Tautness and Kobayashi hyperbolicity.

*(German)*Zbl 1017.32021
Oldenburg: Univ. Oldenburg, Institut für Mathematik (Diss.). i, 100 p. (2003).

Summary: The main objects studied in this work are hyperbolicity, tautness and completeness for domains in \(\mathbb{C}^n\).

In the first chapter we introduce necessary notions and facts. In particular, in §1.1 we give some properties related to the Minkowski functions. In §1.3 and §1.4 we recall Royden’s criterion for taut domains. Applying it we give new proofs for some known results. We also present a motivation to study \(\widetilde k\)-hyperbolicity.

The second chapter is devoted to studying the hyperbolic (mainly \(k\)-, \(\widetilde k\)-, Brody hyperbolic) Hartogs type domains. In §2.1 we give a full characterization of \(k\)-hyperbolicity for a class of Hartogs domains with balanced fibers. Moreover, we find some sufficient conditions for a Hartogs domain with balanced fibers to be \(\widetilde k\)-hyperbolic. In §2.2 we discuss conditions for a Hartogs-Laurent domain to be hyperbolic. We also prove that for any \(n\geq 4\) there is a Hartogs-Laurent domain in \(C^n\) which has a local plurisubharmonic antipeak function at infinity, but which does not have a local plurisubharmonic peak function at infinity.

The main goal of the third chapter is to study some new applications of Royden’s criterion. In particular, we find a full characerization of tautness for the class of Hartogs type domains with balanced fibers. Moreover, a new relationship between (global) tautness and local tautness is established, which may be considered as a generalization of the Kerzmann Rosay Theorem. In §3.4 and §3.5 we study hyperconvexity and \(k\)-completeness of some Hartogs type domains.

The last chapter contains results on \(\widetilde k\)-hyperbolicity and \(k\)-completeness for the class of balanced domains. In §4.1 we point out that there is a big difference between \(\widetilde k\)-hyperbolicity and Brody hyperbolicity in the class of balanced domains. We prove that for any \(n\geq 3\) there is a pseudoconvex non-\(\widetilde k\)-hyperbolic balanced domain in \(\mathbb{C}^n\) with positive-definite Minkowski function that is Brody hyperbolic. In §4.2 we give another example of the following result obtained by Jarnicki-Pflug: For \(n\geq 3\) there is a bounded pseudoconvex balanced domain in \(\mathbb{C}^n\) with continuous Minkowski function that is not \(k\)-complete.

In the first chapter we introduce necessary notions and facts. In particular, in §1.1 we give some properties related to the Minkowski functions. In §1.3 and §1.4 we recall Royden’s criterion for taut domains. Applying it we give new proofs for some known results. We also present a motivation to study \(\widetilde k\)-hyperbolicity.

The second chapter is devoted to studying the hyperbolic (mainly \(k\)-, \(\widetilde k\)-, Brody hyperbolic) Hartogs type domains. In §2.1 we give a full characterization of \(k\)-hyperbolicity for a class of Hartogs domains with balanced fibers. Moreover, we find some sufficient conditions for a Hartogs domain with balanced fibers to be \(\widetilde k\)-hyperbolic. In §2.2 we discuss conditions for a Hartogs-Laurent domain to be hyperbolic. We also prove that for any \(n\geq 4\) there is a Hartogs-Laurent domain in \(C^n\) which has a local plurisubharmonic antipeak function at infinity, but which does not have a local plurisubharmonic peak function at infinity.

The main goal of the third chapter is to study some new applications of Royden’s criterion. In particular, we find a full characerization of tautness for the class of Hartogs type domains with balanced fibers. Moreover, a new relationship between (global) tautness and local tautness is established, which may be considered as a generalization of the Kerzmann Rosay Theorem. In §3.4 and §3.5 we study hyperconvexity and \(k\)-completeness of some Hartogs type domains.

The last chapter contains results on \(\widetilde k\)-hyperbolicity and \(k\)-completeness for the class of balanced domains. In §4.1 we point out that there is a big difference between \(\widetilde k\)-hyperbolicity and Brody hyperbolicity in the class of balanced domains. We prove that for any \(n\geq 3\) there is a pseudoconvex non-\(\widetilde k\)-hyperbolic balanced domain in \(\mathbb{C}^n\) with positive-definite Minkowski function that is Brody hyperbolic. In §4.2 we give another example of the following result obtained by Jarnicki-Pflug: For \(n\geq 3\) there is a bounded pseudoconvex balanced domain in \(\mathbb{C}^n\) with continuous Minkowski function that is not \(k\)-complete.

##### MSC:

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |

32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |

32A19 | Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.) |

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

32T40 | Peak functions |