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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101201.png" /> the unit disc in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101202.png" />.
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Denote by $\Delta$ the unit disc in the complex plane $\mathbb{C}$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101203.png" /> be a [[Complex manifold|complex manifold]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101204.png" /> a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101206.png" /> a tangent vector. Consider any [[Holomorphic mapping|holomorphic mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101207.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k1101209.png" />. The infinitesimal Kobayashi pseudo-metric is defined by
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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
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$$
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ds(p,\chi) = \inf_f \left\lbrace{\frac{1}{c}}\right\rbrace\ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012010.png" /></td> </tr></table>
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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, $\chi \neq 0$
 
 
The complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012011.png" /> is said to be Kobayashi hyperbolic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012012.png" /> is locally bounded below by a strictly positive constant as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012013.png" /> varies over the tangent bundle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012014.png" />
 
  
 
General references for this area are: [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a3]]].
 
General references for this area are: [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a3]]].
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1) The unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012015.png" />. In this case the Kobayashi pseudo-metric coincides with the Poincaré metric (cf. also [[Poincaré model|Poincaré model]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012016.png" /> is Kobayashi hyperbolic.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012017.png" /> as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012018.png" /> vanishes identically.
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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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012019.png" /> is again Kobayashi hyperbolic. The reason for this is that there is a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012020.png" /> by the unit disc, and coverings are isometries.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012021.png" /> of any dimension is Kobayashi hyperbolic if and only if there is no non-constant holomorphic mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012022.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012023.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012024.png" /> with derivatives at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012025.png" /> converging to infinity. Next one does a suitable scaling to normalize to a sequence which has derivative of length one at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012026.png" /> and which converges to a mapping on the whole plane.
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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110120/k11012028.png" /> are Kobayashi hyperbolic.
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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====
 
====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></table>
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<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>
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<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>
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<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>
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</table>

Latest revision as of 09:26, 21 December 2014

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)
[b1] 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
How to Cite This Entry:
Kobayashi hyperbolicity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kobayashi_hyperbolicity&oldid=16571
This article was adapted from an original article by J.E. Fornæss (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article