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

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 $L_{r}$ operator associated to immersed hypersurfaces with locally bounded $(r+1)$-th mean curvature $H_{r+1}$ of the space forms $\mathbb{N}^{n+1}(c)$ of curvature $c$. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of $\mathbb{N}^{n+1}(c)$ with $H_{r+1}>0$ 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 $L_{r}$-eigenvalue of a closed hypersurface of $\mathbb{N}^{n+1}(c)$.


Introduction
Let Ω be a domain in a smooth Riemannian manifold M and let Φ : Ω → End (T Ω) be a smooth symmetric and positive definite section of the bundle of all endomorphisms of T Ω. Each such section Φ is associated to a second order self-adjoint elliptic operator L Φ (f ) = div (Φ grad f ), f ∈ C 2 (Ω) so that when Φ is the identity section then L Φ = △, the Laplace operator. The L Φ -fundamental tone of Ω is defined by If Ω is bounded with smooth boundary ∂Ω = ∅, the L Φ -fundamental tone of Ω coincides with the first eigenvalue λ L Φ 1 (Ω) of the Dirichlet eigenvalue problem L Φ u + λ u = 0 on Ω, with u|∂Ω = 0, u ∈ C 2 (Ω) ∩ C 0 (Ω) \ {0}. If Ω is bounded with ∂Ω = ∅ then λ L Φ (Ω) = 0. A basic problem is what lower and upper bounds for the fundamental tone of a given domain Ω in a smooth Riemannian manifold can be obtained in terms of Riemannian invariants of Ω. In the first part of this paper we show that the method for giving lower bounds for the △-fundamental tone established in [7] can be extended for self-adjoint elliptic operators L Φ . The lower bounds for the L Φ -fundamental tone of a domain Ω are given in terms of the divergence of vector fields. By carefully choosing a test vector field, we can obtain lower bounds for the L Φ -fundamental tone in terms of geometric invariants. This is done in Theorem (2.1). We consider an immersed hypersurface M into the (n + 1)-dimensional simply connected space form N n+1 (c) of constant sectional curvature c ∈ {1, 0, −1} with locally bounded (r + 1)-th mean curvature and such that a certain differential operator L r , r ∈ {0, 1, . . . , n} is elliptic, see [22]. Then we give lower bounds for the L r -fundamental tone of domains Ω ⊂ ϕ −1 (B N n+1 (c) (p, R)) in terms of the r-th and (r + 1)-th mean curvatures H r and H r+1 . This is done in Theorem (3.2). We then derive from this estimates three geometric corollaries (3.4, 3.5, 3.8) that should be viewed as an extension of Theorem 1 of [16]. There are related results due to Fontenele-Silva [12]. To finish the first part of the paper we consider immersed hypersurfaces M into N n+1 (c) such that the operators L r and L s , 0 ≤ r, s ≤ n are elliptic and we compare the L r and L s fundamental tones λ Lr (Ω), λ Ls (Ω) of domains Ω ⊂ M ⊂ N n+1 (c). In the second part of the paper we make an observation (Theorem 3.11) on the first nonzero eigenvalues of closed hypersurfaces. It follows that in order to get bounds for the eigenvalues of a self-adjoint elliptic differential operator L Φ 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 L r -eigenvalue of a closed hypersurface of N n+1 (c).

L Φ -fundamental tone estimates
Our main estimate is the following method for giving lower bounds for L Φ -fundamental tone of arbitrary domains of Riemannian manifolds. It extends the version of Barta's theorem [5] proved by Cheng-Yau in [11]. It is the same proof (with proper modifications) of a generalization of Barta's theorem proved in [7].

Theorem 2.1
Let Ω be a domain in a Riemannian manifold M and let Φ : Ω → End (T Ω) be a smooth symmetric and positive definite section of T Ω. Then the L Φ -fundamental tone of Ω has the following lower bound If Ω is bounded and with smooth boundary ∂Ω = ∅ then we have equality in (2).
Where X (Ω) is the set of all smooth vector fields on Ω.

Geometric applications
Let us consider the linearized operator L r of the (r + 1)-mean curvature H r+1 = S r+1 /( n r + 1 ) arising from normal variations of a hypersurface M immersed into the (n + 1)-dimensional simply connected space form N n+1 (c) of constant sectional curvature c ∈ {1, 0, −1} where S r+1 is the (r + 1)-th elementary symmetric function of the principal curvatures k 1 , k 2 , . . . , k n . Recall that the elementary symmetric function of the principal curvatures are given by Letting A = −(∇η) be the shape operator of M, where ∇ is the Levi-Civita connection of N n+1 (c) and η a globally defined unit vector field normal to M, we can recursively define smooth symmetric sections P r : M → End (T M ), for r = 0, 1, . . . , n, called the Newton operators, setting P 0 = I and P r = S r Id − AP r−1 so that P r (x) : T x M → T x M is a self-adjoint linear operator with the same eigenvectors as the shape operator A. The operator L r is the second order self-adjoint differential operator associated to the section P r . However, the sections P r may be not positive definite and then the operators L r may not be elliptic, see [22]. However, there are geometric hypothesis that imply the ellipticity of L r , see [9], [18], [4]. Here we will not impose geometric conditions to guarantee ellipticity of the L r , except in corollary (3.5). Instead we will ask the ellipticity on the set of hypothesis in the following way. It is known, see [17], that there is an open and dense subset U ⊂ M where the ordered eigenvalues {µ r 1 (x) ≤ . . . ≤ µ r n (x)} of P r (x) depend smoothly on x ∈ U and continuously on x ∈ M. In addition, the respective eigenvectors {e 1 (x), . . . , e n (x)} form a smooth orthonormal frame in a neighborhood of every point of U. Set ν(P r ) = sup x∈M {µ r n (x)} and µ(P r ) = inf x∈M {µ r 1 (x)}. Observe that if µ(P r ) > 0 then P r is positive definite, thus L r 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 ϕ : M ֒→ N of a Riemannian manifold N is said to have locally bounded (r + 1)-th mean curvature H r+1 if for any p ∈ N and R > 0, the Our next result generalizes in some aspects the main application of [6]. There the first and fourth authors give lower bounds for △-fundamental tone of domains in submanifolds with locally bounded mean curvature in complete Riemannian manifolds.
we have that iii. If c ≤ 0, h r+1 (p, R) = 0 and R > 0 we have that Moreover, there is a point x 0 ∈ N called the barycenter of ϕ(M) in N such that R e (M) = r(x 0 ).
be a complete oriented hypersurface with bounded (r + 1)-th mean curvature H r+1 for some r ≤ n − 1, R chosen as in Theorem (3.2). Suppose that µ(P r ) > 0 so that the L r operator is elliptic. Then M is not closed.
Remark 3.7 Jorge and Xavier, (Theorem 1 of [16]), 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. These inequalities should be also compared with a related result proved by Fontenele-Silva in [12]. (1), be an oriented closed hypersurface with µ r 1 (M) > 0 and H r+1 = 0. Then the extrinsic radius R e (M) ≥ π/2. Remark 3.9 An interesting question is: Is it true that any closed oriented hypersurface with µ r 1 (M) > 0 and H r+1 = 0 intersect every great circle? For r = 0 it is true and it was proved by T. Frankel [13].
We now consider immersed hypersurfaces ϕ : M ֒→ N n+1 (c) with L r and L s elliptic. We can compare the L r and L s fundamental tones of a domain Ω ⊂ M. In particular we can compare with its L 0 -fundamental tone.
Let Ω ⊂ M be a domain with compact closure and piecewise smooth non-empty boundary. Then the L r and L s fundamental tones satisfies the following inequalities Where λ Ls (Ω) and λ Lr (Ω) are respectively the first L s -eigenvalue and L r -eigenvalue of Ω. From (9) we have in particular that

Closed eigenvalue problem
Let M be a closed hypersurface of a simply connected space form N n+1 (c). Similarly to the eigenvalue problem of closed Riemannian manifolds, the interesting problem is what bounds can one obtain for the first nonzero L r -eigenvalue λ Lr 1 (M) in terms of the geometries of M and of the ambient space. Upper bounds for the first nonzero △-eigenvalue or even for the first nonzero L r -eigenvalue, r ≥ 1 have been obtained by many authors in contrast with lower bounds that are rare. For instance, Reilly [23] extending earlier result of Bleecker and Weiner [8] obtained upper bounds for λ △ 1 (M) of a closed submanifold M of R m in terms of the total mean curvature of M. Reilly's result applied to compact submanifolds of the sphere M ⊂ S m+1 (1), this later viewed as a hypersurface of the Euclidean space S m+1 (1) ⊂ R m+2 obtains upper bounds for λ △ 1 (M), see [2]. Heintze, [15] extended Reilly's result to compact manifolds and Hadamard manifolds M . In particular for the hyperbolic space H n+1 . The best upper bounds for the first nonzero △-eigenvalue of closed hypersurfaces M of H n+1 in terms of the total mean curvature of M was obtained by El Soufi and Ilias [25]. Regarding the L r operators, Alencar, Do Carmo, and Rosenberg [2] obtained sharp (extrinsic) upper bound the first nonzero eigenvalue λ Lr 1 (M) of the linearized operator L r of compact hypersurfaces M of R m+1 with S r+1 > 0. Upper bounds for λ Lr 1 (M) of compact hypersurfaces of S n+1 , H n+1 under the hypothesis that L r is elliptic were obtained by Alencar, Do Carmo, Marques in [1] and by Alias and Malacarne in [3] see also the work of Veeravalli [27]. On the other hand, lower bounds for λ Lr 1 (M) of closed hypersurfaces M ⊂ N n+1 (c) are not so well studied as the upper bounds, except for r = 0 in which case L 0 = △. In this paper we make a simple observation (Theorem 3.11) that to obtain lower and upper bounds for the L Φ -eigenvalues (Dirichlet or Closed eigenvalue problem) it is enough to obtain lower and upper bounds for the eigenvalues of Φ and for the eigenvalues for the Laplacian in the respective problem. When applied to the L r operators (supposing them elliptic) we obtain lower bounds for closed hypersurfaces of the space forms via Cheeger's lower bounds for the first △-eigenvalue of closed manifolds. Let {µ 1 (x) ≤ . . . ≤ µ n (x)} be the ordered eigenvalues of Φ(x). Setting ν(Φ) = sup x∈Ω {µ n (x)} and µ(Φ) = inf x∈Ω {µ 1 (x)} we have the following theorem.
if Ω is a closed manifold. Then λ L Φ (Ω) satisfies the following inequalities, where λ △ (Ω) is the △-fundamental tone of Ω or the first nonzero △-eigenvalue of Ω.

Proof of Theorem 2.1
Let Ω be an arbitrary domain, X be a smooth vector field on Ω and f ∈ C ∞ 0 (Ω). The vector field f 2 ΦX has compact support supp(f 2 ΦX) ⊂ supp(f ) ⊂ Ω. 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 λ Lr (Ω) this inequality above implies that When Ω is a bounded domain with smooth boundary ∂Ω = ∅ then λ Lr (Ω) = λ Lr 1 (Ω). This proof above shows that λ Lr Let v ∈ C 2 (Ω) ∩ C 0 (Ω) be a positive first L r -eigenfunction 3 of Ω and if we set X 0 = −grad log(v) we have that This proves (3).
4.2 Proof of Theorem 3.2 and Corollaries 3.4, 3.5, 3.8 We start this section stating few lemmas necessary to construct the proof of Theorem (3.2). The first lemma was proved in [19] for the Laplace operator and for the L r operator in [20] and [21]. We reproduce its proof to make the exposition complete.
Proof: Each P r is also associated to a second order self-adjoint differential operator defined by ✷f = Trace (P r Hess (f )) see [11], [14]. We have that ✷f = Trace (P r Hess (f )) = div (P r grad f ) − trace (∇P r ) grad f.
Rosenberg [24] proved that when the ambient manifold is the simply connected space form N n+1 (c) then Trace (∇P r ) grad ≡ 0, see also [22]. Therefore L r f = Trace (P r Hess (f )). Using Gauss equation to compute Hess (f ) we obtain where α(X, Y ), η = A(X), Y . Let {e i } be an orthonormal frame around p that diagonalize the section P r so that P r (x)(e i ) = µ r i (x)e i . Thus Substituting (19) into (20) we have that Here Hess f (X) = ∇ X grad f and Hess f (X, Y ) = ∇ X grad f, Y . 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.  ρ(x)), if sup γ K = k 2 1 and ρ < π/2k 1 .
In particular, if the Newton operator P r is positive definite then S r > 0.
Therefore we have that Setting e T i = grad ρ, e i grad ρ and e ⊥ i = e i − e T i , by the Hessian Comparison Theorem we have that and From (28) and (29) wee have that If c ≤ 0 then ρ · υ(ρ) ≥ 1 thus from (30) we have that If c > 0 then ρ · υ(ρ) = ρ · √ c · cot[ √ c ρ] ≤ 1 thus from (30) we have that To prove the Corollaries (3.4) and (3.5), observe that the hypotheses µ(P r )(M) > 0 (in Corollary 3.4) and H r+1 > 0 (in Corollary 3.5) imply that the L r 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 L r -fundamental tone would be zero and by Theorem

Proof of Theorem 3.10
Let ϕ : W ֒→ N 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 ⊂ W be a domain with compact closure and piecewise smooth nonempty boundary and suppose that the Newton operators P r and P s , 0 ≤ s, r ≤ n − 1 are positive definite when restricted to M. Let µ(r) = µ(P r , M), µ(s) = µ(P s , M) and ν(r) = ν(P r , M), ν(s) = ν(P s , M). Given a vector field X on M we can find a vector field Y on M such that P r X = κ · P s Y , κ constant. Now div (P r X) − |P 1/2 r X| 2 = κ · div (P s Y ) − P r X, X = κ · div (P s Y ) − κ 2 P s Y, P −1 r P s Y (33) = κ · div (P s Y ) − |P 1/2 Consider {e i } be an orthonormal basis such that P r e i = µ r i e i and P s e i = µ s i e i . Letting Y = n i=1 y i e i then ≥ 0, if κ ≤ µ(r) ν(s) Combining (33) with (34) and by Theorem (2.1) we have that for every 0 < κ ≤ µ(r) ν(s) . This proves (9).

Proof of Theorem 3.11
Recall that for any smooth symmetric section Φ : Ω → End (T Ω) there is an open and dense subset U ⊂ Ω where the ordered eigenvalues {µ 1 (x) ≤ . . . ≤ µ n (x)} of Φ(x) depend smoothly on x ∈ U and continuously in all Ω. In addition, the respective eigenvectors {e 1 (x), . . . , e n (x)} form a smooth orthonormal frame in a neighborhood of every point of U, see [17]. Let f ∈ C 2 0 (Ω)\{0} (f ∈ C 2 (Ω) with ∫ Ω f = 0) be an admissible function for (the closed L Φ -eigenvalue problem if Ω is a closed manifold) the Dirichlet L Φ -eigenvalue problem. It is clear that f is an admissible function for the respective △-eigenvalue problem. Writing grad f (x) = n i=1 e i (f )e i (x) we have that From (36) we have that and Taking the infimum over all admissible functions in (38) we obtain (11).