Haisheng LIU

Abstract If V is an irreducible quasi-Kähler complex variety and E is a vector bundle over reg(V),the author proves that (reg(V),E)=W1,2(reg(V),E),and that for dimC reg(V)> 1,the natural inclusion W1,2(reg(V),E)L2(reg(V),E)is compact,the natural inclusion W1,2(reg(V),E)(reg(V),E)is continuous.

Keywords Quasi-Kähler variety,Sobolev spaces

1 Introduction

In[4],Bei proved several results on Sobolev spaces of irreducible complex projective varieties.The one we concern here in this paper is the following theorem.

Theorem 1.1(cf.[4,Theorem 4.1])Let V ⊂CPnbe an irreducible complex projective variety of complex dimension v.Let E be a vector bundle over reg(V)and let h be a metric on E,Riemannian if E is a real vector bundle.Hermitian if E is a complex vector bundle.Let g be the Kähler metric on reg(V)induced by the Fubini-study metric of CPn.Finally,let ∇: C∞(reg(V),E)→C∞(reg(V),T∗reg(V)⊗E)be a metric connection.We have the following properties:

(1)W1,2(reg(V),E)=

(2)Assume that v > 1.Then there exists a continuous inclusion W1,2(reg(V),E)

(3)Assume that v >1.Then the inclusion W1,2(reg(V),E)L2(reg(V),E)is a compact operator.

These results have many applications,for example,on the L2-theory of Kähler manifolds with singularities.The situation of non-Kähler complex spaces also interests us a lot.The geometry on non-Kähler manifolds has long been studied in many papers.

In this paper,we generalize the results on Sobolev space of projective varieties in [4]and get the corresponding results on quasi-Kähler complex subvarieties,using basically the same method that Bei [4]used.Let M be a differential manifold of dimension n with a Hermitian metric h.The metric h is often identified with the associated positive Hermitian (1,1)-form ωhon X,called the Kähler form (or the fundamental form)of h.The metric ωhis said to be quasi-Kähler if ∇X(J)(Y)+∇JX(J)(JY)=0.Quasi-Kähler manifolds are important because they include the classes A K and N K of almost Kähler and nearly Kähler manifolds.A large part of the theory of the geometry and topology of Kähler manifolds can be carried over to the class N K (cf.[10]).As a generalization of Kähler manifolds,we are interested in quasi-Kähler manifolds naturally.

In Section 2,we recall some basic concepts and notations that will be used in this paper,such as almost complex manifold,Hermitian manifold(or complex manifold),metric connection,etc.We also recall the fundamental knowledge of Sobolev space and operator extension theory of differential operators which are densely defined on a Hilbert space.

In Section 3,we prove several results analogous to the corresponding results in [4].Firstly,we prove a proposition which states that there exists a certain sequence of cut-off functions on an irreducible quasi-Kähler complex variety.

Proposition 1.1Let V be an irreducible quasi-Kähler complex variety of M and let g be the metric on reg(V)induced by the Hermitian metric h of M.Then there exists a sequence of Lipschsitz functions with compact support {φj}j∈Nsuch that

(1)0 ≤φj≤1 for each j;

(2)φj→1 point-wise;

(3)φj∈D(d0,min)for each j ∈N and

In particular,1 ∈D(d0,min).

Secondly,we show some results on the Sobolev space of an irreducible quasi-Kähler complex variety.

Theorem 1.2Let M be a compact quasi-Kähler complex manifold with a Hermitian metric h and V be an irreducible complex analytic subvariety of dimension v of M.Let E be a vector bundle over reg(V)and let h be a metric on E,Riemannian if E is a real vector bundle,Hermitian if E is a complex vector bundle.Let g be the metric on reg(V)induced by the metric of M.Finally let ∇:C∞(reg(V),E)→C∞(reg(V),T∗reg(V)⊗E)be a metric connection.We have the following properties:

(1)W1,2(reg(V),E)=

(2)Assume that v > 1.Then there exists a continuous inclusion W1,2(reg(V),E)

(3)Assume that v > 1,then the inclusion W1,2(reg(V),E)L2(reg(V),E)is a compact operator.

As in[4],the proof of this theorem depends on Kato’s inequality,Sobolev inequality and the existence of a suitable sequence of cut-off functions.When V is an irreducible complex projective variety,the regular part of V,reg(V),is a Kähler manifold endowed with the incomplete Kähler metric,i.e.,the Fubini-study metric.The fact that any Kähler submanifold is minimal yields the mean curvature equals 0,i.e.,H=0.This enables us to use the results in [14,16].Meanwhile,we would like to point out that some of those results only require the condition that the mean curvature is bounded on reg(V).

Thirdly,we get the following useful proposition.

Proposition 1.2Let (reg(V),g)be as in Theorem 3.1.Let E and F be two vector bundles over reg(V)endowed respectively with metrics h and ρ,Riemannian if E and F are real vector bundles,Hermitian if E and F are complex vector bundles.Finally,let ∇: C∞(V,E)→C∞(M,T∗reg(V)⊗E)be a metric connection.Consider a first order differential operator of this type:

where θ0∈C∞(reg(V),Hom(T∗reg(V)⊗E,F)).Assume that θ0extends as a bounded operator

Then we have the following inclusion:

In particular,(1.2)holds when D is the de Rham differential dk:Ωkc(reg(V))→Ωk+1c(reg(V)),a Dirac operator D : C∞c(reg(V),E)→ C∞c(reg(V),E),or the Dolbeault operator:

And after this proposition,we have several remarks and corollaries.

In Section 4,we show that Theorem 3.1 can be applied on Schrödinger operators and heat operators on the irreducible quasi-Kähler complex varieties.

2 Preliminary

In this section,we recall some basic concepts and notations that will be used throughout this paper.

2.1 Basic concepts,notations and results on almost complex manifolds and quasi-Kähler manifolds

An almost complex manifold M is a differentiable manifold on which there exists a (1,1)-tensor J,which we may consider as an isomorphism J : X(M)→X(M),satisfying the condition J2=−1.An almost complex manifold is orientable and of even dimension.Denotea Riemannian metric on M and X(M)the real vector fields over M.M is an almost Hermitian manifold (or an almost complex manifold)if J is compatible with the inner product in such a way thatfor all X,Y ∈X(M).There are tow special tensors defined in terms of the almost complex structure J that are very important.Firstly,the Kähler form (or the fundamental form)associated tois defined as

By easy calculations,one can see that

A well-known theorem in [7,17]states that M is a complex manifold if and only if S vanishes identically.If we extend the Riemannian connection ∇Xof M to be a derivation on the tensor algebra of M,then we have the formulae

Let X,Y ∈X(M),M is called

An almost complex structure is said to be integrable if it is induced by a complex structure.Actually,if M is a complex manifold of dimension n.Denote TM the real tangent space,TMCthe complex tangent space.Let (z1,··· ,zn)be a local holomorphic coordinate in a neighborhood U of p ∈M.Write zj=xj+iyj.Then (x1,··· ,xn,y1,··· ,yn)is a smooth real coordinate in U andgives a local frame of the tangent bundle TM.Denote by

for each 1 ≤j ≤n.Then T1,0M is the complex subbundle of TMCspanned bywhile T0,1M is spanned byLet us consider the isomorphism J : TM →TM defined by

for each 1 ≤j ≤n.Then the map J defined above is the almost complex structure induced by the complex structure of M.The following results are very useful.

Proposition 2.1(cf.[9,Proposition 5.2]If M is Kähler,almost Kähler,nearly Kähler,quasi-Kähler,or Hermitian,then any Hermitian submanifold of M has the same property.

Theorem 2.1(cf.[9,Theorem 5.7])A quasi-Kähler submanifold is a minimal variety.

From the above two results,we know that every complex submanifold of a quasi-Kähler complex manifold is quasi-Kähler and minimal.Since the regular part of a complex variety is a complex submanifold,we could use the words “quasi-Kähler variety” without any ambiguity.As we have mentioned above,the fact that the projective variety is minimal plays an important role in the proofs of some theorems in [4,14].Thus,we could expect that similar results could be proved on irreducible quasi-Kähler complex varieties.

2.2 Lp-space,Sobolev space and operator extension

We will recall some basic notations on Lp-spaces,Sobolev spaces and differential operators,all of which will be used in this paper.We refer to [3,5,12],or the appendix in [20]for a thorough discussion on the relative materials.Let (M,g)be an open and possibly incomplete Riemannian manifold of dimension m.Let E be a vector bundle over M of rank r with a metric h,Hermitian if E is a complex vector bundle,Riemannian if E is a real vector bundle.Let dvgdenote the volume element of g.We consider M endowed with Riemannian measure as in[12,p.59]or [5,p.29].A section is called measurable if,for any local trivialization (U,φ)of E,every φrof φ(s|U)=(φ1,··· ,φr)is a measurable function.For a fixed measurable section s,define its local norm(or point-wise norm)respect to h asThen for every p,1 ≤p<∞,we can define the Lp-norm of a measurable section s,

and thus the Lp-space of measurable sections over M,

When 1 ≤p < ∞,Lp(M,E)is a Banach space; furthermore if 1 < p < ∞,Lp(M,E)is a reflexive Banach space,i.e.,Lp(M,E)(Lp(M,E))′.When p=2,L2(M,E)is a Hilbert space with the natural inner product

Another important and useful conclusion is that when 1 ≤p < ∞,Lp(M,E)has a natural dense subset C∞c(M,E),the space of smooth sections with compact support.We can also define L∞(M,E)to be the space of measurable sections whose essential supremum is bounded,i.e.,L∞(M,E):={s |ess sup|s|h< ∞}.Notice that L∞(M,E)is also a Banach space.We would like to clarify the notations a little more.The spaces Lp(M,E)clearly depend on M,E,h,g,whereas we still denote them as Lp(M,E)instead of Lp(M,h,E,g)when there is no danger of confusion.Particularly,if E is the trivial bundle M ×R,we will write Lp(M,g)instead of Lp(M,R)while for the k-th exterior power of the cotangent bundle ΛkT∗M,we will write LpΩk(M,g)instead of Lp(M,ΛkT∗M).Suppose F is another vector bundle over M endowed with a metric ρ and P : C∞c(M,E)→C∞c(M,F)is a differential operator of order d ∈N.The formal adjoint of P,

is the differential operator defined by the following property: For each u ∈C∞c(M,E)and for each v ∈C∞c(M,F),we have the identity

On the basis of the above,we can now recall some knowledge of the extensions of an operator.

One can check that the operator P defined above is an unbounded,densely defined and closable operator from Lp(M,E)to Lp(M,F).In general cases,P may have several different closed extensions between the minimal and maximal extensions.For completeness,we recall the definitions of the minimal and maximal extensions below.The domain of the maximal extension of P :Lp(M,E)→Lp(M,F)is defined,in the distributional sense,as

and the minimal extension as

We put Pmaxs=v,Pmins=w by the definition above.On the other hand,the minimal extension of P is the closure of C∞c(M,E)under the graph norm ||s||L2(M,E)+||Ps||L2(M,F).One can check the following two important identity:

which means that Ptminis the adjoint of Pmaxrespect to the Hilbert space L2(M,E)and L2(M,F),and similarly Ptmaxwith respect to Pmin.Another useful fact is the orthogonal decomposition of the L2-space:

Next,we recall the concepts and notations of Sobolev space associated to a metric connection.Let E and h be defined as above.Let ∇: C∞(M,E)→C∞(M,T∗ME)be a metric connection,i.e.,a connection that is compatible with the differential and the metric in such a way: for each s,u ∈C∞(M)we have d(h(s,u))=h(∇s,u)+h(s,∇u).Let∇t:C∞c(M,T∗M ⊗E)→C∞c(M,E)be the formal adjoint of ∇with respect toand g.Then we can define the Sobolev space W1,2(M,E)as

By the definition of dom(Pmax),we can see that W1,2(M,E)=dom(∇max).We can also define the Sobolev spaceas follows:

Analogously to the case of maximal extension,by the definition of dom(Pmin),we can also conclude that(M,E)=dom(∇min).If E is the trivial bundle M × R,we will write W1,2(M,g)and(M,g)when there is no danger of confusion.We refer the reader to [1]for more information on Sobolev spaces.We would like to recall the following result for the reader’s convenience.

Proposition 2.2Let (M,g)be an open and possibly incomplete Riemannian manifold of dimension m.Let E be a vector bundle over M endowed with a metric h.Let U ⊂M be an open subset with compact closure.Consider the spaces L2(U,E|U)and(M,E)where U is endowed with the metric g|U.Then the natural inclusion

is a compact operator.Therefore the map

given by

is an injective and compact operator.

For the proof of the above proposition,we refer the reader to [15,p.349]and [20,p.179].

The de Rham differential operator acting on the space of smooth k-forms with compact support,dk: Ωkc(M)→Ωk+1c(M).Given a Riemannian metric g on M,we denote byand by |·|gkrespectively the metric and the point-wise norm induced by g on ΛkT∗M for each k=0,··· ,m,where m=dimM.When k=1,we will simply denote the corresponding term byand |·|ginstead ofand |·|g1.We will denote bythe metric that g induces on T∗M ⊗ΛkT∗M.Following the definitions,we denote by dk,max/min:L2Ωk(M,g)→L2Ωk+1(M,g)respectively the maximal and minimal extension of dkacting on the space of L2k-forms.

2.3 Basics on quadratic forms and the Friedrich extension

We will recall some basic results on quadratic forms and the Friedrich extension of a positive and symmetric operator.The reader can find a general description in[15,C.1].One can also find a more thorough discussion on this topic in[18–19].Let H be a complex Hilbert space with inner productand normA quadratic form is a sesquilinear map Q:dom(Q)×dom(Q)→C,where dom(Q)is a dense linear subspace of H.Q is called positive if Q(u,u)≥0 for any u ∈dom(Q).A positive quadratic form Q is called closed if(dom(Q),)is complete,where

Let B : H →H be a linear unbounded densely defined operator.B is called self-adjoint if B=B∗,symmetric if B ⊂B∗,and positive if≥0 for any u ∈dom(B).Now if we assume that B is symmetric and positive,we can define the quadratic form associated to B asDenote byB the inner product given by+QB(,)and by dom(QB)the completion dom(B)throughOne can check that the identity map Id : dom(B)→dom(B)extends as a bounded injective map iQB: dom(QB)→H.By this injective map,dom(QB)can always be identified with its image in H,that is

Following the above discussion,we can define the Friedrich extension of a positive self-adjoint operator B,denoted by BF.The domain of Friedrich extension is given by

and we put BFu:=v.One can check that BFis a positive and self-adjoint operator.What’s more,the above definition is equivalent to

and BF=B∗(u),that is

for u ∈dom(BF).One can also find a more complete discussion of Friedrich extension in [2].

3 Sobolev Spaces on Irreducible Quasi-Kähler Complex Varieties

In this section,we generalize the results of [4]and we use just the same method used in [4].Actually,we just use Bei’s proof and notations with some changes to adapt it to our situation.We work on irreducible quasi-Kähler complex varieties V.This mean that V is locally the zero set of a (finite)family of holomorphic functions of a compact quasi-Kähler complex manifold such that it is impossible to decompose V as a (finite)union of complex varieties which are not equal to V.To be more precise,if V=V1∪V2where V1,V2are complex varieties,then we have Vi=∅or Vi=V for i=1,2.We refer the reader to [11]for more details on this topic.Given an irreducible quasi-Kähler complex variety,we denote the singular subset of V by sing(V)and the regular part by reg(V):=V sing(V).The regular part reg(V)then becomes a quasi-Kähler complex manifold as we can see by [9].Usually,if sing(V)∅,reg(V)is an open and incomplete quasi-Kähler complex manifold with the induced metric from M.Now we prove a proposition which claims the existence of a certain sequence of cut-off functions.This result is similar to that contained in [14,p.871],[21,Theorems 3.1–3.2]and [4,Theorem 4.2].

Proposition 3.1(cf.[4,Proposition 4.1])Let M be a complex manifold and let h and g be two Hermitian metrics on M such that g ≥h.Then for each η ∈Ω1c(M),we haveTherefore the identity map Id : Ω1c(M)→Ω1c(M)extends as a continuous inclusion L2Ω1(M,g)L2Ω1(M,h),so that for each φ ∈L2Ω1(M,g)we have

ProofThe proof is essentially a calculation of linear algebra.One can see more details in[8,p.146]for example.

Proposition 3.2Let M be a compact quasi-Khler complex manifold with a Hermitian metric h and V be an irreducible quasi-Kähler complex variety of M,and let g be the metric on reg(V)induced by the metric of M.Then there exists a sequence of Lipschitz functions with compact support {φj}j∈Nsuch that

(1)0 ≤φj≤1 for each j;

(2)φj→1 point-wise;

(3)φj∈D(d0,min)for each j ∈N and

In Particular,1 ∈D(d0,min).

ProofLet π :→V be a resolution of singularities(which exists thank to the fundamental work in [13]).We recall that π :→V is a holomorphic and surjective map such that

is a biholomorphism where E=π−1(sing(V))is the exceptional set.Moreover,we can assume that E is a divisor with only normal crossings,that is,the irreducible components of E are regular and meet complex transversely.In particular,π−1(reg(V))is a union of finite number compact complex submanifolds,which meansfor some m ∈N+.Therefore,we have codimR(Si)≥2 for each i=1,··· ,m.Suppose that h is a Hermitian metric on.Let us define V′:=π−1(reg(V))and h′=h|V′.Firstly,we will show that there is a sequence of Lipschitz functions {ψj}j∈Nwith compact support on (V′,h′),which satisfies the three properties stated in the proposition.We use the method used in [6,14].Define Mi:=Si.Let ribe the distance function to Siinduced by h.let εj:=andThen we define ψj,Mias

We can easily check that each ψj,Midefined in (3.1)is a Lipschitz function with compact support,partially by the fact that M is compact.Then by [12,Theorem 11.3],[4,Proposition 1.2]and the fact that Mihas finite volume,we get that {ψj,Mi}i∈N⊂D(d0,min)on (Mi,h|Mi).We can also easily check that 0 ≤ψj,Mi≤1 andpoint-wise.Moreover,by [6],we can get

We recall that the previous limit is based on an estimate of the volume of a tubular neighborhood of Si.In this estimate,the lower bound of the real codimension of Siis a key factor.Now we define

For each j ∈N,ψjis defined as a product of a finite number of non negative Lipschitz functions with compact support and bounded above by 1.We can easily check that ψjis a nonnegative Lipschitz function with compact support and bounded above by 1 itself.Thus,arguing as above,we can conclude that ψjj∈N⊂D(d0,min)on (V′,h′).Clearly for each ψjwe have 0 ≤ψj≤1 andpoint-wise.Now we have to show that

By the fact that 0 ≤γi≤1 to establish (3.3),it is enough to show that

for each p,q ∈{1,··· ,m}.

This follows because

and by (3.2),we have

and

These relations allow us to conclude that on (V′,h′)there is a sequence of Lipschitz functions with compact support {ψj}j∈N,which satisfies the three properties stated in this proposition.Now letbe the Kähler metric on V′defined as π∗g.We can seeas the pullback of the metric on M through the map π :→V ⊂M.By the fact that dπ,the differential of π,degenerates onV′,we get that g ≤Ch′,for some positive real constant C > 0.Now as an immediate application of Proposition 3.1,we can conclude that the sequence {ψj}j∈Nsatisfies the three properties stated in this proposition also with respect to the balanced manifold (V′,).Finally,by the fact that π|V′: (V′,)→(reg(V),g)is an isometry,defining φj:=ψj◦(π|V′)−1,we obtain our desired sequence on (reg(V),g).

We would like to point out that in the proof of Proposition 3.2,we use the resolution theorem of Hironaka.Thus we should assume that the background manifold should be both quasi-Kähler and complex,namely,we only consider quasi-Kähler complex varieties.Next we give our main theorem in this paper.

Theorem 3.1Let M be a compact quasi-Kähler complex manifold with a Hermitian metric h and V be an irreducible quasi-Kähler complex subvariety of M.Let E be a vector bundle over reg(V)and let h′be a metric on E,Riemannian if E is a real vector bundle,Hermitian if E is a complex vector bundle.Let g be the metric on reg(V)induced by the metric h.Finally,let ∇: C∞(reg(V),E)→C∞(reg(V),T∗reg(V)⊗E)be a metric connection.We have the following properties:

(1)W1,2(reg(V),E)=W01,2(reg(V),E).

(2)Assume that v >1,then there exists a continuous inclusion

(3)Assume that v > 1,then the inclusion W1,2(reg(V),E)L2(reg(V),E)is a compact operator.

ProofThe first point follows by Proposition 3.2 and [4,Proposition 3.1].The continuous inclusion W1,20 (reg(V),g)Lreg(V),gis established in [14,p.874]or [21,p.113].Now,by the first point of this theorem (or by [14,Theorem 4.1]or by [21,Corollary 3.1]),we know that W1,2(reg(V),g)=(reg(V),g)and therefore we have the continuous inclusion

By [4,Proposition 2.1],we get the continuous inclusion

Finally,by the density of C∞(reg(V),E)∩W1,2(reg(V),E)in W1,2(reg(V),E)(cf.[4,Proposition 1.2]),the continuous inclusionis established.Finally the third point is a consequence of the second point and [4,Proposition 3.3].

Remark 3.1The statement of Theorem 3.1 can be reformulated as follows.dom(∇max)=dom(∇min),there is a continuous inclusion domand the natural inclusion dom(∇max)L2(reg(V),E)is a compact operator where dom(∇max)is endowed with the corresponding graph norm.

Remark 3.2We would like to mention that the key point which enables us to generalize the results of [4]from the irreducible complex projective varieties to the quasi-Kähler complex subvarieties is that every quasi-Kähler complex submanifold is minimal.

Corollary 3.1Under the assumptions of Theorem 3.1,im(∇min)=im(∇max)is a closed subspace of L2(reg(V),T∗reg(V)E).

ProofBy Theorem 3.1,we know that ∇min=∇maxand therefore im(∇min)=im(∇max).Then the corollary follows by [4,Corollary 3.1].

Proposition 3.3Let (reg(V),g)be as in Theorem 3.1.Let E and F be two vector bundles over reg(V)endowed respectively with metrics h and ρ,Riemannian if E and F are real vector bundles,Hermitian if E and F are complex vector bundles.Finally,let ∇: C∞(V,E)→C∞(M,T∗reg(V)⊗E)be a metric connection.Consider a first order differential operator of this type:

where θ0∈C∞(reg(V),Hom(T∗reg(V)⊗E,F)).Assume that θ0extends as a bounded operator,

Then we have the following inclusion:

In particular (3.5)holds when D is the de Rham differential dk:Ωkc(reg(V))→Ωk+1c(reg(V)),a Dirac operator D : C∞c(reg(V),E)→ C∞c(reg(V),E),or the Dolbeault operator:

ProofThis follows from Theorem 3.1 and [4,Proposition 3.2].

4 Schrödinger Operators on Irreducible Quasi-Kähler Complex Varieties

Let V be a quasi-Kähler complex subvariety,reg(V)be its regular part and E be a vector bundle over reg(V)endowed with a metric h,Riemannian if E is a real vector bundle,Hermitian if E is a complex vector bundle.Let h′be the induced metric of h on reg(V)and let∇: C∞(reg(V),E)→C∞(reg(V),T∗reg(V)⊗E)be a metric connection.We consider some Schrödinger type operators

where ∇t: C∞c(reg(V),T∗reg(V)⊗E)→C∞c(reg(V),E)is the formal adjoint of ∇and L ∈C∞(reg(V),End(E)).

Theorem 4.1Let V,E,g,h,and ∇be as described above.Let

be a Schrödinger type operator with L ∈C∞(reg(V),End(E)).Assume that:

(1)P is symmetric and positive.

(2)There is a constant c ∈R such that for each s ∈C∞(reg(V),E),we have

Let PF:L2(reg(V),E)→L2(reg(V),E)be the Friedrich extension of P and let

be the Friedrich extension of ∆0:C∞c(reg(V))→C∞c(reg(V)).Then the heat operator associated to PF,

is a trace class operator and its trace satisfies the following inequality:

where m is the rank of the vector bundle E.

ProofThis follows by [4,Proposition 3.5]and Theorem 3.1.

Let (M,g)be as above.Let kP(t,x,y)be the smooth kernel of the hear operator e−tPF.Denote the pointwise operator norm of the heat operator as

Proposition 4.1Under the assumptions of Theorem 4.1.Assume that dimCV >1.Then the following inequality holds for 0

This implies that:

(1)e−tPFis a ultracontractive operator for each 0 < t < 1.This means that for each 00 such that

for each s ∈L1(reg(V),E).In particular,for each 0 < t < 1,e−tPF: L1(reg(V),E)→L∞(reg(V),E)is continuous.

(2)If s is an eigensection of PF:L2(reg(V),E)→L2(reg(V),E)then s ∈L∞(reg(V),E).

ProofThe proof follows by [4,Proposition 3.6]and Theorem 3.1.

AcknowledgementThe author wants to express great thanks to Prof.Kefeng Liu for many useful discussions and comments.