ENERGY IDENTITY FOR APPROXIMATIONS OF HARMONIC MAPS FROM SURFACES

Transcription

1 TRANSACTIONS OF THE AERICAN ATHEATICAL SOCIETY Volume 362, Number 8, August 21, Pages S 2-997(1)912-3 Article electronically published on arch 23, 21 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS FRO SURFACES TOBIAS LA Abstract. We prove the energy identity for min-max sequences of the Sacs- Uhlenbec and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles. 1. Introduction Let ( 2,g) be a smooth and compact Riemannian surface and let (N n,h)be a smooth and compact Riemannian manifold, both without boundary. We assume that N n R m isometrically. For u W 1,2 (,N) we define the Dirichlet energy (1.1) E(u) = u 2 dv g. Critical points of E are called harmonic maps and they solve the elliptic system (1.2) Δu + A(u)( u, u) =, where A is the second fundamental form of the embedding N R m. The geometric interest in harmonic maps from surfaces comes from the fact that if the harmonic map is additionally conformal (i.e., angle-preserving), then the image of the map is a minimal immersion of in N. For example it is well nown that every harmonic map u : S 2 N is minimal. It is therefore of interest to find critical points of the Dirichlet energy. Since E does not satisfy the Palais-Smale condition the classical variational methods do not apply to E. In order to overcome this difficulty, Sacs & Uhlenbec [2] introduced a regularization of the Dirichlet energy. ore precisely, they considered for every α>1andu W 1,2α (,N) the functional (1.3) E α (u) = (1 + u 2 ) α dv g. Since this functional satisfies the Palais-Smale condition they were able to show the existence of a smooth critical point of E α for every α>1 by classical variational Received by the editors December 17, athematics Subject Classification. Primary 58E2; Secondary 35J6, 53C3. Key words and phrases. Geometric analysis, harmonic maps, energy identity. The author would lie to than Yuxiang Li for pointing out an error in an earlier version of the paper. 77 c 21 American athematical Society Reverts to public domain 28 years from publication

2 78 TOBIAS LA methods. These critical points u α solve the elliptic system (1.) 1 α de α(u) = div((1 + u α 2 ) α 1 u α )+(1+ u α 2 ) α 1 A(u α )( u α, u α )=. Sacs & Uhlenbec then studied sequences of critical points u α (α 1) of E α with uniformly bounded energy E α (u α ) c. They showed that for a subsequence α 1 the maps u α converge wealy in W 1,2 (,N) and strongly away from at most finitely many singular points to a smooth harmonic map u 1 C (,N). oreover they were able to perform a blowup around these finitely many singular points and they showed that the blowups are nontrivial minimal two-spheres. As an application of this analysis Sacs & Uhlenbec proved the existence of a minimal two-sphere in every homotopy class if π 2 (N) =. What was left over in their analysis of sequences of critical points of E α was the question if there is some energy loss occurring during the blow-up process. In [7] the author considered a different regularization of the Dirichlet energy; namely, for every ε> and every u W 2,2 (,N) we studied the functional (1.5) E ε (u) = u 2 dv g + ε Δu 2 dv g. The Euler-Lagrange equation of E ε is given by (1.6) Δu εδ 2 u = A(u)( u, u)+f[u], where f[u] T u N and (1.7) f[u] c( u )ε( u 3 u + 2 u 2 + u ). For every ε>the functional E ε satisfies the Palais-Smale condition, and therefore critical points exist and they are smooth. Hence, as in the case of the Sacs- Uhlenbec approximation, we studied sequences u ε C (,N) (ε ) of critical points of E ε with uniformly bounded energy E ε (u ε ) c. We were able to show that for a subsequence ε the maps u ε converge wealy in W 1,2 (,N) and strongly away from at most finitely many singular points to a smooth harmonic map u : N. oreover, by performing a blowup around the singular points, we showed that at most finitely many minimal two-spheres were separating. Additionally we were able to show that there is no energy lost during the blow-up process if N = S n R n+1. The case of a general target manifold was left open. In the main result of this paper we show that for both approximations and general target manifolds there is no energy loss occurring if we assume an additional entropy-type condition. ore precisely we have the following. Theorem 1.1. Let ( 2,g) be a smooth, compact Riemannian surface without boundary and let N be a smooth and compact Riemannian manifold without boundary, which we assume to be isometrically embedded into R n. Let u α C (,N) (α 1) be a sequence of critical points of E α with uniformly bounded energy. oreover we assume that u α satisfies (1.8) lim inf (α 1) log(1 + u α α 1 2 )(1 + u α 2 ) α dv g =. Then there exists a sequence α 1 and at most finitely many points x 1,...,x l such that u α u 1 wealy in W 1,2 (,N) and in Cloc (\{x1,...,x l },N), where u 1 : N is a smooth harmonic map.

3 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 79 By performing a blowup at each x i, 1 i l, one gets that there exist at most finitely many nontrivial smooth harmonic maps ω i,j : S 2 N, 1 j j i, sequences of points x i,j, xi,j xi, and sequences of radii r i,j R +, r i,j, such that (1.9) max{ ri,j (1.1) (1.11) r i,j, ri,j r i,j ), dist(xi,j,xi,j r i,j + ri,j }, 1 i l, 1 j, j j i, j j, lim sup(r i,j )1 α =1 1 i l, 1 j j i and l lim E j i α (u α )=E(u 1 )+vol()+ E(ω i,j ). i=1 j=1 Remar 1.2. The theorem remains true if we replace everywhere E α by E ε, u α by u ε, u 1 by u, the assumption (1.8) by (1.12) the estimate (1.1) by (1.13) and (1.11) by (1.1) lim sup lim inf ε log(1 ε ε ) Δu ε 2 dv g =, ε (r i,j )2 = 1 i l, 1 j j i, lim E ε (u ε )=E(u )+ l j i E(ω i,j ). i=1 j=1 Remar 1.3. By the results of Duzaar & Kuwert [] (Theorem 2), the above theorem implies that we also have a decomposition in terms of homotopy classes. Of course now one has to as if there exist sequences of critical points of E α, resp. E ε, satisfying (1.8), resp. (1.12). The answer to this question is yes and more precisely we have the following. Lemma 1.. Let α>1and let F P(W 1,2α (,N)) be a collection of sets. Let Φ:[, [ W 1,2α (,N) W 1,2α (,N) be any continuous semi-flow such that Φ(, ) =id, Φ(t, ) is a homeomorphism of W 1,2α (,N) for any t and E α (Φ(t, u)) is nonincreasing in t for any u W 1,2α (,N). We assume that Φ(t, F ) F for all t [, ) and all F F. We define (1.15) β α = inf E α (u) sup F F u F and we assume that β α <. Then for almost every α there exists a critical point u α C (,N) of E α with E α (u α )=β α and such that lim inf (α 1) log( 1 (1.16) α 1 α 1 ) log(1 + u α 2 )(1 + u α 2 ) α dv g =. With the obvious modifications the same conclusion remains true for the energy E ε. Remar 1.5. For examples of subsets F P(W 1,2α (,N)) satisfying the hypotheses of the above lemma we refer the reader to [15] (p. 19) or [2] (p. 88).

4 8 TOBIAS LA As a corollary of the above theorem and lemma, we obtain a new proof of a result of Jost [6] on the energy identity for min-max sequences for the Dirichlet energy. Corollary 1.6. Let ( 2,g) be a smooth, compact Riemannian surface without boundary and let N R n be a smooth and compact Riemannian manifold without boundary. oreover let A be a compact parameter manifold; for simplicity we assume A =, andleth : A N be continuous. Let H be the class of all maps homotopic to h and (1.17) β := inf E(h(,t)). sup h H t A Then there exists a sequence u α C (,N) of critical points of E α, a harmonic map u 1 : N and at most finitely many points x 1,...,x l such that (1.18) E α (u α )=β α = inf E α (h(,t)), (1.19) (1.2) (1.21) β α u α u α sup h H t A β +vol(), u 1 wealy in W 1,2 (,N) and u 1 in Cloc(\{x 1,...,x l },N). oreover there exist at most finitely many nontrivial smooth harmonic maps ω i,j : S 2 N, 1 i l, 1 j j i,sequencesofpointsx i,j, x i,j x i,and sequences of radii r i,j (1.22) max{ ri,j (1.23) (1.2) r i,j, ri,j r i,j R +, r i,j ), dist(xi,j,xi,j r i,j + ri,j, such that }, 1 i l, 1 j, j j i, j j, lim sup(r i,j )1 α =1 1 i l, 1 j j i and l lim E j i α (u α )=E(u 1 )+vol()+ E(ω i,j ). i=1 j=1 Remar 1.7. With the obvious modifications the corollary remains true for the biharmonic approximation E ε. Proof. The proof of this result is quite standard, but we include it here for the sae of completeness. It is obvious that for all α>1wehave β +vol() β α. Let δ>andchoose h H C ( A, N) such that sup E( h(,t)) β + δ. t A Then for (α 1) small enough we have sup E α ( h(,t)) β +vol()+δ + c( h)(α 1) t A β +vol()+2δ. This implies lim α 1 β α = β +vol().

5 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 81 The result now follows from the minimax principle (see [2]), Theorem 1.1 and Lemma 1.. In the existing literature there are already some partial results available for the energy identity for the Sacs-Uhlenbec approximation and there are many more results available for related problems. In the following we want to mention some of these results. For the Sacs-Uhlenbec approximation, Duzaar & Kuwert [] and Chen & Tian [1] proved the energy identity for sequences of minimizers of the energy E α in a given homotopy class. Recently oore [1] proved the energy identity (he actually proved (1.11) with the Dirichlet energy E instead of the full α-energy E α on the left hand side) for min-max sequences of the Sacs-Uhlenbec approximation under the additional assumption that the target manifold has finite fundamental group. The additional assumptions made by Chen & Tian and oore were used to ensure that the sequence of minimizers, respectively the min-max sequence, converges to a geodesic of finite length on the necs connecting the bubbles and the wea limit (or body map) which then implies the energy identity. In our proof we use completely different arguments, but we want to mention that it is not directly clear from our analysis that the sequence of critical points satisfying the entropy condition converges to a geodesic of finite length on the necs. In a recent independent wor, Li & Wang [8] proved Theorem 1.1 in the special case of sequences of minimizers (in their own homotopy class) of E α. For sequences of harmonic maps and min-max sequences for the Dirichlet energy, the energy identity was proved by Jost [6] (see also [16] for an alternative proof of the energy identity for sequences of harmonic maps). Recently Colding & inicozzi [2] proved the energy identity for sequences of maps with bounded Dirichlet energy which are almost conformal and which satisfy a certain replacement property. The energy identity for the harmonic map heat flow and Palais-Smale sequences for the Dirichlet energy with tension field bounded in L 2 was established by Qing [17] (in the case N = S n ) and independently by Ding & Tian [3] and Wang [27] in the general case. Alternative proofs have been given by Qing & Tian [18] and Lin & Wang [1]. See also the paper of Topping [25] for more refined results in this case. Lin & Wang [11], [12] used a Ginzburg-Landau approximation to regularize the Dirichlet energy and proved the energy identity in this situation. The disadvantage of the Ginzburg-Landau approximation is that the approximating maps do not have to map into the target manifold; only in the limit are they forced to do this. For maps from higher-dimensional domains the energy identity for sequences of harmonic maps has been proved by Lin & Rivière [9] for N = S n. For other related problems such as sequences of Yang-ills fields on a four-dimensional manifold, respectively biharmonic maps from a four-dimensional manifold into the sphere, the energy identity has been proved by Rivière [19], respectively Wang [28]. In the following we give a brief outline of the paper. In section 2 we prove Theorem 1.1 for the Sacs-Uhlenbec approximation of harmonic maps. We start by recalling the small-energy regularity estimates and the blow-up procedure of Sacs & Uhlenbec [2] in section 2.1. In Proposition 2.3 we prove the very important estimate for the concentration radius of the bubbles. The advantages of having a good estimate for the concentration radius can also

6 82 TOBIAS LA be seen in the paper of Topping [25]. In the next two sections we prove a Hopfdifferential type estimate and an estimate for the tangential component of solutions of (1.) on annular regions. These estimates are proved in the same way as the corresponding estimates for harmonic maps; see for example [2] and [3]. In section 2. we use the bubbling induction argument of Ding & Tian [3] to reduce the proof of the energy identity to the case of one bubble. In this situation we then combine the previous estimates with the estimate for the concentration radius to complete the proof of the energy identity. In section 3 we treat the case of the biharmonic approximation. For this approximation the estimate for the concentration radius (see (3.7)) has already been proved in [7]. In section 3.1 we review the small-energy estimates and the blow-up process from [7]. In section 3.2 we use the stress-energy tensor of E ε to get a Hopfdifferential type estimate for the biharmonic approximation. The rest of the proof of the energy identity then follows as in the case of the Sacs-Uhlenbec approximation, and in sections 3.3 and 3. we briefly describe the necessary modifications. In section we use variational methods to prove Lemma 1.. We closely follow the wor of Struwe [23]. We use the notation o (1), o R (1) and o R (1) to denote terms which tend to zero as, R andr, respectively. 2. Energy identity for the Sacs-Uhlenbec approximation of harmonic maps In this section we prove Theorem 1.1 for the Sacs-Uhlenbec approximation of harmonic maps Results of Sacs and Uhlenbec and estimates for the concentration radius. We consider sequences of critical points u α C (,N) of the functional E α with uniformly bounded energy E α (u α ) c and which satisfy the condition (1.8). Due to the uniform boundedness of the energy it is easy to see that there exists a subsequence α 1 such that (2.1) (α 1) log(1 + u α 2 )(1 + u α 2 ) α dv g and u α u 1 wealy in W 1,2. In section 3 of [2], Sacs & Uhlenbec proved the following small energy regularity result for solutions of (1.). Theorem 2.1. There exists ε > such that if u α (α close to one) is a critical point of E α with B 2R u α 2 <ε (where R>), then we have for every m N, (2.2) osc BR u α + m u α L (B R )R m c( u α 2 ) 1 2. B 2R With the help of this theorem, Sacs & Uhlenbec were able to show that the sequence u α converges strongly to a smooth harmonic map u 1 : N away from finitely many points. These finitely many singular points x i, 1 i l, are characterized by the condition that (2.3) lim sup E(u α,b R (x i )) ε, for every R> and every 1 i l. Around these finitely many singular points they were able to perform a blowup and show that a nontrivial harmonic twosphere separates. The blowup can be done as follows: Fix R > such that

7 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 83 B R (x i ) B R (x j )= for every i, j {1,...,l}, i j. Because of (2.3) there exists a sequence of points x i xi and radii r i such that (2.) max E(u α,b r i y B R (x i ) (y)) = E(u α,b r i (x i )) = ε 2. Defining v i :B R N, r i (2.5) v(x) i =u α (x i + rx), i we see that v i solves (1.) with 1 replaced by (ri )2 and moreover (2.6) max E(v y B R,B i 1 (y)) = E(v,B i 1 ()) = ε 2. 2r i Therefore we can apply Theorem 2.1 to v i and get that vi converges in C1 to a smooth harmonic map ω i from R 2 into N. By the point removabilty result of Sacs & Uhlenbec we can then extend ω i to a smooth harmonic map from S 2 to N. As a consequence of this blow-up procedure we get the following estimate for the concentration radius. Lemma 2.2. Using the above notation we have that (2.7) 1 lim sup(r) i 1 α <, for every 1 i l. Proof. Because of (2.) and Hölder s inequality we now that ε 2 = E(u α,b r i (x i )) ( c(r) i 2(α 1) α. From this the claim follows. (1 + u α 2 ) α ) 1 α (r) i 2(α 1) α In the next proposition we use (2.1) to improve the above estimate for the concentration radius (see also [23] where this was observed for a similar approximation of a different problem). Proposition 2.3. We have that (2.8) lim (ri ) 1 α =1, for every 1 i l. Proof. We let ε be as above and we assume without loss of generality that l =1. Furthermore we let r 1 = r, x 1 = x and u α = u. For every N we define the set ε (2.9) Ω = {x B r (x ) u (x) 2 } πr and we claim that there exists a constant c> such that for every N we have (2.1) Ω cr. 2

8 8 TOBIAS LA If this is not the case we can find a subsequence m such that (2.11) Ω m r2 m m. From (2.) and Theorem 2.1 we get (2.12) u m L (B r m (x m )) c ε r m. From the definition of Ω m we see that for every x B (x r m m )\Ω m we have the estimate (2.13) u m (x) ε 2 πr m. Using (2.11), (2.12) and (2.13) we get from (2.), ε 2 = u m 2 = u m 2 + B r (x m m ) cε Ω m r 2 m + πr 2 m ε πr 2 m Ω m B r m (x m )\Ω m u m 2 c m + ε ε, as m. This contradiction proves the estimate (2.1). Now we use (2.1), Lemma 2.2, the definition of Ω and (2.1) to estimate = lim (α 1) log(1 + u 2 )(1 + u 2 ) α lim (α 1) log( u 2 ) u 2α Ω c lim (α 1)r 2(1 α ) log( ε πr 2 ) = c lim (1 α )r 2(1 α ) (2 log r log ε +logπ) = c lim r2 2α log r 2 2α c lim log r2 2α, and hence the desired convergence result for the concentration radius follows A Hopf-differential type estimate. In the case of sequences of harmonic maps or Palais-Smale sequences for the Dirichlet energy with tension field bounded in L 2 an important ingredient in the proof of the energy identity was an estimate for the Hopf differential (see e.g. [3], [2]). In the next lemma we show that a related result is true for solutions of (1.).

9 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 85 Lemma 2.. Let B R 2 be the unit ball and let u α C (B,N) be a solution of (1.). Then we have for every <r<1 and every α close to one, (1 + u α 2 ) α 1 (u α ) r 2 c (1 + B r Br (u α) θ 2 r 2 )(1 + u α 2 ) α 1 c(α 1) (2.1) + (1 + u α 2 ) α. r B r Proof. We multiply equation (1.) by x u α and integrate over B r to get (remember that A(u α )( u α, u α ) T uα N for every x B) = div((1 + u α 2 ) α 1 u α )x u α B r = (1 + u α 2 ) α 1 u α 2 + r(1 + u α 2 ) α 1 (u α ) r 2 B r B r 1 (1 + u α 2 ) α 1 x (1 + u α 2 ). 2 B r Next we integrate by parts and get α (1 + u α 2 ) α 1 x (1 + u α 2 )= (1 + u α 2 ) α 2 B r B r r + B r 2 (1 + u α 2 ) α. Using the identity u α 2 = (u α ) r r 2 (u α) θ 2 and combining everything, we end up with r(1 + u α 2 ) α 1 (u α ) r 2 1 c B r B r r (1 + u α 2 ) α 1 (u α ) θ 2 + c(α 1) (1 + u α 2 ) α B r + c r(1 + u α 2 ) α 1. B r 2.3. Estimate for the tangential component. In this section we show that if the Dirichlet energy is small on all annular regions with bounded geometry, then the tangential derivative of u α converges to zero on the annular region which is the union of all the annuli with bounded geometry. The proof of this fact closely follows the previous wor of Sacs & Uhlenbec [2] and Ding & Tian [3]. In the following we use for <a 1 <a 2 < 1 the notation A(a 1,a 2 )={x R 2 a 1 x a 2 }. Lemma 2.5. There exists δ > such that for all δ<δ and all solutions u α C (B,N) of (1.) with A(r,2r) u α 2 <δfor every r (R 1, R 2 2 ), we have for α 1 small enough, (2.15) R 2 2R 1 2π 1 r (u α) θ 2 drdθ c δ(1 + (log R 1 α 1 )).

10 86 TOBIAS LA Proof. Let δ <ε and let y A(2R 1, R 2 ). Then we have that 2 y 3, y 3 (R 1, R 2 2 ) and B y (y) B y \B 2 y. From our assumption and Theorem 2.1 we therefore conclude that (2.16) 2 x i i u α (x) c δ, i=1 for every x A(2R 1, R 2 ). Now we let R 2 R 1 =2 l + q, l N and q, and define A = A(2 R 1, 2 +1 R 1 ) for all 1 l 1 and we let A l = A(2 l R 1, R 2 ). Next we note that equation (1.) can equivalently be written as (2.17) Δu α + A(u α )( u α, u α )= 2(α 1) 2 u α, u α u α 1+ u α 2 =: f α. Now we let h = h(r) be a piecewise linear function which equals the mean value of u α on { R 2 } S 1 and {2 R 1 } S 1 for all 1 l 1. With the help of this we have Δ(u α h)+a(u α )( u α, u α )=f α. Testing this equation with u α h and integrating over A we get (u α h) 2 = (u α h)(a(u α )( u α, u α ) f α ) A A 2π R 1 (u α h)(u α h) r (2 +1 R 1,θ)dθ 2π 2 R 1 (u α h)(u α h) r (2 R 1,θ)dθ. We remar that the boundary integrals of (u α h)h r vanish since h is equal to the mean value of u α on these boundaries and h r is piecewise constant. Because of (2.16) and the Sobolev embedding (which we only apply on the annuli A )we now that for every x A we have (2.18) 2 u α h (x)+ x i i u α cδ 1 2. i=1 This implies that (u α h) 2 cδ 1 2 ( u α 2 + f α ) A A 2π R 1 (u α h)(u α ) r (2 +1 R 1,θ)dθ 2π 2 R 1 (u α h)(u α ) r (2 R 1,θ)dθ.

11 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 87 Taing the sum over we get (u α h) 2 cδ 1 2 A(2R 1, R 2 ) A(2R 1, R 2 ) 2π ( u α 2 + f α ) + R 2 (u α h)(u α ) r ( R 2,θ)dθ 2π 2R 1 (u α h)(u α ) r (2R 1,θ)dθ cδ 1 2 (1+(logR 1 α 1 )), where we used (2.18) to estimate f α c(α 1) 2 u α A(2R 1, R 2 ) (log R1 1 α ). This finishes the proof of the lemma. A(2R 1, R 2 ) 2.. Proof of the energy identity. Proof. Because of the induction argument of Ding & Tian [3] we now that it is enough to prove the energy identity in the presence of one bubble. Since we are dealing with a local problem we assume from now on that u α : R 2 B 1 N and that we have only one energy concentration point x 1 =. Using the notation from section 2.1 we assume that we obtain the bubble by rescaling with the factor r 1 = r. From the smooth convergence u α u 1 away from we conclude that E α (u α,b 1 \B R ) E(u 1,B 1 \B R )+vol(b 1 \B R ), for every < R < 1. Similarly, from the local C 1 -convergence v 1 = v = u α (r ) ω, wehaveforeveryr>, E α (u α,b Rr ) E(ω). oreover this also implies that for every R>and>, (2.19) E α (u α ; B R \B R )+E α (u α ; B r R\B r R), as and R. Therefore it is easy to see that the proof of the energy identity in the case of one bubble is reduced to showing that (2.2) lim lim lim E α (u α,a(rr,r )) =. R R Next we claim that due to the fact that we have only one bubble we can assume that for any δ>thereexists > such that for all > we have (2.21) E(u α,b 2r \B r ) <δ, for every Rr r R 2. To see this we argue by contradiction. If the claim is false, we may assume that as there exists s (Rr, R 2 ) such that E(u α ; B 2s \B s )= max r (Rr, R 2 ) E(u α ; B 2r \B r ) δ.

12 88 TOBIAS LA From (2.19) we get that (2.22) By defining R s and Rr s. ṽ : B R \B r R N s s ṽ (x) =u α (s x) we have that ṽ solves (1.) with 1 replaced by (s ) 2 and ((s ) 2 + ṽ 2 ) α cs 2(α 1) (2.23) B R \B r R s s (2.2) E(ṽ ; B 2 \B 1 ) δ., By (2.23), (2.22), Proposition 2.3 and the arguments of section 2.1 we may assume that ṽ ṽ wealy in W 1,2 loc (R2 \{},N), where ṽ : R 2 N is a harmonic map with finite Dirichlet energy. We have two possibilities. The first one is that there exists r > such that sup sup E(ṽ ; B r (x)) <ε. N x B \B 1 With the help of Theorem 2.1 and a covering argument, this implies that ṽ ṽ in C (B 2 \B 1,N). Since R 2 \{} is conformally equivalent to S 2 \{N,S} we conclude from (2.2) and the point removability result of Sacs & Uhlenbec [2], that ṽ can be lifted to a smooth nontrivial harmonic map from S 2 to N, contradicting the assumption that we have only one bubble ω. The second possibility is that we have at least one energy-concentration point y B \B 1. Now we can apply the blow-up procedure of section 2.1 to conclude that there must exist a nontrivial harmonic two-sphere, again contradicting the assumption that there is only one bubble. This proves (2.21) and hence we can combine Theorem 2.1, Proposition 2.3, Lemma 2. and Lemma 2.5 (with R 1 = Rr and R 2 = R ) to estimate (1 + u α 2 ) α c u α 2 + o R (1) A(2Rr, R ) A(2Rr, R ) 2π R = c 2Rr R 2π (r (u α ) r r (u α ) θ 2 )drdθ + o R (1) 1 c 2Rr r (u α ) θ 2 drdθ + o R (1) R 1 + c(α 1) 2Rr r ( u α 2α dx)dr B r o (1) + o R (1) + c δ + c(1 α )log(rr ) o (1) + o R (1) + c δ,

13 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 89 which, combined with (2.19), proves (2.2) (since δ>was arbitrary) and therefore the main theorem in the case of one bubble. Remar 2.6. By a careful inspection of the above proof it is easy to see that the energy identity remains true for general sequences of critical points of E α if and only if lim (ri,j )1 α =1, for all 1 i l and all 1 j j i. This fact has also been observed by Li & Wang [8]. 3. Energy identity for the biharmonic approximation of harmonic maps In this section we prove Theorem 1.1 for the biharmonic approximation of harmonic maps Estimates and blowup. In the following we consider sequences of critical points u ε C (,N)(ε ) of the functional E ε with uniformly bounded energy E ε (u ε ) c and which satisfy (1.12). First of all we choose a subsequence ε such that ε log( 1 (3.1) ) Δu ε 2 = o (1). ε Due to the uniform bound on the W 1,2 -norm of u ε we get the existence of a further subsequence (still denoted by ε ) such that u ε u wealy in W 1,2 (,N). In [7] we were able to show the following small energy estimate (see Corollary 2.1 in [7]). Theorem 3.1. There exists δ > and c> such that if u ε C (,N) is a solution of (1.6) with B 2R ( u 2 + ε Δu 2 ) <δ, then we have for ε small enough and every m N, (3.2) osc BR u ε + R m m u ε L (B R ) c ( ( u 2 + ε Δu 2 )) 1 2. B 2R Hence, as in section 2.1, the sequence u ε converges strongly to u away from finitely many singular points x i, 1 i l, which are characterized by the condition (3.3) lim sup E ε (u ε,b R (x i )) δ, for every R> and every 1 i l. Around these finitely many singular points we were able to perform a blowup similar to the one of section 2.1 (see section 3 of [7]). Namely, for R > such that B R (x i ) B R (x j )= for every 1 i j l, there exists a sequence of points x i xi and a sequence of radii r i such that (3.) Defining (3.5) max E ε y B R (x i (u ε,b r i ) (y)) = E ε (u ε,b r i (x i )) = δ 2. w i :B R N, r i w(x) i =u ε (x i + rx), i

14 9 TOBIAS LA we see that w i solves (1.6) with ε replaced by ε = (3.6) ε (r i )2 and max E ε (w y B R,B i 1 (y)) = E ε (w,b i 1 ()) = δ 2. 2r i Hence we can apply Theorem 3.1 to w i and conclude that wi converges smoothly to some map ω i C W 1,2 (R 2,N). Then we were able to show (Lemma 3.1 in [7]) that for every 1 i l, ε (3.7) ε = (r i, )2 and therefore ω i is a harmonic map with finite Dirichlet energy and can therefore be lifted to a smooth harmonic map from S 2 to N Stress-energy tensor. For a smooth map u we have the well-nown stressenergy tensor Sαβ 1 (u) givenby (3.8) S 1 αβ(u) = 1 2 u 2 δ αβ α u, β u. An easy calculation shows that if u is a harmonic map, then we have (3.9) α Sαβ(u) 1 = Δu, β u =. Again for a smooth map u we have the stress-energy tensor Sαβ 2 (v) defined by (see [5] and [13]) (3.1) S 2 αβ (u) =1 2 Δu 2 δ αβ + γ u, γ Δu δ αβ α u, β Δu β u, α Δu. By another easy calculation we see that if u is an extrinsic biharmonic map (i.e. a solution of Δ 2 u T u N), then we have (3.11) α Sαβ(u) 2 = β u, Δ 2 u =. Combining (3.9) and (3.11) we see that (3.12) α (Sαβ(u 1 ε ) εsαβ(u 2 ε )) = β u ε, (εδ 2 Δ)u ε = if u ε is a solution of (1.6). As in the case of harmonic maps (see [2]) we use this divergence-free quantity to get a Hopf-differential type estimate for solutions of (1.6). Lemma 3.2. Let u ε C (B,N) be a solution of (1.6). Then we have for all <r<1, (3.13) (u ε ) r 2 1 B r r 2 (u ε ) θ 2 + cε Δu ε 2 + cε ( Δu ε 2 + u ε 3 u ε ). B r r B r B r Proof. ultiplying (3.12) by x β and integrating by parts we get for every <r<1, (Sαβ(u 1 ε ) εsαβ(u 2 ε ))δ αβ = (Sαβ(u 1 ε ) εsαβ(u 2 ε ))x β ν α, B r B r where ν is the outer unit normal to B r. Now we calculate (Sαβ(u 1 ε ) εsαβ(u 2 ε ))δ αβ = ε Δu ε 2

15 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 91 and (Sαβ(u 1 ε ) εsαβ(u 2 ε ))x β ν α = r 2 u ε 2 r (u ε ) r 2 rε( 1 2 Δu ε 2 + u ε, Δu ε 2 (u ε ) r, (Δu ε ) r ) = 1 2r (u ε) θ 2 r 2 (u ε) r 2 rε( 1 2 Δu ε 2 + u ε, Δu ε 2 (u ε ) r, (Δu ε ) r ), where we used the identity u 2 = u r r u 2 θ 2. This finishes the proof of the lemma Estimate for the tangential component. In this subsection we prove an estimate for the biharmonic approximation similar to the one given in section 2.3 for the Sacs-Uhlenbec approximation. Lemma 3.3. There exists δ 1 > such that for all δ<δ 1 and all solutions u ε of (1.6) with A(r,2r) ( u ε 2 +ε Δu ε 2 ) <δfor every r (R 1, R 2 2 ), we have for ε small enough, (3.1) R 2 2R 1 2π 1 r (u ε) θ 2 drdθ c δ(1 + ε (R 1 ) 2 ). Proof. The proof follows directly from that of Lemma 2.5. Namely instead of using Theorem 2.1 we use Theorem 3.1 to conclude that (3.15) x i i u ε c δ i=1 for every x A(2R 1, R 2 ). oreover we note that equation (1.6) can equivalently be written as Δu ε + A(u ε )( u ε, u ε )=εδ 2 u ε + f[u ε ] (3.16) = f ε. Using this form of the equation it is easy to see that the proof of Lemma 2.5 carries over to this situation once we notice that because of (1.7) and (3.15) we have δ f ε c δε ( u ε + u ε 3 u ε + 2 u ε 2 + u ε ) A(2R 1, R 2 ) A(2R 1, R 2 ) c ε δ (R 1 ) Proof of the energy identity. Proof. Following the remars of section 2. (using the results of section 3.1) we can assume that we have only one energy concentration point x 1 = B 1 R 2 and one bubble ω 1 which is obtained by rescaling u ε by the factor r 1 = r. Again the proof of the energy identity is reduced to showing that (3.17) lim R lim lim E ε (u ε,b R \B Rr )=. R

16 92 TOBIAS LA Using similar arguments as in section 2. we can moreover assume that for any δ>thereexists > such that for all > we have (3.18) E ε (u ε,b 2r \B r ) <δ, for every Rr r R 2. Hence we can apply (3.15) with R 1 = Rr and R 2 = R to get ε Δu ε 2 cδε (3.19) A(2Rr, R ) ε cδ R 2 r 2 = o (1), A(2Rr, R ) dx x where we used (3.7) in the last line. Combining (3.1), Lemma 3.2, Lemma 3.3, (3.18), (3.19) and (3.7) we get E ε (u ε,a(2rr, R R )) Rr (3.2) which proves (3.17). c R Rr 2π 2π R + cε (r (u ε ) r r (u ε ) θ 2 )drdθ + o (1) 1 r (u ε ) θ 2 drdθ + cε R Rr ( 1 r B r Δu ε 2 )dr ( Δu ε 2 + u ε 3 u ε )+o (1) Rr B r cε log( 1 ) Δu ε 2 + cε Rr (r ) 2 + o (1) + c δ cε log( 1 ) Δu ε 2 + o (1) + c δ ε o (1) + c δ,. Proof of Lemma 1. We closely follow the wor of Struwe [23] (see also [21], [22] and [2]). Proof. Since the methods are very similar for both approximations we only prove the lemma for E α. First of all we note that the minimax principle (see for example [2], Theorem.2) guarantees the existence of a critical point u α of E α with E α (u α )=β α. The difficult part now consists of showing that we can also find a sequence of critical points satisfying (1.8). We note that it is easy to see that the function α β α = inf E α (u) sup F F u F is nondecreasing and hence differentiable almost everywhere with differential L1 ([1,α 1 ]) for α 1 > 1. Therefore it follows that dβ α dα (.1) 2B = lim inf α 1 (α 1) log( 1 α 1 )dβ α dα =.

17 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 93 To see this we assume that B> and we get for (A 1) very small, A 1 dβ α dα B dα A 1 dα (α 1) log(α 1) =, which contradicts the fact that dβ α dα L1 ([1,α 1 ]). Next we let α>1 be a point of differentiabilty of β α and we choose a sequence α α (α +1 α ). For every N we choose F F such that sup E α (u) β α +(α α). u F Since β α is differentiable in α we get that for sufficiently large we have β α β α +( dβ α dα +1)(α α). Combining the above two estimates we get β α sup E α (u) sup E α (u) β α +(α α) β α +( dβ α u F u F dα +2)(α α). Next we choose v F such that β α (α α) E α (v). Combining all this gives the existence of a map v such that β α (α α) E α (v) E α (v) sup E α (u) β α + α α u F β α +( dβ α (.2) dα +2)(α α). Now we prove three intermediate steps. Step 1. For every v W 1,2α (,N) which satisfies (.2) we have the estimate (.3) From (.2) we get α E α (v) dβ α dα +3. E α (v) E α (v) α α dβ α dα +3, and hence by the mean value theorem there exists a number α α α such that (.) Since moreover α E α (u) = α E α (v) dβ α dα +3. = α E α (u) log(1 + u 2 )(1 + u 2 ) α log(1 + u 2 )(1 + u 2 ) α for every u W 1,2α (,N) we finish the proof of Step 1.

18 9 TOBIAS LA Step 2. We have (.5) sup{ de α (u),v de α (u),v v W 1,2α (u TN) 1}, where W 1,2α (u TN)={v W 1,2α (,R m ) v(x) T u(x) N x }, uniformly for all u W 1,2α (,N) satisfying (.2). Toseethiswenotethatforeveryv W 1,2α (u TN) with v W 1,2α (u TN) 1 we have de α (u),v de α (u),v (2α (1 + u 2 ) α 1 2α(1 + u 2 ) α 1 ) u v =: I. Now we estimate I 2(α α)( (1 + u 2 ) α ) α 1 α ( u 2α ) 1 2α ( v 2α ) 1 2α +2α ((1 + u 2 ) α 1 (1 + u 2 ) α 1 ) u v c(α α)+2α ((1 + u 2 ) α 1 (1 + u 2 ) α 1 ) u v. Next we use the estimate 2 u v 1 δ (1 + u 2 )+δ v 2, (.2) and Young s inequality to get 2α ((1 + u 2 ) α 1 (1 + u 2 ) α 1 ) u v α δ (E α (u) E α (u)) + αδ (1 + u 2 ) α 1 v 2 c(α α) + δα( α 1 E α (u)+ 1 v 2α ). δ α α Choosing δ = α α we conclude that I c α α and this proves (.5). Step 3. There exists a sequence u W 1,2α (,N) satisfying (.2) and (.6) de α (u ) (W 1,2α (,N)). If this is not the case we can find δ> such that de α (u) (W 1,2α (,N)) δ for all u satisfying (.2) and all large enough. For these we let e : W 1,2α (,N) W 1,2α (u TN) be a locally Lipschitz continuous pseudo-gradient vector field for E α with e (u) W 1,2α (u TN) 1and for all u satisfying (.2). de α (u),e (u) 1 2 de α (u) (W 1,2α (,N)) 2δ,

19 ENERGY IDENTITY FOR APPROXIATIONS OF HARONIC APS 95 Let ψ C (R) be a cutoff function such that ψ 1, ψ(s) =fors, ψ(s) =1fors 1andfor large enough we let ( Eα (u) ( β α (α α) ) ) ψ (u) =ψ. α α Since e is Lipschitz continuous the vector field ẽ (u) =ψ (u)e (u) then also defines a Lipschitz continuous tangent vector field. Finally we let φ : R + W 1,2α (,N) W 1,2α (,N) be the flow generated be ẽ : d dt φ (t, u) =ẽ (φ (t, u)), t >, (.7) φ (,u)=u. Let F F be chosen as above and define for v F, v t = φ (t, v). Then we now from the assumptions of the lemma that v t F for all t R + and that sup E α (v t ) sup E α (v) β α +(α α) v F v F for all t. Hence (.8) (t) = supe α (v t ) β α v F is attained only at points v for which (v ) t satisfies (.2). By noting that this implies ψ ((v ) t )=1wecalculate d dt E α((v ) t )= de α ((v ) t ), d dt (v ) t = ψ ((v ) t ) de α ((v ) t ),e ((v ) t ) de α ((v ) t ),e ((v ) t ) + de α ((v ) t ) de α ((v ) t ),e ((v ) t ) 2δ + o (1), where we used (.5) in the last step. This shows that for large enough we get (.9) d (t) δ< dt and hence (t) <β α for large t, contradicting the definition of β α. Altogether this finishes the proof of Step 3. To finish the proof of the lemma we consider a sequence u W 1,2α (,N) satisfying (.2) and (.6). We now that u W 1,2α (,N) c and therefore we may assume that u u α wealy in W 1,2α (,N) and strongly in L 2α C,β (,N) for some <β<1. Since C (,N) isdenseinw 1,2α (,N) wecanmoreover find a sequence u l C (,N) such that u l u α strongly in W 1,2α (,N). Next we define the functional F α : W 1,2α (,R m ) R by (.1) F α (u) = (1 + u 2 ) α. Clearly we have that E α (u) =F α (u) for all u W 1,2α (,N). For v W 1,2α (,N) wedefinetheprojection P v : W 1,2α (,R m ) W 1,2α (v TN).

20 96 TOBIAS LA Following the proof of Lemma 3.26 in [26] we get that (id P u )(u u l ) W 1,2α (,Rm ), as, l. Hence we get from (.6) as in Lemma 3.7 of [26] that (.11) df α (u ),u u l, as, l. By convexity we now that for every, l N we have (1 + u l 2 ) α (1 + u 2 ) α + α (1 + u 2 ) α 1 u (u l u ) + α (u u l ) 2. For any fixed l N we use this together with (.11) to get o (1) = df α (u ),u u l ((1 + u 2 ) α (1 + u l 2 ) α + α (u u l ) 2) ((1 + u 2 ) α (1 + u l 2 ) α + α (u u l ) 2) c( u l L (,R m ))(α α). By letting first and then l we conclude that u u α pointwise a.e. and that E α (u ) E α (u α ). Hence we finally get that u u α strongly in W 1,2α (,N). By (.2) we have E α (u ) E α (u α ) = β α and by Steps 2 and 3 we conclude that u α is a critical point of E α. Since the function s log(1 + s 2 )(1 + s 2 ) α is convex we now that α E α is lower semi-continuous on W 1,2α (,N) and therefore we can use Step 1 to get α E α (u α ) lim inf αe α (u ) dβ α dα +3. Combining all this with (.1) we finish the proof of the lemma. References 1. J. Chen and G. Tian. Compactification of moduli space of harmonic mappings. Comm. ath. Helv., 7:21 237, R (21:582) 2. T. Colding and W. inicozzi. Width and finite extinction time of Ricci flow. Preprint, W. Y. Ding and G. Tian. Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom., 3:53 55, R (97e:5855). F. Duzaar and E. Kuwert. inimization of conformally invariant energies in homotopy classes. Calc. Var. Partial Differ. Equ., 6: , R (99d:585) 5. G.Y. Jiang. The conservation law for 2-harmonic maps between Riemannian manifolds. Acta ath. Sinica, 3:22 225, R (88:5828) 6. J. Jost. Two-dimensional geometric variational problems. John Wiley and Sons, Chichester, R11926 (92h:585) 7. T. Lamm. Fourth order approximation of harmonic maps from surfaces. Calc. Var. Partial Differ. Equation s, 27: , 26. R (27:5823) 8. Y. Li and Y. Wang. A wea energy identity and the length of necs for a Sacs-Uhlenbec α-harmonic map sequence. Preprint, F. Lin and T. Rivière. Energy quantization for harmonic maps. Due ath. J., 111: , 22. R18765 (22:5836) 1. F. Lin and C. Wang. Energy identity of harmonic map flows from surfaces at finite singular time. Calc. Var. Partial Differ. Equ., 6:369 38, R1623 (99:587)

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