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 be a family of locally rectifiable curves on a Riemann surface . The modulus problem is defined for if there is a non-empty class of conformally-invariant metrics (cf. Conformally-invariant metric) given on such that is square integrable in the -plane for every local uniformizing parameter and if

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

is called the modulus of the family of curves . The reciprocal of is called the extremal length of the family of curves .

The modulus problem for a family of curves is often defined as follows: Let be the subclass of such that for and ,

If the set is non-empty, then the quantity

is called the modulus of the family . If is non-empty but is empty, then is assigned the value . It is the latter definition of the modulus that is adopted below.

Let be a family of locally rectifiable curves on a Riemann surface for which the modulus problem is defined, and let . Then every metric from is an admissible metric for the modulus problem for . If in there is a metric for which

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

The fundamental property of the modulus is its conformal invariance.

Theorem 1. Let and be two conformally-equivalent Riemann surfaces, let be a univalent conformal mapping of onto , let be a family of locally rectifiable curves given on , and let be the family of images of the curves in under . If the modulus problem is defined for and the modulus of is , then the modulus problem is also defined for and .

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

Theorem 2. Let be a family of locally rectifiable curves on a Riemann surface , and suppose that the modulus problem is defined for and that . If and are extremal metrics for this modulus problem, then everywhere on except, possibly, on a subset of of measure zero.

Examples of moduli of families of curves.

1) Let be a rectangle with sides and , and let () be a family of locally rectifiable curves in that join the sides of length (). Then

2) Let be the annulus , let be the class of rectifiable Jordan curves in that separate the boundary components of and let be the class of locally rectifiable curves in that join the boundary components of . Then and . In both cases and are characteristic conformal invariants of . Hence, is called the modulus of the domain for the class and is called the modulus of for .

There is a well-known connection between the moduli of families of curves under a quasi-conformal mapping. Let be a family of curves in some domain and let be the image of under a -quasi-conformal mapping of . Then the moduli and of and , respectively, satisfy the inequality

The generalization of the concept of the modulus to several families of curves turns out to be important in applications. Let be families of locally rectifiable curves on a Riemann surface (as a rule, are, respectively, homotopy classes of curves). Let be non-negative real numbers, not all equal to zero, and let be the class of conformally-invariant metrics on for which is integrable for every local parameter and such that

If the set is non-empty, then the modulus problem is said to be defined for the families of curves and the numbers . In this case the quantity

is called the modulus of this problem. If in there is a metric for which

then this metric is called an extremal metric for the modulus problem .

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 has been proved under fairly general assumptions. The above definition extends to the case of families of curves on a surface obtained by removing from finitely many points , where the families , , consist of closed Jordan curves homotopic on 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 relative to a point (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 -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)