Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions

Abstract. We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz Lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results from [2]. The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper.

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“…Let us start with Reinhardt domains, which are seemingly well understood (see [19]). Even in this case we get new results.…”

Section: Circular Domains and Complex Geodesicsmentioning

confidence: 99%

Schwarz lemma for the tetrablock

Edigarian

Zwonek

2009

Bulletin of the London Mathematical Society

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“…We mention that there are a lot of works concerning the completeness of β M (cf. [4], [5], [6], [12], [13], [14], [15], [17], [20]). On the other hand, Diederich and Ohsawa [10] proved that the Bergman distance for a bounded C 2 pseudoconvex domain in C n has a lower bound of a constant multiple log | log δ Ω |.…”

Section: Theoremmentioning

confidence: 99%

On the Bergman metric of pseudoconvex domains in a complex projective space

Chen

2005

Proc. Amer. Math. Soc.

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“…In 1959, Kobayashi [11] began to investigate the completeness of the Bergman metric. After that, there are a lot of papers concerning the Bergman completeness for bounded pseudoconvex domains in C n (see [3], [18] for a review). There are two general results: One says that any bounded hyperconvex domain is Bergman complete (cf.…”

Section: Introductionmentioning

confidence: 99%

The Bergman metric on a Stein manifold with a bounded plurisubharmonic function

Chen

Zhang

2002

Trans. Amer. Math. Soc.

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Abstract. In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature ≤ −A/ρ 2 , which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.

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Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions

Cited by 28 publications

References 0 publications

Schwarz lemma for the tetrablock

Schwarz lemma for the tetrablock

On the Bergman metric of pseudoconvex domains in a complex projective space

The Bergman metric on a Stein manifold with a bounded plurisubharmonic function

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