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First, we give some characterization of hyperbolic embeddedness in the almost complex case. Next, we study the stability of hyperbolically embedded manifolds under a small perturbation of almost complex structures. Finally, we obtain generalizations and extensions of theorems of Kobayashi, Kiernan, Kwack and Noguchi for almost complex manifolds.Keywords Pseudoholomorphic curves · Almost complex manifolds · Hyperbolic embedding IntroductionThe extension of holomorphic maps is one of the fundamental problems of complex analytic geometry. Since Kobayashi [10,11] introduced the notion of Kobayashi pseudodistance, much attention has been paid to this problem.The aim of this paper is twofold. The first is to study the hyperbolic embeddedness and prove several characterizations for almost complex manifolds. Also, we give a sufficient condition for a complement of an hypersurface whenever it exists to be hyperbolically embedded in a given almost complex manifold. We close this topic by proving that the notion of being hyperbolically embedded and complete is stable under small variations of almost complex structures. The second aim is to investigate the extension and convergence theorems for pseudo-holomorphic curves defined over the punctured unit disc in the complex plane into hyperbolically embedded almost complex submanifolds. We prove that such curves extend to the whole unit disc and satisfy the Noguchi's convergence theorem [19].

Using convex subrings of *$\mathbb{C}$, a nonstandard extension of $\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

First, we establish some Schwarz type inequalities for mappings with bounded Laplacian, then we obtain boundary versions of the Schwarz lemma.

Nonstandard mathematics furnishes a remarkable connexion between analytic and algebraic geometry. We describe this interplay for the most basic notions like complex spaces/algebraic schemes, generic points, differential forms etc. We obtain -by this point of view -in particular new results on the prime spectrum of a Stein algebra.2000 Mathematics Subject Classification 32C15, 14A15 (primary); 32E10, 26E35, 58AQ10 (secondary)