Kobayashi hyperbolicity

domain without large complex discs

Kobayashi hyperbolicity describes in a precise sense whether a complex manifold contains arbitrarily large copies of a one-dimensional complex disc. Extreme examples are the complex disc and the whole complex plane. The former is an example of a Kobayashi-hyperbolic manifold while the latter has arbitrarily large discs in it and is not Kobayashi hyperbolic.

Denote by the unit disc in the complex plane .

Let be a complex manifold, a point in and a tangent vector. Consider any holomorphic mapping with , . The infinitesimal Kobayashi pseudo-metric is defined by

The complex manifold is said to be Kobayashi hyperbolic if is locally bounded below by a strictly positive constant as varies over the tangent bundle,

General references for this area are: [a1], [a2] and [a3].

Examples.

1) The unit disc . In this case the Kobayashi pseudo-metric coincides with the Poincaré metric (cf. also Poincaré model).

2) More generally than Example 1), any bounded domain in is Kobayashi hyperbolic.

3) At the opposite end, the Kobayashi pseudo-metric of the complex plane as well as vanishes identically.

4) On the other hand, is again Kobayashi hyperbolic. The reason for this is that there is a covering of by the unit disc, and coverings are isometries.

5) The remarkable Brody theorem states that a compact complex manifold of any dimension is Kobayashi hyperbolic if and only if there is no non-constant holomorphic mapping of to .

The proof of this theorem starts, assuming non-hyperbolicity, with a sequence of holomorphic mappings of the unit disc to with derivatives at converging to infinity. Next one does a suitable scaling to normalize to a sequence which has derivative of length one at and which converges to a mapping on the whole plane.

6) Generalizations of 4) and 5) have been obtained by M. Green, see [a2], who gave some criteria ensuring that the complement of a finite family of complex hypersurfaces in complex projective space is Kobayashi hyperbolic.

7) The hyperbolicity of has traditionally been a useful tool in complex dynamics in one dimension. Recently, Kobayashi hyperbolicity has been used in complex dynamics in higher dimensions. For example, T. Ueda, see [a3], showed that all Fatou components, i.e. sets of normality of iterates, of a holomorphic mapping on are Kobayashi hyperbolic.

References

[a1]	S. Kobayashi, "Hyperbolic manifolds and holomorphic mappings" , M. Dekker (1970)
[a2]	S. Lang, "Introduction to complex hyperbolic spaces" , Springer (1987)
[a3]	J.E. Fornæss, "Dynamics in several complex variables" , CMBS , 87 , Amer. Math. Soc. (1996)