Estimates of the Kobayashi Metric on Almost Complex Manifolds

We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold (M, J) admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in M and to give a sufficient condition for the complete hyperbolicity of a domain in (M, J). Finally we obtain the regularity up to the bounday of J-holomorphic discs attached to a totally real submanifold in M . Introduction The properties of the Kobayashi-Royden infinititesimal pseudometric on complex manifolds (with the standard integrable structure) have been intensively studied by many authors (see, for instance, [15] and references therein). On the contrary, its study on almost complex manifolds began quite recently (see [18], [9])) and many basic questions are still open. The main goal of the present paper is to give a precise lower estimate for the Kobayashi-Royden pseudometric on a strictly pseudoconvex domain in an almost complex manifold. Such estimates are well known in the integrable case (see [11], [22]). However, the standard methods have no direct generalization to the case of an almost complex structure. For instance, the natural and usual technique consisting in osculating the boundary of a strictly pseudoconvex domain by balls cannot be used, since the Kobayashi-Royden metric of the unit ball with a non-standard almost complex structure cannot be computed explicitely. For this reason, we use another approach based on the construction of special classes of plurisubharmonic functions, developped essentially by N. Sibony [22] in the case of the standard complex structure (see also [10] by K. Diederich-J.E. Fornaess and [17] by N. Kerzman-J.P. Rosay). A corollary of our main results states that every point in an almost complex manifold has a complete hyperbolic neighborhood. Related results were obtained in [9] and [16]. Acknowledgments. The authors thank E. Chirka, B. Coupet and S. Ivashkovich for helpful discussions. This work was partially done while the first author was visiting the Université de Lille 1. He thanks this organization for its support. 1. Preliminaries We will sometimes refer to results proved for complex manifolds and valid without substantial modification in the almost complex case. For coherence of exposition we will not present all of them in the Preliminaries. They will be stated with references in the different Sections. Date: 2008-2-15. 2000 Mathematics Subject Classification. Primary: 32V40. Secondary 32V25, 32H02, 32H40, 32V10. 1 2 HERVÉ GAUSSIER AND ALEXANDRE SUKHOV 1.1. Almost complex manifolds. Let M be a smooth (everywhere in this paper this means C∞) real (2n)-dimensional manifold equipped with an almost complex structure J , that is a C∞-field of complex linear structures on the tangent bundle TM of M (we call (M,J) an almost complex manifold). For example, if M = R we may consider the standard complex structure J0 which associates to every point p ∈ R the endomorphism of Tp(M) = R given by the matrix J0 = (