Spectral estimates and discreteness of spectra under Riemannian submersions

For Riemannian submersions, we establish some estimates for the spectrum of the total space in terms of the spectrum of the base space and the geometry of the fibers. In particular, for Riemannian submersions of complete manifolds with closed fibers of bounded mean curvature, we show that the spectrum of the base space is discrete if and only if the spectrum of the total space is discrete.


Introduction
The spectrum of the Laplacian on a Riemannian manifold is an isometric invariant whose relation with the geometry of the manifold is not comprehended completely. In particular, its behavior under maps between Riemannian manifolds, which respect the geometry of the manifolds to some extent, remains largely unclear. In this paper, we study the behavior of the spectrum under Riemannian submersions.
The notion of Riemannian submersion was introduced in the 1960s as a tool to study the geometry of a manifold in terms of the geometry of simpler components, namely, the base space and the fibers. Similarly to other geometric quantities, it is natural to describe the spectrum of the total space in terms of the geometry and the spectrum of the base space and the fibers. Of course, the term geometry of the fibers refers both to the intrinsic and the extrinsic geometry of the fibers as submanifolds of the total space. There are various results on the spectrum of closed (that is, compact and without boundary) total spaces, in case the submersion has totally geodesic, or minimal fibers, or fibers of basic mean curvature (cf. for instance [4] and the references therein). Our results focus mostly on the non-compact case, which is in general more complicated and less understood. To set the stage, let p : M 2 → M 1 be a Riemannian submersion and denote the fiber over x ∈ M 1 by F x := p −1 (x). In the first part of the paper, we establish a lower bound for the bottom of the spectrum of the total space, under the assumption that the (unnormalized) mean curvature H of the fibers is bounded in a specific way. In particular, we extend the recent result of [5] about Riemannian submersions. According to [5,Theorem 1.1], if M 1 is the m-dimensional hyperbolic space H m , and the mean curvature vector field of the fibers is bounded by H ≤ C ≤ m − 1, then the bottom of the spectrum of the Laplacian on M 2 satisfies It should be noticed that m − 1 = 2 √ λ 0 (H m ). This result is extended in [5,Theorem 5.1] to the case where the base manifold is Hadamard with sectional curvature bounded from above by a negative constant, or the base manifold is a warped product of some special form. In its general version, in the assumption and the conclusion of the above formulation, m − 1 is replaced by a certain lower bound for 2 √ λ 0 (M 1 ). Our first result generalizes this estimate in various directions and provides some information in the case where the equality holds and λ 0 (M 1 ) is an isolated point of the spectrum of the Laplacian on M 1 .
It should be emphasized that, in this theorem, there are no assumptions on the geometry or the topology of the base space. In particular, the manifolds involved in Theorem 1.1 do not have to be complete, which in the sequel allows us to derive a similar estimate involving the bottoms of the essential spectra, by exploiting the decomposition principle. Moreover, Theorem 1.1 gives a quite natural (and sharper than [5, Theorem 5.1]) estimate for submersions over negatively curved symmetric spaces and yields an analogous lower bound if the base manifold is a complete, negatively curved, locally symmetric space.
Conceptually, it seems interesting that the last term in the estimate of Theorem 1.1 shows up, while in [5] the intrinsic geometry of the fibers does not play any role. For example, equality in the estimate of Theorem 1.1 holds if M 2 is the Riemannian product M 1 × F for any Riemannian manifold F. For the aforementioned reason, equality in the estimate of [5, In general, it is quite important to establish lower bounds for the bottom of the spectrum, or even deduce whether it is zero or not. It follows from Theorem 1.1 that if p : M 2 → M 1 is a Riemannian submersion with minimal fibers, M 1 is closed, and λ 0 (M 2 ) = 0, then the bottom of the spectrum of almost any fiber is zero, since the spectrum of the Laplacian on M 1 is discrete. As Example 3.3 shows, in this case, the bottom of the spectrum of some fibers may be positive. In principle, in order to deduce that the bottom of the spectrum is positive, one needs information on the global geometry of the underlying manifold. However, in the above setting, we obtain that λ 0 (M 2 ) > 0 if λ 0 (F x ) > 0 for any x in a set of positive measure.
In the second part of the paper, we consider Riemannian submersions p : M 2 → M 1 of complete manifolds with closed fibers. Such submersions under further constraints, are studied in [2]. According to [2,Theorem 1], if the fibers are minimal, then the spectra and the essential spectra of the Laplacians satisfy σ (M 1 ) ⊂ σ (M 2 ) and σ ess (M 1 ) ⊂ σ ess (M 2 ). In this paper, we extend this result in a natural way. Instead of comparing the spectra of the Laplacians, we compare the spectrum of a Schrödinger operator on M 1 , whose potential is determined by the volume of the fibers, with the spectrum of the Laplacian on M 2 . To be more precise, let V (x) be the volume of the fiber over x ∈ M 1 , and consider the Schrödinger operator on M 1 . The potential of this operator measures the deviation of √ V from being harmonic. In particular, if the submersion has minimal fibers (or more generally, fibers of constant volume), then S coincides with the Laplacian on M 1 .
In the case where M 2 is closed and the submersion has fibers of basic mean curvature, Bordoni [3] considered the restriction c of the Laplacian acting on lifted functions and the restriction 0 of the Laplacian acting on functions whose average is zero on any fiber. In [3,Theorem 1.6], he showed that σ (M 2 ) = σ ( c ) ∪ σ ( 0 ). In this situation, the spectrum of S coincides with the spectrum of c . It should be noticed that expressing the latter one as the spectrum of a Schrödinger operator on the base manifold allows us to relate it more easily to the spectrum of the Laplacian on the base manifold.
For submersions of complete manifolds with closed fibers, we compare the bottoms of the (essential) spectra of S and of the Laplacian on the total space. If the submersion has fibers of basic mean curvature, we prove that the (essential) spectrum of S is contained in the (essential, respectively) spectrum of the Laplacian on M 2 . This is formulated in the following generalization of [2, Theorem 1]. If the submersion has closed fibers of bounded mean curvature, then it is easy to estimate the bottom of the (essential) spectrum of S in terms of the bottom of the (essential, respectively) spectrum of the Laplacian on M 1 and the mean curvature of the fibers. Hence, in this setting, we obtain the following application of Theorem 1.2.
Finally, we consider the problem of discreteness of spectra under Riemannian submersions. A Riemannian manifold M has discrete spectrum if the essential spectrum of the Laplacian on M is empty. Although there are many results establishing connections between the geometry of M and the discreteness of the spectrum of M (cf. for instance [2] and the references provided there), their relation is not comprehended completely.
In our context, there are examples of Riemannian submersions p : M 2 → M 1 of complete manifolds with closed fibers, where M 1 has discrete spectrum but M 2 does not, and conversely, where M 2 has discrete spectrum but M 1 does not (cf. [2,Subsection 4.2]). In [2], it is proved that if p has minimal fibers, then M 1 has discrete spectrum if and only if M 2 has discrete spectrum. As an application of Theorems 1.1 and 1.2, we extend this equivalence under the weaker assumption that the fibers have bounded mean curvature. The paper is organized as follows: in Sect. 2, we give some preliminaries involving the spectrum of Schrödinger operators, and recall some basic facts on Riemannian submersions. In Sect. 3, we study Riemannian submersions with fibers of bounded mean curvature and establish Theorem 1.1. In Sect. 4, we consider Riemannian submersions with closed fibers and prove Theorem 1.2. In this section, we also obtain Corollaries 1.3 and 1.4.

Preliminaries
Throughout this paper, manifolds are assumed to be connected and without boundary, unless otherwise stated. For a possibly non-connected Riemannian manifold M, we denote by the nonnegative definite Laplacian on M. A Schrödinger operator on M is an operator of the is densely defined, symmetric and bounded from below. Therefore, it admits Friedrichs extension. We denote the spectrum and the essential spectrum of its Friedrichs extension by σ (S) and σ ess (S), respectively, and their bottoms (that is, their minimums) by λ 0 (S) and λ ess 0 (S), respectively. In case of the Laplacian (that is, V = 0) these sets and quantities are denoted by σ (M), σ ess (M) and λ 0 (M), λ ess 0 (M), respectively. The spectrum of S is called discrete if σ ess (S) is empty. In this case, we have by definition that λ ess 0 (S) = + ∞. For a nonzero, compactly supported, Lipschitz function f on M, the Rayleigh quotient of f with respect to S is defined by The Rayleigh quotient of f with respect to the Laplacian is denoted by R( f ). The next characterization for the bottom of the spectrum of a Schrödinger operator follows from Rayleigh's Theorem and the fact that the Friedrichs extension of an operator preserves its lower bound (cf. for instance [13, Subsection 2.1] and the references therein).

Proposition 2.1 Let S be a Schrödinger operator on a Riemannian manifold M. Then the bottom of the spectrum of S is given by
where the infimum is taken over all f ∈ C ∞ c (M)\{0}, or over all f ∈ Lip c (M)\{0}.

Proposition 2.2 Let S be a Schrödinger operator on a Riemannian manifold M. Then for any sequence
Proof If the right-hand side is infinite, there is nothing to prove. If it is finite, we denote it by λ, and after passing to a subsequence, if necessary, we may assume that R S ( f n ) → λ. For any ε > 0, there exists n 0 ∈ N such that R S ( f n ) < λ + ε for any n ≥ n 0 . Consider the infinite dimensional space H ε spanned by { f n : n ≥ n 0 }. Any element g ∈ H ε \{0} is of the form g = n 0 +k n=n 0 a n f n for some k ∈ N and a n ∈ R, n 0 ≤ n ≤ n 0 + k. The assumption that the supports of f n are pairwise disjoint yields that Since ε > 0 is arbitrary, we conclude from [7, Proposition 2.1] that λ ess 0 (S) ≤ λ. Let ϕ be a positive, smooth function on M such that Sϕ = λϕ for some λ ∈ R. Denote by L 2 ϕ (M) the L 2 -space of M with respect to the measure ϕ 2 d Vol, where d Vol stands for the volume element of M induced from its Riemannian metric. It is immediate to verify that the map μ ϕ : The renormalization of S with respect to ϕ is defined by where S (F) is the Friedrichs extension of S considered as in (2), and D(·) denotes the domain of the operator. More details on the renormalization of Schrödinger operators may be found in [14,Section 7].
which shows that S ϕ is a weighted Laplacian on M. The Rayleigh quotient of a nonzero f ∈ C ∞ c (M) with respect to S ϕ is given by 3 For any f ∈ C ∞ c (M)\{0} and C ∈ R, we have that: Proof Both statements follow easily from the definition of S ϕ and the fact that μ ϕ is an isometric isomorphism.
We now consider Schrödinger operators on complete Riemannian manifolds. According to the next proposition, a Schrödinger operator on a complete Riemannian manifold is essentially self-adjoint; that is, the Friedrichs extension of S coincides with the closure of S considered as in (2). This allows us to characterize the spectrum of the operator in terms of compactly supported smooth functions.

Proposition 2.4 Let S be a Schrödinger operator on a complete Riemannian manifold M. Then S is essentially self-adjoint.
Proof By virtue of [10, Theorem 1] and (1), we have that there exists λ ∈ R and a positive ϕ ∈ C ∞ (M) such that Sϕ = λϕ. Denote by S ϕ the renormalization of S with respect to ϕ. Then [11,Theorem 2.2] implies that the operator is essentially self-adjoint, M being complete. Taking into account that this operator corresponds to S − λ [considered as in (2)] under the isometric isomorphism μ ϕ , we derive that S is essentially self-adjoint.
Let S be a Schrödinger operator on a complete Riemannian manifold M.
The next propositions follow from the decomposition principle [8] and the fact that the spectrum of a self-adjoint operator consists of approximate eigenvalues of the operator.

Proposition 2.5 Let S be a Schrödinger operator on a complete Riemannian manifold M,
and consider λ ∈ R. Then: where λ 0 (S, M\K n ) is the bottom of the spectrum of S on M\K n . In particular, the spectrum of S is discrete if and only if the right-hand side limit is infinite.
The following property of the bottom of the essential spectrum is an immediate consequence of Propositions 2.1 and 2.6.

Corollary 2.7 Let S be a Schrödinger operator on a complete Riemannian manifold M. Then there exists a sequence
, with supp f n pairwise disjoint, such that R S ( f n ) → λ ess 0 (S).

Riemannian submersions
Let M 1 , M 2 be Riemannian manifolds with dim(M 2 ) > dim(M 1 ). A surjective, smooth map p : M 2 → M 1 is called a submersion if its differential is surjective at any point y ∈ M 2 . For any x ∈ M 1 , the fiber F x := p −1 (x) over x is a possibly non-connected submanifold of M 2 . The kernel of p * y is called the vertical space at y, and is denoted by (T y M 2 ) v . Evidently, the vertical space at y is the tangent space of the fiber F p(y) . The horizontal space (T y M 2 ) h at y is defined as the orthogonal complement of the vertical space at y. The submersion p is called Riemannian submersion if the restriction p * y : (T y M 2 ) h → T p(y) M 1 is an isometry for any y ∈ M 2 . For more details on Riemannian submersions, see for example [9]. Let p : M 2 → M 1 be a Riemannian submersion. A vector field Y on M 2 is called horizontal (vertical) if Y (y) belongs to the horizontal (vertical, respectively) space at y for any y ∈ M 2 . It is clear that any vector field Y on M 2 is written uniquely as Y = Y h + Y v , with Y h horizontal and Y v vertical. Any vector field X on M 1 has a unique horizontal lift on M 2 , which is denoted byX ; that is,X is horizontal and p * X = X . A vector field Y on M 2 is called basic if Y =X for some vector field X on M 1 .
We denote by H the (unnormalized) mean curvature of the fibers, which is defined by where α(·, ·) is the second fundamental form of F p(y) , and {e i } k i=1 is an orthonormal basis of (T y M 2 ) v . The Riemannian submersion p has minimal fibers, fibers of basic mean curvature, or fibers of bounded mean curvature if H = 0, H is basic, or H is bounded, respectively. Given Proof Both statements follow from straightforward computations, which may be found for instance in [2, Subsection 2.2].
Recall that the fibers of a Riemannian submersion are submanifolds of the total space. This allows us to consider the spectrum of a fiber, with respect to the Riemannian metric inherited by the ambient space. In particular, we regard the bottom of the spectrum of the fiber as a function on the base space. According to the next lemma, this function is upper semi-continuous, while Example 3.3 demonstrates that it does not have to be continuous, even if the fibers are minimal. Proof Let C > 0 and x ∈ M 1 such that λ 0 (F x ) < C. We know from Proposition 2.1 that there exists f ∈ C ∞ c (F x )\{0} such that R( f ) < C. Observe that f can be extended to an f ∈ C ∞ c (M 2 ), and there exists an open neighborhood U of x such that f is nonzero on F y for any y ∈ U . Then R( f | F y ) depends continuously on y ∈ U , which shows that there exists an open neighborhood U of x such that R( f | F y ) < C for any y ∈ U . Applying Proposition 2.1 to the Riemannian manifold F y gives that λ 0 (F y ) < C for any y ∈ U , as we wished.

Submersions with fibers of bounded mean curvature
The aim of this section is to prove Theorem 1.1. Let p : M 2 → M 1 be a Riemannian submersion of possibly non-complete Riemannian manifolds. As in [2,3] It is worth to mention that in the published version of [2] there is a typo in this definition, which was meant to be as above, and is in this way in the arXiv version of [2]. Using the first variational formula (similarly to [2, Lemma 2.2] and [3, Formula (1.2)]), we have that f av ∈ C ∞ c (M 1 ) and its gradient is related to the gradient of f by for any x ∈ M 1 and X ∈ T x M 1 , whereX is the horizontal lift of X on F x . The pushdown of f on M 1 is given by This quantity was used by Bordoni to establish spectral estimates for submersions with minimal fibers, and M 2 closed (cf. [4, Section 3] and the references provided there). In the context of Riemannian coverings, a similar quantity was introduced in [1] to derive a spectral estimate and was used further in [13] to study coverings preserving the bottom of the spectrum.

Lemma 3.1
Consider f ∈ C ∞ c (M 2 ) and its pushdown g on M 1 . Then for any x ∈ M 1 with g(x) > 0, the gradient of g satisfies

In particular, g is Lipschitz, its gradient is defined almost everywhere and vanishes (if defined) in points where g is zero.
Proof Given x ∈ M 1 such that g(x) > 0, it is evident that g is differentiable at x. Consider an orthonormal basis {e i } m i=1 of T x M 1 , and denote byẽ i the horizontal lift of e i on F x , 1 ≤ i ≤ m. Using (4), we obtain that which proves the asserted inequality, because {ẽ i } m i=1 spans the horizontal space at each point of F x . Bearing in mind that g is continuous, and on the set where g is positive we have that g is differentiable with bounded gradient, it is easy to see that g is Lipschitz. The proof is completed by Rademacher's Theorem and the fact that g is nonnegative.

Proposition 3.2 Let p : M 2 → M 1 be a Riemannian submersion, such that the mean curvature of the fibers satisfies H
, with f L 2 (M 2 ) = 1, and its pushdown g ∈ Lip c (M 1 ). Then the Rayleigh quotients of f and g are related by

Proof
The assumption that f L 2 (M 2 ) = 1 yields that g L 2 (M 1 ) = 1. Lemma 3.1, together with the fact that H ≤ C and f L 2 (M 2 ) = 1, gives the estimate In view of Proposition 2.1, we have that Recall that at any point of M 2 , the tangent space of M 2 splits as the orthogonal sum of the horizontal and the vertical space. Since f L 2 (M 2 ) = 1, we deduce that where we applied Proposition 2.1 to the fibers. The conclusion is now a consequence of (5) and (6).
Suppose now that the equality holds and that λ 0 (M 1 ) / ∈ σ ess (M 1 ). By virtue of Proposition 2.1, we readily see that there exists Consider the sequence (g n ) n∈N ⊂ Lip c (M 1 ) consisting of the pushdowns of f n . It follows from Proposition 3.2 that for any ε > 0 there exists n 0 ∈ N such that for any n ≥ n 0 . Notice that the last term is nonnegative, and Proposition 2.1 gives the estimate √ R(g n ) ≥ √ λ 0 (M 1 ) ≥ C/2. Thus, (7) yields that R(g n ) → λ 0 (M 1 ). Since λ 0 (M 1 ) / ∈ σ ess (M 1 ), we derive from [13, Proposition 3.5] that after passing to a subsequence, if necessary, we may assume that g n → ϕ in L 2 (M 1 ) for some ϕ ∈ C ∞ (M 1 ) satisfying ϕ = λ 0 (M 1 )ϕ. Then [13,Proposition 3.7] implies that ϕ is positive. By Lemma 2.9, we know that for any c > 0 the set is closed, and in particular, measurable. For any n ≥ n 0 , we conclude from (7) that Then A c is of measure zero for any c > 0, since ϕ is positive in M 1 and ε > 0 is arbitrary. Hence, λ 0 (F x ) is equal to its infimum for almost any x ∈ M 1 .
The next example shows that in the second statement of Theorem 1.1, in general, we do not have that λ 0 (F x ) = 0 for any x ∈ M 1 , even if the base manifold is closed, the total space is complete and the fibers are minimal. We may also assume that the distance between any two different C(x n , 3r n ) is at least one. Consider the compactly supported, Lipschitz functions It is clear that grad f n vanishes almost everywhere outside B(x n , 2r n )\C(x n , r n ). The restriction of grad f n in B(x n , 2r n )\C(x n , r n ) can be extended to a nowhere vanishing, smooth vector field X n in B(x n , 3r n )\C(x n , r n /2) (for instance, − r −1 n grad d(·, x n ) is such an extension).
From the fact that the volume element of g t coincides with the volume element of g 0 , we derive that the Rayleigh quotient of f n with respect to the Laplacian corresponding to g t satisfies R g t ( f n ) = B(x n ,2r n )\C(x n ,r n ) grad g t f n 2 g t B(x n ,2r n ) f 2 n ≤ c n r −2 n (1 − χ n (t) + χ n (t)r 2 n /nc n ), (9) where we used that f n = 1 in B(x n , r n ). For t = 0, there exists n 0 ∈ N such that χ n (t) = 1 for any n ≥ n 0 . In view of Proposition 2.1, taking the limit as n → + ∞ in (9) gives that λ 0 (M, g t ) = 0 for any t = 0. Let q : R → S 1 = R/Z be the usual Riemannian covering. Consider the product manifold M × S 1 and the denote by p : M × S 1 → S 1 the projection to the second factor. Endow M × S 1 with the Riemannian metric g(x, y) = g t (x) × g S 1 (y), for x ∈ p −1 (q(t)) for some − 1/2 ≤ t ≤ 1/2, and y ∈ S 1 . Then p : M × S 1 → S 1 is a Riemannian submersion. Since the volume element of g t is independent from t, it is not hard to see that the fibers of p are minimal. Moreover, g coincides with the product metric g 0 × g S 1 outside the union of K n := C(x n , 3r n ) × S 1 with n ∈ N. Taking into account that the distance between different K n 's is at least one, and that (M × S 1 , g 0 × g S 1 ) is complete, it is not difficult to verify that (M × S 1 , g) is also complete.
We now discuss some straightforward applications of Theorem 1.1.
Proof According to McKean's Theorem [12], the bottom of the spectrum of M 1 is bounded by The asserted inequality is a consequence of Theorem 1.1.
It is worth to point out that a similar estimate (without the last term) may be derived from [5, Theorem 5.1]. However, Theorem 1.1 yields a sharper estimate than [5, Theorem 5.1] for submersions over negatively curved symmetric spaces.
It is well known that an m-dimensional negatively curved symmetric space, after rescaling its metric, is isometric to KH n , where K is the algebra of real, complex, quaternionic, or Cayley numbers. Here, n stands for the dimension of KH n over K, while its real dimension is m = nd, where d := dim R K. In the case where K is the algebra of Cayley numbers, we have that n = 2. The sectional curvature of KH n is bounded by − 4 ≤ K ≤ − 1, the exponential growth of KH n is given by μ(KH n ) = m + d − 2, and the bottom of the spectrum satisfies λ 0 (KH n ) = μ(KH n ) 2 /4.

Corollary 3.5 Let p : M 2 → KH n be a Riemannian submersion with fibers of bounded mean curvature H
Proof It follows immediately from Theorem 1.1.
A wider class of examples where Theorem 1.1 is applicable consists of submersions over complete, negatively curved, locally symmetric spaces. Any such space, after rescaling its Riemannian metric, is isometric to a regular quotient M = KH n / , where is a discrete group. According to the formulas of Sullivan [15] and Corlette [6], the bottom of the spectrum of M is given by where μ( ) is the exponential growth of . Theorem 1.1 can be applied to submersions over such a manifold M in the apparent way.

Submersions with closed fibers
Throughout this section, we consider a Riemannian submersion p : M 2 → M 1 of complete manifolds with closed fibers. We denote by V (x) the volume of the fiber F x over x ∈ M 1 , by H the mean curvature vector field of the fibers, and by X the smooth vector field on M 1 defined by Fix an open, bounded domain U of M 1 , and let f ∈ C ∞ c (M 2 ) with f = 1 in F y for any y ∈ U . Then f av = V in U , and (4) shows that grad V = −V X in U . It follows from the fact that U is arbitrary that grad V = −V X in M 1 . It is immediate to verify that Let S be the Schrödinger operator on M 1 defined by Observe that S √ V = 0, which allows us to consider the renormalization S √ V of S with respect to √ V .
Proof of Theorem 1. 2 Let f ∈ C ∞ c (M 1 )\{0} andf its lift on M 2 . By Lemmas 2.3 and 2.8, we deduce that This, together with Proposition 2.1, proves that λ 0 (M 2 ) ≤ λ 0 (S). We know from Corollary 2.7 that there exists ( f n ) n∈N ⊂ C ∞ c (M 1 )\{0}, with supp f n pairwise disjoint, such that R S ( f n ) → λ ess 0 (S). Then the liftsg n of g n := f n / √ V , also have pairwise disjoint supports. Taking into account Proposition 2.2 and (10), it is easy to see that which establishes the first assertion. Suppose now that the submersion has fibers of basic mean curvature. Consider λ ∈ R, f ∈ C ∞ c (M 1 )\{0}, andf its lift on M 2 . Using Lemma 2.8, formula (3), and the fact that grad V = −V p * H , we compute In view of Lemma 2.3, this implies that From Proposition 2.5, we obtain that for any λ ∈ σ ess (S), there exists a characteristic sequence ( f n ) n∈N ⊂ C ∞ c (M 1 ) for S and λ, with supp f n pairwise disjoint. Then (11) yields that the sequence (g n ) n∈N ⊂ C ∞ c (M 2 ), consisting of the lifts of g n := f n / √ V , is a characteristic sequence for and λ, with suppg n pairwise disjoint. We conclude from Proposition 2.5 that λ ∈ σ ess (M 2 ). The proof of σ (S) ⊂ σ (M 2 ) is similar.
Proof of Corollary 1.3 Let X be the vector field and S the Schrödinger operator defined in the beginning of this section. Notice that X ≤ C. Given f ∈ C ∞ c (M 1 ), with f L 2 (M 1 ) = 1, we have that (12), it is straightforward to verify that The proof is completed by Theorem 1.2.
Proof of Corollary 1. 4 Let C be the supremum of the norm of the mean curvature of the fibers. It follows from Corollary 1.3 that if the spectrum of M 1 is not discrete, then the spectrum of M 2 is not discrete. Conversely, suppose that M 1 has discrete spectrum, and let (K n ) n∈N be an exhausting sequence of M 1 consisting of compact subsets of M 1 . From Proposition 2.6, we readily see that (λ 0 (M 1 \K n )) n∈N is an increasing sequence that diverges. In particular, there exists n 0 ∈ N such that C ≤ 2 √ λ 0 (M 1 \K n ) for any n ≥ n 0 . Applying Theorem 1.1 to the restriction of p : M 2 \ p −1 (K n ) → M 1 \K n over any connected component of M 1 \K n gives that for any n ≥ n 0 . Observe that ( p −1 (K n )) n∈N is an exhausting sequence of M 2 consisting of compact subsets of M 2 , because p has closed fibers. In view of Proposition 2.6, taking the limit as n → + ∞ in (13), we derive that M 2 has discrete spectrum.
Finally, we present some basic examples where our results can be applied. We assume that the manifolds involved in these examples are complete. for some positive ψ ∈ C ∞ (M 1 ), where k is the dimension of the fiber (cf. for instance [9,Theorem 1.7]). It is immediate to obtain statements for Clairaut submersions with closed and connected fibers, analogous to the ones we established for warped products.
(iii) Let G be a compact and connected Lie group acting smoothly and freely via isometries on a Riemannian manifold M, with dim(M) > dim(G). Then it is not difficult to see that the projection p : M → M/G is a Riemannian submersion with closed fibers of basic mean curvature.