View source for 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|Extremal metric, method of the]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371701.png" /> be a family of locally rectifiable curves on a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371702.png" />. The modulus problem is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371703.png" /> if there is a non-empty class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371704.png" /> of conformally-invariant metrics (cf. [[Conformally-invariant metric|Conformally-invariant metric]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371705.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371706.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371707.png" /> is square integrable in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371708.png" />-plane for every local uniformizing parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371709.png" /> and if

<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/e/e037/e037170/e03717010.png" /></td> </tr></table>

are not simultaneously equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717012.png" />. (Each of the above integrals is understood as a Lebesgue integral.) In this case the quantity

<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/e/e037/e037170/e03717013.png" /></td> </tr></table>

is called the modulus of the family of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717014.png" />. The reciprocal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717015.png" /> is called the extremal length of the family of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717016.png" />.

The modulus problem for a family of curves is often defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717017.png" /> be the subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717018.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717020.png" />,

<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/e/e037/e037170/e03717021.png" /></td> </tr></table>

If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717022.png" /> is non-empty, then the quantity

<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/e/e037/e037170/e03717023.png" /></td> </tr></table>

is called the modulus of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717025.png" /> is non-empty but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717026.png" /> is empty, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717027.png" /> is assigned the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717028.png" />. It is the latter definition of the modulus that is adopted below.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717029.png" /> be a family of locally rectifiable curves on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717030.png" /> for which the modulus problem is defined, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717031.png" />. Then every metric from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717032.png" /> is an admissible metric for the modulus problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717033.png" />. If in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717034.png" /> there is a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717035.png" /> for which

<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/e/e037/e037170/e03717036.png" /></td> </tr></table>

then this metric is called an extremal metric in the modulus problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717037.png" />.

The fundamental property of the modulus is its conformal invariance.

Theorem 1. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717039.png" /> be two conformally-equivalent Riemann surfaces, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717040.png" /> be a univalent conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717041.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717042.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717043.png" /> be a family of locally rectifiable curves given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717045.png" /> be the family of images of the curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717046.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717047.png" />. If the modulus problem is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717048.png" /> and the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717049.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717050.png" />, then the modulus problem is also defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717052.png" />.

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

Theorem 2. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717053.png" /> be a family of locally rectifiable curves on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717054.png" />, and suppose that the modulus problem is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717055.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717056.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717058.png" /> are extremal metrics for this modulus problem, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717059.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717060.png" /> except, possibly, on a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717061.png" /> of measure zero.

Examples of moduli of families of curves.

1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717062.png" /> be a rectangle with sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717064.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717065.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717066.png" />) be a family of locally rectifiable curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717067.png" /> that join the sides of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717068.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717069.png" />). Then

<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/e/e037/e037170/e03717070.png" /></td> </tr></table>

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717071.png" /> be the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717072.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717073.png" /> be the class of rectifiable Jordan curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717074.png" /> that separate the boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717075.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717076.png" /> be the class of locally rectifiable curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717077.png" /> that join the boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717078.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717080.png" />. In both cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717082.png" /> are characteristic conformal invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717083.png" />. Hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717084.png" /> is called the modulus of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717085.png" /> for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717087.png" /> is called the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717088.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717089.png" />.

There is a well-known connection between the moduli of families of curves under a [[Quasi-conformal mapping|quasi-conformal mapping]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717090.png" /> be a family of curves in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717091.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717092.png" /> be the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717093.png" /> under a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717095.png" />-quasi-conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717096.png" />. Then the moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170100.png" />, respectively, satisfy the inequality

<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/e/e037/e037170/e037170101.png" /></td> </tr></table>

The generalization of the concept of the modulus to several families of curves turns out to be important in applications. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170102.png" /> be families of locally rectifiable curves on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170103.png" /> (as a rule, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170104.png" /> are, respectively, homotopy classes of curves). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170105.png" /> be non-negative real numbers, not all equal to zero, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170106.png" /> be the class of conformally-invariant metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170107.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170108.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170109.png" /> is integrable for every local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170110.png" /> and such that

<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/e/e037/e037170/e037170111.png" /></td> </tr></table>

If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170112.png" /> is non-empty, then the modulus problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170113.png" /> is said to be defined for the families of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170114.png" /> and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170115.png" />. In this case the quantity

<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/e/e037/e037170/e037170116.png" /></td> </tr></table>

is called the modulus of this problem. If in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170117.png" /> there is a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170118.png" /> for which

<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/e/e037/e037170/e037170119.png" /></td> </tr></table>

then this metric is called an extremal metric for the modulus problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170120.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170121.png" /> has been proved under fairly general assumptions. The above definition extends to the case of families of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170122.png" /> on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170123.png" /> obtained by removing from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170124.png" /> finitely many points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170125.png" />, where the families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170127.png" />, consist of closed Jordan curves homotopic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170128.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170129.png" /> relative to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170130.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170131.png" />-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>