# A Note on Constant Mean Curvature Foliations of Noncompact Riemannian Manifolds

Authors: S. Ilias; B. Nelli; M. Soret
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

We aimed to study constant mean curvature foliations of noncompact Riemannian manifolds, satisfying some geometric constraints. As a byproduct, we answer a question by M. P. do Carmo (see Introduction) about the leaves of such foliations.


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## Body

## 1. Introduction

Consider a codimension-one foliation of a Riemannian manifold, whose leaves have a constant mean curvature. When the ambient manifold is compact, there are a bunch of results stating that such a foliation is totally geodesic, provided some geometric assumption is satisfied [1, 2]. This kind of phenomenon is sometimes true in the noncompact case as well. Meeks in [3] proved that any codimension-one constant mean curvature foliation of the three-dimensional Euclidean space is totally geodesic. Oshikiri [4] proved the analogous result in a Riemannian manifold with nonnegative Ricci curvature, provided the leaves have quadratic volume growth.In this article, we prove that the leaves of any codimension-one constant mean curvature foliation of a Riemannian manifold with nonnegative Ricci curvature are totally geodesic, provided they are parabolic (Theorem4). Theorem 4 generalizes Oshikiri’s result, as quadratic volume growth implies they are parabolic.Then, we consider a Riemannian manifoldN with zero volume entropy (Section 2), and we prove that, if ℱ is a codimension-one foliation of N such that any leaf L has constant mean curvature HL, then infHL=0 (Theorem 2). Notice that having zero volume entropy is a weaker assumption than having nonnegative Ricci curvature, by Bishop’s comparison theorem [5] (Corollary 2.1.1).Finally, we point out that if the leaves of a foliationℱ have the same constant mean curvature, then the leaves are stable as shown in [6], Proposition 3, and Section 3, where a sketch of the proof is given.We recall that do Carmo in [7] asked the following question: is a noncompact, complete, stable, constant mean curvature hypersurface of ℝn+1, n≥3, necessarily minimal?Our result yields a positive answer to do Carmo’s question, provided the hypersurface is a leaf of a foliation such that the leaves have the same constant mean curvature.The answer was already known to be positive forn=2 [8] ([9], when the ambient manifold is the hyperbolic space). Later, the answer was proved to be positive for n = 3 and 4 by Elbert et al. [10] and independently by Cheng [11], using a Bonnet–Myers’ type method. Moreover, the authors gave a positive answer to do Carmo’s question in the following cases: (1) a hypersurface with zero volume entropyof a space form of any dimension [12] (Corollary 8). (2) a hypersurface of ℝn+1, ℍn+1, n≤5 with total curvature with polynomial growth [13] (Corollary 6.3).


## 2. Basic Notions

LetN be a complete, noncompact Riemannian manifold. We define the entropy associated with the volume of geodesic balls in N [14, 15].Definition 1.
LetBσNR be a geodesic ball in N of radius R, centered at a fixed point σ∈N, and denoted by BσNRas its volume. The volume entropy of N is(1)μN≔lim supR⟶∞lnBσΝRR.
The notion of volume entropy does not depend on the centerσ of the balls. It is worthwhile to notice that μΝ=0 is equivalent to(2)limsupr⟶∞BσΝReαr=0,∀α>0.
Then, it is natural to say thatNΝ has subexponential growth if μN=0 and exponential growth if μN>0.
We observe that having subexponential growth is a weaker assumption than being bounded by a polynomial of any degree. For example, ifBσΝR=erβ and β<1, then N has subexponential growth.
The volume entropy of a manifoldN is strictly related to the bottom of its essential spectrum and to the Cheeger constant. Let us be more precise about this.
LetΔ be the Laplacian on N, then the bottom of the spectrum σΝ of −Δ is(3)λ0N=infσN=inff∈C0∞Nf≠0∫Ν∇f2∫Νf2.
The bottom of the essential spectrumσessN of −Δ is(4)λ0essN=infσessN=supKλ0NK,where K runs through all compact subsets of N.
The Cheeger constanthN of a Riemannian manifold N is defined as hN=infΩ∂Ω/Ω, where Ω runs over all compact domains of N with piecewise smooth boundary ∂Ω.
Cheeger [16] and Brooks [14] proved the following important comparison result between the bottom of the essential spectrum, the volume entropy, and the Cheeger constant.Theorem 1 (Brooks–Cheeger’s Theorem).
IfN has infinite volume, then(5)hΝ24≤λ0N≤λ0NK≤λ0essN≤μΝ24,where K is any compact subset of N.We finally recall that a Riemannian manifold is called parabolic if it does not admit a nonconstant positive superharmonic function. Parabolicity is strictly related to the volume growth of a manifold. In fact, quadratic area growth implies parabolicity [17] (Corollary 7.4) [6].


## 3. Foliations of Manifolds with Zero Volume Entropy

In this section, we study foliations, with constant mean curvature leaves, of manifolds with zero volume entropy. In the article,ℱ will be a C3 codimension-one foliation of a manifold N and N will be a C3 unit vector field of N normal to the leaves of ℱ.We have the following result.Theorem 2.
LetN be a manifold with zero volume entropy. Let ℱ be a codimension-one C3 foliation of N such that any leaf L has constant mean curvature HL≥0. Then, infHL=0.Before doing the proof of Theorem2, let us state a consequence in the case of a constant mean curvature foliation.Corollary 1.
LetN be a manifold with zero volume entropy. Let ℱ be a codimension-one C3 foliation of NΝ by leaves of constant mean curvature H. Then, H=0.As we remarked in Introduction, the leaves of a constant mean curvature foliation are stable, provided all the leaves have the same mean curvature ([2], Proposition 3).We give a sketch of the proof of the latter for the sake of completeness.By [2] (Proposition 1), one has(6)RicN,N+A2+θ2=divL∇NN,where N is a unit vector field defined in N, perpendicular to ℱ at any point, A is the second fundamental form of L, θ is defined by θX=∇NN,X for every vector field X tangent to L, and θ2=∑i=1nθEi2, where E1,…,En is a local orthonormal base of the tangent space to L.Recall that the second variation of the volume of a leafL is(7)V″0=∫L∇f2−f2RicN,N+A2,where f is any smooth function on N with compact support.Then, by using equality (6) multiplied by f, one gets(8)V″0=∫L∇f2−f2RicN,N+A2=∫L∇f2+f2θ2−f2divL∇NN=∫L∇f2+f2θ2−divLf2∇NN+∇NNf2=∫L∇f+fθ2≥0.Then,V″0≥0; this means that the leaves are stable.The stability of the leaves of such a foliation suggests to inquire if we can answer Carmo’s question [7]: is a noncompact, complete, stable, constant mean curvature hypersurface of ℝn, n≥3, necessarily minimal?Corollary 1 yields a positive answer to Carmo’s question, provided the hypersurface is a leaf of a foliation with constant mean curvature leaves.Proof of Theorem 2.
If there is a minimal leafL, we have nothing to prove. Then, we may assume that infHL>0, and let N be the unit vector field normal to the leaves of ℱ given by the normalized mean curvature vector. Then, there exists c>0 such that HL≥c for every leaf L. By straightforward computation, we obtain the following:(9)nHp=divΝNp,where H is the mean curvature of the leaf passing through p.
Then, we integrate (9) on a ball of radius R of N, say BR,(10)∫BRnHp=∫BRdivNNp=∫∂BRN,ν≤vol∂BR,where the last equality is by the divergence theorem. Keeping into account that Hp≥c at any point p∈Ν, (10) yields(11)nc≤vol∂BRvolBR.
In particular,(12)nc≤infRvol∂BRvolBR=hΝ.
Then, the Brooks–Cheeger’s theorem stated in Section2 yields μΝ≥hΝ>0 that is a contradiction.
Using the Gauss formula, one can prove the analogous of Theorem2 for foliations of constant scalar curvature.Theorem 3.
LetN be a manifold with zero volume entropy and nonpositive sectional curvatures. Let ℱ be a codimension-one C3 foliation of N such that any leaf L has constant scalar curvature SL≥0. Then, infSL=0.Proof.
By the Gauss formula, one has(13)SL=∑i<jK¯ei,ej+HL2−A2,where e1,…,en is a base of the tangent space to a leaf L, K¯ei,ej is the sectional curvature of NΝ along spanei,ej, A is the norm of the second fundamental form of L, and HL is its mean curvature.
Then, being the sectional curvatures ofNN nonpositive,one has that SL<HL2; hence, SL>c implies HL>c, and one can apply Theorem 2.
An immediate consequence of Theorem2 is the following result.Corollary 2.
LetN be a manifold with zero volume entropy and nonpositive sectional curvatures. Let ℱ be a codimension-one C3 foliation of NΝ by leaves of constant scalar curvature S. Then, S=0.Remark 1.
It is worthwhile noticing that all the results of this section hold for foliations given by complete graphs of constant mean or scalar curvature, when they exist.


## 4. Minimal Parabolic Foliations of Manifolds with Nonnegative Ricci Curvature

In this section, we prove a kind of Bernstein’s theorem for the leaves of a minimal foliation of a Riemannian manifold with nonnegative Ricci curvature, provided the leaves are parabolic.Theorem 4.
LetN be a Riemannian manifold with nonnegative Ricci curvature and let ℱ be a codimension-one foliation of N by minimal leaves. If the leaves of the foliation are parabolic, then they are totally geodesic.Theorem4 is a generalization of Theorem 1 in [4]. In fact, quadratic area growth implies parabolicity ([17], Corollary 7.4, and [6]). When the ambient space is ℝn+1, for n=2, the leaves of a minimal foliation must be totally geodesic [3], while for n≥4, there is no parabolic complete minimal hypersurface ([18], Proposition 2.3).Proof of Theorem 4.
We first need a definition. We say thatv is an exhaustion function on L if v is continuous on L and such that all the sets ℬr:=p∈L:vx≤r are precompact. Notice that the latter is equivalent to say that vp⟶∞ as p⟶∞ (that is, p leaves every compact).
By [12] (Theorem 7.6), a manifold L is parabolic if and only if there exists a smooth exhaustion function v on L such that(14)∫∞drFlux∂ℬrv=∞,where Flux∂ℬrv=∫∂ℬr∇v,ν, being the outward unit normal to ∂ℬr.
Equality (6) in Section 3 holds for the minimal leaf L, and integrating it on ℬr yields(15)∫ℬrRicN,N+A2+θ2=∫ℬrdivL∇NN=∫∂ℬrθν,and for the last equality, we used the divergence theorem.
AsRicN,N and A2 are nonnegative, we have(16)∫ℬrθ2≤∫∂ℬrθν≤∫∂ℬrθ2∇v1/2∫∂ℬr∇v1/2,where the last inequality is by the Cauchy–Schwarz inequality.
By definingfr=∫ℬrθ2, the coarea formula yields fr=∫0r∫∂ℬsθ2/∇vds. Then, f′r=∫∂ℬsθ2/∇vds.
Assume thatfr≠0. With this notation, the square of (16) is written as(17)1Flux∂ℬrv≤f′rf2r.
We integrate inequality (17) between a fixed r0 and R (where f is nonzero), and we get(18)∫r0RdrFlux∂ℬrv≤1fr0−1fR.
By lettingR go to ∞, inequality (18) gives a contradiction. In fact, as f is a nondecreasing function, then the right-hand side is bounded, while by hypothesis, the left-hand side tends to infinity.
Then,f≡0, that is ∇N,N≡0, on the leave L, and equality (6) yields RN,N+A2≡0. As the Ricci curvature is nonnegative, we get A2≡0, i.e., L is totally geodesic.



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