Foliations by hypersurfaces with constant mean curvature

Math. Z. 207, 97-108 (1991)

Mathematische

Zeitschrift

Foliations by hypersurfaces

with constant mean curvature*

J.L.M. Barbosa 1,**, K. Kenmotsu 2'*** and G. Oshikiri 2"*** 1 Universidale Federal Ceara, Campus do Pici, Fortaleza-Ce, Brazil

2 Department of Mathematics, College of General Education, Tohoku University, Kawauchi, Sendai 980, Japan

Received June 23, 1989; in final form May 24, 1990

w 1 Introduction

In this work we study a codimension-one C3-foliation ~ of a complete Rieman- nian manifold M whose leaves have constant mean curvature. We show that, when M is compact with nonnegative Ricci curvature, all the leaves of ~,~ must be totally geodesic and Ricci curvature vanishes in the direction normal to the leaves. It then follows from a theorem of Oshikiri [O 1] that M is locally a Riemannian product of a leaf by a normal curve. When M is flat and noncom- pact we prove that, if all the leaves of ~- have the same constant mean curvature H then f " is a minimal foliation. Recently Barbosa, Gomes and Silveira [BGS] and also Meeks [ M ] have studied the case M = R 3 and, under the same set of hypothesis, have that the leaves are totally geodesic. Such result is not true in general as one can readily see, by considering a foliation of R", n > 8 defined as follows: take a smooth function f: R " - - , R that is a counter-example to the Bernstein conjecture and consider the foliation of R "+1 whose leaves are the graphs o f f + c , where c is any real number. We have also considered the case in which M has constant negative curvature, say a. We obtained that, if all the leaves of ~- have the same constant mean curvature H, and I H l > ] / ~ a , then [HI = ] S ~ a . One example of such foliation is the foliation of the hyperbolic space by horospheres.

w 2 Preliminaries

Let M be a ( n + 1)-dimensional orientable Riemannian manifold and ~ be a codimension one C3-foliation on M. ( . , . ) will represent the metric on M. Given a point p of M we may always choose an orthonormal frame field {e~, ..., e,+a} defined in a neighborhood of p such that the vectors e 1 . . . . , e,

* Dedicated to Professor Ichiro Satake on his 60th birthday ** Partially supported by FINEP (BRAZIL)

are t a n g e n t to the leaves of o~ and e,+ 1 is n o r m a l to them. We refer to such frame as an adapted frame field. W e shall m a k e use of the following c o n v e n t i o n on the range of indices:

I < A , B , C . . . . < n + l , l < i , j , k . . . . <n,

and we shall agree that repeated indices are s u m m e d over the respective ranges. F o r a given frame field, let {wl . . . w,+ 1} represent its dual frame. T h e structure equations on M are given by

(2.1) d W A = ~ W ~ ^ W n A , WAR+WBA=O

(2.2) d Wan = ~ W AC /X WCB + f2AB

(2.3) ~'2AB = - - 8 9 W X A WD, R A B C D + R A B D c = O .

T h e Ricci c u r v a t u r e in the direction of e, + 1 is simply

(2.4) Ric(e, + 1) = ~ Ri, + 1 i, + 1.

Let D be the c o v a r i a n t differentiation on M. T h e n we have

(2.5) O eA = ~, Wan es

a n d D veA just m e a n s DeA(V). C o n s i d e r that the chosen frame field is an a d a p t e d one. Let 0a and OAB represent the restriction of the forms w a and WAn to vector fields t a n g e n t to the leaves of ~-. It is then clear that

On + 1 = O, 0 i = Wi,

(2.6)

a n d that

(2.7) 0,+1i = - ~,hi~ Oj.

It follows f r o m (2.1) and (2.6) that

~ 0 . + 11 ^ O i = O

and so, b y Cartari's l e m m a ,

(2.8) hit = hji.

T h e second f u n d a m e n t a l f o r m of the leaves is then given by

(2.9) B = Y',hij Oi Oj

and its n o r m by

(2.10) II B II 2 = ~, h2s"

i , j = l

T h e m e a n c u r v a t u r e function is

Foliations by hypersurfaces with constant mean curvature 99

We observe that the sign of H depends on the choice of e,+ ~. In fact, the vector field defined locally by H G + a is globally defined on each leaf of ~-. As a consequence o f this, if H 4:0 at each point of a leaf, then the leaf is orientable. If N is a unit vector field n o r m a l to a leaf of J~ in some o p e n set, we m a y c h o o s e the a d a p t e d frame field in such w a y that N = e , + a . T h e resulting m e a n c u r v a t u r e for the leaf will be referred as the m e a n c u r v a t u r e ~in the direction of N.

The divergence of a vector field V o n M is locally defined as

(2.12) div (V) = ~ (DeA V, ea).

F o r vector fields that are also tangent to the leaves of ~- we m a y also c o m p u t e their divergence a l o n g the leaves as

(2.13) div L (V) = ~ (De, V, ei).

2.14 Proposition. L e t ~ be a codimension one ca-foliation o f a Riemannian mani- fold M and let N be a unit vector field normal to the leaves o f ~ in some open

set U o f M. Set X = DN N. Then, on U we have

(2.15) div N = - n H

(2.16) divL(X ) = - n N [ H I + [IB [I 2 + R i c ( g ) + IX[ 2

(2.17) div X = div L X - [X[ 2

where H is the mean curvature in the direction o f N.

Proof. C h o o s e an a d a p t e d frame field for a n e i g h b o r h o o d of a point such that e,+ 1 = N . F r o m (2.7) it follows that

(2.18)

where

(2.19)

W.+ li~-" - ~ h i j O j + xi w.+ 1

xi = w. + x i(G + 1) = ( D . . . . e, + 1, ei) = ( X , el).

Consequently, we have

Den+ ~ = - ~ hij Ojei + ~ xi w,+ l ei

(2.20)

a n d so,

div (N) = div (e. + 1 ) = ~ (De., e. + 1, eA ) = ~ (De, e . + l , ei) = - 2 h i i = - n i l .

This proves (2.15). T o prove (2.17) just observe that, if V i s a vector field o n M tangent to the leaves of o~, then

div V = ~ ( D e , V, ei) + ( D . . . . V, e , + l ) =diVL V - ( V , D . . . . e.+ l ) .

We will represent the exterior differentiation on a leaf L of ~ by dL. If f i s a function defined in some open set of M, where we have defined an adapted

frame field as above, then we have

(2.21) d f = d c f + e , . 1 [ f ] w. +1.

Using this, it follows from (2.18) that

dw.+ li = ~ ( d c

hlj +

§ Wn+I § XiWn+lj^ Wj

--- ~ {e. + a [hlk] + ~, hij Wjk (e, + 1)-- ~ hij hjk + d xl (ek)-- Xi Xk} Wk ^ W. + 1 + terms in Wk ^ wl. On thc other hand, from (2.2), we obtain

d W . + l i = ~ W . + 1 j A W j i - - ~ R , + I , , + l k W . + I ^ W k -- 89 lik, A W,

= ~, { -- hjk wji(e. +1) -- ~ x j Wji(ek) + R , +1i, +1 k} Wk ^ W. +1 + terms in Wk ^ Wt.

Comparing the above two expressions for dw.+ l iwe get (2.22) ( d x i + ~ x j w j i ) ( e k ) = X l X k + ~ h l j h j k + R , + l i , + l k

-

(d h~k + y' hjk wj~ + y~ h~j wjk)(e. + 1).

Observe that

(2.23) dxi(ek) =dL xi(ek),

and that

(2.24) (De, X , el) = (dL xl + ~, x j wji)(ei).

It then follows from (2.22) that

i = 1 i,j=l i = 1

- e . + l

h. = l X l a +

i

Therefore (2.16) is proved.

We observe that, for minimal foliations, (2.15) was proved by Oshikiri [O 1] and (2.16) by Oshikiri [ 0 2 ] .

We shall remark that, although N may be only locally defined, the vector field X = D N N and the function N [ H ] are globally defined on M. Therefore formula (2.16) is valid throughout M. Because X is globally defined, it is natural to consider its dual form

Foliations by hypersurfaces with constant mean curvature 101

Since X is tangent to the leaves of J~, then 0 is a 1-form that, at each point p of M, belongs to the cotangent space of the leaf of o~ containing p.

2.26 Proposition. Let ~ be a codimension-one C3-foliation of a Riemannian mani- fold M. Let X be the vector field defined locally by DNN, where N is any local unit vector field normal to the leaves of ~ . Let 0 be its dual form. I f f: M ~ R is any smooth function which is constant along the leaves of o~ then

(2.27) dL N [ f ] = N [ f ] 0,

along any leaf L o f f .

Proof Take an adapted frame field, in some neighborhood of a point p of M, for which e,+ 1 = N . For a function f: M ~ R that is constant along the leaves of ~ we have

(2.28) el [N [ f ] ] = ei [ N [ f ] ] - N [el [ f ] ] = [el, N] I f ] . Observe that

(2.29) [ei, N] =De, N - - D u e i = ~ w , + tj(ei) ej--~Wik(N) ek--Wi,+l (N)N. F r o m (2.28), (2.29) and (2.18) it now follows that

(2.30) Therefore

el I N [ f ] ] = -- w,, +, (N) N [ f ] = xl N [ f ] .

dL N I-f] = N I f ] ~ xi wi = N I f ] O. This proves the proposition.

2.31 Proposition. Let ~ be a codimension-one ca-foliation of a connected Rieman- nian manifold M and let f: M ~ R be a continuous function which is constant along the leaves of J~. l f f is not constant on M then the set A = { x ~ M ; f ( x )

= m a x , ( f (x))} contains at least one compact leaf.

Recall that a codimension-one foliation ~ of a Riemannian manifold M is transversely orientable if we may choose a continuous unit vector field N, defined on M, that is normal to the leaves of Y . If the leaves of ~ are also orientable then we m a y choose the orientation of each leaf of ~ in such way that, for any positively oriented local frame field { e l , ..., e,, e,+l = N } on M, {el . . . e,} is positively oriented on the leaves of ~ . The orientation on each leaf, in such case, can be represented by a "volume element", that is the restric- tion to each leaf of the n-form r defined by

(2.32) r . . . x . ) = < x , ^ ... ^ X , , N > .

For a positively oriented adapted frame field { e I . . . e,, e , + l = N } defined in some open set of M we readily find the following local expression for ~0

(2.33) q~=w 1 A ... A w,.

Observe that

(2.34) O = w l /x ... A W , + I

is a local expression for the volume element of M. The following proposition was proved by Rummler [-Rm].

2.35 Proposition. L e t o ~ be a codimension-one C3-foliation o f a Riemannian mani- fold M . A s s u m e that ~ is orientable and transversely orientable. Then

(2.36) d ~o = ( - 1)" + 1 n H O .

P r o o f We will use the local expressions of ~0 and 9 given above. Making use

of (2.1) we obtain

d q ~ = ~ , ( - 1 ) i + l w x A ... A W i - 1 A w l , + , A w , + l/',Wi+l/', ... ^ w . .

Using (2.18) and moving w,+l to the last position in each term we rewrite the above expression as

d q ~ = ( - 1 ) " + l ~ hu wl /x ... A w,+ l =(--1)"+ l n H O .

Therefore the proposition is proved.

w 3 The main results

Foliations by hypersurfaces with constant mean curvature 103

3.1 Theorem. Let M be a compact Riemannian manifold with nonnegative Ricci curvature and o~, a codimension-one C3-foliation of M whose leaves have constant mean curvature. Then any leaf of o~ is a totally goedesic submanifold of M. Further- more M is locally a Riemannian product of a leaf of ~ and a normal curve, and the Ricci curvature in the direction normal to the leaves is zero.

Proof As we have observed we m a y assume that ~- is transversely orientable. Since the m e a n curvature function H : M - - , R , that associates to each point the value of the mean curvature of the leaf of ~- through that point, is constant on the leaves then, from proposition (2.31), either H is constant on M or there exists a compact leaf L of.~- having the property that

(3.2) HL = maxM H(p).

Assume first that H is nonconstant in M. This implies that N [ H ] = O along L. It then follows from (2.16) that

(3.3) div L X = Ir B [I 2 +

[x[2

Since L is compact and Ric(N)>-0, the divergence theorem, applied to (3.3) yields

(3.4) rlBIl=0, I S l = 0 and R i e ( N ) = 0

along L. Therefore L is totally geodesic and, in particular, HL=O. F r o m (3.2) we conclude that H < 0 . By considering the function - H the same reasoning applies and now the conclusion is H > 0. Therefore H - 0 . Contradiction. Hence H is constant along M. N o w N ( H ) - O and follows from (2.16) and (2.17) that

div (X) = rl B II 2 + Ric(N).

Integration over M yields II B ]l = 0 and R i c ( N ) = 0. The result now follows f r o m a theorem of Oshikiri [O 1].

3.5 Corollary. There is no codimension-one C3-foliation of the Euclidean sphere S"(1) whose leaves have constant mean curvature.

This corollary fully generalizes the compact case of theorem (3.13) of [BGS]. The following corollary generalizes proposition (2.36) of K a m b e r and T o n d e u r [ K T ] .

3.6 Corollary. Let M be a compact oriented fiat Riemannian manifold and ~ , a codimension-one C3-foliation of M whose leaves have constant mean curvature. Then ~ is induced from a hyperplane Joliation on the universal covering of M.

3.7 Proposition. Let M be a Riemannian manifold with positive Ricci curvature. A n y codimension-one C3-foliation of M whose leaves have the same constant mean curvature can not have a compact leaf

This proposition generalizes theorem (3.12) of [BGS].

In what follows Q"(a) will represent a n-dimensional complete Riemannian manifold with constant sectional curvature a.

3.8 Theorem. Let ~ be a codimensional-one Ca-foliation of Qn+l(a) such that each leaf L has constant mean curvature H L. Assume a<O and IHLF>~--a.

Then infl HLI = ]fl-~.

Proof First of all observe that there is no loss of generality in assuming that

Q"+l(a) is simply connected. If not, we consider its universal covering space

and, using the covering projection, we m a y define on it a foliation that satisfies the same set of hypothesis as ~ . It follows that Q" + 1 (a) is the (n + 1)-dimensional Euclidean space E" § ~ (a = 0) or the (n + 1)-dimensional hyperbolic space ~ " § 1 (a) of curvature a < 0 . The foliation ~ will be then orientable and transversely orientable. If there is a leaf L for which H L = ~ - - ~ then there is nothing to be proved. So, we will assume IHLI > ] f Z ~ . We m a y then choose a unit vector field N, normal to the leaves, such that the m e a n curvature HL, c o m p u t e d in the direction of N, satisfies

(3.9) ( - 1)" +1HL > l / _ a.

Set c = inf[HLI. If c = ~ we are done. So, we assume that c > ] / ~ d . According to (2.35) the volume element r of each leaf and the volume element 9 of Q"+l(a)

are related by formula (2.36). Let BR represent a ball of radius R on Q"+l(a).

We then have the following estimation for the volume of B R :

(3.10) vol(BR)= ~ ~ =

BR BR

( __ 1),,+ 1 1

nH dq~=nc aJ R ~o.

Observe that, for proving the above inequality, we have first used (3.9) to guaran- tee that ( - - 1 ) " § and then concluded that (-1)"+lH>inflHL]=C. As before H means the function whose value on each leaf L is HL.

Let w reprent the volume element of ~?BR. Let {X1 . . . X,} be local ortho- n o r m a l frame field tangent to t?BR such that w(X1 .... , X , ) = I . Since

IX1 A ... ^ X , t = I and using (2.32) we obtain q~(X1, ..., X , ) = ( X t A ... A X , , N ) < 1. Therefore

(3.11) ~p<w.

F r o m (3.10) and (3.11) it follows that

(3.12) nc < v~

-- vol(BR )

When a = 0 we obtain from this inequality that

(3.13) 0 < c < ( n + 1)/nR.

Foliations by hypersurfaces with constant mean curvature 105

W h e n a < 0, we have

(3.14)

R

vol(BR) = ~ Vo S".(t) d t

0

vol(SBR) = V o S".(R)

where Vo=vol(S"(1)) and Sa(t) 1 s i n h ( l / ~ t ) . Set C ~ ( t ) = c o s h ( V ~ - a t ) . We then have S'~ (t) = C. (t) and = ~ - - a

R

(3.15) v o l ( S B . ) = ~ n V o S ~ - ' ( t ) C.(t) dt.

0

Substitution of (3.13) and (3.14) in (3.11) yields

(3.16)

R

I I70 S~ - 1 (t) C a (t) d t

c<= ~ _R

V o S~(t) d t

0

T a k i n g the limit of the right h a n d side of this inequality when R goes to oo and m a k i n g use of L ' H o s p i t a l rule one obtains

(3.17) c_< lira C , ( R ) _{= V - a "} , / ~

Since c was a s s u m e d to be larger t h a n / - a , we have reached a contradiction. This shows that c m u s t be ] ~ - - a and completes the p r o o f of the theorem.

3.18 Corollary. I f all leaves o f a codimension-one C3-foliation o f a complete and f i a t manifold M have the same constant mean curvature then its leaves are minimal

submanifolds o f M .

3.19 Corollary. I f all leaves o f a codimension-one C3-foliation o f Q"(a), a<O, have the same constant mean curvature H and [H I -> ~ - a , then [ H I - V ~ .

w 4 Appendix

Let z=F(xl, ..., xn) be a real valued C3-function defined on the disk D={(x:, ..., x,)~R"; x~ + ... +x.2=<r2}. Suppose that the mean curvature H of the non-parametric hypersurface of R" + ~ defined by F satisfies

(4.1) [Hl~c>0.

Then, Heinz-Chern inequality ([H], [C]) is

(4.2) c r < 1.

Consider the foliation ~- of D x R C R "+1, whose leaves are the translations of the graph of F in the z-direction. Let N be the unit normal vector field to the leaves of ~ such that the m e a n curvature of each leaf computed in the direction of N satisfies:

(4.3) (-- 1 ) " + I H > 0 .

This is possible since we are assuming (4.1). According to (2.35) the volume element q~ of each leaf and the volume element 4~ of R" +1 are related by formula (2.36). We then have (4.4) 2 a r " v ~ = S cb< l~ ~ q~.

n _{D • [ - a,a]} n c _{d(D • [ - a,a])}

By the same reasoning used to prove (3.10) we obtain

(4.5) ~ q~ < vol(~](D x [ - a , a])).

O(D x [ - a , a ] )

F"

Since vol(O(D x [ - a , a ] ) ) = 2 a r n- 1 vol(S"- x(1))+ 2 - - vol(S"- 1(1)) we obtain fiom (4.4) and (4.5)

n

that

r

(4.6) c r < 1 + - - .

a n

Making a grow to ~ we conclude that

cr<=l.

This proves (4.2). We observe that if F is defined in the open disk then the above reasoning shows that, for any r ' < r we have c r < 1. Therefore c r < 1 also in this case. We observe that, if we start with an n-dimensional rectangle of sides 211 . . . 2l, the same proof will show that

(4.7) c < l ( / ~ + ... +~,).

Making ( n - k) of the lj go to oo we conclude that

4.8 Proposition. Let z = F ( x l . . . x,) be a real valued C3-function defined on a strip Ixll < ll . . . . , I x~l < 1~. Suppose that the mean curvature H o f the non-parametric hypersurface o f R" + 1 defined by F satisfies [HI > c > 0. Then we have

- - n \ t I "

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