Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Transcription

1 J Eur Math Soc, 37 3 c Springer-Verlag & EMS 999 Fanghua Lin Tristan Rivière Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents Received December 3, 998 / final version received May 0, 999 Abstract There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S -valued function defined on the boundary of a bounded regular domain of R n When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension- minimal current minimizing the area within the homology class induced from the S -valued boundary data The union of this harmonic map and the minimal current is the natural generalization of the harmonic extension I Introduction I Vortex equations Complex Ginzburg-Landau equations originated in the theory of superconductivity [6] When the Ginzburg-Landau parameter is chosen to be a special constant, the equations are called self-dual vortex equations which were carefully studied by Jaffe and Taubes [9] For the vortex equation on a Riemannian surface, one considers an open, smooth domain with, possibly empty, smooth boundary Let L be a complex line bundle over equipped with a Hermitian metric <,> For a section u of L we write u(x) =< u(x), u(x) > Then the Ginzburg-Landau functionals are defined for a section u of L and a unitary connection A on L The self-dual case of this functional is given by E(u, A, ) = [ da + A u + 4 ] ( u ) dx, (I) FH Lin: Courant Institute of Mathematical Sciences, New York University, 5 Mercer Street, New York, NY 00-85, USA FH Lin was partially supported by the NSF-grant DMS # T Rivière: Ecole Normale Supérieure de Cachan, Centre de Mathématiques et de leurs Applications, Unité associée au CNRS URA-6, 6, avenue du président Wilson, F-9435 Cachan Cedex, France Mathematics Subject Classification (99): 35J0, 35J5, 35J60, 35J50, 35Qxx, 49Q05, 49Q5, 49Q0, 53Z05, 58E5, 58E0, 8T3

2 38 Fanghua Lin, Tristan Rivière where dx is the volume form of some fixed Kähler metric on Asusual we adopt the following notations: A u = (d ia)u ; d is the exterior derivative Hence the unitary property simply means d < u,v>=< A u,v>+<u, A v> for sections u, v of L The curvature of A is F = da Thus (I) is the usual Yang-Mills-Higgs functional for this special case In local coordinates (x, x ) on, we write k A = A( )= x k k ia k, k =,, and F kj = k A j j A k = i( k A j A j A k A ) Then the Euler- Lagrange equations for E are A u = u( u ) k F kj = I ( j ia j )u,u (I), where A = k A k A, and where we employ the usual summation convention E has two important properties The first one is called the gauge invariance, ie the value of E is invariant under the gauge transformation (u, A) (u exp(iψ), A + dψ), for a real valued function ψ The second important feature of E is the self duality Namely, decomposing A into its (, 0) and (0, ) parts, A = A + A, in case = R and if u(x), A u(x) 0 sufficiently fast as x,thenecan be written as [ E(u, A) = A u + F R ( u ) ] dx (I3) + πd for some integer d, the so-called vortex number (see [9], page 54) Thus we see that the infimum for E, namely πd, is attained if and only if the vortex equations A u = 0 F = (I4) ( u ) are satisfied Of course, since E is non negative, this is possible only if d 0 (if d < 0, one should consider antiholomorphic sections instead of holomorphic ones) Taubes [33] showed that for any collection of N points x j R with

3 Ginzburg-Landau equations 39 multiplicities N j, there is a solution, unique up to gauge equivalence, of the vortex equations with u(x j ) = 0, j =,, N The situation for a compact Riemannian surface is the same One can rewrite E as [ E(u, A, ) = A u + F ( u ) ] dx (I5) + π deg L, where deg L is the degree of L and denotes the contraction with the Kähler form of Thus the infimum π deg L is achieved by the solution of the vortex equations A u = 0 F = (I6) ( u ) We refer to the works by Bradlow and Garcia-Pradu for the detailed analysis on (I6) ([6], [7], [3], [4]) I The scaling effect On R, for the functional E(u, A) = [ da + A u + 4 ] ( u ) dx, R one can easily introduce the scaling dimensions for u and A in such a way that the term R A u dx is scaling invariant Thus we put u to be of dimension 0, A to be of dimension andso A uis of dimension The scaled functional is [ E(u, A) = da + A u + ] ) dx, (I7) 4 ( u R 0 << It still is self-dual and gauge invariant The Euler-Lagrange equations for (I7) are A u = ) u( u k F kj = I ( j ia j )u,u (I8)

4 40 Fanghua Lin, Tristan Rivière Again, the vortex equations on R are A u = 0 F = ( u ) (I9) On the general Riemannian surface the second equation becomes F = ( u ) (I0) Note that a necessary condition for solving (I0) on is π deg L < Vol (I) The latter will obviously be true when is sufficiently small In [7], Hong-Jost-Struwe studied the asymptotic behavior of minimal solutions of A u = 0 F = (I) ( u ) on a compact Riemannian surface They showed that, for a fixed d = deg L 0, and for some sequence n 0, there are points x j, j =,, l d, such that u, A u 0, da 0 uniformly on compact subsets of \{x,, x l } Moreover, for h = da, one has h π lj = δ x j in the sense of measures, where delta functions have to be counted with multiplicity This yields a method for degenerating a line bundle L on of degree d into a flat line bundle with d singularities (counted with multiplicity) and a covariantly constant section The above described result is a two-dimensional analogue of works by Taubes ([34], [35]) on the Seiberg-Witten equations Taubes used them to relate the Seiberg-Witten and Gromov invariants in four dimensional geometry through a similar change of scales I3 Superconductivity In the theory of superconductivity, particularly for those high T c superconductors, the coupling constant (or the Ginzburg-Landau parameter) is often very large Hence instead of (I7), one has to look at variational integrals: [ E (u, A, ) = da + A u + ] ) dx, (I3) 4 ( u

5 Ginzburg-Landau equations 4 for 0 <<< The energy functional (I3) is, though gauge invariant, no longer self-dual in the sense we discussed before Thus the analysis has to be done on this variational integral and its corresponding second order Euler-Lagrange equations instead of the first order vortex equations When is a two-dimensional domain, and if one ignores the effect of a magnetic field, ie the connection A, then it suffices to study the following model problem: [ min u + ] ( u ) dx (I4) The natural boundary condition for (I4) is the standard Dirichlet boundary condition u = g (I5) Here u is a complex-valued function and g : S is a smooth unit vector field of degree d In [4], Bethuel-Brezis-Helein systematically analysed the problem (I4)- (I5) Then, by taking subsequences if necessary, one has i) ii) in u n (x) u (x) = d j= x a j x a j exp(ih a(x)) C,α loc ( \{a,, a d }), h a = 0 in u = g on ; [ u n + ] ( u n ) dx = πd log + min W(b, g, )+o n b d n () Here W(, g, )is a function defined on d which is called the renormalized energy; iii) a (a,, a d ) is a global minimum of W(, g, );

6 4 Fanghua Lin, Tristan Rivière iv) ( u n ) n u n π log π d j= d δ a j, in the sense of Radon measures We remark that the above statements were shown in [4] under the additional assumption that is star-shaped The key conclusion following from this assumption is the estimate ( u ) dx C(g, ) (I6) Using the approach by Struwe [3], one can drop this additional assumption Indeed the estimate (I6) also follows from [3] Later in [9] an elegant approach showed also this estimate without using the star-shaped property for It turns out, from the point of view of analysis, the variational problem (I3) is a small perturbation of the problem (I4) Indeed, in [5], Bethuel-Rivière established corresponding results to the ones in [4] for the minimization problem associated with (I3) with a suitable boundary condition by using similar analytical arguments See also [0] and [9] for results under a more physical boundary condition and an external applied magnetic field j= δ a j I4 Ginzburg-Landau equations in high dimensions The purpose of this article is to study the asymptotic behavior of minimizers of the Ginzburg-Landau functionals (I3) in high dimensions In [5], the second author first studied the problem when the dimension of is three He proved among other results that the minimizers of (I3) converge (by taking a subsequence if needed) away from a one-dimensional length minimizing current Similar to the situations in the two dimensional case, the analysis in [5] suggest that the essential analytical difficulties in studying such problems lie in the following model problem: [ min u + ] ( u ) dx (I7) subject to the Dirichlet boundary condition u = g : S (I8)

7 Ginzburg-Landau equations 43 For this reason we shall therefore discuss only the model problem (I7)- (I8) I4 The Dirichlet boundary condition To further simplify the presentation we will make use of the following assumptions: (A) is a smooth convex domain in R n, n 3; (A) on we prescribed a family of boundary values g : C,for 0, such that (i) d( g g dθ) = S, wheresis a fixed smooth (n 3)-dimensional current with integer multiplicity (ie it can be represented by a (n 3)-dimensional smooth compact submanifold in with integer multiplicity); (ii) g, g (x) ifr c,and k C g (x) max(r,) on, where C and c are a positive constants independent of k and r = dist(x, spt S) From (A) one deduces in particular g C log (I9) and ( g ) C, (I0) where C is a constant independent of In part IV of this paper we will need to strengthen a bit assumption (A) and prescribe a more precise shape of g close to its zero set: we will add a third hypothesis to i) and ii) (iii) There exists r > 0 such that for any x 0 in spt S there exists a diffeomorphism 0 of B r (x 0 ) and a rotation R of R n such that ( x ) g = f 0 R where 0 (x 0 ) = x 0 and ( 0 Id) (x 0 ) = 0 ( ) x + ix f (x,, x n ) = h χ( x + ix ), x + ix where χ is an increasing function satifying χ(0) = 0andχ on [, + ), moreover h is any function from S into S such that k h are uniformly bounded, independently of (A) enforced by iii) is called (A )

8 44 Fanghua Lin, Tristan Rivière I4 The energy density concentration set When dim = n = 3, S = k j= d jδ a j,forsomea j, j =,, k Since is compact, we have k j= d j = 0 It was shown in [5] that, given n tending to zero, from a subsequence u n of minimizers of (I7) with u n = g n on among H g n (, C) one can extract a subsequence which converges in H loc ( \ spt T) to a harmonic map u into S where T is a length minimizing current supported in with T = k j= d jδ a j Suppose such a length minimizing current is unique, then the whole family u,0<<, converges to u as 0 whenever g does In the beginning of Sect III of this paper we shall give an alternative and much simpler proof of a part of the main result in [5] For the general dimensions we have the following result which is the first part of the main result in this paper: Theorem I Suppose the assumptions (A), (A) are valid and that u, 0 << are minimizers of (I7)-(I8) Let µ = e (u ) dx π log = [ u + 4 ( u ) ] dx π log Then, for any sequence n 0, there is a subsequence of {µ n } that converges weakly (as Radon measures) to a Radon measure µ such that spt µ = spt T, µ( ) = M(T) (mass of T) Here T is an area minimizing codimension two current in R n with T = S In the case that such T is unique, the whole family µ, 0 <<, converges to µ as 0 + We should point out that the proof of the above theorem does not use the existence of area minimizing currents T with T = S The proof of the latter fact often needs the compactness theorem of Federer-Fleming for integral currents [] Thus the paper gives an alternative, though not necessary simple, proof of this useful fact At this point, it is interesting to point out that our arguments can be easily adopted to the problem studied in [34] to show the energy concentration set (the collapsing set for harmonic spinors) are two-dimensional areaminimizing surfaces The holomorphic structure proved in [34] comes from the self-duality property of the Seiberg-Witten functionals considered there Though we have studied here a simple minded variational problem, we believe that we have developed here a very general analytical frame-work that can be used in various applications that latter may be more interesting than the main conclusion of the paper We should also point out that the infinite energy concentration sets have to be area minimizing is not particularly surprising from formal analysis It is also naturally suggested by the results on its associated gradient flow []

9 Ginzburg-Landau equations 45 I43 The limiting map We also give a description of the sequence of minimizers itself This uses a somehow different approach as the one developed in part III to prove Theorem I In this approach we assume hypothesis (A ) above but we do not have to assume anymore that is convex (hypothesis (A)) can be any regular bounded domain of R n In particular can be topologically different from S n We are still interested in the ( situation ) where g / g admits no extension in W, (, S ) So either d g g dθ = S = 0 or we can also have g g C (, S ), π ( ) = and there exists at least a generator γ of π ( ) which is contractible in and such that deg( g g ; γ) = 0 Of course one can also have both situations together We will use the following elementary lemma proved in the appendix: Lemma A7 Let be a bounded regular domain in R n,letgbe a regular map from into C( such that ) g ({0}) is a submanifold of Denote by S the current S = d g g dθ and S = spt S Then, there exists a class L in H n (, S, Z) such that, for any current L representing L one has i) g admits a regular extension from \spt L into S ii) for any closed curve γ in \ g ({0}) such that γ = σ where σ is a -cycle in we have deg (g/ g,γ)=σ L This class L is uniquely determined by S and the degree of g on any closed curve in \ g ({0}) In order to simplify the statement of our second main theorem we will make the following assumption on the boundary condition g (A3) The class L H n (, S, Z) defined by g is independent on, moreover L = 0 The following result generalizes to any dimension the result of F Bethuel, H Brezis and F Hélein in [4] in dimension two and the result of the second author in [5] for the dimension three case Theorem I Let be a bounded domain in R n,let n be a sequence tending to zero and g n be a sequence of boundary conditions from into C verifying (A ) and (A3) Ifu n denotes a sequence of minimizers of E n then one can extract a subsequence (still denoted u n ) which converges in Hloc ( \ spt T, C) to an harmonic map u from \ spt T into S,whereT minimizes the area in the class T Moreover d(u dθ) = T Remark I In view of this result the union of the harmonic map u and the minimal current T is the right object which generalizes the harmonic extension of g = lim g from into S when it does not exist

10 46 Fanghua Lin, Tristan Rivière I5 Description of the paper In Sect II we shall establish two important ingredients of our proofs The first is the energy monotonicity property The second is the η-compactness lemma The η-compactness lemma was first shown also in [5] for dim = n = 3, here we generalized it to arbitrary dimension n 3 It is the starting point of our analysis The Sect III is devoted to the proof of Theorem I In the first part of Sect III we restrict to the dimension 3 case and we give a relatively simplified proof of a part of the main result in [5] It also presents the key idea we use in the second part of Sect III to generalize it to high dimensions Here we first analyze the defect measure µ and establish various properties concerning spt µ, such as its density with respect to H n, (n )-dimensional Hausdorff measure, its rectifiability and orientability Then we use energy arguments to show sptµ = spt T, M(T) = µ( ) and T is an area-minimizing current in with T = S In the final section IV we prove Theorem I This part is independent from part III, in particular we give an alternative proof of Theorem I, using the Federer-Fleming Theorem this time The interest of this approach, which is the high-d version of the approach in [4] for d =, and [5] for d = 3, is that at the same time it gives the convergence of u away from spt µ which could not be deduced directly from the approach in the previous section II Fundamental lemmas II Bounding the energy density II Basic estimates Suppose u,0<<, are minimizers of E () over Hg (, C), then using the maximum principle one has Lemma II [3] u L ( ) (II) Using a Gagliardo-Nirenberg type interpolation inequality, Lemma II and the Euler-Lagrange equation one has Lemma II [3] There exists C > 0 depending only on and the constants in hypothesis (A) such that u L ( ) C (II)

11 Ginzburg-Landau equations 47 Finally a comparison construction yields the following (see also Lemma III): Lemma II3 Let T be a current representing T,then E (u ) π(m(t)+δ) log, for any δ>0and for all sufficiently small >0 Note this estimate is particularly simple when n = 3 II The monotonicity formula Lemma II4 (Monotonicity formula) The following identity holds = r d [ dr r n [ r n 3 B r u + B r u ν n ( u ) ] (n ) + ( ) ] u n (II3) The Euler Lagrange equation for u is u = u ( u ) (II4) We multiply u by the Pohozaev quantity n by parts on B r Weget n u u x i B r x i = r i= B r u ν + u + B r B r i= x i u x i n i,k= and we integrate u u x i x k x k x i (II5) Integrating by parts on i the last integral of the right-hand side of (II5) we obtain n B r i,k= x i u x k u x k x i = B r r u n B r u (II6)

12 48 Fanghua Lin, Tristan Rivière Multiplying the right-hand side of the Euler equation (II4) by the Pohozaev quantity and integrating by parts we obtain u ( ) n u u x B r i = ( ) u r x i= i 4 B r + n ( ) (II7) u 4 Combining (II4)(II7) we get ( ) u + n 4 4 = r B r B r r u ν + r B r B r u n B r ( u ) B r u (II8) We multiply this identity by r n and we get the desired result We also need a boundary version of the energy monotonicity formula Consider a part of of the form B r (x 0 ) where x 0 Assume that r is sufficiently small Assume 0 B r (x 0 ) We can parameterize B r (x 0 ) in the following way B r (x 0 ) ={x B (x 0 ): x n =ψ(x, x n )} such that ψ(0) = ψ(0) = 0andlet ψ C δ 0 Denote by d = dist(x 0,spt S) and r := B r (x 0 ) Let u be a minimizer of E () on r We have the following boundary version of the energy monotonicity formula Lemma II5 (Boundary energy monotonicity) With above notations one has, for r (0, r ) and any 0 <α<, that [ d {e rα r n+ u + dr r r n+ +r n+ B r (x 0 ) B r (x 0 ) u ρ + n (n ) ( u ) ( u ) xν u ν dx C r d e rα ] } dx (II9) Here and C are constants depending only on the upper bound of the ratio r α /d, onδ 0 and the constants in the hypothesis (A), but not on r or

13 Ginzburg-Landau equations 49 Proof Assume first for the simplicity of the presentation that r = B r (x 0 ) is star-shaped around x 0, this implies in particular (x x 0 )ν 0 for any x B r (x 0 ) Multiply the equation u + u( u )=0by the Pohozaev multiplier i (x i xi 0) u x i Integrating on r like in the proof of the interior monotonicity formula one gets d dr [ = r r n u n ( + u ) ] r (n ) [ r n 3 B r u ν + (x x 0 )ν r n B r + r n + ( u ) ] n u ν (x x 0 )τ u B r ν g τ (x x 0 )ν g r n B r, (II0) where B r means B r (x 0 ) and where τ x x 0 is the orthogonal projection of x x 0 on the tangent plane of at x Of course, the worst term to deal with is the last one In order to bound it one uses an idea from [8] One can always find an extension g of g in r such that g C/dand g C/d The idea is to multiply the equation satisfied by u by i (x i xi 0) g integrate it on r This yields B r (x x 0 )τ u ν g τ = + r (x x 0 )ν g r ν u ν ( u x k x k u ( u ) r (x i x 0 i ) g x i ) (x i x 0 i ) g x i The first term of the right-hand side of (II) is bounded by (x x 0 )ν g r ν u ν (x x 0 )ν u d r ν C δ r n d + δ (x x 0 )ν r u ν x i and to (II) (II)

14 50 Fanghua Lin, Tristan Rivière The second term of the right-hand side of (II) can be bounded in the following way ( u (x i xi 0 r x k x ) g ) C u C k x i d r rn α +r α u d r (II3) Finally for the last term of the right-hand side of (II) we write u ( u ) (x r i x 0 ) g Cr u (II4) x i d r Now the difficulty is to handle the term u r dx Here we should point out that u To estimate u r dx, we use the same trick as in [8], ( we multiply the equation u + u u ) =0byφ ( u ) u Here φ(t) is a smooth positive function of t 0 such that φ(0) = 0, φ(t) = for t,andφ (t) 0 Recall that g (x) ifdist(x,spt S) After integration by parts, we obtain ( u r u ) dx u dx + u u r B r (0) ρ + C rn d Therefore ( u r ( u dx r ) u + r u ) +C rn d e (u)+ r B r (0) u ρ + C rn d (II5) combining (II), (II4) and (II5) we get u ( u ) g x r i C r α x i d rα E (u)+ C r n δ d +δ xν u B r ν (II6) Combining now (II0), (II), (II), (II3) and (II6) we get, for δ = /8 Y + α r Y [ u α r r n 3 ν + ( u ) ] n + r n B r xν u ν C r d, (II7)

15 Ginzburg-Landau equations 5 where Y = r n+ r [ u + n ( u ) ] dx (n ) Multiplying (II7) by exp( r α ) we get (II9) We do not assume anymore that r is star-shaped We have to study the perturbation terms induced by omitting this assumption We claim that x B r (x 0 ) (x x 0 )ν (x x 0 )ν cr, (II8) where c is independent of r or x 0 in Wehave ( ) n ψ ν= e (+ ψ ) / n e i x i Since ψ(0) = 0wehave ψ (x) Cr and in order to prove (II8) it suffices to prove i= x B r (x 0 ) (x x 0 )e n cr (II9) One can notice that it suffices to prove the previous identity with y 0 instead of x 0,wherey 0 is the projection of x 0 on along e n (ie y 0 = x 0 + λe n, where λ 0) Since ψ(0) = 0wehave ψ(x) ψ(y 0 ) Cr and this implies (x y 0 )e n cr From this identity and the discussion above we deduce (II8) So, without the star-shapedness assumption for r, instead of (II7) we get Y + α r α Y r [ r n 3 B r u ν + xν u r n B r ν C r n 3 B r C r n 3 + ( u ) ] n u ν C r d (II0) u B r Observe that we have C ν r n 3 C, this is the reason why this term is not r n So we have to bound the term in front of the integral and not so bad For r sufficiently small compared to the C -norm of one can ensure that r is always star-shaped for r r around some point z 0 for

16 5 Fanghua Lin, Tristan Rivière which one has (x z 0 )ν cr, wherecis independent of r or x 0 One can apply all the previous Pohozaev arguments on r but around z 0 Using similar estimates as above one deduces that u ] r n 3 ν C [Y + r n+3 (r n Y) + r (II) d Inserting this estimate in (II0) one gets a similar estimate as (II7) and we can conclude in the same way II3 A uniform bound of the energy density In Part IV we will need the following bound for the density of energy Lemma II6 For any x 0 in the following bound holds r n u + B r (x 0 ) ( u ) Clog where C is independent on, r and x 0, (II) Proof It is clear from the global upper bound of the energy given by Lemma II3 and from the monotonicity formulas (Lemma II4 and Lemma II5) that (II) holds for any x 0 K,whereKis a compact set included in \ spt S and for any r > 0 but the constant a priori could depend on K Let us take x 0 spt S and prove that (II) holds for a C independent of r, x 0 spt S and We use the notations of the proof of Lemma II5, for instance Y still denotes the density of energy on r = B r : Y = E r n (u)(b r ) (II0) implies, using (II9) (in fact since x 0 similar arguments as the ones used to prove (II9) give also (x x 0 )ν Cr ) Y C r n 3 + r n B r u ν C r n 3 (x x 0 )τ u B r ν g τ B r g, (II3) where τ x x 0 is the orthogonal projection of x x 0 on the tangent plane of at x First of all we have g C r n 3 B r r n 3 B r max(dist(x, spt S), ) C log (II4) We have also the following a-priori bound [ u r n ν C Y + Y r + log ] (II5) r B r (x 0 )

17 Ginzburg-Landau equations 53 Indeed (II5) is established in the following way: take a point z 0 in B r (x 0 ) around which B r (x 0 ) is star-shaped with (x z 0 )ν cr where c is some universal constant This is always possible for r small compared to the C -norm of Let us apply the Pohozaev formula in B r (x 0 ) around z 0 This easily implies r n B r (x 0 ) u ν C e r n (u) + C e B r (x 0 ) r n (u) B r (x 0 ) + C r n B r (x 0 ) g + g u ν (II6) Observe that we have from hypothesis (A ) g C log r n B r (x 0 ) r (II7) Thus combining (II6) and (II7) one gets (II5) Observe now that from hypothesis (A ) on g one deduces that (x x 0 )τ g ( ) τ f x i 0 + Cr g, x i (II8) ( ) ( ) x + ix x + ix where f (x) denotes f (x) = h χ, (recall that x + ix from (A f ) we have spt (χ ) [0,]) Thus x i x i 0 has a support in S ={x ; dist(x; spt S) } and is bounded by Cr/ Combining this fact with (II8) we get (x x 0 )τ u r n B r ν g τ C r n S B r + C r n 3 B r g u ν (II9) Using the fact that S B r C r n 3, (II3), (II4), (II5) and (II9) we finally obtain Y Clog CY C r (II30) This differential inequality integrated between and any r gives the result for x 0 spt S (for r (II) is a direct consequence of the L bounds u and u C, see Lemma II and Lemma II)

18 54 Fanghua Lin, Tristan Rivière Now take any point x 0 and any r > 0 Let d = dist(x 0 ; spt S) If r>d/, let z 0 spt S such that d = x 0 z 0 Since we have proven the lemma for any point on spt S we have r n e (u) B r (x 0 ) r n B 3r (z 0 ) e (u) C log and the lemma is proven in this case So we just have to consider the case where r < d/ and we can also assume that x 0 Indeed, if the lemma is proven for the point on the boundary, for any point x 0 one has the estimate (II) for it s projection on We use it for r dist(x 0, ),thisgives the estimate for x 0 and any r > dist (x 0, )/ and the estimate (II) between and dist (x 0, )/ is just a consequence of the interior monotonicity formula Thus we have x 0 and r < d/ On B r (x 0 ) we have g C Thus the Pohozaev identity implies d Y C r n 3 B r u ν C r d C r n d B r u ν (II3) Using (II5) we bound the last term of the right-hand side of (II3) in the following way C r n d B r u ν C r ( d Y + C Y r + log r ) (II3) Combining (II3) and (II3) we get [ Y C Y+ ( log ) r + d d ( Y + C Y ) ] r (II33) So at any point d/ s r one of these 4 possibilities occur Y (s) CY(s) Y (s) C d Y (s) C s d ( log ) Y (s) C d Y (II34) Integrating all these possibilities between r and d/ one get Y(r) CY(d)+ C(log ) Since (II) holds for d/, this implies (II) for r and Lemma II6 is proven

19 Ginzburg-Landau equations 55 II The eta-compactness Lemma This part of our work is devoted to the proof of one of the main properties we use for solutions of the complex Ginzburg-Landau functional: the eta-compactness property This roughly says that if the energy in a ball is sufficiently small then the density of the order parameter u cannot approach 0 on the ball of half radius and if this remains true as the coupling constant tends to infinity we will have compactness on this ball This property is reminiscent of the ɛ-regularity lemma proved by R Schoen and K Uhlenbeck for the minimizing harmonic map (see [8]) The eta-compactness Lemma was proved in the 3-dimensional case for minimizers in [5], it can also be used for the study of similar loss of compactness for the minimizing sequence of the gauge invariant Ginzburg-Landau functional in dimension (see [6]) Here we give a proof of this eta-compactness property in any dimension and for critical points in general This proof follows step by step the one in [5] except at the end, where the comparison argument using the minimality of the solution is replaced by a more refined one requiring only the fact that we have a critical point of the Ginzburg-Landau functional Let be a domain in R n for n > Lemma II7 (eta-compactness) Let u be a critical point of the Ginzburg- Landau functional satisfying u and u C/ where C is independent of, then there exists η, λ and o such that for any < o and for any ball B ρ (x 0 ) where ρ λ, ρ n E (u) ( B ρ (x 0 ) ) η log ρ u (x 0 ) Proof of the η-compactness lemma We introduce the following notations E r = u n ( + ) u, B r (n ) I r = u n ( + u ) de r = B r (n ) dr F r = u B r ν + ( u ), n J r = u ν + ( ) u df r = n dr B r,

20 56 Fanghua Lin, Tristan Rivière Using these notations, Lemma II4 becomes [ ] d Er = df r dr r n r n dr (II35) The hypothesis implies in particular ρ [ ] r r J n 3 r η log ρ Integrating by parts ρ [ r F ρ + (n ) ρn ] ρ J r n 3 r = ρ r [ r n F r df r r n dr we obtain ] η log ρ + F n Using the fact that u C/ and u we obtain F = u n B n ν + ( ) u C n n Thus, if η log λ C (ie λ exp( C η )), since ρ λ, wehavef / n η log ρ/ and finally we get ρ [ r r F n r + ] r J n 3 r Cη log ρ Using the mean value formula we deduce the existence of r [, ρ] such that r n 3 J r + ( r ) n 3 J r + F r r n C η (II36) We make the following change of scale r andu ũ Thus ũ is a minimizer of ( ũ r ) ( + ) ũ B Using Lemma A6 and the fact that ũ is parallel to ũ we have in T = B \ B ( ũ ũ ) [( = ( )d i ũ )] r r ; ( )d ũ + r ( ũ ũ ) + n ( ũ ũ ) r r r r, (II37) where ( ) and d respectively denote the Hodge operator and the external differentiation on B r and (a; b) is the scalar product between two complex

21 Ginzburg-Landau equations 57 numbers a and b Let 0 be the inverse of the Laplace Beltrami operator on n -form in T for the Dirichlet boundary conditions (v T = 0and v T =0) and let v be the following n -forms in T v = 0 ( i ũ r ;( )d ũ ) (II38) Denote also by 0 the inverse operator of the Laplace operator on functions for Dirichlet boundary conditions and by H the following function on T H = 0 (( )d v ( )d v) (II39) We claim that < p < + we have H p C p T v p T (II40) Indeed, let ω S n be the volume form on S r = B r such that dr = ω S n We have ( )d v = d v; ω S n Write v = v ik dx d ˇx i d ˇx k dx n i<k We have dv = for i > k) Thus n k= i =n v ik i ( )i dx d ˇx k dx n (where v ik := v ki ( )d v = ( dv; ω ) = i =k x k r v ik x i = i =k x k r v ik x i + ( derivatives of v of order ) = d v; ω + ( derivatives of v of order ) This proves (II40) Denote by K the following function in T ( ( K = 0 ũ ũ ) + n ũ ũ ) r r r r r (II4) Thus we have ( ũ ) r ( )d v + H K = 0 in T ũ ũ r ( )d v+ H K =ũ ũ r on T

22 58 Fanghua Lin, Tristan Rivière Let ξ =ũ ũ r ( )d v+h Kin T, using standard results on harmonic functions we have, for any domain ω T, ξ C(ω) ξ C(ω) ũ ω T r (II4) T Choose ω = B 7/8 \ B 5/8 Inωwe have ( ũ ũ ) = ξ r r r + ( ) r d v + H r K r = ξ r + ( )d ( ) v r r ( )d v + H r K r (II43) n Let < q <, using standard elliptic estimates and the mean value n formula, we deduce from (II38), (II40), (II4), (II4) and (II43) that there exists t (5/8, 7/8) such that ( ) ( Bt ξ ũ ), T r ( v q) ( q ) ( C Bt r T ũ ũ ), T r ( K ) ( C ũ ), Bt r T r ( H q) ( q ) ( C Bt r T ũ ũ ) (II44), T r ( ) ( Bt ũ C T ũ ), ( ũ + ( r ) ( ) Bt ũ ) ( C ũ + ( r ) ( ) T ũ ) r Combining the previous inequalities, (II4) and (II43) we obtain r ( ũ ũ ) ( ) ( W C ũ r,q ( B t ) T T + C ( T ũ r r ) + C ( ũ r T ) ũ r ) (II45) Using the fact that ũ ũ = 0 we deduce that d ( ) (ũ d ũ) =ũ r ũ=+ ( ũ ũ ) r r + n ũ ũ r r,(ii46)

23 Ginzburg-Landau equations 59 where r denotes the Laplace operator on B r and d ( ) the adjoint of the exterior differentiation d for the scalar product induced on B r Using (II36) and (II44) we have ( r ) ( ũ ) Cη B t (II47) ( r ) ũ C and we deduce that {x: ũ(x) < /} is contained in Cη(r /) n 3 balls of radius /r in B t Letω be this union of balls and let ω be the union of the balls having the same centers and radii /r in B t Let a(x) be a positive function on B t satisfying a(x) = ũ in B t \ ω a(x) in ω (II48) and a(x) C r in B t First observe that ( ) d (a(x) ũ d ũ) = d ũ ũ d ũ ( ) (II49) ũ = d ũ d ũ = 0 in B t \ω ũ Let denote the Hodge operator on forms on B t admits an inverse on - or -forms in B t (for n 4) If n = 3 we restrict ourselves to exact -forms We have a(x)ũ d ũ = d ( ) (d (a(x)ũ d ũ)) + d α on B t, (II50) where α(x) is the function equal to d ( ) (a(x)ũ d ũ) Let K(x, y) = i= λ i ψ i (x) ψi (y) be the kernel of on d ( B t ) ( ) K(x, y) π B t π,whereπ B t (x,y) xand π (x, y) y, where λ i are the eigenvalues of, ψ i the corresponding eigenforms and, for v y B t ψ i (y);v := ψ i (y)v Standard results on kernels imply K(x, y) C x y n 3 and x K(x, y) C x y n (II5)

24 60 Fanghua Lin, Tristan Rivière (Recall dim B t = n ) In view of (II49) we have (d (a(x)ũ d ũ)) = K(x, y)d (a(y)ũ d ũ) (y) B t = K(x, y)d (a(y)ũ d ũ) (y) ω (II5) Using (II48), (II5) and the fact that ũ Cr /,weget d ( ) C (d (a(x)ũ d ũ)) (x) ω x y r (II53) n Letting < p < n n,wehave [ ] p d ( ) (d (a(x)ũ d ũ)) p C B t B t ω x y r n (II54) C p ( r ) p ω p, where we have used Hölder + Fubini and the fact that p(n ) < n =dim B t Combining (II47) and (II54), we have, for any < p < n n, d ( ) (d (a(x)ũ d ũ)) C p η (II55) L p On the other hand, Combining (II43), (II46) and (II50), we have t α = d ( ) (ũ d ũ) + d ( ) ((a )ũ d ũ) = ( )d ( v r ) ξ r + r ( )d v H r + K r n ũ ũ r r + d( ) ((a )ũ d ũ) Letting < q < n n,wehave ( ) q ( (a ) q ũ d ũ q a q q d ũ B t B t B t ( ( C ) ) q ( ) q ũ + ω ũ B t T C η q ( ) q ( r T ũ ) q (II56) )q (II57)

25 Ginzburg-Landau equations 6 Finally, combining (II36), (II44), (II45), (II56) and (II57) we get, for any < q < n n α L q C q η ( ) ũ + Cq η (II58) T Thus, (II50), (II55) and (II58) imply, for any < q < ( If we take B t ũ d ũ q n n, ) q ( ) / Cq η ũ + C q η (II59) T n p <, choose any q < n n ( ) p ( ũ d ũ p B t C q η γ B t ũ d ũ q ( T ũ n,wehave )γ q ( ) + Cq η γ B t ũ d ũ ) γ (II60) where γ q + γ = p Using the mean value formula simultaneously for slices at the same time one can ensure inequality (II60) holds for B t and B 7t/8 in the same time and we have, denoting T t = B t \ B 7t/8 ( T t ũ dũ p ) p Cq η γ ( T ũ ) + Cq η γ (II6) Let a(x) be the function equal to in {x ; u(x) >/}and equal to 4 ũ otherwise and let ω be the set where a(x) = 4 The form a(x)ũ dũ is the solution of h = dd (a(x)ũ dũ) + d d(a(x)ũ dũ) (II6) h = a(x)ũ dũ on T t Observe that the boundary condition means that both normal and tangential components of the forms h and ũ dũ coincide on T t h = h 0 + h + h, where h 0 = 0 in T t h = a(x)ũ dũ on T t

26 6 Fanghua Lin, Tristan Rivière h = d d(a(x)ũ dũ) in T t h = 0 on T t h = dd (a(x)ũ dũ) h = 0 on T t in T t In view of (II6) we have for p (n )/n h 0 L C h 0 W p,p (Tt ) C a(x)ũ dũ L p ( T t ) (II63) Moreover we have d(a(x)ũ dũ) = 4d(χ ũ dũ) where χ is the characteristic function of ω Thus h L C χ ũ dũ L C ω ( T ũ 4 ) ( ) ( r C ( ũ ) ) ( T ( ) ( r C ( ũ ) ) ( ũ T T T ( ) ) ũ 4 r ) (II64) ( ) Cη ũ T Using the fact that ũ ũ=0, we write d (a(x)ũ dũ) = d ((a(x) )ũ dũ) and we get h L C (a(x) )ũ dũ L ( ) ( ) C u L + ω ũ 4 T

27 Ginzburg-Landau equations 63 ( ) ( r C ( ũ ) ) ( T T ( ) ) ũ 4 r ( ) ( r C ( ũ ) ) ( ũ T T ( ) Cη ũ T Thus combining (II63), (II64) and (II65) we get ( ) ũ dũ Cη γ ũ + η ũ + Cη γ T t T T ) (II65) (II66) We can always find a good slice τ between t and 7t such that 8 ( ũ r ) ( + ũ Cη γ ũ + Cη γ B τ T (II67) The monotonicity formula implies (τr ) n B τr e (u) C n (τr ) n 3 B τr e (u) Writing (II67) in the usual scale, using (II44) and the estimate above we E have either r η γ, which permits to conclude the lemma as it is r n explained below, (γ can be so close to zero as we want) or we have E τr E Cηγ (τr ) n r r n + η γ where τ / Thus, because of the monotonicity formula we have Using (II35) we have E r r n E r r ( r ) n r / E r ( r ) n Cη γ E r r n + η γ (II68) r n J rdr C r J r n r dr C F r r / r n and since we have chosen r such that F r /r n Cη (see (II36)), we get

28 64 Fanghua Lin, Tristan Rivière E r r n Cη γ E r r n + η γ + Cη Thus, for η sufficiently small, this implies E r is small, and in particular, r n because of the monotonicity formula, E / n is small Precisely we have ( ) u g(η), n B where g(η) 0asη 0 Since u C/, forηsufficiently small, we necessarily have u (x 0 ) We need a version of the η compactness on the boundary u is a critical point of E verifying u = g on the boundary Assume that g verifies (A), thus we have Lemma II8 (eta-compactness lemma at the boundary) For any 0 <α<, there are positive constants η, λ, ρ, 0 depending only on and the constants in condition (A), such that, for any < 0, for any x 0 \spt S and for any ρ verifying min(ρ, d / α ) ρ λ where d = dist(x 0,spt S) one has [ u ρ n + ( u ) ] η log ρ B ρ (x 0 ) u (x 0 ) Proof of the eta-compactness lemma at the boundary We adapt the proof of Lemma II7 to our present situation Denote by ξ the ratio ρ ξ = dist(x 0, spt S) ρ α ρα ρ α d Observe that, in order to verify the hypothesis of the theorem, ξ has to tend to zero as ρ is taken smaller and smaller So one should think about ξ as something small since ρ (like η) will be chosen to be sufficiently small at the end of the proof From the hypothesis of the lemma ρ α is bounded by so d all the constants in the monotonicity formula at the boundary (Lemma II5) are bounded Using this Lemma we deduce, like in the proof of (II36), the existence of r [, ρ] such that u ν + ( u ) C(η + ξ), r n 3 ( B r B r /)

29 Ginzburg-Landau equations 65 r n 3 r n B r (x 0 ) B r u ν x r ν u ν dx C (η + ξ), + ( u ) C(η + ξ) (II69) We have only to deal with the case where dist(x 0, ) is so small as we want compared to r Indeed if dist(x 0, ) Cr one can apply once again the mean value formula in order to get a possibly smaller r satisfying (II69) and such that B r (x 0 ) and the remaining part of the proof is identical to the proof of Lemma II7 in order to get u(x 0 ) / Since we only consider the case dist(x 0, ) << r and since we can take ρ so small as we want compared to the C -norm of we can be so close as we want to the situation where ( /r ) B n(x0 ) is B n (0) the unit ball of R n once we have made the change of scale r This means that the constants (Sobolev constants, constants for the Dirichlet Problem for the Laplace Beltrami operator on B (x 0 ) /r etc) can be bounded independently of x 0, r and Let ũ(x) = u(r x), g(x) = g(r x) on /r and s = B s (x 0 ) /r ũ is a minimizer of ( ũ r ) + ( ũ ) with ũ = gon /r B (x 0 ) Observe that condition (ii) of (A) implies k g Cξ k on (II70) Denote T + = \ / Working on T + instead of working on T, one can follow similar arguments as the ones used to pass from identity (II37) to identity (II45) in order to decompose ũ (ũ r r ):Let 0 be the inverse of the Laplace Beltrami Operator on T + for n or 0 forms Denote also like in the proof of Lemma II7 ( v = 0 i ũ ) r ;( )d ũ, H= 0 (( )d v ( )d v), ( K = 0 r r ( ũ ũ ) r + n ũ ũ ) r r

30 66 Fanghua Lin, Tristan Rivière Let ζ be the solution of ζ = 0 in T + ζ = 0 on ( B (x 0 ) B / (x 0 )) /r ζ = ( )d v on /r (B (x 0 ) B / (x 0 )) From standard elliptic theory we have ζ q ( )d v q W q,q ( /r (B (x 0 ) B / (x 0 ))) T + T + v q Let ξ be the harmonic extension of ũ ũ r in T + We have clearly on T + ũ ũ r = ξ + ( )d v ζ H + K For x /r denote by e τ (x) the orthogonal projection of (x x 0 )/ x x 0 on the tangent plane to /r at x Wehave ũ ũ r /r B (x 0 ) /r B (x 0 ) r n 3 B r (x 0 ) C(ξ + η), ũ e τ + /r B (x 0 ) g + r n 3 B r (x 0 ) ũ ν ν(x x0 ) x x 0 u x x 0 ν x x 0 ν (II7) where we have used (II69) Since ξ ζ is harmonic there exists a -form γ such that d(ξ ζ) = d γ in T + and γ = dσ where σ T + = 0 Moreover γ verifies γ L q d(ξ ζ) W,q ξ ζ L q C(η + ξ) / (II7) see for instance [8] Thus we have ( ũ ũ ) ( ) v = ι d γ + ( )d r r r r r ( )d v H r + K r,

31 Ginzburg-Landau equations 67 where ι r d γ is the interior product between r and d γ and we have ι r d γ = ( )d γ (II73) Now, like in the proof of Lemma II7, one can find a good slice, a t n (/, ), such that for any q (, n ) (II44) holds (where B t and T are respectively replaced by B t and T + ) and such that also γ q C γ q C(η + ξ) q (II74) B t T + Since ( )d corresponds to tangential derivatives along B t,wehave ( ( ) ũ ũ / W C(η + ξ) r r) u,q ( B t (x 0 ) /r ) T + (II75) + C(η + ξ) Let N = B t (x 0 ) /r OnNwe define a(x) like in (II48) Let α be the function that is the solution of the following problem t α = d ( ) (a(x)ũ d ũ) in N (II76) φ = 0 on N, where t denotes the Laplace Beltrami Operator on p N for any 0 p n From standard results on Hodge decomposition (see for instance [8]), there exists a unique -form β such that a(x)ũ d ũ d α = d ( ) β on N β = dσ on N for some σ Nsatisfying σ N = 0 β is in fact the unique -form satisfying t β = d (a(x)ũ d ũ) d ( ) β N = g g d g g β N = 0 on N in N on β is in fact the minimizer of { } min d ( ) β a(x)ũ d ũ + d α + d β ; s t β N = 0 N (II77) (II78) (II79)

32 68 Fanghua Lin, Tristan Rivière For problems (II76) and (II78) the solutions are given by convolutions with Calderon-Zygmund Kernels and an analysis similar to the one developed for passing from (II50) to (II60) Using (II75) instead of (II45) yields ( B t (x 0 ) /r ũ dũ p ) p Cq (η γ + ξ γ ) ( T + ũ ) + C q (η γ + ξ γ ), (II80) where < q < n n, n n p < andγ is given by γ q + γ = p Now the remaining part of the proof can be established almost identically like the end of the proof of Lemma II7 in order to obtain that ( u ) h(η, ξ), (II8) n B where h(η, ξ) 0asη, ξ 0 And since u C/, forηand ρ sufficiently small, h(η, ξ) is sufficiently small in order to deduce from (II8) that u(x 0 ) / III The energy concentration set as a minimal current III A short proof in dimension 3 We let to be a bounded smooth convex domain in R n and let g : S be a smooth map We consider [ min E (u) min u + ] 4 ( u ) dx 0 <<< (III) for u Hg (, C) {u H (, C) : u = g} Suppose n = and degree of g : S is d 0 Then, one of the principal results proved in [4] can be stated as follows Let [ u + ] µ = ( u ) dx π log, 0 <<<, here u is a minimizer of (III) Then, for any sequence n 0, there is a subsequence (still denoted by { n }) such that µ n µweakly as Radon measures and that d µ = δ a j, (III) j=

33 Ginzburg-Landau equations 69 for some d distinct points a,, a d inside Moreover, the d-tuple point a = (a,, a d ) is a global minimum of the so called renormalized energy W(b, g, ),b d In the case where is a ball there is no topological obstruction for extending a smooth map g : S to a smooth map g : S when n 3 In order to obtain a similar statement as that for dim = n = described above, we want to allow the boundary data g to have some topological non trivial singularities on (see part I43 of the paper and Lemma A7) In [5] the second author considered the case dim = n = 3, and a family of boundary data g,0<<, such that the assumption (A) is valid in particular: ( ) g N d dθ = S = d j δ a g j, (III3) j= for some a,, a N, andd,, d N Z HereC,C are positive constants independent of Since is compact, we must have N j= d j = 0, (dim = 3) We shall now give an alternative proof of the following main result of [5] Theorem III [5] For any sequence n 0,letu [ n be a sequence of min- u n + ( u n ) ] imizers of E n ()Letµ n = e n (u)dx, e π log n n (u)= Then there is a subsequence of µ n (still denoted by µ n ) such that µ n µ as Radon measures Moreover spt µ = spt T, µ( ) = M(T) HereTis a length minimizing current in such that T = S, and M(T) denote the mass of the current T We note that the above formulation of the part of the main result of [5] which concerns the energy concentration set immediately unifies the statements of the results in both -D and 3-D cases Unlike the -D case where the locations of the singularities (the support of µ) is determined by the finite part of the total energy (the so-called renormalized energy, the next term in the energy asymptotic expansions), in 3-D (or high dimensions) case the support of µ is essentially determined by the infinite energy concentrations (the first term in the energy asymptotic expansions) The proof of Theorem III given below is rather different from [5] n Proof of Theorem III Let µ = e (u ) dx π log = [ u + ( u ) ] dx π log

34 70 Fanghua Lin, Tristan Rivière Then µ ( ) M(T)+δ by Lemma II3 Thus for any sequence n 0, we may obtain a subsequence of {µ n } (still denoted by {µ n }) such that µ n µweakly as Radon measures As a consequence of the energy monotonicity Lemma II4, µ(b r(x 0 )) is a monotone nondecreasing function of r,for r r (0,r x 0),x 0,r x 0 = dist(x 0, ) Similarly, by Lemma II5, we see exp( r) µ(b r(x 0 )) is a monotone nondecreasing function of r (0, r r 0 ( )), for any x 0 In particular, (µ, x) = lim r 0 µ(b r (x)) r exists for all x Moreover, (µ, x) is upper semi-continuous in x Let = { x : (µ, x) >0 } Then the general results in the Sect III bellow show that is, in particular, a H rectifiable set But here we will not need this fact Next we assume C = C + ic is a regular value of the map u : C such that 6 < C + C < 4 Let Ɣ = u {C } Suppose, for the moment, that all d j =±, j =,, N ThereN=l, and from the properties ( of g one may obtain k embedded C -curves Ɣ j, j =,, k, such that k ) j= Ɣ j = N j= d jδ a j Blaschke s theorem implies that (by taking subsequences if needed) Ɣ j n Ɣ j as n 0, for j =,, k, in the Hausdorff metric (see [] page 83) Each Ɣ j is a connected, compact subset of Applying the eta-compactness lemmas (Lemma II7 and Lemma II8), we see that (µ, x) η 0 /forallx Ɣ j ( \spt S), j =,, k Indeed, if (µ, x) η 0 /forsomex Ɣ j and x / spt S, then µ(b r(x)) η r 0 / for all sufficiently small r We may assume r is so small that B r (x) spt S = Now for sufficiently small, andsmallr,wemaylet =/r, and v (y) = u (ry + x),thenµ (B (0)) η 0 / Here µ = e (v ) dx π log Note that g (y) = g (ry + x) will satisfy the assumption of Lemma II5 From the latter fact we would conclude that u (x) This contradicts to the fact x Ɣ j,forsome j Therefore k H (Ɣ j ) µ( ) M(T) η 0 η 0 j= the latter estimate implies, in particular, that each Ɣ j is H -rectifiable: indeed a connected -dimensional set of finite H measure is rectifiable (cf [], Theorem )

35 Ginzburg-Landau equations 7 We want to show that (µ, x) forh -ae x k j= Ɣj Thenwe can easily deduce that k j= H (Ɣ j ) M(T) Since T is a multiplicity length minimizing current, and since ( k j= Ɣ j ) = T, we see each Ɣ j must be a line segment since is convex Moreover, µ( ) = M(T), andµ=h ( k j= Ɣj )Thisisthe conclusion of Theorem III Nowwehavetoshow (µ, x), for H -ae x Ɣ j, j =,, klet us give a proof of this fact for x in the interior of Ifx Ɣ j ( \ spt S), the arguments should be modified slightly Since (µ, x) is approximately continuous H -ae x Ɣ j,weget,for H -ae x 0, (µ, x) is H -approximate continuous (as a function defined on Ɣ j )atx 0,andthatƔ j has a unique tangent line at x 0 For simplicity we assume x 0 = 0 and the tangent line at x 0 is the z-axis Let η λ : x x/λ, forλ>0, and µ λ (A) = λ µ(η λ A), for any Borel measurable set A R 3 By the monotonicity formula, we conclude that there is a sequence λ m 0, m such that µ λm νweakly as Radon measures ν is a tangent measure of µ at 0 (cf [30]) Moreover, ν(b r(0)) (ν, 0) r (µ, x 0 ) On the other hand the measure µ = (µ, x)h Ɣ j has a unique tangent measure ν at x 0, ν = (µ, x)h {z axis} Note that µ µ ν ν We apply monotonicity formula again to obtain ν ν Inother words, µ λ ν = (µ, x)h {z axis} as λ 0 Note that we have (µ, 0) η>0 Let C δ = Bδ (0) [,]for >δ>0 Then we may find a suitable δ (0, ) such that and hence ν ( B δ (0) [,]) =0 µ λ ( Bδ (0) [,]) 0asλ 0 (III4) We note that µ n,λ = [ ] v + λ ( v ) n log dx, n here v(x) = u n (λx) It is then clear from the eta-compactness lemma that for ae z [,+],deg(v, Bδ (0) {z}) = dis well defined and is independent of z [,] Indeed (III4) and eta-compactness lemma imply that v on B δ (0) [,]

36 7 Fanghua Lin, Tristan Rivière We claim d = 0 Indeed, if d = 0, we first choose t [ δ, δ], t [ +δ, + δ], such that ] [ v + λ ( v ) S δ (t i ) B δ (0) {t j} n log n << η for j =, Where S δ (t j ) = Bδ (0) {t j} Since we are in dimension, we may construct a new map ṽ on Bδ (0) {t j}such that [ T ṽ + λ ( ṽ ) ] dx << η log n n We now define ṽ as follows: ṽ = v on C δ ṽ be as above on Bδ (0) {t j}, j =,, ṽ minimizes [ u + λ Bδ (0) [t,] n [ u + λ Bδ (0) [,t ] n and B δ (0) [t,t ] ( u ) ] dx ( u ) ] dx [ u + λ ( u ) ] dx n subject to corresponding Dirichlet boundary conditions By the proof of energy-monotonicity formula, we see the first two integrals are bounded by ηcδ log, the last integral is bounded by o(η)cδ log Here o(η) << η By choosing δ suitably small, we can be sure that [ ṽ + λ ( ṽ ) ] dx < C δ n η log n On the other hand µ n,λ(c δ ) 3 η for n large and λ suitably small This contradicts the energy minimizing property of v Thus we can assume d = 0 Then [4] implies that [ T v + λ ( v ) ] dx π d log λ ( ( )) o log Bδ (0) {z} n n n whenever [ ( S δ (z) v + λ v ) ] dx = o(log n n ), and deg(v, S δ (z)) = d Noteλ>0isfixedhere

37 Ginzburg-Landau equations 73 Indeed, if we apply arguments in [] or [7], then, since any (x, y) with v(x, y, z) / is contained in Bδ/ (0), and T v Cλ/ n,then deg(v, S δ (z)) = d implies B δ (0) {z} [ T v + λ n We thus conclude that (µ, 0) d ( v ) ] dx π d log λ K n Remark Let (SuppT) δ be the δ-neighborhood of the support of T in Proposition Then arguments of [] or [7] show that E (u,(suppt) δ ) π M(T) log K Thus u is locally uniformly bounded in Hloc ( \ (Supp T)) and hence u n u in C,α ( \ (SuppT)) as n 0 (cf [3] and []) Now we allow d j to be arbitrary integers such that N j= d j = 0 As before, we have µ n µas Radon measures Let Ɣ = u {C } be as before, and let Ɣ j, j =,, m be m embedded C -curves such that Ɣ j = δ a j δ b j,hereaj,bj, with a j a k b j a l, for some a k, a l, k = l Moreover, m j= Ɣ j = N j= d jδ a j and m N j= d j Apply Blaschke s theorem to each {Ɣ j} so that Ɣ j n Ɣ j in the Hausdorff metric Ɣ j is compact, connected subset of Moreover, by the eta-compactness lemma H (Ɣ j ) η µ( ) < Thus each Ɣ j is rectifiable We may find a lipschitz map f : [0, H (Ɣ j )] into Ɣ j such that f(0) = lim b j n = a l, f(h (Ɣ j )) = lim a j n = a k Moreover, f gives an arc-parameterization of its image Ɣ j = f([0, H (Ɣ j )]) Ɣ j ( Note that k j= Ɣ j k ) Supp(µ), j= Ɣ j = N j= d jδ a j Moreover, the last arguments in the proof of Proposition imply that for H -ae x such that x belongs to d of curves in {Ɣ,, Ɣk },wehave (µ, x) πd Thus k µ( ) π H (Ɣ j ) π M(T) j= The last inequality along with the fact µ( ) π(m(t)+δ),foranyδ>0, implies that µ = k j= H Ɣ j and k j= Ɣ j (with proper orientations) is a length minimizing current with boundary d j δ a j This completes the proof of Theorem III