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# On Hyperbolicity of Domains with Strictly Pseudoconvex Ends

Published:2012-12-04
Printed: Feb 2014
Department of Mathematics and Statistics, School of Science and Technology, University of New England Armidale, NSW 2351 Australia
• Martin Kolář,
Department of Mathematics and Statistics, Masaryk University, Janackovo nam. 2a, 662 95 Brno
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## Abstract

This article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when $\Omega\subset{\mathbb C}^{n}$ corresponds to a sub-level set of a smooth, real-valued function $\Psi$, such that the form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.
 Keywords: Kobayashi-hyperbolicity, Kähler metric, plurisubharmonic function
 MSC Classifications: 32Q45 - Hyperbolic and Kobayashi hyperbolic manifolds 32Q35 - Complex manifolds as subdomains of Euclidean space

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