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''of a family of curves''

A concept which, along with that of the modulus of a family of curves, is a general form of the definition of conformal invariants and lies at the basis of the method of the extremal metric (cf. [[Extremal metric, method of the|Extremal metric, method of the]]).

Let  $  \Gamma $
be a family of locally rectifiable curves on a [[Riemann surface|Riemann surface]]  $  R $. 
The modulus problem is defined for  $  \Gamma $
if there is a non-empty class  $  P $
of conformally-invariant metrics (cf. [[Conformally-invariant metric|Conformally-invariant metric]])  $  \rho ( z)  | d z | $
given on  $  R $
such that  $  \rho ( z) $
is square integrable in the  $  z $-
plane for every local uniformizing parameter  $  z ( = x + i y ) $
and if

$$ 
A _  \rho  ( R)  =  {\int\limits \int\limits } _ { R } \rho  ^ {2} ( z)  d x  d y \ \ 
\textrm{ and } \  L _  \rho  ( \Gamma )  =  \inf _ {\gamma \in \Gamma }
 \int\limits _  \gamma  \rho ( z)  | d z |
$$

are not simultaneously equal to  $  0 $
or  $  \infty $. 
(Each of the above integrals is understood as a Lebesgue integral.) In this case the quantity

$$ 
M ( \Gamma )  =  \inf _ {\rho \in P } \ 

\frac{A _  \rho  ( R) }{[ L _  \rho  ( \Gamma ) ]  ^ {2} }

$$

is called the modulus of the family of curves  $  \Gamma $. 
The reciprocal of  $  M ( \Gamma ) $
is called the extremal length of the family of curves  $  \Gamma $.

The modulus problem for a family of curves is often defined as follows: Let  $  P _ {L} $
be the subclass of  $  P $
such that for  $  \rho \in P _ {L} $
and  $  \gamma \in \Gamma $,

$$ 
\int\limits _  \gamma  \rho ( z)  | d z |  \geq   1 .
$$

If the set  $  P _ {L} $
is non-empty, then the quantity

$$ 
M ( \Gamma )  =  \inf _ {\rho \in P _ {L} }  A _  \rho  ( R)
$$

is called the modulus of the family  $  \Gamma $. 
If  $  P $
is non-empty but  $  P _ {L} $
is empty, then  $  M ( \Gamma ) $
is assigned the value  $  \infty $. 
It is the latter definition of the modulus that is adopted below.

Let  $  \Gamma $
be a family of locally rectifiable curves on a Riemann surface  $  R $
for which the modulus problem is defined, and let  $  M ( \Gamma ) \neq \infty $. 
Then every metric from  $  P _ {L} $
is an admissible metric for the modulus problem for  $  \Gamma $. 
If in  $  P _ {L} $
there is a metric  $  \rho  ^ {*} ( z)  | dz | $
for which

$$ 
M ( \Gamma )  =  \inf _ {\rho \in P _ {L} }  A _  \rho  ( R) ,
$$

then this metric is called an extremal metric in the modulus problem for  $  \Gamma $.

The fundamental property of the modulus is its conformal invariance.

Theorem 1. Let  $  R $
and  $  R _ {1} $
be two conformally-equivalent Riemann surfaces, let  $  f $
be a univalent conformal mapping of  $  R $
onto  $  R _ {1} $, 
let  $  \Gamma $
be a family of locally rectifiable curves given on  $  R $, 
and let  $  \Gamma _ {1} $
be the family of images of the curves in  $  \Gamma $
under  $  f $. 
If the modulus problem is defined for  $  \Gamma $
and the modulus of  $  \Gamma $
is  $  M ( \Gamma ) $, 
then the modulus problem is also defined for  $  \Gamma _ {1} $
and  $  M ( \Gamma _ {1} ) = M ( \Gamma ) $.

The following theorem shows that if there is an extremal metric, then it is essentially unique:

Theorem 2. Let  $  \Gamma $
be a family of locally rectifiable curves on a Riemann surface  $  R $, 
and suppose that the modulus problem is defined for  $  \Gamma $
and that  $  M ( \Gamma ) \neq \infty $. 
If  $  \rho _ {1}  ^ {*} ( z)  | dz | $
and  $  \rho _ {2}  ^ {*} ( z)  | dz | $
are extremal metrics for this modulus problem, then  $  \rho _ {2}  ^ {*} ( z) = \rho _ {1}  ^ {*} ( z) $
everywhere on  $  R $
except, possibly, on a subset of  $  R $
of measure zero.

Examples of moduli of families of curves.

1) Let  $  D $
be a rectangle with sides  $  a $
and  $  b $, 
and let  $  \Gamma $(
$  \Gamma _ {1} $) 
be a family of locally rectifiable curves in  $  D $
that join the sides of length  $  a $(
$  b $). 
Then

$$ 
M ( \Gamma )  =  
\frac{a}{b}
 ,\ \ 
M ( \Gamma _ {1} )  =  
\frac{b}{a}
 .
$$

2) Let  $  D $
be the annulus  $  r < | z | < 1 $, 
let  $  \Gamma $
be the class of rectifiable Jordan curves in  $  D $
that separate the boundary components of  $  D $
and let  $  \Gamma _ {1} $
be the class of locally rectifiable curves in  $  D $
that join the boundary components of  $  D $. 
Then  $  M ( \Gamma ) = (  \mathop{\rm ln}  1 / r ) / 2 \pi $
and  $  M ( \Gamma _ {1} ) = 2 \pi /  \mathop{\rm ln} ( 1 / r ) $. 
In both cases  $  M ( \Gamma ) $
and  $  M ( \Gamma _ {1} ) $
are characteristic conformal invariants of  $  D $. 
Hence,  $  M ( \Gamma ) $
is called the modulus of the domain  $  D $
for the class  $  \Gamma $
and  $  M ( \Gamma _ {1} ) $
is called the modulus of  $  D $
for  $  \Gamma _ {1} $.

There is a well-known connection between the moduli of families of curves under a [[Quasi-conformal mapping|quasi-conformal mapping]]. Let  $  \Gamma $
be a family of curves in some domain  $  D $
and let  $  \Gamma _ {1} $
be the image of  $  \Gamma $
under a  $  K $-
quasi-conformal mapping of  $  D $. 
Then the moduli  $  M ( \Gamma ) $
and  $  M ( \Gamma _ {1} ) $
of  $  \Gamma $
and  $  \Gamma _ {1} $, 
respectively, satisfy the inequality

$$ 
K  ^ {-} 1 M ( \Gamma )  \leq   M ( \Gamma _ {1} )  \leq   K M ( \Gamma ) .
$$

The generalization of the concept of the modulus to several families of curves turns out to be important in applications. Let  $  \Gamma _ {1} \dots \Gamma _ {n} $
be families of locally rectifiable curves on a Riemann surface  $  R $(
as a rule,  $  \Gamma _ {1} \dots \Gamma _ {n} $
are, respectively, homotopy classes of curves). Let  $  \alpha _ {1} \dots \alpha _ {n} $
be non-negative real numbers, not all equal to zero, and let  $  P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
be the class of conformally-invariant metrics  $  \rho ( z)  | dz | $
on  $  R $
for which  $  \rho  ^ {2} ( z) $
is integrable for every local parameter  $  z = x + i y $
and such that

$$ 
\int\limits _ {\gamma _ {j} } \rho ( z)  | d z |  \geq   \alpha _ {j} \ \ 
\textrm{ for }  \gamma _ {j} \in \Gamma _ {j} ,\ 
j = 1 \dots n .
$$

If the set  $  P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
is non-empty, then the modulus problem  $  {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
is said to be defined for the families of curves  $  \{ \Gamma _ {j} \} $
and the numbers  $  \{ \alpha _ {j} \} $. 
In this case the quantity

$$ 
M ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} )  = \ 
\inf _ {\rho \in P ( \{ \Gamma _ {j} \} ,\ 
\{ \alpha _ {j} \} ) } \ 
{\int\limits \int\limits } _ { R } \rho  ^ {2} ( z)  d x  d y
$$

is called the modulus of this problem. If in  $  P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
there is a metric  $  \rho  ^ {*} ( z)  | dz | $
for which

$$ 
{\int\limits \int\limits } _ { R } [ \rho  ^ {*} ( z) ]  ^ {2} \ 
d x  d y  =  M ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) ,
$$

then this metric is called an extremal metric for the modulus problem  $  {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $.

The modulus problem defined in this way is also a conformal invariant. For such moduli a uniqueness theorem analogous to Theorem 2 holds. The existence of an extremal metric for the modulus problem  $  {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
has been proved under fairly general assumptions. The above definition extends to the case of families of curves  $  \Gamma _ {1} \dots \Gamma _ {n} $
on a surface  $  R _ {1} $
obtained by removing from  $  R $
finitely many points  $  a _ {1} \dots a _ {N} $, 
where the families  $  \Gamma _ {1} \dots \Gamma _ {k} $, 
$  k \leq  n $, 
consist of closed Jordan curves homotopic on  $  R _ {1} $
to circles of sufficiently small radii and centres at corresponding selected points. Such an extremal-metric problem in conjunction with the above concept of the modulus of a simply-connected domain  $  D $
relative to a point  $  a \in D $(
see [[Modulus of an annulus|Modulus of an annulus]]) is connected with the theory of [[Capacity|capacity]] of plane sets.

Other generalizations and modifications of the concept of the modulus of a family of curves are also known (see [[#References|[6]]]–[[#References|[10]]]). This concept has been extended to the case of curves and surfaces in space. Uniqueness theorems and a number of properties of such moduli have been established, in particular, an analogue of the inequalities

for  $  K $-
quasi-conformal mappings in space has been obtained (see [[#References|[9]]] and [[#References|[10]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Ahlfors,   A. Beurling,   "Conformal invariants and function-theoretic null-sets"  ''Acta Math.'' , '''83'''  (1950)  pp. 101–129</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Ahlfors,   "Lectures on quasiconformal mappings" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,   "On the existence of certain general extremal metrics"  ''Ann. of Math.'' , '''66''' :  3  (1957)  pp. 440–453</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.V. Kuz'mina,   "Moduli of families of curves and quadratic differentials" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Hersch,   "Longeurs extrémales et théorie des fonctions"  ''Comment. Math. Helv.'' , '''29''' :  4  (1955)  pp. 301–337</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.M. Tamrazov,   "A theorem of line integrals for extremal length"  ''Dokl. Akad. Nauk Ukrain. SSSR'' , '''1'''  (1966)  pp. 51–54  ((in Ukrainian; English summary))</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B. Fuglede,   "Extremal length and functional completion"  ''Acta Math.'' , '''98'''  (1957)  pp. 171–219</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.V. Shabat,   "The modulus method in space"  ''Soviet Math. Dokl.'' , '''1''' :  1  (1960)  pp. 165–168  ''Dokl. Akad. Nauk SSSR'' , '''130''' :  6  (1960)  pp. 1210–1213</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.V. Sychev,   "Moduli and quasi-conformal mappings in space" , Novosibirsk  (1983)  (In Russian)</TD></TR></table>