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Mathematische Zeitschrift

Unbounded Kobayashi Hyperbolic Domains in Cn Hervé Gaussier1 · Nikolay Shcherbina2 Received: 6 February 2020 / Accepted: 7 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space Cn to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in C3 that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in C3 and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary. Keywords Kobayashi hyperbolicity · Plurisubharmonic functions Mathematics Subject Classification 32Q45 · 32U05

Introduction According to the Riemann mapping theorem, every simply-connected domain in C, different from C, is biholomorphically equivalent to the unit disk 1 (0) := {λ ∈ C : |λ| < 1}. It is well known that this result has no direct generalization to higher dimension, since for instance every domain in Cn containing a nonconstant entire curve cannot be biholomorphic to a bounded domain in Cn . There are different tools to distinguish domains, among which invariant metrics (under the action of biholomorphisms) play an important role. We recall that if M is a complex manifold, r (0) := {λ ∈ C : |λ| < r } for every r > 0 and H(r (0), M) denotes the set of holomorphic maps from r (0) to M, then the Kobayashi pseudometric k M is defined on T M by k M (z; v) := inf{1/r > 0 : ∃ f ∈ H(r (0), M), f (0) = z, f  (0) = v}. A complex manifold M of complex dimension n is Kobayashi hyperbolic if for every point p ∈ M, there is a neighbourhood U of p in M and a constant c > 0 such that k M (z, v) ≥ c vg for every z ∈ U and every v ∈ Tz M, where ·g is any Hermitian

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Hervé Gaussier [email protected] Nikolay Shcherbina [email protected]

1

Université Grenoble Alpes, CNRS, IF, 38000 Grenoble, France

2

Department of Mathematics, University of Wuppertal, 42119 Wuppertal, Germany

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H. Gaussier, N. Shcherbina

norm on U induced from Cn . If K M denotes the inner distance induced by k M , then M is Kobayashi hyperbolic if K M is a distance on M. Notice that the topology induced by K M on M is then equivalent to the natural topology of M. From the definition of k M we see that every bounded domain in Cn is Kobayashi hyperbolic, whereas a complex manifold containing a nonconstant entire curve is not Kobayashi hyperbolic. Since the Kobayashi metric is a biholomorphic invariant, it follows that a complex manifold that is not Kobayashi hyperbolic does not admit any bounded representation, i.e., is not biholomorphic to any bounded domain in Cn . The first purpose of the paper is to give a sufficient condition from pluripotential theory for an unbounded domain to be Kobayashi hyperbolic. For r > 0 and z ∈ Cn , n ≥ 1, we denote by Brn (z) the Euclidean open ball centered at z with radius r , i.e. Brn (z) := {w ∈ Cn :