A weak energy identity and the length of necks for a Sacks-Uhlenbeck α -harmonic map sequence
aa r X i v : . [ m a t h . DG ] M a y A weak energy identity and the length of necks for aSacks-Uhlenbeck α -harmonic map sequence Yuxiang Li ∗ Youde Wang † Abstract
Assume that M is a closed surface and N is a compact Riemannian manifold withoutboundary. Let u α : M → N be the critical point of E α with E α ( u α ) < C . Assume u isthe weak limit of u α in W , ( M, N ) and x is the only blow-up point in B σ ( x ) ⊂ M with n bubbles. Then, on a local coordinate system on B σ ( x ) which origin is x , we can findsequences x iα → λ iα → i = 1 , · · · , n ) s.t. u α ( x iα + λ iα x ) → v i , where v i are harmonicmaps from S to N . We define µ i = lim inf α → ( λ iα ) − α . We will prove thatlim α → E α ( u α , B σ ( x )) = E ( u , B σ ( x )) + | B σ ( x ) | + n X j =1 µ j E ( v j ) . Further, when n = 1, we define ν = lim inf α → ( λ α ) −√ α − , then we have:If ν = 1, then u ( B σ ( x )) ∪ v ( S ) is connected;If 1 < ν < + ∞ , then u ( B σ ( x )) and v ( S ) are connected by a geodesic with length L = r E ( v ) π log ν . If ν = + ∞ , the neck contains at least one geodesic with infinite length.We also give an example of neck which shows the neck contains at least one geodesic ofinfinite length. Mathematics Subject Classification:
Let (
M, g ) be a smooth closed Riemann surface, and (
N, h ) ⊂ R K be an n-dimensional smoothcompact Riemannian submanifold. We always assume that N ֒ → R K is an isometric embeddingand has no boundary. ∗ This paper was written while the first author was researching at Mathematisches Institut, Albert-Ludwigs-Universit¨at Freiburg, supported by Alexander von Humboldt Foundation. † Partially supported by 973 project of China, Grant No. 2006CB805902. W , ( M, N ) denote the Sobolev space of W , maps from M into N . If u ∈ W , ( M, N ),locally, we define the energy density e ( u ) of u at x ∈ M by e ( u )( x ) = |∇ g u | = g ij ( x ) h αβ ( u ( x )) ∂u α ∂x i ∂u β ∂x j . It is easy to check that e ( u ) = T race g u ∗ h, where u ∗ h is the pull-back of the metric tensor h . Usually, the energy E ( u ) of u is defined by E ( u ) = Z M e ( u ) dV g , and the critical points of E are called harmonic maps. We know that a harmonic map u satisfiesthe following equation: τ ( u ) = ∆ u + A ( u )( ∇ u, ∇ u ) = 0 , where A is the second fundamental form of N in R K .It is not easy to find a harmonic map, since E does not satisfy the Palais-Smale conditionwhen the dimensions of domain manifold dim( M ) ≥
2. Eells and Sampson first employedthe heat flow method to approach the existence problems of harmonic maps and successfullydeformed a map from a closed manifold into a manifold with nonpositive sectional curvatureinto a homotopic harmonic map. Concretely, they considered the heat flow for harmonic maps(or the negative gradient flow of the energy functional E ( u )): ∂u∂t = τ g ( u ) . If we can establish the global existence of the above flow with respect to the time variable t ,or roughly speaking, the flow flows to infinity smoothly, then we are able to find a sequence u k = u ( x, t k ) s.t. t k → + ∞ and u k converges to a harmonic map (see [E-S]).As dim( M ) = 2, it is well-known that the energy functional is of conformal invariance andharmonic maps for this case are of special importance and interest. In fact, mathematicians paymore attention to this case. To prove the existence of harmonic maps from a closed surface Sacksand Uhlenbeck in their pioneering paper [S-U] employed a perturbed energy functional whichsatisfies the Palais-Smale condition, hence defined the so called α -harmonic map to approximatethe harmonic map. More precisely, for every u ∈ W , α ( M, N ) Sacks and Uhlenbeck definedthe so called α -energy E α as E α ( u ) = Z M (1 + |∇ u | ) α dV g , which can be regarded as a perturbation of energy E , and considered the α -harmonic maps, i.e.the critical points of E α in W , α ( M, N ), which satisfy the following equation:∆ g u α + ( α − ∇ g |∇ g u α | ∇ g u α |∇ g u α | + A ( u α )( du α , du α ) = 0 . If there is a subsequence u k = u α k which converges smoothly as α k → u α k will converge to aharmonic map. 2ater, Struwe used the heat flow method of Eells and Sampson to approach the existenceproblems for harmonic maps from a closed surface and he obtained almost the same results asin [S-U]. Chang showed the same results as in [St] for the case where the domain manifold is acompact surface with smooth boundary (see [C]).However, for both cases, the blow-up might happen. That is to say, we are only sure thatthe convergence is smooth away from finitely many points (which are called blow-up points)to a smooth harmonic map u , which might be a trivial map. Around a blow-up point p , theenergy will concentrate, i.e., we will havelim r → lim inf k → + ∞ Z B r ( p ) |∇ u k | dV g > . And then, we can find sequences lim k → + ∞ x ik → p , lim k → + ∞ λ ik → i = 1 , · · · , n , s.t. u k ( x ik + λ ik x ) → w i in C kloc ( R \ A i ) , where all w i are non-trivial harmonic maps from S to N , and A i is a finite set.Then two problems occur. One is that if we have the energy identity, i.e.lim k → + ∞ Z B σ |∇ u k | dV g = Z B σ |∇ u | dV g + n X i =1 E ( w i ) . The other one is what the neck is if it exists?When u k = u ( x, t k ) is a subsequence of a heat flow for two dimensional harmonic maps, theabove two problems are deeply studied. The energy identities have been proved by Qing [Q] (inthe case N = S n ) and Ding-Tian [D-T] in the general case. In [Lin-W], Lin-Wang gave anotherproof of the energy identity. For the neck, Qing-Tian [Q-T] proved that there is no neck if theblow-up happened at infinite time ( Ding [D] proved a more general case), and Topping [T] gavea surprising example of heat flow blowing up at finite time s.t. the weak limit is not continuous.Unexpectedly, the energy identity for an α -harmonic sequence with bounded energy is stillopen. Now, many people believe that the methods used to solve the identity for heat flow, ormore generally a sequence with tension fields τ bounded in L , are not powerful enough to solvethe energy identity for an α -harmonic map sequence. The reason lies in the identity (2.3) inthis paper. For a sequence with tension fields τ bounded in L , (2.3) becomes Z ∂B r | ∂u k ∂r | ds − Z ∂B r |∇ u k | ds = O ( Z B r | τ ( u k ) ||∇ u k | dV g ) + O (1) . then the right side of the above identity is bounded. However, in (2.3), a very bad term α − r Z B r (1 + |∇ u α | ) α − |∇ u α | dV g appears.The known energy identities for some special α -harmonic sequences are usually obtained bymethods which are completely different with the one of [D-T]. Now we would like to mentionthe following cases.If { u α } is a sequence of minimizing α -harmonic map, i.e. every u α is the minimizer of E α ,which belongs to the same homotopic class, Chen and Tian [C-T] proved that the necks consist3f some geodesics of finite length, and moreover this implies no loss of energy in necks for thesequence (see also [D-K]).Another important case is the energy identity for a minimax sequence. We let M be acompact Riemann surface, A be a parameter manifold. Let h : M × A → N be continuous.Assume H be the class of all maps homotopic to h , and β α ( H ) = inf h ∈ H sup t ∈ A E α ( h ( · , t )) . (1.1)We can deduce from Jost’s result [J] that there is at least one sequence u α k which attained β α k ( H ) satisfies the energy identity as α k → α -harmonic sequence, especially the necks between the bubbles. However,we can not give a final proof on the energy identity for such Sacks-Uhlenbeck sequence, instead,we only show a weaker energy identity and give some observation on this subject. On the otherhand, we exploit the details of the necks. Precisely we provide a new method to show that thenecks converge to geodesics and obtain the formula on the length of the geodesics.Now, we assume that u α is a sequence of α -harmonic maps from ( M, g ) to (
N, h ) with E α ( u α ) < Θ . Then, by the theory of Sacks and Uhlenbeck, we are able to assume that there exists a sequence α k →
1, s.t. u α k converges to a harmonic map u : M → N smoothly away from a finite manypoints { x i } as α k →
1. We assume that there are n bubbles at the point x . Then we are ableto assume that there are x jα k → x and λ jα k → j = 1 , · · · , n , such that v jα k = u α k ( x jα k + λ jα k x )converge in C kloc ( R \ { p , p , · · · , p s j } ) to non-trivial harmonic maps v j : S → N. Moreover, we assume that one of the following holds: H1 . For any fixed R , B Rλ iαk ( x iα k ) ∩ B Rλ jαk ( x jα k ) = ∅ whenever ( α k −
1) are sufficiently small. H2 . λ iαk λ jαk + λ jαk λ iαk → + ∞ as α k → Remark 1.
One is easy to check that if ( λ iα k , x iα k ) and ( λ jα k , x jα k ) do not satisfy H1 and H2 ,then we can find subsequences of λ iα k , x iα k and λ jα k , x jα k s.t. λ iαk λ jαk → λ ∈ (0 , ∞ ) and x iαk − x jαk λ jαk → a ∈ R . Since u α k ( x iα k + λ iα k x ) = u α k ( x jα k + λ jα k ( x iα k − x jα k λ jα k + λ iα k λ jα k x )) , we have v i ( x ) = v j ( a + λx ) , and then v i and v j are in fact the same bubble. R , we have Z B Rλjαk ( x jαk ) \ ( ∪ sji =1 B δλjαk ( x jαk + λ jαk p i )) |∇ g u α k | α k dV g = ( λ jα k ) − α Z B R \ ( ∪ sji =1 B δ ( p i )) |∇ g v jα k | α k dV g ( x jαk + λ jαk x ) . (1.2)Since Z B R \ ( ∪ sji =1 B δ ( p i )) |∇ g v jα k | α k dV g ( x jαk + λ jαk x ) → Z B R \ ( ∪ sji =1 B δ ( p i )) |∇ v j | dx, and λ jα k < , we define µ j = lim inf α → ( λ jα ) − α ≤ lim k →∞ Z B Rλjαk ( x jαk ) \ ( ∪ sji =1 B δλjαk ( x jαk + λ jαk p i )) |∇ g u jα k | α k dV g Z B R \ ( ∪ sji =1 B δ ( p i )) |∇ v j | dx ≤ Θ −| M |− E ( u ) θ , (1.3)where θ = inf { E ( u ) : u is a nontrivial harmonic map from S to N } . Therefore, we know µ j ∈ [1 , Θ − | M | − E ( u ) θ ] . The first task of this paper is to get the following weak energy identity:
Theorem 1.1.
Let M be a smooth closed Riemann surface and N be a smooth compact Rie-mannian manifold without boundary. Assume that u α k ∈ C ∞ ( M, N ) ( α k → is a sequenceof α k -harmonic maps with uniformly bounded energy and x be the only blow-up point of thesequence { u α k } in B σ ( x ) ⊂ M . Then, passing to a subsequence, there exist u : M → N whichis a smooth harmonic map and finitely many bubbles v j : S → N such that u α k → u weaklyin W , ( M, N ) and in C ∞ loc ( B σ ( x ) \ { x } , N ) and the following identity holds lim k → + ∞ E α k ( u α k , B σ ( x )) = E ( u , B σ ( x )) + | B σ ( x ) | + n X j =1 µ j E ( v j ) , (1.4) where µ j is defined by (1.3) and n is the number of bubbles at x . This theorem tells us that the energy identity holds true if and only if µ j = 1. It provides anew route to approach the problem whether the necks contain energy or not. Remark 2.
By Lemma 2.2 in section 2, µ j = 1 implies lim k → + ∞ E ( u α k , B σ ( x )) = E ( u , B σ ( x )) + n X j =1 E ( v j ) , (1.5)5 nd reversely, by Lemma 2.2 and (2.4), (1.5) also implies µ j = 1 . It is our another purpose to study the behavior of the necks connecting bubbles. For thissake, we need to define ν j = lim inf α → ( λ jα ) −√ α − . We will see that the above quantity play an important role in the discussion on the behavior ofblowing up. Our main results are stated as follows:
Theorem 1.2.
Let M be a smooth closed Riemann surface and N be a smooth closed Rieman-nian manifold and u α k ∈ C ∞ ( M, N ) be a sequence of α k -harmonic maps with uniformly boundedenergy and u α k converges to a smooth harmonic map u : M → N in C ∞ loc ( B σ ( x ) \ { x } , N ) as α k → . Assume there is only one bubble in B σ ( x ) ⊂ M for { u α k } and v : S → N is thebubbling map. Let ν = lim inf α → ( λ α ) −√ α − . Then we have1) when ν = 1 , the set u ( B σ ( x )) ∪ v ( S ) is a connected subset of N ;2) when ν ∈ (1 , ∞ ) , the set u ( B σ ( x )) and v ( S ) are connected by a geodesic with Length L = r E ( v ) π log ν ;
3) when ν = + ∞ , the neck contains at least an infinite length geodesic. Remark 3.
Although we state and prove Theorem 1.2 only for one bubble case, it is not difficultto follow the steps in section 3.2 to prove the general case. However, the general case is quitecomplicated, for example, if we have 2 bubbles: u α ( λ α x + x ) → v , and u α ( λ α x + x ) → v which satisfy: λ α /λ α → and ν , ν < ∞ , then u ( B δ ( x )) , v ( S ) are connected by a geodesicwith length L = r E ( v ) + E ( v ) π log ν , and v ( S ) , v ( S ) are connected by a geodesic with length L = r E ( v ) π log ν ν . We should mention that after we completed the paper we found that Moore had proved thatif a neck is of finite length L and ˜ g ≥ M ), then L = q E ( v ) π log ν (note that in[M], E ( u ) is defined to be R M |∇ u | dV g ). However, the arguments to prove Theorem 1.2 in thispaper is completely different from Moore’s proof. The key estimation of us is the Proposition4.1 in section 4, which gives the details of the necks.The Proposition 4.1 also provides a new method to prove that the necks consist of geodesics,which has been already proved by Chen and Tian [C-T]. In this paper, we will make use of thefollowing curve Γ α ( r ) = 12 π Z π u α ( r, θ ) dθ
6o approximate the necks. With the help of Proposition 4.1, one can easily calculate the secondfundamental form of the approximation curve, and then to prove that the limiting curve satisfiesthe equation of geodesic in N .We failed to find a sufficient condition s.t. ν i < + ∞ , but we will show that there are indeedmany cases that the necks contain at least one infinite length geodesic: Corollary 1.3.
Let α k → , and u k : M → N be a minimizer of E α k in the homotopic classcontaining u k . We assume for any i = j , u i and u j are not in the same homotopic class. If sup k E α k ( u k ) < + ∞ , then u k will blow up, and the neck contains at least one infinite length geodesic. Remark 3.
In the last section, by constructing a manifold N we will give an example of aminimizing α -harmonic map sequence, which satisfies the condition in the above corollary. Thisindicates that there exists a neck joining bubbles which is a geodesic of infinite length. We conclude this introduction with showing the following proposition as a consequence of The-orem 1.1, which implies the result due to Chen-Tian that, if the necks consist of some geodesicsof finite length, then the energy identity is true:
Proposition 1.4.
The energy identity holds true for a subsequence of u α if and only if lim inf α → k∇ u α k α − C ( M ) = 1 . (1.6) The limit set of such subsequence has no neck if and only if lim inf α → k∇ u α k √ α − C ( M ) = 1 . The bubbles in limit set of such subsequence are joined by some geodesics of finite length, if andonly if lim inf α → k∇ u α k √ α − C ( M ) < + ∞ . Proof.
We only prove the first claim.First, we prove (1.6) implies µ j = 1. We assume v jα ( x ) = u α ( x jα + λ jα x ) converges to v j in C loc ( R n \ { p , p , · · · , p s } ). Then we have( λ jα ) − α = |∇ u α ( x jα + λ jα x ) | α − |∇ v jα ( x ) | α − for any x with |∇ v j ( x ) | 6 = 0. Hence we get µ j ≤ µ j = 1.Now, we will prove “ µ j = 1 for all j ” implies (1.6). Let x α to be the point s.t. |∇ u α | ( x α ) =max |∇ u α | , and λ α = 1 |∇ u α | ( x α ) . We set v α ( x ) = u α ( x α + λ α x ). One is easy to check that v α will converge to a non-trivialharmonic map v locally. By H1 and H2 we must find a j , s.t. B Rλ jα ( x jα ) ∩ B Rλ α ( x α ) = ∅ , and C λ jα < λ α < Cλ jα C >
0. Hence we get | λ α | α − → ✷ Acknowledgement:
The authors is grateful to thank Professor W. Ding for his help and encour-agement. The first author would like to thank Prof. E. Kuwert for many helpful discussions.
In this section we intend to establish some integral formulas on α -harmonic maps from a closedsurfaces by the variations of domain. Of course, we need to choose some suitable variationalvector fields on M which generate the transformations of M . We will see that these integralrelations will play an important role in the proofs of main theorems.Note that the functional E α is not conformal invariant. For example, on an isothermalcoordinate system around a point p ∈ M , if we set the metric g = e ϕ (( dx ) + ( dy ) )with p = (0 , ϕ (0) = 0 and ˜ u α ( x ) = u α ( λx ), then we will get Z B δ (1 + |∇ g u α | ) α dV g = Z B δλ λ − α ( λ + |∇ g ′ ˜ u α | ) α dV g ′ , where g ′ = e ϕ ( p + λx ) (( dx ) + ( dy ) ). We also ought to note that an α -harmonic map sequence u α may have several bubbles near a blowing up point, for example, there are sequences λ α , λ α ,s.t. λ α λ α → , λ α → , as α →
1, and v α ( x ) = u α ( λ α x ) → v in C kloc ( R ) , v α ( x ) = u α ( λ α x ) → v in C kloc ( R \ { } ) , where v , v are non-trivial harmonic maps from S to N. For this case, we have v α ( x ) = v α (cid:18) λ α λ α x (cid:19) , i.e. v ( x ) is in fact a bubble for the sequence v α . Therefore, we need to consider the equationof v α , and one is easy to check that v α is locally a critical point of the functional F ( v ) = Z B δ (( λ α ) + |∇ v | ) α dV g α , where g α = e ϕ ( λ α x ) (( dx ) + ( dy ) ). For this reason, we need to consider a more general α -energywhich is of the following form: E α,ǫ α ( u ) = Z B δ ( ǫ α + |∇ g u α | ) α dV g α . Let u α be the critical point of the above functional. Then, u α satisfies the following ellipticsystem which is also called the equation of α -harmonic maps:∆ g α u α + ( α − ∇ g α |∇ g α u α | ∇ g α u α ǫ α + |∇ g α u α | + A ( u α )( du α , du α ) = 0 . (2.1)8ere we always assume that the sequence ǫ α ( ǫ α ≤
1) satisfieslim α → ǫ αα − > β > . (2.2)It follows from (1.3) that this assumption is reasonable.From now on, we consider u α to a map sequence from ( B, g ) to (
N, h ) which satisfy equa-tion (2.1). We assume that g = e ϕ α (( dx ) + ( dx ) ) with ϕ α (0) = 0 and ϕ α → ϕ smoothly.Moreover, we assume that u α → u in C kloc ( ¯ B \ { } ).Next, we recall the well-known ǫ -regularity theorem due to Sacks-Uhlenbeck [S-U]: Theorem 2.1.
Let u : B → N satisfies equation (2.1) where B ⊂ M is a ball with radius .There exists ǫ > and α > such that if E ( u, B ) < ǫ and ≤ α ≤ α , then for all smaller r < , we have k∇ u k W ,p ( B r ) ≤ C ( p, r ) E ( u, B ) , here B r ⊂ B is a ball with radius r , < p < ∞ . We also have
Lemma 2.2.
Let u α be the critical point of E α with E α ( u α ) ≤ Θ . We have β < lim inf α → k ( ǫ α + |∇ g u α | ) α − k C ( B ) ≤ lim sup α → k ( ǫ α + |∇ g u α | ) α − k C ( B ) < β , where β is independent of α .Proof. Obviously, we only need to prove k∇ g u α k α − C ( B ) < C . We assume that there is sequence α k →
1, s.t. k∇ g u α k k α k − C ( B ) → + ∞ as k → + ∞ .Let |∇ g u α k | ( x α k ) = max {|∇ g u α k |} , and λ k = |∇ g u αk | , v k ( x ) = u α k ( x α k + λ k x ). Then asubsequence of { v k j } converges to a new nontrivial harmonic map from S to N . Then by (1.3),we obtain the following λ − α kj k j < C , which contradicts with the choice of α k . ✷ Take an 1-parameter family of transformations { φ s } which is generated by the vector field X .If we assume X is supported in B , then we have E α,ǫ α ( u ◦ φ s ) = Z B ( ǫ α + |∇ g ( u ◦ φ s ) | ) α dV g = Z B ( ǫ α + X β | d ( u ◦ φ s )( e β ( x )) | ) α dV g ( x )= Z B ( ǫ α + X β | du ( φ s ∗ ( e β ( x ))) | ) α dV g ( x )= Z B ( ǫ α + X β | du ( φ s ∗ ( e β ( φ − s ( x )))) | ) α J ac ( φ − s ) dV g , { e α } is a local orthonormal basis of T B . Noting dds J ac ( φ − s ) dV g | s =0 = − div ( X ) dV g , we have proved the formula dE f ( u )( u ∗ ( X )) = − Z B ( ǫ α + |∇ g u | ) α div ( X ) dV g +2 α P β Z B ( ǫ α + |∇ g u | ) α − h du ( ∇ e β X ) , du ( e β ) i dV g . Now, we assume u α to be the critical point of E α . For any vector field X on B , we have − Z B ( ǫ α + |∇ g u α | ) α divXdV g + 2 α X β Z B ( ǫ α + |∇ g u α | ) α − h du α ( ∇ e β X ) , du α ( e β ) i dV g = 0 . Next, for 0 < t ′ < t ≤ ρ , we choose a vector field X with compact support in B ρ by X = η ( r ) r ∂∂r = η ( | x | ) x i ∂∂x i , where η is defined by η ( r ) = if r ≤ t ′ t − rt − t ′ if t ′ ≤ r ≤ t if r ≥ t, where r = p ( x ) + ( x ) . By a direct computation we obtain div ( X ) = 2 η + rη ′ + rη ∂ϕ∂r , and ∇ ∂∂x X = η ∂∂x + η ′ ( x ) r ∂∂x + η ′ x x r ∂∂x + ηx Γ ∂∂x + ηx Γ ∂∂x + ηx Γ ∂∂x + ηx Γ ∂∂x . Then, P β h du α ( ∇ e β X ) , du α ( e β ) i dV g = h du α ( ∇ ∂∂x X ) , du α ( ∂∂x ) i dx + h du α ( ∇ ∂∂x X ) , du α ( ∂∂x ) i dx = ( η |∇ u α | + η ′ r | ∂u α ∂r | + O ( | x | ) |∇ u α | ) dx, where ∇ is the Riemannian connection with respect to standard metric. Hence, we derive0 = (2 α − Z B t η ( ǫ α + |∇ g u α | ) α − |∇ u α | dx + Z B t O ( | x | )( ǫ α + |∇ g u α | ) α − |∇ u α | dx − ǫ α Z B t η ( ǫ α + |∇ g u α | ) α − dV g + ǫ α t − t ′ Z B t \ B t ′ r ( ǫ α + |∇ g u | ) α − dV g + 1 t − t ′ Z B t \ B t ′ ( ǫ α + |∇ g u α | ) α − [ |∇ u α | r − αr | ∂u α ∂r | ] dx − Z B t ǫ α ( ǫ α + |∇ g u α | ) α − rη ∂ϕ∂r dV g . t ′ → t in the above identity and using Lemma 2.2, we obtain the following Z ∂B t ( ǫ α + |∇ g u α | ) α − | ∂u α ∂r | ds − α Z ∂B t ( ǫ α + |∇ g u α | ) α − |∇ u α | ds = ( α − αt Z B t ( ǫ α + |∇ g u α | ) α − |∇ u α | dx + O ( t ) , (2.3)where ds is the volume element of ∂B t with respect to the Euclidean metric. We know thatthe metric g can be written as g = e ϕ ( dr + r dθ ) in the polar coordinate system. Set u α,θ = 1 r ∂u α ∂θ . Since |∇ u α | = | ∂u α ∂r | + | u α,θ | , we get from the above identity(1 − α ) Z ∂B t ( ǫ α + |∇ g u α | ) α − (cid:12)(cid:12) ∂u α ∂r (cid:12)(cid:12) ds − α Z ∂B t ( ǫ α + |∇ g u α | ) α − (cid:12)(cid:12) u α,θ (cid:12)(cid:12) ds = ( α − αt Z B t ( ǫ α + |∇ g u α | ) α − |∇ u α | dx + O ( t ) . (2.4) Denote ∆ = ∂ ∂ ( x ) + ∂ ∂ ( x ) . By (2.1), we have the equation:∆ u α + ( α − ∇ |∇ g u α | ∇ u α ǫ α + |∇ g u α | + A ( u α )( du α , du α ) = 0 . As in [Lin-W], we multiply the both sides of the above equation with r ∂u α ∂r to obtain Z B t r ∂u α ∂r ∆ u α dx = − ( α − Z B t ∇ |∇ g u α | ∇ uǫ α + |∇ g u α | r ∂u α ∂r dx. It is easy to see Z B t r ∂u α ∂r ∆ u α dx = Z ∂B t r | ∂u α ∂r | ds − Z B t ∇ ( r ∂u α ∂r ) ∇ u α dx. Since Z B t ∇ ( r ∂u α ∂r ) ∇ u α dx = Z B t ∇ (cid:0) x k ∂u α ∂x k (cid:1) ∇ u α dx = Z B t |∇ u α | dx + Z t Z π r ∂ ( |∇ u α | ) ∂r rdθdr = Z B t |∇ u α | dx + 12 Z ∂B t |∇ u α | tds − Z B t |∇ u α | dx = 12 Z ∂B t |∇ u α | tds , then, we have Z ∂B t ( | ∂u α ∂r | − |∇ u α | ) ds = − α − t Z B t ∇ |∇ g u α | ∇ u α ǫ α + |∇ g u α | r ∂u α ∂r dx. (2.5)11ence, it follows Z ∂B t ( | ∂u α ∂r | − | u α,θ | ) ds = − α − t Z B t ∇ |∇ g u α | ∇ u α ǫ α + |∇ g u α | r ∂u α ∂r dx. (2.6)Thus, we obtain two key variational identities (2.4) and (2.6) which will be used repeatedly inour following argument. In this section, we discuss the weak energy identity on a sequence of α -harmonic maps. Byfollowing the idea of Ding and Tian in [D-T] we will apply (2.4) (2.5) to give the proof ofTheorem 1.1.Let B σ = B σ (0) be a ball in R with the metric g = e ϕ α ( x ) ( dx ⊗ dx + dx ⊗ dx ), where ϕ ∈ C ∞ ( B σ ) and ϕ α (0) = 0, and ϕ α converges smoothly. We set u α : B σ → N be a mapwhich satisfies equation (2.1). Clearly, (2.4), (2.5) and (2.6) hold.We assume that for any α E α,ǫ α ( u α , B σ ) < C , and 0 is the only blow-up point in B σ . Without loss of generality, we assume u α → u in C kloc ( B σ \ { } ), where u is a harmonic map from B σ to N .We can get the first bubble in the following way. Let x α ∈ B δ s.t. |∇ u α ( x α ) | = max B δ |∇ u α | ,and λ α = Bδ |∇ u α | . Then, without loss of generality, we may assume in C kloc ( R ) u α ( x α + λ α x ) → v . Now, we assume there exists another n − v , · · · , v n , and sequences x iα , λ iα s.t. u α ( x iα + λ iα x ) → v i in C kloc ( R \ A i ), where A i are finite sets. Clearly, we may assume λ α = min i ∈{ , ··· ,n } { λ iα } . Moreover, we assume that for any i = j , one of the H1 and H2 holds. First we prove the Theorem 1.1 in the case of n = 1, where n is the number of the bubbles.The general case will be explained in the next subsection.We denote λ α = λ α , x α = x α , and v = v . We defineΛ α ( R ) = Z B Rλα ( x α ) |∇ g u α | α dV g , Λ = lim R → + ∞ lim α → Λ α ( R ) . and µ = lim α → λ − αα .
12y (1.3), we have Λ = µE ( v ). Moreover, we also havelim R → + ∞ lim α → Z B Rλα ( ǫ α + |∇ g u α | ) α − |∇ g u α | dV g = lim R → + ∞ lim α → Z B R ( ǫ α λ α + |∇ g v α | ) α − λ − αα |∇ v α | dx = µ Z R |∇ v | dx = Λ . (3.1)Furthermore, we claim that for any ǫ > δ and R such that, ∀ λ ∈ ( Rλ α , δ ),there holds Z B λ \ B λ ( x α ) |∇ g u α | dV g ≤ ǫ. (3.2)Suppose that the claim is false, then we may assume that there exist α i → λ ′ i → λ ′ i λ αi → + ∞ such that Z B λ ′ i \ B λ ′ i ( x αi ) |∇ g u α i | dV g ≥ ǫ. (3.3)Denote v ′ α i ( x ) = u α i ( λ ′ i x + x α i ), we may assume v ′ α i → v ′ in C kloc ( R \ ( { } ∪ A ) , N ), where A isa finite set which does not contain 0. If A = ∅ then it follows from (3.3) that v ′ is a nonconstantharmonic sphere which is different from v . This contradicts the assumption n = 1. Next, ifthere exists x ∈ A , then, by a similar argument with that we get v = v , we can still obtaina sequence x i → x , ˜ λ i →
0, s.t. v ′ i ( x i + ˜ λ i x ) converges to a harmonic map v . Hence we get u α i ( x α i + ˜ λ i ( λ α i x + x i )) converges to v strongly, and then v is the second harmonic map. Thisproves that the claim (3.2) must be true.Set u ∗ α = 12 π Z π u α ( x α + re iθ ) dθ. One is easy to check that, for any a < b , the following inequality holds true Z B b \ B a ( x α ) | ∂u ∗ α ∂r | dx = Z ba Z π (cid:12)(cid:12)(cid:12)(cid:12) π Z π ∂u α ∂r d ˜ θ (cid:12)(cid:12)(cid:12)(cid:12) dθrdr ≤ π Z ba ( Z π | ∂u α ∂r | d ˜ θ Z π dθ ) rdr = Z ba Z π | ∂u α ∂r | rdrdθ = Z B b \ B a ( x α ) | ∂u α ∂r | dx. (3.4)By applying (3.2) and Sacks-Uhlenbeck ǫ -regularity theorem (Theorem 2.1), we have thefollowing Lemma 3.1.
For any Rλ α < a < b < δ , we have Z B b \ B a ( x α ) |∇ g u α | r |∇ g u α | dV g ≤ C Z B b \ B a ( x α ) |∇ g u α | dV g . nd Z B b \ B a ( x α ) |∇ g u α | · | u α − u ∗ α | dV g ≤ C Z B b \ B a ( x α ) |∇ g u α | dV g , where C does not rely on α .Proof. First, we prove the first inequality in the above lemma. We assume that 2 K a ∈ ( b, b )and set D i = B i a \ B i − a ( x α ) . We rescale D i to B \ B , and u α to ˜ u α . By Theorem 2.1 (the ǫ -regularity theory), we haveon D i |∇ g u α | ≤ i − a |∇ ˜ u α | C ( B \ B ) ≤ C i − a k∇ ˜ u α k L ( B \ B / ) = C i − a k∇ g u α k L ( D i +1 ∪ D i ∪ D i − ) . Hence, it follows k r ∇ g u α k C ( D i ) ≤ C |∇ g u α | ≤ C k∇ g u α k L ( D i +1 ∪ D i ∪ D i − ) Similarly, we have k r ∇ g u α k C ( D i ) ≤ C ′ k∇ g u α k L ( D i +1 ∪ D i ∪ D i − ) . Then we have Z D i |∇ g u α | r |∇ g u α | dV g ≤ C Z D i +1 ∪ D i ∪ D i − |∇ g u α | dV g Z D i dV g r ≤ C ′ Z D i +1 ∪ D i ∪ D i − |∇ g u α | dV g . Therefore, we get the first inequality in this Lemma. The proof of the second inequality goes toalmost the same. ✷ R B δ \ B Rλα ( x α ) | u α,θ | dx The goal of this subsection is to prove the following
Lemma 3.2.
For α -harmonic map sequence u α ( α → ), there holds true lim δ → lim R → + ∞ lim α → Z B δ \ B Rλα ( x α ) | u α,θ | dx = 0 . Proof.
We adopt the technique of Sacks-Uhlenbeck [S-U] and [L-W] to show the lemma. Using(3.2) we have | u ∗ α ( r ) − u α ( r, θ ) | ≤ ǫ . (3.5)We compute 14 B δ \ B Rλα ( x α ) |∇ g u α | dV g = Z B δ \ B Rλα ( x α ) ∇ u α ∇ ( u α − u ∗ α ) dx + Z B δ \ B Rλα ( x α ) ∇ g u α ∇ g u ∗ α dV g = − Z B δ \ B Rλα ( x α ) ∆ u α ( u α − u ∗ α ) dx + Z B δ \ B Rλα ( x α ) ∇ u α ∇ u ∗ α dx + Z ∂ ( B δ \ B Rλα ( x α )) ∂u α ∂r ( u α − u ∗ α ) ds = Z B δ \ B Rλα ( x α ) A ( u α )( du α , du α )( u α − u ∗ α ) dV g +( α − Z B δ \ B Rλα ( x α ) ∇ g |∇ g u α | ∇ g u α ǫ α + |∇ g u α | ( u α − u ∗ α ) dV g + Z ∂ ( B δ \ B Rλα ( x α )) ∂u α ∂r ( u α − u ∗ α ) ds + Z B δ \ B Rλα ( x α ) ∂u α ∂r ∂u ∗ α ∂r dx. (3.6)On the other hand, noting (3.4) we have Z B δ \ B Rλα ( x α ) ∂u α ∂r ∂u ∗ α ∂r dx ≤ sZ B δ \ B Rλα ( x α ) | ∂u α ∂r | dx Z B δ \ B Rλα ( x α ) | ∂u ∗ α ∂r | dx ≤ Z B δ \ B Rλα ( x α ) | ∂u α ∂r | dx. (3.7)Hence, by using Lemma 3.1, (3.5), (3.7) and noting the following fact | ∇ g |∇ g u α | ∇ g u α ǫ α + |∇ g u α | | ≤ |∇ g u α | , we can infer from (3.6) Z B δ \ B Rλα ( x α ) |∇ u α | dx ≤ Z B δ \ B Rλα ( x α ) | ∂u α ∂r | dx + 3 C ( α − Z B δ |∇ g u α | dV g + Z ∂B δ ( x α ) ∂u α ∂r ( u α − u ∗ α ) ds − Z ∂B Rλα ( x α ) ∂u α ∂r ( u α − u ∗ α ) ds + ǫ ′ Z B δ \ B Rλα ( x α ) |∇ u α | dx, where ǫ ′ = ǫ k A k L ∞ ( M ) .Since |∇ u α | = | ∂u α ∂r | + | u α,θ | , we get Z B δ \ B Rλα ( x α ) | u α,θ | dx ≤ − Z ∂B δ ( x α ) ∂u α ∂r ( u α − u ∗ α ) ds + Z ∂B Rλα ( x α ) ∂u α ∂r ( u α − u ∗ α ) ds + C ′ (( α −
1) + ǫ ) . Keeping (3.2) in mind, we have 15im δ → lim α → Z ∂B δ ( x α ) ∂u α ∂r ( u α − u ∗ α ) ds = 0 , and lim R → + ∞ lim α → Z ∂B Rλα ( x α ) ∂u α ∂r ( u α − u ∗ α ) ds = 0 . Hence, we can see the above inequality implies the conclusion of Lemma 3.2. ✷ Immediately we infer from Lemma 2.2 and Lemma 3.2
Corollary 3.3.
There holds true lim δ → lim R → + ∞ lim α → Z B δ \ B Rλα ( x α ) ( ǫ α + |∇ g u α | ) α − | u α,θ | dx = 0 . We set F α ( t ) = Z B λtα ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ u α | dx,E r,α ( t ) = Z B λtα \ B λt α ( x α ) ( ǫ α + |∇ g u α | ) α − | ∂u α ∂r | dx, and E θ,α ( t ) = Z B λtα \ B λt α ( x α ) ( ǫ α + |∇ g u α | ) α − | u α,θ | dx. By (2.3), for t ∈ [ ǫ, t ], we have(1 − α ) E ′ r,α − α E ′ θ,α = α − α log λ α F α ( t ) + O ( λ tα log λ α ) . Then (1 − α ) E r,α ( t ) − α E θ,α ( t ) = 12 Z tt [ 1 α log λ α − α F α ( t ) + O ( λ tα log λ α )] dt. It is easy to check that the sequences { (1 − α ) E r,α ( t ) − α E θ,α ( t ) } and { F α ( t ) } are compact in C ([ ǫ, t ]) topology for any ǫ >
0. Therefore, there exist two functions F and E r which belongto C ([ ǫ, t ]) such that, as α → F α → F, E r,α → E r in C ([ ǫ, t ]) . Hence, we infer from the above integration equality E r ( t ) = − log µ Z tt F dt = − log µ Z tt ( E r ( t ) + F ( t )) dt. This implies that E r ( t ) ∈ C and E ′ r = − log µ ( E r + F ( t )) .
16t follows E r ( t ) = µ t − t F ( t ) − F ( t ) . Next, we prove that lim t → F ( t ) = Λ . (3.8)Integrating (2.3) with respect to t on the interval [ Rλ α , λ t α ] we obtain F α ( t ) − Z B Rλα ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ u α | dx ≤ C Z B λt α \ B Rλα ( x α ) ( ǫ α + |∇ g u α | ) α − | u α,θ | dx + C Z λ t α Rλ α α − r dr + C ( λ t α − Rλ α )Noting the following holds true (from Corollary 3.3)lim t → lim R → + ∞ lim α → Z B λt α \ B Rλα ( x α ) ( ǫ α + |∇ g u α | ) α − | u α,θ | dx = 0 , and lim t → lim R → + ∞ lim α → Z λ t α Rλ α α − r dr = lim t → (1 − t )2 log µ = 0 . Thus, (3.8) follows from the above inequality in view of (3.1).On the other hand side, we have Z B λtα ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ g u α | dV g = E r,α ( t ) + E θ,α ( t ) + F α ( t ) . Noting Corollary 3.3, i.e. lim α → E θ,α ( t ) = 0, we can deduce the followinglim α → Z B λtα ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ g u α | dV g = E r ( t ) + F ( t ) = µ t − t F ( t ) . Thus, we have shown the following
Lemma 3.4.
For any t ∈ (0 , and ǫ α > with lim α → ǫ α − α ≥ β , there holds true lim α → Z B λtα ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ g u α | dV g = µ − t Λ . Proof of Theorem 1.1
Here we restrict us to the case of one bubble. By taking almost thesame argument as we proved (3.8), we obtainlim δ → lim t → lim α → Z B δ \ B λtα ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ g u α | dV g = 0 , (3.9)17hich leads to lim δ → lim α → Z B δ ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ g u α | dV g = µ Λ = µ E ( v ) . Obviously, this implies the required conclusion. So we have completed the proof of Theorem 1.1in the case that n = 1. For the general case that n >
1, the proof can be completed by induction in n , the number ofbubbles.We set λ ′ α = max i {| x iα − x α | + | λ iα |} . Without loss of generality, we assume x α ≡ λ ′ α is attained by the n -th bubble, i.e. λ ′ α = | x n α | + | λ n α | . Let v α ( x ) = u α ( λ ′ α x ). Then v α will converges to v except finite points. Since λ n α and λ α satisfies H1 or H2 , then we have | x n α | λ α → + ∞ , or λ n α λ α → + ∞ , and therefore we have λ ′ α λ α → + ∞ .So, it is easy to check that 0 is a blowup point of the sequence { v α } .Similar to the proof of (3.2), we have for any ǫ >
0, there are δ and R s.t. Z B λ \ B λ ( x α ) |∇ g u α | dV g ≤ ǫ, ∀ λ ∈ ( Rλ ′ α , δ ) . (3.10)We set v α ( x ) = u α ( λ ′ α x ) and assume v α ⇁ v . Then using the arguments in the above subsection(in this case F ( t ) → lim R → + ∞ lim α → E α ( v α , B R ) as t → α → Z B δ ( x α ) ( ǫ α + |∇ u α | ) α − |∇ u α | dV g = lim α → ( λ ′ α ) − α ) lim R → + ∞ lim α → Z B R ( λ ′ α ǫ α + |∇ v α | ) α − |∇ v α | . Moreover, (3.10) implies that all the blowup points lie in B R for some R > v is a non-trivial harmonic map. ii) v is trivial.In case i), v is a bubble, then we can assume v is in fact one of v i ’s for i ∈ { , · · · , n } .We set v m to be equivalent to v , then lim α → ( λ ′ ) − α = µ m , and E ( v ) = E ( v m ). Since thereis only n − { v α } , by induction, we havelim α → Z B R ( λ ′ α ǫ α + |∇ v α | ) α − |∇ v α | dV g = E ( v , B R ) + X i = m ( µ i µ m ) E ( v i ) . In case ii), one is easy to check that | x n α | λ n α → + ∞ . Then x = lim α → x n α λ ′ α which lies on ∂B isa blow-up point. Then there are at least two blowup points 0 and x . So, at any blowup pointof v α , there are most n − α → Z B R ( λ ′ α ǫ α + |∇ v α | ) α − |∇ v α | dV g = n X i =1 ( µ i lim α → ( λ ′ α ) − α ) E ( v i ) . Thus, we complete the proof of Theorem 1.1. 18
Description and further analysis of the necks
In this section, we always assume there is only one bubble on some small ball B δ . ν = 1 In this subsection, we assume ν = 1. Then we have µ = 1, andlim δ → lim R → + ∞ lim α → Z B δ \ B Rλα ( x α ) |∇ g u α | dV g = 0 . (4.1)We will use the arguments of Ding [D].For simplicity, we assume P = log δ − log Rλ α log 2is an integer. For any integer k ∈ [1 , P − Q k ( t ) = B k + t Rλ α \ B k − t Rλ α ( x α ) , where t + k ≤ P and k − t ≥ Q k ( t ) thefollowing inequality holds Z Q k ( t ) |∇ u α | dx ≤ Z Q k ( t ) A ( u α )( du α , du α )( u α − u ∗ α ) dx + C ( α − Z Q k ( t +2) |∇ u α | dx + Z ∂Q k ( t ) ∂u α ∂r ( u α − u ∗ α ) ds + Z Q k ( t ) | ∂u α ∂r | dx. (4.2)Next, we will apply Pohozaev identity (2.5) to controll the last term in the above inequality, i.e. R Q k ( t ) | ∂u α ∂r | dx . For the sake of convenience, we set H ( r ) = − Z B r ( x α ) ∇ g |∇ g u α | ∇ g u α ǫ α + |∇ g u α | r ∂u α ∂r dV g = − Z B r ( x α ) ∇ |∇ g u α | ∇ u α ǫ α + |∇ g u α | r ∂u α ∂r dx. Using Lemma 3.1, we have | H ( r ) | ≤ Z B r \ B Rλα ( x α ) |∇ g u α | r | ∂u α ∂r | dx + | H ( Rλ α ) |≤ C Z B δ ( x α ) |∇ u α | dx + | H ( Rλ α ) | < C ′ , where we use the factlim α → | H ( Rλ α ) | ≤ lim α → Z B Rλα ( x α ) |∇ g u α | r |∇ g u α | = Z B R |∇ v α | r |∇ v α | dx < C ( R ) . Therefore, combining these with (2.5) we obtain19 Q k ( t ) (cid:12)(cid:12) ∂u α ∂r (cid:12)(cid:12) dx − Z Q k ( t ) (cid:12)(cid:12) ∇ u (cid:12)(cid:12) dx ≤ C Z k + t Rλ α k − t Rλ α α − r dr ≤ C ( α − t. It follows (4.2) and the above inequality( 12 − ǫ ) Z Q k ( t ) |∇ u α | dx ≤ C ( α − t + 1) + Z ∂Q k ( t ) ∂u α ∂r ( u α − u ∗ α ) ds . (4.3)On the other hand, we have (cid:12)(cid:12)(cid:12)Z ∂Q k ( t ) ∂u α ∂r ( u α − u ∗ α ) ds (cid:12)(cid:12)(cid:12) ≤ sZ ∂Q k ( t ) | ∂u α ∂r | ds Z ∂Q k ( t ) | u α − u ∗ α | ds ≤ sZ ∂Q k ( t ) | ∂u α ∂r | ds Z ∂Q k ( t ) | u α,θ | r ds ≤ "Z ∂Q k ( t ) | ∂u α ∂r | rds + Z ∂Q k ( t ) | u α,θ | rds = 12 Z ∂Q k ( t ) r |∇ u α | ds = 2 t + k − Rλ α Z ∂B t + kRλα ( x α ) |∇ u α | ds − k − t − Rλ α Z ∂B k − tRλα ( x α ) |∇ u α | ds . (4.4)Let f k ( t ) = Z Q k ( t ) |∇ u α | dx. From (4.4) we know Z ∂Q k ( t ) ∂u α ∂r ( u α − u ∗ α ) ds ≤
12 log 2 f ′ k ( t ) . Hence, by combining (4.3) and the above inequality we have(1 − ǫ ) f k ( t ) ≤ f ′ k ( t ) + C ( α − t + 1) . Multiplying the two sides of the above inequality by 2 − (1 − ǫ ) t and integrating we obtain f k (1) ≤ C − (1 − ǫ ) t f k ( t ) + C ( α − . It is easy to check that, if we set t = L k = (cid:26) k if k − ≤ PP − k if k − > P then, we get p E ( u α , Q k (1)) ≤ C − aL k p E ( u α , B δ \ B Rλ α ( x α )) + C √ α − a and C . 20y the standard L p estimate, we have osc B k +1 Rλα \ B k − Rλα ( x α ) u α ≤ C − aL k p E ( u α , B δ \ B Rλ α ( x α )) + C √ α − . These inequalities imply osc B δ \ B Rλα ( x α ) u α ≤ C p E ( u α , B δ \ B Rλ α ( x α )) P − aL k + C √ α − P ≤ C p E ( u α , B δ \ B Rλ α ( x α )) + C ( R, δ ) √ α − C log λ −√ α − α . Letting α →
1, and then R → + ∞ , δ →
0, we get osc B δ \ B Rλα ( x α ) u α → . Thus we proved Theorem 1.2 in the case ν = 1. ν > The goal of this section is to show the neck converges to a geodesic in N and furthermorecalculate the length of the geodesic.For this sake, we will consider the behaviors of u α on ∂B λ tα ( x α ) with t ∈ [ t , t ], where0 < t < t <
1. By the arguments in section 3.1.2, we can see easily that Z B λtα ( x α ) |∇ g u α | dV g → µ − t E ( v )in C ([ t , t ]). Then, it is easy to yield Z B λtα \ B λtα ( x α ) |∇ g u α | dV g → C ([ t , t ]). Therefore, for any t ∈ [ t , t ], we have osc ∂B λtα ( x α ) u α ≤ C Z B λtα \ B λtα ( x α ) |∇ g u α | dV g → , (4.5)i.e. u α | ∂B λtα ( x α ) will subconverge to a point belonging to N . Especially, we have that, as α → u α ( ∂B λ t α ) → y ∈ N and u α ( ∂B λ t α ) → y ∈ N. For simplicity, we will use “( r, θ )” to denote “ x α + r (cos θ, sin θ )”. Now we can state themain results of this subsection as follows: Proposition 4.1.
When ν > and < t ≤ t α ≤ t < , we have, after passing to asubsequence, lim α → α − Z B Rλtαα \ B R λtαα ( x α ) | u α,θ | dx = 0 (4.6)21 or any R > , and √ α − (cid:16) u α ( λ t α α r, θ ) − u α ( λ t α α , (cid:17) → ~a log r strongly in C k ( S × [ R , R ] , R n ) , where θ is the angle parameter of the ball centered at x α , ~a ∈ T y N is a vector in R n with | ~a | = µ − lim α → t α r E ( v ) π , and y = lim α → u α ( λ t α α , θ ) . To prove Proposition 4.1, we first prove the following
Lemma 4.2.
When ν > and < t ≤ t α ≤ t < , we have lim α → α − Z B Rλtαα \ B R λtαα ( x α ) | u α,θ | dx < C (4.7) where C does not depend on R .Proof. We set Q ( t ) = B t λ tαα ( x α ) \ B − t λ tαα ( x α ) . Here we assume 2 t ≤ λ − ǫα , where ǫ < min { t , − t } . Applying (2.5), we get from (4.2) the following( 12 − ǫ ) Z Q ( t ) |∇ u α | dx ≤ ( α − (cid:0)Z t λ tαα − t λ tαα r H ( r ) dr + C Z B t +2 λtαα \ B − t − λtαα ( x α ) |∇ u α | dx (cid:1) − Z ∂Q ( t ) ∂u α ∂r ( u α − u ∗ α ) ds. (4.8)For any r ∈ [ λ t α + ǫα , λ t α − ǫα ], it is easy to check that | H ( r ) − H ( λ t α α ) | ≤ Z B λtα − ǫα \ B λtα + ǫα ( x α ) |∇ g u α || r ∂u α ∂r | dx. Using Lemma 3.1, we can get Z B λtα − ǫα \ B λtα + ǫα ( x α ) |∇ g u α || r ∂u α ∂r | dx ≤ C Z B λtα − ǫα \ B λtα + ǫα ( x α ) |∇ u α | dx. By integrating (2.3) we obtain(1 − α ) Z λ tα − ǫα λ tα + ǫα ds Z ∂B s ( x α ) ( ǫ α + |∇ g u α | ) α − (cid:12)(cid:12) ∂u α ∂r (cid:12)(cid:12) ds − α Z λ tα − ǫα λ tα + ǫα ds Z ∂B s ( x α ) ( ǫ α + |∇ g u α | ) α − s (cid:12)(cid:12) ∂u α ∂θ (cid:12)(cid:12) ds = Z λ tα − ǫα λ tα + ǫα ( α − αs Z B s ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ u α | dx ! ds + O ( λ t α − ǫ ) α ) . (4.9)22y Corollary 3.3 we know that the second term on the left hand side of the above inequalityvanishes as α →
0. On the other hand, noting the fact ( ǫ α + |∇ g u α | ) α − is bounded, we have Z λ tα − ǫα λ tα + ǫα ( α − αs Z B s ( x α ) ( ǫ α + |∇ g u α | ) α − |∇ u α | dx ! ds ≤ C Z λ tα − ǫα λ tα + ǫα ( α − αs ds = Cǫα log λ − αα . Noting the fact lim α → ǫ α − α = β >
0, we can infer from (4.9) that, as α is close to 1 enough, Z B λtα − ǫα \ B λtα + ǫα ( x α ) |∇ u α | dx ≤ Cǫ (log λ − αα + 1) + O ( λ t α − ǫ ) α ) . So, when α is close to 1 enough we can always choose ǫ such that C Z B λtα − ǫα \ B λtα + ǫα ( x α ) |∇ u α | dx ≤ Cǫ (log µ + 1) < ǫ . Hence, we have H ( λ t α α ) − ǫ ≤ H ( r ) ≤ H ( λ t α α ) + ǫ . (4.10)Let f ( t ) = Z Q ( t ) |∇ g u α | dV g = Z Q ( t ) |∇ u α | dx. By using a similar estimate with (4.4) and (4.10), we infer that as α is close to 1 enough thereholds (1 − ǫ ) f ( t ) ≤ f ′ ( t ) + ( α − at + ǫ ) , where a = 4 log 2 H ( λ t α α ) + ǫ . Then, it is easy to see (2 − (1 − ǫ ) t f ) ′ ≥ − ( α − at + ǫ )2 − (1 − ǫ ) t log 2 . Hence, we get f ( t ) ≤ − (1 − ǫ )( τ − t ) f ( τ ) + α − − ǫ (cid:0) ǫ + at + a log 2 − aτ − (1 − ǫ )( τ − t ) − a − (1 − ǫ )( τ − t ) (1 − ǫ ) log 2 (cid:1) . Then, it follows f ( k ) ≤ C ( k )2 − (1 − ǫ ) τ f ( τ ) + α − − ǫ ( ǫ + ak + a log 2 + aC ( k ) aτ − (1 − ǫ ) τ ) . Let 2 τ = λ − ǫα . Then Z B kλtαα \ B k λtαα ( x α ) |∇ u α | dx ≤ C ( k ) λ ǫ (1 − ǫ ) α + α − − ǫ ( H ( λ t α α )4 k log 2 + a log 2+ C ( k ) λ ǫ (1 − ǫ ) α log λ α ) . (4.11)23n the other hand, by (2.6) and (4.10), we get Z B kλtαα \ B k λtαα ( x α ) ( | ∂u α ∂r | − | u α,θ | ) dx ≥ ( α − k log 2( H ( λ t α α ) − ǫ ) . (4.12)Therefore, subtracting (4.11) by (4.12) we obtain2 Z B kλtαα \ B k λtαα ( x α ) | u α,θ | dx ≤ C ( k ) λ ǫ (1 − ǫ ) α + ( α − − ǫ (2 ǫ H ( λ t α α )4 k log 2 + a log 2+ C ( k ) λ ǫ (1 − ǫ ) α log λ α ) + ǫ ( α − k log 2 . (4.13)Since ν = lim α → λ −√ α − α >
1, it is easy to see that, for any m > λ ǫ (1 − ǫ ) α = o (( α − m ) . (4.14)Then, noting (4.14) and letting ǫ → α → α − Z B kλtαα \ B k λtαα ( x α ) | u α,θ | dx ≤ a ′ , where a ′ is a constant which does not depend on R . Thus, we finish the proof of the lemma. ✷ Now, we are in the position to give the proof of Proposition 4.1.
Proof.
First we show (4.6). Since lemma 4.2 says Z B kλtαα \ B k λtαα ( x α ) | u α,θ | α − dx = Z k λ tαα − k λ tαα r (cid:0)Z π | ∂u α ∂θ | α − dθ (cid:1) dr ≤ a ′ , for any ǫ >
0, we can always find k which is independent of α , s.t. there exist L α ∈ [2 k , k ]such that 1 α − Z ∂B Lαλtαα ( x α ) | u α,θ | rds = 1 α − Z π | ∂u α ∂θ ( L α λ t α α , θ ) | dθ < ǫ, and 1 α − Z ∂B Lα λtαα ( x α ) | u α,θ | rds = 1 α − Z π | ∂u α ∂θ | dθ < ǫ. Then (cid:12)(cid:12)(cid:12)Z ∂Q (log L α / log 2) ∂u α ∂r ( u α − u ∗ α ) ds (cid:12)(cid:12)(cid:12) ≤ sZ ∂Q (log L α / log 2) r | ∂u α ∂r | ds Z π | ∂u α ∂θ | dθ ≤ s ǫ ( α − Z ∂Q (log L α / log 2) r | ∂u α ∂r | ds. Z ∂Q (log L α / log 2) r | ∂u α ∂r | ds ≤ C Z ∂Q (log L α / log 2) r | u α,θ | ds + C ( α −
1) + Cλ t α α ≤ ( C + ǫ )( α − . By (4.8) and (4.10), we get(1 − ǫ ) Z B Lαλtαα \ B Lα λtαα ( x α ) |∇ u α | dx ≤ ǫ ( α −
1) + 2( α − H ( λ t α α ) log L α + ǫ ) . (4.15)Noting (4.10) we can infer from (2.6)1 α − Z B Lαλtαα \ B Lα λtαα ( x α ) (cid:0)(cid:12)(cid:12) ∂u α ∂r (cid:12)(cid:12) − | u α,θ | (cid:1) dx = Z L α λ tαα Lα λ tαα r H ( r ) dr, which implies that1 α − Z B Lαλtαα \ B Lα λtαα ( x α ) (cid:0)(cid:12)(cid:12) ∂u α ∂r (cid:12)(cid:12) − | u α,θ | (cid:1) dx ≥ L α ( H ( λ t α α ) − ǫ ) . Combining (4.15) with the above inequality we conclude the following inequality holds true as α is close to 1 sufficiently 1 α − Z B Lαλtαα \ B Lα λtαα ( x α ) | u α,θ | dx ≤ ǫ + Cǫ.
Thus, we have shown (4.6).Next, we turn to proving the remaining assertions of Proposition 4.1.For { t α } ⊂ [ t , t ], we assume u α ( ∂B λ tαα ) → y, α → . As N is regarded as an embedded submanifold in R K , for simplicity, we may assume y = 0 ∈ N and T y N = R n , where R K = R n × R K − n . We also let λ ′ α = λ t α α , x ′ α = ( λ ′ α ,
0) + x α and u ′ α ( x ) = u α ( λ ′ α x + x α ) , v α ( x ) = 1 √ α − u α ( λ ′ α x + x α ) − u α ( x ′ α )] . By (4.11) and Theorem 2.1, we get k∇ u ′ α k C ( B k \ B − k ) + k∇ u ′ α k C ( B k \ B − k ) < C ( k ) √ α − , and then k∇ v α k C ( B k \ B − k ) + k∇ v α k C ( B k \ B − k ) < C ( k ) . Noting that v α (1 ,
0) = 0, we get k v α k C ( B k \ B − k ) < C ′ ( k ) . v α + √ α − A ( y ) + o (1))( dv α , dv α ) + ( α − O ( |∇ v α | ) = 0 , hence, the sequence v α −→ v in C kloc ( R \ { } )where v satisfies ∆ v = 0 with v = v ( | x | ) . Set v = ( a , a , · · · , a n , , · · · ,
0) log r. We deduce from (2.4) that Z ∂B t ( ǫ α + |∇ u α | ) α − |∇ v α | ds = 2 α α − Z ∂B t ( ǫ α + |∇ u α | ) α − | v α,θ | ds + 2(2 α − t Z B t ( ǫ α + |∇ u α | ) α − |∇ u α | dx. Recalling that F α ( t ) = Z B λtα ( ǫ α + |∇ u α | ) α − |∇ u α | dx and keeping (4.6) in our minds, we infer from the above identity and Lemma 2.2 that, as α → Z B λ ′ α \ B λ ′ α ( ǫ α + |∇ u α | ) α − |∇ v α | dx = 2 α α − Z λ tαα λ tαα t F α (log λ α t ) dt + o (1)= 2 α α − F α ( t α ) + o (1) → F ( lim α → t α ) . On the other hand, we have that, as α →
1, there holds Z B λ ′ α \ B λ ′ α ( ǫ α + |∇ u α | ) α − |∇ v α | dx = Z B \ B (cid:18) ǫ α + |∇ g v α | α − λ ′ α (cid:19) α − |∇ v α | dx → πµ lim α → t α | a | log 2 . Hence, we get lim α → v α = ( a , · · · , a n , , · · · ,
0) log r with m X i =1 a i = Λ π µ − α → t α . As v : S −→ N is the corresponding only bubble, then the above identity can be written as | ~a | = E ( v, S ) π µ − α → t α . Thus, we complete the proof of Proposition 4.1. ✷ orollary 4.3. Let α k be a sequence s.t. E α k ( u α k , B λ tαk ( x α )) → µ − t E ( v ) in C ([ t , t ]) with respect to C -norm. If ν > , then Z λ tαk λ tαk √ α − | ∂u α k ∂r | dr → log 2 µ − t r E ( v ) π in C ([ t , t ]) , and √ α − r | ∂u α k ∂r | )( λ tα , θ ) → µ − t r E ( v ) π in C ([ t , t ] × S ) .Proof. We need only to prove the first claim, since the proof of the second claim is similar. Ifthe first claim was not true, then we assumed that there was a subsequence α k i , t i → t s.t. (cid:12)(cid:12)(cid:12) Z λ tiαki λ tiαki √ α − | ∂u α ki ∂r | dr − log 2 µ − t i r E ( v ) π (cid:12)(cid:12)(cid:12) ≥ ǫ > . On the other hand, from the above arguments on Proposition 4.1 we know that, after passingto a subsequence, there holds u α ki ( λ α ki x ) − u α ki ( λ α ki , √ α − → ~a log r, with | ~a | = (cid:12)(cid:12)(cid:12) µ − t q E ( v ) π (cid:12)(cid:12)(cid:12) . Hence we derive the followinglim i → + ∞ Z λ tiαki λ tiαki √ α − | ∂u α ki ∂r | dr = | ~a | Z r dr = log 2 µ − t r E ( v ) π . This is a contradiction. ✷ ν > First, we need to show the necks for the α -harmonic map sequence converge to some geodesicsin N which join the bubbles. For this goal, we denote the curve12 π Z π u α ( r, θ ) dθ : [ λ t α , λ t α ] −→ R n by Γ α . For simplicity, we set ω α ( r ) = 12 π Z π u α ( r, θ ) dθ. First, we claim that if Γ α → Γ, then Γ must lies on N . This is a direct corollary of (4.5). Next,we will prove a subsequence of Γ α will converges locally to a geodesic of N and then give theformula of length of Γ. 27y computation we have¨ ω α = 12 π Z π ¨ u α ( r, θ ) dθ = 12 π Z π (¨ u α ( r, θ ) + u α,θθ r ) dθ = 12 π Z π ∆ u α dθ − π Z π ˙ u α r dθ = − π Z π A ( u α )( du α , du α ) − α − π Z π ∇ |∇ g u α | ∇ u α ǫ α + |∇ g u α | dθ − ˙ ω α r where we have used the fact Z π u θθ ( r, θ ) dθ = 0 . Let G α = − ¨ ω α − ˙ ω α r . Denote the induced metric of Γ α in R K by h α , and let A Γ α be the second fundamental form ofΓ α in R K .Given λ t α α ∈ [ λ t α , λ t α ]. As before, we always have u α ( λ t α α r, θ ) − u α ( λ t α α , √ α − → ~a log r, where ~a ∈ T y N and y = lim α → u α ( λ t α α , θ ). Therefore, we have˙ ω α ( λ t α α ) = √ α − λ t α α ( ~a + o (1)) , h α ( ddr , ddr ) = | ˙ ω α | = α − λ t α α ( | ~a | + o (1)) . (4.16)where o (1) → α →
1. Moreover, we have G α ( λ t α α ) = 12 π Z π A ( u α )( du α , du α ) dθ + α − π Z π ∇ |∇ g u α | ∇ u α ǫ α + |∇ g u α | dθ = α − λ t α α ( 12 π Z π A ( y )( ~a, ~a ) dθ + o (1)) + ( α − Z π O ( |∇ g u α | ) dθ = α − λ t α α ( A ( y )( ~a, ~a ) + o (1) + O ( √ α − α − λ t α α ( A ( y )( ~a, ~a ) + o (1)) . In the above identity we have made use of the fact ν > m > λ t α α = o (( α − m ) . Noting that h A ( y )( ~a, ~a ) , ~a i = 0, we get − A Γ α ( dω α , dω α ) = ¨ ω α − h ¨ ω α , ˙ ω α i| ˙ ω | ˙ ω α = − G α + h G α , ˙ ω α i ˙ ω α | ˙ ω α | = − α − λ t α α ( A ( y )( ~a, ~a ) + o (1)) . (4.17)28ence, we get k A Γ α k h α ( λ t α α ) < C. Similar to the proof of Corollary 4.3, we have, after passing to a subsequence, k A Γ α k h α ( λ tα ) < C for any t ∈ [ t , t ].Now, we fix y ∈ N , and let s to be the arc length parameter of ω α ( t ) with s ( λ t α α ) = 0. Weassume ω α ( λ t α α ) → y as α →
1. It is well-known that k A Γ α k h α ( λ t α α ) does not depend on thechoice of parameter, and d ω α ds = − A Γ α ( ω α )( dω α ds , dω α ds ) , then ω α ( s ) will converges locally to ω ( s ) in C , where s is still the arc length parameter. Thisimplies that Γ α | [ λ t α ,λ t α ] converges locally to a curve Γ locally. We claim that A Γ α ( ω α )( dω α ds , dω α ds ) → A ( ω )( dωds , dωds )strongly in C ([0 , s ] , R n ) for sufficiently small s . If this was not true, then for any small s wecould find a subsequence of { u α } , still denoted by { u α } , such that s ′ α = Z λ t ′ αα λ tαα | ˙ ω α | dr → s ′ ∈ (0 , s )s.t. (cid:12)(cid:12)(cid:12) A Γ α ( ω α )( dω α ds , dω α ds ) − A ( ω )( dωds , dωds ) (cid:12)(cid:12)(cid:12) s = s ′ α > ǫ. To apply Proposition 4.1, we must ensure that t ′ α ∈ [ t , t ]. For simplicity, we may assume λ t α = 2 P λ t α α where P is an integer. Then, applying Corollary 4.3 we have Z i +1 λ tαα i λ tαα | ˙ ω α | dr = √ α − µ − ( t α − i log λα log 2 r E ( v ) π + o α (1) ! . Therefore, as α is close to 1 enough, Z λ t α λ tαα | ˙ ω α | dr = P − X i =0 Z i +1 λ tαα i λ tαα | ˙ ω α | dr ≥ P √ α − r E ( v ) π log 2 + o α (1) ! ≥ C ( t α − t λ −√ α − α ≥ C t ν > . Therefore, we may always choose s to be very small, for example s < C t log ν , such that t ′ α ∈ [ t , t ]. Then, as before there holds u α ( λ t ′ α α r, θ ) − u α ( λ t ′ α α , √ α − → ~a ′ log r. ω α ( λ t ′ α α ) | ˙ ω α ( λ t ′ α α ) | = dω α ds ( s ′ α ) → dωds ( s ′ ) . Applying (4.16) and (4.17), we get that, after passing a subsequence the following holds A Γ α ( ω α )( dω α ds , dω α ds ) | s = s ′ α = 1 | ˙ ω α ( λ t ′ α α ) | A Γ α ( ω α )( ˙ ω α , ˙ ω α ) | r = λ t ′ αα → A ( ω )( dωds , dωds ) | s = s ′ which contradicts the choice of s ′ α . So, we infer that dωds ( s ) − dωds (0) = − Z s A ( ω )( dωds , dωds ) ds holds near s = 0. This shows ω is smooth near 0 and satisfies d ωds = − A ( ω )( dωds , dωds ) . Therefore, we obtain finally ∇ N dωds dωds = d ωds + A ( ω )( dωds , dωds ) = 0 , which means that Γ is a geodesic.Next, we calculate the length of the geodesic Γ. For simplicity, we assume λ t α = 2 P λ t α forsome integer P . Then we have P = t − t log 2 log λ α . When ν = + ∞ , by Corollary 4.3, we have L (Γ α | B k +1 λt α \ kλt α ( x α ) ) ≥ √ α − r E ( v ) π log 2 + o (1)) . Then L (Γ α ) ≥ CP √ α − ≥ C log λ −√ α − α → + ∞ . This implies L (Γ) = + ∞ . Now, we assume ν < + ∞ . By Corollary 4.3, L (Γ α | B k +1 λt α \ kλt α ( x α ) ) = √ α − r E ( v ) π log 2 + o α (1)) , where o α (1) → α → L (Γ) = lim α → √ α − P r E ( v ) π log 2 = ( t − t ) r E ( v ) π log ν. osc B λtα \ B Rλα ( x α ) u α → , as α → , then R → + ∞ and t → osc B δ \ B λtα ( x α ) u α → , as α → , then δ → and t → . (4.19)Since ν < + ∞ implies µ = 1, from Theorem 1.1 we knowlim t → lim R → + ∞ lim α → Z B λtα \ B Rλα ( x α ) |∇ u α | = 0 . Therefore, we can use the same method as in Subsection 4.1 (we replace δ with λ tα ) to deduce osc B λtα \ B Rλα u α ≤ C q E ( u α , B λ tα \ B Rλ α ( x α )) + C (1 − t ) log ν + C √ α − , then (4.18) follows. Similarly, we can prove (4.19). Hence, we derive the length formula of thegeodesic Γ L = r E ( v ) π log ν. Thus, we finish the proof of Theorem 1.2. ✷ Next, we want to give the proof of Corollary 1.3. However, to prove the corollary we onlyneed to prove the following proposition:
Proposition 4.4. If ν < + ∞ , then when ( α − is sufficiently small, all the u α are in thesame homotopy class.Proof. When α i − α j − k u i − u j k C ≤ i ( N ) where i ( N )is the injective radius of N . Hence, by using exponential map we know that u α i and u α j arehomotopic in M \ B δ , B λ t α \ B λ t α and B Rλ α respectively.Let p = u (0) and q = v (+ ∞ ). By (4.18), we know that u i ( B δ ( x α ) \ B λ t α ( x α )) is containedin a simply connected ball centered at p when α is close to 1 enough, δ and t are small enough.Similarly, by (4.19) we also have u j ( B λ t α \ B Rλ α ( x α )) is contained in a small simply connectedball in N with center q when α − δ and 1 − t are sufficiently small. Hence u i and u j arehomotopic in B δ \ B λ t α and B λ t α \ B Rλ α respectively. So u i and u j are homotopic. ✷ In this paper we only consider the case u α is an α -harmonic maps when the conformal structureof M is fixed. Naturally, one will ask the following problems (i) what could we say in the case u α is an α -harmonic maps and the conformal structure of M varies with α , (ii) whether themethods in this paper can be extended to a class of variational problem which is more generalthan α -energy or not. In a forthcoming paper we will further develop some tools to discuss someissues which relate to the above problems.On the other hand, one want to know whether one can give an example to show there is aneck joining the bubbles in the limit of an α -harmonic map sequence is of infinite length or not.31owever, if we can construct a manifold N and find a minimizing α -harmonic map sequencewhich satisfies the condition of Corollary 1.3, then the corollary tells us that indeed there existsa necks in the limit which if of infinite length. By modifying the example of Duzaar and Kuwert(see page 304 of [D-K]) we can construct such example as following . Example.
Let Z act on R by τ κ ( x, y, z ) = ( x +4 k , y +4 k , z +4 k ), where κ = ( k , k , k ) ∈ Z .Consider ˜ X = R \ ∪ κ τ κ ( B (0))and X is the quotient of ˜ X . Then X is a compact manifold with boundary. Topologically, X is T minus a small ball.Let Φ be a conformal map from R to ∂B (0), s.t. Φ( x ) = (1 , ,
0) when | x | > x ) = ( − , ,
0) when | x | <
1, and deg (Φ) = 1 if we consider Φ be a map from S to S .Moreover, we let γ k : [0 , → ˜ X be a curve connect (4 k − , ,
0) and (1 , , v k = Φ( x ) , | x | ≥ δγ k (cid:18) log r − log Rǫ log δ − log Rǫ (cid:19) , Rǫ < | x | < δτ ( k, , (Φ( xǫ )) . | x | ǫ ≤ R We denote π to be the projection from ˜ X to X , then π ( v k ) ∈ π ( X ). We have Z B δ \ B Rǫ |∇ v k | = 2 π Z δRǫ | ∂γ k ∂r | rdr< c k ˙ γ k L ∞ ( − log Rǫ + log δ ) Z δRǫ drr = c k ˙ γ k L ∞ log δ − log Rǫ , Z R \ B δ |∇ v k | ≤ E (Φ) , and Z B Rǫ |∇ v k | ≤ E (Φ) . So, we can find suitable R and ǫ , s.t. E ( π ( u k )) = E ( u k ) ≤ E (Φ) + 1 . We claim that [ π ( v k )] are different homotopy classes. Assuming this is not true, we can finda continuous map H ( x, t ) : S × [0 , → X s.t. H ( x,
0) = π ( v i ) and H ( x,
1) = π ( v j ) . Since S × [0 ,
1] is simply connected, we are able to lift H to ˜ H which is a map from S × [0 , → ˜ X with ˜ H ( x,
0) = v i . We assume that ˜ H ( x,
1) = τ κ ( v j ). Hence [ v i ] = [ τ κ ( v j )]. Therefore[ ∂B (0) + ∂τ ( i, , ( B (0))] = [ ∂τ κ ( B (0)) + ∂τ ( j, , τ κ ( B (0))] in π ( ˜ X ) , where π ( ˜ X ) is the second homotopy group of ˜ X . However, it is easy to check that π ( ˜ X ) = { } ,then by Hurewicz Theorem, the above identity is not true.Now, we proceed to construct N . Let f be a homeomorphism from X to Y = X . Weconsider the quotient space of X ∪ Y , obtained by gluing every point x ∈ ∂X with f ( x ) ∈ ∂Y N and a projection φ : N → X . Oneis easy to check that π ( v k ) can be also considered as a map from S to N with E ( π ( v k )) < C .We claim that [ π ( v k )] are some different homotopic classes with each other in π ( N ). Assum-ing it is not true. Then, we can find a continuous map H ( x, t ) : S × [0 , → N such that H ( x,
0) = π ( v i ) and H ( x,
1) = π ( v j ). Hence, φ ( H ( x, t )) is just a homotopic map of π ( v i ) and π ( v j ) in X . A contradiction.Finally, we would like to ask the following problems: Problem 1.
Suppose all α -harmonic maps u α belong to the same homotopic class and satisfythe energy identity as α →
1. Do the necks consist of some geodesics of finite length?
Problem 2.
Could we find a sequence α k →
1, and α k harmonic maps u α k , s.t. 1)the Morseindex tends to infinite; 2) sup k E α k ( u α k ) < ∞ ; 3) for any i = j , u α i and u α j are not homotopicto each other. References [C] Chang, K.C.: Heat flow and boundary value problem for harmonic maps. Ann. Inst. H.Poincare Anal. Non Lineaire (1989), no. 5, 363–395.[C-T] Chen, J. and Tian, G.: Compactification of moduli space of harmonic mappings. Com-ment. Math. Helv. (1999), 201-237.[C-M] Colding, T. H. and Minicozzi II, W. P.: Width and finite extinction time of Ricci flow.Preprint.[D] Ding, W.: Lectures on the heat flow of harmonic maps. Manuscript.[D-T] Ding, W. and Tian, G.: Energy identity for a class of approximate harmonic maps fromsurfaces, Comm. Anal. Geom. (1995), 543-554.[D-K] Duzaar, F and Kuwert, E: Minimization of conformally invariant energies in homotopyclasses. Calc. Var. Partial. Differ. Equ., (1998), 285-313.[E-S] Eells, J. and Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer. J.Math. (1964), 109–160.[J] Jost, J.: Two-dimensional geometric variantional problems. John Wiley and Sons, Chich-ester, 1991.[L] Lamm, T.: Energy identy for approximations of harmonic maps from surfaces. Preprint.[L-W] Li, Y. and Wang, Y.: Bubbling location for a sequence of approximate f -harmonic mapsfrom surfaces. Preprint.[Lin-W] Lin, F. and Wang, C.: Energy identity of harmonic map flows from surfaces at finitesingular time. Calc. Var. Partial Differential Equations (1996), 595-633.[Q] Qing, J.: On singularities of the heat flow for harmonic maps from surfaces into spheres.Comm. Anal. Geom., (1995), 297-315.[Q-T] Qing, J. and Tian, G.: Bubbling of the heat flows for harmonic maps from surfaces.Comm. Pure. Apple. Math., (1997), 295-310.[S-U] Sacks, J. and Uhlenbeck, K.: The existence of minimal immersions of 2-spheres, Ann. ofMath. , (1981), 1-24.[St] Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces, Comment.Math. Helv. (1985), 558-581.[T] Topping, P.: Winding behaviour of finite-time singularities of the harmonic map heat flow.Math. Z.,247