Asymptotically Flat Ricci Flows T.A. Oliynyk† 1, E Woolgar‡ 2 8 † Max-Planck-Institutfu¨r Gravitationsphysik (Albert Einstein Institute),Am Mu¨hlenberg 1, D-14476 0 Potsdam, Germany. 0 ‡ Dept of Mathematical and Statistical Sciences, University of Alberta, 2 Edmonton,AB, Canada T6G 2G1. n a J 4 2 Abstract G] We study Ricci flows on Rn, n 3, that evolve from asymptotically flat initial data. ≥ D Under mild conditions on the initial data, we show that the flow exists and remains asymptotically flat for an interval of time. The mass is constant in time along the flow. . h We then specialize to the case of rotationally symmetric, asymptotically flat initial t a data containing no embedded minimal hyperspheres. We show that in this case the m flow is immortal, remains asymptotically flat, never develops a minimal hypersphere, [ and converges to flat Euclidean space as the time diverges to infinity. We discuss the 2 behaviour of quasi-local mass under the flow, and relate this to a conjecture in string v theory. 8 3 4 7 0 6 0 / h t a m : v i X r a [email protected] [email protected] 1 1 Introduction The Ricci flow ∂g ij = 2R . (1.1) ij ∂t − was first introduced in the mathematics literature by Richard Hamilton [20] in 1982. Almost immediately, it was applied to the classification problem for closed 3-manifolds and much subsequent work in the subject in the intervening 25 years has been focused on this application, culminating in the recent celebrated results of Perelman [29]. By contrast, Ricci flow on noncompact manifolds has received somewhat less at- tention. Of course, structures on noncompact manifolds, such as Ricci solitons, are relevant to the compact case, and this has been to now an important motivation for work on the noncompact case. The case of asymptotically flat Ricci flow has remained virtually untouched (nontrivial solitons do not occur in this case [27]). But physics provides considerable motivation to study the asymptotically flat case. Our interest in it arises out of a conjectural scenario in string theory. Equation (1.1) is the leading-order renormalization group flow equation for a nonlinear sigma model that describes quantum strings propagating in a background spacetime [17].3 What is important to understand from this statement is that fixed points of this equation provide geometric backgrounds in which the low energy excitations of quantum strings can propagate (in the approximation that radii of curvature are large and excitation energies small relative to the so-called string scale). The variable t in renormalization group flow is not time: it is (a constant times) the logarithm of the so-called renormalization scale. However, there are conjectured relationships between renormalization group flow and temporal evolution. A specific case concerns tachyon condensation, the scenario wherein an unstable string system is balanced at the top of a hill of potential energy (for a review of tachyon condensation, see[23]). Thesystemfallsoffthehill,radiatingawayenergyingravitationalwaves. The system comes to rest in a valley representing a stable minimum of potential energy. In openstringtheory,amoreelaborateversionofthisscenarioinvolvingtheevaporationof a braneand theformation of closed strings is now well understood,even quantitatively. In closed string theory, much less is known but, conjecturally, the fixed points of the renormalization group flow equation (1.1) are the possible endpoints of this evolution. Sometimes it is further conjectured that time evolution in closed string theory near the fixed points is determined by renormalization group flow, and then t in (1.1) does acquire an interpretation as a time. Comparing both sides of this picture, we see that the radiation of positive energy in the form of gravitational waves as the system comes to rest in the valley should produce a corresponding decrease in the mass of the manifold under the Ricci flow. This suggests that we should endeavor to formulate and test a conjecture that mass decreases under Ricci flow, at least if the initial mass is positive. The asymptotically flat case has a well-defined notion of mass, the ADM mass, so this seems an appropriate setting in which to formulate the conjecture. However, the metric entering the renormalization group flow or Ricci flow in this scenario is not the full spacetime metric, for which (1.1) would not be even quasi-parabolic, but rather 3We ignore thedilaton since it can be decoupled from themetric in renormalization group flow. 2 the induced Riemannian metric on a suitable spacelike submanifold [19]. Now ADM mass is conserved (between Cauchy surfaces, and in the closed string scenario of [19]), even in the presence of localized sources of radiation. This, we will see, is reflected in the Ricci flow. The mass of g will not change during evolution by (1.1). But if energy loss through gravitational radiation occurs, then the quasi-local mass contained within a compact region should change along the flow to reflect this.4 In this paper, we focus first on the asymptotically flat case of Ricci flow in general. Section 2 describes asymptotically flat manifolds, with no assumption of rotational symmetry. Continuing with the general asymptotically flat case, in Subsection 3.1 we state and prove our short-term existence result Theorem 3.1, showing that a general asymptoticallyflatdatasetonRn willalwaysevolveunderRicciflow,remainingsmooth andasymptotically flatonamaximaltimeinterval[0,T ). WewillshowthattheADM M mass remains constant during this interval, at least for non-negative scalar curvature (i.e., the positive mass case, the usual case of physical interest). Moreover, if T < M ∞ then the norm of the Riemann curvature must become unbounded as t T , just as m ր in the compact case. We show this in Subsection 3.2. The short-term existence proof in Section 3 depends on detail provided in the ap- pendices. In Appendix A, we derive weighted versions of standard Sobolev estimates such the Sobolev inequalities and Moser estimates. We then use these estimates in Appendix B to prove local existence and uniqueness in weighted Sobolev spaces for uniformly parabolic systems. We specializetorotational symmetryinSection 4. InSection 4.1, wepasstoacoor- dinate system well suited to our subsequent assumption that no minimal hyperspheres are present initially. We show in Section 4.3 that this coordinate system remains well- defined on the interval [0,T ). This is essentially a consequence of the result, proved m in Section 4.2, that no minimal hyperspheres develop during the flow. Theabsenceofminimalspheresallowsustoanalysetheproblemintermsofasingle PDE, the master equation (4.18). From this equation, we derive a number maximum principles that yield uniform bounds on the curvature which allow us to conclude that T = . We obtain these principles in the first two subsections of Section 5. Even M ∞ better, we obtain not just uniform bounds but decay estimates, from which we can prove convergence to flat Euclidean space. Now given our assumptions, this is the only Ricci-flat fixed point available. That is, the string theory discussion above would lead one to conjecture that: When no minimal hypersphere is present, rotationally symmetric, asymptotically flat Ricci flow is immortal and converges to flat space as t ; → ∞ andthisiswhatweshow. Thoughwehavemotivatedthisconjecturefromstringtheory forthecaseof positiveinitial mass,wewillprovethatitholdswhetheror nottheinitial mass is positive. This is our main theorem, proved in Subsection 5.3, which states: Theorem 1.1. Let xi n be a fixed Cartesian coordinate system on Rn, n 3. Let { }i=1 ≥ gˆ = gˆ dxidxj be an asymptotically flat, rotationally symmetric metric on Rn of class ij 4We prefer not to discuss in terms of the Bondi mass, which would require us to pass back to the Lorentzian setting which is not our focus in this article. See [19] for a discussion in terms of Bondi mass. 3 Hk with k > n/2+4 and δ <0. If (Rn,gˆ) does not contain any minimal hyperspheres, δ then there exists a solution g(t,x) C∞((0, ) Rn) to Ricci flow (1.1) such that ∈ ∞ × (i) g(0,x) = gˆ(x), (ii) g δ C1([0,T],Hk−2) and g δ C1([T ,T ],Hℓ) for any 0 < T < T < ij− ij ∈ δ ij− ij ∈ 1 2 δ 1 2 , 0< T < , ℓ 0, ∞ ∞ ≥ (iii) for each integer ℓ 0 there exists a constant C > 0 such that ℓ ≥ C sup ℓRm(t,x) ℓ t > 0, x∈Rn|∇ |g(t,x) ≤ (1+t)tℓ/2 ∀ (iv) the flow converges to n-dimensional Euclidean space En in the pointed Cheeger- Gromov sense as t , and → ∞ (v) if furthermore k > n/2+6, δ < min 4 n,1 n/2 , Rˆ 0, and Rˆ L1, then { − − } ≥ ∈ the ADM mass of g(t) is well defined and mass(g(t)) = mass(gˆ) for all t 0. ≥ When a minimal hypersphere is present initially, if the neck is sufficiently pinched then we expect long-time existence to fail. To see why, consider rotationally symmetric metrics on Sn. If there is a sufficiently pinched minimal (n 1)-sphere, the curvature − blowsupinfinitetime. Thishasbeenshownbothrigorously(n 3)[3]andnumerically ≥ (n = 3) [18]. Our assumption of no minimal spheres in the initial data is intended to preventthis. Theability tomakethisassumptionandtochoosecoordinatesadaptedto it is a distinct advantage of the noncompact case. However, we also expect (based, e.g., on ([18]) that for initial data with minimal hyperspheres that have only a mild neck pinching, the flow will continue to exist globally in time as well. Thus, when a minimal hypersphere is present, we believe there would be considerable interest in determining a precise criterion for global existence in terms of the degree of neck pinching because of the possibility, raised in [18], that the critical case on the border between singularity formation and immortality may exhibit universal features such as those observed in critical collapse in general relativity [10]. Theconstancy of theADMmassinstatement (v)isnotatoddswiththeconclusion that the flow converges to a flat and therefore massless manifold. This constancy was also noted in [13] but we draw different conclusions concerning the limit manifold, owing to our use of the pointed Cheeger-Gromov sense of convergence of Riemannian manifolds.5 InSubsection4.4wedefinethreedifferentkindsofmetricballsin(Rn,g(t)), n 3; balls of fixed radius, fixed volume, and fixed surface area of the bounding ≥ hypersphere. To clarify the behaviour of the mass in the limit t , we express → ∞ the Brown-York quasi-local mass of these balls in terms of sectional curvature and, by anticipating thedecay rate for sectional curvaturederived in Section 5, show that these quasi-local masses go to zero as t , even though the ADM mass, as measured at → ∞ 5The rotationally symmetric, expanding soliton of [19] can be used to illustrate this phenomenon explicitly (albeit in 2 dimensions, whereas our results are for n≥3 dimensions). For this soliton, one can easily compute the Brown-York quasi-local mass on any ball whose proper radius is fixed in time and see that for each such ballthequasi-local mass tendstozero as t→∞,and theflowconverges to Euclidean 2-space. But the mass at infinity of the soliton (the deficit angle of the asymptotic cone in 2 dimensions) is a constant of themotion which can beset by initial conditions to takeany value. 4 infinity, is constant. The picture is not strongly dependent on the definition of quasi- local mass, of which the Brown-York definition is but one among many. In rotational symmetry in any dimension, the metric has only one “degree of freedom”. The study of the evolution of quasi-local mass then reduces to the study of this single degree of freedom, no matter which definition of quasi-local mass one prefers.6 Although local existence, uniqueness, and a continuation principle for Ricci flow on non-compact manifolds with bounded curvature are known [31, 7], it does not follow immediately from these results that Ricci flow preserves the class of asymptotically flat metrics. One of the main results of this paper is to show that Ricci flow does in fact preserve the class of asymptotically flat metrics. Independent of our work, Dai and Ma have recently announced that they have also been able to establish this result [13], as has List in his recent thesis [25]. Our approach to the problems of local existence, uniqueness, continuation, and asymptotic preservation is to prove a local existence and uniqueness theorem for quasi- linear parabolic equations with initial data lying in a weighted Sobolev space, and then use it to show that Ricci flow preserves the class of asymptotically flat metrics. An important advantage of this approach rather than appealing to the results of [31, 7, 25, 13] is that we obtain a local existence and uniqueness theorem on asymptotically flat manifolds that is valid for other types of geometric flows to which the results of [31, 7, 25, 13] do not immediately apply, and which are of interest in their own right. For example, our local existence results contained in appendix B combined with the DeTurck trick will yield local existence, uniqueness, and a continuation principle for the following flows on asymptotically flat manifolds: ∂ g = 2R +4 u u t − ij ∇i ∇j (static Einstein flow), ∂ u= ∆u t (cid:27) ∂tgij = −α′ Rij +∇i∇jΨ+ 14HjpqHjpq ∂ Ψ = α′(∆Ψ Ψ 2+ H 2) (1st order sigma model RG flow), t 2 (cid:0) −|∇ | | | (cid:1)  ∂ B = α′( kH H kΨ) (H := dB)  t ij 2 ∇ kij − kij∇ and   α′ ∂ g = α′ R + R R klm (2nd order sigma model RG flow with B = Φ = 0). t ij ij iklm j − 2 (cid:0) (cid:1) We note that the static Einstein flow has been previously considered in the thesis [25]. There a satisfactory local existence theory on noncompact manifolds is developed and an also a continuation principle for compact manifolds is proved. Theproblem of global existence for rotationally symmetricmetrics on R3 has previ- ouslybeeninvestigated in[24]. Theretheassumptionsontheinitialmetricaredifferent than ours. Namely, the initial metric in [24] has positive sectional curvature and the 6The assumption of spherical symmetry in general relativity precludes gravitational radiation, ac- cording to the Birkhoff theorem. But on the string side of our scenario, the picture is one of closed stringsexistingasperturbationsthatbreakthesphericalsymmetryofthebackgroundmetric(aswell, we should include a dilaton background field that modifies general relativity). Viewed in the string picture,theseperturbationscreatetheradiationthatisdetectedasachangeinthequasi-localmassof thespherically symmetric Ricci flow. 5 manifold opens up as least as fast as a paraboloid. Under these assumptions, it is shown that Ricci flow exists for all future times and converges to either a flat metric or a rotationally symmetric Ricci soliton. Finally, throughout we fix the dimension of the manifold to be n 3. As well, we ≥ usually work with the Hamilton-DeTurck form of the Ricci flow ∂g ij = 2R + ξ + ξ , (1.2) ij i j j i ∂t − ∇ ∇ which is obtained from the form (1.1) by allowing the coordinate basis in which g is ij written to evolve by a t-dependent diffeomorphism generated by the vector field ξ. Acknowledgments. We thank Suneeta Vardarajan for discussions concerning the string theory motivation for this work. EW also thanks Barton Zwiebach for his ex- planation of the rolling tachyon. This work was begun during a visit by TO to the Dept of Mathematical and Statistical Sciences of the University of Alberta, which he thanks for hospitality. The work was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. 2 Asymptotically flat manifolds The definition of asymptotically flat manifolds that we employ requires the use of weighted Sobolev spaces, which we will now define. Let V be a finite dimensional vector space with inner product ( ) and corresponding norm . For u Lp (Rn,V), ·|· |·| ∈ loc 1 p , and δ R, the weighted Lp norm of u is defined by ≤ ≤ ∞ ∈ σ−δ−n/pu Lp if 1 p < k k ≤ ∞ kukLpδ :=  (2.1) σ−δu if p =  L k k ∞ ∞ with  σ(x) := 1+ x2 . (2.2) | | The weighted Sobolev norms are then givepn by 1/p DIu p if 1 p < k kLp ≤ ∞ δ I  (cid:16)|XI|≤k −| |(cid:17) u :=  (2.3) k kWδk,p   DIu if p = k kL∞δ I ∞ −| | |I|≤k  X    where k N , I = (I ,...,I ) Nn is a multi-index and DI = ∂I1...∂In. Here ∈ 0 1 n ∈ 0 1 n ∂ = ∂ and (x1,...,xn) are the standard Cartesian coordinates on Rn. The weighted i ∂xi Sobolev spaces are then defined as Wk,p = u Wk,p(Rn,V) u < . δ { ∈ loc |k kWδk,p ∞} Note that we have the inclusion k,p ℓ,p W W for k ℓ, δ δ (2.4) δ1 ⊂ δ2 ≥ 1 ≤ 2 6 k,p k−1,p and that differentiation ∂ : W W is continuous. In the case p = 2, we will i δ → δ−1 usethealternative notation Hk = Wk,2. Thespaces L2 andHk areHilbertspaces with δ δ δ δ inner products uv := (uv)σ−2δ−ndnx (2.5) h | iL2δ Rn | Z and uv := DIuDIv , (2.6) h | iHδk h | iL2δ I |I|≤k −| | X respectively. As with the Sobolev spaces, we can define weighted version of the bounded Ck function spaces Ck := Ck(Rn,V) Wk,∞ spaces. For a map u C0(Rn,V) and δ R, b ∩ ∈ ∈ let u := sup σ(x)−δu(x) . k kCδ0 x∈Rn| | Using this norm, we define the norm in the usual way: k·kCδk u := ∂Iu . k kCδk k kCδ0 I |I|≤k −| | X So then Ck := u Ck(Rn,V) u < . δ ∈ |k kCδk ∞ (cid:8) (cid:9) We are now ready to define asymptotically flat manifolds. Definition 2.1. Let M be a smooth, connected, n-dimensional manifold, n 3, with ≥ a Riemannian metric g and let E be the exterior region x Rn x > R . Then R { ∈ | | | } for k > n/2 and δ < 0, (M,g) is asymptotically flat of class Hk if δ (i) g Hk (M), ∈ loc (ii) there exists afinitecollection U m of opensubsets of M and diffeomorphisms { α}α=1 Φ :E U such that M U is compact, and α R α α α → \∪ (iii) for each α 1,...,m , there exists an R > 0 such that (Φ∗g) δ ∈ { } α ij − ij ∈ Hk(E ), where(x1,...,xn)arestandardCartesiancoordinates onRn andΦ∗g = δ R α (Φ∗g) dxidxj. α ij The integer m counts the number of asymptotically flat “ends” of the manifold M. As discussed in the introduction, we are interested in manifolds where M = Rn and ∼ hence m = 1. In this case, we can assume that g = g dxidxj is a Riemannian metric ij on Rn such that g δ , gij δij Hk (2.7) ij − ij − ∈ δ where gij are the components of the inverse metric, satisfying gijg = δi. We note jk k that results of this section and Theorems 3.1, 3.4, and 3.5 of the next section are are easily extended to the general case. We leave the details to the interested reader. In the following section, we will need to use diffeomorphisms generated by the flows of time-dependent vector fields and also their actions on the metric and other 7 geometrical quantities. Therefore, we need to understand the effect of composing a map in Hk(Rn,V) with a diffeomorphism on Rn. Following Cantor [5], we define δ k := ψ : Rn Rn ψ 1I Hk, ψ is bijective, and ψ−1 1I Hk Dδ { → | − ∈ δ − ∈ δ} which is the group of diffeomorphisms that are asymptotic to the identity at a rate fast enough so that the difference lies in Hk. We will need to understand not only δ when composition preserves the Hk spaces but also when composition (ψ,u) u ψ is δ 7→ ◦ continuousasamapfrom k Hk toHk. In[5], Cantorstudiedthisproblemunderthe Dδ× δ δ assumption that δ n/2. He assumed δ n/2 because that was what he needed ≤ − ≤ − to prove the weighted multiplication lemma (see Lemma A.3). However, it is clear from his arguments that the proofs of his results are valid whenever the multiplication lemma holds and Hk C1. Therefore, by Lemmata A.2 and A.3, his results are valid δ ⊂ b for δ 0. ≤ Theorem 2.2. [Corollary 1.6,[5]] For k > n/2 + 1 and δ 0, the map induced by ≤ composition Hk k Hk : (u,ψ) u ψ δ ×Dδ −→ δ 7−→ ◦ is continuous. Cantor also proved the following three useful results: Lemma 2.3. [Lemma 1.7.2,[5]] If k > n/2+1, δ 0, and f is a C1 diffeomorphism ≤ b such that f 1I Hk then f k. − ∈ δ ∈ Dδ Theorem 2.4. [Theorem 1.7,[5]] For k > n/2+1 and δ 0, k is an open subset of ≤ Dδ f :Rn Rn f 1I Hk . { → | − ∈ δ } Theorem 2.5. [Theorem 1.9,[5]] For k > n/2+1 and δ 0, k is a topological group ≤ Dδ under composition and a smooth Hilbert manifold. Also, right composition is smooth. The following proposition is a straightforward extension of Cantor’s work. Proposition 2.6. If k > n/2+1, δ 0, and u Hk+ℓ (ℓ 0) then the map ≤ ∈ δ ≥ k Hk : ψ u ψ Dδ −→ δ 7−→ ◦ is of class Cℓ. Using these results, it is not difficult to see that the proof of Theorem 3.4 of [15] generalizes to the Hk spaces with the result being: δ Theorem 2.7. Suppose δ 0, k > n/2+2 and X :( κ,κ) Rn Rn (κ > 0) defines ≤ − × → a continuous map X :( κ,κ) Hk+ℓ(Rn,Rn) : t X(t, ) (ℓ 0). − −→ δ 7−→ · ≥ Let ψ denote the flow of the time dependent vector field X(t,x) on Rn that satisfies t ψ = 1I . Then there exists a κ (0,κ) such that ψ (t ( κ ,κ )) defines a C1+ℓ 0 ∗ t ∗ ∗ ∈ ∈ − curve in k. Dδ 8 3 Local Existence 3.1 Existence of General Asymptotically Flat Ricci Flows We now prove a local existence result for Ricci flow on asymptotically flat manifolds. Theorem 3.1. Let gˆ be an asymptotically flat metric of class Hk with δ < 0 and δ k >n/2+3. Then there exists a T > 0 and a family g(t),t [0,T) of asymptotically { ∈ } flat metrics of class Hk−2 such that g(0) = gˆ, δ g δ , gij δij C1([0,T),Hk−2), ij − ij − ∈ δ and ∂ g = 2R for all t [0,T). Moreover, g(t,x) C∞((0,T) M) and g δ , t ij ij ij ij − ∈ ∈ × − gij δij C1([T ,T ],Hℓ) for any ℓ 0 and 0 < T < T < T. − ∈ 1 2 δ ≥ 1 2 Proof. Let Γ˜k denote the Christoffel symbols for the Euclidean Levi-Civita connection ij on M = Rn. Following the now standard method, see [9] Sec. 3.3, we first solve the ∼ Hamilton-DeTurck flow ∂ g = 2R + W + W , g(0) = gˆ, (3.1) t ij ij i i j j − ∇ ∇ where W = g Wk := g gpq(Γk Γ˜k ), (3.2) j jk jk pq − pq and Γk are the Christoffel symbols for the Levi-Civita connection derived from g. ij Since M = Rn, we can use global Cartesian coordinates (x1,...,xn) where Γ˜k = 0. ∼ ij With respect to the Cartesian coordinates, the initial value problem (3.1) becomes, see Lemma 2.1 in [31], ∂ h = gij∂ ∂ h + 1gpqgrs ∂ h ∂ h +2∂ h ∂ h 2∂ h ∂ h t ij i j ij 2 i pr j qs p jp q is− p jp s iq 2∂ h ∂ h 2∂ h ∂ h , (3.3) j pr s iq(cid:0) i pr s jq − − hij(0) = gˆij −δij ∈ Hδk, (cid:1) (3.4) where g = δ +h . But k > n/2+3 and δ < 0, so we can apply Theorem B.3 to ij ij ij conclude that the quasi-linear parabolic initial value problem (3.3)–(3.4) has a local solution h (t,x) that satisfies ij h , gij δij C0([0,T),Hk) C1([0,T),Hk−2) (3.5) ij − ∈ δ ∩ δ for some T > 0, h (t,x), gij(t,x) C∞((0,T) Rn), (3.6) ij ∈ × and h C1([T ,T ],Hℓ) for any ℓ 0 and 0 < T < T < T. The time-dependent ij ∈ 1 2 δ ≥ 1 2 vector field Wk is given by Wk = gijΓk = 1gijgkp ∂ h +∂ h ∂ h , (3.7) ij 2 i jp j ip − p ij and Wk defines a continuous map from [0,(cid:0)T) to Hk(Rn,Rn) by ((cid:1)3.5) and Lemma A.3. δ Note also that Wk C∞((0,T) Rn). Letting ψ (x) = (ψ1(x),...,ψn(x)) denote the ∈ × t t t flow of Wk where ψ = 1I , Theorem 2.7 implies that the map, shrinkingT if necessary, 0 9 [0,T) t ψ k−1 is C1. In particular, this implies that ψi(x) = xi+φi(x) where ∋ 7→ t ∈ Dδ t t the map [0,T) t φ Hk−1 is C1. But Wk C∞((0,T) Rn), so we also get that ∋ 7→ t ∈ δ ∈ × ψ(t,x) C∞((0,T) Rn). ∈ × Let h¯ denote the pullback of h by the diffeomorphism ψ so that t h¯ = ψ∗h = h ψ ∂ ψp∂ ψq. (3.8) ij t ij pq ◦ t i t j t (cid:0) (cid:1) (cid:0) (cid:1) Then h¯ C0([0,T),Hk−2) by Proposition 2.2 and Lemma A.3. Also, h¯ (t,x) ij ∈ δ ij ∈ C∞((0,T) Rn) by (3.6). Differentiating (3.8) with respect to t yields × ∂ ¯h = ∂ h ψ ∂ ψp∂ ψq + ∂ h ψ ∂ ψr∂ ψp∂ ψq t ij t pq ◦ t i t j t r pq ◦ t t t i t j t p q p q + h ψ ∂ ∂ ψ ∂ ψ +∂ ψ ∂ ∂ ψ . (3.9) (cid:0) pq ◦ (cid:1)t i t t j t(cid:0) i t j(cid:1)t t (cid:0) (cid:1)(cid:0) (cid:1) Using the same arguments as above, we also find that ∂ h¯ C0([0,T),Hk−2). t ij ∈ δ Finally, let g¯= ψ∗g. Then g¯ is a solution to the Ricci flow equation, see Ch. 3.3 of t [9], ∂ g¯ = 2R¯ with initial data g¯(0) = gˆ. Furthermore, t ij ij − g¯ δ = h¯ +δ ∂ ψp∂ ψq δ = h¯ +δ ∂ φp∂ φq (3.10) ij − ij ij pq i t j t − ij ij pq i t j t andhenceg¯ δ C1([0,T),Hk−2)sinceweshowedabovethat∂ φi,h¯ C1([0,T),Hk−2). ij− ij ∈ δ j t ij ∈ δ Similar arguments show that g¯ δ , g¯ij δij C1([T ,T ],Hℓ) follows from ij − ij − ∈ 1 2 δ h C1([T ,T ],Hℓ). Also,g¯ C∞((0,T) Rn)followseasilyfromh (t,x),ψ(t,x) ij ∈ 1 2 δ ij ∈ × ij ∈ C∞((0,T) Rn). × Corollary 3.2. Let k > n/2 + 4 and g(t) be the Ricci flow solution from Theorem 3.1. Then R C1([0,T),Hk−4) and g (t) = gˆ +f (t) where f C1([0,T),Hk−4). ij ∈ δ−2 ij ij ij ij ∈ δ−2 Moreover, if k > n/2+6, δ < 4 n and Rˆ L1 then R(t) C1([0,T),L1). − ∈ ∈ Proof. Leth = g δ . ThentheRiccicurvatureofghastheformR = B (gpq,∂ ∂ h ) ij ij ij ij ij ℓ m rs − +C (gpq,∂ h )whereB andC areanalytic functions thatarelinear and quadratic, ij q rs ij ij respectively, in their second variables. It follows from the weighted multiplication Lemma A.3 that the map Hℓ (gij δij,h ) R Hℓ is well defined and δ ∋ − ij 7→ ij ∈ δ−2 analytic for η 0 and ℓ > n/2. This proves the first statement. ≤ t Integrating ∂ g = 2R with respect to t yields g (t) gˆ = 2 R (s)ds. t ij − ij ij − ij − 0 ij But R C1([0,T),Hk−4), and thus the map [0,T) t 2 tR (s)ds Hk−4 is ij ∈ δ−2 ∋ 7→ − 0 ij R∈ δ−2 well defined and continuously differentiable. This completes the proof of the second R statement. The Ricci scalar satisfies the equation ∂ R = ∆R+ Ric 2. (3.11) t | | Integrating this yields R(t) = Rˆ+ t ∆R(s)+ Ric2(s) ds . From Corollary 3.2 and 0 | | the weighted multiplication lemma A.3, we see that ∆R + Ric 2 C1([0,T),Hk−6). R (cid:0) (cid:1) | | ∈ δ−4 By the weighted H¨older and Sobolev inequalities (Lemmata A.1 and A.2), we have Hk−6 L∞ L1. Thus t ∆R(s)+ Ric 2(s) ds L1 for all t [0,T). δ−4 ⊂ δ−4 ⊂ 0 | | ∈ ∈ R (cid:0) (cid:1) 10