## 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 $$-\rho$$.

### MSC:

 32F45 Invariant metrics and pseudodistances in several complex variables 32Q60 Almost complex manifolds 32Q65 Pseudoholomorphic curves 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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### References:

 [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
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