Namespaces
Variants
Actions

Difference between revisions of "Hyperbolic metric, principle of the"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
Suppose that domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482802.png" /> lie in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482803.png" />-plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482804.png" />-plane, respectively, and suppose they have at least three boundary points each; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482805.png" /> be a holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482806.png" /> taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482807.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h0482809.png" /> be the line elements of the hyperbolic metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828011.png" /> at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828013.png" />, respectively. The following inequality will then be true:
+
<!--
 +
h0482801.png
 +
$#A+1 = 60 n = 0
 +
$#C+1 = 60 : ~/encyclopedia/old_files/data/H048/H.0408280 Hyperbolic metric, principle of the
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/h/h048/h048280/h04828014.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
At any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828015.png" /> equality holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828017.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828018.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828019.png" /> conformally onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828020.png" />, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828021.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828022.png" /> conformally onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828023.png" />. The principle of the hyperbolic metric generalizes the [[Schwarz lemma|Schwarz lemma]] to multiply-connected domains in which a hyperbolic metric can be defined.
+
Suppose that domains  $  D $
 +
and  $  G $
 +
lie in the  $  z $-
 +
plane and  $  w $-
 +
plane, respectively, and suppose they have at least three boundary points each; let  $  w = f( z) $
 +
be a holomorphic function in  $  D $
 +
taking values in $  G $,  
 +
and let  $  d \sigma _ {z} $
 +
and  $  d \sigma _ {w} $
 +
be the line elements of the hyperbolic metric of  $  D $
 +
and  $  G $
 +
at points  $  z $
 +
and  $  w = f( z) $,  
 +
respectively. The following inequality will then be true:
  
In formulating the principle of the hyperbolic metric it is permissible to replace the assumption on the analyticity of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828025.png" /> by a more general assumption, i.e. that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828026.png" /> is an analytic function which is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828027.png" /> by any one of its elements and which can be analytically continued in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828028.png" /> along any path.
+
$$
 +
d \sigma _ {w}  \leq  d \sigma _ {z} .
 +
$$
  
The same principle can also be formulated about the behaviour of the hyperbolic length of curves, the hyperbolic distance or the hyperbolic area for a given mapping. In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828029.png" /> is a rectifiable curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828030.png" />, then (for the meaning of the symbols see [[Hyperbolic metric|Hyperbolic metric]])
+
At any point  $  z _ {0} \in D $
 +
equality holds if  $  f( z) \equiv w [ \zeta ( z)] $
 +
in  $  D $,  
 +
where the function  $  \zeta = \zeta ( z) $
 +
maps  $  D $
 +
conformally onto the disc  $  E = \{  \zeta  : {| \zeta | < 1 } \} $,
 +
while the function  $  w = w ( \zeta ) $
 +
maps  $  E $
 +
conformally onto  $  G $.  
 +
The principle of the hyperbolic metric generalizes the [[Schwarz lemma|Schwarz lemma]] to multiply-connected domains in which a hyperbolic metric can be defined.
  
<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/h/h048/h048280/h04828031.png" /></td> </tr></table>
+
In formulating the principle of the hyperbolic metric it is permissible to replace the assumption on the analyticity of the function  $  f( z) $
 +
in  $  D $
 +
by a more general assumption, i.e. that  $  f( z) $
 +
is an analytic function which is defined in  $  D $
 +
by any one of its elements and which can be analytically continued in  $  D $
 +
along any path.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828033.png" /> are two points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828034.png" />, then
+
The same principle can also be formulated about the behaviour of the hyperbolic length of curves, the hyperbolic distance or the hyperbolic area for a given mapping. In fact, if  $  L $
 +
is a rectifiable curve in $  D $,  
 +
then (for the meaning of the symbols see [[Hyperbolic metric|Hyperbolic metric]])
  
<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/h/h048/h048280/h04828035.png" /></td> </tr></table>
+
$$
 +
\mu _ {G} ( f ( L))  \leq  \mu _ {D} ( L).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828036.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828037.png" />, then
+
If $  z _ {1} $
 +
and  $  z _ {2} $
 +
are two points in $  D $,  
 +
then
  
<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/h/h048/h048280/h04828038.png" /></td> </tr></table>
+
$$
 +
r _ {G} ( f ( z _ {1} ), f ( z _ {2} ))  \leq  \
 +
r _ {D} ( z _ {1} , z _ {2} ).
 +
$$
  
Equality in these inequalities holds only in the above-mentioned case.
+
If  $  B $
 +
is a domain in $  D $,
 +
then
  
The above result as applied to the hyperbolic distance shows that under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828039.png" /> the image of the hyperbolic disc with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828040.png" /> is contained in the hyperbolic disc with its centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828041.png" /> of the same hyperbolic radius.
+
$$
 +
\Delta _ {G} ( f ( B))  \leq  \Delta _ {D} ( B).
 +
$$
  
This result is a generalization to the case of multiply-connected domains of the following fact in the theory of [[Conformal mapping|conformal mapping]] (the invariant form of Schwarz' lemma): Under the mapping of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828042.png" /> by a regular function
+
Equality in these inequalities holds only in the above-mentioned case.
 
 
<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/h/h048/h048280/h04828043.png" /></td> </tr></table>
 
 
 
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828044.png" />, the hyperbolic distance between the images of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828047.png" /> does not exceed the hyperbolic distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828049.png" />, and is equal to that distance only for a bilinear transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828050.png" /> onto itself.
 
  
The principle of the hyperbolic metric is connected with the [[Lindelöf principle|Lindelöf principle]] as follows. If the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828052.png" /> have a Green function and are simply connected, both these principles are identical. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828053.png" /> is simply connected, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828054.png" /> is multiply connected, the principle of the hyperbolic metric yields a more precise estimate of the domain containing the image of a hyperbolic disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828055.png" />, defined by an inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828056.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828058.png" /> denotes the Green function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828059.png" /> with logarithmic pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048280/h04828060.png" />. The principle of the hyperbolic metric is also applicable to cases in which Lindelöf's principle does not apply — e.g. to domains having at least three boundary points but not having a Green function.
+
The above result as applied to the hyperbolic distance shows that under the mapping  $  w = f( z) $
 +
the image of the hyperbolic disc with centre at the point  $  z _ {0} \in D $
 +
is contained in the hyperbolic disc with its centre at the point  $  w _ {0} = f( z _ {0} ) $
 +
of the same hyperbolic radius.
  
 +
This result is a generalization to the case of multiply-connected domains of the following fact in the theory of [[Conformal mapping|conformal mapping]] (the invariant form of Schwarz' lemma): Under the mapping of the disc  $  E $
 +
by a regular function
  
 +
$$
 +
w  =  f ( z),\ \
 +
| f ( z) |  <  1
 +
$$
  
====Comments====
+
in  $  E $,
 +
the hyperbolic distance between the images of the points  $  z _ {1} $
 +
and  $  z _ {2} $
 +
of  $  E $
 +
does not exceed the hyperbolic distance between  $  z _ {1} $
 +
and  $  z _ {2} $,
 +
and is equal to that distance only for a bilinear transformation of  $  E $
 +
onto itself.
  
 +
The principle of the hyperbolic metric is connected with the [[Lindelöf principle|Lindelöf principle]] as follows. If the domains  $  D $
 +
and  $  G $
 +
have a Green function and are simply connected, both these principles are identical. If  $  D $
 +
is simply connected, while  $  G $
 +
is multiply connected, the principle of the hyperbolic metric yields a more precise estimate of the domain containing the image of a hyperbolic disc in  $  D $,
 +
defined by an inequality  $  g _ {D} ( z, z _ {0} ) > \lambda $
 +
under the mapping  $  w = f( z) $,
 +
where  $  g _ {D} ( z, z _ {0} ) $
 +
denotes the Green function of  $  D $
 +
with logarithmic pole at  $  z _ {0} \in D $.
 +
The principle of the hyperbolic metric is also applicable to cases in which Lindelöf's principle does not apply — e.g. to domains having at least three boundary points but not having a Green function.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Conformal invariants. Topics in geometric function theory" , McGraw-Hill  (1973)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Conformal invariants. Topics in geometric function theory" , McGraw-Hill  (1973)</TD></TR>
 +
</table>

Latest revision as of 11:51, 1 May 2023


Suppose that domains $ D $ and $ G $ lie in the $ z $- plane and $ w $- plane, respectively, and suppose they have at least three boundary points each; let $ w = f( z) $ be a holomorphic function in $ D $ taking values in $ G $, and let $ d \sigma _ {z} $ and $ d \sigma _ {w} $ be the line elements of the hyperbolic metric of $ D $ and $ G $ at points $ z $ and $ w = f( z) $, respectively. The following inequality will then be true:

$$ d \sigma _ {w} \leq d \sigma _ {z} . $$

At any point $ z _ {0} \in D $ equality holds if $ f( z) \equiv w [ \zeta ( z)] $ in $ D $, where the function $ \zeta = \zeta ( z) $ maps $ D $ conformally onto the disc $ E = \{ \zeta : {| \zeta | < 1 } \} $, while the function $ w = w ( \zeta ) $ maps $ E $ conformally onto $ G $. The principle of the hyperbolic metric generalizes the Schwarz lemma to multiply-connected domains in which a hyperbolic metric can be defined.

In formulating the principle of the hyperbolic metric it is permissible to replace the assumption on the analyticity of the function $ f( z) $ in $ D $ by a more general assumption, i.e. that $ f( z) $ is an analytic function which is defined in $ D $ by any one of its elements and which can be analytically continued in $ D $ along any path.

The same principle can also be formulated about the behaviour of the hyperbolic length of curves, the hyperbolic distance or the hyperbolic area for a given mapping. In fact, if $ L $ is a rectifiable curve in $ D $, then (for the meaning of the symbols see Hyperbolic metric)

$$ \mu _ {G} ( f ( L)) \leq \mu _ {D} ( L). $$

If $ z _ {1} $ and $ z _ {2} $ are two points in $ D $, then

$$ r _ {G} ( f ( z _ {1} ), f ( z _ {2} )) \leq \ r _ {D} ( z _ {1} , z _ {2} ). $$

If $ B $ is a domain in $ D $, then

$$ \Delta _ {G} ( f ( B)) \leq \Delta _ {D} ( B). $$

Equality in these inequalities holds only in the above-mentioned case.

The above result as applied to the hyperbolic distance shows that under the mapping $ w = f( z) $ the image of the hyperbolic disc with centre at the point $ z _ {0} \in D $ is contained in the hyperbolic disc with its centre at the point $ w _ {0} = f( z _ {0} ) $ of the same hyperbolic radius.

This result is a generalization to the case of multiply-connected domains of the following fact in the theory of conformal mapping (the invariant form of Schwarz' lemma): Under the mapping of the disc $ E $ by a regular function

$$ w = f ( z),\ \ | f ( z) | < 1 $$

in $ E $, the hyperbolic distance between the images of the points $ z _ {1} $ and $ z _ {2} $ of $ E $ does not exceed the hyperbolic distance between $ z _ {1} $ and $ z _ {2} $, and is equal to that distance only for a bilinear transformation of $ E $ onto itself.

The principle of the hyperbolic metric is connected with the Lindelöf principle as follows. If the domains $ D $ and $ G $ have a Green function and are simply connected, both these principles are identical. If $ D $ is simply connected, while $ G $ is multiply connected, the principle of the hyperbolic metric yields a more precise estimate of the domain containing the image of a hyperbolic disc in $ D $, defined by an inequality $ g _ {D} ( z, z _ {0} ) > \lambda $ under the mapping $ w = f( z) $, where $ g _ {D} ( z, z _ {0} ) $ denotes the Green function of $ D $ with logarithmic pole at $ z _ {0} \in D $. The principle of the hyperbolic metric is also applicable to cases in which Lindelöf's principle does not apply — e.g. to domains having at least three boundary points but not having a Green function.

References

[a1] L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973)
How to Cite This Entry:
Hyperbolic metric, principle of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_metric,_principle_of_the&oldid=11504
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article