Minimization of an area

The problem of finding the minimum of the area $A(F)$ of a Riemann surface to which a given domain $B$ of the $z$-plane is mapped by a one-to-one regular function $F$ of a given class $R$, that is, the problem of finding

$$\begin{array}{}\text{(*)}& \underset{F\in R}{min}A(F)=\underset{F\in R}{min}{\iint }_B|F^{\prime }(z){|}^{2}d\sigma \end{array}$$

( $d\sigma$ is the surface element). The integral in (*), taken over $B$, is understood as the limit of integrals over domains $B_n$, $n=1,2\dots$ which exhaust the domain $B$, that is, are such that ${\bar{B}}_n\subset B$, $B_n\subset B_{n+1}$ and such that any closed set $E\subset B$ lies in $B_n$ from some $n$ onwards.

When $R$ is the class of functions $F(z)$, $F(0)=0$, $F^{\prime }(0)=1$, regular in a given simply-connected domain $B$ containing $z=0$ and having more than one boundary point, the minimum $A$ of the areas $A(F)$ of the images of $B$ in the class $R$ is given by the unique function univalently mapping $B$ onto the full disc $|z|<r$, where $r$ is the conformal radius of $B$ at $z=0$( cf. Conformal radius of a domain); moreover, $A=\pi r^2$.

The problem of finding the minimal area of the image of a multiply-connected domain has also been considered (see [1]).

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