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

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 $n = 2$.