© Association des Annales de l'institut Fourier, 2004, tous droits réservés. L'accès aux articles de la revue « Annales de l'institut Fourier » (http://aif.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://aif.cedram.org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ji (1) J2.

No abstract

We prove a version of the reflection principle for pseudoholomorphic disks with boundary on totally real submanifolds in almost-complex manifolds. Furthermore, we give a proof of the Gromov compactness theorem for pseudoholomorphic curves with boundary on immersed totally real submanifolds. As a corollary we show that a complex disk can be attached to any immersed Lagrangian submanifold with only transversal double points in a complex linear space.

Introduction Statement of the main result.Denote by ∆(r) the disk of radius r in C, ∆ := ∆(1), and for 0 < r < 1 denote by A(r, 1) := ∆ \∆(r) an annulus in C. Let ∆ n (r) denote the polydisk of radius r in C n and ∆ n := ∆ n (1). Let X be a compact complex manifold and consider a meromorphic mapping f from the ring domain ∆ n × A(r, 1) into X. In this paper we shall study the following:Question. Suppose we know that for some nonempty open subset U ⊂ ∆ n our map f extends onto U ×∆. What is the maximalÛ ⊃ U such that f extends meromorphically ontoÛ × ∆? This is the so-called Hartogs-type extension problem. IfÛ = ∆ n for any f with values in our X and any initial (nonempty!) U then one says that the Hartogs-type extension theorem holds for meromorphic mappings into this X. For X = C, i.e., for holomorphic functions, the Hartogs-type extension theorem was proved by F. Hartogs in [Ha]. If X = CP 1 , i.e., for meromorphic functions, the result is due to E. Levi, see [Lv]. Since then the Hartogs-type extension theorem has been proved in at least two essentially more general cases than just holomorphic or meromorphic functions. Namely, for mappings into Kähler manifolds and into manifolds carrying complete Hermitian metrics of nonpositive holomorphic sectional curvature, see [Gr], The goal of this paper is to initiate the systematic study of extension properties of meromorphic mappings with values in non-Kähler complex manifolds. Let h be some Hermitian metric on a complex manifold X and let ω h be the associated (1, 1)-form. We call ω h (and h itself) pluriclosed or dd c -closed if dd c ω h = 0. In the sequel we shall not distinguish between Hermitian metrics and their associated forms. The latter we shall call simply metric forms. *This research was partially done during the author's stays at MSRI (supported in part by NSF grant DMS-9022140) and at MPIM. I would like to give my thanks to both institutions for their hospitality.

The aim of this note is to prove a result on extension of meromorphic mappings, which can be considered as a direct generalisation of the Hartogs extension theorem for holomorphic functions.