# Extremal length

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).

Let $\Gamma$ be a family of locally rectifiable curves on a 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) $\rho ( z) | d z |$ 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. 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) is connected with the theory of capacity of plane sets.

Other generalizations and modifications of the concept of the modulus of a family of curves are also known (see [6][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 [9] and [10]).

#### References

 [1] L.V. Ahlfors, A. Beurling, "Conformal invariants and function-theoretic null-sets" Acta Math. , 83 (1950) pp. 101–129 [2] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) [3] L.V. Ahlfors, "Lectures on quasiconformal mappings" , v. Nostrand (1966) [4] J.A. Jenkins, "On the existence of certain general extremal metrics" Ann. of Math. , 66 : 3 (1957) pp. 440–453 [5] G.V. Kuz'mina, "Moduli of families of curves and quadratic differentials" , Amer. Math. Soc. (1982) (Translated from Russian) [6] J. Hersch, "Longeurs extrémales et théorie des fonctions" Comment. Math. Helv. , 29 : 4 (1955) pp. 301–337 [7] 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)) [8] B. Fuglede, "Extremal length and functional completion" Acta Math. , 98 (1957) pp. 171–219 [9] 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 [10] A.V. Sychev, "Moduli and quasi-conformal mappings in space" , Novosibirsk (1983) (In Russian)
How to Cite This Entry:
Extremal length. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extremal_length&oldid=46892
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