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)