The definition of the Kobayashi-Royden pseudo-metric for almost complex manifolds is similar to its definition for complex manifolds. We study the question of completeness of some domains for this metric. In particular, we study the completeness of the complement of submanifolds of co-dimension 1 or 2. The paper includes a discussion, with proofs, of basic facts in the theory of pseudo-holomorphic discs.

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-{\tau }^2$ with $\tau >0$. Let $N\left(\phi \right)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N\left(\phi \right)$ and $\gamma$ in terms of $\tau$. When $Y$ is compact and $\gamma$ a geodesic circle, or when $Y$ has finite volume and $\gamma$ is a closed horocycle, we prove that $\gamma$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N\left(\phi \right)$ and $\gamma$ is $O\left(\tau \right)$. This bound is sharp.

First, we give some characterizations of the Kobayashi hyperbolicity of almost complex manifolds. Next, we show that a compact almost complex manifold is hyperbolic if and only if it has the Δ*-extension property. Finally, we investigate extension-convergence theorems for pseudoholomorphic maps with values in pseudoconvex domains.

Dans cet article, je montre qu’un domaine $D$ est hyperbolique pour la pseudodistance intégrée de Carathéodory $c_D^i$ (c’est-à-dire que $c_D^i$ est une distance sur $D$) si et seulement si la pseudodistance de Carathéodory $c_D$ vérifie la propriété de séparation faible suivante : tout point $x$ de $D$ possède un voisinage $V$ tel que, pour tout point $y$ de $V$, $y\ne x$, $c_D(x,y))\ne 0$. Je construis aussi un exemple d’un domaine $c_D^i$-hyperbolique et non $c_D$-hyperbolique.