Spectral estimates and discreteness of spectra under Riemannian submersions
aa r X i v : . [ m a t h . DG ] O c t Spectral estimates and discreteness of spectra underRiemannian submersions
Panagiotis Polymerakis
Abstract
For Riemannian submersions, we establish some estimates for the spectrum ofthe total space in terms of the spectrum of the base space and the geometry of thefibers. In particular, for Riemannian submersions of complete manifolds with closedfibers of bounded mean curvature, we show that the spectrum of the base space isdiscrete if and only if the spectrum of the total space is discrete.
The spectrum of the Laplacian on a Riemannian manifold is an isometric invariant whoserelation with the geometry of the manifold is not comprehended completely. In particular,its behavior under maps between Riemannian manifolds, which respect the geometry ofthe manifolds to some extent, remains largely unclear. In this paper, we study the behaviorof the spectrum under Riemannian submersions.The notion of Riemannian submersion was introduced in the sixties as a tool to studythe geometry of a manifold in terms of the geometry of simpler components, namely, thebase space and the fibers. Similarly to other geometric quantities, it is natural to describethe spectrum of the total space in terms of the geometry and the spectrum of the basespace and the fibers. Of course, the term geometry of the fibers refers both to the intrinsicand the extrinsic geometry of the fibers as submanifolds of the total space. There arevarious results on the spectrum of closed total spaces, in case the submersion has totallygeodesic, or minimal fibers, or fibers of basic mean curvature (cf. for instance [4] and thereferences therein). Our results focus mostly on the non-compact case, which is in generalmore complicated and less understood.To set the stage, let p : M → M be a Riemannian submersion and denote by F x := p − ( x ) the fiber over x ∈ M . In the first part of the paper, we establish a lower Date:
October 3, 2019
Key words and phrases.
Bottom of spectrum, discrete spectrum, Riemannian submersion. H of the fibers is bounded in a specific way. Inparticular, we extend the recent result of [5] about Riemannian submersions. Accordingto [5, Theorem 1.1], if M is the m -dimensional hyperbolic space H m , and the meancurvature vector field of the fibers is bounded by k H k ≤ C ≤ m −
1, then the bottom ofthe spectrum of the Laplacian on M satisfies λ ( M ) ≥ ( m − − C ) . It should be noticed that m − p λ ( H m ). This result is extended in [5, Theorem5.1] to the case where the base manifold is Hadamard with sectional curvature boundedfrom above by a negative constant, or the base manifold is a warped product of somespecial form. In its general version, in the assumption and the conclusion of the aboveformulation, m − p λ ( M ). Our first resultgeneralizes this estimate in various directions, and provides some information in the casewhere the equality holds and λ ( M ) is an isolated point of the spectrum of the Laplacianon M . Theorem 1.1.
Let p : M → M be a Riemannian submersion, such that the mean cur-vature of the fibers satisfies k H k ≤ C ≤ p λ ( M ) . Then λ ( M ) ≥ ( p λ ( M ) − C/ + inf x ∈ M λ ( F x ) . If, in addition, the equality holds and λ ( M ) / ∈ σ ess ( M ) , then λ ( F x ) is almost everywhereequal to its infimum. It should be emphasized that, in this theorem, there are no assumptions on thegeometry or the topology of the base space. In particular, Theorem 1.1 gives a quitenatural (and sharper than [5, Theorem 5.1]) estimate for submersions over negativelycurved symmetric spaces, and yields an analogous lower bound if the base manifold is acomplete, negatively curved, locally symmetric space. Moreover, the manifolds involved inTheorem 1.1 do not have to be complete, which in the sequel allows us to derive a similarestimate involving the bottoms of the essential spectra, by exploiting the DecompositionPrinciple.Conceptually, it seems interesting that the last term in the estimate of Theorem 1.1shows up, while in [5] the intrinsic geometry of the fibers does not play any role. Forexample, equality in the estimate of Theorem 1.1 holds if M is the Riemannian product M × F for any Riemannian manifold F . For the aforementioned reason, equality in theestimate of [5, Theorem 1] holds for M = H k × F only for Riemannian manifolds F with λ ( F ) = 0.In general, it is quite important to establish lower bounds for the bottom of thespectrum, or even deduce whether it is zero or not. It follows from Theorem 1.1 that2f p : M → M is a Riemannian submersion with minimal fibers, M is closed, and λ ( M ) = 0, then the bottom of the spectrum of almost any fiber is zero, since thespectrum of the Laplacian on M is discrete. As Example 3.3 shows, in this case, thebottom of the spectrum of some fibers may be positive. In principle, in order to deducethat the bottom of the spectrum is positive, one needs information on the global geometryof the underlying manifold. However, in the above setting, we obtain that λ ( M ) > λ ( F x ) > x in a set of positive measure.In the second part of the paper, we consider Riemannian submersions p : M → M of complete manifolds with closed fibers. Such submersions under further constraints, arestudied in [2]. According to [2, Theorem 1], if the fibers are minimal, then the spectra andthe essential spectra of the Laplacians satisfy σ ( M ) ⊂ σ ( M ) and σ ess ( M ) ⊂ σ ess ( M ).In this paper, we extend this result in a natural way. Instead of comparing the spectraof the Laplacians, we compare the spectrum of a Schr¨odinger operator on M , whosepotential is determined by the volume of the fibers, with the spectrum of the Laplacianon M . To be more precise, let V ( x ) be the volume of the fiber over x ∈ M , and considerthe Schr¨odinger operator S := ∆ − ∆ √ V √ V on M . 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 constantvolume), then S coincides with the Laplacian on M .In the case where M is closed and the submersion has fibers of basic mean curvature,Bordoni [3] considered the restriction ∆ c of the Laplacian acting on lifted functions andthe restriction ∆ of the Laplacian acting on functions whose average is zero on anyfiber. In [3, Theorem 1.6], he showed that σ ( M ) = σ (∆ c ) ∪ σ (∆ ). In this situation, thespectrum of S coincides with the spectrum of ∆ c . It should be noticed that expressingthe latter one as the spectrum of a Schr¨odinger operator on the base manifold allows usto 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 ofthe (essential) spectra of S and of the Laplacian on the total space. If the submersion hasfibers of basic mean curvature, we prove that the (essential) spectrum of S is containedin the (essential, respectively) spectrum of the Laplacian on M . This is formulated inthe following generalization of [2, Theorem 1]. Theorem 1.2.
Let p : M → M be a Riemannian submersion of complete manifolds,with closed fibers. Then λ ( M ) ≤ λ ( S ) and λ ess0 ( M ) ≤ λ ess0 ( S ) . If, in addition, thefibers have basic mean curvature, then σ ( S ) ⊂ σ ( M ) and σ ess ( S ) ⊂ σ ess ( M ) . Finally, we consider the problem of discreteness of spectra under Riemannian sub-mersions. A Riemannian manifold M has discrete spectrum if the essential spectrum ofthe Laplacian on M is empty. Although there are many results establishing connections3etween the geometry of M and the discreteness of the spectrum of M (cf. for example[2] and the references provided there), their relation is not comprehended completely.In our context, there are examples of Riemannian submersions p : M → M ofcomplete manifolds with closed fibers, such that M has discrete spectrum and M doesnot have discrete spectrum, or M does not have discrete spectrum and M has discretespectrum (cf. [2, Subsection 4.2]). In [2], it is proved that if p has minimal fibers, then M has discrete spectrum if and only if M has discrete spectrum. As an application ofTheorems 1.1 and 1.2, we extend this equivalence under the weaker assumption that thefibers have bounded mean curvature. Corollary 1.3.
Let p : M → M be a Riemannian submersion of complete manifolds,with closed fibers of bounded mean curvature. Then M has discrete spectrum if and onlyif M has discrete spectrum. The paper is organized as follows: In Section 2, we give some preliminaries involvingthe spectrum of Schr¨odinger operators, and recall some basic facts on Riemannian sub-mersions. In Section 3, we study Riemannian submersions with fibers of bounded meancurvature and establish Theorem 1.1. In Section 4, we consider Riemannian submersionswith closed fibers and prove Theorem 1.2 and Corollary 1.3.
Acknowledgements.
I would like to thank Werner Ballmann and Dorothee Sch¨uthfor their helpful comments and remarks. I am also grateful to the Max Planck Institutefor Mathematics in Bonn for its support and hospitality.
Throughout this paper manifolds are assumed to be connected and without boundary,unless otherwise stated. For a possibly non-connected Riemannian manifold M , we denoteby ∆ the non-negative definite Laplacian on M . A Schr¨odinger operator on M is anoperator of the form S = ∆ + V , with V ∈ C ∞ ( M ), such that h Sf, f i L ( M ) ≥ c k f k L ( M ) (1)for some c ∈ R and any f ∈ C ∞ c ( M ). Then the operator S : C ∞ c ( M ) ⊂ L ( M ) → L ( M ) (2)is densely defined, symmetric and bounded from below. Therefore, it admits Friedrichsextension. We denote the spectrum and the essential spectrum of its Friedrichs extensionby σ ( S ) and σ ess ( S ), respectively, and their bottoms (that is, their minimums) by λ ( S )and λ ess0 ( S ), respectively. In case of the Laplacian (that is, V = 0) these sets and quantitiesare denoted by σ ( M ), σ ess ( M ) and λ ( M ), λ ess0 ( M ), respectively. The spectrum of S iscalled discrete if σ ess ( S ) is empty. In this case, we have by definition that λ ess0 ( S ) = + ∞ .4or a non-zero, compactly supported, Lipschitz function f on M , the Rayleigh quo-tient of f with respect to S is defined by R S ( f ) := R M ( k grad f k + V f ) R M f . The Rayleigh quotient of f with respect to the Laplacian is denoted by R ( f ). The nextcharacterization for the bottom of the spectrum of a Schr¨odinger operator follows fromRayleigh’s Theorem and the fact that the Friedrichs extension of an operator preservesits lower bound (cf. for instance [13, Subsection 2.1] and the references therein). Proposition 2.1.
Let S be a Schr¨odinger operator on a Riemannian manifold M . Thenthe bottom of the spectrum of S is given by λ ( S ) = inf f R S ( f ) , where the infimum is taken over all f ∈ C ∞ c ( M ) r { } , or over all f ∈ Lip c ( M ) r { } . Proposition 2.2.
Let S be a Schr¨odinger operator on a Riemannian manifold M . Thenfor any sequence ( f n ) n ∈ N ⊂ C ∞ c ( M ) r { } , with supp f n pairwise disjoint, we have that λ ess0 ( S ) ≤ lim inf n R S ( f n ) . Proof:
If the right hand side is infinite, there is nothing to prove. If it is finite, we denoteit 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 ∈ N such that R S ( f n ) < λ + ε for any n ≥ n . Consider theinfinite dimensional space H ε spanned by { f n : n ≥ n } . Any element g ∈ H ε r { } is ofthe form g = P n + kn = n a n f n for some k ∈ N and a n ∈ R , n ≤ n ≤ n + k . The assumptionthat the supports of f n are pairwise disjoint yields that R S ( g ) = P n + kn = n a n R M ( k grad f n k + V f n ) P n + kn = n a n R M f n ≤ max n ≤ n ≤ n + k R S ( f n ) < λ + ε. Since ε > λ ess0 ( S ) ≤ λ .Let ϕ be a positive, smooth function on M such that Sϕ = λϕ for some λ ∈ R .Denote by L ϕ ( M ) the L -space of M with respect to the measure ϕ d Vol, where d Volstands for the volume element of M induced from its Riemannian metric. It is immediateto verify that the map µ ϕ : L ϕ ( M ) → L ( M ), given by µ ϕ ( u ) = ϕu , is an isometricisomorphism. The renormalization of S with respect to ϕ is defined by S ϕ := µ − ϕ ◦ ( S ( F ) − λ ) ◦ µ ϕ , with D ( S ϕ ) := µ − ϕ ( D ( S ( F ) )) , S ( F ) is the Friedrichs extension of S considered as in (2), and D ( · ) denotes thedomain of the operator. More details on the renormalization of Schr¨odinger operatorsmay be found in [14, Section 7]. Given f ∈ C ∞ c ( M ), it is straightforward to compute S ϕ f = ∆ f − ϕ h grad ϕ, grad f i , (3)which shows that S ϕ is a weighted Laplacian on M . The Rayleigh quotient of a non-zero f ∈ C ∞ c ( M ) with respect to S ϕ is given by R S ϕ ( f ) := h S ϕ f, f i L ϕ ( M ) k f k L ϕ ( M ) = R M k grad f k ϕ R M f ϕ . Lemma 2.3.
For any f ∈ C ∞ c ( M ) r { } and C ∈ R , we have that:(i) R S ϕ ( f ) = R S ( ϕf ) − λ, (ii) k ( S ϕ − C ) f k L ϕ ( M ) = k ( S − λ − C )( ϕf ) k L ( M ) .Proof: Both statements follow easily from the definition of S ϕ and the fact that µ ϕ is anisometric isomorphism.We now consider Schr¨odinger operators on complete Riemannian manifolds. Accord-ing to the next proposition, a Schr¨odinger operator on a complete Riemannian manifoldis essentially self-adjoint; that is, the Friedrichs extension of S coincides with the closureof S considered as in (2). This allows us to characterize the spectrum of the operator interms of compactly supported smooth functions. Proposition 2.4.
Let S be a Schr¨odinger 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 apositive ϕ ∈ C ∞ ( M ) such that Sϕ = λϕ . Denote by S ϕ the renormalization of S withrespect to ϕ . Then [11, Theorem 2.2] implies that the operator S ϕ : C ∞ c ( M ) ⊂ L ϕ ( M ) → L ϕ ( M )is essentially self-adjoint, M being complete. Taking into account that this operatorcorresponds to S − λ (considered as in (2)) under the isometric isomorphism µ ϕ , wederive that S is essentially self-adjoint.Let S be a Schr¨odinger operator on a complete Riemannian manifold M . For λ ∈ R ,a sequence ( f n ) n ∈ N ⊂ C ∞ c ( M ) r { } is called characteristic sequence for S and λ , if k ( S − λ ) f n k L ( M ) k f n k L ( M ) → , as n → + ∞ . The next propositions follow from the Decomposition Principle [8] and the fact that thespectrum of a self-adjoint operator consists of approximate eigenvalues of the operator.6 roposition 2.5.
Let S be a Schr¨odinger operator on a complete Riemannian manifold M , and consider λ ∈ R . Then:(i) λ ∈ σ ( S ) if and only there exists a characteristic sequence for S and λ ,(ii) λ ∈ σ ess ( S ) if and only if there exists a characteristic sequence ( f n ) n ∈ N for S and λ ,with supp f n pairwise disjoint. Proposition 2.6 ([2, Proposition 3.2]) . Let S be a Schr¨odinger operator on a completeRiemannian manifold M , and ( K n ) n ∈ N an exhausting sequence of M consisting of compactsubsets of M . Then the bottom of the essential spectrum of S is given by λ ess0 ( S ) = lim n λ ( S, M r K n ) , where λ ( S, M r K n ) is the bottom of the spectrum of S on M r K n . In particular, thespectrum 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 immediateconsequence of Propositions 2.1 and 2.6.
Corollary 2.7.
Let S be a Schr¨odinger operator on a complete Riemannian manifold M . Then there exists ( f n ) n ∈ N ⊂ C ∞ c ( M ) r { } , with supp f n pairwise disjoint, such that R S ( f n ) → λ ess0 ( S ) . Let M , M be Riemannian manifolds with dim( M ) > dim( M ). A surjective, smoothmap p : M → M is called a submersion if its differential is surjective at any point y ∈ M .For any x ∈ M , the fiber F x := p − ( x ) over x is a possibly non-connected submanifoldof M . The kernel of p ∗ y is called the vertical space at y , and is denoted by ( T y M ) v .Evidently, the vertical space at y is the tangent space of the fiber F p ( y ) . The horizontalspace ( T y M ) h at y is defined as the orthogonal complement of the vertical space at y . Thesubmersion p is called Riemannian submersion if the restriction p ∗ y : ( T y M ) h → T p ( y ) M is an isometry for any y ∈ M . For more details on Riemannian submersions, see forexample [9].Let p : M → M be a Riemannian submersion. A vector field Y on M is called horizontal ( vertical ) if Y ( y ) belongs to the horizontal (vertical, respectively) space at y for any y ∈ M . It is clear that any vector field Y on M is written uniquely as Y = Y h + Y v , with Y h horizontal and Y v vertical. Any vector field X on M has a uniquehorizontal lift on M , which is denoted by ˜ X ; that is, ˜ X is horizontal and p ∗ ˜ X = X . Avector field Y on M is called basic if Y = ˜ X for some vector field X on M .We denote by H the (unnormalized) mean curvature of the fibers, which is definedby H ( y ) := k X i =1 α ( e i , e i ) , α ( · , · ) is the second fundamental form of F p ( y ) , and { e i } ki =1 is an orthonormalbasis of ( T y M ) v . The Riemannian submersion p has minimal fibers , fibers of basic meancurvature , or fibers of bounded mean curvature if H = 0, H is basic, or k H k is bounded,respectively.Given a function f : M → R , the function ˜ f := f ◦ p is called the lift of f on M .The next lemma provides a simple expression for the gradient and the Laplacian of alifted smooth function on M . Lemma 2.8.
Let p : M → M be a Riemannian submersion. Consider f ∈ C ∞ ( M ) andits lift ˜ f on M . Then we have that:(i) grad ˜ f = ^ grad f ,(ii) ∆ ˜ f = f ∆ f + h ^ grad f , H i .Proof: Both statements follow from straightforward computations, which may be foundfor 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 metricinherited by the ambient space. In particular, we regard the bottom of the spectrum ofthe fiber as a function on the base space. According to the next lemma, this functionis upper semi-continuous, while Example 3.3 demonstrates that it does not have to becontinuous, even if the fibers are minimal.
Lemma 2.9.
Let p : M → M be a Riemannian submersion. Then the function λ ( F x ) is upper semi-continuous with respect to x ∈ M .Proof: Let
C > x ∈ M such that λ ( F x ) < C . We know from Proposition 2.1 thatthere exists f ∈ C ∞ c ( F x ) r { } such that R ( f ) < C . Observe that f can be extended toan f ∈ C ∞ c ( M ), and there exists an open neighborhood U of x such that f is non-zero on F y for any y ∈ U . Then R ( f | F y ) depends continuously on y ∈ U , which shows that thereexists an open neighborhood U ′ of x such that R ( f | F y ) < C for any y ∈ U ′ . ApplyingProposition 2.1 to the Riemannian manifold F y gives that λ ( F y ) < C for any y ∈ U ′ , aswe wished. The aim of this section is to prove Theorem 1.1. Let p : M → M be a Riemanniansubmersion of possibly non-complete Riemannian manifolds. As in [2, 3], for f ∈ C ∞ c ( M ),its average f av on M is defined by f av ( x ) := Z F x f.
8t 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]. Usingthe first variational formula (similarly to [2, Lemma 2.2] and [3, Formula (1.2)]), we havethat f av ∈ C ∞ c ( M ) and its gradient is related to the gradient of f by h grad f av ( x ) , X i = Z F x h grad f − f H, ˜ X i (4)for any x ∈ M and X ∈ T x M , where ˜ X is the horizontal lift of X on F x . The pushdown of f on M is given by g ( x ) := p ( f ) av ( x ) = (cid:18)Z F x f (cid:19) / . This quantity was used by Bordoni to establish spectral estimates for submersions withminimal fibers, and M closed (cf. [4, Section 3] and the references provided there). Inthe context of Riemannian coverings, a similar quantity was introduced in [1] to derive aspectral estimate, and was used further in [13] to study coverings preserving the bottomof the spectrum. Lemma 3.1.
Consider f ∈ C ∞ c ( M ) and its pushdown g on M . Then for any x ∈ M with g ( x ) > , the gradient of g satisfies k grad g ( x ) k ≤ Z F x (cid:13)(cid:13) (grad f ) h − f H (cid:13)(cid:13) . In particular, g is Lipschitz, and its gradient vanishes at almost any point where g is zero.Proof: Given x ∈ M such that g ( x ) >
0, it is evident that g is differentiable at x .Consider an orthonormal basis { e i } mi =1 of T x M , and denote by ˜ e i the horizontal lift of e i on F x , 1 ≤ i ≤ m . Using (4), we obtain that h grad g ( x ) , e i i = 14 g ( x ) (cid:18)Z F x h grad f − f H, ˜ e i i (cid:19) = 1 g ( x ) (cid:18)Z F x f (cid:10) grad f − f H , ˜ e i (cid:11)(cid:19) ≤ g ( x ) (cid:18)Z F x f (cid:19) (cid:18)Z F x (cid:10) grad f − f H , ˜ e i (cid:11) (cid:19) = Z F x (cid:10) grad f − f H , ˜ e i (cid:11) , which proves the asserted inequality, because { ˜ e i } mi =1 spans the horizontal space at eachpoint of F x . Bearing in mind that g is continuous, and on the set where g is positive wehave 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 non-negative.9 roposition 3.2. Let p : M → M be a Riemannian submersion, such that the meancurvature of the fibers satisfies k H k ≤ C ≤ p λ ( M ) . Consider f ∈ C ∞ c ( M ) , with k f k L ( M ) = 1 , and its pushdown g ∈ Lip c ( M ) . Then the Rayleigh quotients of f and g are related by R ( f ) ≥ ( p R ( g ) − C/ + Z M λ ( F x ) g ( x ) dx. Proof:
The assumption that k f k L ( M ) = 1 yields that k g k L ( M ) = 1. Lemma 3.1,together with the fact that k H k ≤ C and k f k L ( M ) = 1, gives the estimate R ( g ) ≤ Z M (cid:13)(cid:13) (grad f ) h − f H (cid:13)(cid:13) ≤ Z M k (grad f ) h k + C Z M | f |k (grad f ) h k + C ≤ Z M k (grad f ) h k + C (cid:18)Z M k (grad f ) h k (cid:19) / + C . In view of Proposition 2.1, we have that C/ ≤ p λ ( M ) ≤ p R ( g ), which shows that Z M k (grad f ) h k ≥ ( p R ( g ) − C/ . (5)Recall that at any point of M , the tangent space of M splits as the orthogonal sumof the horizontal and the vertical space. Since k f k L ( M ) = 1, we deduce that Z M k (grad f ) h k = Z M k grad f k − Z M k (grad f ) v k = R ( f ) − Z M Z F x k grad( f | F x ) k dx ≤ R ( f ) − Z M λ ( F x ) Z F x f dx = R ( f ) − Z M λ ( F x ) g ( x ) dx, (6)where we applied Proposition 2.1 to the fibers. The conclusion is now a consequence of(5) and (6). Proof of Theorem 1.1:
Given x ∈ M , set Λ( x ) := λ ( F x ). From Proposition 2.1, weobtain that for any ε >
0, there exists f ∈ C ∞ c ( M ), with k f k L ( M ) = 1, such that R ( f ) < λ ( M ) + ε . Let g ∈ Lip c ( M ) be the pushdown of f . Propositions 2.1, 3.2, andthe fact that k g k L ( M ) = 1, imply that λ ( M ) + ε > ( p R ( g ) − C/ + Z M Λ g ≥ ( p λ ( M ) − C/ + inf M Λ , ε > λ ( M ) / ∈ σ ess ( M ). By virtue ofProposition 2.1, we readily see that there exists ( f n ) n ∈ N ⊂ C ∞ c ( M ), with k f n k L ( M ) = 1,such that R ( f n ) → λ ( M ) = ( p λ ( M ) − C/ + inf M Λ . Consider the sequence ( g n ) n ∈ N ⊂ Lip c ( M ) consisting of the pushdowns of f n . It followsfrom Proposition 3.2 that for any ε > n ∈ N such that ε > ( p R ( g n ) − C/ − ( p λ ( M ) − C/ + Z M (Λ − inf M Λ) g n (7)for any n ≥ n . Notice that the last term is non-negative, and Proposition 2.1 gives theestimate p R ( g n ) ≥ p λ ( M ) ≥ C/
2. Thus, (7) yields that R ( g n ) → λ ( M ). Accordingto [13, Propositions 3.5 and 3.7], after passing to a subsequence, if necessary, we mayassume that g n → ϕ in L ( M ) for some positive ϕ ∈ C ∞ ( M ) with ∆ ϕ = λ ( M ) ϕ .By Lemma 2.9, we know that for any c > A c := { x ∈ M : Λ( x ) ≥ inf M Λ + c } is closed, and in particular, measurable. For any n ≥ n , we conclude from (7) that ε ≥ Z M (Λ − inf M Λ) g n ≥ Z A c (Λ − inf M Λ) g n ≥ c Z A c g n → c Z A c ϕ , as n → + ∞ . Then A c is of measure zero for any c >
0, since ϕ is positive in M and ε > λ ( F x ) is equal to its infimum for almost any x ∈ M .The next example shows that in the second statement of Theorem 1.1, in general, wedo not have that λ ( F x ) = 0 for any x ∈ M , even if the base manifold is closed and thefibers are minimal. Example 3.3.
Let (
M, g ) be an m -dimensional, non-compact, complete Riemannianmanifold with m ≥ λ ( M, g ) >
0. Fix a diverging sequence ( x n ) n ∈ N ⊂ M and r n > C ( x n , r n ) are disjoint and the exponential map restrictedto the corresponding open ball exp : B (0 , r n ) ⊂ T x n M → M is injective for any n ∈ N .Consider the compactly supported, Lipschitz functions f n ( y ) = d ( y, x n ) < r n , − d ( y, x n ) /r n if r n ≤ d ( y, x n ) ≤ r n , d ( y, x n ) > r n . It is clear that grad f n vanishes almost everywhere outside B ( x n , r n ) r C ( x n , r n ). Therestriction of grad f n in B ( x n , r n ) r C ( x n , r n ) can be extended to a nowhere vanishing,11mooth vector field X n in B ( x n , r n ) r C ( x n , r n /
2) (for instance, − r − n grad d ( · , x n ) is suchan extension).For n ∈ N , consider a positive ϕ n ∈ C ∞ ( M ), with ϕ n ( y ) = 1 if d ( y, x n ) < r n / d ( y, x n ) > r n /
2, and ϕ n ( y ) = r n /nc n if r n < d ( y, x n ) < r n , where c n := max (cid:26) , Vol B ( x n , r n )Vol B ( x n , r n ) − (cid:27) . Let χ n : [ − / , / → [0 ,
1] be an even, smooth function with χ n (0) = 0 and χ n ( t ) =1 for | t | ≥ t n := min { , r n } / nc n . For each non-zero − / ≤ t ≤ /
2, define theRiemannian metric g t on M , which coincides with the original metric g outside theunion of B ( x n , r n ) r C ( x n , r n /
2) with n ∈ N , and in any B ( x n , r n ) r C ( x n , r n /
2) isgiven by g t ( Y, Z ) = (1 − χ n ( t ) + χ n ( t ) ϕ n ) − g ( X n , X n ) if Y = Z = X n , Z = X n and g ( Y, X n ) = 0 , (1 − χ n ( t ) + χ n ( t ) ϕ n ) / ( m − g ( Y, Z ) if g ( Y, X n ) = g ( Z, X n ) = 0 . for any tangent vectors Y, Z . It is elementary to compute k grad g t f n k g t = r − n (1 − χ n ( t ) + χ n ( t ) r n /nc n ) in B ( x n , r n ) r C ( x n , r n ) . (8)From the fact that the volume element of g t coincides with the volume element of g , wederive that the Rayleigh quotient of f n with respect to the Laplacian corresponding to g t satisfies R g t ( f n ) = R B ( x n , r n ) r C ( x n ,r n ) k grad g t f n k g t R B ( x n , r n ) f n ≤ c n r − n (1 − χ n ( t ) + χ n ( t ) r n /nc n ) , (9)where we used that f n = 1 in B ( x n , r n ). For t = 0, there exists n ∈ N such that χ n ( t ) = 1for any n ≥ n . In view of Proposition 2.1, taking the limit as n → + ∞ in (9) gives that λ ( M, g t ) = 0 for any t = 0.Let q : R → S = R / Z be the usual Riemannian covering. Consider the productmanifold M × S endowed with the Riemannian metric g ( x, y ) = g t ( x ) × g S ( y ), for x ∈ p − ( q ( t )) for some − / ≤ t ≤ /
2, and y ∈ S . Then the projection to the secondfactor p : M × S → S is a Riemannian submersion. Since the volume element of g t isindependent from t , it is not hard to see that the fibers of p are minimal.It remains to show that λ ( M × S , g ) = 0. To this end, consider h n ∈ Lip c ( M × S )defined by h n ( x, y ) := f n ( x ). Similarly to (9), using that χ n is even, 0 ≤ χ n ≤
1, and χ n ( t ) = 1 for | t | ≥ t n , we obtain that R ( h n ) R / − / R M k grad g t f n k g t dt R / − / R M f n dt ≤ c n r − n Z / (1 − χ n ( t ) + χ n ( t ) r n /nc n ) dt → , as n → + ∞ . We conclude from Proposition 2.1 that λ ( M × S , g ) = 0, while we havethat λ ( F q (0) ) >
0, and λ ( F y ) = 0 for any y ∈ S r { q (0) } .12e now discuss some straightforward applications of Theorem 1.1. Corollary 3.4.
Let p : M → M be a Riemannian submersion, where M is an m -dimensional Hadamard manifold of sectional curvature K ≤ − a for some a > . If themean curvature of the fibers satisfies k H k ≤ C ≤ ( m − a , then λ ( M ) ≥ (( m − a − C ) x ∈ M λ ( F x ) . Proof:
According to McKean’s Theorem [12], the bottom of the spectrum of M isbounded by λ ( M ) ≥ ( m − a . 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 derivedfrom [5, Theorem 5.1]. However, Theorem 1.1 yields a sharper estimate than [5, Theorem5.1] for submersions over negatively curved symmetric spaces.It is well known that an m -dimensional negatively curved symmetric space, afterrescaling its metric, is isometric to KH m , where K is the algebra of real, complex, quater-nionic, or Cayley numbers. In the latter case, we have that m = 2. The sectional curvatureof KH m is bounded by − ≤ K ≤ −
1, the exponential growth of KH m is given by µ ( KH m ) = m + d − , where d := dim R K , and the bottom of the spectrum satisfies λ ( KH m ) = µ ( KH m ) / Corollary 3.5.
Let p : M → KH m be a Riemannian submersion with fibers of boundedmean curvature k H k ≤ C ≤ m + d − . Then λ ( M ) ≥ ( m + d − − C/ + inf x ∈ M λ ( F x ) . Proof:
It follows immediately from Theorem 1.1.A wider class of examples where Theorem 1.1 is applicable consists of submersionsover complete, negatively curved, locally symmetric spaces. Any such space, after rescal-ing its Riemannian metric, is isometric to a regular quotient M = KH m / Γ, where Γ is adiscrete group. According to the formulas of Sullivan [15] and Corlette [6], the bottom ofthe spectrum of M is given by λ ( M ) = (cid:26) λ ( KH m ) if µ (Γ) ≤ µ ( KH m ) / ,µ (Γ)( µ ( KH m ) − µ (Γ)) if µ (Γ) ≥ µ ( KH m ) / , where µ (Γ) is the exponential growth of Γ. Theorem 1.1 can be applied to submersionsover such a manifold M in the apparent way.13 Submersions with closed fibers
Throughout this section we consider a Riemannian submersion p : M → M of completemanifolds with closed fibers. We denote by V ( x ) the volume of the fiber F x over x ∈ M ,by H the mean curvature vector field of the fibers, and by X the smooth vector field on M defined by X ( x ) := 1 V ( x ) Z F x p ∗ H for any x ∈ M .Fix an open, bounded domain U of M , and let f ∈ C ∞ c ( M ) with f = 1 in F y forany y ∈ U . Then f av = V in U , and (4) shows that grad V = − V X in U . It follows fromthe fact that U is arbitrary that grad V = − V X in M . It is immediate to verify that − ∆ √ V √ V = 14 k X k −
12 div X. Let S be the Schr¨odinger operator on M defined by S := ∆ − ∆ √ V √ V = ∆ + 14 k X k −
12 div X. Observe that S √ V = 0, which allows us to consider the renormalization S √ V of S withrespect to √ V . Proof of Theorem 1.2:
Let f ∈ C ∞ c ( M ) r { } and ˜ f its lift on M . By Lemmas 2.3 and2.8, we deduce that R ( ˜ f ) = R M k grad ˜ f k R M ˜ f = R M k grad f k V R M f V = R S √ V ( f ) = R S ( f √ V ) . (10)This, together with Proposition 2.1, proves that λ ( M ) ≤ λ ( S ). We know from Corollary2.7 that there exists ( f n ) n ∈ N ⊂ C ∞ c ( M ) r { } , with supp f n pairwise disjoint, such that R S ( f n ) → λ ess0 ( S ). Then the lifts ˜ g 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 λ ess0 ( M ) ≤ lim inf n R ( ˜ g n ) = lim inf n R S ( f n ) = λ ess0 ( S ) , which establishes the first assertion.Suppose now that the submersion has fibers of basic mean curvature. Consider λ ∈ R , f ∈ C ∞ c ( M ) r { } , and ˜ f its lift on M . Using Lemma 2.8, formula (3), and that14rad V = − V p ∗ H , we compute k (∆ − λ ) ˜ f k L ( M ) = Z M ( f ∆ f + h ^ grad f , H i − λ ˜ f ) = Z M (∆ f + h grad f, p ∗ H i − λf ) V = Z M (cid:18) ∆ f − √ V h grad f, grad √ V i − λf (cid:19) V = k ( S √ V − λ ) f k L √ V ( M ) . In view of Lemma 2.3, this implies that k (∆ − λ ) ˜ f k L ( M ) k ˜ f k L ( M ) = k ( S √ V − λ ) f k L √ V ( M ) k f k L √ V ( M ) = k ( S − λ )( f √ V ) k L ( M ) k f √ V k L ( M ) . (11)From Proposition 2.5, we obtain that for any λ ∈ σ ess ( S ), there exists a characteristicsequence ( f n ) n ∈ N ⊂ C ∞ c ( M ) for S and λ , with supp f n pairwise disjoint. Then (11)yields that the sequence (˜ g n ) n ∈ N ⊂ C ∞ c ( M ), consisting of the lifts of g n := f n / √ V , isa characteristic sequence for ∆ and λ , with supp ˜ g n pairwise disjoint. We conclude fromProposition 2.5 that λ ∈ σ ess ( M ). The proof of σ ( S ) ⊂ σ ( M ) is similar. Corollary 4.1.
Let p : M → M be a Riemannian submersion of complete manifolds,with closed fibers of bounded mean curvature k H k ≤ C . Then λ ( M ) ≤ ( p λ ( M ) + C/ and λ ess0 ( M ) ≤ ( p λ ess0 ( M ) + C/ . Proof:
Let X be the vector field and S the Schr¨odinger operator defined in the beginningof this section. Notice that k X k ≤ C . Given f ∈ C ∞ c ( M ), with k f k L ( M ) = 1, we havethat R S ( f ) = R ( f ) + 14 Z M k X k f − Z M f div X ≤ R ( f ) + C Z M |h grad f , X i|≤ R ( f ) + C C Z M | f |k grad f k≤ ( p R ( f ) + C/ , (12)where we used the divergence formula and the Cauchy-Schwarz inequality. By virtue ofProposition 2.1, estimate (12) shows that λ ( S ) ≤ ( p λ ( M )+ C/ . The first statementis now a consequence of Theorem 1.2. 15ccording to Corollary 2.7, there exists ( f n ) n ∈ N ⊂ C ∞ c ( M ), with supp f n pairwisedisjoint, k f n k L ( M ) = 1, and R ( f n ) → λ ess0 ( M ). Bearing in mind Proposition 2.2 and(12), it is straightforward to verify that λ ess0 ( S ) ≤ lim inf n R S ( f n ) ≤ ( p λ ess0 ( M ) + C/ . The proof is completed by Theorem 1.2.
Proof of Corollary 1.3:
Let C be the supremum of the norm of the mean curvature ofthe fibers. It follows from Corollary 4.1 that if the spectrum M is not discrete, then thespectrum of M is not discrete.Conversely, suppose that M has discrete spectrum, and let ( K n ) n ∈ N be an exhaustingsequence of M consisting of compact subsets of M . From Proposition 2.6, we readily seethat ( λ ( M r K n )) n ∈ N is an increasing sequence that diverges. In particular, there exists n ∈ N such that C ≤ p λ ( M r K n ) for any n ≥ n . Applying Theorem 1.1 to therestriction of p : M r p − ( K n ) → M r K n over any connected component of M r K n gives that λ ( M r p − ( K n )) ≥ ( p λ ( M r K n ) − C/ (13)for any n ≥ n . Observe that ( p − ( K n )) n ∈ N is an exhausting sequence of M consistingof compact subsets of M , because p has closed fibers. In view of Proposition 2.6, takingthe limit as n → + ∞ in (13), we derive that M has discrete spectrum.Finally, we present some basic examples where our results can be applied. We assumethat the manifolds involved in these examples are complete. Examples 4.2. (i) The warped product M = M × ψ F is the product manifold en-dowed with the Riemannian metric g N × ψ g F , where ψ ∈ C ∞ ( M ) is positive. Theprojection to the first factor p : M → M is a Riemannian submersion with fibers ofbasic mean curvature H = − k grad(ln ˜ ψ ) , where k = dim( F ). Suppose that F is closed, and consider the Schr¨odinger operator S := ∆ − ∆ ψ k/ ψ k/ on M . Taking into account Theorem 1.2, we deduce that σ ( S ) ⊂ σ ( M ) and σ ess ( S ) ⊂ σ ess ( M ). If, in addition, grad(ln ψ ) is bounded, then Corollary 1.3 im-plies that σ ess ( M ) = ∅ if and only if σ ess ( M ) = ∅ . It is worth to point out thatsurfaces of revolution are warped products of the form R × ψ S .(ii) A wider class of Riemannian submersions than warped products, consists of Clairautsubmersions , which were introduced by Bishop motivated by a result of Clairaut onsurfaces of revolution. A Riemannian submersion p : M → M is called Clairautsubmersion if there exists a positive f ∈ C ∞ ( M ), such that for any geodesic c M , the function ( f ◦ c ) sin θ is constant, where θ ( t ) is the angle between c ′ ( t )and ( T c ( t ) M ) h . Bishop showed that a Riemannian submersion p : M → M withconnected fibers, is a Clairaut submersion if and only if the fibers are totally umbilicalwith mean curvature H = − k grad(ln ˜ ψ )for some positive ψ ∈ C ∞ ( M ), where k is the dimension of the fiber (cf. for instance[9, Theorem 1.7]). It is immediate to obtain statements for Clairaut submersionswith closed and connected fibers, analogous to the ones we established for warpedproducts.(iii) Let G be a compact and connected Lie group acting smoothly and freely via isome-tries on a Riemannian manifold M , with dim( M ) > dim( G ). Then the projection p : M → M/G is a Riemannian submersion with closed fibers of basic mean curva-ture.
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