The method of negative curvature: the Kobayashi metric on 𝑃₂ minus 4 lines

Abstract

Bloch, and later H. Cartan, showed that if H 1 , … , H n + 2 {H_1}, \ldots ,{H_{n + 2}} are n + 2 n + 2 hyperplanes in general position in complex projective space P n {{\mathbf {P}}_n} , then P n − H 1 ∪ ⋯ ∪ H n + 2 {{\mathbf {P}}_n} - {H_1} \cup \cdots \cup {H_{n + 2}} is (in current terminology) hyperbolic modulo Δ \Delta , where Δ \Delta is the union of the hyperplanes ( H 1 ∩ ⋯ ∩ H k ) ⊕ ( H k + 1 ∩ ⋯ ∩ H n + 2 ) ({H_{^1}} \cap \cdots \cap {H_k}) \oplus ({H_{k + 1}} \cap \cdots \cap {H_{n + 2}}) for 2 ⩽ k ⩽ n 2 \leqslant k \leqslant n and all permutations of the H i {H_i} . Their results were purely qualitative. For n = 1 n = 1 , the thrice-punctured sphere, it is possible to estimate the Kobayashi metric, but no estimates were known for n ⩾ 2 n \geqslant 2 . Using the method of negative curvature, we give an explicit model for the Kobayashi metric when n = 2 n = 2 .