Namespaces
Variants
Actions

Difference between revisions of "Glueing theorems"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
g0445001.png
 +
$#A+1 = 48 n = 0
 +
$#C+1 = 48 : ~/encyclopedia/old_files/data/G044/G.0404500 Glueing theorems
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
Theorems that establish the existence of analytic functions subject to prescribed relations on the boundary of a domain.
 
Theorems that establish the existence of analytic functions subject to prescribed relations on the boundary of a domain.
  
Lavrent'ev's glueing theorem [[#References|[1]]]: Given any analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445001.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445002.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445004.png" />, then one can construct two analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445008.png" />, mapping the rectangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g0445009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450012.png" /> univalently and conformally onto disjoint domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450014.png" />, respectively, in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450015.png" />. This theorem was used (see [[#References|[6]]]) to prove the existence of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450018.png" />, realizing a quasi-conformal mapping of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450019.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450020.png" /> and possessing almost-everywhere a given characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450021.png" />, where
+
Lavrent'ev's glueing theorem [[#References|[1]]]: Given any analytic function $  x _ {1} = \phi ( x) $
 +
on $  [ - 1 , 1 ] $
 +
with $  \phi ( \pm  1 ) = \pm  1 $
 +
and  $  \phi  ^  \prime  ( x) > 0 $,  
 +
then one can construct two analytic functions $  f _ {1} ( z , h ) $
 +
and $  f _ {2} ( z , h ) $,  
 +
where $  z = x + i y $
 +
and $  h = \textrm{ const } $,  
 +
mapping the rectangles $  | x | < 1 $,  
 +
$  - h < y < 0 $
 +
and $  | x | < 1 $,
 +
0 < y < h $
 +
univalently and conformally onto disjoint domains $  D _ {1} $
 +
and $  D _ {2} $,  
 +
respectively, in such a way that $  f _ {1} ( x , h ) = f _ {2} ( \phi ( x) , h ) $.  
 +
This theorem was used (see [[#References|[6]]]) to prove the existence of a function $  w = f ( z) $,
 +
$  f ( 0) = 0 $,  
 +
$  f ( 1) = 1 $,  
 +
realizing a quasi-conformal mapping of the disc $  | z | \leq  1 $
 +
onto the disc $  | w | \leq  1 $
 +
and possessing almost-everywhere a given characteristic $  h ( z) $,
 +
where
 +
 
 +
$$
 +
h ( z)  =
 +
\frac{w _ {\overline{z}\; }  }{w _ {z} }
 +
,\ \
 +
| h ( z) |  \leq  h _ {0}  <  1 ,
 +
$$
  
<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/g/g044/g044500/g04450022.png" /></td> </tr></table>
+
and  $  h ( z) $
 +
is a measurable function defined for almost-all  $  z= x+ iy $,
 +
$  | z | \leq  1 $.  
 +
A modified form of Lavrent'ev's theorem was also used to solve the problem of mapping a simply-connected Riemann surface conformally onto the disc [[#References|[5]]].
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450023.png" /> is a measurable function defined for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450025.png" />. A modified form of Lavrent'ev's theorem was also used to solve the problem of mapping a simply-connected Riemann surface conformally onto the disc [[#References|[5]]].
+
Other glueing theorems (with weaker restrictions on the functions of type  $  x _ {1} = \phi ( x) $,
 +
see [[#References|[2]]]) have played a major role in the theory of Riemann surfaces. Another example is as follows (see [[#References|[3]]], [[#References|[5]]]): Suppose one is given an arc  $  \gamma _ {1} $
 +
on the circle  $  | z | = 1 $
 +
with end points  $  a $
 +
and  $  b $,
 +
$  a \neq b $,
 +
and a function $  g ( z) $
 +
on  $  \gamma _ {1} $
 +
with the properties: 1) at all the interior points of  $  \gamma _ {1} $,
 +
$  g ( z) $
 +
is regular and  $  g ^  \prime  ( z) \neq 0 $;
 +
2) the function  $  z _ {1} = g ( z) $
 +
establishes a one-to-one mapping of $  \gamma _ {1} $
 +
onto the complementary arc  $  \gamma _ {2} $
 +
on  $  | z | = 1 $
 +
leaving  $  a $
 +
and  $  b $
 +
invariant. Then there is a function
  
Other glueing theorems (with weaker restrictions on the functions of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450026.png" />, see [[#References|[2]]]) have played a major role in the theory of Riemann surfaces. Another example is as follows (see [[#References|[3]]], [[#References|[5]]]): Suppose one is given an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450027.png" /> on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450028.png" /> with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450031.png" />, and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450033.png" /> with the properties: 1) at all the interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450035.png" /> is regular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450036.png" />; 2) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450037.png" /> establishes a one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450038.png" /> onto the complementary arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450040.png" /> leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450042.png" /> invariant. Then there is a function
+
$$
 +
= F ( z) =
 +
\frac{1}{z}
  
<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/g/g044/g044500/g04450043.png" /></td> </tr></table>
+
+ a _ {1} z + \dots ,
 +
$$
  
regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450044.png" /> except at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450045.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450046.png" /> at the interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450047.png" />.
+
regular for $  | z | \leq  1 $
 +
except at 0, a , b $,
 +
such that $  F ( z) = F ( g ( z) ) $
 +
at the interior points of $  \gamma _ {1} $.
  
It has also been proved that there is a univalent function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044500/g04450048.png" /> with these properties (see [[#References|[4]]], Chapt. 2).
+
It has also been proved that there is a univalent function $  F( z) $
 +
with these properties (see [[#References|[4]]], Chapt. 2).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  "Sur une classe de répresentations continues"  ''Mat. Sb.'' , '''42''' :  4  (1935)  pp. 407–424</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.I. Volkovyskii,  "On the problem of the connectedness type of Riemann surfaces"  ''Mat. Sb.'' , '''18''' :  2  (1946)  pp. 185–212  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.C. Schaeffer,  D.C. Spencer,  "Variational methods in conformal mapping"  ''Duke Math. J.'' , '''14''' :  4  (1947)  pp. 949–966</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.C. Schaeffer,  D.C. Spencer,  "Coefficient regions for schlicht functions" , ''Amer. Math. Soc. Coll. Publ.'' , '''35''' , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.P. Belinskii,  "General properties of quasi-conformal mappings" , Novosibirsk  (1974)  pp. Chapt. 2, Par. 1  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  "Sur une classe de répresentations continues"  ''Mat. Sb.'' , '''42''' :  4  (1935)  pp. 407–424</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.I. Volkovyskii,  "On the problem of the connectedness type of Riemann surfaces"  ''Mat. Sb.'' , '''18''' :  2  (1946)  pp. 185–212  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.C. Schaeffer,  D.C. Spencer,  "Variational methods in conformal mapping"  ''Duke Math. J.'' , '''14''' :  4  (1947)  pp. 949–966</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.C. Schaeffer,  D.C. Spencer,  "Coefficient regions for schlicht functions" , ''Amer. Math. Soc. Coll. Publ.'' , '''35''' , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.P. Belinskii,  "General properties of quasi-conformal mappings" , Novosibirsk  (1974)  pp. Chapt. 2, Par. 1  (In Russian)</TD></TR></table>

Revision as of 19:42, 5 June 2020


Theorems that establish the existence of analytic functions subject to prescribed relations on the boundary of a domain.

Lavrent'ev's glueing theorem [1]: Given any analytic function $ x _ {1} = \phi ( x) $ on $ [ - 1 , 1 ] $ with $ \phi ( \pm 1 ) = \pm 1 $ and $ \phi ^ \prime ( x) > 0 $, then one can construct two analytic functions $ f _ {1} ( z , h ) $ and $ f _ {2} ( z , h ) $, where $ z = x + i y $ and $ h = \textrm{ const } $, mapping the rectangles $ | x | < 1 $, $ - h < y < 0 $ and $ | x | < 1 $, $ 0 < y < h $ univalently and conformally onto disjoint domains $ D _ {1} $ and $ D _ {2} $, respectively, in such a way that $ f _ {1} ( x , h ) = f _ {2} ( \phi ( x) , h ) $. This theorem was used (see [6]) to prove the existence of a function $ w = f ( z) $, $ f ( 0) = 0 $, $ f ( 1) = 1 $, realizing a quasi-conformal mapping of the disc $ | z | \leq 1 $ onto the disc $ | w | \leq 1 $ and possessing almost-everywhere a given characteristic $ h ( z) $, where

$$ h ( z) = \frac{w _ {\overline{z}\; } }{w _ {z} } ,\ \ | h ( z) | \leq h _ {0} < 1 , $$

and $ h ( z) $ is a measurable function defined for almost-all $ z= x+ iy $, $ | z | \leq 1 $. A modified form of Lavrent'ev's theorem was also used to solve the problem of mapping a simply-connected Riemann surface conformally onto the disc [5].

Other glueing theorems (with weaker restrictions on the functions of type $ x _ {1} = \phi ( x) $, see [2]) have played a major role in the theory of Riemann surfaces. Another example is as follows (see [3], [5]): Suppose one is given an arc $ \gamma _ {1} $ on the circle $ | z | = 1 $ with end points $ a $ and $ b $, $ a \neq b $, and a function $ g ( z) $ on $ \gamma _ {1} $ with the properties: 1) at all the interior points of $ \gamma _ {1} $, $ g ( z) $ is regular and $ g ^ \prime ( z) \neq 0 $; 2) the function $ z _ {1} = g ( z) $ establishes a one-to-one mapping of $ \gamma _ {1} $ onto the complementary arc $ \gamma _ {2} $ on $ | z | = 1 $ leaving $ a $ and $ b $ invariant. Then there is a function

$$ w = F ( z) = \frac{1}{z} + a _ {1} z + \dots , $$

regular for $ | z | \leq 1 $ except at $ 0, a , b $, such that $ F ( z) = F ( g ( z) ) $ at the interior points of $ \gamma _ {1} $.

It has also been proved that there is a univalent function $ F( z) $ with these properties (see [4], Chapt. 2).

References

[1] M.A. Lavrent'ev, "Sur une classe de répresentations continues" Mat. Sb. , 42 : 4 (1935) pp. 407–424
[2] L.I. Volkovyskii, "On the problem of the connectedness type of Riemann surfaces" Mat. Sb. , 18 : 2 (1946) pp. 185–212 (In Russian)
[3] A.C. Schaeffer, D.C. Spencer, "Variational methods in conformal mapping" Duke Math. J. , 14 : 4 (1947) pp. 949–966
[4] A.C. Schaeffer, D.C. Spencer, "Coefficient regions for schlicht functions" , Amer. Math. Soc. Coll. Publ. , 35 , Amer. Math. Soc. (1950)
[5] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[6] P.P. Belinskii, "General properties of quasi-conformal mappings" , Novosibirsk (1974) pp. Chapt. 2, Par. 1 (In Russian)
How to Cite This Entry:
Glueing theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Glueing_theorems&oldid=47102
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article