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 $\Delta$ the unit disc in the complex plane $\mathbb{C}$.

Let $M$ be a complex manifold, $p$ a point in $M$ and $\chi \in T_pM$ a tangent vector. Consider any holomorphic mapping $f : \Delta \rightarrow M$ with $f(0) = p$, $f'_*(\partial/\partial z) = c\chi$. The infinitesimal Kobayashi pseudo-metric is defined by $$ ds(p,\chi) = \inf_f \left\lbrace{\frac{1}{c}}\right\rbrace\ . $$

The complex manifold $M$ is said to be Kobayashi hyperbolic if $ds(p,\chi)$ is locally bounded below by a strictly positive constant as $(p,\chi)$ varies over the tangent bundle, $\chi \neq 0$

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

Examples.

1) The unit disc $\Delta$. 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 $\mathbb{C}^n$ is Kobayashi hyperbolic.

3) At the opposite end, the Kobayashi pseudo-metric of the complex plane $\mathbb{C}$ as well as $\mathbb{C} \setminus \{0\}$ vanishes identically.

4) On the other hand, $\mathbb{C} \setminus \{0,1\}$ is again Kobayashi hyperbolic. The reason for this is that there is a covering of $\mathbb{C} \setminus \{0,1\}$ by the unit disc, and coverings are isometries.

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

The proof of this theorem starts, assuming non-hyperbolicity, with a sequence of holomorphic mappings of the unit disc to $M$ with derivatives at $0$ converging to infinity. Next one does a suitable scaling to normalize to a sequence which has derivative of length one at $0$ 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 $\mathbb{C} \setminus \{0,1\}$ 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 $\mathbf{P}^n$ 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)