Minimal immersions of Riemannian manifolds in products of space forms

LOWER BOUNDS ESTIMATES OF THE LAPLACIAN

SPECTRUM ON COMPLETE SUBMANIFOLDS

M. P. CAVALCANTE AND F. MANFIO

Abstract. In this paper we obtain a lower bound estimate of the spec-trum of Laplacian operator on complete submanifolds with boundedmean curvature, whose ambient space admits a Riemannian submersionover a complete Riemannian manifold with bounded negative sectionalcurvature. This estimate generalizes many previous known estimates.

1. Introduction

Let Mm be a complete non compact m-dimensional Riemannian manifold.Fixed a point p ∈ M , let B(p; r) denote the open geodesic ball of radius rcentered at p. Let λ1(r) > 0 denote the first eigenvalue of the Dirichlet valueproblem

∆φ+ λφ = 0 in B(p; r),φ = 0 in ∂B(p; r),

where ∆ denotes the Laplace-Beltrami operator on M . The fundamentaltone of M is defined by

λ1(M) = limr→∞

λ1(r).

Of course it does not depend on the choice of the point p and coincides withthe first non zero eigenvalue when M is compact. Following some authors westill call λ1(M) the first eigenvalue of M when M is not compact. Moreover,λ1(M) can be characterized variationally as following:

λ1(M) = inf

− ∫M ϕ∆ϕ∫M ϕ2

: ∀ϕ ∈ C∞0 (M)

.

In particular, λ1(M) ≥ 0 and it is the bottom of the spectrum of −∆ on M .Above, and along this paper, we omit the volume element in the integralsfor the sake of simplicity.

The problem to estimate the first eigenvalue λ1(M) has been extensivelystudied. Of course, it is much harder to give a lower bound for λ1(M) than

Date: December 18, 2013.2010 Mathematics Subject Classification. Primary 35P15, 53C42; Secondary 53A10.Key words and phrases. Eigenvalues, Mean Curvature, Isometric Immersions, Rie-

mannian submersions.The first author was supported by CNPq/Brazil, grant 306131/2012-9.The second author was supported by Fapesp/Brazil, grant 2012/15970-2.

1

2 M. P. CAVALCANTE AND F. MANFIO

an upper bound. According to Schoen and Yau (see §III.4 in [11]), it is animportant question to find conditions on M which will imply λ1(M) > 0.

In this direction, McKean [9] showed that if M is simply connected andits sectional curvature satisfies KM ≤ −1, then

λ1(M) ≥ (m− 1)2

4= λ1(Hm),

where Hm denotes the m-dimensional hyperbolic space of sectional curvature−1. This estimate was extended by Veeravalli [12] for a quite general classof manifolds.

In the context of submanifolds, Castillon (see Theoreme 2.3 in [5]) con-sidered a complete submanifold Mm immersed in a Hadamard manifold M

n

with bounded sectional curvature K ≤ −b2 < 0 and bounded mean curva-ture vector |H| ≤ α < b and he were able to proved that

λ1(M) ≥ (m− 1)2(b− α)2

4.

Latter, Cheung and Leung [6] found lower bounds estimates when Mis complete and isometrically immersed in the hyperbolic space Hn withbounded mean curvature vector field |H| ≤ α < m − 1 (see also Theorem4.3.(4) of [3]). Namelly,

λ1(M) ≥ (m− 1− α)2

4.

Recently, Berard, Castillon and the first author [1], using a different ap-proach, obtained a sharp lower bound estimate for λ1(M), when M is ahypersurface immersed into Hn × R with constant mean curvature. Wepoint out that Bessa and co-authors, using other techniques, obtained in [2]eigenvalue estimates for minimal submanifolds of warped product spaces.

In this work we apply the ideas of [1] to find a general lower boundfor λ1(M) on complete submanifolds with bounded mean curvature, whoseambient space only admits a Riemannian submersion over a complete Rie-mannian manifold with bounded negative sectional curvature or over theclass of Riemannian manifolds consided in [12], (see Section 5). In partic-ular, when the base manifold of the submersion is the hyperbolic space Hk

we obtain:

Theorem 1.1. Let f : Mm → Mn be an isometric immersion of a complete

Riemannian manifold Mm into a Riemannian manifold Mn, which admits

a Riemannian submersion π : M → Hk. Let H be the mean curvature of M ,

αF the second fundamental form of the fibers of M , HF its mean curvature

and A the O’Neill tensor of M . If

c = infk − 1− ‖H‖ − ‖HF‖ − (n−m)(2‖A‖+ ‖αF‖+ 1

) > 0,

LOWER BOUND ESTIMATES FOR COMPLETE SUBMANIFOLDS 3

then

λ1(M) ≥ c2

4.

This paper is organized as follows. In Sections 2 we present the prelimi-naries results for latter use. In particular we recall an useful condition on aRiemannian manifold which implies a positive lower bound estimate for thefirst eigenvalue. In Sections 3 and 4 we present some results on Riemann-ian submersions and on Busemann functions in order to make the papermore clear and self-contained. A main step in the our approach is to usea comparison theorem for the Hessian of Busemann functions. In Section5, we state and prove the main theorems which generalizes Theorem 1.1 intwo cases, namely, when the base manifold has bounded negative sectionalcurvature and when the base manifold is a Riemannian warped-product ofa complete manifold by the real line (see Theorem 5.1). We finish this Sec-tion describing new examples of submersions where the constant in the maintheorem is positive.

The authors are grateful to Professors P. Piccione, H. Rosenberg and D.Zhou for helpful comments about this work.

2. Preliminaries

In this section we present two general lemmas, which will be used latter.The first result is known in the literature, but we present here its proof forthe sake of completeness.

Lemma 2.1. Let Mm be a complete Riemannian manifold that carries asmooth function F : M → R satisfying

‖grad F‖ ≤ 1 and |∆F | ≥ c,for some constant c > 0. Then, for any smooth and relatively compactdomain Ω ⊂M we have

λ1(Ω) ≥ c2

4,

where λ1(Ω) is the first eigenvalue of the Laplacian operator ∆ in Ω, withDirichlet boundary condition.

Proof. Let ϕ ∈ C∞0 (Ω), where Ω ⊂M is a pre-compact domain. Multiplyingthe inequality c ≤ ∆F by ϕ2, integrating over Ω and applying the Green’sfirst identity, we get:

c

∫Ωϕ2 ≤

∫Ωϕ2 ·∆F = −

∫Ω〈grad ϕ2, grad F 〉.

Using the Cauchy-Schwarz geometric-inequality and the fact that ‖grad F‖ ≤1, we have:

|〈grad ϕ2, grad F 〉| ≤ 2|ϕ| · ‖grad ϕ‖ ≤ εϕ2 +1

ε‖grad ϕ‖2,


for any ε > 0. It follows that

ε(c− ε)∫

Ωϕ2 ≤

∫Ω‖grad ϕ‖2.

We can maximize the constant in the left-hand side by choosing ε to be c2

and the conclusion follows from the min-max principle.

Given an isometric immersion f : Mm → Mn between Riemannian mani-

folds M and M , let α denote its second fundamental form. Then, the meancurvature vector (not normalized) H of M is defined by H = trα.

The following well-known result relates the Laplacian of a function on Mand its restriction to M .

Lemma 2.2. Let f : Mm → Mn be an isometric immersion with mean

curvature vector H. Let F : M → R be a smooth function and let F = F |Mbe its restriction to M . Then, on M , we have:

∆F = ∆F +n−m∑i=1

Hess F (Ni, Ni)−H(F ),

where N1, . . . , Nn−m is an orthonormal frame of TM⊥.

Proof. See, for example, [7, Lemma 2].

3. Riemannian Submersions

Let M and B be differentiable manifolds of dimensions n and k respec-

tively. A smooth map π : M → B is a submersion if it is surjective and its

differential dπ(p) has maximal rank at every point p ∈ M . It follows that forall x ∈ B the fiber Fx = π−1(x) is a (n− k)-dimensional embedded smooth

submanifold of M . Moreover, for every point p ∈ Fx, the tangent space ofFx at point p coincides with the kernel of dπ(p), i.e., TpFx = ker dπ(p).

If M and B are Riemannian manifolds, then a submersion π : M → Bis called a Riemannian submersion if for all x ∈ B and for all p ∈ Fx,the differential map dπ restricted to the orthogonal subspace TpF⊥x is an

isometry onto TxB. A vector field on M is called vertical if it is alwaystangent to fibers, and it is called horizontal if it is always orthogonal tofibers. Let V denote the vertical distribution consisting of vertical vectorsand H denote the horizontal distribution consisting of horizontal vectors on

M . The corresponding projections from TM to V and H are denoted bythe same symbols.

For any given vector field X ∈ X(B), there exists a unique horizontal

vector field X ∈ X(M) which is π-related to X, that is, dπ(p) · X(p) = X(x)for all x ∈ B and all p ∈ Fx, called horizontal lifting of X. A horizontal

vector field X ∈ X(M) is called basic if it is π-related to some vector fieldX ∈ X(B).


The following proposition, which can be found in [10, Lemma 1], summa-rize some basic proprieties about π-related fields.

Proposition 3.1. Let π : M → B be a Riemannian submersion and let X

and Y be basic vector fields, π-related to X and Y , respectively. Then:

(a) 〈X, Y 〉 = 〈X,Y 〉 π,

(b) [X, Y ]H is basic and it is π-related to [X,Y ],

(c) (∇XY )H is basic and it is π-related to ∇XY ,

(d) ∇XY = ∇XY + 1

2 [X, Y ]V ,

where ∇ and ∇ are the Levi-Civita connections of B and M , respectively.

Let D ⊂ TM denote the smooth distribution on M consisting of verticalvectors; D is clearly integrable, the fibers of the submersion being its max-imal integral leaves. The orthogonal distribution D⊥ is the smooth rank k

distribution on M consisting of horizontal vectors. The second fundamentalform of the fibers is a symmetric tensor αF : D ×D → D⊥, defined by

αF (v, w) = (∇vW )H,

where W is a vertical extension of w. The mean curvature vector of thefiber is the horizontal vector field HF defined by HF = trαF . In terms ofa orthonormal frame, we have

HF (p) =

n−k∑i=1

αF (ei, ei) =

n−k∑i=1

(∇eiei)H,(3.1)

where e1, . . . , en−k is a local orthonormal frame to the fiber at p. The fibers

are minimal submanifolds of M when HF ≡ 0, and are totally geodesic whenαF = 0.

We need some formulas relating the derivatives of π-related objects in Mand B. Let us start with the divergence of vector fields.

Lemma 3.2. Let X ∈ X(M) be a basic vector field, π-related to X ∈ X(B).

The following relation holds between the divergence of X and X at x ∈ Nand p ∈ Fx:

divX(p) = divX(x)− 〈X(p), HF (p)〉.

Proof. Let X1, . . . , Xk, Xk+1, . . . , Xn be a local orthonormal frame of TM ,

where X1, . . . , Xk are basic fields. The equality follows from identities (a)and (c) in Proposition 3.1, and formula (3.1) using this frame.

Giving a smooth function F : B → R, denote by F = F π : M → R its

lifting to M . It is easy to see that the gradient of F is the horizontal liftingof the gradient of F , i.e.,

grad F = grad F .(3.2)


The Laplace operator in B of a smooth function F : B → R and the

Laplace operator in M of its lifting F = F π are related by the followingformula.

Lemma 3.3. Let F : B → R be a smooth function and set F = F π. Then,for all x ∈ B and all p ∈ Fx:

∆F (p) = ∆F (x) + 〈grad F (p), HF (p)〉.

Proof. It follows easily from (3.2) and Lemma 3.2 applied to the vector fields

X = grad F and X = grad F .

Associated with a Riemannian submersion π : M → B, there are two

natural (1, 2)−tensors T and A on M , introduced by O’Neill in [10], and

defined as follows: for vector fields X, Y tangent to M , the tensor T isdefined by

TXY =(∇XVY V

)H+(∇XVY H

)V.

Note that π : M → B has totally geodesic fibers if and only if T vanishesidentically. The tensor A, known as the integrability tensor, is defined by

AXY =(∇XHY H

)V+(∇XHY V

)H.

The tensor A measures the obstruction to integrability of the horizontaldistribution H. In particular, for any horizontal vector field X and anyvertical vector field V , we have:

AXV =(∇XV

)H.(3.3)

The following lemma gives useful expressions for the Hessian of the lifting

F : M → R of a smooth function F : B → R, when we consider horizontaland vertical vector fields.

Lemma 3.4. If X and Y are basic, and V and W are vertical vector fields,

we have the following expressions for the Hessian of the lifting F = F π of

F to M :

(a) Hess F (X,Y ) = HessF (π∗X,π∗Y ) π,

(b) Hess F (V,W ) = −⟨αF (V,W ), grad F

⟩,

(c) Hess F (X,V ) = −⟨AXV, grad F

⟩.

Proof. The first assertion follows from (3.2) and item (c) in the Proposition3.1. The second one is a straightforward calculation, and the third assertionfollows directly from (3.3).


4. Busemann Functions

In this section we describe comparison results for the Hessian of Busemannfunctions on two classes of Riemannian manifolds, both are generalizationof the hyperbolic space. These classes of manifolds will be used as the basespace of the Riemannian submersions we will consider in our main theorem.

4.1. Busemann functions on manifolds with bounded negative sec-tional curvature. Given a > 0, let Hk(−a2) denote de k-dimensionalhyperbolic space with constant sectional curvature −a2. We consider thewarped-product model, that is

Hk(−a2) = (Rk−1 × R, h),

where

h = e−2asdx2 + ds2.

In this model, the curve γ : R → Hk(−a2), given by γ(s) = (x0, s), is ageodesic for any x0 ∈ Rk−1, and the function F : Hk(−a2)→ R, given by

F (x, s) = s,(4.1)

is its associated Busemann function. By a direct computation we getHessF = e−2asdx2,

∆F = (k − 1)a.

Now we will estimate the Hessian of the Busemann function F defined ina complete Riemannian manifold Bk with sectional curvature between twonegative constants. In order to obtain the Hessian of F , one takes a point pon a geodesic sphere of radius r, and let the center of the sphere go to infinity.In this case, the sphere converges to a horosphere, and the Hessian of thedistance function will converge to the Hessian of the Busemann function.So, a comparison theorem for the hessian of a Busemann function followsfrom the comparison theorem for the Hessian of the distance function. See[4] for a proof.

Lemma 4.1. Let Bk be a complete Riemannian manifold with sectionalcurvature K satisfying −a2 ≤ K ≤ −b2, for some constants a, b > 0. IfF : B → R is a Busemann function, then

b‖X‖2 ≤ HessF (X,X) ≤ a‖X‖2,

for any vector X orthogonal to grad F .


4.2. Busemann functions on manifolds with warped product struc-ture. Let (Nk−1, g) be a complete Riemannian manifold and let w : R→ Rbe a smooth function. Inspired in the hyperbolic space, we consider theRiemannian warped-product manifold

B = (N × R, h),(4.2)

where

h = e2w(s)g + ds2.

Considere now the Busemann function F : B → R defined by F (x, s) = s.As above, a direct computation gives

HessF = w′(s)e2w(s)g,

∆F = w′(s)(k − 1).

In particular we have the following lemma:

Lemma 4.2. Let Bk be a Riemannian manifold as in (4.2) and assumethat the function w satisfies b ≤ w′ ≤ a, for some constants a, b > 0. IfF : B → R is the Busemann function defined as above, then

b‖X‖2 ≤ HessF (X,X) ≤ a‖X‖2

for any vector X orthogonal to grad F .

In particular the following consequence will be use in the main theorem.

Corollary 4.3. Under the conditions of Lemma 4.1 or Lemma 4.2 we have

∆F ≥ (k − 1)b.

Remark 4.4. It is important to point out that Riemannian manifolds givenby (4.2) form a wide class. In particular, we may choose the manifold Nin such way that B has positive sectional curvature in some directions (see[12]).

5. Main result and examples

In this section we use all the above previous results to prove a lower boundestimates for the first eigenvalue of the Laplace operator on manifolds iso-metrically immersed on Riemannian manifolds which carries a Riemanniansubmersion on the two classes of manifolds described above. In particular,using Lemmas 4.1 and 4.2 and its corollary above, we are able to presenta unified proof to both cases. Bellow, we use a bar for geometric objectsrelated with the metric of B and a tilde for geometric objects related with

the metric of M .


Theorem 5.1. Let Bk be a complete Riemannian manifold as in Lemma 4.1

or as in Lemma 4.2, and let π : Mn → Bk be a Riemannian submersion.

Let Mm be a complete Riemannian manifold and f : Mm → Mn be anisometric immersion. Assume that F : B → R is a Busemann function and

consider its lifting F : M → R. If F = F |M is its restriction to M , then

∆F ≥ (k − 1)b+HF (F )− (n−m)(a+ 2‖A‖+ ‖αF‖

)+H(F ).

In particular, if

c = inf(k − 1)b− ‖HF‖ − (n−m)(a+ 2‖A‖+ ‖αF‖

)− ‖H‖ > 0,

then

λ1(M) ≥ c2

4.

Proof. From Lemma 3.3 and Corollary 4.3 we have:

∆F = ∆F + 〈grad F ,HF 〉 ≥ (k − 1)b+HF (F ).(5.1)

On the other hand, from Lemma 2.2,

∆F = ∆F +n−m∑i=1

Hess F (Ni, Ni)−H(F ),(5.2)

where N1, . . . , Nn−m is an orthonormal frame of TM⊥. For each 1 ≤ i ≤n−m, we write

Ni = NHi +NVi ,

where NHi and NVi denote the horizontal and vertical projection of Ni onto

TM , respectively. Moreover, since (5.2) is a tensorial equation, we mayassume that each NHi is basic. Thus, using Lemmas 3.4, 4.1 and 4.2 we get

∆F ≤ ∆F + (n−m)(a+ 2‖A‖+ ‖αF‖

)−H(F ).

So, plugging this in (5.1) we obtain

∆F ≥ (k − 1)a+HF (F )− (n−m)(b+ 2‖A‖+ ‖αF‖

)+H(F ).

The result follows from Lemma 2.1.

5.1. Lower bounds in warped products. Suppose that the ambient

space Mn = Hk ×ρ Fn−k admits a warped product structure, where thewarped function ρ satisfies ‖grad ρ‖/ρ ≤ 1.

By considering the projection on the first factor π : Hk ×ρ Fn−k → Hk

as a Riemannian submersion, we have that the tensor A is identically zero,‖αF‖ ≤ 1, and in particular ‖HF‖ ≤ n− k.

Let Mm be a complete Riemannian manifold and f : Mm → Mn be anisometric immersion such that its mean curvature vector H satisfies ‖H‖ ≤α, where α is a positive constant to be determined . If F : Hk → R is the


Busemann function given in (4.1), a lower bound estimates for the infimumin (5.1) goes as follows:

c = infk − 1− ‖HF‖ − (n−m)(1 + ‖αF‖)− ‖H‖≥ infk − 1− n+ k − 2(n−m)− ‖H‖= 2(k +m)− 3n− 1− α.

In particular, λ1(M) > 0 if we take 0 < α < 2(k +m)− 3n− 1.

5.2. Lower bounds in submersions with totally geodesic fibbers.

Let Mn be a Riemannian manifold with nonpositive sectional curvature and

π : Mn → Hk be a Riemannian submersion with totally geodesic fibers.This means that αF = 0, and thus HF = 0. Furthermore, the submersion πis integrable in the sense that the horizontal distribution is integrable (cf. [8,

Proposition 3.1]). Thus, if f : Mm → Mn is an isometric immersion, whosemean curvature vector H satisfies ‖H‖ ≤ α, for some positive constantα < k +m− n− 1, we have

c ≥ k − 1− (n−m)− ‖H‖≥ k +m− n− 1− α > 0,

and thus λ1(M) > 0.

References

1. P. Berard, P. Castillon, M. Cavalcante, Eigenvalue estimates for hypersurfaces inHm × R and applications, Pacific J. Math. 253 no. 1, 19–35, (2011).

2. G. P. Bessa, S.C. Garcıa-Martınez, L. Mari, H.F. Ramirez-Ospina, Eigenvalue esti-mates for submanifolds of warped product spaces, to appear in Math. Proc. CambridgePhilos. Soc.

3. G. P. Bessa, J. F. Montenegro, Eigenvalue estimates for submanifolds with locallybounded mean curvature, Ann. Global Anal. Geom. 24, 279–290, (2003).

4. G. P. Bessa, J. H. de Lira, S. Pigola, A. Setti, Curvature Estimates for Submanifoldsin Horocylinder, arXiv:1308.5926 [math.DG].

5. P. Castillon, Sur l’operateur de stabilite des sous-varietes a courbure moyenne con-stante dans l’espace hyperbolique, Manuscripta Math., 94, 385–400, (1997).

6. L.-F. Cheung, P.-F. Leung, Eigenvalue estimates for submanifolds with boundary meancurvature in the hyperbolic space, Math. Z. 236, 525–530, (2001).

7. J. Choe, R. Gulliver, Isoparametric inequalities on minimal submanifolds of spaceforms, Manuscripta Math. 77: 2-3, 169–189, (1992).

8. R. H. Escobales, Jr., Riemannian submersions with totally geodesic fibers, J. Diff.Geom., 10, 253–276, (1975).

9. H. P. McKean, An upper bound to the spectrum of ∆ on a manifold of negative cur-vature, J. Diff. Geom., 4, 359–366, (1970).

10. B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13, 459–469,(1966),.

11. R. Schoen, S.T. Yau, Lectures on Differential Geometry, 414 p., International Press,Cambridge, MA (2010).

12. A. R. Veeravalli, Une remarque sur l’inegalite de McKean, Comment. Math. Helv. 78,884–888, (2003).


IM, Universidade Federal de Alagoas, Maceio, AL, CEP 57072-970, BrazilE-mail address: [email protected]

ICMC, Universidade de Sao Paulo, Sao Carlos, SP, CEP 13561-060 , BrasilE-mail address: [email protected]

Minimal immersions of Riemannian manifolds in products of space forms

Documents