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Asymptotic behaviour of holomorphic strips

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PII: S0294-1449(00)00066-4 Ann. I. H. Poincaré – AN 18, 5 (2001) 573–612  2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0294-1449(00)00066-4/FLA ASYMPTOTIC BEHAVIOUR OF HOLOMORPHIC STRIPS Joel W. ROBBIN a, Dietmar A. SALAMON b a University of Wisconsin, Department of Methematics, Madison, WI 53 706, USA b ETH-Zürich, Zürich, Switzerland Received 10 May 2000 ABSTRACT. – The asymptotic behaviour of a finite energy pseudoholomorphic strip with Lagrangian boundary conditions in a symplectic manifold is determined by an eigenfunction of the linearized operator at the (transverse) intersection.  2001 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Le comportement asymptotique d’une bande pseudoholomorphe d’énergie finie, à frontière dans une sous-variété Lagrangienne d’une variété symplectique, est déterminé par une fonction propre du problème linéarisé le long de l’intersection (transverse).  2001 Éditions scientifiques et médicales Elsevier SAS Introduction This paper deals with the asymptotic behaviour of pseudoholomorphic strips in symplectic manifolds that satisfy Lagrangian boundary conditions. More precisely, let (M,ω) be a symplectic manifold and L0,L1 ⊂M be closed (not necessarily compact) Lagrangian submanifolds that intersect transversally. Fix a t-dependent family of almost complex structures Jt on M that are compatible with ω. We consider smooth maps u :R+ i[0,1] →M that satisfy the boundary value problem ∂su+ Jt(u)∂tu= 0, u(R)⊂ L0, u(R+ i)⊂ L1. Such holomorphic strips were studied by Floer [7,8] and he used them in his definition of the Floer homology of Lagrangian intersections. The standard theory of such holomorphic strips shows that if u has finite energy then the limit p = lim s→∞u(s, t) exists and is an intersection point of L0 and L1. E-mail address: [email protected] (J.W. Robbin). 574 J.W. ROBBIN, D.A. SALAMON / Ann. I. H. Poincaré – AN 18 (2001) 573–612 Our main result (Theorem B) asserts that the l

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