The Yamabe flow on asymptotically flat manifolds
aa r X i v : . [ m a t h . DG ] F e b The Yamabe flow on asymptotically flatmanifolds
Eric Chen ∗ Yi Wang † Abstract
We study the Yamabe flow starting from an asymptotically flat man-ifold ( M n , g ). We show that the flow converges to an asymptoticallyflat, scalar flat metric in a weighted global sense if Y ( M, [ g ]) >
0, andshow that the flow does not converge otherwise. If the scalar curvatureis nonnegative and integrable, then the ADM mass at time infinity dropsby the limit of the total scalar curvature along the flow.
In this article we study the long-time existence and convergence of the Yamabeflow ( ∂g∂t = − Rg,g (0) = g , (1.1)starting from an asymptotically flat manifold ( M n , g ). Here R denotes thescalar curvature of the Riemannian metric g = g ( t ). This flow preserves the con-formal class of g and is the natural analogue of the volume-normalized Yamabeflow on compact manifolds introduced by Hamilton [Ham89]. He was motivatedby the resolution of the Yamabe problem due to Yamabe, Trudinger, Aubin,and Schoen [Yam60; Tru68; Aub76; Sch84], which showed that every conformalclass of Riemannian metrics on a compact manifold admits at least one metricof constant scalar curvature, known as a Yamabe metric, which minimizes theEinstein–Hilbert functional. Hamilton proposed the volume-normalized Yam-abe flow, which can be viewed as the gradient flow of the Einstein–Hilbertfunctional within a fixed conformal class, as a natural evolution equation whichcould potentially evolve a given metric on a compact manifold to one of constantscalar curvature within the same conformal class.Hamilton was already able to prove the long-time existence of the volume-normalized Yamabe flow on compact manifolds when he introduced it. Conver-gence of the flow to a metric of constant scalar curvature has now been mostly ∗ University of California, Santa Barbara, [email protected]; research partially sup-ported by an AMS–Simons Travel grant † Johns Hopkins University, [email protected]; research partially supported by NSF CA-REER Award DMS-1845033 C ∞ loc convergence to a scalar flat limit metric, using crucially theassumption of initially non-negative scalar curvature.Our work adds to the study of (1.1) in the non-compact case by providing along-time existence result in the setting of asymptotically flat manifolds withoutrequiring any additional curvature assumptions, and by showing that the flow inthis setting converges in a strong, global weighted sense (in C ∞− τ ′ ) to a scalar flat,asymptotically flat metric whenever one might hope for this — namely wheneverthere exists a scalar flat, asymptotically flat metric lying in the conformal classof the initial metric. Asymptotically flat metrics are interesting to study underYamabe flow because as shown by Cheng–Zhu, who were motivated by similarresults in the Ricci flow setting [DM07], asymptotic flatness is preserved underthe Yamabe flow and moreover the ADM mass is monotonically decreasing[CZ15]. As a consequence of our convergence results, the drop from the ADMmass along the flow to the mass of the limiting scalar flat metric is accountedfor by the total scalar curvature pushed out to spatial infinity by the flow.While we were completing this manuscript we learned of recent new work byMa [Ma21] in some directions similar to ours. His result considers the long-timeexistence of the flow, which overlaps with some of the content in our Lemma 3.4and its corollary Theorem 1.1. But the focus of our paper is to prove the strongglobal C ∞− τ ′ convergence when Y ( M n , [ g ]) >
0. Towards this goal, the centralissue is the sufficiently fast decay rate estimate of k R k L ∞ , which we achieve inseveral steps. One of the first observations is the monotonicity of the integral k R k L p for p = n/
2. This allows us to prove that k R k L p tends to zero as t → ∞ for p in a neighborhood of n/
2. Another key step is to obtain the convergenceof k R k L npn − → k R k L ∞ ≤ O ( t − − δ ) — here δ > R ∞ k R k L ∞ dt leads to the unweighted and weighted convergenceof the flow in Section 5. To better understand the complete picture, we alsostudy the flow when Y ( M n , [ g ]) ≤
0. In this remaining case, we exclude thepossibility of convergence. 2 .1 Main results
Below we describe our main results. The definition of a C k + α − τ asymptoticallyflat (AF) manifold below is the same as in [CZ15; Ma19] and this along withrelated notions of weighted H¨older and Sobolev spaces are stated precisely inSection 2.1.Our first result is the long-time existence of the Yamabe flow on all AF manifolds. On the order of asymptotic flatness we always assume that τ > Theorem 1.1.
Given any C k + α − τ AF manifold ( M n , g ) , k ≥ , there existsa Yamabe flow starting from it defined for all positive times with ( M n , g ( t )) remaining C k + α − τ ′ asymptotically flat for all τ ′ ≤ min { τ, n − } . We will not be concerned in this work with the uniqueness of Yamabe flowin the AF setting; above, Theorem 1.1 refers to any fine solution of the Yamabeflow whose short-time existence on AF manifolds was shown in [CZ15, Corollary2.5] — see Definition 2.3 for details. These are always the solutions which westudy in this work, but below we will often write the Yamabe flow with thismeaning implicit for ease of presentation.The special case R g ≥ Y ( M, [ g ]) > C α − τ version of the long-time existence result of Theorem 1.1.Here, Y ( M, [ g ]) denotes the following conformally invariant quantity whichwe call the Yamabe constant, motivated by the definition of the Yamabe con-stant in the compact case: Y ( M, [ g ]) := inf v ∈ C ∞ ( M ) ,v =0 n − n − R M |∇ v | + R g v dV g (cid:16)R | v | nn − dV g (cid:17) n − n , (1.2)which plays an important role in the prescribed scalar curvature problem on con-formal classes of AF metrics [CB81; Max05; DM18]. For instance, by [Max05,Proposition 3], which gives the correct version of a result first claimed in [CB81,Theorem 2.1], the conformal class ( M, [ g ]) admits a scalar flat, AF metric if andonly if it is Yamabe-positive — we state a version of this result in Proposition3.2 below.Having obtained long-time existence of the Yamabe flow on AF manifolds,we next study its convergence properties. On compact manifolds, proving theconvergence of the Yamabe flow required much more work than proving long-time existence — see [Ye94; SS03; Bre05; Bre07]. In our setting, if the conformalclass ( M, [ g ]) admits a scalar flat, asymptotically flat metric (which must beunique), this gives additional information with which we are able to obtainstrong quantitative decay of the scalar curvature. By our earlier discussion,we can equivalently formulate such a condition in terms of the positivity of Y ( M, [ g ]), and we have two distinct possibilities for the behavior of the flow as t → ∞ as below. 3 heorem 1.2. Let ( M n , g ) be a C k + α − τ AF manifold with k ≥ .(1) If Y ( M n , [ g ]) > , then the Yamabe flow ( M n , g ( t )) starting from ( M n , g ) converges uniformly in C k + α to the unique C k + α − τ AF metric g ∞ ∈ [ g ] as t → ∞ .(2) If Y ( M n , [ g ]) ≤ , then then the Yamabe flow ( M n , g ( t )) starting from ( M n , g ) does not converge. In particular, g ( t ) = u ( t ) n − g will fail toremain uniformly equivalent to g as t → ∞ , and both k u ( t ) k L ∞ and the L Euclidean-type Sobolev constant of g ( t ) will tend to positive infinity. The second part of Theorem 1.2 raises the question of what more can be de-termined regarding the non-convergence along the Yamabe flow on a large classof AF manifolds. In the compact case, Schwetlick–Struwe showed that a failureof convergence must imply a particular kind of infinite time bubbling behaviorfor the volume-normalized Yamabe flow [SS03]. In the non-compact, confor-mally flat setting Choi–Daskalopoulos have produced examples of Yamabe flowswith infinite-time Type II singularities, which satisfy sup M × [0 , ∞ ) | Rm( x, t ) | = ∞ [CD18]. Finite-time singularities in the non-compact, conformally flat settinghave also been studied in [CDK18; DKS19].The first part of Theorem 1.2 above gives a uniform global convergence whichis strong enough to allow us to identify the limiting metric g ∞ . If we imposesome mild restrictions on the decay order τ (which are natural for instance if wewish to consider the ADM mass under the flow), then we can further improveto weighted convergence in Theorems 1.3 and 1.4 below. Theorem 1.3.
Let ( M n , g ) be a C k + α − τ AF manifold with Y ( M, [ g ]) > , k ≥ , and τ > . Then there exists a Yamabe flow ( M n , g ( t )) starting from ( M n , g ) defined for all positive times and a metric g ∞ on M n which is C k + α − τ ′ AF for all τ ′ < min { τ, n − } so that for any such τ ′ we have k g ( t ) − g ∞ k C k + α − τ ′ = O ( t − δ ) , as t → ∞ , (1.3) for some δ > . In particular, this Yamabe flow converges in C k + α − τ ′ to theasymptotically flat, scalar flat metric g ∞ . As noted earlier, the work of Cheng–Zhu shows that under appropriate ini-tial conditions, the ADM mass of an asymptotically flat manifold with non-negative and integrable scalar curvature is nonincreasing under Yamabe flow[CZ15, Theorem 1.5], which suggests further study of the mass along the flow.Theorem 1.3 will suffice for this purpose when n ≥
4, since mass is well-definedfor τ > n − ≥
1; however, when n = 3 the condition τ > R g ≥ R g ∈ L (if we areconcerned with the mass) we can still obtain a weighted convergence result thatwill allow us to study the mass in dimension n = 3.4 heorem 1.4. Let ( M , g ) be a C k + α − τ AF manifold with Y ( M, [ g ]) > , k ≥ , τ > , R g ≥ , and R g ∈ L . Then there exists a Yamabe flow ( M n , g ( t )) starting from ( M n , g ) defined for all positive times and a metric g ∞ on M n which is C k + α − τ ′ AF for all τ ′ < min { τ, } so that for any such τ ′ we have k g ( t ) − g ∞ k C k + α − τ ′ = O ( t − δ ) , as t → ∞ , (1.4) for some δ > . In particular, this Yamabe flow converges in C k + α − τ ′ to theasymptotically flat, scalar flat metric g ∞ . It is well-known that m ( g ) ≥ m ( g ∞ ), since a scalar flat AF metric minimizesthe mass among scalar nonnegative metrics within its conformal class [SY79],and one can moreover compute this difference as a multiple of the integral of R g against u ∞ with respect to the volume form of g , where g ∞ = u n − ∞ g . Byour weighted convergence results in Theorems 1.3 and 1.4 combined with themonotonicity of mass along Yamabe flow [CZ15] and the lower semicontinuityof the ADM mass under C − τ convergence when τ > n − [Li18; MS12, Theorem14; Jau19, Theorem 13], this difference is also the mass drop at time infinity ofthe Yamabe flow starting from ( M n , g ), and is therefore controlled by the L norm of the scalar curvature as it escapes to infinity. Corollary 1.5.
For n ≥ , let ( M n , g ) be a C k + α − τ AF manifold with non-negative scalar curvature and k ≥ , τ > n − , along with R g ∈ L ( M n , g ) .Then along the Yamabe flow ( M n , g ( t )) starting from ( M n , g ) , lim t →∞ (cid:18) m ( g ( t )) − n − ω n − Z R g ( t ) dV t (cid:19) = m ( g ∞ ) . (1.5) In particular if n = 3 , , or or if τ > n − so that m ( g ( t )) is constant alongthe flow, then m ( g ) − m ( g ∞ ) = 12( n − ω n − lim t →∞ Z R g ( t ) dV t . (1.6)Since for any compact region K ⊂ M , lim t →∞ R K R g ( t ) dV g ( t ) = 0, we see in(1.6) that along the Yamabe flow, the difference between the initial mass and thelimit mass is accounted for by the total scalar curvature pushed out to infinityby the flow. Such a phenomenon has also been shown by Li to occur for theRicci flow on asymptotically flat spaces if long-time existence is assumed [Li18].We also note that although [CZ15] does not show that the mass is constantalong the Yamabe flow in general when n = 5, our convergence results give usadditional curvature control that allows us to deduce this as well from theirarguments. Theorem 1.1 is proved straightforwardly from a maximum principle argumentusing a result of Dilts–Maxwell which states that one can prescribe any strictly5egative scalar curvature in the conformal class of AF metrics associated to anyasymptotically flat ( M n , g ) [DM18], along with standard parabolic regularitytheory. The bulk of the paper which follows is devoted to the proof of Theorems1.2, 1.3, and 1.4.To prove our convergence results in Theorems 1.2, 1.3, and 1.4 we will needto establish the decay of scalar curvature at suitably fast rates. Towards thispurpose we again start with bounds on the conformal factor u ( x, t ) along Yam-abe flow using the assumption that there exists a scalar flat, AF metric in theconformal class ( M, [ g ]) (or equivalently, that Y ( M, [ g ]) > k R k L ∞ tends to zero as t → ∞ . Proposition 1.6.
Along the Yamabe flow ( M n , g ( t )) starting from a C k + α − τ AFmanifold with Y ( M n , [ g ]) > , k ≥ , and τ > , we have that sup x ∈ M n | R ( x, T ) | T →∞ −−−−→ . (1.7)After this we obtain a quantitative decay rate estimate of the L ∞ -norm of R g ( t ) along the flow. This gives rise to the following result, which is an importantstep towards deriving our desired convergence of the flow. Proposition 1.7.
Let ( M n , g ) be a C k + α − τ AF manifold with Y ( M, [ g ]) > , k ≥ , and τ > . Then for any δ < τ there exists C > such that k R k L ∞ ≤ Ct − − δ . This decay rate estimate will allow us to derive our first uniform convergenceresult, Theorem 1.2. In fact, it will also be strong enough to allow us to concludeour first weighted convergence result, Theorem 1.3, once we further assume onthe order of asymptotic flatness that τ >
1. However, as mentioned before thiscondition is too restrictive when n = 3 because it excludes many manifolds withwell-defined mass. So in this case we need another estimate which gives fasterdecay of the scalar curvature than in Proposition 1.7. By adding the natu-ral conditions of nonnegative, integrable scalar curvature, we can satisfactorilyweaken the restriction on τ . Although analogues of the following estimate canbe proven for other dimensions as well, we will only state it for dimension n = 3since Proposition 1.7 already sufficiently covers dimensions n ≥ Proposition 1.8.
In the setting of Proposition 1.7, if n = 3 , τ > , R g ≥ ,and R g ∈ L ( M n , g ) , then for all α < there exists C > such that k R k ∞ ≤ Ct − α . Together, Propositions 1.7 and 1.8 allow us to establish that u ( t ) must con-verge to some u ∞ which is asymptotic to 1 and is a conformal factor correspond-ing to a scalar flat deformation of g . We can then conclude that u ∞ must bethe conformal factor corresponding to the unique C k + α − τ scalar flat metric in theconformal class of g , and proceed to prove the weighted convergence results ofTheorems 1.3 and 1.4. 6 .3 Organization of the article The organization of the article is as follows: In Section 2 we start by recallingsome preliminaries on the short time existence of Yamabe flow on AF manifoldsas well as definitions of the relevant weighted H¨older spaces and Sobolev spaces,before proceeding in Section 3 to prove the bounds on the conformal factors u ( x, t ) needed in the rest of the paper. These bounds then allow us to prove thegeneral long-time existence result, Theorem 1.1, as well as the uniform scalarcurvature estimate of Proposition 3.6 when Y ( M n , [ g ]) >
0. In Section 4 westudy the decay of the scalar curvature under the flow when Y ( M, [ g ]) > We begin with a brief general discussion of Yamabe flow, before proceeding tointroduce background results on the short-time existence of Yamabe flow onasymptotically flat manifolds and conformal deformations.Suppose that ( M n , g ( t )) evolves according to the Yamabe flow with initialmetric g , satisfying ( ∂∂t g = − Rg,g (0) = g . (2.1)Since g ( t ) remains in the same conformal class along the flow, we may write g ( t ) = u ( t ) n − g . Then we have the following relation between R g and R g : − a ( n )∆ g u + R g u = R g u n +2 n − , (2.2)where a ( n ) = n − n − . Thus the Yamabe flow can be rewritten as an evolutionequation for the conformal factor u ( t ): ∂∂t u n +2 n − = n + 24 ( a ( n )∆ g u − R g u ) . (2.3)Below, we will often denote N = n +2 n − . We first recall some standard function spaces and related definitions used in theanalysis and definition of asymptotically flat (AF) manifolds. See for instance[Bar86; DM18].
Definition 2.1.
Let M n be a complete differentiable manifold such that thereexists a compact K ⊂ M n and a diffeomorphism Φ : M n \ K → R n \ B R (0) ,for some R > . Let r ≥ be a smooth function on M n that agrees under he identification Φ with the Euclidean radial coordinate | x | in a neighborhoodof infinity, and let ˆ g be a smooth metric on M n which is equal to the Euclideanmetric in a neighborhood of infinity under the identification Φ . Then with allquantities below computed with respect to the metric ˆ g , we have the followingfunction spaces:The weighted Lebesgue spaces L qβ ( M ) , for q ≥ and weight β ∈ R , consist ofthose locally integrable functions on M such that the following respective normsare finite: k v k L qβ ( M ) = ( (cid:0)R M | v | q r − βq − n dx (cid:1) q , q < ∞ , ess sup M (cid:0) r − β | v | (cid:1) , q = ∞ . The weighted Sobolev spaces W k,qβ ( M ) are then defined in the usual way withthe norms k v k W k,qβ ( M ) = k X j =0 (cid:13)(cid:13) D jx v (cid:13)(cid:13) L qβ − j ( M ) . The weighted C k spaces C kβ ( M ) consist of the C k functions for which thefollowing respective norms are finite: k v k C kβ ( M ) = k X j =0 sup M r − β + j (cid:12)(cid:12) D jx v (cid:12)(cid:12) . The weighted H¨older spaces C k + αβ ( M ) , α ∈ (0 , , consist of those v ∈ C kβ ( M ) for which the following respective norms are finite: k v k C k + αβ ( M ) = k v k C kβ ( M ) + sup x = y ∈ M min( r ( x ) , r ( y )) − β + k + α (cid:12)(cid:12) D kx v ( x ) − D kx v ( y ) (cid:12)(cid:12) d ( x, y ) α . Remark.
The function spaces defined above are independent of the choices of ˆ g and r . In fact, different choices of the metric ˆ g and the positive function r will produce equivalent norms. Since ˆ g agrees with the Euclidean metric in aneighborhood of infinity, we will often use δ ij to denote a choice of metric ˆ g . We can now define our precise notions of asymptotically flat metrics. Anasymptotically flat manifold is then a smooth manifold with an asymptoticallyflat metric.
Definition 2.2 (Asymptotically flat metrics) . Given M n as in Definition 2.1,a metric g is said to be a W k,q − τ (respectively C k − τ , C k + α − τ ) asymptotically flat(AF) metric if τ > and g − ˆ g ∈ W k,q − τ ( M ) (respectively C k − τ ( M ) , C k + α − τ ( M ) ) . (2.4) The number τ > is called the order of the asymptotically flat metric. .2 Short-time existence We now recall the short-time existence results for Yamabe flow on asymptoti-cally flat manifolds which will be used in this paper. The short-time existence ofthe Yamabe flow starting from an asymptotically flat manifold and the preserva-tion of asymptotic flatness along the flow have been established by Cheng–Zhufor C α − τ AF metrics (quoted in Theorem 2.4 below). However, we will studycertain higher order C k + α − τ AF metrics along the Yamabe flow; hence we willalso check that results analogous to those of Cheng–Zhu hold in these cases aswell.We start by recalling the definition of the particular kind of solution of theYamabe flow which we will consider throughout. As mentioned earlier, whetherthe Yamabe flows defined below are unique in general remains open.
Definition 2.3 ([CZ15, Definition 1.2]) . We say that g ( t ) is a fine solution ofthe Yamabe flow on a complete manifold ( M n , g ) on a maximal time interval [0 , T ) if g ( t ) = u ( t ) n − g with u (0) ≡ satisfies (1.1) and for any T ∈ (0 , T ) there exists δ = δ ( T ) and C = C ( T ) such that on [0 , T ] , < δ ≤ | u ( x, t ) | ≤ C , sup [0 ,T ] × M n |∇ g u ( x, t ) | ≤ C , sup [0 ,T ] × M n | Rm ( g ) | ( x, t ) ≤ C , and moreovereither T < ∞ and lim t → T | Rm | ( · , t ) = ∞ , or T = ∞ . Remark.
In fact by [MCZ12, Theorem 1], the blowup alternative in the defini-tion above also holds true when rewritten in terms of the scalar curvature: wemust have either T < ∞ and lim t → T | R | ( · , t ) = ∞ or T = ∞ . We now quote Cheng–Zhu’s results on the existence of fine solutions to theYamabe flow starting from C α − τ AF manifolds.
Theorem 2.4 ([CZ15, Corollary 2.5]) . If ( M n , g ) is a C α − τ AF manifold, τ > , then there exists a fine solution of the Yamabe flow starting from ( M n , g ) on a maximal time interval [0 , T ) with T > . Theorem 2.5 ([CZ15, Theorem 1.3]) . Let u ( x, t ) on ≤ t < T be the confor-mal factor corresponding to a fine solution to the Yamabe flow on a C α − τ AFmanifold ( M n , g ) with u ( · , ≡ , and let v = 1 − u . Then v ( x, t ) ∈ C α − τ ( M ) .Hence g ij ( t ) − δ ij ∈ C α − τ ( M ) for t ∈ [0 , T ) , and in particular ( M n , g ( t )) re-mains a C α − τ AF manifold along the Yamabe flow for t ∈ [0 , T ) . As mentioned before, we require the analogues of the above two results forAF manifolds with estimates also on higher order derivatives — in particular, for C k + α − τ AF manifolds, k ≥
2. Clearly if we replace C α − τ with C k + α − τ in the state-ment of Theorem 2.4 the statement remains true. Thus we can conclude withthe analogue of Theorem 2.5 below. The proof is a straightforward adaptationof Cheng–Zhu’s proof of Theorem 2.5 and is presented in Appendix A. Theorem 2.6.
Let u ( x, t ) on ≤ t < T be the conformal factor correspondingto a fine solution to the Yamabe flow on a C k + α − τ AF manifold ( M n , g ) , k ≥ , with u ( · , ≡ , and let v = 1 − u . Then v ( x, t ) ∈ C k + α − τ ( M ) . Hence g ij ( t ) − δ ij ∈ C k + α − τ ( M ) for t ∈ [0 , T ) , and in particular ( M n , g ( t )) remains a C k + α − τ AF manifold along the Yamabe flow for t ∈ [0 , T ) . Bounds on the conformal factor and long-timeexistence
In this section we recall the results of [CB81; DM18] on conformal deformationsof asymptotically flat metrics in order to obtain upper and lower bounds onthe conformal factor u ( t ) as it evolves along Yamabe flow. These bounds willthen imply the long-time existence of any fine Yamabe flow starting from anasymptotically flat manifold. Observe that if ˜ g = v n − g is a fixed metric conformal to the initial metric g on M n from which we start a Yamabe flow, then the conformal factor u ( t ) of(2.3) also satisfies for w = u ( t ) v − , ∂∂t w n +2 n − = n + 24 ( a ( n )∆ ˜ g w − R ˜ g w ) , (3.1)which is exactly (2.3) but with ˜ g and w ( t ) replacing g and u ( t ), respectively.This suggests making an advantageous choice of background metric ˜ g withwhich to study (3.1). First, Dilts–Maxwell showed that for suitable W k,p − τ AFmanifolds, it is always possible to deform to negative scalar curvature [DM18].For our purposes it is more convenient to work with C k + α − τ AF manifolds, andthe analogous statement holds in this setting as well.
Proposition 3.1.
Let ( M n , g ) be a C k + α − τ AF manifold, k ≥ with τ ∈ (0 , n − .Suppose R ′ ∈ C k + α − − τ satisfies R ′ ≤ R g . Then there exists a positive function φ with φ − ∈ C k + α − τ such that the scalar curvature of g ′ = φ n − g is R ′ . Inparticular g ′ is also a C k + α − τ AF metric.
If instead we want to conformally deform to R ′ ≡
0, then work of Cantor–Brill [CB81] (corrected and completed by Maxwell [Max05]) tells us that wecan do so for AF metrics belonging to suitable W k,p − τ classes if and only if theyare Yamabe positive [CB81; Max05]. Again, the analogous statement holds for C k + α − τ AF manifolds.
Proposition 3.2.
Let ( M n , g ) be a C k + α − τ AF manifold, k ≥ , with τ ∈ (0 , n − . Then the following are equivalent:(1) We have Y ( M, [ g ]) > .(2) There exists a positive function φ with φ − ∈ C k + α − τ such that ˜ g = φ n − g is conformally equivalent to ˜ g and R ˜ g ≡ . We will describe how Propositions 3.1 and 3.2 follow from their W k,p − τ versionsin Appendix B. 10 .2 Conformal factor bounds By making a suitable choice of background metric as detailed earlier, with theresults of Section 3.1 we can obtain some control of the conformal factor u ( t ) of(2.3) along the Yamabe flow of an asymptotically flat metric. In turn we canthen achieve some control of the Sobolev constant as defined below. Definition 3.3. If ( M n , g ) is a C − τ AF manifold, then there exists a smallestconstant C g > such that for every u ∈ W , ( M, g ) , the following L Euclidean-type Sobolev inequality holds: (cid:18)Z | u | nn − dV g (cid:19) n − n ≤ C g Z |∇ u | dV g . (3.2) We call C g the Sobolev constant of the metric g . Remark.
It is well known that C n,e ≤ C g , where C n,e is the Sobolev constantof the flat metric on R n ; see for instance [Heb99, Proposition 4.2]. We will consider two cases — first, the general case where ( M n , g ) is anarbitrary C k + α − τ AF manifold, and second, the case when moreover Y ( M n , [ g ]) >
0. In the general case, below we show that for any finite time
T >
0, theconformal factor u ( t ) is bounded away from both 0 and ∞ . Lemma 3.4. If u ( x, t ) is a solution of (2.3) corresponding to a fine solutionof the Yamabe flow starting from the C k + α − τ AF manifold ( M n , g ) , k ≥ , thenfor any T for which the Yamabe flow exists on [0 , T ] , there exists a C ( T ) > depending only on T and g such that we have the bounds < C ( T ) − ≤ u ( x, t ) ≤ C ( T ) < ∞ , (3.3) for any ( x, t ) ∈ [0 , T ] × M .Proof. First, observe that ∂∂t u = − n − R g ( t ) u ≤ − n − (cid:18) inf x ∈ M R g (0) ( x ) (cid:19) u, since inf x ∈ M R g ( t ) ( x ) is nondecreasing under the Yamabe flow. Therefore wehave u ( x, t ) ≤ C ( T ) < ∞ if the flow exists on [0 , T ] for C ( T ) > T and inf x ∈ M R g (0) ( x ).Next, using Proposition 3.1 we may write w = u ( t ) v − as in (3.1), with v corresponding to a suitable choice of prescribed R ˜ g < w , and hence u also, thereby completing the proof.Let U ⊂ M be an open set, and let U t = (0 , t ] × U , Γ t = ( { } × U ) ∪ ([0 , t ] × ∂U ). We claim that for ǫ >
0, if the Yamabe flow exists on [0 , t ],11hen the minimum of w + ǫt cannot be achieved on U t . Otherwise, at such aspace-time point ( x, t ) in U t , we have ∂∂t ( w + ǫt ) ≤
0, and0 ≥ ∂∂t ( w + ǫt ) = w − N (cid:18) n − (cid:19) ( a ( n )∆ ˜ g ( w + ǫt ) − R ˜ g w ) + ǫ> w − N (cid:18) n − (cid:19) a ( n )∆ ˜ g ( w + ǫt ) . But this is impossible, since ∆ ˜ g ( w + ǫt ) ≥ w ( x, t ) ≥ inf x ∈ M w ( x, >
0, giving us the desired lower bound.Indeed, suppose at some ( x, t ) that w ( x, t ) < inf x ∈ M w ( x, ǫ > w ( x, t ) + ǫt < inf x ∈ M w ( x,
0) at this samespace-time point. But w is asymptotic to 1 at spatial infinity in the interval[0 , t ], so by taking U sufficiently large we see that this means w + ǫt achieves aminimum in U t , which we have seen is impossible.Lemma 3.4 immediately implies the long-time existence of the Yamabe flowunder the hypotheses of Theorem 1.1. Proof of Theorem 1.1.
Suppose on the contrary that ( M n , g ( t )) is a fine solu-tion of the Yamabe flow starting from a C α − τ AF manifold ( M n , g ) whichexists up to a finite-time singularity T >
0. But Lemma 3.4 implies that theconformal factor u ( t ) is uniformly bounded on [0 , T ), and by estimating as inthe proof of Lemma 5.2 it follows by standard parabolic regularity theory that u ( t ) is uniformly bounded in C α . Hence the scalar curvature of ( M n , g ( t ))is uniformly bounded on [0 , T ), contradicting the blowup alternative for finesolutions of the Yamabe flow given in the remark following Definition 2.3.If we additionally suppose that Y ( M n , [ g ]) >
0, then we can uniformlybound the conformal factor u ( t ) both from above and from below. Lemma 3.5.
Let u ( x, t ) be the solution of (2.3) given by Theorems 2.4 and 2.6starting from the C k + α − τ AF manifold ( M n , g ) , k ≥ , and further suppose that Y ( M n , [ g ]) > . Then for any time interval [0 , T ] on which u is defined andany ( x, t ) ∈ [0 , T ] × M , there exist C , C > depending only on g such that < C ≤ u ( x, t ) ≤ C < ∞ . (3.4) Proof.
We already saw in the proof of Lemma 3.4 how to obtain the lower bound0 < C ≤ u ( x, t ), so we only need to justify the upper bound.Write w = u ( t ) v − as in (3.1), with v given by the conformal factor cor-responding to R ˜ g ≡ ǫ >
0, the max-imum of w − ǫt cannot be achieved on U t . Otherwise, at such a point in U t ,12e have ∂∂t ( w − ǫt ) ≥
0, and therefore0 ≤ ∂∂t ( w − ǫt ) = w − N (cid:18) n − (cid:19) a ( n )∆ ˜ g ( w − ǫt ) − ǫ< w − N (cid:18) n − (cid:19) a ( n )∆ ˜ g ( w − ǫt ) . But ∆( w − ǫt ) ≤ x, t ) such that w ( x, t ) > sup y ∈ M w (0 , y ), then u ( x, t ) − ǫt > sup y ∈ R n u (0 , y ) for ǫ > u is asymptotic to 1 at spatial infinity, if we take U sufficiently large,then u − ǫt achieves a maximum in U t , which we have seen is impossible.Lemma 3.5 then implies the uniform boundednes of the scalar curvature forall positive times by the same argument used to prove Theorem 1.1. We willneed this fact later. Proposition 3.6.
Let ( M n , g ) be a C k + α − τ AF manifold with Y ( M, [ g ]) > , k ≥ , and τ > . Then the Yamabe flow starting from ( M n , g ) has scalarcurvature uniformly bounded in time for all t > .Proof. Because of the uniform bounds from Lemma 3.5, standard parabolicregularity theory applied to the evolution equation of u ( t ) implies that thescalar curvature R is uniformly bounded for all t > Y ( M n , [ g ]) > M n , g ( t )) for all positive times. This will allow usto study the convergence of the flow as t → ∞ . Corollary 3.7. If u ( x, t ) is the solution of (2.3) corresponding to a fine solutionof the Yamabe flow starting from the C k + α − τ AF manifold ( M n , g ) , k ≥ , andmoreover Y ( M, [ g ]) > , then there exists a constant D = D ( g ) such that forany T for which the Yamabe flow exists on [0 , T ] the following Sobolev inequalityholds for every u ∈ W , ( M, g ( t )) , : (cid:18)Z | u | nn − dV g ( t ) (cid:19) n − n ≤ D Z |∇ u | dV g ( t ) . (3.5) We now study the evolution of the scalar curvature R along the Yamabe flowstarting from an asymptotically flat manifold in the Y ( M n , [ g ]) > Y ( M n , [ g ]) > L R , and then proceed to establishthe decay rate estimates of Proposition 1.7 and 1.8 on the L ∞ norm of R intime. We have the following equations for the evolution of the scalar curvature andthe volume form under the Yamabe flow: ∂∂t R = ( n − R + R , ∂∂t dV t = − n R dV t Under the hypotheses of Theorem 1.3, we can therefore compute the evolu-tion of k R k L p along the Yamabe flow for p sufficiently large. Lemma 4.1.
Let ( M n , g ) be a C k + α − τ AF manifold with Y ( M, [ g ]) > , k ≥ and τ ∈ (0 , n − . Then for all p > n τ , along the Yamabe flow starting from ( M n , g ) we have ddt Z | R | p dV t ≤ − n − p − p Z |∇| R | p | dV t + (cid:16) p − n (cid:17) Z | R | p R dV t . (4.1) Proof.
This follows from ∂∂t R = ( n − R − n − |∇ R | + 2 R , and ddt Z | R | p dV t = Z p R ) p − ∂∂t R − n | R | p R dV t , (4.2)since by our assumptions on the asymptotic decay of ( M n , g ) which are pre-served along the flow by Theorem 2.6 (and indeed for all positive times byTheorem 1.1) these integral quantities are well-defined.Since C k + α − ˜ τ AF manifolds are also C k + α − τ if ˜ τ > τ , for the purposes of provingTheorem 1.3 it suffices to assume that τ < n −
2, in which case we always have p > n τ >
1. Therefore we will always be assuming that τ < n − k R k p is a monotonically nonincreasingquantity along the flow, for appropriate p . Corollary 4.2.
We have that ddt R | R | n dV t ≤ . As a result we also have monotonicity of ddt R | R | p dV t , for p close to n , aswell as the integrability in time of certain other L p norms of R in space, whichwill be important when we establish the L ∞ decay of R .14 emma 4.3. There exists ǫ = ǫ ( g ) > such that ddt R | R | p dV t ≤ for all p ∈ (cid:0) n − ǫ, n + ǫ (cid:1) . Moreover for all such p , Z ∞ (cid:18)Z | R | p nn − dV t (cid:19) n − n dt < ∞ . (4.3) Proof.
Applying the Sobolev inequality (3.5) to the first term and the H¨olderinequality to the second term on the right-hand side of (4.1), we obtain ddt Z | R | p dV t ≤ − C ( n, p ) D (cid:18)Z | R | p nn − dV t (cid:19) n − n (4.4)+ (cid:12)(cid:12)(cid:12) p − n (cid:12)(cid:12)(cid:12) (cid:18)Z | R | p nn − dV t (cid:19) n − n (cid:18)Z | R | n dV t (cid:19) n . Hence there exists an ǫ > p − n < ǫ then − C ( n, p ) K + (cid:12)(cid:12)(cid:12) p − n (cid:12)(cid:12)(cid:12) k R g k L n < , (4.5)which implies that ddt R | R | p dV t ≤
0, since we know that ddt R | R | p dV t is nonin-creasing.For such p , we additionally see that (4.4) implies ddt Z | R | p dV t + C (cid:18)Z | R | p nn − dV t (cid:19) n − n ≤ , for some C which may depend on n , p , and g . Since R | R | p dV t is nonincreasingand nonnegative, we may integrate this inequality to deduce (4.3). Remark.
In order to prove Proposition 1.7 we actually need the monotonicity inCorollary 4.3 to hold for all p > n τ (for sufficiently large times). But to justifythis fact we will need to demonstrate the decay of k R k ∞ first in Proposition 4.4below, before returning to this in Corollary 4.6. In the above discussions, if p < n is small) then one maywish to be careful with the |∇| R | p | integrand in (4.1). We check that thediscussion in this section leading up to Lemma 4.3 still holds in this context inAppendix C. R Having obtained the decay of appropriate integral norms of R , we can now pro-ceed to control the L ∞ norm of R along the flow by Moser iteration argumentssimilar to those in [Yan88]. The central ideas are standard, but we include somedetails for clarity because we need to adapt them to obtain precise control onthe decay rate of R , similar to the situation for | Rm | on certain asymptoticallyflat Ricci flows studied by the first named author in [Che19].In fact we will need to pass from L p to L ∞ control of R several times, so webegin with the following estimate with somewhat general assumptions.15 roposition 4.4. Let ( M n , g ( t )) be a Yamabe flow starting from C k + α − τ AFmanifold with Y ( M, [ g ]) > , k ≥ , and τ ∈ (0 , n − , and suppose that k R k L q ≤ α t − γ , for q > n , (4.6) k R k L p ≤ α t − γ , for p > n τ , (4.7) for some nonnegative constants α , α , γ , γ . Then there exists a constant C = C ( n, q, p , α , α , g ) > such that sup x ∈ M n | R ( x, T ) | ≤ C max n T p − γ q ( n +2) p q − n ) − γ , T − n p − γ o . (4.8) Proof.
We selectively denote f = | R | below in order to distinguish the roles thatdifferent factors of | R | play. Applying H¨older’s inequality to the second term onthe right in (4.1), we see that for any p > n τ , q > n , and δ > p ddt Z f p dV t ≤ − p − p ( n − Z |∇ f p | dV t + (cid:18)Z | R | q dV t (cid:19) q (cid:18) δ − n q Z f p dV t (cid:19) − n q (cid:18) δ ( − n q ) nn − Z f p nn − dV t (cid:19) n − n n q . Note that the above in fact holds for all p >
1, assuming the integrability ofall terms involved. We restrict to p > n τ because of the spatial decay of f = | R | on a C k + α − τ AF manifold, recalling that we assume τ < n −
2. Let β = β ( t ) = α t − γ , so that k R k L q ≤ β ( t ). We apply Young’s inequality to thelast term on the right to see that1 p ddt Z f p dV t ≤ − p − p ( n − Z |∇ ( f p ) | + βδ − n q Z f p dV t + βδ − n q (cid:18)Z f p nn − dV t (cid:19) n − n . Applying the Sobolev inequality (3.5) to the last term on the right aboveand setting δ = (cid:16) p − n − βDp (cid:17) q q − n , we obtain for all p ≥ p ddt Z f p dV t + C p Z |∇ ( f p ) | dV t ≤ C p,q,β Z f p dV t , (4.9)where C p = ( p − n − p and C p,q,β = pβ (cid:16) βDp p − n − (cid:17) n q − n . Now define, for T < τ < τ ′ < T , the function ψ : [0 , T ] → [0 , ψ ( t ) = , ≤ t ≤ τ, t − ττ ′ − τ , τ ≤ t ≤ τ ′ , , τ ′ ≤ t ≤ T. ψ and find that ddt (cid:18) ψ Z f p dV t (cid:19) + ψC p Z |∇ ( f p ) | dV t ≤ ( C p,q,β ψ + ψ ′ ) Z f p dV t , so that integrating and using the fact that β ( t ) is decreasing, for any ˜ t ∈ [ τ ′ , T ]we have Z f p dV ˜ t + C p Z ˜ tτ ′ Z |∇ ( f p ) | dV t dt ≤ (cid:18) C p,q,β ( τ ) + 1 τ ′ − τ (cid:19) Z Tτ Z f p dV t dt. (4.10)We now define for τ ∈ [0 , T ], H ( p, τ ) = Z Tτ Z f p dV t dt, and let ν = 1 + n . We claim that for p > n τ and 0 ≤ τ < τ ′ ≤ T , H ( νp, τ ′ ) ≤ DC p (cid:18) C p,q,β ( τ ) + 1 τ ′ − τ (cid:19) ν H ( p, τ ) ν . (4.11)Indeed, Z Tτ ′ Z f νp dV t dt ≤ Z Tτ ′ (cid:18)Z f p dV t (cid:19) n (cid:18)Z f p nn − dV t (cid:19) n − n dt ≤ D (cid:18) sup τ ′ ≤ t ≤ T Z f p dV t (cid:19) n Z Tτ ′ Z |∇ ( f p ) | dV t dt, so that (4.10) implies the claim. Now we iterate (4.11) to obtain L ∞ control.Let p be as assumed in (4.7), and define η = ν q q − n , p k = ν k p , τ k = T − η − k ) T , Φ k = H ( p k , τ k ) pk . (4.12)We apply (4.11) to see thatΦ k +1 = H ( νp k , τ k +1 ) νpk (4.13) ≤ (cid:18) DC p (cid:19) νpk (cid:18) C p k ,q,β ( τ k ) + 1 τ k +1 − τ k (cid:19) pk H ( p k , τ k ) pk ≤ (cid:18) DC p (cid:19) νpk (cid:16) C ( n, q, p ) D n q − n β ( τ k ) q q − n + C ( n, q ) ηT (cid:17) pk η kpk Φ k ≤ (cid:18) DC p (cid:19) νpk (cid:18) C ( n, q, p ) D n q − n (cid:16) α (cid:17) q q − n T − γ q q − n + C ( n, q ) ηT (cid:19) pk η kpk Φ k , C ( n, q, p ) = p q q − n (cid:16) p p − n − (cid:17) n q − n and C ( n, q ) = η − . Since P ∞ k =0 1 p k = p n +22 < ∞ and P ∞ k =0 kp k < ∞ , we can iterate (4.13) to obtainsup x ∈ M n | R ( x, T ) | ≤ C ( n, q, p , α , D ) max( T − γ q q − n , T − ) p n +22 Φ . (4.14)Finally, we have thatΦ = Z T T Z | R | p dV t dt ! p ≤ C ( α ) T − γ + p . Putting things together we obtain the claimed estimate for | R | .Next, by slightly modifying the proof of Proposition 4.4 we will establishProposition 1.6, which shows that k R k L ∞ decays to zero in time. Proof of Proposition 1.6.
Substituting p = n into (4.3) from Lemma 4.3, wehave Z ∞ (cid:18)Z | R | n nn − dV t (cid:19) n − n dt < ∞ . (4.15)Moreover, R | R | n nn − dV t is uniformly bounded along the flow, since k R k L ∞ and k R k L n are both uniformly bounded by Proposition 3.6 and Corollary 4.2,respectively. Hence Z ∞ Z | R | n nn − dV t < ∞ . (4.16)We now refer back to the proof of Proposition 4.4. Choose q > n so that k R k L q ≤ α for some constant α = α ( g ), which is possible by Lemma 4.3,and set p = n nn − . Following the same steps up to (4.14) we obtain, with γ = 0, that sup x ∈ M n | R ( x, T ) | ≤ C ( n, q, α , D ) max(1 , T − n +22 p )Φ . Then, since Φ = Z T T Z | R | p dV t dt ! p T →∞ −−−−→ , we find that indeed k R k L ∞ → k R k L p is uniformly bounded along the flow for p closeto n , Proposition 1.6 allows us interpolate to obtain decay of integral norms of R as well. 18 orollary 4.5. For ǫ = ǫ ( g ) > as in Lemma 4.3 and p > n − ǫ , we have k R k L p t →∞ −−−→ . In particular, we have k R k L n t →∞ −−−→
0. This allows us to strengthen therange of exponents covered by Lemma 4.3.
Corollary 4.6.
For any p > n τ , we have ddt R | R | p dV t ≤ for t ≥ T ( p ) sufficiently large. Moreover for all such p , (cid:18)Z | R | p nn − dV t (cid:19) n − n = o ( t − ) . (4.17) Proof.
In the proof of Lemma 4.3 observe that for any fixed p , the inequality(4.5) will hold when k R g k n is replaced by k R g ( t ) k n for all t sufficiently large,since we now know that k R k n decays to zero in time. And because we now knowthat R | R | p nn − dV t is monotonically nonincreasing in time for t sufficiently large,the integrability expressed in (4.2) implies (4.17).We will use these consequences of Proposition 1.6 to strengthen our controlof R and obtain our decay rate estimates on k R k ∞ . R Although Proposition 1.6 only told us that k R k ∞ tends to zero, we can use theconsequent improved integral decay estimates of Corollary 4.5 and 4.6 to showthat the L ∞ norm of R must in fact decay at a particular rate, thus provingProposition 1.7. Proof of Proposition 1.7.
By Corollary 4.6, we have that for any p > n τ and α >
0, for sufficiently large times it holds that k R k L p nn − ≤ αt − p . Choose p , p > n τ with p nn − > n , and apply the estimate (4.8) of Proposition 4.4with q = p nn − and p = p nn − . We thus obtain the estimate (for T sufficientlylarge),sup x ∈ M n | R ( x, T ) | ≤ C ( n, q, p , α , α , g ) max (cid:26) T n qnp n − q ) , T − n p (cid:27) . (4.18)We now compare the two T exponents above to see that we can achieve thedecay exponent in the statement of the Proposition. For the second exponent,clearly as p approaches n τ from above the exponent − n p approaches − − τ from above. For the first exponent, notice that for fixed p > q → n n +4 qnp ( n − q ) = −∞ . If τ ≥ n − then n ≥ nn − · n τ , so we will alwaysbe able to choose q > n sufficiently close to n such that n +4 qnp ( n − q ) < − n p .On the other hand, if 0 < τ < n − then the following inequality holds at q = n τ nn − > n : n + 4 q np ( n − q ) < − n p . q with a q slightly larger than q , finishing the proof.As described in the Introduction, the decay in Proposition 1.7 is not quitegood enough for our purposes when n = 3, so we need the improved estimate ofProposition 1.8 for dimension n = 3 under additional assumptions of nonnega-tive, integrable scalar curvature. We conclude this section with its proof. Proof of Proposition 1.8.
By [CZ15, Theorem 1.5], under the assumptions ofProposition 1.8, the scalar curvature remains integrable along the flow andmoreover R R dV t is nonincreasing; in fact for dimension n = 3 we have ddt Z R dV t ≤ − Z R dV t ≤ . So we may interpolate with the L ∞ estimate of Proposition 1.7 to find that k R k p < ∞ for all p ≥
1, and moreover k R k p = O (cid:16) t − (1+ δ ) ( − p ) (cid:17) , (4.19)for all δ < τ . By our hypothesis on τ then the evolution equation for ddt R R p dV t is well-defined whenever p >
1, so we can apply the estimate (4.8) of Proposition4.4 with q = p nn − = 3 p and p = p , for some p > τ and p > k R k q ≤ α t − p and (4.19) to estimate k R k p , weobtainsup x ∈ M n | R ( x, T ) |≤ C ( n, q, p, α , , g ) max n T p − n ( n +2)( n − p (2 q − n ) − (1+ δ ) ( − p ) , T − n p − (1+ δ ) ( − p ) o . (4.20)It suffices to show that given any α < we can choose q, p so that bothexponents of T are less than − α . For the first exponent, setting p = 1 and n = 3 we obtain 1 − q − n < − < − , if we set q = 3 · τ , using that τ > . For the second exponent, setting p = 1gives us exactly − . Therefore by setting p > q > · τ closeto 3 · τ we obtain our desired estimate. Using the decay rate estimates on the scalar curvature of ( M n , g ( t )) evolvingunder the Yamabe flow from an initial Yamabe-positive asymptotically flat met-ric satisfying the hypotheses of Proposition 1.7 or 1.8, we proceed in this section20o prove Theorem 1.3. Below we start with the unweighted convergence of theconformal factor u ( t ) before proceeding to study its weighted convergence as t → ∞ . We first show that the decay rate estimate of Proposition 1.7 gives us uniformconvergence to a limiting continuous function u ∞ . Lemma 5.1.
Under a Yamabe flow satisfying the hypotheses of Proposition 1.7,there exists a continuous function u ∞ ( x ) > on M such that k u ( x, t ) − u ∞ ( x ) k L ∞ ( M ) ≤ Ct δ , (5.1) and u ∞ ( x ) − → , as r → ∞ , (5.2) for all δ < τ . If the flow additionally satisfies the hypotheses of Proposition 1.8,then this holds for all δ < . (Recall from Definition 2.1 that r ≥ is a smoothfunction that agrees with | x | in asymptotic coordinates on M n .)Proof. Since ∂∂t u = − n − Ru , we have for all x ∈ M n that u ( x, t ) = e R t − n − R ( x,t ) dt u ( x, . By Proposition 1.7, for any 0 < δ < τ , | R ( x, t ) | ≤ C (1 + t ) δ . Thus for t , t >> | u ( x, t ) − u ( x, t ) | ≤ C | e c − δ · tδ − e c − δ · tδ | · u ( x, ≤ C | t δ − t δ | · | u ( x, | → , as t , t → ∞ . This implies u ( x, t ) is a Cauchy sequence in L ∞ ( M ) as t → ∞ . We then imme-diately conclude that there exists a limiting function u ∞ ( x ) ∈ L ∞ ( M ) whichsatisfies k u ( x, t ) − u ∞ ( x ) k L ∞ ( M ) ≤ Ct δ . And since for each time t > | u ( x, t ) − | ≤ Cr τ , as r → ∞ , (5.3)we deduce (5.2).If we are in the setting of Proposition 1.8, then the entire argument abovecarries through but with δ < . 21ote that we could not yet conclude above that | u ∞ ( x ) − | ≤ Cr τ , as r → ∞ , because the bound C in (5.3) is not uniform in t >
0. But in the next twopropositions we will be able to first show that u n − ∞ g is a scalar flat metric, andthen show that u ∞ ( x ) − Proposition 5.2.
Under a Yamabe flow satisfying the hypotheses of Proposition1.7, for all < α ′ < α , k u ( x, t ) − u ∞ ( x ) k C k + α ′ loc ≤ Ct δ , (5.4) for any < δ < τ / . Moreover, u ∞ ∈ C k + αloc ( M ) and g ∞ = u ∞ ( x ) n − g is ascalar flat metric, i.e. ∆ g u ∞ ( x ) − a ( n ) R g ( x ) u ∞ ( x ) = 0 . Recall a ( n ) = n − n − . If the flow additionally satisfies the hypotheses of Propo-sition 1.8, then (5.4) holds for all δ < .Proof. First recall that ∂∂t u = ( n − u N − ∆ g u − a ( n ) R g u N − ! . Since we have uniform and positive upper and lower bounds on u by Lemma3.5, and since R g ∈ C k − α ( M ) we see that (cid:13)(cid:13)(cid:13)(cid:13) R g u N − (cid:13)(cid:13)(cid:13)(cid:13) L ∞ loc ( M ) ≤ C, for all t ∈ [0 , ∞ ) . Thus for all p ≥
1, on any compact subset Ω ⊂⊂ M , k R g u N − k L p (Ω × [ t ,t +1]) ≤ C, for all t ∈ [0 , ∞ ) . k u k L ∞ (Ω × [ t ,t +1]) ≤ C, for all t ∈ [0 , ∞ ) . We now apply the standard Schauder estimates for parabolic equations to seethat u is in W , ,ploc (Ω × [ t , t + 1]) and satisfies k u k W , ,p (Ω × [ t ,t +1]) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) R g u N − (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω × [ t ,t +1]) + k u k L ∞ (Ω × [ t ,t +1]) ! ≤ C ,p , for all t ∈ [0 , ∞ ) . k u k C α, α (Ω × [ t ,t +1]) ≤ C ,α . (5.5)where we have chosen p > n + 2 so that 1 − n +2 p = α . The embedding constantdepends on n, p, diam(Ω) − , ˆ g and the length of time interval (which is 1 in ourcase). In particular, it is independent of t .To derive higher order local regularity of u , we use an induction argument toprove that u ∈ C k + α, k + α (Ω × [ t , t +1]). Assuming u ∈ C l + α, l + α (Ω × [ t , t +1]),we want to show that u ∈ C l +2+ α, l +2+ α (Ω × [ t , t + 1]) if l ≤ k −
2. From (5.5),this is already proved for l = 0 and 1. Since R g ∈ C k − αloc , by a productestimate for parabolic H¨older spaces we find for such l that (cid:13)(cid:13)(cid:13)(cid:13) R g u N − (cid:13)(cid:13)(cid:13)(cid:13) C l + α, l + α (Ω × [ t ,t +1]) ≤ C, for all t ∈ [0 , ∞ ) . Therefore by higher order Schauder estimates for parabolic equations, u ∈ C l +2+ α, l +2+ α (Ω × [ t , t + 1]) and satisfies k u k C l +2+ α,l +2+ α (Ω × [ t ,t +1]) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) R g u N − (cid:13)(cid:13)(cid:13)(cid:13) C l + α, l + α (Ω × [ t ,t +1]) + k u k L ∞ (Ω × [ t ,t +1]) ! ≤ C l +2 ,α , for all t ∈ [0 , ∞ ) . The induction argument stops when l = k − u ∈ C k + α, k + α (Ω × [ t , t + 1])). Taking j = 0and t = t in Definition A.1, we obtain k u ( x, t ) k C k + α (Ω) ≤ C k + α , for all t ∈ [0 , ∞ ) . Note that C k,α depends on Ω , k, α , and is independent of t , so for every 0 <α ′ < α and every sequence { t j } → ∞ , there exist a subsequence { t j k } → ∞ and a limiting function ˜ u ∞ ( x ) such that u ( x, t j k ) → ˜ u ∞ ( x ) in C k + α ′ loc , as t j → ∞ , (5.6)by the Arzela–Ascoli theorem. But by Lemma 5.1, we also have u ( x, t ) → u ∞ ( x ) in L ∞ ( M ) , as t → ∞ . Thus u ∞ ≡ ˜ u ∞ , and this implies u ( x, t ) → u ∞ ( x ) in C k + α ′ loc , as t → ∞ , k ≥
2, we can pass the limit in the scalar curvature equation to seethat u ∞ ( x ) n − g is a scalar flat metric, i.e.∆ g u ∞ − a ( n ) R g u ∞ = 0 . Finally, we claim that u ∞ ( x ) ∈ C k + αloc , even though the sequence does notconverge in this space as in (5.6). This is because the C k + α (Ω) norm is lowersemicontinuous. Namely, for any compact subset Ω ⊂⊂ M , k u ∞ ( x ) k C k + α (Ω) ≤ lim t →∞ k u ( x, t ) k C k + α (Ω) ≤ C k,α . We now have enough information to conclude that u ∞ is in fact the conformalfactor φ from Proposition 3.1 corresponding to the scalar-flat metric ˜ g . Proposition 5.3.
We have u ∞ − ∈ C k + α − τ . In particular, | u ∞ ( x ) − | ≤ Cr τ , as r → ∞ , (5.7) and | ∂ j u ∞ ( x ) | ≤ Cr τ , as r → ∞ . (5.8) Proof.
By Proposition 3.2 there exists a positive function φ with φ − ∈ C k + α − τ such that ˜ g = φ n − g scalar flat. Since u n − ∞ g is also scalar flat, the function w := u ∞ · φ − will satisfy ∆ ˜ g w ( x ) = 0 . Moreover, by Lemma 5.1. w ( x ) − → r → ∞ . Thus the maximum principleasserts that w ≡
1, so that u ∞ − φ − ∈ C k + α − τ .We can now prove Theorem 1.2. Proof of Theorem 1.2.
Part (1) of Theorem 1.2 follows immediately from com-bining the results of Propositions 5.2 and 5.3.To see Part (2), notice that all we have used to prove Part (1) is that u ( t ) remains uniformly bounded from above and below for all times. So if Y ( M, [ g ]) ≤ u ( t ) remains uniformly bounded from above and below,then Lemma 5.1 and Proposition 5.2 again show that u ( t ) converges uniformlyto some u ∞ ∈ C k + αloc ( M ), asymptotic to 1 at spatial infinity. But by the confor-mal invariance of the Yamabe constant, we have Y ( M, [ g ]) = inf v ∈ C ∞ ( M ) ,v =0 n − n − R M |∇ g ∞ v | dV g ∞ (cid:16)R | v | nn − dV g ∞ (cid:17) n − n = 1 C g ∞ > , (5.9)24here the L Euclidean-type Sobolev constant C g ∞ of g ∞ exists because u ∞ isasymptotic to 1 at spatial infinity. This contradicts our initial assumption onthe Yamabe constant of ( M, [ g ]). Therefore from the proof of Lemma 3.4 wesee that we must in fact have sup x ∈ M u ( x, t ) t →∞ −−−→ ∞ .It remains to show that the L Euclidean-type Sobolev constant of (
M, g ( t ))also blows up as t → ∞ . If not, then we have uniform control of this constant,which allows us to carry out nearly all the arguments of Section 4 (without anyassumption on the boundedness of u ( t )) until Proposition 1.6, where we passedfrom (4.15) to (4.16) using the L ∞ bound of R , which was a consequence of theuniform bounds on u established when Y ( M, [ g ]) >
0. But we can recover theboundedness of R simply with the bound on the Sobolev constant by pluggingthe uniform bound of k R k L p for p slightly larger than n from Lemma 4.3 into theestimate of Proposition 4.4. In fact the constant in that estimate only dependson the Sobolev constant bound D of g , so since we are now assuming controlof the Sobolev constant we obtain the same estimates for k R k ∞ on the intervals[ T, T + 1] for all T ∈ Z ≥ , and hence uniform bound of R for all times. As aresult, we can continue through the same arguments we used to prove u ( t ) isunbounded earlier. Note that although Proposition 5.2 uses the boundednessof u ( t ), we do actually obtain this beforehand when we integrate in provingLemma 5.1. We will now prove the weighted convergence of Yamabe flows starting fromAF manifolds with Y ( M n , [ g ]) >
0, and need to assume for the rest of thissection that τ > n = 3, τ > , R g ≥
0, and R g ∈ L ( M n , g ) as in Theorem 1.4. The difference between the proofs ofTheorems 1.3 and 1.4 occurs with the distinct decay rate estimates, Propositions1.7 and 1.8, required to prove Proposition 5.4 below in the respective settings ofthe two Theorems. In particular, the decay rate of R from Proposition 1.7 aloneis not enough to give us the estimates of Lemma 5.5 when n = 3 and τ > , sofor this case we must bring in the improved estimate of Proposition 1.8.Using a strategy similar to that used to prove [Li18, Lemma 5.2] in a Ricciflow setting, we now prove below Proposition 5.4, a decay estimate for the scalarcurvature in both space and time. In fact, once we have this, the rest of theproofs of Theorems 1.3 and 1.4 are the same, and we will complete the proofsof both together at the end of this section. Proposition 5.4.
Under the hypotheses of either Theorem 1.3 or Theorem 1.4,for any τ ′ < τ there exists some δ > and C > depending only on g suchthat | R ( x, t ) | ≤ Cr τ ′ (1 + t ) δ (5.10) for all ( x, t ) ∈ M × [0 , ∞ ) along the flow. roof. Given τ ′ < τ , choose σ , σ such that τ ′ < σ < σ < τ . Let δ = σ ,and consider D := { ( x, t ) ∈ M × [0 , ∞ ) : r ( x ) ≥ t a } where a > / x, t ) = D , by Proposition 1.7 | R | ≤ Ct − − δ ≤ Ct − − η r − τ ′ for some η >
0, when a > / / | R | ≤ Cr − − σ for ( x, t ) ∈ D holds. In fact, we define h := r σ and w = h · | R | . Then w satisfies theevolution equation (cid:18) ∂∂t − ( n − (cid:19) w ≤ ( n − Bw − ∇ log h · ∇ w ) + 2 h | R | . (5.11)Here B := (2 |∇ h | − h ∆ h ) h . From Lemma 5.5 below, for any β < we can bound B uniformly by C ( r − + r − t − β ) ≤ Ct − a − on D , by choosing β suffiently closeto . By Proposition 1.7, h | R | ≤ w · t − − δ = w · t − − σ . Thus (5.11) becomes (cid:18) ∂∂t − ( n − (cid:19) w ≤ − n − ∇ log h · ∇ w + Cw · t − − δ ′ . where δ ′ := min { a − , σ } > ∂D , we have | R | ≤ Ct − − σ = Cr − (1+ σ ) /a ≤ Cr − − σ for a sufficiently close to 1 /
2. So we can apply the maximum principle on thenoncompact manifold as in [Cho+08, Theorem 12.14] to conclude the claimholds. (See also the statement in [Li18, Theorem 2.1].)Therefore on D we have | R | ≤ Cr − − σ ≤ Ct − − δ r − τ ′ , for some δ > a sufficiently close to 1 / Lemma 5.5.
Under the hypotheses of either Theorem 1.3 or Theorem 1.4, for r ( x ) >> , t >> and any β < , |∇ g ( t ) h | g ( t ) ≤ C |∇ g h | g (0) ≤ Chr , (5.12) | ∆ g ( t ) h | ≤ Chr · (cid:18) r + 1 t β (cid:19) . (5.13)26 roof. (5.12) is straightforward as u is uniformly bounded by Lemma 3.5 and |∇ g ( t ) h | g ( t ) = u − n − |∇ g h | g (0) . Regarding (5.13), we consider∆ g ( t ) h = u − n − ∆ g h + 2 u − n +2 n − g ij ∂ i h∂ j u. (5.14)We know from calculation ∆ g h ≤ Chr , (5.15)and ∂ i h ≤ Chr . (5.16)The next step is to estimate ∂ j u . From the proof of Proposition 5.2, formula(5.4), we see k u ( x, t ) − u ∞ ( x ) k C k + α ′ loc ≤ Ct δ , for any 0 < δ < τ /
2. On the other hand, by (5.7) from Proposition 5.3, we have | ∂ j u ∞ ( x ) | ≤ Cr τ . Recall that in the n = 3 case where we imposed additionalassumptions on R g we can replace the restriction on δ by 0 < δ < , and thisalso holds when n ≥ τ > n ≥ β < that | ∂ j u ( x, t ) | ≤ C (cid:18) t β + 1 r τ (cid:19) ≤ C (cid:18) t β + 1 r (cid:19) . (5.17)Then, plugging (5.15), (5.16), (5.17) into (5.14), we complete the proof of(5.13).Now that we have weighted decay of the scalar curvature, by integrating wecan start to prove the weighted convergence of the conformal factor u ( t ). Proposition 5.6.
Under the hypotheses of either Theorem 1.3 or Theorem 1.4,for any τ ′ < τ we have u ( x, t ) − u ∞ ( x ) → in C − τ ′ , as t → ∞ , where u ∞ ∈ C k + α − τ is as in Proposition 5.3. Moreover, there is a δ > suchthat k u ( x, t ) − u ∞ ( x ) k C − τ ′ ≤ Ct δ . (5.18)27 roof. Since ∂∂t u = − n − Ru , we have u ( x, t ) = e R t − n − R ( x,t ) dt u ( x, . (5.19)By Proposition 5.4, | R ( x, t ) | ≤ Cr τ ′ (1 + t ) δ . (5.20)Thus for t , t >> | u ( x, t ) − u ( x, t ) | ≤ Ce rτ ′ · | e c − δ · tδ − e c − δ · tδ | · u ( x, ≤ C r τ ′ · | t δ − t δ | · | u ( x, | . This implies r τ ′ · | u ( x, t ) − u ( x, t ) | ≤ C | t δ − t δ | → , as t , t → ∞ . Note that u ( x, t ) − ∈ C − τ ′ for every t > u ( x, t ) − C − τ ′ as t → ∞ . We may then conclude that k u ( x, t ) − u ∞ ( x ) k C − τ ′ ≤ Ct δ . Before proceeding to the proof of Theorem 1.3, we need to establish someuniform control of the conformal factor u ( t ) in some parabolic weighted Sobolevand H¨older spaces. Lemma 5.7.
Under the hypotheses of either Theorem 1.3 or Theorem 1.4, andfor any τ ′ < τ :(1) For any p > , there exists a constant C > such that for any < τ < τ ′ and all t ≥ . k u − k W k,k/ ,p − τ ( M × [ t ,t +1]) ≤ C. (2) There exists a constant C > , such that for all t ≥ , k u − k C k + α, k + α − τ ′ ( M × [ t ,t +1]) ≤ C. Here C is independent of time t . roof. To estimate k u − k W k,k/ ,p − τ ( M × [ t ,t +1]) , we observe that u − N u N − ∂∂t ( u −
1) = ∆ g ( u − − a ( n ) R g ( u −
1) + a ( n ) R g , where a ( n ) = n − n − . By Proposition A.5 in the Appendix, k u − k W k,k/ ,p − τ ( M × [ t ,t +1]) ≤ C (cid:18) k R g k W k − ,k/ − ,p − τ − ( M × [ t ,t +1]) + k u − k L p − τ ( M × [ t ,t +1]) (cid:19) . Since R g ∈ C k − α − τ − ( M ) and R g is independent of t , k R g k W k − ,k/ − ,p − τ − ( M × [ t ,t +1]) ≤ C. Thus it suffices to prove that k u − k L p − τ ( M × [ t ,t +1]) ≤ C .Recall (5.19) and (5.20) from the proof of Theorem 5.6: u ( x, t ) = e R t − n − R ( x,t ) dt u ( x, , and | R ( x, t ) | ≤ Cr τ ′ (1 + t ) δ . (5.21)We also know from the AF assumption on g that | u ( x, − | ≤ Cr τ . Thus | u ( x, t ) − | ≤ e R t − n − R ( x,t ) dt · | u ( x, − | + | e R t − n − R ( x,t ) dt − | (5.22) ≤ C · | u ( x, − | + C (cid:12)(cid:12)(cid:12)(cid:12)Z t − R ( x, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cr τ ′ . Above in the second line of (5.22), we have used | e s − | ≤ e | ¯ s | · | s | , where ¯ s issome point in [0 , s ], so that | e s − | ≤ e | s | · | s | . Letting s = R t − n − R ( x, t ) dt so that | s | ≤ − Cr τ ′ (1+ t ′ ) δ (cid:12)(cid:12) tt ′ =0 ≤ C by (5.21) gives us the estimate.As a result we see that u − ∈ C − τ ′ ( M × [ t , t + 1]) ֒ → L p − τ ( M × [ t , t + 1],for any τ < τ ′ , completing the proof of (1).To prove (1), again by Proposition A.5 in the Appendix we have k u − k C k + α,k + α − τ ′ ( M × [ t ,t +1]) ≤ C k R g k C k − α, k − α − τ ′− ( M × [ t ,t +1]) + k u − k C − τ ′ ( M × [ t ,t +1]) ! . u − ∈ C − τ ′ ( M × [ t , t +1]) and R g ∈ C k − α − τ − ( M ) we obtain our desiredestimate.Now we are ready to prove Theorems 1.3 and 1.4. Proof of Theorems 1.3 and 1.4.
We will bootstrap to inductively prove higherglobal regularities of φ in weighted parabolic H¨older spaces. More precisely, wewant to show the following estimate: for all l ≤ k and τ ′ < τ (assuming asthroughout this section that τ < n − k φ ( x, t ) k C l + α, l + α − τ ′ ( M × [ t ,t +1]) ≤ Ct δ . (5.23) Claim 1: (5.23) holds for l = 0.Let φ ( x, t ) = u ( x, t ) − u ∞ ( x ). Subtracting the two equations that u ( x, t ) and u ∞ ( x ) satisfy, ∂∂t u N = ( n − N [∆ g u − a ( n ) R g ( x ) u ]and ∆ g u ∞ ( x ) − a ( n ) R g ( x ) u ∞ ( x ) = 0 , we obtain ∂∂t φ ( x, t ) = n − u N − [∆ g φ ( x, t ) − a ( n ) R g ( x ) φ ( x, t )] . (5.24)By (5.18) of Proposition 5.6, there is a δ > k φ ( x, t ) k C − τ ′ ( M ) ≤ Ct δ . Therefore for all p ≥ τ < τ ′ , k φ ( x, t ) k L p − τ ( M × [ t ,t +1]) ≤ Ct δ , (5.25)where C depends only on p ≥ g (but is independent of t ).Since u ( x, t ) − ∈ C − τ ′ ( M ) and 0 < C ≤ u ( x, t ) ≤ C < ∞ , we have u N − ( x,t ) − ∈ C − τ ′ ( M ) as well. We also have R g ∈ C k − α − τ − ( M ). Hence, bythe triangle inequality and the product estimate from Lemma A.3, R g ( x ) u N − ( x, t ) = R g ( x ) · (cid:18) u N − ( x, t ) − (cid:19) + R g ( x ) ∈ C − τ ′ − ( M ) , and thus for all 0 < τ < τ ′ , R g ( x ) u N − ( x, t ) ∈ L p − τ − ( M × [ t , t + 1]).30utting everything together, we obtain that for any t > k R g φu N − k L p − τ − ( M × [ t ,t +1]) (5.26) ≤k R g ( x ) u N − ( x, t ) k L p − τ − ( M × [ t ,t +1]) · k φ ( x, t ) k C ( M ) ≤ Ct δ . Now by the weighted Schauder estimate for parabolic equations from [CZ15,Theorem 5.3] (or see Proposition A.5) applied to (5.24), we have for each t > k φ k W , ,p − τ ( M × [ t ,t +1]) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) R g φu N − (cid:13)(cid:13)(cid:13)(cid:13) L p − τ − ( M × [ t ,t +1]) + k φ k L p − τ ( M × [ t ,t +1]) ! , (5.27)so substituting (5.26) and (5.25) into (5.27), we derive k φ ( x, t ) k W , ,p − τ ( M × [ t ,t +1]) ≤ Ct δ . (5.28)Since (5.28) holds for all p ≥
1, we have by Sobolev embedding results forparabolic weighted spaces that (see statement (iii) of Lemma A.2), k φ ( x, t ) k C α, α − τ ( M × [ t ,t +1]) ≤ Ct δ . where C = C ( α, τ ) and we have chosen p > n + 2 so that 1 − n +2 p = α . Claim 2:
If (5.23) is valid for l ≥
0, then it holds for l + 2 when l ≤ k − φ and the weighted Schauder estimatesof Proposition A.5, we have k φ ( x, t ) k C l +2+ α, l +2+ α − τ ′ ( M × [ t ,t +1]) ≤ C k R g φu N − k C l + α, l + α − τ ′− ( M × [ t ,t +1]) + k φ k C − τ ′ ( M × [ t ,t +1]) ! . (5.29)From (5.18) we have k φ k C − τ ′ ( M × [ t ,t +1]) ≤ Ct δ , So it remains to show that k R g φu N − k C l + α,l + α − τ ′− ( M × [ t ,t +1]) ≤ Ct δ . (5.30)31n fact, by the product estimate of Lemma A.3 along with Lemma 5.7, and thefact that u − ∈ C l + α, l + α − τ ′ ( M × [ t , t + 1]) ֒ → C l + α, l + α ( M × [ t , t + 1]) for l ≤ k − k R g ( 1 u N − − k C l + α, l + α − τ ′− ( M × [ t ,t +1]) ≤ C k R g k C l + α − τ ′− ( M ) · k u − k C l + α,l + α ( M × [ t ,t +1]) ≤ C, and also have R g ∈ C k − α − τ − ( M ) ֒ → C l + α − τ ′ − ( M ) for all l ≤ k − R g u N − = R g ( 1 u N − −
1) + R g ∈ C l + α, l + α − τ ′ − ( M × [ t , t + 1]) . Hence (5.30) follows as k R g φu N − k C l + α, l + α − τ ′− ( M × [ t ,t +1]) ≤k R g u N − k C l + α, l + α − τ ′− ( M × [ t ,t +1]) · k φ ( x, t ) k C l + α, l + α ( M × [ t ,t +1]) ≤ Ct δ . (5.31)Above, the last inequality uses the inductive hypothesis (5.23). By substituting(5.31) into (5.29) we have proved Claim 2.Claim 1 and 2 together complete the proof of (5.23). To conclude, in thedefinition of the C k + α, k + α − τ ( M × [ t , t + 1]) spaces as stated in Definition A.1of the Appendix we can take j = 0 and t = t , and derive for all t > k φ ( x, t ) k C k + α − τ ′ ( M ) ≤ k φ ( x, t ) k C k + α, k + α − τ ′ ( M × [ t ,t +1]) ≤ A k t δ , and thus u ( x, t ) − → u ∞ ( x ) − C k + α − τ ′ ( M ) as t → ∞ . A C k + α − τ AF metrics along the Yamabe flow
Here we prove Theorem 2.6 — namely, we check that a C k + α − τ AF metric, k ≥ C k + α − τ AF metric along a fine solution of the Yamabe flow.First we need to recall the definitions of some parabolic H¨older spaces from[CZ15].
Definition A.1 (see [CZ15, Definition 4.1]) . Let ( M n , g ) be a complete Rieman-nian manifold such that there exists a compact K ⊂ M n and a diffeomorphism Φ : M n \ K → R n \ B R (0) , for some R > . Let r ≥ be a smooth function on n that agrees under the identification Φ with the Euclidean radial coordinate | x | in a neighborhood of infinity, and let ˆ g be a smooth metric on M n which isequal to the Euclidean metric in a neighborhood of infinity under the identifica-tion Φ . Let M = M n × [0 , T ] . Then with all quantities below computed withrespect to the metric ˆ g , we have the following function spaces:The weighted Lebesgue spaces L qβ ( M ) , for q ≥ and weight β ∈ R , consist ofthose locally integrable functions on M for which the following respective normsare finite: k v k L qβ ( M ) = (cid:16)R T R M | v | q r − βq − n dx (cid:17) q , q < ∞ , ess sup M (cid:0) r − β | v | (cid:1) , q = ∞ . The weighted Sobolev spaces W k,k/ ,qβ ( M ) are then defined in the usual waywith the norms k v k W k,k/ ,qβ ( M ) = X i +2 j ≤ k (cid:13)(cid:13)(cid:13) D ix D jt v (cid:13)(cid:13)(cid:13) L qβ − i − j ( M ) . The weighted C k spaces C kβ ( M ) consist of the C k functions for which thefollowing respective norms are finite: k v k C kβ ( M ) = X i +2 j ≤ k sup M r − β + i +2 j (cid:12)(cid:12)(cid:12) D ix D jt v (cid:12)(cid:12)(cid:12) . The weighted H¨older spaces C k + α, ( k + α ) / β ( M ) , α ∈ (0 , , consist of those v ∈ C kβ ( M ) for which the following respective norms are finite: k v k C k + α, ( k + α ) / β ( M ) = k v k C kβ ( M ) + [ v ] C k + αβ ( M ) + h v i C k + αβ ( M ) . Here, [ v ] C k + αβ ( M ) = X i +2 j = k sup ( x,t ) =( y,s ) ∈M min( r ( x ) , r ( y )) − β + i +2 j + α (cid:12)(cid:12)(cid:12) D ix D jt v ( x, t ) − D ix D jt v ( y, s ) (cid:12)(cid:12)(cid:12) δ (( x, t ) , ( y, s )) α , (A.1) where δ (( x, t ) , ( y, s )) = d ( x, y ) + | t − s | , and for k ≥ , h v i C k + αβ ( M ) = X i +2 j = k − sup ( x,t ) =( y,s ) ∈M r ( x ) − β + i +2 j + α +1 (cid:12)(cid:12)(cid:12) D ix D jt v ( x, t ) − D ix D jt v ( x, s ) (cid:12)(cid:12)(cid:12) | t − s | α +12 . (A.2)33e now record some embedding results for the spaces above from [CZ15,Theorem 4.5]. Note that (iii) below is a sharpened version of an embeddingresult which appears in the proof of [CZ15, Theorem 1.3] Lemma A.2.
Under the hypotheses of Definition A.1, the following inequalitieshold:(i) For ≤ p ≤ q ≤ ∞ and β < β , we have k v k L pβ ( M ) ≤ C k v k L qβ ( M ) (ii) For β = β + β , ≤ p, q, s ≤ ∞ , and p = q + s , we have k v k L pβ ( M ) ≤ k v k L qβ ( M ) | v k L sβ ( M ) and k v k C α,α/ β ( M ) ≤ k v k C α,α/ β ( M ) | v k C α,α/ β ( M ) ′ (A.3) (iii) For p > n + 2 and α = 1 − n +2 p , we have k v k C α,α β ( M ) ≤ C k v k W , ,pβ ( M ) . We next state some generalizations of product inequalities for the weightedparabolic Sobolev and H¨older norms found in [CZ15, Theorem 4.5]. They followfrom the definitions of these weighted norms. In the the H¨older case one mustcheck that C k + α, ( k + α ) / β ֒ → C j + α, ( j + α ) / β for all 0 ≤ j ≤ k . Lemma A.3.
Let β = β + β with β , β ≤ .(i) For ≤ p, q, s ≤ ∞ with p = q + s and a given nonnegative integer k , wehave k uv k W k,k/ ,pβ ( M ) ≤ C k u k W k,k/ ,qβ ( M ) k v k W k,k/ ,sβ ( M ) , where C depends on M and k .(ii) For k ≥ , we have k uv k C k + α, ( k + α ) / β ( M ) ≤ C k u k C k + α, ( k + α ) / β ( M ) k v k C k + α, ( k + α ) / β ( M ) , where C depends on M and k . We now check that an unweighted H¨older estimate holds for the conformalfactor u which evolves along the Yamabe flow.34 emma A.4. Let g ( t ) be a fine solution of the Yamabe flow starting from theasymptotically flat manifold ( M n , g ) on the maximal time interval [0 , T ) , givenby g ( t ) = u ( t ) n − g with u (0) ≡ , and let T < T . Then given r > , thereexists a sequence A k > for k = 1 , , . . . , such that k u k C k + α, ( k + α ) / ( B g ( p,r ) × [0 ,T ] ) ≤ A k , independently of the point p ∈ M . Hence there exist uniform bounds on [0 , T ] for the curvature Rm ( x, t ) and all of its derivatives.Proof. By [CZ15, Theorem 2.4], on a given [0 , T ] the conformal factor u ( x, t )satisfies 0 < c ≤ u ( x, t ) ≤ c for some constants c , c . Therefore we may applythe Krylov-Safonov estimate for parabolic equations to (2.3) and then repeatedlyapply the Schauder estimates for parabolic equations (see for instance [Lie96,Theorem 4.9]) to obtain the conclusion.With the help of the product inequalities of Lemma A.3 we can then adaptthe arguments in the proof of [CZ15, Theorem 5.4] to establish higher-orderweighted Sobolev and H¨older estimates for the conformal factor u ( t ). Proposition A.5.
Let u ( x, t ) be a fine solution of the Yamabe flow startingfrom an asymptotically flat manifold ( M n , g ) on the maximal time interval [0 , T ) , given by g ( t ) = u ( t ) n − g with u (0) ≡ . Further let v = 1 − u , andsuppose for a fixed T < T that < δ ≤ u ( x, t ) ≤ C ′ on [0 , T ] .(i) If ( M n , g ) is W k,p − τ asymptotically flat for some k ≥ , then there exists C = C ( n, k, p, τ, δ, C ′ ) such that k v k W k,k/ ,p − τ ( M ) ≤ C (cid:16) k R g k W k − ,k/ − ,p − − τ ( M ) + k v k L p − τ ( M ) (cid:17) . (ii) If ( M n , g ) is C k + α − τ asymptotically flat for some k ≥ , and we have k v k C k − α, ( k − α ) / ( M ) ≤ C ′′ , then there exists C = C ( n, k, τ, δ, C ′ , C ′′ ) such that k v k C k + α, ( k + α ) / − τ ( M ) ≤ C (cid:16) k R g k C k − α, ( k − α ) / − − τ ( M ) + k v k C − τ ( M ) (cid:17) (A.4) Proof.
The proofs of (i) and (ii) similar and moreover are straightforward adap-tations of the proofs of Theorems 5.3 and 5.4 respectively in [CZ15], once wehave the multiplicative inequalities of Lemma A.3. Indeed the statements of[CZ15, Theorems 5.3, 5.4] are exactly the statements of Proposition A.5 in thecase k = 2. As a result we will only give the proof of (ii) below, making thischoice because in this work we are mainly concerned with C k + α − τ asymptoticallyflat manifolds.Following the proof of [CZ15, Theorem 5.4], we first use a scaling argumenton annuli along with the Schauder estimates for parabolic equations as in [Lie96,35heorem 4.9] to obtain k v k C k + α, ( k + α ) / − τ ( E R × [0 ,T ]) ≤ C ( k ( ∂ t − ∆ ) v k C k − α, ( k − α ) / − − τ ( E R × [0 ,T ]) + k v k C − τ ( E R × [0 ,T ]) ) , (A.5)for R > R , where R > E R = Φ − ( R n \ B R (0)) ⊂ M . Here ∆ denotes the Laplacian with respect to the flat metric defined by Φon E R . Define the operator P = h (∆ g − a ( n ) R g ) = h ( g ij ∂ ∂x i ∂x j + b j ∂∂x j − a ( n ) R g ) , where h = N (1 − v ) N − and in the last expression above we have rewritten ∆ g in terms of the Euclidean coordinates given by Φ. We then have( ∂ t − P ) v = a ( n ) R g . We will now compare ( ∂ t − ∆ ) and ( ∂ t − P ) on E R × [0 , T ]. Decompose v = v + v ∞ using a suitable cutoff function so that supp( v ∞ ) ⊂ E R ; thenwith the help of the product inequality of Lemma A.3 we can estimate k (∆ − P ) v ∞ k C k − α, ( k − α ) / − − τ ( E R × [0 ,T ]) ≤ k hg ij − δ ij k C k − α, ( k − α ) / ( E R × [0 ,T ]) k D x v ∞ k C k − α, ( k − α ) / − − τ ( E R × [0 ,T ]) + k h k C k − α, ( k − α ) / ( E R × [0 ,T ]) k b k C k − α, ( k − α ) / − ( E R × [0 ,T ]) k D x v ∞ k C k − α, ( k − α ) / − − τ ( E R × [0 ,T ]) + k h k C k − α, ( k − α ) / ( E R × [0 ,T ]) k R g k C k − α, ( k − α ) / − ( E R × [0 ,T ]) k v ∞ k C k − α, ( k − α ) / − τ ( E R × [0 ,T ]) ≤ C ( k hg ij − δ ij k C k − α, ( k − α ) / ( E R × [0 ,T ]) + k b k C k − α, ( k − α ) / − ( E R × [0 ,T ]) + k R g k C k − α, ( k − α ) / − ( E R × [0 ,T ]) ) k v ∞ k C k + α, ( k + α ) / − τ ( E R × [0 ,T ]) . (A.6)Moreover, k hg ij − δ ij k C k − α, ( k − α ) / ( E R × [0 ,T ]) + k b k C k − α, ( k − α ) / − ( E R × [0 ,T ]) + k R g k C k − α, ( k − α ) / − ( E R × [0 ,T ]) → R → ∞ because of the asymptotic decay of v and g . Thus, by taking R sufficiently large we obtain by combining (A.5), (A.6), and (A.7) that k v ∞ k C k + α, ( k + α ) / − τ ( M ) ≤ C ( k ( ∂ t − P ) v ∞ k C k − α, ( k − α ) / − − τ ( M ) + k v ∞ k C − τ ( M ) . (A.8)36o deal with the first term on the right, if we let ζ R be a suitably chosen cutofffunction so that v = ζ R v , v ∞ = (1 − ζ R ) v , then we have k ( ∂ t − P ) v ∞ k C k − α, ( k − α ) / − − τ ≤ k ( ∂ t − P ) v k C k − α, ( k − α ) / − − τ + k ( ∂ t − P )( ζ R v ) k C k − α, ( k − α ) / − − τ ≤ k ( ∂ t − P ) v k C k − α, ( k − α ) / − − τ + C R k v + |∇ g v |k C k − α, ( k − α ) / (( E R \ E R ) × [0 ,T ]) . (A.9)We leave this for now and turn to consider v . On the bounded space-time do-main supp( v ) ⊂ M we can directly apply the Schauder estimates for parabolicequations as in [Lie96, Theorem 4.9], to obtain, similarly to (A.8), k v k C k + α, ( k + α ) / − τ ( M ) ≤ C ( k ( ∂ t − P ) v k C k − α, ( k − α ) / − − τ ( M ) + k v k C − τ ( M ) , (A.10)and similar to (A.9) we also have k ( ∂ t − P ) v k C k − α, ( k − α ) / − − τ ≤ k ( ∂ t − P ) v k C k − α, ( k − α ) / − − τ + C R k v + |∇ g v |k C k − α, ( k − α ) / (( E R \ E R ) × [0 ,T ]) . (A.11)Finally, putting (A.8), (A.9), (A.10), and (A.11) together, applying a H¨oldernorm interpolation inequality to deal with the |∇ g v | terms, and recalling that( ∂ t − P ) v = a ( n ) R g , we obtain the desired estimate (A.4).We are now in a position to prove Theorem 2.6. Proof of Theorem 2.6.
The case k = 2 is proved in [CZ15, Theorem 1.3]. So weneed only concern ourselves with k ≥
3, and by induction we may assume that k v k C k − α, ( k − α ) / − τ is bounded on M .Clearly by our hypotheses v is bounded in C k − α, ( k − α ) / ( M ). And since( M n , g ) is C k + α − τ asymptotically flat, we also have R g ∈ C k − α, ( k − α ) / − − τ ( M ).Proposition A.5 therefore implies that v ∈ C k + α, ( k + α ) / − τ , which in turn gives u ( x, t ) − ∈ C k + α − τ for all t ∈ [0 , T ] as desired. B C k + α − τ conformal deformations We will indicate here how the statements Propositions 3.1 and 3.2 on conformaldeformations of AF manifolds with metrics lying in weighted H¨older spacesfollow from the the corresponding results given in [CB81; Max05; DM18], whichwork with weighted Sobolev spaces. We begin by recalling those results.Corresponding to Proposition 3.1, we have:37 roposition B.1 ([DM18, Lemma 4.3]) . Let ( M n , g ) be a W k,p − τ AF manifoldwith k ≥ , k > np , and τ ∈ (0 , n − . Suppose R ′ ∈ W k − − − τ satisfies R ′ ≤ R g .Then there exists a positive function φ with φ − ∈ W k,p − τ such that the scalarcurvature of g ′ = φ n − g is R ′ . In particular g ′ is also a W k,p − τ AF metric.
Corresponding to Proposition 3.2 we have:
Proposition B.2 ([DM18, Proposition 3]) . Let ( M n , g ) be a W k,p − τ AF manifoldwith k ≥ , k > np , and τ ∈ (0 , n − . Then the following are equivalent:(1) We have Y ( M, [ g ]) > .(2) There exists a positive function φ with φ − ∈ W k,p − τ such that ˜ g = φ n − g is conformally equivalent to ˜ g and R ˜ g ≡ . We note that in [DM18], Proposition B.1 is actually stated in the W ,p − τ setting, but the proof uses the W k,p − τ results established earlier in [DM18], andthe more general W k,p − τ statement given above follows by the same argument.Now starting from the hypotheses of Propositions B.1 and B.2, we firstobserve that C k + α − τ ֒ → W k,p − τ ′ for any p < ∞ if τ ′ < τ . Therefore Propositions3.1 and 3.2 immediately give us the existence of metrics g ′ and ˜ g belonging to W k,p − τ ′ , and we just need to establish that they additionally belong to C k + α − τ .Moreover, by taking p sufficiently large we have W k,p − τ ′ ֒ → C α − τ ′ , for any τ ′ < τ .We can then conclude the results of both Propositions 3.1 and 3.2 by proving aregularity estimate arising from this information. Indeed, this is essentially anapplication of the elliptic theory of weighted H¨older spaces on punctured regionsof R n as for instance in [PR00]. Lemma B.3.
Let ( M n , g ) be a C k + α − τ ′ AF manifold, k ≥ , with τ ∈ (0 , n − .If R ∈ C k − α − − τ and φ ∈ C k + α − τ ′ for all τ ′ < τ satisfy − a ( n )∆ g φ + R g φ = Rφ N , (B.1) then φ − ∈ C k + α − τ .Proof. Let ψ = φ −
1. Elliptic weighted Schauder estimates can be derivedby scaling on annuli as in the proof of Proposition A.5 (essentially the sameprocedure as in [CZ15; Bar86]), which when applied to (B.1) then imply k ψ k C k + α − τ ′ ≤ C (cid:16) k R g k C k − α − − τ ′ + k R k C k − α − − τ ′ + k ψ k C − τ ′ (cid:17) . Above we have used the product inequalities of Proposition A.3 along with ψ ∈ C α in order to deal with the multiplications of R g and R against powersof ψ . The same estimate would then show us that ψ ∈ C k + α − τ once we establishthat ψ ∈ C − τ , and we shall now do so.38or ease of notation we assume that we have asymptotic coordinates Φ : M \ K → R n \ B (0). Then we can rewrite (B.1) in the Euclidean coordinates as − ∆ ψ + ( δ ij − g ij ) D ij ψ + b j D j ψ + R g (1 + ψ ) = R (1 + ψ ) N (B.2)where ∆ is the Euclidean Laplacian and − ∆ g = − g ij D ij + b j D j . Using that g ∈ C k + α − τ , D ψ ∈ C k − α − − τ ′ ֒ → C α − , Dψ ∈ C k − α − − τ ′ ֒ → C α − , and R ∈ C k − α − − τ ,all together with the product inequality from Proposition A.3, we see that (B.2)takes the form − ∆ ψ = f ∈ C α − − τ , on R n \ B (0) . (B.3)We now apply the Kelvin transform from R n \ B (0) to B (0) \{ } . Follow-ing the definitions in [PR00], we will denote weighted spaces on B (0) \{ } byfor instance C ∗ k + α − n + τ , in order to distinguish them from weighted spaces on R n \ B (0), which we will still denote by C k + α − τ . We will similarly denote theKelvin transform of ψ by ψ ∗ : B (0) \{ } → R , and similarly for other func-tions.It is then possible to check that ψ ∈ C k + α − τ implies ψ ∗ ∈ C ∗ k + α − n + τ . Inparticular, since τ ∈ (0 , n −
2) we also have 2 + n − τ ∈ (0 , n − : C ∗ α − n + τ, D → C ∗ α − n + τ − is invertible for all τ ∈ (0 , n − C ∗ α − n + τ, D means those functions of C ∗ α − n + τ which vanishon ∂B (0). Thus, since we already know that ψ ∈ C k + α − τ ′ for any τ ′ < τ , we seethat ψ ∗ ∈ C ∗ α − n + τ ′ and − ∆ ψ ∗ = | x ∗ | − f ∗ ∈ C ∗ α − n + τ ′ − , while using (B.3) and [PR00, Proposition 2.4], there exists a w ∗ ∈ C ∗ α − n + τ, D solving − ∆ w ∗ = | x ∗ | − f ∗ ∈ C ∗ α − n + τ − . Hence ψ ∗ − w ∈ C ∗ α − n + τ ′ must be harmonic in the entire ball B (0) since theorder 2 − n + τ ′ > − n implies the singularity at zero can be removed. Thus wesee that ψ ∗ ∈ C ∗ α − n + τ ֒ → C ∗ − n + τ as well, and transforming back to R n \ B (0)we obtain as desired that ψ ∈ C − τ . C L p control of R for small p Here we address regularity issuse that might potentially arise in proving Lemma4.3, and show that they do not pose any problems.We first look at a lemma about auxiliary function w constructed in order todeal with the set the scalar curvature vanishes { x ∈ M n , R ( x, t ) = 0 } at time t . Lemma C.1.
There exists a solution w to ∂ t w = ( n − w + Rw with theappropriate spatial decay w ( x, ∼ | x | − s and |∇ w ( x, | ∼ | x | − s − at the initialtime and w ( x, > on M n , and w ( x, t ) > for all x ∈ M n , t > . roof of of Lemma C.1. We have already got L ∞ control of R from the Moseriteration, so there exists λ > sup R n × [0 , ∞ ) | R | . By results of [CTY11] the funda-mental solution Ψ( x, t ) to operator ∂ t − ( n − − R exists, and satisfying heatkernel type estimate: Thus define w ( x, t ) := Z M Ψ( x − y, t ) w ( y, dy satisfying ∂ t w = ( n − w + Rw with initial condition. We now want to show (1) w ( x, t ) > w ( x, >
0; and (2) w ( x, t ) decays of order s for all 0 < t < t max if w ( x,
0) decays of order s . It is not hard to see w satisfies conditions ofthe maximum principle Theorem 4.3 of Ecker and Huisken [EH91]. In fact,conditions (i), (ii), (iv) in Theorem 4.3 [EH91] are obvious. To show condition(iii) R ,T R M exp ( − α r t ( p, y ) ) |∇ w | ( y ) dµ t ( y ) dt < ∞ for some α >
0, we notethat since ∇ w ( x,
0) decays, and thus L ∞ on M n , R M Ψ( x − y, t ) ∇ w ( y, dµ t ( y )is L ∞ on M n . Thus Z M exp ( − α r t ( p, y ) ) |∇ w | ( y ) dµ t ( y ) ≤ Z M exp ( − α r t ( p, y ) ) dµ t ( y ) < ∞ . To prove (2) w ( x, t ) decays of order s if w ( x,
0) decays of order s , we use theidea of Cheng-Zhu [CZ15], and consider the function f ( x, t ) := h ( x ) w ( x, t ) − C where h ( x ) := | x | s , and C is a large enough constant so that f ( x, ≤ w ( x, f ( x, t ) satisfies an evolution equationin which its coefficients satisfy all the conditions of the maximum principle ofEcker-Huisken [EH91] that we used before. Thus f ( x, t ) ≤ f ( x, ≤ w ( x, t ) ≤ C | x | s . Note | x | is equivalent to the distance function with respect metric g ( t ), so w ( x, t ) ≤ Cd g ( t ) ( x, s . This allows us to prove a version of Lemma 4.3 without needing to worryabout regularity issues at points when R = 0. We can control | R | from aboveby a positive function which satisfies the same decay estimates along with ∂∂t A ≤ ( n − A + RA, so we can carry out our estimates on A instead. Proof of Lemma 4.3.
Now we can run the rest of the argument to control L p norms of R when p < n . Let A := R + w ; ∂ t w = ( n − w + Rw . By theabove explanation, w >
0. Hence
A > w satisfies ∂ t w = ( n − w ) − n − |∇ w | + 2 Rw . Weset the initial condition on u so that it has the same asymptotic decay rate as R , ie. w ∼ ǫr − − τ . Then by Cauchy-Schwarz |∇ ( R + w ) | ≤ | R | + w )( |∇| R || + |∇ w | ) . Thus A = R + w satisfies the evolution inequaity ∂ t A ≤ ( n − A − n − |∇ A | A + 2 RA and therefore (since we have the strict inequality A > ∂ t A ≤ ( n − A + RA.
The evolution inequality A satisfies is exactly the form of inequality satisfied by R , except now we know that A > | R | ≥
0. Thus for any p such that R A p dV t is integrable (in particular p < n but close to n ) ddt Z A p dV t ≤ Z pA p − ∂ t A − n RA p dV t . Hence ddt Z A p dV t ≤ Z pA p − [( n − A + RA ] − n RA p dV t , and ddt Z A p dV t ≤ − p ( n − p − Z A p − |∇ A | dV t + (cid:16) p − n (cid:17) Z RA p dV t . (C.1)Note n ≥ p < n but close to n , p − >
0. So the first term onthe right has a negative sign. In the meanwhile recall for the second term that (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) p − n (cid:17) Z RA p (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) p − n (cid:12)(cid:12)(cid:12) k R k L n (cid:18)Z A p nn − dV t (cid:19) n − n . (C.2)Since we have proved k R k L n is monotonic decreasing, it is bounded dependingonly on g , so for p < n but very close to n this second term can be absorbedby the gradient term, using the Sobolev inequality. Therefore we see as desiredthat for such p , ddt Z A p dV t ≤ . eferences [Aub76] Thierry Aubin. “´Equations diff´erentielles non lin´eaires et probl`emede Yamabe concernant la courbure scalaire”. In: J. Math. PuresAppl. (9) issn : 0021-7824.[Bar86] Robert Bartnik. “The mass of an asymptotically flat manifold”. In:
Comm. Pure Appl. Math. issn : 0010-3640.[Bre05] Simon Brendle. “Convergence of the Yamabe flow for arbitrary ini-tial energy”. In:
J. Differential Geom. issn :0022-040X.[Bre07] Simon Brendle. “Convergence of the Yamabe flow in dimension 6and higher”. In:
Invent. Math. issn : 0020-9910.[BV19] Eric Bahuaud and Boris Vertman. “Long-time existence of the edgeYamabe flow”. In:
J. Math. Soc. Japan issn : 0025-5645.[CB81] Murray Cantor and Dieter Brill. “The Laplacian on asymptoticallyflat manifolds and the specification of scalar curvature”. In:
Com-positio Math. issn : 0010-437X.[CD18] Beomjun Choi and Panagiota Daskalopoulos. “Yamabe flow: steadysolitons and type II singularities”. In:
Nonlinear Anal.
173 (2018),pp. 1–18. issn : 0362-546X.[CDK18] Beomjun Choi, Panagiota Daskalopoulos, and John King. “TypeII Singularities on complete non-compact Yamabe flow”. In: arXive-prints , arXiv:1809.05281 (Sept. 2018), arXiv:1809.05281. arXiv: .[Che03] Yazhe Chen.
Second Order Parabolic Equations (in Chinese) . BeijingUniversity Mathematics Series. Beijing University Press, Beijing,2003.[Che19] Eric Chen.
Convergence of the Ricci flow on asymptotically flat man-ifolds with integral curvature pinching . 2019. arXiv: .[Cho+08] Bennett Chow et al.
The Ricci flow: techniques and applications.Part II . Vol. 144. Mathematical Surveys and Monographs. Ana-lytic aspects. American Mathematical Society, Providence, RI, 2008,pp. xxvi+458. isbn : 978-0-8218-4429-8.[CTY11] Albert Chau, Luen-Fai Tam, and Chengjie Yu. “Pseudolocality forthe Ricci flow and applications”. In:
Canad. J. Math. issn : 0008-414X.[CZ15] Liang Cheng and Anqiang Zhu. “Yamabe flow and ADM mass onasymptotically flat manifolds”. In:
J. Math. Phys. issn : 0022-2488.42DKS19] Panagiota Daskalopoulos, John King, and Natasa Sesum. “Extinc-tion profile of complete non-compact solutions to the Yamabe flow”.In:
Comm. Anal. Geom. issn : 1019-8385.[DM07] Xianzhe Dai and Li Ma. “Mass under the Ricci flow”. In:
Comm.Math. Phys. issn : 0010-3616.[DM18] James Dilts and David Maxwell. “Yamabe classification and pre-scribed scalar curvature in the asymptotically Euclidean setting”.In:
Comm. Anal. Geom. issn : 1019-8385.[EH91] Klaus Ecker and Gerhard Huisken. “Interior estimates for hypersur-faces moving by mean curvature”. In:
Invent. Math. issn : 0020-9910.[Ham89] Richard S. Hamilton. “Lectures on geometric flows”. Unpublished.1989.[Heb99] Emmanuel Hebey.
Nonlinear analysis on manifolds: Sobolev spacesand inequalities . Vol. 5. Courant Lecture Notes in Mathematics.New York University, Courant Institute of Mathematical Sciences,New York; American Mathematical Society, Providence, RI, 1999,pp. x+309.[Jau19] Jeffrey L. Jauregui. “Lower semicontinuity of the ADM mass in di-mensions two through seven”. In:
Pacific J. Math. issn : 0030-8730.[Li18] Yu Li. “Ricci flow on asymptotically Euclidean manifolds”. In:
Geom.Topol. issn : 1465-3060.[Lie96] Gary M. Lieberman.
Second order parabolic differential equations.
English. Singapore: World Scientific, 1996, pp. xi + 439. isbn : 981-02-2883-X.[LV20] Jørgen Olsen Lye and Boris Vertman.
Long-time existence of Yam-abe flow on singular spaces with positive Yamabe constant . 2020.arXiv: .[Ma19] Li Ma. “Yamabe flow and metrics of constant scalar curvature ona complete manifold”. In:
Calc. Var. Partial Differential Equations issn : 0944-2669.[Ma21] Li Ma. “Global Yamabe flow on asymptotically flat manifolds”.In: arXiv e-prints , arXiv:2102.02399 (Feb. 2021), arXiv:2102.02399.arXiv: .[Max05] David Maxwell. “Solutions of the Einstein constraint equations withapparent horizon boundaries”. In:
Comm. Math. Phys. issn : 0010-3616.[MCZ12] Li Ma, Liang Cheng, and Anqiang Zhu. “Extending Yamabe flow oncomplete Riemannian manifolds”. In:
Bull. Sci. Math. issn : 0007-4497.43MS12] Donovan McFeron and G´abor Sz´ekelyhidi. “On the positive masstheorem for manifolds with corners”. In:
Comm. Math. Phys. issn : 0010-3616.[PR00] Frank Pacard and Tristan Rivi`ere. “Elliptic Operators in WeightedH¨older Spaces”. In:
Linear and Nonlinear Aspects of Vortices . Birkh¨auserBoston, 2000, pp. 21–49.[Sch19] Mario B. Schulz. “Yamabe Flow on Non-compact Manifolds withUnbounded Initial Curvature”. In:
The Journal of Geometric Anal-ysis (July 2019).[Sch20] Mario B. Schulz. “Unconditional existence of conformally hyperbolicYamabe flows”. In:
Analysis & PDE
J. Differential Geom. issn : 0022-040X.[SS03] Hartmut Schwetlick and Michael Struwe. “Convergence of the Yam-abe flow for “large” energies”. In:
J. Reine Angew. Math.
562 (2003),pp. 59–100. issn : 0075-4102.[SY79] Richard Schoen and Shing Tung Yau. “On the proof of the positivemass conjecture in general relativity”. In:
Comm. Math. Phys. issn : 0010-3616.[Tru68] Neil S. Trudinger. “Remarks concerning the conformal deformationof Riemannian structures on compact manifolds”. In:
Ann. ScuolaNorm. Sup. Pisa Cl. Sci. (3)
22 (1968), pp. 265–274. issn : 0391-173X.[Yam60] Hidehiko Yamabe. “On a deformation of Riemannian structures oncompact manifolds”. In:
Osaka Math. J.
12 (1960), pp. 21–37. issn :0388-0699.[Yan88] Deane Yang. “ L p pinching and compactness theorems for compactRiemannian manifolds”. eng. In: S´eminaire de Th´eorie Spectraleet G´eom´etrie, No. 6, Ann´ee 1987–1988 . Univ. Grenoble I, Saint-Martin-d’H`eres, 1988, pp. 81–89.[Ye94] Rugang Ye. “Global existence and convergence of Yamabe flow”. In:
J. Differential Geom. issnissn