An Existence Theorem for Stationary Discs in Almost Complex Manifolds

An existence theorem for stationary discs of strongly pseudo-convex domains in almost complex manifolds is proved. More precisely, it is shown that, for all points of a suitable neighborhood of the boundary and for any vector belonging to certain open subsets of the tangent spaces, there exists a unique stationary disc passing through that point and tangent to the given vector. Introduction Analysis on almost complex manifolds recently became a rapidly growing research area in modern geometric analysis, due to the impulse given by the fundamental paper of Gromov [4], where deep connections between almost complex and symplectic structures have been discovered. One of the main tools in Gromov’s approach to the study of almost complex manifold is given by the pseudo-holomorphic discs, namely the holomorphic maps from the unit disc ∆ ⊂ C into the considered almost complex manifold (M,J). When the almost complex structure J is integrable, such pseudo-holomorphic discs coincide with the well known holomorphic discs of a complex manifold and they represent one of the central objects of complex analysis in several variables. The theory of classical holomorphic discs is by now well developed, while for what concerns the pseudo-holomorphic disc in a generic almost complex manifold, several quite natural questions are still open. In the classical case, the theory of holomorphic discs is tightly related with the studies on the Kobayashi-Royden metric of complex domains. Such metric can be easily defined also for domains in a general almost complex manifold, but, up to now, relatively few thing are known on its properties in this more general setting. Quite recently, new results have been obtained by Gaussier and the second author in [2] and by Ivashkovich and Rosay in [6], concerning the boundary behavior of the Kobayashi metric on strictly pseudoconvex domains in almost complex manifolds, a class of domains which plays a basic role in Gromov’s theory. In this paper we continue those studies focusing on the almost complex analogue of Lempert’s idea of stationary discs of pseudoconvex domains. The family of stationary discs consists of a special class of holomorphic discs that are attached to the boundary of a bounded domain. They were considered for the first time in the celebrated paper [9]. In that paper, Lempert shows that the extremal discs for the Kobayashi metric of a strongly convex domain D ⊂ C, coincide with the stationary discs, i.e. with the holomorphic discs admitting a meromorphic lift to the cotangent bundle T C, with the boundary attached to the conormal bundle of ∂D and with exactly one pole of order one at the origin. Lempert’s idea of stationary discs turned out to be a very important and fruitful tool in a variety of topics of 2000 Mathematics Subject Classification. 32H02, 53C15.