ) = 0 and (/~2 + 3 ~ ' ) E 2 ( 7 ' ) = 0

which implies t ha t V(~') 2 is an eigenvector of the shape opera to r A, and its

eigenvalue is - 3 ~ ' (which obviously occurs at any point of M). Define P = {pE M / V ( ~ ' ) 2 = 0}, the set of critical points of (c() 2. The

following result describes the subset P of M:

T H E O R E M 1. Let M be a 2-type surface in S 3 c R 4. Then V = M - P is

empty or is dense in M.

P r o o f If c( is not constant on M, then V is a non-empty open subset of M. Let p b.e any point of V, then f rom (10) we have

#1 + 3c( = 0 or /*2 + 3~' = 0 (but not simultaneously).

2-TYPE SURFACES IN S 3 333

Consequent ly, the Gauss ian curvature of M on V is given by

(11) G = 1 - 15(~') 2.

Of course, this formula also holds on the closure ~" of V in M. If ~" ~ M, let

U be any c o m p o n e n t of M , l?. Because (c() 2 is constant on U, so is f, which proves that U is contained in a hyperplane of R 4 perpendicular to Xo.

Therefore, G on U (and on [7) is given by

(12) G = 1 + (c() 2.

Since V ~ 0 is not empty, one has c( = 0 on it and also on/.7. This proves

that U is contained in a hyperplane of R 4 perpendicular to x o and passing

through the origin of R 4. S o f vanishes identically on U, and x o is parallel

to ~ at any point of U. N o w from (1) and (2) one has

(13) ( A H , ¢ ) = A 0 ( + {IAI 2 + 2 } ~ ' = b e ( - C ( X o , ~ )

which gives c(xo, ~5 = 0 on U, which is impossible as c ¢ 0.

R E M A R K S ; (a) If M has a constant Gauss ian curvature G, then f rom

Theorem 1 we obtain that V is empty and thus M has a constant (non-zero)

mean curvature. Now, because xo = 0, formula (13) implies I A I 2 is constant, and we can use (5) to get 6o 2 = 0. Since M is flat it is not difficult to deduce

that M is the Riemannian product of two plane circles of different radii (see,

for instance, [3]).

(b) The last fact can be considered as a na tura l extension of a result due

to Lawson [6], which stated that the Clifford torus is the only non-tota l ly

geodesic minimal surface with constant Gauss ian curvature in S 3 c R 4.

(c) Define C = {p ~ M/c((p)= 0}, the set of zeros of c(. Certainly C is

contained in P. One can prove that C consists only of a finite number of

points. T o do this, one only needs to see that if a critical point p c C o f f is

degenerate, then f >~ 0 which can be proved to be impossible by using a

simple a rgument on the min imum of./, unless x o was the origin of R 4. (d) Now we can use (5) and (10) to get the following information: Fo r any

point of V we can choose locally {El,/~2} such that/~1 = 5c(; Pz = - 3 c ( ;

E 1(c() = 0 and

(14) ~ 'cof(E, ) = 5 , gE2(~ ); ~ ' 6 o 2 ( E 2 ) = O.

Therefore we can take differentiation in (14) to get

(15) G = - ~ A log(c() 2

and this formula holds in M except at most in a finite number of points. Consequently, f rom the G a u s s - B o n n e t theorem, one can conclude that the

334 MANUEL BARROS AND OSCAR J. GARAY

Eu le r n u m b e r of M is non -pos i t i ve . So the s i t ua t i on is qu i t e different to the

m i n i m a l case (cf. L a w s o n [7-1). Ac tua l ly , one can use some f o r m u l a s in the

nex t sec t ion to see t ha t C is indeed e m p t y and so M is a t o p o l o g i c a l t o m s .

Therefore , o u r m a i n resul t in the next sec t ion can be c o n s i d e r e d to be

a n a l o g o u s to the resul t expec ted in L a w s o n ' s w e l l - k n o w n open p r o b l e m for

m i n i m a l t o r u s in S 3 c2 R 4 (see, for ins tance, [10]).

3. T H E MAIN RESULT

Since M is a 2 - type surface in R 4, we can use (1), (2), and (8) to write:

(16) c x o = 2V(o() 2 + [b~' - {IAI 2 + 2}c( - Ac( ]¢

+ [2{1 + (a') 2 } + c - b]x .

F o r any p o i n t on V, we can c h o o s e loca l ly { E l , E2 } such t h a t

(17) (5O 0) 01) A = - 3 ~ ' ; Ax =

~'~o~ (E~) = ~E~ (¢) ; , 2 0{ O 1 (E2) = 0

V(~X') 2 = E2(cx')2E2; E1 ( c ( ) = 0

VEE1 = (.02(E1)E2;Vel E 2 = - 0 9 2 (E 1 )E 1 ; Ve2E1 = Ve2E2 = O.

There fore , we have

0 = E 1 (CXo) = 2E 1E2(~') 2 E 2 - ~ I W ' I 2

+ E l {b~' - (IAI 2 + 2 ) ¢ - A~'}~

- { 5 b ( ¢ ) 2 - 5 1 A I 2 ( ~ ' ) 2 + 10(c() / -~ 5c (Ac( }E1

+ {2(~'): + 2 + c - b}E1,

C o n s e q u e n t l y , we o b t a i n

(18) E1 E2 (~,)2 = 0

(19) El{be'-(IAI 2 + 2 ) a ' - A ~ ' } = 0

(20) -~ - ]Va ' [ 2 - 5b(c() 2 + 12(c() 2 + 5a 'Ac( + 2 + c - b

+ 51AI 2 (~,)2 = 0.

Simi la r ly , f rom E2 (cxo) = 0, we can get

(21) - -6o(Ei(~x ' ) 2 + E 2 { b o ( -- ([AI 2 + 2 ) c ( - A T ' } = 0

(22) 2 E 2 E 2 ( ~ ' ) 2 + 3b(~') 2 - 31A12(~') 2 -- 4(0() 2

- 3 ~ ' A ~ ' + 2 + c - b = 0 .

2-TYPE SURFACES IN S 3 3 3 5

Now, by using (17) it is not difficult to see that

(23) 2 E 2 E 2 (~,)2 = ~ ]V~ ' [ 2 . 2A(~ , )2 .

which together with a'Aa' = ½A(c~') z + [V~'] z allow us to rewrite (20) and (22) as follows:

( 2 4 ) -2S-[VT'j 2 - - 5 A ( g ' ) 2 + 5b (~ ' ) 2 - 5 1 A [ 2 ( ~ ' ) 2

- 1 2 ( ~ ' ) 2 + b - c - 2 = 0 .

(25 ) -½[V~'[ 2 - ~A(~') 2 + 3b(cr') 2 - 3[a[ 2 (C¢') 2

-4(~') 2 - b + c + 2 = 0 .

Finally, (24) and (25) give:

(26) 51V~'[ 2 - 10A(~') 2 + 16(~') z + 8 ( c - b +2) =0 .

This formula holds on V, but actually it holds anywhere because of Theorem 1.

Now we have:

THEOREM 2. The Riemannian product of two plane circles of different and suitable radii are the only 2-type surfaces in S 3 c R 4.

Proof. First we use (7) to obtain

2 f A 4 ( ~ ' ) 2 d V + ( c - b + 2 ) v o l ( M ) = c I , jvfdV (27)

but now

(28)

because

fM fdV = f~a (X0, X) dV = [x o [2 vo1 (M)

(Xo,X)= fM(XO,Xv)dV = f (Xo,xq)dV =(xo,Xq)=O.

Therefore, from (26), (27), and (28) we have

5fM[V~'lz dV + 8ClXo 12 vol (M) = 0,

which gives x o = 0, and as e' is constant, M is the Riemannian product of two plane circles of suitable different radii.

REMARK. Theorem 2 has been proved by B. Y. Chen under the additional assumption that the surface was mass-symmetric.

336 M A N U E L B A R R O S AND O S C A R J. G A R A Y

R E F E R E N C E S

1. Barros, M. and Chen, B. Y., 'Stationary 2-Type Surfaces in a Hypersphere'. To appear in J. Math. Soc. Japan.

2. Barros, M. and Chen, B. Y., 'Spherical Submanifolds which are of 2-type via the Second Standard Imbedding of the Sphere'. To appear in Nagoya Math. J.

3. Chen, B. Y., 'A Characterization o1 the Standard Flat Tori'. Proc. Amer. Math. Soc. 37 (1973), 564-567.

4. Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Math. Vol. 1. World Scientific, 1984.

5. Hoffmann, D. A., 'Surfaces in Constant Curvature Manifolds with Parallel Mean Curvature Vector Field', Bull. Amer. Math. Soe. 78 (1972), 247-250.

6. Lawson, H. B., 'Local Rigidity Theorems for Minimal Hypersurfaces', Ann. Math. 89 (1969), 187-197.

7. Lawson, H. B., 'Complete Minimal Surfaces in S 3', Ann. Math. 92 (1970), 335-374. 8. Ros. A., 'On Spectral Geometry of Kaehler Submanifolds', J. Math. Soc. Japan 36 (1984),

433-448. 9. Takahashi, T., 'Minimal Immersions of Riemannian Manifolds', J. Math. Soc. Japan 18

(1966), 380-385. 10. Yau, S. T., 'Problem Section. Seminar on Differential Geometry', Ann. Math. Studies, vol.

102, Princeton Univ. Press. 1982.

(Received. August 27. 1986)

A u t h o r s ' address:

M a n u e l B a r r o s and O s c a r J. G a r a y ,

D e p a r t a m e n t o de G e o m e t r i a y T o p o l o g i a ,

U n i v e r s i d a d de G r a n a d a ,

18071-Granada ,

Spain.