On Gromov's theorem and L 2 Hodge decomposition

PII. S0161171204210365 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON GROMOV’S THEOREM AND

L

_{-HODGE DECOMPOSITION}

FU-ZHOU GONG and FENG-YU WANG

Received 22 October 2002

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 correspondingL2-harmonic sections. In particular, some known results concerning Gromov’s theorem and theL2_{-Hodge decomposition are}

considerably improved.

2000 Mathematics Subject Classification: 58J50, 58J65.

1. Introduction. Recall that Hodge’s decomposition theorem provides a represen-tation of the de Rham cohomology by the space of harmonic forms over a compact Riemannian manifold. A useful consequence of this theorem is that thepth Betti num-berbp coincides with the space dimension of harmonicp-forms. This enables one to estimatebpusing analytic approaches. A very famous result in the literature is the fol-lowing Gromov’s theorem [15] (see [5] for extensions to Riemannian vector bundles). Throughout the paper, letMbe a connected complete Riemannian manifold of dimen-siond.

Theorem1.1(Gromov’s theorem). IfM is compact and oriented with diameterD,

then there is a positive constantηdepending only ondsuch that

b1≤d providedD2Ric≥ −η. (1.1)

This theorem has already been extended to a Riemannian vector bundle of rankl

(cf. [5] and the references therein). Furthermore, an explicitηhas been provided by Gallot in [12,13] for such a theorem to hold. It is not difficult to see that theηgiven there decays at least exponentially fast inl1/2_{asl→ ∞(for the first Betti number, it} decays exponentially fast ind3/2_{asd→ ∞), seeRemark 1.3for details. In this paper,} we provide a more explicit numberηof orderl−2_{(see (1.5)).}

LetΩbe a Riemannian vector bundle of ranklover the manifoldM. Denote byᏹ, Γ(Ω), andΓ0(Ω), respectively, the measurable (with respect to thedx-complete Borel

σ-field), the smooth, and the compactly supported smooth sections of Ω. Let ·,·x denote the inner product onΩx, and forf∈ᏹ, let|f|(x)= |f (x)|x:= f (x), f (x)1/2

x . ForV∈C2_{(M), we consider the operator}

˜

L:=+∇∇V−R, (1.2)

where dx denotes the Riemannian volume element. We assume that (˜L,Ᏸ(˜L)) is a bounded-above selfadjoint operator onL2

Ω(µ):= {f ∈ᏹ:µ(|f|2):=

M|f|2dµ <∞} with the usual inner product. In particular, ˜Lis bounded above when R is bounded below. Finally, let

R(x)=infRω, ωx:ω∈Ωx,|ω|x=1

, x∈M. (1.3)

Our first result is the following theorem.

Theorem_1.2. _{LetMbe compact with diameterD. Letb(}˜_{L)denote the space}

dimen-sion of the˜L-harmonic spaceker ˜L:= {f∈Ᏸ(L)˜ : ˜Lf=0}. Assume thatRic−HessV≥ −K for someK≥0. IfR≥ −cfor somec∈R, then

bL˜≤linf t>0e

cD2_(t+1) 1+ 1

4texp

_KD2

8 +

KD2

1−e−KD2 . (1.4)

Consequently,b(˜L)≤lprovidedD2_{infR >−η, where}

η:=sup t>0

1 1+tlog

l+1

l1+(1/4t)expKD2_1/8+1/1−e−KD2

≥ 1

1+lexpKD2_1/8+1/1−e−KD2log l+1

l+1/4.

(1.5)

In particular, whenMis oriented,b1≤dif there existsV∈C2(M)such that

D2_{Ric−HessV≥ −}log

(d+1)/(d+1/4)

1+d(1+1/d)d+9/8 . (1.6)

Remark1.3. In the case whereV=0, it was proved by Bérard et al. [5] that there isη >0 depending only onl, KD2_{, anddsuch thatb(˜L)≤lprovidedD}2_{infR >−η.} More precisely, [13, Corollary 3.2] provided an explicitη:=ε2_c(l)−2_{, whereε≤1/2 is a} positive constant depending only ond, and (see [12, page 333] and [13, page 365])

c(l) >2l−1/2

l1/2_/2

0 (cosht)

d−1_dt=Ol−1/2_exp(d−1)l1/2

2 (1.7)

for large l. Therefore, ifd >1, thenc(l)−2 _{is at least exponentially small inl}1/2_{. In} particular, for the first Betti number, one hasl=d, and thus theηgiven in [12,13] has the main order exp[−d3/2_{]. On the other hand,Theorem 1.2provides more explicitηof} orderl−2_{. Finally, we mention that there exist examples to show thatb}

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 27

The proof ofTheorem 1.2is based on lower bound estimates of eigenvalues of−L˜. Indeed, when, for example,σess(˜L)= ∅, lettingλ1≤λ2≤ ···denote the eigenvalues of −˜_{Lcounting multiplicity, one has}

b˜L≤infn−1 :λn>0. (1.8)

Here and in what follows,σ (·)andσess(·)denote, respectively, the spectrum and the essential spectrum of a linear operator. This leads us to study the eigenvalue estimation inSection 2. In fact, this study should be interesting in itself.

On the other hand, however, whenM is noncompact, it is interesting to study the finiteness ofb(L)˜. To show that b(˜L)is finite, it suffices to prove 0∉σess(˜L). More-over, when −˜Lis a weighted Hodge Laplacian on differential forms, the feature that 0∉σess(˜L)implies anL2-Hodge decomposition (see, e.g., [7, Theorem 5.10, Corollary 5.11]). Therefore, the results obtained inSection 2also imply the following theorem which improves a result by Ahmed and Stroock [1] who used a different approach. For orientedM, letΩ=Λp_:=Λp_T∗_{Mbe the bundle ofp-forms (i.e., the exteriorp-bundle).}

Consider∆pµ=d∗µd+ddµ∗, whered∗µis theL2Ω(µ)-adjoint of the exterior derivatived. Let

(∆pµ,Ᏸ(∆pµ)),((d∗µ)p,Ᏸ((d∗µ)p)), and(dp,Ᏸ(dp))denote, respectively, the correspond-ing operators onL2

Λp(µ)with domains.

Theorem1.4. LetM be noncompact and oriented. Let᏾be the curvature term in

the Weitzenböck formula onΩ:=Λp_{. Assume thatµ(dx):=e}V_{dx is a finite measure}

and᏾−HessV is bounded below. If there exists a positive functionU∈C2_{(M)such that}

U+Vis bounded,{U≤N}is compact for eachN >0,|∇U| → ∞asU→ ∞, and

lim sup U→∞

∆U

|∇U|2<1, (1.9)

thenσess(∆ p

µ)= ∅. Consequently,imdp−1is closed and

L2_Λp(µ)=imd∗µ

p+1

Ᏸ((d∗µ)p+1₎⊕imdp−1_Ᏸ(dp−1₎⊕ker∆µp_Ᏸ(∆pµ), α:=infφ,∆pµφ_L2

Λp(µ):φ⊥ker∆

p µ_Ᏸ(∆p

µ),φL2_Λp(µ)=1

>0.

(1.10)

Remark _1.5. _{Ahmed and Stroock have proved (1.10) under some stronger} condi-tions (cf. [1, Theorem 5.1]). Indeed, their conditions (e.g., (1.1) and the second part of (2.8) in [1]) imply that lim supU→∞(∆U /|∇U|2)≤0 which is stronger than (1.9).

More-over, their conditions also imply the ultracontractivity of the semigroup generated by ∆+∇VonM, which is rather restrictive so that some important models are excluded. For instance,Theorem 1.4applies toV= −|x|2_onM=Rn_{, but [1, Theorem 5.1] does} not since it is well known that the Ornstein-Uhlenbeck semigroup is not ultracontrac-tive (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.

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 Theorems1.2and1.4are presented inSection 3.

2. Spectrum estimates on Riemannian vector bundles. Let{Xi}be a locally normal frame and∇Xi the usual covariant derivative alongXi. Then the horizontal Laplacian

reads=d

i=1∇2Xi which is naturally defined onΓ(Ω). Letµ(dx)=e

V (x)_{dxfor some}

V∈C2_{(M), wheredxdenotes the Riemannian volume element.} Consider the operator

˜

L=+∇∇V−R, (2.1)

where R is a symmetric measurable endomorphism of Ω such that (˜L,Γ0(Ω))is es-sentially selfadjoint onL2_Ω(µ)and is bounded above, that is,−˜Ᏹ(f , f ):=µ(f ,˜Lf)≤ Cµ(|f|2_{)for someC ∈Rand allf∈Γ}

0(Ω). Recall thatf , gx:= f (x), g(x)x and

µ(u):=Mudµ for any x ∈M, f , g ∈ Ω, and any u∈ L1(µ). Let (˜L,Ᏸ(˜L)) be the unique selfadjoint extension of (˜L,Γ0(Ω))which is also bounded above. Let R(x)= inf{Rω, ωx:ω∈Ωx,|ω| =1}, x∈M. We assume thatR∈L1loc(dx)and there exists

C≥0 such thatµ(|∇u|2_)+µ(Ru2_{)≥ −Cµ(u}2_{)for allu∈C}∞

0(M). Then the following form is closable and let(Ᏹ,Ᏸ(Ᏹ))denote its closure (see, e.g., [17, Corollary VI.1.28]):

Ᏹ(u, v):=µ∇u,∇v+µRuv, u, v∈C₀∞(M). (2.2)

Note that the boundedness from below of(Ᏹ, C0∞(M))does not imply that ofR, see, for example, [19, Remark 2.4]. Let(L,Ᏸ(L))be the smallest closed extension (i.e., the Friedrichs extension) of (∆+ ∇V−R, C₀∞(M)) (which is selfadjoint by [17, Theorem VI.2.6]) andPtRthe corresponding strongly continuous semigroup. Below, we will write

L=∆+∇V−Rfor simplicity.

To study the essential spectrum of ˜L, we follow the line of [28] to use the following.

Donnelly-Li’s decomposition principle. _{IfRis bounded below, thenσess(}˜_L)=

σess(˜L|Bc)for any compact domainB, whereL˜|Bcdenotes the restriction of˜LonBcwith 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{fn}∞

n=1⊂Ᏸ(˜L)be such that

µ(|fn|2_{)=1 and ˜Ᏹ(f}

n, fn)≤c1for somec1>0 and alln≥1; we have supnµ(|∇fn|2) < ∞ sinceR is bounded from below. Moreover, sinceV is locally bounded, we obtain supn

B|∇fn|2dx <∞. Therefore, by Sobolev embedding theorem,{1Bfn}is relatively compact onL2

Ω(µ)(recall that suppfn⊂BandVis bounded onB). Hence Donnelly-Li’s argument applies.

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 29

in the study of the Hodge Laplacian on differential forms over compact manifolds, see, for instance, [3,4] and the references therein.

Theorem2.1. (1)For anyt >0,

P˜tf≤PtR|f|, f∈L2Ω(µ). (2.3)

(2)For anyt >0,P˜tandP R

t have smooth transition densities with respect toµdenoted

byp˜t(x, y)andpRt(x, y), respectively, which satisfyp˜t(x, y)op≤ptR(x, y),x, y∈M,

wherep˜topdenotes the operator norm of the linear operatorp˜t(x, y):Ωy→Ωx. (3)For anyf∈Γ0(Ω), one hasᏱ(f , f )˜ ≥Ᏹ(|f|,|f|).

Proof. _{(1) By [3, Theorem 16] (cf.Theorem A.5below), it suffices to show that for} anyf , g∈Γ0(Ω)satisfyingf , gL2

Ω(µ)=µ(|f|·|g|), one has

_{Ᏹ(f , g)}˜ _{≥Ᏹ|f|,|g|. (2.4)}

Sincef , g ≤ |f|·|g|butf , gL2

Ω(µ)=µ(|f|·|g|), we havef , g = |f|·|g|,µ-a.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 allg∈Γ0(Ω)(cf. [3, Lemma VI.31] and its proof). Moreover, since any order derivatives ofg are zero on{|g| =0}, we obtain

˜

Ᏹ(f , g)= −µf ,˜Lg≥ −µf ,+∇∇V

g+µR|f|·|g|

=µ

1_{|g|>0}

∇|f_|g|g_| ,∇g +µR|f|·|g|

=µ

1{|g|>0}|_|f_g|_||∇g|2

+1₂µ1{|g|>0}∇∇(|f|/|g|)g, g+µR|f|·|g|

≥µ

1{|g|>0}|f|·|∇|g||

2

|g| +µ

|g|

∇|f|

|g|,∇|g| +µ

R|f|·|g|

=µ∇|f|,∇|g|+µR|f|·|g| =Ᏹ|f|,|g|.

(2.5)

Therefore (2.4) holds.

with supports contained inNysuch thathnµ→δyweakly asn→ ∞. By (2.3), we obtain

_Mp˜t(x, z)f (z)hn(z)µ(dz)

_x

=_P˜_thnf(x)x≤PR

thn(x)=

Mp R

t(x, z)hn(z)µ(dz).

(2.6)

By letting n→ ∞, we arrive at |p˜t(x, y)ω|x ≤ p R

t(x, y). Therefore, p(x, y)˜ op ≤

ptR(x, y).

(3) Recall thatµ(f ,P˜tf)=µ(P˜t/2f ,P˜t/2f)for anyf∈Γ0(Ω); it follows from (2.3) that

˜

Ᏹ(f , f )= −1 2limt↓0

µP˜tf 2

−µ|f|2

t

≥ −1₂lim t↓0

µPtR|f|

2

−µ|f|2

t

=Ᏹ(f , f ).

(2.7)

From now on, we letPt0denote the strongly continuous semigroup onL2(µ) gener-ated by∆+∇V, andp0t(x, y)its transition density with respect toµwhich is positive since M is connected. Let ¯λ=infσess(−˜L), where we put inf∅ = ∞ as usual. When ¯

λ > λ1:=infσ (−L)˜ , letλ1≤λ2≤ ··· be all the eigenvalues (counting multiplicity) of −˜_{Lcontained in[λ1,}¯_λ).

To estimate the numberb(˜L)by usingTheorem 2.1, we need the following lemma.

Lemma_2.2. _Let{_fi}_{be an orthonormal family in}Ᏸ(_L)˜ _{such thatLf}˜ _i= −_δifiwith {δi}n

i=1satisfyingδ1≤δ2≤ ··· ≤δn. Then

l

M

p˜t/2(·, y)op2 µ(dy)≥e−δnt n

i=1

_fi2

. (2.8)

Proof. For any 1≤j≤landx∈M, let

gj(y)= n

i=1

fi(x), ej(x)

xfi(y), y∈M, (2.9)

where{ej}1≤j≤lis an orthonormal basis onΩx. We have

e−δnt/2 n

i=1

fi 2

(x)≤ l

j=1

_˜

Pt/2gj, ej

(x)

= l

j=1

M

˜

pt/2(x, y)gj(y), ej(x)

xµ(dy)

≤l

M

p˜t/2(x, y)2opµ(dy) 1/2

l

j=1

_gj2 L2_Ω(µ)

 

1/2

.

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 31

The proof is completed by noting that

l

j=1

gj2L2 Ω(µ)=

l

j=1 n

i=1

fi, ej2(x)= n

i=1

fi2(x). (2.11)

Theorem2.3. (1)If infσ (R−∆−∇V ) >0, thenb(˜L)=0.

(2)Assume thatµis a probability measure. IfR≥0, thenb(L)˜ ≤l. Moreover, for any f∈ker ˜L,|f|is constant.

(3)Assume thatR is bounded from below. Then ¯λ≥infσess(−(∆+ ∇V )+R).

Con-sequently, letρ(x)be the Riemannian distance betweenx and a fixed pointo∈M, if δ:=limρ→∞R+infσess(−∆−∇V ) >0, then¯λ≥δ >0and henceb(˜L) <∞.

Proof. _{(1) Letf}∈_ker(L)˜_{. Ifa:}=_{infσ (}−∆−∇_V+_{R) >0, then}

µ|f|2_=µP˜

tf2≤µ

P_tR|f|2

≤e−2at_µ|f|2 _{→0 (2.12)}

ast→ ∞, hencef=0,µ-a.e.

(2) For fixedn≤b(˜L), let{f1, . . . , fn} ⊂ker ˜Lbe an orthonormal family. IfR≥0, by the proof ofTheorem 2.1we have|P˜tf| ≤Pt0|f|for allf∈L2Ω(µ)and hencep˜t(x, y)op≤

pt0(x, y). Then byLemma 2.2(note thatδn=λn=0 in the present case), for any com-pact setB⊂M,

n

i=1

B

fi2dµ≤l

B

p0

t(x, x)µ(dx) (2.13)

holds for allt >0 and any compact setB⊂M. We now intend to show thatpt0(x, x)↓1 ast↑ ∞for allx∈M. Observing that

d dtp

0

t(x, x)= −

M

_∇p0t/2(x,·) 2

(y)µ(dy)≤0, (2.14)

then pt0(x, x)is decreasing in t. Next, noting that the Dirichlet form for∆+ ∇V is irreducible since M is connected, we havePt0u−µ(u)L2_(µ)→0 as t→ ∞ for any u∈L2_{(µ)(see, e.g., the appendix in [2]). For fixedx∈M, lettingu(y)=p}0

1(x, y), we obtain

P0

tu−µ(u) 2 L2_(µ)=µ

p0

t+1(·, x)−1

2 =p0

2(t+1)(x, x)−1. (2.15)

Therefore,p0t(x, x)→1 ast→ ∞. Now by first lettingt→ ∞and thenB→M, we obtain from (2.13) thatn≤l, henceb(˜L)≤lsincenis arbitrary. Moreover, forf∈ker ˜L, we have (note that|f|is continuous)

|f|(x)2_=P˜

tf(x)2≤p02t(x, x)µ

|f|2_{. (2.16)}

(3) By Donnelly-Li’s decomposition principle mentioned above, we have

infσess

−(∆+∇V )+R= lim n→∞infσ

−(∆+∇V )+R_Bo(n)c

(2.17)

and the same formula holds for ˜Lin place of−(∆+ ∇V )+R. Then the proof is com-pleted byTheorem 2.1(3) and by noting that infσ ([−(∆+∇V )+R]|Bo(n)c)≥infB_o(n)cR+

infσ (−(∆+∇V )|Bo(n)c).

We remark thatTheorem 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)=land R ≥0; for instance, the Betti numbers on torus (see, e.g., [15]).

We are now ready to estimateλn and then use the basic estimate (2.18) to obtain more estimates ofb(˜L).

Theorem2.4. Assume thatµis a probability measure andR≥ −cfor somec∈R. If

pt0(x, x)is integrable with respect toµfor somet >0, then¯λ= ∞,σess(˜L)= ∅, and

λn≥sup t>0 1

tlog

ne−ct

lMpt0(x, x)µ(dx)

, n≥1. (2.18)

In the case thatp0t(x, x)is not integrable, letδs,t=µ({x:p0t(x, x) > s}). If there exists

some positiveβdefined on(0,∞)such that

µu2_{≤r µ|∇u|}2_{+β(r )µ|u|}2

, r >0, u∈C₀∞(M), (2.19)

then¯λ= ∞and

λn≥sup

₁

tlog nεe−ct

sl ∧supr >0 1

r

1−cr−2β(r )ε+δs,t

:ε∈(0,1), s, t >0

.

(2.20)

Especially, if (2.19) holds forβ(r )=exp[α(1+r−1/δ_{)]for someα >0andδ >1, one has}

λn≥λ([logn−θ]+)δfor someλ, θ >0and alln≥1.

Proof. _Ifpt0_{(x, x)is integrable with respect toµ, thenPt}0_{(and hence Pt}R_{) is} uni-formly integrable inL2_{(µ). Recall that a linear operatorP onL}2_{(µ)is called uniformly} integrable if supu2≤1µ(|P u|

2₁

{|P u|>r})→0 asr→ ∞. Therefore, by [29, Theorems 2.2

and 3.1] (see also [14]), we haveσess(−(∆+ ∇V )+R)= ∅. Thus, ¯λ= ∞ according to

Theorem 2.3(3). Next, let ˜Lc_=+∇

∇V+c. For the same reason, we haveσess(˜Lc)= ∅. Letλc

1≤λc2≤ ··· denote all eigenvalues of−L˜ccounting multiplicity. Since−L˜≥ −L˜c, it follows from the max-min principle thatλn≥λcnfor alln≥1 (see, e.g., [20, problem 1, page 364]). Therefore, it suffices to prove (2.18) forR≡ −c. In this case, iffiare the

L2_{-unit eigenvectors forλi, then, byLemma 2.2,}

l

M

p˜t/2(x, y)2opµ(dy)≥e−λnt n

i=1

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 33

ThenTheorem 2.3(2) yields that

ne−λnt_≤lect

M

p0

t(x, x)µ(dx). (2.22)

This proves (2.18).

If (2.19) holds forβ(r )=exp[α(1+r−1/δ_{)]for someα >0 andδ >1, by [28, Corollary} 5.2] we havep0

t(x, x)≤exp[λ(1+t−1/(δ−1))]for someλ >0 and allt >0. Then by (2.18),

λn≥sup t>0

1

tlog

ne−ct

lexpλ1+t−1/(δ−1). (2.23)

Takingt=ε−1_(logn)1−δ_{for smallε >0 andn≥2, we obtain}

λn≥ε

1−λε1/(δ−1)logn1−δ−c−εlogl+λlognδ−1, n≥2. (2.24)

Therefore, the last assertion follows by takingε∈(0, λ1−δ_).

It remains to prove (2.20). The proof is similar to that of [29, Theorem 3.2]. LetAs,t= {x:p0t(x, x)≤s}. ByTheorem 2.1(2) and inequality (2.21), we obtain

e−λnt n

i=1

1A_s,tfi 2

≤lect_p0

t(·,·)1As,t≤le

ct_{s. (2.25)}

This implies that

λn≥ 1

tlog

  1

lsect n

i=1

µ1A_s,tfi 2

 _≥1

tlog nε

lsect (2.26)

providedµ(1A_s,t|fi|2_{)≥εfor all 1≤i≤n. On the other hand, if there existsisuch that}

µ(1A_s,t|fi|2_{) < ε, we have}

µfi 2

=µ1A_s,tfi+µ

1Ac_s,tfi 2

≤2ε+δs,t

. (2.27)

Combining this with (2.19), we obtain

1=µfi 2

≤r µ∇fi 2

+β(r )µfi 2

≤r˜_Ᏹfi, fi+cr+2β(r )ε+δs,t =r λi+cr+2β(r )

ε+δs,t

, r >0.

(2.28)

Therefore,

λn≥λi≥sup r >0 1

r

1−cr−2β(r )ε+δs,t. (2.29)

To estimatep0

t(x, x), we assume that

Ric−HessV(X, X)≥ −K|X|2_{, X∈T M, (2.30)}

for someK≥0. By the dimension-free Harnack inequality obtained in [25], we have (cf. [29, page 277])

pt0(x, x)≤ 1

µBo(r ) exp

#

Kρ(o, x)+r2

1−e−Kt

$

, r >0, t >0, o, x∈M. (2.31)

Therefore, there exist c1, c2 > 0 such that As,1= {x :p01(x, x)≤ s} ⊃ {x : ρ(x)≤

c1

%

logs−c2}for alls >1.

Corollary_2.5. _{Assume thatµis a probability measure and (2.30) holds. IfR}≥ −_c

and (2.19) holds, then

λn≥ sup ε∈(0,1), s>1

lognεe− c

sl ∧supr >0 1

r

&

1−cr−2β(r )ε+µρ > c1

'

logs−c2

(

(2.32)

for somec1, c2>0and alln≥1. Consequently, if, in addition, (2.19) holds forβ(r )= exp[α(1+r−1/δ_{)]for someα, δ >0, thenλ}

n≥λ([logn−θ]+)δfor someλ, θ >0and all

n≥1.

Proof. By Theorem 2.4, it suffices to check the second assertion for δ ∈(0,1]. By [28, Corollary 6.3], there exists α1, α2 > 0 such that µ(ρ ≥ c1

%

logs−c2) ≤

α1exp[−α2(logs)1/(2−δ)] for alls >1. Then the proof is completed by some simple calculations.

Now we come back to estimateb(˜L)by using (2.18).

Corollary2.6. Assume thatµis a probability measure andR≥ −cfor somec∈R.

Then

b˜L≤inf t>0e

ct_l

Mp 0

t(x, x)µ(dx). (2.33)

If (2.30) holds, then

b˜L≤ inf t>0, r >0

lect

µBo(r )

M exp

#

Kρ(x)+r2

1−e−Kt

$

µ(dx). (2.34)

If, in particular,Mis compact, then

b(˜L)≤linf t>0exp

ct+ KD

2

1−e−Kt

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 35

Proof. Assume that the right-hand side of (2.33) is finite; byTheorem 2.3one has ¯

λ= ∞and hence b(˜L) <∞. Letn=b(˜L), we haveλn=0. Then (2.33) follows from (2.18). Moreover, (2.34) follows from (2.33) and (2.31). Finally, by (2.31) withr=Dand

o=x, we obtain

p0

t(x, x)≤exp

_KD2 1−e−Kt

, x∈M. (2.36)

Then (2.35) follows from (2.33).

To conclude this section, we consider the following two examples on noncompact manifolds.

Example2.7. LetMbe noncompact andµ=eVdxa probability measure. If there ex-istsα >0 such that limρ→∞(∆+∇V )ρ≤ −α, where the limit is taken outside of the cut

locus ofo, and limρ→∞R >−α2/4, then 0∉σess(˜L)and henceb(˜L) <∞. Indeed (see, e.g., [27, (2.8)]), we have limn_→∞infσ (−(∆+∇V )|Bo(n)c)≥α2/4. By Donnelly-Li’s

decompo-sition principle [10], we have infσess(−(∆+∇V ))≥α2/4. Therefore, byTheorem 2.3(3), 0∉σess(˜L)provided limρ→∞R >−α2/4.

Example_2.8. _{LetMbe noncompact with Ricci curvature bounded from below and}

δ >1 a constant. LetV = −αρ˜for some α >0 with ˜ρ∈C∞(M)such thatρδ_−ρ˜is bounded andµ is a probability measure, whereρis as above. The existence of ˜ρ is guaranteed by a classical approximation theorem and the volume comparison theorem. By [28, Corollary 2.5], (2.19) holds withβ(r )=exp[c1(1+r−δ/[2(δ−1)])]for somec1>0. Therefore byTheorem 2.4, ifRis bounded from below, then there existλ, θ >0 such that

λn≥λ

logn−θ+2(δ−1)/δ, n≥1, (2.37)

providedδ >2. If, in addition, (2.30) holds, then (2.37) holds for allδ >1.

3. Proofs of Theorems1.2and1.4

Proof ofTheorem_1.2. _{Obviously, we may assume thatµis a probability measure} since it is finite. Letx∈M be fixed. For anyy∈M, letu(y)=p0

D2(x, y). We have p_D02_(1+t)(y, x)=P_D02_tu(y). Since, by (2.36), u∞ ≤exp[KD2/(1−e−KD

2

)], applying [26, Theorem 4.4 ] withλ=0 we obtain

∇p0_D2_(t+1)(·, x)(y)≤ 1 4D2_texp

_KD2 1−e−KD2

D

0 exp

_Kr2

8

dr

≤_4Dt1 exp

KD2

₁

8+ 1 1−e−KD2 .

Since_Mp0

D2(t+1)(y, x)µ(dy)=1, there existsy∈Msuch thatp 0

D2(t+1)(y, x)≤1. We obtain

p0_D2_(t+1)(x, x)≤1+ 1 4texp

KD2

₁

8+ 1

1−e−KD2 . (3.2)

Then (1.4) follows from (2.33). IfD2_{infR >−η, then there existt}

0>0 andε >0 such that

D2infR≥ε−_t 1 0+1

log l+1

l1+1/4t0

expKD2_1/8+1/1−e−KD2. (3.3)

Therefore, we may apply (1.4) whent=t0and

c:= −_Dε2+ 1

D2_(t 0+1)

log l+1

l1+1/4t0

expKD2_/8+KD2_/1−e−KD2 (3.4)

to obtain

bL˜≤le−ε(1+t0)l+1

l < l+1. (3.5)

This implies b(L)˜ ≤l sinceb(˜L)∈Z+. Moreover, taking t=lexp[KD2/8+KD2/(1−

eKD2_{)], we obtain the desired lower bound ofηfrom its definition.} Finally, letΩ=Λ1 _{and ˜L= −∆}1

µ; we haveR=Ric−HessV (cf. (3.7) and (3.10)) and

l=d. Letηbe defined by (1.5); we haveη <log((d+1)/d). If−KD2_:=D2_{infR≥ −η,} thenKD2_{<log((d+1)/d)and hence}

η > log

(d+1)/d+1/4

1+dexplog(d+1)/d(1/8+1+d)

=log

(d+1)/d+1/4 1+d1+d−1d+9/8 =:η

_.

(3.6)

Therefore, ifD2_{infR≥ −η, thenD}2_{infR >−ηand henceb(−∆}1

µ)≤d. Then the proof is completed by noting thatb1=b(−∆1µ)sinceM is compact (see [7, Theorem 5.12]).

To prove Theorem 1.4, we fix p ∈[0, d]∩Z+, and letΩ= Λp be the exterior p

-bundle over the oriented manifoldM. We havel=d!/(p!(d−p)!). Consider−˜_L=∆p µ:= d∗_µd+dd∗_µ. We have (see [7]) d∗_µ=δ−i_∇V and hence

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 37

where∆p_{:=δd+dδand᏾denote, respectively, the usual Hodge Laplacian and the} cur-vature term involved in the Weitzenböck formula onp-forms, and LX=diX+iXd is the Lie differentiation in the directionX. Moreover,(∆pµ,Γ0(Λp))is essentially selfadjoint, see, for example, [7, page 692].

Let{Ej}d

j=1be a locally normal frame with dual{ωj} ⊂Λ1. We have, for any differ-ential formφ,

i∇Vdφ= d

i,j=1

i∇V ,EiEi

ωj∧∇Ejφ

= ∇∇Vφ− d

i,j=1

∇V , Ei

ωj∧

iE_i∇Ejφ

,

di∇Vφ= d

i,j=1

HessVEi, Ej

ωi∧

iE_jφ+

d

i,j=1

∇V , Ej

ωi∧

iE_j∇E_iφ.

(3.8)

Therefore,

L∇V= d

i,j=1

HessVEi, Ej

ωi∧iEj+∇∇V:=HessV+∇∇V. (3.9)

Combining this with (3.7), we obtain

∆pµ= −−∇∇V+᏾−HessV. (3.10)

Let Cµ∞,p := ∩∞n=1Ᏸ((∆ p

µ)n). By [7, Theorem 5.3], Cµ∞,p ⊂ Γ(Λp). Let d|_C∞,p µ (resp.,

d∗µ|C∞_µ,p) denote the restriction ofd(resp.,d∗µ) onC∞ ,p

µ . We have the following general result.

Theorem3.1. LetR=᏾−HessV. If either

infσ−∆−∇V+R>0 or limρ→∞R >supσess(∆+∇V ), (3.11)

where we putsup∅ = −∞as usual, thenim dp−1_{is closed and}

L2

Λp(µ)=imd∗µ

p+1

Ᏸ((d∗µ)p+1)⊕im d

p−1

Ᏸ(dp−1₎⊕ker∆pµ_Ᏸ(∆pµ), Cµ∞,p=im dµ∗_C∞,p+1

µ ⊕im d

Cµ∞,p−1⊕ker∆

p µ_Ᏸ(∆p

µ),

ker∆p

µ_Ᏸ(∆pµ)

ker dp

Ᏸ(dp₎

im dp−1

Ᏸ(dp−1₎

ker d|C∞µ,p

im d|_C∞,p−1 µ

.

(3.12)

Proof. _{By Theorems2.1(1) and2.3(3), each condition inTheorem 3.1implies 0}∉

σess(∆pµ). Then the proof is completed by some classical results (see, e.g., [7, Theorem 5.10, Corollary 5.11] and [8, Corollary 10]).

Proof ofTheorem_1.4. _{We have}

If (1.9) holds, then there existsε∈(0,1)such that∆U−(1−ε)|∇U|2_{→ −∞asU→ ∞.} By the proof of Theorem 1.2 in [25] (or the paragraph after it), we have

inf

)

M|∇u|2e−Udx Mu2e−Udx

: 0=u∈C₀∞(M), u=0 on{U≤n}

*

≥εinf U≥n

(1−ε)|∇U|2_{−∆U→ ∞}

(3.14)

asn→ ∞. Since U+V is bounded, we have infσ (−(∆+ ∇V )|{U >n})→ ∞asn→ ∞.

Therefore,σess(∆+∇V )= ∅by Donnelly-Li’s decomposition principle, and henceα >0. The proof is completed byTheorem 3.1.

Finally, we would like to introduce one more example to checkTheorem 3.1, for which

Theorem 3.1applies butTheorem 1.4(hence [1, Theorem 5.1]) does not.

Example_3.2. _{LetM be oriented noncompact with a poleo, andρthe Riemannian} distance function too. Assume that the Ricci curvature is bounded below by−kfor somek≥0, and the sectional curvatures are nonpositive. TakeV∈C2_{(M)such thatV=} −c1ρδ+c2outside a neighborhood ofo, wherec1>0,δ≥1, and letc2∈Rbe such thatµ is a probability measure. By Hessian comparison theorem, we have lim supρ→∞HessV≤

0. Moreover, by Laplacian comparison theorem, one has lim supρ→∞Lρ≤ %

k(d−1)−c1 forδ=1 but lim supρ→∞Lρ= −∞forδ >1, whereL:=∆+ ∇V. Then, byExample 2.7

andTheorem 3.1, we have 0∉σess(∆pµ)and hence the decompositions in (3.12) hold provided at least one of the following is fulfilled:

(1) δ >1 and᏾is bounded below.

(2) δ=1 and lim infρ→∞᏾>−(1/4)(c1−%k(d−1))+2.

Appendix

Spectrum estimates on Hilbert bundles. Let(E,Ᏺ, µ)be a complete measure space andH:= {(Hx,·,·x):x∈E}a family of separable real Hilbert spaces (i.e., a Hilbert bundle overE). Assume that there is a (possibly finite) sequence{ej} ⊂+x∈EHxsuch that forµ-a.e.x∈E,{ej(x)}is an orthonormal basis inHx. Set

ᏹ=

)

f∈, x∈E

Hx:f ,ejisᏲ-measurable for allj

*

, (A.1)

wheref ,ej(x):= f (x), ej(x)x. We callᏹthe space ofᏲ-measurable sections ofH. Forp≥1, letLpH(µ)= {f ∈ᏹ:|f| ∈Lp(µ)}, whereLp(µ)denotes theLp-space of real-valued functions. As usual, we denoteµ(u)=Eudµforu∈L1(µ)and regardf=

ginLpH(µ)providedf=g µ-a.e. ThenL2H(µ)is a Hilbert space with the inner product f , gL2

H(µ):=µ(f , g). We refer to [24] for more information on Hilbert bundles.

Recall thatA⊂Lp_H(µ)is said to beLp_{-uniformly integrable if}

lim r→∞sup

µ|f|p₁

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 39

A linear operatorP onLp_H(µ)is calledLp_{-uniformly integrable if{P f:µ(|f|}p_)≤1}is so. Moreover, denote byσ (P )andσess(P ), respectively, the spectrum and the essential spectrum of a linear operatorP.

Let(_Ᏹ˜,Ᏸ(˜_Ᏹ))be a positive-definite symmetric closed form onL2

H(µ), and let ˜Pt and

(˜_L,Ᏸ(˜L))denote, respectively, the associated contraction semigroup and its generator. It is well known that ˜Lis selfadjoint and negative defined onL2H(µ), and (cf. [17] or [18])

˜

Ᏹ(f , g)= −µf ,˜Lg, f∈Ᏸ(_Ᏹ˜), g∈Ᏸ(˜L),

d

dtP˜tf=˜LP˜tf=P˜t˜Lf , t≥0, f∈Ᏸ(˜L).

(A.3)

We will study the essential spectrum of ˜Lby using the following Poincaré-type in-equality:

µ|f|2≤rᏱ(f , f )˜ +β(r )µ|f|2, f∈Ᏸ(Ᏹ), r > r˜ 0, (A.4)

wherer0≥0 is a constant andβ is a positive function defined on(r0,∞). We may assume thatβin (A.4) is decreasing since the inequality remains true withβreplaced by ¯β(r ):=inf{β(s):s∈(r0, r ]}forr > r0. 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].

Lemma _A.1. _{Assume that µ is a probability measure andp}≥_{1 is fixed. Let P be}

a bounded linear operator onLpH(µ)with transition densityp(x, y), that is, forµ-a.e.

x, y∈E,p(x, y):Hy→Hxis a bounded linear operator such that

P f (x)=

Ep(x, y)f (y)µ(dy), f∈L p

H(µ). (A.5)

Suppose that forµ-a.e.x∈E,

j

E

_{p(x, y)}∗_ej(x)

yµ(dy) 2

<∞, (A.6)

wherep(x, y)∗is the adjoint operator ofp(x, y). If{P f:f∞≤1}isLp-uniformly in-tegrable, thenP (A):= {P f:f∈A}is relatively compact inLpH(µ), for anyLp-uniformly

integrable setA⊂LpH(µ).

Proof. We will use the following Bourbaki theorem (see [6, page 112]).“A bounded

set in the dual spaceBof a separable Banach spaceBis compact and metrisable with respect to the weak topologyσ (B, B).”IfP (A)is not relatively compact inLpH(µ), then there existε >0 and a sequence{fn}∞

n=1⊂Asuch thatP fn−P fmLp

H(µ)≥ε,n=m.

SinceP is bounded andAisLp_{-uniformly integrable, we may takeK >0 such that}

_{P fn,K−P fm,K}

wherefn,K=fn1{|fn|≤K}. We fix a version offnfor eachnand letᏲbe theµ-completion

of theσ-fieldσ ({fn,K,ej:n, j≥1})which isµ-separable. Let

ᏹ₌ )

f∈, x∈E

Hx:

f ,ejisᏲ-measurable for allj≥1

*

⊂ᏹ. (A.8)

LetLpH(µ)be defined asL p

H(µ)forᏲandᏹin place ofᏲandᏹ, respectively, which is separable sinceᏲisµ-separable. Forf∈LH1(µ), letµ(f|Ᏺ):=

jµ(f , ej|Ᏺ)ej, whereµ(·|Ᏺ_{)is the conditional expectation with respect toµunderᏲ. By Bourbaki}

theorem withB=L1

H(µ), there existsf∈L∞H(µ)such thatfni,K→f weakly for some ni↑ ∞, that is, for anyg∈L1H(µ), one hasµ(fni,K, g)→µ(f , g). Noting that for

anyg∈L1

H(µ)and anyf∈L∞H(µ), one hasµ(f, g)=µ(µ(g|Ᏺ), f), we obtain

µ(fni,K, g)→µ(f , g)for allg∈L

1

H(µ). Then forµ-a.e.x,

_{P f (x)−P fn}

i,K(x)

2 x

=

j

Pn_i,Kf (x)−P f (x), ej(x)2x

=

j

E

fni,K(y)−f (y), p(x, y)∗ej(x)

yµ(dy) 2

≤K+f∞2

j

E

p(x, y)∗ej(x)yµ(dy) 2

.

(A.9)

By (A.6) and the dominated convergence theorem, we obtain, forµ-a.e.x,

lim

n_i→∞P f (x)−P fni,K(x)

2 x=

j lim n_i→∞µ

fn_i,K−f , p(x,·)∗ej(x)2=0. (A.10)

Since{|P fn_i,K|p:i≥1}is uniformly integrable andµis a probability measure,P fn_i,K→

P f inLpH(µ). This is a contradiction to (A.7).

Directly following an argument in [14] (see also [29, Theorem 3.1]), we obtain the following result.

Theorem A.2. Assume that µ is a probability measure. Ifσ_ess(−L)˜ ⊂[r₀−1,∞)for

somer0≥0, then (A.4) holds for some β∈C(r0,∞). Conversely, ifP˜t has transition

density satisfying (A.6) for eacht >0, then (A.4) impliesσess(−˜L)⊂[r0−1,∞).

Next, we turn to estimate eigenvalues of ˜L. Let ¯λ=infσess(−˜L), where we put inf∅ = ∞. Assume thatσ (−˜L)∩[0,λ)¯ = ∅, whereσ (−˜L)denotes the spectrum of−˜L. We list all eigenvalues of−˜L(counting multiplicity) in[0,λ)¯ as follows:λ1≤λ2≤ ··· ≤λn≤ ···. We will follow the line of [29] where the eigenvalues estimation was studied on real-valued function spaces.

LemmaA.3. Let¯λand{λn}be as above. Assume thatP˜_thas transition densityp˜_t(x, y)

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 41

eigenvectors for{λi}. Then

l

E

p˜t/2(·, y)op2 µ(dy)≥e−λnt n

i=1

_fi2

(A.11)

for anyn≤#{i:λi<¯λ}.

Proof. _Letx∈_{E be such thatl}=_dimHxand{_ej(x)}_{is an orthonormal basis in}

Hx. The proof is then similar to that ofLemma 2.2.

TheoremA.4. In the situation ofLemma A.3, assume thatµis a probability measure.

If there existst >0such that

C(t):=

E×E

p˜t/2(x, y)2opµ(dx)µ(dy) <∞, (A.12)

thenσess(˜L)= ∅and hence¯λ= ∞; equivalently, (A.4) holds forr0=0and someβ∈

C(0,∞)byTheorem A.2. Moreover,

λn≥ 1

tlog n

lC(t), n≥1. (A.13)

Consequently,#{i:λi≤λ} ≤lC(t)eλtfor eachλ≥0.

Proof. IfC(t) <∞, then ˜P_t/2isL2-uniformly integrable. ByLemma A.1, ˜P_t/2is com-pact and henceσess(˜L)= ∅. Next, by (A.11) we obtainlC(t)≥ne−λnt.

A natural way to apply the above results is to compare ˜Ᏹand ˜ptwith the correspon-dences on theL2_{-space of functions, so that known results on functions can be used.} The next result is implied by [3, Theorem 16] (see also [23] for related results), and the final one is a result on the existence of transition density for operators onL2H(µ).

Theorem_A.5. _Let(Ᏹ,Ᏸ(Ᏹ))_{be a symmetric closed form onL}2_{(µ)which is bounded}

from below. Assume that the associated semigroupPtis positivity-preserving. Then the

following two statements are equivalent:

(1) |P˜tf| ≤Pt|f|for allt >0andf∈L2H(µ);

(2) iff∈Ᏸ(˜_Ᏹ), one has|f| ∈Ᏸ(Ᏹ)andᏱ(|f|,|g|)≤ |_Ᏹ˜(f , g)|for allf,gin a core of˜Lsuch thatf , gL2

H(µ)=µ(|f|·|g|).

PropositionA.6. Assume thatµis a probability measure. LetP˜be a bounded linear

operator onL2H(µ)andpa nonnegative measurable function onE×Esuch that

P u=

p(·, y)u(y)µ(dy) (A.14)

provides a bounded linear operatorP onL2_{(µ). Ifl:=#{ej}<∞and|P f˜ | ≤P|f|for}

anyf∈L2

H(µ), thenP˜has transition densityp˜satisfyingp(x, y)˜ op≤lp(x, y).

Proof. For anyi, j∈Nwithi, j≤l, letµij be a set function onᏲ×Ᏺdefined by

µij(A)=

A1

ei(x),P˜1A(x)ej(x)xµ(dx), A∈Ᏺ×Ᏺ, (A.15)

where A1= {x ∈E : there existsy ∈Esuch that(x, y)∈ A}and A(x)= {y ∈E :

(x, y)∈A}. Since ˜Pis bounded, it is easy to check thatµij _{is a signed measure.} More-over, by Jordan’s decomposition theorem and that|_{P f}˜ _{| ≤P|f|for anyf∈L}2_{(µ), we} have|µij_|:=(µij₎+_+(µij₎−_{≤pµ×µ. Thenµ}ij_{is absolutely continuous with respect to}

µ×µwith densitypij _satisfying|pij_{| ≤p. Define ˜p(x, y):H}

y→Hxby

˜

p(x, y)ω= l

i,j=1

pij_{(x, y)ω, e}

j(y)yei(x), ω∈Hy. (A.16)

It is easy to check thatp(x, y)˜ op≤lp(x, y)for anyx, y∈E and ˜pis a transition density of ˜P. Indeed, for anyf , g∈L2

H(µ), we have

g,P f˜ _L2 H(µ)=

l

i,j=1

E

_˜

Pf , ej

ej

,g, ei

ei

dµ

= l

i,j=1

E×E

f , ej

(y)g, ei

(x)µij(dxdy)

= l

i,j=1

E×Ep

ij_{(x, y)f , e} j

(y)g, ei

(x)µ(dx)µ(dy)

=

g,

Ep(˜ ·, y)f (y)µ(dy)

L2_H(µ)

.

(A.17)

Next, let the additional conditions hold. Forx∈E,y in the support ofµ, and any

ω∈Hy with|ω|y =1, lete∈ᏹbe such thate(y)=ω,|e| =1, and ˜p(x,·)e(·)and

p(x,·)are bounded and continuous in a neighborhoodNyofy. Let{fn}be a sequence of nonnegative continuous functions with supports contained inNysuch thatfnµ→δy weakly asn→ ∞. We have

_Ep(x, z)e(z)˜ fn(z)µ(dz)

_x=P˜fne(x)x≤P fn(x)

=

Ep(x, z)fn(z)µ(dz).

(A.18)

By lettingn→ ∞, we obtain|p(x, y)ω˜ |x≤p(x, y). Therefore,p˜op≤psince we may take ˜p(x,·)=0 outside of the support ofµ.

ON GROMOV’S THEOREM ANDL -HODGE DECOMPOSITION 43

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Fu-Zhou Gong: Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China

E-mail address:fzgong@mail.amt.ac.cn

Feng-Yu Wang: Department of Mathematics, Beijing Normal University, Beijing 100875, China