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

$$ \tag{* } \min _ {F \in R } A ( F ) = \min _ {F \in R } \ {\int\limits \int\limits } _ { B } | F ^ { \prime } ( z) | ^ {2} d \sigma $$

( $ 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 $ \overline{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]).

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)