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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 \(-\rho\).

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 \(-\rho\).

Reviewer: Vasile Oproiu (Iaşi)

### MSC:

32F45 | Invariant metrics and pseudodistances in several complex variables |

32Q60 | Almost complex manifolds |

32Q65 | Pseudoholomorphic curves |

32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |

### Keywords:

Kobayashi-Royden pseudo-norm; almost complex manifolds; Schwarz lemmas; complete hyperbolicity
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\textit{S. Ivashkovich} and \textit{J.-P. Rosay}, Ann. Inst. Fourier 54, No. 7, 2387--2435 (2004; Zbl 1072.32007)

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