aa r X i v : . [ m a t h . DG ] F e b GLOBAL YAMABE FLOW ON ASYMPTOTICALLY FLAT MANIFOLDS
LI MAA bstract . In this paper, we study the existence of global Yamabe flow on asymptotically flat(in short, AF or ALE) manifolds. Note that the ADM mass is preserved in dimensions 3,4and 5. We present a new general local existence of Yamabe flow on a complete Riemannianmanifold with the initial metric quasi-isometric to a background metric of bounded scalarcurvature. Asymptotic behaviour of the Yamabe flow on ALE manifolds is also addressedprovided the initial scalar curvature is non-negative and there is a bounded subsolution to thecorresponding Poisson equation. We also present a maximum principle for a very generalparabolic equations on the complete Riemannian manifolds.
Mathematics Subject Classification 2010 : 53E99, 35A01, 35K55, 35R01, 53C21.
Keywords : Yamabe flow, global existence, scalar curvature, asymptotic behaviour
1. I ntroduction
The goal of this paper is to consider the global Yamabe flow on complete manifolds.This topic has recently been studied in the works [23], [28], [9] and [24, 25]. Since theYamabe flow is degenerate, the expected global flow is rare, however, the Yamabe flow onasymptotically flat (in short, AF or ALE) manifolds is widely believed to be global. We shallconfirm this in this paper. Hamilton [19] [18] have introduced Yamabe flow which describesa family of Riemannian metrics g ( t ) subject to the evolution equation ∂∂ t g = − R ( g ) g , where R ( g ) denotes the scalar curvature corresponding to the metric g . Hamilton proved localin time existence of Yamabe flows on compact manifolds without boundary. Asymptoticbehaviour of the Yamabe flow was subsequently analysed by B. Chow [10], R. Ye [36],Schwetlick and M. Struwe [34] and S. Brendle [7]. The discrete Morse flow method for 2-d Li Ma’s research was partially supported by the National Natural Science Foundation of China(No.11771124).
Yamabe flow was developed in [29]. The theory of Yamabe flows on non-compact manifoldswas addressed by Ma and An [1]. Daskalopoulos and Sesum [14] analysed the profilesof self-similar solutions (Yamabe solitons). More recently, Bahuaud and Vertman [4, 5]constructed Yamabe flows on spaces with incomplete edge singularities such that the singularstructure is preserved along the Yamabe flow. Choi, Daskalopoulos, and King [8] were ableto find solutions to the Yamabe flow on the Euclidean space R n which develop a type IIsingularity in finite time. In the interesting work [17], Gregor Giesen and Peter M. Toppingobtained remarkable results about Yamabe flow incomplete surfaces. In [32, 33], assumingthat the initial metric is conformally hyperbolic with conformal factor and scalar curvaturebounded from above, Schulz had obtained existence of instantaneously complete Yamabeflows on hyperbolic space of arbitrary dimension n ≥
3. The study of Yamabe flow on R n may be included in the class of porous-media equations [3] [13] [20].Given an n -dimensional complete Riemannian manifold ( M n , g ), n ≥
3. The Yamabeflow on ( M n , g ) is a family of Riemannian metrics { g ( · , t ) } on M defined by the evolutionequation(1) ∂ g ∂ t = − Rg in M n × [0 , T ) , g ( · , = g in M n , where R is the scalar curvature of the metric g : = g ( · , t ) = u n − g , where u : M n → R + isa positive smooth function on M n . Let p = n + n − , L g u = ∆ g u − aR g u and a = n − n − . Bychanging time by a constant scale, (1) can be written in the equivalent form(2) ∂ u p ∂ t = L g u , in M n × [0 , T ) , u ( · , = , in M n . To understand the local existence result of the Yamabe flow on the Riemannian manifold( M , g ), we may choose a base metric g M on M and write the Yamabe flow equation [32] as LOBAL YAMABE FLOW 3 follows. Let g ( t ) = w ( x , t ) g M with w = w ( x , t ) > M . Then the Yamabe flow equation is1 m − w t = − wRn − = − R n − + ∆ g M ww + ( n − |∇ w | g M w , with w (0) = w . We may denote the terms involving w in the equation above by B [ w ] : = ( n − (cid:16) − R n − + ∆ g M ww + ( n − |∇ w | g M w (cid:17) . We shall apply inverse function theorem to this form of equation to get local existence ofsolutions.Before presenting the main result of this paper, we need the following two definitions. Thefirst one is the definition of asymptotically flat (AF or ALE) manifold of order τ >
Definition 1.
A Riemannian manifold M n , n ≥ , with C ∞ metric g is called asymptoticallyflat of order τ if there exists a decomposition M n = M ∪ M ∞ (for simplicity we deal onlywith the case of one end and the case of multiple ends can be dealt with similarly) with M compact and a di ff eomorphism M ∞ (cid:27) R n − B ( o , R ) for some constant R > such thatg i j − δ i j ∈ C + α − τ ( M )(3) (defined in Definition 6 below) in the coordinates { x i } induced on M ∞ . And the coordinates { x i } are called asymptotic coordinates. The second one is about the fine solution to Yamabe flow ([9] [26]).
Definition 2.
We say that u ( x , t ) ∈ C ( M × [0 , t max ) is a fine function if < δ ≤ u ( x , t ) ≤ Cfor ≤ t ≤ T with any < T < t max and sup M n × [0 , T ] |∇ g u ( x , t ) | ≤ C. We say that u ( x , t ) ∈ C ( M × [0 , t max ) is a fine solution of the Yamabe flow, ≤ t < t max , on a complete manifold ( M n , g ) if it is a fine function solution to the Yamabe flow and sup M n × [0 , T ] | Rm ( g ) | ( x , t ) ≤ C forany T < t max , such that either lim t → t max sup M | Rm | ( · , t ) = ∞ for t max < ∞ or t max = ∞ , where Rm ( g ) is the Riemannian curvature of the metric g : = g ( t ) = u / ( n − g . LI MA
We remark that in the language from (2.5.2) in [31] ( see also [27]), the fine solution isuniformly quasi-isometric to the initial metric g on every interval [0 , T ] for 0 < T < t max .Our main result is in below. Theorem 3.
Given an n-dimensional asymptotically flat manifold ( M n , g ) of any order τ > n − . There exists an unique global Yamabe flow g ( x , t ) = u ( x , t ) / ( n − g with the initial metricg (0) = g and the solution u ( x , t ) , ≤ t < t < ∞ , is a fine solution to the Yamabe flow (2)and the flow preserves the AF property of the initial metric. In other word, for v = − u, wehave v ( x , t ) ∈ C + α − τ ( M ) and g i j ( x , t ) − δ i j ∈ C + α − τ ( M ) for t ∈ [0 , t max ) . The uniqueness part follows from the standard argument and we shall omit the detail. Weshall present a general local existence result in Theorem 7 in section 2.As a direct application of the computation as showed in [9] and [23], we have
Theorem 4.
Let u ( x , t ) , ≤ t < t < ∞ , be the fine solution to the Yamabe flow (2) onan n-dimensional asymptotically flat manifold ( M n , g ) of order τ > n − with u (0) = .Assume that R g ≥ and R g ∈ L ( M ) , where R g is the scalar curvature of g . Denotedby g ( t ) = u n − g . Then for n = , , or , ADM mass m ( g ( t )) (see [31] [22] or below forthe definition) is well-defined under the Yamabe flow (2) for ≤ t < ∞ (i.e. ADM mass isindependent of the choices of the coordinates),i.e, m ( g ( t )) ≡ m ( g ) . Recall here that the ADM mass of n -dimensional AF Riemannian manifolds [22] is definedas m ( g ) = lim r →∞ ω Z S r ( ∂ j g i j − ∂ i g j j ) dS i , (4)where ω denotes the volume of unit sphere in R n , S r denotes the Euclidean sphere withradius r and dS i is the normal surface volume element to S r with respect to Euclidean metric.Similar results for Ricci flow were established in [6] and [12]. LOBAL YAMABE FLOW 5
With an application of Theorem 5 in [26], we get the convergent result of the globalYamabe flow below.
Theorem 5.
Let u ( x , t ) , ≤ t < ∞ , be the global solution to the Yamabe flow (2) on ann-dimensional asymptotically flat manifold ( M n , g ) of order τ > n − with u (0) = . Assumethat R g ≥ and there exists a bounded sub-solution w to the Poisson equationL g w = ∆ g w − aR g w ≥ , in M . Then the Yamabe flow g ( t ) converges in C ∞ loc ( M ) to a Yamabe metric of scalar curvature zero. We now recall the definition of weighted spaces (see [22]) for elliptic operators on asymp-totically flat manifolds.
Definition 6.
Suppose ( M n , g ) is an n-dimensional asymptotically flat manifold with asymp-totic coordinates { x i } . Denote D jx v = sup | α | = j | ∂ | α | ∂ x i ··· ∂ x ij v | . Let r ( x ) = | x | on M ∞ (defined in Defi-nition 1) and extend r to a smooth positive function on all of M n . For q ≥ and β ∈ R , theweighted Lebesgue space L q β ( M ) is defined as the set of locally integrable functions v withthe norm given by || v || L q β ( M ) = ( R M | v | q r − β q − n dx ) q , q < ∞ ;ess sup M ( r − β | v | ) , q = ∞ .Then the weighted Sobolev space W k , q β ( M ) is defined as the set of functions v for which | D jx v | ∈ L q β − j ( M ) with the norm || v || W k , q β ( M ) = k X j = || D jx v || L q β − j ( M ) . For a nonnegative integer k, the weighted C k space C k β ( M ) is defined as the set of C k functionsv with the norm || v || C k β ( M ) = k X j = sup M r − β + j | D jx v | . LI MA
The weighted H¨older space C k + αβ ( M ) is defined as the set of functions v ∈ C k β ( M ) with thenorm || v || C k + αβ ( M ) = || v || C k β ( M ) + sup x , y ∈ M min( r ( x ) , r ( y )) − β + k + α | D kx v ( x ) − D kx v ( y ) || x − y | α . We end the introduction with a brief outline of the paper. We discuss the local existencetheory of Yamabe flow on a complete Riemannian manifold with bounded scalar curvaturein section 2, this part may be well-known to experts. In section 3, we obtain the globalYamabe flows on AF manifolds. We show or give an outline proof of Theorems 4 and 5. Inthe appendix section 4, we discuss the general version of maximum principle, which may beused in the space decaying argument of the Yamabe flows on AF manifolds.2. Y amabe flow : local existence Let ( M , g M ) be a complete Riemannian manifold of dimension n = dim ( M ). Given aninitial metric g = w g M where w > M . Let R = R ( g ) be the scalarcurvature of the initial Riemannian metric g . The following local existence results of thesolutions to the Yamabe flow (5) may be well-known for experts [1] (see also Theorem 2.4in [9]), but it is new. Theorem 7.
Let ( M , g M ) be an n-dimensional complete manifold with bounded scalar cur-vature and let g = w g M , where w > is a fine function on M. Then Yamabe flow (5)below with initial metric g has a smooth solution on a maximal time interval [0 , T max ) withT max > such that either T max = + ∞ or the evolving metric contracts to a point at finite timeT max . Since the assumption above is weaker than previous existence result, one can not expectsthe uniqueness of the Yamabe flow. We shall use the formulation from the interesting paper[32]. The plan of the proof is to get the local existence result of the Yamabe flow on theRiemannian manifold ( M , g M ) by considering the evolution equation in the following form LOBAL YAMABE FLOW 7 ([1]) 1 n − w t = − wRn − = − R n − + ∆ g M ww + ( n − |∇ w | g M w , (5)with w (0) = w . We denote the terms of right side of the equation (5) by B [ w ] : = ( n − (cid:16) − R n − + ∆ g M ww + ( n − |∇ w | g M w (cid:17) . We first set up the local existence result on any bounded domain with uniform time inter-val. Given a smooth, bounded domain Ω ⊂ M and T >
0, we may assume that − R ≥ nc forsome constant c and we consider the problem ∂ w ∂ t = B [ w ] in Ω × [0 , T ], w = φ on ∂ Ω × [0 , T ], w = w on Ω × { } (6)for given 0 < w ∈ C ,α ( Ω ) and φ ∈ C ,α ;1 , α ( ∂ Ω × [0 , T ]) satisfying φ ( · , t ) = w on ∂ Ω × [0 , T ]Since w and R g are bounded on the compact set ∂ Ω and w >
0, the nonlinear term B [ w ] iswell-defined at the initial time.By the standard parabolic theory [21] we may solve the linear parabolic problem n − ∂ ˜ u ∂ t − ∆ g M ˜ uw − ( n − < ∇ ˜ u , ∇ w > g M w = − R n − Ω × [0 , T ],˜ u = φ on ∂ Ω × [0 , T ],˜ u = w on Ω × { } .(7)to get the solution ˜ u . Since Ω is bounded and since w > M , there exists some δ > Ω and w such that w ≥ δ in Ω × [0 , T ]. Therefore, equation (7) is uniformlyparabolic with regular coe ffi cients and the initial-boundary conditions are satisfied. Accord-ing to linear parabolic theory [21][IV.5, Theorem 5.2], problem (7) has a unique solution˜ u ∈ C ,α ;1 , α ( Ω × [0 , T ]). Since w > φ ( · , t ) = w for all t ∈ [0 , T ], the parabolic maxi-mum principle applied to ˜ u ( · , t ) implies ˜ u ≥ ε on Ω × [0 , T ] for some ε > Ω LI MA and ˜ u . For the short time t >
0, we want to get the solution w to (6) which will be close tothe function ˜ u .We shall use the inverse function theorem to construct the short time solution to (6) on anybounded domain. Lemma 8 (Short-time existence on bounded domains) . Let Ω ⊂ M be a smooth boundeddomain in ( M , g M ) . Then there exists T > such that problem (6) has a unique solution.Proof. We shall construct a solution w to (6) is of the form w = ˜ u + v , where ˜ u solves (7) and ∂ v ∂ t = B [ ˜ u + v ] − ∂ ˜ u ∂ t in Ω × [0 , T ], v = ∂ Ω × [0 , T ], v = Ω × { } .(8)For the H¨older exponent 0 < α <
1, we define the working space X : = { v ∈ C ,α ;1 , α ( Ω × [0 , T ]) | v = Ω × { } ) ∪ ( ∂ Ω × [0 , T ]) } , Y : = { f ∈ C ,α ;0 , α ( Ω × [0 , T ]) | f = ∂ Ω × { }} . Notice that the map F : X → Y , F : v ∂∂ t ( ˜ u + v ) − B [ ˜ u + v ] . is well-defined because the initial-boundary conditions imply that at every p ∈ ∂ Ω for every v ∈ X , we have ( Fv )( p , = (cid:16) ∂ ˜ u ∂ t − B [ ˜ u ] (cid:17) ( p , = (cid:16) ∂φ∂ t ( · , − B [ u ] (cid:17) ( p ) = . The linearization of B [ ˜ u ] around ˜ u ∈ C ,α ;1 , α ( Ω × [0 , T ]) gives the linear operator˘ L ( ˜ u ) = ( n − (cid:16) − ∆ g M ˜ u ˜ u − ( n − |∇ ˜ u | g M ˜ u + ( n − u < ∇ ˜ u , ∇ · > g M + ∆ g M ˜ u (cid:17) . LOBAL YAMABE FLOW 9
We claim that the map F is Fr´echet di ff erentiable at 0 ∈ X . The reason is below. First, themap F is Gˆateaux di ff erentiable at 0 ∈ X with derivative DF (0) : X → Yw ∂∂ t w − L ( ˜ u ) w . Second, the mapping u ˘ L ( u ) is continuous near ˜ u because ˜ u is bounded away from zero.Hence, DF (0) is the Fr´echet-derivative of S at 0 ∈ X . Note that the linear operator ∂∂ t − ˘ L ( ˜ u )is uniformly parabolic.Let f ∈ Y be an arbitrary element. By definition, 0 = f ( · ,
0) on ∂ Ω . We consider the linearparabolic problem ∂ w ∂ t − ˘ L ( ˜ u ) w = f in Ω × [0 , T ], w = ∂ Ω × [0 , T ], w = Ω × { } .(9)As before, linear parabolic theory guarantees that (9) has a unique solution w ∈ X . Hence,the continuous linear map DF (0) : X → Y is invertible.By the Inverse Function Theorem, F is invertible in some neighborhood V ⊂ Y of F (0).Claim that V contains an element h such that h ( · , t ) = ≤ t ≤ ε and su ffi ciently small ε >
0. Fix f : = F (0) = ∂∂ t ˜ u − B [ ˜ u ]. Choose η : [0 , T ] → [0 , ff function suchthat η ( t ) = , for t ≤ ε, , for t > ε, ≤ d η dt ≤ ε . Note that η f ∈ V for su ffi ciently small ε >
0. In fact, since ˜ u is smooth in Ω × [0 , T ], wehave f ∈ C ( Ω × [0 , T ]). Noting at t =
0, we have f ( · , = ∂ ˜ u ∂ t ( · , − B [ w ] = Ω ,(10) we may estimate | f ( · , s ) | = | f ( · , s ) − f ( · , | ≤ s | f | C ( Ω × [0 , T ]) . (11)Take t , s ∈ [0 , T ] and t > s . For s > ε , we have( f − η f )( · , s ) = ( f − η f )( · , t ) = . Then we may assume s ≤ ε . In this case we may estimate the time di ff erence of the function( f − η f ) in the following way. | ( f − η f )( · , t ) − ( f − η f )( · , s ) |≤ | f ( · , t ) − f ( · , s ) | + | η f ( · , t ) − η f ( · , s ) |≤ (cid:0) + | η ( t ) | (cid:1) | f ( · , t ) − f ( · , s ) | + | f ( · , s ) || η ( t ) − η ( s ) |≤ | f | C | t − s | + s | f | C | η ′ | C | t − s |≤ (cid:0) + s ε (cid:1) | f | C | t − s | (12) ≤ | f | C | t − s | . (13)By (10), we may reduce the special case s = | ( f − η f )( · , t ) | ≤ t | f | C . (14)Since the left-hand side of (14) vanishes for t > ε , we may have | f − η f | C ≤ ε | f | C . If | t − s | < ε , the estimate (13) implies that | ( f − η f )( · , t ) − ( f − η f )( · , s ) | ≤ ε − α | f | C | t − s | α . LOBAL YAMABE FLOW 11 If | t − s | ≥ ε , we may replace the estimate by the fact that | ( f − η f )( · , t ) − ( f − η f )( · , s ) | ≤ | f − η f | C ≤ ε | f | C ≤ ε − α | f | C | t − s | α . Then, [ f − η f ] α , t ≤ ε − α | f | C . For the estimation of the spatial H¨older seminorm, we may obtain a similar estimate from(11) and estimate the space di ff erence: | ( f − η f )( x , t ) − ( f − η f )( y , t ) |≤ | − η ( t ) || f ( x , t ) − f ( y , t ) | α | f ( x , t ) − f ( y , t ) − α |≤ | f | α C d ( x , y ) α (cid:0) ε | f | C (cid:1) − α = (4 ε ) − α | f | C d ( x , y ) α , where d ( x , y ) is the Riemannian distance between x and y in ( M , g M ). Then, | f − η f | Y ≤ C ε β − α | f | C . This implies that η f belongs to the neighborhood V of f if ε > ffi ciently small. Bythe construction above, F − ( η f ) is a solution to (8) in Ω × [0 , ε ]. Setting T = ε >
0, we thenobtain the desired result. (cid:3)
We are now going to prove Theorem 7.
Proof.
We now may obtain the local in time solution to (5) on the whole Riemannian man-ifold ( M , g M ). Recall that we have assumed the scalar curvature of g = w g M is bounded.Let Ω ⊂ Ω ⊂ · · · be the smooth compact domain exhaustion of M (the existence of suchdomain exhaustion was used in [15]). Recall that p = n + n − , L g v = ∆ g v − aR g v and a = n − n − . We may write g = v / ( n − g M for w = v / ( n − and look for for solution of the form g ( x , t ) = ˇ u / ( n − g = ( ˇ uv ) / ( n − g M = v / ( n − g M = ug M . Then the Yamabe flow equation may be written as ∂ v p ∂ t = L g M v , x ∈ M , t > , with the initial data v (0) = v . Recall that L g M v = ∆ g M v − aR g M v in M . For shortening thenotation, we may assume v = g = g M . Then, the solution ˇ u ( x , t ) to Yamabeflow (5) may be obtained by a sequence of approximation solutions u m ( x , t ) = ˇ u m ( x , t ) / ( n − obtained above. Note that ˇ u m ( x , t ) satisfies(15) ∂ ˇ u pm ∂ t = L g ˇ u m , x ∈ Ω m , t > , ˇ u m ( x , t ) > , x ∈ Ω m , t > , ˇ u m ( x , t ) = , x ∈ ∂ Ω m , t > , ˇ u m ( · , = , x ∈ Ω m . Since ˇ u m ( x , t ) = ∂ Ω m , by the maximum principle, we may conclude thatmax Ω m ˇ u m ( t ) ≤ (1 + n − n − n +
2) sup M n | R g | t ) n − . and min Ω m ˇ u m ( t ) ≥ (1 − n − n − n +
2) sup M n | R g | t ) n − . We see that ˇ u m ( t ) has an uniformly upper bound on [0 , t ) for any t > , ( n − n + n −
2) sup Mn | R g | ]. Let T = ( n − n + n −
2) sup Mn | R g | . Then every local solution { u m } is well-defined on the time interval [0 , T ].Applying Trudinger’s estimate [35] (or the Krylov-Safonov estimate) and Schauder esti-mate of parabolic equations to (5) on any ball B g ( p , r ) ⊂ ( M , g ), we have || u m || C + α, + α ( B g ( p , r ) × [0 , T ]) ≤ C , where C is independent of the point p . Using the diagonal subsequence of { u m } , wemay extract a C + α, + α loc convergent sequence with its positive limit u ( x , t ) = ˇ u n − on whole LOBAL YAMABE FLOW 13 M , which is the desired local in time solution to (5). Since g ( · , t ) = ˇ u n − g , we also havesup B g ( p , r ) × [0 , T ] | Rm ( x , t ) | ≤ C . With this understanding, we may extend the solution to the maxi-mal time solution as we wanted.This completes the proof of the result. (cid:3) global Y amabe flows on ALE manifolds
Assume that ( M , g ) is an ALE manifold. We shall show that the Yamabe flow existsglobally. Assume by contrary that the maximal time of Yamabe flow is finite, i.e., T max < ∞ .Then according to AF property of the solution in Theorem 5.1 [9] (and its argument is basedon the generalized maximum principle Theorem 9 as showed in appendix), we know thatthere exists a compact set S ⊂ M such that12 ≤ u ( x , t ) ≤ / , ∀ ( x , t ) ∈ ( M \ S ) × [0 , T max )and there is a point x ∈ S such that u is non-trivial in a neighborhood of x . As in [36],using DiBennedetto’s estimate we may extend the solution u continuously to T max . UsingYe’s argument, we know that u can not have any zero point in S at T max . Therefore, thereexist two positive constants c and c such that c ≤ u ( x , t ) ≤ c , ∀ ( x , t ) ∈ M × [0 , T max ] uniformly . Then we may use the standard parabolic theory [35] to extend the solution beyond T max ,which is a contradiction with T max < ∞ . The decaying property of the Yamabe flow followsfrom Theorem 5.1 in [9]. This then completes the proof of Theorem 3.The proof of Theorem 4 follows from the application of Theorem 7, Theorem 5.1 in [9],and Theorem 6 in [23].Since the proof of Theorem 5 is by now easy to give and the proof is below. Proof.
We choose δ > u = δ w <
1. Note that ˜ u is the lower solutionof the Yamabe flow. Let ˜ g = ˜ u / ( n − g . Then the scalar curvature of the metric ˜ g is non-negative and we also have g ≥ ˜ g . Note that along the Yamabe flow g ( t ) ≥ ˜ g and by the maximum principle we know that thescalar curvature R = R ( g ( t )) ≥ M . Since ∂ g ∂ t = − Rg ≤ , We then know that the Yamabe flow g ( t ) converges in C ∞ loc ( M ) to a Yamabe metric of scalarcurvature zero. (cid:3)
4. A ppendix : the maximum principle In this section, we present a generalized version of the maximum principle (see Theorem4.3 in [16] and Theorem 2.6 in [9]), where they consider the maximum principle for theparabolic equation ∂∂ t v − ∆ v ≤ b · ∇ v + cv or ∂∂ t v − div ( a ∇ v ) ≤ b · ∇ v + cv on noncompactmanifolds, where ∆ and ∇ depend on g ( t ). Our maximum principle is about the more generalequation m ( x ) ∂∂ t v − div ( a ∇ v ) ≤ b · ∇ v + cv , M × [0 , T )where m ( x ) is a positive regular function on M . Theorem 9.
Suppose that the complete noncompact manifold M n with Riemannian metricg ( t ) satisfies the uniformly volume growth conditionvol g ( t ) ( B g ( t ) ( p , r )) ≤ exp ( k (1 + r )) for some point p ∈ M and a uniform constant k > for all t ∈ [0 , T ] . Let v be a di ff erentiablefunction on M × (0 , T ] and continuous on M × [0 , T ] . Assume that v and g ( t ) satisfy LOBAL YAMABE FLOW 15 (i) The di ff erential inequalitym ( x ) ∂∂ t v − div ( a ∇ v ) ≤ b · ∇ v + cv , where m ( x ) is a positive continuous function on M such that < m ≤ m ( x ) ≤ m for someconstant m > and m > , the vector field b and the function a and c are uniformlybounded < α ′ ≤ a ≤ α , sup M × [0 , T ] | b | ≤ α , sup M × [0 , T ] | c | ≤ α , for some constants α ′ , α , α < ∞ . Here ∆ and ∇ depend on g ( t ) .(ii) The initial data v ( p , ≤ , for all p ∈ M.(iii) The growth condition Z T ( Z M exp [ − α d g ( t ) ( p , y ) ] |∇ v | ( y ) d µ t ) dt < ∞ . for some constant α > .(iv) Bounded variation condition in metrics in the sense that sup M × [0 , T ] | ∂∂ t g ( t ) | ≤ α for some constant α < ∞ .Then we have v ≤ on M × [0 , T ] . Remark 10.
Note that the conditions (iii) and (iv) are satisfied if the sectional curvature ofg ( t ) and ∇ v are uniformly bounded on [0 , T ] . There are many versions of maximum princi-ples, one may prefer to [2] and [11] . Proof of Theorem 9:
Fix K > θ > h ( y , t ) = − θ d g ( t ) ( p , y )4(2 η − t ) , < t < η, where d g ( t ) ( p , y ) is the distance between p and y at time t and 0 < η < min( T , K , α , α ).Then ddt h = − θ d g ( t ) ( p , y )4(2 η − t ) − θ d g ( t ) ( p , y )2(2 η − t ) ddt d g ( t ) ( p , y ) . By (iv), we have | ddt d g ( t ) ( p , y ) | ≤ α d g ( t ) ( p , y ) . Then we have that ddt h ≤ − θ − |∇ h | + θ − α |∇ h | (2 η − t ) , We choose θ = α . Using η ≤ α we have m ddt h + a |∇ h | ≤ . (16)Let K >
0, which will be a very large constant. Taking f K = max { min( f , K ) , } and 0 < ǫ <η , we have Z ηǫ e − β t ( Z M φ e h f K ( div ( a ∇ f ) − ∂ f ∂ t ) d µ t ) dt ≥ − α Z ηǫ e − β t ( Z M φ e h f K |∇ f | d µ t ) dt − α Z ηǫ e − β t ( Z M φ e h f K f d µ t ) dt LOBAL YAMABE FLOW 17 for some smooth time independent compactly supported function φ on M n , where β > ≤ − Z ηǫ e − β t ( Z M φ e h a < ∇ f K , ∇ f K > d µ t ) dt − Z ηǫ e − β t ( Z M φ e h f K a < ∇ h , ∇ f > d µ t ) dt − Z ηǫ e − β t ( Z M φ e h f K a < ∇ φ, ∇ f > d µ t ) dt − Z ηǫ e − β t ( Z M m ( x ) φ e h f K ∂ f ∂ t d µ t ) dt + α Z ηǫ e − β t ( Z M φ e h f K f d µ t ) dt + α Z ηǫ e − β t ( Z M φ e h f K |∇ f | d µ t ) dt = I + II + III + IV + V + VI . By Schwartz’ inequality, we deriveII ≤ Z ηǫ e − β t ( Z M φ e h a |∇ f | d µ t ) dt + Z ηǫ e − β t ( Z M φ e h f K a |∇ h | d µ t ) dt , III ≤ Z ηǫ e − β t ( Z M φ e h a |∇ f | d µ t ) dt + Z ηǫ e − β t ( Z M e h f K a |∇ φ | d µ t ) dt , and VI ≤ Z ηǫ e − β t ( Z M φ e h a |∇ f | d µ t ) dt + α Z ηǫ e − β t ( Z M e h f K a |∇ φ | d µ t ) dt ≤ Z ηǫ e − β t ( Z M φ e h a |∇ f | d µ t ) dt + α α ′ Z ηǫ e − β t ( Z M e h f K |∇ φ | d µ t ) dt . Since − e h f K ∂ f ∂ t ≤ − e h f K ∂ f K ∂ t + ∂∂ t ( e h f K ( f K − f )) , and f K ( f K − f ) ≤ , we obtainIV + V ≤ − Z ηǫ e − β t ( Z M m ( x ) φ e h ∂ f K ∂ t d µ t ) dt + Z ηǫ e − β t ( Z M m ( x ) φ ∂∂ t ( e h f K ( f K − f )) d µ t ) dt − α Z ηǫ e − β t ( Z M φ e h f K ( f K − f ) d µ t ) dt + α Z ηǫ e − β t ( Z M φ e h f K d µ t ) dt . Moreover, we have | ddt ( d µ t ) | ≤ n α d µ t by (iv). Now we choose β > m β ≥ n α + α + α α ′ . ThenIV + V ≤ − e − β t Z M m ( x ) φ e h f K d µ t | t = η + e − β t Z M m ( x ) φ e h f K d µ t | t = ǫ + Z ηǫ e − β t ( Z M m ( x ) φ e h f K ∂ h ∂ t d µ t ) dt − m β Z ηǫ e − β t ( Z M φ e h f K d µ t ) dt + e − β t Z M φ e h f K ( f K − f ) d µ t | t = η − e − β t Z M φ e h f K d µ t | t = ǫ . Combining the estimates of I − VI and letting ǫ →
0, we obatin0 ≤ − Z η e − β t ( Z M φ e h a |∇ f K | d µ t ) dt + Z η e − β t ( Z M φ e h a |∇ f | d µ t ) dt + Z η e − β t ( Z M e h f K a |∇ φ | d µ t ) dt − e − β t Z M m ( x ) φ e h f K d µ t | t = η . by f K ≡ t = ≤ φ ≤ φ ≡ B g ( p , R ), φ ≡ B g ( p , R +
1) and |∇ g φ | g ≤
2. Then we have12 e − βη Z B g ( p , R ) m ( x ) φ e h f K d µ t | t = η ≤ Z η e − β t ( Z B g ( p , R + φ e h a ( |∇ f | − |∇ f K | ) d µ t ) dt + C ( α ) Z η e − β t ( Z B g ( p , R + \ B g ( p , R ) e h f K ad µ t ) dt , LOBAL YAMABE FLOW 19 where C ( α ) is a constant only depending on α . By 0 < η < min( K , α ) and volumegrowth assumptions on M n , we have Z η e − β t ( Z B g ( p , R + \ B g ( p , R ) e h f K ad µ t ) dt → , as R → ∞ . Then we derive12 e − βη Z M φ e h f K d µ t | t = η ≤ Z η e − β t ( Z M φ e h a ( |∇ f | − |∇ f K | ) d µ t ) dt . Letting K → ∞ , we conclude that12 e − βη Z M m ( x ) φ e h (max( f , d µ t | t = η ≤ , where 0 < η < min( T , K , α , α ). That implies that f ≤ M n × [0 , η ]. By the inductionargument, we then have that f ≤ M n × [0 , T ]. (cid:3) R eferences [1] Y.An, L.Ma, The Maximum Principle and the Yamabe Flow , Partial Di ff erential Equations and TheirApplications, World Scientific, Singapore, pp211-224, 1999.[2] T. Aubin, Nonlinear Analysis on Manifolds, Monge-Amper´e Equations . Springer, 1982.[3] D.G. Aronson,
The porous medium equation . Nonlinear di ff usion problems (Montecatini Terme, 1985),1-46, Lecture Notes in Math., 1224, Springer, Berlin, 1986.[4] Eric Bahuaud and Boris Vertman. Yamabe flow on manifolds with edges . Math. Nachr., 287(2-3):127–159,2014.[5] Eric Bahuaud and Boris Vertman.
Long-time existence of the edge Yamabe flow . J. Math. Soc. Japan 71.2(Apr. 2019), pp. 651-688[6] T.Balehowsky, E. Woolgar,
The Ricci flow of asymptotically hyperbolic mass and applications ,http: // arxiv.org / abs / Convergence of the Yamabe flow for arbitrary initial energy , J. Di ff erential Geometry 69(2005), 217-278[8] B. Choi, P. Daskalopoulos, and J. King. Type II Singularities on complete non-compact Yamabe flow .ArXiv e-prints, September 2018. [9] L.Cheng, A.Zhu,
Yamabe flow and ADM mass on asymptotically flat manifolds , J.Math. Phys., 56,101507(2015); doi:10.1063 / Yamabe flow on locally conformally flat manifolds , Comm. pure appl. math.,Vol.XLV,(1992)1003-1014[11] B. Chow and D. Knopf,
The Ricci Flow: An Introduction . Math. Surveys Monogr., vol. 110 (2004).[12] X.Dai, L.Ma,
Mass under Ricci flow , Commun. Math. Phys., 274, 65-80 (2007).[13] P. Daskalopoulos and C.E. Kenig,
Degenerate Di ff usion. Initial Value Problems and Local RegularityTheory , EMS Tracts Math., Europ. Math. Soc., Zurich, 2007.[14] Panagiota Daskalopoulos and Natasa Sesum. The classification of locally conformally flat Yamabe soli-tons . Adv. Math., 240:346–369, 2013.[15] J.Dodziuk,
Maximum principle for parabolic inequalities and the heat flow on open manifolds . IndianaUniv. Math. J. 32, 703-716 (1983).[16] K.Ecker, G.Huisken,
Interior estimates for hypersurfaces moving by mean curvature , Invent. math.105,547-569(1991)[17] Gregor Giesen and Peter M. Topping.
Existence of Ricci flows of incomplete surfaces . Comm. PartialDi ff erential Equations 36.10 (2011), pp. 1860-1880.[18] R. Hamilton, The Ricci flow on surfaces . In: Mathematics and General Relativity, Contemporary Mathe-matics 71, AMS, 237-261. (1988)[19] Richard S. Hamilton.
Lectures on geometric flows. unpublished, 1989.[20] M.Herrero, M.Pierre,
The Cauchy problem for u t = ∆ u m when < m <
1, Trans. A.M.S.,291(1)(1985)145-158.[21] O. A. Ladyzhenskaja, V. A. Solonnikov, and N. N. Ural´ceva.
Linear and Quasi-linear Equations of Para-bolic Type . American Mathematical Society, translations of mathematical monographs. American Math-ematical Society, 1988.[22] John M. Lee, Thomas H. Parker,
The Yamabe problem , Bull. Amer. Math. Soc. (N.S.) 17 (1987), 37-91.[23] Li Ma,
Yamabe flow and metrics of constant scalar curvature on a complete manifold , Calc.Var. PartialDi ff erential Equations, (2019) 58:30[24] Li Ma, Convergence of Ricci flow on R2 to the plane . Di ff erential Geometry and its Applications 31 (2013)388-392[25] Li Ma, Gap theorems for locally conformally flat manifolds . J. Di ff erential Equations 260 (2016), no. 2,1414-1429 LOBAL YAMABE FLOW 21 [26] Li Ma,
Yamabe metrics, fine solutions to Yamabe flow, and local L -stability , arxiv: 2020,[27] Li Ma, The Dirichlet Problem at Infinity on a Quasi-hyperbolic manifold , Proc. international conferenceon pure and appl. math., P195-207, ed. K.S.Chang and K.C.Chang, South Korea Press, 1994.[28] Li Ma, L.Cheng,
Yamabe flow and Myers type theorem . Journal of Geom. Anal., 24(2014)246-270.[29] Li Ma; Ingo Witt,
Discrete Morse flow for Ricci flow and porous medium equation . Commun. NonlinearSci. Numer. Simul. 59 (2018), 158-164. (Reviewer: Kin Ming Hui) 53C44 (35K55)[30] R.Schoen,
Variation theory for the total scalar curvature functional for Riemannian metric and relatedtopics , Lecture notes in Math 1365 (Springer, Berlin,1987), 120-154[31] R. Schoen, S.T.Yau, lectures on Di ff erential geometry , Academic Press, 1986.[32] M.Schulz, Instantaneously complete Yamabe flow on hyperbolic space , Calc. Var. Partial Di ff erentialEquations. 58 (2019), no. 6, Paper No. 190, 30 pp.[33] M. Schulz, Incomplete Yamabe flows and removable singularities . J. Funct. Anal. 278 (2020), no. 11,108475, 18 pp.[34] H.Schwetlick, M.Struwe,
Convergence of the Yamabe flow for ’large’ energies , J. Reine Angew. Math.562 (2003), 59-100.[35] Neil S. Trudinger.
Pointwise estimates and quasilinear parabolic equations . Comm. Pure Appl. Math.,21:205-226, 1968.[36] R. Ye,
Global existence and convergence of the Yamabe flow . J.Di ff .Geom. 39 (1994), 35-50.L i MA, S chool of M athematics and P hysics , U niversity of S cience and T echnology B eijing , 30 X ueyuan R oad , H aidian D istrict B eijing , 100083, P.R. C hina D epartment of M athematics , H enan N ormal university , X inxiang , 453007, C, 453007, C