pact we prove that, if all the leaves of ~- have the same constant mean curvature. H then f" is a ... We shall make use of the following convention on the range of indices ..... This shows that c must be ]~--a and completes the proof of the theorem. .... Hector, G., Hirsch, U.: Introduction to the Geometry of Foliations, Part B. Friedr.,.
Math. Z. 207, 97-108 (1991)
Mathematische Zeitschrift 9 Springer-Verlag 1991
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 Riemannian 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 noncompact 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.
w2 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) *** Partially supported by the Alexander von Humboldt Foundation
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are tangent 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. We shall m a k e use of the following convention on the range of indices:
I .
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 manifold M . Assume that ~ is orientable and transversely orientable. Then (2.36)
d ~o= ( - 1)" + 1n 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 nHO.
Therefore the proposition is proved.
w3 The main results We now study a codimension-one C3-foliation ~ of a Riemannian manifold M such that each leaf L has constant mean curvature H L. We may assume that ~- is transversely orientable, otherwise we will work on the orientable double covering of the plane field T~- tangent to ~ . We then have on M a globally defined unit vector field N normal to the leaves of ~ . The mean curvature H L of a leaf L will always represent the mean curvature in the direction of N. We will consider the function H: M--, R whose value at each point p is H r , being L the leaf of ~ passing through the point p.
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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. Furthermore 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 may assume that ~- is transversely orientable. Since the mean 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 = IrB [I2 +
[x[2 +
Ric(N).
Since L is compact and Ric(N)>-0, the divergence theorem, applied to (3.3) yields (3.4)
rlBIl=0,
ISl=0
and
Rie(N)=0
along L. Therefore L is totally geodesic and, in particular, HL=O. From (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. Now N ( H ) - O and follows from (2.16) and (2.17) that div (X) = rlB II2 + Ric(N). Integration over M yields IIB ]l = 0 and R i c ( N ) = 0. The result now follows from 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 Tondeur [KT]. 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. Any codimension-one C3-foliation of M whose leaves have the same constant mean curvature can not have a compact leaf Proof Assume that we have on M a foliation with the above-mentioned properties and that it has a compact leaf L. Since all leaves have the same constant mean curvature then N [ H ] = 0. Formula (3.3) is then true on L. Since R i c ( N ) > 0 the divergence theorem applied to (3.3) yields a contradiction.
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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~--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 mean curvature HL, computed in the direction of N, satisfies ( - 1)" +1HL > l / _ a.
(3.9)
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)
( __ 1),,+ 1
vol(BR)= ~ ~ = BR
nH
BR
1
dq~=nc aJR ~o.
Observe that, for proving the above inequality, we have first used (3.9) to guarantee 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 orthonormal 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
