Some characterizations of hyperbolic almost complex manifolds

Abstract

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∆∗-extension property. Finally, we investigate extensionconvergence theorems for pseudoholomorphic maps with values in pseudoconvex domains. Introduction. The concept of Kobayashi hyperbolicity has recently been extended to almost complex manifolds by several authors (see [7]–[9]). The main problem which interested those authors was to characterize the Kobayashi hyperbolicity of almost complex manifolds. The main goal of this article is to investigate criterions for the Kobayashi hyperbolicity of almost complex manifolds and for extensions of pseudoholomorphic curves. The paper is organized as follows: Section 1, which is essentially preliminary, contains the properties of almost complex manifolds used in the proofs of the main results. In Section 2, we give some characterizations of the Kobayashi hyperbolicity of almost complex manifolds. The first is furnished by a local estimate of the Kobayashi–Royden metric, which was proved by Royden [15] in the complex case. Then we define the Landau property and we prove that it is equivalent to hyperbolicity; this generalizes Hahn–Kim’s [5] results. In Section 3, we investigate the relationship between the ∆∗-extension property and hyperbolicity for compact almost complex manifolds. An almost complex manifold (M,J) is said to have the ∆∗-extension property if every pseudoholomorphic curve f : ∆∗ → (M,J) extends to a pseudoholomorphic curve f : ∆→ (M,J). It is known that each hyperbolic compact almost complex manifold has the ∆∗-extension property (see [6]); 2010 Mathematics Subject Classification: 32Q45, 32Q60, 32Q65.