Some vanishing and niteness results on complete manifolds: a - - PowerPoint PPT Presentation

Some vanishing and …niteness results on complete manifolds: a generalization of the Bochner technique

Stefano Pigola Convegno Nazionale di Analisi Armonica, 22-25 maggio 2007

In this talk we present some results, recently obtained in collaboration with M. Rigoli and A.G. Setti, that extend the original Bochner technique to the case

• f Lp harmonic forms on geodesically complete manifolds and in the presence
• f an amount of negative curvature.

Basic references.

[1] S. P., M. Rigoli, A.G. Setti, Vanishing theorems on Riemannian manifolds and applications. J. Funct. Anal. 229 (2005), 424–461. [2] S. P., M. Rigoli, A.G. Setti, A …niteness theorem for the space of Lp harmonic sections. To appear in Rev. Mat. Iberoamer. [3] S. P., M. Rigoli, A.G. Setti, Topics in geometric analysis: vanishing and …niteness results on complete manifolds. Book in preparation.

We will move in the realm of Geometric Analysis. Roughly speaking: you are given a geometric problem. Summarize it into a family of functions (of geometric content) which, in turn, are governed by a system of di¤erential (in)equalities. Obtain information on the qualitative and quantitative properties of solutions of these di¤erential systems. Geometry, in general, will impose some further constrains and guide the analysis of solutions. Apply this information to the given geometric functions and get a conclusion about the original problem. A prototypical example: the celebrated Bochner technique, originally intro- duced by S. Bochner in the ’50s to investigate the relation between the topology and the curvature of a closed (i.e. compact and without boundary) Riemannian manifold.

The case of a closed manifold

Bochner original argument

Original question: may one prescribe the sign of the curvature on a generic smooth, closed manifold? Let M be a smooth, compact manifold. Then, there is a contractible, open set E M, with E = M, such that E supports a metric with constant curvature

• f a prescribed sign. Simply …x any metric (; ) on M, a reference origin p 2 M

and delete from M the corresponding cut-locus cut (p), which is a closed (hence compact) set of zero-measure. Thus E = M cut (p) is di¤eomorphic to the star-shaped, relatively compact, open set 0 2 E TpM Rm via the exponential map expp. To conclude, …x a constant curvature metric on E and pull it back on E.

• Remark. In a (quite strong) sense, the topology of M is contained in the

(apparently evanescent) removed set cut (p), e.g., the inclusion i : cut (p) , ! M induces isomorphisms between homology (and cohomology) groups Hk (cut (p) ; Z) ' Hk (M; Z) at least for k 6= m; m 1. Now, closing M cut (p) by addition of cut (p) produces a non-trivial topology that, in general, may represent an obstruction for M to support a Riemannian metric with some curvature bound, e.g., given sign. Bochner result goes pre- cisely in this direction. Let us recall the argument.

Theorem 1 (Bochner) Let (M; h; i) be a connected, closed, oriented, Rie- mannian manifold, m = dim M. Set b1 (M; R) for the …rst (real) Betti num- ber of M: Then Ric 0 on M = ) b1 (M; R) m the equality holding if and only if M is a ‡at torus. Furthermore, Ric > 0 at some p 2 M = ) b1 (M; R) = 0:

• Proof. (From Geometry to Analysis) De…ne the Hodge-Laplacian as

H! = (d + d) ! = 0: where d is the exterior di¤erential and stands for the (formal) adjoint of d with respect to the L2 inner product of k-forms. Set Hk (M) = fk-forms ! : H! = 0g ; the vector space of harmonic k-forms on M. By Hodge-de Rham theory b1 (M; R) = dim H1 (M) : Weitzenbock-Bochner formula states that, for ! 2 H1 (M), (BW) 1 2 j!j2 = jD!j2 + Ric

• !#; !#

;

where is the Laplace-Beltrami operator (+d2=dx2 on R) and D denotes the extension to 1-forms of the Levi-Civita connection of M. Suppose Ric 0. By assumptions and (BW) j!j2 0; i.e,. j!j2 subharmonic Note that: M closed implies j!j =const. Two di¤erent viewpoints: (a) L1 viewpoint. The smooth function j!j attains its maximum at some point and, therefore, by the Hopf maximum principle we conclude that j!j =const. (b) Lp viewpoint. Use the divergence theorem: 0 =

Z

M div

• j!j2 r j!j2

=

Z

M

• r j!j2
• 2+j!j2 j!j2

Z

M

• r j!j2
• 2 0:

This again implies j!j =const.

Use this information into (BW) formula: (BW) = ) D! = 0 = ) ! is determined by its value at any p 2 M. Fix p. The evaluation map "p (!) = !p : H1 (M) ! 1 T

p M

• is an

injective homomorphism. Therefore dim H1 (M) m: Note that (BW) = ) Ric

• !#

p ; !# p

• = 0; at p:

Therefore, Ric (p) > 0 = ) !p = 0 = ) ! 0 = ) dim H1 (M) = 0:

• Remark. Crucial fact in the above proof:

Ric 0 = ) j!j 0. Question. What happens in the presence of an amount of negative curvature?

• Answer. In general, there is no uniform bound of dim H1 (M), i.e., no uniform

control on the topology.

• Example. Let S be an orientable, closed Riemann surface of genus g 2, by

uniformization (and recalling the Gauss-Bonnet theorem) we can endow S with a Riemannian metric of Gauss curvature 1.

Analytical counterpart. Set R (x) = min

v2Sm1TxM

Ricx (v; v) ; the pointwise lower bound of the Ricci tensor. From (BW) we have the Bochner inequality (*) 1 2 j!j2 + R (x) j!j2 jD!j2 0: We would like to get LHS () = 0: But, in general, the maximum principle fails to hold for inequalities of this type. Divergence theorem does not help us.

• Remark. A fundamental result by M. Gromov [Comm. Math. Helv. 1981]

states that a uniform limitation on the Betti numbers of a close manifold is

• btained by requiring a control on a further Riemannian invariant, namely, the
• diameter. From a di¤erent (more analytic) perspective, we shall see momen-

tarily how one could think of extending Bochner estimating theorem in the presence of (a little amount of) negative curvature.

Generalized maximum principle

We are given a solution 0 of (*) + q (x) 0: Main assumption. Assume M supports a function ' > 0 such that ' + q (x) ' 0:

• Remark. The existence of ' is related to spectral properties of the operator

q (x).

• Idea. To absorb the linear term of (*) using a combination of the solutions

and '.

Set u = ' = ) u + hru; r log 'i 0: There is no linear (i.e. zero-order) term in u. Therefore, the Hopf maximum principle applies. M closed = ) u attains maximumHopf = ) u const: Equivalently, = C'; C 0: Use the di¤erential inequalities satis…ed by and ' and deduce that, in fact, + q (x) = 0:

Special case: ! 2 H1 (M) ; = j!j2 ; q (x) = 2R (x) ; with R (x) the lower Ricci bound. Bochner inequality yields 0 = 1 2 j!j2 + R (x) j!j2 jD!j2 0: Once again, ! is parallel, thus extending the original Bochner result.

• Remark. Let M be closed (parabolic su¢ces).

If R (x) 0, then ' 2R (x) ' 0: The superharmonic function ' > 0 must be contstant. Hence R (x) 0. As a consequence (in the compact - parabolic - setting) the Main assuption represents a genuine extension of Bochner condition Ric 0 only in case R (x) changes its sign.

The setting of open manifolds

• Question. What does of the previous picture survive in the case of a non-

compact manifold (M; h; i)? Examples help us to understand the situation. We shall consider the general case of harmonic k-forms, any k. First, we need to introduce some more notations and inequalities.

Bochner(-type) and Kato inequalities

Let (M; h; i) be any manifold, m = dim M. We consider k-forms, any k. Assume: (a) case k = 1; Ric R (x) : (b) case k > 1, x R (x) ; where x : 2 (TxM) ! 2 (TxM) is the curvature operator. Take ! 2 Hk (M). Then, by Gallot-Meyer [J. Math. pures et appl. 1973], the following Bochner inequality holds 1 2 j!j2 + CR (x) j!j2 jD!j2 0; for a suitable C = C (k; m) > 0. E.g. C = k (m k) if M = Hm

1.

Direct computations show that j!j f j!j + CR (x) j!jg jD!j2 jr j!jj2 ; The sign of the RHS: in general, one has the Kato inequality jD!j2 jr j!jj2 0: In case ! is both closed and co-closed, i.e., d! = 0; ! = 0; then we have the re…ned Kato inequality jD!j2 jr j!jj2 A jr j!jj2 ; for a suitable constant A = A (m; k) > 0. E.g. k = 1 = ) A = 1= (m 1)

Notation. LpHk (M) =

n

! 2 Hk (M) : j!j 2 Lpo : Remarks.

• 1. Alexandru-Rugina [Rend. Sem. Mat. Univ. Politec. Torino 1996]

! 2 Lp6=2Hk (M) 6= ) d! = 0 nor ! = 0:

• 2. Ga¤ney [Annals 1954] Global integration by parts:

! 2 L2Hk (M) + (M; h; i) complete

9 > = > ; =

) d! = 0; ! = 0 = ) Re…ned Kato.

Conclusion: take ! 2 Hk (M). Then = j!j 0 satis…es an inequality of the form (*) f + q (x) g A jr j2 ; with q (x) 2 C0, and A 2 R: We shall refer to (*) as the general Bochner-type inequality.

General existence result

Bochner result is essentially a vanishing&estimating theorem for the space of harmonic forms. Harmonic forms on a closed manifold represent cohomology classes: therefore their vanishing or their presence is a topological question. In contrast, in the non-compact setting, harmonic forms may represent nothing, even for a geodesically complete manifold (M; h; i).

• Example. On the ‡at Euclidean space Rm every di¤erential k-form h (x) dxi1^

::: ^ dxik is harmonic provided h (x) is a harmonic function. Now, the space

• f harmonic functions on Rm is not …nitely generated (e.g. harmonic polyno-

mials). Therefore dim Hk (Rm) = +1: Remark. Let (M; h; i) be a generic open manifold. Then, for every k, dim Hk ( M) 6= 0.

Fix M, @ 2 C1. Fix !0 2 k

• with non-zero tangential (or

normal) part on @: Du¤ and Spencer [Annals 1952]= ) 9!! 6= 0 solution of

(

H! = 0;

• n

! = !0,

• n @:

Choose D so that M D has no compact components. H! = 0 uniquer continuation = ) ! 6 0 on D. Malgrange [Ann. Inst. Fourier 1955-1956]= )we can uniformly approximate !

• n D by a harmonic k-form on M, say 2 Hk (M) :

Take su¢ciently close to ! on D = ) 6 0. Thus 2 Hk (M) 6= 0:

The need of some integrability condition

Analytic counterpart of the above non-vanishing. Suppose the curvature

• perator of (M; h ; i) satis…es R (x). Then, a harmonic k-form ! 2

Hk (M) satis…es 1 2 j!j2 + CR (x) j!j2 jD!j2 0; with C = C (m; k) > 0. Restrict now to 0, so that R (x) 0 and the above reduces to j!j2 0: Following Bochner original argument, one tries to reach the Liouville-type con- clusion j!j = const and, hence, that ! is parallel. However, in general, j!j is neither (i) L1, nor (ii) in some Lp<+1 integrability class and we have no Liouville property at all.

It happens that the geometry of the underlying manifold (M; h; i) enters the game when we consider harmonic functions and forms with these special inte- grability properties. The L1 and Lp<+1 situations are substantially di¤erent. They are dealt with completely di¤erent methods. The L1 case is often con- sidered in the perspective of the weak maximum principle at in…nity. We shall not consider this situation here. For an extensive study of the subject, we refer to a paper by P.-Rigoli-Setti, [Memoirs AMS 2005]. In this talk we will focus our attention on the Lp<1 case.

• Example. The case of Rm is easy to handle and, therefore, can be used to

exemplify the situation. Let LpHk (Rm) be the vector space of the harmonic k-forms ! on Rm satisfying j!j 2 Lp (Rm). We show that dim LpHk (Rm) = 0: To see this, take any ! 2 LpHk (Rm). Then, by Bochner-Weitzenbock, j!j 2 Lp (Rm) is a subharmonic function. Since non-negative subharmonic functions in Rm enjoy the Lp mean-value property we get, for any …xed x0 2 Rm and for every R > 0, j!jp (x0)

R

BR(x0) j!jp

vol (BR (x0)): Letting R ! +1 we deduce j!j (x0) = 0. Since x0 was an arbitrary point, we conclude ! 0, as claimed. The same proof works for a geodesically complete manifold (M; h; i) satisfying 0.

Controlling the negative part of the curvature

Example. Let Hm be the standard hyperbolic space of dimension m and constant curvature 1. We realize it as the Poincarè disc

B @Bm

1 (0) ; 4 P dxi dxi

• 1 jxj22

1 C A ;

where Bm

1 (0) is the Euclidean unit ball of Rm.

Note that the hyperbolic metric is a pointwise conformal deformation of the Euclidean one. Note that, in general, if g h; i = 2 (x) h; i then L2Hk (M; h; i) ' L2Hk M; g h; i

• provided

2k = m = dim M:

Specifying the situation to the Hyperbolic space we deduce (*) L2Hk (Hm) ' L2Hk (Bm

1 (0)) :

Now, we have already observed that dim Hk (Rm) = +1. Since Bm

1 has …nite Euclidean volume, every ! 2 Hk (Rm) restricts to a form

!0 2 L2Hk

Bm

1 (0)

• :

By unique continuation, the restriction map is injective. It follows that dim L2Hk

Bm

1 (0)

• = +1. According to (*), we conclude

dim L2Hk

H2k

= +1:

Bottom of the spectrum and Morse index

We introduce a “measure” of the negative part of the curvature in such a way that, “small” negative curvature implies …nite dimensionality and, furthermore, small enough gives vanishing. This will be done using spectral properties of a suitable Schrodinger operator. From now on, (M; h; i) will denote a geodes- ically complete, non-compact, connected Riemannian manifold of dimension m = dim M. Suppose the curvature operator of M satis…es R (x), R (x) 0, so that, for every ! 2 Hk (M), its length = j!j satis…es f + CR (x) g A jr j2 ; where C = C (k; m) > 0 and A = A (k; m) 0, with A > 0 if ! is both closed and co-closed. Let us consider the Schrodinger operator L = q (x) , q (x) = CR (x)

For any smooth M, the operator (L; C1

c ()) has discrete spectrum

1 (L) < 2 (L) 3 (L) ::: N0 (L) < 0 N0+1 (L) ::: De…nition. Ind (L) = N0 = #negative eigenvalues. By domain monotonicity of eigenvalues 1 2 = ) k

• L2
• k
• L1
• =

) Ind

• L1
• Ind
• L2
• ;

De…nition. Ind (LM) = sup%M Ind (L) +1: De…nition. 1 (LM) = inf

%M 1 (L) =

inf

'2Lipc(M)

R jr'j2 R q (x) '2 R '2

1:

Finiteness and vanishing of the Morse index essentially re‡ect the fact that the (positive part of the) potential is small in some integral sense. Theorem 2 (G.V. Rosenbljum, W. Cwikel, E. Lieb) Let (Mm; h; i) be com- plete and support the L2-Sobolev inequality S

Z

M v2

1 Z

M jrvj2 , 8v 2 C1 0 (M)

for some > 1 and some constant S > 0. Let L = q (x) where q+ (x) = max (q (x) ; 0) 2 L

• 1 (M) ;

Then, L is a semi-bounded, essentially self-adjoint operator on L2 (M) with non-negative essential spectrum. Let N0 be the number (counting multiplicity)

• f strictly negative eigenvalues of L. Then, 9C = C (m; S) > 0 such that

N0 = Ind (LM) C kq+k

L

• 1(M)

:

PDE counterparts of the spectral properties

Theorem 3 Let L = q (x). Then (a) R. Gulliver 1984, Fisher Colbrie [Invent. 1985]. For any domain W M, 1 (LW) 0 ( ) 9' > 0 solution of L' 0 on W: (b) Moss-Piepenbrink [Paci…c Math. J. 1978], Fisher Colbrie-Schoen [C.P.A.M. 1980]. Ind (LM) < +1 = ) 9K M such that 1

LMK

In particular Ind (LM) < +1 = ) 9' > 0 : L' 0 on M K; for some K M:

• Remark. The bottom-of-the-spectrum condition on M K is slightly weaker

than the …niteness of the Morse index. Moreover, it is easier to handle. For instance, suppose (M; h; i) enjoys a global L2-Sobolev inequality

Z

juj2

1

S1

• Z

jruj2 ; 8u 2 C1

c (M)

with > 1, S > 0: Then, direct application of Hölder inequality gives kqk

L

• 1(M)

< +1 = ) 1

LMK 0;

for a su¢ciently large K M. Furthermore kqk

L

• 1(M)

S = ) 1 (LM) 0: Remark. The PDE reformulation of the spectral properties is very suitable for an analytic approach to vanishing and …niteness results in the spirit of the generalized maximum principle.

Main theorems: Bochner generalized

Vanishing Vanishing results for L2 harmonic sections on complete manifolds under spec- tral assumptions go back to a paper by W. Elworthy and S. Rosenberg, [Acta

• Appli. Math. 1988]. Their technique relies on very re…ned probabilistic tools

and requires the additional curvature assumption infM Ric > 1 in order to guarantee that Brownian paths do not explode (a.s.) in a …nite time. Soon later, P. Berard, [Manuscripta Math. 1990], generalized Elworthy-Rosenberg results by removing the curvature condition. Moreover, his proof is completely elementary and makes a direct use of the spectral assumption. He gets conclu- sion only in case of L2 energies.

Theorem 4 (P.-Rigoli-Setti, J.F.A. 2005) Let (M; h ; i) be a complete man- ifold, a(x) 2 L1

loc(M) and let 0 2 Liploc(M) satisfy the di¤erential

inequality (*) + a(x) 2 Ajr j2 weakly on M: for some A 2 R. Suppose that there exists ' 2 Liploc (M) satisfying ' + Ha (x) ' 0 weakly on M for some H such that H A + 1; H > 0 If 2 L2p (M) for some A + 1 p H, p > 0,

then there exist a constant C 0 such that C' = H: Further, (i) If H 1 > A; then is constant on M, and if in addition, a(x) does not vanish identically, then is identically zero; (ii) If H 1 = A; and does not vanish identically, then ' and therefore H satisfy (*) with equality sign.

Finiteness In case of L2 harmonic sections, …niteness results have been extensively in- vestigated by many authors under di¤erent assumptions. We limit ourselves to quote the “minimal hypersurfaces" papers [J. reine angew. Math. 2004;

• Math. Res. Let. 2002] by P. Li and J. Wang, where Morse index assump-

tions are used in a way similar to the present note, and the “L2-cohomology paper"[Math. Ann. 1999], by G. Carron where quantitative dimensional esti- mates are obtained assuming that the underlying manifold supports a global Sobolev inequality; see also [Duke 1998, G.A.F.A. 2003].

• Remark. The classical minimal (hyper)surface theory in Euclidean space greatly

in‡uenced the investigation of …niteness results on non-compact spaces under spectral properties of the relevant Schrodinger operator (the stability operator L = jIIj2). According to the harmonic function theory by P. Li and L.F. Tam, [J.D.G. 1992], the dimension of (suitable subspaces of) L2H1 (M) re‡ects in some sense the topology at in…nity of the underlying manifold (num- ber of non-parabolic ends). This applies in particular to minimal submanifolds

• f …nite index (where all ends are non-parabolic). Also, on a generic complete

manifold, according to the decomposition theorem by Hodge-de Rham-Kodaira, L2 harmonic forms completely represent the (reduced) L2 cohomology of the manifold. It has been recently observed by D. Alexandru-Rugina, [Tensors, 1996], that, in case of manifolds with bounded geometry (e.g. co-compact coverings), the spaces Lp<2Hk (M) imbeds continuously into the correspond- ing Lp cohomology spaces. Furthermore, it is known from works by J. Dodziuk, [Topology 1977], and V.M. Gol’dshtein, V.I. Kuz’minov, I.A. Shvedov, [Sibirsk. Mat. Zh. 29 1988], that the Lp cohomology (non reduced, in fact) of a co-compact covering is a homotopy invariant of the base (compact!) manifold.

Theorem 5 (P.-Rigoli-Setti, Rev. Mat. Iberoam.) Let (M; h; i) be a con- nected, complete, m-dimensional Riemannian manifold and E a Riemannian (Hermitian) vector bundle of rank l over M. The space of its smooth sections is denoted by (E). Having …xed a (x) 2 C0 (M) , A 2 R, H p satisfying the further restrictions (1) p A + 1; p > 0; let V = V (a; A; p; H) (E) be any vector space with the following property: (P) Every 2 V has the unique continuation property, i.e., is the null section whenever it vanishes on some domain; furthermore the locally-Lipschitz

function u = jj satis…es (2)

8 > > < > > :

u (u + a (x) u) A jruj2 weakly on M

R

Br u2p = o

• r2

as r ! +1: If there exists a solution 0 < ' 2 Liploc of the di¤erential inequality (3) ' + Ha (x) ' 0 weakly on M K for some compact set K M, then (4) dim V d; for some d < +1 depending only on the geometry of M in a neighborhood

• f K.

The following consequence extends on previous work by Carron and Li-Wang. Corollary 6 Let (M; h ; i) be a complete manifold satisfying the global Sobolev inequality

Z

juj2

1

S1

• Z

jruj2 ; 8u 2 C1

c (M) ;

with > 1. Assume that x R (x) for some R (x) 0: Then R (x) 2 L

• 1 (M) =

) dim L2pHk (M) < +1: for every k = 1; :::; m and for every p 1. Furthermore, kR (x)k

L

• 1(M)

S Cp = ) L2pHk (M) = 0 with C = C (k; m) > 0 the constant in the Bochner-Witzenbock formula for harmonic k-forms.

Some Geometric applications

Theorem 7 (Topology at in…nity of submanifolds of CH spaces) Let f : (M; h ; i) ! (N; ( ; )) be an isometric immersion of a complete manifold M

• f dimension m 3 into a Cartan-Hadamard manifold N whose sectional

curvature (along f) satis…es (0 ) NSecf(x) NR (x) for some NR 2 C0 (M). Assume that the mean curvature vector …eld satis…es jHj 2 Lm(M). Let a (x) = (m 1) NR (x) + jIIj (jIIj + m jHj) (x) : with II the second fundamental tensor of f. Set L = a(x): Ind(LM) = 0 = ) M has only one end. Ind(LM) < +1 = ) M has …nitely many ends.

Theorem 8 (Reduction of codimension) Let (M; h ; i) be a complete Rie- mannian manifold of dimension m 3. Assume Ricci R(x)

• n M:

Set HL = HR(x). If Ind

HLM

• < +1

for some H m2

m1, then, there exists a compact set K M and an integer

N = N (H; K) m such that the following holds. Let f : M ! Rd, d > N be a harmonic immersion whose energy density satis…es the growth condition

Z

BR

jd fj2p = o

• R2

, as R ! +1; for some m2

m1 p H. Then, there is an N-dimensional a¢ne subspace

AN of Rd such that f(M) AN.

Outline of the proof of the vanishing theorem

Combine the solutions and ' so to obtain a new function u = ' p

H p

which, in turn, satis…es the easy to handle inequality udiv

• '

2p H ru

• 0:

Note that '

2p H u2 2 L1 (M) :

Now, the key step is to obtain a general Liouville theorem for (possibly changing sign!) solutions of the problem

(

vdiv (wrv) 0 w jvjq 2 L1 (M) :

In our special case, deduce u const. Equivalently C' = H: Use this information into inequality (*) and get H (H 1 + A) H2 jr j2 0: This latter produces the dichotomy in the statement of the theorem and gives the desired conclusions.

Outline of the proof of the …niteness theorem.

Choose R >> 1 in such a way that K BR (o) and, therefore, inequality (3) holds on M BR (o) : Note that, by unique continuation, the restriction map V !

• EjBR
• 7!

jBR is an injective homomorphism. Use the same symbol V to denote the image of V in

• EjBR
• . An extension of a classical result by P. Li states that if T V

be any …nite dimensional subspace then, there exists a (non-zero) section 2 T such that, setting =

• ; it holds

(5) (dim T)min(1;p)

Z

BR

• 2p vol (BR) min fl; dim T gmin(1;p) sup

BR

• 2p:

Now, observe that, on every su¢ciently small closed ball, 1

H

LB(x)

• > 0,

where H L = Ha (x) ; and therefore there exists w > 0 solution of w + Ha (x) w = 0: As above deduce that u = w p

H p

satis…es udiv

• w

2p H ru

• 0 on B (x) :

Obtain a local Lq-mean value inequality for solutions u of this inequality and apply it to get sup

B

2p C

Z

B2

2p:

The local inequalities patches together and, in the special case of , give sup

BR

• 2p C0

Z

BR+1

• 2p:

Inserting into (5) we obtain (6) (dim T)min(1;p) R

BR

2p Cvol (BR) min fl; dim T gmin(1;p) nR

BR

2p +

R

A(R;R+1)

2po where A (a; b) is the annulus Bb Ba. Now, using once again the combina- tion of and ' and a careful cut-o¤ analysis inspired by Li-Wang estimating technique obtain the estimate

Z

A(R;R+1)

• 2p C00

Z

BR

• 2p:

Inserting into (6) and simplifying gives dim T C000 min fl; dim T g for some C000 depending only on the geometry of BR. This proves that any …nitely generated subspace T of V have a dimension which is bounded by a universal constant, depending only on the rank l of E and on the geometry of

• BR. The same bound must work for the dimension of the whole V .