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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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