# 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" , Oeuvres , 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)