Erratum. Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents. Fang-Hua Lin Tristan Rivière

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1 J. Eur. Math. Soc., 87 9 c SringerVerlag & EMS Erratum FangHua Lin Tristan Rivière omlex GinzburgLanau equations in high imensions an coimension two area minimizing currents J. Eur. Math. Soc., 7 (999) In our revious aer [] omlex GinzburgLanau equations in high imensions an coimension two area minimizing currents, we stuie the asymtotic behavior of energy minimizing solutions of the Ginzburg Lanau equations. But the comactness Lemma was for arbitrary solutions which may not be energy minimizing. We foun there is a ga in this version of the roof of the comactness Lemma (Lemma II.7). This comactness Lemma as well as asymtotic behavior of arbitrary solutions are treate in our forthcoming aer []. Here we shall simly resent our earlier roof of the comactness for energy minimizing solutions. All the statements in the rest of the aer [] are not affecte by this moification. The arguments from (II.6) to (II.68) have to be moifie in the following way. Starting from (II.6) we say: In fact we will be mainly intereste in such that W H, that is n. We will now exten the ma u into a ma energy as ossible. in all of by using as less We will ecomose into a union of isjoint cubes having eges of length, where will be taen very small (to be fixe later). We will first exten u between an. As it is rove in [] it is ossible to choose this union of cubes such that, if enotes the corresoning seleton for this union of cubes (for n ), we F.H. Lin: ourant Institute of Mathematical Sciences, New Yor University, 5 Mercer Street, New Yor, NY 85, USA T. Rivière: Ecole Normale Suèrieure e achan, entre e Mathèmatiques et e leurs Alications, Unitè associeè au NRS URA6, 6, avenue u rèsient Wilson, F945 achan eex, France

2 88 FangHua Lin, Tristan Rivière have in the same time u K n u n r u K n r u n () u u K n u u n Where K is a constant eening only on n not on nor on. First of all, will be chosen such that u in. ombining (II.47) of [] an () we get " r u r u K n Thus, this is the case if # $ n is sufficiently small (ineenantly of ). Denote by the seleton homothetic to, containe in, for the homothetie of rate t an let % t & be the seleton in the interior of ( having ) as bounary. we will construct the extension of u in ( * on % by inuction on. For + we tae We clearly have x +,u t x x in %. u K u r K r u () / K u u

3 GinzburgLanau equations 89 Before to construct the extension of u in the interior of ( we construct the extension of u in, on the by inuction on. On each cell of the seleton, exten raially from the bounary of the cell, where x +,u t x x, to the center of the cell. One verifies that we have u r 4 5 a. e. in Reeating these extensions on u r u until + n we obtain that u 6 u r r u u u () a. e. in Now we construct in % by inuction on. We have % 4 +8% ) ). Here also, in each cell, we choose to be the raial extension, relative to the center of the cell, from it s value on the bounary. We establish in the same way inequalities lie (), relacing % by % 4 an n 4 by n 5. Reeating this roceure until + n, we get in articular r n9 : 6 : 6 u r u (4)

4 W Y > > 9 FangHua Lin, Tristan Rivière Finally we exten in in the following way. We have a. e. an <; W =. Thus > > ; L *= an since an H R = + Thus + an () we euce that +?, there exists A; W = such that + an 6 B+ in ex id i, where ;FE? G, an from (II.68) of [] W H I 6 KJ q ML T for some fixe NOPN, where + have H I 6 QJ L T R Let be the harmonic extension of in *, we have 6 H I 6 KJ TS n u u q ML. By Sobolev embeing we L T R u #S (5) We tae + ex iu i in. For the moulus of in we choose V+XW, where W is the solution of the following roblem r W/ZW[+ in * WV+ in In [4] we rove that min 6 \W< min 6 6 r [W u ] u an r u (6) (7)

5 GinzburgLanau equations 9 ombining (4), (5), (7) an the minimality of u in we get u r u S T u u #S hoose ^+ n6 (recall that we ha alreay chosen _ n6 ). Using (II.44) of [] an going bac to the usual scale we get, for à+ inf n?"o, E tr tr n #b E r r n a #b where t. Thus, because of the monotonicity formula we have (II.68) of [] an the roof can be ene lie in []. References. F. Bethuel: The aroximation roblem for Sobolev mas between two manifols. Acta Math. 67, 5 6 (99). F.H. Lin, T. Rivière: omlex GinzburgLanau equations in high imensions an coimension two area minimizing currents. JEMS, 7 (999). F.H. Lin, T. Rivière: omlex GinzburgLanau Equations in High imensions an minimal submanifols II. In rearation 4. T. Rivière: Line vortices in the Uc Higgs moel..o..v (996)