Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1990-0958888-X
MathSciNet review: 958888
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [1] L. Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938), 359-364. MR 1501949
  • [2] -, The theory of meromorphic curves, Acta Soc. Sci. Fenn. Ser. A 3 (1941), 3-31. MR 0004309 (2:357b)
  • [3] A. Bloch, Sur les systèmes de fonctions holomorphes à variétiés linéaires lacunaires, Ann. Ecole Norm 43 (1926), 309-362. MR 1509274
  • [4] E. Borel, Sur les zéros des fonctions entières, Acta Math. 20 (1896), 357-396.
  • [5] R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213-219. MR 0470252 (57:10010)
  • [6] H. Cartan, Sur les systèmes de fonctions holomorphes a variétés lacunaires et leurs applications, Ann. Ecole Norm. 45 (1928), 255-346. MR 1509288
  • [7] J. Carlson and P. Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. (2) 95 (1972), 557-584. MR 0311935 (47:497)
  • [8] S. S. Chern, Holomorphic curves in the plane, Differential Geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 73-94. MR 0422693 (54:10679)
  • [9] M. Cowen, The Kobayashi metric on $ {{\mathbf{P}}_n} - ({2^n} + 1)$ hyperplanes, Value Distribution Theory, Marcel Dekker, New York, 1974, pp. 205-223. MR 0352543 (50:5030)
  • [10] M. Cowen and P. Griffiths, Holomorphic curves and metrics of negative curvature, J. Anal. Math. 29 (1976), 93-153. MR 0508156 (58:22677)
  • [11] P. Kiernan and S. Kobayashi, Holomorphic mappings into projective space with lacunary hyperplanes, Nagoya Math. J. 50 (1973), 199-216. MR 0326007 (48:4353)
  • [12] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, 1970. MR 0277770 (43:3503)
  • [13] S. Lang, Hyperbolic and diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), 159-205. MR 828820 (87h:32051)
  • [14] -, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677 (88f:32065)
  • [15] H. Royden, Remarks on the Kobayashi metric, Several Complex Variables. II (Proc. Internat. Conf. Univ. Maryland, 1970), Lecture Notes in Math., vol. 185, Springer-Verlag, New York, 1971, pp. 125-137. MR 0304694 (46:3826)
  • [16] H. Weyl and J. Weyl, Meromorphic functions and analytic curves, Princeton Univ. Press, Princeton, N.J., 1943. MR 0009057 (5:94d)
  • [17] H. Wu The equidistribution theory of holomorphic curves, Princeton Univ. Press, Princeton, N.J., 1970. MR 0273070 (42:7951)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32H15, 32H25

Retrieve articles in all journals with MSC: 32H15, 32H25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0958888-X
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society