L p -stability and positive scalar curvature rigidity of Ricci-flat ALE manifolds
aa r X i v : . [ m a t h . DG ] S e p L p -STABILITY AND POSITIVE SCALAR CURVATURE RIGIDITY OFRICCI-FLAT ALE MANIFOLDS KLAUS KR ¨ONCKE AND OLIVER L. PETERSEN
Abstract.
We prove stability of integrable ALE manifolds with a parallel spinor under Ricciflow, given an initial metric which is close in L p ∩ L ∞ , for any p ∈ (1 , n ), where n is thedimension of the manifold. In particular, our result applies to all known examples of 4-dimensional gravitational instantons. Our decay rates are strong enough to prove positivescalar curvature rigidity in L p , for each p ∈ h , nn − (cid:17) , generalizing a result by Appleton. Contents
1. Introduction 21.1. Geometric setup 31.2. Main results 41.3. Outline of the proof of stability 71.4. Structure of the paper 102. The space of gauged Ricci-flat metrics 102.1. Asymptotic structure 102.2. A projection map onto F L p -maximum principle 183.3. Short-time estimates for the heat flow of the modified Lichnerowicz Laplacian 243.4. Short-time estimates for the Ricci-de-Turck flow 304. The Ricci-de Turck flow and a mixed evolution problem 334.1. A Ricci-de Turck flow with moving gauge 334.2. A mixed evolution operator 344.3. The Ricci flow as a mixed evolution problem 355. The iteration map 365.1. Estimates for the linear problem 365.2. The Banach space 375.3. Mapping properties of the iteration map 395.4. Contraction properties of the iteration map 466. Long-time existence and convergence 596.1. Establishing a fixed point of the iteration map 596.2. Optimal convergence rates of the modified Ricci-de Turck flow 626.3. Decay of the de Turck vector field and the Ricci curvature 696.4. Convergence of the Ricci-flow 726.5. Positive scalar curvature rigidity 73References 76 Introduction
A one-parameter family { g t } t ∈ I of Riemannian metrics on a manifold M n , n ≥ ∂ t g t = − g t . The Ricci flow was introduced in the eighties by Hamilton [Ham82] and it has become an impor-tant tool in Riemannian geometry ever since. Its success culminated in Perelman’s proof of thePoincar´e and Geometrization Conjectures about the classification of closed three-dimensionalmanifolds [Per02]. A natural question in geometric analysis is the stability of stationary pointsof the Ricci flow on the space of metrics (modulo homotheties), which we call Ricci solitons.This problem is relevant for the formation of singularities under the Ricci flow. Any singularityadmits a blowup limit which is a Ricci soliton, and its instability would exclude it as a possiblesingularity model for generic initial data [IKˇS19].On compact manifolds, the stability problem is by now well understood in terms of Perelman’sentropies due to work by Haslhofer-M¨uller and the first author [HM14, Kr¨o15, Kr¨o20]. In thispaper, we are interested in the stability problem on non-compact manifolds . As singularitymodels for the 4-dimensional Ricci flow on compact manifolds, only non-compact Ricci-flat
ALEspaces can appear if the scalar curvature is bounded along the flow, see [Bam18]. In fact, suchsingularities have recently been shown to exist by Appleton [App19] (however, with unboundedscalar curvature). We expect that stability questions of ALE spaces are deeply connected to theformation of singularites under 4-dimensional Ricci flow.The main result of this paper states that integrable ALE spaces with a parallel spinor (henceRicci-flat) are dynamically stable to perturbations in a small L p ∩ L ∞ -neighbourhood, for any p < n . In terms of fall-off conditions on the perturbations, our result require a fall-off of order O (cid:0) r − − ǫ (cid:1) . See Theorem 1.8 for the precise statement. Our main result applies in particular to allknown 4-dimensional Ricci-flat ALE spaces (which all belong to the Kronheimer classification ofgravitational instantons). An interesting application of our result is the scalar curvature rigidityresult of integrable ALE spaces with a parallel spinor with respect to perturbations in L p ∩ L ∞ for p < nn − , see Theorem 1.12. We construct counterexamples to scalar curvature rigidity for p > nn − , showing that our scalar curvature rigidity result is at least almost sharp, see Theorem1.13.Our result is a significant improvement over the L -stability result of Ricci-flat ALE spaces,by Deruelle and the first the first author [DK20], where the initial data was assumed to be L ∩ L ∞ -close and no convergence rate was established. As a consequence, only convergence ofthe Ricci-de Turck flow was shown in [DK20], not convergence of the Ricci flow. In this paper,we establish sharp convergence rates for the Ricci-de Turck, allowing us to conclude convergenceof the actual Ricci flow. Analogous to these results, the stability of R n was proven by Schulze,Schn¨urer and Simon [SSS08]. Their proof relies heavily on the explicit geometry of R n andcannot be generalized to the ALE setting.There are several further results in the literature on stability of Ricci flow on certain non-compact manifolds, including the stability of hyperbolic space [SSS11], hyperbolic spaces withcusps [Bam14], symmetric spaces of non-compact type [Bam15], complex hyperbolic space[Wu13], cosh-cylinders by the first author [Kr¨o18] and further non-trivial non-compact expand-ing Ricci solitons [Der15, DL17, WW16]. All these do not appear as blowup limits of Ricci flows,hence these results are not relevant for Ricci flow singularities. Moreover, the main technicaldifference is that the continuous spectrum of the linearized operator in the ALE case is the non-negative real axis , whereas the continuous spectrum in the above mentioned results (apart from R n ) is bounded way from zero . TABILITY OF RICCI-FLAT ALE MANIFOLDS 3
Geometric setup.
Before we explain the main results of this paper, let us introduce thegeometric setting we are working in.
Definition 1.1 (ALE manifold) . A complete Riemannian manifold ( M n , g ) is called asymptot-ically locally Euclidean with one end of order τ > K ⊂ M and adiffeomorphism φ : M ∞ := M \ K → ( R n \ B ) / Γ, where Γ is a finite subgroup of SO ( n ) actingfreely on R n , such that (cid:12)(cid:12) ( ∇ eucl ) k ( φ ∗ g − g eucl ) (cid:12)(cid:12) eucl = O ( r − τ − k )holds on ( R n \ B ) / Γ. The diffeomorphism φ will also be called “coordinate system at infinity”.We are particularly interested in Ricci-flat
ALE manifolds, of which many examples do exist:
Example 1.2 (Ricci-flat ALE manifolds) . The simplest example of a Ricci-flat ALE manifold(different from R n ) is the Eguchi-Hanson manifold. Let α , α , α be the standard left-invariantone-forms on S . For each ǫ >
0, define the Eguchi-Hanson metric g eh,ǫ := r ( r + ǫ ) (cid:0) dr ⊗ dr + r α ⊗ α (cid:1) + (cid:0) r + ǫ (cid:1) ( α ⊗ α + α ⊗ α ) , for r >
0. After we quotient by Z , we can smoothly glue in an S at r = 0 to get the(complete) Eguchi-Hanson manifold ( T S , g eh,ǫ ), which is ALE with Γ = Z and hyperk¨ahler,hence Ricci-flat. This is an example in Kronheimer’s classification of hyperk¨ahler ALE manifolds[Kro89]: Each 4-dimensional hyperk¨ahler ALE manifold is diffeomorphic to a minimal resolutionof ( R \ { } ) / Γ, where Γ ⊂ SU(2) be a discrete subgroup acting freely on S .These examples satisfy an important assumption, which can be defined under the followingcondition: Definition 1.3 (Spin ALE manifold) . We say that an ALE manifold is spin if it carries a spinstructure which is compatible with the Euclidean spin structure on M ∞ .The main assumption is now that ( M, h ) is an ALE spin manifold with a parallel spinor . Thisassumption has various consequences, c.f. also the discussion in [KP20, Section 6]: • ( M, h ) is Ricci-flat. • If (
M, h ) has irreducible holonomy unless it is flat. Consequently,Hol(
M, h ) ∈ { SU( n/ , Sp( n/ , Spin(7) } . (1) • M is even-dimensional (we therefore excluded the case of holonomy G ). • ( M, h ) has at most finite fundamental group.
Remark 1.4.
All known Ricci-flat ALE manifolds satisfy (1) and thus carry a parallel spinor.Moreover, all these groups actually appear as holonomy groups of Ricci-flat ALE manifolds,see [Kro89, Joy99, Joy00, Joy01]. It is an open question whether there are other examples, c.f.[BKN89, p. 315].Up to a gauge term, the linearization of the Ricci curvature is given by an elliptic operator,called the Lichnerowicz Laplacian:
Definition 1.5.
The
Lichnerowicz Laplacian on a Ricci-flat manifold (
M, h ) is defined as∆ L := ∇ ∗ ∇ − C ∞ ( M, S M ) → C ∞ ( M, S M ) , where Rm( k ) ij = R imnj k mn , for any k ∈ C ∞ ( M, S M ). The manifold ( M, h ) is called linearly stable if ∆ L ≥ KLAUS KR ¨ONCKE AND OLIVER L. PETERSEN
It is well known that a Ricci-flat manifold with a parallel spinor is linearly stable [DWW05,Wan91]. The reason is that there is a parallel bundle endomorphismΦ : C ∞ ( S M ) → C ∞ ( S ⊗ T ∗ M ) , such that Φ ◦ ∆ L = D T ∗ M ◦ Φ , (2)where D T ∗ M is the twisted Dirac operator on vector spinors.Another nessecary notion we need is the one of integrability which we define in the following.For 1 ≤ p < q ≤ ∞ , we use the notation L [ p,q ] := L p ∩ L q = ∩ r ∈ [ p,q ] L r . Furthermore, for a fixedmetric ˆ h , we define M [ p,q ] as the set of metrics g such that g − ˆ h ∈ L [ p,q ] ( S M ). Definition 1.6.
A spin ALE manifold ( M, ˆ h ) with a parallel spinor is called L [ p, ∞ ] -integrableif there exists an L [ p, ∞ ] -neighbourhood U ⊂ M [ p, ∞ ] such that the set F U := n h ∈ U | Ric h = 0 , h ˆ h − d (tr h ˆ h ) = 0 o is a finite-dimensional submanifold of M [ p, ∞ ] only containing metrics with a parallel spinor andsatisfying T ˆ h F U = ker L [ p, ∞ ] (∆ L, ˆ h ) := n k ∈ ker(∆ L,h ) | k ∈ L [ p, ∞ ] ( S M ) o . We call it integrable, if it is L [ p, ∞ ] -integrable for all p ∈ (1 , ∞ ). Remark 1.7.
The additional condition 2div h ˆ h − d (tr h ˆ h ) = 0 serves as a gauge condition. Insuitable weighted Sobolev spaces, it defines a slice of the action of the diffeomorphism group onthe space of metrics, see [DK20, Proposition 2.11].The integrability condition has been shown to hold for K¨ahler and hence also for hyperk¨ahlermanifolds, see [DK20]. Therefore, the integrability is in fact known to be automatic , given aparallel spinor, unless the holonomy is Spin(7). However, also in the Spin(7) case integrabilityis widely expected to be true.1.2. Main results.
We formulate the main theorem of this paper. The appearing norms andcovariant derivatives are taken with respect to ˆ h . Theorem 1.8.
Let ( M n , ˆ h ) be an ALE manifold, which carries a parallel spinor and is integrable.Then for each q ∈ (1 , n ) and each L [ q, ∞ ] -neighbourhood U ⊂ M of ˆ h in the space of metrics,there exists another L [ q, ∞ ] -neighbourhood V ⊂ U of ˆ h with the following property:For each metric g ∈ V on M , the Ricci flow { g t } t ≥ starting at g exists for all time and thereis a family of diffeomorphisms { φ t } t ≥ such that φ ∗ t g t ∈ U for all t ≥ and φ ∗ t g t converges to aRicci-flat metric h ∞ as t → ∞ .Moreover, if g − ˆ h ∈ L p for some p ∈ (1 , q ] , there exists a smooth family of Ricci-flat metrics h t , such that follwing convergence rates do hold: (i) For each k ∈ N and τ > , there exists a constant C = C ( τ ) such that for all t ≥ , wehave k h t − h ∞ k C k ≤ C · t − np + τ . (3)(ii) For r ∈ [ p, ∞ ] and k ∈ N such that n (cid:16) p − r (cid:17) + k < n p , there exists a constant C = C ( p, q, k ) such that for all t ≥ , we have (cid:13)(cid:13) ∇ k ( φ ∗ t g t − h t ) (cid:13)(cid:13) L r ≤ C · t − n ( p − r ) − k . (4) TABILITY OF RICCI-FLAT ALE MANIFOLDS 5 (iii)
For r ∈ [ p, ∞ ] and k ∈ N such that n (cid:16) p − r (cid:17) + k ≥ n p , there exists for each τ > aconstant C = C ( p, q, k, τ ) such that for all t ≥ , we have (cid:13)(cid:13) ∇ k ( φ ∗ t g t − h t ) (cid:13)(cid:13) L r ≤ C · t − n p + τ . (5)The technical difficulty of this geometric situation can be read off already from spectral prop-erties of the Lichnerowicz Laplacian. All the other results on non-compact manifolds (except R n )mentioned in the final paragraph of the introduction use property strict linear stability , whichmeans P ≥ c > c ∈ R and the associated linear operator P . This property gives nice decayestimates for the heat kernel and causes exponential convergence of the flow. In contrast, thecontinuous spectrum of ∆ L on Ricci-flat ALE manifolds is always [0 , ∞ ). Thus (6) can neverhold in this setting, not even on ker L (∆ L ) ⊥ . Instead, one can prove the weaker inequality∆ L | ker L (∆ L ) ⊥ ≥ ∇ ∗ ∇ > , which was the central ingredient for proving the aforementioned L -stability result in [DK20].In this paper, we use novel estimates for the heat kernel of ∆ L and its derivatives, which theauthors developed in a recent paper [KP20]. Here, we follow an approach by Koch and Lamm[KL12] and establish the Ricci-de Turck flow as the fixed point of a contraction map. In thepresent geometric situation, we had to overcome some technical obstacles, which we explain inSubsection 1.3 below. Remark 1.9.
The diffeomorphisms φ t are coming from the de Turck vector field: The family φ ∗ t g t is a Ricci-de Turck flow. For t ∈ [0 , h as the reference metric. For t ≥ φ ∗ t g t is a Ricci-de Turck flow with moving reference metric h t . This choice of gauge turned out to bemore convenient in our setting.The assumption of having a parallel spinor is pitoval for the following two reasons: • We showed in [KP20] that under this assumption, ker L (∆ L, ˆ h ) ⊂ O ∞ ( r − n ), which im-proves the result in [DK20], where we only showed ker L (∆ L, ˆ h ) ⊂ O ∞ ( r − n ). Thisallows us to have a better control on the reference metrics h t , as we will then have h t − ˆ h ∈ O ∞ ( r − n ) as well. • In [KP20], we computed (with the help of (2)) optimal estimates on the heat kernelof the Lichnerwicz Laplacian and its derivatives. These estimates are strong enough toestablish the Ricci flow as the fixed point of an iteration map.
Remark 1.10.
The decay rates for φ ∗ t g t − h t in (4) and (5) coincide with the decay rates of thenorm of the map e − ∆ L, ˆ h : L p ( S M ) ∩ ker L (∆ L, ˆ h ) ⊥ → L r ( S M ) . The convergence rate of h t in (3) comes from integrating the inequality k ∂ t h k C k ≤ C k φ ∗ t g t − h t k W ,r ≤ Ct − n ( p − r ) , (7)where we can pick r ∈ ( p, ∞ ] as large as we want. Here, the first estimate follows from theconstruction of h t and the second one follows from (5). We could also replace the C k -norm byany W k,s -norm with s ∈ (1 , ∞ ]. Note that the right hand side of (7) is not integrable, for any r ∈ ( p, ∞ ], if p ∈ [ n, ∞ ]. This rate therefore explains why we cannot take p ∈ [ n, ∞ ]. This is insharp contrast to the Euclidean case, where one can take h t ≡ h R n . There, one also expects therate in (4) to hold for all p ∈ [1 , ∞ ], see [App18] for partial results. KLAUS KR ¨ONCKE AND OLIVER L. PETERSEN
If we restrict to p < n , we can get rid of the diffeomorphisms: Theorem 1.11.
Let ( M n , ˆ h ) be an ALE manifold, which carries a parallel spinor and is in-tegrable. Then for each q ∈ (1 , n ) , there exists an L [ q, ∞ ] -neighbourhood V with the followingproperty: For each metric g ∈ V satisfying g − ˆ h ∈ L p for some p ∈ (1 , n ) , the Ricci flow g t starting at g exists for all time and converges to a Ricci-flat limit metric h ∞ as t → ∞ .Moreover, there exists a smooth family of Ricci-flat metrics h t such that for each τ > , we havea constant C = C ( τ ) such that for all t ≥ , k h t − h ∞ k C k ≤ C · ( t − n p + τ , if p ∈ (cid:0) , n (cid:1) ,t − n p + τ , if p ∈ (cid:2) n , n (cid:1) , k g t − h t k C k ≤ C · t − n p + τ , with the norms taken with respect to the limit metric h ∞ . Our third main result is an application of the previous ones and reads as follows:
Theorem 1.12.
Let ( M n , ˆ h ) be an ALE Ricci-flat spin manifold which is integrable and carriesa parallel spinor. Then for each q ∈ (1 , n ) , there exists a L q, ∞ -neighbourhood U of ˆ h in the spaceof metrics such that each smooth metric g ∈ U on M satisfying scal g ≥ , and (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L p < ∞ for some p ∈ h , nn − (cid:17) is Ricci-flat. This theorem generalizes a corresponding result for Euclidean space [App18] which also holdsfor q = ∞ . It is related to the rigidity part of the positive mass theorem, for which there existsalso a version on ALE spin manifolds [Dah97]. The ADM mass of an ALE manifold ( M n , g ) is m ( g ) = lim r →∞ Z S n − ( r ) / Γ ( ∂ j g ij − ∂ i g jj ) dV S n − ( r ) , where the components of g are taken with respect to an asymptotic coordinate system. Now if g − ˆ h ∈ L p , we heuristically expect g − ˆ h ∈ o (cid:16) r − np (cid:17) , ∂ ( g − ˆ h ) ∈ o (cid:16) r − np − (cid:17) . Due to [BKN89], we know that ˆ h − h R n ∈ O ∞ (cid:0) r − n +1 (cid:1) in suitable coordinates. We get ∂g ∈ o (cid:16) r − np − (cid:17) , if p > nn − , ∂g ∈ O (cid:0) r − n (cid:1) , if p ∈ (cid:18) , nn − (cid:19) . Thus for p < nn − we expect m ( g ) = 0 and g has to be Ricci-flat by the rigidity statement of thepositive mass theorem.For p = nn − , it is unclear what happens. In [App18], a partial result was shown for Euclideanspace, which we are not able to reproduce here. For the remaining cases for p , the converse holdsunder even milder assumptions on the background metric. Theorem 1.13.
Let ( M, ˆ h ) be a Ricci-flat manifold. Then for every p > nn − there exists asequence ( g i ) i ∈ N with scal g t > such that g i → ˆ h in L [ p, ∞ ] as i → ∞ . Moreoever, the g i can bechosen to be conformal to ˆ h . This assertion is a simple consequence of the implicit function theorem. Nevertheless, we couldnot find it in this form in the literature. Therefore we state it here to complement Theorem 1.12.
TABILITY OF RICCI-FLAT ALE MANIFOLDS 7
Outline of the proof of stability.
For proving stability of a given Ricci-flat metric h , itis more convenient to use the Ricci-de Turck flow ∂ t g t = − g t + L V ( g t ,h ) g t , V ( g, h ) k = g ij (Γ( g ) kij − Γ( h ) kij ) . (8)instead of the Ricci flow as it has the advantage of being strictly parabolic. More precisely, itcan be written in terms of the difference k = g − h as ∂ t k + ∆ L,h k = g − ∗ R h ∗ k ∗ k + g − ∗ g − ∗ ∇ k ∗ ∇ k + ∇ (( g − − h − ) ∗ ∇ k ) . (9)In integral form, the latter equation reads k t = e − t ∆ L,h k + Z t e − ( t − s )∆ L,h [ g − s ∗ R h ∗ k s ∗ k s + g − s ∗ g − s ∗ ∇ k s ∗ ∇ k s ] ds + Z t e − ( t − s )∆ L,h [ ∇ (( g − s − h − s ) ∗ ∇ k s )] ds. (10)It is now tempting to find a solution of this integral equation by picking an initial k , setting k (1) t = e − t ∆ L,h k and defining inductively k ( i +1) t = e − t ∆ L,h k + Z t e − ( t − s )∆ L,h [( g ( i ) s ) − ∗ R h ∗ k ( i ) s ∗ k ( i ) s + g − s ∗ ( g ( i ) s ) − ∗ ∇ k ( i ) s ∗ ∇ k ( i ) s ] ds + Z t e − ( t − s )∆ L,h [ ∇ ((( g ( i ) s ) − − h − ) ∗ ∇ k ( i ) s )] ds for i ∈ N . One would now hope that as i → ∞ , k ( i ) t converges in a suitable Banach space to asolution of (10) and hence of (9). In fact, this strategy was successfully carried out in [KL12],to prove L ∞ -stability of R n under Ricci flow.Carrying out these steps in the general ALE case is far more complicated. Here, we willexplain the main technical issues and outline the ideas how to overcome these problems.1.3.1. Controlling the linear part.
Problem 1.14.
The operator ∆
L,h will in general have a nontrivial ( L )-kernel and hence, e − t ∆ L,h admits stationary points. However, we need some decay for k in order to bound theconvolution integral.In [KP20], we were able to derive optimal polynomial decay rates of the heat kernel on theorthogonal complement of the kernel, which means we have to assume k ⊥ ker L (∆ L,h ). Theseestimates can be used if we find a projection map Φ which maps from a neighbourhood U of agiven Ricci-flat metric ˆ h onto a set of Ricci-flat metrics F , and comes with the property that g − Φ( g ) ⊥ L ker L (∆ L, Φ( g ) ) for g ∈ U . The Ricci-de Turck flow is then slightly modified to ∂ t g t = − g t + L V ( g t , Φ( g t )) g t . (11)The evolution equation on k t = g t − Φ( g t ) now looks slightly different than (9): ∂ t k + ∆ L,h k = (1 − D g Φ)( g − ∗ R h ∗ k ∗ k + g − ∗ g − ∗ ∇ k ∗ ∇ k + ∇ (( g − − h − ) ∗ ∇ k )) , (12)where h = h t = Φ( g t ) and the additional D g Φ-term describes the evolution of h . In view of(10), the next problem arises: Problem 1.15.
Compute the heat kernel of ∆
L,h for a time-dependent family h t of Ricci-flatmetrics. KLAUS KR ¨ONCKE AND OLIVER L. PETERSEN
We will solve this problem by assuming that h t converges to a limit h ∞ . We are thenrewriting (12) as an equation on k t (the part of k t orthogonal to ker L (∆ L,h ∞ )), with lefthand side given by ∂ t k + ∆ L,h ∞ k and an appropriate modified right hand side containing(∆ L,h t − ∆ L,h ∞ )( k t ). Controlling k t is already good enough to control k t : The orthogonalprojection Π ⊥ t : ker L (∆ L,h t ) ⊥ → ker L (∆ L,h ∞ ) ⊥ is an isomorphism for h t close enough to h ∞ .1.3.2. Finding the right Banach space.
The heat kernel of ∆
L,h admits mapping properties ofthe form (cid:13)(cid:13)(cid:13) ∇ i ◦ e − t ∆ L,h | ker ⊥ L (cid:13)(cid:13)(cid:13) L p ,L q ≤ Ct − α ( p,q,i ) , for some α ( p, q, i ) ≥
0. This suggests a suitable linear combination of termssup t ≥ (cid:16) t α ( p,q,i ) · (cid:13)(cid:13) ∇ i k t (cid:13)(cid:13) L q (cid:17) , to define a norm controlling k t . Since at most second derivatives apper in the evolution equation,it is not nessecary to use terms with i ≥
3. We also need a norm controlling h t . The evolutionequation ∂ t h = D g Φ( g − ∗ R h ∗ k ∗ k + g − ∗ g − ∗ ∇ k ∗ ∇ k + ∇ (( g − − h − ) ∗ ∇ k )) (13)suggests a combination ofsup t ≥ (cid:13)(cid:13)(cid:13) h t − ˆ h (cid:13)(cid:13)(cid:13) L p , sup t ≥ (cid:16) t β ( p,q ) · k ∂ t h t k L q (cid:17) . The first norm controls the distance to a Ricci-flat reference metric ˆ h . The second part determinesa possible convergence to a limit metric h ∞ . Here, β ( p, q ) ≥ k t and its derivatives.1.3.3. Controlling the inhomogeneous part in the iteration process.
In the iteration process, wewill have tripels ( h ( i ) t , h ( i ) ∞ , k ( i ) t ) consisting of an evolving family of Ricci-flat shadow metrics h ( i ) t with a limit h ( i ) ∞ and an evolving family of symmetric 2-tensors k ( i ) t orthogonal to the respectivekernels of h ( i ) t , which form a family of evolving metrics g ( i ) t := k ( i ) t + h ( i ) t that should eventuallyconverge to a Ricci flow. In the iteration process, we will have to control terms of the form Z t e − ( t − s )∆ L,h ( i ) ∞ H ( k ( i ) s , h ( i ) s ) ds. The polynomial decay rates appering so far suggest to control integrals of the form Z t s − α ( t − s ) − β ds. (14)If min { α, β } ≥
1, this integral is not finite. However, we can at least control the interior part ofthe integral by an elementary lemma.
Lemma 1.16.
Let α, β >
0, and define γ = min { α, β } , δ := max { α, β } . Then there exists aconstant C = C ( α, β ) such that for all t ≥
2, we have Z t − s − α ( t − s ) − β ds ≤ C · t − γ if δ > ,t − γ log( t ) if δ = 1 ,t − α − β if δ < , (15)and the rates on the right hand side are optimal. In particular, we have Z t − s − α ( t − s ) − β ds ≤ C · t − θ TABILITY OF RICCI-FLAT ALE MANIFOLDS 9 for every θ < min { α, β, α + β − } and also for θ = min { α, β, α + β − } , if max { α, β } 6 = 1. Proof.
If we substitute s = r + t/
2, the left hand side of the inequality can be written andestimated from above and below as( t − − β Z − t/ ( r + t/ − α dr + ( t − − α Z t/ − ( t/ − r ) − β dr ≤ Z − t/ ( r + t/ − α ( t/ − r ) − β dr + Z t/ − ( r + t/ − α ( t/ − r ) − β dr ≤ ( t/ − β Z − t/ ( r + t/ − α dr + ( t/ − α Z t/ − ( t/ − r ) − β dr. The rest of the proof follows from elementary calculus and a case by case analysis. (cid:3)
Treating boundary terms of the time integral.
In our proof, we will let an initial metric g (which is L p ∩ L ∞ -close to ˆ h ) evolve under the Ricci-de Turck flow (with gauge metric ˆ h ) evolveup to time t = 1. The metric g and the tensors h = Φ( g ), k := g − h are smooth and wecan bound all derivatives in terms of the initial data. By starting the iteration argument fromtime 1 instead of 0, we get sequences of metrics and tensors k ( i ) t , h ( i ) t , whose norms do not blowup as t ց
1. In this way, we can just get rid of the integral in (14) from 0 to 1.For the integral from t − t , there is one term that causes troubles: Problem 1.17.
We need to control the term Z tt − e − ( t − s )∆ L,h [ ∇ (( g − s − h − s ) ∗ ∇ k s )] ds. (16)In the iteration process, we need to control up to second derivatives of k ( i +1) s by using onlyup to second derivatives of k ( i ) s . Short-time estimates for parabolic equations show that (cid:13)(cid:13)(cid:13) ∇ e − ( t − s )∆ L,h Θ (cid:13)(cid:13)(cid:13) L p ≤ C ( t − s ) − k Θ k W ,p ∈ L ([ t − , t ]) (cid:13)(cid:13)(cid:13) ∇ e − ( t − s )∆ L,h Θ (cid:13)(cid:13)(cid:13) L p ≤ C ( t − s ) − k Θ k L p / ∈ L ([ t − , t ]) . Therefore, we can only estimate (cid:13)(cid:13)(cid:13)(cid:13) ∇ Z tt − e − ( t − s )∆ L,h [ ∇ (( g − s − h − s ) ∗ ∇ k s )] ds (cid:13)(cid:13)(cid:13)(cid:13) L p ≤ C sup s ∈ [ t − ,t ] (cid:13)(cid:13) ∇ (( g − s − h − s ) ∗ ∇ k s ) (cid:13)(cid:13) W ,p , but the right hand side contains third derivatives of k and thus, the iteration argument can notbe closed. Instead, we put this part of the integral to the left hand side of the equation as follows:For an initial tensor k , and a fixed time t >
1, we solve the equation ∂ s k + ∆ L,h ( i ) ∞ k = 0 , for s ∈ [1 , max { t − , } ]and afterwards the equation ∂ t k + ∆ L,g ( i ) ,h ( i ) k = 0 , for s ∈ [max { t − , } , t ] . Here, ∆
L,g ( i ) ,h ( i ) is a slightly modified Lichnerowicz Laplacian which captures exactly the criticalterms just discussed. It turns out that the associated evolution operator admits the same mappingproperties as the Lichnerwicz Laplacian. For large t , the short-time estimates for s ∈ [ t − , t ] donot destroy the decay rates generated by the Lichnerowicz Laplacian for s ∈ [1 , t − Structure of the paper.
In Section 2, we describe the space of gauged Ricci-flat metrics F in detail, study the asymptotics of its elements and derive sharp estimates for projectionmaps defined by elements in F . In Section 3, we derive novel Shi-type estimates for L p -normsunder parabolic equations. In Section 4, a suitable integral expression for solutions of the Ricci-de Turck flow is derived. Section 5 is the technical core of the paper, in which we study theprecise mapping properties of the iteration map which comes from the aforementioned integralexpression. These estimates are used to establish the Ricci-de Turck flow as a fixed point of thismap in Section 6 and to conclude the main Theorem 1.8. We conclude by proving the remainingtheorems of the introduction at the end of Section 6. Acknowledgements.
Part of this work was carried out while the authors were visiting theInstitut Mittag-Leffler during the program
General Relativity, Geometry and Analysis in Fall2019, supported by the Swedish Research Council under grant no. 2016-06596. We wish tothank the institute for their hospitality and for the excellent working conditions provided. Thework of the authors is supported through the DFG grant KR 4978/1-1 in the framework of thepriority program 2026:
Geometry at infinity .2.
The space of gauged Ricci-flat metrics
In order to set up the stability problem, we introduce the space of metrics we are consideringand the space of gauged Ricci-flat metrics, which we show convergence to.2.1.
Asymptotic structure.
Let from now on M denote the space of smooth Riemannianmetrics on M . As explained in the introduction, we will prove dynamical stability , i.e. that anyRicci flow starting close to ˆ h converges to a Ricci-flat metric near ˆ h . Due to the diffeomorphisminvariance, the space of Ricci-flat metrics near ˆ h is infinite dimensional within M . In order to geta finite dimensional space of Ricci-flat metrics near ˆ h , we therefore need to impose the de-Turckgauge. This corresponds to considering the following set F := { h ∈ M | h = L V ( h, ˆ h ) h } , where V ( h, ˆ h ) is the de-Turck vector field, defined byˆ h ( V ( h, ˆ h ) , · ) := div h ˆ h − d (tr h ˆ h )or locally by V ( h, ˆ h ) l = h ij (Γ( h ) lij − ˆΓ(ˆ h ) lij ) . In [DK20, Section 2], it was shown that for any Ricci-flat ALE manifold ( M, ˆ h ) and any p ∈ (1 , ∞ ), there exists a L [ p, ∞ ] -neighbourhood U such that F ∩ U = { h ∈ U | Ric h = 0 , V ( h, ˆ h ) = 0 } = F U . In particular, if ˆ h is integrable, then F ∩ U is a smooth manifold.For the analysis performed in this subsection we need weighted Sobolev spaces which aredefined as follows: Fix a point x ∈ M and pick a smooth function such that ρ ( y ) = p d ( x, y ) , for all y ∈ M outside a compact set, where d is the Riemannian distance. Definition 2.1.
Let E be a Riemannian vector bundle over M . For any k ∈ N , p ∈ [1 , ∞ ) andany δ ∈ R , the weighted Sobolev space W k,pδ ( V ) is the space of V -valued sections u ∈ W k,ploc ( V ) TABILITY OF RICCI-FLAT ALE MANIFOLDS 11 such that k u k k,p,δ := k X j =0 (cid:18)Z M (cid:12)(cid:12) ( ρ ∇ ) j u (cid:12)(cid:12) p ρ − δp − n dx (cid:19) /p is finite. We also use the notation L pδ := W ,pδ . Proposition 2.2.
Let ( M, ˆ h ) be an integrable Ricci-flat ALE manifold with a parallel spinor and p ∈ (1 , ∞ ) . Then, there is an open L [ p, ∞ ] -neighbourhood U of ˆ h , such that if h ∈ U∩F , then h − ˆ h ∈ O ∞ (cid:0) r − n (cid:1) . Proposition 2.2 is an improvement of Theorem 2.7 and Remark 2.8 in [DK20], where it wasproven that h − ˆ h ∈ O ∞ (cid:0) r − n (cid:1) . The tool to improve this result is a result of our companion paper [KP20], which says that underthe assumption of a parallel spinor, elements in the L -kernel of ∆ L decay as r − n , i.e.ker L (∆ L ) ⊂ O ∞ (cid:0) r − n (cid:1) . (17)By linearizing the defining equation in F , we note that any k ∈ T h F satisfies0 = ∆ L,h k + L h k, Γ( h ) − Γ(ˆ h ) i h, where h k, Γ( h ) − Γ(ˆ h ) i m := h iq h jl k ij (Γ( h ) mql − Γ(ˆ h ) mql )and from the proof of [DK20, Thm. 2.7], it follows that T h F ⊂ O ∞ (cid:0) r − n (cid:1) . (18)The key to prove Proposition 2.2 is to first improve (18) as follows: Proposition 2.3.
Let ( M, ˆ h ) be an integrable Ricci-flat ALE manifold with a parallel spinor.Then there is a small L [ p, ∞ ] -neighbourhood U with the following properties (i) dim ker L (∆ L ) is constant for all h ∈ U ∩ F . In particular, we can choose for each h ∈ U ∩ F a set of tensors { e ( h ) , . . . e m ( h ) } smoothly depending on h that forms an L ( g ) -orthonormal basis of ker L (∆ L ) . (ii) For all h ∈ U ∩ F , we have T h F ⊂ O ∞ (cid:0) r − n (cid:1) . Proof.
Let us start with the proof of (i): Since ˆ h is integrable, U ∩ F is a smooth manifold andall tangent spaces T h F have the same dimension for h ∈ U ∩ F . We first construct an injection i : T h F → ker L (∆ L,h ). Let k ∈ T h F . Then k satisfies0 = ∆ L,h k + L h k, Γ( h ) − Γ(ˆ h ) i h We now add a gauge term to k to get an element in ker L (∆ L,h ). More precisely, let X be avector field and consider the tensor k = k + L X g . Then we have, due to standard commutationformulas for operators on Ricci-flat metrics that∆ L,h k = ∆ L,h ( L X h ) − L h k, Γ( h ) − Γ(ˆ h ) i h = L ∆ X h − L h k, Γ( h ) − Γ(ˆ h ) i h. Thus we have to solve the equation ∆ X = h k, Γ( h ) − Γ(ˆ h ) i in a suitable function space. Observe that Γ( h ) − Γ(ˆ h ) ∈ H iδ for δ > − n and k ∈ H iδ for δ > − n (for any i ∈ N in both cases). Therefore, h k, Γ( h ) − Γ(ˆ h ) i ∈ H iδ for δ > − n and i > n + 1 due to the estimate (cid:13)(cid:13)(cid:13) h k, Γ( h ) − Γ(ˆ h ) i (cid:13)(cid:13)(cid:13) H iδ δ ≤ C (cid:13)(cid:13)(cid:13) Γ( h ) − Γ(ˆ h ) (cid:13)(cid:13)(cid:13) H iδ k k k H iδ Now the connection Laplacian ∆ : H iδ ( T M ) → H i − δ − ( T M ) is a Fredholm operator for thenonexceptional values δ ∈ R \ { , , , . . . } ∪ { − n, − n, − n . . . } . Due to the identity ∆ | X | = − |∇ X | for harmonic vector fields, the maximum principle implies that every bounded harmonicvector field is parallel. On the other hand, since ( M, g ) is ALE and Ricci-flat, it cannot containparallel vector fields, due to the Cheeger-Gromoll splitting theorem. Therefore, we even haveker(∆) ∩ H iδ ( T M ) = { } for δ <
1. As a consequence, duality arguments (see e.g. [Pac13, Section10]) imply that ∆ : H iδ ( T M ) → H i − δ − ( T M ) is an isomorphism for all δ ∈ (1 − n, \ { − n, } .Therefore, we find for each δ ∈ (1 − n,
1) and i > n + 1 a unique solution X ∈ H i +2 δ ( T M ) of theequation ∆ X = h k, Γ( h ) − Γ(ˆ h ) i . Moreover, the vector field X is the same for all possible choices for i and δ .Now, ∆ L k = 0 and k ∈ ker L (∆ L ) because L X h ∈ H i − δ − ( S M ) ⊂ H i − − n ( S M ) = L ( S M ).Therefore we can define the desired map by i : k k + L X h where X ∈ H i − n ( T M ) is definedas the unique solution of the above equation. This map is injective because kL X k k H i − − n ≤ C k X k H i − n ≤ C k ∆ X k H i − − − n ≤ C (cid:13)(cid:13)(cid:13) Γ( h ) − Γ(ˆ h ) (cid:13)(cid:13)(cid:13) H i − − k k k H i − − n and therefore, k i ( k ) k H i − − n ≥ k k k H i − − n − kL X h k H i − − n ≥ k k k H i − − n , provided that the neighbourhood U is chosen small enough. We getdim ker L (∆ L,h ) ≥ dim T h F = dim T ˆ h F = dim ker L (∆ L, ˆ h ) . Because of (17), dim ker L (∆ L,h ) = dim ker(∆
L,h ) ∩ H iδ ( S M ) for all i ∈ N and δ ∈ ( − n, δ nonexpectional. Due to a standard fact from functional analysis, the function A dim ker( A ) is upper semi-continuous with respect to the operator norm on a fixed space ofFredholm operators . Therefore,dim ker L (∆ L,h ) = dim ker H iδ (∆ L,h ) ≤ dim ker H iδ (∆ L, ˆ h ) = dim ker L (∆ L, ˆ h )as well. We conclude (i).For the proof of (ii), let k, X and k as above. We know that k ∈ T h F ⊂ O ∞ (cid:0) r − n (cid:1) , k + L X h = k ∈ ker L (∆ L ) ⊂ O ∞ (cid:0) r − n (cid:1) . However, we also have seen that X ∈ H iδ ( T M ) for all δ > − n and i > n + 3 so that Sobolevembedding implies X ∈ O ∞ (cid:0) r δ (cid:1) for all δ > − n . Standard arguments (c.f. [KP20, Proposition 4.3]), using the equation∆ X = h k, Γ( h ) − Γ(ˆ h ) i ∈ O ∞ (cid:0) r − n (cid:1) show that we actually have X ∈ O ∞ (cid:0) r − n (cid:1) , TABILITY OF RICCI-FLAT ALE MANIFOLDS 13 hence L X h ∈ O ∞ (cid:0) r − n (cid:1) . Therefore, k ∈ T h F satisfies k = k − L X h ∈ O ∞ (cid:0) r − n (cid:1) , which finishes the proof. (cid:3) Proof of Proposition 2.2.
This is now an immediate consequence of Proposition 2.3 (ii). (cid:3)
A projection map onto F . In the previous subsection, we developed some understandingof the space F of gauged Ricci-flat metrics. We would like to construct a smooth mapΦ : U ⊂ M → F on some open neighbourhood U of ˆ h , provided that F is a smooth manifold. The constructiongoes as follows. First define the mapΨ : ker L (∆ L, ˆ h ) ⊥ ˆ h × F → M , ( k, h ) h + k − m X j =1 h k, e j ( h ) i L ( h ) e j ( h ) , where ( e j ( h )) mj =1 is an L ( h )-orthonormal basis of ker L (∆ L,h ) (depending smoothly on h ). Remark 2.4.
Note that Ψ | F = id F . and thatΨ( k, h ) ∈ h + ker L (∆ L,h ) ⊥ h . In fact, we have the following lemma:
Lemma 2.5.
For every p ∈ (1 , ∞ ) , there is an open L [ p, ∞ ] -neighbourhood V of (0 , ˆ h ) such that Ψ | V : V → M is a diffeomorphism of Banach manifolds onto its image.Proof.
Due to (17), ker L (∆ L, ˆ h ) ⊂ L [ p, ∞ ] for every p ∈ (1 , ∞ ). Now it is straightforward to seethat D Ψ (0 , ˆ h ) = id with respect to the decomposition(ker L (∆ L, ˆ h ) ⊥ ∩ L [ p, ∞ ] ) ⊕ ker L (∆ L, ˆ h ) ∼ = L [ p, ∞ ] ( S M ) , where we have identified T ˆ h F = ker(∆ L, ˆ h ) . The inverse function theorem now proves the claim. (cid:3)
Using the (smooth) projection map π : ker(∆ L, ˆ h ) ⊥ × F → F , and the neighbourhood U := Ψ( V ) , we may use the previous lemma to define the smooth mapΦ : U → F g π ◦ Ψ − ( g ) . When we later will consider a Ricci flow g t , we will through the map Φ always have a “shadowing”curve of Ricci-flat metrics h t := Φ( g t ) which is such that k t := g t − h t ⊥ h t ker L (∆ L,h t ) . The goal will then be to show that k t converges (fast enough) to 0 and that h t converges to alimit metric, i.e. that the Ricci flow converges to a metric g ∞ ∈ F .2.3. Properties of projection maps.
Throughout this subsection, let ( M, ˆ h ) be an integrableRicci-flat ALE manifold with a parallel spinor. Let further p ∈ (1 , ∞ ) and U be an L [ p, ∞ ] -neighbourhood which is so small that the projection map Φ : U → U ∩ F from the previoussubsection is defined. Since h − ˆ h ∈ O ∞ ( r − n ), the appearing norms and covariant derivativescan be taken with respect to any h ∈ U ∩ F .Here, we collect a few properties of projection maps. For h ∈ U ∩ F , let { e i ( h ) } ≤ i ≤ m be anorthonormal basis of ker L (∆ L,h ) which depends smoothly on h . For p ∈ (1 , ∞ ), we define thenatural projection mapsΠ k h : L [ p, ∞ ] ( S M ) → ker L (∆ L,h ) , k m X i =1 ( k, e i ( h )) L ( h ) e i ( h ) , Π ⊥ h : L [ p, ∞ ] ( S M ) → ker L (∆ L,h ) ⊥ ∩ L [ p, ∞ ] ( S M ) , k k − m X i =1 ( k, e i ( h )) L ( h ) e i ( h ) . If we choose U small enough, the matrix A ij = ( e i ( h ) , e j (¯ h )) L (¯ h ) is invertible for every pair h, ¯ h ∈ U ∩ F . In other words, the mapΠ k h, ¯ h := Π k ¯ h | ker L (∆ L,h ) : ker L (∆ L,h ) → ker L (∆ L, ¯ h )is invertible. Lemma 2.6.
If Π k h, ¯ h is invertible, the mapΠ ⊥ h, ¯ h := Π ⊥ ¯ h | ker L (∆ L,h ) ⊥ : ker L (∆ L,h ) ⊥ ∩ L [ p, ∞ ] ( S M ) → ker L (∆ L, ¯ h ) ⊥ ∩ L [ p, ∞ ] ( S M )is also invertible for every p ∈ (1 , ∞ ) and its inverse is given by(Π ⊥ h, ¯ h ) − = id − (Π k ¯ h,h ) − ◦ Π k h . (19) Proof.
For ¯ k ∈ ker L (∆ L, ¯ h ) ⊥ ∩ L [ p, ∞ ] ( S M ), let k := ¯ k − (Π k ¯ h,h ) − ◦ (Π k h (¯ k )) . Then, k ∈ ker L (∆ L,h ) ⊥ ∩ L [ p, ∞ ] ( S M ), becauseΠ k h ( k ) = Π k h (¯ k ) − Π k h ◦ (Π k ¯ h,h ) − ◦ (Π k h (¯ k )) = Π k h (¯ k ) − Π k h (¯ k ) = 0 . Moreoever, Π ¯ h ( k ) = k − Π k ¯ h ( k )= ¯ k − (Π k ¯ h,h ) − ◦ (Π k h (¯ k )) − Π k ¯ h [¯ k − (Π k ¯ h,h ) − ◦ (Π k h (¯ k )]= ¯ k − Π k ¯ h (¯ k ) − (Π k ¯ h,h ) − ◦ (Π k h (¯ k )) + Π k ¯ h ◦ (Π k ¯ h,h ) − ◦ Π k h (¯ k ) = ¯ k. In the last equation, we used that Π k ¯ h (¯ k ) = 0 and that (Π k ¯ h,h ) − ◦ Π k h (¯ k ) ∈ ker L (∆ L, ¯ h ). Hence,Π ⊥ h, ¯ h is invertible because we constructed its inverse explicitly. (cid:3) TABILITY OF RICCI-FLAT ALE MANIFOLDS 15
Remark 2.7.
Later, we need to obtain estimates on the difference (Π ⊥ h , ¯ h ) − − (Π ⊥ h , ¯ h ) − ,which a priori doesn’t make sense as the operators are defined on different spaces. We canhowever make sense of this difference on all of L q ( S M ) and q ∈ (1 , ∞ ) by using the right handside of (19) as a definition. For convenience, we don’t change the notation for this extension. Lemma 2.8.
For g ∈ U , D g Φ vanishes on ker L (∆ L,h ) ⊥ ∩ L [ p, ∞ ] ( S M ). Proof.
Let g ∈ U be fixed and consider a curve g t in U with g = g . We may split g t = h t + k t where h t is a curve in F and k t ∈ ker L (∆ L,h t ) ⊥ . Let h = h and k = k . By the previouslemma, we can write k t = Π ⊥ h t (˜ k t ) with ˜ k t ∈ ker L (∆ L,h ) ⊥ . Differentiating at t = 0 yields g ′ = h ′ + Π ⊥ h (˜ k ′ t ) | t =0 + D Π ⊥ ( h, ˜ k t ) ( h ′ t ) | t =0 = h ′ + k ′ + D Π ( h,k ) ( h ′ ) , where D Π ( h,k ) is the Fr´echet derivative of Π ( . ) ( . ) : U ∩ F × L [ p, ∞ ] ( S M ) → L [ p, ∞ ] ( S M ) in thefirst component at ( h, k ). It is a map D Π ( h,k ) : T h F → L [ p, ∞ ] ( S M ) . With respect to the decomposition L [ p, ∞ ] ( S M ) = T h F ⊕ ker L (∆ L,h ) ⊥ ∩ L [ p, ∞ ] ( S M )and corresponding projection mapsproj T h F : L [ p, ∞ ] ( S M ) → T h F , proj ker L (∆ L,h ) ⊥ : L [ p, ∞ ] ( S M ) → ker L (∆ L,h ) ⊥ ∩ L [ p, ∞ ] ( S M ) , the differential D h,k Ψ reads D h,k Ψ = (cid:18) id T h F + proj T h F ◦ D Π ( h,k ) ( . ) 0proj ker L (∆ L,h ) ⊥ ◦ D Π ( h,k ) ( . ) id ker L (∆ L,h ) ⊥ (cid:19) , where h = Φ( g ) and k = g − h . The differential of Φ is therefore given by D g Φ = proj T h F ◦ ( D h,k Ψ) − = (id T h F + proj T h F ◦ D Π ( h,k ) ( . )) − ◦ proj T h F , (20)and the assertion is immediate. (cid:3) Lemma 2.9. If U was chosen small enough, then for every h ∈ U ∩ F , the mapΠ kF ,h = Π k h | T h F : T h F → ker L (∆ L,h )is invertible. In this case the projection maps proj T h F and proj ker L (∆ L,h ) ⊥ are given byproj T h F ( k ) = (Π kF ,h ) − (Π k h ( k )) , proj ker L (∆ L,h ) ⊥ ( k ) = Π ⊥ h ( k − (Π k F,h ) − (Π k h ( k )) . Proof.
At first, we clearly have (Π kF ,h ) − (Π k h ( k )) ∈ T h F and Π ⊥ h ( k − (Π k F,h ) − (Π k h ( k )) ⊥ ker L (∆ L,h ) by construction. It thus remains that they add up to k . We have(Π kF ,h ) − (Π k h ( k )) + Π ⊥ h ( k − (Π k F,h ) − (Π k h ( k ))= Π ⊥ h ( k ) + (Π kF ,h ) − (Π ⊥ h ( k )) − Π ⊥ h (Π k F,h ) − (Π k h ( k ))= Π ⊥ h ( k ) + Π k h (Π kF ,h ) − (Π k h ( k ))= Π ⊥ h ( k ) + Π k h ( k ) = k, which finishes the proof. (cid:3) Lemma 2.10.
Let g, ¯ g ∈ U and h, ¯ h, h , ¯ h , h , ¯ h ∈ U ∩ F . (i) For all q ∈ (1 , ∞ ) , r ∈ (1 , ∞ ] and l ∈ N , there exists a constant C = C ( q, r, l, U ) suchthat (cid:13)(cid:13)(cid:13) ∇ l ◦ Π k h ( k ) (cid:13)(cid:13)(cid:13) L r ≤ C k k k L q . (ii) For all q, r ∈ (1 , ∞ ) and l ∈ N , there exists a constant C = C ( q, r, l, U ) such that (cid:13)(cid:13) ∇ l ◦ Π ⊥ h ( k ) (cid:13)(cid:13) L q ≤ C ( (cid:13)(cid:13) ∇ l k (cid:13)(cid:13) L q + k k k L r ) . (iii) For all q ∈ (1 , ∞ ) , r ∈ (1 , ∞ ] and l ∈ N , there exists a constant C = C ( q, r, l, U ) suchthat (cid:13)(cid:13) ∇ l ◦ D Φ g ( k ) (cid:13)(cid:13) L r ≤ C k k k L q . (iv) For all q, r ∈ (1 , ∞ ) and l ∈ N , there exists a constant C = C ( q, r, l, U ) such that (cid:13)(cid:13)(cid:13) ∇ l ◦ (Π ⊥ h, ¯ h ) − ( k ) (cid:13)(cid:13)(cid:13) L q ≤ C ( (cid:13)(cid:13) ∇ l k (cid:13)(cid:13) L q + k k k L r ) . (v) For all q ∈ (1 , ∞ ) , r ∈ (1 , ∞ ] and l ∈ N , there exists a constant C = C ( q, r, l, U ) suchthat (cid:13)(cid:13) (Π ⊥ h − Π ⊥ ¯ h )( k ) (cid:13)(cid:13) L r ≤ C (cid:13)(cid:13) h − ¯ h (cid:13)(cid:13) L p k k k L q . (vi) For all q ∈ (1 , ∞ ) , r ∈ (1 , ∞ ] and l ∈ N , there exists a constant C = C ( q, r, l, U ) suchthat k ( D Φ g − D Φ ¯ g )( k ) k L r ≤ C k g − ¯ g k L [ p, ∞ ] k k k L q . (vii) For all q ∈ (1 , ∞ ) , r ∈ (1 , ∞ ] and l ∈ N , there exists a constant C = C ( q, r, l, U ) suchthat (cid:13)(cid:13)(cid:13) ∇ l ◦ [(Π ⊥ h , ¯ h ) − − (Π ⊥ h , ¯ h ) − ]( k ) (cid:13)(cid:13)(cid:13) L r ≤ C ( k h − h k L p + (cid:13)(cid:13) ¯ h − ¯ h (cid:13)(cid:13) L p ) k k k L q Proof.
Recall that ker L (∆ L,h ) ⊂ O ∞ ( r − n ), so that ker L (∆ L,h ) ⊂ W l,q for all l ∈ N and q ∈ (1 , ∞ ]. This, (i) and (ii) follow immediately from the definition of the projection maps Π k h and Π ⊥ h . From Lemma 2.9 and (20), we see that D g Φ = A ◦ Π k Φ( h ) , where A : ker L (∆ L, Φ( g ) ) → T Φ( g ) F is a linear map between finite dimensional spaces. On bothspaces, all elements are in O ∞ ( r − n ) and all W l,q -norms are equivalent for l ∈ N and q ∈ (1 , ∞ ].Therefore by using (i), we get (cid:13)(cid:13) ∇ l D Φ g ( k ) (cid:13)(cid:13) L r ≤ (cid:13)(cid:13)(cid:13) A ◦ Π k Φ( h ) ( k ) (cid:13)(cid:13)(cid:13) W l,r ≤ C (cid:13)(cid:13)(cid:13) Π k Φ( h ) ( k ) (cid:13)(cid:13)(cid:13) L r ≤ C k k k L q , which proves (iii). For (iv), recall from the proof of Lemma 2.6 that(Π ⊥ h, ¯ h ) − = id − (Π k ¯ h,h ) − ◦ Π k h . The map A = (Π k ¯ h,h ) − is a linear map between finite-dimensional spaces on which all W l,q -normsare equivalent for l ∈ N and q ∈ (1 , ∞ ]. Therefore, again by using (i), we get (cid:13)(cid:13)(cid:13) ∇ l (Π ⊥ h, ¯ h ) − (Π k h ( k )) (cid:13)(cid:13)(cid:13) L q ≤ (cid:13)(cid:13)(cid:13) A ◦ Π k h ( k ) (cid:13)(cid:13)(cid:13) W l,q ≤ C (cid:13)(cid:13)(cid:13) Π k h ( k ) (cid:13)(cid:13)(cid:13) L q ≤ C k k k L r , which implies (iv). For (v), observe first that(Π ⊥ h − Π ⊥ ¯ h )( k ) = (Π k h − Π k ¯ h )( k ) . By (i), we have a family of linear bounded maps ∇ l ◦ Π ⊥ h : L q → L r which depends smoothlyon h , and in particular, the dependence is Lipschitz. This implies (v). From the constructionof D g Φ and (i), we also have a family of linear bounded maps D g Φ : L q → L r which depends TABILITY OF RICCI-FLAT ALE MANIFOLDS 17 smoothly on g , and in particular, the dependence is Lipschitz. The estimate in (vi) is immediate.For the final point, we remark that using Remark 2.7, we may write ∇ l ◦ [(Π ⊥ h , ¯ h ) − − (Π ⊥ h , ¯ h ) − ] = ∇ l ◦ (Π k ¯ h ,h ) − ◦ Π k h − ∇ l ◦ (Π k ¯ h ,h ) − ◦ Π k h . By construction and the proof of part (iv), we have a family of bounded maps ∇ l ◦ (Π ⊥ h, ¯ h ) − ◦ Π k h : L q → L r which is smooth in h and ¯ h , in particular Lipschitz in both entries with respect to the L p norm.This proves part (vii). (cid:3) Short-time estimates for parabolic equations
Various expansions for the Ricci-de Turck flow.
Let h be a fixed Ricci-flat metricand consider h -gauged Ricci-de Turck flow, i.e. the evolution equation ∂ t g = − g + L V ( g,h ) g, V ( g, h ) k = g ij (Γ( g ) kij − Γ( h ) kij ) . Let ∇ denote the Levi-Civita connection and |·| the norm with respect to h . Lemma 3.1.
The Ricci-de Turck flow can be written with respect to the difference k = g − h as ∂ t k + ∆ L,g,h k = F ( g − , g − , ∇ k, ∇ k ) , (21) ∂ t k + ∆ L,h k = F ( g − , g − , ∇ k, ∇ k ) + F ( g − , R, k, k ) + F ( g − , k, ∇ k ) , (22) ∂ t k + ∆ h k = F ( g − , g − , ∇ k, ∇ k ) + F ( g − , g, R, k ) + ∇ a (( g ab − h ab ) ∇ b k ij ) , (23) where ∆ L,g,h k ij = − g ab ∇ ab k ij − k ab g ka h lb g ip h pq R jklq − k ab g ka h lb g jp h pq R iklq , ∆ L,h k ij = − h ab ∇ ab k ij , and the F i are h -parallel maps which are C ∞ ( M ) -linear in all entries.Proof. According to [Shi89, Lemma 2.1], this evolution equation can be rewritten as ∂ t g ij = g ab ∇ ab g ij − g kl g ip h pq R jklq − g kl g jp h pq R iklq + g ab g pq (cid:18) ∇ i g pa ∇ j g qb + ∇ a g jp ∇ q g ib (cid:19) − g ab g pq ( ∇ a g jp ∇ b g iq − ∇ j g pa ∇ b g iq − ∇ i g pa ∇ b g jq ) , where the curvature and the covariant derivatives are taken with respect to h . If h is Ricci-flat,this equation can be rewritten in terms of the difference k = g − h as ∂ t k ij = g ab ∇ ab k ij + k ab g ka h lb g ip h pq R jklq + k ab g ka h lb g jp h pq R iklq + g ab g pq (cid:18) ∇ i k pa ∇ j k qb + ∇ a k jp ∇ q k ib (cid:19) − g ab g pq ( ∇ a k jp ∇ b k iq − ∇ j k pa ∇ b k iq − ∇ i k pa ∇ b k jq ) . Then (21) follows from setting F ( g − , g − , ∇ k, ∇ k ) := g ab g pq (cid:18) ∇ i k pa ∇ j k qb + ∇ a k jp ∇ q k ib (cid:19) − g ab g pq ( ∇ a k jp ∇ b k iq − ∇ j k pa ∇ b k iq − ∇ i k pa ∇ b k jq ) . For (22), we first write the Lichnerowicz Laplacian as∆
L,h k ij = − h ab ∇ ab k ij − k ab h ka h lb R jkli − k ab h ka h lb R iklj . Note that the last two terms are equal but their separate treatment allows a better comparisonwith ∆
L,g,h from the previous lemma. We compute g ka g ip h pq R jklq − h ka R jkli = g ka g ip h pq R jklq − h ka h ip h pq R jklq = g ka ( g ip − h ip ) h pq R jklp + ( g ka − h ka ) h ip h pq R jklq = g ka k ip h pq R jklp − k mn g km h an h ip h pq R jklq = g ka k ip h pq R jklp − k mn g km h an R jkli and by exchanging i and j , g ka g jp h pq R iklq − h ka R iklj = g ka k jp h pq R iklp − k mn g km h an R iklj . By summing up, we obtain∆
L,h k − ∆ L,g,h k = ( g ab − h ab ) ∇ ab k ij + k ab h lb ( g ka k ip h pq R jklq − k mn g km h an R jkli )+ k ab h lb ( g ka k jp h pq R iklq − k mn g km h an R iklj )=: ( g ab − h ab ) ∇ ab k ij + F ( g − , R, k, k )and ( g ab − h ab ) ∇ ab k ij = − k pq g ap h bq ∇ ab k ij =: F ( g − , k, ∇ k ) . Then, (22) follows from (21). Finally, (23) follows from from computing( g ab − h ab ) ∇ ab k ij = ∇ a (( g ab − h ab ) ∇ b k ij ) − ∇ a ( g ab − h ab ) ∇ b k ij = ∇ a (( g ab − h ab ) ∇ b k ij ) + g ap g bq ∇ a k pq ∇ b k ij , setting F ( g − , g − , ∇ k, ∇ k ) := F ( g − , g − , ∇ k, ∇ k ) + g ap g bq ∇ a k pq ∇ b k ij ,F ( g − , g, R, k ) := k ab g ka h lb g ip h pq R jklq − k ab g ka h lb g jp h pq R iklq and using (21) again. (cid:3) An L p -maximum principle. A standard tool for parabolic equations are short-time de-rivative estimates of the form (cid:13)(cid:13) ∇ k u t (cid:13)(cid:13) L ∞ ≤ C · t − k k u k L ∞ . The main purpose of this chapteris to develop analogous estimates for the L p norm. The main tool for doing this is the followingtheorem which we call the L p -maximum principle. Theorem 3.2.
Let ( M, h t ) t ≥ be a smooth 1-parameter family of ALE manifolds. Let g t bea second 1-parameter family of complete Riemannian metrics on M such that C h t ≤ g t ≤ C · h t , |∇ h t g t | ≤ C < ∞ for all t ≥ and a time-independent constant C > .Let E, F, G be tensor bundles over M equipped with the natural family of Riemannian metricsand connetions induced by h t . Let u ( t ) ∈ C ∞ ( E ) and H ( t ) ∈ L ∞ (End( E )) , H ( t ) ∈ L ∞ (Hom( T ∗ M ⊗ E, E )) ,H ( t ) ∈ L ∞ (Hom( T ∗ M ⊗ E, T M ⊗ E )) , H ( t ) ∈ L ( E ) ,H ( t ) ∈ L ∞ (Hom( E, F )) , H ( t ) ∈ L ∞ (Hom( E, G )) be time-dependent sections. TABILITY OF RICCI-FLAT ALE MANIFOLDS 19 (i)
Suppose that u satisfies the evolution inequality ∂ t | u | ≤ g ab ∇ ab | u | + 2 h H ( u ) + H ( ∇ u ) + ∇ a (( H ) ab ∇ b u ) + H , u i− − δ ) g ab h∇ a u, ∇ b u i for some δ ∈ [0 , Then for every p ∈ (1 + δ, ∞ ) , there exists an ǫ > such that thefollowing holds: If k H k L ∞ < ǫ , H ∈ L p ( E ) and u (0) ∈ L p for some p ∈ [ p , ∞ ) , wehave u ( t ) ∈ L p for all t ≥ and the estimate k u ( t ) k L p ≤ e R t ψ ( s ) ds k u (0) k L p + (cid:13)(cid:13)(cid:13) e R ts ψ ( r ) dr · H ( s ) (cid:13)(cid:13)(cid:13) L p ([0 ,t ] × M ) , where ψ ( t ) = C (cid:16) k H k L ∞ + k∇ g k L ∞ + k H k L ∞ + 1 (cid:17) and C = C ( p , ǫ, g t , h t , n ) but independent of p . (ii) Suppose that u satisfies the evolution inequality ∂ t | u | ≤ g ab ∇ ab | u | + h H ( u ) + H ( ∇ u ) + ∇ a (( H ) ab ∇ b u ) + H , u i− δ ) g ab h∇ a u, ∇ b u i + h∇ ∗ ( H ( u )) + ∇ ∗ ( H ( ∇ u )) , ∇ u i . for some δ ∈ [0 , . Then for δ < p < p < ∞ , there exists an ǫ > such thatthe following holds: If k H k L ∞ + k H k L ∞ < ǫ , H ∈ L p ( E ) and u (0) ∈ L p for some p ∈ [ p , p ] , we have u ( t ) ∈ L p for all t ≥ and the estimate k u ( t ) k L p ≤ e R t ψ ( s ) ds k u (0) k L p + (cid:13)(cid:13)(cid:13) e R ts ψ ( r ) dr · H ( s ) (cid:13)(cid:13)(cid:13) L p ([0 ,t ] × M ) , where ψ ( t ) = C (cid:16) k H k L ∞ + k∇ g k L ∞ + k H k L ∞ + k H k L ∞ + 1 (cid:17) and C = C ( p , p , ǫ, g t , h t , n ) but independent of p .Proof. We start with the proof of (i) and first establish the desired estimate for p < ∞ . Let q = p and ρ > F = | k | and F ρ = | u | + ρ . Then we get ∂ t F qρ ≤ g ab ∇ ab F qρ − q ( q − g ab h∇ a F, ∇ b F i F q − ρ − − δ ) q · g ab h∇ a u, ∇ b u i F q − ρ + 2 q · h H ( u ) + H ( ∇ u ) + ∇ a (( H ) ab ∇ b u ) + H , u i F q − ρ . (24)Choose for each x ∈ M and large R > φ R,x such that φ R,x ≡ B R ( x ) , φ R,x ≡ M \ B R ( x ) , |∇ φ R,x | ≤ /R, |∇ φ R,x | ≤ /R . Let us define the quantity A ( R, ρ, t ) = sup x ∈ M Z M F qρ ( t ) · φ R,x dV.
In order to do this, we multiply (24) by φ := φ R,x and integrate over M . Then we get ∂ t Z M F qρ φ dV = Z M g ab ∇ ab F qρ φ dV − q ( q − Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV − − δ ) q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + 2 q Z M h H ( u ) + H ( ∇ u ) , u i F q − ρ φ dV + 2 q Z M h∇ a (( H ) ab ∇ b u ) + H , u i F q − ρ φ dV. Performing integration by parts with the first term yields Z M g ab ∇ ab F qρ φ dV = − q Z M ∇ b F · F q − ρ ∇ a g ab · φ dV − q Z M ∇ b F · F q − ρ g ab ∇ a φ · φ dV We get, using the Peter-Paul inequality − q Z M ∇ b F · F q − ρ ∇ a g ab · φ dV ≤ C · q Z M |∇ u || u ||∇ g | F q − ρ · φ dV ≤ ǫ · q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + C ( ǫ ) · q Z M F qρ · |∇ g | · φ dV Because g ab ∇ a F ∇ b φ ≤ C |∇ u || u ||∇ φ | , another application of the Peter-Paul inequality yields − q Z M ∇ b F · F q − ρ g ab ∇ a φ · φ dV ≤ ǫ · q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + C ( ǫ ) · q Z M F qρ · |∇ φ | dV. Similarly 2 q Z M h H ( ∇ u ) , u i F q − ρ φ dV ≤ ǫ · q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + C ( ǫ ) k H k L ∞ · q Z M F qρ φ dV. We easily get 2 q Z M h H ( u ) , u i F q − ρ φ dV ≤ q k H k L ∞ Z M F qρ · φ dV. Using Young’s inequality ab ≤ p ′ a p ′ + q ′ b q ′ for a = | H | , b = F q − ρ , p ′ = 2 q and q ′ = q q − yields2 q Z M h H , u i F q − ρ φ dV ≤ Z M | H | q φ dV + (2 q − Z M F qρ φ dV. Let us now look at the remaining term. Integration by parts yields2 q Z M h∇ a (( H ) ab ∇ b u ) , u i F q − ρ φ dV = − q Z M h ( H ) ab ∇ b u ) , ∇ a u i F q − ρ φ − q ( q − Z M h ( H ) ab ∇ b u ) , u i∇ a F · F q − ρ φ dV − q Z M h ( H ) ab ∇ b u ) , u i F q − ρ ∇ φ · φ dV. We have − q ( q − Z M h ( H ) ab ∇ b u ) , u i∇ a F · F q − ρ φ dV = − q ( q − Z M ( H ) ab ∇ b F ∇ a F · F q − ρ φ dV ≤ q ( q − Z M | H ||∇ F | F q − ρ φ dV. Using the Peter Paul inequality again, we get − q Z M h ( H ) ab ∇ b u ) , u i F q − ρ ∇ φ · φ dV = − q Z M ( H ) ab ∇ b F · F q − ρ ∇ φ · φ dV TABILITY OF RICCI-FLAT ALE MANIFOLDS 21 ≤ q Z M | H ||∇ F | F q − ρ φ dV + q Z M | H ||∇ φ | · F qρ dV. Summing up and using |∇ u | ≤ Cg ab h∇ a u, ∇ b u i , we get2 q Z M h∇ a (( H ) ab ∇ b u ) , u i F q − ρ φ dV ≤ Cq Z M | H | g ab h∇ a F, ∇ b F i F q − ρ φ dV + 2 q Z M | H ||∇ φ | · F qρ dV. Summarizing all the terms, we obtain ∂ t Z M F qρ φ dV ≤ − [ q ( q − − Cq k H k L ∞ ] Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV − − δ − ǫ ) q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + C ( k H k L ∞ + k∇ g k L ∞ + k H k L ∞ + 1) · q Z M F qρ φ dV + C (1 + k H k L ∞ ) · q Z M |∇ φ | F qρ dV + Z M | H | q φ dV We now claim that the sum of the first two terms is nonpositive, provided that ǫ and k H k L ∞ are chosen small enough. If q ≥
2, it is immediate. Before proceeding with the other cases, notefirst that by the Cauchy-Schwarz inequality, we get g ab h∇ a F, ∇ b F i ≤ g ab h∇ a u, u ih∇ b u, u i ≤ g ab h∇ a u, ∇ b u i F ρ , so that − [ q ( q − − Cq k H k L ∞ ] Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV − − δ − ǫ ) q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV ≤ − q ( q − Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV − − δ − ǫ − Cq k H k L ∞ ) · q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV. Thus if q ∈ [1 , ǫ and k H k L ∞ are smallenough. If p ∈ (1 ,
2) and q ∈ [ p , − q ( q − Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV − − δ − ǫ − Cq k H k L ∞ ) · q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV ≤ − q − − δ − ǫ − Cq k H k L ∞ ) · q Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV. The right hand side is nonpositive, provided that ǫ + 2 C k H k L ∞ is smaller than a constantwhich depends on p but is independent of q . We arrive at the estimate ∂ t Z M F qρ φ dV ≤ C ( k H k L ∞ + k∇ g k L ∞ + k H k L ∞ + 1) · q Z M F qρ φ dV + C (1 + k H k L ∞ ) R − · q Z B R F qρ dV + k H k qL q Abbreviate ψ ( t ) = C k H k L ∞ + k∇ g k L ∞ + k H k L ∞ + 1) . Integrating this differential inequality in time, we obtain Z M F qρ ( t ) φ dV ≤ exp (cid:18) q · Z t ψ ( s ) ds (cid:19) Z M F qρ (0) φ dV + Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) (cid:18) C (1 + k H k L ∞ ) R − · q Z B R F qρ dV + k H k qL q (cid:19) ds ≤ exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) A ( R, ρ,
0) + C doubl · CR − Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) A ( R, ρ, s ) ds + Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) k H k qL q ds, where we used the definition of A and the fact that we can cover B R ( x ) by C doubl balls of radius R . By taking the supremum over all x ∈ M on the left hand side, we conclude A ( R, ρ, t ) ≤ exp (cid:18) q · Z t ψ ( r ) dr (cid:19) A ( R, ρ, C doubl · CR − Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) A ( R, ρ, s ) ds + Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) k H k qL q ds. By a variant of Gronwall’s lemma (c.f. [MPF91, p. 356]), we get A ( R, ρ, t ) ≤ exp (cid:18) q · Z t ψ ( r ) dr (cid:19) A ( R, ρ,
0) + Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) k H k qL q ds + γ ( t ) Z t α ( t ) β ( s ) exp (cid:18)Z t β ( r ) γ ( r ) dr (cid:19) ds, where α ( t ) = exp (cid:18) q · Z t ψ ( r ) dr (cid:19) A ( R, ρ,
0) + Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) k H k qL q ds,β ( t ) = C doubl · CR − exp (cid:18) − q · Z s ψ ( r ) dr (cid:19) ds,γ ( t ) = exp (cid:18) q · Z t ψ ( r ) dr (cid:19) . Letting ρ → R → ∞ and using p = 2 q , we get k u ( t ) k pL p = k F ( t ) k qL q ≤ exp (cid:18) q · Z t ψ ( r ) dr (cid:19) k F (0) k qL q + Z t exp (cid:18) q · Z ts ψ ( r ) dr (cid:19) k H k qL q ds ≤ exp (cid:18) p · Z t ψ ( r ) dr (cid:19) k u (0) k pL p + Z t exp (cid:18) p · Z ts ψ ( r ) dr (cid:19) k H k pL p ds. With x ( t ) = exp (cid:18)Z t ψ ( r ) dr (cid:19) k u (0) k L p , y ( t ) = Z t (cid:20) (exp (cid:18)Z ts ψ ( r ) dr (cid:19) k H k L p (cid:21) p ds ! p TABILITY OF RICCI-FLAT ALE MANIFOLDS 23 and the elementary inequality ( x ( t ) p + y ( t ) p ) p ≤ | x ( t ) | + | y ( t ) | , we finally get k u ( t ) k L p ≤ exp (cid:18)Z t ψ ( r ) dr (cid:19) k u (0) k L p + (cid:18)Z t exp (cid:18) p · Z ts ψ ( r ) dr (cid:19) k H k pL p ds (cid:19) p for all p < ∞ with the function ψ chosen independently of p . This finishes the proof of (i).For the proof of (ii), we proceed as in the first part and we also use the notation from thebeginning of the proof. We have to deal with two additional terms2 q Z M h∇ ∗ ( H ( u )) + ∇ ∗ ( H ( ∇ u )) , u i F q − ρ φ dV. For the first term, we proceed as follows:2 q Z M h∇ ∗ ( H ( u )) , u i F q − ρ φ dV = 2 q Z M h H ( u ) , ∇ u i F q − ρ φ dV + 2 q ( q − Z M h H ( u ) , u i∇ F · F q − ρ φ dV + 4 q Z M h H ( u ) , u i F q − ρ ∇ φ · φ dV ≤ qǫ Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + C ( ǫ ) q Z M | H | F qρ φ dV + ǫ · q ( q − Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV + C ( ǫ ) q ( q − Z M | H | F qρ φ dv + 2 q Z M | H | F qρ φ dV + 2 q Z M F qρ |∇ φ | dV = 2 qǫ Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + ǫ · q ( q − Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV + C ( ǫ ) q ( q + 1) Z M | H | F qρ φ dV + 2 q Z M F qρ |∇ φ | dV. The second term is treated as2 q Z M h∇ ∗ ( H ∗ ∇ u ) , u i F q − ρ φ dV = 2 q Z M h H ∗ ∇ u, ∇ u i F q − ρ φ dV + 2 q ( q − Z M h H ( ∇ u ) , u i∇ F · F q − ρ φ dV + 4 q Z M h H ( ∇ u ) , u i F q − ρ ∇ φ · φ dV ≤ Cq ( q + 1) k H k L ∞ Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + 2 q k H k L ∞ Z M F qρ |∇ φ | dV Summarizing with the terms of part (i), we obtain ∂ t Z M F qρ φ dV ≤ − [(1 − ǫ ) q ( q − − Cq k H k L ∞ ] Z M g ab h∇ a F, ∇ b F i F q − ρ φ dV − [2(1 − δ − ǫ ) q − Cq ( q + 1) k H k L ∞ ] Z M g ab h∇ a u, ∇ b u i F q − ρ φ dV + C ( k H k L ∞ + k∇ g k L ∞ + k H k L ∞ + q · k H k L ∞ + 1) · q Z M F qρ φ dV + C (1 + k H k L ∞ + k H k L ∞ ) · q Z M |∇ φ | F qρ dV + Z M | H | q φ dV. Because there are terms containing H and H which are quadratic in q , we are not able to provean estimate uniform in q for all large q . However, assuming additionally a bound 2 q = p ≤ p < ∞ , we may proceed as in part (i) to finish the proof of part (ii). (cid:3) Short-time estimates for the heat flow of the modified Lichnerowicz Laplacian.
In this section, we establish shortime estimates for solutions of the linear heat equation ∂ t k + ∆ L,g,h k = 0 . (25)We assume that both g and h depend on time and that h t ∈ U ∩ F where U and F are asin Proposition 2.2. The involved scalar products and covariant derivatives appearing here areinduced by h and hence also depend on time. Lemma 3.3.
Let k t , t ∈ [0 , T ] be a solution of the above evolution equation with initial data k and 1 < p < p < ∞ . and p ∈ [ p , ∞ ). Assume further that sup t k g k W , ∞ < C < ∞ .(i) If p ∈ [ p , ∞ ), l ∈ { , } and k ∈ W l,p , then k ( t ) ∈ W l,p for all t ≥ k k t k W l,p ≤ e C · t k k k W l,p (26)for some constant C = C ( n, g, h, ǫ, p , T, l ).(ii) If p ∈ [ p , p ] and k ∈ W ,p , then k t ∈ W ,p for all t ≥ k k t k W ,p ≤ e C · t k k k W ,p (27)for some constant C = C ( n, g, h, ǫ, p , T, l ).(iii) If p ∈ [ p , p ] and k ∈ W ,p , then k t ∈ W ,p for all t > k k t k W ,p + C · t / · (cid:13)(cid:13) ∇ k t (cid:13)(cid:13) L p ≤ e C · t k k k W ,p (28)for some constants C i = C i ( n, g , ǫ, p , T ), i = 2 , p ∈ [ p , ∞ ), q ∈ [ p, ∞ ), k ∈ L q and ∇ k ∈ L p , then ∇ k t ∈ L p for all t ∈ [0 , T ] and wehave k∇ k t k L p ≤ e C · t ( k∇ k k L p + k k k L q )for some constant C = C ( n, g , ǫ, T, p ).(v) If p ∈ [ p , p ], q ∈ [ p, ∞ ), k ∈ W ,q and ∇ k ∈ L p , then ∇ k t ∈ L p for all t ∈ [0 , T ]and we have (cid:13)(cid:13) ∇ k t (cid:13)(cid:13) L p ≤ e C · t ( (cid:13)(cid:13) ∇ k (cid:13)(cid:13) L p + k k k W ,q )for some constant C = C ( n, g , ǫ, T, p ). Proof.
We start by proving part (i). Under the evolution equation, we have ∂ t | k | = g ab ∇ ab | k | − g ab h∇ a k, ∇ b k i + 2 h ˆ R [ k ] + ∂ t h ∗ k, k i ∂ t |∇ k | = g ab ∇ ab |∇ k | − g ab h∇ a ∇ k, ∇ b ∇ k i + 2 h [ ∇ , g ab ∇ ab ] k + ∇ ˆ R [ k ] , ∇ k i + h ∂ t h ∗ ∇ k + ∇ ∂ t h ∗ k, ∇ k i . Because we have [ ∇ , g ab ∇ ab ] k = ∇ g − ∗ ∇ k + g − ∗ R ∗ ∇ k + g − ∗ ∇ R ∗ k ˆ R [ k ] = g − ∗ g ∗ R ∗ k, part (i) follows from Theorem 3.2 (i) applied to u = k ∈ C ∞ ( S M ) , u = ( k, ∇ k ) ∈ C ∞ ( S M ⊕ T ∗ M ⊗ S M ) . TABILITY OF RICCI-FLAT ALE MANIFOLDS 25
It is convenient to prove (iv) now. In this case, we apply Theorem 3.2 to u = ∇ k ∈ C ∞ ( S M )and regard the terms containing k as part of the inhomogeneity. All these terms are of the form ∇ ∂ t h ∗ k and ∇ R ∗ k . Because k∇ ∂ t h k L r + k∇ R k L r < ∞ , for all r ∈ (1 , ∞ ], ∇ ∂ t h ∗ k and ∇ R ∗ k are actually in L p . Therefore, an application of Theorem 3.2together with short-time estimates for the L q -norm of k proves part (iv). For (ii), we additionallycompute ∂ t |∇ k | = g ab ∇ ab |∇ k | − g ab h∇ a ∇ k, ∇ b ∇ k i + 2 h [ ∇ , g ab ∇ ab ] k + ∇ ˆ R [ k ] , ∇ k i + h ∂ t h ∗ ∇ k + ∇ ∂ t h ∗ k + ∇ ∂ t h ∗ ∇ k, ∇ k i . Using [ ∇ , g ab ∇ ab ] k = [ ∇ , g ab ] ∇ ab k + g ab [ ∇ , ∇ ab ] k = ∇ ( ∇ g − ∗ ∇ k ) + ∇ g − ∗ ∇ k + ∇ R ∗ k + ∇ R ∗ ∇ k + R ∗ ∇ k, we can rewrite the latter equation as ∂ t |∇ k | = g ab ∇ ab |∇ k | − g ab h∇ a ∇ k, ∇ b ∇ k i + 2 h∇ ( ∇ g − ∗ ∇ k + ∇ ˆ R [ k ]) , ∇ k i + 2 h∇ g − ∗ ∇ k + ∇ R ∗ k + ∇ R ∗ ∇ k + R ∗ ∇ k, ∇ k i + h ∂ t h ∗ ∇ k + ∇ ∂ t h ∗ k + ∇ ∂ t h ∗ ∇ k, ∇ k i . Part (ii) follows from Theorem 3.2 (ii), applied to u = ( k, ∇ k, ∇ k ) ∈ C ∞ ( S M ⊕ T ∗ M ⊗ S M ⊕ T ∗ M ⊗ T ∗ M ⊗ S M ) . We have to apply the extended version of the maximum principle because we have to deal withthe term ∇ ( ∇ g − ∗ ∇ k + ∇ ˆ R [ k ]) without getting second derivatives of g − . Now we prove (v).Similarly as in the proof of (iv), we regard the terms in the evolution of ∇ k which contain k and ∇ k as a part of the inhomogeneity. Alle these terms are of the form ∇ ∂ t h ∗ k, ∇ ∂ t h ∗ ∇ k, ∇ R ∗ k ∇ R ∗ ∇ k. Because k∇ ∂ t h k W ,r + k∇ R k W ,r < ∞ , for all r ∈ (1 , ∞ ], these products are all in L p . We can thus apply Theorem 3.2 (i) together withshort-time estimates for the W ,q -norm of k to get (v).For part (iii), we first compute ∂ t |∇ ( At k ) | = ∂ t ( A t |∇ k | )= A |∇ k | + A t (cid:0) g ab ∇ ab |∇ k | − g ab h∇ a ∇ k, ∇ b ∇ k i (cid:1) + 2 A t (cid:16) h∇ ( ∇ g − ∗ ∇ k + ∇ ˆ R [ k ]) + ∇ g − ∗ ∇ k + ∇ R ∗ k + R ∗ ∇ k, ∇ k i (cid:17) + A t (cid:0) h ∂ t h ∗ ∇ k + ∇ ∂ t h ∗ k + ∇ ∂ t h ∗ ∇ k, ∇ k i (cid:1) = A |∇ k | + (cid:16) g ab ∇ ab |∇ ( At k ) | − g ab h∇ a ∇ ( At k ) , ∇ b ∇ ( At k ) i (cid:17) + 2 h∇ ( ∇ g − ∗ ∇ ( At k ) + At ∇ ˆ R [ k ]) , ∇ ( At k ) i + h∇ g − ∗ ∇ ( At k ) + At ∇ R ∗ k + At R ∗ ∇ k, ∇ ( At k ) i + h ∂ t h ∗ ∇ ( At k ) + At ∇ ∂ t h ∗ k + At ∇ ∂ t h ∗ ∇ k, ∇ ( At k ) i . Part (iii) then also follows from Theorem 3.2 (ii), applied to u = ( k, ∇ k, ∇ ( At k )) ∈ C ∞ ( S M ⊕ T ∗ M ⊗ S M ⊕ T ∗ M ⊗ T ∗ M ⊗ S M ) , where A has to be chosen small in dependence of p . (cid:3) Lemma 3.4.
Let t ∈ [0 , T ], g t , ˜ g t be families of Riemannian metrics on M and h t , ˜ h t be familiesof Ricci-flat metrics in U ∩ F where U and F are as in Proposition 2.2. Take all covariantderivatives and norms with respect to h t and suppose that k g t k W , ∞ + k ˜ g t k W , ∞ < ∞ .Let 1 < p ≤ p ≤ p < ∞ and suppose that k g − ˜ g k L ∞ < ǫ where ǫ = ǫ ( p , p ) > k t ˜ k t , t ∈ [0 , T ] be solutions of the evolution equations ∂ t k + ∆ L,g,h k = 0 , ∂ t ˜ k + ∆ L, ˜ g, ˜ h ˜ k = 0with initial data k , ˜ k , respectively.(i) If l ∈ { , , } and k , ˜ k ∈ W l,p , then k t − ˜ k t ∈ W l,p for all t ≥ (cid:13)(cid:13)(cid:13) k t − ˜ k t (cid:13)(cid:13)(cid:13) W l,p ≤ e C · t (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W l,p + sup s ∈ [0 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W l, ∞ (cid:17) e C · t (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W l,p for some constant C = C ( n, g, h, ǫ, p , p , l ).(ii) If k , ˜ k ∈ W ,p , then k t − ˜ k t ∈ W ,p for all t > (cid:13)(cid:13)(cid:13) k t − ˜ k t (cid:13)(cid:13)(cid:13) W ,p + C · t / · (cid:13)(cid:13)(cid:13) ∇ ( k t − ˜ k t ) (cid:13)(cid:13)(cid:13) L p ≤ e C · t (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W ,p + sup s ∈ [0 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) e C · t (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W l,p for some constants C i = C i ( n, g , ǫ, p , T ), i = 1 , , k , ˜ k ∈ L p , then k t − ˜ k t ∈ W ,p for all t > (cid:13)(cid:13)(cid:13) k t − ˜ k t (cid:13)(cid:13)(cid:13) W ,p + C · t / · (cid:13)(cid:13)(cid:13) ∇ ( k t − ˜ k t ) (cid:13)(cid:13)(cid:13) L p + C · t · (cid:13)(cid:13)(cid:13) ∇ ( k t − ˜ k t ) (cid:13)(cid:13)(cid:13) L p ≤ e C · t (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L p + sup s ∈ [0 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) e C · t (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) L p for some constants C i = C i ( n, g , ǫ, p , T ), i = 1 , , q ∈ [ p, p ], k (0) , ˜ k ∈ L q and ∇ k , ∇ ˜ k ∈ L p , then ∇ ( k t − ˜ k t ) ∈ L p for all t ∈ [0 , T ]and we have (cid:13)(cid:13)(cid:13) ∇ ( k t − ˜ k t ) (cid:13)(cid:13)(cid:13) L p ≤ e C · t (cid:16)(cid:13)(cid:13)(cid:13) ∇ ( k − ˜ k ) (cid:13)(cid:13)(cid:13) L p + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L q (cid:17) + sup s ∈ [0 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) e C · t (cid:13)(cid:13)(cid:13) ∇ ˜ k (cid:13)(cid:13)(cid:13) L p + sup s ∈ [0 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) e C · t (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) L q for some constant C = C ( n, g , ǫ, p , p ).(v) If q ∈ [ p, p ], k , ˜ k ∈ W ,q and ∇ k , ∇ ˜ k ∈ L p , then ∇ ( k t − ˜ k t ) ∈ L p for all t ∈ [0 , T ]and we have (cid:13)(cid:13)(cid:13) ∇ ( k t − ˜ k t ) (cid:13)(cid:13)(cid:13) L p ≤ e C · t (cid:16)(cid:13)(cid:13)(cid:13) ∇ ( k − ˜ k ) (cid:13)(cid:13)(cid:13) L p + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W ,q (cid:17) + sup s ∈ [0 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) e C · t (cid:13)(cid:13)(cid:13) ∇ ˜ k (cid:13)(cid:13)(cid:13) L p TABILITY OF RICCI-FLAT ALE MANIFOLDS 27 + sup s ∈ [0 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) e C · t (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,q for some constant C = C ( n, g , ǫ, p , p ). Proof.
We have the evolution equations ∂ t k = g ab ∇ ab k + R g,h [ k ] ,∂ t ˜ k = ˜ g ab ˜ ∇ ab ˜ k + R ˜ g, ˜ h [˜ k ] = g ab ∇ ab ˜ k + (˜ g ab − g ab ) ∇ ab ˜ k + ˜ g ab ( ˜ ∇ ab − ∇ ab )(˜ k ) + R ˜ g, ˜ h [˜ k ] , = g ab ∇ ab ˜ k + ∇ a [(˜ g ab − g ab ) ∇ b ˜ k ] − [ ∇ a (˜ g ab − g ab )] ∇ b ˜ k + ˜ g ab ( ˜ ∇ ab − ∇ ab )(˜ k ) + R ˜ g, ˜ h [˜ k ] ∂ t ( k − ˜ k ) = g ab ∇ ab ( k − ˜ k ) + ( g ab − ˜ g ab ) ∇ ab ˜ k + ˜ g ab ( ∇ ab − ˜ ∇ ab )(˜ k ) + R g,h [ k ] − R ˜ g, ˜ h [˜ k ]= g ab ∇ ab ( k − ˜ k ) + ∇ a [( g ab − ˜ g ab ) ∇ b ˜ k ] − [ ∇ a ( g ab − ˜ g ab )] ∇ b ˜ k + ˜ g ab ( ∇ ab − ˜ ∇ ab )(˜ k ) + R g,h [ k ] − R ˜ g, ˜ h [˜ k ] . Note that R g,h [ k ] − R ˜ g, ˜ h [˜ k ] = ( g − − ˜ g − ) ∗ ˜ g ∗ ˜ R ∗ ˜ k + g − ∗ ( g − ˜ g ) ∗ ˜ R ∗ ˜ k + g − ∗ g ∗ ( R − ˜ R ) ∗ ˜ k + g − ∗ g ∗ R ∗ ( k − ˜ k )and ∇ ˜ k − ˜ ∇ ˜ k = ∇ ( h − ˜ h ) ∗ ˜ g − ∗ ˜ k + ˜ g − ∗ ∇ ( h − ˜ h ) ∗ ∇ ( h − ˜ h ) ∗ ˜ k + ˜ g − ∗ ∇ ( h − ˜ h ) ∗ ∇ ˜ k,R − ˜ R = ∇ ( h − ˜ h ) ∗ ˜ g − + ˜ g − ∗ ∇ ( h − ˜ h ) ∗ ∇ ( h − ˜ h ) . Thus all these terms are easy to handle because they are at most first order in ˜ k and k − ˜ k . Wehave the evolution equations ∂ t | ˜ k | ≤ g ab ∇ ab | ˜ k | − g ab h∇ a ˜ k, ∇ b ˜ k i + 2 h∇ a [(˜ g ab − g ab ) ∇ b ˜ k ] , ˜ k i + 2 h∇ a [ g ab − ˜ g ab ] ∇ b ˜ k + ˜ g ab ( ˜ ∇ ab − ∇ ab )(˜ k ) + R ˜ g, ˜ h [˜ k ] , ˜ k i ,∂ t | k − ˜ k | ≤ g ab ∇ ab | k − ˜ k | − g ab h∇ a ( k − ˜ k ) , ∇ b ( k − ˜ k ) i + h∇ a [( g ab − ˜ g ab ) ∇ b ˜ k ] , k − ˜ k i + h [ ∇ a (˜ g ab − g ab )] ∇ b ˜ k + ˜ g ab ( ∇ ab − ˜ ∇ ab )(˜ k ) + R g,h [ k ] − R ˜ g, ˜ h [˜ k ] , k − ˜ k i . The crucial point in applying Theorem 3.2 is to handle the off diagonal terms appropriately. Forthis purpose, we write u = ( u , u ) = ( k − ˜ k, AB ˜ k ) ∈ C ∞ ( S M ⊕ S M ) , where A := sup t ∈ [0 ,T ] n k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ o , and B > | u | = | u | + | u | = | k − ˜ k | + A B | ˜ k | , reads ∂ t | u | ≤ g ab ∇ ab | u | − g ab h∇ a u, ∇ b u i + 2 h∇ a [(˜ g ab − g ab ) ∇ b u ] , u i + 2 h∇ a [ g ab − ˜ g ab ] ∇ b u + ˜ g ab ( ˜ ∇ ab − ∇ ab )( u ) + R ˜ g, ˜ h [ u ] , u i + 1 AB h∇ a [( g ab − ˜ g ab ) ∇ b u ] + [ ∇ a (˜ g ab − g ab )] ∇ b u + ˜ g ab ( ∇ ab − ˜ ∇ ab )( u ) , u i + h R g,h [ k ] − R ˜ g, ˜ h [˜ k ] , u i . Observe that for every ǫ >
0, we can choose
B > AB | ( g ab − ˜ g ab ) ∇ b u | ≤ B |∇ u | ≤ ǫ |∇ u | , where we used the definition of A . It is also straightforward to see that1 AB (cid:16) | [ ∇ a (˜ g ab − g ab )] ∇ b u | + | ˜ g ab ( ∇ ab − ˜ ∇ ab )( u ) | (cid:17) ≤ C ( |∇ u | + | u | )where C is independent of A by the definition of A . Similarly, one also shows that | R g,h [ k ] − R ˜ g, ˜ h [˜ k ] | ≤ C ( | u | + | u | ) ≤ C | u | with C independent of A . For this reason, we can write schematically2 h∇ a [(˜ g ab − g ab ) ∇ b u ] , u + 1 AB u i = h∇ a (( H ) ab ∇ b u ) , u ih∇ a [ g ab − ˜ g ab ] ∇ b u , u + 1 AB u i = h H ( u ) , u ih ˜ g ab ( ˜ ∇ ab − ∇ ab )( u ) , u + 1 AB u i = h H ( u ) + H ( ∇ u ) , u ih R ˜ g, ˜ h [ u ] , u i + h R g,h [ k ] − R ˜ g, ˜ h [˜ k ] , u i = h H ( u ) , u i , where the H i satisfy the conditions of Theorem 3.2, with E = S M ⊕ S M . Note that the scalarproducts are taken on S M on the left hand side and on S M ⊕ S M on the right hand side.Applying Theorem 3.2 yields (i) for l = 0.Before we continue estimating derivatives, we remark that for any (2 , v , we have[ ∇ , v ab ∇ ab ] k = ∇ v ∗ ∇ k + v ∗ R ∗ ∇ k + v ∗ ∇ R ∗ k [ ∇ , v ab ∇ ab ] = ∇ ( ∇ v ∗ ∇ k ) + ∇ v ∗ ∇ k + v ∗ ∇ R ∗ k + v ∗ ∇ R ∗ ∇ k + v ∗ R ∗ ∇ k from which we conclude ∂ t ∇ ˜ k = g ab ∇ ab ∇ ˜ k + ∇ n ˜ g ab ( ˜ ∇ ab − ∇ ab )(˜ k ) + R ˜ g, ˜ h [˜ k ] o + ∇ a [(˜ g ab − g ab ) ∗ ∇ b ∇ ˜ k ]+ ∇ ∂ t h ∗ ˜ k + ∂ t h ∗ ∇ ˜ k + ∇ ( g − − ˜ g − ) ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ R ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ ∇ R ∗ ˜ k∂ t ∇ ( k − ˜ k ) = g ab ∇ ab ∇ ( k − ˜ k ) + ∇ n ˜ g ab ( ∇ ab − ˜ ∇ ab )(˜ k ) + R g,h [ k ] − R ˜ g, ˜ h [˜ k ] o + ∇ a [( g ab − ˜ g ab ) ∗ ∇ b ∇ ˜ k ] + ∇ ∂ t h ∗ ( k − ˜ k ) + ∂ t h ∗ ∇ ( k − ˜ k )+ ∇ g − ∗ ∇ ( k − ˜ k ) + g − ∗ R ∗ ∇ ( k − ˜ k ) + g − ∗ ∇ R ∗ ( k − ˜ k )+ ∇ ( g − − ˜ g − ) ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ R ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ ∇ R ∗ ˜ k∂ t ∇ ˜ k = g ab ∇ ab ∇ ˜ k + ∇ n ˜ g ab ( ˜ ∇ ab − ∇ ab )(˜ k ) + R ˜ g, ˜ h [˜ k ] o + ∇ a [(˜ g ab − g ab ) ∗ ∇ b ∇ ˜ k ]+ ∇ ∂ t h ∗ ˜ k + ∇ ∂ t h ∗ ∇ ˜ k + ∂ t h ∗ ∇ ˜ k + ∇ ( ∇ ( g − − ˜ g − ) ∗ ∇ ˜ k ) + ∇ ( g − − ˜ g − ) ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ ∇ R ∗ ˜ k + ( g − − ˜ g − ) ∗ ∇ R ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ R ∗ ∇ ˜ k∂ t ∇ ( k − ˜ k ) = g ab ∇ ab ∇ ( k − ˜ k ) + ∇ n ˜ g ab ( ∇ ab − ˜ ∇ ab )(˜ k ) + R g,h [ k ] − R ˜ g, ˜ h [˜ k ] o + ∇ a [( g ab − ˜ g ab ) ∗ ∇ b ∇ ˜ k ]+ ∇ ∂ t h ∗ ( k − ˜ k ) + ∇ ∂ t h ∗ ∇ ( k − ˜ k ) + ∂ t h ∗ ∇ ( k − ˜ k )+ ∇ g − ∗ ∇ ( k − ˜ k ) + g − ∗ R ∗ ∇ ( k − ˜ k ) + g − ∗ ∇ R ∗ ( k − ˜ k ) TABILITY OF RICCI-FLAT ALE MANIFOLDS 29 + ∇ ( g − − ˜ g − ) ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ R ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ ∇ R ∗ ˜ k + ∇ ( ∇ ( g − − ˜ g − ) ∗ ∇ ˜ k ) + ∇ ( g − − ˜ g − ) ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ ∇ R ∗ ˜ k + ( g − − ˜ g − ) ∗ ∇ R ∗ ∇ ˜ k + ( g − − ˜ g − ) ∗ R ∗ ∇ ˜ k + ∇ ( ∇ g − ∗ ∇ ( k − ˜ k )) + ∇ g − ∗ ∇ ( k − ˜ k )+ g − ∗ ∇ R ∗ ( k − ˜ k ) + g − ∗ ∇ R ∗ ∇ ( k − ˜ k ) + g − ∗ R ∗ ∇ ( k − ˜ k )In the following, we sketch to which expressions we have to apply Theorem 3.2 in order to getall the other cases of the Lemma. The details are left to the reader.The cases l = 1 , u = ( k − ˜ k, AB ˜ k, ∇ ( k − ˜ k ) , AB ∇ ˜ k ) ∈ C ∞ ( S M ⊕ , ( T ∗ M ⊗ S M ) ⊕ ) ,A := sup t ∈ [0 ,T ] n k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ o , and to u = ( k − ˜ k, AB ˜ k, ∇ ( k − ˜ k ) , AB ∇ ˜ k, ∇ ( k − ˜ k ) , AB ∇ ˜ k ) ∈ C ∞ (cid:0) ( R ⊕ T ∗ M ⊕ T ∗ M ⊗ ) ⊗ S M ⊕ (cid:1) , where A := sup t ∈ [0 ,T ] n k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ o , and in both cases, B > u = ( k − ˜ k, AB ˜ k, ∇ ( k − ˜ k ) , AB ∇ ˜ k, AB t ∇ ( k − ˜ k ) , AB B t ∇ ˜ k ) ,A := sup t ∈ [0 ,T ] n k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ o , where B > B > u = ( k − ˜ k, AB ˜ k, B t ∇ ( k − ˜ k ) , AB B t ∇ ˜ k, B t ∇ ( k − ˜ k ) , AB B t ∇ ˜ k ) ,A := sup t ∈ [0 ,T ] n k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ o , where B > B , B > u = ( ∇ ( k − ˜ k ) , AB ∇ ˜ k ) ,A := sup t ∈ [0 ,T ] n k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ o , where B > k − ˜ k, AB ˜ k are treated as inhomogeneities,which can be bounded using (i), c.f. also the proof of the previous lemma. Similarly, (v) followsfrom applxing Theorem 3.2 to u = ( ∇ ( k − ˜ k ) , AB ∇ ˜ k ) ,A := sup t ∈ [0 ,T ] n k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ o , with a large constant B >
0. Here, the terms k − ˜ k, AB ˜ k, ∇ ( k − ˜ k ) , AB ∇ ˜ k ) are treated asinhomogeneities. (cid:3) Short-time estimates for the Ricci-de-Turck flow.
Consider a Ricci-flat backgroundmetric h and the h -gauged Ricci-de Turck flow g ( t ) written with respect to k ( t ) = g ( t ) − h as in(23). We abbreviate R [ k ] = F ( g − , g, R, k ) Q [ k ] = F ( g − , g − , ∇ k, ∇ k ) Q [ k ] = (( h + k ) ab − h ab ) ∇ b k ij with covariant derivatives, Laplacians and curvature of h . Thus if k evolves according to (23),we have ∂ t | k | + ∆ | k | = − |∇ k | + 2 h R [ k ] + Q [ k ] + ∇ Q [ k ] , k i and for covariant derivatives, ∂ t |∇ l k | + ∆ |∇ l k | = 2 h [ ∇ l , ∆] k + ∇ l ( R [ k ] + Q [ k ]) + [ ∇ l , ∇ Q ][ k ] + ∇ Q [ ∇ l k ] , ∇ l k i− |∇ l +1 k | . Lemma 3.5.
Let g ( t ), t ∈ [0 , T ] be a solution of the Ricci-de Turck flow with Ricci-flat back-ground metric h and k ( t ) := g ( t ) − h . Then there exists an ǫ > k k ( t ) k L ∞ < ǫ ∀ t ∈ [0 , T ] , , then for every m ∈ N there exists a constant C m = C ( m, ǫ, g , T ) such that k∇ m k ( t ) k L ∞ ≤ C m · t − m/ · k k (0) k L ∞ . Proof.
This is a standard shorttime existence result, see c.f. [Bam14, Proposition 2.8], whicheasily carries over to the present situation. (cid:3)
Lemma 3.6.
For every δ >
T >
0, there exists an ǫ > C l , l ∈ N suchthat for each m ∈ N , the function F m ( t ) := m X l =0 C l · t l · |∇ l k ( t ) | , t ∈ [0 , T ]satisfies the evolution inequality ∂ t F m + ∆ F m ≤ − (2 − δ ) G m + D m · F m + 2 m X l =0 C l · t l · h∇ Q [ ∇ l k ] , ∇ l k i , as long as k is a solution of the Ricci-de Turck flow with k k ( t ) k L ǫ for all t ∈ [0 , T ]. Here, G m = m X l =0 C l · t l · |∇ l +1 k | and D m = D m ( δ, T, C , . . . , C k ) > Proof.
Standard computations and estimates show that h [ ∇ l , ∆] k, ∇ l k i = l X m =0 ∇ m R ∗ ∇ l − m k ∗ ∇ l k ≤ C ( l, g ) l X m =0 |∇ m k | . Now by Lemma 3.5, we can choose ǫ > (cid:13)(cid:13) ∇ l k ( t ) (cid:13)(cid:13) L ∞ ≤ C ( l, h ) · t − l/ k k ( t ) k L ∞ for all l ∈ N and t ∈ (0 , (cid:13)(cid:13) g − (cid:13)(cid:13) L ∞ ≤ C ( n ) and (cid:13)(cid:13) ∇ l g − (cid:13)(cid:13) L ∞ ≤ C ( l, n ) · t − l/ k k ( t ) k L ∞ . We now write R [ k ] as R [ k ] = k ∗ g − ∗ g ∗ R so that h R [ k ] , ∇ l k i ≤ C ( l, n ) X l + l + l + l = l |∇ l k ||∇ l g − |∇ l g ||∇ l R ||∇ l k | TABILITY OF RICCI-FLAT ALE MANIFOLDS 31 ≤ C ( l, n ) l X m =0 |∇ l − m R ||∇ m k ||∇ l k | + ǫ · C ( l, n ) X l + l ≤ ll ≥ t − l / |∇ l k ||∇ l k |≤ C ( l, n ) l X m =0 |∇ m k | + ǫ · C ( l, n ) l X j =1 t − ( l − j ) |∇ j k | ≤ C ( l, n ) l X m =0 |∇ m k | + ǫ · C ( l, n ) l X j =1 t − ( l +1) − j |∇ j k | . We now write Q [ k ] as Q [ k ] = Φ ∗ ∇ k ∗ ∇ k so that ∇ l Q [ k ] = ∇ l (Φ ∗ ∇ k ∗ ∇ k ) = X ≤ l ,l ,l ≤ ll + l + l = l ∇ l Φ ∗ ∇ l +1 k ∗ ∇ l +1 k. Because Φ = g − ∗ g − , we have that k Φ k L ∞ ≤ C ( n ) and (cid:13)(cid:13) ∇ l Φ (cid:13)(cid:13) L ∞ ≤ C ( l, n ) · t − l/ k hk ( t ) k L ∞ for l ∈ N . Therefore, h∇ l Q [ h ] , ∇ l k i ≤ C ( l, n ) X ≤ j ≤ i ≤ l |∇ l − i Φ | · |∇ j +1 h | · |∇ i − j +1 k | · |∇ l h |≤ C ( l, n ) · t · ǫ − X ≤ j ≤ i ≤ l |∇ l − i Φ | · |∇ j +1 k | · |∇ i − j +1 k | + 14 t − ǫ |∇ l k | ≤ C ( l, n ) · k k k L ∞ ǫ − l X j =0 t − l + j · |∇ j +1 k | + 14 t − ǫ |∇ l k | ≤ C ( l, n ) · ǫ · l +1 X j =1 t − ( l +1)+ j · |∇ j k | . The second commutator term is of the form ∇ l ∇ a (Ψ ab ∇ b k ) − ∇ a Ψ ab ∇ b ∇ l hk where Ψ ab = ( h + k ) ab − h ab . We rewrite this as ∇ l ∇ a (Ψ ab ∇ b k ) − ∇ a Ψ ab ∇ b ∇ l k = [ ∇ l , ∇ a ]Ψ ab ∇ b k + ∇ a ([ ∇ l , Ψ ab ] ∇ b k ) + ∇ a (Ψ ab [ ∇ l , ∇ b ] k )= l − X m =0 ∇ m ∇ Ψ ∗ ∇ l +1 − m k + X l + l + l = l ∇ l R ∗ ∇ l Ψ ∗ ∇ l k. We have that (cid:13)(cid:13) ∇ l Ψ (cid:13)(cid:13) L ∞ ≤ C ( l, n ) · t − l/ k k ( t ) k L ∞ for all l ∈ N . Therefore, h [ ∇ l , ∇ Q ][ k ] , ∇ l k i = l − X m =0 ∇ m ∇ Ψ ∗ ∇ l +1 − m k ∗ ∇ l k + X l + l + l = l ∇ l R ∗ ∇ l Ψ ∗ ∇ l k ∗ ∇ l k and the first of these two expressions is estimated by l − X m =0 ∇ m ∇ Ψ ∗ ∇ l +1 − m k ∗ ∇ l k ≤ C ( l, n ) ǫ − · t l − X m =0 |∇ m +1 Ψ | |∇ l +1 − m k | + ǫ · t − · |∇ l k | ! ≤ C ( l, n ) · ǫ l − X m =0 t − m |∇ l +1 − m k | + t − · |∇ l k | ! ≤ C ( l, n ) · ǫ · l +1 X j =2 t − ( l +1)+ j |∇ j k | while the other one is estimated by X l + l + l = l ∇ l R ∗ ∇ l Ψ ∗ ∇ l k ∗ ∇ l k ≤ C ( l, n ) ǫ − X l + l + l = l |∇ l R | |∇ l Ψ | |∇ l k | + ǫ |∇ l k | ! ≤ C ( l, n ) · ǫ X l + l ≤ l t − l |∇ l k | + |∇ l k | ≤ C ( l, n ) · ǫ l X j =0 t − l + j |∇ j k | ≤ C ( l, n ) · ǫ l X j =1 t − ( l +1)+ j |∇ j k | + C ( l, n ) t − l | k | . By a standard computation and putting together the above estimates, we conclude that (with C := C + C + C and C = C + C + C + C ) ∂ t F m + ∆ F m ≤ m X l =0 l · C l · t l − |∇ l k | − m X l =0 C l · t l · |∇ l +1 k | + 2 m X l =0 C l t l · C ( l, h ) l X j =0 |∇ j k | + 2 · ǫ m X l =0 C l · C ( l, n ) · l +1 X j =1 t j − · |∇ j k | + 2 m X l =0 C l · t l h∇ Q [ ∇ l k ] , ∇ l k i = m X l =0 [( l + 1) C l +1 − C l + 2 ǫ m X j = l C j · C ( j, n )] · t l · |∇ l +1 k | + m X l =0 ( k X j = l C j · C ( j, g ) · t j − l ) · t l · |∇ l k | + 2 m X l =0 C l · t l h∇ Q [ ∇ l k ] , ∇ l k i which proves the lemma provided the C l are chosen accordingly. (cid:3) Remark 3.7.
In Lemma 3.6, we can choose C = 1 so that F k (0) = | k | . We assume that thisconvention holds from now on. Lemma 3.8.
Let g ( t ), t ∈ [0 , T ] be a solution of the Ricci-de Turck flow with Ricci-flat back-ground metric h and k ( t ) := g ( t ) − h . Then there exists an ǫ > k k ( t ) k L ∞ < ǫ ∀ t ∈ [0 , T ] , k k (0) k L p ( ∞ ) < ∞ for some p ∈ [ p , ∞ ) , then for every m ∈ N there exists a constant C m = C ( m, ǫ, g , T, p ) (but independent from p )such that k∇ m k ( t ) k L p ≤ C m · t − m/ · k k (0) k L p . Proof.
Let T be as in the lemma, ǫ > δ > δ < p . Given this data, choosethe constants C l from Lemma 3.6, consider the time-dependent section u m ( t ) := ( p C l · t l ∇ l k ( t )) ml =0 ∈ C ∞ m M l =0 ( T ∗ M ) ⊗ l ⊗ S M ! , TABILITY OF RICCI-FLAT ALE MANIFOLDS 33 for which we have | u m | = F m , |∇ u m | = G m . According to Lemma 3.6, we have ∂ t | u m | ≤ h ab ∇ ab + D m | u m | + h∇ a (( h + k ) ab − h ab ) ∇ b u m , u m i − − δ ) |∇ u m | . Now if ǫ > H ab := ( h + k ) ab − h ab is so small that we can apply Theorem3.2. This yields p C l · t l (cid:13)(cid:13) ∇ l k ( t ) (cid:13)(cid:13) L p ≤ k u m ( t ) k L p ≤ C k u m (0) k L p = C k k (0) k L p and the result is immediate. (cid:3) The Ricci-de Turck flow and a mixed evolution problem
A Ricci-de Turck flow with moving gauge.
Let g t be a solution of the h t -gauged Ricci-de Turck flow, where h t be a curve of Ricci-flat metrics. In Subsection 3.1, we have seen thatthe evolution equation on g t can be written in two different ways: ∂ t g + ∆ L,h k = H := F ( g − , g − , ∇ k, ∇ k ) + F ( g − , R, k, k ) + F ( g − , k, ∇ k ) , (29) ∂ t g + ∆ L,g,h k = H := F ( g − , g − , ∇ k, ∇ k ) . (30)Here, k = g − h and the F i are tensor fields, viewed as C ∞ -multilinear maps. Definition 4.1.
Let b h be an integrable Ricci-flat ALE metric with a parallel spinor, and U aneighbourhood of b h in the space of metrics, on which the projection mapΦ : U → F is defined as in Subsection 2.2. Then a family of metrics g t in U is called a Ricci-de Turck flowwith moving gauge, if it satisfies the evolution equation ∂ t g = − g + L V ( g, Φ( g )) g (31)Since Φ( g ) is Ricci-flat, the Ricci-de Turck flow with moving gauge expands as in (29) and (30)with respect to h = Φ( g ) and k = g − h . We will work with both expressions in the following. Proposition 4.2.
Let g t be a smooth family of Riemannian metrics in U and let h ∞ be somefixed metric in U ∩ F . Suppose that U is so small that Π h, ¯ h is invertible for all h, ¯ h ∈ U ∩ F (c.f.Lemma 2.6). Then g t is a Ricci-de Turck flow with moving gauge if and only if the components h t = Φ( g t ) , k t = g t − h t , k t = Π ⊥ h t ,h ∞ ( k t ) satisfy the coupled system ∂ t h = D g Φ( H ) ,∂ t k + ∆ L, ∞ k = Π ⊥∞ [(∆ L, ∞ − ∆ L,h )( k ) + (1 − D g Φ)( H )] , where H is defined in (29) and Π ⊥∞ := Π ⊥ h ∞ . On the other hand, it is also equivalent to thesystem ∂ t h = D Φ g ( H ) ,∂ t k + ∆ L,g,h k = [∆ L,g,h , Π ⊥∞ ]( k ) + Π ⊥∞ [ D g Φ((∆
L,g,h − ∆ L,h )( k )) + (1 − D g Φ)( H )] . Proof.
Splitting (29) up into g = h + k = Φ( g ) + (1 − Φ)( g ) yields the equivalent system ∂ t h = D g Φ( ∂ t g ) = D g Φ( − ∆ L,h k + H ) = D g Φ( H ) ,∂ t k = (1 − D g Φ)( ∂ t g ) = (1 − D g Φ)( − ∆ L,h k + H ) = − ∆ L,h k + (1 − D g Φ)( H ) . Here we used that D g Φ vanishes on ker L (∆ L,h ) ⊥ = im(∆ L,h ), by Lemma 2.8. Applying Π ⊥∞ tothe second equation yields ∂ t k = Π ⊥∞ ( ∂ t k ) = Π ⊥∞ ( − ∆ L,h k + (1 − D g Φ)( H ))= − Π ⊥∞ (∆ L, ∞ k ) + Π ⊥∞ ((∆ L, ∞ − ∆ L,h )( k ) + (1 − D g Φ)( H ))= − ∆ L, ∞ k + Π ⊥∞ ((∆ L, ∞ − ∆ L,h )( k ) + (1 − D g Φ)( H )) . By construction, k = Π h,h ∞ ( k ). By assumption, Π h,h ∞ is invertible, so that k is determined by k . Therefore, the evolution equations on k and k are actually equivalent.It remains to show equivalence of the Ricci flow to the second system. This is done similarly.The first equation of the system is the same as the first equation from the first system. For thesecond equation, we get from (30) and the chain rule that ∂ t k = (1 − D g Φ)( ∂ t g ) = (1 − D g Φ)( − ∆ L,g,h k + H ) . Applying Π ⊥∞ yields ∂ t k = Π ⊥∞ ( ∂ t k ) = Π ⊥∞ [(1 − D g Φ)( − ∆ L,g,h k + H )]= − Π ⊥∞ (∆ L,g,h k ) + Π ⊥∞ [ D g Φ(∆
L,g,h k ) + (1 − D g Φ)( H )]= − ∆ L,g,h k + [∆ L,g,h , Π ⊥∞ ]( k ⊥ ) + Π ⊥∞ [ D g Φ((∆
L,g,h − ∆ L,h )( k )) + (1 − D g Φ)( H )] . Note again that by assumption, k and k contain the same information. Therefore, the evolutionequations on k and k are equivalent. (cid:3) We will use both forms of the Ricci-de Turck flow equation at once to obtain the Ricci flowas a fixed point argument.4.2.
A mixed evolution operator.
Let h ∞ ∈ F be a fixed metric and g t ∈ U and h t ∈ U ∩ F , t ≥ L, ∞ = ∆ L,h ∞ and ∆ L,g,h where we suppressed the dependence of g, h on t . We build now a mixed evolutionoperator, depending on both operators, and hence on the metrics g t , h t and h ∞ . For t ≥ s ∈ [1 , t ], we consider the evolution problem ∂ r k + ∆ L, ∞ k = 0 , for r ∈ [ s, max { t − , s } ] ,∂ r k + ∆ L,g,h k = 0 , for r ∈ [max { t − , s } , t ] ,k | r = s = k ′ . We now define for 1 ≤ s ≤ t the operator P ( g, h, h ∞ ) s → t as the map which associates to giveninital data k ′ the solution of one of the above initial value problems. Remark 4.3. • Note that by construction, the mixed solution operator P ( g, h, h ∞ ) s → t depends continuously on all involved parameters. • Note that if t > s ∈ [1 , t − P ( g, h, h ∞ ) s → t = P ( g, h, h ∞ ) t − → t ◦ e − ( t − − s )∆ L,h ∞ . Let us now extend this construction to the inhomogeneous problem. For t ≥ s ∈ [1 , t ],we consider the inhomogeneous evolution problem ∂ r k + ∆ L,g,h k = F r , for r ∈ [ s, max { t − , s } ] , TABILITY OF RICCI-FLAT ALE MANIFOLDS 35 ∂ r k + ∆ L, ∞ k = F r , for r ∈ [max { t − , s } , t ] ,k | r = s = k ′ . We now define for 1 ≤ s ≤ t the operator Q ( g, h, h ∞ ) s → t ( k ′ , F ) as the map which associates togiven inital data k = k s the solution of one of the above initial value problems. Observe that theDuhamel principle also holds in a very general setting so that Q ( g, h, h ∞ ) s → t ( k ′ , F ) = P ( g, h, h ∞ ) s → t ( k ′ ) + Z ts P ( g, h, h ∞ ) r → t ( F r ) dr. (32)4.3. The Ricci flow as a mixed evolution problem.
Now let us turn back to the Ricci-deTurck flow with moving gauge. Let g t be such a flow and h ∞ ∈ U ∩ F be arbitrary. Introducethe quantities h t = Φ( g t ) , k t = g t − h t , k t = Π ⊥ h t ,h ∞ ( k t )Due to Proposition 4.2, the flow equation is equivalent to ∂ t h = D g Φ( H ) ,∂ t k + ∆ L, ∞ k = Π ⊥∞ [(∆ L, ∞ − ∆ L,h )( k ) + (1 − D g Φ)( H )] (33)and ∂ t h = D g Φ( H ) ,∂ t k + ∆ L,g,h k = [∆ L,g,h , Π ⊥∞ ]( k ) + Π ⊥∞ [ D g Φ((∆
L,g,h − ∆ L,h )( k )) + (1 − D g Φ)( H )] . (34)Let g be an initial metric and g t , t ∈ [0 ,
1] be a b h -gauged Ricci-de Turck flow, starting at g .Split g = Φ( g ) + (1 − Φ)( g ) and let h = Φ( g ) and k = (1 − Φ)( g ). Continue now withthe Ricci-de Turck flow with moving gauge and let h t , k t as above. For fixed t ∈ [1 , h t , k t ) as a solution (33). For t >
2, we regard ( h t , k t ) as the tuple obtained from solving (33)for time s ∈ [1 , t −
1] and (34) for time s ∈ [ t − , t ]. Due to (32), this implies that h t = h + Z t D g Φ( H ( s )) ds,k t = P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) + Z t P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds, (35)with I ( t, s, k, h, h ∞ ) = χ [1 , max { t − , } ] ( s ) · (cid:8) Π ⊥∞ [(∆ L, ∞ − ∆ L,h )( k ) + (1 − D g Φ)( H )] (cid:9) + χ (max { t − , } ,t ] ( s ) · (cid:8) [∆ L,g,h , Π ⊥∞ ]( k ) + Π ⊥∞ [ D g Φ((∆
L,g,h − ∆ L,h )( k )) + (1 − D g Φ)( H )] (cid:9) for t > s ∈ [1 , t ]. Remark 4.4.
This complicated construction resolves the regularity problem that was adressedin Subsection 1.3.4.Now we use (35) to identify the map of which the Ricci-de Turck flow with moving gauge isa fixed point. For this purpose, let h t , t ∈ [1 , ∞ ] a smooth curve in U ∩ F which converges toa limit metric h ∞ . Furthermore, let k t , t ∈ [1 , ∞ ) be family of symmetric 2-tensors such that k t ⊥ ker L (∆ L,h t ). Finally, assume that g t := h t + k t ∈ U . Define ψ ( h, k ) t := h + Z t D g Φ( H ( s )) ds, t ∈ [1 , ∞ ] ,ψ ( h, k ) t := P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) + Z t P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds. In order to get again a curve in
U ∩ F and family of symmetric 2-tensors which are orthogonalto ker L (∆ L,h t ), we define as correction terms ψ ( h, k ) = Φ( ψ ( h, k )) ,ψ ( h, k ) = (Π ⊥ h,h ∞ ) − ( ψ ( h, k )) . These two maps unify to the map ψ ( h, k ) = ( ψ ( h, k ) , ψ ( h, k )) and due to (35), the Ricci-deTurck flow with moving gauge (viewed as the tuple ( h t , k t )) is a fixed point of this map. Ourgoal in the next section is to identify this map as a contraction map in a suitable Banach spaceso that a Ricci-de Turck flow with moving gauge can be found via an iteration procedure.5. The iteration map
In this section, we are going to study the map ψ we just defined in detail.5.1. Estimates for the linear problem.
Let us summarize some results for the linearizedversion of the problem. The following result is Theorem 6.11 in our companion paper [KP20].
Theorem 5.1 (Heat kernel and derivative estimates) . Let ( M n , h ) be an ALE manifold with aparallel spinor. (i) For each < p ≤ q < ∞ , there exists a constant C = C ( p, q ) such that for all t > , wehave (cid:13)(cid:13) e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13) L p ,L q ≤ Ct − n ( p − q )(ii) For each i ∈ { , . . . , n − } and p ∈ (1 , ni ) , there exists a constant C = C ( p, i ) such thatfor all t > , we have (cid:13)(cid:13) ∇ i ◦ e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13) L p ,L p ≤ Ct − k k h k L p . (iii) For each i ∈ N , p ∈ [ ni , ∞ ) ∩ (1 , ∞ ) , t > and ǫ > , there exists C = C ( p, i, t , ǫ ) suchthat for all t > t , we have (cid:13)(cid:13) ∇ k ◦ e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13) L p ,L p ≤ Ct − n p + ǫ k h k L p . We can also state the result more generally as follows.
Corollary 5.2 (Heat kernel and derivative estimates) . Let ( M n , h ) be an ALE manifold with aparallel spinor, < p ≤ q < ∞ and i ∈ N . (i) If n (cid:16) p − q (cid:17) + i < n p there exists a constant C = C ( p, q, i ) such that for all t > , wehave (cid:13)(cid:13) ∇ i ◦ e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13) L p ,L q ≤ Ct − n ( p − q ) − i . (ii) If n (cid:16) p − q (cid:17) + i ≥ n p there exists for each t > and ǫ > a constant C = C ( p, q, i, t , ǫ ) such that for all t > t , we have (cid:13)(cid:13) ∇ i ◦ e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13) L p ,L q ≤ Ct − n p + ǫ . Proof.
Writing ∇ i ◦ e − t ∆ L ◦ Π ⊥ h = ∇ i ◦ e − t ∆ L ◦ e − t ∆ L ◦ Π ⊥ h ◦ Π ⊥ h = ∇ i ◦ e − t ∆ L ◦ Π ⊥ h ◦ e − t ∆ L ◦ Π ⊥ h , we immediately get (cid:13)(cid:13) ∇ i ◦ e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13) L p ,L q ≤ (cid:13)(cid:13)(cid:13) ∇ i ◦ e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13)(cid:13) L q ,L q (cid:13)(cid:13)(cid:13) e − t ∆ L ◦ Π ⊥ h (cid:13)(cid:13)(cid:13) L p ,L q , and the estimate follows from Theorem 5.1 and a case by case analysis. (cid:3) TABILITY OF RICCI-FLAT ALE MANIFOLDS 37
These estimates are in sharp contrast to the Euclidean Laplacian, where Theorem 5.1 (i) holdsfor any choice of 1 ≤ p ≤ q ≤ ∞ and i ∈ N . However, for particular differential operators, weobtained better results. Let DV ( k ) = ddt | t =0 V ( h + tk, h ) , D Ric( k ) = ddt | t =0 Ric h + tk , be the Fr´echet derivatives of the de Turck vector field in the first component and the Ricci tensor,respectively. The following result is [KP20, Theorem 6.13]. Theorem 5.3 (Special derivative estimates) . Let ( M n , h ) be an ALE manifold with a parallelspinor. Then for each p ∈ (1 , ∞ ) there exists a constant C = C ( p ) such that (cid:13)(cid:13) DV ◦ e − t ∆ L h (cid:13)(cid:13) L p ≤ Ct − k h k L p , (cid:13)(cid:13) D Ric ◦ e − t ∆ L h (cid:13)(cid:13) L p ≤ Ct − k h k L p . The Banach space.
Let ( M n , ˆ h ) be an integrable ALE manifold with a parallel spinor.Let q ∈ (1 , n ) and r be an auxiliary H¨older exponent satisfying r ∈ ( n, ∞ ) , n (cid:18) q − r (cid:19) > . We equip the space of maps k : [1 , ∞ ) → C ∞ ( S M ) with the norms k k k X q,r (ˆ h ) := sup t ≥ n k k k L q (ˆ h ) + t k∇ k k L q (ˆ h ) + t min { , n ( q − r ) } ( (cid:13)(cid:13) ∇ k (cid:13)(cid:13) L q (ˆ h ) + k k k W ,r (ˆ h ) ) o , k k k Z q,r (ˆ h ) := sup t ≥ n k k k L q (ˆ h ) + t n ( q − r ) k ∂ t k k L q (ˆ h ) o , and the set of maps ( h, k ) : [1 , ∞ ) → C ∞ ( S M ⊕ S M ) with the norms k ( h, k ) k Y q,r (ˆ h ) := k k k X q,r (ˆ h ) + k h k Z q,r ()ˆ h ) In the first slot, we will typically insert h − ˆ h , where h = h t is a family of Ricci-flat metrics in U ∩ F with U being a small L [ q, ∞ ] -neighbourhood of ˆ h . Lemma 5.4.
Let U be an L [ q, ∞ ] -neighbourhood of ˆ h which is so small that Proposition 2.2holds. Let h t , t ∈ [1 , ∞ ) be a family of metrics in U ∩ F and define X q,r , Y q,r with respect to h t .Then there exist a constant C = C ( U ) only depending on U such that1 C k k k X q,r (ˆ h ) ≤ k k k X q,r ( h ) ≤ C k k k X q,r (ˆ h ) , C k h k Z q,r (ˆ h ) ≤ k h k Z q,r ( h ) ≤ C k h k Z q,r (ˆ h ) . Proof.
By standard estimates, this follows easily from Proposition 2.2, as h − ˆ h ∈ O ∞ ( r − n ). (cid:3) Due to this lemma, we will from now on suppress the dependence of the norm on the metricfor notational convenience and allow any curve h t in U ∩ F . The first lemma justifies the X -partof the norm Lemma 5.5.
Let ∆
L,h be the Lichnerwicz Laplacian of a Ricci-flat metric with a parallel spinorand k ∈ L q ( S M ). Then, (cid:13)(cid:13) e − t ∆ L,h ◦ Π ⊥ h ( k ) | [1 , ∞ ) (cid:13)(cid:13) X q,r ≤ C k k k L q . Proof.
This is immediate from Theorem 5.1. (cid:3)
In the following we introduce the norm k k k L [ q, ∞ ] = k k k L q + k k k L ∞ which is the natural norm on L [ q, ∞ ] = L q ∩ L ∞ . From now on, let U be an L [ q, ∞ ] -neighbourhoodof ˆ h , which is so small that Proposition 2.2 holds and that the projection map Φ of Subsection h t ) t ≥ be a family of metrics in U ∩ F and ( k t ) t ≥ be a familyof symmetric 2-tensors. We finally set g t = h t + k t , t ∈ [1 , ∞ ). Lemma 5.6.
We have (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L [ q, ∞ ] + (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) W , ∞ ≤ C (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r . Proof.
By Sobolev embedding, Proposition 2.2 and definition of the norms, we have (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L [ q, ∞ ] + (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) W , ∞ ≤ k k k L q + k k k W , ∞ + (cid:13)(cid:13)(cid:13) h − b h (cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13) h − b h (cid:13)(cid:13)(cid:13) W , ∞ ≤ k k k L q + k k k W ,r + (cid:13)(cid:13)(cid:13) h − b h (cid:13)(cid:13)(cid:13) L q ≤ C k k k X q,r + C (cid:13)(cid:13)(cid:13) h − b h (cid:13)(cid:13)(cid:13) Z q,r = C (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r , which proves the lemma. (cid:3) Lemma 5.7.
There exists an ǫ > ψ is well-defined if (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ . Proof.
In order to show that ψ is well defined, we have to ensure that the terms involving theprojection map Φ make sense. These terms are given by D g Φ and ψ ( h, k ) = Φ( ψ ( h, k )) = Φ (cid:18) h + Z t D g Φ( H ( s )) ds (cid:19) . In order to do so, we have to show that g ∈ U , h + Z t D g Φ( H ( s )) ds ∈ U , here U is the L [ q, ∞ ] -neighbourhood which is the domain of definition of Φ. At first, Lemma 5.6shows that g ∈ U if (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r is chosen small enough. Let now q ′ ∈ (1 , q ], r ′ ∈ [ r, ∞ ) and q ′′ ∈ (1 , ∞ ]. Using Lemma 2.10 (iii), − n (cid:18) q ′ − r ′ (cid:19) ≤ − n (cid:18) q − r (cid:19) < − , and (29), we establish the estimate (cid:13)(cid:13)(cid:13)(cid:13)Z t D g Φ( H ( s )) ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′′ ≤ C Z t k H ( s ) k L r ′ ds ≤ C Z t [ k∇ k k L r ′ + (cid:13)(cid:13) ∇ k (cid:13)(cid:13) L r ′ k k k L r ′ + k R k L ∞ k k k L r ′ ] ds ≤ C Z t s − n ( q ′ − r ′ ) ds · k k k X q ′ ,r ′ ≤ C k k k X q ′ r ′ . (36)Here, we use here only for q ′′ ∈ { q, ∞} , q ′ = q and r ′ = r , but later, we will make use of thisinequality in full generality. We get (cid:13)(cid:13)(cid:13)(cid:13) h + Z t D g Φ( H ( s )) ds − ˆ h (cid:13)(cid:13)(cid:13)(cid:13) L [ q, ∞ ] ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t D g Φ( H ( s )) ds (cid:13)(cid:13)(cid:13)(cid:13) L [ q, ∞ ] + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L [ q, ∞ ] ≤ C k k k X q,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r TABILITY OF RICCI-FLAT ALE MANIFOLDS 39 so that h + Z t D g Φ( H ( s )) ds ∈ U , if (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r is chosen small enough. (cid:3) Mapping properties of the iteration map.
Let ( M n , ˆ h ) be an integrable ALE manifoldwith a parallel spinor and q ∈ (1 , n ), r ∈ ( n, ∞ ) so that n (cid:18) q − r (cid:19) > U be an L [ q, ∞ ] -neighbourhood of ˆ h , which is so small that Proposition 2.2 holdsand that the projection map Φ of Subsection 2.2 is defined. Let furthermore ( h t ) t ≥ be a familyof metrics in U ∩ F and ( k t ) t ≥ be a family of symmetric 2-tensors. We finally set g t = h t + k t , t ∈ [1 , ∞ ). The goal of this subsection is to derive the following mapping property: Theorem 5.8.
There exists an ǫ > such that the map ψ satisfies the estimate (cid:13)(cid:13)(cid:13) ψ ( h, k ) − (ˆ h, (cid:13)(cid:13)(cid:13) Y q,r ≤ C (cid:18) k k k W ,q + k k k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q + k k k X q,r (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r (cid:19) as long as (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ . The proof of this theorem is split up in Propositions 5.9 and 5.10 below, in which the estimatesfor the components of ψ are established. In fact we will also prove estimates for certain otherH¨older exponents q ′ ∈ (1 , q ] and r ′ ∈ [ r, ∞ ). These more general estimates will be importantlater in the paper for detecting the optimal convergence behaviour for the Ricci-de Turck flowwith moving gauge. Proposition 5.9.
There exists an ǫ > such that we have k ψ ( h, k ) k Z q ′ ,r ′ ≤ C (cid:16)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q + k k k X q ′ ,r ′ (cid:17) for any q ′ ∈ (1 , q ] and r ′ ∈ [ r, ∞ ) , as long as (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ .Proof. Due to Lemma 5.7, ψ is well defined under the assumption of the proposition. Due toLemma 2.10 (iii), D g Φ is bounded on L q ′ for each q ′ ∈ (1 , ∞ ) and g ∈ U . Therefore, Φ is alsoLipschitz with respect to the L q ′ -norm. Using ˆ h = Φ(ˆ h ), h = Φ( h ) and the estimate in (36),we then get (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ˆ h (cid:13)(cid:13)(cid:13) L q ′ ≤ (cid:13)(cid:13)(cid:13)(cid:13) Φ (cid:18) h + Z t D g Φ( H ( s )) ds (cid:19) − Φ( h ) (cid:13)(cid:13)(cid:13)(cid:13) L q ′ + (cid:13)(cid:13)(cid:13) Φ( h ) − Φ(ˆ h ) (cid:13)(cid:13)(cid:13) L q ′ ≤ C (cid:13)(cid:13)(cid:13)(cid:13)Z t D g Φ( H ( s )) ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′ + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q ′ ≤ C k k k X q ′ r ′ + C (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q ′ . Secondly, we estimate, similarly as in (36) t n (cid:16) q ′ − r ′ (cid:17) k ∂ t ψ ( h, k ) k L q ′ ≤ Ct n (cid:16) q ′ − r ′ (cid:17) (cid:13)(cid:13)(cid:13) D ψ ( h,k ) Φ( H ) (cid:13)(cid:13)(cid:13) L q ′ ≤ Ct n (cid:16) q ′ − r ′ (cid:17) h k∇ k k L r ′ + (cid:13)(cid:13) ∇ k (cid:13)(cid:13) L r ′ k k k L r ′ + k R k L ∞ k k k L r ′ i ≤ C k k k X q ′ ,r ′ , which finishes the proof of the lemma. (cid:3) Proposition 5.10.
There exists and ǫ > such that we have k ψ ( h, k ) k X q,r ≤ C h k k k W ,q + k k k W ,r + ( k k k X q + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z ) k k k X q,r i as long as (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) X q,r < ǫ . Moreover, for any q ′ ∈ (1 , q ] satisfying n (cid:18) q ′ − q (cid:19) ≤ min (cid:26) n r , n (cid:18) q − r (cid:19) − , (cid:27) , we have k ψ ( h, k ) k X q ′ ,r ≤ C (cid:20) k k k W ,q ′ + k k k W ,r + (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) k k k X q,r (cid:21) . We prove the second inequality as the first one is a special case of the second. It will followfrom a series of lemmas.
Lemma 5.11.
For any q ′ ∈ (1 , q ] and r ′ ∈ [ r, ∞ ), we have k ψ ( h, k ) k X q ′ ,r ′ = (cid:13)(cid:13) (Π ⊥ h,h ∞ ) − ( ψ ( h, k )) (cid:13)(cid:13) X q ′ ,r ′ ≤ C (cid:13)(cid:13) ψ ( h, k ) (cid:13)(cid:13) X q ′ ,r ′ Proof.
Let us abbreviate ψ := ψ ( h, k ) t for fixed h, k, t in the proof. Due to the assumptionson q and r , we have β := min (cid:26) n (cid:18) q ′ − r ′ (cid:19) , (cid:27) ≥ min (cid:26) n (cid:18) q − r (cid:19) , (cid:27) ≥ . Therefore, using Lemma 2.10 (iv), we get (cid:13)(cid:13) (Π ⊥ h,h ∞ ) − ( ψ ) (cid:13)(cid:13) L q ′ ≤ C (cid:13)(cid:13) ψ (cid:13)(cid:13) L q ′ ,t (cid:13)(cid:13) ∇ (Π ⊥ h,h ∞ ) − ( ψ ) (cid:13)(cid:13) L q ′ ≤ Ct (cid:13)(cid:13) ∇ ψ (cid:13)(cid:13) L q ′ + Ct n (cid:16) q ′ − r ′ (cid:17) (cid:13)(cid:13) ψ (cid:13)(cid:13) L r ′ ,t β (cid:13)(cid:13) ∇ (Π ⊥ h,h ∞ ) − ( ψ ) (cid:13)(cid:13) L q ′ ≤ Ct β (cid:13)(cid:13) ∇ ψ (cid:13)(cid:13) L q ′ + t n (cid:16) q ′ − r ′ (cid:17) (cid:13)(cid:13) ψ (cid:13)(cid:13) L r ′ ,t n (cid:16) q ′ − r ′ (cid:17) (cid:13)(cid:13) (Π ⊥ h,h ∞ ) − ( ψ ) (cid:13)(cid:13) W ,r ′ ≤ Ct n (cid:16) q ′ − r ′ (cid:17) (cid:13)(cid:13) ψ (cid:13)(cid:13) W ,r ′ , and the lemma follows from the definition of the X q ′ ,r ′ -norm. (cid:3) Proof of Proposition 5.10.
By Lemma 5.11 and the triangle inequality, we have k ψ ( h, k ) k X q ′ ,r ≤ (cid:13)(cid:13) P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) (cid:13)(cid:13) X q ′ ,r ′ + (cid:13)(cid:13)(cid:13)(cid:13)Z t P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ . The first term on the right hand side is treated in Lemma 5.12 below. The integrand will besplit up in two terms which are esimated in Lemma 5.15 and Lemma 5.18 below. (cid:3)
Lemma 5.12.
There exists an ǫ > (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) X q,r < ǫ , we have (cid:13)(cid:13) P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) (cid:13)(cid:13) X q ′ ,r ′ ≤ C ( k k k W ,q ′ + k k k W ,r ′ ) . (37)for all q ′ ∈ (1 , q ] and r ′ ∈ [ r, ∞ ). Moreover, if t ≥ s ∈ [1 , t −
2] and p ′ ∈ (1 , q ′ ], we have (cid:13)(cid:13) ∇ i ◦ P ( g, h, h ∞ ) s → t (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13) L q ′′ ≤ C ( t − s ) − n (cid:16) p ′ − p (cid:17) − α ( q ′′ ,i ) k Θ k L p ′ , (38) TABILITY OF RICCI-FLAT ALE MANIFOLDS 41 where Θ is an arbitrary symmetric 2-tensor and α : { q ′ , r ′ }×{ , , } → R is defined by α ( q ′ ,
0) =0, α ( q ′ ,
1) = , α ( q ′ ,
2) = min n , n ( q ′ − r ′ ) o and α ( r ′ , i ) = n ( q ′ − r ′ ) otherwise. Proof.
By Lemma 5.6, the assumption (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) X q,r < ǫ implies smallness of k k k W , ∞ . Inparticular, (cid:13)(cid:13) g − (cid:13)(cid:13) W , ∞ < ∞ (recall that g = h + k ). Therefore, for t ∈ [1 , (cid:13)(cid:13) P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) (cid:13)(cid:13) W ,q ≤ C k k k W ,q ′ , (cid:13)(cid:13) P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) (cid:13)(cid:13) W ,r ′ ≤ C k k k W ,r ′ . From now on, let t ≥ s ∈ [1 , t − t − t ) and Theorem 5.1 (to pass from 1 to t − k , s = 1, p ′ = q ′ and using the definition of the X q ′ ,r ′ -norm.At first, we clearly have (cid:13)(cid:13) P ( g, h, h ∞ ) → t (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13) L q ′ = (cid:13)(cid:13)(cid:13) P ( g, h, h ∞ ) t − → t ◦ e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L q ′ ≤ C (cid:13)(cid:13)(cid:13) e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L q ′ ≤ C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) k k s k L p ′ Because we again have β := min (cid:26) n (cid:18) q ′ − r ′ (cid:19) , (cid:27) ≥ min (cid:26) n (cid:18) q − r (cid:19) , (cid:27) ≥ , we can estimate (cid:13)(cid:13) ∇ ◦ P ( g, h, h ∞ ) s → t (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13) L q ′ = (cid:13)(cid:13)(cid:13) ∇ ◦ P ( g, h, h ∞ ) t − → t ◦ e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L q ′ ≤ C (cid:16)(cid:13)(cid:13)(cid:13) ∇ ◦ e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L q ′ + (cid:13)(cid:13)(cid:13) e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L r ′ (cid:17) ≤ C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) − k Θ k L p ′ + C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) − n (cid:16) q ′ − r ′ (cid:17) k Θ k L p ′ ≤ C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) − k Θ k L p ′ and (cid:13)(cid:13) ∇ ◦ P ( g, h, h ∞ ) s → t (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13) L q ′ = (cid:13)(cid:13)(cid:13) ∇ ◦ P ( g, h, h ∞ ) t − → t ◦ e − ( t − s − L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L q ′ ≤ C (cid:16)(cid:13)(cid:13)(cid:13) ∇ ◦ e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L q ′ + (cid:13)(cid:13)(cid:13) e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) W ,r ′ (cid:17) ≤ C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) − β k Θ k L p ′ + C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) − n (cid:16) q ′ − r ′ (cid:17) k Θ k L p ′ ≤ C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) − β k Θ k L p ′ . Finally, we have (cid:13)(cid:13) P ( g, h, h ∞ ) s → t (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13) W ,r ′ = (cid:13)(cid:13)(cid:13) P ( g, h, h ∞ ) t − → t ◦ e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) W ,r ′ ≤ C (cid:13)(cid:13)(cid:13) e − ( t − − s )∆ L (Π ⊥ h ∞ (Θ)) (cid:13)(cid:13)(cid:13) L r ′ ≤ C ( t − − s ) − n (cid:16) p ′ − q ′ (cid:17) − n (cid:16) q ′ − r ′ (cid:17) k Θ k L p ′ , which finishes the proof of the lemma. (cid:3) Lemma 5.13.
For p ′ ∈ (1 , r ′ ], q ′ ∈ (1 , q ], r ′ ∈ [ r, ∞ ), t ∈ [1 , ∞ ) and t ′ ∈ [ t, ∞ ], we have (cid:13)(cid:13) (∆ L,h t ′ − ∆ L,h t )( k t ) (cid:13)(cid:13) L p ′ ≤ Ct − n (cid:16) q ′ − r ′ (cid:17) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q ′ ,r ′ k k k X q ′ ,r ′ Proof.
Choose p ′′ ∈ ( p ′ , ∞ ] so that p ′ = p ′′ + r ′ . Then by the H¨older inequality, (cid:13)(cid:13) (∆ L,h t ′ − ∆ L,h t )( k t ) (cid:13)(cid:13) L p ′ ≤ C k h t − h ∞ k W ,p ′′ k k t k W ,r ′ ≤ C k h t − h t ′ k L q ′ k k t k W ,r ′ ≤ C Z t ′ t k ∂ s h s k L q ′ ds · k k t k W ,r ′ ≤ C Z ∞ t s − n (cid:16) q ′ − r ′ (cid:17) ds · (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q ′ ,r ′ t − n (cid:16) q ′ − r ′ (cid:17) k k k X q ′ ,r ′ ≤ Ct − n ( q − r ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q ′ ,r ′ k k k X q ′ ,r ′ , (39)which finishes the proof. (cid:3) Lemma 5.14.
For q ′ ∈ (1 , q ] satisfying n (cid:18) q ′ − q (cid:19) < n r , we have k (1 − D g Φ)( H )( t ) k L q ′ ≤ Ct − β − n ( q − r ) k k k X q,r , with β = min n , n (cid:16) q − r (cid:17)o > . Proof.
By the condition on q ′ , we may now choose r ′ ∈ ( r, ∞ ], such that1 q ′ = 1 q + 1 r ′ . By Lemma 2.10 (iii), the H¨older inequality, interpolation and Sobolev embedding, we get k (1 − D g Φ)( H )( t ) k L q ′ ≤ C k H ( t ) k L q ′ ≤ C ( (cid:13)(cid:13) ∇ k t (cid:13)(cid:13) L q k k t k L r ′ + k∇ k t k L q ′ + k R k L q ′ k∇ k t k L ∞ ) ≤ C ( (cid:13)(cid:13) ∇ k t (cid:13)(cid:13) L q k k t k L r ′ + k R k L q ′ k∇ k t k L r ′ ) ≤ C ( (cid:13)(cid:13) ∇ k t (cid:13)(cid:13) L q k k t k W ,r + k R k L q ′ k∇ k t k W ,r ) ≤ C (cid:16) t − β − n ( q − r ) + t − n ( q − r ) (cid:17) k k k X q,r ≤ Ct − β − n ( q − r ) k k k X q,r . with β = min n , n (cid:16) q − r (cid:17)o > . (cid:3) Lemma 5.15.
Let t ≥
3. Then we have the estimate (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ ≤ C ( k k k X q,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r ) k k k X q,r for all q ′ ∈ (1 , q ] , r ′ ∈ [ r, ∞ ) which satisfy n (cid:18) q ′ − q (cid:19) + n (cid:18) r − r ′ (cid:19) < min (cid:26) n (cid:18) q − r (cid:19) − , (cid:27) , n (cid:18) q ′ − q (cid:19) < n r , (40) TABILITY OF RICCI-FLAT ALE MANIFOLDS 43
Proof.
By the triangle inequality, we first split the term further up as (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t Π ⊥∞ [(∆ L, ∞ − ∆ L,h )( k ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ + (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t Π ⊥∞ (1 − D g Φ)( H ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ . Now let us consider the first of these terms. We estimate each of the terms of the X q ′ ,r ′ -normseparately. Let q ′′ ∈ { q ′ , r ′ } , i ∈ { , , } and α : { q ′ , r ′ } × { , , } → R be the function fromLemma 5.12. Let p ′ ∈ (1 , q ) be small. By Lemma 5.12 and Lemma 5.13, we get (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − P ( g, h, h ∞ ) s → t Π ⊥∞ [(∆ L, ∞ − ∆ L,h )( k ) ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′′ ≤ Z t − (cid:13)(cid:13) ∇ i ◦ P ( g, h, h ∞ ) s → t Π ⊥∞ (cid:13)(cid:13) L p ′ ,L q ′′ k (∆ L, ∞ − ∆ L,h )( k ) k L p ′ ds ≤ C Z t − ( t − s ) − n ( p ′ − q ′ ) − α ( q ′′ ,i ) s − n ( q − r ) ds · (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r k k k X q,r ≤ Ct − α ( q ′′ ,i ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r k k k X q,r . The last inequality is justified by Lemma 1.16: We have n (cid:18) p ′ − q ′ (cid:19) + α ( q ′′ , i ) > α ( q ′′ , i ) , n (cid:18) q − r (cid:19) − > n (cid:18) q − r (cid:19) ≥ α ( q ′′ , i )and (40) implies n (cid:18) p ′ − q ′ (cid:19) + α ( q ′′ , i ) + 3 n (cid:18) q − r (cid:19) − > n (cid:18) p ′ − r ′ (cid:19) − α ( q ′′ , i ) > α ( q ′′ , i ) , for all choices of q ′′ and i , provided that p ′ is chosen small enough. We have thus shown that (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t Π ⊥∞ [(∆ L, ∞ − ∆ L,h )( k ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ ≤ C (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r k k k X q,r . For the other part of the integral, the estimate is slightly different. By Lemma 5.12 and Lemma5.14 (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − P ( g, h, h ∞ ) s → t Π ⊥∞ (1 − D g Φ)( H ) ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′′ ≤ Z t − (cid:13)(cid:13) ∇ i P ( g, h, h ∞ ) s → t Π ⊥∞ (cid:13)(cid:13) L q ′ ,L q ′′ k (1 − D g Φ)( H ) k L q ′ ds ≤ Z t − ( t − s ) − α ( q ′′ ,i ) s − − n ( q − r ) ds · k k k X q,r ≤ Ct − α ( q ′′ ,i ) · k k k X q,r . The last inequality is again justified by Lemma 1.16 and (40), since12 + n (cid:18) q − r (cid:19) > ,
12 + n (cid:18) q − r (cid:19) > n (cid:18) q ′ − r ′ (cid:19) ≥ α ( q ′′ , i ) . This proves (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t Π ⊥∞ (1 − D g Φ)( H ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ ≤ C k k k X q,r and we conclude (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ′ ≤ C (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) k k k X q,r , as desired (cid:3) Lemma 5.16.
Let t ≥
2. Then we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t − { t − , } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ≤ C ( k k k X q,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r ) k k k X q,r , for all q ′ ∈ (1 , q ] with n (cid:18) q ′ − q (cid:19) < n r . Proof.
By Lemma 3.3 (iii), we first have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t − { t − , } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ,q ′′ ≤ C sup s ∈ [min { t − , } ,t − k I ( t, s, k, h, h ∞ ) ds k L q ′′ for q ′′ ∈ { q ′ , r } . Recall that I ( t, s, k, h, h ∞ ) = (∆ L, ∞ − ∆ L,h )( k ) + (1 − D g Φ)( H )for s ∈ [min { t − , } , t − s ∈ [min { t − , } ,t − k (∆ L, ∞ − ∆ L,h s )( k s ) k L q ′′ ≤ Ct − n ( q − r ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r k k k X q,r , Lemma 5.13 yields sup s ∈ [min { t − , } ,t − k (1 − D g Φ)( H )( s ) k L q ′ ≤ Ct − β − n ( q − r ) k k k X q,r , and standard estimates similar as in Lemma 5.14 yieldsup s ∈ [min { t − , } ,t − k (1 − D g Φ)( H )( s ) k L r ≤ sup s ∈ [min { t − , } ,t − k k s k W ,r ≤ Ct − n ( q − r ) k k k X q,r . The function α defined in Lemma 5.12 satisfies α ( q ′′ , i ) ≤ β + n (cid:18) q − r (cid:19) ≤ n (cid:18) q − r (cid:19) , α ( q ′′ , i ) ≤ n (cid:18) q − r (cid:19) − (cid:3) In the remainder of this subsection, we estimate the integral from max { t − , } to t . Westart with a technical lemma. Lemma 5.17.
For s ∈ [max { t − , } , t ], q ′ ∈ (1 , q ] and r ′ ∈ [ r, ∞ ) satisfying1 q ′ < q + 1 r ′ , we have k I ( t, s, k, h, h ∞ ) k W ,r ′ ≤ C ( k k k W ,r ′ + k h − h ∞ k L q ) k k k W ,r ′ , TABILITY OF RICCI-FLAT ALE MANIFOLDS 45 k I ( t, s, k, h, h ∞ ) k W ,q ′ ≤ C ( k k k W ,r ′ + k∇ k k W ,q + k h − h ∞ k L q ) k k k W ,r ′ . Proof.
Let p ′ ∈ { q ′ , r ′ } . Recall that for s ∈ [ t − , t ], we have I ( t, s, k, h, h ∞ ) = [∆ L,g,h , Π ⊥∞ ]( k ) + Π ⊥∞ [ D g Φ((∆
L,g,h − ∆ L,h )( k )) + (1 − D g Φ)( H )]We have [∆ L,g,h , Π ⊥∞ ]( k ) = [∆ L,g,h − ∆ L,h , Π ⊥∞ ]( k ) + [∆ L,h − ∆ L, ∞ , Π ⊥∞ ]( k ) , Using g = h + k , the first of these terms can be written as[∆ L,g,h − ∆ L,h , Π ⊥∞ ]( k ) = (∆ L,g,h − ∆ L,h )Π ⊥∞ ( k ) − Π ⊥∞ ((∆ L,g,h − ∆ L,h )( k ))= k ∗ ∇ Π ⊥∞ ( k ) + R ∗ k ∗ Π ⊥∞ ( k ) − Π ⊥∞ ( k ∗ ∇ k + R ∗ k ∗ k )and the second one is[∆ L,h − ∆ L, ∞ , Π ⊥∞ ]( k ) = (∆ L,h − ∆ L, ∞ )Π ⊥∞ ( k ) − Π ⊥∞ ((∆ L,h − ∆ L, ∞ )( k ))= ( h − h ∞ ) ∗ ∇ Π ⊥∞ ( k ) + ∇ ( h − h ∞ ) ∗ ∇ Π ⊥∞ ( k )+ ∇ ( h − h ∞ ) ∗ Π ⊥∞ ( k ) + R ∗ ( h − h ∞ ) ∗ Π ⊥∞ ( k ) − Π ⊥∞ (( h − h ∞ ) ∗ ∇ k + ∇ ( h − h ∞ ) ∗ ∇ k ) − Π ⊥∞ ( ∇ ( h − h ∞ ) ∗ k + R ∗ ( h − h ∞ ) ∗ k ) . A suitable combination of Lemma 2.10 (ii), the H¨older inequality and elliptic regularity for h − h ∞ yields (cid:13)(cid:13) [∆ L,g,h , Π ⊥∞ ]( k ) (cid:13)(cid:13) W ,p ′ ≤ C ( k k k W ,r ′ + k h − h ∞ k L q ) k k k W ,r ′ . Similarly, we get (cid:13)(cid:13) Π ⊥∞ [ D g Φ((∆
L,g,h − ∆ L,h )( k )) (cid:13)(cid:13) W ,p ′ ≤ C k (∆ L,g,h − ∆ L,h )( k ) k L r ′ = C (cid:13)(cid:13) k ∗ ∇ k + R ∗ k ∗ k (cid:13)(cid:13) L r ′ ≤ C k k k W ,r ′ . Recall that H = g − ∗ g − ∗ ∇ k ∗ ∇ k which yields pointwise bounds | H | ≤ C |∇ k | , |∇ H | ≤ C ( |∇ k ||∇ k | + |∇ k | ) . Because q ≤ q ′ < q + r , we may find r ′′ ∈ ( r ′ , ∞ ] such that1 q ′ = 1 q + 1 r ′′ We then get, using the H¨older inequality and Sobolev embedding, (cid:13)(cid:13) Π ⊥∞ (1 − D g Φ)( H ) (cid:13)(cid:13) W ,q ′ ≤ C k H k W ,q ′ ≤ C k∇ k k W ,q k∇ k k L r ′′ ≤ C k∇ k k W ,q k k k W ,r . Similarly, (cid:13)(cid:13) Π ⊥∞ (1 − D g Φ)( H ) (cid:13)(cid:13) W ,r ′ ≤ C k H k W ,r ′ ≤ C k∇ k k W ,r ′ k∇ k k L ∞ ≤ C k k k W ,r ′ , which finishes the proof. (cid:3) Lemma 5.18.
Let t ≥
1. Then for q ′ ∈ (1 , q ] satisfying1 q ′ < q + 1 r , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t max { t − , } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X q ′ ,r ≤ C ( k k k X q,r + k h − h ∞ k Z q,r ) k k k X q,r . Proof.
Let p ′ ∈ { q ′ , r } . By short-time estimates (Lemma 3.3 (iii)), we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t max { t − , } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ,p ′ ≤ C sup s ∈ [ t − ,t ] k I ( t, s, k, h, h ∞ ) k W ,p ′ . From Lemma 5.17, we have k I ( t, s, k, h, h ∞ ) k W ,p ′ ≤ C ( k k k W ,r + k∇ k k W ,q + k h − h ∞ k L q ) k k k W ,r . By definition of the norms, we havesup s ∈ [ t − ,t ] k h − h ∞ k L r k k k W ,r ≤ Ct − n ( q − r ) − n ( q − r ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r k k k X q,r , sup s ∈ [ t − ,t ] k k k W ,r ≤ Ct − n ( q − r ) k k k X q,r , sup s ∈ [ t − ,t ] k∇ k k W ,q k k k W ,r ≤ Ct − − n ( q − r ) k k k X q,r . Combining all these estimates and using the conditions on q ′ , we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t max { t − , } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ,p ′ ≤ Ct − n (cid:16) q ′ − r (cid:17) (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) k k k X q,r , which yields the desired estimate. (cid:3) Contraction properties of the iteration map.
Let us assume the same as at the be-ginning of Subsection 5.3. Additionally, we demand that n (cid:18) q − r (cid:19) = 1 . The goal of this subsection is to derive the following mapping property:
Theorem 5.19.
There exists an ǫ > such that the operator ψ satisfies the estimate (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r ≤ C ( k k k W ,q + k k k W ,r )+ C (cid:18)(cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r (cid:19) (cid:13)(cid:13)(cid:13) ( h − ˜ h, k − ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r as long as (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ . The proof of this theorem is split up in Propositions 5.22 and 5.23 below, in which the estimatesfor the components of ψ are established. At first, recall that H = F ( g − , g − , ∇ k, ∇ k ) + F ( g − , R, k, k ) + F ( g − , k, ∇ k )and abbreviate F := F ( g − , g − , ∇ k, ∇ k ) , ˜ F := F (˜ g − , ˜ g − , ˜ ∇ ˜ k, ˜ ∇ ˜ k ) ,F := F ( g − , R, k, k ) , ˜ F := F (˜ g − , ˜ R, ˜ k, ˜ k ) ,F := F ( g − , k, ∇ k ) , ˜ F := F (˜ g − , ˜ k, ˜ ∇ ˜ k ) . TABILITY OF RICCI-FLAT ALE MANIFOLDS 47
Lemma 5.20.
There exists an ǫ > (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ , we havethe pointwise estimates | F − ˜ F | ≤ C | k − ˜ k ||∇ k | + |∇ ( k − ˜ k ) | ( |∇ k | + |∇ ˜ k | )+ C ( | h − ˜ h | + |∇ ( h − ˜ h ) | )( |∇ k | + |∇ ˜ k | ) , | F − ˜ F | ≤ C | k − ˜ k | ( | k | + | ˜ k | )( | R | + | ˜ R | ) + ( | h − ˜ h | + |∇ ( h − ˜ h ) | + |∇ ( h − ˜ h ) | ) | k | , | F − ˜ F | ≤ C ( | k − ˜ k ||∇ k | + | ˜ k ||∇ ( k − ˜ k ) | )+ C ( | h − ˜ h || k ||∇ k | + |∇ ( h − ˜ h ) ||∇ ˜ k || ˜ k | + |∇ ( h − ˜ h ) | | ˜ k | ) . Proof.
Note that Lemma 5.6 ensures that k k k W , ∞ + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W , ∞ is small. We first look at thedifference F ( g − , g − , ∇ k, ∇ k ) − F (˜ g − , ˜ g − , ˜ ∇ ˜ k, ˜ ∇ ˜ k ) = F ( g − − ˜ g − , g − , ∇ k, ∇ k )+ F (˜ g − , g − − ˜ g − , ∇ k, ∇ k )+ F (˜ g − , ˜ g − , ∇ k − ˜ ∇ ˜ k, ∇ k )+ F (˜ g − , ˜ g − , ˜ ∇ ˜ k, ∇ k − ˜ ∇ ˜ k ) . Note that | g − − ˜ g − | ≤ C ( | k − ˜ k | + | h − ˜ h | ) . Note further, that by using the tensor T kij := Γ kij − ˜Γ kij , we can write ∇ ˜ k − ˜ ∇ ˜ k = T ∗ ˜ k . Thetensor T is schematically of the form T = h − ∗ ∇ ( h − ˜ h ). Therefore we get |∇ k − ˜ ∇ ˜ k | ≤ C ( |∇ ( k − ˜ k ) | + | T ||∇ ˜ k | ) , | T | ≤ C |∇ ( h − ˜ h ) || , and the first inequality is obtained from combining all these estimates. The other estimates areperformed similarly, using in addition R − ˜ R = ∇ T + T ∗ T, ∇ ˜ k − ˜ ∇ ˜ k = ∇ T ∗ ˜ k + T ∗ ∇ ˜ k + T ∗ T ∗ ˜ k. The details are left to the reader. (cid:3)
Lemma 5.21.
There exists an ǫ > (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ , we have (cid:13)(cid:13)(cid:13) H − ˜ H (cid:13)(cid:13)(cid:13) L q ≤ Ct − n ( q − r ) (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) (cid:20) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) . Proof.
By Lemma 5.20 and standard estimates, we conclude (cid:13)(cid:13)(cid:13) F − ˜ F (cid:13)(cid:13)(cid:13) L q ≤ C (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L q k∇ k k L ∞ + C (cid:13)(cid:13)(cid:13) ∇ ( k − ˜ k ) (cid:13)(cid:13)(cid:13) L q (cid:16) k∇ k k L ∞ + (cid:13)(cid:13)(cid:13) ∇ ˜ k (cid:13)(cid:13)(cid:13) L ∞ (cid:17) + C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W ,q (cid:16) k k k W , ∞ + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) ≤ C (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L q k k k W ,r + C (cid:13)(cid:13)(cid:13) ∇ ( k − ˜ k ) (cid:13)(cid:13)(cid:13) L q (cid:16) k k k W ,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r (cid:17) + C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W ,q (cid:16) k k k W ,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r (cid:17) ≤ Ct − n ( q − r ) (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r k k k X q,r + t − − n ( q − r ) (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) + Ct − n ( q − r ) (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) X q,r (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19)
28 KLAUS KR ¨ONCKE AND OLIVER L. PETERSEN ≤ Ct − n ( q − r ) (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) . Furthermore, we have (cid:13)(cid:13)(cid:13) F − ˜ F (cid:13)(cid:13)(cid:13) L q ≤ C (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L ∞ (cid:16) k k k L ∞ + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) L ∞ (cid:17) (cid:16) k R k L q + (cid:13)(cid:13)(cid:13) ˜ R (cid:13)(cid:13)(cid:13) L q (cid:17) + C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W ,q k k k L ∞ ≤ C (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W ,r (cid:16) k k k W ,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r (cid:17) + C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q k k k W ,r ≤ Ct − n ( q − r ) (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q (cid:19) (cid:18) k k k X q + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q (cid:19) . Choose r ′ ∈ ( q, ∞ ) such that q = r ′ + r . Then the H¨older inequality yields (cid:13)(cid:13)(cid:13) F − ˜ F (cid:13)(cid:13)(cid:13) L q ≤ C (cid:16)(cid:13)(cid:13) ∇ k (cid:13)(cid:13) L q (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) ∇ ( k − ˜ k ) (cid:13)(cid:13)(cid:13) L q (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) L ∞ (cid:17) + C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W ,r ′ (cid:16) k k k L ∞ (cid:13)(cid:13) ∇ k (cid:13)(cid:13) L r + (cid:13)(cid:13)(cid:13) ∇ ˜ k (cid:13)(cid:13)(cid:13) L r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) L r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) L ∞ (cid:17) ≤ C (cid:16)(cid:13)(cid:13) ∇ k (cid:13)(cid:13) L q (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W ,r + (cid:13)(cid:13)(cid:13) ∇ ( k − ˜ k ) (cid:13)(cid:13)(cid:13) L q (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r (cid:17) + C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q (cid:18) k k k W ,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r (cid:19) ≤ Ct − n ( q − r ) (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) (cid:20) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) , which finishes the proof of the lemma. (cid:3) Proposition 5.22.
There exists an ǫ > such that if (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ , wehave (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) Z q,r ≤ C (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) . Proof.
We estimate, using Lemma 2.10 (iii) and (vi) and Lemma 5.21, (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L q ≤ C (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L q ≤ C Z t (cid:13)(cid:13)(cid:13) D g Φ( H ( h, k )) − D ˜ g Φ( H (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) L q ds ≤ C Z t (cid:13)(cid:13)(cid:13) ( D g Φ − D ˜ g Φ)( H (˜ h, ˜ k )) + D g Φ(( H ( h, k ) − H (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) L q ds ≤ C Z t [ k g − ˜ g k L q (cid:13)(cid:13)(cid:13) H (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L r/ + (cid:13)(cid:13)(cid:13) H ( h, k ) − H (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L q ] ds ≤ C Z t [ k g − ˜ g k L q (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r + (cid:13)(cid:13)(cid:13) H ( h, k ) − H (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L q ] ds ≤ C Z t s − n ( q − r ) ds (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) ≤ C (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) For the other term of the norm, we estimate using Lemma 2.10 (iii) and (vi) again, t n ( q − r ) (cid:13)(cid:13)(cid:13) ∂ t ( ψ ( h, k ) − ψ (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) L q TABILITY OF RICCI-FLAT ALE MANIFOLDS 49 = t n ( q − r ) (cid:13)(cid:13)(cid:13) D ψ ( h,k ) Φ( D g Φ( H ( h, k ))) − D ψ (˜ h, ˜ k ) Φ( D ˜ g Φ( H (˜ h, ˜ k ))) (cid:13)(cid:13)(cid:13) L q ≤ t n ( q − r ) (cid:13)(cid:13)(cid:13) ( D ψ ( h,k ) Φ − D ψ (˜ h, ˜ k ) Φ)( D ˜ g Φ( H (˜ h, ˜ k ))) (cid:13)(cid:13)(cid:13) L q + t n ( q − r ) (cid:13)(cid:13)(cid:13) D ψ ( h,k ) Φ( D g Φ − D ˜ g Φ)( H (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) L q + t n ( q − r ) (cid:13)(cid:13)(cid:13) D ψ ( h,k ) Φ( D g Φ( H ( h, k ) − H (˜ h, ˜ k ))) (cid:13)(cid:13)(cid:13) L q ≤ Ct n ( q − r ) (cid:13)(cid:13)(cid:13) H (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L r/ (cid:16)(cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L q + k g − ˜ g k L q (cid:17) + Ct n ( q − r ) (cid:13)(cid:13)(cid:13) H ( h, k ) − H (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) L q ≤ Ct n ( q − r ) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r (cid:16)(cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q (cid:17) + Ct n ( q − r ) (cid:13)(cid:13)(cid:13) H ( h, k ) − H (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) L q ≤ C (cid:18) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) . In the last inequality, we used Lemma 5.21 again and the estimate from the first part of theproof. (cid:3)
Proposition 5.23.
There exists an ǫ > sucn that if (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q,r < ǫ ,we have (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:20) k k k W ,r + k k k W ,q + k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) · (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) Proof.
At first, we have (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q,r ≤ (cid:13)(cid:13)(cid:13) (Π ⊥ h,h ∞ ) − ( ψ ( h, k ) − ψ (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) [(Π ⊥ h,h ∞ ) − − (Π ⊥ ˜ h, ˜ h ∞ ) − ] ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q,r By Lemma 5.11, we know that (cid:13)(cid:13)(cid:13) (Π ⊥ h,h ∞ ) − ( ψ ( h, k ) − ψ (˜ h, ˜ k )) (cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q,r . Furthermore, by Lemma 2.10 (vii) and Proposition 5.10, (cid:13)(cid:13)(cid:13) [(Π ⊥ h,h ∞ ) − − (Π ⊥ ˜ h, ˜ h ∞ ) − ] ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:18) k k k W ,q + k k k W ,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r . Thus, to finish the proof, it suffices to establish the estimate (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) X q ≤ C (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) · (cid:20) k k k W ,r + k k k W ,q + k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) We rewrite this difference as ψ ( h, k ) − ψ (˜ h, ˜ k )= P ( g, h, h ∞ ) → t ◦ Π ⊥ h ∞ ( k ) − P (˜ g, ˜ h, ˜ h ∞ ) → t ◦ Π ⊥ ˜ h ∞ ( k )+ Z t [ P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) − P (˜ g, ˜ h, ˜ h ∞ ) s → t I ( t, s, ˜ k, ˜ h, ˜ h ∞ )] ds = P ( g, h, h ∞ ) → t ◦ Π ⊥ h ∞ ( k ) − P (˜ g, ˜ h, ˜ h ∞ ) → t ◦ Π ⊥ ˜ h ∞ ( k )+ Z t − P ( g, h, h ∞ ) s → t [ I ( t, s, k, h, h ∞ ) − I ( t, s, ˜ k, ˜ h, ˜ h ∞ )] ds + Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ] I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) ds + Z t − t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ] I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) ds + Z tt − [ P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) − P (˜ g, ˜ h, ˜ h ∞ ) s → t I ( t, s, ˜ k, ˜ h, ˜ h ∞ )] ds The terms on the right hand side are estimated in Lemma 5.24, 5.27, 5.29, 5.30, 5.31 and 5.32below. (cid:3)
Lemma 5.24.
We have the estimate (cid:13)(cid:13)(cid:13) P ( g, h, h ∞ ) → t ◦ Π ⊥ h ∞ ( k ) − P (˜ g, ˜ h, ˜ h ∞ ) → t ◦ Π ⊥ ˜ h ∞ ( k ) (cid:13)(cid:13)(cid:13) X q,r ≤ C ( k k k W ,q + k k k W ,r ) (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) ˜ k − k (cid:13)(cid:13)(cid:13) X q,r (cid:19) . For t ≥
5, and any p ′ ∈ (1 , q ] satisfying n (cid:16) p ′ − r (cid:17) = 1, we even have (cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ( P ( g, h, h ∞ ) → t ◦ Π ⊥ h ∞ ( k ) − P (˜ g, ˜ h, ˜ h ∞ ) → t ◦ Π ⊥ ˜ h ∞ ( k )) (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C k k k L p ′ (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) ˜ k − k (cid:13)(cid:13)(cid:13) X q,r (cid:19) . Proof.
We abbreviate, for each s ∈ [1 , t ], ϕ s = P ( g, h, h ∞ ) → s ◦ Π ⊥ h ∞ ( k ) , ˜ ϕ s = P (˜ g, ˜ h, ˜ h ∞ ) → s ◦ Π ⊥ ˜ h ∞ ( k )For t ≤
5, Lemma 3.4 (i), Lemma 2.10 (v), Lemma 5.6 and elliptic regularity show that for q ′ ∈ { q, r } , k ϕ t − ˜ ϕ t k W ,q ′ ≤ C (cid:13)(cid:13)(cid:13) (Π ⊥ h ∞ − Π ⊥ ˜ h ∞ )( k ) (cid:13)(cid:13)(cid:13) W ,q ′ + C sup s ∈ [1 ,t ] (cid:16) k g s − ˜ g s k W , ∞ + (cid:13)(cid:13)(cid:13) h s − ˜ h s (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) (cid:13)(cid:13)(cid:13) Π ⊥ ˜ h ∞ ( k ) (cid:13)(cid:13)(cid:13) W ,q ′ ≤ C k k k W ,q ′ (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) , and the desired estimate of the lemma follows. For t > ϕ t − ϕ t by ϕ t − − ˜ ϕ t − which we do by short-time estimates. The term ϕ t − − ˜ ϕ t − isthen easier to estimate in terms of the initial data because we have by construction ϕ t − = e − ( t − L,h ∞ ◦ Π ⊥ h ∞ ( k ) , ˜ ϕ t − = e − ( t − L, ˜ h ∞ ◦ Π ⊥ ˜ h ∞ ( k ) . TABILITY OF RICCI-FLAT ALE MANIFOLDS 51
Again by Lemma 3.4 (i), we estimate at first k ϕ t − ˜ ϕ t k W ,r ≤ C k ϕ t − − ˜ ϕ t − k W ,r + k ˜ ϕ t − k W ,r sup s ∈ [ t − ,t ] (cid:16) k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W ,r (cid:17)! . For the W ,q -norm, we proceed a little bit differently. In this case, we obtain from Lemma 3.4(i), (iv) and (v) that k ϕ t − ˜ ϕ t k L q ≤ C k ϕ t − − ˜ ϕ t − k L q + k ˜ ϕ t − k L q sup s ∈ [ t − ,t ] (cid:16) k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ (cid:17)! , k∇ ( ϕ t − ˜ ϕ t ) k L q ≤ C ( k∇ ( ϕ t − − ˜ ϕ t − ) k L q + k ϕ t − − ˜ ϕ t − k L r )+ C ( k∇ ˜ ϕ t − k L q + k ˜ ϕ t − k L r ) sup s ∈ [ t − ,t ] (cid:16) k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) , (cid:13)(cid:13) ∇ ( ϕ t − ˜ ϕ t ) (cid:13)(cid:13) L q ≤ C (cid:0)(cid:13)(cid:13) ∇ ( ϕ t − − ˜ ϕ t − ) (cid:13)(cid:13) L q + k∇ ( ϕ t − − ˜ ϕ t − ) k L r k ϕ t − − ˜ ϕ t − k L r (cid:1) + C (cid:0)(cid:13)(cid:13) ∇ ˜ ϕ t − (cid:13)(cid:13) L q + k∇ ˜ ϕ t − k L r + k ˜ ϕ t − k L r (cid:1) · sup s ∈ [ t − ,t ] (cid:16) k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) . By using g = h + k , ˜ g = ˜ h + ˜ k and elliptic regularity for h and ˜ h , we findsup s ∈ [ t − ,t ] (cid:16) k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) , Putting these estimates together and multiplying by t n (cid:16) p ′ − q (cid:17) , we get (cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ( ϕ t − ˜ ϕ t ) (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ( ϕ t − − ˜ ϕ t − ) (cid:13)(cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ˜ ϕ t − (cid:13)(cid:13)(cid:13)(cid:13) X q,r (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) ˜ k − k (cid:13)(cid:13)(cid:13) X q,r (cid:19) ≤ C ((cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ( ϕ t − − ˜ ϕ t − ) (cid:13)(cid:13)(cid:13)(cid:13) X q,r + k k k L p ′ (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) ˜ k − k (cid:13)(cid:13)(cid:13) X q,r (cid:19)) From now on we abbreviate for notational convenience∆ L := ∆ L,h ∞ , ˜∆ L = ∆ L, ˜ h ∞ , Π ⊥ = Π ⊥ h ∞ , ˜Π ⊥ = Π ⊥ ˜ h ∞ . We can rewrite the difference ϕ t − − ˜ ϕ t − as ϕ t − − ˜ ϕ t − = ϕ t − − ˜Π ⊥ ( ϕ t − ) + ˜Π ⊥ ( ϕ t − − ˜ ϕ t − )= (Π ⊥ − ˜Π ⊥ )( ϕ t − ) + (Π ⊥ ˜ h ∞ ,h ∞ ) − ◦ Π ⊥ ( ϕ t − − ˜ ϕ t − ) , where we used hat ϕ t − = Π ⊥ ( ϕ t − ) and ˜ ϕ t − = ˜Π ⊥ ( ˜ ϕ t − ). Let us use the notation ψ t − = Π ⊥ ( ϕ t − − ˜ ϕ t − ) . From Lemma 2.10 (v), we get the estimates k ϕ t − − ˜ ϕ t − k W ,r + ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k ϕ t − k L r + k ψ t − k W ,r (cid:19) , and k ϕ t − − ˜ ϕ t − k L q ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k ϕ t − k L r + k ψ t − k L q (cid:19) , k∇ ( ϕ t − − ˜ ϕ t − ) k L q ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k ϕ t − k L r + k∇ ψ t − k L q + k ψ t − k L r (cid:19) , (cid:13)(cid:13) ∇ ( ϕ t − − ˜ ϕ t − ) (cid:13)(cid:13) L q ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k ϕ t − k L r + (cid:13)(cid:13) ∇ ψ t − (cid:13)(cid:13) L q + k ψ t − k W ,r (cid:19) . By Corollary 5.2, k ϕ t − k L r ≤ Ct − n (cid:16) p ′ − r (cid:17) k k k L p ′ and we conclude (cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ( ϕ t − − ˜ ϕ t − ) (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C k k k L p ′ (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ψ t − (cid:13)(cid:13)(cid:13)(cid:13) X q,r ! . Thus to finish the proof, it suffices to show (cid:13)(cid:13)(cid:13)(cid:13) t n (cid:16) p ′ − q (cid:17) ψ t − (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C k k k L p ′ (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r . We are going to establish this estimate for the remainder of this proof. Because for s ∈ [1 , t − ϕ s and ˜ ϕ s are solutions of the evolution problems ∂ s ϕ s + ∆ L ϕ s = 0 , ϕ = Π ⊥ ( k ) ,∂ s ˜ ϕ s + ˜∆ L ˜ ϕ s = 0 , ˜ ϕ = ˜Π ⊥ ( k ) , the quantity ψ s := Π ⊥ ( ϕ s − ˜ ϕ s ) is a solution of the problem ∂ s ψ s + ∆ L ψ s = Π ⊥ ◦ ( ˜∆ L − ∆ L )( ˜ ϕ s ) , ψ = Π ⊥ ◦ (Π ⊥ − ˜Π ⊥ )( k ) , which is then written by the Duhamel principle as ψ t − = e − ( t − L ψ + Z t − e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds = e − ( t − L ψ + Z e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds + Z t − t − e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds + Z t − e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds. We are now going to estimate these terms separately. At first, we obtain t n (cid:16) p ′ − q (cid:17) (cid:16)(cid:13)(cid:13)(cid:13) e − ( t − L ψ (cid:13)(cid:13)(cid:13) L q + t (cid:13)(cid:13)(cid:13) ∇ e − ( t − L ψ (cid:13)(cid:13)(cid:13) L q + t n ( q − r ) (cid:13)(cid:13)(cid:13) ∇ e − ( t − L ψ (cid:13)(cid:13)(cid:13) L q (cid:17) ≤ C ( t − n (cid:16) p ′ − q (cid:17) (cid:16) (cid:13)(cid:13)(cid:13) e − ( t − L ψ (cid:13)(cid:13)(cid:13) L q + ( t − (cid:13)(cid:13)(cid:13) ∇ e − ( t − L ψ (cid:13)(cid:13)(cid:13) L q + C ( t − n (cid:16) p ′ − q (cid:17) ( t − n ( q − r ) (cid:13)(cid:13)(cid:13) ∇ e − ( t − L ψ (cid:13)(cid:13)(cid:13) L q (cid:17) ≤ C k ψ k L p ′ ≤ C (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k k k L p ′ and similarly t n (cid:16) p ′ − q (cid:17) · t n (cid:16) p ′ − r (cid:17) (cid:13)(cid:13)(cid:13) e − ( t − L ψ (cid:13)(cid:13)(cid:13) W ,r ≤ C ( t − n (cid:16) p ′ − r (cid:17) (cid:13)(cid:13)(cid:13) e − ( t − L ψ (cid:13)(cid:13)(cid:13) W ,r ≤ C k ψ k L p ′ ≤ C (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k k k L p ′ TABILITY OF RICCI-FLAT ALE MANIFOLDS 53 by Corollary 5.2. For the next term Z e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds = e − ( t − L (cid:18)Z e − (2 − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds (cid:19) we need to use Sobolev spaces of negative order to get rid of the second derivatives of ˜ ϕ . Let usfirst abbreviate η = Z e − (2 − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds,η = e − ∆ L η , η t − = e − ( t − L η = e − ( t − L η . Then η t − is the term we wish to estimate. Similarly as above, we have t n (cid:16) p ′ − q (cid:17) (cid:16) k η t − k L q + t k η t − k L q + t n ( q − r ) k η t − k L q (cid:17) ≤ C ( t − n (cid:16) p ′ − q (cid:17) (cid:16) k η t − k L q + ( t − k η t − k L q + ( t − n ( q − r ) k η t − k L q (cid:17) ≤ C k η k L p ′ and similarly t n (cid:16) p ′ − q (cid:17) · t n ( q − r ) k η t − k W ,r ≤ C k η k L p ′ by mapping properties of the heat operator. Now because e − ∆ L extends to bounded maps e − ∆ L : L q ′ → W ,q ′ , ∀ q ′ ∈ (1 , ∞ ) ,e − s ∆ L : W ,q ′ → W ,q ′ , ∀ q ′ ∈ (1 , ∞ ) , s ∈ [0 , , duality implies that it is also a bounded map e − ∆ L : W − ,q ′ → L q ′ , ∀ q ′ ∈ (1 , ∞ ) ,e − s ∆ L : W − ,q ′ → W − ,q ′ , ∀ q ′ ∈ (1 , ∞ ) , s ∈ [0 , . Because Π ⊥ : W ,q ′ → W ,q ′ is bounded and self-adjoint on L , it also admits a boundedextension Π ⊥ : W − ,q ′ → W − ,q ′ . These observations imply k η k L q ≤ C k η k W − ,q = C (cid:13)(cid:13)(cid:13)(cid:13)Z e − (2 − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds (cid:13)(cid:13)(cid:13)(cid:13) W − ,q ≤ C sup s ∈ [1 , (cid:13)(cid:13)(cid:13) ( ˜∆ L − ∆ L )( ˜ ϕ ) (cid:13)(cid:13)(cid:13) W − ,q . Using the tensor T kij = Γ kij − ˜Γ kij , the difference of two Lichnerwicz Laplacians can be written as( ˜∆ L − ∆ L )( χ ) = ( h − − ˜ h − ) ∗ ( ∇ ˜ ϕ + R ∗ ˜ ϕ ) + ˜ h − ∗ ( ∇ T ∗ ˜ ϕ + T ∗ ∇ ˜ ϕ + T ∗ T ∗ ˜ ϕ ) . Let now χ ∈ C ∞ cs ( S M ) be a compactly supported test tensor. Since we have the schematic form T = ( h − − ˜ h − ) ∗ ∇ ( h − ˜ h ), suitable integration by parts yields(( ˜∆ L − ∆ L ) ˜ ϕ, χ ) L = (( h − − ˜ h − ) ∗ ( ∇ ˜ ϕ + R ∗ ˜ ϕ ) , χ ) L + (˜ h − ∗ ( ∇ T ∗ ˜ ϕ + T ∗ ∇ ˜ ϕ + T ∗ T ∗ ˜ ϕ ) , χ )) L ≤ C k ˜ ϕ k L q k χ k W ,q ∗ (cid:13)(cid:13)(cid:13) ˜ h ∞ − h ∞ (cid:13)(cid:13)(cid:13) W , ∞ . Here, q ∗ is the conjugate H¨older exponent of q . Using the definition of negative Sobolev spaces,we obtain (cid:13)(cid:13)(cid:13) ( ˜∆ L − ∆ L )( ˜ ϕ ) (cid:13)(cid:13)(cid:13) W − ,q ≤ C k ˜ ϕ k L q (cid:13)(cid:13)(cid:13) ˜ h ∞ − h ∞ (cid:13)(cid:13)(cid:13) W , ∞ ≤ C k ˜ ϕ k L q (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r , so that k η k L q ≤ C sup s ∈ [1 , (cid:13)(cid:13)(cid:13) ( ˜∆ L − ∆ L )( ˜ ϕ ) (cid:13)(cid:13)(cid:13) W − ,q ≤ C (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z sup s ∈ [1 , k ˜ ϕ k L q ≤ C (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k k k L q . Consequently, (cid:13)(cid:13)(cid:13)(cid:13)Z e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q ≤ C (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k k k L q . Let q ′ = p, r . Then, (cid:13)(cid:13)(cid:13)(cid:13)Z t − t − e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds (cid:13)(cid:13)(cid:13)(cid:13) W ,q ′ ≤ C sup s ∈ [ t − ,t − (cid:13)(cid:13)(cid:13) ( ˜∆ L − ∆ L )( ˜ ϕ s ) (cid:13)(cid:13)(cid:13) W ,q ′ ≤ C (cid:13)(cid:13)(cid:13) h ∞ − ˜ h ∞ (cid:13)(cid:13)(cid:13) W ,q ′ sup s ∈ [ t − ,t − k ˜ ϕ s k W , ∞ ≤ C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r k ˜ ϕ t − k L r ≤ C ( t − − n (cid:16) p ′ − r (cid:17) (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r k k k L p ′ ≤ Ct − n (cid:16) p ′ − r (cid:17) (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r k k k L p ′ . Now we estimate the final term. Let α : { q, r } × { , , } → R be the function from Lemma 5.12.Choose p ′′ ∈ (1 , p ′ ) small and let q ′′ ∈ ( p ′′ , ∞ ) such that p ′′ = q ′′ + r . Under these assumptions,1 = n (cid:18) p ′′ − q (cid:19) + α ( q ′ , i ) > n (cid:18) p ′ − q (cid:19) + α ( q ′ , i )1 = n (cid:18) p ′ − r (cid:19) = n (cid:18) p ′ − q (cid:19) + n (cid:18) q − r (cid:19) ≥ n (cid:18) p ′ − q (cid:19) + α ( q ′ , i ) ,n (cid:18) p ′′ − q (cid:19) + α ( q ′ , i ) + n (cid:18) p ′ − r (cid:19) − > n p ′ − q ) + α ( q ′ , i ) . By the H¨older inequality, Corollary 5.2 and Lemma 1.16, we therefore get (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − e − ( t − − s )∆ L ◦ Π ⊥ ( ˜∆ L − ∆ L )( ˜ ϕ ) ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′ ≤ Z t − (cid:13)(cid:13)(cid:13) ∇ i e − ( t − − s )∆ L ◦ Π ⊥ (cid:13)(cid:13)(cid:13) L p ′′ → L q ′ (cid:13)(cid:13)(cid:13) ˜ h ∞ − h ∞ (cid:13)(cid:13)(cid:13) W ,q ′′ k ˜ ϕ k W ,r ds ≤ C Z t − ( t − − s ) − n (cid:16) p ′′ − q (cid:17) − α ( q ′ ,i ) ( s − − n (cid:16) p ′ − r (cid:17) ds · (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k k k L p ′ ≤ Ct − n (cid:16) p ′ − q (cid:17) − α ( q ′ ,i ) · (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r k k k L p ′ , which finishes the proof. (cid:3) TABILITY OF RICCI-FLAT ALE MANIFOLDS 55
Remark 5.25.
By shifting the time parameter, we also get under the conditions of Lemma 5.24for t > s ≥ t − s ≥ (cid:13)(cid:13)(cid:13) ∇ i ◦ P ( g, h, h ∞ ) s → t ◦ Π ⊥ h ∞ ( k ) − ∇ i ◦ P (˜ g, ˜ h, ˜ h ∞ ) s → t ◦ Π ⊥ ˜ h ∞ ( k ) (cid:13)(cid:13)(cid:13) L q ′ ≤ C ( t − s ) − n (cid:16) p ′ − p (cid:17) − α ( q ′ ,i ) (cid:18)(cid:13)(cid:13)(cid:13) ˜ k − k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) k k k L p ′ , with α being the function of Lemma 5.12.Recall that χ [1 ,t − [ I ( t, s, k, h, h ∞ ) − I ( t, s, ˜ k, ˜ h, ˜ h ∞ )] = χ [1 ,t − [∆ L,h ∞ − ∆ L,h )( k ) − (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k )]+ χ [1 ,t − [(1 − D g Φ)( H ) − (1 − D ˜ g Φ)( ˜ H )] , so that Z t − P ( g, h, h ∞ ) s → t [ I ( t, s, k, h, h ∞ ) − I ( t, s, ˜ k, ˜ h, ˜ h ∞ )] ds = Z t − P ( g, h, h ∞ ) s → t [∆ L,h ∞ − ∆ L,h )( k ) − (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k )] ds + Z t − P ( g, h, h ∞ ) s → t [(1 − D g Φ)( H ) − (1 − D ˜ g Φ)( ˜ H )] ds We deal with these two terms in the next four lemmas.
Lemma 5.26.
We have for every p ′ ∈ (1 , r ] (cid:13)(cid:13)(cid:13) (∆ L,h ∞ − ∆ L,h )( k ) − (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k ) (cid:13)(cid:13)(cid:13) L p ′ ≤ Cs − n ( q − r ) (cid:18) k h k Z q,r (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) Proof.
We first rewrite(∆
L,h ∞ − ∆ L,h )( k ) − (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k ) = (∆ L,h ∞ − ∆ L, ˜ h ∞ )(˜ k ) + (∆ L,h − ∆ L, ˜ h )(˜ k )+ (∆ L,h ∞ − ∆ L,h )( k − ˜ k )Standard estimates using the H¨older inequality imply (cid:13)(cid:13)(cid:13) (∆ L,h ∞ − ∆ L, ˜ h ∞ )(˜ k ) (cid:13)(cid:13)(cid:13) L p ′ + (cid:13)(cid:13)(cid:13) (∆ L,h − ∆ L, ˜ h )(˜ k ) (cid:13)(cid:13)(cid:13) L p ′ ≤ C (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r ≤ C · s − n ( q − r ) (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r and it is shown as in (39) that (cid:13)(cid:13)(cid:13) (∆ L,h ∞ − ∆ L,h )( k − ˜ k ) (cid:13)(cid:13)(cid:13) L p ′ ≤ C · s − n ( q − r ) − n ( q − r ) k h k Z q,r (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r ≤ C · s − n ( q − r ) k h k Z q,r (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r , which yields the desired result. (cid:3) Lemma 5.27.
We have (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t [(∆ L,h ∞ − ∆ L,h )( k ) − (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k )] ds (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) (cid:18) k h k Z q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) Proof.
Let α : { q, r } × { , , } → R be the function from Lemma 5.12, q ′ ∈ { q, r } and choose p ′ ∈ (1 , q ) small. Then we have 1 = n (cid:18) p ′ − q (cid:19) + α ( q ′ , i ) > α ( q ′ , i ) , = n q − r ) ≥ α ( q ′ , i ) ,n (cid:18) p ′ − q (cid:19) + α ( q ′ , i ) + n (cid:18) q − r (cid:19) > α ( q ′ , i ) , and Lemma 5.12, Lemma 5.26 and Lemma 1.16 yield (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − P ( g, h, h ∞ ) s → t ◦ Π ⊥∞ [(∆ L,h ∞ − ∆ L,h )( k ) − (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k )] ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′ ≤ Z t − (cid:13)(cid:13) ∇ i ◦ P ( g, h, h ∞ ) s → t ◦ Π ⊥∞ (cid:13)(cid:13) L p ′ ,L q ′ (cid:13)(cid:13)(cid:13) (∆ L,h ∞ − ∆ L,h )( k ) − (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k ) (cid:13)(cid:13)(cid:13) L p ′ ds ≤ Z t − ( t − s ) − n (cid:16) p ′ − q (cid:17) − α ( q ′ ,i ) s − n ( q − r ) ds (cid:18) k h k Z q,r (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) ≤ Ct − α ( q ′ ,i ) (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) (cid:18) k h k Z q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:19) , as desired. (cid:3) Lemma 5.28.
We have (cid:13)(cid:13)(cid:13) (1 − D g Φ)( H ) − (1 − D ˜ g Φ)( ˜ H ) (cid:13)(cid:13)(cid:13) L q ≤ Ct − n ( q − r ) (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) · (cid:20) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) Proof.
We first write(1 − D g Φ)( H ) − (1 − D ˜ g Φ)( ˜ H ) = H − ˜ H + ( D g Φ − D ˜ g Φ)( H ) + D ˜ g Φ( ˜ H − H )By Lemma 5.21, we already know (cid:13)(cid:13)(cid:13) H − ˜ H (cid:13)(cid:13)(cid:13) L q ≤ Ct − n ( q − r ) (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) (cid:20) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) . Moreover by Lemma 2.10 (iii), (cid:13)(cid:13)(cid:13) D ˜ g Φ( ˜ H − H ) (cid:13)(cid:13)(cid:13) L q ≤ (cid:13)(cid:13)(cid:13) ˜ H − H (cid:13)(cid:13)(cid:13) L q ≤ Ct − n ( q − r ) (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) (cid:20) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) . Finally by Lemma 2.10 (vi), k ( D g Φ − D ˜ g Φ)( H ) k L q ≤ C (cid:16)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L q (cid:17) k H k L r ≤ C (cid:16)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) L q (cid:17) k k k W ,r ≤ Ct − n ( q − r ) (cid:18)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q (cid:19) k k k X q . TABILITY OF RICCI-FLAT ALE MANIFOLDS 57
This finishes the proof of the lemma. (cid:3)
Lemma 5.29.
We have (cid:13)(cid:13)(cid:13)(cid:13)Z t − P ( g, h, h ∞ ) s → t ◦ Π ⊥∞ [(1 − D g Φ)( H ) − (1 − D ˜ g Φ)( ˜ H )] ds (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:20)(cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r + (cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) (cid:20) k k k X q,r + (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r (cid:21) Proof.
Let α : { q, r } × { , , } → R be the function from Lemma 5.12 and q ′ ∈ { q, r } . Then weget by Lemma 5.28 that (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − P ( g, h, h ∞ ) s → t ◦ Π ⊥∞ [(1 − D g Φ)( H ) − (1 − D ˜ g Φ)( ˜ H )] ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′ ≤ Z t − (cid:13)(cid:13) P ( g, h, h ∞ ) s → t ◦ Π ⊥∞ (cid:13)(cid:13) L q ,L q ′ (cid:13)(cid:13)(cid:13) (1 − D g Φ)( H ) − (1 − D ˜ g Φ)( ˜ H ) (cid:13)(cid:13)(cid:13) L q ds ≤ C Z t − ( t − s ) − α ( q ′ ,i ) s − n ( q − r ) ds ≤ C · t − α ( q ′ ,i ) . The last inequality here follows from Lemma 1.16 and n (cid:18) q − r (cid:19) > , n (cid:18) q − r (cid:19) > n (cid:18) q − r (cid:19) ≥ α ( q ′ , i ) . The result is immediate from the definition of the norm. (cid:3)
Lemma 5.30.
We have (cid:13)(cid:13)(cid:13)(cid:13)Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ] I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r Proof.
We split up Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ] I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) ds = Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ][(∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k )] ds + Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ][(1 − D ˜ g Φ)( ˜ H )] ds Let us now estimate the term Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ][(∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k )] ds Let α : { q, r } × { , , } → R be the function from Lemma 5.12, q ′ ∈ { q, r } and p ′ ∈ (1 , q ) small.Then we can use Lemma 5.13, Lemma 5.24 and Remark 5.25 to obtain (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ][(∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k )] ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′ ≤ Z t − (cid:13)(cid:13)(cid:13) ∇ i [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ] (cid:13)(cid:13)(cid:13) L p ′ ,L q ′ (cid:13)(cid:13)(cid:13) (∆ L, ˜ h ∞ − ∆ L, ˜ h )(˜ k ) (cid:13)(cid:13)(cid:13) L p ′ ds ≤ C Z t − ( t − s ) − n (cid:16) p ′ − q (cid:17) − α ( q ′ ,i ) s − n ( q − r ) ds (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r k h k X q,r ≤ Ct − α ( q ′ ,i ) (cid:13)(cid:13)(cid:13) ˜ h − h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r k h k X q,r The last inequality is justified by Lemma 1.16, since n (cid:18) p ′ − q (cid:19) + α ( q ′ , i ) > α ( q ′ , i ) , n (cid:18) q − r (cid:19) − > n (cid:18) q − r (cid:19) ≥ α ( q ′ , i ) ,n (cid:18) p ′ − q (cid:19) + α ( q ′ , i ) + 3 n (cid:18) q − r (cid:19) − n (cid:18) p ′ − r (cid:19) + α ( q ′ , i ) − > α ( q ′ , i ) . For the term Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ][(1 − D ˜ g Φ)( ˜ H )] ds, we have, using Lemma 5.24 and Remark 5.25, (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ][(1 − D ˜ g Φ)( ˜ H )] ds (cid:13)(cid:13)(cid:13)(cid:13) L q ′ ≤ Z t − (cid:13)(cid:13)(cid:13) ∇ i ( P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ) (cid:13)(cid:13)(cid:13) L q ,L q ′ (cid:13)(cid:13)(cid:13) (1 − D ˜ g Φ)( ˜ H ) (cid:13)(cid:13)(cid:13) L q ds ≤ C Z t − ( t − s ) − α ( q ′ ,i ) s − n ( q − r ) ds (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r ≤ Ct − α ( q ′ ,i ) (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r , and the last inequality follows from Lemma 1.16 since n (cid:18) q − r (cid:19) > , n (cid:18) q − r (cid:19) > n (cid:18) q − r (cid:19) ≥ α ( q ′ , i ) . This finishes the proof. (cid:3)
Lemma 5.31.
We have (cid:13)(cid:13)(cid:13)(cid:13)Z t − t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ] I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r Proof.
Let q ′ = { q, r } . Lemma 3.4 (iii), elliptic regularity and Sobolev embedding yield (cid:13)(cid:13)(cid:13)(cid:13)Z t − t − [ P ( g, h, h ∞ ) s → t − P (˜ g, ˜ h, ˜ h ∞ ) s → t ] I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13) W ,q ′ ≤ C sup s ∈ [ t − ,t ] (cid:16) k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) (cid:13)(cid:13)(cid:13) I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) (cid:13)(cid:13)(cid:13) L q ′ ≤ C sup s ∈ [ t − ,t ] (cid:16)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W ,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q (cid:17) (cid:13)(cid:13)(cid:13) I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) (cid:13)(cid:13)(cid:13) L q ′ ≤ C (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) sup s ∈ [ t − ,t ] (cid:13)(cid:13)(cid:13) I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) (cid:13)(cid:13)(cid:13) L q ′ TABILITY OF RICCI-FLAT ALE MANIFOLDS 59
Exacly as in the proof of Lemma 5.18, we get the estimatesup s ∈ [ t − ,t ] (cid:13)(cid:13)(cid:13) I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) (cid:13)(cid:13)(cid:13) L q ′ ≤ Ct − n ( q − r ) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r , which yields the result by definition of the norm. (cid:3) Lemma 5.32.
We have the estimate (cid:13)(cid:13)(cid:13)(cid:13)Z tt − [ P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) − P (˜ g, ˜ h, ˜ h ∞ ) s → t I ( t, s, ˜ k, ˜ h, ˜ h ∞ )] ds (cid:13)(cid:13)(cid:13)(cid:13) X q,r ≤ C (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r Proof.
By Lemma 3.4 (ii), we get for q ′ = { q, r } that (cid:13)(cid:13)(cid:13)(cid:13)Z tt − [ P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) − P (˜ g, ˜ h, ˜ h ∞ ) s → t I ( t, s, ˜ k, ˜ h, ˜ h ∞ )] ds (cid:13)(cid:13)(cid:13)(cid:13) W ,q ′ ≤ C sup s ∈ [ t − ,t ] (cid:16) k g − ˜ g k W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) W , ∞ (cid:17) (cid:13)(cid:13)(cid:13) I ( t, s, ˜ k, ˜ h, ˜ h ∞ ) (cid:13)(cid:13)(cid:13) W ,q ′ ≤ C sup s ∈ [ t − ,t ] (cid:16)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W , ∞ + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q (cid:17) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,q ′ (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W , ∞ ≤ C sup s ∈ [ t − ,t ] (cid:16)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) W ,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) L q (cid:17) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,q ′ (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) W ,r ≤ Ct − n ( q − r ) (cid:18)(cid:13)(cid:13)(cid:13) k − ˜ k (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) h − ˜ h (cid:13)(cid:13)(cid:13) Z q,r (cid:19) (cid:13)(cid:13)(cid:13) ˜ k (cid:13)(cid:13)(cid:13) X q,r which proves the statement. (cid:3) Long-time existence and convergence
In this section, we are going to prove the main results of the paper. Throughout the section,let the metric ˆ h satisfy the assumptions of the Theorems.6.1. Establishing a fixed point of the iteration map.Definition 6.1.
Let
U ⊂ M be a neighbourhood of ˆ h on which the map Φ of Subsection 2.2 isdefined. We call a family of metrics g t , t ∈ [0 , ∞ ) in U a modified Ricci-de Turck flow startingat g if g t satisfies ∂ t g t = − g t + L V ( g t , Φ( g t )) g t , t > ,∂ t g t = − g t + L V ( g t , ˆ h ) g t , t ∈ [0 , . In other words, for t ∈ [0 , g t evolves under the Ricci-de Turck flow with reference metric ˆ h while for t > g t evolves under the Ricci-de Turck flow with moving reference metric Φ( g t ). Theorem 6.2.
Let q ∈ (1 , n ) and r ∈ ( n, ∞ ) so large that n (cid:16) q − r (cid:17) > and n (cid:16) q − r (cid:17) = 1 .Then for any ǫ > , we can choose δ > so small that if a metric g satisfies (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L ∞ < δ, the modified Ricci-de Turck flow g t starting at g is well-defined, exists for all t ≥ and suchthat for t ≥ , the tensors h t := Φ( g t ) , k t := g t − h t satisfy (cid:13)(cid:13)(cid:13)(cid:16) h t − ˆ h, k t (cid:17)(cid:13)(cid:13)(cid:13) Y q,r < ǫ. Proof.
For any given ǫ >
0, Lemma 3.8 enables us to choose δ > g t with background metric ˆ h with initial data g exists up to time t = 1 and satisfies (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) W ,q + (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) W , ∞ < ǫ . Due to interpolation, this also implies (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) W ,r < ǫ for r ∈ ( q, ∞ ). For any given ǫ >
0, we may choose ǫ > M ⊃ U → F from Subsection 2.2 can be applied to g and such that the tensors h := Φ( g ) , k := g − Φ( g )satisfy k k k W ,q + k k k W ,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q < ǫ We now define for t ≥ h (1) t := h , k (1) t := e − ( t − L,h k . It follows from Lemma 5.12 (applied to the special case where g = h = h ∞ are all equal to h )that (cid:13)(cid:13)(cid:13) k (1) t (cid:13)(cid:13)(cid:13) X q,r ≤ C ( k k k W ,q + k k k W ,r )and it is clear from the definition of the norm that (cid:13)(cid:13)(cid:13) h (1) t − ˆ h (cid:13)(cid:13)(cid:13) Z q,r = (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z q,r = (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q . Therefore, for any given ǫ >
0, we may choose ǫ > (cid:13)(cid:13)(cid:13)(cid:16) h (1) t − ˆ h, k (1) t (cid:17)(cid:13)(cid:13)(cid:13) Y q,r < ǫ . Inductively, we define the tuple (cid:16) h ( i +1) t , k ( i +1) t (cid:17) := ψ (cid:16) h ( i ) , k ( i ) (cid:17) . Now we claim that we can choose ǫ and ǫ so small that (cid:13)(cid:13)(cid:13)(cid:16) h ( i ) t − ˆ h, k ( i ) t (cid:17)(cid:13)(cid:13)(cid:13) Y q,r < ǫ for all i ∈ N . We prove this by induction on i . The claim obivously holds for i = 1. Now observethat due to Theorem 5.8, there exists an ǫ such that the estimate (cid:13)(cid:13)(cid:13) ψ ( h, k ) − (ˆ h, (cid:13)(cid:13)(cid:13) Y q,r ≤ C (cid:18) k k k W ,q + k k k W ,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r (cid:19) (41)holds for some constant C >
0, as long as (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ . If we choose ǫ and ǫ suchthat ǫ < ǫ C , ǫ < min (cid:26) C , ǫ (cid:27) , then the induction assumption implies (cid:13)(cid:13)(cid:13)(cid:16) h ( i ) − ˆ h, k ( i ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r < ǫ < ǫ . TABILITY OF RICCI-FLAT ALE MANIFOLDS 61
Now (41) implies (cid:13)(cid:13)(cid:13)(cid:16) h ( i +1) − ˆ h, k ( i +1) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r = (cid:13)(cid:13)(cid:13) ψ (cid:16) h ( i ) , k ( i ) (cid:17) − (cid:16) ˆ h, (cid:17)(cid:13)(cid:13)(cid:13) Y q,r ≤ C (cid:18) k k k W ,q + k k k W ,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) X q,r + (cid:13)(cid:13)(cid:13)(cid:16) h ( i ) − ˆ h, k ( i ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r (cid:19) ≤ C ( ǫ + ǫ ) ≤ C ǫ + ( C ǫ ) ǫ ≤ ǫ ǫ ǫ and the claim is shown by induction.Now due to Theorem 5.19, there exists an ǫ > ψ satisfies the estimate (cid:13)(cid:13)(cid:13) ψ ( h, k ) − ψ (˜ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r ≤ C ( k k k W ,q + k k k W ,r ) (cid:13)(cid:13)(cid:13) ( h − ˜ h, k − ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r + C (cid:18)(cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r (cid:19) (cid:13)(cid:13)(cid:13) ( h − ˜ h, k − ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r as long as (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13) (˜ h − ˆ h, ˜ k ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ . If we now choose ǫ and ǫ so small that C ǫ + 2 C ǫ ≤ , ǫ ≤ ǫ , we obtain (cid:13)(cid:13)(cid:13)(cid:16) h ( i +2) − h ( i +1) , k ( i +2) − k ( i +1) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r ≤ C (cid:18) k k k W ,q + k k k W ,r + (cid:13)(cid:13)(cid:13)(cid:16) h ( i +1) − ˆ h, k ( i +1) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r + (cid:13)(cid:13)(cid:13)(cid:16) h ( i ) − ˆ h, k ( i ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r (cid:19) · (cid:13)(cid:13)(cid:13)(cid:16) h ( i +1) − h ( i ) , k ( i +1) − k ( i ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r ≤ ( C ǫ + 2 C ǫ ) (cid:13)(cid:13)(cid:13)(cid:16) h ( i +1) − h ( i ) , k ( i +1) − k ( i ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r ≤ (cid:13)(cid:13)(cid:13)(cid:16) h ( i +1) − h ( i ) , k ( i +1) − k ( i ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r for all i ∈ N . Thus by induction, the sequence n(cid:16) h ( i ) , k ( i ) (cid:17)o i ∈ N is a Cauchy sequence in Y q,r . By construction, it converges to an element (cid:0) h ( ∞ ) , k ( ∞ ) (cid:1) ∈ Y q,r which satisfies ψ (cid:16) h ( ∞ ) , k ( ∞ ) (cid:17) = (cid:16) h ( ∞ ) , k ( ∞ ) (cid:17) and thus is by construction the (unique) fixed point of ψ . In additon, if for the ǫ > ǫ is chosen so small that ǫ < ǫ , we get (cid:13)(cid:13)(cid:13)(cid:16) h ( ∞ ) − ˆ h, k ( ∞ ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r ≤ lim i →∞ (cid:13)(cid:13)(cid:13)(cid:16) h ( i ) − ˆ h, k ( i ) (cid:17)(cid:13)(cid:13)(cid:13) Y q,r ≤ ǫ < ǫ. By the discussion in Section 4, (cid:16) g ( ∞ ) t = h ( ∞ ) t + k ( ∞ ) t (cid:17) t ≥ (which is for each fixed time an elementin W ,q ∩ W ,r ⊂ W , ∞ ) is a (weak) solution of the Ricci-de Turck flow with moving gauge,starting at g . On the other hand, a solution of the Ricci-de Turck flow with moving gauge g is uniquely obtained by solving the ˆ h -gauged Ricci-de Turck flow and pulling back by a suitablefamily of diffeomorphisms. By construction, the resulting flow ( g t ) t ≥ is W ,q ∩ W ,r -close toˆ h at least for small times [ t, t + ǫ ]. By uniqueness, g t = g ( ∞ ) t as long as g t does not leave a small neighbourhood. A bootstrapping argument then implies that g t = g ( ∞ ) t for all time whichfinishes the proof. (cid:3) Lemma 6.3.
Let g t ⊂ U , t ∈ [0 , T ] be a solution of the Ricci-de Turck flow with moving gauge h t = Φ( g t ) and k t := g t − h t . Then for all r ∈ ( n, ∞ ), there exists an ǫ > k k t k W ,r < ǫ ∀ t ∈ [0 , T ] , k k k L p < ∞ for some p ∈ [ p , ∞ ) , then for every m ∈ N there exists a constant C m = C ( m, ǫ, h, T, p ) (but independent from p )such that k∇ m k t k L p ≤ C m · t − m/ · ( k k k L p + sup s ∈ [0 ,t ] k k s k W ,r ) . Sketch of proof.
This is very similar to the proof of Lemma 3.8. We only have to deal with theadditional term D g Φ( g − ∗ g − ∗ ∇ k ∗ ∇ k + g − ∗ g ∗ R ∗ k + ∇ (( g − − h − ) ∗ ∇ k )) , in the evolution equation for k . However, a combination of Lemma 2.10 (iii) and standardestimates shows that (cid:13)(cid:13) ∇ m ◦ D g Φ( g − ∗ g − ∗ ∇ k ∗ ∇ k + g − ∗ g ∗ R ∗ k + ∇ (( g − − h − ) ∗ ∇ k )) (cid:13)(cid:13) L p ≤ C · k k k W ,r for all m ∈ N . The result then follows again from an application of Theorem 3.2. Note that the L ∞ -norm of k is small by assumption due to Sobolev embedding. (cid:3) Optimal convergence rates of the modified Ricci-de Turck flow.Proposition 6.4.
Assume the same as in Theorem 6.2 and additionally, (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L p < ∞ , for some p ∈ (1 , q ) . Then the tensors h t , k t , t ≥ in Theorem 6.2 satisfy ( h, k ) ∈ Y p,r ′ for every r ′ ∈ [ r, ∞ ) .Proof. We are first going to show that ( h, k ) ∈ Y p,r . For this purpose, let h , k be as in theprevious proof. Recall that we have k k k W ,q + k k k W ,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q < ǫ . short-time estimates under the Ricci-de Turck flow (Lemma 3.8) yields (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) W ,p < ∞ and therefore, k k k L p ≤ (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) W ,p + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) W ,p < ∞ . By interpolation, k k k W ,q ′ < ∞ for all q ′ ∈ [ p, q ]. Let now q = q > q > . . . > q N = p be afinite sequence of H¨older exponents satisfying n (cid:18) q i − q i − (cid:19) ≤ min (cid:26) n r , n (cid:18) q − r (cid:19) − , (cid:27) , for all i ∈ { , . . . , N } . As q ≥ q i − , n (cid:18) q i − q i − (cid:19) ≤ min (cid:26) n r , n (cid:18) q i − − r (cid:19) − , (cid:27) , TABILITY OF RICCI-FLAT ALE MANIFOLDS 63
A repetitive application of Propostion 5.9 and Proposition 5.10 applied to h = ψ ( h, k ) and k = ψ ( h, k ) yields k k k X qi,r ≤ C (cid:20) k k k W ,qi − + k k k W ,r + (cid:18) k k k X qi − ,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z qi − ,r (cid:19) k k k X qi − ,r (cid:21) , (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z qi,r ≤ C (cid:16)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L qi + k k k X qi,r (cid:17) ≤ C (cid:16)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L q + k k k X qi,r (cid:17) . and after a finite number of steps, we obtain (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y p,r = (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r + k k k X p,r < ∞ , as desired.Now we are goint to show that ( h, k ) ∈ Y p,r ′ for any r ′ ∈ [ r, ∞ ). The argumentation is similarto the above but slightly more involved. Pick a finite sequence r = r < r < . . . < r N = r ′ suchthat min (cid:26) n (cid:18) p − r (cid:19) − , (cid:27) > n (cid:18) r i − − r i (cid:19) . Note that this also impliesmin (cid:26) n (cid:18) p − r i − (cid:19) − , (cid:27) > n (cid:18) r i − − r i (cid:19) , as r i − ≥ r . Note that by interpolation, k k k W ,ri < ∞ . Let us first show that k k k X p,ri ≤ C (cid:20) k k k W ,p + k k k W ,ri + (cid:18) k k k X p,ri − + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − (cid:19) k k k X p,ri − (cid:21) . To do so we estimate for k = ψ ( h, k ), using Lemma 5.11 and the triangle inequality, k ψ ( h, k ) k X p,ri ≤ (cid:13)(cid:13) ψ ( h, k ) (cid:13)(cid:13) X p,ri ≤ (cid:13)(cid:13) P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) (cid:13)(cid:13) X p,ri + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z max { ,t − } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X p,ri + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t max { ,t − } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X p,ri . Lemma 5.12 yields (cid:13)(cid:13) P ( g, h, h ∞ ) → t (Π ⊥ h ∞ ( k )) (cid:13)(cid:13) X p,ri ≤ C ( k k k W ,p + k k k W ,ri )and Lemma 5.15 yields (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z max { ,t − } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X p,ri ≤ C (cid:18) k k k X p,ri − + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − (cid:19) k k k X p,ri − . Now due to the definition of the norms, we have for( A ) := Z t max { ,t − } P ( g, h, h ∞ ) s → t I ( t, s, k, h, h ∞ ) ds the inequality k ( A ) k X p,ri ≤ k ( A ) k X p,ri − + sup t ≥ t n (cid:16) p − ri (cid:17) k ( A ) k W ,ri . From Lemma 5.18, we know already that k ( A ) k X p,ri − ≤ C (cid:18) k k k X p,ri − + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − (cid:19) k k k X p,ri − , and thus it suffices to consider the second term. From Lemma 3.3 (iii) and Lemma 5.17, we get k ( A ) k W ,ri ≤ C sup s ∈ [max { ,t − } ,t ] k I ( t, s, k, h, h ∞ ) k W ,ri + C sup s ∈ [max { ,t − } , max { ,t − } ] k I ( t, s, k, h, h ∞ ) k L ri ≤ C sup s ∈ [max { ,t − } ,t ] ( k k k W ,ri + k h − h ∞ k L p ) k k k W ,ri . Let us distinguish between large times and small times. For t ≤
3, shortime existence resultsyield sup s ∈ [max { ,t − } ,t ] ( k k k W ,ri + k h − h ∞ k L p ) k k k W ,ri ≤ C (cid:18) k k k W ,ri + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − (cid:19) k k k W ,ri . Now let us consider large times t ≥
3. By Sobolev embedding and smoothing Lemma 6.3, wehave sup s ∈ [ t − ,t ] k k k W ,ri ≤ C k k t − k W ,ri ≤ C k k t − k W ,ri − ≤ C k k t − k L ri − ≤ Ct − n (cid:16) p − ri − (cid:17) k k k X p,ri − . Since p < r i − , we getsup s ∈ [ t − ,t ] k k k W ,ri ≤ Ct − n (cid:16) p − ri − (cid:17) k k k X p,ri − ≤ Ct − n (cid:16) ri − − ri (cid:17) k k k X p,ri − and since n (cid:16) p − r i − (cid:17) − > n (cid:16) r i − − r i (cid:17) , we get in addition k h − h ∞ k L p ≤ Ct − n (cid:16) p − ri − (cid:17) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − ≤ Ct − n (cid:16) ri − − ri (cid:17) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − . Combining these results, we get in all cases k ( A ) k W ,ri ≤ Ct − n (cid:16) p − ri (cid:17) (cid:18) k k k X p,ri − + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − (cid:19) k k k X p,ri − . Thus we have shown k k k X p,ri ≤ C (cid:20) k k k W ,p + k k k W ,ri + (cid:18) k k k X p,ri − + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri − (cid:19) k k k X p,ri − (cid:21) < ∞ and from Proposition 5.9, we have (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,ri ≤ C (cid:16)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L p + k k k X p,ri (cid:17) ≤ C (cid:16)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) L p + k k k X p,i (cid:17) < ∞ . Thus after a finite number of steps, we get (cid:13)(cid:13)(cid:13) ( h − ˆ h, k ) (cid:13)(cid:13)(cid:13) Y p,r ′ = (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ + k k k X p,r ′ < ∞ , as desired (cid:3) TABILITY OF RICCI-FLAT ALE MANIFOLDS 65
Corollary 6.5.
Under the conditions of Proposition 6.4, there exists for each τ > a constant C = C ( τ ) such that k h t − h ∞ k C k ≤ C · t − np + τ , k g t − h t k C k ≤ C · t − n p + τ for the tensors h t , k t , t ≥ of Theorem 6.2.Proof. We get from Proposition 6.4 that (cid:13)(cid:13)(cid:13) ( h t − ˆ h, k t ) (cid:13)(cid:13)(cid:13) Y p,r ′ < ∞ , for all r ′ ∈ [ r, ∞ ). Let τ > r ′ so large that n r ′ < nr ′ < τ . Then we get k h t − h ∞ k C k ≤ C k h t − h ∞ k L p ≤ C Z ∞ t k ∂ s h k L p ds ≤ C Z ∞ t s − n ( p − r ′ ) ds (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ ≤ Ct − n ( p − r ′ ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ ≤ Ct − np + τ (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ and by Lemma 6.3, we get k k t k C k ≤ C k k t − k L ∞ ≤ C k k t − k W ,r ′ ≤ t − n (cid:16) p ′ − r ′ (cid:17) k k k X p ′ ,r ′ ≤ Ct − n p + τ k k k X p ′ ,r ′ , and the proof of the theorem is finished. (cid:3) For proving a refinement of the decay of k t , let us fix some time t ≥
1. Define k s = Π ⊥ h s ,h t ( k s ) , s ∈ [1 , t ] . By Proposition 4.2, the evolution on k can be written as ∂ s k s + ∆ L,h t k s = Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] . Observe also that k ( t ) = Π ⊥ h ( t ) ,h ( t ) ( k ( t )) = k ( t ). Therefore, by the Duhamel principle we get analternative formula for k ( t ) which is k t = k t = e − ( t − L,ht k + Z t e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] ds. (42)To obtain estimates for this expression, we have to derive estimates on the integrand, which wasdone in Lemma 5.13 and Lemma 5.14. These estimates will enable us to control the part of theintegral from 1 to max { , t − } . To treat also the part from max { , t − } to t , we need anotherlemma. Lemma 6.6.
For r ′ ∈ [ r, ∞ ) and q ′ ∈ [ p, r ′ ], i ∈ N and t ≥
1, we havesup s ∈ [max { ,t − } ,t ] k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k W i,q ′ ≤ Ct − n ( p − r ′ ) − n (cid:16) p − q ′ (cid:17) (cid:18)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ + k k k X p,r ′ (cid:19) k k k X p,r ′ . Proof.
At first, sup s ∈ [max { ,t − } ,t ] k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k W i,q ′ ≤ C sup s ∈ [max { ,t − } ,t ] ( k h t − h s k W i +2 , ∞ + k k s k W i +2 , ∞ ) k k s k W i +2 ,q ′ . From the end of the proof of Proposition 6.5, we already know thatsup s ∈ [max { ,t − } ,t ] k k s k W i +2 , ∞ ≤ Ct − n ( p − r ′ ) k k k X p,r ′ . In addition,sup s ∈ [max { ,t − } ,t ] k h t − h s k W i +2 , ∞ ≤ sup s ∈ [max { ,t − } ,t ] Z ts k ∂ s ′ h k L p ds ′ ≤ C sup s ∈ [max { ,t − } ,t ] Z ts ( s ′ ) − n ( p − r ′ ) ds ′ · (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ ≤ Ct − n ( p − r ′ ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ ≤ Ct − n ( p − r ′ ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ . Let Θ ∈ [0 ,
1] be defined by the equation1 p − q ′ = Θ (cid:18) p − r ′ (cid:19) . By interpolation,sup s ∈ [max { ,t − } ,t ] k k s k W i +2 ,q ′ ≤ C sup s ∈ [max { ,t − } ,t ] k k s k − θW i +2 ,p k k s k θW i +2 ,r ′ ≤ C k k t − k − θL p k k t − k θL r ′ ≤ C k k k − θX p,r ′ (cid:16) ( t − − n ( p − r ′ ) k k k X p,r ′ (cid:17) θ ≤ Ct − n (cid:16) p − q ′ (cid:17) k k k X p,r ′ , and the statement follows from putting these estimates together. (cid:3) Proposition 6.7.
Assume the same as in Proposition 6.4. Let k t be as there, q ′ ∈ [ p, ∞ ) and i ∈ N . (i) If n (cid:16) p − q ′ (cid:17) + i < n p there exists a constant C = C ( p, q, i ) such that for all t > , wehave (cid:13)(cid:13) ∇ i k t (cid:13)(cid:13) L q ′ ≤ Ct − n (cid:16) p − q ′ (cid:17) − i (43)(ii) If n (cid:16) p − q ′ (cid:17) + i ≥ n p there exists for each t > and τ > a constant C = C ( p, q, i, t , τ ) such that for all t > t , we have (cid:13)(cid:13) ∇ i k t (cid:13)(cid:13) L q ′ ≤ Ct − n p + τ . (44) Proof.
We have, using (42) and the triangle inequality, (cid:13)(cid:13) ∇ i k t (cid:13)(cid:13) L q ′ ≤ (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − L,ht k (cid:13)(cid:13)(cid:13) L q ′ + Z max { ,t − } (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] (cid:13)(cid:13)(cid:13) L q ′ ds + Z t max { ,t − } (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] (cid:13)(cid:13)(cid:13) L q ′ ds. TABILITY OF RICCI-FLAT ALE MANIFOLDS 67
Let α = α ( p, q ′ , i, ǫ ) > p ′ ∈ (1 , p ) small and r ∈ ( n, ∞ ) lrage. Furthermore, we denote α ′ = α ( p ′ , q ′ , i, ǫ ) >
0. Note that α ′ − α = n ( p ′ − p ) in all cases. By Corollary 5.2, we already know (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − L,ht k (cid:13)(cid:13)(cid:13) L q ′ ≤ Ct − α (cid:13)(cid:13) k (cid:13)(cid:13) L p ≤ Ct − α k k k L p . Therefore, by Lemma 5.13, we get Z max { ,t − } (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] (cid:13)(cid:13)(cid:13) L q ′ ds ≤ Z max { ,t − } (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − s )∆ L,ht Π ⊥ h t (cid:13)(cid:13)(cid:13) L p ′ ,L q ′ k (∆ L,h t − ∆ L,h s )( k s ) k L p ′ ds ≤ C Z max { ,t − } ( t − s ) − α ′ s − n ( p − r ) ds · ( (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r + k k k Z p,r ) k k k Z p,r We have α ′ = n (cid:18) p ′ − p (cid:19) + α > α n (cid:18) p − r (cid:19) − > n (cid:18) p − r (cid:19) > αα ′ + 3 n (cid:18) p − r (cid:19) − α + n (cid:18) p ′ − p (cid:19) + 3 n (cid:18) p − r (cid:19) − α + n (cid:18) p ′ − r (cid:19) + n (cid:18) p − r (cid:19) − > α so that Lemma 1.16 implies Z max { ,t − } ( t − s ) − α ′ s − n ( p − r ′ ) ds ≤ Ct − α . From Corollary 5.2 and Lemma 5.14, we have (with β = min n , n (cid:16) p − r ′ (cid:17)o ) that Z max { ,t − } (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(1 − D g s Φ)( H ( s ))] (cid:13)(cid:13)(cid:13) L q ′ ds ≤ Z max { ,t − } (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − s )∆ L,ht Π ⊥ h t (cid:13)(cid:13)(cid:13) L p ,L q ′ k (1 − D g s Φ)( H ( s )) k L p ds ≤ C Z max { ,t − } ( t − s ) − α s − β − n ( p − r ′ ) ds · (cid:18)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ + k k k Z p,r ′ (cid:19) k k k Z p,r ′ , where we have chosen r ′ so large that n (cid:16) p − r ′ (cid:17) >
1. Since any of the given α satisfies α < n p ,we may always choose r ′ so large that α < n (cid:16) p − r ′ (cid:17) ≤ n (cid:16) p − r ′ (cid:17) + β . In addition, we alwayshave n (cid:16) p − r ′ (cid:17) + β >
1. Then we get from Lemma 1.16 that Z max { ,t − } ( t − s ) − α s − β − n ( p − r ′ ) ds ≤ Ct − α . Finally, we get from Lemma 6.6 that Z t max { ,t − } (cid:13)(cid:13)(cid:13) ∇ i ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] (cid:13)(cid:13)(cid:13) L q ′ ds ≤ C sup s ∈ [max { ,t − } ,t ] k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k L q ′ ≤ Ct − n ( p − r ′ ) − n (cid:16) p − q ′ (cid:17) (cid:18)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ + k k k X p,r ′ (cid:19) k k k X p,r ′ ≤ Ct − α (cid:18)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ + k k k X p,r ′ (cid:19) k k k X p,r ′ , because for given α , r ′ was chosen so large that n (cid:18) p − r ′ (cid:19) + n (cid:18) p − q ′ (cid:19) ≥ n (cid:18) p − r ′ (cid:19) ≥ α. Putting all the estimates together, we get (cid:13)(cid:13) ∇ i k t (cid:13)(cid:13) L q ′ ≤ Ct − α (cid:20) k k k L p + (cid:18)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r ′ + k k k X p,r ′ (cid:19) k k k X p,r ′ (cid:21) , which proves the proposition. (cid:3) Proof of Theorem 1.8.
For an initial metric g which is L [ q, ∞ ] -close to ˆ h , we denote by ˜ g t themodified Ricci-de Turck flow starting at g = ˜ g . We are first goint to show that all the assertionsof Theorem 1.8 hold with φ ∗ t g t replaced by ˜ g t . Let C > U , choose ǫ > B [ q, ∞ ] C · ǫ (ˆ h ) := n g ∈ M | (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L ∞ < C · ǫ o ⊂ U . Now if g ∈ V := B [ ∐ , ∞ ] δ (ˆ h ), for a δ > g t ∈ B [ q, ∞ ] C · ǫ (ˆ h ) for all t ∈ [0 , g t exists for all time and for t ≥ h t = Φ(˜ g t ) and k t = ˜ g t − h t satisfy (cid:13)(cid:13)(cid:13) ( h t − ˆ h, k t ) (cid:13)(cid:13)(cid:13) Y q,r < ǫ, which by Lemma 5.6 implies ˜ g t ∈ B [ q, ∞ ] C · ǫ (ˆ h ) ⊂ U for all t ≥
1. The decay and convergence ratesin (i)-(iii) for h t and k t follow from Propositions 6.5 and 6.7. To finish the proof, it suffices toshow that we can write ˜ g t = φ ∗ t g t , where g t is the standard Ricci flow starting at g . For thispurpose, let V t = − ( V (˜ g t , ˆ h ) , t ∈ (0 , ,V (˜ g t , h t ) , t ∈ [1 , ∞ )For t ∈ (0 , V t is of the form V t = (˜ g t ) − ∗ (˜ g t ) − ∗ ∇ ( g t − ˆ h ) while for t ≥
1, we have V t = (˜ g t ) − ∗ (˜ g t ) − ∗ ∇ k t . Therefore, by Lemma 3.8 and Proposition 6.5, V t is bounded in allderivatives for t > k V t k L ∞ ≤ Ct − (cid:13)(cid:13)(cid:13) g − ˆ h (cid:13)(cid:13)(cid:13) L ∞ ∈ L ([0 , t ≤
1. Therefore, V t actually generates a family of diffeomorphisms ( ϕ t ) t ≥ with ϕ = id M .A standard computation shows that the family g t = ϕ ∗ t ˜ g t is a Ricci flow starting at g and theproof is completed with φ t := ϕ − t . (cid:3) Remark 6.8.
Note that the bound in (45) implies the following: For given ǫ >
0, there exists δ > g ∈ B ∞ δ (ˆ h ), the Ricci-de Turck flow ˜ g t ) as well as the standard Ricci flow ( g t )starting at g exist up to time 1 and stay in B ∞ ǫ (ˆ h ) for all t ∈ [0 , TABILITY OF RICCI-FLAT ALE MANIFOLDS 69
Decay of the de Turck vector field and the Ricci curvature.
Throughout this sub-section, we assume the same as in Proposition 6.4. Let as in Theorem 6.2 the family g t be themodified Ricci-de Turck flow starting at g which for t ≥ g t = h t + k t , where h t is Ricci-flat and k t ∈ ker L (∆ L,h t ) ⊥ . The goal of this subsection is to get improvedestimates for the de Turck vector field and the Ricci curvature which are as follows: Proposition 6.9.
For each i ∈ N and τ > , there exists a constant C = C ( i, τ ) such that k V ( g t , k t ) k C i ≤ ( Ct − n p − + τ , if p ∈ (cid:0) , n (cid:1) ,Ct − n p + τ , if p ∈ (cid:2) n , n (cid:1) . k Ric g t k C i ≤ ( C · t − n p − τ , if p ∈ (cid:0) , n (cid:1) C · t − n p + τ , if p ∈ (cid:2) n , n (cid:1) . for all t ≥ . By Taylor expansion along the curve [0 , ∋ s h t + s · k t , we get V ( g t , h t ) = V ( g t , h t ) − V ( h t , h t ) = DV h t ( k t ) + 12 Z (1 − s ) D V h t + s · k t ( k t , k t ) ds, Ric g t = Ric g t − Ric h t = D Ric h t ( k t ) + 12 Z (1 − s ) D Ric g t + s · k t ( k t , k t ) ds. where D i denotes the i th Fr´echet derivative (for V , just in the first variable). The propositionnow follows from analyzing the respective parts on the right hand side. We first need someestimates on the pure linear part of the equations. Lemma 6.10.
For t ≥ i ∈ N and 1 < p ≤ r < ∞ , we have (cid:13)(cid:13) ∇ i ◦ DV ◦ e − t ∆ L (cid:13)(cid:13) L p ,L r ≤ Ct − n ( p − r ) − , (cid:13)(cid:13) ∇ i ◦ D Ric ◦ e − t ∆ L (cid:13)(cid:13) L p ,L r ≤ Ct − n ( p − r ) − . (46)For t ∈ [0 , i ∈ N and r ∈ (1 , ∞ ), we have (cid:13)(cid:13) DV ◦ e − t ∆ L (cid:13)(cid:13) W i +1 ,p ,W i,p ≤ C, (cid:13)(cid:13) D Ric ◦ e − t ∆ L (cid:13)(cid:13) W k +2 ,p ,W k,p ≤ C. (47) Proof.
We consider the case of the de Turck vector field first. By Theorem 5.1 and Theorem 5.3. (cid:13)(cid:13) e − t ∆ L (cid:13)(cid:13) L p ,L r ≤ Ct − n ( p − r ) , (cid:13)(cid:13) DV ◦ e − t ∆ L (cid:13)(cid:13) L r ,L r ≤ Ct − , for all t ≥ DV ◦ e − t ∆ L = DV ◦ e − t ∆ L ◦ e − t ∆ L already implies (46) in thecase i = 0. For derivatives, we first recall the commutation formula ∆ V F ◦ DV = DV ◦ ∆ L ,where ∆ V F is the connection Laplacian on vector fields. Therefore, ∇ i ◦ DV ◦ e − t ∆ L = ∇ i ◦ e − ∆ V F ◦ DV ◦ e − ( t − ) ∆ L By standard estimates (similar as in Lemma 3.3, ∇ i ◦ e − ∆ V F is a bounded map on L r and (46)follows from the case i = 0. Again by standard estimates, e − t ∆ L is bouded on W i,p for t ∈ [0 , DV is a linear first order operator, (47) is immediate. The case of the Ricci curvature iscompletely analogous. Here we use that D Ric is a linear second order operator, the commutatorformula ∆ L ◦ D Ric = D Ric ◦ ∆ L and (cid:13)(cid:13) D Ric ◦ e − t ∆ L (cid:13)(cid:13) L r ,L r ≤ Ct − , which holds due to Theorem 5.3 as well. (cid:3) The next step is to apply these linearizations to k t instead of e − t ∆ L . Lemma 6.11.
For each i ∈ N and τ >
0, there exists a constant C = C ( i, τ ) such that k DV h t ( k t ) k C i ≤ ( Ct − n p + τ − , if p ∈ (cid:0) , n (cid:1) ,Ct − n p + τ , if p ∈ (cid:2) n , n (cid:1) . k D Ric h t ( k t ) k C i ≤ ( Ct − n p + τ − , if p ∈ (cid:0) , n (cid:1) ,Ct − n p + τ , if p ∈ (cid:2) n , n (cid:1) . for all t ≥ Proof.
We will just carry out the proof for DV . The other case is completely analogous and leftas an exercise to the reader. For t ∈ [1 , k DV h t ( k t ) k C i ≤ C k k t k C i +1 ≤ C. Therefore, we may assume t > DV h t to (42), we write DV h t ( k t ) = DV h t ◦ e − ( t − L,ht k + Z t − DV h t ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] ds + Z tt − DV h t ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] ds. and we estimate these three terms separately. Choose a H¨older exponent r ∈ ( n, ∞ ) whoseprecise value is yet to determine but which is so large that n (cid:18) p − r (cid:19) > . For the first term, Lemma 6.10 and Lemma 2.10 (ii) yield (cid:13)(cid:13)(cid:13) ∇ i ◦ DV h t ◦ e − ( t − L,ht k (cid:13)(cid:13)(cid:13) L r ≤ C ( t − − n ( p − r ) − (cid:13)(cid:13) k (cid:13)(cid:13) L p ≤ Ct − n ( p − r ) − k k k L p To estimate the second term, we first deal with the integrands. For 1 ≤ s ≤ t , Lemma 5.13 yields k (∆ L,h t − ∆ L,h s )( k s ) k L p ≤ Cs − n ( p − r ) (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r k k s k X p,r and Lemma 5.14 yields k (1 − D g s Φ)( H ( s )) k L p ≤ Cs − β − n ( p − r ) k k k X p,r , where β = min n , n (cid:16) p − r (cid:17)o > . We now distinguish between two cases. If p < n , we pick r ∈ ( n, ∞ ) so large that n (cid:16) p − r (cid:17) >
1. Then we get β + n (cid:18) p − r (cid:19) >
12 + n (cid:18) p − r (cid:19) , n (cid:18) p − r (cid:19) − > n (cid:18) p − r (cid:19) + 12 . By the triangle inequality and the above estimates, we thus get k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k L p ≤ Cs − n ( p − r ) − k k k X p,r (cid:18) k k k X p,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r (cid:19) . TABILITY OF RICCI-FLAT ALE MANIFOLDS 71 for some small ǫ >
0. Consequently, Lemma 6.10 and Lemma 1.16 imply that (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − DV h t ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] ds (cid:13)(cid:13)(cid:13)(cid:13) L r ≤ C Z t (cid:13)(cid:13)(cid:13) ∇ i ◦ DV h t ◦ e − ( t − s )∆ L,ht Π ⊥ h t (cid:13)(cid:13)(cid:13) L p ,L r k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k L p ds ≤ C Z t − ( t − s ) − n ( p − r ) − s − n ( p − r ) − ds · k k k X p,r (cid:18) k k k X p,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r (cid:19) ≤ Ct − n ( p − r ) − k k k X p,r (cid:18) k k k X p,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r (cid:19) , because n (cid:16) p − r (cid:17) + >
1. In the case p ∈ (cid:2) n , n (cid:1) , we have n (cid:18) p − r (cid:19) < r ∈ ( n, ∞ ) so that β + n (cid:18) p − r (cid:19) > n (cid:18) p − r (cid:19) + 12 > n (cid:18) p − r (cid:19) − . (48)By the triangle inequality and the above estimates, we thus get k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k L p ≤ Cs − n ( p − r ) k k k X p,r (cid:18) k k k X p,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r (cid:19) . Consequently, by Lemma 1.16, (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − DV h t ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] ds (cid:13)(cid:13)(cid:13)(cid:13) L r ≤ C Z t (cid:13)(cid:13)(cid:13) ∇ i ◦ DV h t ◦ e − ( t − s )∆ L,ht Π ⊥ h t (cid:13)(cid:13)(cid:13) L p ,L r k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k L p ds ≤ C Z t − ( t − s ) − n ( p − r ) − s − n ( p − r ) ds · k k k X p,r (cid:18) k k k X p,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r (cid:19) ≤ Ct − n ( p − r ) k k k X p,r (cid:18) k k k X p,r + (cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r (cid:19) due to (48) and the inequality n (cid:16) p − r (cid:17) + >
1. For the last term, we get, using the last partof Lemma 6.10 and Lemma 6.6, (cid:13)(cid:13)(cid:13)(cid:13) ∇ i Z t − DV h t ◦ e − ( t − s )∆ L,ht Π ⊥ h t [(∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s ))] ds (cid:13)(cid:13)(cid:13)(cid:13) L r ≤ C sup s ∈ [ t − ,t ] k (∆ L,h t − ∆ L,h s )( k s ) + (1 − D g s Φ)( H ( s )) k W i +2 ,r ≤ Ct − n ( p − r ) (cid:18)(cid:13)(cid:13)(cid:13) h − ˆ h (cid:13)(cid:13)(cid:13) Z p,r + k k k X p,r (cid:19) . Summing up the inequalities, we get for all i ∈ N that k DV h t ( k t ) k L r ≤ ( Ct − n ( p − r ) − , if p ∈ (cid:0) , n (cid:1) ,Ct − n ( p − r ) , if p ∈ (cid:2) n , n (cid:1) . Combining this with the Sobvolev type inequality k DV h t ( k t ) k C i ≤ C k DV h t ( k t ) k W i +1 ,r and choosing r so large that n r ≤ nr < τ, we obtain the desired result. (cid:3) It remains to consider the error terms in the Taylor expansion.
Lemma 6.12.
For each i ∈ N and τ >
0, there exists a constant C = C ( i, τ ) such that (cid:13)(cid:13)(cid:13)(cid:13)Z (1 − s ) D V h t + s · k t ,h t ( k t , k t ) ds (cid:13)(cid:13)(cid:13)(cid:13) C i ≤ Ct − np + τ , (cid:13)(cid:13)(cid:13)(cid:13)Z (1 − s ) D Ric h t + s · k t ,h t ( k t , k t ) ds (cid:13)(cid:13)(cid:13)(cid:13) C i ≤ Ct − np + τ for all t ≥ Proof.
First note that due to short-time estimates g t is C k -close to ˆ h for all t ≥
1. Therefore,all C k -norms of the metrics g t,s := h t + s · k t are equivalent for t ≥ s ∈ [0 ,
1] and we maysupress the dependence of the norms on the metric. Due to the schematic expression We havethe schematic expressions D V g t + s · k t ,h t ( k t , k t ) = ∇ k t ∗ k t , D Ric g t,s ( k t , k t ) = ∇ k t ∗ k t + ∇ k t ∗ ∇ k t + R g t,s ∗ k t ∗ k t . from which we conclude (cid:13)(cid:13)(cid:13)(cid:13)Z (1 − s ) D V g t + s · k t ,h t ( k t , k t ) ds (cid:13)(cid:13)(cid:13)(cid:13) C i ≤ C k k t k C i +1 ≤ C (cid:16) t − n p + τ (cid:17) = Ct − np + τ , (cid:13)(cid:13)(cid:13)(cid:13)Z (1 − s ) D Ric g t + s · k t ( k t , k t ) ds (cid:13)(cid:13)(cid:13)(cid:13) C i ≤ C k k t k C i +2 ≤ C (cid:16) t − n p + τ (cid:17) = Ct − np + τ . The inequalities on the right hand sides follow from Proposition 6.5 above. (cid:3)
Proof.
This follows now directly from Lemma 6.11 and Lemma 6.12. Note that the terms inLemma 6.11 are dominating for any p ∈ (1 , n ). (cid:3) Convergence of the Ricci-flow.
Proof of Theorem 1.11.
Choose an arbitrary U and let V = V ( U ) as in Theorem 1.8. Then themodified Ricci-de Turck flow g t starting at g exists for all time and converges to a Ricci-flat limit h ∞ as t → ∞ . Moreover, g t → h ∞ as t → ∞ and the tensors h t = Φ( g t ) and k t = g t − h t satisfythe convergence rates of the Propositions 6.5 and 6.7. The family of vector fields V t = − V ( g t , h t ), t ≥ p < n , we have (cid:13)(cid:13) V t (cid:13)(cid:13) C k (ˆ h ) ≤ Ct − α for all t ≥ α >
1. Therefore, the family of diffeomorphisms ( ϕ t ) t ≥ with ϕ = id M generated by V t converges in all derivatives to a limit diffeomorphism ϕ ∞ as t → ∞ . Now let g t = ϕ ∗ t g t , h t = ϕ ∗ t h t , k t = ϕ ∗ t k t , t ∈ [1 , ∞ ) . Observe that ( g t ) t ≥ is a standard Ricci flow starting at g and h t is a family of Ricci-flat metrics.Because h t → h ∞ in C i , we also have h t = ϕ ∗ t h t → ϕ ∗∞ h ∞ =: h ∞ in C i . Therefore, the C i -normsinduced by the Ricci-flat metrics ( ϕ − t ) ∗ h ∞ t ∈ [1 , ∞ ] are equivalent. Recall also that the C i TABILITY OF RICCI-FLAT ALE MANIFOLDS 73 norms of h ∞ = ( ϕ − ∞ ) ∗ h ∞ and ˆ h are equivalent as h ∞ , ˆ h ∈ U ∩ F . Thus we get for each τ > k k t k C i ( h ∞ ) = (cid:13)(cid:13) k t (cid:13)(cid:13) C k (( ϕ − t ) ∗ h ∞ ) ≤ k k t k C i (ˆ h ) ≤ Ct − n p + τ due to Theorem 1.8. To obtain the convergence rate of h t , we compute ∂ t h t = ϕ ∗ t ( ∂ t h t ) − ϕ ∗ t (cid:16) L V ( g t ,h t ) h t (cid:17) , which yields k ∂ t h t k C i ( h ∞ ) ≤ (cid:13)(cid:13) ϕ ∗ t ( ∂ t h t ) (cid:13)(cid:13) C i ( h ∞ ) + (cid:13)(cid:13)(cid:13) ϕ ∗ t ( L V ( g t ,h t ) h t ) (cid:13)(cid:13)(cid:13) C i ( h ∞ ) = (cid:13)(cid:13) ∂ t h t (cid:13)(cid:13) C i (( ϕ − t ) ∗ h ∞ ) + (cid:13)(cid:13)(cid:13) L V ( g t ,h t ) h t (cid:13)(cid:13)(cid:13) C i (( ϕ − t ) ∗ h ∞ ) ≤ C ( (cid:13)(cid:13) ∂ t h t (cid:13)(cid:13) C i (ˆ h ) + (cid:13)(cid:13) V ( g t , h t ) (cid:13)(cid:13) C i +1 (ˆ h ) ) . Proposition 6.4 yields by definition of the Y p,r ′ -norm that (cid:13)(cid:13) ∂ t h t (cid:13)(cid:13) C i (ˆ h ) ≤ Ct − np + τ , where C = C ( τ ) and τ > h t now follows fromProposition 6.9 and integrating in time. (cid:3) Positive scalar curvature rigidity.
In this subsection, we will prove the scalar curvaturerigidity statement using our stability result. We will use that the Ricci curvature (and hence thescalar curvature as well) decay of order O ( t − n p − τ ) for small p . On the other hand, becausethe scalar curavture satisfies the super heat equation ∂ t scal g t + ∆ g t scal g t = 2 | Ric g t | g t along the Ricci flow, we expect a decay rate of at most of order O ( t − n ), which is the L ∞ decayrate of the heat kernel on ALE spaces. We will follow the same strategy as [App18].Let g be a metric satisfying the assumptions of Theorem 1.12. Let ˜ g t be the ˆ h -gauged Ricci-deTurck flow and g t the standard Ricci flow starting from g = g , both defined up to time 1. Weneed to understand the heat kernel of the evolving backgrounds. For 0 ≤ s < t and x, y ∈ M ,let K ( x, t, y, s ) be the heat kernel associated to g t , i.e. u ( t, x ) := Z M K ( x, t, y, s ) u s ( y ) dV g s is the solution of the initial value problem ∂ t u + ∆ g t u = 0 , u ( s, x ) = u s ( x ) . Let ˜ K ( x, t, y, s ) be the heat kernel associated to ˜ g t , then we have the relation K ( x, t, y, s ) = ˜ K ( φ t ( x ) , t, φ t ( y ) , s ) , where φ t are the diffeomorphisms such that φ ∗ t ˜ g t = g t . For 0 < s < t ≤
1, [Zhu16, Theorem 4.2]yields the Gaussian bounds˜ K ( x, t, y, s ) ≤ C ( t − s ) − n exp( C Λ + C ( t − s ) κ + C p ( t − s ) κ ) exp (cid:18) − d ˜ g t ( x, y ) κT )( t − s ) (cid:19) , where Λ = R ts (cid:13)(cid:13) Ric ˜ g t ′ (cid:13)(cid:13) C ( ˜ g t ) dt ′ and Ric ˜ g t ′ ≥ − κ for t ′ ∈ [ s, t ]. By Remark 6.8, g t stays L ∞ -closeto ˆ h up to time 1, so that the induced distance functions d g t , d ˜ g t and d ˆ h are all equivalent. By diffeomorphism invariance, we thus get K ( x, t, y, s ) ≤ C ( t − s ) − n exp( C Λ + C ( t − s ) κ + C p ( t − s ) κ ) · exp (cid:18) − d ˆ h ( x, y ) C exp(4 κT )( t − s ) (cid:19) , (49)where Λ = R ts (cid:13)(cid:13) Ric g t ′ (cid:13)(cid:13) C ( g t ) dt ′ and Ric g t ′ ≥ − κ for t ′ ∈ [ s, t ]. Lemma 6.13.
If scal g ≥
0, then scal g ≥ Proof.
This lemma has been shown in the case of R n in [App18], based on the analysis in[Bam16] and a parabolic scaling argument which does not work on general ALE manifolds. Forthis reason, we present the details here although the ideas are similar as in [Bam16]. Let θ ∈ (0 , t i = θ i , i ∈ N . Due to short-time estimates for the Ricci-deTurck flow, (cid:13)(cid:13) Ric g t ′ (cid:13)(cid:13) C ( g t ) ≤ C t − for t ∈ (0 , K ( x, t i , y, t i +1 ) ≤ C θ − C · i exp (cid:18) − d ˆ h ( x, y ) C θ i (cid:19) . Now let β > ( √ θ, R > r i = R · (1 − β i ) . Fix a point x ∈ M and set a i := inf { scal g t ( y ) | y ∈ B ( r i , x ) } , where B ( r i , x ) is the ball of radius r i around x , measured with respect to ˆ h . Standard regularitytheory of the Ricci-de Turck flow (see e.g. [Bam16]) shows that g t ∈ C loc ([0 , , M ). Therefore,lim inf i →∞ a i ≥ inf { scal g ( y ) | y ∈ B ( R, x ) } ≥ y ∈ M ,scal g ti ( y ) ≥ Z M K ( y, t i , z, t i +1 ) · scal g ti +1 ( z ) dV g ti +1 ≥ a i +1 Z B ( r i +1 − r i ,x ) K ( y, t i , z, t i +1 ) dV g ti +1 − C t i +1 Z M \ B ( r i +1 − r i ,y ) K ( y, t i , z, t i +1 ) dV g ti +1 ≥ a i +1 − C t i +1 θ − C · i exp (cid:18) − ( r i +1 − r i ) C (1 − θ ) i (cid:19) ≥ a i +1 − C · θ − ( C +1) · i exp (cid:18) − R C β i θ i (cid:19) . Because β > θ , we may fix some j ∈ N such that (cid:18) θβ (cid:19) j ≤ θ C +1 . Using x j exp ( − x ) ≤ C for x >
0, we thus getscal g ti ( y ) ≥ a i +1 − C R j · i . We conclude a i ≥ a i +1 − C R j · i , TABILITY OF RICCI-FLAT ALE MANIFOLDS 75 and therefore, scal g ( x ) ≥ a ≥ lim inf i →∞ a i − · C R j ≥ − · C R j . Because x ∈ M was taken arbitrarily, the result follows from letting R → ∞ . (cid:3) Now, we continue with our analysis on large times. Let ( g t ) t ≥ be the standard Ricci flowstarting from g and ( g t ) t ≥ be the Ricci flow with moving gauge, also starting from g . Again,we have diffeomorphisms ( ϕ t ) t ≥ such that g t = ϕ ∗ t g t . Definition 6.14.
Let 1 ≤ s < t and x, y ∈ M . Then the L -length of a curve γ : [ s, t ] → M is L ( γ ) := Z ts √ t − t ′ (scal g t ′ ( γ ( t ′ )) + | γ ′ ( t ′ ) | g t ) dt ′ and the reduced distance between ( x, t ) and ( y, s ) is ℓ ( x, t, y, s ) := 12 √ t − s inf {L ( γ ) | γ : [ s, t ] → M is a smooth curve with γ ( s ) = y and γ ( t ) = x } . Lemma 6.15.
With the same notation as above, we have K ( x, t, y, s ) ≥ π ( t − s )) n exp( − ℓ ( x, t, y, s )) Proof.
In the compact case, this result is [CCG +
08, Lemma 16.49]. The proof of this lemma ison the one hand based on [CCG +
08, Lemma 16.48] (whose proof in turn builds up on results in[CCG +
07] which do also hold for Ricci flows of complete manifolds of bounded curvature) andon the other hand on the weak maximum principle which does also hold in the present situationsituation (see e.g. [CCG +
08, Theorem 12.10]). Therefore, the assertion of [CCG +
08, Lemma16.49] also holds. (cid:3)
Lemma 6.16.
There exist constants C , C > K ( x, t, y, s ) ≥ C (4 π ( t − s )) n exp (cid:18) − ( d ˆ h ( x, y )) C ( t − s ) (cid:19) Proof.
For x, y ∈ M , let γ x,y : [ s, t ] → M be a ˆ h -geodesic joining x and y . Due to the parametriza-tion interval, | γ ′ x,y ( t ′ ) | ˆ h = d ˆ h ( x,y )( t − s ) . Therefore, ℓ ( x, t, y, s ) ≤ L ( γ x,y ) = 1 √ t − s Z ts √ − t ′ (scal g t ′ ( γ x,y ( t ′ )) + | γ ′ x,y ( t ′ ) | g t ′ ) dt ′ ≤ C + C √ t − s Z ts √ t − t ′ | γ ′ x,y ( t ′ ) | h dt ′ = C + 2 C d ˆ h ( x, y )) t − s . Thus, we getexp( − ℓ ( x, t, y, s )) ≥ exp (cid:18) − C − C d ˆ h ( x, y )) t − s (cid:19) = exp( − C ) exp (cid:18) − C d ˆ h ( x, y )) t − s (cid:19) and the result follows from Lemma 6.15. (cid:3) Proof of Theorem 1.12.
By the Duhamel principle, we have, forscal g t ( x ) = Z M K ( x, t, y, s )scal g s ( y ) dV g t + Z t Z M K ( x, t, y, t ′ ) | Ric g t ′ | g t ′ dV g t ′ . Now suppose that Ric g = 0. Then we also have Ric g = 0. Because scal g ≥
0, we thus getscal g t ( x ) > t > x ∈ M . Now fix a point x ∈ M and a ball B r ( x ) ⊂ M such thatscal g ( y ) ≥ R > y ∈ B r ( x ). Then for t > g t ( x ) ≥ Z M K ( x, t, y, s )scal g s ( y ) dV g t ≥ R C (4 π ( t − n Z B r ( x ) exp (cid:18) − ( d ˆ h ( x, y )) C ( t − s ) (cid:19) dV g t ≥ C ( t − − n ≥ Ct − n . on the other hand, by Proposition 6.9, we have for any τ > g t ( x ) = scal g t ( ϕ t ( x )) ≤ C (cid:13)(cid:13) Ric g t (cid:13)(cid:13) C ≤ Ct − n p + τ − , (50)which leads to a contradiction since p < nn − . (cid:3) Remark 6.17.
In [App18, Lemma 6.6], proves that under the present assumptions, scal g t ∈ L for t >
1, if p = nn − . However, we are not able to reproduce this result because we do not have(50) with τ = 0. Proof of Theorem 1.13.
For k ∈ N and δ ∈ R , letConf k,pδ (ˆ h ) = (1 + W k,pδ ( M )) · ˆ h = n g | g conformal to ˆ h and g − ˆ h ∈ W k,pδ ( S M ) o If k > np and δ = − np , we have a mapscal : Conf k,pδ (ˆ h ) → W k − ,pδ − ( M ) , g scal g . Its linearization at ˆ h is given by( n − · ∆ : W k +2 ,pδ ( M ) → W k − ,pδ − ( M ) , see e.g. [Bes08, Theorem 1.174]. Due to the condition on p , we have δ > − n and this map isindeed an isomorphism (see e.g. [Bar86, Proposition 2.2]). Let now f ∈ W k − ,pδ − ( M ) Due to theinverse function theorem for Banach manifolds, scal restricts to a diffeomophismscal : Conf k,pδ (ˆ h ) ⊃ U → V ⊂ W k − ,pδ − ( M ) , for some small neighbourhoods U of ˆ h and V of 0, respectively. Therefore, we find for eachsequence of positive functions f i ∈ V converging to 0 in W k − ,pδ − a sequence of metrics g i ∈ Conf k,pδ (ˆ h ) with scal g i = f i converging to ˆ h in W k,pδ . By Sobolev embedding, we have (cid:13)(cid:13)(cid:13) g i − ˆ h (cid:13)(cid:13)(cid:13) L [ p, ∞ ] = (cid:13)(cid:13)(cid:13) g i − ˆ h (cid:13)(cid:13)(cid:13) L p + (cid:13)(cid:13)(cid:13) g i − ˆ h (cid:13)(cid:13)(cid:13) L ∞ ≤ (cid:13)(cid:13)(cid:13) g i − ˆ h (cid:13)(cid:13)(cid:13) L pδ + C (cid:13)(cid:13)(cid:13) g i − ˆ h (cid:13)(cid:13)(cid:13) L ∞ δ ≤ C (cid:13)(cid:13)(cid:13) g i − ˆ h (cid:13)(cid:13)(cid:13) H k +2 ,pδ → , which proves the result. (cid:3) References [App18] A. Appleton,
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