View source for Kobayashi hyperbolicity

''domain without large complex discs''

Kobayashi hyperbolicity describes in a precise sense whether a [[Complex manifold|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: [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a3]]].

===Examples.===


1) The unit disc $\Delta$. In this case the Kobayashi pseudo-metric coincides with the Poincaré metric (cf. also [[Poincaré model|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 [[#References|[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 [[#References|[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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   "Hyperbolic manifolds and holomorphic mappings" , M. Dekker  (1970)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lang,   "Introduction to complex hyperbolic spaces" , Springer  (1987)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.E. Fornæss,   "Dynamics in several complex variables" , ''CMBS'' , '''87''' , Amer. Math. Soc.  (1996)</TD></TR>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Serge Lang, "Hyperbolic and Diophantine analysis", ''Bulletin of the American Mathematical Society'' '''14''' (1986): 159–205. {{DOI|10.1090/s0273-0979-1986-15426-1}} {{ZBL|0602.14019}}</TD></TR>
</table>