Namespaces
Variants
Actions

Difference between revisions of "Quadratic differential"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
Line 1: Line 1:
''on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760401.png" />''
+
<!--
 +
q0760401.png
 +
$#A+1 = 74 n = 0
 +
$#C+1 = 74 : ~/encyclopedia/old_files/data/Q076/Q.0706040 Quadratic differential
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A rule which associates to each local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760402.png" /> (cf. [[Local uniformizing parameter|Local uniformizing parameter]]) mapping a parametric neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760403.png" /> into the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760404.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760405.png" />), a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760406.png" /> such that for any local parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760408.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760409.png" /> non-empty, the following holds in this intersection:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/q/q076/q076040/q07604010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
''on a Riemann surface  $  R $''
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604011.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604013.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604014.png" />. A quadratic differential is often denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604015.png" />, to which is attributed the invariance with respect to the choice of the local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604016.png" />, as indicated by (1). In other words, a quadratic differential is a non-linear differential of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604017.png" /> on a Riemann surface.
+
A rule which associates to each local parameter $  z $(
 +
cf. [[Local uniformizing parameter|Local uniformizing parameter]]) mapping a parametric neighbourhood  $  U \subset  R $
 +
into the extended complex plane  $  \overline{\mathbf C}\; $(
 +
$  z :  U \rightarrow \overline{\mathbf C}\; $),  
 +
a function  $  Q _ {z} :  z ( U) \rightarrow \overline{\mathbf C}\; $
 +
such that for any local parameters  $  z _ {1} :  U _ {1} \rightarrow \overline{\mathbf C}\; $
 +
and  $  z _ {2} :  U _ {2} \rightarrow \overline{\mathbf C}\; $
 +
with  $  U _ {1} \cap U _ {2} $
 +
non-empty, the following holds in this intersection:
  
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604018.png" /> entering into the definition of a quadratic differential are ordinarily assumed to be measurable or even analytic. In the latter case the quadratic differential is called analytic. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604019.png" /> is called a zero (or pole) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604020.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604021.png" /> if for each local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604022.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604023.png" /> has a zero (or pole) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604024.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604025.png" />. The zeros and poles of a quadratic differential are called critical points of it. The zeros and simple poles are called finite critical points and their totality is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604026.png" />. The set of all poles of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604027.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604028.png" />. If a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604029.png" /> has a tangent at each of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604030.png" /> with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604031.png" />, with tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604032.png" />, and
+
$$ \tag{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/q/q076/q076040/q07604033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{Q _ {z _ {2}  } ( z _ {2} ( p) ) }{Q _ {z _ {1}  } ( z _ {1} ( p) ) }
 +
  = \
 +
\left (
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604034.png" /> is said to be positive, and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604035.png" />, on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604036.png" />. If (2) holds with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604037.png" /> sign replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604039.png" /> is called negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604041.png" />. Each maximal regular curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604042.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604043.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604044.png" />) is called a trajectory (or orthogonal trajectory) of the quadratic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604045.png" />.
+
\frac{d z _ {1} ( p) }{d z _ {2} ( p) }
  
A quadratic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604046.png" /> defined on a finite Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604047.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604048.png" /> if the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604050.png" /> is either empty or consists of a finite number of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604051.png" /> and arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604052.png" /> on each of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604053.png" /> is regular and positive or negative. If, furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604054.png" /> is empty or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604055.png" /> is regular and positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604057.png" /> is called a positive quadratic differential on the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604058.png" />. The metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604059.png" />, called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604061.png" />-metric, is single-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604062.png" /> and invariant with respect to the choice of the local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604063.png" />.
+
\right )  ^ {2} ,\ \
 +
p \in U _ {1} \cap U _ {2} ;
 +
$$
  
In some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604064.png" /> of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604065.png" />, the function
+
here  $  z ( U) $
 +
is the image of  $  U $
 +
in  $  \overline{\mathbf C}\; $
 +
under  $  z $.  
 +
A quadratic differential is often denoted by the symbol  $  Q ( z )  d z  ^ {2} $,
 +
to which is attributed the invariance with respect to the choice of the local parameter  $  z $,
 +
as indicated by (1). In other words, a quadratic differential is a non-linear differential of type  $  ( 2 , 0 ) $
 +
on a Riemann surface.
  
<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/q/q076/q076040/q07604066.png" /></td> </tr></table>
+
The functions  $  Q _ {z} ( \cdot ) $
 +
entering into the definition of a quadratic differential are ordinarily assumed to be measurable or even analytic. In the latter case the quadratic differential is called analytic. A point  $  p \in R $
 +
is called a zero (or pole) of  $  Q ( z)  d z  ^ {2} $
 +
of order  $  k $
 +
if for each local parameter  $  z $
 +
the function  $  Q _ {z} ( \cdot ) $
 +
has a zero (or pole) of order  $  k $
 +
at  $  p $.
 +
The zeros and poles of a quadratic differential are called critical points of it. The zeros and simple poles are called finite critical points and their totality is denoted by  $  C $.
 +
The set of all poles of order  $  k \geq  2 $
 +
is denoted by  $  H $.
 +
If a curve  $  \gamma \subset  R $
 +
has a tangent at each of its points  $  q $
 +
with respect to the parameter  $  z $,
 +
with tangent vector  $  a _ {z} ( q) $,
 +
and
  
is regular, single-valued and univalent for each choice of the sign of the integrand; furthermore, each maximal arc of a trajectory (or orthogonal trajectory) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604067.png" /> is converted under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604068.png" /> into a horizontal (or vertical) line interval. Therefore, through each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604069.png" /> passes a trajectory which is either an open arc or a Jordan curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604070.png" />. The topological and conformal structures of the family of trajectories in a small neighbourhood of each critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604071.png" /> are completely classified in their dependence on the order of the critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604072.png" /> and (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604073.png" /> is a pole of the second order and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604074.png" />) on
+
$$ \tag{2 }
 +
Q _ {z} ( z ( q) ) ( a _ {z} ( q) ) ^ {2}  > 0 ,\ \
 +
q \in \gamma ,
 +
$$
  
<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/q/q076/q076040/q07604075.png" /></td> </tr></table>
+
then  $  Q ( z)  d z  ^ {2} $
 +
is said to be positive, and one writes  $  Q ( z)  d z  ^ {2} > 0 $,
 +
on the curve  $  \gamma $.
 +
If (2) holds with the  $  > $
 +
sign replaced by  $  < $,
 +
then  $  Q ( z)  d z  ^ {2} $
 +
is called negative  $  ( Q ( z)  d z  ^ {2} < 0 ) $
 +
on  $  \gamma $.
 +
Each maximal regular curve on  $  R $
 +
for which  $  Q ( z)  d z  ^ {2} > 0 $(
 +
or  $  Q ( z)  d z  ^ {2} < 0 $)
 +
is called a trajectory (or orthogonal trajectory) of the quadratic differential  $  Q ( z)  d z  ^ {2} $.
 +
 
 +
A quadratic differential  $  Q ( z)  d z  ^ {2} $
 +
defined on a finite Riemann surface  $  R $
 +
belongs to  $  R $
 +
if the boundary  $  \partial  R $
 +
of  $  R $
 +
is either empty or consists of a finite number of points  $  p \notin H $
 +
and arcs  $  \gamma $
 +
on each of which  $  Q ( z)  d z  ^ {2} $
 +
is regular and positive or negative. If, furthermore,  $  \partial  R $
 +
is empty or if  $  Q ( z)  d z  ^ {2} $
 +
is regular and positive on  $  \partial  R $,
 +
then  $  Q ( z)  d z  ^ {2} $
 +
is called a positive quadratic differential on the Riemann surface  $  R $.  
 +
The metric  $  | Q ( z)  | ^ {1 / 2 } |  dz | $,
 +
called a  $  Q $-
 +
metric, is single-valued on  $  R $
 +
and invariant with respect to the choice of the local parameter  $  z $.
 +
 
 +
In some neighbourhood  $  U $
 +
of any point  $  p \in R \setminus  ( C \cup H ) $,
 +
the function
 +
 
 +
$$
 +
\zeta ( q)  = \
 +
\int\limits _ { z( } p) ^ { z( }  q)
 +
Q ( z)  ^ {1/2}  d z
 +
$$
 +
 
 +
is regular, single-valued and univalent for each choice of the sign of the integrand; furthermore, each maximal arc of a trajectory (or orthogonal trajectory) of  $  U $
 +
is converted under  $  \zeta ( q) $
 +
into a horizontal (or vertical) line interval. Therefore, through each point  $  p \in R \setminus  ( C \cup H ) $
 +
passes a trajectory which is either an open arc or a Jordan curve on  $  R $.  
 +
The topological and conformal structures of the family of trajectories in a small neighbourhood of each critical point  $  r $
 +
are completely classified in their dependence on the order of the critical point  $  r $
 +
and (if  $  r $
 +
is a pole of the second order and  $  z ( r) = 0 $)
 +
on
 +
 
 +
$$
 +
\mathop{\rm arg}  \lim\limits _ {q \rightarrow r }  Q _ {z} ( z ( q) ) z ( q)  ^ {2}
 +
$$
  
 
(see [[Local structure of trajectories|Local structure of trajectories]]). A description of the [[Global structure of trajectories|global structure of trajectories]] is known for finite Riemann surfaces and has many important applications (see also [[#References|[1]]]).
 
(see [[Local structure of trajectories|Local structure of trajectories]]). A description of the [[Global structure of trajectories|global structure of trajectories]] is known for finite Riemann surfaces and has many important applications (see also [[#References|[1]]]).
Line 29: Line 127:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) {{MR|0096806}} {{ZBL|0083.29606}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) {{MR|0065652}} {{ZBL|0059.06901}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.V. Ahlfors, L. Bers, , ''Spaces of Riemann surfaces and conformal mapping'' , Moscow (1961) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.M. Tamrazov, "On the general coefficient theorem" ''Math. USSR Sb.'' , '''1''' : 1 (1967) pp. 49–59 ''Mat. Sb.'' , '''72''' : 1 (1967) pp. 59–71 {{MR|0217284}} {{ZBL|0164.37902}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.A. Jenkins, "Some area theorems and a special coefficient theorem" ''Illinois J. Math.'' , '''8''' : 1 (1964) pp. 80–99 {{MR|0160888}} {{ZBL|0131.07601}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) {{MR|0096806}} {{ZBL|0083.29606}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) {{MR|0065652}} {{ZBL|0059.06901}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.V. Ahlfors, L. Bers, , ''Spaces of Riemann surfaces and conformal mapping'' , Moscow (1961) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.M. Tamrazov, "On the general coefficient theorem" ''Math. USSR Sb.'' , '''1''' : 1 (1967) pp. 49–59 ''Mat. Sb.'' , '''72''' : 1 (1967) pp. 59–71 {{MR|0217284}} {{ZBL|0164.37902}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.A. Jenkins, "Some area theorems and a special coefficient theorem" ''Illinois J. Math.'' , '''8''' : 1 (1964) pp. 80–99 {{MR|0160888}} {{ZBL|0131.07601}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Strebel, "Quadratic differentials" , Springer (1984) (Translated from German) {{MR|0766263}} {{MR|0753331}} {{MR|0743423}} {{ZBL|0547.30038}} {{ZBL|0547.30001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Strebel, "Quadratic differentials" , Springer (1984) (Translated from German) {{MR|0766263}} {{MR|0753331}} {{MR|0743423}} {{ZBL|0547.30038}} {{ZBL|0547.30001}} </TD></TR></table>

Revision as of 08:08, 6 June 2020


on a Riemann surface $ R $

A rule which associates to each local parameter $ z $( cf. Local uniformizing parameter) mapping a parametric neighbourhood $ U \subset R $ into the extended complex plane $ \overline{\mathbf C}\; $( $ z : U \rightarrow \overline{\mathbf C}\; $), a function $ Q _ {z} : z ( U) \rightarrow \overline{\mathbf C}\; $ such that for any local parameters $ z _ {1} : U _ {1} \rightarrow \overline{\mathbf C}\; $ and $ z _ {2} : U _ {2} \rightarrow \overline{\mathbf C}\; $ with $ U _ {1} \cap U _ {2} $ non-empty, the following holds in this intersection:

$$ \tag{1 } \frac{Q _ {z _ {2} } ( z _ {2} ( p) ) }{Q _ {z _ {1} } ( z _ {1} ( p) ) } = \ \left ( \frac{d z _ {1} ( p) }{d z _ {2} ( p) } \right ) ^ {2} ,\ \ p \in U _ {1} \cap U _ {2} ; $$

here $ z ( U) $ is the image of $ U $ in $ \overline{\mathbf C}\; $ under $ z $. A quadratic differential is often denoted by the symbol $ Q ( z ) d z ^ {2} $, to which is attributed the invariance with respect to the choice of the local parameter $ z $, as indicated by (1). In other words, a quadratic differential is a non-linear differential of type $ ( 2 , 0 ) $ on a Riemann surface.

The functions $ Q _ {z} ( \cdot ) $ entering into the definition of a quadratic differential are ordinarily assumed to be measurable or even analytic. In the latter case the quadratic differential is called analytic. A point $ p \in R $ is called a zero (or pole) of $ Q ( z) d z ^ {2} $ of order $ k $ if for each local parameter $ z $ the function $ Q _ {z} ( \cdot ) $ has a zero (or pole) of order $ k $ at $ p $. The zeros and poles of a quadratic differential are called critical points of it. The zeros and simple poles are called finite critical points and their totality is denoted by $ C $. The set of all poles of order $ k \geq 2 $ is denoted by $ H $. If a curve $ \gamma \subset R $ has a tangent at each of its points $ q $ with respect to the parameter $ z $, with tangent vector $ a _ {z} ( q) $, and

$$ \tag{2 } Q _ {z} ( z ( q) ) ( a _ {z} ( q) ) ^ {2} > 0 ,\ \ q \in \gamma , $$

then $ Q ( z) d z ^ {2} $ is said to be positive, and one writes $ Q ( z) d z ^ {2} > 0 $, on the curve $ \gamma $. If (2) holds with the $ > $ sign replaced by $ < $, then $ Q ( z) d z ^ {2} $ is called negative $ ( Q ( z) d z ^ {2} < 0 ) $ on $ \gamma $. Each maximal regular curve on $ R $ for which $ Q ( z) d z ^ {2} > 0 $( or $ Q ( z) d z ^ {2} < 0 $) is called a trajectory (or orthogonal trajectory) of the quadratic differential $ Q ( z) d z ^ {2} $.

A quadratic differential $ Q ( z) d z ^ {2} $ defined on a finite Riemann surface $ R $ belongs to $ R $ if the boundary $ \partial R $ of $ R $ is either empty or consists of a finite number of points $ p \notin H $ and arcs $ \gamma $ on each of which $ Q ( z) d z ^ {2} $ is regular and positive or negative. If, furthermore, $ \partial R $ is empty or if $ Q ( z) d z ^ {2} $ is regular and positive on $ \partial R $, then $ Q ( z) d z ^ {2} $ is called a positive quadratic differential on the Riemann surface $ R $. The metric $ | Q ( z) | ^ {1 / 2 } | dz | $, called a $ Q $- metric, is single-valued on $ R $ and invariant with respect to the choice of the local parameter $ z $.

In some neighbourhood $ U $ of any point $ p \in R \setminus ( C \cup H ) $, the function

$$ \zeta ( q) = \ \int\limits _ { z( } p) ^ { z( } q) Q ( z) ^ {1/2} d z $$

is regular, single-valued and univalent for each choice of the sign of the integrand; furthermore, each maximal arc of a trajectory (or orthogonal trajectory) of $ U $ is converted under $ \zeta ( q) $ into a horizontal (or vertical) line interval. Therefore, through each point $ p \in R \setminus ( C \cup H ) $ passes a trajectory which is either an open arc or a Jordan curve on $ R $. The topological and conformal structures of the family of trajectories in a small neighbourhood of each critical point $ r $ are completely classified in their dependence on the order of the critical point $ r $ and (if $ r $ is a pole of the second order and $ z ( r) = 0 $) on

$$ \mathop{\rm arg} \lim\limits _ {q \rightarrow r } Q _ {z} ( z ( q) ) z ( q) ^ {2} $$

(see Local structure of trajectories). A description of the global structure of trajectories is known for finite Riemann surfaces and has many important applications (see also [1]).

O. Teichmüller has investigated the role of quadratic differentials in the theory of extremal conformal and quasi-conformal mapping and in the solution of moduli problems of Riemann surfaces (see [1][3]). He formulated a principle according to which certain quadratic differentials can be associated with extremal problems in the geometric theory of functions, where to each type of extremal problem correspond specific singularities of the quadratic differential (poles), and the geometric properties of the solution are related in a suitable fashion to the structure of the trajectories of the quadratic differential. Inequalities for the coefficients of univalent functions (cf. Univalent function) have been proved in terms of quadratic differentials. A more general inequality for the coefficients of univalent functions in families of domains distributed on a finite Riemann surface is called the general coefficient theorem and is a concrete realization of the Teichmüller principle for a wide class of problems (see [1], [4]). The Teichmüller principle also enables one to establish a special coefficient theorem and to solve a large number of concrete extremal problems (see [1], [5]).

References

[1] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) MR0096806 Zbl 0083.29606
[2] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) MR0065652 Zbl 0059.06901
[3] L.V. Ahlfors, L. Bers, , Spaces of Riemann surfaces and conformal mapping , Moscow (1961) (In Russian)
[4] P.M. Tamrazov, "On the general coefficient theorem" Math. USSR Sb. , 1 : 1 (1967) pp. 49–59 Mat. Sb. , 72 : 1 (1967) pp. 59–71 MR0217284 Zbl 0164.37902
[5] J.A. Jenkins, "Some area theorems and a special coefficient theorem" Illinois J. Math. , 8 : 1 (1964) pp. 80–99 MR0160888 Zbl 0131.07601

Comments

References

[a1] K. Strebel, "Quadratic differentials" , Springer (1984) (Translated from German) MR0766263 MR0753331 MR0743423 Zbl 0547.30038 Zbl 0547.30001
How to Cite This Entry:
Quadratic differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_differential&oldid=48360
This article was adapted from an original article by P.M. Tamrazov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article