zbMATH — the first resource for mathematics

Schwarz-type lemmas for solutions of \(\bar\partial \)-inequalities and complete hyperbolicity of almost complex manifolds. (English) Zbl 1072.32007
The authors study some problems of completeness of certain domains for the Kobayashi-Royden pseudo-metric.
Theorem 1. Let \(D\) be a domain in an almost complex manifold \((X,J)\), \(J\) of class \(\mathcal{C}^1\). Let \(p\in\partial D\). If the boundary of \(D\) is strictly \(J\)-pseudoconvex at \(p\), the point \(p\) is at infinite Kobayashi distance from the points in \(D\).
Theorem 2. Let \(D\) be a hyperbolic domain in an almost complex manifold \((X,J)\), \(J\in \mathcal{C}^2\). Let \(M\) be a closed submanifold of \(D\) of real codimension \(2\) and of class \(\mathcal{C}^3\). If \(M\) is a \(J\)-complex hypersurface, then for every \(p\in M\) and \(q\in D\setminus M\), the Kobayashi distance from \(q\) to \(p\) is infinite. Conversely, if \(p\in M\) and if the tangent space to \(M\) at \(p\) is not \(J\)-complex, then for any neighborhood \(D_1\) of \(p\) in \(D\), there exists \(p^\prime\in D_1\cap M\) that is at finite distance from points in \(D\setminus M\).
Let a real hypersurface \(M\subset D\) be defined by \(\rho =0,\;(\nabla \rho\neq 0\)).
Theorem 3. Let \(D\) be a domain in an almost complex manifold \((X,J)\). Assume that the closed real hypersurface \(M\subset D\) is of class \(\mathcal{C}^2\), and \(J\) is of class \(\mathcal{C}^{3,\alpha}\) (for some \(0< \alpha <1\)). If there exists a complex tangent vector \(Y\) to \(M\) at a point \(p\) such that \(dd_J^c(Y,JY)>0\), then that point \(p\) is at finite distance, in \(D\setminus M\) from points in a region defined by \(\rho >0\). If \(dd_J^c\rho(Y,JY)<0\), simply replace \(\rho\) by \(-\rho\).

32F45 Invariant metrics and pseudodistances in several complex variables
32Q60 Almost complex manifolds
32Q65 Pseudoholomorphic curves
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text: DOI Numdam EuDML
[1] Regular type of real hypersurfaces in (almost) complex manifolds, (2003) · Zbl 1082.32017
[2] Existence of a complex line in tame almost complex tori, Duke Math. J, 94, 1, 29-40, (1998) · Zbl 0981.53084
[3] Characterization of models in \(\C^2\) by their automorphism groups, Int. J. Math, 5, 619-634, (1994) · Zbl 0817.32010
[4] Complex Dynamics, (1993), Springer Verlag · Zbl 0782.30022
[5] Complete hyperbolic neighborhoods in almost complex surfaces, Int. J. Math, 12, 211-221, (2001) · Zbl 1110.32306
[6] Kobayashi hyperbolicity of almost complex manifolds, (1998)
[7] Variétés hyperboliques presque-complexes, (2001)
[8] Symplectic submanifolds and almost-complex geometry, J. Differential Geom, 44, 4, 666-705, (1996) · Zbl 0883.53032
[9] Un théorème de Green presque complexe
[10] Estimates of the Kobayashi metric on almost complex manifolds · Zbl 1083.32011
[11] Pseudoholomorphic curves in symplectic manifolds, Invent. Math, 82, 307-347, (1985) · Zbl 0592.53025
[12] Fonctions PSH sur une variété presque complexe, C. R. Acad. Sci. Paris, Sér. I, 335, 509-514, (2002) · Zbl 1013.32019
[13] The Analysis of Linear Partial Differential Operators III, 274, (1985), Springer-Verlag, Berlin Heidelberg · Zbl 0601.35001
[14] Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures · Zbl 1091.32009
[15] Structure of the moduli space in a neighborhood of a cusp curve and meromorphic hulls, Invent. Math, 136, 571-602, (1999) · Zbl 0930.32017
[16] Complex curves in almost-complex manifolds and meromorphic hulls, Publication Series of Graduiertenkollegs “Geometrie und Mathematische Physik” of the Ruhr-University Bochum, 36, 1-186, (1999)
[17] Existence of close pseudoholomorphic disks for almost complex manifolds and an application to the Kobayashi-Royden pseudonorm, Funct. Anal. and Appl, 33, 38-48, (1999) · Zbl 0967.32024
[18] Pseudoholomorphic mappings and Kobayashi hyperbolicity, Differential Geom. Appl, 11, 265-277, (1999) · Zbl 0954.32019
[19] Hyperbolically imbedded spaces and big Picard theorem, Math. Ann, 204, 203-209, (1973) · Zbl 0244.32010
[20] Multidimensional singular equations and integral equations, Pergamon Press, (1955)
[21] Symplectic manifolds with contact type boundaries, Invent. Math, 103, 651-671, (1991) · Zbl 0719.53015
[22] \(J\)-holomorphic curves and quantum cohomology, Univ. Lect. Series AMS, 6, (1994) · Zbl 0809.53002
[23] Some integration problems in almost complex and complex manifolds, Ann. of Math, 77, 424-489, (1963) · Zbl 0115.16103
[24] Fonctions plurisousharmoniques et courants positifs de type (1,1) sur une variété presque complexe · Zbl 1089.32033
[25] The extension of regular holomorphic maps, Proc. A.M.S, 43, 306-310, (1974) · Zbl 0292.32019
[26] Some properties of holomorphic curves in almost complex manifolds, Holomorphic Curves in Symplectic Geometry, 351-361, (1994), Birkhauser
[27] Singular Integrals and Differentiability Properties of Functions, (1970), Princeton U.P · Zbl 0207.13501
[28] Picard’s theorem and hyperbolicity, Siberian Math. J., 24, 858-867, (1983) · Zbl 0579.32039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.