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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The method of negative curvature: the Kobayashi metric on $ {\bf P}\sb 2$ minus $ 4$ lines


Author: Michael J. Cowen
Journal: Trans. Amer. Math. Soc. 319 (1990), 729-745
MSC: Primary 32H15; Secondary 32H25
MathSciNet review: 958888
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Abstract: Bloch, and later H. Cartan, showed that if $ {H_1}, \ldots ,{H_{n + 2}}$ are $ n + 2$ hyperplanes in general position in complex projective space $ {{\mathbf{P}}_n}$, then $ {{\mathbf{P}}_n} - {H_1} \cup \cdots \cup {H_{n + 2}}$ is (in current terminology) hyperbolic modulo $ \Delta $, where $ \Delta $ is the union of the hyperplanes $ ({H_{^1}} \cap \cdots \cap {H_k}) \oplus ({H_{k + 1}} \cap \cdots \cap {H_{n + 2}})$ for $ 2 \leqslant k \leqslant n$ and all permutations of the $ {H_i}$. Their results were purely qualitative. For $ n = 1$, the thrice-punctured sphere, it is possible to estimate the Kobayashi metric, but no estimates were known for $ n \geqslant 2$. Using the method of negative curvature, we give an explicit model for the Kobayashi metric when $ n = 2$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0958888-X
PII: S 0002-9947(1990)0958888-X
Article copyright: © Copyright 1990 American Mathematical Society