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Anais da Academia Brasileira de Ciências

versão impressa ISSN 0001-3765versão On-line ISSN 1678-2690

An. Acad. Bras. Ciênc. v.78 n.3 Rio de Janeiro set. 2006

http://dx.doi.org/10.1590/S0001-37652006000300001 

MATHEMATICAL SCIENCES

 

Fundamental tone estimates for elliptic operators in divergence form and geometric applications

 

 

Gregório P. BessaI; Luquésio P. JorgeI; Barnabé P. LimaII; José F. MontenegroI

IUniversidade Federal do Ceará - UFC, Departamento de Matemática - Campus do Pici, 60455-760 Fortaleza, CE, Brasil
IIUniversidade Federal do Piauí - UFPi, Departamento de Matemática - Campus Petrônio Portela, 64049-550 Teresina, PI, Brasil

 

 


ABSTRACT

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn + 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn + 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn + 1(c).

Key words: fundamental tone, Lr operator, r-th mean curvature, extrinsic radius, Cheeger's constant.


RESUMO

Estabelecemos um método para obter limites inferiores para o tom fundamental de operadores elípticos em forma divergente em termos do divergente de campos de vetores. Aplicamos esse método para os operadores Lr associados a hipersuperfícies imersas nas formas espaciais Nn(c) de curvatura seccional constante c com (r + 1)-curvatura média Hr + 1 localmente limitada. Obtemos como corolário limites inferiores para o raio extrínseco de hipersuperfícies compactas das formas espaciais Nn(c) com Hr + 1 > 0 em termos das r-ésima e r + 1-ésima curvatura médias. Finalmente, observamos que limites para os autovalores do Laplaciano essencialmente limitam os autovalores dos operadores elípticos em forma divergente. Isso permite mostrar que a constante de Cheeger limita inferiormente o primeiro autovalor não-nulo dos operadores Lr em hypersuperfícies compactas de Nn(c).

Palavras-chave: tom fundamental, operador Lr, r-curvatura média, raio extrínseco, constante de Cheeger.


 

 

INTRODUCTION

Let W be a domain in a smooth Riemannian manifold M and let F : W ® End (TW) be a smooth symmetric and positive definite section of the bundle of all endomorphisms of TW. Each section F is associated to a second order self-adjoint elliptic operator LF(f) = div (F grad f), f Î C2(W). Observe that when F is the identity section then LF = D, the Laplace operator. Recall that the LF-fundamental tone of W is given by

If W is bounded with smooth boundary ¶W ¹ Æ, the LF-fundamental tone of W coincides with the first eigenvalue (W) of the Dirichlet eigenvalue problem LF u + l u = 0 on W, with u|¶W = 0, u Î C2(W) Ç Cº \ {0}. If W is bounded with empty boundary ¶W = Æ then (W) = 0.

A basic problem in Riemannian geometry is what lower and upper bounds for (W) can be obtained in terms of Riemannian invariants. In this paper we show that the method established by Bessa and Montenegro (Bessa and Montenegro 2004) that gives lower bounds for the D-fundamental tone can be extended for self-adjoint elliptic operators LF, (Theorem 2.1). Then we consider immersed hypersurfaces j : M ® n+1(c) of the (n + 1)-dimensional simply connected space form n+1(c) of constant sectional curvature c Î {1,0,-1} with locally bounded (r + 1)-th mean curvature such that the differential operators Lr, r Î { 0,1,..., n} are elliptic. We give lower bounds for the Lr-fundamental tone of domains W Ì j-1 ((p, R)), in terms of the r-th and (r + 1)-th mean curvatures Hr, Hr + 1, (Theorem 3.2), where (p, R) is the geodesic ball of n+1(c) centered at p with radius R. From these estimates we derive three geometric corollaries 3.4, 3.5 and 3.8 that should be viewed as an extension of a result of Jorge and Xavier (Jorge and Xavier 1981). It should be mentioned that these corollaries are related to results due to Vlachos (Vlachos 1997) and to Fontenele and Silva (Fontenele and Silva 2001), see Remark 3.7. In Theorem 3.10 we consider immersed hypersurfaces M of n+1(c) such that the operators Lr and Ls, 0 < r, s < n are elliptic and we compare the Lr and Ls fundamental tones (W)(W) of domains W Ì M Ì n+1(c). In section 4 we observe (Theorem 4.1) that in order to get bounds for the eigenvalues of a self-adjoint elliptic differential operator LF we essentially need bounds for the Laplace operator eigenvalues. This allows us to use Cheeger's constant to give lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of n+1(c). The results are stated and discussed in Sections 2, 3 and 4 and the proofs are given in Section 5.

 

LF-FUNDAMENTAL TONE ESTIMATES

Our main estimate is the following method for giving lower bounds for LF-fundamental tone of arbitrary domains of Riemannian manifolds. It extends the version of Barta's theorem (Barta 1937) proved by Cheng-Yau in (Cheng and Yau 1977). It is the same proof (with proper modifications) of a generalization of Barta's theorem proved in (Bessa and Montenegro 2004).

THEOREM 2.1. Let W be a domain in a Riemannian manifold and let F : End (TW) be a smooth symmetric and positive definite section of End (TW). Then the LF-fundamental tone of W has the following lower bound

If W is bounded and with piecewise smooth boundary ¶W ¹ Æ then we have equality in (2).

Where (W) is the set of all smooth vector fields on W.

 

GEOMETRIC APPLICATIONS

Consider the linearized operator Lr of the (r + 1)-mean curvature

arising from normal variations of a hypersurface M immersed into the (n + 1)-dimensional simply connected space form n+1(c) of constant sectional curvature c Î {1,0,-1}, where Sr + 1 is the (r + 1)-th elementary symmetric function of the principal curvatures k1, k2,..., kn, see (Reilly 1973) and (Rosenberg 1993) for details. Recall that the elementary symmetric function of the principal curvatures are given by

Letting A = - (h) be the shape operator of M, where is the Levi-Civita connection of n+1(c) and h a globally defined unit vector field normal to M, we can recursively define smooth symmetric sections Pr : M ® End (TM), for r = 0, 1,..., n, called the Newton operators, setting P0 = I and Pr = Sr Id - APr - 1 so that Pr(x) : Tx M ® Tx M is a self-adjoint linear operator with the same eigenvectors as the shape operator A. The operator Lr is the second order self-adjoint differential operator

associated to the section Pr. However, the sections Pr may be not positive definite and then the operators Lr may not be elliptic. However, there are geometric hypothesis that imply the ellipticity of Lr, see for instance, Reilly 1973, Caffarelli et al. 1985, Korevaar 1988 or Barbosa and Colares 1997. Here we will not impose geometric conditions to guarantee ellipticity of the Lr, except in corollary 3.5. Instead we will ask the ellipticity on the set of hypothesis. It is known the ordered eigenvalues {(x) < ... < (x)} of Pr (x) depend continuously on x Î M. (Kato 1976 pages 106-109). In fact, this proof can be pushed to prove that they are Lipschitz thus differentiable almost everywhere. In addition, the respective eigenvectors {e1(x),..., en(x)} form a smooth orthonormal frame in a neighborhood of every point. Set n(Pr) = supx Î M{(x)} and µ(Pr) = infx Î M{(x)} . Observe that if µ(Pr) > 0 then Pr is positive definite, thus Lr is elliptic.

We need the following definition of locally bounded (r + 1)-th mean curvature hypersurface in order to state our next result.

DEFINITION 3.1. An oriented immersed hypersurface j : M N of a Riemannian manifold N is said to have locally bounded (r + 1)-th mean curvature Hr + 1 if for any p Î N and R > 0, the number

is finite. Here BN(p, R) Ì N is the geodesic ball of radius R with center at p Î N.

Our next result generalizes in some aspects the main application of (Bessa and Montenegro 2003). There the first and fourth authors give lower bounds for D-fundamental tone of domains in submanifolds with locally bounded mean curvature in complete Riemannian manifolds.

THEOREM 3.2. Let j : M n+1(c) be an oriented hypersurface immersed with locally bounded (r + 1)-th mean curvature Hr + 1 for some r < n - 1 and with µ(Pr ) > 0. Let (p, R) be the geodesic ball centered at p Î n+1(c) with radius R and W Ì j-1 be a connected component. Then the Lr-fundamental tone (W) of W has the following lower bounds.

i. For c = 1 and 0 < R < cot-1 we have that

ii. For c < 0, hr + 1(p, R) ¹ 0 and 0 < R < we have that

iii. If c < 0, hr + 1(p, R) = 0 and R > 0 we have that

DEFINITION 3.3. Let j : M N be an isometric immersion of a closed Riemannian manifold into a complete Riemannian manifold N. For each x Î N, let r(x) = supy Î M distN(x, j(y)). The extrinsic radius Re(M) of M is defined by

Moreover, there is a point x0 Î N called the barycenter of j(M) in N such that Re(M) = r(x0).

COROLLARY 3.4. Let j : M (R) Ì n+1(c) be a complete oriented hypersurface with bounded (r + 1)-th mean curvature Hr + 1 for some r < n - 1, R chosen as in Theorem (3.2). Suppose that µ(Pr ) > 0 so that the Lr operator is elliptic. Then M is not closed.

COROLLARY 3.5. Let j : M n+1(c)1, c Î {1,0,-1} be an oriented closed hypersurface with Hr + 1 > 0. Then there is an explicit constant Lr = Lr (c, infM Sr , supM Sr + 1) > 0 such that the extrinsic radius Re(M) > Lr.

i. For c = 1, Lr = cot-1.

ii. For c Î {0, -1}, Lr = .

REMARK 3.6. The hypothesis Hr + 1 > 0 implies that Hj > 0 and Lj are elliptic for j = 0, 1,...r, see Barbosa and Colares 1997, Caffarelli et al. 1985 or Korevaar 1988. Thus in fact have that Re > max{L0,..., Lr}.

REMARK 3.7. Jorge and Xavier, (Jorge and Xavier 1981) proved the inequalities of Corollary 3.5 when r = 0 for complete submanifolds with scalar curvature bounded from below contained in a compact ball of a complete Riemannian manifold. Moreover, for c = -1 their inequality is slightly better. It is possible to give sharp estimates for the extrinsic radius of a closed hypersurface of n+1(c) in terms of supM|Hr| alone. Vlachos (Vlachos 1997) proved a result that implies that, for each

if c = 0, c = 1 or c = -1 respectively, and that in any case the equality holds if and only if M is a geodesic sphere of the ambient space. The result of Vlachos was extended to any ambient space by Fontenele and Silva (Fontenele and Silva 2001).

REMARK 3.8. An interesting question is: Is it true that any closed oriented hypersurface with (M) > 0 and Hr + 1 = 0 intersect every great circle? For r = 0 it is true and it was proved by T. Frankel, (Frankel 1966).

We now consider immersed hypersurfaces j : M n+1(c) with Lr and Ls elliptic. We can compare the Lr and Ls fundamental tones of a domain W Ì M. In particular we can compare with its L0-fundamental tone.

THEOREM 3.9. Let j : M n+1(c) be an oriented n-dimensional hypersurface M immersed into the (n + 1)-dimensional simply connected space form of constant sectional curvature c and µ(Pr ) > 0 and µ(Ps ) > 0, 0 < s, r < n - 1. Let W Ì M be a domain with compact closure and piecewise smooth nonempty boundary. Then the Lr and Ls fundamental tones satisfies the following inequalities

From (9) we have in particular that

 

CLOSED EIGENVALUE PROBLEM

Let M be a closed hypersurface of a simply connected space form n+1(c). The interesting problem is what bounds can one obtain for the first nonzero Lr-eigenvalue (M) in terms of the geometries of M and of the ambient space. Upper bounds for the first nonzero D-eigenvalue or even for the first nonzero Lr-eigenvalue, r > 1 have been obtained by many authors in contrast with lower bounds that are rare. For instance, Reilly (Reilly 1977) extending earlier result of Bleecker and Weiner (Bleecker and Weiner 1976) obtained upper bounds for (M) of a closed submanifold M of m in terms of the total mean curvature of M. Reilly's result applied to compact submanifolds of the sphere M Ì m+1(1), this latter viewed as a hypersurface of the Euclidean space m+1(1) Ì m+2 obtains upper bounds for (M), see Alencar, Do Carmo and Rosenberg in Alencar et al. 1993. Heintze (Heintze 1988) extended Reilly's result to compact manifolds and Hadamard manifolds . In particular for the hyperbolic space n+1. The best upper bounds for the first nonzero D-eigenvalue of closed hypersurfaces M of n+1 in terms of the total mean curvature of M was obtained by El Soufi and Ilias (Soufi and Ilias 1992). Regarding the Lr operators, Alencar, Do Carmo and Rosenberg (Alencar et al. 1993) obtained sharp (extrinsic) upper bound the first nonzero eigenvalue (M) of the linearized operator Lr of compact hypersurfaces M of m+1 with Sr+1 > 0. Upper bounds for (M) of compact hypersurfaces of n+1, n+1 under the hypothesis that Lr is elliptic were obtained by Alencar, Do Carmo, Marques in (Alencar et al. 2001) and by Alias and Malacarne in (Alias and Malacarne 2004) see also the work of Veeravalli (Veeravalli 2001). On the other hand, lower bounds for (M) of closed hypersurfaces M Ì n+1(c) are not so well studied as the upper bounds, except for r = 0 in which case L0 = D. In this paper we make a simple observation (Theorem 4.1) that to obtain lower and upper bounds for the LF-eigenvalues (Dirichlet or Closed eigenvalue problem) it is enough to obtain lower and upper bounds for the eigenvalues of F and for the eigenvalues for the Laplacian in the respective problem. When applied to the Lr operators (supposing them elliptic) we obtain lower bounds for closed hypersurfaces of the space forms via Cheeger's lower bounds for the first D-eigenvalue of closed manifolds. Let {µ1(x) < ... < µn(x)} be the ordered eigenvalues of F(x). Setting n(F) = supx Î Wn(x)} and µ(F) = infx Î W1(x)} we have the following theorem.

THEOREM 4.1. Let (W) denote the LF-fundamental tone of W if W is unbounded or ¶W ¹ Æ and the first nonzero LF-eigenvalue (W) if W is a closed manifold. Then (W) satisfies the following inequalities,

where lD(W) is the D-fundamental tone of W or the first nonzero D-eigenvalue of W.

Let M be a closed n-dimensional Riemannian manifold, Cheeger (Cheeger 1970) defined the following constant given by

where S Ì M ranges over all connected closed hypersurfaces dividing M in two connected components, i.e. M = W1È W2, W1Ç W2 = Æ such that S = ¶W1 = ¶W2 and he proved that the first nonzero D-eigenvalue (M) > h(M)2/4.

COROLLARY 4.2. Let j : M n+1(c), c Î {1, 0, -1}2 be an oriented closed hypersurface with Hr + 1 > 0. Then the first nonzero Lr-eigenvalue of M has the following lower bound

 

PROOF OF THE RESULTS

PROOF OF THEOREM 2.1.

Let W be an arbitrary domain, X be a smooth vector field on W and f Î (W). The vector field f 2FX has compact support supp (f 2FX) Ì supp (f) Ì W. Let S be a regular domain containing the support of f. We have by the divergence theorem that

Therefore

By the variational formulation (1) of (W) this inequality above implies that

When W is a bounded domain with smooth boundary ¶W ¹ Æ then (W) = (W). This proof above shows that

Let v Î C2(W) Ç Cº be a positive first LF-eigenfunction3 of W and if we set X0 = -grad log(v) we have that

This proves (3).

PROOF OF THEOREM 3.2 AND COROLLARIES 3.4, 3.5 AND 3.8

We start this section stating few lemmas necessary to construct the proof of Theorem 3.2. The first lemma was proved in (Jorge and Koutrofiotis 1980) for the Laplace operator and for the Lr operator in (Lima 2000). We reproduce its proof to make the exposition complete.

LEMMA 5.1. Let j : M n+1(c) be a hypersurface immersed in (n + 1)-dimensional simply connected space form n+1(c) of constant sectional curvature c. Let g : n+1(c) ® be a smooth function and set f = g j. Identify X Î TpM with dj(p) X Î Tj(p) j(M) then we have that

PROOF. Each Pr is also associated to a second order self-adjoint differential operator defined by f = Trace (Pr Hess (f)) see (Cheng and Yau 1977, Hartmann 1978). We have that

Rosenberg (Rosenberg 1993) proved that when the ambient manifold is the simply connected space form n+1(c) then Trace (ÑPr) grad º 0 , see also (Reilly 1973). Thus one has that Lrf = Trace (Pr Hess (f)). Using Gauss equation to compute Hess (f) we obtain

where áa(X, Y), = áA(X), Yñ. Let {ei} be an orthonormal frame around p that diagonalize the section Pr so that Pr(x)(ei) = (x)ei. Thus

Substituting (19) into (20) we have that

Here Hess f (X) = ÑX grad f and Hess f(X, Y) = áÑX grad f, Yñ. The next two lemmas we are gong to present are well known and their proofs are easily found in the literature thus we will omit them here.

LEMMA 5.2 [Hessian Comparison Theorem]. Let M be a complete Riemannian manifold and x0, x1 Î M. Let g : [0, r(x1) ] ® M be a minimizing geodesic joining x0 and x1 where r(x) is the distance function distM(x0, x). Let K be the sectional curvatures of M and u(r), defined below.

Let X = X^ + XT Î TxM, XT = áX, g'ñg' and áX^, g'ñ = 0. Then

See (Schoen and Yau 1994) for a proof.

LEMMA 5.3. Let p Î M and 1 < r < n - 1, let {ei} be an orthonormal basis of TpM such that Pr(ei) = ei and A(ei ) = kiei. Then

i. Trace (Pr) = = (n - r)Sr

ii. Trace (A Pr) = ki = (r + 1)Sr + 1

In particular, if the Newton operator Pr is positive definite then Sr > 0.

To prove Theorem (3.2) set g : B(p, R) Ì n+1(c) ® given by g = R2 - r2 , where r is the distance function (r(x) = dist (x, p)) of n+1(c). Setting f = g j we obtain by (17) that

since Trace (A Pr) = (r + 1) · Sr + 1. Letting X = -grad log f we have that

then by Theorem (2.1) we have that

Computing the Hessian of g we have that

Therefore we have that

Setting = ágrad r, eiñgrad r and = ei - , by the Hessian Comparison Theorem we have that

and

From (28) and (29) we have that

If c < 0 then r · u(r) > 1 thus from (30) we have that

If c > 0 then r · u(r) = r · · cot[ r] < 1 thus from (30) we have that

To prove the Corollaries (3.4) and (3.5) observe that the hypotheses µ(Pr)(M) > 0 (in Corollary 3.4) and Hr + 1 > 0 (in Corollary 3.5) imply that the Lr is elliptic. If the immersion is bounded (contained in a ball of radius R, for those choices of R) and M is closed we would have by one hand that the Lr-fundamental tone would be zero and by Theorem (3.2) that it would be positive. Then M can not be closed if the immersion is bounded. On the other hand if M is closed a ball of radius R centered at the barycenter of M could not contain M because the fundamental tone estimates for any connected component W Ì j-1(j(M) Ç (p, R) is positive. Showing that M ¹ W.

PROOF OF THEOREM 3.9.

Let j : W n+1(c) be an isometric immersion of an oriented n-dimensional Riemannian manifold W into a (n + 1)-dimensional simply connected space form of sectional curvature c. Let M Ì W be a domain with compact closure and piecewise smooth nonempty boundary and suppose that the Newton operators Pr and Ps, 0 < s, r < n - 1 are positive definite when restricted to M. Given a vector field X on M we can find a vector field Y on M such that PrX = k · PsY, k constant. Now

Consider {ei} be an orthonormal basis such that Prei = ei and Psei = ei. Letting Y = yiei then

Combining (33) with (34) and by Theorem (2.1) we have that

for every 0 < k < . This proves (9).

PROOF OF THEOREM 4.1.

Recall that for any smooth symmetric section F : W ® End (TW) there is an open and dense subset U Ì W where the ordered eigenvalues {µ1(x) < ... < µn(x)} of F(x) depend continuously in all W. In addition, the respective eigenvectors {e1(x),..., en(x)} form a smooth orthonormal frame in a neighborhood of every point of W, see (Kato 1976). Let f Î (W) \ {0} (f Î C2(W) with fWf = 0) be an admissible function for (the closed LF-eigenvalue problem if W is a closed manifold) the Dirichlet LF-eigenvalue problem. It is clear that f is an admissible function for the respective D-eigenvalue problem. Writing grad f (x) = ei(f)ei(x) we have that

From (36) we have that

and

Taking the infimum over all admissible functions in (38) we obtain (11).

 

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Manuscript received on October 31, 2005; accepted for publication on April 17, 2006; presented by MANFREDO DO CARMO

 

 

Correspondence to: Gregório P. Bessa
E-mail: bessa@mat.ufc.br
1 If c = 1 suppose that n+1(c) is the open hemisphere of .
2 If c = 1 suppose that n+1(c) is the open hemisphere of .
3 v Î C2(W) Ç (W) if ¶W is not smooth.

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