aa r X i v : . [ m a t h . DG ] F e b NEW YAMABE-TYPE FLOW IN A COMPACTRIEMANNIAN MANIFOLD
LI MA ∗ Abstract.
In this paper, we set up a new Yamabe type flow on acompact Riemannian manifold (
M, g ) of dimension n ≥
3. Let ψ ( x ) beany smooth function on M . Let p = n +2 n − and c n = n − n − . We studythe Yamabe-type flow u = u ( t ) satisfying u t = u − p ( c n ∆ u − ψ ( x ) u ) + r ( t ) u, in M × (0 , T ) , T > r ( t ) = Z M ( c n |∇ u | + ψ ( x ) u ) dv/ Z M u p +1 , which preserves the L p +1 ( M )-norm and we can show that for any initialmetric u >
0, the flow exists globally. We also show that in some cases,the global solution converges to a smooth solution to the equation c n ∆ u − ψ ( x ) u + r ( ∞ ) u p = 0 , on M and our result may be considered as a generalization of the result ofT.Aubin, Proposition in p.131 in [1]. Introduction
Nonlocal evolution equations arise naturally from geometry. The mostfamous one is the normalized Ricci flow preserving the volume [15], intro-duced by R.Hamilton in 1982. The evolutions of planar curves preservingthe length or area enclosed [14] [10] [19] are in this category. To solve theYamabe problem from the view point of evolution equation, Hamilton hasalso proposed the normalized Yamabe flow to approaching a Yamabe metricon a closed manifold. In this paper, we introduce a Yamabe type flow (which preserves the L nn − ( M )-norm, see below for the definition) on a com-pact Riemannian manifold and study its global existence and convergence insome cases. We point out that some arguments in [25] and [6] about Yam-abe flow can be used to handle such a general norm-preserving flow. Wealso notice that many heat flow methods may be introduced to functionalsrelated to Yamabe problem on ( M, g ) ([1] [5], [3], [18]).
Date : Feb. 6th, 2020.2020
Mathematics Subject Classification.
Key words and phrases.
Yamabe-type flow, global existence, norm-preserving flow,scalar curvature, asymptotic behavior. ∗ The research of Li Ma is partially supported by the National Natural Science Foun-dation of China No. 11771124 and a research grant from USTB, China. ∗ We now introduce a new Yamabe-type flow on a compact Riemannianmanifold (
M, g ) of dimension n ≥
3. Let p = n +2 n − , c n = n − n − , and let R ( g ) be the scalar curvature. Assume that ψ is a given smooth function in M . The Yamabe-type flow u = u ( t ) is defined such that u ( t ) satisfies theevolution equation(1) u t = u − p ( c n ∆ u − ψ ( x ) u ) + r ( t ) u, in M × (0 , T ) , T > r ( t ) := r ψ ( t ) := Z M ( c n |∇ u | + ψ ( x ) u ) dv/ Z M u p +1 with initial data u (0) >
0. The local in time solution of this problem (1)is by now standard [20] and can be obtained by the fixed point method orthe method such as the implicit function theorem. This flow preserves thenorm of the evolving function u ( t ), Z M u ( t ) p +1 dv = Z M u (0) p +1 dv which may be assumed to be one for simplicity. In fact, we have1 p + 1 ddt Z M u p +1 dv = Z M u p u t = Z M u ( c n ∆ u − ψ ( x ) u ) + r ( t ) Z M u p +1 = 0 . Hence, Z M u ( t ) p +1 dv = Z M u (0) p +1 dv = 1 . Since, for u = u ( t ), r ( t ) = Z M ( c n |∇ u | + ψ ( x ) u ) dv, Z M u ( t ) p +1 = 1 , we have 12 ddt r ( t ) = Z M c n ( ∇ u, ∇ u t ) + ψ ( x ) uu t = Z M ( − c n ∆ u + ψ ( x ) u ) u t = Z M u p ( − u t u + r ( t )) u t = − Z M u p u t u ≤ . So, r ( t ) is non-increasing in t along the flow.To understand this flow well, we introduce the pseudo-scalar curvature(2) R ψ = u − p ( − c n ∆ u + ψ ( x ) u ) . EW YAMABE-TYPE FLOW IN A MANIFOLD 3
Then the equation (1) can be written as(3) u t = u ( − R ψ + r ( t )) , in M × (0 , T ) , T > . We remark that one may study the flow(4) u t = − R ψ u on any complete non-compact Riemannian manifold of dimension n ≥ M, g )of dimension n ≥
0, we can show that for any initial data u (0) >
0, the flowexists globally.
Theorem 1.
For any initial data u (0) > , the Yamabe-type flow ( u ( t )) to (1) above exists globally and r ∞ = lim t →∞ r ( t ) = r ( ∞ ) exists. One may give a proof of this result using similar arguments as in section4 in [25] or as in [6], which is a local in natural argument for the solution u to Yamabe flow. Here, we prefer to give a direct proof to control thenorm growth of pseudo-scalar curvature along the flow. In the case when( M, g ) is a closed surface, we may also introduce the ψ -Gauss flow flow. Let g ( t ) = e u ( t ) g , where u ( t ) : M → R is a smooth function, and let ψ : M → R be a given smooth function. The ψ -Gauss flow is defined by(5) e u u t = ∆ u − ψ ( x ) + r ( t ) e u , in M × (0 , T ) , T > r ( t ) := r ψ ( t ) := Z M K ψ dv/ Z M e u where K ψ = e − u ( − ∆ u + ψ ( x )) with initial data u (0) >
0. By similarmethod, we know that there is a global flow for (5). Interesting questionsare to find similar results to Chang-Yang [8], [9], and Ding-Liu [11]. RelatedYamabe type flow with boundary data may also be studied.We can also get the convergence result of the flow ( g ( t )) to (1) as in theYamabe-scalar negative and zero cases. So we may define the Yamabe-typeinvariant below. Define, for p = n +2 n − , for u ∈ H ( M ); u = 0, E ( u ) = R M ( c n |∇ u | + ψu ) dv ( R M | u | p +1 dv ) / ( p +1) and Y ψ ( M ) = inf { E ( u ) , u ∈ H ( M ); u = 0 } which is called the Yamabe-type invariant of M . We denote for M = S n and ψ = n ( n − Y ( S n ) = inf { E ( u ); u ∈ H ( M ); u = 0 } for the Yamabe constant on S n . Using Aubin’s argument (see p.131 in [1],seealso [21] and [12]), we know that Y ψ ( M ) ≤ Y ( S n ). In [2], Aubin proved LI MA ∗ that if n ≥ ψ ( x ) < R ( g )( x ) somewhere, then there is a minimizer for Y p si ( M ). Of course, one may use the argument of Brezis-Nirenberg [7] toknow Y ψ ( M ) < Y ( S n ) provided ψ < n ( n −
1) for M n = S n with n ≥ λ ( ψ ) = inf { u ∈ H ( M ); u =0 } R M ( c n |∇ u | + ψu ) dv R M | u | dv . Then it is standard to know that there is a positive function u such that λ ( ψ ) = Z M ( c n |∇ u | + ψu ) dv, Z M u dv = 1 , and − c n ∆ u + ψ ( x ) u = λ ( ψ ) u , in M. We remark that for λ ( ψ ) ≥
0, we have Y ψ ( M ) ≥
0. In fact, for ant u = 0, R M ( c n |∇ u | + ψu ) dv ( R M | u | p +1 dv ) / ( p +1) = R M ( c n |∇ u | + ψu ) dv R M u dv R M u dv ( R M | u | p +1 dv ) / ( p +1) ≥ λ ( M ) R M u dv ( R M | u | p +1 dv ) / ( p +1) ≥ . For λ ( ψ ) <
0, we have Y ψ ( M ) <
0. In fact, taking u = u above, we have Y ψ ( M ) ≤ R M ( c n |∇ u | + ψu ) dv R M u dv R M u dv ( R M | u | p +1 dv ) / ( p +1) = λ ( M ) R M u dv ( R M | u | p +1 dv ) / ( p +1) < . The relation between λ ( M ) and Y ψ ( M ) can be given below. Since, bythe Holder inequality, we have Z M | u | dv ≤ ( V ol ( M )) p − p +1 ( Z M | u | p +1 dv ) / ( p +1) . We then have Y ψ ( M ) ≤ ( V ol ( M )) − p − p +1 λ ( ψ ) , for λ ( ψ ) ≥ Y ψ ( M ) ≥ ( V ol ( M )) − p − p +1 λ ( ψ ) , for λ ( ψ ) <
0. Note that if ψ ( x ) ≤ M , then λ ( M ) <
0. In fact, wemay take u = 1 / p vol ( M ). Then, λ ( M ) ≤ col ( M ) R M ψdv < Theorem 2.
Assume < Y ψ ( M ) < Y ( S n ) and assume, for the initialmetric g = u / ( n − g with u > on M , E ( u ) ≤ Y ( S n ) . Then along the EW YAMABE-TYPE FLOW IN A MANIFOLD 5
Yamabe-type flow ( u ( t )) to (1) , we have a convergent subsequence u ( t j ) → u ∞ > , t j → ∞ , and u ∞ is a smooth function satisfies − c n ∆ u + ψ ( x ) u = r ∞ u p , in M, Z M u p +1 = 1 . Our result may be considered as a generalization of Proposition in p.131in [1]. We remark that with more detailed analysis (see [4]), one may obtainsimilar result to Theorem 1.1 in [6] and we leave this open for interestedreaders for pleasure. Whether the Yamabe-type invariant on (
M, g ) can beachieved by some smooth function u > M , generally speaking, is stillan open problem and may be discussed in latter chances.Assume that λ ( ψ ) < ψ ( x ) < M , we can show that the flowconverges at time infinity. Theorem 3.
Assume that λ ( ψ ) < and ψ ( x ) < on M . The Yamabe-type flow ( u ( t )) converges to a metric of constant pseudo-scalar curvature at t = ∞ . We remark that the results may be extended to the case when p >
1. Infact, assuming that λ ( ψ ) = 0 and ψ ( x ) = 0 on M , we have the followingresult. Theorem 4.
Assume that λ ( ψ ) = 0 and ψ ( x ) = 0 on M . Fix any p > .The Yamabe-type flow ( u ( t )) satisfying (6) u t u = u − p c n ∆ u + r ( t ) , in M × (0 , T ) , T > with r ( t ) = Z M c n |∇ u | dv/ Z M u p +1 exists globally and converges to a positive constant at t = ∞ . Via a use of bubble analysis, we can handle more complicated case as in[6], since the proof is lengthy, we prefer to present it elsewhere.The plan of this note is below. The proof of Theorem 1 will be given insection 2 below. Using Struwe’s compactness result, we prove Theorem 2 insection 3. The proofs of Theorem 3 and Theorem 4 will be given in section4. 2. global existence of Yamabe-type flows
We treat the difficulty case when ψ ( x ) > r (0) > Y ψ ( M ) > r > R = R ψ . Recall r − R = u t u . By (2) we know that − R = u − p ( c n ∆ u − ψ ( x ) u ) . LI MA ∗ Taking the time derivative on both sides, we have ddt ( − R ) = − pu − p − u t ( c n ∆ u − ψ ( x ) u ) + u − p ( c n ∆ u t − ψ ( x ) u t )= p u t u R + u − p ( c n ∆[( r − R ) u ] − ψ ( x )[( r − R ) u ])= p ( r − R ) R + u − p ( c n ∆[( r − R ) u ] − ψ ( x )[( r − R ) u ]) , which can be written as ∂∂t R = u − p ( c n ∆[( R − r ) u ] − ψ ( x )[( R − r ) u ]) + p ( R − r ) R. Define L u v = u − p ( c n ∆[ vu ] − ψ ( x )[ vu ]) , and then we have ∂∂t R = − L u ( r − R ) + p ( R − r ) R. Then we may use the maximum principle to obtain
Proposition 5.
We have the pseudo-scalar curvature lower bound (7) inf M R ψ ( t ) ≥ min (cid:26) inf M R ψ (0) , (cid:27) along the Yamabe-type flow.Proof. Note that at the minimum point of R ( t ), we have Recall that ∂∂t R ≥ u − p (( R − r ) c n ∆ u − ψ ( x )[( R − r ) u ]) + p ( R − r ) R. Recall that R = u − p ( − ∆ u + ψ ( x ) u ) . Then we have ∂∂t R ≥ ( R − r )( − R ) + p ( R − r ) R = ( p − R ( R − r ) . We may write it as ∂∂t ( R exp(( p − Z t ( r − R ))) ≥ , which implies that R exp(( p − Z t ( r − R )) ≥ inf M R (0) . Then we have the conclusion. (cid:3)
Similarly, at the maximum point of R , we have ∂∂t R ≤ ( R − r )( − R ) + p ( R − r ) R = ( p − R ( R − r ) , EW YAMABE-TYPE FLOW IN A MANIFOLD 7 which implies that R exp(( p − Z t ( r − R )) ≤ sup M R (0) , which is useful in the case when sup M R (0) ≤ σ ≥ σ = max (cid:26) sup M (1 − R (0)) , (cid:27) . Then applying the maximum principle again, we have(9) R ψ ( t ) + σ ≥ t ≥ Lemma 6. (10) sup M u ( t ) ≤ C ( T ) and (11) inf M u ( t ) ≥ c ( T ) > . Proof.
Note that(12) u − ∂∂t u ( t ) = − ( R ψ ( t ) − r ( t )) ≤ ( r (0) + σ )Thus,(13) sup M u ( t ) ≤ C ( T )for t ∈ [0 , T ]. Hence, for(14) P := ψ ( x ) + σ sup ≤ t ≤ TM u ( t ) p − , we have(15) − c n ∆ u ( t ) + P u ( t ) ≥ − c n ∆ u ( t ) + ψ ( x ) u ( t ) + σu ( t ) n +2 n − = ( R ψ ( t ) + σ ) u ( t ) p ≥ t ∈ [0 , T ]. Then using a Moser iteration argument (or by Cor. A.5 in[6]) we have Z M u ( t ) ≤ C ( T ) inf M u ( t ) . Then by the volume constrain condition, we have(16) 1 ≤ C ( T ) inf M u ( t ) (cid:18) sup M u ( t ) (cid:19) p . LI MA ∗ for all t ∈ [0 , T ]. Since sup M u ( t ) ≤ C ( T ), we get the conclusion. (cid:3) By now it is standard to set the global existence of the flow. Using theresult from [24] (see also [16]), we have the result below.
Proposition 7.
For
T > , there exist < α < and a constant C ( T ) such that (17) | u ( x , t ) − u ( x , t ) | ≤ C ( T ) (cid:16) ( t − t ) α + d ( x , x ) α (cid:17) for all x , x ∈ M and any t , t ∈ [0 , T ] with < t − t < .Proof. Set v = u p . The Yamabe type flow equation (1) on a compact sub-domain D ⊂ M can be written as the divergence form that ∂ t v = c n div ( v /p − ∇ v ) − pψ ( x ) v /p − v + pr ( t ) v. One can see that the structure conditions (2.1) and (1.2-1.3) in [24] aresatisfied. Then we can invoke Theorem 4.2 in [24] to get the locally uniformlyHolder estimate in D for solutions u up to the initial time t = 0. Namely,for any x ∈ M , r >
0, and
T >
0, there exists uniform positive constants β = β ( B r ( x ) , sup B r ( x ) u (0)) ∈ (0 , , and C ( B r ( x ) , β, u (0) , T ) such that for any j ≥ B r ( x ) ⊂ D , thereholds | u | C β ; β/ ( B r/ ( x ) × [0 ,T ]) ≤ C ( B r ( x ) , sup B r ( x ) u (0) , T ) . As always, we have used the Holder spaces C α ; α/ ( B r/ ( x ) × [0 , T ]) in par-abolic distance defined by g + dt . Using the covering argument we canextend the estimate above to whole parabolic region M × [0 , T ]. (cid:3) We now prove Theorem 1, which is the global existence result of the flowfor any initial data.
Proof.
Fix any
T >
0. With the understanding of Proposition 7, we may usethe standard regularity theory for parabolic equations (see [17], Theorem 5on p. 64 or the book [16]) to conclude that all higher order derivatives of thesolution u are uniformly bounded on every fixed time interval [0 , T ]. Hence,we can extend the flow beyond T and then the flow exists for all time. Thisthen completes the proof of Theorem 1. (cid:3) P.S.sequence of Yamabe-type flows and proof of Theorem 2
To understand the asymptotic behavior of the Yamabe type flow (1), weneed some integral estimates about the scalar curvature R . Then we con-sider the P.S. Sequence of the functional E ( u ). Note that once we have theP.S.sequence along the flow, we may invoke the by now standard argument,that is, Struwe’s compactness result [23] to get the partial compactness ofthe sequence. EW YAMABE-TYPE FLOW IN A MANIFOLD 9
Let ˘ R = R − r . Then12 ddt ˘ R = ˘ R ˘ R t = ˘ RL u ( ˘ R ) + pR ˘ R − ˘ Rr t . Recall that for g ( t ) = u n − g for some fixed background metric g and p = n +2 n − , we have R ψ ( t ) = u − p ( − c n ∆ g u + ψ ( x ) u ) . Let g ( t ) = u ( t ) n − g . The Yamabe-type flow (1) may be written as(18) ∂ t u p = c n ∆ g u − ψ ( x ) u + r ψ ( t ) u p = − R ψ ( t ) u p + r ψ ( t ) u p . We may denote by ∆ = ∆ g for simplicity. Then ∂ t u p +1 = − ( p + 1) ˘ Ru p +1 . We now compute ∂ t [ ˘ R u p +1 ] = [ ˘ RL u ( ˘ R ) + pR ˘ R − ˘ Rr t ] u p +1 − ( p + 1) ˘ R u p +1 . Using R ˘ Ru p +1 = 1, we have ∂ t Z M [ ˘ R u p +1 ] = Z M [ c n ˘ Ru ∆( ˘ Ru ) − ψ ( ˘ Ru ) ] − ˘ R ] u p +1 + pr ˘ R u p +1 , which is= − Z M [ c n |∇ ( ˘ Ru ) | + ψ ( ˘ Ru ) ] − Z M ˘ R u p +1 + pr Z M ˘ R u p +1 . Since ψ ≥ ∂ t Z M [ ˘ R u p +1 ] ≤ − Z M ˘ R u p +1 + pr Z M ˘ R u p +1 . Let f ( t ) = Z M [ ˘ R u p +1 ] := Z M [ ˘ R u p +1 ] dv g . By f ( t ) = Z M [ ˘ R u p +1 ] ≤ ( Z M [ ˘ R u p +1 ]) / , we know that f ′ ( t ) ≤ − f ( t ) / + prf ( t ) . Recall that ddt r ( t ) = − n − f ( t ) . Then for any t > Z ∞ t f ( s ) ds < ∞ . Hence, lim t →∞ f ( t ) = 0 . By this and the differential inequality of f , we obtain that as t → ∞ , f ( t ) →
0. Define r ∞ = lim t →∞ r ( t ) . ∗ Then we have lim t →∞ Z M | R − r ∞ | u p +1 dv g = 0 . We now let t k → ∞ , u k = u ( t k ), and g k = g ( t k ) = u / ( n − k g . Then(19) Z M dv g k == Z M u n/ ( n − k dv g = 1 , and Z M | R g k − r ∞ | n/ ( n +2) dv g k → , that is, as t k → ∞ ,(20) Z M | c n ∆ u k − ψ ( x ) u k + r ∞ u n +2 n − k | n/ ( n +2) dv g → . We then apply Struwe’s compactness result [23] (see Theorem 3.1 in [13] forthe detailed proof) to conclude the following result.
Proposition 8.
Let u k be as above with (19) and (20) . After passing to asubsequence, we may find a non-negative integer m , a non-negative smoothfunction u ∞ and a sequence of m − tuplets ( x j,k , ǫ j,k ) with the following prop-erties(i) The limiting function u ∞ satisfies c n ∆ u ∞ − ψ ( x ) u ∞ + r ∞ u n +2 n − ∞ = 0 . (ii) For i = j , we have, as k → ∞ , ǫ j,k ǫ i,k + ǫ i,k ǫ j,k + d ( x i,k , x j,k ) ǫ i,k ǫ i,k → ∞ (iii) We have as t k → ∞ , k u k − u ∞ − m X i =1 w ( x i,k ,ǫ i,k ) k H ( M ) → , where w ( x i,k ,ǫ i,k ) ( z ) = η x i,k ( z )( 4 n ( n − r ∞ ) n − / ǫ n − / i,k ( ǫ i,k + d ( x i,k , z ) ) − n − with η x i,k ( z ) is the cut-off function defined inside of the ball of the radius δ smaller than the injectivity radius on M , namely, η x i,k ( z ) = η δ ( exp − x i,k ( z )) , η δ ∈ C ( B (2 δ )) , where B (2 δ ) ⊂ R n the ball with center and with radius δ > . As the consequence of Proposition 8, we know that(21) r ∞ = ( E ( u ∞ ) n/ + mY ( S n ) n/ ) /n . Once we have this result, we may easily prove Theorem 2.
EW YAMABE-TYPE FLOW IN A MANIFOLD 11
Proof.
First, we remark that u ∞ is a smooth solution so that if it has a zeropoint in M , by the maximum principle we know that it is identically zero.Second, by (21), along the flow (1) we have Y ψ ( M ) ≤ r ∞ ≤ r ( t ) < r (0) ≤ Y ( S n ) . Third, this then implies that m = 0. Then we have u ∞ > M , whichis the limit of the flow with initial data u . Thus we have proved Theorem2. (cid:3) Convergence part of Yamabe-type flows for negative orflat cases
In this section we prove Theorem 3. In the proof below, we may let p > λ ( ψ ) < Proof.
Define the Yamabe-type quotient Q ( g ( t )) = R M R ψ ( t ) dv t ( R M u ( t ) p +1 dv ) / ( p +1) , where g ( t ) = u ( t ) p − g and we may let dv t = u ( t ) p +1 dv. Along the Yamabe-type flow, we may assume that Z M dv t = 1 . In this case we have Q ( g ( t )) = r ψ ( t ) = Z M ( c n |∇ u | + ψ ( x ) u ) dv/ Z M u p +1 dv, i.e., Q ( g ( t )) = r ψ ( t ) = Z M ( c n |∇ u | + ψ ( x ) u ) dv ≥ λ ( ψ ) Z u dv. Then by R u ≤
1, we know that r ψ ( t ) ≥ λ ( ψ )for all t > r ψ ( t ) is decreasing in t . Hence, as expected, in case λ ( ψ ) ≤ u ( t ) is uniformly bounded above andbelow and as t → ∞ , the flow converges to a metric of constant pseudo-scalarcurvature. This will be done below.Recall that ψ ( x ) < M . Then λ ( ψ ) ≤ r ψ ( t ) ≤ r ψ (0) . Let u m ( t ) = inf M u ( t ). Then by (18) we have(22) ∂ t u pm ≥ − sup M ψ ( x ) u m + r ψ ( t ) u pm . ∗ Then ∂ t u pm ≥ − sup M ψ ( x ) u m + λ ( ψ ) u pm . This can be considered in two cases.
Case 1 . If − sup M ψ ( x ) u m + λ ( ψ ) u pm ≥
0, then u pm ( t ) ≥ u pm (0) . Case 2 . If − sup M ψ ( x ) u m + λ ( ψ ) u pm ≤
0, then we have − sup M ψ ( x ) u m ≤ − λ ( ψ ) u pm . and u p − m ( t ) ≥ sup M ψ ( x ) /λ ( ψ ) . This implies that u m ( t ) is uniformly bounded away from zero that(23) u p − m ( t ) ≥ min { sup M ψ ( x ) /λ ( ψ ) , u p − m (0) } . Similarly for u M ( t ) = sup M u ( t ), we have(24) ∂ t u pM ≤ − inf M ψ ( x ) u M + r ψ ( t ) u pM , and by r ψ ( t ) ≤ r ψ (0), ∂ t u pM ≤ − inf M ψ ( x ) u M + r ψ (0) u pM . We may write this differential inequality as pp − ∂ t u p − M ≤ − inf M ψ ( x ) + r ψ (0) u p − M . When − inf M ψ ( x ) + r ψ (0) u p − M ≤
0, we have ∂ t u p − M ≤ . When − inf M ψ ( x ) + r ψ (0) u p − M ≥
0, we have u p − M ≤ inf M ψ ( x ) /r ψ (0) . Then we have u p − M ≤ max (inf M ψ ( x ) /r ψ (0) , u p − M (0)) . Then we can get the convergent of the flow g ( t ) at t = ∞ .We claim that r ψ ( t ) will eventually become negative, even if this may notbe the case at the initial time. Assume that r ψ ( t ) ≥ ∂ t u pm ≥ − sup M ψ ( x ) u m > t → ∞ , u pm ( t ) → ∞ , which is impossible since the volume of g ( t ) is fixed. Hence, we may choose t > r ψ ( t ) < EW YAMABE-TYPE FLOW IN A MANIFOLD 13 and then r ψ ( t ) ≤ r ψ ( t ) < t ≥ t . By (24) we have for t ≥ t , u pM ( t ) ≤ max { u pM ( t ) , − r ψ ( t ) sup ψ ( x ) } . This together with (23) implies that u is uniformly bounded from above andaway from zero. Then as in [25], we can show that g ( t ) converges smoothly atan exponential rate to a limit metric with negative pseudo-scalar curvature. (cid:3) For comparison, we give an outline of the proof of global existence andconvergence result, Theorem 4, in case for any p > λ ( ψ ) = 0 and ψ ( x ) = 0 on M . Proof.
The global existence of the flow can be done as in the proof of The-orem 1. So we need only consider the convergence part. Note that in thiscase, we have r ψ ( t ) ≥
0. If r ψ (0) = 0, then we have r ψ ( t ) = 0 for all t > r ψ ( t ) = 0 for all time. Thus we may assume that r ψ (0) > u pm ( t ) u pm (0) ≥ exp( Z t r ψ ( t )) . We need to get an uniform upper bound for u M ( t ) = sup M u ( t ). Recall that(26) ∂ t u pM ≤ r ψ ( t ) u pM . We use the Gronwall inequality to get u pM ( t ) /u pM (0) ≤ exp( Z t r ψ ( s ) ds ) . Then we have u pM ( t ) /u pM (0) ≤ exp( Z t r ψ ( s ) ds ) ≤ u pm ( t ) u pm (0) . This is the global Harnack inequality. Therefore, using R M u ( t ) p +1 dv = 1,we have the uniform smooth estimations for u . Using (25) we know that r ψ ( t ) → t → ∞ .Multiplying (18) by u − p ∆ u and integrating, we have(27) p ddt Z M c n |∇ u | dv + 2 Z c n | ∆ u | u − p dv = 2 r ψ ( t ) Z M c n |∇ u | dv Using the inequality(28) Z M c n | ∆ u | dv ≥ c Z M |∇ u | dv ∗ for some uniform constant c > r ψ ( t ) → t → ∞ wehave(29) Z M |∇ u | dv ≤ Ce − ct Integrating (27) in time variable we have for any
T > Z ∞ T Z M | ∆ u | dv ≤ Ce − cT which says that(31) Z ∞ T Z M R ψ dv ≤ Ce − cT Then for t ∈ [ T, T + 1), Z M R ψ dv ≤ Ce − ct . Since r ψ ( t ) decreases we have(32) r ψ ( t ) ≤ Ce − ct for all t >
0. By (29) and Poincare inequality we know that u p convergesto its average exponentially in the L norm. It follows that u ( t ) convergesexponentially to some limit positive constant. (cid:3) References [1] T. Aubin,
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Li Ma, School of Mathematics and Physics, University of Science and tech-nology Beijing, Xueyuan Road 30, Haidian, Beijing 100083, China
Email address : [email protected]@ustb.edu.cn