© Hindawi Publishing Corp. ON GROMOV’S THEOREM AND L 2-HODGE DECOMPOSITION

Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the corresponding L 2-harmonic sections. In particular, some known results concerning Gromov's theorem and the L 2-Hodge decomposition are considerably improved.


Introduction.
Recall that Hodge's decomposition theorem provides a representation of the de Rham cohomology by the space of harmonic forms over a compact Riemannian manifold.A useful consequence of this theorem is that the pth Betti number b p coincides with the space dimension of harmonic p-forms.This enables one to estimate b p using analytic approaches.A very famous result in the literature is the following Gromov's theorem [15] (see [5] for extensions to Riemannian vector bundles).Throughout the paper, let M be a connected complete Riemannian manifold of dimension d. (1.1) This theorem has already been extended to a Riemannian vector bundle of rank l (cf.[5] and the references therein).Furthermore, an explicit η has been provided by Gallot in [12,13] for such a theorem to hold.It is not difficult to see that the η given there decays at least exponentially fast in l 1/2 as l → ∞ (for the first Betti number, it decays exponentially fast in d 3/2 as d → ∞), see Remark 1.3 for details.In this paper, we provide a more explicit number η of order l −2 (see (1.5)).
Theorem 1.2.Let M be compact with diameter D. Let b( L) denote the space dimension of the L-harmonic space ker L := {f ∈ Ᏸ( L) : Lf = 0}.Assume that Ric − Hess V ≥ −K for some . (1.4) Consequently, b( L) ≤ l provided D 2 inf R > −η, where Remark 1.3.In the case where V = 0, it was proved by Bérard et al. [5] that there is η > 0 depending only on l, KD 2 , and More precisely, [13,Corollary 3.2] provided an explicit η := ε 2 c(l) −2 , where ε ≤ 1/2 is a positive constant depending only on d, and (see [12, page 333] and [13, page 365]) for large l.Therefore, if d > 1, then c(l) −2 is at least exponentially small in l 1/2 .In particular, for the first Betti number, one has l = d, and thus the η given in [12,13] has the main order exp[−d 3/2 ].On the other hand, Theorem 1.2 provides more explicit η of order l −2 .Finally, we mention that there exist examples to show that b 1 can be as big as one likes in the absence of any restriction on D 2 K. Also, the above theorem of Here and in what follows, σ (•) and σ ess (•) denote, respectively, the spectrum and the essential spectrum of a linear operator.This leads us to study the eigenvalue estimation in Section 2. In fact, this study should be interesting in itself.
On the other hand, however, when M is noncompact, it is interesting to study the finiteness of b( L).To show that b( L) is finite, it suffices to prove 0 ∉ σ ess ( L).Moreover, when − L is a weighted Hodge Laplacian on differential forms, the feature that 0 ∉ σ ess ( L) implies an L 2 -Hodge decomposition (see, e.g., [7,Theorem 5.10,Corollary 5.11]).Therefore, the results obtained in Section 2 also imply the following theorem which improves a result by Ahmed and Stroock [1] who used a different approach.For oriented M, let Ω = Λ p := Λ p T * M be the bundle of p-forms (i.e., the exterior p-bundle).
Theorem 1.4.Let M be noncompact and oriented.Let be the curvature term in the Weitzenböck formula on Ω := Λ p .Assume that µ(dx) := e V dx is a finite measure and − Hess V is bounded below.If there exists a positive function (1.9) (1.10) Remark 1.5.Ahmed and Stroock have proved (1.10) under some stronger conditions (cf.[1,Theorem 5.1]).Indeed, their conditions (e.g., (1.1) and the second part of (2.8) in [1]) imply that lim sup U →∞ (∆U/|∇U | 2 ) ≤ 0 which is stronger than (1.9).Moreover, their conditions also imply the ultracontractivity of the semigroup generated by ∆ + ∇V on M, which is rather restrictive so that some important models are excluded.For instance, Theorem 1.4 applies to V = −|x| 2 on M = R n , but [1, Theorem 5.1] does not since it is well known that the Ornstein-Uhlenbeck semigroup is not ultracontractive (see, e.g., [21] and the references therein).On the other hand, however, the Gaussian measure is crucial in infinite-dimensional analysis; in particular, it plays a role as the Riemann-Lebesgue measure does in finite dimensions, see [16,22] for details.
In Section 2, by virtue of semigroup domination and the super Poincaré inequality introduced in [28], estimates of eigenvalues obtained in [29] are extended to the present setting.Indeed, we are able to establish analogous results on Hilbert bundles which are included in the appendix at the end of the paper.For readers who do not care about Hilbert bundles, the appendix may be ignored since the account for vector bundles is self-contained.Nevertheless, the study of Hilbert bundles possesses its own interest from the perspective of functional analysis and operator algebra (cf.[24]).The proofs of Theorems 1.2 and 1.4 are presented in Section 3.

Spectrum estimates on Riemannian vector bundles.
Let {X i } be a locally normal frame and ∇ X i the usual covariant derivative along X i .Then the horizontal Laplacian reads = d i=1 ∇ 2 X i which is naturally defined on Γ (Ω).Let µ(dx) = e V (x) dx for some V ∈ C 2 (M), where dx denotes the Riemannian volume element.
To study the essential spectrum of L, we follow the line of [28] to use the following.

Donnelly-Li's decomposition principle. If R is bounded below, then σ ess ( L) = σ ess ( L| B c ) for any compact domain B, where L| B c denotes the restriction of L on B c with Dirichlet boundary conditions.
Although the principle in [10] was given for the Laplacian on functions, its proof indeed works also for our present case.To see this, let In order to study the spectrum of L, we compare Ᏹ(f , f ) with Ᏹ(|f |, |f |) and then use known results for functions.A convenient way to do so is to compare Pt with a semigroup on L 2 (µ).This trick has been widely used in spectral geometry, especially in the study of the Hodge Laplacian on differential forms over compact manifolds, see, for instance, [3,4] and the references therein.
(2) For any t > 0, Pt and P R t have smooth transition densities with respect to µ denoted by pt (x, y) and p R t (x, y), respectively, which satisfy pt (x, y) op ≤ p R t (x, y), x, y ∈ M, where pt op denotes the operator norm of the linear operator pt (x, y) : Ω y → Ω x . ( Proof. (1) By [3, Theorem 16] (cf.Theorem A.5 below), it suffices to show that for any f ,g e. and hence pointwise since f and g are continuous.Thus, f = |f |g/|g| on {|g| > 0}.By Kato's inequality, we have |∇|g|| ≤ |∇g|, µ-a.e. for all g ∈ Γ 0 (Ω) (cf.[3,Lemma VI.31] and its proof).Moreover, since any order derivatives of g are zero on {|g| = 0}, we obtain , by the argument in the proof of [9, Theorem 5.2.1], we conclude that Pt and P R t have smooth transition densities.Moreover, for any x, y ∈ M and any ω ∈ Ω y with |ω| y = 1, let f ∈ ᏹ be such that f (y) = ω and |f | = 1.Then pt (x, •)f (•) and p R t (x, •) are bounded and continuous in a neighborhood N y of y.Let {h n } be a sequence of nonnegative continuous functions with supports contained in N y such that h n µ → δ y weakly as n → ∞.By (2.3), we obtain (2.6) By letting n → ∞, we arrive at | pt (x, y)ω| x ≤ p R t (x, y).Therefore, p(x, y) op ≤ p R t (x, y). ( (2.7) From now on, we let P 0 t denote the strongly continuous semigroup on L 2 (µ) generated by ∆ + ∇V , and p 0 t (x, y) its transition density with respect to µ which is positive since To estimate the number b( L) by using Theorem 2.1, we need the following lemma.
Lemma 2.2.Let {f i } be an orthonormal family in Ᏸ( L) such that (2.8) Proof.For any 1 ≤ j ≤ l and x ∈ M, let where {e j } 1≤j≤l is an orthonormal basis on Ω x .We have Pt/2 g j ,e j (x) pt/2 (x, y)g j (y), e j (x) x µ(dy) The proof is completed by noting that (2.11) (3) Assume that R is bounded from below.Then λ ≥ inf σ ess (−(∆ + ∇V ) + R).Consequently, let ρ(x) be the Riemannian distance between x and a fixed point holds for all t > 0 and any compact set B ⊂ M. We now intend to show that p 0 t (x, x) ↓ 1 as t ↑ ∞ for all x ∈ M. Observing that then p 0 t (x, x) is decreasing in t.Next, noting that the Dirichlet form for ∆ + ∇V is irreducible since M is connected, we have P 0 t u − µ(u) L 2 (µ) → 0 as t → ∞ for any u ∈ L 2 (µ) (see, e.g., the appendix in [2]).For fixed x ∈ M, letting u(y) = p 0 1 (x, y), we obtain Therefore, p 0 t (x, x) → 1 as t → ∞.Now by first letting t → ∞ and then B → M, we obtain from (2.13) that n ≤ l, hence b( L) ≤ l since n is arbitrary.Moreover, for f ∈ ker L, we have (note that |f | is continuous) (3) By Donnelly-Li's decomposition principle mentioned above, we have and the same formula holds for L in place of −(∆ + ∇V ) + R. Then the proof is completed by Theorem 2.1(3) and by noting that inf σ ([−(∆+∇V We remark that Theorem 2.3(1) has already been known by Elworthy and Rosenberg [11] for differential forms.Moreover, as is well known in Hodge's theory, Theorem 2.3(2) is optimal in the sense that there exist examples such that b( L) = l and R ≥ 0; for instance, the Betti numbers on torus (see, e.g., [15]).
We are now ready to estimate λ n and then use the basic estimate (2.18) to obtain more estimates of b( L).
To conclude this section, we consider the following two examples on noncompact manifolds.
To prove Theorem 1.4, we fix p ∈ [0,d] ∩ Z + , and let Ω = Λ p be the exterior pbundle over the oriented manifold M. We have We have (see [7]) d * µ = δ − i ∇V and hence where ∆ p := δd+dδ and denote, respectively, the usual Hodge Laplacian and the curvature term involved in the Weitzenböck formula on p-forms, and L X = di X + i X d is the Lie differentiation in the direction X.Moreover, (∆ p µ , Γ 0 (Λ p )) is essentially selfadjoint, see, for example, [7, page 692].
Let {E j } d j=1 be a locally normal frame with dual {ω j } ⊂ Λ 1 .We have, for any differential form φ, Hess Hess V E i ,E j ω i ∧ i E j +∇ ∇V := Hess V +∇ ∇V . (3.9) Combining this with (3.7), we obtain . We have the following general result.
where we put sup ∅ = −∞ as usual, then im d p−1 is closed and (3.13) A linear operator P on L p H (µ) is called L p -uniformly integrable if {P f : µ(|f | p ) ≤ 1} is so.Moreover, denote by σ (P) and σ ess (P ), respectively, the spectrum and the essential spectrum of a linear operator P .
Let ( Ᏹ, Ᏸ( Ᏹ)) be a positive-definite symmetric closed form on L 2 H (µ), and let Pt and ( L, Ᏸ( L)) denote, respectively, the associated contraction semigroup and its generator.It is well known that L is selfadjoint and negative defined on L 2 H (µ), and (cf.[17] or [18]) We will study the essential spectrum of L by using the following Poincaré-type inequality: where r 0 ≥ 0 is a constant and β is a positive function defined on (r 0 , ∞).We may assume that β in (A.4) is decreasing since the inequality remains true with β replaced by β(r ) := inf{β(s) : s ∈ (r 0 ,r ]} for r > r 0 .A key step of the study is the following lemma, which extends Lemma 3.1 in [14] and hence an earlier result due to Wu [30,31].where p(x, y) * is the adjoint operator of p(x, y).
Proof.We will use the following Bourbaki theorem (see [6, page 112])."A bounded set in the dual space B of a separable Banach space B is compact and metrisable with respect to the weak topology σ (B ,B)."If P (A) is not relatively compact in L p H (µ), then there exist ε > 0 and a sequence {f n } ∞ n=1 ⊂ A such that P f n − P f m L p H (µ) ≥ ε, n = m.Since P is bounded and A is L p -uniformly integrable, we may take K > 0 such that where f n,K = f n 1 {|fn|≤K} .We fix a version of f n for each n and let Ᏺ be the µ-completion of the σ -field σ ({ f n,K , e j : n, j ≥ 1}) which is µ-separable.Let Let L p H (µ) be defined as L p H (µ) for Ᏺ and ᏹ in place of Ᏺ and ᏹ, respectively, which is separable since Ᏺ is µ-separable.For f ∈ L 1 H (µ), let µ(f |Ᏺ ) := j µ( f ,e j |Ᏺ )e j , where µ(•|Ᏺ ) is the conditional expectation with respect to µ under Ᏺ .By Bourbaki theorem with H (µ). Then for µ-a.e.x, P f (x) − P f n i ,K (x) (A.9) By (A.6) and the dominated convergence theorem, we obtain, for µ-a.e.x, . This is a contradiction to (A.7).
Directly following an argument in [14] (see also [29,Theorem 3.1]), we obtain the following result.
Lemma A.3.Let λ and {λ n } be as above.Assume that Pt has transition density pt (x, y) and there exists l ∈ N such that dim H x = l, µ-a.e.x.Let {f i } be the family of normalized Proof.For any i, j ∈ N with i, j ≤ l, let µ ij be a set function on Ᏺ × Ᏺ defined by where (A.17) Next, let the additional conditions hold.For x ∈ E, y in the support of µ, and any ω ∈ H y with |ω| y = 1, let e ∈ ᏹ be such that e(y) = ω, |e| = 1, and p(x, •)e(•) and p(x, •) are bounded and continuous in a neighborhood N y of y.Let {f n } be a sequence of nonnegative continuous functions with supports contained in N y such that f n µ → δ y weakly as n → ∞.We have

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Theorem 1 . 1 (
Gromov's theorem).If M is compact and oriented with diameter D, then there is a positive constant η depending only on d such that b 1 ≤ d provided D 2 Ric ≥ −η.
denote all eigenvalues of − Lc counting multiplicity.Since − L ≥ − Lc , it follows from the max-min principle that λ n ≥ λ c n for all n ≥ 1 (see, e.g., [20, problem 1, page 364]).Therefore, it suffices to prove (2.18) for R ≡ −c.In this case, if f i are the L 2 -unit eigenvectors for λ i , then, by Lemma 2.2, A.1.Assume that µ is a probability measure and p ≥ 1 is fixed.Let P be a bounded linear operator on L A 1 = {x ∈ E : there exists y ∈ E such that (x, y) ∈ A} and A(x) = {y ∈ E : (x, y) ∈ A}.Since P is bounded, it is easy to check that µ ij is a signed measure.Moreover, by Jordan's decomposition theorem and that| P f | ≤ P |f | for any f ∈ L 2 (µ), we have |µ ij | := (µ ij ) + +(µ ij ) − ≤ pµ ×µ.Then µ ij is absolutely continuous with respect to µ × µ with density p ij satisfying |p ij | ≤ p. Define p(x, y) : H y → H x by op ≤ lp(x, y) for any x, y ∈ E and p is a transition density of P .Indeed, for any f ,g ∈ L 2H (µ), we haveg, P f L 2 H (µ) = E p(•, y)f (y)µ(dy)