Hyperbolic embeddedness and extension–convergence theorems of J-holomorphic curves

Abstract

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].

Hyperbolic embeddability of locally complete almost complex submanifolds

“…Using a special covering, we introduce a differential metric on M called the KobayashiRoyden-Green (KRG) metric. We should mention, that using the KRG-metric, the author and Haggui proved the stability of hyperbolic embeddedness under small perturbation of the almost complex structure, see Theorem 3.3 in [4]. G be a fixed length function on N. Cover M by open subsets U 1 , . .…”

“…We assume that there exists h a limiting J -complex line of ∂ M. Let p, q be any pair of points in h(C), then p = h(a 1 ) and q = h(a 2 ) where a 1 , a 2 ∈ R for some positive radius R and there exists a sequence of J -holomorphic curves f n : R → M which converges uniformly to h| R . By [4] (see Theorem 2.2), there exists a length a 2 ), let R tend to ∞, it follows that p = q and h is a constant.…”

“…His proof is based on the so-called Gromov's monotonicity lemma [9]. In [4], the authors gave an elementary proof of that result by showing that the considered pseudo-holomorphic curve defined on the punctured disk has a finite energy and then extends. We provide here another proof based on "Chirka's functions" with prescribed poles.…”

Normal pseudoholomorphic curves

“…This allows a characterization of the hyperbolic embeddedness of an almost complex submanifold (M, J) (not necessarily relatively compact) in an almost complex manifold (N, J), and extensions of pseudoholomorphic curves. We recall that several characterizations of hyperbolic embeddedness were obtained in [7] under the assumption of relative compactness. Mainly, we proved that a relatively compact almost complex submanifold (M, J) is hyperbolically embedded in (N, J) if and only if O J (∆, M ) is relatively compact in O J (∆, N ).…”

Some characterizations of hyperbolic almost complex manifolds

“…Noguchi [13] proved a remarkable theorem about the extension and convergence of holomorphic maps. In [4], the authors proved several Noguchi type extension-convergence theorems in the almost complex case. We give some other variants of such theorems for almost complex maps; this generalizes the result of [17] proved in the complex case.…”