Parametric representation of univalent functions

A representation of univalent functions (cf. Univalent function) that effect a conformal mapping of planar domains onto domains of a canonical form (for example, a disc with concentric slits); it usually arises in the following manner. One chooses a one-parameter family of domains $ Q _ {t} $, $ 0 \leq t < T $, included in one another, $ Q _ {t ^ \prime } \subset Q _ {t} $, $ 0 \leq t ^ \prime < t < T $. For $ Q _ {0} $ one assumes known a conformal mapping $ f _ {0} $ onto some canonical domain $ B _ {0} $. From a known mapping $ f _ {t} $ of $ Q _ {t} $ onto a domain of canonical form one constructs such a mapping $ f _ {t+ \epsilon } $ for $ Q _ {t+ \epsilon } $, where $ \epsilon > 0 $ is small. Under a continuous change of the parameter $ t $ there arise in this way differential equations. The best known of these are the Löwner equation and the Löwner–Kufarev equation. In the discrete case — for lattice domains $ Q _ {t} $ and a natural number $ t $— the transition from $ f _ {t} $ to $ f _ {t + \epsilon } $, $ \epsilon = 1 $, is effected by recurrence formulas. The source of these formulas and equations is usually the Schwarz formula (see [1]) and its generalizations (see [2]). An equally important source of parametric representations are the Hadamard variations (see [3], [4]) for the Green functions $ G _ {t} ( z, z ^ \prime ) $, $ z, z ^ \prime \in Q _ {t} $, of the family of domains mentioned above. For elliptic differential equations Hadamard's method is also called the method of invariant imbedding (see [5]). Below, the connection between parametric representations, Hadamard variations and invariant imbedding are exhibited in the simplest (discrete) case.

Suppose that $ Q $ is a collection of complex integers (a lattice domain) and that the Green function $ G _ {z} ( z, z ^ \prime ) $ is an extremal of the Dirichlet–Douglas functional

$$ I _ {t} ( g) = 2g( z ^ \prime ) + \sum _ { k= } 0 ^ { l } \sum _ {z \in Q _ {0} } \rho _ {k} ( t) | \nabla _ {k} g( z) | ^ {2} $$

in the class $ R _ {0} $ of all real-valued functions $ g( z) $ on $ Q $. Here $ Q _ {0} = \{ {z } : {z, z- 1, z- i, z- 1- i \in Q } \} $,

$$ \tag{1 } \left . \begin{array}{c} {\nabla _ {0} g( z) = g( z) - g( z- 1- i), } \\ {\nabla _ {1} g( z) = g( z- 1)- g( z- i), } \end{array} \right \} $$

$$ \rho _ {k} ( 0) \equiv 1,\ \rho _ {k} ( t+ 1) = \rho _ {k} ( t) + N \delta _ {\zeta _ {t} } , $$

$ N $ is a natural number, $ \delta _ {\zeta _ {t} } $ is the Kronecker symbol, and $ \zeta _ {t} = ( k _ {t} , z _ {t} ) $, $ t = 0 \dots T- 1 $, is a certain collection of pairs of numbers; $ \{ {z _ {t} } : {t = 1 \dots T } \} $ is the boundary of $ Q _ {t} $, and $ k _ {t} = 0 $ or 1. To find an extremum of the functional $ I _ {t} ( g) $ is a problem of quadratic programming. A comparison of its solutions for $ t $ and $ t+ 1 $ gives the basic formula of invariant imbedding (Hadamard variation):

$$ \tag{2 } G _ {t+} 1 ( z, z ^ \prime ) = G _ {t} ( z, z ^ \prime ) - \frac{1}{c _ {t} } \nabla _ {k _ {t} } G _ {t} ( z _ {t} , z) \nabla _ {k _ {t} } G _ {t} ( z _ {t} , z ^ \prime ), $$

where $ c _ {t} = N ^ {-} 1 - \nabla _ {k _ {t} } \nabla _ {k _ {t} } ^ \prime G _ {t} ( z _ {t} , z _ {t} ) > 0 $ and the symbol $ \nabla _ {k} ^ \prime $ denotes the difference operators (1) in the second argument of the Green function. Knowing $ G _ {0} ( z, z ^ \prime ) $ one can obtain step-by-step (recurrently) from (2) all the functions $ G _ {t} ( z, z ^ \prime ) $, $ t = 1 \dots T $. By constructing the Green function, one obtains from the lattice analytic function $ f _ {T} ( z) = G _ {T} ( z, z ^ \prime ) + iH _ {T} ( z, z ^ \prime ) $ according to the equation of Cauchy–Riemann type

$$ (- 1) ^ {k} \nabla _ {1-} k H = \rho _ {k} \nabla _ {k} G, $$

a univalent lattice quasi-conformal mapping $ w = \mathop{\rm exp} [ 2 \pi f( z)] $ of $ Q _ {t} $ into the unit disc. Closest to the origin of coordinates is the image of $ z ^ \prime $. In the limit, as $ n \rightarrow \infty $, the mapping is lattice conformal and the image of $ Q _ {T} $ is a disc with concentric slits. The result is a continuous analogue of (2) (see [6]). When all the domains $ G _ {t} $ are simply connected and the canonical domain is the unit disc $ B $, one succeeds by using a fractional-linear automorphism of $ B $ to represent the Green function in the explicit form

$$ G _ {t} ( z, z ^ \prime ) = \mathop{\rm ln} | 1- f _ {t} ( z) \overline{ {f _ {t} ( z ^ \prime ) }}\; | - \mathop{\rm ln} | f _ {t} ( z) - f _ {t} ( z ^ \prime ) | $$

in terms of the function $ f _ {t} ( z) $ mapping $ Q _ {t} $ onto $ B $ with the normalization $ f( 0) = 0 $, $ 0 \in Q _ {t} $ for all $ t \in [ 0, T) $.

In terms of $ w = f _ {t} ( z) $ the Hadamard variation reduces to an ordinary (Löwner) differential equation. In comparison with the Hadamard variation this equation is considerably simpler; however, information on the boundary of $ Q _ {t} $ is only implicit in it — in terms of the control parameter $ \alpha ( t) = \mathop{\rm arg} f _ {t} ( z _ {t} ) $, because $ f _ {t} ( z _ {t} ) $ is not known beforehand. Nevertheless, the Löwner equation is a basic instrument in the parametric representation.

More general one-parameter families of domains $ Q _ {t} $, $ 0 \leq t < T $, not necessarily imbedded in one another, have also been treated (see [7]). The equations arising in such parametric representations are called Löwner–Kufarev equations. There is also a modification of the Löwner and Löwner–Kufarev equations to the case when the domains $ Q _ {t} $ have a different kind of symmetry or other geometric peculiarities (see [1]).

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	I.A. Aleksandrov, A.S. Sorokin, "The problem of Schwarz for multiply connected circular domains" Sib. Math. J. , 13 : 5 (1972) pp. 671–692 Sibirsk. Mat. Zh. , 13 : 5 (1972) pp. 971–1000
[3]	J. Hadamard, "Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées" , Œuvres , 2 , CNRS (1968) pp. 515–642
[4]	J. Hadamard, "Leçons sur le calcul des variations" , 1 , Hermann (1910)
[5]	R. Bellma, E. Angel, "Dynamic programming and partial differential equations" , Acad. Press (1972)
[6]	V.I. Popov, "Quantization of control systems" Soviet Math. Dokl. , 13 : 6 (1972) pp. 1668–1672 Dokl. Akad. Nauk. SSSR , 207 : 5 (1972) pp. 1048–1050
[7]	P.P. Kufarev, "On one-parameter families of analytic functions" Mat. Sb. , 13 : 1 (1943) pp. 87–118 (In Russian) (English abstract)
[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11